Categoricity in Quasiminimal Pregeometry Classes

TL;DR

This paper provides a direct proof that quasiminimal pregeometry classes are categorical in all uncountable cardinalities without relying on the excellence axiom, simplifying the understanding of their model-theoretic properties.

Contribution

It offers a new proof of categoricity in quasiminimal pregeometry classes that does not depend on the excellence axiom, enhancing theoretical clarity.

Findings

Quasiminimal pregeometry classes are categorical in all uncountable cardinalities.

A direct proof of categoricity is achieved without assuming excellence.

The result simplifies the theoretical framework of quasiminimal classes.

Abstract

Quasiminimal pregeometry classes were introduces by Zilber [2005a] to isolate the model theoretical core of several interesting examples. He proves that a quasiminimal pregeometry class satisfying an additional axiom, called excellence, is categorical in all uncountable cardinalities. Recently Bays et al. [2014] showed that excellence follows from the rest of axioms. In this paper we present a direct proof of the categoricity result without using excellence.