23 Janeiro 1997, 14:30
G. M. Kelly, University of Sydney
On the reflectiveness of coverings in topology, geometry and algebra

Janelidze has defined a very general notion of Galois theory, where the basic data consist of a category C, a reflective full subcategory X, and a class S of the maps in C, with these data subject to mild assumptions. We can define the TRIVIAL COVERINGS f: A --> B in C to be those maps in S which are the pullbacks of their reflexions into X; then a map f: A --> B in C is a COVERING if it lies in S and is LOCALLY a trivial covering, in the sense that its pullback along some effective descent map p: E --> B is a trivial covering. This notion of covering includes many classical concepts. When C is the topos ofsheaves on a connected and locally-connected topological space and K is the category Set of constant sheaves, the coverings are the covering spaces in the classical geometrical sense; when C is a more general connected and locally- -connected topos, we get the covering theory of Barr and Diaconescu; when C is the dual of the category of commutative rings, the coverings are the quasi- -separable ring extensions; when C and X are suitable varieties of universal algebras, the coverings are called CENTRAL EXTENSIONS, generalizing the usual such concepts for groups and algebras; and so on. The central point of Janelidze's general Galois theory is a description of the coverings of B - or rather of those made trivial by pullback along a GIVEN effective descent map p: E --> B - as the actions on X of a certain Galois pregroupoid of the extension (E,p). It is often, but not always, the case that the category of coverings of B is reflective in the category of all S-maps into B. The aim of the present talk is to give sufficient conditions for this, and to prove them satisfied in each of the important examples above.