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authorGregor Kleen <gkleen@yggdrasil.li>2016-02-03 18:22:56 +0100
committerGregor Kleen <gkleen@yggdrasil.li>2016-02-03 18:22:56 +0100
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CatT 1.2.5
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@@ -106,10 +106,20 @@ Topological spaces $(S, \mathcal{O}_S)$, where $S$ is a Set and $\mathcal{O}_S \
106</div> 106</div>
107 107
108<div class="exercise"> 108<div class="exercise">
109Let $A$ be an object of a category $\ca{C}$. 109Show that $\End{\ca{C}}{A}$ is a monoid under composition, where $A$ is an object of a category $\ca{C}$.
110Show that $\hom{\ca{C}}{A}{A}$ is a monoid under composition.
111 110
112<div class="proof"> 111<div class="proof">
112$\circ$ is associative and total on $\End{\ca{C}}{A}$ by the definition of category and $\End{\ca{C}}{A}$ is obviously closed under $\circ$.
113 113
114$\idarr{A}$ is required to exist by the definition of category and indeed an identity of $\circ$ on $\End{\ca{C}}{A}$.
115</div>
116
117<div class="corollary">
118Each monoid $(M, \cdot)$ is a category ([again](./1.1.md)).
119
120<div class="proof">
121We construct a category with exactly one object $\ast$ and associate to every element $m \in M$ an arrow $\arr{\ast}{m}{\ast}$.
122We further define $m \circ m = m \ast m$.
123</div>
114</div> 124</div>
115</div> 125</div>