Condensed Mathematics: The internal Hom of condensed sets and condensed abelian groups and a prismatic construction of the real numbers

TL;DR

This paper explores condensed mathematics, focusing on the internal Hom of condensed abelian groups and sets, providing new proofs and constructions related to the real numbers within this framework.

Contribution

It introduces a construction for the internal Hom in condensed categories and applies it to reprove key theorems and construct the real numbers from discrete spaces.

Findings

New proof of a theorem by Clausen and Scholze

Construction of the real numbers from discrete spaces

Detailed overview of condensed categories and internal Hom

Abstract

We give an overview of the basic definitions of condensed categories, as well as the internal Hom of condensed abelian groups. We give a construction for the internal Hom of condensed sets and apply it to obtain a new proof of a theorem of Clausen and Scholze. Finally, we give a detailed account of a construction of the real numbers from discrete spaces, which is an intermediate step of a theorem by Clausen and Scholze.