Homotopy Theory of Diagrams and CW-Complexes Over a Category

Piacenza, Robert J.

Canadian journal of mathematics, Tome 43 (1991) no. 4, pp. 814-824

Résumé

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.

Détail

DOI : 10.4153/CJM-1991-046-3

Mots-clés : 55U35, Topological category, homotopy, J-cofibration, weak equivalence, J-CW complex, cohomology, equivariant homotopy, fixed point set

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