Long limit models are isomorphic assuming a splitting-like relation

Abstract

We prove the uniqueness of high cofinality limit models in stable abstract elementary classes (AECs) with amalgamation, assuming the existence of a rather weak independence relation. $\textbf{Theorem.}$ Suppose $\mathbf{K}$ is a $\lambda$-stable AEC, where $\operatorname{LS}(\mathbf{K}) \leq \lambda$, $\kappa < \lambda^+$ is regular, and $\mathbf{K}_\lambda$ satisfies the amalgamation property. Let $\mathbf{K}'$ is the class of all $(\lambda, \delta)$-limit models where $\operatorname{cf}(\delta) \geq \kappa$ (or any AC where $\mathbf{K}' \subseteq \mathbf{K}_\lambda$ contains all such $(\lambda, \delta)$-limit models when $\operatorname{cf}(\delta) \geq \kappa$). Suppose also that there is an independence relation on $\mathbf{K}'$ satisfying weak uniqueness, weak existence, universal continuity* in $\mathbf{K}_\lambda$, $(\geq \kappa)$-local character, and $(\lambda, \theta)$-weak non-forking amalgamation in some regular $\theta \in [\kappa, \lambda^+)$. Let $\delta_1, \delta_2 < \lambda^+$ be limit with $\operatorname{cf}(\delta_l) \geq \kappa$ for $l = 1, 2$. Then for all $M, N_1, N_2 \in \mathbf{K}_\lambda$, if $N_l$ is $(\lambda, \delta_l)$-limit over $M$ for $l = 1, 2$, then $N_1 \underset{M}{\cong} N_2$. Moreover, if $K_\lambda$ also satisfies the joint embedding property, then for all $N_1, N_2 \in \mathbf{K}_\lambda$, if $N_l$ is $(\lambda, \delta_l)$-limit for $l = 1, 2$, then $N_1 {\cong} N_2$. This generalises both Theorem 3.1 of arXiv:2503.11605 and Theorem 1.2 of arXiv:1508.04717 - the former to apply to independence relations that satisfy much weaker forms of uniqueness, extension, and non-forking amalagamation, and the latter to independence relations other than $\lambda$-non-splitting. As such, this generalises all other positive isomorphism results of limit models known to the author.