Hey, is there any hope that the ∞-category of analytic rings in the sense of Clausen and Scholze is presentable (up to the cardinality problem)?

I was thinking about it, and you have the ∞,1-category of (animated) condensed rings, which you can then show using Morita-equivalence is equivalent to a full subcategory of the ∞,2-category of condensed stably symmetric monoidal presentable stable ∞-categories of the form Mod_R. Then analytic rings are an extension of this ∞-category by certain kinds of Verdier quotients, but I don't know if it will end up being presentable at that point

I wonder, if it's not true that analytic rings do indeed form a presentable category, maybe at that point one might be better off working with prestably symmetric monoidal Grothendieck prestable ∞-categories (a kind of theory of 'connective 2-rings')

@EdoardoLanari It might be discussed in this paper of Aaron Mazel-Gee: arxiv.org/abs/1510.02402

One of the Riehl-Verity papers also has a comparison of their definition of cocartesian fibrations with Lurie's

Hi Harry,

About the solidification of R^{disc}, it's just R^{disc} again, because anything discrete is already solid (being generated under colimits by Z)

So, e.g., the derived solidification of R/R^{disc} is R^{disc}[1]. Btw you can interpret the former as "closed one-forms" and the solidification as passing to homotopy classes of such

About the presentability thing, at first glance it's not clear to me, but I'll think about it when I get the chance and let you know

@DustinClausen Great! I've been thinking about it all day, and none of the ideas I had could directly prove it. You have a poset of normalized analytic structures on any given condensed connective E_∞-ring, and I think that these things glue together into a bicartesian fibration over the ∞-category of condensed connective E_∞-rings. It would suffice to show that these posets are presentable.

I'm not sure how Kosher this part is, but I've heard that the Balmer spectrum is always a locale for a small stable ⊗-triangulated category, and I think that the poset of normal analytic structures should be a subposet of this lattice. Then you would have to check that it's a sufficiently nice subposet. Warning: I haven't checked anything, but it's an idea.