From 9d3ff4605d30b434cdc0f302bb565606f1bcf722 Mon Sep 17 00:00:00 2001
From: Gregor Kleen <gkleen@yggdrasil.li>
Date: Wed, 3 Feb 2016 18:22:56 +0100
Subject: CatT 1.2.5

---
 provider/posts/simmons-intro-to-cat-t/1.2.md | 14 ++++++++++++--
 1 file changed, 12 insertions(+), 2 deletions(-)

(limited to 'provider')

diff --git a/provider/posts/simmons-intro-to-cat-t/1.2.md b/provider/posts/simmons-intro-to-cat-t/1.2.md
index 3e83b5f..b91cac4 100644
--- a/provider/posts/simmons-intro-to-cat-t/1.2.md
+++ b/provider/posts/simmons-intro-to-cat-t/1.2.md
@@ -106,10 +106,20 @@ Topological spaces $(S, \mathcal{O}_S)$, where $S$ is a Set and $\mathcal{O}_S \
 </div>
 
 <div class="exercise">
-Let $A$ be an object of a category $\ca{C}$.
-Show that $\hom{\ca{C}}{A}{A}$ is a monoid under composition.
+Show that $\End{\ca{C}}{A}$ is a monoid under composition, where $A$ is an object of a category $\ca{C}$.
 
 <div class="proof">
+$\circ$ is associative and total on $\End{\ca{C}}{A}$ by the definition of category and $\End{\ca{C}}{A}$ is obviously closed under $\circ$.
 
+$\idarr{A}$ is required to exist by the definition of category and indeed an identity of $\circ$ on $\End{\ca{C}}{A}$.
+</div>
+
+<div class="corollary">
+Each monoid $(M, \cdot)$ is a category ([again](./1.1.md)).
+
+<div class="proof">
+We construct a category with exactly one object $\ast$ and associate to every element $m \in M$ an arrow $\arr{\ast}{m}{\ast}$.
+We further define $m \circ m = m \ast m$.
+</div>
 </div>
 </div>
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