Universal Spaces for Homotopy Limits of Modules over Coaugmented Functions II

A contraction for a cosimplicial resolution X-1 --> X-. is an "extra codegeneracy map", and the existence of such, is well known to induce a homotopy equivalence between the augmentation and the total space of the resolution. We generalise and strengthen this result by considering cofacial cosimplicial resolutions of length n of diagrams of spaces. We show that if X-1 is a P-diagram and dim P less than or equal to n, and the cofacial resolution X-. admits termwise contractions, then holim X-1 is a retract of tot, holim(p)X(.), and that the tower map {holimX(-1)} --> {tot(n)holim(p)X(.)}(n) is a pro-equivalence in the homotopy category of spaces. (C) 2002 Elsevier Science Ltd. All rights reserved.