A Characterization of Uniqueness of Limit Models in Categorical Abstract Elementary Classes

TL;DR

This paper investigates the conditions under which limit models are unique in certain categorical abstract elementary classes, replacing previous assumptions with a weaker condition and establishing its necessity for specific cardinalities.

Contribution

It introduces a weaker assumption about unions of chains of limit models and proves its necessity and sufficiency for uniqueness in certain cases.

Findings

The union of an increasing and continuous chain of limit models is an amalgamation base.

The weaker assumption is both necessary and sufficient for limit model uniqueness when mbda=mbda^{+n} for 0<n<mbda.

The approach clarifies the conditions needed for the uniqueness of limit models in categorical AECs.

Abstract

In this paper we examine the task set forth by Shelah and Villaveces in \cite{ShVi} of proving the uniqueness of limit models of cardinality $\mu$ in $\lambda$-categorical abstract elementary classes with no maximal models, where $\lambda$ is some cardinal larger than $\mu$. In \cite{Va} and \cite{Va-errata} we identified several gaps in the approach outlined in \cite{ShVi}, and we added the assumption that the union of an increasing chain of limit models is a limit model. Here we replace this assumption with the seemingly weaker statement that the union of an increasing and continuous chain of limit models is an amalgamation base. Moreover, we prove that this assumption is not only sufficient but is necessary to settle the uniqueness of limit models problem attempted in \cite{ShVi} for $\lambda=\mu^{+n}$ when $0<n<\omega$.