Cosimplicial resolutions and homotopy spectral sequences in model categories

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, modelcategories, Bendersky–Thompson completion, Bousfield–Kan completion

Mathematical Subject Classification 2000

Primary: 55U35

Secondary: 18G55, 55P60, 55T15

References

Forward citations

Publication

Received: 26 November 2003

Accepted: 25 December 2003

Published: 26 December 2003

Proposed: Haynes Miller

Seconded: Thomas Goodwillie, Gunnar Carlsson

Authors

A K Bousfield
Department of Mathematics University of Illinois at Chicago Chicago Illinois 60607 USA

Volume 7, issue 2 (2003)

A K Bousfield