Abstract¶

Galois theory is one of the main branches of modern algebra. Categorical Galois theory is a purely categorical approach to it. It was initiated by Grothendieck, then pursued by Joyal and Tirney in the context of toposes and finally by George Janelidze form University of Cape Town. It has not been applied before to the theory of Hopf algebras, and we propose a research plan to find a connection between Categorical Galois Theory and (commutative) Hopf Galois extensions. This topic extends my Ph.D. thesis in which I constructed a Galois connection for Hopf Galois extensions and I proved interesting criteria for closedness. We also propose to extend the results in another direction: towards bialgebroids and Hopf algebroids, which theory erupted in recent years and is one of the main topics in the theory of Hopf algebras today. Third and final domain of study that I want to propose is an abstract Galois theory in categorical homotopy theories. This is probably the most advanced and open idea in the three subtopics I am proposing. We want to find a suitable categorical generalisation of Stasheff results on classification of fibrations to the context of categories of fibrant objects, in the sens of K.Brown (possibly satisfying additional assumptions). We believe that this will lead to a categorical Galois theory in abstract homotopy categories.