representability

Representability is a concept in category theory and its applications in algebraic geometry and logic. A functor F: C^op -> Set is representable if there exists an object R in C together with a natural isomorphism F ≅ Hom_C(-, R). In this case R is called the representing object. By the Yoneda lemma, representability means F is naturally isomorphic to the hom-functor out of a fixed object, and such an R is unique up to a unique isomorphism.

Examples help illustrate the idea. In the category of sets, for any fixed set A, the functor Hom(-, A) is representable by A. In algebraic geometry, if S is a base scheme, a functor F: Sch/S^op -> Set is representable by an S-scheme X precisely when F ≅ Hom_S(-, X). The functor of points associated to a scheme X assigns to every S-scheme T the set of S-morphism T -> X; this functor is representable by X. More generally, representability is a guiding principle for moduli problems: many functors are representable by schemes, algebraic spaces, or stacks, or only locally representable.

Significance of representability lies in translating a functorial description into a concrete object with universal properties. It underpins the construction of moduli spaces, deformation theory, and universal properties, providing a bridge between abstract functorial data and concrete geometric or algebraic objects. Representability criteria and the related concept of the functor of points are central tools in modern algebraic geometry and related fields.