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 $p^*: \mathcal{S} \to \mathcal{E}$ and $p^!: \mathcal{S} \to \mathcal{E}$ exists, and that it coincides with the least subtopos containing $p^*2$, where 2 denotes the subobject classifier of $\mathcal{S}$.