Representable functor

Let C be a locally small category and let Set be the category of sets. For each object A of C let Hom(A,–) be the hom functor that maps object X to the set Hom(A,X).

A functor F : C → Set is said to be representable if it is naturally isomorphic to Hom(A,–) for some object A of C. A representation of F is a pair (A, Φ) where

Φ : Hom(A,–) → F

is a natural isomorphism.

A contravariant functor G from C to Set is the same thing as a functor G : C^op → Set and is commonly called a presheaf. A presheaf is representable when 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 u ∈ F(A) is given by

${\displaystyle u=\Phi _{A}(\mathrm {id} _{A}).\,}$

Conversely, given any element u ∈ F(A) we may define a natural transformation Φ : Hom(A,–) → F via

${\displaystyle \Phi _{X}(f)=(Ff)(u)\,}$

where f is an element of Hom(A,X). In order to get a representation of F we want to know when the natural transformation induced by u is an isomorphism. This leads to the following definition:

A universal element of a functor F : C → Set is a pair (A,u) consisting of an object A of C and an element u ∈ F(A) such that for every pair (X,v) with v ∈ F(X) there exists a unique morphism f : A → X such that (Ff)u = v.

A universal element may be viewed as a universal morphism from the one-point set {•} to the functor F or as an initial object in the category of elements of F.

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.

• 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 Φ_X(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 n^th 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.[1]

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 Hom_C(X,G–) from D to Set. It follows that G has a left-adjoint F if and only if Hom_C(X,G–) is representable for all X in C. The natural isomorphism Φ_X : Hom_D(FX,–) → Hom_C(X,G–) yields the adjointness; that is

${\displaystyle \Phi _{X,Y}\colon \mathrm {Hom} _{\mathcal {D}}(FX,Y)\to \mathrm {Hom} _{\mathcal {C}}(X,GY)}$

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 Hom_D(F–,Y) from C to Set. It follows that F has a right-adjoint G if and only if Hom_D(F–,Y) is representable for all Y in D.

• Subobject classifier
• Density theorem

• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer. ISBN 0-387-98403-8.