Decidable objects and molecular toposes

Abstract

We study several sufficient conditions for the molecularity/local-connectedness of geometric morphisms. In particuar, we show that if $\mathcal{S}$ is a Boolean topos then, for every hyperconnected essential geometric morphism ${p : \mathcal{E} \rightarrow \mathcal{S}}$ such that the leftmost adjoint $p_!$ preserves finite products, $p$ is molecular and ${p^* : \mathcal{S} \rightarrow \mathcal{E}}$ coincides with the full subcategory of decidable objects in $\mathcal{E}$. 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.