Infinitary stability theory

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: $\mathbf{T}\mathbf{h}\mathbf{e}\mathbf{o}\mathbf{r}\mathbf{e}\mathbf{m}$ (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: $\mathbf{T}\mathbf{h}\mathbf{e}\mathbf{o}\mathbf{r}\mathbf{e}\mathbf{m}$ 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}$(K)})^+}$suchthat$K$isGaloisstableinany$\lambda \ge \lambda_0$with$\lambda = \lambda^{<\mu}$\mathrm{.}$$\mathbf{T}\mathbf{h}\mathbf{e}\mathbf{o}\mathbf{r}\mathbf{e}\mathbf{m}$$Let$K$beafully$(<\kappa)$-tameandtypeshortAECwithamalgamation,$\kappa = \beth_{\kappa} > $\text{LS}$ (K)$\mathrm{.}If$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.