monoid (category theory)
{{Short description|Mathematical concept in category theory}}
{{for|the algebraic structure|Monoid}}
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) {{nowrap|(M, μ, η)}} in a monoidal category {{nowrap|(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
: Image:Monoid multiplication.svg
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 Cop.
Suppose that the monoidal category C has a braiding γ. A monoid M in C is commutative when {{nowrap|1=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 {{nowrap|(Ab, ⊗Z, Z)}}, the category of abelian groups, is a ring.
- For a commutative ring R, a monoid object in
- {{nowrap|(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 {{nowrap|[C, C]}} of its endofunctors has a monoidal structure induced by the composition and the identity functor IC. A monoid object in {{nowrap|[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 {{nowrap|ΔX : X → X × X}}. Dually in a category with an initial object and finite coproducts every object becomes a monoid object via {{nowrap|idX ⊔ idX : X ⊔ X → X}}.
Categories of monoids
Given two monoids {{nowrap|(M, μ, η)}} and {{nowrap|(M′, μ′, η′)}} in a monoidal category C, a morphism {{nowrap|f : M → M′}} is a morphism of monoids when
- f ∘ μ = μ′ ∘ (f ⊗ f),
- f ∘ η = η′.
In other words, the following diagrams
File:Category monoids mu.svg, File:Category monoids eta.svg
commute.
The category of monoids in C and their monoid morphisms is written MonC.Section VII.3 in {{cite book|last1=Mac Lane|first1=Saunders|title=Categories for the working mathematician|date=1988|publisher=Springer-Verlag|location=New York|isbn=0-387-90035-7|edition=4th corr. print.}}
See also
- Act-S, the category of monoids acting on sets
References
{{Reflist}}
- {{cite book |first1=Mati |last1=Kilp |first2=Ulrich |last2=Knauer |first3=Alexander V. |last3=Mikhalov |title=Monoids, Acts and Categories |date=2000 |publisher=Walter de Gruyter |isbn=3-11-015248-7}}