Category of functors: Difference between revisions

Latest revision as of 16:00, 25 July 2024

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

Objects are functors $F:C^{op}\to D$
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

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

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 $F(X)$ .
If the functors $Mor_{C}(-,X)$ and $Mor_{C}(-,X')$ are isomorphic, then $X$ and $X'$ are isomorphic in $C$ . More generally, the functor $h:C\to Funct(C^{op},Sets)$ , 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 X\mapsto h_X} , is an equivalence of categories between 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 the full subcategory of representable functors 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 Funct(C^{op},Sets)} .

@@ Line 1: / Line 1: @@
-This article focuses on the category of contravariant functors between two categories.
+{{subpages}}
+This article focuses on the category of contravariant functors between two [[categories]].
 ==The category of functors==
@@ Line 6: / Line 8: @@
 #Objects are '''functors''' <math>F:C^{op}\to D</math>
-#A morphism of functors <math>F,G</math> is a '''natural transformations''' <math>\eta:F\to G</math>; i.e., for each object <math>U</math> of <math>C</math>, a morphism in <math>D</math> <math>\eta_U:F(U)\to G(U)</math> such that for all morphisms <math>f:U\to V</math> in <math>C^{op}</math>, the diagram
+#A morphism of functors <math>F,G</math> is a '''natural transformation''' <math>\eta:F\to G</math>; i.e., for each object <math>U</math> of <math>C</math>, a morphism in <math>D</math> <math>\eta_U:F(U)\to G(U)</math> such that for all morphisms <math>f:U\to V</math> in <math>C^{op}</math>, the diagram (DIAGRAM) commutes.
-commutes.
+A ''natural isomorphism'' is a natural transformation <math>\eta</math> such that <math>\eta_U</math> is an isomorphism in <math>D</math> for every object <math>U</math>.  One can verify that natural isomorphisms are indeed isomorphisms in the category of functors.
-A ''natural isomorphism'' is a natural tranformation <math>\eta</math> such that <math>\eta_U</math> is an isomorphism in <math>D</math> for every object <math>U</math>.  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 <math>h_X=Mor_C(-,X)</math>.
 ==Examples==
@@ Line 21: / Line 23: @@
 # If <math>F</math> is any contravariant functor <math>F:C^{op}\to Sets</math>, then the natural transformations of <math>Mor_C(-,X)</math> to <math>F</math> are in correspondence with the elements of the set <math>F(X)</math>.
-# If the functors <math>Mor_C(-,X)</math> and <math>Mor_C(-,X')</math> are isomorphic, then <math>X</math> and <math>X'</math> are isomorphic in <math>C</math>. More generally, the functor <math>h:C\to Funct(C^{op},Sets)</math>, <math>X\mapsto h_X</math>, is an equivalence of categories between <math>C</math> and the full subcategory of ''representable'' functors in <math>Funct(C^{op},Sets)</math>.
+# If the functors <math>Mor_C(-,X)</math> and <math>Mor_C(-,X')</math> are isomorphic, then <math>X</math> and <math>X'</math> are isomorphic in <math>C</math>. More generally, the functor <math>h:C\to Funct(C^{op},Sets)</math>, <math>X\mapsto h_X</math>, is an equivalence of categories between <math>C</math> and the full subcategory of ''representable'' functors in <math>Funct(C^{op},Sets)</math>.[[Category:Suggestion Bot Tag]]
-==References==
-*{{cite book
- | author = [[David Eisenbud]]
- | coauthors = [[Joe Harris]]
- | year = 1998
- | title = The Geometry of Schemes
- | publisher = [[Springer Science+Business Media|Springer-Verlag]]
- | id = ISBN 0-387-98637-5
-}}

Category of functors: Difference between revisions

Latest revision as of 16:00, 25 July 2024

The category of functors

Examples

The Yoneda lemma

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