Monoid (category theory)

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In category theory, a monoid (or monoid object) in a monoidal category is an object M together with two morphisms

such that the pentagon diagram

Monoid mult.png

and the unitor diagram

Monoid unit.png

commute. In the above notations, Template:Serif is the unit element and , and 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 . A monoid in C is symmetric when



Categories of monoids

Given two monoids and in a monoidal category C, a morphism is a morphism of monoids when

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

See also

  • monoid (non-categorical definition)
  • Act-S, the category of monoids acting on sets


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