Category of functors

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

The category of functors

Let 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} and $D$ 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 $C$, a morphism in $D$ $\eta _{U}:F(U)\to G(U)$ such that for all morphisms $f:U\to V$ in $C^{op}$, the diagram (DIAGRAM) commutes.

A natural isomorphism is a natural transformation $\eta$ 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 $h_{X}=Mor_{C}(-,X)$.

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 $Funct(C^{op},Sets)$.