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 $\kappa$.



Are condensed sets (locally) cartesian closed?
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](https://arxiv.org/abs/2206.02022). (Some incomplete discussion along those lines is also after Lecture 2 of [Condensed Mathematics](https://people.mpim-bonn.mpg.de/scholze/Condensed.pdf).) 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.]


