  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 $\Sigma^\infty X_{+}$-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
$\Sigma^\infty X_{+}$-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 $\Sigma^{\infty}(\Omega X)_+$-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 $\Sigma^{\infty}H_{+}$-comodule
algebras admits an induced model structure, providing a setting for defining
homotopy coinvariants of the coaction of $\Sigma^{\infty}H_{+}$ on a
$\Sigma^{\infty}H_{+}$-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]).

Algebraic Topology;K-Theory and Homology