From 162d9a5fc6437e13cc774cb1056c7c8ee11a9d1d Mon Sep 17 00:00:00 2001 From: favonia Date: Sat, 11 Feb 2023 08:01:22 -0600 Subject: [PATCH 1/4] xy to tikz: triangle in sec 5.2.1 The style is slightly different, but now we have a real equilateral triangle. --- subgroups.tex | 13 +++++++++++-- 1 file changed, 11 insertions(+), 2 deletions(-) diff --git a/subgroups.tex b/subgroups.tex index 9e7e691..37bd86c 100644 --- a/subgroups.tex +++ b/subgroups.tex @@ -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} From 68465eb03892183dd6c72cfa416e9b6ea58f3cc9 Mon Sep 17 00:00:00 2001 From: favonia Date: Sat, 11 Feb 2023 08:47:49 -0600 Subject: [PATCH 2/4] xy to tikz: triangular commuting diagram of two monos --- subgroups.tex | 14 ++++++++++++-- 1 file changed, 12 insertions(+), 2 deletions(-) diff --git a/subgroups.tex b/subgroups.tex index 37bd86c..8d0cb4b 100644 --- a/subgroups.tex +++ b/subgroups.tex @@ -85,8 +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} From 7166e658ded47017fa9b21ac794f72af547adefb Mon Sep 17 00:00:00 2001 From: favonia Date: Sun, 12 Feb 2023 11:15:03 -0600 Subject: [PATCH 3/4] xy to tikz: convert the Ker f diagram --- subgroups.tex | 20 +++++++++++++++++--- 1 file changed, 17 insertions(+), 3 deletions(-) diff --git a/subgroups.tex b/subgroups.tex index 8d0cb4b..f35b204 100644 --- a/subgroups.tex +++ b/subgroups.tex @@ -312,9 +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. From 61484624778535f47eb0567c96e01e38ba1ce5fd Mon Sep 17 00:00:00 2001 From: favonia Date: Sun, 12 Feb 2023 12:17:13 -0600 Subject: [PATCH 4/4] xy to tikz: convert the preimage diagram --- subgroups.tex | 31 ++++++++++++++++++++++++++----- 1 file changed, 26 insertions(+), 5 deletions(-) diff --git a/subgroups.tex b/subgroups.tex index f35b204..9e170c1 100644 --- a/subgroups.tex +++ b/subgroups.tex @@ -340,11 +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