Geometry & Topology, Vol. 7 (2003)
Paper no. 29, pages 1001--1053.
Cosimplicial resolutions and homotopy spectral sequences in model categories
A K Bousfield
Abstract.
We develop a general theory of cosimplicial resolutions, homotopy
spectral sequences, and completions for objects in model categories,
extending work of Bousfield-Kan and Bendersky-Thompson for ordinary
spaces. This is based on a generalized cosimplicial version of the
Dwyer-Kan-Stover theory of resolution model categories, and we are
able to construct our homotopy spectral sequences and completions
using very flexible weak resolutions in the spirit of relative
homological algebra. We deduce that our completion functors have
triple structures and preserve certain fiber squares up to
homotopy. We also deduce that the Bendersky-Thompson completions over
connective ring spectra are equivalent to Bousfield-Kan completions
over solid rings. The present work allows us to show, in a subsequent
paper, that the homotopy spectral sequences over arbitrary ring
spectra have well-behaved composition pairings.
Keywords.
Cosimplicial resolutions, homotopy spectral sequences, model categories, Bendersky-Thompson completion, Bousfield-Kan completion
AMS subject classification.
Primary: 55U35.
Secondary: 18G55, 55P60, 55T15.
DOI: 10.2140/gt.2003.7.1001
E-print: arXiv:math.AT/0312531
Submitted to GT on 26 November 2003.
Paper accepted 25 December 2003.
Paper published 26 December 2003.
Notes on file formats
A K Bousfield
Department of Mathematics
University of Illinois at Chicago
Chicago, Illinois 60607, USA
Email: bous@uic.edu
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