Monoid (category theory) explained

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

• μ: M ⊗ M → M called multiplication,
• η: I → M called unit,

such that the pentagon diagram

and the unitor diagram

commute. In the above notation, 1 is the identity morphism of M, I is the unit element and α, λ and ρ 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 C^op.

Suppose that the monoidal category C has a symmetry γ. A monoid M in C 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.
• 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.^[1]

See also

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

References

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

Notes and References

1. 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..