Coexact completion of profinite Heyting algebras and uniform interpolation

2026-04-09

AI Summary

This study investigates the sheaf representation of finitely presented Heyting algebras from an algebraic perspective and its connection to profinite completion, extending the uniform interpolation property. By constructing the dual category of profinite Heyting algebras, the authors demonstrate that its exact/regular (ex/reg) completion coincides with Ghilardi and Zawadowski’s K-topos, thereby establishing an algebraic equivalence between profinite completion and sheaf representation. Leveraging the internal logic of the K-topos, the uniform interpolation property is generalized to arbitrary profinite Heyting algebras, revealing its semantic origin in the structure of the K-topos. The work also provides a foundational characterization of the K-topos as an infinitary extensive regular category, offering a novel categorical semantic framework for uniform interpolation.

Abstract

This paper shows that the sheaf representation of finitely presented Heyting algebras constructed by Ghilardi and Zawadowski is, from an algebraic perspective, equivalent to the construction of profinite completion. We show that the dual category of profinite Heyting algebras is an infinitary extensive regular category, and its ex/reg-completion is exactly the aforementioned sheaf topos, which we refer to as the K-topos. We show how certain properties of uniform interpolation can be generalised to the context of arbitrary profinite Heyting algebras, and are consequences of the internal logic of the K-topos. Along the way we also establish various topos-theoretic properties of the K-topos.

Problem

• profinite Heyting algebras
• sheaf representation
• uniform interpolation
• coexact completion
• K-topos

Innovation

• profinite completion
• Heyting algebras
• sheaf topos
• uniform interpolation
• ex/reg-completion