Categoricity and multidimensional diagrams

TL;DR

This paper proves the eventual categoricity conjecture for abstract elementary classes using multidimensional diagrams, under assumptions of large cardinal axioms or the generalized continuum hypothesis, advancing understanding of model uniqueness in high cardinalities.

Contribution

It introduces the use of multidimensional diagrams in AECs to prove the categoricity conjecture under large cardinal and GCH assumptions, providing new tools for model theory.

Findings

Proves categoricity in high cardinalities assuming large cardinals.

Establishes categoricity under GCH for AECs with amalgamation.

Uses multidimensional diagrams to analyze model uniqueness.

Abstract

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.