In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. They are related to the notion of correspondences.
A profunctor (also named distributor by the French school and module by the Sydney school) from a category to a category , written
is defined to be a functor
where denotes the opposite category of and denotes the category of sets. Given morphisms respectively in and an element , we write to denote the actions.
Using the cartesian closure of , the category of small categories, the profunctor can be seen as a functor
where denotes the category of presheaves over .
A correspondence from to is a profunctor .
Composition of profunctors
The composite of two profunctors
is given by
where is the left Kan extension of the functor along the Yoneda functor of (which to every object of associates the functor ).
It can be shown that
where is the least equivalence relation such that whenever there exists a morphism in such that
- and .
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 categories,
- 1-cells between two small categories are the profunctors between those categories,
- 2-cells between two profunctors are the natural transformations between those profunctors.
Lifting functors to profunctors
A functor can be seen as a profunctor by postcomposing with the Yoneda functor:
It can be shown that such a profunctor has a right adjoint. Moreover, this is a characterization: a profunctor has a right adjoint if and only if factors through the Cauchy completion of , i.e. there exists a functor such that .