Profunctor Explained

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 by the Sydney school)

\phi from a category C to a category D , written\phi : C\nrightarrow D ,is defined to be a functor\phi : D ^{op} x C\to Set where D ^{op} denotes the opposite category of D and Set denotes the category of sets. Given morphisms f : d\to d', g : c\to c' respectively in D, C and an element x\in\phi(d',c) , we write xf\in \phi(d,c), gx\in\phi(d',c') to denote the actions.

Using the cartesian closure of

Cat , the category of small categories, the profunctor \phi can be seen as a functor\hat{\phi} : C\to\hat{D} where \hat{D} denotes the category Set ^{D ^{op}} of presheaves over D .

A correspondence from

C to D is a profunctor D\nrightarrow C .

Profunctors as categories

An equivalent definition of a profunctor

\phi : C\nrightarrow D is a category whose objects are the disjoint union of the objects of C and the objects of D , and whose morphisms are the morphisms of C and the morphisms of D , plus zero or more additional morphisms from objects of D to objects of C . The sets in the formal definition above are the hom-sets between objects of D and objects of C . (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 \phi ^{op x} \phi \to Set to D ^{op x} C .

This also makes it clear that a profunctor can be thought of as a relation between the objects of

C and the objects of D , 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

\psi\phi of two profunctors\phi : C\nrightarrow D and \psi : D\nrightarrow E is given by\psi\phi = Lan _{Y _{D}} (\hat{\psi})\circ\hat\phi where Lan _{Y _{D}} (\hat{\psi}) is the left Kan extension of the functor \hat{\psi} along the Yoneda functor Y _{D} : D\to\hat D of D (which to every object d of D associates the functor D( - ,d) : D ^{op} \to Set ).

It can be shown that

(\psi\phi)(e,c) = \left(\coprod _{d\in} \psi(e,d) x \phi(d,c)\right) / \sim where \sim is the least equivalence relation such that (y',x')\sim(y,x) whenever there exists a morphism v in D such thaty' = vy \in\psi(e,d') and x'v = x \in\phi(d,c) .Equivalently, profunctor composition can be written using a coend(\psi\phi)(e,c) = \int ^{d\colon} \psi(e,d) x \phi(d,c)

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

A functor

F : C\to D can be seen as a profunctor \phi _{F} : C\nrightarrow D by postcomposing with the Yoneda functor:\phi _{F=Y} _{D\circ} F .

It can be shown that such a profunctor

\phi _{F} has a right adjoint. Moreover, this is a characterization: a profunctor \phi : C\nrightarrow D has a right adjoint if and only if \hat\phi : C\to\hat D factors through the Cauchy completion of D , i.e. there exists a functor F : C\to D such that \hat\phi = Y _{D\circ} F .

See also

• Anafunctor

References

• • Book: Borceux , Francis . Handbook of Categorical Algebra . CUP . 1994.
• Book: Lurie , Jacob . . Princeton University Press . 2009.