On Supercompactly and Compactly Generated Toposes

TL;DR

This paper characterizes Grothendieck toposes with sufficient supercompact or compact objects, exploring their subcategories, related morphisms, and canonical site classes, advancing understanding of their structural properties.

Contribution

It introduces a detailed classification of toposes based on supercompact and compact objects, including their subcategories and generating sites, providing new insights into their structure.

Findings

Characterization of toposes with enough supercompact objects

Characterization of toposes with enough compact objects

Identification of classes of sites generating these toposes

Abstract

We present and characterize the classes of Grothendieck toposes having enough supercompact objects or enough compact objects. In the process, we examine the subcategories of supercompact objects and compact objects within such toposes and classes of geometric morphism which interact well with these objects. We also present canonical classes of sites generating such toposes.