Profunctor – Wikipedia

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

Definition[edit]

A profunctor (also named distributor by the French school and module by the Sydney school)

${displaystyle ,phi }$

from a category

${displaystyle C}$

to a category

${displaystyle D}$

, written

${displaystyle phi colon Cnrightarrow D}$

,

is defined to be a functor

${displaystyle phi colon D^{mathrm {op} }times Cto mathbf {Set} }$

where

${displaystyle D^{mathrm {op} }}$

denotes the opposite category of

${displaystyle D}$

and

${displaystyle mathbf {Set} }$

denotes the category of sets. Given morphisms

${displaystyle fcolon dto d’,gcolon cto c’}$

respectively in

${displaystyle D,C}$

and an element

${displaystyle xin phi (d’,c)}$

, we write

${displaystyle xfin phi (d,c),gxin phi (d’,c’)}$

to denote the actions.

Using the cartesian closure of

${displaystyle mathbf {Cat} }$

, the category of small categories, the profunctor

${displaystyle phi }$

can be seen as a functor

${displaystyle {hat {phi }}colon Cto {hat {D}}}$

where

${displaystyle {hat {D}}}$

denotes the category

${displaystyle mathrm {Set} ^{D^{mathrm {op} }}}$

of presheaves over

${displaystyle D}$

.

A correspondence from

${displaystyle C}$

to

${displaystyle D}$

is a profunctor

${displaystyle Dnrightarrow C}$

.

Profunctors as categories[edit]

An equivalent definition of a profunctor

${displaystyle phi colon Cnrightarrow D}$

is a category whose objects are the disjoint union of the objects of

${displaystyle C}$

and the objects of

${displaystyle D}$

, and whose morphisms are the morphisms of

${displaystyle C}$

and the morphisms of

${displaystyle D}$

, plus zero or more additional morphisms from objects of

${displaystyle D}$

to objects of

${displaystyle C}$

. The sets in the formal definition above are the hom-sets between objects of

${displaystyle D}$

and objects of

${displaystyle C}$

. (These are also known as het-sets, since the corresponding morphisms can be called heteromorphisms.^[1]) The previous definition can be recovered by the restriction of the hom-functor

${displaystyle phi ^{text{op}}times phi to mathbf {Set} }$

to

${displaystyle D^{text{op}}times C}$

.

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

${displaystyle C}$

and the objects of

${displaystyle D}$

, 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[edit]

The composite

${displaystyle psi phi }$

of two profunctors

${displaystyle phi colon Cnrightarrow D}$

and

${displaystyle psi colon Dnrightarrow E}$

is given by

${displaystyle psi phi =mathrm {Lan} _{Y_{D}}({hat {psi }})circ {hat {phi }}}$

where

${displaystyle mathrm {Lan} _{Y_{D}}({hat {psi }})}$

is the left Kan extension of the functor

${displaystyle {hat {psi }}}$

along the Yoneda functor

${displaystyle Y_{D}colon Dto {hat {D}}}$

of

${displaystyle D}$

(which to every object

${displaystyle d}$

of

${displaystyle D}$

associates the functor

${displaystyle D(-,d)colon D^{mathrm {op} }to mathrm {Set} }$

).

It can be shown that

${displaystyle (psi phi )(e,c)=left(coprod _{din D}psi (e,d)times phi (d,c)right){Bigg /}sim }$

where

${displaystyle sim }$

is the least equivalence relation such that

${displaystyle (y’,x’)sim (y,x)}$

whenever there exists a morphism

${displaystyle v}$

in

${displaystyle D}$

such that

${displaystyle y’=vyin psi (e,d’)}$

and

${displaystyle x’v=xin phi (d,c)}$

.

Equivalently, profunctor composition can be written using a coend

${displaystyle (psi phi )(e,c)=int ^{dcolon D}psi (e,d)times phi (d,c)}$

The bicategory of profunctors[edit]

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[edit]

Lifting functors to profunctors[edit]

A functor

${displaystyle Fcolon Cto D}$

can be seen as a profunctor

${displaystyle phi _{F}colon Cnrightarrow D}$

by postcomposing with the Yoneda functor:

${displaystyle phi _{F}=Y_{D}circ F}$

.

It can be shown that such a profunctor

${displaystyle phi _{F}}$

has a right adjoint. Moreover, this is a characterization: a profunctor

${displaystyle phi colon Cnrightarrow D}$

has a right adjoint if and only if

${displaystyle {hat {phi }}colon Cto {hat {D}}}$

factors through the Cauchy completion of

${displaystyle D}$

, i.e. there exists a functor

${displaystyle Fcolon Cto D}$

such that

${displaystyle {hat {phi }}=Y_{D}circ F}$

.

References[edit]