Profunctor

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

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

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

Using the cartesian closure of $\mathbf {Cat}$, a profunctor

$\phi :C\nrightarrow D$

can be seen as a functor

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

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

Composition of profunctors

The composite $\psi \phi$ of two profunctors

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

is given by

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

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

It can be shown that

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

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

$y'=vy$ and $x'v=x$.

Composition of profunctors is associative only up to isomorphism. It is therefore natural 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.

References

Jean Bénabou (2000). "Distributors at Work". {{cite journal}}: Cite journal requires |journal= (help)