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Replace xy with tikz in subgroups.tex #180

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Replace xy with tikz in subgroups.tex #180

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162d9a5
xy to tikz: triangle in sec 5.2.1
favonia Feb 11, 2023
68465eb
xy to tikz: triangular commuting diagram of two monos
favonia Feb 11, 2023
7166e65
xy to tikz: convert the Ker f diagram
favonia Feb 12, 2023
6148462
xy to tikz: convert the preimage diagram
favonia Feb 12, 2023
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78 changes: 66 additions & 12 deletions subgroups.tex
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Expand Up @@ -59,8 +59,17 @@ \subsection{Subgroups as monomorphisms}
\label{ex:sigma2inSigma3}
\marginnote{
That $i:\Sigma_2\to\Sigma_3$ is a monomorphism can visualized as follows: if $\Sigma_3$ represent all symmetries of an equilateral triangle in the plane (with vertices $1$, $2$, $3$), then $i$ is represented by the inclusion of the symmetries leaving $3$ fixed; \ie reflection through the line marked with dots in the picture.
$$\xymatrix{&3\ar@{.}[dd]&\\&&\\
1\ar@{-}[uur]\ar@{-}[rr]&&2\ar@{-}[uul]}$$}
\[
\begin{tikzpicture}[scale=1.5]
\path (0:0) node (one) {$1$}
(0:2) node (two) {$2$}
(60:2) node (three) {$3$};
\draw (one) -- (two);
\draw (two) -- (three);
\draw (three) -- (one);
\draw[dotted] (three) -- (0:1);
\end{tikzpicture}
\]}
Consider the homomorphism $i:\Sigma_2\to\Sigma_3$ of permutation groups corresponding to sending $A:\BSG_2\defequi \FinSet_2$ to $A+\bn1:\BSG_3$.
%This is a monomorphism since $\US i:\USym\Sigma_2\to\USym\Sigma_3$ is an injection.
\end{example}
Expand All @@ -76,8 +85,18 @@ \subsection{Subgroups as monomorphisms}
\begin{lemma}
\label{lem:setofsubgroups}
Let $G$ be a group and $(H,i_H,!),(H',i_{H'},!):\typemono_G$ be two monomorphisms into $G$. The identity type $(H,i_H,!)\eqto{}(H',i_{H'},!)$ is equivalent to
\marginnote{$$\xymatrix{H\ar[rr]^f_\simeq\ar[dr]_{i_H}&&H'\ar[dl]^{i_{H'}}\\
&G&}$$}
\marginnote{
\[
\begin{tikzpicture}[scale=1.5]
\path (-1,0) node (H) {$H$}
(1,0) node (H') {$H'$}
(0,-1) node (G) {$G$};
\draw[->] (H) -- node[above] {$f$} node[below] {$\simeq$} (H');
\draw[->] (H) -- node[below left] {$i_H$} (G);
\draw[->] (H') -- node[below right] {$i_{H'}$} (G);
\end{tikzpicture}
\]
}
$$\sum_{f:\Hom(H,H')}\isEq(\US f)\times (i_{H'}\eqto{}i_H f)$$ and is a proposition.
In particular, the type $\typemono_G$ of monomorphisms into $G$ is a set.
\end{lemma}
Expand Down Expand Up @@ -293,9 +312,23 @@ \subsection{Kernels and cokernels}
\begin{xca}
Given a homomorphism $f:\Hom(G,G')$, prove that
\marginnote{Hint: consider the corresponding property of the preimage of $\Bf$.
$$\xymatrix{L\ar[drr]^h\ar@{.>}[dr]^{k}\ar[ddr]&&\\
&\Ker f\ar[r]_{\incl_{\ker f}}\ar[d]&G\ar[d]^f\\
&{1}\ar[r]&\,G'.}$$}
\[
\begin{tikzpicture}[scale=1.5]
\path (-1,1) node (L) {$L$}
(0,0) node (Ker) {$\Ker f$}
(1,0) node (G) {$G$}
(0,-1) node (one) {$1$}
(1,-1) node (G') {$G'$};
\draw[->,dotted] (L) -- node[above right] {$k$} (Ker);
\draw[->] (L) to[bend left] node[above right] {$h$} (G);
\draw[->] (L) to[bend right] (one);
\draw[->] (Ker) -- node[below] {$\incl_{\ker f}$} (G);
\draw[->] (Ker) -- (one);
\draw[->] (G) -- node[right] {$f$} (G');
\draw[->] (one) -- (G');
\end{tikzpicture}
\]
}
\begin{enumerate}
\item $f$ is a monomorphism if and only if the kernel is trivial
\item $f$ is an epimorphims if and only if the cokernel is contractible.
Expand All @@ -307,11 +340,32 @@ \subsection{Kernels and cokernels}
\end{xca}


The kernel, cokernel and image constructions satisfy a lot of important relations which we will review in a moment, but in our setup many of them are just complicated ways of interpreting the following fact about preimages (see the illustration\footnote{$$\xymatrix{
F_2^{-1}(x_1,p_2)\ar[r]^H_\simeq\ar[d]_{\fst}&f_1^{-1}(x_1)\ar[d]^{\fst}\ar[dl]_{F_1}&\\
(f_2f_1)^{-1}(x_2)\ar[r]^{\fst}\ar[d]^{F_2}&X_0\ar[r]^{f_2f_1}\ar[d]^{f_1}&X_2\ar@{=}[d]\\
f_2^{-1}(x_2)\ar[r]^{\fst}&X_1\ar[r]^{f_2}&X_2.}
$$} in the margin for an overview)
The kernel, cokernel and image constructions satisfy a lot of important relations which we will review in a moment, but in our setup many of them are just complicated ways of interpreting the following fact about preimages (see the illustration\footnote{
\[
\begin{tikzpicture}[scale=1.5]
\path (-.5,2) node (02) {$F_2^{-1}(x_1,p_2)$}
(1,2) node (12) {$f_1^{-1}(x_1)$}
(-.5,1) node (01) {$(f_2f_1)^{-1}(x_2)$}
(1,1) node (11) {$X_0$}
(2,1) node (21) {$X_2$}
(-.5,0) node (00) {$f_2^{-1}(x_2)$}
(1,0) node (10) {$X_1$}
(2,0) node (20) {$X_2$};
\draw[->]
(02) edge node[left] {$\fst$} (01)
(01) edge node[left] {$F_2$} (00)
(12) edge node[right] {$\fst$} (11)
(11) edge node[right] {$f_1$} (10)
(21) edge[-,double] (20)
(02) edge node[above] {$H$} node[below] {$\simeq$} (12)
(01) edge node[above] {$\fst$} (11)
(11) edge node[above] {$f_2f_1$} (21)
(00) edge node[above] {$\fst$} (10)
(10) edge node[above] {$f_2$} (20)
(12) edge node[above left] {$F_1$} (01);
\end{tikzpicture}
\]
} in the margin for an overview)
\begin{lemma}
\label{lem:fibersofcomposites}
Consider pointed functions $(f_1,p_1):(X_0,x_0)\to_*(X_1,x_1)$ and $(f_2,p_2):(X_1,x_1)\to_*(X_2,x_2)$ and the resulting functions
Expand Down