Läuchli's Completeness Theorem from a Topos-Theoretic Perspective

We prove a variant of Läuchli’s completeness theorem for intuitionistic predicate calculus. The formulation of the result relies on the observation (due to Lawvere) that Läuchli’s theorem is related to the logic of the canonical indexing of the atomic topos of Z-sets. We show that the process that transforms Kripke-counter-models into Läuchli-counter-models is (essentially) the inverse image of a geometric morphism. Completeness follows because this geometric morphism is an open surjection.