Condensed vs pyknotic vs consequential

Question

As is probably clear from my previous questions, I am coming to "condensed mathematics" from the naive perspective of a category theorist, without much knowledge of the intended applications in algebraic geometry and functional analysis. I hope therefore that this question is not too naive, but I haven't found a place yet where it is addressed clearly for a "lay" audience.

According to my current understanding, there are four categories that achieve something similar:

• The category of $\kappa$-condensed sets, for some cardinal $\kappa$, which are sheaves on the site of compact Hausdorff spaces of cardinality $<\kappa$.
• The category of pyknotic sets, which are the $\kappa$-condensed sets when $\kappa$ is an inaccessible.
• In the opposite direction (making the site smaller), Johnstone's topological topos, which is the category of sheaves on the full subcategory of spaces containing the point and the one-point compactification of $\mathbb{N}$.
• The category of condensed sets, which is the colimit of the categories of $\kappa$-condensed sets over all $\kappa$, or equivalently the category of "small sheaves" on the large site of all compact Hausdorff spaces.

I want to understand the differences between these categories, and why and in what situations one might choose one over the others. Specifically:

1. Johnstone's topological topos seems closely related to the category of $\aleph_1$-condensed sets. (The references I've seen restrict $\kappa$ to be a strong limit cardinal, but at least the definition seems to make sense for any cardinal.) It seems too much to hope for that they would be equivalent, but are they related by an adjunction at least? How "close" are they?

2. On that note, why is $\kappa$ usually restricted to be a strong limit cardinal?

3. To a pure category theorist, the first three categories have the obvious advantage over the fourth that they are toposes. In fact they are even local toposes: their forgetful functor to sets has a right adjoint as well as a left adjoint. Why might one nevertheless choose to work with the non-local non-topos of condensed sets instead of one of these three toposes?

4. Relatedly, for what applications is it not sufficient to work with $\kappa$-condensed sets for a fixed small $\kappa$, like the smallest uncountable strong limit $\beth_\omega$? Or, for that matter, Johnstone's topological topos? In particular, are there desirable abstract properties that these categories lack? Or are there constructions that would give the "wrong" result when performed in these categories (and if so, in what sense)? Or is it that there are important examples of compactly generated spaces that aren't "$\kappa$-compactly generated" for small $\kappa$ and thus don't embed fully-faithfully in these categories?

Answer 1

Some comments:

Regarding 1): They are quite different. Johnstone actually uses a very general notion of "cover" in his sequential topos -- his site is a full subcategory of metrizable profinite sets (=$\aleph_1$-small profinite sets=countable limits of finite sets=sequential Pro-category of finite sets), but not all of his covers are covers in the condensed/pyknotic sense. I'm not sure the more general covers he allows are of much relevance for his positive results, but they preclude his topos from having enough points. (All the other choices have enough points.) It also means that the generating object $\mathbb N\cup\{\infty\}$ is not actually a quasicompact object in his topos.

So a better comparison would be between the version of Johnstone's topos that restricts to the finitary covers. This admits a geometric morphism from $\aleph_1$-condensed sets. (Edit (Nov 14, 2023): It doesn't: Pullback does not preserve finite limits. One can correct this by enlarging the defining site of Johnstone's topos to include all closed subsets of $(\mathbb N\cup \{\infty\})^n$, for all $n$.) Now $\aleph_1$-condensed sets actually admit a description very similar to this version of Johnstone's topos, but replacing $\mathbb N \cup \{\infty\}$ with the Cantor set, which is the universal metrizable profinite set (i.e. surjects onto any other). But the Cantor set is much bigger than $\mathbb N\cup\{\infty\}$! This has some important consequences, for example $[0,1]$ is quasicompact in $\aleph_1$-condensed sets, but very much fails to be so in Johnstone's topos. Such quasicompactness is used all over the place in our arguments. For example, an extremely important property is the following:

You can consider CW complexes as ($\aleph_1$-)condensed sets. Now any topos has its inherent notion of cohomology, so you can take the resulting cohomology of CW complexes. Then, in the condensed world:

Applied to CW complexes, the internal notion of cohomology agrees with singular cohomology.

I believe this would fail in Johnstone's topos (correct me if I'm wrong!). (Edit (Nov 14, 2023): I believe in the version of Johnstone's topos sketched above, using all closed subsets of $(\mathbb N\cup\{\infty\})^n$ as defining objects and only finitary covers, the computation does come out correctly, as a slightly curious consequence of our new discussion of solid modules.) And I hope you agree that this is a very desirable property. It's the starting point for seeing that the internal notion of group cohomology of all sorts of topological/condensed groups agrees with the various (ad hoc!) notions of continuous group cohomology you can find in the literature.

On the other hand, almost everything we do in condensed sets could also be done already with $\aleph_1$-condensed sets; in fact, I'm contemplating switching to that setting for some things. One nasty issue is that while condensed abelian groups have enough projectives, there are no nontrivial ones that are internally projective. But in $\aleph_1$-condensed abelian groups, $\mathbb Z[\mathbb N\cup\{\infty\}]$ is internally projective!

Regarding 2): As you observe, the theory basically works for any $\kappa$. One thing one might like is that as you increase $\kappa$, the pullback functors are fully faithful. While I don't know whether that's always true, it's true at least when the cardinals are either regular or strong limits. And the reason for choosing strong limits is that in that case, one has enough compact projective objects (the extremally disconnected profinite sets), which are very useful (even if not ultimately necessary) for building the theory.

Regarding 3): The main reason is the desire to avoid artificial choices. Let me elaborate by switching to the next question:

Regarding 4): One thing we prove early on is a general Pontrjagin duality on locally compact abelian groups (even on the derived level). But this requires that there are as many discrete abelian groups as there are compact abelian groups. If you work with $\kappa$-condensed abelian groups, the Pontrjagin duality would force one to restrict not only to $\kappa$-small compact abelian groups, but also to $\kappa$-small discrete abelian groups.

Also, if you really want to say that compactly generated topological spaces embed into condensed sets, without implicitly actually talking about $\kappa$-compactly generated ones, again you need to go this colimit over all $\kappa$.

But in practice, mostly everything is $\aleph_1$-compactly generated, and you can just work with $\aleph_1$-condensed sets (or the much larger category of $\beth_\omega$-condensed sets, where you have enough compact projectives).

But as I said above, you absolutely cannot work with Johnstone's topos, the space $\mathbb N\cup\{\infty\}$ is just too small.