corepresenting

Corepresenting is a concept in category theory describing how a functor can be described in terms of morphisms from a fixed object. Specifically, a functor F: C -> Set is corepresentable if there exists an object A in C and a natural isomorphism F ≅ Hom(A, -): C -> Set. The object A is called the corepresenting object for F. Under this identification, F assigns to each object X a set that is naturally in bijection with the morphisms from A to X, with the bijection varying naturally with X.

The notion is dual to representability. For contravariant functors F: C^op -> Set, the corresponding corepresentability is F ≅ Hom(-, A), with A a representing object in the opposite sense. Thus corepresentability and representability are dual notions, distinguished by the variance of the functor involved.

The Yoneda lemma provides a practical lens on corepresentability. For a covariant F: C -> Set, F is corepresentable by A if and only if there is a natural isomorphism Nat(Hom(A, -), F) ≅ F(A). This gives a concrete criterion: natural transformations from Hom(A, -) to F correspond to elements of F(A). In particular, if such a universal natural isomorphism exists, F and Hom(A, -) encode the same informational content.

Examples help illustrate the concept. In the category of vector spaces over a field k, the functor F(V) = V (viewed as a set) is naturally isomorphic to Hom(k, V); hence F is corepresented by the one-dimensional space k. Another simple instance: if C has a terminal object 1, the functor Hom(1, -) is naturally isomorphic to a constant singleton functor and is thus corepresentable by 1.