Universally Baire sets in $2^{\kappa}$

TL;DR

This paper extends the concept of universally Baire sets from the classical setting of reals to the generalized setting of $2^$, providing equivalent definitions and characterizations for all infinite cardinals.

Contribution

It introduces a generalized theory of universally Baire sets for $2^$, including multiple equivalent definitions for any infinite cardinal $$, broadening the foundational understanding of these sets.

Findings

Established four equivalent definitions for universally Baire sets in $2^$

Generalized the classical notion of universally Baire sets to higher cardinalities

Connected the new definitions to the original concept for $=\omega$

Abstract

We generalize the basic theory of universally Baire sets of $2^\omega$ to a theory of universally Baire subsets of $2^\kappa$. We show that the fundamental characterizations of the property of being universally Baire have natural generalizations that can be formulated also for subsets of $2^\kappa$, in particular we provide four equivalent uniform definitions in the parameter $\kappa$ (for $\kappa$ an infinite cardinal) characterizing for each such $\kappa$ the class of universally Baire subsets of $2^\kappa$. For $\kappa=\omega$, these definitions bring us back to the original notion of universally Baire sets of reals given by Feng, Magidor and Woodin [2].