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 $κ$-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) $\mathcal{C}at(\mathcal{K})=\emptyset$. 2) $\mathcal{C}at(\mathcal{K})=[α, β]$ for some $α, β\in [κ, \beth_ω(κ))$. 3) $\mathcal{C}at(\mathcal{K})=[χ, \infty)$ for some $χ\in [κ, \beth_{(2^κ)^+})$. 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.