# Monoid (category theory)

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

$\mu \circ \gamma =\mu$ .

## 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

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