Timeline for Cohesion relative to a pyknotic/condensed base
Current License: CC BY-SA 4.0
10 events
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| Feb 28, 2021 at 15:28 | comment | added | David Corfield | For a later date, the question whether differential cohesion appears in this vicinity. Perhaps, nonabelian Hodge theory done this way may be a lead: ncatlab.org/nlab/show/nonabelian+Hodge+theory#statement | |
| Feb 26, 2021 at 13:38 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Feb 26, 2021 at 7:24 | vote | accept | David Corfield | ||
| Feb 25, 2021 at 19:28 | comment | added | Peter Scholze | Ah, yes, you can also do that, it's implicit in the "etc." above. For the moment, I don't see any interesting examples of these squares, but I'll keep them in mind. | |
| Feb 25, 2021 at 16:50 | comment | added | David Corfield | Peter, you mentioned elsewhere that ∞-sheaves over the big pro-etale site on all schemes over a separably closed field k are relatively cohesive over condensed anima. What if you took a stable object in the former and formed the pullback square, using the corresponding Π and ♭dR? Does that provide a more interesting factorization? | |
| Feb 25, 2021 at 13:45 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Feb 25, 2021 at 12:31 | comment | added | David Corfield | I believe the construction works even when that left adjoint doesn't preserve finite products. So then choosing a stable object in some ∞ -topos which is relatively cohesive over condensed anima should allow such a fracture square, I think. | |
| Feb 25, 2021 at 12:31 | comment | added | David Roberts♦ | FWIW I think the only place "tangent spaces" could be hiding in the statement ncatlab.org/nlab/show/… would be in the "flat modality". Everything else there seems to me to be rather general cohesion-type technology (certainly the shape modality might be a bit special, but doesn't seem to be anything to do with tangent spaces). (Edit: cross-posted with David C) | |
| Feb 25, 2021 at 12:28 | comment | added | David Corfield | Thanks, Peter. To add to the thicket of tangent notions, there's the work of Ching et al.. I started a conversation about that here: golem.ph.utexas.edu/category/2021/02/…". The differential cohomology hexagon construction on the nLab is meant in a very general sense. Hence the warning "Beware that this is a very general conceptualization of de Rham coefficients". It just needs a cohesive ∞ -topos, and then the choice of a stable object in the latter. As far as I know, such resulting fracture squares should work for relative cohesion too. | |
| Feb 25, 2021 at 9:31 | history | answered | Peter Scholze | CC BY-SA 4.0 |