Closed category

In , a branch of mathematics, a closed category is a special kind of . In a , the external hom (x, y) maps a pair of objects to a set of s. So in the , 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. A closed category can be defined as a with a so-called internal Hom functor with left Yoneda arrows natural in and and dinatural in , and a fixed object of with a natural isomorphism and a dinatural transformation all satisfying certain coherence conditions. are closed categories. In particular, any topos is closed. The canonical example is the . are closed categories. The canonical example is the FdVect with finite-dimensional vector spaces as objects and linear maps as morphisms. More generally, any is a closed category. In this case, the object is the monoidal unit.