Categoricity and Universal Classes

Abstract

Let $(\mathcal{K} ,\subseteq )$ be a universal class with $LS(\mathcal{K})=\lambda$ categorical in regular $\kappa >\lambda^+$ with arbitrarily large models, and let $\mathcal{K}^*$ be the class of all $\mathcal{A}\in\mathcal{K}_{>\lambda}$ for which there is $\mathcal{B} \in \mathcal{K}_{\ge\kappa}$ such that $\mathcal{A}\subseteq\mathcal{B}$. We prove that $\mathcal{K}^*$ is categorical in every $\xi >\lambda^+$, $\mathcal{K}_{\ge\beth_{(2^{\lambda^+})^+}} \subseteq \mathcal{K}^{*}$, and the models of $\mathcal{K}^*_{>\lambda^+}$ are essentially vector spaces (or trivial i.e. disintegrated).