Timeline for Is there a condensed / pyknotic refinement of the shape of an $\infty$-topos?
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12 events
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| Feb 13, 2021 at 17:50 | comment | added | user20948 | @TimCampion The "obvious" way is to take the limit of the pro-object in the category of condensed objects (this should be lossy). Maybe it is helpful to look at the comments in this answer. I believe him that in general, pro-objects don't seem to be well-behaved, and condensed objects are much better behaved. | |
| Feb 13, 2021 at 17:44 | comment | added | Tim Campion | @Yai0Phah I don't know. The only way I know is to take the profinite topology. I don't know how that's supposed to relate to the "condensed direction". | |
| Feb 13, 2021 at 17:44 | comment | added | user20948 | @TimCampion How do you get a topology from being a pro-system (without passing to the pro-finite one)? | |
| Feb 8, 2021 at 20:05 | vote | accept | Tim Campion | ||
| Feb 8, 2021 at 20:05 | comment | added | Tim Campion | So the missing ingredient seems to be Wolf's paper that Clark pointed out, which does show (unless the terminology is misleading me) that the pro-etale topos at any rate is a presheaf topos in the pyknotic sense. Possibly the answer should be that in the condensed / pyknotic setting, Lurie's notion of shape is not necessary because most toposes of interest are now presheaf toposes anyway. I think the question still remains, but I will re-accept this answer for now at least. Thanks everyone! | |
| Feb 8, 2021 at 16:43 | comment | added | Tim Campion | @SimonHenry Thanks, I should have remembered / realized that. In fact, it would make perfect sense if the pro-etale topos is a pyknotic presheaf topos. It's still mysterious to me that I don't see how the topology one gets from being a pro-system relates to the topology that one gets from being pyknotic... | |
| Feb 8, 2021 at 16:38 | comment | added | Simon Henry | @TimCampion : A naive comment as I would need to spend much more time reading this paper to understand all the notation. But an obvious connection between shape and realization is that the shape of Prsh(C) is an infinity groupoid, and corresponds to the realization of the category C. So given that what they call Gal(X) is (if I understand correctly) a topologized version category of points, that does makes a lot of sense. | |
| Feb 8, 2021 at 16:32 | comment | added | Tim Campion | This certainly validates the idea that the pro-etale fundamental group is indeed the pyknotic fundamental group of a pyknotic space, but hang on -- the construction described in 13.8.10 is the pyknotic analog of the geometric realization of a category -- not the shape of an $\infty$-topos. (This actually makes it potentially even more interesting to me for unrelated reasons...) I ought to know the relation between geometric realization and shape in ordinary $\infty$-categories, but I don't... | |
| Feb 8, 2021 at 13:11 | vote | accept | Tim Campion | ||
| Feb 8, 2021 at 16:32 | |||||
| Feb 8, 2021 at 13:09 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
added link to v7 of Exodromy on the arXiv
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| Feb 8, 2021 at 11:25 | comment | added | Clark Barwick | See also Sebastian Wolf's beautiful paper arxiv.org/abs/2012.10502. | |
| Feb 8, 2021 at 10:46 | history | answered | M L | CC BY-SA 4.0 |