  We interpret different constructions of the algebraic $K$-theory of spaces as
an instance of derived Koszul (or bar) duality and also as an instance of
Morita equivalence. We relate the interplay between these two descriptions to
the homotopy involution. We define a geometric analog of the Swan theory
$G^{\bZ}(\bZ[\pi])$ in terms of $\Sigma^{\infty}_{+} \Omega X$ and show that it
is the algebraic $K$-theory of the $E_{\infty}$ ring spectrum $DX=S^{X_{+}}$.
