We interpret different constructions of 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 $DX=S^{X_{+}}$.