In category theory, 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) $\backslash ,\backslash phi$ from a category $C$ to a category $D$, written

$\backslash phi:C\backslash nrightarrow\; D$,

is defined to be a functor

$\backslash phi:D^\{\backslash mathrm\{op\}\}\backslash times\; C\backslash to\backslash mathbf\{Set\}$.

Using the cartesian closure of $\backslash mathbf\{Cat\}$, the profunctor $\backslash phi$ can be seen as a functor

$\backslash hat\{\backslash phi\}:C\backslash to\backslash hat\{D\}$

where $\backslash hat\{D\}$ denotes the category $\backslash mathrm\{Set\}^\{D^\backslash mathrm\{op\}\}$ of presheaves over $D$.

Composition of profunctors

The composite $\backslash psi\backslash phi$ of two profunctors

$\backslash phi:C\backslash nrightarrow\; D$ and $\backslash psi:D\backslash nrightarrow\; E$

is given by

$\backslash psi\backslash phi=\backslash mathrm\{Lan\}\_\{Y\_D\}(\backslash hat\{\backslash psi\})\backslash circ\backslash phi$

where is $\backslash mathrm\{Lan\}\_\{Y\_D\}(\backslash hat\{\backslash psi\})$ the left Kan extension of the functor $\backslash hat\{\backslash psi\}$ along the Yoneda functor $Y\_D:D\backslash to\backslash hat\; D$ of $D$ (which to every object $d$ of $D$ associates the functor $D(-,d):D^\{\backslash mathrm\{op\}\}\backslash to\backslash mathrm\{Set\}$).

It can be shown that

$(\backslash psi\backslash phi)(e,c)=\backslash left(\backslash coprod\_\{d\backslash in\; D\}\backslash psi(e,d)\backslash times\backslash phi(d,c)\backslash right)/\backslash sim$

where $\backslash sim$ is the least equivalence relation such that $(y\text{'},x\text{'})\backslash sim(y,x)$ whenever there exists a morphism $v$ in $B$ such that

$y\text{'}=vy$ and $x\text{'}v=x$.

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

• 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

A functor $F:C\backslash to\; D$ can be seen as a profunctor $\backslash phi\_F:C\backslash nrightarrow\; D$ by postcomposing with the Yoneda functor:

$\backslash phi\_F=Y\_D\backslash circ\; F$.

It can be shown that such a functor $\backslash phi\_F$ has a right adjoint. Moreover, this is a characterization: a profunctor $\backslash phi:C\backslash nrightarrow\; D$ has a right adjoint if and only if $\backslash hat\backslash phi:C\backslash to\backslash hat\; D$ factors through the Cauchy completion of $D$, i.e. there exists a functor $F:C\backslash to\; D$ such that $\backslash hat\backslash phi=Y\_D\backslash circ\; F$.

References

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Category theory