# Difference between revisions of "Monoid (category theory)"

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:[[Image:Monoid unit.png]] | :[[Image:Monoid unit.png]] | ||

− | commute. In the above notations, ''I'' is the unit element and <math>\alpha</math>, <math>\lambda</math> and <math>\rho</math> are respectively the associativity, the left identity and the right identity of the monoidal category '''C'''. | + | commute. In the above notations, {{serif|''I''}} is the unit element and <math>\alpha</math>, <math>\lambda</math> and <math>\rho</math> 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]] < | + | Dually, a '''comonoid''' in a monoidal category '''C''' is a monoid in the [[dual category]] '''C'''<sup>op</sup>. |

Suppose that the monoidal category '''C''' has a [[symmetric monoidal category|symmetry]] <math>\gamma</math>. A monoid <math>M</math> in '''C''' is '''symmetric''' when | Suppose that the monoidal category '''C''' has a [[symmetric monoidal category|symmetry]] <math>\gamma</math>. A monoid <math>M</math> in '''C''' is '''symmetric''' when | ||

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== Examples == | == Examples == | ||

− | * A monoid object in '''[[category of sets|Set]]''' (with the monoidal structure induced by the | + | * A monoid object in '''[[category of sets|Set]]''' (with the monoidal structure induced by the Cartesian product) is a [[monoid]] in the usual sense. |

* A monoid object in '''[[category of topological spaces|Top]]''' (with the monoidal structure induced by the [[product topology]]) is a [[topological monoid]]. | * A monoid object in '''[[category of topological spaces|Top]]''' (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 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 '''[[ | + | * A monoid object in the category of complete join-semilattices '''[[Complete lattice#Morphisms of complete lattices|Sup]]''' (with the monoidal structure induced by the Cartesian product) is a unital [[quantale]]. |

− | * A monoid object in ('''[[category of abelian groups|Ab]]''', ⊗<sub>'''Z'''</sub>, '''Z''') is a [[ring (mathematics)|ring]]. | + | * A monoid object in ('''[[category of abelian groups|Ab]]''', ⊗<sub>'''Z'''</sub>, [[integer|'''Z''']]) is a [[ring (mathematics)|ring]]. |

− | * For a commutative ring ''R'', a monoid object in ( | + | * For a [[commutative ring]] ''R'', a monoid object in ([[category of modules|''R''-'''Mod''']], ⊗<sub>''R''</sub>, ''R'') is an [[R-algebra|''R''-algebra]]. |

− | * A monoid object in | + | * A monoid object in [[category of vector spaces|''K''-'''Vect''']] (again, with the tensor product) is a ''K''-[[algebra over a field|algebra]], a comonoid object is a ''K''-[[coalgebra]]. |

− | * For any category ''C'', the category '' | + | * For any category ''C'', the category [''C'',''C''] of its [[endofunctor]]s has a monoidal structure induced by the composition. A monoid object in [''C'',''C''] is a [[monad (category theory)|monad]] on ''C''. |

== Categories of monoids == | == Categories of monoids == | ||

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* <math>f\circ\eta = \eta'</math>. | * <math>f\circ\eta = \eta'</math>. | ||

− | The category of monoids in '''C''' and their monoid morphisms is written < | + | The category of monoids in '''C''' and their monoid morphisms is written '''Mon'''<sub>'''C'''</sub>. |

== See also == | == See also == |

## Latest revision as of 12:43, 14 February 2014

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

and the unitor diagram

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 **C**^{op}.

Suppose that the monoidal category **C** has a symmetry . A monoid in **C** is **symmetric** when

## Examples

- A monoid object in
**Set**(with the monoidal structure induced by the Cartesian product) is a monoid in the usual sense. - A monoid object in
**Top**(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**) is a ring. - For a commutative ring
*R*, a monoid object in (*R*-**Mod**, ⊗_{R},*R*) is an*R*-algebra. - A monoid object in
*K*-**Vect**(again, with the tensor product) is a*K*-algebra, 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. A monoid object in [*C*,*C*] is a monad on*C*.

## 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 **Mon**_{C}.

## See also

## References

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