Waldhausen K-theory of spaces via comodules

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Let X be a simplicial set. We construct a novel adjunction between the categories of retractive spaces over X and of X₊-comodules, then apply recent work on left-induced model category structures (arXiv:1401.3651v2 [math.AT],arXiv:1509.08154 [math.AT]) to establish the existence of a left proper, simplicial model category structure on the category of X+-comodules, with respect to which the adjunction is a Quillen equivalence after localization with respect to some generalized homology theory. We show moreover that this model category structure stabilizes, giving rise to a model category structure on the category of Σ∞ X₊-comodule spectra. The Waldhausen K-theory of X, A(X), is thus naturally weakly equivalent to the Waldhausen K-theory of the category of homotopically finite Σ∞ X₊-comodule spectra, with weak equivalences given by twisted homology. For X simply connected, we exhibit explicit, natural weak equivalences between the K-theory of this category 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 localized versions of these results. For H a simplicial monoid, the category of Σ∞H₊-comodule algebras admits an induced model structure, providing 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 in arXiv:math/0502183v2 and generalized in arXiv:0902.3393v2 [math.AT]. An algebraic analogue of this was only recently developed, and then only over a field (arXiv:1401.3651v2 [math.AT]).