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By Libman Assaf (-aber


Universal spaces for homotopy limits of modules over coaugmented functors. I. (English summary) Topology 42 (2003), no. 3, 555–568. Let J be a functor from simplicial sets to itself equipped with a natural augmentation X → JX. A J-module is a space X for which this augmentation has a homotopy retraction. As example, one might take JX = Ω ∞ (E ∧ X+) where E is a ring spectrum. This paper considers homotopy inverse limits of diagrams in spaces of J-modules, where the morphisms on the diagram need not be morphisms of modules. The standing assumption seems to be that J is a simplicial functor, which implies that it preserves mapping spaces. The main result is this. Let J • X be the standard cofacial (i.e., cosimplicial without the codegeneracies) resolution of X and let JnX be the homotopy inverse limit of this resolution truncated at n. Then, if X is a P diagram of spaces, where P is a category of dimension at most n and each of the spaces X(p), with p ∈ P, is a J-module, then holimP X is a JnX module. Further applications are given to the unstable Adams spectral sequence. This has the corollary that if X is a J-module then X is pro-equivalent to the tower {JnX}. The definition of the dimension of smal

Year: 2013
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