Definition:Monoid Category
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Definition
Let $\struct {S, \circ}$ be a monoid with identity $e_S$.
One can interpret $\struct {S, \circ}$ 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.
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Also see
- Monoid Category is Category
- Results about monoid categories can be found here.
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.13$