Categoricity transfer for short AECs with amalgamation over sets

TL;DR

This paper proves that in certain short abstract elementary classes with amalgamation over sets, categoricity in one high cardinal implies categoricity in all higher cardinals, generalizing and simplifying previous results.

Contribution

It removes the successor cardinal requirement for categoricity transfer by using shortness and amalgamation over sets, providing a new proof approach and insights into the main gap theorem.

Findings

Categoricity transfers upward without successor assumption

Simplifies Vasey's construction of primes for saturated models

Provides an alternative proof of Morley's categoricity theorem

Abstract

Let ${\bf K}$ be an $\mathrm{LS}({\bf K})$-short abstract elementary class and assume more than the existence of a monster model (amalgamation over sets and arbitrarily large models). Suppose ${\bf K}$ is categorical in some $\mu>\mathrm{LS}({\bf K})$, then it is categorical in all $\mu'\geq\mu$. 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 $\mathrm{LS}({\bf K})$-superstability, ${\bf K}$ 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.