2-filteredness and the point of every Galois topos

TL;DR

This paper investigates the structure of Galois objects within locally connected topoi, demonstrating that they form an inversely 2-filtered 2-category and establishing that every Galois topos possesses a point.

Contribution

It introduces the concept of Galois objects forming an inversely 2-filtered 2-category and proves that all Galois topoi have a point, advancing the understanding of their structural properties.

Findings

Galois objects form an inversely 2-filtered 2-category

Every Galois topos has a point

Construction of 2-filtered bi-limits of topoi is applied

Abstract

A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bi-limits of topoi, we show that every Galois topos has a point.