# Monoid (category theory)

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (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 notation, 1 is the identity morphism of M, 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.