{"paper":{"title":"On the structure of categorical abstract elementary classes with amalgamation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Monica M. VanDieren, Sebastien Vasey","submitted_at":"2015-09-04T15:14:34Z","abstract_excerpt":"For $K$ an abstract elementary class with amalgamation and no maximal models, we show that categoricity in a high-enough cardinal implies structural properties such as the uniqueness of limit models and the existence of good frames. This improves several classical results of Shelah.\n  $\\mathbf{Theorem}$\n  Let $\\mu \\ge \\text{LS} (K)$. If $K$ is categorical in a $\\lambda \\ge \\beth_{\\left(2^{\\mu}\\right)^+}$, then:\n  1) Whenever $M_0, M_1, M_2 \\in K_\\mu$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \\cong_{M_0} M_2$.\n  2) If $\\mu > \\text{LS} (K)$, the model of size $\\lambda$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01488","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}