Profunctor

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^{\mathrm{op}}\times C\to\mathbf{Set} where D^\mathrm{op} denotes the opposite category of D and \mathbf{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 that the category of small categories \mathbf{Cat} is cartesian closed, the profunctor \phi can be seen as a functor : \hat{\phi} : C\to\hat{D} where \hat{D} denotes the category \mathrm{Set}^{D^\mathrm{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^\text{op}\times \phi \to \mathbf{Set} to D^\text{op}\times 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=\mathrm{Lan}{Y_D}(\hat{\psi})\circ\hat\phi where \mathrm{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^{\mathrm{op}}\to\mathrm{Set}).

It can be shown that : (\psi\phi)(e,c)=\left(\coprod_{d\in D}\psi(e,d)\times\phi(d,c)\right)\Bigg/\sim where \sim is the least equivalence relation such that (y',x')\sim(y,x) whenever there exists a morphism v in D such that : y'=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 D}\psi(e,d)\times\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.

References

• {{citation | author-link = Jean Bénabou
• {{cite book
• {{cite book