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[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/profunctor-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki3\/profunctor-wikipedia\/","headline":"Profunctor – Wikipedia","name":"Profunctor – Wikipedia","description":"From Wikipedia, the free encyclopedia In category theory, a branch of mathematics, profunctors are a generalization of relations and also","datePublished":"2018-07-29","dateModified":"2018-07-29","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki3\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2691650573917bbe9b3d1c28ecfb49275110d16c","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2691650573917bbe9b3d1c28ecfb49275110d16c","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki3\/profunctor-wikipedia\/","wordCount":7743,"articleBody":"From Wikipedia, the free encyclopediaIn 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) \u03d5{displaystyle ,phi } from a category C{displaystyle C} to a category D{displaystyle D}, written\u03d5:C\u219bD{displaystyle phi colon Cnrightarrow D},is defined to be a functor\u03d5:Dop\u00d7C\u2192Set{displaystyle phi colon D^{mathrm {op} }times Cto mathbf {Set} }where Dop{displaystyle D^{mathrm {op} }} denotes the opposite category of D{displaystyle D} and Set{displaystyle mathbf {Set} } denotes the category of sets. Given morphisms f:d\u2192d\u2032,g:c\u2192c\u2032{displaystyle fcolon dto d’,gcolon cto c’} respectively in D,C{displaystyle D,C} and an element x\u2208\u03d5(d\u2032,c){displaystyle xin phi (d’,c)}, we write xf\u2208\u03d5(d,c),gx\u2208\u03d5(d\u2032,c\u2032){displaystyle xfin phi (d,c),gxin phi (d’,c’)} to denote the actions.Using the cartesian closure of Cat{displaystyle mathbf {Cat} }, the category of small categories, the profunctor \u03d5{displaystyle phi } can be seen as a functor\u03d5^:C\u2192D^{displaystyle {hat {phi }}colon Cto {hat {D}}}where D^{displaystyle {hat {D}}} denotes the category SetDop{displaystyle mathrm {Set} ^{D^{mathrm {op} }}} of presheaves over D{displaystyle D}.A correspondence from C{displaystyle C} to D{displaystyle D} is a profunctor D\u219bC{displaystyle Dnrightarrow C}.Profunctors as categories[edit]An equivalent definition of a profunctor \u03d5:C\u219bD{displaystyle phi colon Cnrightarrow D} is a category whose objects are the disjoint union of the objects of C{displaystyle C} and the objects of D{displaystyle D}, and whose morphisms are the morphisms of C{displaystyle C} and the morphisms of D{displaystyle D}, plus zero or more additional morphisms from objects of D{displaystyle D} to objects of C{displaystyle C}. The sets in the formal definition above are the hom-sets between objects of D{displaystyle D} and objects of C{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 \u03d5op\u00d7\u03d5\u2192Set{displaystyle phi ^{text{op}}times phi to mathbf {Set} } to Dop\u00d7C{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 C{displaystyle C} and the objects of D{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 \u03c8\u03d5{displaystyle psi phi } of two profunctors\u03d5:C\u219bD{displaystyle phi colon Cnrightarrow D} and \u03c8:D\u219bE{displaystyle psi colon Dnrightarrow E}is given by\u03c8\u03d5=LanYD(\u03c8^)\u2218\u03d5^{displaystyle psi phi =mathrm {Lan} _{Y_{D}}({hat {psi }})circ {hat {phi }}}where LanYD(\u03c8^){displaystyle mathrm {Lan} _{Y_{D}}({hat {psi }})} is the left Kan extension of the functor \u03c8^{displaystyle {hat {psi }}} along the Yoneda functor YD:D\u2192D^{displaystyle Y_{D}colon Dto {hat {D}}} of D{displaystyle D} (which to every object d{displaystyle d} of D{displaystyle D} associates the functor D(\u2212,d):Dop\u2192Set{displaystyle D(-,d)colon D^{mathrm {op} }to mathrm {Set} }).It can be shown that(\u03c8\u03d5)(e,c)=(\u2210d\u2208D\u03c8(e,d)\u00d7\u03d5(d,c))\/\u223c{displaystyle (psi phi )(e,c)=left(coprod _{din D}psi (e,d)times phi (d,c)right){Bigg \/}sim }where \u223c{displaystyle sim } is the least equivalence relation such that (y\u2032,x\u2032)\u223c(y,x){displaystyle (y’,x’)sim (y,x)} whenever there exists a morphism v{displaystyle v} in D{displaystyle D} such thaty\u2032=vy\u2208\u03c8(e,d\u2032){displaystyle y’=vyin psi (e,d’)} and x\u2032v=x\u2208\u03d5(d,c){displaystyle x’v=xin phi (d,c)}.Equivalently, profunctor composition can be written using a coend(\u03c8\u03d5)(e,c)=\u222bd:D\u03c8(e,d)\u00d7\u03d5(d,c){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 whose0-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 F:C\u2192D{displaystyle Fcolon Cto D} can be seen as a profunctor \u03d5F:C\u219bD{displaystyle phi _{F}colon Cnrightarrow D} by postcomposing with the Yoneda functor:\u03d5F=YD\u2218F{displaystyle phi _{F}=Y_{D}circ F}.It can be shown that such a profunctor \u03d5F{displaystyle phi _{F}} has a right adjoint. Moreover, this is a characterization: a profunctor \u03d5:C\u219bD{displaystyle phi colon Cnrightarrow D} has a right adjoint if and only if \u03d5^:C\u2192D^{displaystyle {hat {phi }}colon Cto {hat {D}}} factors through the Cauchy completion of D{displaystyle D}, i.e. there exists a functor F:C\u2192D{displaystyle Fcolon Cto D} such that \u03d5^=YD\u2218F{displaystyle {hat {phi }}=Y_{D}circ F}.References[edit]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki3\/profunctor-wikipedia\/#breadcrumbitem","name":"Profunctor – Wikipedia"}}]}]