Tower Techniques for Cosimplicial Resolutions

Let M be a simplicial model category and J : M --> M a simplicial coaugmented functor. Given an object X, the assignment n bar right arrow 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 = tot(s)([n] bar right arrow 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)}(sgreater than or equal to0)-->{tot(S)X}(s greater than or equal to 0).