A characterization theorem for geometric logic

TL;DR

This paper provides a criterion to identify classes of structures as models of geometric theories within Grothendieck toposes, solving a longstanding problem about infinitary first-order theories and their geometric nature.

Contribution

It introduces a new characterization theorem that determines when a class of structures corresponds to models of a geometric theory in Grothendieck toposes, addressing a question from 1989.

Findings

Established a criterion for geometric theories in Grothendieck toposes

Characterized infinitary first-order theories that are geometric

Solved a problem posed by Ieke Moerdijk in 1989

Abstract

We establish a criterion for deciding whether a class of structures is the class of models of a geometric theory inside Grothendieck toposes; then we specialize this result to obtain a characterization of the infinitary first-order theories which are geometric in terms of their models in Grothendieck toposes, solving a problem posed by Ieke Moerdijk in 1989.