Ebook: Homotopy Limits, Completions and Localizations

The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite completion of Quillen and Sullivan; for R a subring of the rationals, the R-completion coincides up to homotopy, with the localizations of Quillen, Sullivan and others. In part II of these notes, the authors have assembled some results on towers of fibrations, cosimplicial spaces and homotopy limits which were needed in the discussions of part I, but which are of some interest in themselves.

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The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite completion of Quillen and Sullivan; for R a subring of the rationals, the R-completion coincides up to homotopy, with the localizations of Quillen, Sullivan and others. In part II of these notes, the authors have assembled some results on towers of fibrations, cosimplicial spaces and homotopy limits which were needed in the discussions of part I, but which are of some interest in themselves.

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The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite completion of Quillen and Sullivan; for R a subring of the rationals, the R-completion coincides up to homotopy, with the localizations of Quillen, Sullivan and others. In part II of these notes, the authors have assembled some results on towers of fibrations, cosimplicial spaces and homotopy limits which were needed in the discussions of part I, but which are of some interest in themselves.
Content:
Front Matter....Pages I-V
Front Matter....Pages 1-9
The R-completion of a space....Pages 10-47
Fibre lemmas....Pages 48-69
Tower lemmas....Pages 70-98
An R-completion of groups and its relation to the R-completion of spaces....Pages 99-125
R-localizations of nilpotent spaces....Pages 126-162
p-completions of nilpotent spaces....Pages 163-201
A glimpse at the R-completion of non-nilpotent spaces....Pages 202-223
Front Matter....Pages 224-227
Simplicial sets and topological spaces....Pages 228-248
Towers of fibrations....Pages 249-264
Cosimplicial spaces....Pages 265-286
Homotopy inverse limits....Pages 287-324
Homotopy direct limits....Pages 325-340
Erratum to: The R-completion of a space....Pages 349-349
Erratum to: Tower lemmas....Pages 349-349
Erratum to: p-completions of nilpotent spaces....Pages 349-349
Back Matter....Pages 341-348

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The main purpose of part I of these notes is to develop for a ring R a functional notion of R-completion of a space X. For R=Zp and X subject to usual finiteness condition, the R-completion coincides up to homotopy, with the p-profinite completion of Quillen and Sullivan; for R a subring of the rationals, the R-completion coincides up to homotopy, with the localizations of Quillen, Sullivan and others. In part II of these notes, the authors have assembled some results on towers of fibrations, cosimplicial spaces and homotopy limits which were needed in the discussions of part I, but which are of some interest in themselves.
Content:
Front Matter....Pages I-V
Front Matter....Pages 1-9
The R-completion of a space....Pages 10-47
Fibre lemmas....Pages 48-69
Tower lemmas....Pages 70-98
An R-completion of groups and its relation to the R-completion of spaces....Pages 99-125
R-localizations of nilpotent spaces....Pages 126-162
p-completions of nilpotent spaces....Pages 163-201
A glimpse at the R-completion of non-nilpotent spaces....Pages 202-223
Front Matter....Pages 224-227
Simplicial sets and topological spaces....Pages 228-248
Towers of fibrations....Pages 249-264
Cosimplicial spaces....Pages 265-286
Homotopy inverse limits....Pages 287-324
Homotopy direct limits....Pages 325-340
Erratum to: The R-completion of a space....Pages 349-349
Erratum to: Tower lemmas....Pages 349-349
Erratum to: p-completions of nilpotent spaces....Pages 349-349
Back Matter....Pages 341-348
....