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 Quil
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304 A. K. Bousfield D.M. Kan
Homotopy Limits, Completions and Localizations
Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo
Authors
Aldridge K. Bousfield University of Illinois, Department of Mathematics Chicago, Illinois 60680, USA Daniel M. Kan Massachusetts Institute of Technology Cambridge, MA 02139, USA
1st Edition 1972 2nd corrected Printing 1987
Mathematics Subject Classification (1970): 18A30, 18A35, 55-02, 55005, 55010,55099 ISBN 3-540-06105-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-06105-3 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Serlin Heidelberg 1972 Printed in Germany Printing and binding: Druckhaus Seltz, Hemsbach/Sergstr. 2146/3140-543210
Contents
Part I.
Completions and localizations §O.
Chapter I.
§8. §9.
10 13 17 20 24 30 34 40 44
Introduction................................. The principal fibration lemma •.•..••...•.•... Proof of the principal fibration lemma •.••... Nilpotent fibrations •..••.•.•••..•.••.••.•..• The mod-R fibre lemma........................ Proof of the mod-R fibre lemma ••••..•.••.•...
48 50 54 58 62 67
Tower lemmas §l. §2. § 3. §4. §5. § 6. §7. §8.
Chapter IV.
Introduction................................. The triple {R,~,W} on the category of spaces. The total space of a cosimplicial space .•.... The R-completion of a space ..•••.•.•.•..•..•. R-complete, R-good and R-bad spaces ...•....•. Low dimensional behaviour .................... Disjoint unions, finite products and mUltiplicative structures ...•.•......••.....• The fibre-wise R-completion .......••.••.•.•.. The role of the ring R .......................
Fibre lemmas §l. §2. §3. §4. §5. §6.
Chapter III.
1
The R-completion of a space §l. §2. §3. §4. §5. § 6. §7.
Chapter II.
Introduction to Part I •.••.••.••..••.•••..•..
Introduction................................. Pro-isomorphisms of towers of groups •••••...• Iveak pro-homotopy equivalences •.••••.•••••..• Proof of 3.3 ••.•••.•.•...••••.•••...••..•.... R-nilpotent spaces .......•...•.•.•..••••.•..• The tower lemmas •.•••••.••.••.•••..•.•..••.•. Tower version of the mod-R fibre lemma .••..•. An Artin-Mazur-like interpretation of the R-completion ••.••..••.••..•.•.•.•.......•...•
70 73 76 79 82 87 91 94
An R-completion of groups and its relation to the R-completion of spaces §1. §2. §3. §4. §5. §6. §7.
Introduction. • •• . . .. • • . . ••••• . . .• . . . • . . . . . • . . The R-completion of a group ..•.•.....••..•... A variation o
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