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.

[edit] 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 natural arrows

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

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

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

and a natural transformation

$j_A : I \to \left[A\ A\right].\,$

[edit] 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.