Condensed set

Summary

A condensed set is a sheaf of sets on the category of profinite sets (Stone spaces). The coverings are the finite families of maps that are jointly surjective. Condensed sets were introduced by Dustin Clausen and Peter Scholze as a replacement for topological spaces in algebra. Condensed abelian groups form an abelian category with good properties, while topological abelian groups do not. Each topological space $X$ gives a condensed set $S \mapsto C(S, X)$, and for compactly generated spaces no information is lost.