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

Abstract

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})=\{λ\geq κ: \mathcal{K} \text{ is $λ$-categorical}\}$ is one of the following: 1) $\mathcal{C}at(\mathcal{K})=\varnothing$. 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.

Sequel to arXiv:1906.09169 with explicit threshold cardinals