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We extend Freyd's construction to all étale-finite Heyting algebras, in the sense of Evgeny Kuznetsov. These are the Heyting algebras satisfying a generalisation of the law of excluded middle relative to some finite Heyting subalgebra. For every étale-finite Heyting algebra $H$, we use Esakia duality to construct an elementary topos whose lattice of truth values is isomorphic to $H$, thereby extending the class of Heyting algebras for which a positive answer to Pitts' question is known.
The toposes we construct are categories of certain compact étale spaces. As a consequence, they are finitely propositional: every object has a finite cover by subterminal objects. We show that a Heyting algebra occurs as the lattice of truth values of some finitely propositional topos if and only if it is étale-finite. This exhibits an obstruction to extending our use of compact étale spaces beyond the étale-finite case.
From: Igor Arrieta [view email]
[v1]
Tue, 2 Jun 2026 16:34:52 UTC (5,805 KB)
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