A complete classification of categoricity spectra of accessible categories with directed colimits

We provide a complete classification of all the possible categoricity spectra, in terms of internal size, that can appear in a large accessible category with directed colimits, assuming the Singular Cardinal Hypothesis ($SCH$), and providing as well explicit threshold cardinals for eventual categoricity. This includes as a particular case the first complete classification of categoricity spectra of abstract elementary classes (AEC's) entirely in $ZFC$. More specifically, we have the following theorem: Let $\mathcal{K}$ be a large $\kappa$-accessible category with directed colimits. Assume the Singular Cardinal Hypothesis $SCH$ (only if the restriction to monomorphisms is not an AEC). Then the categoricity spectrum \mathcal{C}at(\mathcal{K})=\{\lambda\geq \kappa: \mathcal{K} \text{ is \lambda-categorical}\} is one of the following: 1) $\mathcal{C}at(\mathcal{K})=\emptyset$. 2) $\mathcal{C}at(\mathcal{K})=[\alpha, \beta]$ for some $\alpha, \beta \in [\kappa, \beth_{\omega}(\kappa))$. 3) $\mathcal{C}at(\mathcal{K})=[\chi, \infty)$ for some $\chi \in [\kappa, \beth_{(2^{\kappa})^+})$. This solves in particular Shelah categoricity conjecture for AEC's. There are examples of each of the three cases of the classification, showing that they indeed occur.