Topological completeness for higher-order logic

Abstract: "Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces -- so-called 'topological semantics'. The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic."