This is to announce the posting of my paper "Covering Morphisms in Topos Theory" intended for the Fields Institute Workshop in September. It can be found in: http://www.math.mcgill.ca/~bunge/coveringmorphisms.dvi (.ps) Abstract: In the work of Janelidze, a formal (or pure) notion of locally trivial (or locally constant) covering morphism arises from an abstract categorical framework given by a pair of adjoint functors satisfying certain conditions. Under additional hypotheses on the splitting covers (normality), a groupoid is obtained which encapsulates the Galois theory; this theory has many different examples. In this paper, we concentrate on toposes bounded over an elementary topos S. We show that the construction of the fundamental groupoid of a locally connected S-topos E given by the author is an instance (in the locally simply connected case) of the pure theory of Janelidze and that, unlike the latter, no additional hypothesis is needed on the splitting cover in order to obtain a Galois theory. The explanation is simple: whereas in the pure theory only a pair of adjoint functors is involved, with the left adjoint playing the role of connected components, the Galois theory for toposes arises from a further right adjoint which is available from the beginning. Nevertheless, it is interesting to note that the Galois groupoids obtained in each case are the same for all practical purposes. In connection with the Galois theory that is implicit in the construction and properties of the fundamental groupoid of a locally connected topos over S, we introduce the notion of an S-Galois topos which generalizes the Galois toposes given by Moerdijk (and Grothendieck) for Grothendieck toposes. S>-Galois toposes are generated by S-Galois families and they are precisely the classifying toposes of prodiscrete localic groupoids in S. In addition to the locally constant covering morphisms which arise in connection with the fundamental groupoid of a topos, also the unramified morphisms (local homeomorphisms determined by complete spread objects), and more generally all complete spreads over the topos, deserve to be regarded as (generalized) covering morphisms; these are not instances of the pure theory. Among all coverings we single out the topological ones, by which we mean those coverings which are local homeomorphisms defined by objects with the unique path-lifting property. Marta Bunge 15-Aug-2002 09:17:07 -0300,6965;000000000000-00000000