Site characterizations for geometric invariants of toposes

TL;DR

This paper develops techniques to characterize geometric invariants of Grothendieck toposes via their sites, providing explicit criteria for properties like being localic, atomic, locally connected, or equivalent to a presheaf topos.

Contribution

It introduces general methods for translating topos invariants into site-based criteria and applies these to several key properties, offering explicit characterizations.

Findings

Explicit site characterizations for localic, atomic, and locally connected toposes.

Methodologies for establishing site-based criteria for topos invariants.

Application of techniques to identify when a topos is equivalent to a presheaf topos.

Abstract

We discuss the problem of characterizing the property of a Grothendieck topos to satisfy a given 'geometric' invariant as a property of its sites of definition, and indicate a set of general techniques for establishing such criteria. We then apply our methodologies to specific invariants, notably including the property of a Grothendieck topos to be localic (resp. atomic, locally connected, equivalent to a presheaf topos), obtaining explicit site characterizations for them.