Cofibrance and *completion

Abstract

We construct the Bousfield-Kan completion with respect to a triple, for a model category. In the pointed case, we construct a Bousfield-Kan spectral sequence that computes the relative homotopy groups of the completion of an object.

These constructions are based on the existence of a diagonal for the cofibrant-replacement functor constructed using the small object argument.

A central result that we use, due to Dwyer, Kan and Hirschhorn, is that in a model category homotopy limits commute with the function complex. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax [phone].)