On the representation theory of Galois and atomic topoi

Abstract

In this paper we consider Galois theory as it was interpreted by Grothendieck in SGA1 (Lecture Notes in Mathematics 224 (1971)) and SGA4 (Lecture Notes in Mathematics 269 (1972)) and later extended by Joyal–Tierney in Memoirs of AMS 151 (1984). Grothendieck conceived Galois theory as the axiomatic characterization of the classifying topos of a progroup in terms of a representation theorem for pointed Galois Topoi. Joyal–Tierney extended this to the axiomatic characterization of the classifying topos of a localic group in terms of a representation theorem for pointed Atomic Topoi. Classical Galois theory corresponds to discrete groups (the point is essential), and the representation theorem can be proved by elementary category-theory. This was developed by Barr–Diaconescu (Cahiers Top. Geo. Diff. cat 22-23 (1981) 301). Grothendieck theory corresponds to progroups or prodiscrete localic groups (the point is proessential, a concept we introduce in this paper), and the representation theorem is proved by inverse limit of topoi techniques. This was developed by Moerdijk (Proc. Kon. Nederl. Akad. van Wetens. Series A 92 (1989)). Joyal–Tierney theory corresponds to general localic groups (the point is a general point), and the representation theorem is proved by descent techniques. It can also be proved by the methods of localic Galois theory developed by Dubuc (Advances in Mathematics 175 (2003)). Joyal–Tierney also consider the case of a general localic groupoid (in particular, it includes unpointed Atomic Topoi), which needs a sophisticated change of base. Bunge (Category Theory ’91, CMS Conf. Proc. 13 (1992)) and Kennison (J. Pure Appl. Algeb. 77 (1992)) consider in particular the case of prodiscrete groupoids, and develop an unpointed Grothendieck theory. We consider these contributions, make an original description, development and survey of the whole theory (but do not touch the representation of cohomology aspects), and present our own results.