representable functor




For a locally small category CC, a presheaf on CC or equivalently a functor

F:C opSetF: C^{op} \to Set

on the opposite category of CC with values in Set is representable if it is naturally isomorphic to a hom-functor h X:=hom C(,X):C opSeth_X := \hom_C(-, X): C^{op} \to Set, which sends an object UCU \in C to the hom-set Hom C(U,X)Hom_C(U,X) in CC

and which sends a morphism α:UU\alpha : U' \to U in CC to the function which sends each morphism UXU \to X to the composite UαUXU' \stackrel{\alpha}{\to} U \to X

If we picture Hom C(U,X)Hom_C(U,X) as strands of morphisms as above, then the morphism α:UU\alpha:U'\to U serves to “comb” the strands back from Hom C(U,X)Hom_C(U,X) to Hom C(U,X)Hom_C(U',X), i.e.

h Xα:Hom C(U,X)Hom C(U,X).h_X\alpha: Hom_C(U,X)\to Hom_C(U',X).

The object XX is determined uniquely up to isomorphism in CC, and is called a representing object for FF.

Representability is one of the most fundamental concepts of category theory, with close ties to the notion of adjoint functor and to the Yoneda lemma. It is the crucial concept underlying the idea of universal property; thus for example crucial concepts such as “limit”, “colimit”, “exponential object”, “Kan extension” and so on are naturally expressed in terms of representing objects. The concept permeates much of algebraic geometry and algebraic topology.


In ordinary category theory

For a functor F:C opSetF: C^{op} \to Set (also called a presheaf on CC), a representation of FF is a specified natural isomorphism

θ:hom C(,c)F\theta: \hom_C(-, c) \stackrel{\sim}{\to} F

By the Yoneda lemma, any such transformation θ\theta (isomorphism or not) is uniquely determined by an element ξF(c)\xi \in F(c). As above, the object cc is called a representing object (or often, universal object) for FF, and the element ξ\xi is called a universal element for FF. Again, it follows from the Yoneda lemma that the pair (c,ξ)(c, \xi) is determined uniquely up to unique isomorphism.

Following the proof of the Yoneda lemma, representability means precisely this: given any object xx of CC and any element αF(x)\alpha \in F(x), there exists a unique morphism f:xcf: x \to c such that the function F(f)F(f) carries the universal element ξF(c)\xi \in F(c) to αF(x)\alpha \in F(x). Such a dry formulation fails to convey the remarkable power of this concept, which can really only be appreciated through the myriad examples which illustrate it.

In enriched category theory

The above definition generalizes straightforwardly to enriched category theory.

Let VV be a closed monoidal category and CC a VV-enriched category.

Then for every object cCc \in C there is a VV-enriched functor

C(c,):CV C(c,-) : C \to V

from CC to VV regarded canonically as a VV-enriched category.

This is defined

  • on objects by C(c,):dC(c,d)VC(c,-) : d \mapsto C(c,d) \in V

  • on morphisms between dd and dd' by

C(c,) d,d:C(d,d)[C(c,d),C(c,d)], C(c,-)_{d,d'} : C(d,d') \to [C(c,d), C(c,d')] \,,

being the adjunct of the composition morphism

c,d,d:C(d,d)C(c,d)C(c,d). \circ_{c,d,d'} : C(d,d') \otimes C(c,d) \to C(c,d') \,.

A VV-enriched functor F:CVF : C \to V is representable if there is cCc \in C and a VV-enriched natural transformation η:FC(c,)\eta : F \to C(c,-).

If VV is symmetric monoidal one can form the opposite category C opC^{op} and have the analogous definition for representable functors F:C opVF : C^{op} \to V.

In higher category theory

The notion of representable functors has its straightforward analogs also in higher category theory.


The central point about examples of representable functors is:

Representable functors are ubiquitous .

To a fair extent, category theory is all about representable functors and the other universal constructions: Kan extensions, adjoint functors, limits, which are all special cases of representable functors – and representable functors are special cases of these.

Listing examples of representable functors in category theory is much like listing examples of integrals in analysis: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see coend for more).

Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.


If F:JCF:J\to C is a diagram in CC, we can construct a diagram hom C(,F):JSet C op\hom_C(-,F):J\to Set^{C^{op}} in the functor category Set C opSet^{C^{op}} as the composite of FF with the curried hom-functor CSet C opC\to\Set^{C^{op}} (the Yoneda embedding). The object-wise limit of this diagram in Set, that is, the functor C opSetC^{op}\to\Set sending an object xx to the set which is the limit of the diagram hom C(x,F):JSet\hom_C(x,F):J\to\Set, is representable iff the diagram FF has a limit in CC; in fact, a representing object for that limit functor is exactly limF\lim F, and we obtain a natural isomorphism

limhom C(,F)hom C(,limF).\lim \hom_C(-,F)\cong\hom_C(-,\lim F).


For an example in the case of the product, let c,dc, d be objects of CC, and consider the presheaf given by a product of hom-functors

hom C(,c)×hom C(,d):C opSet;\hom_C(-, c) \times \hom_C(-, d): C^{op} \to Set;

that is, the functor which takes an object xx of CC to the set hom C(x,c)×hom C(x,d)\hom_C(x, c) \times \hom_C(x, d). A product c×dc \times d is precisely a representing or universal object for this presheaf, where the universal element is precisely the pair of projection maps

(π c,π d)hom(c×d,c)×hom(c×d,d)(\pi_c, \pi_d) \in \hom(c \times d, c) \times \hom(c \times d, d)

We leave it to the reader to check that the representability here means precisely that given a pair of maps

(f,g)hom(x,c)×hom(x,d)(f, g) \in \hom(x, c) \times \hom(x, d)

there exists a unique element in hom(x,c×d)\hom(x, c \times d), denoted f,g\langle f, g \rangle, such that

π cf,g=fπ df,g=g.\pi_c \langle f, g \rangle = f \qquad \pi_d \langle f, g \rangle = g.

Weighted limits

The above example has an important straightforward generalization.

Noticing that the limit over the functor H:JSetH : J \to Set is just the collection of cones over HH whose tip is the point

limH=[J,Set](Δpt,H) lim H = [J,Set](\Delta pt, H)

the above expression limhom C(,F)\lim\hom_C(-,F) can be rewritten equivalently as [J,Set](Δpt,C(,F()))[J,Set](\Delta pt, C(-,F(-))). Replacing in this expression the constant terminal functor Δpt:JSet\Delta pt : J \to Set by any other functor leads to the notion of weighted limit, as described there.

Exponential objects

Suppose CC is a category which admits finite products; given objects c,dc, d, consider the presheaf

hom C(×c,d):C opSet.\hom_C(- \times c, d): C^{op} \to Set.

A representing or universal object for this presheaf is an exponential object d cd^c; the universal element

ehom C(d c×c,d)e \in \hom_C(d^c \times c, d)

is a morphism called the evaluation map eval:d c×cdeval: d^c \times c \to d.

Classifying bundles

Consider a category TopTop of ‘nice’ spaces (just to fix the discussion, let’s say paracompact spaces, although this is a technical point), and a topological group GG therein, i.e., a group internal to TopTop. There is a presheaf

GBund:Top opSetG\Bund: Top^{op} \to Set

which assigns to each space XX the set of isomorphism classes of GG-bundles over XX, and assigns to each continuous map f:XYf: X \to Y the function

GBund(f):GBund(Y)GBund(X)G\Bund(f): G\Bund(Y) \to G\Bund(X)

which carries a (class of a) GG-bundle EYE \to Y to the (class of the) pullback bundle f *EXf^*E \to X. It is well-known that the pullback construction is invariant with respect to homotopic deformations; that is, this presheaf descends to a functor on the homotopy category,

GBund:Ho Top opSet.G\Bund: Ho_{Top}^{op} \to Set.

A classifying space G\mathcal{B}G is precisely a representing object for this functor; the universal element is the (isomorphism class of the) classifying GG-bundle [π:GG][\pi: \mathcal{E}G \to \mathcal{B}G].

These general considerations are quite commonplace in algebraic topology, where they crop up for example in the connection between generalized cohomology theories and spectra; cf. Brown’s representability theorem.


Early accounts:

  • Alexander Grothendieck, Section A.1 of: Technique de descente et théorèmes d’existence en géométrie algébriques. II. Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p. (numdam:SB_1958-1960__5__369_0)

A discussion of representable functors in the context of enriched category theory is in section 1.6 and section 1.10 of

  • Max Kelly, Basic concepts of enriched category theory (pdf)

A query discussion on differences between representable functor and representation of a functor is archived here.

Last revised on June 18, 2021 at 14:35:33. See the history of this page for a list of all contributions to it.