Chains of saturated models in AECs

Abstract

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: Theorem 0.1.IfKis a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enoughλ: (1)The union of an increasing chain ofλ-saturated models isλ-saturated.(2)There exists a type-full goodλ-frame with underlying class the saturated models of sizeλ.(3)There exists a unique limit model of sizeλ. Our proofs use independence calculus and a generalization of averages to this non first-order context.