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Cosimplicial resolutions and homotopy spectral sequences in model categories

A K Bousfield

Geometry & Topology 7 (2003) 1001–1053

arXiv: math.AT/0312531


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.

cosimplicial resolutions, homotopy spectral sequences, modelcategories, Bendersky–Thompson completion, Bousfield–Kan completion
Mathematical Subject Classification 2000
Primary: 55U35
Secondary: 18G55, 55P60, 55T15
Forward citations
Received: 26 November 2003
Accepted: 25 December 2003
Published: 26 December 2003
Proposed: Haynes Miller
Seconded: Thomas Goodwillie, Gunnar Carlsson
A K Bousfield
Department of Mathematics
University of Illinois at Chicago
Illinois 60607