representable functor

A representable functor is one that is secretly just “maps out of (or into) a fixed object.” In any locally small category C, each object A determines a functor Hom(A, -) that takes an object X to the set of morphisms A -> X. A functor F : C -> Set is called representable when it is naturally isomorphic to one of these — when there is an object A, the representing object, that completely encodes F as the operation of mapping out of A.

Precisely, a covariant functor F : C -> Set is representable if there is an object A and a natural isomorphism Hom(A, -) ≅ F. By the Yoneda lemma such a representation is essentially unique: A is determined up to isomorphism, and the isomorphism corresponds to a distinguished universal element of F(A). A contravariant functor C -> Set is representable when it is isomorphic to Hom(-, A) for some A. Many “moduli” or “solution” functors of algebra and geometry are defined exactly by asking that they be representable.

Representability is the categorical face of universal properties. To say a construction is a representing object is to say it satisfies a universal property, and conversely. For example the functor sending a ring R to the set of its units is representable by the ring Z[x, y]/(xy − 1); the polynomial ring Z[x] represents the underlying-set functor on commutative rings. Recognizing a functor as representable instantly supplies a universal object.