xy to tikz: convert the preimage diagram

Author: favonia

subgroups.tex

@@ -337,11 +337,32 @@ \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