Galois theory and a general notion of central extension

AbstractWe 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 Fröhlich 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”