Categoricity transfer for short AECs with amalgamation over sets

Abstract

Let $\mathbf{K}$ be an $\mathrm{LS}(\mathbf{K})$-short abstract elementary class and assume more than the existence of a monster model (amalgamation over sets and arbitrarily large models). Suppose $\mathbf{K}$ is categorical in some $\mu >\mathrm{LS}(\mathbf{K})$, then it is categorical in all ${\mu }^{\prime }\ge \mu$. Our result removes the successor requirement of $\mu$ made by Grossberg-VanDieren, at the cost of using shortness instead of tameness; and of using amalgamation over sets instead of over models. It also removes the primes requirement by Vasey which assumes tameness and amalgamation over models. As a corollary, we obtain an alternative proof of the upward categoricity transfer for first-order theories by Morley and Shelah. In our construction, we simplify Vasey's results to build a weakly successful frame. This allows us to use Shelah-Vasey's argument to obtain primes for sufficiently saturated models. If we replace the categoricity assumption by $\mathrm{LS}(\mathbf{K})$-superstability, $\mathbf{K}$ is already excellent for sufficiently saturated models. This sheds light on the investigation of the main gap theorem for uncountable first-order theories within ZFC.

arXiv comment34 pages