Timeline for Higher $\infty$-categories
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 2, 2019 at 16:03 | vote | accept | Alec Rhea | ||
| Mar 2, 2019 at 11:25 | answer | added | Simon Henry | timeline score: 14 | |
| Mar 2, 2019 at 7:32 | history | edited | Alec Rhea | CC BY-SA 4.0 |
corrected error pointed out by Mike, added open problem mentioned by Harry
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| Mar 2, 2019 at 7:18 | comment | added | Alec Rhea | @ShayBenMoshe That is a compelling reason to consider the infinite-dimensional sphere under its product topology, but is it true because colimits categorify disjoint unions and quotients which yield an 'only finitely many coordinates are nonzero' behavior? For example is the infinite-dimensional sphere also the limit of that same sequence of embeddings of spheres as equators, and if so would this correspond to the box topology on it? | |
| Mar 2, 2019 at 5:31 | comment | added | Mike Shulman | @HarryGindi It's easy to define a strict $\mathbb{Z}$-category. You can try to define a weak one by adapting Batanin's algebraic approach. Alternatively you can try to define some kind of "spectra" in $\omega$-categories. But I don't know of anyone who's seriously pursued it. | |
| Mar 1, 2019 at 10:03 | comment | added | Denis Nardin | @ShayBenMoshe As soon as you leave the nice world of spaces homotopy equivalent to CW complexes, you've got to be more specific with what you mean by "contractible" (weakly contractible? of contractible weak shape? of contractible strong shape? With a homotopy between the identity and a constant map?) I suspect it is weakly contractible (i.e. that maps from $S^n$s "cannot" see the difference between the product and the box topology) but I'll confess not having thought about it. | |
| Mar 1, 2019 at 8:47 | comment | added | Shay Ben Moshe | @AlecRhea I rewrote this as an answer. Regarding the topology, yes I work with the product topology. The infinite-dimensional sphere is also the colimit of the embeddings of spheres as equators. In those two respects this is the more natural choice. I don't know the answer for the box topology, never thought about this, maybe someone else does. | |
| Mar 1, 2019 at 8:44 | answer | added | Shay Ben Moshe | timeline score: 6 | |
| Mar 1, 2019 at 4:13 | comment | added | Eric Peterson | I would also like to know if there’s such a definition available! I’m aware of Kan’s “semisimplicial spectra”, so I understand the remark about Z-groupoids, but I myself have never seen a Z-non-groupoid. | |
| Mar 1, 2019 at 1:14 | comment | added | Harry Gindi | @MikeShulman has anyone ever come up with such a definition? | |
| Mar 1, 2019 at 1:08 | comment | added | Alec Rhea | @MikeShulman Probably a good call, thanks for indulging me :^). | |
| Mar 1, 2019 at 1:03 | comment | added | Mike Shulman | @AlecRhea I think you should probably start by trying to think of examples, and then, if you find any, formulate the definition so as to describe them. | |
| Mar 1, 2019 at 0:30 | comment | added | Alec Rhea | @MikeShulman Very interesting! My intuition in response to Harry was that the Grothendieck ring of a $\delta$-number ordinal (or all of $O_n$) has predecessors for all limit ordinals and so might admit a better definition but I thought it was far fetched, but $\mathbb{Z}$ is the Grothendieck ring of $\omega$, so maybe this generalization could also make sense? | |
| Mar 1, 2019 at 0:23 | comment | added | Mike Shulman | A different generalization that I think is interesting is the notion of "$\mathbb{Z}$-category", in which negative (but finite) dimensional cells are also allowed. In particular, a "$\mathbb{Z}$-groupoid" is, or should be, the same as a spectrum in the sense of homotopy theory. | |
| Feb 28, 2019 at 23:40 | comment | added | Alec Rhea | @ShayBenMoshe Possibly another dumb question but are you working in the product topology when you say that the infinite dimensional sphere is contractible? Is it still contractible in uniform/box topologies or would $\infinity$-categories only be sensitive to product topology considerations? | |
| Feb 28, 2019 at 23:18 | comment | added | Alec Rhea | @ShayBenMoshe It seems like this answers the question if you’d like to post it as an answer, assuming that any ‘good’ definition of $\infty$-category for higher infinities would match the homotopy definition at $\omega$ and thus become trivial. | |
| Feb 28, 2019 at 23:12 | comment | added | Alec Rhea | @ShayBenMoshe Fascinating, thank you. (second part of comment deleted due to dumb mistake) | |
| Feb 28, 2019 at 23:08 | comment | added | Harry Gindi | @ShayBenMoshe Yes, precisely. | |
| Feb 28, 2019 at 22:59 | comment | added | Shay Ben Moshe | I was thinking about this at some point, and I had some thoughts (although I might be completely wrong). The numbers appearing can be seen as dimensions of cells, and infinite-dimensional sphere is contractible. So it seems, at least from that naive perspective, that infinite ordinals will contribute no homotopical information. | |
| Feb 28, 2019 at 22:33 | answer | added | Harry Gindi | timeline score: 10 | |
| Feb 28, 2019 at 22:27 | comment | added | Alec Rhea | @DenisNardin I can understand that sentiment, but for me $\infty$-category theory looks like a beautiful tower of abstraction and I wonder why it stops at the first infinity. (of course I am looking up from the bottom of the tower and not constructing it at the top, so some of those construction workers who frequent this site may be able to correct my misapprehension) | |
| Feb 28, 2019 at 22:19 | comment | added | Denis Nardin | At least for me the reason I care about $(\omega,n)$-categories is that they are the one which show up in practice. But I'd love to see examples of more exotic things around! | |
| Feb 28, 2019 at 22:16 | history | asked | Alec Rhea | CC BY-SA 4.0 |