is defined to be 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 . (These are also known as het-sets, since the corresponding morphisms can be called heteromorphisms.) 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
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 .