Decidable objects and molecular toposes

TL;DR

This paper investigates conditions under which geometric morphisms are molecular or locally connected, providing characterizations and applying these to specific toposes, including Gaeta toposes, to deepen understanding of their structural properties.

Contribution

It offers new criteria for molecularity and local-connectedness of geometric morphisms, especially in Boolean toposes, and characterizes reflections inducing molecular maps.

Findings

If al Sa0is a Boolean topos, certain hyperconnected morphisms are molecular.

Characterization of reflections between categories with finite limits that induce molecular maps.

Establishment of molecularity for specific geometric morphisms between Gaeta 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.