Are condensed sets (locally) cartesian closed?

Question

The category of condensed sets is the colimit of the toposes of $\kappa$-condensed sets over all cardinals $\kappa$, or equivalently the category of "small sheaves" on the large site of all compact Hausdorff spaces. As such, it is not itself a topos, although it is an infinitary pretopos. I have encountered claims that it is additionally locally cartesian closed, but I have not found a proof written down.

Is the category of condensed sets (locally) cartesian closed?

Based on general results about small (pre)sheaves that I have found, I suspect that this ought to be equivalent to some statement such as "For any compact Hausdorff spaces $A,B$ there exists a set of compact Hausdorff spaces $C_i$ and maps $e_i : A\times C_i \to B$ such that for any extremally disconnected compact Hausdorff space $X$, any map $A\times X\to B$ factors through a map $X\to C_i$ for some $i$." But I have not checked the details.

Of course, the category of $\kappa$-condensed sets is locally cartesian closed, since it is a topos. If the category of all condensed sets is also locally cartesian closed, I would also like to know under what conditions the embedding of $\kappa$-condensed sets in condensed sets is a locally cartesian closed functor. My guess would be that this is some kind of closure property of $\kappa$ that ought to be true for arbitrarily large values of $\kappa$.

Answer 1

Condensed sets are indeed locally cartesian closed. On the other hand, for no cardinal $\kappa$ (no matter how inaccessible) the functor from $\kappa$-condensed sets to condensed sets preserves all internal Hom's. Indeed, consider the internal Hom from a discrete set $I$ towards the discrete set $\{0,1\}$: This is evidently $\prod_I \{0,1\}$, and within all condensed sets it is represented by this profinite set. But if $|I|>\kappa$, this is not a $\kappa$-condensed set.

To prove that internal Hom's exist, one uses the following criterion. See for example Proposition 2.1.9 of Lucas Mann's Thesis. (Some incomplete discussion along those lines is also after Lecture 2 of Condensed Mathematics.) I should mention that we define a profinite set to be $\kappa$-small if the corresponding Boolean algebra is $\kappa$-small. (This is in general slightly different than asking about the size of underlying set of the profinite set. If $\kappa$ is a strong limit, the distinction disappears, but for regular $\kappa$ it is relevant.)

Lemma. Let $X$ be a functor from profinite sets to sets that satisfies the hypersheaf condition, but is not necessarily small. Let $\kappa$ be a regular cardinal. Then $X$ is a $\kappa$-condensed set if and only if $X$ takes $\kappa$-cofiltered limits (of profinite sets) to $\kappa$-filtered colimits (of sets), i.e. $$ X(\varprojlim_i S_i) = \varinjlim_i X(S_i) $$ for $\kappa$-cofiltered diagrams of profinite sets $S_i$.

In particular, $X$ is a condensed set if and only if the functor $$ X: \mathrm{ProFin}^{\mathrm{op}} = \mathrm{Ind}(\mathrm{Fin}^{\mathrm{op}})\to \mathrm{Set} $$ is accessible, i.e. commutes with $\kappa$-filtered colimits for some large enough $\kappa$.

Now one checks that this applies to the internal Hom $\mathrm{Hom}(X,Y)$ between any condensed sets $X$ and $Y$. More precisely, note first that this criterion implies that condensed sets admit arbitrary small limits (as any $\lambda$-small limit commutes with $\kappa$-filtered colimits for $\kappa>\lambda$). Resolving $X$ by disjoint unions of profinite sets, this reduces the existence of internal Hom's to the case that $X=S$ is a profinite set. In that case, the internal Hom $\mathrm{Hom}(S,Y)$ is given by the functor taking any profinite set $T$ to $Y(S\times T)$. But if $T\mapsto Y(T)$ sends $\kappa$-cofiltered limits to $\kappa$-filtered colimits, then so does $T\mapsto Y(S\times T)$. See also Corollary 2.1.10 and Remark 2.1.12 in Lucas Mann's Thesis.

Essentially the same argument works in a slice (over some profinite set, and then in general), showing that condensed sets are locally cartesian closed.

[Edit: In the comments, Mike Shulman asked whether the internal Hom commutes with the inclusion of $\lambda$-small $\kappa$-condensed sets into all condensed sets. And indeed it does when $\kappa$ is regular and $\lambda<\kappa$, and in fact this is what the argument above proves. Here, a condensed set is $\lambda$-small $\kappa$-condensed if it is a $\lambda$-small colimit of $\kappa$-small profinite sets. Even better, see Z.M's question in the comments, the proof shows that if $\kappa$ is regular and $\lambda<\kappa$, and $X$ is a $\lambda$-small condensed set (of any condensedness) and $Y$ is a $\kappa$-condensed set (but of any cardinality) then $\mathrm{Hom}(X,Y)$ is a $\kappa$-condensed set.]