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 \mathcal{C} with a so-called internal Hom functor with left Yoneda arrows natural in B and C and dinatural in A, and a fixed object I of \mathcal{C} with a natural isomorphism and a dinatural transformation all satisfying certain coherence conditions.

Examples