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.