Санат теориясындағы өкілдік функционалдар

In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i. e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. 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.

Consider the contravariant functor P : Set → Set which maps each set to its power set and each function to its inverse image map. To represent this functor we need a pair (A,u) where A is a set and u is a subset of A, i. e. an element of P(A), such that for all sets X, the hom set Hom(X,A) is isomorphic to P(X) via ΦX(f) = (Pf)u = f−1(u). Take A = {0,1} and u = {1}. Given a subset S ⊆ X the corresponding function from X to A is the characteristic function of S.
Forgetful functors to Set are very often representable. In particular, a forgetful functor is represented by (A, u) whenever A is a free object over a singleton set with generator u. The forgetful functor Grp → Set on the category of groups is represented by (Z, 1). The forgetful functor Ring → Set on the category of rings is represented by (Z[x], x), the polynomial ring in one variable with integer coefficients. The forgetful functor Vect → Set on the category of real vector spaces is represented by (R, 1). The forgetful functor Top → Set on the category of topological spaces is represented by any singleton topological space with its unique element. A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds to a G set. The unique hom functor Hom(•,–) from G to Set corresponds to the canonical G set G with the action of left multiplication. Standard arguments from group theory show that a functor from G to Set is representable if and only if the corresponding G set is simply transitive (i. e. a G torsor or heap). Choosing a representation amounts to choosing an identity for the heap. Let C be the category of CW complexes with morphisms given by homotopy classes of continuous functions. For each natural number n there is a contravariant functor Hn : C → Ab which assigns each CW complex its nth cohomology group (with integer coefficients). Composing this with the forgetful functor we have a contravariant functor from C to Set. Brown's representability theorem in algebraic topology says that this functor is represented by a CW complex K(Z,n) called an Eilenberg–MacLane space. Let R be a commutative ring with identity, and let R Mod be the category of R modules. If M and N are unitary modules over R, there is a covariant functor B: R Mod → Set which assigns to each R module P the set of R bilinear maps M × N → P and to each R module homomorphism f : P → Q the function B(f) : B(P) → B(Q) which sends each bilinear map g : M × N → P to the bilinear map f∘g : M × N→Q. The functor B is represented by the R module M ⊗R N.

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.

Any functor K : C → Set with a left adjoint F : Set → C is represented by (FX, ηX(•)) where X = {•} is a singleton set and η is the unit of the adjunction. 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.

The categorical notions of universal morphisms and adjoint functors can both be expressed using representable functors. 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

is a bijection for all X and Y. 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.