Profunctor


In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules.

Definition

A profunctor 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.

Profunctors as categories

An equivalent definition of a profunctor is a category whose objects are the disjoint union of the objects of and the objects of, and whose morphisms are the morphisms of and the morphisms of, plus zero or more additional morphisms from objects of to objects of. The sets in the formal definition above are the hom-sets between objects of and objects of. The previous definition can be recovered by the restriction of the hom-functor to.
This also makes it clear that a profunctor can be thought of as a relation between the objects of and the objects of, where each member of the relation is associated with a set of morphisms. A functor is a special case of a profunctor in the same way that a function is a special case of a relation.

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 .
It can be shown that
where is the least equivalence relation such that whenever there exists a morphism in such that

The bicategory of profunctors

Composition of profunctors is associative only up to isomorphism. The best one can hope is therefore to build a bicategory Prof whose

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.