Timeline for Are higher categories useful?
Current License: CC BY-SA 3.0
28 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 26, 2015 at 18:11 | comment | added | Ryan Budney | @DmitriPavlov: There's a whole spectrum of ways to split hairs here. It's probably best to indicate, concretely, how higher category theory is valued by people who would not be considered members of that community. Perhaps it is not vital to the proofs of any theorems, but this question is about how (from outside the field) one might value the subject. So it's not meant to be an appeal to people that are already invested in the tools. Perhaps I should have said something more like that to begin with. | |
| Jan 26, 2015 at 15:33 | comment | added | Dmitri Pavlov | @RyanBudney: The answer depends on your definitions of “used” and “need”. Using (higher) categories in this context is a bit like using complex numbers instead of pairs of real numbers: one can always rewrite a result with complex numbers using pairs of real numbers, so in the strictest possible sense complex numbers are not “needed”. | |
| Jan 26, 2015 at 15:17 | comment | added | Ryan Budney | @DmitriPavlov: I suspect you and Andy are using different criteria for "play a role". Likely Andy is setting the bar at "is used to prove a foundational result in the theory" or something like that. For example, Teichmuller theory and geometric flows are used to prove foundational results in 3-manifold theory. Saying that I can (if I choose) use subject X to study subject Y is a far lower bar. Do people in subject Y need that tool for something? Answering that would get you somewhere but it would still be significantly below Andy's bar. | |
| Dec 11, 2011 at 15:42 | answer | added | Dmitri Pavlov | timeline score: 11 | |
| Dec 10, 2011 at 10:22 | comment | added | Dmitri Pavlov | @Andy: I gave you some very concrete examples (C*-dynamical systems, free probability theory) that disprove your claims. Also, I never claimed that higher categories are “central” in mathematics. | |
| Dec 9, 2011 at 19:23 | comment | added | Andy Putman | (and with that, I'm going to bow out of this conversation). | |
| Dec 9, 2011 at 19:23 | comment | added | Andy Putman | @Dmitri : You're welcome to whatever opinion you want; however, your take on the centrality of higher categories in mathematics is a bit outside the mainstream, and simply loudly asserting it is unlikely to convince the skeptics. | |
| Dec 9, 2011 at 19:18 | comment | added | André Henriques | @Dmitri, Mariano, Andy: Please don't let this discussion get overheated. Bicategories might one day play an essential role in all of math,... but I don't think that's the case today. | |
| Dec 9, 2011 at 18:11 | comment | added | Dmitri Pavlov | @Andy: Even if we assume for a moment that higher categories play no role in the areas you listed (which is clearly false, bimodules/bicategories are used in C*-dynamical systems, free probability theory, etc.), I still fail to see how they constitute “most of mathematics”, as Mariano claims. As for “extraordinary claims”, I regard claims that higher categories have no applications whatsoever to a certain area as extraordinary, not vice versa. | |
| Dec 9, 2011 at 16:57 | comment | added | Mariano Suárez-Álvarez | +1 Andy; Indeed! I am pretty sure there would be much rejoicing if higher categories end up being useful and fundamental in dealing with weighted weak inequalities for singular integral operators or in establishing existence results for smooth solutions to two or three non linear evolution equations (say Euler equations, to keep our optimism to reasonable levels) I am quite sure this is not the case and that no one expects it to be so in the near future. | |
| Dec 9, 2011 at 16:30 | comment | added | Andy Putman | @Dmitri : I think that the onus of providing evidence is on someone making an extraordinary claim (eg that higher category theory is useful in most of mathematics). That being said, it is easy to enumerate huge swaths of math in which higher category theory has hitherto played no role. For example, analysis, Riemannian geometry, geometric topology (other than quantum topology, which is really a different subject), dynamics, geometric group theory, analytic number theory, probability theory, etc. And even in areas where it plays some role (eg algebraic geometry), most people don't use it. | |
| Dec 9, 2011 at 11:17 | comment | added | Dmitri Pavlov | @Mariano: Could you please supply some evidence to your claim “most of mathematics has absolutely no use whatsoever of higher categories”? I feel that exactly the opposite is true, at least if one considers bicategories as higher categories. Also, how about this claim: Most of science has absolutely no use whatsoever of higher mathematics? Here by “higher mathematics” I mean everything except calculus and other elementary mathematics and by “science” in the expression “most of science” I mean sciences that collect empirical evidence (i.e., not mathematics or computer science). | |
| Dec 9, 2011 at 9:09 | comment | added | Tim Porter | I remember in the 1970s going to conferences where all the talks were on stable homotopy theory and we, as postgrads, were being told that it was the most useful part of homotopy theory. That sort of claim is always over the top. Stable homotopy theory is very useful, and also has its limitations! | |
| Dec 9, 2011 at 4:10 | comment | added | Andy Putman | @Buschi and @Andre : I think that statements like that are part of the reason that some mathematicians are a bit skeptical about higher category theory. | |
| Dec 9, 2011 at 0:51 | comment | added | Dylan Wilson | How about this: arxiv.org/abs/0810.4535 ? | |
| Dec 9, 2011 at 0:37 | comment | added | user9072 | Mariano, might be some scientists think or at least thought (some decades ago) this about science and math, too. (At least if the latter is interpreted in a nontrivial way.) | |
| Dec 8, 2011 at 22:05 | answer | added | Mike Shulman | timeline score: 17 | |
| Dec 8, 2011 at 21:46 | comment | added | Mariano Suárez-Álvarez | André, your comparison is very very skewed: it is a safe bet that most of mathematics has absolutely no use whatsoever of higher categories! | |
| Dec 8, 2011 at 20:52 | comment | added | Jonathan Chiche | "Are there any results where higher categories or the higher categorical perspectives play an essential role?" Higher categories play an essential role in making me thrill. (Although I must admit that this result may be reached by other means.) | |
| Dec 8, 2011 at 19:54 | comment | added | André Henriques | Here's a thought: [Higher category theory : Mathematics] = [Mathematics : Science]. I was inspired by xkcd.com/435. More seriously, is it maybe the case that the reactions of generic mathematicians towards higher category theory are similar to those of generic scientists towards mathematics?... | |
| Dec 8, 2011 at 19:46 | answer | added | Konrad Waldorf | timeline score: 14 | |
| Dec 8, 2011 at 19:45 | comment | added | Buschi Sergio | Higher stegory theory is a higher (and more deep) point of view to less higher mathematical structures (categories), then question become : is the mathematic useful? | |
| Dec 8, 2011 at 19:37 | answer | added | Konrad Waldorf | timeline score: 42 | |
| Dec 8, 2011 at 19:04 | answer | added | Tim Porter | timeline score: 12 | |
| Dec 8, 2011 at 18:17 | answer | added | Kevin Walker | timeline score: 23 | |
| Dec 8, 2011 at 17:42 | comment | added | Eitan Chatav | I think they play a big role in low dimensional topology. Braided monoidal categories, which are really 3-categories, are a big deal in quantum invariants of knots and 3-manifolds. | |
| Dec 8, 2011 at 17:20 | answer | added | André Henriques | timeline score: 21 | |
| Dec 8, 2011 at 17:00 | history | asked | Chris Schommer-Pries | CC BY-SA 3.0 |