Tower techniques for cosimplicial resolutions

Abstract

Let ℳ be a simplicial model category and J : ℳ → ℳ a simplicial coaugmented functor. Given an object X, the assignment n↦J n+1 X defines a cofacial resolution (an augmented cosimplicial space without its codegeneracy maps). Following Bousfield and Kan we define J s X = tots([n] ↦ J n+1 X). An object X is called J-injective if it is a retract of JX in Ho(ℳ) via the natural map. We show that certain homotopy limits of J-injective objects are J s -injective. Our method is to use the notion of pro-weak equivalences which was first introduced in a different language and context by David Edwards and Harold Hastings. The key observation is that a cofacial resolution X (-1) → X which admits a left contraction gives rise to a pro-weak equivalence of towers {X(-1)}s≥0→{totS X}s ≥ 0.