• Source: 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 Cop.
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 IC. 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 ΔX : X → X × X. Dually in a category with an initial object and finite coproducts every object becomes a monoid object via idX ⊔ idX : 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 MonC.




See also


Act-S, the category of monoids acting on sets




References



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

Kata Kunci Pencarian:

• Monoid
• Kategori (matematika)
• Kategori gelanggang
• Monad (teori kategori)
• Komposisi fungsi
• Produk (teori kategori)
• Rak dan ganjalan
• Monoid (category theory)
• Monoid
• Category (mathematics)
• Monad (category theory)
• Kernel (category theory)
• Monoidal category
• Pushout (category theory)
• Free monoid
• Semigroup
• Product (category theory)