From 61484624778535f47eb0567c96e01e38ba1ce5fd Mon Sep 17 00:00:00 2001 From: favonia Date: Sun, 12 Feb 2023 12:17:13 -0600 Subject: [PATCH] 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