Monoid (category theory) explained

together with two morphisms

\mu\colonM ⊗ M\toM

called multiplication,

η\colonI\toM

called unit,such that the pentagon diagram

and the unitor diagram

commute. In the above notation,

is the identity morphism of

is the unit element and

\alpha,λ

and

\rho

are respectively the associator, the left unitor and the right unitor of the monoidal category

Dually, a comonoid in a monoidal category

is a monoid in the dual category

lC^op

Suppose that the monoidal category

has a braiding

\gamma

. A monoid

is commutative when .

Examples

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:
- the unit object

of the monoidal category can be taken to be any singleton.

\mu

corresponds to the monoid operation in the usual sense.

corresponds to the function that maps the single member of

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 I_C. 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 Mon_C.^[1]

Book: Mati . Kilp . Ulrich . Knauer . Alexander V. . Mikhalov . Monoids, Acts and Categories . 2000 . Walter de Gruyter . 3-11-015248-7.

Section VII.3 in Book: Mac Lane. Saunders. Categories for the working mathematician. 1988. Springer-Verlag. New York. 0-387-90035-7. 4th corr. print..