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Good question! You essentially already give the answer, but let me spell it out. First, in the solid case, as you say one can compute the derived solidification $\mathbb Z[S]^\blacksquare$ for any com …

Like any profinite group (or much more general types of topological groups, such as compactly generated ones), you can consider it as a condensed group in the sense of condensed mathematics. In fact, …

I will prove that the result is true if $F$ is a finitely generated field, but fails if $F$ is countably generated field that is not finitely generated. Let me first discuss the case $F=\mathbb Q$. Fo …

I'm sorry for being cryptic. The subtle point in the construction is that the maps $T_n\to T_{n,j}$ are not all surjective, i.e. one cannot construct this pro-system as a system of quotients. By induc …

Good question! We've been trying to figure this out as we went along, but so far unsuccessfully. Some more precise points: For many (but definitely not all) applications to geometry over the real num …

Here is an essentially tautological answer. The notion of uniformity makes sense also for condensed sets -- it is a condensed set $X$ together with certain subsets $U\subset X\times X$ termed entourag …

This is not true in general, the most important observation being that it fails already when $V$ is discrete. In that case $V\otimes^{\blacksquare} \mathbb Z((T))$ is just the usual algebraic tensor p …

Great question! So far, we haven't been able to produce analytic ring structures on $\mathbb C$-algebras that are not induced. Similarly, if we equip $\mathbb Q_p$ with a liquid analytic ring structur …

Great question! The answer is Yes. Let me elaborate a little. The question is more generally about the left adjoint to the inclusion $\mathrm{An}\to \mathrm{CondAn}$ from anima to condensed anima. Thi …

The sum $M=\bigoplus_{\mathbb N} \mathbb Z_p$ is not first-countable, but it is Cauchy complete. More precisely, $M$ maps isomorphically to $\varprojlim_{U\subset M} M/U$ where $U$ runs over open subg …

Let me add to Z. M's answer, and note that Dustin has no reason to apologize at all: What he said is literally correct. Namely, one can directly show that $L_{F/\mathbb Z}^\blacksquare$ is isomorphic …

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 in …

Flatness in analytic geometry is an interesting question! As Dustin says, it comes with several important caveats. First, open immersions may not be flat even in the weakest sense of the word. Here is …

Let me recall a little bit of the background. The question is about the relation between topological spaces and pyknotic sets, and properties of the topos of pyknotic sets. Recall that pyknotic sets a …

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 ( …