Categoricity transfer in simple finitary abstract elementary classes

Abstract

We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of ℵ₀-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let (𝕂,≼_𝕂) be a simple finitary AEC, weakly categorical in some uncountable κ. Then (𝕂,≼_𝕂) is weakly categorical in each λ≥min{κ,ℶ_(2^ℵ0)⁺}. If the class (𝕂,≼_𝕂) is also LS(𝕂)-tame, weak κ-categoricity is equivalent with κ-categoricity in the usual sense.

We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.