Functor Category - Examples

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.
• An arrow category (whose objects are the morphisms of, and whose morphisms are commuting squares in ) is just, 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 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 spaces over the field k).
• Any ring R can be considered as a one-object preadditive category; the category of left modules over R is the same as the additive functor category Add(R,Ab) (where Ab denotes the category of abelian groups), 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 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(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 on a general category C, one needs more structure: a Grothendieck topology on C. (Some authors refer to categories that are equivalent to SetC as presheaf categories.)