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'''.
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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]] <math>\mathbf{C}^{\mathrm{op}}</math>.
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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 cartesian product) is a [[monoid]] in the usual sense.
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* 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 '''[[Complete_lattice#Morphisms_of_complete_lattices|Sup]]''' (with the monoidal structure induced by the cartesian product) is a unital [[quantale]].
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* 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]]''', &otimes;<sub>'''Z'''</sub>, '''Z''') is a [[ring (mathematics)|ring]].
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* A monoid object in ('''[[category of abelian groups|Ab]]''',&otimes;<sub>'''Z'''</sub>, [[integer|'''Z''']]) is a [[ring (mathematics)|ring]].
* For a commutative ring ''R'', a monoid object in ('''[[category of modules|''R''-Mod]]''', &otimes;<sub>''R''</sub>, ''R'') is an [[R-algebra|''R''-algebra]].
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* For a [[commutative ring]] ''R'', a monoid object in ([[category of modules|''R''-'''Mod''']],&otimes;<sub>''R''</sub>,''R'') is an [[R-algebra|''R''-algebra]].
* 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]].
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* 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 ''[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''.
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* 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 <math>\mathbf{Mon}_\mathbf{C}</math>.
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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

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

.

Examples

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

References

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