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An \'etale structure over a topological space $X$ is a continuous family ofstructures (in some first-order language) indexed over $X$. We give anexposition of this fundamental concept from sheaf theory and its relevance tocountable model theory and invariant descriptive set theory. We show that manyclassical aspects of spaces of countable models can be naturally framed andgeneralized in the context of \'etale structures, including the Lopez-Escobartheorem on invariant Borel sets, an omitting types theorem, and variouscharacterizations of Scott rank. We also present and prove the countableversion of the Joyal-Tierney representation theorem, which states that theisomorphism groupoid of an \'etale structure determines its theory up tobi-interpretability; and we explain how special cases of this theorem recoverseveral recent results in the literature on groupoids of models and functorsbetween them.

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