Categoricity and amalgamation for AEC and $ \kappa $ measurable

Abstract

In the original version of this paper, we assume a theory $T$ that the logic $\mathbb L_{\kappa, \aleph_{0}}$ is categorical in a cardinal $\lambda > \kappa$, and $\kappa$ is a measurable cardinal. There we prove that the class of model of $T$ of cardinality $<\lambda$ (but $\geq |T|+\kappa$) has the amalgamation property; this is a step toward understanding the character of such classes of models. In this revised version we replaced the class of models of $T$ by $\mathfrak k$, an AEC (abstract elementary class) which has LS-number ${<} \, \kappa,$ or at least which behave nicely for ultrapowers by $D$, a normal ultra-filter on $\kappa$. Presently sub-section \S1A deals with $T \subseteq \mathbb L_{\kappa^{+}, \aleph_{0}}$ (and so does a large part of the introduction and little in the rest of \S1), but otherwise, all is done in the context of AEC.