On the structure of categorical abstract elementary classes with amalgamation

Abstract

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. $\mathbf{Theorem}$ Let $\mu \ge \text{LS}(K)$. If $K$ is categorical in a $\lambda \ge {\beth }_{{\left(2^{\mu }\right)}^{+}}$, then: 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 $M_1{\cong }_{M_0}{M}_2$. 2) If $\mu >\text{LS}(K)$, the model of size $\lambda$ is $\mu$-saturated. 3) If $\mu \ge {\beth }_{(2^{\text{LS}(K)}{)}^{+}}$ and $\lambda \ge {\beth }_{{\left(2^{{\mu }^{+}}\right)}^{+}}$, then there exists a type-full good $\mu$-frame with underlying class the saturated models in $K_{\mu }$. Our main tool is the symmetry property of splitting (previously isolated by the first author). The key lemma deduces symmetry from failure of the order property.

19 pages. This has since been merged with arXiv:1508.03252

Cited by in corpus (1)

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