Category of functors

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

This article focuses on the category of contravariant functors between two categories.

## The category of functors[edit]

Let C and D be two categories. The category of functors Funct(Cop,Sets) has

1. Objects are functors F:Cop->D
2. A morphism of functors F,G is a natural transformation η:F->G; i.e., for each object U of C, a morphism in D ηU:F(U)->G(U) such that for all morphisms f:U->V in Cop, the diagram (DIAGRAM) commutes.

A natural isomorphism is a natural transformation η such that ηU is an isomorphism in D for every object U. 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 hX=MorC(−,X).

## Examples[edit]

1. In the theory of schemes, the presheaves hX 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[edit]

Let C be a category and let X,X′ be objects of C. Then

1. If F is any contravariant functor F:Cop->Sets, then the natural transformations of MorC(−,X) to F are in correspondence with the elements of the set F(X).
2. If the functors MorC(−,X) and MorC(−,X′) are isomorphic, then X and X′ are isomorphic in C. More generally, the functor h:C->Funct(Cop,Sets), X↦hX, is an equivalence of categories between C and the full subcategory of representable functors in Funct(Cop,Sets).