Timeline for Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
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21 events
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| Feb 11, 2019 at 7:46 | comment | added | Tyler Lawson | @DmitriPavlov There are also Rezk's "Stuff about quasicategories" and Groth's "A short course course on $\infty$-categories" available online. | |
| Feb 11, 2019 at 7:45 | comment | added | Tyler Lawson | @DylanWilson Yes, I think it's a good idea to moderate the teaching of both. I'm not sure how to do this entirely. I certainly don't want to teach a definition of homotopy colimits that requires cofibrant replacement in the projective model structure, but I also want concrete methods to calculate left derived functors of colim. (Also, I look forward to seeing a version of G-spectra that is not confusing in any way whatsover...) | |
| Feb 10, 2019 at 22:17 | comment | added | Peter May | Dylan: In the end it is actual examples that interest us, and the theory comes along for the ride. RIGHT. By far the most interesting theorem in stable homotopy theory in the past decade is the HHR solution of the Kervaire invariant problem, which of course makes no use of $\infty$ categories. It seems unimaginable that it would have been found starting from such non-concrete foundations. Tyler: I totally agree with your answer, and would add that your beautiful paper on BP not being an $E_{12}$ spectrum is another such example. | |
| Feb 10, 2019 at 20:39 | comment | added | Kevin Carlson | @DmitriPavlov I'm not aware of any other comprehensive introductions to quasicategories. Riehl and Verity cover a lot of the same ground model independently, but even they need some model categorical machinery to construct their examples. | |
| Feb 10, 2019 at 19:42 | comment | added | Dmitri Pavlov | @KevinCarlson: And not just HTT, but also Cisinski's book and Joyal's notes rely on model categories very heavily. (Are there any other introductory texts on quasicategories?) | |
| Feb 10, 2019 at 17:49 | comment | added | Kevin Carlson | @DenisNardin I might add that the introductory material to quasicategory theory, at least as presented in HTT, relies for numerous fundamental results on a virtuosic command of model category theory. | |
| Feb 10, 2019 at 17:17 | comment | added | Denis Nardin | @MikeMiller Admittedly I did have the advantage of an advisor guiding me. Alas, the introductory material to quasi-categories is, I feel, not good enough for independent study (especially its almost total lack of examples). I think this is a big problem, but this is maybe a conversation to be had elsewhere... | |
| Feb 10, 2019 at 16:41 | comment | added | mme | @DenisNardin As an outsider who tried to get a little understanding of both (or either) model categories and quasicategories, the time investment to get a basic level of comfort/understanding of quasicategories seemed much higher to me. (To be clear, I make no claims about whether it is nonetheless important/worthwhile/etc; I am too far removed to have opinions, much less knowledge.) | |
| Feb 10, 2019 at 16:16 | comment | added | Denis Nardin | @DavidWhite Regarding realizability of $N_∞$-operads. That's, uh, my fault. I was supposed to write the ∞-categorical version of that stuff down for our parametrized homotopy theory project. Sorry, I'll finish it at some point... | |
| Feb 10, 2019 at 15:01 | comment | added | Dylan Wilson | ... but if the point is just that these things played an important role in the historical development of the subject, I’d agree. I just don’t agree that these developments wouldn’t have happened with some other set of foundations. In the end it is actual examples that interest us, and the theory comes along for the ride. Topological abelian groups were an inevitable object of study since they appear in nature. Strictly commutative ring spectra hardly ever appear in nature, so I think their invention was less inevitable. | |
| Feb 10, 2019 at 14:54 | comment | added | Dylan Wilson | You also raise another question, about historical developments. I’m not sure I believe this argument about G-spectra. I would argue that norms remained undiscovered for many years perhaps precisely because the notion of strictly commutative ring was obscuring them. In another timeline, they probably would’ve been invented when the need arose for HHR to compute differentials in the slice tower, regardless of the foundations that existed at the time. You also mention some other strict structures, which occur in nature. Infty-category theory has plenty of room for these questions, and... | |
| Feb 10, 2019 at 14:46 | comment | added | Dylan Wilson | rarely exist in nature. The difficult nature of homotopy coherence is replaced by (often inexplicit) cofibrant replacements and ‘natural zigzags’. So again- I think model categories are fine as a pedagogical tool, but maybe we should revise the way we teach them a bit? Perhaps take a more Dwyer-Kan approach to these things? (a compromise between full-on infty categories and model categories.) | |
| Feb 10, 2019 at 14:45 | history | edited | David White | CC BY-SA 4.0 |
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| Feb 10, 2019 at 14:42 | comment | added | Dylan Wilson | A great answer! You raise yet another important question- the pedagogical value of model categories. I agree that with strict models we may more quickly state definitions and theorems. But the cost is a host of other pitfalls for the student. I’ve had lots of confusing conversations with model category theorists talking about something like “G-spectra” and saying things that didn’t seem right, only to discover that they were secretly using some notion of weak equivalences which produced a completely different homotopy theory. The other cost is that strict models are easy to define but (contd) | |
| Feb 10, 2019 at 14:21 | comment | added | David White | Wonderful answer! On the topic of commutative $G$-spectra, I'll point out that this area provides counterpoints to Dylan's claim that he knows of "no homotopically meaningful results...that requires one of the model structures". For instance, my proof (with Javier Gutierrez) that $N_\infty$-operads exist for any realizable collection $F = (F_n)$ of families of subgroups of $G\times \Sigma_n$, is extremely simple with model categories. The $N_\infty$ operads are simply cofibrant replacements of Com in appropriate model structures on $G$-operads. I don't know an $\infty$-categorical proof. | |
| Feb 10, 2019 at 14:07 | comment | added | Yemon Choi | +1 (as an outsider) for "Not least, part of the problem is that it is difficult to appreciate the development of higher category theory before you have some familiarity with the problems that it solves" -- this applies with HTC replaced by many other things, some of which I perpetrate | |
| Feb 10, 2019 at 13:43 | comment | added | Tyler Lawson | @Denis That may be, and I don't want to suggest that model categories are intuitive. But there is a level of operation before model categories, or an alternative, become needed, and I want to say that working coherently makes that step much harder. | |
| Feb 10, 2019 at 11:36 | comment | added | Denis Nardin | From the didactic point of view, I have to say that people sometime forget how hard learning model categories can be. From the perspective of someone who has had to learn model categories and quasicategories at the same time, model categories were a lot harder and less motivated. I did learn them, in the end, but I'm not 100% sure they are as intuitive as some people suggest. | |
| Feb 10, 2019 at 11:08 | history | edited | Tyler Lawson | CC BY-SA 4.0 |
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| S Feb 10, 2019 at 10:19 | history | answered | Tyler Lawson | CC BY-SA 4.0 | |
| S Feb 10, 2019 at 10:19 | history | made wiki | Post Made Community Wiki by Tyler Lawson |