A proof of Shelah's eventual categoricity conjecture and an extension to accessible categories with directed colimits

TL;DR

This paper proves Shelah's eventual categoricity conjecture for AECs in ZFC and extends it to accessible categories with directed colimits under SCH, establishing a broad categoricity transfer principle and confirming a conjecture on eventual amalgamation.

Contribution

It provides the first ZFC proof of Shelah's conjecture for AECs and generalizes the result to accessible categories with directed colimits assuming SCH, also confirming Grossberg's conjecture.

Findings

Proves Shelah's eventual categoricity conjecture in ZFC for AECs.

Extends categoricity transfer results to accessible categories with directed colimits under SCH.

Establishes that high enough categoricity implies eventual amalgamation in AECs and accessible categories.

Abstract

We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more general context of accessible categories with directed colimits. If $\mathcal{K}$ is such a category, we show that there is a cardinal $\mu$ such that if $\mathcal{K}$ is $\lambda$-categorical for some $\lambda \geq \mu$ (i.e., it has only one object of internal size $\lambda$ up to isomorphism), then $\mathcal{K}$ is eventually categorical (i.e., it is $\lambda'$-categorical for every $\lambda' \geq \mu$). When considering cardinalities of models of infinitary theories $\mathbb{T}$ of $\mathcal{L}_{\kappa, \theta}$ that axiomatize $\mathcal{K}$, the result implies, under $SCH$, the following infinitary version of Morley's categoricity theorem: let $S$ be the class of cardinals $\lambda$ which are of cofinality at least $\theta$ but are not successors of cardinals of cofinality less than $\theta$. Then, if $\mathbb{T}$ is a $\mathcal{L}_{\kappa, \theta}$ theory whose models have directed colimits and it is $\lambda$-categorical for some $\lambda \geq \mu$ in $S$, then it is $\lambda'$-categorical for every $\lambda' \geq \mu$ in $S$; moreover, we also exhibit an example that shows that the exceptions in the class $S$ are needed. Along the way we also prove Grossberg conjecture, according to which categoricity in a high enough cardinal implies eventual amalgamation. We establish this result in AEC's and, assuming in addition $SCH$, in the more general context of accessible categories whose morphisms are monomorphisms.