Categoricity Transfer in Abstract Elementary Classes and Shelah's Main Gap Theorem

This monograph investigates the structural properties of Abstract Elementary Classes (AECs) with a specific focus on Shelah's Categoricity Conjecture and its interplay with the Main Gap Theorem. We analyze the conditions under which categoricity in a high-enough cardinal transfers to all subsequent cardinals, employing the machinery of Galois types (orbital types), tameness, and good -frames. We provide a rigorous examination of the stability spectrum and the existence of prime models within the context of tame AECs satisfying the amalgamation property. Furthermore, we explore the dichotomy between structure and non-structure theorems, extending the classical first-order Main Gap results to the non-elementary context. The paper culminates in a synthesis of recent advancements by Grossberg, VanDieren, and Shelah, establishing a partial upward categoricity transfer theorem under the assumption of -tameness and type-shortness.