Notions of Amalgamation for Abstract Elementary Classes

Motivated by the free products of groups, the direct sums of modules, and Shelah’s (λ, 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 M₀ ≤ M₁,M₂, and we show that if a weak AEC K designates a unique strong amalgam to every triple M₀ ≤ M₁,M₂, then K satisfies categoricity transfer at cardinals ≥ θ(K) + 2^LS(K), where θ(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 ∈ 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.

We present a framework of a “notion of amalgamation” for a given abstract elementary class. Abstracting from the examples of free amalgamation of groups and direct sum of modules, we isolate the axiomatic properties of absolute minimality, regularity, continuity, and admitting decomposition (Definition 2.6), which we assume throughout the paper. We also define the uniqueness property of amalgams, which intuitively states that for any triple of models there is a unique amalgam (up to isomorphism) which is “nice”. We refer to a notion satisfying all of the above as a notion of free amalgamation, and establish that when a class K has a notion of free amalgamation and is categorical in a sufficiently large cardinal, then it behaves analogously to the models of a unidimensional first order theory.