nLab
profunctor

Contents

Idea

The concept of profunctor is a generalization of the concept of functor in much the same way that the concept of bimodule generalizes that of algebra homomorphism (in fact, this may be understood as a special case of enriched profunctors).

Definition

If CC and DD are categories, then a profunctor from CC to DD is a functor of the form

H F:D op×CSet. H_F \colon D^{op}\times C \to Set \,.

Such a profunctor is usually written as F:CDF\colon C ⇸ D. For dDd \in D and cCc \in C the set H FH_F is also called the set of heteromorphisms from dd to cc.

Every functor f:CDf\colon C\to D induces two profunctors D(1,f):CDD(1,f)\colon C ⇸ D and D(f,1):DCD(f,1)\colon D ⇸ C, defined by D(1,f)(d,c)=D(d,f(c))D(1,f)(d,c) = D(d,f(c)) and D(f,1)(c,d)=D(f(c),d)D(f,1)(c,d) = D(f(c),d). (Here D(,)D(-,-) denotes the hom functor of DD and 11 denotes the identity functor on the respective category.) Since this construction may be thought of as the adjunct of the composition of ff with the Yoneda embedding CfDYoneda[D op,Set]C \stackrel{f}{\longrightarrow} D \stackrel{Yoneda}{\longrightarrow} [D^{op},Set], these profunctors are called representable (or sometimes one of them is called corepresentable) and this way profunctors subsume and generalize ordinary functors.

In particular the identity profunctor Id:CCId \colon C ⇸ C is represented by the identity functor and hence is given by the hom-functor C(,):C op×CSetC(-,-) : C^{op} \times C \to Set itself.

The notion generalizes to many other kinds of categories. For instance, if CC and DD are enriched over some symmetric closed monoidal category VV, then a profunctor from CC to DD is a VV-functor D opCVD^{op} \otimes C\to V. If they are internal categories, then a profunctor CDC ⇸ D is an internal diagram on D op×CD^{op}\times C, and so on. There are also other equivalent definitions in each case; see below.

A profunctor is also sometimes called a (bi)module or a distributor or a correspondence, though the latter word is also used for a span. The term “module” tends to be common in Australia, especially in the enriched case; here the intuition is that for one-object VV-categories, i.e. monoids in VV, profunctors really are the same as bimodules between such monoids in the usual sense. “Profunctor” is perhaps more common in the Set-based and internal cases (but is also used in the enriched case); here the intuition is that a profunctor is a generalization of a functor, via the construction of “representable” profunctors. Jean Bénabou, who invented the term and originally used “profunctor,” now prefers “distributor,” which is supposed to carry the intuition that a distributor generalizes a functor in a similar way to how a distribution generalizes a function.

Note that the convention that a profunctor is a functor D op×CSetD^{op}\times C \to Set is not universal; some authors reverse CC and DD and/or put the “op” on the other one. See the discussion below.

The bicategory of profunctors

Profunctors are composed by using a coend to “trace out” the middle variable. Specifically, for profunctors F:CDF : C ⇸ D and G:DEG : D ⇸ E, their composite GF:CEG \circ F: C ⇸ E is defined to be

GF:= dDF(d,)G(,d). G \circ F := \int^{d \in D} F(d,-)\otimes G(-,d) \,.

This yields a bicategory in which

This bicategory is variously denoted ProfProf, ModMod, or DistDist, according to one’s chosen name for profunctors. In the VV-enriched case, it is written VProfV Prof or VModV Mod or VDistV Dist.

The construction of the “representable” profunctors D(1,f)D(1,f) and D(f,1)D(f,1) from a functor f:CDf\colon C\to D yield two identity-on-objects functors CatProfCat \to Prof and Cat opProfCat^{op}\to Prof. Moreover, it is easy to check that D(1,f)D(f,1)D(1,f) \vdash D(f,1) in the bicategory ProfProf; thus CatProfCat\to Prof is a proarrow equipment in the sense of Wood (in fact, the prototypical one). This same fact can also be expressed by defining a (pseudo) double category in which functors and profunctors are the two kinds of arrows; the construction of representable profunctors is then given by companions and conjoints in this double category, which make it a framed bicategory, hence an equivalent representation of a proarrow equipment.

The full sub-2-category Prof repProf_{rep} on representable profunctors is equivalent to Cat anaCat_{ana}, the 2-category of anafunctors. See there for more details.

Alternative definitions

In terms of colimit-preserving functors on presheaf categories

A basic fact (e.g. Kashiwara, Schapira, Categories and Sheaves, corollary 2.7.4, page 63) is that for AA a cocomplete category, colimit-preserving functors from presheaves on some category CC to AA are canonically equivalent to functors from CC to AA: we have an equivalence of functor categories

Cocont(PSh(C),A)Func(C,A). Cocont(PSh(C),A) \simeq Func(C,A) \,.

This may be thought of as a consequence of the co-Yoneda lemma (and hence, of course, of the Yoneda lemma) which says that every presheaf is a colimit over representables, i.e. over objects in the image of the Yoneda embedding Y:CPSh(C)Y : C \to PSh(C). This immediately implies that a colimit-preserving functor on PSh(C)PSh(C) is already determined by its restriction along YY to CC.

Now, profunctors D opCVD^{op} \otimes C \to V are adjunct to functors C[D op,V]PSh(D)C \to [D^{op}, V] \simeq PSh(D). Hence by the above, profunctors are equivalent to colimit-preserving functors

PSh(C)PSh(D). PSh(C) \to PSh(D) \,.

Indeed, there is an equivalence of bicategories between VProfV Prof and the 2-category of categories and colimit-preserving functors and natural transformations between their presheaf categories. Note that the latter is a strict 2-category which can thus serve as a “natural” strictification of VProfV Prof.

An explicit statement of this can be found for instance as prop. 4.2.4 in

  • Gian Luca Cattani, PhD thesis from BRICS, University of Aarhus (pdf)

From this perspective, the representable profunctor induced by an ordinary VV-functor f:CDf : C \to D is the adjunct of the postcomposition

CfDY[D op,V] C \stackrel{f}{\to} D \stackrel{Y}{\to} [D^{op},V]

with the Yoneda embedding under the Hom-adjunction.

The formulation of profunctors as colimit-preserving functors on presheaf categories plays a big role also in the context of (∞,1)-categories. A presentable (∞,1)-category is one equivalent to a localization of some (∞,1)-category of (∞,1)-presheaves (i.e. some reflective (∞,1)-subcategory of the latter). The collection of all presentable (∞,1)-categories and colimit-preserving (∞,1)-functors betweem them forms the symmetric monoidal (∞,1)-category of presentable (∞,1)-categories, whose tensor product is the “bilinear” tensor product coming from interpreting colimit-preserving functors as “linear” (reading: colimit \sim sum).

This (,1)(\infty,1)-category Pr LPr^L therefore is an (,1)(\infty,1)-analog of Set-ModSet\text{-}Mod. In geometric ∞-function theory one finds (see section 4 there) that morphisms in Pr LPr^L encode the “correspondence operations” such as Fourier-Mukai and its generalizations. See in that context also the examples below.

In terms of two-sided discrete fibrations

Recall that a functor D opSetD^{op}\to Set can equivalently be described as a discrete (Grothendieck) fibration, and similarly a functor CSetC\to Set can be described as a discrete opfibration. Thus, a profunctor D op×CSetD^{op}\times C\to Set could be described by a discrete opfibration over D op×CD^{op}\times C, or a discrete fibration over D×C opD\times C^{op}, but there is also a more directly “two-sided” fibrational description. A two-sided fibration from CC to DD is a functor EC×DE\to C\times D which is a fibration over DD and an opfibration over CC in a compatible way. Such a fibration represents a pseudofunctor D op×CCatD^{op}\times C\to Cat, and hence if it is discrete it represents a profunctor D op×CSetD^{op}\times C\to Set.

This definition/characterization of profunctors works for internal categories as well, but not for enriched ones. It is sometimes called the graph of a profunctor (although this is sometimes also used for the other fibrational representations mentioned above).

In terms of two-sided codiscrete cofibrations

Yet another way of representing profunctors is via their collages, also called cographs. The collage of a profunctor H:CDH\colon C ⇸ D is, in particular, a category H¯\bar{H} equipped with functors CH¯C\to \bar{H} and DH¯D\to\bar{H} which are fully faithful and jointly bijective on objects.

In fact, the objects of the undercategory (CD)/Cat(C\sqcup D)/Cat which are collages of profunctors CDC ⇸ D can be characterized, up to equivalence, as the two-sided codiscrete cofibrations, i.e. the two-sided discrete fibrations in Cat opCat^{op}. In simpler and more explicit language, these are the categories MM which contain CC and DD as disjoint full subcategories which are jointly-wide (i.e. together contain all the objects), and such that there are no morphisms from an object of CC to an object of DD. Equivalently, they are the categories which admit a functor to the interval category II such that DD is the fiber over 00 and CC is the fiber over 11.

When viewing a profunctor H:CDH\colon C ⇸ D in this way, one may sometimes speak of elements of H(d,c)H(d,c) as heteromorphisms from dd to cc, since they are morphisms in the category H¯\bar{H} and can be “composed” with morphisms of CC and DD (this corresponds to the “action” of CC and DD on HH in the other formulations), but they go between objects of two different categories (namely CC and DD).

This characterization works just as well in both the internal and enriched case. Perhaps surprisingly, it also tends to give the “right” notion of profunctor starting with many other, even more exotic, 2-categories. However, it is trickier to figure out how to define the composite of profunctors viewed as codiscrete cofibrations; see codiscrete cofibration.

Comparing fibrations and cofibrations

If CgH¯fDC \overset{g}{\to} \bar{H} \overset{f}{\leftarrow} D is a codiscrete cofibration representing a profunctor HH from CC to DD, then the two-sided discrete fibration representing the same profunctor can be obtained as the comma category (fg)(f\downarrow g) with its two projections to CC and DD.

Dually, if CpEqDC \overset{p}{\leftarrow} E \overset{q}{\to} D is a two-sided discrete fibration representing a profunctor from CC to DD, then the codiscrete cofibration representing the same profunctor can be obtained as the cocomma object? (qp)(q\uparrow p) with the two inclusions of CC and DD.

In fact, in any 2-category with comma and cocomma objects, we have an adjunction

cocomma:Span(C,D)Cospan(C,D):comma. cocomma: Span(C,D) \;\rightleftarrows\; Cospan(C,D) : comma.

One can show that comma objects are always discrete fibrations, and dually cocomma objects are always codiscrete cofibrations. In CatCat and other similar 2-categories, this adjunction is idempotent and restricts to an equivalence between the categories of discrete fibrations and codiscrete cofibrations (both of which are of course equivalent to the category of profunctors from CC to DD). This is a two-sided version of the Grothendieck construction.

Examples

  • Recall that a one-object Vect-enriched category is just an algebra, while a general Vect-enriched category is an algebroid. The full sub-bicategory of VectModVect\Mod on one-object VectVect-enriched categories is the familiar category of algebras, bimodules and bimodule homomorphisms. In this case, the “representable” profunctors correspond to the way in which every morphism ABA \to B of algebras induces the AA-BB bimodule which as a vector space is BB with obvious right BB action and left AA-action induced by first mapping AA to BB via ff and then using multiplication in BB.

  • The full sub-bicategory of SetProfSet Prof on discrete categories is the bicategory of sets, spans of sets and morphisms of spans:

    SetMod discSpan(Set). Set\Mod_{disc} \simeq Span(Set) \,.

    In particular a relation between sets is a special case of this. From this point of view the 2-category Prof of profunctors is a categorification of the category Rel of sets and relations.

    Moreover, the representable profunctor between discrete category induced by a function f:CDf : C \to D of sets is the span

    C Id f C D. \array{ && C \\ & {}^{Id}\swarrow && \searrow^{f} \\ C &&&& D } \,.
  • Similarly, the full sub-bicategory of internal profunctors in S=(Set op,×)S = (Set^{op}, \times) on “discrete categories” is the bicategory of cospans

    Set opMod discSpan(Set op)=Cospan(Set). Set^{op}\Mod_{disc} \simeq Span(Set^{op}) = Cospan(Set) \,.
  • for enrichment over a category of chain complexes an enriched category is a dg-category and a profunctor is now a dg-bimodule of dg-categories. This appears notably in the definition of noncommutative motives.

References

Original sources

The original published source for profunctors is

  • Jean Bénabou, Les distributeurs, Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, rapport 33, 1973

based on several series of lectures starting in 1969, but these notes are hard to come by. They are available from the author by request. Much (if not most) of the existing work on profunctors has been developed by Bénabou.

Marta Bunge independently considered profunctors in chapter 3 of her unpublished 1966 PhD thesis

  • Marta Bunge, Categories of Set-Valued Functors, University of Pennsylvania, 1966

based on suggestions by Bill Lawvere, including the monograph

  • Michel Andre?, Categories of Functors and Adjoint Functors, Batelle Institute at Geneva, 1964

Some of these ideas were exposed at Oberwolfach in 1966. There are extant notes taken by Anders Kock of a talk by Bill Lawvere, but there is only a passing mention of ‘generalised functors’ (what are now called profunctors) and their ‘generalised matrix multiplication’.

More recent accounts

A standard reference is

  • Francis Borceux, Handbook of Categorical Algebra I, Cambridge UP 1994. sec. 7.7.

Excellent notes from a course on distributors given by Jean Bénabou in June 2000 at TU Darmstadt, and prepared by Thomas Streicher:

A nice example of profunctors between Lawvere metric spaces can be found in this comment.

See the Joyal's CatLab for the theory of Set-valued distributors:

Profunctors play an important in categorical shape theory. The original source is

  • D. Bourn, J.-M. Cordier, Distributeurs et théorie de la forme, Cah.Top.Géom.Diff.Cat. 21 no.2 (1980) pp.161-189. (pdf)

The material together with a general discussion of profunctors is also available in English in the recently reprinted standard reference

  • J.-M. Cordier and Tim Porter, Shape Theory: Categorical Methods of Approximation , (1989), Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008).

M. Batanin discusses simplicial enriched profunctors in the context of strong shape theory in

  • M. Batanin , Categorical Strong Shape Theory , Cah.Top.Géom.Diff.Cat. 38 no.1 (1997) pp.3-66. (pdf)

Expositions

There is some exposition of profunctors at

The common generalization of bimodules and spans in terms of profunctors has been discussed on the blog at

Notation

Profunctors are often notated with a slashed or barred arrow, as in CDC ⇸ D, which is U+21F8 in Unicode. It is not always obvious how to draw this character, so here are some hints.

  • On the nLab (or anywhere that accepts SGML character entities, including raw HTML on the web), it can be found using a Unicode entity:

    ⇸
  • In LaTeX, one can use \nrightarrow (producing ‘\nrightarrow’) in a pinch, but a nice-looking extensible barred arrow command \xslashedrightarrow can also be produced with the following preamble code (modified from amsmath’s \xrightarrow). It requires the packages amsmath and mathtools to be loaded.

    \makeatletter
    \def\slashedarrowfill@#1#2#3#4#5{%
      $\m@th\thickmuskip0mu\medmuskip\thickmuskip\thinmuskip\thickmuskip
       \relax#5#1\mkern-7mu%
       \cleaders\hbox{$#5\mkern-2mu#2\mkern-2mu$}\hfill
       \mathclap{#3}\mathclap{#2}%
       \cleaders\hbox{$#5\mkern-2mu#2\mkern-2mu$}\hfill
       \mkern-7mu#4$%
    }
    \def\rightslashedarrowfill@{%
      \slashedarrowfill@\relbar\relbar\mapstochar\rightarrow}
    \newcommand\xslashedrightarrow[2][]{%
      \ext@arrow 0055{\rightslashedarrowfill@}{#1}{#2}}
    \makeatother

    The command \xslashedrightarrow can then be used with one required argument and one optional argument, just like \xrightarrow. A version taking no arguments can of course be defined with

    \def\slashedrightarrow{\xslashedrightarrow{}}

    A simpler barred arrow taking no arguments can be created with

    \def\slashedrightarrow{\relbar\joinrel\mapstochar\joinrel\rightarrow}
  • In Xypic, a barred arrow (to the right, in this example) can be produced with

    \ar[r]|-@{|}

Discussion

The following discussion is about where to put the “op” in the definition of a profunctor.

Todd: There is an inevitable debate here about whether one should use C opDVC^{op} \otimes D \to V or CD opVC \otimes D^{op} \to V. My own convention is to use the latter. For example, every functor CDC \to D yields a profunctor by composition with the Yoneda embedding on DD.

Mike: My convention is D opCD^{op}\otimes C. I agree with your reasoning for why DD should be contravariant; I like to put it first because in the hom-functor C(,)C(-,-) the contravariant variable appears first.

Sridhar Ramesh: But surely, just as well, a functor from CC to DD yields a contravariant functor from CC to Set DSet^D and thus a profunctor C opDVC^{op} \otimes D \to V, by composition with the contravariant Yoneda embedding of DD into Set DSet^D? At the moment, I still do not see why there is reason to prefer in the abstract general one to the other of (c,d)Hom D(F(c),d)(c, d) \mapsto Hom_D(F(c), d) and (d,c)Hom D(d,F(c))(d, c) \mapsto Hom_D(d, F(c)), though it’s not an issue I’ve thought very much about or have strong emotions regarding. Are there further reasons beyond the above?

Mike Shulman: Well, the covariant Yoneda embedding is arguably more natural and important than the contravariant one. If a profunctor CDC⇸ D is a functor CSet D opC\to Set^{D^{op}}, then we can think of it as assigning to every cCc\in C a presheaf on DD, which may or may not be representable. The profunctor “is” a functor just when all its values are representable presheaves. Of course, if instead a profunctor CDC⇸ D were a functor C opSet DC^{op}\to Set^D, i.e. C(Set D) opC\to (Set^D)^{op}, then we could think of it as assigning to each cCc\in C a functor DSetD\to Set, which might or might not be (co)representable. However, for a bunch of reasons it’s often more natural to think of an object of DD as determined by the maps into it, rather than the maps out of it—in other words by its generalized elements, or in yet other words by the presheaf it represents. Although of course formally, there is a complete duality.

Revised on November 28, 2014 08:54:35 by Urs Schreiber (82.224.164.72)