# 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 of 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 in
*nLab*