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).
If $C$ and $D$ are categories, then a profunctor from $C$ to $D$ is a functor of the form
Such a profunctor is usually written as $F\colon C$⇸$D$. For $d \in D$ and $c \in C$ the set $H_F(d,c)$ is also called the set of heteromorphisms from $d$ to $c$.
Every functor $f\colon C\to D$ induces two profunctors $D(1,f)\colon C$⇸$D$ and $D(f,1)\colon D$⇸$C$, defined by $D(1,f)(d,c) = D(d,f(c))$ and $D(f,1)(c,d) = D(f(c),d)$. (Here $D(-,-)$ denotes the hom functor of $D$ and $1$ denotes the identity functor on the respective category.) Since this construction may be thought of as the adjunct of the composition of $f$ with the Yoneda embedding $C \stackrel{f}{\longrightarrow} D \stackrel{Yoneda}{\longrightarrow} [D^{op},Set]$, 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 $Id \colon C$⇸$C$ is represented by the identity functor and hence is given by the hom-functor $C(-,-) : C^{op} \times C \to Set$ itself.
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 $D^{op} \otimes C\to V$. If they are internal categories, then a profunctor $C$⇸$D$ is an internal diagram on $D^{op}\times C$, 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 $D^{op}\times C \to Set$ is not universal; some authors reverse $C$ and $D$ and/or put the “op” on the other one. See the discussion below.
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
This yields a bicategory in which
objects are categories,
morphisms are profunctors with the above composition, and
This bicategory is variously denoted Prof, $Mod$, or $Dist$, according to one’s chosen name for profunctors. In the $V$-enriched case, it is written $V Prof$ or $V Mod$ or $V Dist$.
The construction of the “representable” profunctors $D(1,f)$ and $D(f,1)$ from a functor $f\colon C\to D$ yield two identity-on-objects functors $Cat \to Prof$ and $Cat^{op}\to Prof$. Moreover, it is easy to check that $D(1,f) \vdash D(f,1)$ in the bicategory $Prof$; thus $Cat\to Prof$ 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 $Prof_{rep}$ on representable profunctors is equivalent to $Cat_{ana}$, the 2-category of anafunctors. See there for more details.
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 small category $C$ to $A$ are canonically equivalent to functors from $C$ to $A$: we have an equivalence of functor categories
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 $Y : C \to PSh(C)$. This immediately implies that a colimit-preserving functor on $PSh(C)$ is already determined by its restriction along $Y$ to $C$.
Now, profunctors $D^{op} \otimes C \to V$ are adjunct to functors $C \to [D^{op}, V] \simeq PSh(D)$. Hence by the above, profunctors are equivalent to colimit-preserving functors
Indeed, there is an equivalence of bicategories between $V Prof$ 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 $V Prof$ (e.g. Cattani 1999, Prop. 4.2.4).
From this perspective, the representable profunctor induced by an ordinary $V$-functor $f : C \to D$ is the adjunct of the postcomposition
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 $(\infty,1)$-category $Pr^L$ therefore is an $(\infty,1)$-analog of $Set\text{-}Mod$. In geometric ∞-function theory one finds (see section 4 there) that morphisms in $Pr^L$ encode the “correspondence operations” such as Fourier-Mukai and its generalizations. See in that context also the examples below.
Recall that a functor $D^{op}\to Set$ can equivalently be described as a discrete (Grothendieck) fibration, and similarly a functor $C\to Set$ can be described as a discrete opfibration. Thus, a profunctor $D^{op}\times C\to Set$ could be described by a discrete opfibration over $D^{op}\times C$, or a discrete fibration over $D\times C^{op}$, 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 $D^{op}\times C\to Cat$, and hence if it is discrete it represents a profunctor $D^{op}\times C\to Set$.
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).
Yet another way of representing profunctors is via their collages, also called cographs. The collage of a profunctor $H\colon C$⇸$D$ is, in particular, a category $\bar{H}$ equipped with functors $C\to \bar{H}$ and $D\to\bar{H}$ which are fully faithful and jointly bijective on objects.
In fact, the objects of the undercategory $(C\sqcup D)/Cat$ 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 $Cat^{op}$. 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 $H\colon C$⇸$D$ 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 $\bar{H}$ 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.
If $C \overset{g}{\to} \bar{H} \overset{f}{\leftarrow} D$ 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 $C \overset{p}{\leftarrow} E \overset{q}{\to} D$ 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
One can show that comma objects are always discrete fibrations, and dually cocomma objects are always codiscrete cofibrations. In $Cat$ 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.
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 $Vect\Mod$ on one-object $Vect$-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 $Set Prof$ on discrete categories is the bicategory of sets, spans of sets and morphisms of spans:
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
Similarly, the full sub-bicategory of internal profunctors in $S = (Set^{op}, \times)$ on “discrete categories” is the bicategory of cospans
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.
If a functor represents a given profunctor, then the action of the functor on morphisms is determined by the action of the profunctor and the representation isomorphism. For details see at representability determines functoriality
If $\mathcal{D}$ is Cauchy complete, then the profunctors that correspond to functors (via Yoneda, which here looks like $F\mapsto F_*$, with $F_*(d,c)=\mathcal{D}(d,F c)$) are exactly those that admit right adjoints. In general, right adjoint profunctors correspond to profunctors into the Cauchy completion of their codomain.
Original articles:
Maren Justesen, Bikategorien af Profunktorer, Aarhus 1968 (pdf)
Marta Bunge, Chpater 3 of: Categories of Set-Valued Functors, University of Pennsylvania, 1966
based on suggestions by Bill Lawvere
Michel André, Categories of Functors and Adjoint Functors, Batelle Institute at Geneva, 1964, see also Categories of Functors and Adjoint Functors,American Journal of Mathematics, Vol. 88, No. 3 (Jul., 1966), pp. 529-543
Jean Bénabou, Les distributeurs, Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, rapport 33, 1973
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’.
Texbook account:
Lecture notes:
Exposition:
A nice example of profunctors between Lawvere metric spaces can be found in this comment.
The following classic paper is a good appetizer
See the Joyal's CatLab for the theory of Set-valued distributors:
Internal profunctors are considered in
Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint Minneola 2014; sect. 2.4)
Peter Johnstone, Sketches of an Elephant I, Oxford UP 2002. (pp.359-367)
The relation to locally presentable categories is almost explicit in:
Profunctors play an important in categorical shape theory. The original source is
The material together with a general discussion of profunctors is also available in English in the reprinted monograph
Discussion of simplicial enriched profunctors in the context of strong shape theory in
Overview and application in computer science:
The common generalization of bimodules and spans in terms of profunctors has been discussed on the blog at
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]|-@{|}