Notions of amalgamation for AECs and categoricity

Abstract

Motivated by the free products of groups, the direct sums of modules, and Shelah's $(\lambda,2)$-goodness, we study strong amalgamation properties in Abstract Elementary Classes. Such a notion of amalgamation consists of a selection of certain amalgams for every triple $M_0\leq M_1, M_2$, and we show that if $K$ designates a unique strong amalgam to every triple $M_0\leq M_1, M_2$, then $K$ satisfies categoricity transfer at cardinals $\geq\theta(K)+2^{\text{LS}(K)}$, where $\theta(K)$ is a cardinal associated with the notion of amalgamation. We also show that if such a unique choice does not exist, then there is some model $M\in K$ having $2^{|M|}$ many extensions which cannot be embedded in each other over $M$. Thus, for AECs which admit a notion of amalgamation, the property of having unique amalgams is a dichotomy property in the sense of Shelah's classification theory.