nLab
profunctor

Definition

If $C$ and $D$ are categories, a profunctor from $C$ to $D$ is a functor $D^{\mathrm{op}}\times C\to \mathrm{Set}$. Such a profunctor is usually written as $F:C⇸D$.

Every functor $f:C\to D$ induces two profunctors $D(1,f):C⇸D$ and $D(f,1):D⇸C$, defined by $D(1,f)(d,c)=D(d,f(c))$ and $D(f,1)(c,d)=D(f(c),d)$. These profunctors are called representable (or sometimes one of them is corepresentable).

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

The notion generalizes to many other kinds of categories. For instance, if $C$ and $D$ are enriched over some symmetric closed monoidal category $V$, then a profunctor from $C$ to $D$ is a $V$-functor $D^{\mathrm{op}}\otimes C\to V$. If they are internal categories, then a profunctor $C⇸D$ is an internal diagram on $D^{\mathrm{op}}\times C$, 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 $V$-categories, i.e. monoids in $V$, 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 $D^{\mathrm{op}}\times C\to \mathrm{Set}$ is not universal; some authors reverse $C$ and $D$ 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 $F:C⇸D$ and $G:D⇸E$, their composite $G\circ F:C⇸E$ is defined to be

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

This yields a bicategory in which

• objects are categories,

• morphisms are profunctors with the above composition, and

• 2-morphisms are natural transformations.

This bicategory is variously denoted $\mathrm{Prof}$, $\mathrm{Mod}$, or $\mathrm{Dist}$, according to one’s chosen name for profunctors. In the $V$-enriched case, it is written $V\mathrm{Prof}$ or $V\mathrm{Mod}$ or $V\mathrm{Dist}$.

The construction of the “representable” profunctors $D(1,f)$ and $D(f,1)$ from a functor $f:C\to D$ yield two identity-on-objects functors $\mathrm{Cat}\to \mathrm{Prof}$ and ${\mathrm{Cat}}^{\mathrm{op}}\to \mathrm{Prof}$. Moreover, it is easy to check that $D(1,f)⊢D(f,1)$ in the bicategory $\mathrm{Prof}$; thus $\mathrm{Cat}\to \mathrm{Prof}$ 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 ${\mathrm{Prof}}_{\mathrm{rep}}$ on representable profunctors is equivalent to ${\mathrm{Cat}}_{\mathrm{ana}}$, 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 $A$ a cocomplete category, colimit-preserving functors from presheaves on some category $C$ to $A$ are canonically equivalent to functors from $C$ to $A$: we have an equivalence of functor categories

$$\mathrm{Cocont}(\mathrm{PSh}(C),A)\simeq \mathrm{Func}(C,A).$$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 colimit over representables, i.e. over objects in the image of the Yoneda embedding $Y:C\to \mathrm{PSh}(C)$. This immediate implies that a colimit-preserving functor on $\mathrm{PSh}(C)$ is already determined by its restriction along $Y$ to $C$.

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

$$\mathrm{PSh}(C)\to \mathrm{PSh}(D).$$PSh(C) \to PSh(D) \,.

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

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

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

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

$$C\stackrel{f}{\to }D\stackrel{Y}{\to }[D^{\mathrm{op}},V]$$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 $(\mathrm{\infty },1)$-category ${\mathrm{Pr}}^L$ therefore is an $(\mathrm{\infty },1)$-analog of $\mathrm{Set}\text{-}\mathrm{Mod}$. In geometric ∞-function theory one finds (see section 4 there) that morphisms in ${\mathrm{Pr}}^L$ 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 $D^{\mathrm{op}}\to \mathrm{Set}$ can equivalently be described as a discrete (Grothendieck) fibration, and similarly a functor $C\to \mathrm{Set}$ can be described as a discrete opfibration. Thus, a profunctor $D^{\mathrm{op}}\times C\to \mathrm{Set}$ could be described by a discrete opfibration over $D^{\mathrm{op}}\times C$, or a discrete fibration over $D\times C^{\mathrm{op}}$, but there is also a more directly “two-sided” fibrational description. A two-sided fibration from $C$ to $D$ is a functor $E\to C\times D$ which is a fibration over $D$ and an opfibration over $C$ in a compatible way. Such a fibration represents a pseudofunctor $D^{\mathrm{op}}\times C\to \mathrm{Cat}$, and hence if it is discrete it represents a profunctor $D^{\mathrm{op}}\times C\to \mathrm{Set}$.

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 $H:C⇸D$ is, in particular, a category $\overline{H}$ equipped with functors $C\to \overline{H}$ and $D\to \overline{H}$ which are fully faithful and jointly bijective on objects.

In fact, the objects of the undercategory $(C\bigsqcup D)/\mathrm{Cat}$ which are collages of profunctors $C⇸D$ can be characterized, up to equivalence, as the two-sided codiscrete cofibrations, i.e. the two-sided discrete fibrations in ${\mathrm{Cat}}^{\mathrm{op}}$. In simpler and more explicit language, these are the categories $M$ which contain $C$ and $D$ 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 $C$ to an object of $D$. Equivalently, they are the categories which admit a functor to the interval category $I$ such that $D$ is the fiber over $0$ and $C$ is the fiber over $1$.

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

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 $C\stackrel{g}{\to }\overline{H}\stackrel{f}{\leftarrow }D$ is a codiscrete cofibration representing a profunctor $H$ from $C$ to $D$, then the two-sided discrete fibration representing the same profunctor can be obtained as the comma category $(f\downarrow g)$ with its two projections to $C$ and $D$.

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

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

$$\mathrm{cocomma}:\mathrm{Span}(C,D)\rightleftarrows \mathrm{Cospan}(C,D):\mathrm{comma}.$$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 $\mathrm{Cat}$ 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 $C$ to $D$). 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 $\mathrm{Vect}Mod$ on one-object $\mathrm{Vect}$-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 $A\to B$ of algebras induces the $A$-$B$ bimodule which as a vector space is $B$ with obvious right $B$ action and left $A$-action induced by first mapping $A$ to $B$ via $f$ and then using multiplication in $B$.

• The full sub-bicategory of $\mathrm{Set}\mathrm{Prof}$ on discrete categories is the bicategory of sets, spans of sets and morphisms of spans:

  $$\mathrm{Set}{Mod}_{\mathrm{disc}}\simeq \mathrm{Span}(\mathrm{Set}).$$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 $f:C\to D$ of sets is the span

  $$\begin{array}{ccc}& & C\\ & {}^{\mathrm{Id}}\swarrow & & {\searrow }^f\\ C& & & & D\end{array}.$$\array{ && C \\ & {}^{Id}\swarrow && \searrow^{f} \\ C &&&& D } \,.

• Similarly, the full sub-bicategory of internal profunctors in $S=({\mathrm{Set}}^{\mathrm{op}},\times )$ on “discrete categories” is the bicategory of cospans

  $${\mathrm{Set}}^{\mathrm{op}}{Mod}_{\mathrm{disc}}\simeq \mathrm{Span}({\mathrm{Set}}^{\mathrm{op}})=\mathrm{Cospan}(\mathrm{Set}).$$Set^{op}\Mod_{disc} \simeq Span(Set^{op}) = Cospan(Set) \,.

Related entries

• module

• 2-vector space

• double profunctor

References

Original sources

The original published source for profunctors is

• Jean Bénabou, Les distributeurs, Univesité 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

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

• Jean Bénabou, Distributors at work, (2000), pdf

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

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

• Andre Joyal, Distributors and barrels

There is a discussion of profunctors in the recently republished:

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

Expositions

There is some exposition of profunctors at

• John Baez, Re: Klein 2-Geometry VII (blog)

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

• John Baez, Bimodules versus spans (blog)

Notation

Profunctors are often notated with a slashed or barred arrow, as in $C⇸D$, 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 $C^{\mathrm{op}}\otimes D\to V$ or $C\otimes D^{\mathrm{op}}\to V$. My own convention is to use the latter. For example, every functor $C\to D$ yields a profunctor by composition with the Yoneda embedding on $D$.

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

Sridhar Ramesh: But surely, just as well, a functor from $C$ to $D$ yields a contravariant functor from $C$ to ${\mathrm{Set}}^D$ and thus a profunctor $C^{\mathrm{op}}\otimes D\to V$, by composition with the contravariant Yoneda embedding of $D$ into ${\mathrm{Set}}^D$? At the moment, I still do not see why there is reason to prefer in the abstract general one to the other of $(c,d)\mapsto {\mathrm{Hom}}_D(F(c),d)$ and $(d,c)\mapsto {\mathrm{Hom}}_D(d,F(c))$, 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 $C⇸D$ is a functor $C\to {\mathrm{Set}}^{D^{\mathrm{op}}}$, then we can think of it as assigning to every $c\in C$ a presheaf on $D$, 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 $C⇸D$ were a functor $C^{\mathrm{op}}\to {\mathrm{Set}}^D$, i.e. $C\to ({\mathrm{Set}}^D{)}^{\mathrm{op}}$, then we could think of it as assigning to each $c\in C$ a functor $D\to \mathrm{Set}$, 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 $D$ 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. Athough of course formally, there is a complete duality.