The Classifying Topos of a Continuous Groupoid. I

We investigate some properties of the functor B which associates to any continuous groupoid G its classifying topos BG of equivariant G-sheaves. In particular, it will be shown that the category of toposes can be obtained as a localization of a category of continuous groupoids. If G is a group, the category of G-sets (sets equipped with a right G-action) is a topos BG, which classifies principal G-bundles: for instance, if A is a topological space there is an equivalence between topos morphisms Sheaves(A) —♦ BG and principal G-bundles over X. The construction of BG also applies to the case where G is a topological group, or more generally, a topological groupoid. It is a rather surprising result that this essentially exhausts the range of toposes: Joyal and Tierney (1984) have shown that any topos is equivalent to one of the form BG for a topological group G, provided one works with a slightly generalized notion of topological space, by taking the lattice of open sets as the primitive notion, rather than the set of points (one sometimes speaks of "pointless" spaces). The continuous groupoids of this paper are the groupoid objects in this category of generalized spaces. The aim of this paper is threefold. First, G >-> BG is a functor, and we wish to investigate how the properties of the topos BG depend on those of the continuous groupoid G, and more generally how the properties of a geometric morphism BG-► BH depend on those of the map of continuous groupoids G —► H. The second aim is to extend the Joyal-Tierney result, and not only represent toposes in terms of continuous groupoids, but also the geometric morphisms from one topos to another. There are several possible solutions to this problem. In this paper, I present one approach, and show that the category of toposes can be obtained as a category of fractions from a category of continuous groupoids. Another approach, somewhat similar in spirit to the Morita theorems for categories of modules, will be presented elsewhere. The third aim of this paper is of a more methodological nature: in presenting many arguments concerning generalized, "pointless" spaces, I have tried to convey the idea that by using change-of-base techniques and exploiting the internal logic of a Grothendieck topos, point-set arguments are perfectly suitable for dealing with pointless spaces (at least as long as one stays within the "stable" part of the theory). Although the general underlying idea is very clear (see e.g. the discussion in 5.3 below), it is a challenging open problem to express this Received by the editors December 8, 1986 and, in revised form, July 14, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 18B35, 18B40, 18F10, 18F20.