Timeline for Is the canonical model structure on strict $\infty$-Cat left proper?
Current License: CC BY-SA 4.0
11 events
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Apr 12, 2021 at 14:59 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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Apr 12, 2021 at 14:48 | history | edited | David White | CC BY-SA 4.0 |
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Apr 10, 2021 at 15:44 | history | edited | Simon Henry | CC BY-SA 4.0 |
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Apr 10, 2021 at 15:32 | comment | added | Tim Campion | I think maybe the transferred model structure does not exist, though. I think there are strict $\omega$-categories $C$ not weakly equivalent to a strict $\omega$-category $C'$ with fibrant Street nerve. | |
Apr 10, 2021 at 15:16 | comment | added | Tim Campion | If the argument makes sense to you, then you're ahead of me -- your first objection is compelling to me! One bit of wiggle room that could be used is the fact that left properness is a property that depends only on the weak equivalences and not the cofibrations. So if the transferred model structure exists, is left proper, and has the same weak equivalences as the canonical model structure, that implies the canonical model structure is also left proper. | |
Apr 10, 2021 at 15:13 | comment | added | Simon Henry | @TimCampion : Ah I understand your argument now, indeed the localization part was important. the problem is in the left transfered: the left adjoint isn't a Quillen functor. In the verity model structure you have a cofibration that "makes a cell marked", which is send to the map in $Cat_\omega$ that collaps a cell to an identity, which isn't a cofibration. | |
Apr 10, 2021 at 14:57 | comment | added | Simon Henry | @TimCampion Assuming it is true that the model structure is transfered from Verity's model structure (because that does sounds reasonable), I don't think this implies that left properness is preserved does it ? I think for example any tractable simplicial model structure is left transfered from the Ching-Riehl model structure on its coalgebraic cofibrant object which is always left proper. (or maybe the "localization" part is important in your argument ? ) | |
Apr 10, 2021 at 14:53 | comment | added | Tim Campion | The Street nerve is a fully faithful functor $Cat_\omega \hookrightarrow Strat$ from strict omega-categories to stratified simplicial sets, which has a left adjoint exhibiting it as a localization. I suspect the canonical model structure on $Cat_\omega$ is projectively-transferred along this adjunction from the Verity model structure for complicial sets. Verity's model structure is left proper, so this should imply that the canonical model structure is also left proper, right? | |
Apr 10, 2021 at 14:28 | answer | added | Mike Shulman | timeline score: 6 | |
Apr 10, 2021 at 13:36 | history | edited | Simon Henry | CC BY-SA 4.0 |
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Apr 10, 2021 at 13:31 | history | asked | Simon Henry | CC BY-SA 4.0 |