nLab
representable functor

Overview

If $C$ is a locally small category, then we say a functor

$$F:C^{\mathrm{op}}\to \mathrm{Set}$$F: C^{op} \to Set

is representable if it is naturally isomorphic to a hom-functor ${\mathrm{hom}}_C(-,c):C^{\mathrm{op}}\to \mathrm{Set}$. The object $c$ is determined uniquely up to isomorphism, and is called a representing object for $F$.

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.

Definition

More precisely, given a functor $F:C^{\mathrm{op}}\to \mathrm{Set}$ (also called a presheaf on $C$), a representation of $F$ is a specified natural isomorphism

$$\theta :{\mathrm{hom}}_C(-,c)\stackrel{\sim }{\to }F$$\theta: \hom_C(-, c) \stackrel{\sim}{\to} F

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

Following the proof of the Yoneda lemma, representability means precisely this: given any object $x$ of $C$ and any element $\alpha \in F(x)$, there exists a unique morphism $f:x\to c$ such that the function $F(f)$ carries the universal element $\xi \in F(c)$ to $\alpha \in F(x)$. 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.

Definition in enriched category theory

The above definition generalizes straightforwardly to enriched category theory.

Let $V$ be a closed monoidal category and $C$ a $V$-enriched category.

Then for every object $c\in C$ there is a $V$-enriched functor

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

from $C$ to $V$ regarded canonically as a $V$-enriched category.

This is defined

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

• on morphisms between $d$ and $d\prime$ by

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

being the adjunct of the composition morphism

$${\circ }_{c,d,d\prime }:C(d,d\prime )\otimes C(c,d)\to C(c,d\prime ).$$\circ_{c,d,d'} : C(d,d') \otimes C(c,d) \to C(c,d') \,.

A $V$-enriched functor $F:C\to V$ is representable if there is $c\in C$ and a $V$-enriched natural transformation? $\eta :F\to C(c,-)$.

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

Examples

Example 1: limits

If $F:J\to C$ is a diagram in $C$, we can construct a diagram ${\mathrm{hom}}_C(-,F)$ in the functor category ${\mathrm{Set}}^{C^{\mathrm{op}}}$ as the composite of $F$ with the curried hom-functor $C\to {Set}^{C^{\mathrm{op}}}$ (the Yoneda embedding). The object-wise limit of this diagram in Set, that is, the functor $C^{\mathrm{op}}\to Set$ sending on object $x$ to the set which is the limit of the diagram ${\mathrm{hom}}_C(x,F):J\to Set$, is representable iff the diagram $F$ has a limit in $C$; in fact, a representing object for that limit functor is exactly $\lim F$, and we obtain a natural isomorphism

$$\lim {\mathrm{hom}}_C(-,F)\cong {\mathrm{hom}}_C(-,\lim F).$$\lim\hom_C(-,F)\cong\hom_C(-,\lim F).

Sub-Example: Products

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

$${\mathrm{hom}}_C(-,c)\times {\mathrm{hom}}_C(-,d):C^{\mathrm{op}}\to \mathrm{Set};$$\hom_C(-, c) \times \hom_C(-, d): C^{op} \to Set;

that is, the functor which takes an object $x$ of $C$ to the set ${\mathrm{hom}}_C(x,c)\times {\mathrm{hom}}_C(x,d)$. A product $c\times d$ is precisely a representing or universal object for this presheaf, where the universal element is precisely the pair of projection maps

$$({\pi }_c,{\pi }_d)\in \mathrm{hom}(c\times d,c)\times \mathrm{hom}(c\times d,d)$$(\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

$$(f,g)\in \mathrm{hom}(x,c)\times \mathrm{hom}(x,d)$$(f, g) \in \hom(x, c) \times \hom(x, d)

there exists a unique element in $\mathrm{hom}(x,c\times d)$, denoted $\mathrm{langle}f,g⟩$, such that

$${\pi }_c⟨f,g⟩=f{\pi }_d⟨f,g⟩=g.$$\pi_c \langle f, g \rangle = f \qquad \pi_d \langle f, g \rangle = g.

Example 1 a: weighted limits

The above example has an important straightforward generalization.

Noticing that the limit over the functor $H:J\to \mathrm{Set}$ is just the collection of cones over $H$ whose tip is the point

$$\lim H=[J,\mathrm{Set}](\Delta \mathrm{pt},H)$$lim H = [J,Set](\Delta pt, H)

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

Example 2: exponential objects

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

$${\mathrm{hom}}_C(-\times c,d):C^{\mathrm{op}}\to \mathrm{Set}.$$\hom_C(- \times c, d): C^{op} \to Set.

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

$$e\in {\mathrm{hom}}_C(d^c\times c,d)$$e \in \hom_C(d^c \times c, d)

is a morphism called the evaluation map $\mathrm{eval}:d^c\times c\to d$.

Example 3: classifying bundles

Consider a category $\mathrm{Top}$ of ‘nice’ spaces (just to fix the discussion, let’s say paracompact spaces, although this is a technical point), and a topological group $G$ therein, i.e., a group internal to $\mathrm{Top}$. There is a presheaf

$$GBund:{\mathrm{Top}}^{\mathrm{op}}\to \mathrm{Set}$$G\Bund: Top^{op} \to Set

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

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

which carries a (class of a) $G$-bundle $E\to Y$ to the (class of the) pullback bundle $f^{*}E\to X$. 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,

$$GBund:{\mathrm{Ho}}_{\mathrm{Top}}^{\mathrm{op}}\to \mathrm{Set}.$$G\Bund: Ho_{Top}^{op} \to Set.

A classifying space $ℬG$ is precisely a representing object for this functor; the universal element is the (isomorphism class of the) classifying $G$-bundle $[\pi :ℰG\to ℬG]$.

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.

References

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

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

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Discussion

I am pretty unhappy that all entries related to limits, colimits and representable things at nlab say that the limit, colimit and representing functors are what normally in strict treatment are just the vertices of the corresponding universal construction. A representable functor is not a functor which is naturally isomorphic to Hom(-,c) but a pair of an object and such isomorphism! Similarly limit is the synonym for limiting cone (= universal cone), not just its vertex. Because if it were most of usages and theorems would not be true. For example, the notion and usage of creating limits under a functor, includes the words about the behaviour of the arrow under the functor, not only of the vertex. Definitions should be the collections of the data and one has to distinguish if the existence is really existence or in fact a part of the structure.–Zoran

Mike: I disagree (partly). First of all, a functor $F$ equipped with an isomorphism $F\cong {\mathrm{hom}}_C(-,c)$ is not a representable functor, it is a represented functor, or a functor equipped with a representation. A representable functor is one that is “able” to be represented, or admits a representation.

Second, the page limit says “a limit of a diagram $F:D\to C$ … is an object $\lim F$ of $C$ equipped with morphisms to the objects $F(d)$ for all $d\in D$…” (emphasis added). It doesn’t say “such that there exist” morphisms. (Prior to today, it defined a limit to be a universal cone.) It is true that one frequently speaks of “the limit” as being the vertex, but this is an abuse of language no worse than other abuses that are common and convenient throughout mathematics (e.g. “let $G$ be a group” rather than “let $(G,\cdot ,e)$ be a group”). If there are any definitions you find that are wrong (e.g. that say “such that there exists” rather than “equipped with”), please correct them! (Thanks to your post, I just discovered that Kan extension was wrong, and corrected it.)

Zoran Skoda I fully agree, Mike that “equipped with” is just a synonym of a “pair”. But look at entry for limit for example, and it is clear there that the limiting cone/universal cone and limit are clearly distinguished there and the term limit is used just for the vertex there. Unlike for limits where up to economy nobody doubt that it is a pair, you are right that many including the very MacLane representable take as existence, but then they really use term “representation” for the whole pair. Practical mathematicians are either sloppy in writing or really mean a pair for representable. Australians and MacLane use indeed word representation for the whole thing, but practical mathematicians (example: algebraic geometers) are not even aware of term “representation” in that sense, and I would side with them. Let us leave as it is for representable, but I do not believe I will ever use term “representation” in such a sense. For limit, colimit let us talk about pairs: I am perfectly happy with word “equipped” as you suggest.

Mike: I’m not sure what your point is about limits. The definition at the beginning very clearly uses the words “equipped with.” Later on in the page, the word “limit” is used to refer to the vertex, but this is just the common abuse of language.

Regarding representable functors, since representations are unique up to unique isomorphism when they exist, it really doesn’t matter whether “representable functor” means “functor such that there exists an isomorphism $F\cong {\mathrm{hom}}_C(-,c)$” or “functor equipped with an isomorphism $F\cong {\mathrm{hom}}_C(-,c)$.” (As long as it doesn’t mean something stupid like “functor equipped with an object $c$ such that there exists an isomorphism $F\cong {\mathrm{hom}}_C(-,c)$.”) In the language of stuff, structure, property, we can say that the Yoneda embedding is fully faithful, so that “being representable” is really a property, rather than structure, on a functor.

Zoran Skoda For limit look at the section Unwrapping. This section clearly distiguishes between the limiting cone and the limit. It says roughly if we have a limit then we can define a special cone, and then it is called limiting cone. So it strongly suggests that the limiting cone is something different from a limit, but in fact they are synonyms, except that it is common to abuse the language and say a limit for the vertex only: however one does not do that abuse when introducing the notion. Then one has to be clear that it is the same thing. Even it uses a terminology “universal limiting cone” what is double notation as limiting cone and universal cone and universal initial cone and initial cone and limit are equivalent terms. One allows “universal initial” because sometimes one says “universal terminal”.

For representable I agreed, though I will of course continue using it with the structure whenever needed.

Mike: Actually, that page has not defined limits using cones, but using an isomorphism of hom-sets. This is, of course, equivalent, but the section “Unwrapping” is about that equivalence. So when it says “if the limit $\lim F\in C$ of $F$ exists, then it singles out a special cone” it means that given an object $\lim F$ equipped with an isomorphism $\mathrm{Hom}(c,\lim F)\cong \mathrm{Hom}(\mathrm{pt},\mathrm{Hom}(c,F-))$ natural in $c$, there exists a particular cone and so on. On the next line it defines that cone, using the isomorphism with which $\lim F$ is equipped.

That is, a limit is an object equipped with data which makes it a limit. That data might be a limiting cone, or it might be a certain natural isomorphism; it doesn’t matter since either can be constructed from the other.

I don’t think anyone would argue, though, that that page is a work in progress. It’s certainly not all that clear which parts are the definitions, which parts are alternate definitions, and which parts are theorems.

Toby: Hum … If you think of mathematics as done in type theory with the ‘propositions as types’ paradigm, then there is no literal difference between ‘such that there exists’ and ‘equipped with’. At best, you can distinguish these as locutions for describing what the notion of equality is (or what the morphisms are) between the things being described (in which case ‘equipped with’ is what you usually want), but in definitions one ought to be explicit about that too.

I would normally say ‘equipped with’ (or just ‘with’), and I think that you, Zoran, should feel free to change ‘such that there exists’ to ‘equipped with’ when you find it appropriate. But I also think that mathematicians should learn to read ‘such that there exists’ to mean ‘equipped with’ unless it's explicitly stated that one is using a more loose notion of equality (or morphism).

Mike: Perhaps unsurprisingly, I disagree. In all the working mathematics I have ever seen, types and propositions are not the same. The set of all $X$s is not the same as the property of existence of $X$s; the latter is the image of the former. It’s fine and fun to view your types as propositions and propositions as types and there are real insights to be gained in that way, but I think that when applied to ordinary mathematics, one should use a type theory in which propositions (aka (-1)-types) are distinguished from types (aka 0-types).

Toby: It's fine and fun to have a hierarchy of $n$-types (although $1$-types should be types, of course), but this is just terminology to handle the equalities/isomorphisms/equivalences automatically, so it doesn't really disagree with me. Each convention is an abuse of language with respect to the other (and each is used, often thoughtlessly, in practice) that disappears when one is more explicit.