Category of functors

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

The category of functors

Let and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} be two categories. The category of functors has

  1. Objects are functors
  2. A morphism of functors is a natural transformation ; i.e., for each object of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} , a morphism in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_U:F(U)\to G(U)} such that for all morphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f:U\to V} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^{op}} , the diagram (DIAGRAM) commutes.

A natural isomorphism is a natural transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_X=Mor_C(-,X)} .

Examples

  1. 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

  1. If is any contravariant functor , then the natural transformations of to are in correspondence with the elements of the set .
  2. 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 .