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

TL;DR

This paper explores the algebraic K-theory of spaces through derived Koszul duality and Morita equivalence, revealing connections to homotopy involutions and defining a geometric analog of Swan theory.

Contribution

It introduces a geometric analog of Swan theory in terms of spectra and relates different constructions of algebraic K-theory via duality and Morita equivalence.

Findings

Identifies derived Koszul duality as a framework for algebraic K-theory of spaces

Defines a geometric analog of Swan theory using spectra

Shows the algebraic K-theory of the spectrum DX as a key result

Abstract

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 $DX=S^{X_{+}}$.