Decidable objects and molecular toposes

Abstract

We study several sufficient conditions for the molecularity/localconnectedness of geometric morphisms. In particular, we show that if S is a Boolean topos, then, for every hyperconnected essential geometric morphism p : E → S such that the leftmost adjoint p! preserves finite products, p is molecular and p ∗ : S → E coincides with the full subcategory of decidable objects in E. We also characterize the reflections between categories with finite limits that induce molecular maps between the respective presheaf toposes. As a corollary we establish the molecularity of certain geometric morphisms between Gaeta toposes.