  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.
