Closed category

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In category theory, a branch of mathematics, objects have set of morphisms between them, a closed category is a category where each set of morphisms is underlying to an object of the category itself.

[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.