Hostname: page-component-7bb8b95d7b-lvwk9 Total loading time: 0 Render date: 2024-09-17T11:30:27.246Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopy Theory of Diagrams and CW-Complexes Over a Category

Published online by Cambridge University Press:  20 November 2018

Robert J. Piacenza
Robert J. Piacenza*
Affiliation:
Department of Mathematical Sciences, University of Alaska Fairbanks, Fairbanks, Alaska 99775-1110, USA
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

55U35Topological categoryhomotopyJ-cofibrationweak equivalenceJ-CW complexcohomologyequivariant homotopyfixed point set
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 4 , 01 August 1991 , pp. 814 - 824
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bredon, Glen E., Sheaf theory. McGraw-Hill, 1967.Google Scholar
2. Dubuc, Eduardo J., Kan extensions in enriched category theory, . Springer lecture notes in Math 145(1970).Google Scholar
3. Dwyer, W.G. and Kan, D.M., Singular functors and realization functors, Proceed, of the Koninlijke Nederlandse Akademie Van Wetenschappen, Series A., 87(1984), 147–153.Google Scholar
4. Elmendorf, A.D., Systems of fixed point sets, Trans. Amer. Math. Soc. 277(1983), 275–284.Google Scholar
5. Heller, Alex, Homotopy in functor categories, Trans. Amer. Math. Soc. 272 (1982), 185–202.Google Scholar
6. May, Peter, The homotopical foundations of algebraic topology, mimeographed notes, University of Chicago.Google Scholar
7. Piacenza, R., Cohomology of diagrams and equivariant singular theory, Pacific J. Math. 91(1981), 435–444.Google Scholar
8. Quillen, D., Homotopical algebra, . Springer lecture notes in Math 43(1967).Google Scholar
9. Spanier, E., Algebraic topology. McGraw-Hill, 1966.Google Scholar
10. Vogt, R., Convenient categories of topological spaces for homotopy theory, Archiv der Mathematik, XXII (1971), 545–555.Google Scholar
11. Vogt, R., Equivariant singular homology and cohomology, preprint.Google Scholar
12. Waner, S., Equivariant homotopy theory and Milnor's theorem, Trans. Amer. Math. Soc. 258(1980), 351–368.Google Scholar