Profunctor

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

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

where $D^\backslash mathrm\{op\}$ denotes the opposite category of $D$ and $\backslash mathbf\{Set\}$ denotes the category of sets. Given morphisms $f\; :\; d\backslash to\; d\text{'},\; g\; :\; c\backslash to\; c\text{'}$ respectively in $D,\; C$ and an element $x\backslash in\backslash phi(d\text{'},c)$, we write $xf\backslash in\; \backslash phi(d,c),\; gx\backslash in\backslash phi(d\text{'},c\text{'})$ to denote the actions.

Using that the category of small categories $\backslash mathbf\{Cat\}$ is cartesian closed, 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 $D$.

A correspondence from $C$ to $D$ is a profunctor $D\backslash nrightarrow\; C$.

An equivalent definition of a profunctor $\backslash phi\; :\; C\backslash nrightarrow\; D$ 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 $\backslash phi^\backslash text\{op\}\backslash times\; \backslash phi\; \backslash to\; \backslash mathbf\{Set\}$ to $D^\backslash text\{op\}\backslash times\; C$.

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.

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

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

where $\backslash mathrm\{Lan\}\_\{Y\_D\}(\backslash hat\{\backslash psi\})$ is 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

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

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 $D$ such that

: $y\text{'}=vy\; \backslash in\backslash psi(e,d\text{'})$ and $x\text{'}v=x\; \backslash in\backslash phi(d,c)$.

Equivalently, profunctor composition can be written using a coend

: $(\backslash psi\backslash phi)(e,c)=\backslash int^\{d\backslash colon\; D\}\backslash psi(e,d)\backslash times\backslash phi(d,c)$

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.

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

• Anafunctor

{{reflist}}

{{refbegin}}

• {{citation

| first = Jean

| last = Bénabou

| author-link = Jean Bénabou

| year = 2000

| title = Distributors at Work

| url = http://www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf

}}

• {{cite book

| first = Francis

| last = Borceux

| title = Handbook of Categorical Algebra

| publisher = CUP

| year = 1994

}}

• {{cite book

| first = Jacob

| last = Lurie

| title = Higher Topos Theory

| publisher = Princeton University Press

| year = 2009

}}

• {{nlab|id=profunctor|title=Profunctor}}
• {{nlab|id=heteromorphism|title=Heteromorphism}}

{{refend}}

Category:Functors