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What are the topos points of $\mathrm{CondSet}$, i.e. the geometric morphisms $\mathrm{Set} \to \mathrm{CondSet}$?

More generally, is there a concise description of the geometric morphisms $\mathcal{E} \to \mathrm{CondSet}$ for any Grothendieck topos $\mathcal{E}$, for example in the form of a geometric theory classified by $\mathrm{CondSet}$?

When asking these questions, we have to keep in mind that the category $\mathrm{CondSet}$ is not actually an honest topos, but is "approximated" by the toposes of $\kappa$-condensed sets (by my understanding of Peter Scholzes Lectures on Condensed Mathematics). So the above questions each have two variants: (1) What is the answer for $\mathrm{CondSet}_\kappa$? (2) How should we even interpret the notion of points and generalized points in the case of the full $\mathrm{CondSet}$?

Partial answers are very much welcome, of course!

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    $\begingroup$ I have no clue, but maybe a couple of comments. The category $\mathsf{CHaus}$ of compact Hausdorff spaces is a dear friend of the category of condensed sets and there are not many cocontinuous functors $\mathsf{CHaus} \to \mathsf{Set}$. This follows from the following observation. The composition of such a functor with the Stone-Čech compactification would give a cocontinuous functor $\mathsf{Top} \to \mathsf{CHaus} \to \mathsf{Set}$, and it was shown that $\mathsf{Cocont}(\mathsf{Top}, \mathsf{Set}) \simeq \mathsf{Set}$. (Thm F here: arxiv.org/pdf/2106.11115.pdf). $\endgroup$ Jun 28, 2021 at 11:57
  • $\begingroup$ @IvanDiLiberti You would get a functor CHaus $\to$ Set from a point, but I don't think it would cocontinuous. $\endgroup$ Jun 28, 2021 at 14:09
  • $\begingroup$ @SimonHenry I agree, but the argument that shows that there aren't many might be recycled. Afterall condensed sets feel like spaces. $\endgroup$ Jun 28, 2021 at 14:32
  • $\begingroup$ Indeed, points of $\mathbf{CondSet}$ are equivalent to coherent functors from $\mathbf{CHaus}$ to $\mathbf{Set}$. $\endgroup$ Jun 29, 2021 at 6:23
  • $\begingroup$ @IvanDiLiberti Actually, cocontinuous functors $\mathbf{CHaus} \to \mathcal{C}$ are described in Example 6.9 of that paper. They correspond to "co-(compact Hausdorff spaces)" internal to $\mathcal{C}$. I haven't thought much about what happens for $\mathcal{C}=\mathbf{Set}$, though. $\endgroup$ Jun 29, 2021 at 7:12

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The category $\mathbf{Cond}$ of condensed sets is equivalent to the category of small sheaves over any of the following three large sites. (For small sheaves, see Mike Shulman's paper Exact completions and small sheaves in TAC.)

  1. The category $\mathbf{Comp}$ of compact Hausdorff spaces, equipped with the coherent topology (i.e. finite jointly epimorphic families are covering);
  2. The category $\mathbf{ProFin}$ of profinite sets, equipped with the coherent topology;
  3. The category $\mathbf{Proj}$ of projective compact Hausdorff spaces, equipped with the disjunctive topology (i.e. families of finite coproduct inclusions are covering).

It follows that for any Grothendieck topos (and more generally any infinitary-pretopos) $\mathscr{E}$, the category of cocontinuous left exact functors from $\mathbf{Cond}$ to $\mathscr{E}$ is equivalent to the following three categories:

  1. The category of coherent functors from $\mathbf{Comp}$ to $\mathscr{E}$ (i.e. functors that preserve finite limits, finite coproducts, and epimorphisms);
  2. The category of coherent functors from $\mathbf{ProFin}$ to $\mathscr{E}$;
  3. The category of functors from $\mathbf{Proj}$ to $\mathscr{E}$ that preserve finite coproducts and weak finite limits.

These are all instances of the universal property of the category of small sheaves on a (possibly large) site (for which, see Shulman's paper cited above). Namely, the category of condensed sets is the infinitary-pretopos completion of (i) the pretopos $\mathbf{Comp}$, (ii) the coherent category $\mathbf{ProFin}$, (iii) the weakly lextensive category $\mathbf{Proj}$.

Now, essentially because the category $\mathbf{Cond}$ of condensed sets is "too big" (e.g. it is not locally presentable), cocontinuous functors $\mathbf{Cond} \to \mathscr{E}$ need not have right adjoints. For that reason, in place of geometric morphisms (defined as adjoint pairs with left exact left adjoint) from a Grothendieck topos $\mathscr{E}$ to $\mathbf{Cond}$, one should instead consider cocontinuous left exact functors from $\mathbf{Cond}$ to $\mathscr{E}$. In particular, the "correct" notion of point of $\mathbf{Cond}$ is a cocontinuous left exact functor $\mathbf{Cond} \to \mathbf{Set}$.

Thus, by the above, points of $\mathbf{Cond}$ are equivalent to (i) coherent functors $\mathbf{Comp} \to \mathbf{Set}$, (ii) coherent functors $\mathbf{ProFin} \to \mathbf{Set}$, and (iii) functors $\mathbf{Proj} \to \mathbf{Set}$ that preserve finite coproducts and weak finite limits.

Finally, for $\kappa$ a strong limit cardinal, the category of $\kappa$-condensed sets is equivalent to be the category of sheaves over the sites $\mathbf{Comp}_{\kappa}$, $\mathbf{ProFin}_{\kappa}$, and $\mathbf{Proj}_{\kappa}$, defined to be the full subcategories of the above sites spanned by the spaces of cardinality $< \kappa$, with the induced topologies. As above, it follows that points of the topos of $\kappa$-condensed sets are equivalent to certain kinds of functors (the same kinds as above) from these full subcategories to $\mathbf{Set}$.

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    $\begingroup$ "one should instead consider cocontinuous left exact functors" ah, excellent. I agree with this, from a forcing point of view. One can get non-locally-small examples of toposes of small sheaves that nonetheless still have the inverse image part of what would normally be the functor of global points. $\endgroup$ Jun 29, 2021 at 7:53
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Here is a way to construct a jointly surjective family of points for the topos of $\kappa$-condensed sets. This gives a partial answer to question (1). The calculation is based on discussions I had with Aurélien Sagnier. Today at Toposes Online, Dustin Clausen mentioned the same jointly surjective family of points, maybe based on similar calculations.

One way to calculate this, is to notice that every $\kappa$-condensed set has a restriction to a sheaf on $\beta(S)$, where $S$ is a set of cardinality $< \kappa$ and $\beta(S)$ is its Stone–Čech compactification. The process of restricting a $\kappa$-condensed set to sheaf on $\beta(S)$ is the inverse image part of a geometric morphism $\mathbf{Sh}(\beta(S)) \longrightarrow \mathbf{CondSet}_\kappa.$

The family of all geometric morphisms $\mathbf{Sh}(\beta(S)) \longrightarrow \mathbf{CondSet}_\kappa$, where $S$ goes over all sets of cardinality $<\kappa$, is a jointly surjective family of geometric morphisms, because you can recover a condensed set by looking at the values it takes on each $\beta(S)$. So if you take a jointly surjective family of points of $\mathbf{Sh}(\beta(S))$ for each $S$, then all points taken together give a jointly surjective family of points for the topos of $\kappa$-condensed sets.

Because $\beta(S)$ is a topological space, $\mathbf{Sh}(\beta(S))$ has enough points, and because $\beta(S)$ is sober, the points of $\mathbf{Sh}(\beta(S))$ are precisely the elements of $\beta(S)$. Note that the construction of $\beta(S)$ uses the axiom of choice.

So the conclusion is that you get a jointly surjective family of points given by the pairs $(S,x)$, where $S$ is a set of cardinality $< \kappa$ and $x$ is a point of $\beta(S)$, or alternatively, an ultrafilter on $S$.

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