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In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any model category and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.

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rdf:type	yago:WikicatCategory-theoreticCategories yago:Abstraction100002137 yago:Class107997703 yago:Collection107951464 yago:Group100031264 television station
rdfs:label	Homotopie-Kategorie (de) Homotopy category (en) 호모토피 범주 (ko) 同倫範疇 (zh)
rdfs:comment	In der Mathematik ist die Homotopie-Kategorie die Kategorie, deren Objekte die topologischen Räume und deren Morphismen die Homotopieklassen stetiger Abbildungen sind. Sie wird mit hTop bezeichnet. (de) In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any model category and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra. (en) 호모토피 이론에서 호모토피 범주(homotopy範疇, 영어: homotopy category)는 주어진 모형 범주에서, 모든 약한 동치를 동형 사상으로 만들어 얻는 범주이다. (ko) 在數學的拓撲學領域中，同倫範疇是處理同倫問題時格外便利的範疇論語言。它的對象是拓撲空間，態射是連續函數的同倫類，這是的一個例子；由於同倫關係在映射的合成下不變，同倫範疇的定義是明確的。所有拓撲空間構成的同倫範疇通常記為或；有時也會考慮較小一類的空間，例如或CW複形。兩空間在同倫範疇中同構的充要條件是它們同倫等價。設為拓撲空間，它們在同倫範疇中的態射集記為。同倫理論的基本課題之一便是研究，例如當是球面時，的計算就歸結到同倫群的計算。 (zh)
dct:subject	Homotopy theory Categories in category theory
Wikipage page ID	2040756 (xsd:integer)
Wikipage revision ID	1111690067 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Module (mathematics) Homeomorphic Homotopy class Homotopy equivalence Derived category Quasi-isomorphism Two-dimensional space Continuous map Mathematics Circle Full subcategory Morphism Möbius strip Concrete category Equivalence of categories Compactly generated space Functor Spectrum (topology) Suspension (topology) Mathematical Proceedings of the Cambridge Philosophical Society Adjoint functors Category of sets Localization of a category Pointed space American Mathematical Society Daniel Quillen Faithful functor Isomorphism Ring (mathematics) Smash product Homotopy theory Abelian group Bijection Cofibration Eilenberg–MacLane space Homotopy group Homotopy theory Model category Categories in category theory CW complex Fibration Grothendieck category Natural isomorphism Category (mathematics) Category of topological spaces Chain complex Loop space Sheaf (mathematics) Singular homology University of Chicago Press Subspace topology Stable homotopy category Weak equivalence (homotopy theory) Real line Simplicial set Topological space Representable functor Weak Hausdorff space J.H.C. Whitehead Peter Freyd Springer-Verlag Whitehead's theorem Path component Singular cohomology Weak homotopy equivalence
Link from a Wikipage to an external page	http://www.math.cornell.edu/~hatcher/AT/ATpage.html http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html http://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf https://web.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf http://hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf
sameAs	Homotopy category Homotopy category Homotopy category Homotopy category Homotopy category Homotopy category Homotopy category
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has abstract	In der Mathematik ist die Homotopie-Kategorie die Kategorie, deren Objekte die topologischen Räume und deren Morphismen die Homotopieklassen stetiger Abbildungen sind. Sie wird mit hTop bezeichnet. (de)

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