# Monoid (category theory)

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For other uses, see Monoid (disambiguation).

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

• μ: MMM called multiplication,
• η: IM 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 Cop.

Suppose that the monoidal category C has a symmetry γ. A monoid M in C is commutative when μ o γ = μ.

## Categories of monoids

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

• f o μ = μ' o (ff),
• f o η = η'.

In other words, the following diagrams

,

commute.

The category of monoids in C and their monoid morphisms is written MonC.[1]

## See also

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

## References

1. ^ 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