nLab
profunctor

Definition

Let $V$ be a symmetric closed monoidal category and $C$, $D$, enriched categories over $V$. Then a profunctor or (bi)module or distributor from $C$ to $D$ is a $V$-functor from $C\otimes D^{\mathrm{op}}$ to $V$, we write

$$F:C⇸D:C\otimes D^{\mathrm{op}}\to V.$$F : C ⇸ D : C\otimes D^{op} \to V \,.

Such profunctors are composed by using a coend to “trace out” the middle variable:

for $F:C⇸D$ and $G:D⇸E$ profunctors, 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 $VMod$ with

• objects are $V$-enriched categories;

• morphisms are profunctors with the above composition;

• 2-morphisms are natural transformations.

At least for $V=\mathrm{Set}$ a morphism in $\mathrm{Set}\mathrm{Mod}$ is often called a profunctor and the bicategory $\mathrm{Set}\mathrm{Mod}$ is then often denoted $\mathrm{Prof}$.

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 $C\otimes D^{\mathrm{op}}\to V$ 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 $V\mathrm{Mod}$ and that of categories and colimit-preserving functors and natural transformation between their presheaf categories.

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)

This formulation 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 $(\mathrm{\infty }\mathrm{,1})$-category ${\mathrm{Pr}}^L$ therefore is an $(\mathrm{\infty }\mathrm{,1})$-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.

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.

• For $V=(\mathrm{Set},\times )$, $\mathrm{SetMod}$ is the bicategory of locally small categories, profunctors and transformations.

  The full sub-bicategory on discrete categories is that 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) \,.

• Accordingly for $S=({\mathrm{Set}}^{\mathrm{op}},\times )$ we get 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) \,.

Remarks

• Every ordinary $V$-functor $f:C\to D$ yields a profunctor $\hat{F}:C⇸D$ which 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. This extends to a bifunctor

  $$VCat\to VMod.$$V\Cat \to V\Mod \,.

  • For $V=\mathrm{Vect}$ this is the generalization of how 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 $B$.

  • For $V=\mathrm{Set}$ this is the fact that every map $f:C\to D$ of sets induces the span

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

• The bifunctor $VCat\to VMod$ exhibits $VMod$ as a framed bicategory, and makes $VCat$ into a 2-category equipped with proarrows.

Related entries

• module

• 2-vector space

References

(what is a good comprehensive reference?)

Some exposition 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)

There is a discussion of profunctors in the recently republished:

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

Excellent notes from a course on distributors given by Jean Bénabou in June 2000 at TU Darmstadt, and prepared by Thomas Streicher, are available from his website http://www.mathematik.tu-darmstadt.de/~streicher.

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

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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 can also be produced with the following preamble code (modified from amsmath’s \xrightarrow):

  \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

• In Xypic, a barred arrow (to the right, in this example) can be produced with

  \ar[r]|-@{|}

Discussion

On a previous version of this entry with opposite convention on where to put the ${}^{\mathrm{op}}$ Todd has remarked

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.