Category of functors

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

The category of functors

Let $C$ 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 $Funct(C^{op},Sets)$ has

1. Objects are functors $F:C^{op}\to D$
2. A morphism of functors $F,G$ is a natural transformation $\eta :F\to G$; i.e., for each object $U$ 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 $\eta _{U}$ 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 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 $h_{X}$ 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 $C$ be a category and let $X,X'$ be objects of $C$. Then

1. If $F$ is any contravariant functor $F:C^{op}\to Sets$, then the natural transformations of $Mor_{C}(-,X)$ to $F$ are in correspondence with the elements of the set $F(X)$.
2. If the functors $Mor_{C}(-,X)$ and $Mor_{C}(-,X')$ are isomorphic, then $X$ and $X'$ are isomorphic in $C$. More generally, the functor $h:C\to Funct(C^{op},Sets)$, $X\mapsto h_{X}$, is an equivalence of categories between $C$ and the full subcategory of representable functors in $Funct(C^{op},Sets)$.