Universal spaces for homotopy limits of modules over coaugmented functors (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⩽n, and the cofacial resolution X• admits termwise contractions, then holim X−1 is a retract of totn holimP X•, and that the tower map {holim X−1}→{totn holimP X•}n is a pro-equivalence in the homotopy category of spaces.