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

TL;DR

This paper investigates the minimal subtopos in a pre-cohesive topos that contains both the direct and inverse image subcategories, revealing its equivalence to the subtopos generated by the subobject classifier.

Contribution

It establishes the existence and characterization of the least subtopos containing specific subcategories in a pre-cohesive geometric morphism.

Findings

The least subtopos containing both $p^*$ and $p^!$ exists.

This subtopos coincides with the one generated by the subobject classifier 2.

The result clarifies the structure of subtopoi in pre-cohesive toposes.

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 $\mathcal{S}$.