Categoricity transfer for short AECs with amalgamation over sets [article]

Samson Leung
2022 arXiv   pre-print
Let K be an LS( K)-short abstract elementary class and assume more than the existence of a monster model (amalgamation over sets and arbitrarily large models). Suppose K is categorical in some μ>LS( K), then it is categorical in all μ'≥μ. Our result removes the successor requirement of μ 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
more » ... 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 LS( K)-superstability, 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:2203.08956v1 fatcat:racxdwyzsfe5zcro7h2wov2pry