Homotopic Hopf–Galois extensions revisited

Abstract

In this article we revisit the theory of homotopic Hopf–Galois extensions introduced in [9], in light of the homotopical Morita theory of comodules established in [3].We generalize the theory to a relative framework, which we believe is new even in the classical context and which is essential for treating the Hopf–Galois correspondence in [19]. We study in detail homotopic Hopf–Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterization analogous to the one Rognes provided in the context of ring spectra [26]. An interesting feature in the differential graded setting is the close relationship between homotopic Hopf–Galois theory and Koszul duality theory. We show that nice enough principal fibrations of simplicial sets give rise to homotopic Hopf–Galois extensions in the differential graded setting, for which this Koszul duality has a familiar form.