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

TL;DR

This paper classifies all possible categoricity spectra in large accessible categories with directed colimits under SCH, including a complete classification for AECs in ZFC, and provides explicit thresholds for eventual categoricity.

Contribution

It offers a complete classification of categoricity spectra in accessible categories with directed colimits, including the first ZFC classification for AECs, and confirms the Shelah categoricity conjecture.

Findings

Categoricity spectra are either empty, an interval [α, β], or [χ, ∞).

Provides explicit thresholds for categoricity in accessible categories.

Includes examples for each case of the classification.

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})=\{\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.