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

Definition

A profunctor (also named distributor by the French school and module from a category, written

1. ,

is defined to be a functor

denotes the opposite category denotes the category of sets to denote the actions.

Using that the category of small categories is cartesian closed can be seen as a functor

of presheaves.

A correspondence.

Profunctors as categories

is a category whose objects are the disjoint union. (These are also known as het-sets, since the corresponding morphisms can be called heteromorphisms.

, 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

of two profunctors

is given by

is the left Kan extension along the Yoneda functor).

It can be shown that

is the least equivalence relation such that

1. .

Equivalently, profunctor composition can be written using a coend

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.

Properties

Lifting functors to profunctors

by postcomposing with the Yoneda functor:

1. .

has a right adjoint factors through the Cauchy completion.

See also

• Anafunctor

References