Generalized String Topology and Derived Koszul Duality

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 $M^{-TM}\simeq \text{bbD}M$. Then, using an $\mathrm{\infty }$-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 ${\mathrm{\Sigma }}_{+}^{\mathrm{\infty }}\mathrm{\Omega }M$ and $\text{bbD}M$.