Calculus of Fractions and Homotopy Theory
Calculus of Fractions and Homotopy Theory
复制标题
DOI：
10.1007/978-3-642-85844-4
复制
发表时间：
1967
期刊：
影响因子：
--
通讯作者：
P. Gabriel;M. Zisman
中科院分区：
文献类型：
--
作者：
P. Gabriel;M. Zisman
本工作的主要目的是向读者介绍一个特别好的同伦研究范畴，即同伦主题范畴（IV）。这个范畴，事实上，-根据第七章和一个著名的定理的JHC WHITEHEAD-相当于范畴的CW-复模同伦，即范畴的对象是空间的同伦类型的CW-复和其态射是同伦类的连续映射之间的空间。它也是等价的（I，1.3）一类分数的范畴的拓扑空间模同伦，并范畴的Kan复合模同伦（IV）。为了定义我们的同伦范畴，似乎有用的是尽可能密切地遵循在同调代数中已被证明有效的方法。因此，我们的范畴是阿贝尔范畴（VERDIER）的派生范畴的“拓扑”类似物。代数机器后，这项工作基本上是基于包括通常的接地理论范畴总结在字典和理论范畴的分数，形成的主题第一章的书。仅仅是拓扑机器就归结为凯利空间的一些性质（第一章和第三章）。我们研究的出发点是单纯集（CSS复合体或以前术语中的半单纯集）的类别。
The main purpose of the present work is to present to the reader a particularly nice category for the study of homotopy, namely the homo topic category (IV). This category is, in fact,-according to Chapter VII and a well-known theorem of JHC WHITEHEAD-equivalent to the category of CW-complexes modulo homotopy, ie the category whose objects are spaces of the homotopy type of a CW-complex and whose morphisms are homotopy classes of continuous mappings between such spaces. It is also equivalent (I, 1.3) to a category of fractions of the category of topological spaces modulo homotopy, and to the category of Kan complexes modulo homotopy (IV). In order to define our homotopic category, it appears useful to follow as closely as possible methods which have proved efficacious in homo logical algebra. Our category is thus the" topological" analogue of the derived category of an abelian category (VERDIER). The algebraic machinery upon which this work is essentially based includes the usual grounding in category theory-summarized in the Dictionary-and the theory of categories of fractions which forms the subject of the first chapter of the book. The merely topological machinery reduces to a few properties of Kelley spaces (Chapters I and III). The starting point of our study is the category, 10 Iff of simplicial sets (CSS complexes or semi-simplicial sets in a former terminology).