An extension of the Galois theory of Grothendieck. (English) Zbl 0541.18002

Mem. Am. Math. Soc. 309, 71 p. (1984).
Grothendieck has formulated classical Galois theory as equivalence of categories: the étale topos over a field k is equivalent to the category of topological spaces with G-action, where G is the Galois group of the separable closure of k. This can be generalized by taking for G a groupoid \(G_ 1\rightrightarrows G_ 0\) (the arrows are the domain and codomain). Moreover, for a general topos, the notion of space has to be extended.
Topological spaces are given locally by their lattices of open sets, which is here abstracted to arbitrary complete lattices with distributive property, called locales, and so the category of extended spaces is defined as the dual of that of locales. Then, for any topos, there is a groupoid \(G_ 1\rightrightarrows G_ 0\) in the category of extended spaces such that the given topos is equivalent to that of sheaves on \(G_ 0\) with \(G_ 1\)-action.
For demonstration a descent theory for locales is developed. For instance, the notion of an open mapping of extended spaces (and also of topoi) is defined, and it is shown that open surjections are effective descent morphisms for sheaves.
Reviewer: J.H.de Boer


18B25 Topoi
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
12F10 Separable extensions, Galois theory
18B30 Categories of topological spaces and continuous mappings (MSC2010)
14F20 Étale and other Grothendieck topologies and (co)homologies
18F10 Grothendieck topologies and Grothendieck topoi
06D99 Distributive lattices
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
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