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:
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(,).
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 :
We also have:
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.
In mathematics, specifically category theory, a functor is a map between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that 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.
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. It allows the embedding of any category into a category of functors defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.
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 can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.
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 abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.
In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.
In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-categoryC is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.
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