Generalized String Topology and Derived Koszul Duality

TL;DR

This paper extends generalized string topology to a functor linking the homotopy theories of parametrized spaces and spectra, and relates it to derived Koszul duality via $$-categorical Morita theory.

Contribution

It rigidifies and broadens the string topology construction, connecting it to derived Koszul duality through an $$-categorical framework.

Findings

Provides a functorial extension of string topology operations.

Establishes a connection between parametrized homotopy theory and module spectra.

Reinterprets the construction via derived Koszul duality and Morita theory.

Abstract

The generalized string topology construction of Gruher and Salvatore assigns to any bundle of $E_n$-algebras $A$ over a closed oriented manifold $M$ a collection of intersection-type operations on the homology of the total space. These operations are realized by an $H_n$-ring structure on the Thom spectrum $A^{-TM}$ under the Thom isomorphism. We rigidify and extend this construction to a functor connecting the homotopy theory of spaces and spectra parametrized by $M$ to the homotopy theory of module spectra over the Atiyah-Milnor-Spanier-Whitehead dual $M^{-TM} \simeq \bbD M$. Then, using an $\infty$-categorical version of Morita theory, we give an alternative description of our construction in terms of the derived Koszul duality (alias bar-cobar duality) between $\Sigma^\infty_+ \Omega M$ and $\bbD M$.