Timeline for Why do we need model categories?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 28, 2017 at 9:40 | comment | added | Denis Nardin | @SimonHenry I'm pretty sure that Dylan means the former. There are two big insights in Voevodsky's work on the Milnor conjecture: the usage of 'sheaves up to homotopy' and the application of standard (in homotopy theory!) properties of the Steenrod algebra to solve an algebro-geometric problem. Neither of them is reliant on the homotopical framework of choice, and the former in particular is a lot easier to do with ∞-categories than with model categories. | |
| Nov 28, 2017 at 8:43 | comment | added | Simon Henry | @DylanWilson: Could you clarify what you mean (I just want to be sure I understand): what you say is that the new and fundamental contribution of Voevodsky's work on Milnor conjecture was the use of homotopy theoretic methods, but the specific the form of these methods (model categories, Derivators, quasi-categories, homotopical categories, fibration categories and so one) is not essential, or that the homotopy theoretic point of view itself is not an essential part in the proof of the Milnor conjecture and the main ideas can be rephrased without it ? | |
| Nov 27, 2017 at 22:39 | comment | added | Dylan Wilson | @LennartMeier I guess I'm not actually disagreeing with you. It is true that we need some framework to work with, and which one we choose or find easiest is up to us. But it's hard to point to one and say "that's the best one" from a practical point of view, since that is a matter of personal skill and familiarity with a given framework. That said, I think it's possible (but not so mathematically relevant) to make arguments about which framework most closely captures the 'Platonic' notion of what a 'homotopy theory' is. | |
| Nov 27, 2017 at 22:34 | comment | added | Dylan Wilson | @LennartMeier: I think it's a matter of history mixed with personal preference. For example, had the foundations of equivariant homotopy theory been written down using, say, derivators first instead of model categories, I'm sure that wouldn't have slowed HHR down. They used what was available to them in the literature (and even had to rewrite and revisit some of that- they also specifically de-emphasized model categories in their appendix and preferred to use homotopical categories; model category structures were instead viewed as a way of producing flat objects in sufficient supply.) | |
| Nov 27, 2017 at 22:20 | comment | added | Lennart Meier | @DylanWilson: While the concrete choice of homotopical framework is neither essential for motivic homotopy theory nor for HHR, without the existence of such frameworks both would have been as hard to carry out als quantum mechanics without Hilbert spaces or even matrices (doable, as Heisenberg did, but probably not communicable). The existence of such frameworks is content expressed as form and maybe not the point, but maybe the plane we draw in on. | |
| Nov 27, 2017 at 16:53 | comment | added | Denis Nardin | I'm sorry, but I have to agree with Dylan. As a matter of fact, last year in the Thursday seminar we went through the proof of the Milnor conjecture more-or-less completely and I don't think we mentioned model categories once (and we could have easily done the same with HHR). Model categories are a useful tool, but not for these things, and in fact using them tends to obscure the more fundamental ideas here | |
| Nov 27, 2017 at 15:47 | comment | added | Dylan Wilson | without them (see, for example, Nikolaus-Scholze on the construction of the Tate diagonal, which is essentially equivalent.) | |
| Nov 27, 2017 at 15:47 | comment | added | Dylan Wilson | the actual content in both works lies in a place that has little to do with what formalism you use. For Voevodsky, one has the (honestly) fundamental computations of the Steenrod operations and the non-formal input due to Rost about the motive of a quadric. For HHR one has the construction of the slice spectral sequence and the computation of the slices of norms of real cobordism. The construction of the norm, while seemingly dependent on the formalism of model categories, can be done... | |
| Nov 27, 2017 at 15:47 | comment | added | Dylan Wilson | Oy, I'm trying not to get pulled in but I can't help it... I think it is very misleading to call the use of model categories in Voevodsky's work and Hill-Hopkins-Ravenel's paper "fundamental", for two reasons. First, the formalism of model categories in both cases is in fact not essential, in the sense that it is possible (no claims on easier!!) to utilize a different way of doing homotopy theory than using model categories. Second, whether you use model categories or infinity categories or derivators or whatever that is not the point: ... | |
| Nov 27, 2017 at 15:36 | history | answered | David White | CC BY-SA 3.0 |