Category of functors
This article focuses on the category of contravariant functors between two categories.
The category of functors
Let and be two categories. The category of functors has
- Objects are functors
- A morphism of functors is a natural transformation ; i.e., for each object of , a morphism in such that for all morphisms in , the diagram (DIAGRAM) commutes.
A natural isomorphism is a natural transformation such that is an isomorphism in for every object . One can verify that natural isomorphisms are indeed isomorphisms in the category of functors.
An important class of functors are the representable functors; i.e., functors that are naturally isomorphic to a functor of the form .
- In the theory of schemes, the presheaves are often referred to as the functor of points of the scheme X. Yoneda's lemma allows one to think of a scheme as a functor in some sense, which becomes a powerful interpretation; indeed, meaningful geometric concepts manifest themselves naturally in this language, including (for example) functorial characterizations of smooth morphisms of schemes.
The Yoneda lemma
Let be a category and let be objects of . Then
- If is any contravariant functor , then the natural transformations of to are in correspondence with the elements of the set .
- If the functors and are isomorphic, then and are isomorphic in . More generally, the functor , , is an equivalence of categories between and the full subcategory of representable functors in .