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 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 .
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 MonC.Section VII.3 in
See also Act-S, the category of monoids acting on sets
References