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.
A contravariant functor G : C → Set is said to representable if 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 of u ∈ F(A) is given by
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.
Representations of functors are unique up to a unique isomorphism. That is, if (A1,Φ1) and (A2,Φ2) represent the same functor, then there exists a unique isomorphism φ : A1 → A2 such that
Stated in terms of universal elements: if (A1,u1) and (A2,u2) represent the same functor, then there exists a unique isomorphism φ : A1 → A2 such that
Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covariant) representable functors preserve all limits. It follows that any functor which fails to preserve some limit is not representable.
Contravariant representable functors take colimits to limits.
Conversely, if K is represented by a pair (A, u) and all small copowers of A exist in C then K has a left adjoint F which sends each set I to the Ith copower of A.
Therefore, if C is a category with all small copowers, a functor K : C → Set is representable if and only if it has a left adjoint.
Let G : D → C be a functor and let X be an object of C. Then (A,φ) is a universal morphism from X to G if and only if (A,φ) is a representation of the functor HomC(X,G–) from D to Set. It follows that G has a left-adjoint F if and only if HomC(X,G–) is representable for all X in C. The natural isomorphism ΦX : HomD(FX,–) → HomC(X,G–) yields the adjointness; that is
The dual statements are also true. Let F : C → D be a functor and let Y be an object of D. Then (A,φ) is a universal morphism from F to Y if and only if (A,φ) is a representation of the functor HomD(F–,Y) from C to Set. It follows that F has a right-adjoint G if and only if HomD(F–,Y) is representable for all Y in D.
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