On equivariant homotopy equivalences

1981 ◽  
Vol 12 (2) ◽  
pp. 83-94
Author(s):  
Italo José Dejter
Keyword(s):  
Equivariant Homotopy

Homotopy Theory of Diagrams and CW-Complexes Over a Category

1991 ◽  
Vol 43 (4) ◽  
pp. 814-824 ◽  
Author(s):  
Robert J. Piacenza
Keyword(s):  
Classical Theory ◽  
Homotopy Theory ◽  
Homotopy Groups ◽  
Weak Equivalence ◽  
Equivariant Homotopy ◽  
Topological Category ◽  
Sheaf Theory ◽  
Closed Model ◽  
Base Category ◽  
Closed Model Category

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.

scholarly journals On equivariant homotopy type

Topology ◽  
1978 ◽  
Vol 17 (3) ◽  
pp. 267-272 ◽  
Author(s):  
I.M. James ◽  
G.B. Segal
Keyword(s):  
Homotopy Type ◽  
Equivariant Homotopy

Equivariant Homotopy and Cohomology Theory

1996 ◽  
Author(s):  
J. May ◽  
Michael Cole ◽  
Gustavo Comezana ◽  
Steven Costenoble ◽  
Anthony Elmendorf ◽  
...  
Keyword(s):  
Cohomology Theory ◽  
Equivariant Homotopy
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