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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: De …

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 …

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 …

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 ( …

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 …

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 …

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 …

There are several questions (implicit) here. In the texts as they are written, how much knowledge on derived categories (as triangulated categories, or as stable $\infty$-categories) is assumed? Doe …

Good question! I think the real context for the question was whether certain objects that are implicit in work of Darmon (and collaborators) could exist within this framework of analytic geometry. The …

Yes, that is true. I'm not sure I'm explaining the step that's confusing you, but you are asking about a special case of the statement that the functor $S\mapsto\underline{S}$ from profinite sets to c …

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 …

Isn't the dualizing complex defined in general in the proper case by taking applying the right adjoint of $\pi_\ast$ to $k$? That's what I'll take as the definition anyways. The Gorenstein property ju …

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 …

Here's a rule of thumb: As long as the construction is nonarchimedean and does not involve noncompleted tensor products, it's solid. More precisely, anything you can build from discrete abelian groups …