Timeline for Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
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26 events
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| Feb 11, 2019 at 3:31 | comment | added | David White | @DenisNardin thanks for making it publicly available. I also fully support and encourage you to write up your work on $N_\infty$-operads. I think fighting each other about which framework to use is a waste of productivity, and runs the risk of discouraging people from working in homotopy theory. So let's all get back to real work! | |
| Feb 10, 2019 at 17:05 | comment | added | Denis Nardin | @DavidWhite Here it is, although I feel it goes on and on to prove what is, at heart, a fairly immediate idea. I hope it can be of help to someone, however. | |
| Feb 10, 2019 at 16:01 | comment | added | Denis Nardin | DylanWilson Yes, I realized it later. @DavidWhite I'm going to write it down with all references, following Dylan's suggestion which is cleaner but it's really short so I'll probably add a few remarks to make it at least a page long... | |
| Feb 10, 2019 at 15:29 | comment | added | Dylan Wilson | @DenisNardin Another quick construction is: by the universal property of Sp amongst stable, symmetric monoidal, presentable $\infty$-categories, we get, for free, a colimit-preserving symmetric monoidal functor $\mathsf{Sp} \to \mathsf{D}(\mathbb{Z})$. The right adjoint is what we're after, which is automatically lax symmetric monoidal by nonsense. | |
| Feb 10, 2019 at 14:36 | comment | added | David White | @DenisNardin: If you don't mind putting it somewhere accessible (say, for posterity), I think that would help people who just read this comment thread. By the way, are the discussions in chat saved permanently anywhere? | |
| Feb 10, 2019 at 8:38 | comment | added | Denis Nardin | @DmitriPavlov It's really trivial. Say, a paragraph's worth of arguments. I can write it down and send it to you if you want. | |
| Feb 10, 2019 at 8:38 | comment | added | Denis Nardin | Ah, or more trivially, you do get a lax symmetric monoidal functor $\mathrm{Ch}(\mathbb{Z})[w^{-1}]\to \mathrm{Space}$ and then you use the universal property of $\mathrm{Sp}$ to provide the lax symmetric monoidal lift to spectra | |
| Feb 10, 2019 at 8:38 | comment | added | Dmitri Pavlov | @DenisNardin: I think that filling in the details will make this argument far more involved than it appears to be. Is this variant written up anywhere? As for not having conversations in comments, I never understood the point of that policy, and it is never enforced anyway. | |
| Feb 10, 2019 at 8:34 | comment | added | Denis Nardin | The enrichment sends a pair of objects $x,y$ to the functor $T\mapsto \mathrm{Map}_{\mathcal{C}}(x,T\otimes y)$ where $T$ is the natural tensoring with spaces. If you want further details, pop in the homotopy theory chatroom 'cause I don't think we are supposed to have conversations here. | |
| Feb 10, 2019 at 8:33 | comment | added | Dmitri Pavlov | @DenisNardin: How does knowing that the suspension functor is a self-equivalance help us construct the enrichment over Sp? | |
| Feb 10, 2019 at 8:32 | comment | added | Denis Nardin | The suspension functor is a self-equivalence (by explicitely providing a homotopy inverse). | |
| Feb 10, 2019 at 8:31 | comment | added | Dmitri Pavlov | @DenisNardin: How do you intend to prove that the underlying quasicategory of a stable model category is enriched over Sp? | |
| Feb 10, 2019 at 8:08 | comment | added | Denis Nardin | @DmitriPavlov I'd have to think about the Quinn-Quillen ring spectra (their construction seems quite model-independent to me), but constructing the symmetric monoidal Eilenberg-MacLane functor from dg-rings is quite trivial: it is the functor $\mathrm{Ch}(\mathbb{Z})[w^{-1}]→\mathrm{Sp}$ corepresented by the unit. This does use that the model structure on chain complexes is a symmetric monoidal stable model structure, but crucially doesn't even use that there is a model category presenting spectra. | |
| Feb 10, 2019 at 4:46 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Feb 10, 2019 at 0:42 | comment | added | Dmitri Pavlov | @DylanWilson: The traditional notion of a Grothendieck fibration of 1-categories is also not invariant under equivalence of categories; the invariant notion is known as a Street fibration: ncatlab.org/nlab/show/Grothendieck%20fibration#StreetFibration. | |
| Feb 10, 2019 at 0:32 | comment | added | Dmitri Pavlov | @DylanWilson: I am well aware of the paper by Ayala-Francis, and you do not need to point it out to me repeatedly. Your claims about fibrations being model-independent are incorrect: in order to recover Lurie's notion of a Cartesian fibration (say) from the Ayala-Francis model-independent notion one must impose an additional condition that the map involved is a fibration in the Joyal model structure. This additional condition has no model-independent meaning. | |
| Feb 10, 2019 at 0:28 | comment | added | Dmitri Pavlov | @DylanWilson: I made no claims about fibrations (other than the length of text involved in writing them down) in this post. To what statement are you referring? | |
| Feb 9, 2019 at 23:56 | comment | added | Dylan Wilson | As I stated in a different comment, your claims about fibrations are incorrect. See this paper for a model independent development of the basic theory: arxiv.org/abs/1702.02681 | |
| Feb 9, 2019 at 20:28 | comment | added | Dmitri Pavlov | @DenisNardin: Yes, the Eilenberg–MacLane functor from the category of differential-graded rings to the category of symmetric ring spectra in simplicial sets is constructed immediately using the lax monoidal structure of the Dold–Kan functor, whereas the analogous construction in the quasicategorical world is far more involved. A more sophisticated example is given by the Quinn–Quillen ring spectra constructed out of bordisms of manifolds. | |
| Feb 9, 2019 at 20:04 | comment | added | Denis Nardin | What I mean is that I'm taking the full of HTT and HA as a given (suppose we are developing them anyway for our own reasons), do we have applications that use the specific models for spectra? Sorry if I'm a bit unclear. Do you have specific examples of strict monoids in spectra that are "easier" to construct? | |
| Feb 9, 2019 at 20:02 | comment | added | Dmitri Pavlov | @DenisNardin: How exactly are ∞-operads not relevant when you work with A_∞- and E_∞-ring spectra? It is certainly much easier (in a precise sense: you need fewer lines of text) to construct a strict monoid in symmetric spectra than to construct an analog in quasicategories using fibrations. | |
| Feb 9, 2019 at 19:57 | comment | added | Denis Nardin | I'm sorry, I am a bit confused. Constructing the monoidal structure on the ∞-category Sp seems very easy to me (there are at least two ways of doing that, and one is exactly the same as for the model categories of diagram spectra, using Day convolution). Or do you mean the development of the theory of ∞-operads? this does not seem too relevant to the present question, that is specifically about the usefulness of particular models for spectra | |
| Feb 9, 2019 at 19:54 | comment | added | Dmitri Pavlov | @DenisNardin: I already pointed out such examples in my answer: monoidal categories and operads. Look at Lurie's Higher Algebra, for example, Chapter 4. | |
| Feb 9, 2019 at 19:50 | comment | added | Denis Nardin | I'd be really interested if you could point out an example where some model category of spectra is actually easier to work with (not a rhetorical question!). I've thought a bit about it, but I cannot, although that may be also because I'm very familiar with q-categories and constructions that seem trivial to me might seem very complicated to someone else. | |
| Feb 9, 2019 at 19:43 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 576 characters in body
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| Feb 9, 2019 at 19:28 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |