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 $(𝒞, \otimes, I)$ is an object $M$ together with two morphisms

$μ : M \otimes M \to M$ called multiplication,
$η : I \to M$ 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 associator, the left unitor and the right unitor of the monoidal category $𝒞$ .

Dually, a comonoid in a monoidal category $𝒞$ is a monoid in the dual category $𝒞^{o p}$ .

Suppose that the monoidal category $𝒞$ has a braiding $γ$ . A monoid $M$ in $𝒞$ is commutative when $μ \circ γ = μ$ .

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. In this context:
- the unit object $I$ of the monoidal category can be taken to be any singleton.
- the multiplication $μ$ corresponds to the monoid operation in the usual sense.
- the unit $η$ corresponds to the function that maps the single member of $I$ 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 (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.

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.

Kilp, Mati; Knauer, Ulrich; Mikhalov, Alexander V. (2000). Monoids, Acts and Categories. Walter de Gruyter. ISBN 3-11-015248-7.

[1] 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.

Monoid (category theory)

Examples

Categories of monoids

See also

References