Editing Functor category (section)

== Examples ==
* If <math>I</math> is a small [[discrete category]] (i.e. its only morphisms are the identity morphisms), then a functor from <math>I</math> to <math>C</math> essentially consists of a family of objects of <math>C</math>, indexed by <math>I</math>; the functor category <math>C ^I</math> can be identified with the corresponding product category: its elements are families of objects in <math>C</math> and its morphisms are families of morphisms in <math>C</math>.

* An [[arrow category]] <math>\mathcal{C}^\rightarrow</math> (whose objects are the morphisms of <math>\mathcal{C}</math>, and whose morphisms are commuting squares in <math>\mathcal{C}</math>) is just <math>\mathcal{C}^\mathbf{2}</math>, where '''2''' is the category with two objects and their identity morphisms as well as an arrow from one object to the other (but not another arrow back the other way).
* A [[graph theory|directed graph]] consists of a set of arrows and a set of vertices, and two functions from the arrow set to the vertex set, specifying each arrow's start and end vertex. The category of all directed graphs is thus nothing but the functor category <math>\textbf{Set}^C</math>, where <math>C</math> is the category with two objects connected by two parallel morphisms (source and target), and '''Set''' denotes the [[category of sets]].
* Any [[group (mathematics)|group]] <math>G</math> can be considered as a one-object category in which every morphism is invertible. The category of all [[Group action (mathematics)#Definition|<math>G</math>-sets]] is the same as the functor category '''[[category of sets|Set]]'''<sup><math>G</math></sup>. Natural transformations are [[Group action (mathematics)#Morphisms and isomorphisms between G-sets|<math>G</math>-maps]].
* Similar to the previous example, the category of ''K''-linear [[group representation|representations]] of the group <math>G</math> is the same as the functor category '''Vect'''<sub>''K''</sub><sup><math>G</math></sup> (where '''Vect'''<sub>''K''</sub> denotes the category of all [[vector space]]s over the [[field (mathematics)|field]] ''K'').
* Any [[ring (mathematics)|ring]] <math>R</math> can be considered as a one-object preadditive category; the category of left [[module (mathematics)|modules]] over <math>R</math> is the same as the additive functor category Add(<math>R</math>,<math>\textbf{Ab}</math>) (where <math>\textbf{Ab}</math> denotes the [[category of abelian groups]]), and the category of right <math>R</math>-modules is Add(<math>R^\text{op}</math>,<math>\textbf{Ab}</math>). Because of this example, for any preadditive category <math>C</math>, the category Add(<math>C</math>,<math>\textbf{Ab}</math>) is sometimes called the "category of left modules over <math>C</math>" and Add(<math>C^\text{op}</math>,<math>\textbf{Ab}</math>) is the "category of right modules over <math>C</math>".
* The category of [[presheaf|presheaves]] on a topological space <math>X</math> is a functor category: we turn the topological space into a category <math>C</math> having the open sets in <math>X</math> as objects and a single morphism from <math>U</math> to <math>V</math> if and only if <math>U</math> is contained in <math>V</math>. The category of presheaves of sets (abelian groups, rings) on <math>X</math> is then the same as the category of contravariant functors from <math>C</math> to <math>\textbf{Set}</math> (or <math>\textbf{Ab}</math> or <math>\textbf{Ring}</math>). Because of this example, the category Funct(<math>C^\text{op}</math>, <math>\textbf{Set}</math>) is sometimes called the "[[Presheaf (category theory)|category of presheaves]] of sets on <math>C</math>" even for general categories <math>C</math> not arising from a topological space. To define [[sheaf (mathematics)|sheaves]] on a general category <math>C</math>, one needs more structure: a [[Grothendieck topology]] on <math>C</math>. (Some authors refer to categories that are  [[equivalence (category theory)|equivalent]] to <math>\textbf{Set}^C</math> as ''[[presheaf (category theory)|presheaf]] categories''.<ref>{{cite book
 |author      = Tom Leinster
 |year        = 2004
 |title       = Higher Operads, Higher Categories
 |publisher   = Cambridge University Press
 |bibcode = 2004hohc.book.....L
 |url         = http://www.maths.gla.ac.uk/~tl/book.html
 |url-status     = dead
 |archiveurl  = https://web.archive.org/web/20031025120434/http://www.maths.gla.ac.uk/~tl/book.html
 |archivedate = 2003-10-25
}}</ref>)