From 15e7a756f4bd0c06259c7c97c8191c15927c67c6 Mon Sep 17 00:00:00 2001
From: Philip Kaludercic <philip.kaludercic@fau.de>
Date: Fri, 22 Mar 2024 14:50:51 +0100
Subject: [PATCH] Improve typesetting in the digraph example

---
 background.tex | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/background.tex b/background.tex
index 93b720c..3a88efc 100644
--- a/background.tex
+++ b/background.tex
@@ -110,15 +110,16 @@
 
 \begin{example}\label{ex:gra}
   The \de{category} \(\Gra\) has
-  \href{https://en.wikipedia.org/wiki/Directed_graph}{digraphs}
+  \href{https://en.wikipedia.org/wiki/Directed_graph}{di(-rected )graphs}
   \((V, E)\) as \des{object} and
   \href{https://en.wikipedia.org/wiki/Graph_homomorphism}{graph
     homomorphisms} as \des{morphism}.  That means that a \de{morphism}
-  \(f : \map{X}{Y}\) have to preserve
+  \(f : \map{\mathfrak{A}}{\mathfrak{B}}\) have to preserve
   \href{https://en.wikipedia.org/wiki/Strongly_connected_component}{strongly
     connected components}, i.e.\
-  \[ \forall a, b \in V(X) \st a ~_{E(X)} b \implies f(a) ~_{E(Y)}
-    f(b).\]
+  \[ \forall a, b \in V(X) \st a \sim_{E(\mathfrak{A})} b \implies f(a)
+    \sim_{E(\mathfrak{B})} f(b),\] where \(x \sim_{E{\mathfrak{G}}} y\) says that
+  there is a path from \(x\) to \(y\) in the digraph \(\mathfrak{G}\).
 
   The \defn{initial}{initial object} are therefore the empty graph
   \(V = \{\}\), since there are no components to be preserved, and the
-- 
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