is a category where the objects are the functors
and the morphisms are natural transformations
Functor categories are of interest for two main reasons: Suppose
is a small category (i.e. the objects and morphisms form a set rather than a proper class) and
, has as objects the covariant functors from
, and as morphisms the natural transformations between such functors.
Note that natural transformations can be composed: if
is a natural transformation from the functor
defines a natural transformation from
With this composition of natural transformations (known as vertical composition, see natural transformation),
satisfies the axioms of a category.
In a completely analogous way, one can also consider the category of all contravariant functors from
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
by performing them "componentwise", separately for each object in
As a consequence we have the general rule of thumb that the functor category
shares most of the "nice" properties of
: We also have: So from the above examples, we can conclude right away that the categories of directed graphs,
-sets and presheaves on a topological space are all complete and cocomplete topoi, and that the categories of representations of
, and presheaves of abelian groups on a topological space
are all abelian, complete and cocomplete.
in a functor category that was mentioned earlier uses the Yoneda lemma as its main tool.
The Yoneda lemma states that the assignment is a full embedding of the category
naturally sits inside a topos.
The same can be carried out for any preadditive category
: Yoneda then yields a full embedding of
naturally sits inside an abelian category.
The intuition mentioned above (that constructions that can be carried out in
) can be made precise in several ways; the most succinct formulation uses the language of adjoint functors.
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