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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 …

This is an interesting question. First, I think the [PS] reference does not give the "correct" Betti stack. In my notes on 6 functors, I define a different stack $X_B$ such that $D_{\mathrm{qc}}(X_B)$ …

I have finally found some time to write up the $6$-functor formalisms in coherent cohomology (a la Gaitsgory--Rozenblyum) and for $D$-modules, see Lecture 8 and its appendix. A short synopsis is that …

When defining a homotopy-coherent structure, you have to strike the correct balance between supplying enough data (so that all isomorphisms (between isomorphisms, ...) that you need later are actually …

Let me add there is now a reference for the claims in Dan Peterson's answer, namely Marco Volpe has worked out the Topological $6$-functor formalism. I also gave some (brief) account of this in Lectur …

I should point out that Joshi's paper does not falsify Remark 9 of our note. In Joshi's Theorem 4.8 (which he claims to falsify our Remark 9) the curve $X/E$ stays the same (and hence of course its te …

Briefly (I will elaborate below): One expects that their fully faithful functor from (roughly) $p$-adic representations of $G(\mathbb Q_p)$ to (roughly) coherent sheaves on the Emerton--Gee stack exte …

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 …

In Lemma 4.1.7, we actually assume that $R$ is f-semiperfect (i.e. a quotient of a perfect ring by a finitely generated ideal); I doubt the result is true without this assumption. Note that $W_{PD}$ i …

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 …

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 …

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 …

Somehow that question slipped my radar, sorry! The truth is that shamefully I'm not able to say much, as I don't have a strong knowledge of resolution of singularities. But at least so far, the flow o …

Classically, the functor $f^!$ is indeed not a right adjoint in general. Clausen and I have recently found a way to make it a right adjoint in general, by enlarging the category of modules to that of …

The question is local on $X$, so we may assume that $\mathcal E$ is finite free, of rank $n$, say. In that case, as also SashaP points out, the question amounts to the question whether $\mathbb A^n_X$ …