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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

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

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

$L:\left[BC\right]\to \left[\left[AB\right]\left[AC\right]\right]$

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

$i_A:A\cong \left[IA\right]$

and a dinatural transformation

$j_A:I\to \left[AA\right].$

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
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