Coexact Completion of Profinite Heyting Algebras and Uniform Interpolation

This paper shows that the sheaf representation of finitely generated free Heyting algebras constructed by Ghilardi and Zawadowski can be factored as the profinite completion of Heyting algebras, followed by identifying the dual category of profinite Heyting algebras as a full subcategory of a sheaf topos. 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 that they are consequences of the internal logic of the K-topos. Along the way we also establish various topos-theoretic properties of the K-topos.