Functor category explained

In category theory, a branch of mathematics, a functor category

D^C

is a category where the objects are the functors

F:C\toD

and the morphisms are natural transformations

η:F\toG

between the functors (here,

G:C\toD

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.

Definition

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

, written as Fun(

), Funct(

[C,D]

, or

D^C

, has as objects the covariant functors from

, and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if

\mu(X):F(X)\toG(X)

is a natural transformation from the functor

F:C\toD

to the functor

G:C\toD

, and

η(X):G(X)\toH(X)

is a natural transformation from the functor

to the functor

, then the composition

η(X)\mu(X):F(X)\toH(X)

defines a natural transformation from

. 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

; we write this as Funct(

C^op,D

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

, denoted by Add(

Examples

is a small discrete category (i.e. its only morphisms are the identity morphisms), then a functor from

essentially consists of a family of objects of

, indexed by

; the functor category

C^I

can be identified with the corresponding product category: its elements are families of objects in

and its morphisms are families of morphisms in

l{C}^→

(whose objects are the morphisms of

l{C}

, and whose morphisms are commuting squares in

l{C}

) is just

l{C}²

, 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

bf{Set}^C

, where

is the category with two objects connected by two parallel morphisms (source and target), and Set denotes the category of sets. When viewed from this perspective, such directed graphs are often called quivers.

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^G. Natural transformations are

-maps.

Similar to the previous example, the category of K-linear representations of the group

is the same as the functor category Vect_K^G (where Vect_K denotes the category of all vector spaces over the field K).

can be considered as a one-object preadditive category; the category of left modules over

is the same as the additive functor category Add(

bf{Ab}

) (where

bf{Ab}

denotes the category of abelian groups), and the category of right

-modules is Add(

R^op

bf{Ab}

). Because of this example, for any preadditive category

, the category Add(

bf{Ab}

) is sometimes called the "category of left modules over

" and Add(

C^op

bf{Ab}

) 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

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

bf{Set}

(or

bf{Ab}

bf{Ring}

). Because of this example, the category Funct(

C^op

bf{Set}

) 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

bf{Set}^C

as presheaf categories.^[1])

Facts

Most constructions that can be carried out in

can also be carried out in

D^C

by performing them "componentwise", separately for each object in

. For instance, if any two objects

and

have a product

X x Y

, then any two functors

and

D^C

have a product

F x G

, defined by

(F x G)(c)=F(c) x G(c)

for every object

. Similarly, if

η_c:F(c)\toG(c)

is a natural transformation and each

η_c

has a kernel

K_c

in the category

, then the kernel of

in the functor category

D^C

is the functor

with

K(c)=K_c

for every object

As a consequence we have the general rule of thumb that the functor category

D^C

shares most of the "nice" properties of

is complete (or cocomplete), then so is

D^C

;

is an abelian category, then so is

D^C

;

We also have:

is any small category, then the category

bf{Set}^C

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

, let

Hom(-,X)

be the contravariant representable functor from

bf{Set}

. The Yoneda lemma states that the assignment

X\mapsto\operatorname{Hom}(-,X)

is a full embedding of the category

into the category Funct(

C^op

bf{Set}

). 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(

C^op

bf{Ab}

). So

naturally sits inside an abelian category.

The intuition mentioned above (that constructions that can be carried out in

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\toE

induces a functor

F^C:D^C\toE^C

(by composition with

). If

and

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

E x C\toD

stand in a natural one-to-one correspondence with the functors from

D^C

. The category

bf{Cat}

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

Notes and References

Book: Tom Leinster . 2004 . Higher Operads, Higher Categories . Cambridge University Press . 2004hohc.book.....L . dead . https://web.archive.org/web/20031025120434/http://www.maths.gla.ac.uk/~tl/book.html . 2003-10-25.

Functor category explained

Definition

Examples

Facts

See also

Notes and References