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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,
modelcategories, Bendersky–Thompson completion,
Bousfield–Kan completion
Mathematical Subject Classification 2000
Primary: 55U35
Secondary: 18G55, 55P60, 55T15
Publication
Received: 26 November 2003
Accepted: 25 December 2003
Published: 26 December 2003
Proposed: Haynes Miller
Seconded: Thomas Goodwillie, Gunnar Carlsson