Home > Functor category
Category theoryIn category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations 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;
- a standard construction embeds a given category in a functor category; the functor category has much nicer properties than the original category, allowing to perform certain operations that were not available in the original setting.
1 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 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, 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 (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 C to D, denoted by Add(C,D).
2 Examples
- If I is a small discrete 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 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.
- 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 SetG.
- Similar to the previous example, the category of k-linear representations of the group G is the same as the functor category k-VectG (where k-Vect denotes the category of all vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (fors over the fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil k).
- Any ringIn ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. History See Ring theory Definition and notation A ring is an a R can be considered as a one-object preadditive category; the category of left modulesAlgebra In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of over R is the same as the additive functor category Add(R,Ab) (where Ab denotes the category of abelian groupsIn mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group), 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 iff 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 sheavesAlternate meanings: River Sheaf, King Sceaf, sheaf toss In mathematics, a sheaf ''F on a given topological space X gives a set or richer structure F ''U for each open set U of X''. The structures F ''U are compatible with the operations of restricting the on a general category C, one needs more structure: a Grothendieck topology on C.
