Monoid (category theory)

In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) 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 μ ∘ γ = μ.

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 (Ab, ⊗_Z, Z), the category of abelian groups, is a ring.
For a commutative ring R, a monoid object in
- (R-Mod, ⊗_R, R), 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 [C, C] of its endofunctors has a monoidal structure induced by the composition and the identity functor I_C. A monoid object in [C, C] 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 Δ_X : X → X × X. Dually in a category with an initial object and finite coproducts every object becomes a monoid object via id_X ⊔ id_X : X ⊔ X → X.

Categories of monoids

Given two monoids (M, μ, η) and (M′, μ′, η′) in a monoidal category C, a morphism f : M → M′ 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]

References

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