nLab
condensed set

Idea

Condensed sets are basic objects in condensed mathematics, whose aim is to provide a convenient setting in the framework for working with algebraic objects that are equipped with sort of a topology. A related alternative is provided by pyknotic sets.

Definition

Definition

A condensed set is a sheaf of sets on the pro-étale site of a point — in other words, on the category of profinite spaces with finite jointly surjective families of maps as covers — which is the colimit of a small diagram of representables (a small sheaf?).

That is, a condensed set is a functor

$$ProfiniteSet^op \longrightarrow Set$$

such that the natural maps

$$T(\emptyset) \longrightarrow *$$

and

$$T(S\sqcup S') \longrightarrow T(S) \times T(S')$$

are bijections for any profinite sets $S$ and $S'$, whereas the natural fork

$$T(S)\to T(S') \rightrightarrows T(S'\times_S S')$$

is an equalizer for any surjection of profinite sets $S'\to S$.

Scholze, p.7 modifies this definition to deal with size issues:

For any uncountable strong limit cardinal $\kappa$, the category of $\kappa$-condensed sets is the category of sheaves on the site of profinite sets of cardinality less than $\kappa$, with finite jointly surjective families of maps as covers.

The category of condensed sets is then the (large) colimit of the category of $\kappa$-condensed sets along the filtered poset of all uncountable strong limit cardinals $\kappa$, hence is the category of small sheaves?.

Properties

Condensed sets form a pretopos.

Related concepts

• pyknotic set

• pro-étale site

• profinite set

• condensed mathematics

References

• Peter Scholze, Lectures on condensed mathematics, pdf
• Peter Scholze, Lectures on analytic geometry, pdf
• Dustin Clausen, Peter Scholze, Masterclass in condensed mathematics, YouTube playlist, website (including pdf notes)