Definition:Monoid Category

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

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


 * objects: Only one, say $*$;
 * morphisms: For all $a \in S$, a morphism $a: * \to *$;
 * identities: The morphism $e_S: * \to *$ is $\operatorname{id}_*: * \to *$;
 * composition: The monoid operation $\circ$ defines composition.

The category that so arises is called a monoid category.