Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces

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 $G^{\text{bZ}}(\text{bZ}[\pi ])$ in terms of ${\mathrm{\Sigma }}_{+}^{\mathrm{\infty }}\mathrm{\Omega }X$ and show that it is the algebraic $K$-theory of the $E_{\mathrm{\infty }}$ ring spectrum $DX=S^{X_{+}}$.