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.

Topological spaces formalize the idea of spaces with a notion of “nearness” of points. However, they fail to handle the idea of “points that are infinitely near, but distinct” in a useful way. Condensed sets handle this idea in a useful way. (Scholze 21)

For instance, $\mathbb{R}/\mathbb{Q}$ and $\ell^2(\mathbb{N})/\ell^1(\mathbb{N})$ are indiscrete as topological spaces but retain structure as condensed sets.

Condensed sets contain information about limits of images of any set, $A$, along the ultrafilters of $A$.

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 locally small, well-powered, locally cartesian closed infinitary-pretopos, that is neither a Grothendieck topos nor an elementary topos – since it lacks both a small separator (indeed, it is not even total) and a subobject classifier. It has a large separator of finitely presentable projectives, and hence is algebraically exact. (Campbell 20)

See \cite[Proposition 1.7]{ScholzeLCM} for the following proposition.

Proposition

The forgetful functor from the category of topological spaces to condensed sets is a faithful functor. It becomes fully faithful when restricted to compactly generated spaces. (In the case of κ-condensed sets, one must take κ-compactly generated spaces instead.)

This functor admits a left adjoint, which sends a condensed set $T$ to the topological space given by the underlying set $T(*)$ of $T$ equipped with the quotient topology induced by the map

$$\coprod_{S\to T}S\to T(*),$$

where $S$ runs over all (κ-small) profinite sets mapping into $T$. The counit of this adjunction coincides with the counit $X^{cg}\to X$ of the adjunction between (κ-small) compactly generated spaces and topological spaces.

The left adjoint exists also for topological groups and other algebraic structures, but in this case, the underlying set is not $T(*)$.

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)
• Peter Scholze, Math Stack Exchange answer.
• Alexander Campbell, How nice is the category of condensed sets?, talk abstract.