A short proof of Shelah's eventual categoricity conjecture for AEC's with interpolation, under $GCH$

Abstract

We provide a short proof of Shelah's eventual categoricity conjecture, assuming the Generalized Continuum Hypothesis ($GCH$), for abstract elementary classes (AEC's) with interpolation, a strengthening of amalgamation which is a necessary and sufficient condition for an AEC categorical in a high enough cardinal to satisfy eventual categoricity. The proof builds on an earlier topos-theoretic argument which was syntactic in nature and recurred to $\kappa$-classifying toposes. We carry out here the same proof idea but from the semantic perspective, making use of a connection between $\kappa$-classifying toposes on one hand and the Scott adjunction on the other hand, this latter developed independently.

We study eventual categoricity with the hypothesis of an interpolation condition. The advantage is that it can be shown to be necessary and sufficient over $ZFC$, since $GCH$ can be removed through forcing arguments. arXiv admin note: text overlap with arXiv:1906.09169