# Monoid (category theory)

In category theory, a monoid (or monoid object) ${\displaystyle (M,\mu ,\eta )}$ in a monoidal category ${\displaystyle (\mathbf {C} ,\otimes ,I)}$ is an object M together with two morphisms

such that the pentagon diagram

and the unitor diagram

commute. In the above notations, I is the unit element and ${\displaystyle \alpha }$, ${\displaystyle \lambda }$ and ${\displaystyle \rho }$ 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 ${\displaystyle {\mathbf {C} }^{\mathrm {op} }}$.

Suppose that the monoidal category C has a symmetry ${\displaystyle \gamma }$. A monoid ${\displaystyle M}$ in C is symmetric when

${\displaystyle \mu \circ \gamma =\mu }$.

## Examples

• A monoid object in Set (with the monoidal structure induced by the cartesian product) is a monoid in the usual sense.
• A monoid object in Top (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) is a ring.
• For a commutative ring R, a monoid object in (R-Mod, ⊗R, R) is an R-algebra.
• A monoid object in K-Vect (again, with the tensor product) is a K-algebra, 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. A monoid object in [C,C] is a monad on C.

## Categories of monoids

Given two monoids ${\displaystyle (M,\mu ,\eta )}$ and ${\displaystyle (M',\mu ',\eta ')}$ in a monoidal category C, a morphism ${\displaystyle f:M\to M'}$ is a morphism of monoids when

The category of monoids in C and their monoid morphisms is written ${\displaystyle {\mathbf {Mon} }_{\mathbf {C} }}$.