{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/infinitary-stability-theory","title":"Infinitary stability theory","arxiv_id":"1412.3313","date":"2014-12-10","proceeding":null,"authors":["Sebastien Vasey"],"abstract":"We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois type of length less than a fixed cardinal $\\kappa$. We show: $\\mathbf{Theorem}$ (The semantic-syntactic correspondence) An AEC $K$ is fully $(<\\kappa)$-tame and type short if and only if Galois types are syntactic in the Galois Morleyization. This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are: $\\mathbf{Theorem}$ Let $K$ be a $\\text{LS}(K)$-tame AEC with amalgamation. The following are equivalent: * $K$ is Galois stable in some $\\lambda \\ge \\text{LS}(K)$. * $K$ does not have the order property (defined in terms of Galois types). * There exist cardinals $\\mu$ and $\\lambda_0$ with $\\mu \\le \\lambda_0 < \\beth_{(2^{\\text{LS}(K)})^+}$ such that $K$ is Galois stable in any $\\lambda \\ge \\lambda_0$ with $\\lambda = \\lambda^{<\\mu}$. $\\mathbf{Theorem}$ Let $K$ be a fully $(<\\kappa)$-tame and type short AEC with amalgamation, $\\kappa = \\beth_{\\kappa} > \\text{LS} (K)$. If $K$ is Galois stable, then the class of $\\kappa$-Galois saturated models of $K$ admits an independence notion ($(<\\kappa)$-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.","url_abs":"http://arxiv.org/abs/1412.3313v6","url_pdf":"http://arxiv.org/pdf/1412.3313v6.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"infinitary-stability-theory","repo_url":"https://github.com/julianmendez/tabulas","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}