On the structure of categorical abstract elementary classes with amalgamation

Abstract: For $K$ an abstract elementary class with amalgamation and no maximal models, we show that categoricity in a high-enough cardinal implies structural properties such as the uniqueness of limit models and the existence of good frames. This improves several classical results of Shelah. $\mathbf{Theorem}$ Let $\mu \ge \text{LS} (K)$. If $K$ is categorical in a $\lambda \ge \beth_{\left(2^{\mu}\right)^+}$, then: 1) Whenever $M_0, M_1, M_2 \in K_\mu$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \cong_{M_0} M_2$. 2) If $\mu > \text{LS} (K)$, the model of size $\lambda$ is $\mu$-saturated. 3) If $\mu \ge \beth_{(2^{\text{LS} (K)})^+}$ and $\lambda \ge \beth_{\left(2^{\mu^+}\right)^+}$, then there exists a type-full good $\mu$-frame with underlying class the saturated models in $K_\mu$. Our main tool is the symmetry property of splitting (previously isolated by the first author). The key lemma deduces symmetry from failure of the order property.