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An étale 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 étale 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 étale 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.
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A Measurable Selector in Kadison’s Carpenter’s Theorem
Abstract: We show the existence of a measurable selector in Carpenter’s Theorem due to Kadison. This solves a problem posed by Jasper and the first author in an earlier work. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of $$L^{2}(\mathbb{R}^{d})$$ and Carpenter’s Theorem for type $$\text{I}_{\infty }$$ von Neumann algebras.
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- Award ID(s):
- 1665056
- PAR ID:
- 10167277
- Date Published:
- Journal Name:
- Canadian Journal of Mathematics
- ISSN:
- 0008-414X
- Page Range / eLocation ID:
- 1 to 24
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4153/S0008414X19000373
