Functor category

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

  • many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable;
  • every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.

DefinitionEdit

Suppose   is a small category (i.e. the objects and morphisms form a set rather than a proper class) and   is an arbitrary category. The category of functors from   to  , written as Fun( ,  ), Funct( , ),  , or  , has as objects the covariant functors from   to  , and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if   is a natural transformation from the functor   to the functor  , and   is a natural transformation from the functor   to the functor  , then the collection   defines a natural transformation from   to  . 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   to  ; we write this as Funct( ).

If   and   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   to  , denoted by Add( , ).

ExamplesEdit

  • If   is a small discrete category (i.e. its only morphisms are the identity morphisms), then a functor from   to   essentially consists of a family of objects of  , indexed by  ; the functor category   can be identified with the corresponding product category: its elements are families of objects in   and its morphisms are families of morphisms in  .
  • An arrow category   (whose objects are the morphisms of  , and whose morphisms are commuting squares in  ) is just  , where 2 is the category with two objects and their identity morphisms as well as an arrow from one object to the other (but not another arrow back the other way).
  • A directed graph consists of a set of arrows and a set of vertices, and two functions from the arrow set to the vertex set, specifying each arrow's start and end vertex. The category of all directed graphs is thus nothing but the functor category  , where   is the category with two objects connected by two parallel morphisms (source and target), and Set denotes the category of sets.
  • Any group   can be considered as a one-object category in which every morphism is invertible. The category of all  -sets is the same as the functor category Set .
  • Similar to the previous example, the category of  -linear representations of the group   is the same as the functor category k-Vect  (where k-Vect denotes the category of all vector spaces over the field  ).
  • Any ring   can be considered as a one-object preadditive category; the category of left modules over   is the same as the additive functor category Add( , ) (where   denotes the category of abelian groups), and the category of right  -modules is Add( ). Because of this example, for any preadditive category  , the category Add( , ) is sometimes called the "category of left modules over   and Add( , ) is the category of right modules over  .
  • The category of presheaves on a topological space   is a functor category: we turn the topological space into a category   having the open sets in  as objects and a single morphism from   to   if and only if   is contained in  . The category of presheaves of sets (abelian groups, rings) on   is then the same as the category of contravariant functors from   to   (or   or  ). Because of this example, the category Funct( ,  ) is sometimes called the "category of presheaves of sets on   even for general categories   not arising from a topological space. To define sheaves on a general category  , one needs more structure: a Grothendieck topology on  . (Some authors refer to categories that are equivalent to   as presheaf categories.[1])

FactsEdit

Most constructions that can be carried out in   can also be carried out in   by performing them "componentwise", separately for each object in  . For instance, if any two objects   and   in   have a product  , then any two functors   and   in   have a product  , defined by   for every object   in  . Similarly, if   is a natural transformation and each   has a kernel   in the category  , then the kernel of   in the functor category   is the functor   with   for every object   in  .

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

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

We also have:

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

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  , modules over the ring  , and presheaves of abelian groups on a topological space   are all abelian, complete and cocomplete.

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

 

is a full embedding of the category   into the category Funct( , ). So   naturally sits inside a topos.

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

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

The functor category   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   to  . The category   of all small categories with functors as morphisms is therefore a cartesian closed category.

See alsoEdit

ReferencesEdit

  1. ^ Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. Bibcode:2004hohc.book.....L. Archived from the original on 2003-10-25. CS1 maint: discouraged parameter (link)