# Definition:Monoid Category

## Definition

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.