Representable functor

Inmathematics,particularlycategory theory,arepresentable functoris a certainfunctorfrom an arbitrarycategoryinto thecategory of sets.Such functors give representations of an abstract category in terms of known structures (i.e.setsandfunctions) 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 categoryCare the functorsgivenwithC.Their theory is a vast generalisation ofupper setsinposets,andYoneda's representabilitytheorem generalizesCayley's theoremingroup theory.

Definition

LetCbe alocally small categoryand letSetbe thecategory of sets.For each objectAofClet Hom(A,–) be thehom functorthat maps objectXto the set Hom(A,X).

AfunctorF:C→Setis said to berepresentableif it isnaturally isomorphicto Hom(A,–) for some objectAofC.ArepresentationofFis a pair (A,Φ) where

Φ: Hom(A,–) →F

is a natural isomorphism.

Acontravariant functorGfromCtoSetis the same thing as a functorG:C^op→Setand is commonly called apresheaf.A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(–,A) for some objectAofC.

Universal elements

According toYoneda's lemma,natural transformations from Hom(A,–) toFare in one-to-one correspondence with the elements ofF(A). Given a natural transformation Φ: Hom(A,–) →Fthe corresponding elementu∈F(A) is given by

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

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

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

wherefis an element of Hom(A,X). In order to get a representation ofFwe want to know when the natural transformation induced byuis an isomorphism. This leads to the following definition:

Auniversal elementof a functorF:C→Setis a pair (A,u) consisting of an objectAofCand an elementu∈F(A) such that for every pair (X,v) consisting of an objectXofCand an elementv∈F(X) there exists a unique morphismf:A→Xsuch that (Ff)(u) =v.

A universal element may be viewed as auniversal morphismfrom the one-point set {•} to the functorFor as aninitial objectin thecategory of elementsofF.

The natural transformation induced by an elementu∈F(A) is an isomorphism if and only if (A,u) is a universal element ofF.We therefore conclude that representations ofFare in one-to-one correspondence with universal elements ofF.For this reason, it is common to refer to universal elements (A,u) as representations.

Examples

Thefunctor represented by a schemeAcan sometimes describe families of geometric objects.For example,vector bundlesof rankkover a given algebraic variety or schemeXcorrespond to algebraic morphisms $X\to A$ whereAis theGrassmannianofk-planes in a high-dimensional space. Also certain types of subschemes are represented byHilbert schemes.
LetCbe the category ofCW-complexeswith morphisms given by homotopy classes of continuous functions. For each natural numbernthere is a contravariant functorHⁿ:C→Abwhich assigns each CW-complex itsn^thcohomology group(with integer coefficients). Composing this with theforgetful functorwe have a contravariant functor fromCtoSet.Brown's representability theoremin algebraic topology says that this functor is represented by a CW-complexK(Z,n) called anEilenberg–MacLane space.
Consider the contravariant functorP:Set→Setwhich maps each set to itspower setand each function to itsinverse imagemap. To represent this functor we need a pair (A,u) whereAis a set anduis a subset ofA,i.e. an element ofP(A), such that for all setsX,the hom-set Hom(X,A) is isomorphic toP(X) via Φ_X(f) = (Pf)u=f⁻¹(u). TakeA= {0,1} andu= {1}. Given a subsetS⊆Xthe corresponding function fromXtoAis thecharacteristic functionofS.
Forgetful functorstoSetare very often representable. In particular, a forgetful functor is represented by (A,u) wheneverAis afree objectover asingleton setwith generatoru.
- The forgetful functorGrp→Seton thecategory of groupsis represented by (Z,1).
- The forgetful functorRing→Seton thecategory of ringsis represented by (Z[x],x), thepolynomial ringin onevariablewithinteger coefficients.
- The forgetful functorVect→Seton thecategory of real vector spacesis represented by (R,1).
- The forgetful functorTop→Seton thecategory of topological spacesis represented by any singleton topological space with its unique element.
AgroupGcan be considered a category (even agroupoid) with one object which we denote by •. A functor fromGtoSetthen corresponds to aG-set.The unique hom-functor Hom(•,–) fromGtoSetcorresponds to the canonicalG-setGwith the action of left multiplication. Standard arguments from group theory show that a functor fromGtoSetis representable if and only if the correspondingG-set is simply transitive (i.e. aG-torsororheap). Choosing a representation amounts to choosing an identity for the heap.
LetRbe a commutative ring with identity, and letR-Modbe the category ofR-modules. IfMandNare unitary modules overR,there is a covariant functorB:R-Mod→Setwhich assigns to eachR-modulePthe set ofR-bilinear mapsM×N→Pand to eachR-module homomorphismf:P→Qthe functionB(f):B(P) →B(Q) which sends each bilinear mapg:M×N→Pto the bilinear mapf∘g:M×N→Q.The functorBis represented by theR-moduleM⊗_RN.^[1]

Analogy: Representable functionals

Consider a linear functional on a complexHilbert spaceH,i.e. a linear function $F:H\to \mathbb {C}$ .TheRiesz representation theoremstates that ifFis continuous, then there exists a unique element $a\in H$ which representsFin the sense thatFis equal to the inner product functional $\langle a,-\rangle$ ,that is $F(v)=\langle a,v\rangle$ for $v\in H$ .

For example, the continuous linear functionals on thesquare-integrable function space $H=L^{2}(\mathbb {R} )$ are all representable in the form $\textstyle F(v)=\langle a,v\rangle =\int _{\mathbb {R} }a(x)v(x)\,dx$ for a unique function $a(x)\in H$ .The theory ofdistributionsconsiders more general continuous functionals on the space of test functions $C=C_{c}^{\infty }(\mathbb {R} )$ .Such a distribution functional is not necessarily representable by a function, but it may be considered intuitively as a generalized function. For instance, theDirac delta functionis the distribution defined by $F(v)=v(0)$ for each test function $v(x)\in C$ ,and may be thought of as "represented" by an infinitely tall and thin bump function near $x=0$ .

Thus, a function $a(x)$ may be determined not by its values, but by its effect on other functions via the inner product. Analogously, an objectAin a category may be characterized not by its internal features, but by itsfunctor of points,i.e. its relation to other objects via morphisms. Just as non-representable functionals are described by distributions, non-representable functors may be described by more complicated structures such asstacks.

Properties

Uniqueness

Representations of functors are unique up to a unique isomorphism. That is, if (A₁,Φ₁) and (A₂,Φ₂) represent the same functor, then there exists a unique isomorphism φ:A₁→A₂such that

\Phi _{1}^{-1}\circ \Phi _{2}=\mathrm {Hom} (\varphi,-)

as natural isomorphisms from Hom(A₂,–) to Hom(A₁,–). This fact follows easily fromYoneda'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:A₁→A₂such that

(F\varphi )u_{1}=u_{2}.

Preservation of limits

Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covariant) representable functorspreserve all limits.It follows that any functor which fails to preserve some limit is not representable.

Contravariant representable functors take colimits to limits.

Left adjoint

Any functorK:C→Setwith aleft adjointF:Set→Cis represented by (FX,η_X(•)) whereX= {•} is asingleton setand η is the unit of the adjunction.

Conversely, ifKis represented by a pair (A,u) and all smallcopowersofAexist inCthenKhas a left adjointFwhich sends each setIto theIth copower ofA.

Therefore, ifCis a category with all small copowers, a functorK:C→Setis representable if and only if it has a left adjoint.

Relation to universal morphisms and adjoints

The categorical notions ofuniversal morphismsandadjoint functorscan both be expressed using representable functors.

LetG:D→Cbe a functor and letXbe an object ofC.Then (A,φ) is a universal morphism fromXtoGif and only if(A,φ) is a representation of the functor Hom_C(X,G–) fromDtoSet.It follows thatGhas a left-adjointFif and only if Hom_C(X,G–) is representable for allXinC.The natural isomorphism Φ_X:Hom_D(FX,–) → Hom_C(X,G–) yields the adjointness; that is

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

is a bijection for allXandY.

The dual statements are also true. LetF:C→Dbe a functor and letYbe an object ofD.Then (A,φ) is a universal morphism fromFtoYif and only if (A,φ) is a representation of the functor Hom_D(F–,Y) fromCtoSet.It follows thatFhas a right-adjointGif and only if Hom_D(F–,Y) is representable for allYinD.^[2]

References

^Hungerford, Thomas.Algebra.Springer-Verlag. p. 470.ISBN 3-540-90518-9.
^Nourani, Cyrus (19 April 2016).A Functorial Model Theory: Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos.CRC Press. p. 28.ISBN 978-1482231502.

Mac Lane, Saunders(1998).Categories for the Working Mathematician.Graduate Texts in Mathematics5(2nd ed.). Springer.ISBN 0-387-98403-8.

[1] Hungerford, Thomas.Algebra.Springer-Verlag. p. 470.ISBN 3-540-90518-9.

[2] Nourani, Cyrus (19 April 2016).A Functorial Model Theory: Newer Applications to Algebraic Topology, Descriptive Sets, and Computing Categories Topos.CRC Press. p. 28.ISBN 978-1482231502.

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