-
Notifications
You must be signed in to change notification settings - Fork 13
/
index.pug
264 lines (238 loc) · 15.1 KB
/
index.pug
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
include header
html
head
meta(property='og:title' content='GROUPOЇD')
meta(property='og:description' content='Groupoїd Infinity')
meta(property='og:url' content='https://groupoid.space/')
block title
title GROUPOЇD
block content
article.main
.exe
section
p.
<a href="https://groupoid.space/institute/">Groupoid Infinity Institute</a> is doing formalization of mathematics
in the formal programming language called <a href="https://anders.groupoid.space/">Anders</b>
<a href="https://opam.ocaml.org/packages/anders/">1.3.0</a>,
a CCHM/HTS variant of cubical type systems.
br
<hr size=1>
br
.exe
section
h1 MATHEMATICS
p.
Mathematical theories, models and languages.
.types
-
var logic = {
'Category Theory': { 'Categories': 'https://anders.groupoid.space/mathematics/categories/category/',
'Functors, Adjunctions': 'https://anders.groupoid.space/mathematics/categories/functor/',
'Natural Transformations': '',
'Kan Extensions' : '',
'(Co)limits': '',
'Universal Properties' : '',
'Structure Identity Principle' : '' },
'Topos Theory': { 'Topology' : '',
'Coverings' : '',
'Grothendieck Topology' : 'https://anders.groupoid.space/mathematics/topoi/topos/',
'Grothendieck Topos' : '',
'Geometric Morphisms': '',
'Elementary Topos': '' },
'Geometry': { 'Nisnevich Site' : '',
'Zariski Site' : '',
'Theory of Schemes' : '',
'Noetherian Scheme' : '',
'A¹-Homotopy Theory' : '',
'Cohesive Topos' : '',
'Etale Topos': '',
'Local Homotopy Theory': '' }}; var algebra = {
'Algebraic Systems': { 'Algebraic structure' : 'https://anders.groupoid.space/mathematics/algebra/algebra/',
'Group, Subgroup' : '',
'Normal Group' : '',
'Factorgroup' : '',
'Abelian Group' : '',
'Ring' : '',
'Module' : '',
'Field Theory' : '',
'Linear Algebra' : '',
'Lie, Leibniz Algebra' : '',
},
'(Co)Homotopy Theory': { 'Pullback' : 'https://anders.groupoid.space/mathematics/homotopy/pullback/',
'Pushout' : 'https://anders.groupoid.space/mathematics/homotopy/pushout/',
'Limit, Fiber' : 'https://anders.groupoid.space/mathematics/geometry/bundle/',
'Suspension, Loop' : '',
'Smash, Wedge, Join' : '',
'H-(co)spaces' : '',
'Eilenberg-MacLane Spaces' : '',
'Cell Complexes' : '',
'(Co)Homotopy' : '',
'Hopf Invariant' : '' },
'(Co)Homology Theory': { 'Chain Complex' : '',
'(Co)Homology' : 'https://anders.groupoid.space/mathematics/algebra/homology/',
'Stinrod Axioms' : '',
'Hom and Tensor' : '',
'Resolutions' : '',
'Derived Cat, Fun' : '',
'Tor, Ext and Local Cohomology' : '',
'Homological Algebra' : '',
'Cohomology Operations' : '' } }; var foundations = {
'Categorical Models': { 'Homotopical Algebra' : '',
'Presheaf Type Theory' : 'https://anders.groupoid.space/mathematics/topoi/presheaf/',
'Set Quillen model' : 'https://tonpa.guru/stream/2020/2020-05-03%20%D0%9C%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B5%20%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8.htm',
'Top Quillen model' : 'https://tonpa.guru/stream/2020/2020-05-03%20%D0%9C%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B5%20%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8.htm',
'sSet Quillen model' : 'https://tonpa.guru/stream/2020/2020-05-03%20%D0%9C%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B5%20%D0%BA%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D0%B8.htm',
'Pi Presheaf' : 'https://anders.groupoid.space/mathematics/topoi/presheaf/',
'Sigma Presheaf' : 'https://anders.groupoid.space/mathematics/topoi/presheaf/',
'sSet Presheaf' : 'https://anders.groupoid.space/mathematics/topoi/presheaf/',
'I Presheaf' : 'https://anders.groupoid.space/mathematics/topoi/presheaf/' },
'Type Systems': { 'METAMATH' : 'https://bertrand.groupoid.space/',
'STLC' : 'https://alonzo.groupoid.space/',
'PTS' : 'https://henk.groupoid.space/',
'CCHM HTS' : 'https://anders.groupoid.space/',
'HoTT-I' : 'https://github.com/groupoid/valery'
}
};
br
br
.type
for v, k in algebra
ol.type__col
h3= k
for j, i in v
if j === ""
li: b()= i
else
li: a(href= j)= i
br
br
.type
for v, k in logic
ol.type__col
h3= k
for j, i in v
if j === ""
li: b()= i
else
li: a(href= j)= i
br
br
.type
for v, k in foundations
ol.type__col
h3= k
for j, i in v
if j === ""
li: b()= i
else
li: a(href= j)= i
.exe
section
h1 HOMOTOPY LIBRARY
p.
Homotopy Library for <a href="https://anders.groupoid.space/">Anders</a>
theorem prover consists of two parts Foundations and Mathematics just
as <a href="https://homotopytypetheory.org/book/">HoTT Book</a>.
h2 Foundations
p.
MLTT, Modal and Univalent Foundations represent a basic
language primitives of <b>Anders</b> and its base library.
section
.macro
.macro__col
h3#mltt <b>MLTT</b>
ol
li: <a href='https://anders.groupoid.space/foundations/mltt/pi/'>Π</a>, <a href='https://anders.groupoid.space/foundations/mltt/sigma/'>Σ</a>, <a href='https://anders.groupoid.space/foundations/mltt/id/'>=</a>
li: <a href='https://anders.groupoid.space/foundations/mltt/inductive/'>0, 1, 2, W</a>
li: <a href='https://anders.groupoid.space/foundations/mltt/either/'>+</a>, <a href='https://anders.groupoid.space/foundations/mltt/maybe/'>1+</a>
li: <a href='https://anders.groupoid.space/foundations/mltt/nat/'>N</a>, <a href='https://anders.groupoid.space/foundations/mltt/list/'>LIST</a>
li: <a href='https://anders.groupoid.space/foundations/mltt/fin/'>FIN</a>, <a href='https://anders.groupoid.space/foundations/mltt/vec/'>VEC</a>
.macro__col
h3#modal <b>MODAL</b>
ol
li: a(href='https://anders.groupoid.space/foundations/modal/derham/') DERHAM
li: a(href='https://anders.groupoid.space/foundations/modal/process/') PROCESS
li: a(href='https://anders.groupoid.space/foundations/modal/infinitesimal/') INFINITESIMAL
.macro__col
h3#univalent <b>UNIVALENT</b>
ol
li: a(href='https://anders.groupoid.space/foundations/univalent/equiv/') EQUIV
li: a(href='https://anders.groupoid.space/foundations/univalent/iso/') ISO
li: a(href='https://anders.groupoid.space/foundations/univalent/funext/') FUNEXT
li: a(href='https://anders.groupoid.space/foundations/univalent/path/') PATH
li: a(href='https://anders.groupoid.space/foundations/univalent/glue/') GLUE
h2 Mathematics
p.
The second part is dedicated to mathematical models and theories internalized in this language.
section
.macro
.macro__col
h3#algebra <b>ALGEBRA</b>
ol
li: a(href='https://anders.groupoid.space/mathematics/algebra/algebra/') ALGEBRA
li: a(href='https://anders.groupoid.space/mathematics/algebra/homology/') HOMOLOGY
.macro__col
h3#geometry <b>GEOMETRY</b>
ol
li: a(href='https://anders.groupoid.space/mathematics/geometry/etale') ETALE
li: a(href='https://anders.groupoid.space/mathematics/geometry/bundle') BUNDLE
li: a(href='https://anders.groupoid.space/mathematics/geometry/manifold') MANIFOLD
.macro__col
h3#homotopy <b>HOMOTOPY</b>
ol
li: a(href='https://anders.groupoid.space/mathematics/homotopy/coequalizer/') COEQUALIZER
li: a(href='https://anders.groupoid.space/mathematics/homotopy/pushout/') PUSHOUT
li: a(href='https://anders.groupoid.space/mathematics/homotopy/pullback/') PULLBACK
li: a(href='https://anders.groupoid.space/mathematics/homotopy/hopf/') HOPF
li: a(href='https://anders.groupoid.space/mathematics/homotopy/cw/') CW
section
.macro
.macro__col
h3#categories <b>CATEGORIES</b>
ol
li: a(href='https://anders.groupoid.space/mathematics/categories/category/') CATEGORY
li: a(href='https://anders.groupoid.space/mathematics/categories/functor/') FUNCTOR
li: a(href='https://anders.groupoid.space/mathematics/categories/groupoid/') GROUPOID
.macro__col
h3#categories <b>TOPOI</b>
ol
li: a(href='https://anders.groupoid.space/mathematics/topoi/topos/') TOPOS
li: a(href='https://anders.groupoid.space/mathematics/topoi/presheaf/') PRESHEAF
br.
p.
The base library for <b>cubicaltt</b> is given on separate page:
<a href='https://groupoid.space/misc/library/'>Formal Mathematics: The Cubical Base Library</a><br>
h1 Languages
p.
Non-dependent languages are targets or runtimes, and dependently typed language are provers.
h2 Verification
p.
Theorem provers for logic, lambda encodings exploration and homotopy calculus.
p.
19 JAN 2010 — <a href='https://bertrand.groupoid.space'>Bertrand: Metamath</a><br>
05 APR 2022 — <a href='https://alonzo.groupoid.space/'>Alonzo: STLC</a><br>
10 OCT 2016 — <a href='https://henk.groupoid.space/'>Henk: System P<sub>ω</sub></a><br>
02 JUN 2022 — <a href='https://per.groupoid.space/'>Per: MLTT</a><br>
26 JAN 2022 — <a href='https://anders.groupoid.space/'>Anders: Modal HoTT</a><br>
h2 Runtime
p.
Runtimes and effect type systems.
p.
31 DEC 2016 — <a href='runtime/cps/'>Certified CPS</a><br>
31 DEC 2016 — <a href='runtime/intercore/'>Intercore MP</a><br>
10 OCT 2016 — <a href='runtime/effects/'>Effects IOI/IO</a><br>
10 OCT 2016 — <a href='runtime/ftl/'>Formal Tensor FTL</a><br>
h1 Lecture Notes
p.
The series of articles on foundation and mathematics of Homotopy Type Theory.
p.
— <a href='articles/mltt/mltt.pdf'>Issue I: Internalizing MLTT</a><br>
— <a href='articles/hott/hott.pdf'>Issue III: Homotopy Type Theory</a><br>
— <a href='articles/topos/topos.pdf'>Issue VIII: Formal Set Topos</a><br>
— <a href='articles/pts/pts.pdf'>Addendum I: Pure Type System</a><br>
— <a href='articles/equ/equ.pdf'>Addendum II: Many Faces of Equality</a><br>
p.
<!-- Groupoid Infinity granted <a href="https://groupoid.space/misc/internship/">Internship Program</a>. <br -->
Groupoid Infinity is opening <a href="https://groupoid.space/institute/">Applied Math Institute</a>.
hr
include footer