A Categorical Construction of Ultrafilters

TL;DR

This paper presents a new categorical method to construct ultrafilters using inverse limits of finite partitions, providing an elementary perspective and addressing a specific open question in the field.

Contribution

It introduces a novel categorical construction of ultrafilters via inverse limits, offering an elementary approach and applying it to resolve an open problem.

Findings

Ultrafilters can be constructed as inverse limits of finite partitions.

The categorical construction provides an elementary and intuitive understanding.

The approach answers a specific open question negatively.

Abstract

Ultrafilters are useful mathematical objects having applications in nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms of the inverse limit of an inverse family of finite partitions; this is an elementary and intuitive presentation of a consequence of the profiniteness of Stone spaces. We then apply this construction to answer a question of Rosinger posed in arXiv:0709.0084v2 in the negative.