TL;DR
This paper introduces a method to determine whether a Grothendieck topos satisfies De Morgan's law or the law of excluded middle, providing syntactic and model-theoretic criteria for classifying toposes.
Contribution
It offers a general decision procedure for De Morgan's law in toposes and characterizes theories with classifying toposes satisfying these logical laws.
Findings
A syntactic characterization of theories with De Morgan classifying toposes
Model-theoretic criteria for localizations of presheaf toposes
Applications to classifying toposes of geometric theories
Abstract
We present a general method for deciding whether a Grothendieck topos satisfies De Morgan's law (resp. the law of excluded middle) or not; applications to the theory of classifying toposes follow. Specifically, we obtain a syntactic characterization of the class of geometric theories whose classifying toposes satisfy De Morgan's law (resp. are Boolean), as well as model-theoretic criteria for theories whose classifying toposes arise as localizations of a given presheaf topos.
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