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

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