Homotopic Hopf-Galois extensions: foundations and examples

Abstract

Hopf‐Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group G are the Hopf‐Galois extensions with respect to the dual of the group algebra of G . Rognes recently defined an analogous notion of Hopf‐Galois extensions in the category of structured ring spectra, motivated by the fundamental example of the unit map from the sphere spectrum to MU . This article introduces a theory of homotopic Hopf‐Galois extensions in a monoidal category with compatible model category structure that generalizes the case of structured ring spectra. In particular, we provide explicit examples of homotopic Hopf‐Galois extensions in various categories of interest to topologists, showing that, for example, a principal fibration of simplicial monoids is a homotopic Hopf‐ Galois extension in the category of simplicial sets. We also investigate the relation of homotopic Hopf‐Galois extensions to descent. 16W30, 55U35; 13B05, 55P42, 57T05, 57T30