From 5a23c59d2e60ea2210240574701f17b08dddba69 Mon Sep 17 00:00:00 2001 From: Ulrik Buchholtz Date: Thu, 17 Aug 2023 14:41:15 +0100 Subject: [PATCH] two more figures for the intro --- intro.tex | 55 +++++++++++++++++++++++++++++++++++++++++++++++-------- 1 file changed, 47 insertions(+), 8 deletions(-) diff --git a/intro.tex b/intro.tex index 9309abf..3499308 100644 --- a/intro.tex +++ b/intro.tex @@ -103,16 +103,55 @@ \chapter{Introduction to the topic of this book} The setup we adopt, ``HoTT'' or ``univalent foundations'', seems custom-built for such an approach. -In practice, one of the most important things is to be able to \emph{compare} symmetries of ``thing 1'' and ``thing 2''. In our case this amounts to nothing but a function the takes thing 1 to thing 2. -\begin{quote} - Picture of the function as in the original Goodnotes, putting some names to things. -\end{quote} +In practice, one of the most important things is to be able to \emph{compare} symmetries of ``thing~1'' and ``thing~2''. In our case this amounts to nothing but a function, $f$, that takes thing~1, $x_1$, to thing~2, $x_2$. +\begin{center} +\begin{tikzpicture} + \begin{scope}[scale=0.8] + \node (X) at (1,2) {$X_1$}; + \node[dot,label=left:$x_1$] (x1) at (0,0) {}; + \draw (0,-2) + .. controls ++(150:-1) and ++(180:1) .. (3,-2) + .. controls ++(180:-1) and ++(-100:1.3) .. (4.5,0) + .. controls ++(-100:-1.3) and ++(-10:2) .. (2,1.5) + .. controls ++(-10:-2) and ++(90:1) .. (-1,0) + .. controls ++(90:-1) and ++(150:1) .. (0,-2); + \draw[->] (x1) .. controls ++(80:-1) and ++(170:1) .. (.5,-1.5) + .. controls ++(170:-1) and ++(200:1) .. (3,-1.4) + .. controls ++(200:-1) and ++(-80:.5) .. (3.8,0) + .. controls ++(-80:-.5) and ++(-10:.3) .. (3,1) + .. controls ++(-10:-.3) and ++(80:1) .. (x1); + \draw (1,-1) arc(210:330:.8 and .5); + \draw (2.09,-1.18) arc(60:120:.8 and .7); + \draw (1.5,0) arc(210:330:.8 and .5); + \draw (2.59,-0.18) arc(60:120:.8 and .7); + \draw[->] (4.8,0) -- node[auto] {$f$} (6.3,0); + \end{scope} + \begin{scope}[xshift=6cm,scale=0.8] + \node (X) at (1,2) {$X_2$}; + \draw (0,-1) + .. controls ++(200:-1) and ++(180:1) .. (2,-2) + .. controls ++(180:-1) and ++(270:1) .. (4,0) + .. controls ++(270:-1) and ++(20:2) .. (2,2) + .. controls ++(20:-2) and ++(90:1) .. (-1,0) + .. controls ++(90:-1) and ++(200:1) .. (0,-1); + \node[dot,label=below:$x_2$] (x2) at (0,0) {}; + \draw[->] (x2) .. controls ++(-20:1.5) and ++(170:1) .. + (2,-1) .. controls ++(170:-1) and ++(-70:1) .. + (3.1,0) .. controls ++(-70:-1) and ++(90:.5) .. + (3.5,0) .. controls ++(90:-.5) and ++(-120:2) .. + (3,1) .. controls ++(-120:-2) and ++(-20:-1.5) .. (x2); + \draw (1,0) arc(210:330:.8 and .5); + \draw (2.09,-.18) arc(60:120:.8 and .7); + \end{scope} +\end{tikzpicture} +\end{center} While such comparisons of symmetries are traditionally handled by something called a ``group homomorphism'' which is a function satisfying a rather long list of axioms, in our case the only thing we need to know of the function is that it really does take thing 1 to thing 2 -- everything else then follows naturally. -Some important examples have provocatively simple representations in this framework. For instance, consider the circle -\begin{quote} - picture of the pointed circle -\end{quote} +Some important examples have provocatively simple representations in this framework. For instance, consider the circle shown in the margin.\marginnote{% + \begin{tikzpicture} + \node[dot,label=right:$x$] (base) at (1,0) {}; + \draw (0,0) circle (1); + \end{tikzpicture}} Since symmetries are interpreted as loops, you see that you have a loop for every integer -- the number $7$ can be represented by looping seven times counterclockwise. As we shall see, in our setup any loop is naturally identified with a unique integer (the ``winding number'' if you will). Everything you can wish to know about the structure of the ``group of integers'' is encoded in the circle. Another example is the ``free group of words in two letters $a$ and $b$''. This is represented by