Closed category
In category theory, a branch of mathematics, a closed category is a special kind of category.
In a locally small category, the external hom (x, y) maps a pair of objects to a set of morphisms. So in the category of sets, this is an object of the category itself. In the same vein, in a closed category, the (object of) morphisms from one object to another can be seen as lying inside the category. This is 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 with a so-called internal Hom functor
- ,
with left Yoneda arrows natural in and and dinatural in
and with 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
- Closed category at the nLab