Timeline for Inductive folk model structure on strict ω-categories
Current License: CC BY-SA 4.0
6 events
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| Nov 5, 2018 at 12:23 | comment | added | Harry Gindi | @SimonHenry No, no, I was agreeing! My comment was about constructing a nerve that is a right-Quillen functor to Rezk Θ-spaces. When markings are introduced, I was thinking maybe you could do what I said there, but I just realized that the construction I just defined fails to be functorial, so it's back to the drawing board. Anyway, yes, the idea of using markings is definitely the right idea! | |
| Nov 5, 2018 at 12:17 | comment | added | Simon Henry | I don't really follow you last comment, I probably don't know the construction you are refering too, so maybe your idea is better. My idea was more to essentially follow the original strategy of Lafont, Metayer, Worytkiewicz appropriately corrected to take the marking into account. Probably the only place that require some works is in adapting the construction of cylinder/path objects in a way that treat the marking correctly. | |
| Nov 5, 2018 at 12:05 | comment | added | Harry Gindi | @SimonHenry Yes, indeed, this looks like the correct way to do it! The obvious adjunction to construct in this case comes from the functor $\Theta\times \Delta \to \operatorname{Cat}^+_\omega$ defined by the rule $([t],[n])\mapsto [t] \times \overline{\mathcal{O}^n}$ where $\overline{\mathcal{O}^n}$ is the nth oriental with its unique nontrivial globular n-cell marked. Interesting question: Does Ara's rigidity result still apply to this functor with markings introduced? Edit: Yes, probably. This functor still does not preserve cofibrations, so we end up back at the other question. | |
| Nov 5, 2018 at 10:51 | comment | added | Simon Henry | I don't know the answer to the question you ask (and I don't really have an intuition on whether the answer should be yes or no). But what should defintitely exists and somehow represents "inductive strict $\infty$-categories" is a model strucutre on "maked strict $\infty$-categories" where fibrant objects are those where the markings satisfies some stability property and where marked cell can be reversed up to higher marked cells, and equivalence between bifibrant objects are the things you can invert modulo a marked natural transformation. | |
| Nov 5, 2018 at 10:36 | answer | added | Maxime Lucas | timeline score: 4 | |
| Nov 4, 2018 at 23:08 | history | asked | Harry Gindi | CC BY-SA 4.0 |