HOME

TheInfoList



OR:

In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a functor category D^C is a category where the objects are the
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s F: C \to D and the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s are
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s \eta: F \to G between the functors (here, G: C \to D 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 In mathematics, the Yoneda lemma is arguably the most important result in category theory. It 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 (vie ...
); 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 Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
) and D is an arbitrary category. The category of functors from C to D, written as Fun(C, D), Funct(C,D), ,D/math>, 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 \mu (X) : F(X) \to G(X) is a natural transformation from the functor F : C \to D to the functor G : C \to D, and \eta(X) : G(X) \to H(X) is a natural transformation from the functor G to the functor H, then the collection \eta(X)\mu(X) : F(X) \to H(X) defines a natural transformation from F to H. With this composition of natural transformations (known as vertical composition, see
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
), 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^\text,D). If C and D are both preadditive categories (i.e. their morphism sets are
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
s and the composition of morphisms is bilinear), 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 In mathematics, in the field of category theory, a discrete category is a category whose only morphisms are the identity morphisms: :hom''C''(''X'', ''X'') = {id''X''} for all objects ''X'' :hom''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ '' ...
(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. * An arrow category \mathcal^\rightarrow (whose objects are the morphisms of \mathcal, and whose morphisms are commuting squares in \mathcal) is just \mathcal^\mathbf, 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 \textbf^C, where C is the category with two objects connected by two parallel morphisms (source and target), and Set denotes the
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 the total functions from ''A'' to ''B'', and the composition o ...
. * Any group G can be considered as a one-object category in which every morphism is invertible. The category of all G-sets is the same as the functor category
Set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
G. Natural transformations are G-maps. * Similar to the previous example, the category of ''K''-linear representations of the group G is the same as the functor category Vect''K''G (where Vect''K'' denotes the category of all
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over the field ''K''). * Any ring R can be considered as a one-object preadditive category; the category of left
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over R is the same as the additive functor category Add(R,\textbf) (where \textbf denotes the
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: indeed, every small abelian category can be embedded in Ab. Properties The zero object of ...
), and the category of right R-modules is Add(R^\text,\textbf). Because of this example, for any preadditive category C, the category Add(C,\textbf) is sometimes called the "category of left modules over C" and Add(C^\text,\textbf) is the "category of right modules over C". * The category of
presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
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 \textbf (or \textbf or \textbf). Because of this example, the category Funct(C^\text, \textbf) is sometimes called the "
category of presheaves In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on ...
of sets on C" even for general categories C not arising from a topological space. To define sheaves on a general category C, one needs more structure: a Grothendieck topology on C. (Some authors refer to categories that are equivalent to \textbf^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 X\times Y, then any two functors F and G in D^C have a product F\times G, defined by (F \times G)(c) = F(c)\times G(c) for every object c in C. Similarly, if \eta_c : F(c) \to G(c) is a natural transformation and each \eta_c has a kernel K_c in the category D, then the kernel of \eta 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 In English, the phrase ''rule of thumb'' refers to an approximate method for doing something, based on practical experience rather than theory. This usage of the phrase can be traced back to the 17th century and has been associated with various t ...
that the functor category D^C shares most of the "nice" properties of D: * if D is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
(or cocomplete), then so is D^C; * if D is an
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 prototypical example of an abelian category is the category of ...
, then so is D^C; We also have: * if C is any small category, then the category \textbf^C of
presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
is a
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
. 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 In mathematics, the Yoneda lemma is arguably the most important result in category theory. It 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 (vie ...
as its main tool. For every object X of C, let \text(-,X) be the contravariant
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets a ...
from C to \textbf. The Yoneda lemma states that the assignment :X \mapsto \operatorname(-,X) is a full embedding of the category C into the category Funct(C^\text,\textbf). 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^\text,\textbf). 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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
. Every functor F : D \to E induces a functor F^C : D^C \to 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; in particular the functors from E \times C \to D stand in a natural one-to-one correspondence with the functors from E to D^C. The category \textbf of all small categories with functors as morphisms is therefore a cartesian closed category.


See also

*
Diagram (category theory) In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets i ...


References

{{Category theory Functors Categories in category theory