Superstability and symmetry

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.