Universal spaces for homotopy limits of modules over coaugmented functors. II (Q1811542)

A contraction for a cosimplicial resolution \(X^{-1} \to X^\bullet\) induces a homotopy equivalence between the augmentation and the total space of the resolution. The author generalizes this result by considering cofacial (i.e., cosimplicial without the codegeneracies) resolutions of length \(n\) of diagrams of spaces: Let \(P\) be a small category and let \(X^{-1} \to X^\bullet\) be a termwise trivial cofacial resolution of \(P\)-diagrams. If dim \(P\leq n\) and \(X^{-1}\) is fibrant then holim\(_{P}X^{-1}\) is, up to homotopy, a retract of tot\(_n\)holim\(_PX^\bullet\). This paper is complemented by [ibid., 555-568 (2003; Zbl 1025.55007)].