Aspects of derived Koszul duality
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This thesis comprises two distinct chapters. In the first, we rigidify constructions of generalized string topology Thom spectra due to Gruher--Salvatore into lax symmetric monoidal functors from spaces parametrized by a closed manifold to a highly structured category of spectra. We then use derived Koszul duality to show that our functor satisfies a homotopy-theoretic universal property, resulting in independence from all auxiliary choices used in its construction. In the second, we initiate a program to understand the positive-characteristic analogue of Ben-Zvi--Nadler's categorified Hochschild--Kostant--Rosenberg theorem by giving an interpretation of the E-infinity-algebra of mod-p cochains on the circle in spectral algebraic geometry, building on work of Mandell. We then apply derived Koszul duality to indicate a connection to a conjectural form of Cartier duality in the spectral setting.