Submission date:

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:
Theorem
If a universal class is categorical in cardinals of arbitrarily high cofinality, then it is categorical on a tail of cardinals.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55 (Secondary)

Keywords and phrases:

Full text arXiv 1512.00060: pdf, ps.