Waldhausen K-theory of Spaces via Comodules

Let X be a simplicial set. We construct a novel adjunction between the categories RX of retractive spaces over X and ComodX+ of X+comodules, then apply recent work on left-induced model category structures [5], [16] to establish the existence of a left proper, simplicial model category structure on ComodX+ with respect to which the adjunction is a Quillen equivalence after localization with respect to some generalized homology theory E∗. We show moreover that this model category structure on ComodX+ stabilizes, giving rise to a model category structure on ComodΣ∞X+ , the category of Σ∞X+-comodule spectra. It follows that the Waldhausen K-theory of X, A(X), is naturally weakly equivalent to the Waldhausen K-theory of Comod Σ∞X+ , the category of homotopically finite Σ∞X+-comodule spectra, where the weak equivalences are given by twisted homology. For X simply connected, we exhibit explicit, natural weak equivalences between the K-theory of Comod Σ∞X+ and that of the category of homotopically finite Σ∞(ΩX)+-modules, a more familiar model for A(X). For X not necessarily simply connected, we have E∗-local versions of these results for any generalized homology theory E∗. For H a simplicial monoid, ComodΣ∞H+ admits a monoidal structure and induces a model structure on the category AlgΣ∞H+ of Σ ∞H+-comodule algebras. This provides a setting for defining homotopy coinvariants of the coaction of Σ∞H+ on a Σ∞H+-comodule algebra, which is essential for homotopic Hopf-Galois extensions of ring spectra as originally defined by Rognes [27] and generalized in [15]. An algebraic analogue of this was only recently developed, and then only over a field [5].