Homotopic Hopf-Galois extensions revisited

TL;DR

This paper revisits and generalizes the theory of homotopic Hopf-Galois extensions, connecting it with homotopical Morita theory, and explores its applications in differential graded algebras and simplicial sets.

Contribution

It introduces a new relative framework for homotopic Hopf-Galois extensions, applicable even in classical contexts, and links the theory with Koszul duality and descent in differential graded algebras.

Findings

Established a descent-type characterization for dg algebra extensions.

Demonstrated the relationship between homotopic Hopf-Galois extensions and Koszul duality.

Showed how principal fibrations induce such extensions in the dg setting.

Abstract

In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in arXiv:0902.3393v2 [math.AT], in light of the homotopical Morita theory of comodules established in arXiv:1411.6517 [math.AT]. 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 forthcoming work of the second author and Karpova. 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. 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.