Condensed mathematics through compactological spaces

TL;DR

This paper explores the relationship between condensed mathematics and compactological spaces, providing a rigorous proof of their equivalence and framing condensed sets as a categorical completion of compactological spaces.

Contribution

It offers a detailed survey of compactological spaces, proves their equivalence to a subclass of condensed sets, and clarifies the categorical completion linking these concepts.

Findings

Condensed sets are a categorical completion of compactological spaces.

The equivalence between quasiseparated condensed sets and compactological spaces is established.

Provides an elementary perspective on condensed mathematics through compactological sets.

Abstract

In their 2022 lecture notes on condensed sets, Clausen and Scholze mentioned in a remark that the important subclass of quasiseparated condensed sets is equivalent to the category of so-called compactological spaces defined by Waelbroeck in the 1960s. In this paper we survey the latter category in detail, we give a rigorous proof of Clausen and Scholze's claim, and we establish that condensed sets are a formal categorical completion of Waelbroeck's compactological spaces. The latter answers a question asked by Hanson in 2023 and permits the interpretation of compactological sets as an 'elementary' approach to condensed mathematics.