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An application of descent to a classification theorem for toposes. (English) Zbl 0698.18003
In “An extension of the Galois theory of Grothendieck” [Mem. Am. Math. Soc. 309 (1984; Zbl 0541.18002)], A. Joyal and M. Tierney proved a classification theorem for bounded toposes over a base topos \({\mathcal S}\) using descent theory. Their result states that every such topos \({\mathcal E}\) is equivalent to one of the form \({\mathcal B}G\), the topos of √©tale G-spaces, where G is a localic groupoid. These toposes \({\mathcal B}G\) have been called the “classifying topos of G” by I. Moerdijk [“The classifying topos of a continuous groupoid. I, Trans. Am. Math. Soc. 310, No.2, 629-668 (1988)], for if G is a discrete group in the topos of sets, then \({\mathcal B}G\) classifies principal G-bundles. This cannot always be the case, because if G is a connected topological group, then \({\mathcal B}G\) reduces to Sets.
The paper under review considers the case of an arbitrary spatial groupoid G. To G one can associate in a canonical fashion a new spatial groupoid \(\hat G\) such that \({\mathcal B}G\simeq {\mathcal B}\hat G\). Then, it is shown how \({\mathcal B}G\) can be viewed as classifying principal \(\hat G- \)bundles, where the notion of G-bundle is suitably generalized from groups to spatial groupoids. The techniques employed are those of descent theory and the theory of stacks and stack completions. Two different descriptions of the stack completion of G are utilized to analyze the topos \({\mathcal B}G\) and establish the connection with principal bundles.
Reviewer: K.I.Rosenthal

MSC:
18B25 Topoi
18D35 Structured objects in a category (MSC2010)
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
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