# Definition:Monoid Category

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## 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**.

## Also see

- Monoid Category is Category
- Results about
**monoid categories**can be found here.

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 1.4.13$