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: *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.

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 is a dg-algebra.
A monoid object in *K*-**Vect**, the category of 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
Δ
X
:
X
→
X
×
X
. Dually in a category with finite coproducts every object becomes a monoid object via
i
d
X
⊔
i
d
X
:
X
⊔
X
→
X
.

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}.