# Monoid (category theory)

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In category theory, a **monoid** (or **monoid object**) in a monoidal category is an object *M* together with two morphisms

such that the pentagon diagram

and the unitor diagram

commute. In the above notations, Template:Serif is the unit element and , and are respectively the associativity, the left identity and the right identity of the monoidal category **C**.

Dually, a **comonoid** in a monoidal category **C** is a monoid in the dual category **C**^{op}.

Suppose that the monoidal category **C** has a symmetry . A monoid in **C** is **symmetric** when

## Examples

- A monoid object in
**Set**(with the monoidal structure induced by the Cartesian product) is a monoid in the usual sense. - A monoid object in
**Top**(with the monoidal structure induced by the product topology) is a topological monoid. - A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton theorem.
- A monoid object in the category of complete join-semilattices
**Sup**(with the monoidal structure induced by the Cartesian product) is a unital quantale. - A monoid object in (
**Ab**, ⊗_{Z},**Z**) is a ring. - For a commutative ring
*R*, a monoid object in (*R*-**Mod**, ⊗_{R},*R*) is an*R*-algebra. - A monoid object in
*K*-**Vect**(again, with the tensor product) is a*K*-algebra, a comonoid object is a*K*-coalgebra. - For any category
*C*, the category [*C*,*C*] of its endofunctors has a monoidal structure induced by the composition. A monoid object in [*C*,*C*] is a monad on*C*.

## Categories of monoids

Given two monoids and in a monoidal category **C**, a morphism is a **morphism of monoids** when

The category of monoids in **C** and their monoid morphisms is written **Mon**_{C}.

## See also

## References

- Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov,
*Monoids, Acts and Categories*(2000), Walter de Gruyter, Berlin ISBN 3-11-015248-7