Sufficient cohesion over atomic toposes

Abstract

Let (D, Jat) be an atomic site and j : Sh(D, Jat) → Db be the associated sheaf topos. Any functor φ : C → D induces a geometric morphism C →b Db and, by pulling-back along j, a geometric morphism q : F → Sh(D, Jat). We give a sufficient condition on φ for q to satisfy the Nullstellensatz and Sufficient Cohesion in the sense of Axiomatic Cohesion. This is motivated by the examples where Dop is a category of finite field extensions.