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 (orbital) type of length less than a fixed cardinal κ. We show: Theorem 0.1 (The semantic-syntactic correspondence). An AEC K is fully (< κ)-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: Theorem 0.2. Let K be a LS(K)-tame AEC with amalgamation. The following are equivalent: (1) K is Galois stable in some λ ≥ LS(K). (2) K does not have the order property (defined in terms of Galois types). (3) There exist cardinals μ and λ0 with μ ≤ λ0 < i(2LS(K))+ such that K is Galois stable in any λ ≥ λ0 with λ = λ. Theorem 0.3. Let K be a fully (< κ)-tame and type short AEC with amalgamation, κ = iκ > LS(K). If K is Galois stable, then the class of κ-Galois saturated models of K admits an independence notion ((< κ)-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.