Homotopic Hopf-Galois extensions revisited (Q1747033)

Summary: In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in [the second author, Geom. Topol. Monogr. 16, 79--132 (2009; Zbl 1196.55010)], in light of the homotopical Morita theory of comodules established in [the authors, Isr. J. Math. 227, No. 1, 239--287 (2018; Zbl 1433.16038)]. 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 [\textit{V. Karpova}, Homotopic Hopf-Galois extensions of commutative differential graded algebras. Lausanne: EPFL (PhD Thesis) (2014)]. 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 [\textit{J. Rognes}, Galois extensions of structured ring spectra. Stably dualizable groups. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1166.55001)]. 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.