Koszul duality and the bar spectral sequence

Abstract

The bar spectral sequence for algebras over a spectral operad relates Koszul duality phenomena in several contexts. In this thesis, we apply this classical tool to the Koszul dual pair given by the (non-unital) E subscript ∞-operad and the spectral Lie operad over Fₚ. The bar spectral sequence for E subscript ∞-algebras yields the structure of operations on mod p Topological André-Quillen cohomology and the homotopy groups of spectral partition Lie algebras, building on the work of Brantner-Mathew. In the colimit, the unary operations are Koszul dual to the Dyer-Lashof algebra. On the other hand, the bar construction against certain spectral Lie algebras models labeled configuration spaces by a theorem of Knudsen. The associated bar spectral sequence yields new results on their mod p homology at low weights, as well as interesting patterns of universal differentials. We also record an attempt with Andrew Senger on detecting these differentials via deformation of the bar comonad.