De Morgan classifying toposes

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. © 2009 Elsevier Inc. All rights reserved

Atomic Toposes and Countable Categoricity

“…Proof As it is remarked in [3], the Booleanization T of T axiomatizes the homogeneous T-models. If T has no models in Set then our thesis is trivially true, so we may suppose that T has at least one model in Set.…”

“…Remark 3.7 We recall from [3] that a geometric theory T over a signature is a Boolean if and only if for every geometric formula φ(x) over there exists a geometric formula ψ(x) over in the same context, denoted ¬φ(x), such that…”

“…Proof Consider the Booleanization T of the theory T (as it was defined in [3]). T is a geometric theory over , and our hypotheses say precisely that T has a model M in Set.…”

Universal models and definability

“…De Morgan) that we proved in [4] is expressed in terms of the 'reduced site' (C, J |C), whereC is the full subcategory of C on the objects which are not J -covered by the empty sieve, and asserts that Sh(C, J ) is Boolean (resp. De Morgan) if and only if J |C = DC (resp.…”

“…Now, let us recall from [4] that the Booleanization (resp. DeMorganization) of an elementary topos E is the largest dense Boolean subtopos of E (resp.…”