representable

Representable is a term used in category theory to describe a functor that is naturally isomorphic to a Hom-functor. Let C be a category and F: C -> Set be a functor. F is said to be representable if there exists an object c in C and a natural isomorphism F ≅ Hom_C(c, -). The object c is called a representing (or representing) object for F, and the isomorphism identifies F with the set of morphisms from c to the varying argument.

Yoneda lemma: For any object c in C, there is a natural isomorphism Nat(Hom_C(c, -), F) ≅ F(c). In particular, if F is representable by c, the representing object is unique up to unique isomorphism, and F is completely determined by its value at c.

Examples: In Sets, taking a fixed set S, the functor X ↦ Hom(S, X) ≅ X^S is representable by S. In the category of groups, the functor A ↦ Hom(G, A) for a fixed group G is representable by G. In algebraic geometry, a functor from rings to sets that is isomorphic to Hom_R(R, -) is representable by the ring R; more generally, representable functors are the “points” of a space in the functor-of-points language.

Non-examples and caveats: Not every functor is representable; for instance, the functor that assigns to a set X its set of finite subsets Fin(X) is not representable by any fixed set, as no fixed S can satisfy X^S ≅ Fin(X) for all X. Representability is a strong structural property and often reflects a universal property of the representing object.

Properties: Representable functors preserve limits that exist in C, because Hom(c, -) preserves limits in the argument. The representing object is unique up to unique isomorphism, by the Yoneda-related arguments.