---
tag: 9NM376YWF2
kind: concept
name: Condensed set
msc: [18F10, 18F20]
specializes: [RT8A8FJ566]
sources:
  - url: https://en.wikipedia.org/wiki/Condensed_mathematics
---

## Summary

A condensed set is a [[sheaf|RT8A8FJ566]] of sets on the category of profinite sets ([[Stone spaces|ZV38HA5PAQ]]).
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|Y9CZ3MTNF1]] 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.

## Explanations

### L1 human

A topological space can be probed by mapping small test spaces into it.
A condensed set keeps only this probing data.
For each profinite set $S$, such as a finite set or the Cantor set, it gives the set of "maps from $S$".
It also says how these maps restrict along maps of test spaces.
This is enough to recover most spaces of interest, and it behaves much better with algebra.
For example, the quotient $\R / \Q$ is a useless indiscrete space, but as a condensed group it keeps real information.