Representable functor

In mathematics, particularly , a representable functor is a certain functor from an arbitrary into the . Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory. Let C be a and let Set be the . For each object A of C let Hom(A,–) be the hom functor that maps object X to the set Hom(A,X). A functor F : C → Set is said to be representable if it is naturally isomorphic to Hom(A,–) for some object A of C. A representation of F is a pair (A, Φ) where Φ : Hom(A,–) → F is a natural isomorphism. A contravariant functor G from C to Set is the same thing as a functor G : Cop → Set and is commonly called a . A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(–,A) for some object A of C. According to Yoneda's lemma, natural transformations from Hom(A,–) to F are in one-to-one correspondence with the elements of F(A). Given a natural transformation Φ : Hom(A,–) → F the corresponding element u ∈ F(A) is given by Conversely, given any element u ∈ F(A) we may define a natural transformation Φ : Hom(A,–) → F via where f is an element of Hom(A,X). In order to get a representation of F we want to know when the natural transformation induced by u is an isomorphism. This leads to the following definition: A universal element of a functor F : C → Set is a pair (A,u) consisting of an object A of C and an element u ∈ F(A) such that for every pair (X,v) consisting of an object X of C and an element v ∈ F(X) there exists a unique morphism f : A → X such that (Ff)(u) = v. A universal element may be viewed as a universal morphism from the one-point set {•} to the functor F or as an initial object in the of F.