Representable Functor - Universal Elements

Universal Elements

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) with 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 category of elements of F.

The natural transformation induced by an element u ∈ F(A) is an isomorphism if and only if (A,u) is a universal element of F. We therefore conclude that representations of F are in one-to-one correspondence with universal elements of F. For this reason, it is common to refer to universal elements (A,u) as representations.