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| In [[category theory]], a branch of [[mathematics]], '''profunctors''' are a generalization of [[binary relation|relations]] and also of [[bimodule]]s. They are related to the notion of [[correspondence (mathematics)|correspondence]]s.
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| == Definition ==
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| A '''profunctor''' (also named '''distributor''' by the French school and '''module''' by the Sydney school) <math>\,\phi</math> from a [[category (mathematics)|category]] <math>C</math> to a category <math>D</math>, written
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| :<math>\phi \colon C\nrightarrow D</math>,
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| is defined to be a [[functor (category theory)|functor]]
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| :<math>\phi \colon D^{\mathrm{op}}\times C\to\mathbf{Set}</math>
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| where <math>D^\mathrm{op}</math> denotes the [[opposite category]] of <math>D</math> and <math>\mathbf{Set}</math> denotes the [[category of sets]]. Given morphisms <math> f\colon d\to d', g\colon c\to c'</math> respectively in <math> D, C </math> and an element <math> x\in\phi(d',c)</math>, we write <math>xf\in \phi(d,c), gx\in\phi(d',c')</math> to denote the actions.
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| Using the [[cartesian closed category|cartesian closure]] of <math>\mathbf{Cat}</math>, the [[category of small categories]], the profunctor <math>\phi</math> can be seen as a functor
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| :<math>\hat{\phi} \colon C\to\hat{D}</math>
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| where <math>\hat{D}</math> denotes the category <math>\mathrm{Set}^{D^\mathrm{op}}</math> of [[presheaf (category theory)|presheaves]] over <math>D</math>.
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| A [[correspondence (mathematics)|correspondence]] from <math> C </math> to <math> D</math> is a profunctor <math> D\nrightarrow C</math>.
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| === Composition of profunctors ===
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| The composite <math>\psi\phi</math> of two profunctors
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| :<math>\phi\colon C\nrightarrow D</math> and <math>\psi\colon D\nrightarrow E</math>
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| is given by
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| :<math>\psi\phi=\mathrm{Lan}_{Y_D}(\hat{\psi})\circ\hat\phi</math>
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| where <math>\mathrm{Lan}_{Y_D}(\hat{\psi})</math> is the left [[Kan extension]] of the functor <math>\hat{\psi}</math> along the [[Yoneda functor]] <math>Y_D \colon D\to\hat D</math> of <math>D</math> (which to every object <math>d</math> of <math>D</math> associates the functor <math>D(-,d) \colon D^{\mathrm{op}}\to\mathrm{Set}</math>).
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| It can be shown that
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| :<math>(\psi\phi)(e,c)=\left(\coprod_{d\in D}\psi(e,d)\times\phi(d,c)\right)\Bigg/\sim</math>
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| where <math>\sim</math> is the least equivalence relation such that <math>(y',x')\sim(y,x)</math> whenever there exists a morphism <math>v</math> in <math>D</math> such that
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| :<math>y'=vy \in\psi(e,d')</math> and <math>x'v=x \in\phi(d,c)</math>.
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| === The bicategory of profunctors ===
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| 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
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| * 0-cells are [[small category|small categories]],
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| * 1-cells between two small categories are the profunctors between those categories,
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| * 2-cells between two profunctors are the [[natural transformation]]s between those profunctors.
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| == Properties ==
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| === Lifting functors to profunctors ===
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| A functor <math>F \colon C\to D</math> can be seen as a profunctor <math>\phi_F \colon C\nrightarrow D</math> by postcomposing with the Yoneda functor:
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| :<math>\phi_F=Y_D\circ F</math>.
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| It can be shown that such a profunctor <math>\phi_F</math> has a right adjoint. Moreover, this is a characterization: a profunctor <math>\phi \colon C\nrightarrow D</math> has a right adjoint if and only if <math>\hat\phi \colon C\to\hat D</math> factors through the [[Karoubi envelope|Cauchy completion]] of <math>D</math>, i.e. there exists a functor <math>F \colon C\to D</math> such that <math>\hat\phi=Y_D\circ F</math>.
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| == References ==
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| *{{cite journal
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| | first = Jean
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| | last = Bénabou
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| | year = 2000
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| | title = Distributors at Work
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| | url = http://www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf
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| }}
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| *{{cite book
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| | first = Francis
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| | last = Borceux
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| | title = Handbook of Categorical Algebra
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| | publisher = CUP
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| | year = 1994
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| }}
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| *{{cite book
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| | first = Jacob
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| | last = Lurie
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| | title = Higher Topos Theory
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| | publisher = Princeton University Press
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| | year = 2009
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| }}
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| * {{nlab|id=profunctor|title=Profunctor}}
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| == See also ==
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| * [[Categorical bridge]]
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| * [[Correspondence_(mathematics)]]
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| [[Category:Functors]]
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38 year old Valuer Cristopher Paterno from Eganville, has pastimes for instance quilting, como ganhar dinheiro na internet and volunteer. Always loves visiting places like Sangay National Park.
Feel free to visit my web site - ganhando dinheiro na internet