Universal spaces for homotopy limits of modules over coaugmented functors (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 SP∞-modules. Finite SP∞-limits are nilpotent spaces with a very strong finiteness property. We show that the cofacial Bousfield–Kan construction of the functors Jn 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→JnX, for some n<∞.