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Functor category

In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. Functor categories are of interest for two main reasons:

Table of contents
1 Definition
2 Examples
3 Facts


Suppose C is a small category (i.e. the objects form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Funct(C,D) or DC, has as objects the covariant functors from C to D, and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if μ(X) : F(X) → G(X) is a natural transformation from the functor F : CD to the functor G : CD, and η(X) : G(X) → H(X) is a natural transformation from the functor G to the functor H, then the collection η(X)μ(X) : F(X) → H(X) defines a natural transformation from F to H. With this composition of natural transformations, DC satisfies the axioms of a category.

In a completely analogous way, one can also consider the category of all contravariant functors from C to D; we write this as Funct(Cop,D).

If C and D are both preadditive categories (i.e. their morphism sets are abelian groups and the composition of morphisms is bilinear), then we can consider the category of all additive functors from C to D, denoted by Add(C,D).



Most constructions that can be carried out in D can also be carried out in DC by performing them "componentwise", separately for each object in C. For instance, if any two objects X and Y in D have a product X×Y, then any two functors F and G in DC have a product F×G, defined by (F×G)(c) = F(cG(c) for every object c in C. Similarly, if ηc : F(c)→G(c) is a natural transformation and each ηc has a kernel Kc in the category D, then the kernel of η in the functor category DC is the functor K with K(c) = Kc for every object c in C.

As a consequence we have the general rule of thumb that the functor category DC shares most of the "nice" properties of D:

We also have: So from the above examples, we can conclude right away that the categories of directed graphs, G-sets and presheaves on a topological space are all complete and cocomplete topoi, and that the categories of representations of G, modules over the ring R, and presheaves of abelian groups on a topological space X are all abelian, complete and cocomplete.

The embedding of the category C in a functor category that was mentioned earlier uses the Yoneda lemma as its main tool. For every object X of C, let Hom(-,X) be the contravariant representable functor from C to Set. The Yoneda lemma states that the assignment

is a full embedding of the category C into the category Funct(Cop,Set). So C naturally sits inside a topos, which is nice.

The same can be carried out for any preadditive category C: Yoneda then yields a full embedding of C into the functor category Add(Cop,Ab). So C naturally sits inside an abelian category, which is nice.

The intuition mentioned above (that constructions that can be carried out in D can be "lifted" to DC) can be made precise in several ways; the most succinct formulation uses the language of adjoint functors. Every functor F : DE induces a functor FC : DCEC (by composition with F). If F and G is a pair of adjoint functors, then FC and GC is also a pair of adjoint functors.

The functor category DC has all the formal properties of an exponential object; in particular the functors from E × CD stand in a natural one-to-one correspondence with the functors from E to DC. The category Cat of all small categories with functors as morphisms is therefore a cartesian closed category.