Superstability and Symmetry

TL;DR

This paper explores superstability in abstract elementary classes, introducing symmetry over limit models to understand the uniqueness of limit models and its implications for Shelah's Categoricity Conjecture.

Contribution

It introduces a new formulation involving towers to analyze symmetry over limit models in $$-superstable AECs, advancing the understanding of model uniqueness.

Findings

Symmetry over limit models is characterized using towers.

The formulation provides insights into the uniqueness of limit models.

Results contribute to the study of Shelah's Categoricity Conjecture.

Abstract

This paper continues the study of superstability in abstract elementary classes (AECs) satisfying the amalgamation property. In particular, we consider the definition of $\mu$-superstability which is based on the local character characterization of superstability from first order logic. Not only is $\mu$-superstability a potential dividing line in the classification theory for AECs, but it is also a tool in proving instances of Shelah's Categoricity Conjecture. In this paper, we introduce a formulation, involving towers, of symmetry over limit models for $\mu$-superstable abstract elementary classes. We use this formulation to gain insight into the problem of the uniqueness of limit models for categorical AECs.