# 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

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 **C**^{op}.

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

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

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