Galois theory and a general notion of central extension

Abstract We propose a theory of central extensions for universal algebras, and more generally for objects in an exact category C , centrality being defined relatively to an “admissible” full subcategory X of C . This includes not only the classical notions of central extensions for groups and for algebras, but also their generalization by Frohlich to a pair consisting of a variety C of ω-groups and a subvariety X . Our notion of central extension is adapted to the generalized Galois theory developed by the first author, the use of which enables us to classify completely the central extensions of a given object B, in terms of the actions of an “internal Galois pregroupoid”.

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