000134306 001__ 134306 000134306 005__ 20190117191946.0 000134306 037__ $$aARTICLE 000134306 245__ $$aHomotopic Hopf-Galois extensions: foundations and examples 000134306 269__ $$a2009 000134306 260__ $$c2009 000134306 336__ $$aJournal Articles 000134306 520__ $$aHopf-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. 000134306 6531_ $$aAlgebraic Topology 000134306 6531_ $$aRings and Algebras 000134306 700__ $$0240499$$aHess, Kathryn$$g105396 000134306 773__ $$j16$$q79-132$$tGeometry and Topology Monographs 000134306 8564_ $$uhttp://arxiv.org/abs/0902.3393$$zURL 000134306 8564_ $$s426902$$uhttps://infoscience.epfl.ch/record/134306/files/HHG-published.pdf$$yn/a$$zn/a 000134306 909C0 $$0252139$$pUPHESS$$xU10968 000134306 909CO $$ooai:infoscience.tind.io:134306$$pSV$$pGLOBAL_SET$$particle 000134306 917Z8 $$x105396 000134306 937__ $$aGR-HE-ARTICLE-2009-001 000134306 973__ $$aEPFL$$rREVIEWED$$sPUBLISHED 000134306 980__ $$aARTICLE