Existentially closed models and locally zero-dimensional toposes

TL;DR

This paper generalizes the concept of existentially closed models to geometric morphisms between toposes, exploring their properties, differences from classical notions, and classifying toposes in the new setting.

Contribution

It extends the notion of existentially closed models to topos theory, analyzing properties, differences, and classifying toposes of these generalized morphisms.

Findings

Existentially closed geometric morphisms exist for all models.

Classical equivalences of definitions do not hold in the topos-theoretic generalization.

Characterization of classifying toposes when conditions coincide.

Abstract

The notion of an existentially closed model is generalised to a property of geometric morphisms between toposes. We show that important properties of existentially closed models extend to existentially closed geometric morphisms, such as the fact that every model admits a homomorphism to an existentially closed one. Other properties do not generalise: classically, there are two equivalent definitions of an existentially closed model, but this equivalence breaks down for the generalised notion. We study the interaction of these two conditions on the topos-theoretic level, and characterise the classifying topos of the e.c. geometric morphisms when the conditions coincide.