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Boolean Galois theories. (English) Zbl 1031.18003
Let \((I,H,\eta, \varepsilon):{\mathcal C}\to{\mathcal X}\) be an adjunction between categories with pullbacks where \(I:{\mathcal C}\to{\mathcal X}\) is left adjoint to \(H:{\mathcal X}\to{\mathcal C}\). Let \(p:E\to B\) be a given fixed effective descent morphism in \({\mathcal C}\). The adjunction induces an adjunction \((I^E,H^E,\eta^E, \varepsilon^E):{\mathcal C}\downarrow E\to{\mathcal X}\downarrow I(E)\) between slices categories with \(I^E:\mathbb{C}\downarrow E\to \mathbb{X}\downarrow I(E)\) defined by \(I^E(A,\alpha)= (I^E(A), I^E(\alpha))\) and \(I^E(f)= I(f)\). The object \(E\) is said to be \(I\)-admissible if \(\varepsilon^E\) is an isomorphism. Different constructions and examples of admissible objects are given in the paper and, in particular in the so-called “geometrical categories” where they are related to connected objects. What is called: “The fundamental theorem of Galois theory” in this paper, is expressed as an equivalence of categories, which, under the above admissibility condition, can be written as \({\mathcal S} pl_I(E,p)\simeq{\mathcal X}^{\text{Gal}_I(E,p)}\) where
– \({\mathcal S} pl_I(E,p)\) is the full subcategory of objects \((A,\alpha)\) in \({\mathcal C}\downarrow B\) split over \(p\),
– \({\mathcal X}^{\text{Gal}_I(E, p)}\) is the category of internal actions of the profinite Galois pregroupoid \(\text{Gal}_I(E,p)\) of \((E,p)\).
All the notions which are used are defined and illustrated and some examples are given.
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18A22 Special properties of functors (faithful, full, etc.)
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
18A25 Functor categories, comma categories
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
06E15 Stone spaces (Boolean spaces) and related structures
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