Profunctor

Summary

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

Definition

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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  .

Profunctors as categories

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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

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The composite   of two profunctors

  and  

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  .

Equivalently, profunctor composition can be written using a coend

 

Bicategory of profunctors

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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

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Lifting functors to profunctors

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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  .

See also

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References

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  • Bénabou, Jean (2000), Distributors at Work (PDF)
  • Borceux, Francis (1994). Handbook of Categorical Algebra. CUP.
  • Lurie, Jacob (2009). Higher Topos Theory. Princeton University Press.
  • Profunctor at the nLab
  • Heteromorphism at the nLab