Atomic toposes and countable categoricity

TL;DR

This paper characterizes geometric theories classified by atomic toposes with enough points, showing that such complete theories are countably categorical, and discusses related applications in model theory.

Contribution

It provides a model-theoretic characterization linking atomic toposes with countable categoricity of geometric theories, expanding understanding of their structure.

Findings

Complete geometric theories classified by atomic toposes are countably categorical.

Atomic toposes with enough points correspond to specific model-theoretic properties.

Applications of these characterizations are discussed.

Abstract

We give a model-theoretic characterization of the class of geometric theories classified by an atomic topos having enough points; in particular, we show that every complete geometric theory classified by an atomic topos is countably categorical. Some applications are also discussed.