- Closed category
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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 B and C and dinatural in A
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 I 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
Categories:- Closed categories
- Category theory stubs
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