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

• called multiplication,
• 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 and 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 .

Suppose that the monoidal category has a braiding . A monoid in 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.
• the multiplication corresponds to the monoid operation in the usual sense.
• the unit 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 .

Categories of monoids

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

• f ∘ μ = μ′ ∘ (f ⊗ f),
• f ∘ η = η′.

In other words, the following diagrams

,

commute.

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

See also

• Act-S, the category of monoids acting on sets

References