Categoricity in multiuniversal classes

TL;DR

This paper extends Shelah's categoricity conjecture results from universal classes to multiuniversal classes, showing that categoricity in these classes can be characterized similarly, with types determined by finite restrictions.

Contribution

It generalizes categoricity results to multiuniversal classes, introducing algebraic closure as a key concept and proving types are determined by finite restrictions.

Findings

Categoricity transfer in multiuniversal classes.

Galois types are determined by finite restrictions.

Extension of Shelah's conjecture to broader classes.

Abstract

The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a result of the second author.