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 — that 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 $CondSet_\kappa$ 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 $CondSet$ 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.

Another possible way to deal with set theoretic issue is presented in Analytic Stacks.

The category of light profinite sets is the full subcategory of $ProfiniteSet$ consisting of countable sequential limits of finite sets. The category of light condensed sets $LightCondSet$ is the category of sheaves on the site of light profinite sets with finite jointly surjective families of maps as covers.

Equivalence of sites

Several different sites can be used to define condensed sets, and, more generally, condensed ∞-groupoids:

• compact Hausdorff spaces;

• Stone spaces (compact Hausdorff totally disconnected spaces);

• Stonean spaces (compact Hausdorff extremally disconnected spaces).

In all three cases, morphisms are given by continuous maps and covering families are given by finite families of jointly surjective continuous maps.

The equivalence of sites is established in Yamazaki. See also Proposition 2.3, 2.7.

The category of light profinite sets is also equivalent to the opposite of the category of countable Boolean algebras (Clausen-Scholze 2023, Lecture 2).

Properties

Condensed sets form a locally small, well-powered, locally cartesian closed infinitary-pretopos $CondSet$, 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) In a talk by Alexander Campbell, it is claimed that the category of condensed sets is the infinitary-pretopos completion of the pretopos of compact Hausdorff spaces.

According to Peter Scholze in this comment on the nCafé and Mike Shulman in this comment on the nCafé, condensed sets satisfy external CoSHEP and WISC, but internal CoSHEP fails.

See Proposition 1.7 for the following proposition.

Proposition

The forgetful functor from the category of topological spaces to κ-condensed sets is a faithful functor. It becomes fully faithful when restricted to κ-compactly generated spaces.

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 $T(*)$.

For the category of light condensed sets, there is a functor that reflects back onto the category of sequential topological spaces

Related concepts

• pyknotic set

• light condensed set

• pro-étale site

• profinite set

• condensed infinity-groupoid

• condensed abelian group

• 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.
• Alexander Campbell, The universal property of the category of condensed sets, talk abstract
• Dustin Clausen, Peter Scholze, Analytic Stacks, YouTube playlist, website

The equivalence of various sites for condensed sets is established in

• Koji Yamazaki, Condensed Sets on Compact Hausdorff Spaces, arXiv:2211.13855.