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 objectss 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 with a so-called internal Hom functor

with left Yoneda arrows

natural and dinatural with a natural isomorphism

and a dinatural transformation

1. ,

all satisfying certain coherence conditions.

Examples

• Cartesian closed categories are closed categories. In particular, any elementary 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.

• More generally, any monoidal closed category is the monoidal unit.

References