Waldhausen K-theory of spaces via comodules
Let X be a simplicial set. We construct a novel adjunction be- tween the categories RX of retractive spaces over X and ComodX+ of X+- comodules, then apply recent work on left-induced model category structures ,  to establish the existence of a left proper, simplicial model category structureonComodX+ withrespecttowhichtheadjunctionisaQuillenequiv- alence 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 hf equivalent to the Waldhausen K-theory of ComodΣ∞X+, the category of ho- motopically finite Σ∞X+-comodule spectra, where the weak equivalences are given by twisted homology. For X simply connected, we exhibit explicit, nat- hf ural weak equivalences between the K-theory of ComodΣ∞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 al- gebras. This provides a setting for defining homotopy coinvariants of the coaction of Σ∞H+ on a Σ∞H+-comodule algebra, which is essential for ho- motopic Hopf-Galois extensions of ring spectra as originally defined by Rognes  and generalized in . An algebraic analogue of this was only recently developed, and then only over a field .