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About: Homotopy category

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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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  • 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)
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  • 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)
rdfs:label
  • Homotopie-Kategorie (de)
  • Homotopy category (en)
  • 호모토피 범주 (ko)
  • 同倫範疇 (zh)
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