The categoricity spectrum of large abstract elementary classes

TL;DR

This paper classifies the possible categoricity spectrums of large abstract elementary classes under certain set-theoretic assumptions, providing a comprehensive understanding of when such classes are categorical.

Contribution

It offers a complete characterization of categoricity spectrums for AECs with amalgamation, addressing longstanding questions in Shelah's categoricity conjecture.

Findings

Categoricity spectrum can be empty, an end segment, or a finite interval.

Under stronger GCH, spectrum is either bounded or contains an end segment.

Answers key questions related to Shelah's categoricity conjecture.

Abstract

The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically, the categoricity spectrum is either empty, an end segment starting below the Hanf number, or a closed interval consisting of finite successors of the L\"owenheim-Skolem-Tarski number (there are examples of each type). We also prove (assuming a strengthening of the GCH) that the categoricity spectrum of an abstract elementary class with no maximal models is either bounded or contains an end segment. This answers several longstanding questions around Shelah's categoricity conjecture.