Descriptive properties of higher Kurepa trees

TL;DR

This paper explores the properties of higher Kurepa trees at uncountable regular cardinals, using descriptive set theory to analyze the representation of cofinal branches as continuous images and retracts of generalized Baire spaces.

Contribution

It provides a complete characterization of the consistent scenarios for representing cofinal branches of $ ext{k}$-Kurepa trees as continuous images and retracts of ${}^ ext{kappa} ext{kappa}$ for various uncountable regular cardinals.

Findings

Complete picture of consistent scenarios for representations

Characterization of cofinal branches as continuous images

Results applicable to the consistency of related statements

Abstract

We use generalizations of concepts from descriptive set theory to study combinatorial objects of uncountable regular cardinality, focussing on higher Kurepa trees and the representation of the sets of cofinal branches through such trees as continuous images of function spaces. For different types of uncountable regular cardinals $\kappa$, our results provide a complete picture of all consistent scenarios for the representation of sets of cofinal branches through $\kappa$-Kurepa trees as retracts of the generalized Baire space ${}^\kappa\kappa$ of $\kappa$. In addition, these results can be used to determine the consistency of most of the corresponding statements for continuous images of ${}^\kappa\kappa$.