In category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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 homomorphisms between categories, or morphisms when in the category of small categories....

s between two given categories form a category, where the objects are the functors and the morphisms 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. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

s between the 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.

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 Funct(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 μ(X) : F(X) → G(X) is a natural transformation from the functor F : C → D to the functor G : C → D, and η(X) : G(X) → H(X) is a natural transformation from the functor G to the functor H, then the collection η(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. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

), 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(C^op,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 over the monoidal category of abelian groups...

 (i.e. their morphism sets are abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s and the composition of morphisms is bilinear

Bilinear operator

In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments. Matrix multiplication is an example.-Definition:...

), 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 C^I 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 graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
  
   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 Set^C, where C is the category with two objects connected by two morphisms, and Set denotes the category of sets
  Category of sets
  In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
  
  .
• Any group
  Group (mathematics)
  In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
  
   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 symmetries of objects using groups. 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 transformations of the set...
  
   is the same as the functor category Set
  Category of sets
  In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
  
  ^G.
• Similar to the previous example, the category of k-linear representations
  Group representation
  In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices 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
  A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
  
  s over the field
  Field (mathematics)
  In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it 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 an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
  
   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, wherein the corresponding scalars are allowed to 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 Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....
  
  ), and the category of right R-modules is Add(R^op,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(C^op,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 into 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(C^op, 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...
  
   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. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
  
   on C. (Some authors refer to categories that are equivalent to Set^C as presheaf categories.)

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

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 of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

 X×Y, then any two functors F and G in D^C have a product F×G, defined by (F×G)(c) = F(c)×G(c) for every object c in C. Similarly, if η_c : F(c)→G(c) is a natural transformation and each η_c has a kernel K_c in the category D, then the kernel of η 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

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 D^C 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 products and inverse limits....
  
   (or cocomplete), then so is D^C;
• 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 kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
  
  , then so is D^C;

We also have:

• if C is any small category, then the category Set^C of presheaves is a topos
  Topos
  In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
  
  .

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. It is a vast generalisation of Cayley's theorem from group theory...

 as its main tool. For every object X of C, let Hom(-,X) be the contravariant representable functor

Representable functor

In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...

 from C to Set. The Yoneda lemma states that the assignment
is a full embedding of the category C into the category Funct(C^op,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(C^op,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

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 F^C : D^C → E^C (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

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 finite products and exponential objects are called cartesian closed categories...

; in particular the functors from E × C → D stand in a natural one-to-one correspondence with the functors from E to D^C. 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. These categories are particularly important in mathematical logic and the theory of programming, in...

.