\'Etale structures and the Joyal-Tierney representation theorem in countable model theory

Abstract

An \'etale structure over a topological space $X$ is a continuous family of structures (in some first-order language) indexed over $X$. We give an exposition of this fundamental concept from sheaf theory and its relevance to countable model theory and invariant descriptive set theory. We show that many classical aspects of spaces of countable models can be naturally framed and generalized in the context of \'etale structures, including the Lopez-Escobar theorem on invariant Borel sets, an omitting types theorem, and various characterizations of Scott rank. We also present and prove the countable version of the Joyal-Tierney representation theorem, which states that the isomorphism groupoid of an \'etale structure determines its theory up to bi-interpretability; and we explain how special cases of this theorem recover several recent results in the literature on groupoids of models and functors between them.