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Monoid (category theory)

Mathematical concept in category theory

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Source updated August 22, 2026

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) $(M,\mu ,\eta )$ in a monoidal category $({\mathcal {C}},\otimes ,I)$ is an object $M$ together with two morphisms

$\mu \colon M\otimes M\to M$ called multiplication,
$\eta \colon I\to M$ 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 $\alpha ,\lambda$ and $\rho$ are respectively the associator, the left unitor and the right unitor of the monoidal category ${\mathcal {C}}$ .

Dually, a comonoid in a monoidal category ${\mathcal {C}}$ is a monoid in the dual category ${\mathcal {C}}^{\mathrm {op} }$ .

Suppose that the monoidal category ${\mathcal {C}}$ has a braiding $\gamma$ . A monoid $M$ in ${\mathcal {C}}$ is commutative when $\mu \circ \gamma =\mu$ .

01Examples

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. In this context:
- the unit object $I$ of the monoidal category can be taken to be any singleton.
- the multiplication $\mu$ corresponds to the monoid operation in the usual sense.
- the unit $\eta$ corresponds to the function that maps the single member of $I$ to the identity element in the monoid.
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 argument.
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 a 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 unital associative 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 a terminal object and finite products, every object becomes a comonoid object via the diagonal morphism Δ_X : X → X × X. Dually in a category with an initial object and finite coproducts every object becomes a monoid object via id_X ⊔ id_X : X ⊔ X → X.

02Categories 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 ∘ μ = μ′ ∘ (f ⊗ f),
f ∘ η = η′.

In other words, the following diagrams

commute.

The category of monoids in C and their monoid morphisms is written Mon_C.

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Sources and credits

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Monoid unit svg.svg by IkamusumeFan, CC BY-SA 4.0

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Monoid (category theory)

01Examples

02Categories of monoids

Sources and credits

Related topics

Semigroup action