[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/end-category-theory-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/end-category-theory-wikipedia\/","headline":"End (category theory) – Wikipedia","name":"End (category theory) – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia This article is about the type of transformation. For the category of morphisms denoted","datePublished":"2014-01-08","dateModified":"2014-01-08","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cd810e53c1408c38cc766bc14e7ce26a?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\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bd2920909bbd7d8cbabc9f12da14b3956ac587ed","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bd2920909bbd7d8cbabc9f12da14b3956ac587ed","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/end-category-theory-wikipedia\/","wordCount":6487,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopediaThis article is about the type of transformation. For the category of morphisms denoted as End, see Endomorphism.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In category theory, an end of a functor S:Cop\u00d7C\u2192X{displaystyle S:mathbf {C} ^{mathrm {op} }times mathbf {C} to mathbf {X} } is a universal extranatural transformation from an object e of X to S.       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4More explicitly, this is a pair (e,\u03c9){displaystyle (e,omega )}, where e is an object of X and \u03c9:e\u2192\u00a8S{displaystyle omega :e{ddot {to }}S} is an extranatural transformation such that for every extranatural transformation \u03b2:x\u2192\u00a8S{displaystyle beta :x{ddot {to }}S} there exists a unique morphism        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4h:x\u2192e{displaystyle h:xto e}of X with \u03b2a=\u03c9a\u2218h{displaystyle beta _{a}=omega _{a}circ h}for every object a of C.By abuse of language the object e is often called the end of the functor S (forgetting \u03c9{displaystyle omega }) and is writtene=\u222bcS(c,c)\u00a0or just\u00a0\u222bCS.{displaystyle e=int _{c}^{}S(c,c){text{ or just }}int _{mathbf {C} }^{}S.}Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram\u222bcS(c,c)\u2192\u220fc\u2208CS(c,c)\u21c9\u220fc\u2192c\u2032S(c,c\u2032),{displaystyle int _{c}S(c,c)to prod _{cin C}S(c,c)rightrightarrows prod _{cto c’}S(c,c’),}where the first morphism being equalized is induced by S(c,c)\u2192S(c,c\u2032){displaystyle S(c,c)to S(c,c’)} and the second is induced by S(c\u2032,c\u2032)\u2192S(c,c\u2032){displaystyle S(c’,c’)to S(c,c’)}.The definition of the coend of a functor S:Cop\u00d7C\u2192X{displaystyle S:mathbf {C} ^{mathrm {op} }times mathbf {C} to mathbf {X} } is the dual of the definition of an end.Thus, a coend of S consists of a pair (d,\u03b6){displaystyle (d,zeta )}, where d is an object of  X and \u03b6:S\u2192\u00a8d{displaystyle zeta :S{ddot {to }}d}is an extranatural transformation, such that for every extranatural transformation \u03b3:S\u2192\u00a8x{displaystyle gamma :S{ddot {to }}x} there exists a unique morphismg:d\u2192x{displaystyle g:dto x} of X with \u03b3a=g\u2218\u03b6a{displaystyle gamma _{a}=gcirc zeta _{a}} for every object a of C.The coend d of the functor S is writtend=\u222bcS(c,c)\u00a0or\u00a0\u222bCS.{displaystyle d=int _{}^{c}S(c,c){text{ or }}int _{}^{mathbf {C} }S.}Characterization as colimit: Dually, if X is cocomplete and C is small, then the coend can be described as the coequalizer in the diagram\u222bcS(c,c)\u2190\u2210c\u2208CS(c,c)\u21c7\u2210c\u2192c\u2032S(c\u2032,c).{displaystyle int ^{c}S(c,c)leftarrow coprod _{cin C}S(c,c)leftleftarrows coprod _{cto c’}S(c’,c).}Examples[edit]Natural transformations:Suppose we have functors F,G:C\u2192X{displaystyle F,G:mathbf {C} to mathbf {X} } thenHomX(F(\u2212),G(\u2212)):Cop\u00d7C\u2192Set{displaystyle mathrm {Hom} _{mathbf {X} }(F(-),G(-)):mathbf {C} ^{op}times mathbf {C} to mathbf {Set} }.In this case, the category of sets is complete, so we need only form the equalizer and in this case\u222bcHomX(F(c),G(c))=Nat(F,G){displaystyle int _{c}mathrm {Hom} _{mathbf {X} }(F(c),G(c))=mathrm {Nat} (F,G)}the natural transformations from F{displaystyle F} to G{displaystyle G}.  Intuitively, a natural transformation from F{displaystyle F} to G{displaystyle G} is a morphism from F(c){displaystyle F(c)} to G(c){displaystyle G(c)} for every c{displaystyle c} in the category with compatibility conditions.  Looking at the equalizer diagram defining the end makes the equivalence clear.Geometric realizations:Let T{displaystyle T} be a simplicial set.  That is, T{displaystyle T} is a functor \u0394op\u2192Set{displaystyle Delta ^{mathrm {op} }to mathbf {Set} }.  The discrete topology gives a functor Set\u2192Top{displaystyle mathbf {Set} to mathbf {Top} }, where Top{displaystyle mathbf {Top} } is the category of topological spaces.  Moreover, there is a map \u03b3:\u0394\u2192Top{displaystyle gamma :Delta to mathbf {Top} } sending the object [n]{displaystyle [n]} of \u0394{displaystyle Delta } to the standard n{displaystyle n}-simplex inside Rn+1{displaystyle mathbb {R} ^{n+1}}.  Finally there is a functor Top\u00d7Top\u2192Top{displaystyle mathbf {Top} times mathbf {Top} to mathbf {Top} } that takes the product of two topological spaces.Define S{displaystyle S} to be the composition of this product functor with T\u00d7\u03b3{displaystyle Ttimes gamma }.  The coend of S{displaystyle S} is the geometric realization of T{displaystyle T}.References[edit]Mac Lane, Saunders (2013). Categories For the Working Mathematician. Springer Science & Business Media. pp.\u00a0222\u2013226.Loregian, Fosco (2015). “This is the (co)end, my only (co)friend”. arXiv:1501.02503 [math.CT].External links[edit]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/end-category-theory-wikipedia\/#breadcrumbitem","name":"End (category theory) – Wikipedia"}}]}]