Definition:Monoid Category

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Let $\left({S, \circ}\right)$ be a monoid with identity $e_S$.

One can interpret $\left({S, \circ}\right)$ as being a category, with:

Objects:         Only one, say $*$
Morphisms: $a: * \to *$, for all $a \in S$
Composition: $a \circ b: * \to *$ is defined using the operation $\circ$ of the monoid $S$
Identity morphisms: $\operatorname{id}_* := e_S: * \to *$

The category that so arises is called a monoid category.

Also see