Independence in abstract elementary classes

We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. $\mathbf{Theorem}$ (Superstability from categoricity) Let $K$ be a $(<\kappa)$-tame AEC with amalgamation. If $\kappa = \beth_\kappa > \text{LS} (K)$ and $K$ is categorical in a $\lambda > \kappa$, then: * $K$ is stable in all cardinals $\ge \kappa$. * $K$ is categorical in $\kappa$. * There is a type-full good $\lambda$-frame with underlying class $K_\lambda$. Under more locality conditions, we prove that the frame extends to a global independence notion (for types of arbitrary length). $\mathbf{Theorem}$ (A global independence notion from categoricity) Let $K$ be a densely type-local, fully tame and type short AEC with amalgamation. If $K$ is categorical in unboundedly many cardinals, then there exists $\lambda \ge \text{LS} (K)$ such that $K_{\ge \lambda}$ admits a global independence relation with the properties of forking in a superstable first-order theory. Modulo an unproven claim of Shelah, we deduce that Shelah's categoricity conjecture follows from the weak generalized continuum hypothesis and the existence of unboundedly many strongly compact cardinals.