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

References

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