Timeline for Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
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| Feb 10, 2019 at 22:14 | comment | added | Clark Barwick | Oh, this you mean. Yes. You can definitely state it that way. | |
| Feb 10, 2019 at 19:48 | comment | added | Dmitri Pavlov | @ClarkBarwick: My fault, I should have mentioned that the space of choices is B(Z/2)^n, of course. As for your comment about spaces of model categories: can we not state uniqueness in terms of spaces of relative categories, using the model structure constructed in your paper with Kan? (And passing to the next Grothendieck universe to allow for large categories.) | |
| Feb 10, 2019 at 17:58 | comment | added | Clark Barwick | @DylanWilson I did indeed; thank you. I probably also should have said that the choice of orientation is a discrete choice, or contractible once you choose a component. | |
| Feb 10, 2019 at 15:25 | comment | added | Dylan Wilson | @ClarkBarwick Thanks Clark! I also dislike the tone of the debate. And did you mean B(Z/2)^n? | |
| Feb 10, 2019 at 13:19 | comment | added | Clark Barwick | I know no purely model-category-theoretic formulation of this, because I know no way to speak accurately about a space of model categories. The result Dmitri states is however a consequence thereof. | |
| Feb 10, 2019 at 13:19 | comment | added | Clark Barwick | I don't understand this debate, and I dislike its internet-y tone. I would like however to correct a misapprehension about the Unicity Theorem of Schommer-Pries and me. We do more than just construct an equivalence of homotopy theories between any model and ϴ_n-spaces. We show that the space of homotopy theories of (∞,n)-categories is a BZ/2. That means that not only are there equivalences between any two models, but they are unique up to orientation, which is in turn a contractible choice. | |
| Feb 10, 2019 at 4:46 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Feb 10, 2019 at 1:14 | comment | added | Dmitri Pavlov | @DylanWilson: “Far removed” is a stretch: the OP just edited his question to indicate that he is interested in (∞,1)-categories, so discussing to what extent quasicategories can be substituted for (∞,1)-categories is directly relevant to determining what fits the scope of OP's question and what doesn't. As for your request to move this to chat: sorry, but no way. This discussion will be read by hundreds of people in the future, whereas the chat will be seen only by a few, and will then be buried forever. | |
| Feb 10, 2019 at 1:05 | comment | added | Dylan Wilson | Is there some reason you have ignored my repeated requests to move this discussion (which is by now far removed from the OP’s question) to chat? | |
| Feb 10, 2019 at 0:38 | comment | added | Dmitri Pavlov | @DylanWilson: Yes, I agree with your last claim. Lurie's books, as well as many (most?) papers that use the term “∞-category” make use of such model-dependent statements every once in a while, as you can see. That's why “∞-category” can only mean quasicategory and not “(∞,1)-category”. | |
| Feb 10, 2019 at 0:34 | comment | added | Dylan Wilson | That statement in HTT asks for a map of simplicial sets to be an isomorphism, which is not a model independent claim. Obviously it is not possible to give model independent proofs of model dependent claims! | |
| Feb 10, 2019 at 0:11 | comment | added | Dmitri Pavlov | @DylanWilson: Yes, any (∞,1)-functor classifies a lax (∞,1)-functor to the category of (∞,1)-categories and (∞,1)-profunctors. (I think I misinterpreted what you were saying about inner fibrations.) As for your claim about all theorems in Lurie's book, what would the procedure that you described do to Proposition 2.3.3.7 in HTT, for example? (This proposition is one of many examples of “Many theorems in Lurie's books”.) | |
| Feb 9, 2019 at 23:51 | comment | added | Dylan Wilson | (All the theorems you are worried about in Lurie remain true by replacing pullbacks/pushouts of ssets with htpy pullbacks/pushouts of quasicategories and using the model independent definitions I referenced. The first line of the new proof in HTT would then read “begin by replacing all maps by inner fibrations...” etc.) | |
| Feb 9, 2019 at 23:49 | comment | added | Harry Gindi | By the way, there was a really long period of time before HTT where ∞-category meant strict or some algebraic model of weak omega category. If it were up to me, I'd use the term "weak n-category" for (∞,n)-category (including in dimension 1), since there will never be a usable algebraic model for weak n-categories above dimension 2. | |
| Feb 9, 2019 at 23:46 | comment | added | Dylan Wilson | Yes, actually, they can... cf Ayala-Francis “fibrations” paper for model-independent definitions. When implemented inside the model of quasicategories they will be equivalent to saying “a map of quasicategories homotopic to a ??-fibration”. I don’t understand what you mean about inner fibrations: every map between quasicategories is homotopic to an inner fibration, so the model independent notion would just be “any functor”. | |
| Feb 9, 2019 at 23:34 | comment | added | Dmitri Pavlov | Left fibrations cannot be given a model-independent definition, because they require (1) a model-independent lifting property and (2) a model-dependent property (i.e., being a fibration in the Joyal model structure). Many theorems in Lurie's books will be false if you drop (2) from the definition of a left fibration. Your claim about inner fibrations being different from left fibrations is not correct: inner fibrations also have a model-independent component, they classify (∞,1)-functors to the (∞,1)-category of (∞,1)-categories and (∞,1)-profunctors between them. | |
| Feb 9, 2019 at 23:26 | comment | added | Dylan Wilson | Left/right fibrations, (co)cartesian fibrations and exponentialble (“flat inner”) fibrations can all be given model independent definitions. Inner fibrations and categorical fibrations can’t but these are not fundamental for the theory (precisely for this reason). | |
| Feb 9, 2019 at 23:26 | comment | added | Dmitri Pavlov | Yes, I am aware of Lurie's notes. In Lecture 6 he defines (in Definition 6) L(C,Q) using a specific model. In Lecture 7, he defines Kan fibrations of simplicial spaces in Definition 3. These are specific models. Of course, Quinn's construction can also be declared quasicategorical: after all, one can take the homotopy colimit of a simplicial set as a functor from Δ^op to spaces, which yields an object in the quasicategory of spaces. But this only further validates my point. | |
| Feb 9, 2019 at 23:20 | comment | added | Dmitri Pavlov | I'd say that such usage of the term “∞-category” only creates more confusion and should be avoided. Most people that use quasicategories refer to them as ∞-categories and they talk about various flavors of fibrations of ∞-categories, which only makes sense if you use a specific model, since any map of simplicial sets is weakly equivalent to a fibration in the Joyal model structure. Myself, I am not too excited about this change of terminology, but in any case things can be made unambiguous if one uses “(∞,1)-category” to refer to the model-independent notion. | |
| Feb 9, 2019 at 22:45 | comment | added | Dylan Wilson | Okay but really: I'm no longer gonna comment on this thread, it's way too long! So any further discussion should really happen in a chat room. | |
| Feb 9, 2019 at 22:44 | comment | added | Dylan Wilson | @DmitriPavlov re quinn spectra: I'll look into it. But have you seen math.harvard.edu/~lurie/287x.html ? In particular Lecture 20? | |
| Feb 9, 2019 at 22:43 | comment | added | Dylan Wilson | @DmitriPavlov re infty cats: I dunno if it's as standard as you say. I, and many others, use "$\infty$-categories" without picking a model, and try to make only model-independent statements. It just so happens that the only model for which some of the foundational results have so far been proven is in quasicategories. Sometimes the world reacts poorly to the phrase "model-independent" and so we say "we're using quasicategories" so as to avoid an argument. (Other times people are doing further foundational work within a model, and choose quasicategories for convenience.) | |
| Feb 9, 2019 at 22:30 | comment | added | Dmitri Pavlov | Quinn's own expository account of his construction can be found in his paper “Assembly maps in bordism-type theories”. | |
| Feb 9, 2019 at 22:28 | comment | added | Dmitri Pavlov | It would be helpful if you clarified the terminology used in your answer (maybe even edit the answer itself). Before Lurie's Higher Topos Theory came out in 2006, ∞-categories were synonymous with (∞,1)-categories and referred to any model, not just quasicategories. After 2006, terminology rapidly changed, and today ∞-categories are overwhelmingly used to mean quasicategories. If you indeed meant (∞,1)-categories instead of quasicategories, it would be helpful to point this out, since many people probably understood quasicategories where you said ∞-categories. | |
| Feb 9, 2019 at 22:19 | comment | added | Dmitri Pavlov | The OP does mean specifically quasicategories (alias ∞-categories, as in Lurie's books), as opposed to (∞,1)-categories: he refers to another question, which references the “∞-category of spectra defined in Higher Algebra”, which is a quasicategorical construction. Also, these days (meaning after Lurie's books came out) ∞-categories are a synonym for quasicategories, whereas the “model-independent” version is now referred to as (∞,1)-categories. For instance, when Lurie is talking about cartesian fibrations of ∞-categories, he means specifically quasicategories, and not just any model. | |
| Feb 9, 2019 at 22:11 | comment | added | Dylan Wilson | @DmitriPavlov either way, I think this discussion has perhaps exceeded a reasonable length for a comment thread. If you want to talk more maybe we should move to the homotopy theory chat room. | |
| Feb 9, 2019 at 22:09 | comment | added | Dylan Wilson | @DmitriPavlov regarding "Quinn-Quillen bordism spectra": I'd be happy to go learn what those are and get back to you, but I think we need to somehow fix some parameters for this "debate" ahead of time. After all, my claim to you would be that the original construction has heavy extra burdens regarding compatibility/comparison/extra structure, and your claim to me (it feels) will be that whatever I do it will be "less efficient" or "essentially the same". | |
| Feb 9, 2019 at 22:07 | comment | added | Dylan Wilson | @DmitriPavlov The original question asked about $\infty$-categories, not quasicategories, so I don't know why we keep coming back to this point about choosing a model for $\infty$-categories. And when you say that model category theorists could use relative categories throughout that is precisely my point: we should use $\infty$-categories to characterize and construct things with universal properties. The experimental and observable fact is: people that construct things using model categories usually don't do this! | |
| Feb 9, 2019 at 22:02 | comment | added | Dmitri Pavlov | For the record, I do prefer “model-independent” constructions, such as the one described by you for the Eilenberg–MacLane functor. But to reiterate what I said before, such constructions make sense in any model, not just quasicategories, but also relative categories, etc. The original question, however, asked specifically if sticking exclusively to quasicategories is possible (and, by extension, makes things easier/better). | |
| Feb 9, 2019 at 21:57 | comment | added | Dmitri Pavlov | “you seem very focused on the particular example of dgas”: Not at all, I also mentioned the Quinn–Quillen model for bordism spectra, as used in surgery theory, and I haven't seen any attempts to explain how this model can be encoded in quasicategories with similar efficiency. | |
| Feb 9, 2019 at 21:53 | comment | added | Dmitri Pavlov | Almost all of what you are saying applies to any model of (∞,1)-categories, whether it is quasicategories, relative categories, simplicial categories, etc. (The arguments probably also work in the framework of Riehl–Verity.) This is true, for instance, for your paragraph about “stronger theorems”: all arguments there are model-independent, so it is not clear how quasicategories make any difference here. Recall, however, that one of the interpretations of the original question is whether one can use only quasicategories, and it is here that you arguments cease to make sense. | |
| Feb 9, 2019 at 21:49 | comment | added | Dmitri Pavlov | Concerning Dold–Kan: as you yourself said, such results are established at the level of foundations, so I am not sure what your objection is here. The E_∞-structure on the Dold-Kan functor that I mentioned is the terminal such structure (its operations are natural transformation between Γ^{⊗n} and Γ), so comparing it to other structures is trivial. | |
| Feb 9, 2019 at 21:48 | comment | added | Dylan Wilson | @DmitriPavlov Anyway, you seem very focused on the particular example of dgas, which I was using only to illustrate a larger point. This is beginning to take on the tone of the song "Anything you can do I can do better" which was not my intent. Please read the first few paragraphs: in the end, we should use whatever we personally like in order to prove theorems! The remaining paragraphs were an explanation of my personal feelings on the matter. | |
| Feb 9, 2019 at 21:47 | comment | added | Dmitri Pavlov | The comparisons of model-categorical constructions also occur at the level of foundations, so it is not clear what you are trying to say here. The Barwick–Schommer-Pries result, in particular, can be just as easily stated for model categories: the model category of (∞,n)-categories must be Quillen equivalent to a certain left Bousfield localization of simplicial presheaves on Θ_n. | |
| Feb 9, 2019 at 21:44 | comment | added | Dylan Wilson | @DmitriPavlov Proving that the Dold-Kan functor is "E_infty lax" and, moreover, producing a specific E_infty lax structure and then comparing it to all the others is, you must admit, more involved than you are letting on... | |
| Feb 9, 2019 at 21:43 | comment | added | Dylan Wilson | @DmitriPavlov Again your target is moving. These types of comparison results are occurring at the level of foundations. (I notice you didn't ask about comparing ZFC to other nonstandard models of set theory- but surely that should be added to your list of things we need to make sure don't mess with our theorems, right?) But if you are interested in comparison results at the foundational level, we already have a nice success story in the Unicity theorem of Barwick-Schommer-Pries. The theory is young- I imagine similar theorems will eventually be proven about operads and enriched theories. | |
| Feb 9, 2019 at 21:42 | comment | added | Dmitri Pavlov | “It is not really clear how to immediately deduce that with the set-up you give...”: Actually, it is: the Dold–Kan functor is an E_∞-lax functor, so it immediately induces a functor from CDGA to Alg_{E_∞}(Sp^Σ(sSet)). This is just as formal as the case of CDGAs. | |
| Feb 9, 2019 at 21:38 | comment | added | Dmitri Pavlov | “2. Check that this particular construction agrees with the other particular construction produced by some other model category theorist.”: How is this any different in the quasicategorical world? One quasicategory theorist uses cocartesian fibrations and ∞-preoperads, another uses dendroidal sets and dendroidal fibrations, and then these models must be compared, which is quite involved. Even defining things is often unreasonably more difficult, e.g., for a long time we did not have a theory of enriched colored ∞-operads. | |
| Feb 9, 2019 at 21:38 | comment | added | Dylan Wilson | @DmitriPavlov I was trying to explain a general paradigm. It is sometimes (maybe even relatively often) the case in practice, with model categories, that the functor you write down is not a Quillen functor for whatever model structure you are using, and that you need to change model structures or otherwise justify the existence of enough objects to 'resolve' by. This happens often when you need to do something with tensor products in a situation when you have enough flat objects but not enough projectives, say, but you also want to do something else that requires enough injectives, etc. | |
| Feb 9, 2019 at 21:36 | comment | added | Dmitri Pavlov | “1. Check that this particular construction can be appropriately derived.”: this is completely automatic, because such constructions are Quillen functor, which can always be derived. For example, the Eilenberg–MacLane functor mentioned above is automatically a left Quillen functor. So it is not accurate for you to say that proving this is a “burden”. | |
| Feb 9, 2019 at 21:33 | comment | added | Dmitri Pavlov | “(i) choose a model for what 'dgas' are”: You do not get to choose what dgas are, dgas are by far the most commonly used model for ring spectra outside of homotopy theory proper. So either your theory connects to all other branches of mathematics that use dgas (e.g., representation theory, algebraic geometry, etc.), or it doesn't. | |
| Feb 9, 2019 at 21:20 | history | answered | Dylan Wilson | CC BY-SA 4.0 |