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**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.

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

### MSC:

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) |