
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 formula_1 with a so-called internal Hom functor
with left Yoneda arrows
natural in formula_4 and formula_5 and dinatural in formula_6, and a fixed object formula_7 of formula_1 with a natural isomorphism
and a dinatural transformation
all satisfying certain coherence conditions.