Timeline for Higher $\infty$-categories
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 2, 2019 at 8:56 | comment | added | Harry Gindi | @MikeShulman I may be misremembering, then! | |
| Mar 2, 2019 at 5:33 | comment | added | Mike Shulman | @HarryGindi I don't see anything in that discussion that leads me to believe that the source and target of an $\omega$-cell must be the same. | |
| Mar 1, 2019 at 1:17 | comment | added | Harry Gindi | @MikeShulman At least part of the discussion was at the end of this long thread nforum.ncatlab.org/discussion/3185/… Unfortunately I think another part was on the cat theory mailing list. | |
| Mar 1, 2019 at 1:08 | comment | added | Alec Rhea | @MikeShulman Ah, I was afraid something like this might be the case, much appreciated. I’ll look into Paoli’s paper and try to make sense of things. | |
| Mar 1, 2019 at 1:02 | comment | added | Mike Shulman | @HarryGindi ss=st and ts=tt doesn't mean that the source and target are the same, that would be something like ss = ts or st = tt. If you can remember any English words that might have appeared in the forum discussion, we can search for those. | |
| Mar 1, 2019 at 1:00 | comment | added | Mike Shulman | @AlecRhea a 3-categorical internalization would only give you 2-dimensionally-weak equivalences, and similarly for any finite $n$, whereas to get to fully weak higher categories you have to be maximally weak at every dimension. | |
| Mar 1, 2019 at 0:59 | comment | added | Harry Gindi | @AlecRhea I believe so, yes. I never read through it in complete detail, but I did see one of Paoli's talks at a conference, and it seems to be what you are describing. | |
| Mar 1, 2019 at 0:54 | comment | added | Alec Rhea | Thank you for the reference Harry, do you mean this paper arxiv.org/pdf/1707.01868.pdf? | |
| Mar 1, 2019 at 0:45 | comment | added | Harry Gindi | @AlecRhea I think you need to use Simpson-Tamsamani/Paoli's formulation. | |
| Mar 1, 2019 at 0:41 | comment | added | Harry Gindi | @MikeShulman ss=st and ts=tt at all levels. We discussed this on the nforum somewhere around a decade ago. If you know how to search there efficiently, I would be in your debt. | |
| Mar 1, 2019 at 0:35 | comment | added | Alec Rhea | @MikeShulman Thanks for catching the error — would a $3$-categorical internalization do the trick, where the component $2$-categories between objects can now internally define a notion of equivalence as arrows and $2$-cells? | |
| Mar 1, 2019 at 0:21 | comment | added | Mike Shulman | @HarryGindi What does it mean for an $\omega$-cell to have "the same n-source and n-target for all n", and why do you say that is true? | |
| Mar 1, 2019 at 0:19 | comment | added | Mike Shulman | @AlecRhea I don't think that iterated 2-categorical internalization gives you a fully weak definition of higher category either. It makes everything hold up to isomorphism, but when you get to $n$-categories for $n>2$ you need things to hold up to equivalence instead. | |
| Feb 28, 2019 at 23:10 | comment | added | Alec Rhea | I’m not familiar with the notion of a globular set but that’s my own ignorance, thank you for elaborating and I’ll look into the notion to try and understand the obstruction. (if no other answers appear in the next few days I’ll accept) | |
| Feb 28, 2019 at 23:06 | comment | added | Harry Gindi | @AlecRhea You can define a $\gamma$-globular set for any ordinal $\gamma$. You can try to extend higher categories to $\omega^+=\omega+1$, but things become trivial for the reason I described here. | |
| Feb 28, 2019 at 22:54 | comment | added | Alec Rhea | Ah, so the issue is that $n$-cells usually map between $n-1$ cells but $\omega$ has no predecessor so the algebraic definition goes wonky at the first limit ordinal stage? Are you taking an $\omega$-cell to be a collection of $n$-cells where $n$ ranges over $\omega$? | |
| Feb 28, 2019 at 22:45 | comment | added | Harry Gindi | @AlecRhea An infinite dimensional cell must have the same n-source and n-target for all n. | |
| Feb 28, 2019 at 22:40 | comment | added | Alec Rhea | Thank you, but if I understand the nlab article linked above correctly the notion of strict $\omega$-categories is what you get if you internalize at the $1$-categorical level repeatedly, but if you internalize inside a $2$-category with prior knowledge of $2$-cells you can loosen the strict horizontal equalities internalization usually gives to isos as desired and correctly iterate the recursion all the way to $\omega$ to get the same thing as the homotopy-based definition (if you also add 'completeness' to guarantee that horizontal and vertical composition 'play well'). What am I missing? | |
| Feb 28, 2019 at 22:33 | history | answered | Harry Gindi | CC BY-SA 4.0 |