Definitions

# Representable functor

In mathematics, especially in category theory, a representable functor is a functor of a special form 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.

## Definition

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 which maps objects X to the set Hom(A,X).

A functor F : CSet 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 : CSet is said to representable if it is naturally isomorphic to the contravariant hom-functor Hom(–,A) for some object A of C.

## Universal elements

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 of uF(A) is given by

$u = Phi_A\left(mathrm\left\{id\right\}_A\right).,$
Conversely, given any element uF(A) we may define a natural transformation Φ : Hom(A,–) → F via
$Phi_X\left(f\right) = \left(Ff\right)\left(u\right),$
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 : CSet is a pair (A,u) consisting of an object A of C and an element uF(A) such that for every pair (X,v) with vF(X) there exists a unique morphism f : AX 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 uF(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.

## Examples

• Consider the contravariant functor P : SetSet 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 SX the corresponding function from X to A is the characteristic function of S.
• Forgetful functors to Set are very often representable. It particular, a forgetful functor is represented by (A, u) whenever A is a free object over a singleton set with generator u.
• 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). Choosing a representation amounts to choosing an identity for the group structure.
• 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 : CAb 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-Mac Lane space.

## Properties

### Uniqueness

Representations of functors are unique up to a unique isomorphism. That is, if (A11) and (A22) represent the same functor, then there exists a unique isomorphism φ : A1A2 such that

$Phi_1^\left\{-1\right\}circPhi_2 = mathrm\left\{Hom\right\}\left(varphi,-\right)$
as natural isomorphisms from Hom(A2,–) to Hom(A1,–). This fact follows easily from Yoneda's lemma.

Stated in terms of universal elements: if (A1,u1) and (A2,u2) represent the same functor, then there exists a unique isomorphism φ : A1A2 such that

$\left(Fvarphi\right)u_1 = u_2.$

### Preservation of limits

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 : CSet with a left adjoint F : SetC 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 : CSet is representable if and only if it has a left adjoint.

## Relation to universal morphisms and adjoints

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

Let G : DC 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

$Phi_\left\{X,Y\right\}colon mathrm\left\{Hom\right\}_\left\{mathcal D\right\}\left(FX,Y\right) to mathrm\left\{Hom\right\}_\left\{mathcal C\right\}\left(X,GY\right)$
is a bijection for all X and Y.

The dual statements are also true. Let F : CD 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.

## References

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