Infinitary stability theory

TL;DR

This paper introduces Galois Morleyization to connect AECs with syntactic stability theory, enabling new results on stability, tameness, and independence in abstract elementary classes.

Contribution

It establishes a semantic-syntactic correspondence for tame AECs and advances stability theory within this framework using Galois Morleyization.

Findings

Galois types are syntactic in the Galois Morleyization for fully tame and type short AECs.

Equivalence of Galois stability, absence of order property, and stability in certain cardinals.

Existence of an independence notion similar to forking in stable first-order theories for Galois saturated models.

Abstract

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois type of length less than a fixed cardinal $\kappa$. We show: $\mathbf{Theorem}$ (The semantic-syntactic correspondence) An AEC $K$ is fully $(<\kappa)$-tame and type short if and only if Galois types are syntactic in the Galois Morleyization. This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are: $\mathbf{Theorem}$ Let $K$ be a $\text{LS}(K)$-tame AEC with amalgamation. The following are equivalent: * $K$ is Galois stable in some $\lambda \ge \text{LS}(K)$. * $K$ does not have the order property (defined in terms of Galois types). * There exist cardinals $\mu$ and $\lambda_0$ with $\mu \le \lambda_0 < \beth_{(2^{\text{LS}(K)})^+}$ such that $K$ is Galois stable in any $\lambda \ge \lambda_0$ with $\lambda = \lambda^{<\mu}$. $\mathbf{Theorem}$ Let $K$ be a fully $(<\kappa)$-tame and type short AEC with amalgamation, $\kappa = \beth_{\kappa} > \text{LS} (K)$. If $K$ is Galois stable, then the class of $\kappa$-Galois saturated models of $K$ admits an independence notion ($(<\kappa)$-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.