Categoricity transfer in simple finitary abstract elementary classes

We continue to study nitary abstract elementary classes, de ned in [7]. We introduce a concept of weak κ -categoricity and an f-primary model in a א0 -stable simple nitary AEC with the extension property, and gain the following theorem: Let (K,4K) be a simple nitary AEC, weakly categorical in some uncountable κ . Then (K,4K) is weakly categorical in each λ ≥ min{κ,i(2א0 )+} . We have that if the class (K,4K) is also LS(K) -tame, weak κ -categoricity is equivalent with κ -categoricity in the usual sense. We also discuss the relation between nitary AECs and some other non-elementary frameworks and give several examples.