Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $ -nonsplitting. The following holds: 1.When $\mu $ -nonforking is restricted to $(\mu,\geq \chi )$ -limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension, and continuity. It also has local character $\chi $. This generalizes (...) Vasey’s result [37, Corollary 13.16] which assumed $\mu $ -superstability to obtain same properties but with local character $\aleph _0$.2.There is $\lambda \in [\mu,h(\mu ))$ such that if $\mathbf {K}$ is stable in every cardinal between $\mu $ and $\lambda $, then $\mathbf {K}$ has $\mu $ -symmetry while $\mu $ -nonforking in (1) has symmetry. In this case:(a) $\mathbf {K}$ has the uniqueness of $(\mu,\geq \chi )$ -limit models: if $M_1,M_2$ are both $(\mu,\geq \chi )$ -limit over some $M_0\in K_{\mu }$, then $M_1\cong _{M_0}M_2$ ;(b)any increasing chain of $\mu ^+$ -saturated models of length $\geq \chi $ has a $\mu ^+$ -saturated union. These generalize [31] and remove the symmetry assumption in [10, 38].Under $(<\mu )$ -tameness, the conclusions of (1), (2)(a)(b) are equivalent to $\mathbf {K}$ having the $\chi $ -local character of $\mu $ -nonsplitting.Grossberg and Vasey [18, 38] gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the U-rank. (shrink)

The classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP $_1$ theories it must come from Kim-dividing. We generalise this work to the framework of Abstract Elementary Categories (AECats) (...) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple, and NSOP $_1$ -like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking, and long Kim-dividing. (shrink)