Notions of amalgamation for AECs and categoricity

TL;DR

This paper explores strong amalgamation properties in Abstract Elementary Classes, showing that uniqueness of amalgams leads to categoricity transfer, while non-uniqueness results in many non-embeddable extensions, establishing a dichotomy in classification theory.

Contribution

It introduces a notion of strong amalgamation in AECs, proves that uniqueness implies categoricity transfer, and demonstrates a dichotomy between unique and non-unique amalgamation.

Findings

Unique amalgamation implies categoricity transfer at large cardinals.

Non-unique amalgamation leads to models with many non-embeddable extensions.

The property of having unique amalgams is a dichotomy in Shelah's classification theory.

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.