Monoid (category theory)

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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: MMM called multiplication,
  • η: IM called unit,

such that the pentagon diagram

Monoid multiplication.svg

and the unitor diagram

Monoid unit svg.svg

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 o γ = μ.


Categories of monoids[edit]

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

Category monoids mu.svg, Category monoids eta.svg


The category of monoids in C and their monoid morphisms is written MonC.[1]

See also[edit]

  • Act-S, the category of monoids acting on sets


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