Homotopy Theory of Diagrams and CW-Complexes Over a Category
Homotopy Theory of Diagrams and CW-Complexes Over a Category
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范畴上的图和 CW 复形的同伦理论
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发表时间：
1991
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通讯作者：
R. Piacenza
中科院分区：
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作者：
R. Piacenza
本文的目的是引入拓扑范畴上的CW复形的概念。本文的主要定理给出了基于拓扑范畴的空间图的同伦理论与相同基范畴上的CW复形的同伦理论之间的等价性。在第一节中，我们引入了基于固定拓扑范畴的空间图的同伦范畴。在第二节中，定义了图的同伦群。它们被用来定义推广了经典定义的弱等价和J-n等价的概念。在第三节中，我们采用CW络合物的经典理论来发展用于图的细胞理论。在第四节中，我们使用层理论定义了一个合理的图的上同调理论，并将它与前面定义的理论进行了比较。在第五节中，我们为图的同伦理论定义了一个封闭的模型范畴结构。证明了这种Quillen型同伦理论与J-CW复形的同伦理论是等价的。在第六节中，我们应用我们的构造和结果证明了Elmendorf最初用不同的方法证明的等变同伦理论中的一个有用的结果。
The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category. A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.