Functor Category - Facts

Facts

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(c)×G(c) for every object c in C. Similarly, if η_c : F(c)→G(c) is a natural transformation and each η_c has a kernel K_c in the category D, then the kernel of η in the functor category DC is the functor K with K(c) = K_c 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:

• if D is complete (or cocomplete), then so is DC;
• if D is an abelian category, then so is DC;

We also have:

• if C is any small category, then the category SetC of presheaves is a topos.

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.

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.

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 : D → E induces a functor FC : DC → EC (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 × C → D 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.