Monoid (category theory) on arvau

Wikipedia / Monoid (category theory)

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object together with two morphisms

such that the pentagon diagram

and the unitor diagram

Dually, a comonoid is a monoid in the dual category.

has a braiding is commutative when .

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:

of the monoidal category can be taken to be any singleton.
corresponds to the monoid operation in the usual sense.
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 , the category of abelian groups, is a ring.

For a commutative ring R, a monoid object in

, 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 of its endofunctors has a monoidal structure induced by the composition and the identity functor IC. A monoid object in 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 . Dually in a category with an initial object and finite coproducts every object becomes a monoid object via .

Given two monoids and in a monoidal category C, a morphism is a morphism of monoids when

In other words, the following diagrams

commute.

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