About: Dold–Kan correspondence

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In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the th homology group of a chain complex is the th homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.)

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rdf:type	yago:WikicatSimplicialSets yago:Abstraction100002137 yago:Collection107951464 yago:Group100031264 yago:Set107996689
rdfs:label	Dold–Kan correspondence (en) 정규화 사슬 복합체 (ko)
rdfs:comment	호몰로지 대수학에서 정규화 사슬 복합체(正規化사슬複合體, 영어: normalized chain complex)는 아벨 범주의 단체 대상에 대하여 정의되는 사슬 복합체이다. 이는 아벨 범주의 단체 대상의 범주와 자연수 등급 사슬 복합체의 범주 사이의 동치를 정의하며, 이 동치를 돌트-칸 대응(Dold–Kan對應, 영어: Dold–Kan correspondence)이라고 한다. (ko) In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the th homology group of a chain complex is the th homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.) (en)
dcterms:subject	Theorems in abstract algebra Simplicial sets
Wikipage page ID	39431504 (xsd:integer)
Wikipage revision ID	1100797006 (xsd:integer)
Link from a Wikipage to another Wikipage	Cubical set Equivalence of categories Simplicial abelian group Functor Albrecht Dold Daniel Kan Jacob Lurie Theorems in abstract algebra Simplicial sets Eilenberg–MacLane space Homotopy group Model category Chain complex European Mathematical Society Simplicial set Simplicial homotopy Chain homotopy ∞-category Homology group
Link from a Wikipage to an external page	http://people.fas.harvard.edu/~amathew/doldkan.pdf https://web.archive.org/web/20160913201635/http:/people.fas.harvard.edu/~amathew/doldkan.pdf%7Carchive-date= http://www.math.harvard.edu/~lurie/papers/DAG-I.pdf
sameAs	Dold–Kan correspondence Dold–Kan correspondence Dold–Kan correspondence Dold–Kan correspondence
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id	Dold-Kan+correspondence (en)
title	Dold-Kan correspondence (en)
has abstract	In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the th homology group of a chain complex is the th homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.) Example: Let C be a chain complex that has an abelian group A in degree n and zero in all other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space . There is also an ∞-category-version of the Dold–Kan correspondence. The book "Nonabelian Algebraic Topology" cited below has a Section 14.8 on cubical versions of the Dold–Kan theorem, and relates them to a previous equivalence of categories between cubical omega-groupoids and crossed complexes, which is fundamental to the work of that book. (en) 호몰로지 대수학에서 정규화 사슬 복합체(正規化사슬複合體, 영어: normalized chain complex)는 아벨 범주의 단체 대상에 대하여 정의되는 사슬 복합체이다. 이는 아벨 범주의 단체 대상의 범주와 자연수 등급 사슬 복합체의 범주 사이의 동치를 정의하며, 이 동치를 돌트-칸 대응(Dold–Kan對應, 영어: Dold–Kan correspondence)이라고 한다. (ko)
prov:wasDerivedFrom	wikipedia-en:Dold–Kan_correspondence?oldid=1100797006&ns=0
page length (characters) of wiki page	5357 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Dold–Kan_correspondence
is Link from a Wikipage to another Wikipage of	Double groupoid Simplicial group Pursuing Stacks Albrecht Dold Daniel Kan Differential graded module Kan fibration Cotriple homology Simplicial commutative ring Bloch's higher Chow group Homotopy theory Dold-Kan correspondence Chain complex Simplicial set Simplicial Lie algebra Simplicial homotopy Stable ∞-category
is Wikipage redirect of	Dold-Kan correspondence
is known for of	Albrecht Dold Daniel Kan
is known for of	Albrecht Dold Daniel Kan
is foaf:primaryTopic of	wikipedia-en:Dold–Kan_correspondence

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