# Monoid (category theory)

In category theory, a branch of mathematics, a **monoid** (or **monoid object**) (*M*, *μ*, *η*) in a monoidal category (**C**, ⊗, *I*) is an object *M* together with two morphisms

*μ*:*M*⊗*M*→*M*called*multiplication*,*η*:*I*→*M*called*unit*,

such that the pentagon diagram

and the unitor diagram

commute. In the above notations, *I* is the unit element and α, λ and ρ 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 *γ*. A monoid *M* in **C** is **commutative** when *μ* o *γ* = *μ*.

## Examples

- A monoid object in
**Set**, the category of sets (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense. - A monoid object in
**Top**, the category of topological spaces (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**), the category of abelian groups, is a ring. - For a commutative ring
*R*, a monoid object in- (
*R*-**Mod**, ⊗_{R},*R*), the category of modules over*R*, is an*R*-algebra. - the category of graded modules is a graded
*R*-algebra. - the category of chain complexes of
*R*-modules is a differential graded algebra.

- (
- A monoid object in
*K*-**Vect**, the category of*K*-vector spaces (again, with the tensor product), is a*K*-algebra, and 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 and the identity functor*I*_{C}. A monoid object in [*C*,*C*] is a monad on*C*. - For any category with finite products, every object becomes a comonoid object via the diagonal morphism . Dually in a category with finite coproducts every object becomes a monoid object via .

## Categories of monoids

Given two monoids (*M*, *μ*, *η*) and (*M'*, *μ'*, *η'*) in a monoidal category **C**, a morphism *f* : *M* → *M* ' is a **morphism of monoids** when

*f*o*μ*=*μ'*o (*f*⊗*f*),*f*o*η*=*η'*.

In other words, the following diagrams

commute.

The category of monoids in **C** and their monoid morphisms is written **Mon**_{C}.[1]

## See also

- Act-S, the category of monoids acting on sets

## References

- Section VII.3 in Mac Lane, Saunders (1988).
*Categories for the working mathematician*(4th corr. print. ed.). New York: Springer-Verlag. ISBN 0-387-90035-7.

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