Difference between revisions of "Monoid (category theory)"
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| − | 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
- called multiplication,
- and called unit,
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 Cop.
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 MonC.
See also
References
- Mati Kilp, Ulrich Knauer, Alexander V. Mikhalov, Monoids, Acts and Categories (2000), Walter de Gruyter, Berlin ISBN 3-11-015248-7