What does the topos of (light) condensed sets classify?

Question

Recall that $\mathrm{Pro}(\mathbf{FinSet}) = *_{\text{proét}}$, the category of profinite sets, forms a site with finite jointly surjective families as covers, and that the category of sheaves on this set is the category $\mathbf{Cond}(\mathbf{Set}) = \mathbf{Sh}(\mathrm{Pro}(\mathbf{FinSet}))$ of condensed sets. One expects this to be a topos, but it is not because of size issues (the site of profinite sets is too large).

Clausen and Scholze proposed in 2023 to replace the site of profinite sets with that of light profinite sets (that of countable formal cofiltered limits of finite sets), which resolves the size issues and makes the the category of light condensed sets a topos.

Now, every (Grothendieck) topos is the classifying topos of some geometric theory, and more precisely that of cover-preserving flat functors out of its site of definition. What does the topos light condensed sets classify? Equivalently, what are the cover-preserving functors out of the site of (light) profinite sets?

This question could be trivial, but I currently just can't wrap my head around it.

But, even more generally, what does the pro-étale topos of a scheme classify?

Answer 1

The topos of light condensed sets is generated by the Cantor set $\Delta = \prod_{\mathbb N} \{0,1\}$. So it classifies "Cantor space objects". Here is what this gives, essentially tautologically:

Definition. A Cantor space object $X$ in a topos $T$ is an object $X\in T$ together with an action of $\mathrm{End}(\Delta)$ on $X$, with the following two properties:

1. If $i_0: \Delta\cong \{0\}\times \Delta\hookrightarrow \Delta$ and $i_1: \Delta\cong \{1\}\times \Delta\hookrightarrow \Delta$ are the two maps reembedding the Cantor set into itself, then the induced map $(i_0,i_1): X\sqcup X\to X$ (via letting $i_0,i_1\in \mathrm{End}(\Delta)$ act on $X$) is an isomorphism.
2. Any surjective map $f: \Delta\to \Delta$ yields a surjective map $f: X\to X$.
3. Some condition ensuring "commutation with finite limits".

I'm not quite happy with this. One issue is that the Cantor space wants to be the infinite limit $\prod_{\mathbb N} \{0,1\}$, but this infinite limit need not be preserved by geometric morphisms. In particular, in $T$ there is still a map $X\to \prod_{\mathbb N} \{0,1\}$ (using the isomorphism from 1. in an inductive way) but it does not have to be an isomorphism. If it were, then functoriality in $\mathrm{End}(\Delta)$ would be automatic, and not a weird structure.

Now there is something peculiar about morphisms of pro-etale topoi: The pullback functor commutes with all limits, not just finite limits. If one restricts to the light setting, it still commutes with countable limits.

So maybe, if one is looking for a good "universal property" of (light) condensed sets, one should be looking in a different category: That of topoi, with morphisms those geometric morphisms that commute with (countable) limits. In fact, it may be that any geometric morphism I have ever constructed out of light condensed sets has this property!

I believe in this category, light condensed sets are sub-initial: To any topos, there is always at most one morphism from light condensed sets that commutes with all colimits and countable limits. Indeed, we know where a point goes, then we know where finite sets go, then we know where countable limits of finite sets go, i.e. all light profinite sets, and then we know where all light condensed sets must go. In particular, the category of sets is no longer initial in this variant of the category of topoi. Indeed, sending any light condensed set to the underlying set defines a morphism in this category, but it does not have a section.

I do wonder what the new initial object is (which, I presume, must exist for abstract reasons?). I guess one can also define a variant of the category of (macro)topoi where pullback functors commute with all limits, and I wonder how the category of all condensed sets relates to the initial object thereof (it should still be sub-initial).