View source for SO(5)

In [[category theory]], a branch of [[mathematics]], '''profunctors''' are a generalization of [[binary relation|relations]] and also of [[bimodule]]s.  They are related to the notion of [[correspondence (mathematics)|correspondence]]s.

== Definition ==

A '''profunctor''' (also named '''distributor''' by the French school and '''module''' by the Sydney school) <math>\,\phi</math> from a [[category (mathematics)|category]] <math>C</math> to a category <math>D</math>, written 

:<math>\phi \colon C\nrightarrow D</math>, 

is defined to be a [[functor (category theory)|functor]]

:<math>\phi \colon D^{\mathrm{op}}\times C\to\mathbf{Set}</math>

where <math>D^\mathrm{op}</math> denotes the [[opposite category]] of <math>D</math> and <math>\mathbf{Set}</math> denotes the [[category of sets]]. Given morphisms <math> f\colon d\to d', g\colon c\to c'</math> respectively in <math> D, C </math> and an element <math> x\in\phi(d',c)</math>, we write <math>xf\in \phi(d,c), gx\in\phi(d',c')</math> to denote the actions.  

Using the [[cartesian closed category|cartesian closure]] of <math>\mathbf{Cat}</math>, the [[category of small categories]], the profunctor <math>\phi</math> can be seen as a functor

:<math>\hat{\phi} \colon C\to\hat{D}</math>

where <math>\hat{D}</math> denotes the category <math>\mathrm{Set}^{D^\mathrm{op}}</math> of [[presheaf (category theory)|presheaves]] over <math>D</math>.

A [[correspondence (mathematics)|correspondence]] from <math> C </math> to <math> D</math> is a profunctor <math> D\nrightarrow C</math>.

=== Composition of profunctors ===

The composite <math>\psi\phi</math> of two profunctors

:<math>\phi\colon C\nrightarrow D</math> and <math>\psi\colon D\nrightarrow E</math>

is given by

:<math>\psi\phi=\mathrm{Lan}_{Y_D}(\hat{\psi})\circ\hat\phi</math>

where <math>\mathrm{Lan}_{Y_D}(\hat{\psi})</math> is the left [[Kan extension]] of the functor <math>\hat{\psi}</math> along the [[Yoneda functor]] <math>Y_D \colon D\to\hat D</math> of <math>D</math> (which to every object <math>d</math> of <math>D</math> associates the functor <math>D(-,d) \colon D^{\mathrm{op}}\to\mathrm{Set}</math>).

It can be shown that

:<math>(\psi\phi)(e,c)=\left(\coprod_{d\in D}\psi(e,d)\times\phi(d,c)\right)\Bigg/\sim</math>

where <math>\sim</math> is the least equivalence relation such that <math>(y',x')\sim(y,x)</math> whenever there exists a morphism <math>v</math> in <math>D</math> such that

:<math>y'=vy \in\psi(e,d')</math>  and  <math>x'v=x \in\phi(d,c)</math>.

=== The bicategory of profunctors ===

Composition of profunctors is associative only up to isomorphism (because the product is not strictly associative in '''Set'''). The best one can hope is therefore to build a [[bicategory]] '''Prof''' whose
* 0-cells are [[small category|small categories]],
* 1-cells between two small categories are the profunctors between those categories,
* 2-cells between two profunctors are the [[natural transformation]]s between those profunctors.

== Properties ==
=== Lifting functors to profunctors ===
A functor <math>F \colon C\to D</math> can be seen as a profunctor <math>\phi_F \colon C\nrightarrow D</math> by postcomposing with the Yoneda functor:

:<math>\phi_F=Y_D\circ F</math>.

It can be shown that such a profunctor <math>\phi_F</math> has a right adjoint. Moreover, this is a characterization: a profunctor <math>\phi \colon C\nrightarrow D</math> has a right adjoint if and only if <math>\hat\phi \colon C\to\hat D</math> factors through the [[Karoubi envelope|Cauchy completion]] of <math>D</math>, i.e. there exists a functor <math>F \colon C\to D</math> such that <math>\hat\phi=Y_D\circ F</math>.

== References ==
*{{cite journal
 | first = Jean
 | last = Bénabou
 | year = 2000
 | title = Distributors at Work
 | url = http://www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf
 }}
*{{cite book
 | first = Francis
 | last = Borceux
 | title = Handbook of Categorical Algebra
 | publisher = CUP
 | year = 1994
}}
*{{cite book
| first = Jacob
| last = Lurie
| title = Higher Topos Theory
| publisher = Princeton University Press
| year = 2009
}}
* {{nlab|id=profunctor|title=Profunctor}}

== See also ==
* [[Categorical bridge]]
* [[Correspondence_(mathematics)]]

[[Category:Functors]]