Amalgamation from categoricity in universal classes

Sebastien Vasey

Published 2015-06-23Version 1

We prove that, in universal classes, categoricity in a high-enough cardinal implies amalgamation. The proof stems from ideas of Adi Jarden and Will Boney. $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in a high-enough cardinal, then there exists $\lambda$ such that: * $K_{\ge \lambda}$ has amalgamation. * $K_{\ge \lambda}$ is fully good (i.e. it admits a global forking-like notion). We obtain several categoricity transfers: $\mathbf{Corollary}$ Let $K$ be a universal class. * If $K$ is categorical in a high-enough successor cardinal, then $K$ is categorical on a tail of cardinals. * Assume that an unpublished claim of Shelah holds, and that $2^{\lambda} < 2^{\lambda^+}$ for all cardinals $\lambda$. If $K$ is categorical in a high-enough cardinal, then $K$ is categorical on a tail of cardinals. We work in a more general context than universal classes: abstract elementary classes which admit intersections, a notion introduced by Baldwin and Shelah.