Functor Category

Let C be a small category (its objects and morphisms form a set) and D an arbitrary category. Then the category of functors from C to D , denoted by Fun ( C , D ), Funct ( C , D ) or D ^C , is defined as follows: objects are covariant functors from C to D , morphisms are natural transformations between these functors. Since the composition of natural transformations is natural (see. Natural transformation ) and the identity transformation is natural, D ^C satisfies the axioms of the category.

Similarly, the category of contravariant functors from C to D is defined, denoted by Funct ( C ^op , D ).

• If I is a small discrete category (all morphisms are identical), then a functor from I to C is just a family of objects C indexed by I. Category C ^I in this case corresponds to a certain category of the work .
• Arrow category${\displaystyle {\mathcal{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">C</span></span>}}^{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">\to </span></span>}}$ {\ displaystyle {\ mathcal {C}} ^ {\ rightarrow}}   (objects are morphisms of C , morphisms are commutative squares) is a category${\displaystyle {\mathcal{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">C</span></span>}}^{\mathbf{<span\; class="mwe-math-element"><span\; class="mwe-math-mathml-inline\; mwe-math-mathml-a11y"\; style="display:\; none;">2</span></span>}}}$ {\ displaystyle {\ mathcal {C}} ^ {\ mathbf {2}}}   , where 2 denotes a category of two objects, identical morphisms, as well as one morphism from the first object to the second.
• a directed graph is a set of arrows and many vertices that map each arrow to the top-start and top-end. The category of directed graphs is nothing more than the category Set ^C , where C is the category with two objects and two morphisms between them, and Set is the category of sets .

• If D is a complete category (or cofolder), then so is D ^C ;
• If D is an Abelian category , then so is D ^C ;
• If C is a small category, then the category of presheafs Set ^C is topos .
• Each functor F : D → E induces a functor F ^C : D ^C → E ^C (by composition with F ). If F and G are a pair of conjugate functors , then such are F ^C and G ^C.
• The category D ^C satisfies all the properties of the exponential ; in particular, the functors E × C → D are in one-to-one correspondence with functors from E to D ^C. The Cat category of small categories is therefore Cartesian closed .

• Tom Leinster. Higher Operads, Higher Categories . - Cambridge University Press, 2004. Archived December 6, 2013 to Wayback Machine