Revista de la
Unión Matemática Argentina

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Vol. 67, no. 2 (2024)

Decidable objects and molecular toposes. Matías Menni
We study several sufficient conditions for the molecularity/local-connectedness of geometric morphisms. In particular, 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.