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Jan 20, 2021 at 9:47 comment added Denis Nardin @Yai0Phah No you're right, profinite animas embed fully faithfully into condensed animas. But if you want to continue this discussion we should move to chat, since this is getting long for the comments.
Jan 20, 2021 at 9:45 comment added user20948 Condensed world is much more general. In some sense, I am asking how lossy when passing these pro-anima (well, in their terms) to condensed anima. For pro-anima coming from profinite spaces, this should be fully faithful. I don't know whether I am mistaken?
Jan 20, 2021 at 9:27 comment added Denis Nardin @Yai0Phah If you're interested in spectral formal geometry, I suggest looking into the recent developments in condensed mathematics. I suspect this will be the right home for the theory, clarifying various issues that appeared even in the classical theory. Unfortunately there are still not many references for it, but the recent masterclass in Copenhagen can be a useful starting point.
Jan 20, 2021 at 9:21 comment added user20948 Thank you very much. I am more interested in objects coming from formal geometry. The setup of adic $\mathbb E_\infty$-rings on SAG seems to be ugly. However, according to your response, pro-$\mathbb E_\infty$-rings are probably not the correct objects. Do you know anything between profinite spaces and pro-spaces which is still well-behaved in some sense?
Jan 19, 2021 at 20:10 comment added Denis Nardin @Yai0Phah No that's false. This is one of the reason to restrict to profinite spaces, where it is true (a reference for this fact is theorem E.3.1.6 in Lurie's Spectral algebraic geometry, appendix E in general is a thorough study of profinite homotopy theory)
Jan 19, 2021 at 19:25 comment added user20948 Is it true that equivalences of pro-spaces are detected by the equivalences of underlying pro-system of homotopy groups? Is there any $\infty$-categorical proof?
Dec 28, 2018 at 13:43 history edited Denis Nardin CC BY-SA 4.0
Added references
Dec 27, 2018 at 19:44 comment added Denis Nardin @TimCampion Yes, I forgot a lot of the theory was also there. The comparison with Friedlander should follow immediately from the description of the sheafification functor in the hypercomplete case (where the sheafification can be computed by a colimit over all the hypercovers). I'll try to hunt down a few more precise references and add them to this answer this evening or tomorrow
Dec 27, 2018 at 19:04 comment added Tim Campion I think I'm happy with the shape theory in HTT, it's really the part where one passes from a pro-space to a pro-($\pi$-finite)-space that confuses me (which I don't think is to be found in HTT). For instance, has anybody actually proved in the literature that this construction agrees with what Friedlander does? I see now from the citation in the exodromy paper that Appendix E of SAG seems to be about this, maybe it's there?
Dec 27, 2018 at 18:56 comment added Denis Nardin The theory in this language can be found in Higher Topos Theory, section 7.1.6. Another related paper where a generalization of this theory is considered is arxiv.org/abs/1807.03281
Dec 27, 2018 at 18:55 comment added Denis Nardin @TimCampion What finiteness conditions do you mean? The profinite completion is just for convenience (it's kind of hard to understand what information is actually contained in a prospace, a profinite space is more accessible)
Dec 27, 2018 at 18:53 comment added Tim Campion I've always been confused about how the finiteness conditions enter in. Is there a place where this modern perspective is written up carefully?
Dec 27, 2018 at 11:21 history edited Denis Nardin CC BY-SA 4.0
Remark about why hypercovers appear
Dec 27, 2018 at 8:34 history edited Denis Nardin CC BY-SA 4.0
changed notation
Dec 27, 2018 at 8:17 history edited Denis Nardin CC BY-SA 4.0
Added remark
Dec 27, 2018 at 8:10 vote accept geometer
Dec 27, 2018 at 8:10 history answered Denis Nardin CC BY-SA 4.0