The least subtopos containing the discrete skeleton of $\Omega$

Abstract

Let $p:\mathcal{E}\to \mathcal{S}$ be a pre-cohesive geometric morphism. We show that the least subtopos of $\mathcal{E}$ containing both the subcategories $p^{\ast }:\mathcal{S}\to \mathcal{E}$ and $p^{!}:\mathcal{S}\to \mathcal{E}$ exists, and that it coincides with the least subtopos containing $p^{\ast }2$, where 2 denotes the subobject classifier of $\mathcal{S}$.