Monoid (category theory)

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

$\mu :M\otimes M\to M$ called multiplication,
and $\eta :I\to M$ called unit,

such that the diagrams

and

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 $\mathbf {C} ^{\mathrm {op} }$ .

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

\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.