In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations between the functors (here, G: C \to D is another object in the category). Functor categories are of interest for two main reasons:

Definition

Suppose C is a small category (i.e. the objects and morphisms form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Fun(C, D), Funct(C,D), [C,D], or D ^C, 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 is a natural transformation from the functor F : C \to D to the functor G : C \to D, and is a natural transformation from the functor G to the functor H, then the composition defines a natural transformation from F to H. With this composition of natural transformations (known as vertical composition, see natural transformation), D^C 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. 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).

Examples

Facts

Most constructions that can be carried out in D can also be carried out in D^C 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\times Y, then any two functors F and G in D^C have a product F\times G, defined by for every object c in C. Similarly, if is a natural transformation and each \eta_c has a kernel K_c in the category D, then the kernel of \eta in the functor category D^C 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 D^C 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 be the contravariant representable functor from C to. The Yoneda lemma states that the assignment is a full embedding of the category C into the category Funct(C^\text{op},). 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(C^\text{op},\textbf{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 D^C) can be made precise in several ways; the most succinct formulation uses the language of adjoint functors. Every functor F : D \to E induces a functor (by composition with F). If F and G is a pair of adjoint functors, then F^C and G^C is also a pair of adjoint functors. The functor category D^C has all the formal properties of an exponential object; in particular the functors from stand in a natural one-to-one correspondence with the functors from E to D^C. The category of all small categories with functors as morphisms is therefore a cartesian closed category.