The Gleason Cover of a Realizability Topos

Recently Benno van den Berg [1] introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the ‘constant object’ functor from the topos of sets preserves finite coproducts, and that De Morgan’s law is satisfied. In this paper we show that these properties are no accident: for any Schönfinkel algebra Λ, the Herbrand realizability topos over Λ may be obtained as the Gleason cover (in the sense of [8]) of the ordinary realizability topos over Λ. As a corollary, we obtain the functoriality of the Herbrand realizability construction on the category of Schönfinkel algebras and computationally dense applicative morphisms.