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Monoid (category theory)

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Monoid (category theory)

In category theory, a monoid (or monoid object) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms


  • μ: MMM called multiplication,
  • η: IM called unit,
  • such that the pentagon diagram

    and the unitor diagram

    commute. In the above notations, 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 o γ = μ.


  • 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 theorem.
  • 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 an R-algebra.
  • the category of graded modules is a graded R-algebra.
  • the category of chain complexes is a dg-algebra.
  • A monoid object in K-Vect, the category of vector spaces (again, with the tensor product), is a 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 finite products, every object becomes a comonoid object via the diagonal morphism Δ X : X X × X . Dually in a category with finite coproducts every object becomes a monoid object via i d X i d 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 : MM ' is a morphism of monoids when

  • f o μ = μ' o (ff),
  • f o η = η'.
  • In other words, the following diagrams



    The category of monoids in C and their monoid morphisms is written MonC.


    Monoid (category theory) Wikipedia

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