Vlll A. JOYAL § M. TIERNEY along a continuous map. The category of (extended) spaces and continuous maps is then defined to be the dual of the category of locales. This idea has a fairly long history in the literature, going back at least to Wallman (1938) [20]. For an exhaustive bibliography, see Johnstone [14]. It seems fair to say, however, that these authors were primarily intrigued by the possibility of doing "topology without points" (Papert 1964 [17]) rather than being forced to make the extension as one is in topos theory. We try here to lay a systematic foundation for this study, giving special attention to the concept of an open mapping of spaces, a notion we need for the proof of our principal theorem. Secondly, we find that the difference between a general topos and sheaves on our new notion of space resides precisely in the possibility of action by a spatial groupoid. That is, if G, J G. is a groupoid in the category of (extended) spaces, then we prove that the general notion of topos is captured by considering sheaves on Gn together with a continuous action by G1. Results of this kind should be thought of as a general type of Galois theory (hence the title of the paper), extending Grothendieck's interpretation [9] of classical Galois theory, in which he shows that the 6tale topos of a field k, which is the classifying topos for the theory of separably closed extensions of k, is equivalent to the category of continuous G-sets, where G is the profinite Galois group of a particular separable closure of k. Indeed, this turns out to be a special case of another structure theorem: any connected atomic topos with a point is continuous G-sets for an appropriate spatial group G. Our basic technique is descent theory for morphisms of topoi and locales. Locales are commutative ringlike objects: supremum plays the role of addition, whereas infimum is the product. The structure corresponding to abelian group is sup-lattice. Once this is recognized, it is clear that the notion of module will play a central role. In fact, our first descent theorem for modules is completely analogous to the usual descent theorems of commutative algebra. After treating descent, we establish various facts about locales, and interpret them geometrically in the dual category, i.e. the category of spaces. The principal result states that open surjections of spaces are effective descent morphisms for sheaves. Later we extend the notion of open map to morphisms of topoi, and again prove that open surjections are effective descent morphisms for sheaves. This, together with the observation that any topos can be spatially covered by such a morphism yields the representation theorems. A powerful method for studying a morphism of topoi f: E~ * E, is to regard E~ as a Grothendieck topos relative to E- , . That is, we treat E2 as a category of E-,-valued sheaves on a site in E.. The feasibility of this approach is due to the fact that we know how to interpret such set-theoretical notions as "site" and "sheaf" in E- . . The possibility of interpreting general set-theoretical notions in an arbitrary Grothendieck topos was discovered by Lawvere and Tierney in their attempt to give an elementary axiomatization of the concept of topos. This resulted in the
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