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arXiv · 2606.03861

A topos for \'etale-finite Heyting algebras

Marco Abbadini·Rodrigo Nicolau Almeida·Igor Arrieta
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Abstract

A longstanding open problem is whether every Heyting algebra is the lattice of truth values (i.e., of subterminal objects) of some elementary topos. A positive answer is known for complete Heyting algebras (i.e., locales) via sheaves, and for Boolean algebras via a construction due to Peter Freyd. We extend Freyd's construction to all \'etale-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 \'etale-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 the Heyting-to-topos problem is known. The toposes we construct are categories of certain compact \'etale 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 \'etale-finite. This exhibits an obstruction to extending the use of compact \'etale spaces beyond the \'etale-finite case.

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BibTeXRIS

Marco Abbadini, Rodrigo Nicolau Almeida, Igor Arrieta. 2026-06-02. A topos for \'etale-finite Heyting algebras. https://arxiv.org/abs/2606.03861

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