Mathematics – Algebraic Topology
Scientific paper
2003-12-31
Geom. Topol. 7(2003) 1001-1053
Mathematics
Algebraic Topology
Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper29.abs.html
Scientific paper
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.
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