Timeline for Does the cubical nerve preserve weak equivalences of simplicial sets?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 19, 2019 at 4:13 | comment | added | Tim Campion | @AlexanderCampbell as I recall yes. I read about them in Grandis and Mauri ‘s “cubical sets and their site “ | |
| Feb 19, 2019 at 4:08 | comment | added | Alexander Campbell | @TimCampion Is there a relationship with the symmetric cubical sets studied in this paper of Isaacson? | |
| Feb 18, 2019 at 19:51 | comment | added | D.-C. Cisinski | It seems I was overconfident: Prop. 4.3.12 does not imply that forgetting connections preserves weak equivalences (all other assertions are correct though). It is true that forgetting connections preserves weak equivalences, but we have to adapt the argument (a monoidal (as opposed to Cartesian) version of 4.3.12). | |
| Feb 18, 2019 at 14:55 | comment | added | Tim Campion | @AlexanderCampbell There's no reference that I'm aware of, but I had in mind adapting Theorem 8.4.13 in Asterisque 308. And a model structure on cubical sets with connections and extensions with the expected generating cofibrations and generating acyclic cofibrations can be transferrred from the non-extension world by mimicking Prop 8.3.8 in Asterisque 308, which does something similar for symmetric simplicial sets. | |
| Feb 18, 2019 at 14:50 | vote | accept | Tim Campion | ||
| Feb 18, 2019 at 14:48 | comment | added | Tim Campion | It seems I didn't read carefully enough, because I had the impression that 4.3.12 required the nerve to be induced by a functor between the sites (which this one is not). But no, it is more powerful than I realized -- thanks! | |
| Feb 18, 2019 at 8:48 | comment | added | D.-C. Cisinski | The main tool here is Prop. 4.3.12 in Astérisque 308. This proves that the "cubical nerve" as above, but also its version with connections do preserve and detect weak equivalences. It also implies right away that there is a symmetric version (from symmetric simplicial sets to cubical sets with reversals), and that forgetting connections also preserve and detect weak equivalences. Since the link between simplicial sets and symmetric simplicial sets is also documented (prop. 8.3.8 in loc. cit.), this says that all versions you could naturally come with will be homotopically well behaved. | |
| Feb 18, 2019 at 8:38 | comment | added | Alexander Campbell | Is there a reference for cubical sets with extensions? | |
| Feb 18, 2019 at 7:22 | answer | added | Alexander Campbell | timeline score: 6 | |
| Feb 18, 2019 at 5:13 | comment | added | Dylan Wilson | Of course, there is a natural collection of simplicial sets containing Kan complexes and nerves of categories ;) | |
| Feb 18, 2019 at 3:33 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 18, 2019 at 3:27 | history | asked | Tim Campion | CC BY-SA 4.0 |