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 ${\mathcal{K}}^{\ast }$ 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 ${\mathcal{K}}^{\ast }$ is categorical in every $\xi >{\lambda }^{+}$, ${\mathcal{K}}_{\ge {\beth }_{(2^{{\lambda }^{+}}{)}^{+}}}\subseteq {\mathcal{K}}^{\ast }$, and the models of ${\mathcal{K}}_{>{\lambda }^{+}}^{\ast }$ are essentially vector spaces (or trivial i.e. disintegrated).