Universal Spaces for Homotopy Limits of Modules over Coaugmented Functions I

For a coaugmented functor J on spaces, we consider J-modules and finite J-limits. The former are spaces X which are retracts of JX via the natural map. The latter are homotopy limits of J-modules arranged in diagrams whose shape is finite dimensional. Familiar examples are generalised Eilenberg MacLane spaces, which are the SPinfinity-modules. Finite SPinfinity-limits are nilpotent spaces with a very strong finiteness property. We show that the cofacial Bousfield-Kan construction of the functors J(n) is universal for finite J-limits in the sense that every map X --> Y where Y is a finite J-limit, factors through such natural map X --> J(n)X, for some n < infinity. (C) 2002 Elsevier Science Ltd. All rights reserved.