On the structure of categorical abstract elementary classes with amalgamation

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. $\mathbf{T}\mathbf{h}\mathbf{e}\mathbf{o}\mathbf{r}\mathbf{e}\mathbf{m}$ 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 $\mu \ge {\mathrm{\beth }}_{(2^{\text{LS}(K)}{)}^{+}}$ and $\lambda \ge \beth_{\left(2^{\mu^+}\right)^+}$,thenthereexistsatype-fullgood$\mu$-framewithunderlyingclassthesaturatedmodelsin$K_\mu$. 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.