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In [[category theory]], a branch of [[mathematics]], a '''closed category''' is a special kind of [[category (mathematics)|category]]. | |||
In any category (more precisely, in any [[locally small category]]), the morphisms between any two given objects ''x'' and ''y'' comprise a [[set (mathematics)|set]], the ''external hom'' (''x'', ''y''). In a closed category, these morphisms can be seen as comprising an object of the category itself, the ''internal hom'' [''x'',''y'']. | |||
Every closed category has a [[forgetful functor]] to the [[category of sets]], which in particular takes the internal hom to the external hom. | |||
==Definition== | |||
A '''closed category''' can be defined as a [[category (mathematics)|category]] V with a so-called [[internal Hom functor]] | |||
:<math>\left[-\ -\right] : V^{op} \times V \to V</math> , | |||
left [[Yoneda arrow]]s [[natural transformation|natural]] in <math>B</math> and <math>C</math> and [[dinatural transformation|dinatural]] in <math>A</math> | |||
:<math>L : \left[B\ C\right] \to \left[\left[A\ B\right] \left[A\ C\right]\right]</math> | |||
and a fixed object I of V such that there is a [[natural isomorphism]] | |||
:<math>i_A : A \cong \left[I\ A\right]</math> | |||
and a [[dinatural transformation]] | |||
:<math>j_A : I \to \left[A\ A\right].\, </math> | |||
==Examples== | |||
*[[Cartesian closed category|Cartesian closed categories]] are closed categories. In particular, any [[topos]] is closed. The canonical example is the [[category of sets]]. | |||
*[[Compact closed category|Compact closed categories]] are closed categories. The canonical example is the [[category (mathematics)|category]] '''FdVect''' with finite dimensional [[vector spaces]] as objects and [[linear maps]] as morphisms. | |||
*More generally, any [[monoidal closed category]] is a closed category. In this case, the object <math>I</math> is the monoidal unit. | |||
== References == | |||
* Eilenberg, S. & Kelly, G.M. ''Closed categories'' Proceedings of the Conference on Categorical Algebra. (La Jolla, 1965) Springer. 1966. pp. 421–562 | |||
* {{nlab|id=closed+category|title=Closed category}} | |||
{{Portal|Category theory}} | |||
[[Category:Closed categories| ]] | |||
{{categorytheory-stub}} | |||
Revision as of 16:59, 24 April 2013
In category theory, a branch of mathematics, a closed category is a special kind of category.
In any category (more precisely, in any locally small category), the morphisms between any two given objects x and y comprise a set, the external hom (x, y). In a closed category, these morphisms can be seen as comprising an object of the category itself, the internal hom [x,y].
Every closed category has a forgetful functor to the category of sets, which in particular takes the internal hom to the external hom.
Definition
A closed category can be defined as a category V with a so-called internal Hom functor
left Yoneda arrows natural in and and dinatural in
and a fixed object I of V such that there is a natural isomorphism
and a dinatural transformation
Examples
- Cartesian closed categories are closed categories. In particular, any topos is closed. The canonical example is the category of sets.
- Compact closed categories are closed categories. The canonical example is the category FdVect with finite dimensional vector spaces as objects and linear maps as morphisms.
- More generally, any monoidal closed category is a closed category. In this case, the object is the monoidal unit.
References
- Eilenberg, S. & Kelly, G.M. Closed categories Proceedings of the Conference on Categorical Algebra. (La Jolla, 1965) Springer. 1966. pp. 421–562
- Template:Nlab
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