In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ...
In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. ...
A profunctor (also named distributor by the French school and module by the Sydney school) from a category C to a category D, written
is defined to be a functor
Using the cartesian closure of , the profunctor φ can be seen as a functor In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. ...
where denotes the category of presheaves over D.
Composition of profunctors
The composite ψφ of two profunctors
is given by
where is the left Kan extension of the functor along the Yoneda functor of D (which to every object d of D associates the functor ). Kan extensions are universal constructs in category theory, a branch of mathematics. ...
It can be shown that
where is the least equivalence relation such that whenever there exists a morphism v in B such that
- y' = vy and x'v = x.
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 In mathematics, a bicategory is a concept in category theory used to extend the notion of sameness (i. ...
- 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.
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...
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 functor φF 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 D, i.e. there exists a functor such that . The Karoubi envelope is a classification of the idempotents of a category. ...
- Bénabou, Jean (2000). "Distributors at Work".
- Borceux, Francis (1994). Handbook of Categorical Algebra. CUP.