Closed category

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

$\left[-\ -\right] : V^{op} \times V \to V$ ,

left Yoneda arrows natural in $B$ and $C$ and dinatural in $A$

$L : \left[B\ C\right] \to \left[\left[A\ B\right] \left[A\ C\right]\right]$

and a fixed object I of V such that there is a natural isomorphism

$i_A : A \cong \left[I\ A\right]$

and a dinatural transformation

$j_A : I \to \left[A\ A\right].\,$

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