Monoid (category theory)
- called multiplication,
- and called unit,
such that the pentagon diagram
and the unitor diagram
commute. In the above notations, I is the unit element and , and 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 Cop.
Suppose that the monoidal category C has a symmetry . A monoid in C is symmetric when
- 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 and in a monoidal category C, a morphism is a morphism of monoids when
The category of monoids in C and their monoid morphisms is written MonC.
- Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, Monoids, Acts and Categories (2000), Walter de Gruyter, Berlin ISBN 3-11-015248-7