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Relatively Boolean and De Morgan toposes and locales. (English) Zbl 0806.18002

It is well known that the topos of presheaves on a small category \({\mathcal C}\) is Boolean (resp. satisfies De Morgan’s law) iff \({\mathcal C}\) is a groupoid (resp. satisfies the Ore condition). Of course, both these results depend on the Booleanness of the base topos of sets in which the presheaves take values (and in which \({\mathcal C}\) lives); but the authors of this paper felt that they were “too good to be abandoned” over a non-Boolean base topos. Here they introduce “relative” notions of clopen and regular open elements of an internal frame in a topos, and use them to define relative versions of Booleanness and De Morgan’s law for a geometric morphism \(\gamma : {\mathcal E} \to {\mathcal S}\). The latter reduce to the traditional notions (applied to \({\mathcal E})\) in the case when \({\mathcal S}\) is Boolean. They also show that the results quoted above remain valid in the relative setting, when \({\mathcal E}\) is the topos of presheaves on an internal category in \({\mathcal S}\); and they characterize when the conditions hold for the topos of sheaves on an internal locale in \({\mathcal S}\), at least in the case when \(\gamma\) is open.

MSC:

18B25 Topoi
03G30 Categorical logic, topoi
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References:

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