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 $\ge |T|+\kappa$) 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.

We now use the framework of AECs rather than models of a fixed theory