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.
Keywords
model category, monoidal category,
Hopf–Galois extension, descent, co-ring
