Geometry & Topology

Cosimplicial resolutions and homotopy spectral sequences in model categories

A K Bousfield

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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.

Article information

Geom. Topol., Volume 7, Number 2 (2003), 1001-1053.

Received: 26 November 2003
Accepted: 25 December 2003
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55U35: Abstract and axiomatic homotopy theory
Secondary: 18G55: Homotopical algebra 55P60: Localization and completion 55T15: Adams spectral sequences

cosimplicial resolutions homotopy spectral sequences modelcategories Bendersky–Thompson completion Bousfield–Kan completion


Bousfield, A K. Cosimplicial resolutions and homotopy spectral sequences in model categories. Geom. Topol. 7 (2003), no. 2, 1001--1053. doi:10.2140/gt.2003.7.1001.

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