A survey on tame abstract elementary classes

TL;DR

This survey reviews recent advances in the classification theory of tame abstract elementary classes, highlighting the role of independence relations and categoricity transfer results in understanding their structure.

Contribution

It compiles and discusses recent progress in developing a classification theory for tame AECs, emphasizing the importance of Shelah's good frames and categoricity transfer theorems.

Findings

Categoricity in high cofinality cardinals implies categoricity on a tail of cardinals.

Development of independence relations like Shelah's good frames is central to the theory.

Several new categoricity transfer results have been established for tame AECs.

Abstract

Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification theory for them have been made. Abstract independence relations such as Shelah's good frames have been found to be key objects. Several new categoricity transfers have been obtained. We survey these developments using the following result (due to the second author) as our guiding thread: $\mathbf{Theorem}$ If a universal class is categorical in cardinals of arbitrarily high cofinality, then it is categorical on a tail of cardinals.