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

TL;DR

This paper offers a concise proof of Shelah's eventual categoricity conjecture for AECs with interpolation under GCH, connecting topos theory and the Scott adjunction from a semantic perspective.

Contribution

It provides a simplified, semantic proof of Shelah's conjecture for AECs with interpolation, extending previous syntactic approaches using topos theory.

Findings

Proof under GCH for AECs with interpolation

Connection established between classifying toposes and Scott adjunction

Simplification of previous syntactic proof approach

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.