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This paper continues the study of superstability in abstract elementary classes (AECs) satisfying the amalgamation property. In particular, we consider the definition of μ -superstability which is based on the local character characterization of superstability from first order logic. Not only is μ -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 μ -superstable abstract elementary classes. We use this formulation to gain insight into the problem of the uniqueness of limit models for categorical AECs. [ABSTRACT FROM AUTHOR]

Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for nonsplitting, a particular notion of independence. We generalize their result as follows: given any abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah–Villaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah–Villaveces proof. [ABSTRACT FROM AUTHOR]