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

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 $μ$ in $λ$-categorical abstract elementary classes with no maximal models, where $λ$ is some cardinal larger than $μ$. 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 $λ=μ^{+n}$ when $0<n<ω$.