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Functor category
 
 
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In category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
, a branch of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the functor
Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories....
s between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformation
Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved....
s between functors. Functor categories are of interest for two main reasons: An element of a functor category is called a diagram
Diagram (category theory)

In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well: an indexed family of sets is a collection of sets, indexed by a fixed set , while a diagram is a collection of objects and morphisms, indexed...
.

ose C is a small category (i.e. the objects form a set rather than a proper class) and D is an arbitrary category.






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In category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
, a branch of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the functor
Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories....
s between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformation
Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved....
s between functors. 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.
An element of a functor category is called a diagram
Diagram (category theory)

In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well: an indexed family of sets is a collection of sets, indexed by a fixed set , while a diagram is a collection of objects and morphisms, indexed...
.

Definition

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

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved....
), DC 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(Cop,D).

If C and D are both preadditive categories
Preadditive category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched category over the monoidal category of abelian groups....
 (i.e. their morphism sets are abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
s and the composition of morphisms is bilinear
Bilinear operator

In mathematics, a bilinear map is a function of two arguments that is linear map in each. An example of such a map is multiplication of integers....
), then we can consider the category of all additive functors from C to D, denoted by Add(C,D).

Examples

  • If I is a small discrete category
    Discrete category

    In mathematics, especially category theory, a discrete category is a category whose only morphisms are the identity morphisms. It is the simplest kind of category....
     (i.e. its only morphisms are the identity morphisms), then a functor from I to C essentially consists of a family of objects of C, indexed by I; the functor category CI can be identified with the corresponding product category: its elements are families of objects in C and its morphisms are families of morphisms in C.
  • A directed graph
    Graph theory

    In mathematics and computer science, graph theory is the study of graph : mathematical structures used to model pairwise relations between objects from a certain collection....
     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 SetC, where C is the category with two objects connected by two morphisms, and Set denotes the category of sets
    Category of sets

    In mathematics, the category of sets, denoted as Set, is the Category theory whose Category theory are all Set and whose morphisms are all function s....
    .
  • Any group
    Group (mathematics)

    In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
     G can be considered as a one-object category in which every morphism is invertible. The category of all G-sets
    Group action

    In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
     is the same as the functor category Set
    Category of sets

    In mathematics, the category of sets, denoted as Set, is the Category theory whose Category theory are all Set and whose morphisms are all function s....
    G.
  • Similar to the previous example, the category of k-linear representations
    Group representation

    In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
     of the group G is the same as the functor category k-VectG (where k-Vect denotes the category of all vector space
    Vector space

    File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
    s over the field
    Field (mathematics)

    In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
     k).
  • Any ring
    Ring (mathematics)

    In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
     R can be considered as a one-object preadditive category; the category of left modules
    Module (mathematics)

    In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalar to lie in a field , the "scalars" may lie in an arbitrary ring....
     over R is the same as the additive functor category Add(R,Ab) (where Ab denotes the category of abelian groups
    Category of abelian groups

    In mathematics, the category theory Ab has the abelian groups as object and group homomorphisms as morphisms. This is the prototype of an abelian category....
    ), and the category of right R-modules is Add(Rop,Ab). Because of this example, for any preadditive category C, the category Add(C,Ab) is sometimes called the "category of left modules over C" and Add(Cop,Ab) is the category of right modules over C.
  • The category of presheaves on a topological space X is a functor category: we turn the topological space in a category C having the open sets in X as objects and a single morphism from U to V if and only if U is contained in V. The category of presheaves of sets (abelian groups, rings) on X is then the same as the category of contravariant functors from C to Set (or Ab or Ring). Because of this example, the category Funct(Cop, Set) is sometimes called the "category of presheaves of sets on C" even for general categories C not arising from a topological space. To define sheaves
    Sheaf (mathematics)

    In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one....
     on a general category C, one needs more structure: a Grothendieck topology
    Grothendieck topology

    In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space....
     on C. (Some authors refer to categories that are equivalent to SetC as presheaf
    Presheaf (category theory)

    In category theory, a branch of mathematics, a -valued presheaf on a category is a functor . Often presheaf is defined to be a Set-valued presheaf....
     categories    .)


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
Product (category theory)

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product, the direct product of groups, the direct product of rings and the product topology....
 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 ?c : F(c)?G(c) is a natural transformation and each ?c has a kernel Kc in the category D, then the kernel of ? in the functor category DC is the functor K with K(c) = Kc for every object c in C.

As a consequence we have the general rule of thumb
Rule of thumb

A rule of thumb is a principle with broad application that is not intended to be strictly accurate or reliable for every situation. It is an easily learned and easily applied procedure for approximately calculating or recalling some value, or for making some determination....
 that the functor category DC shares most of the "nice" properties of D:
  • if D is complete
    Limit (category theory)

    In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as product and inverse limits....
     (or cocomplete), then so is DC;
  • if D is an abelian category
    Abelian category

    In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernel s and cokernels exist and have desirable properties....
    , then so is DC;


We also have:
  • if C is any small category, then the category SetC of presheaves
    Presheaf (category theory)

    In category theory, a branch of mathematics, a -valued presheaf on a category is a functor . Often presheaf is defined to be a Set-valued presheaf....
     is a topos
    Topos

    In mathematics, a topos is a type of category that behaves like the category of sheaf theory of Set on a topological space. For a discussion of the history of topos theory, see the article Background and genesis of topos theory....
    .


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
Yoneda lemma

In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object....
 as its main tool. For every object X of C, let Hom(-,X) be the contravariant representable functor
Representable functor

In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets....
 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
Adjoint functors

In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency....
. 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
Exponential object

In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all product and exponential objects are called cartesian closed category....
; 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
Cartesian closed category

In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors....
.

See also

  • Diagram (category theory)
    Diagram (category theory)

    In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well: an indexed family of sets is a collection of sets, indexed by a fixed set , while a diagram is a collection of objects and morphisms, indexed...