Timeline for Limit of weak equivalences in a Bousfield localization
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21 events
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| Dec 28, 2018 at 8:27 | comment | added | Lao-tzu | Just to record—I realize that Morel-Voevodsky's homotopy (co)limits for simplicial sheaves is the same as Hirschhorn's chapter 18 (a bad one), by using that (co)limits of diagrams (like (co)ends) of presheaves is just taken section-wise; but they use functorial fibrant replacements (a good one) at some later point. | |
| Dec 26, 2018 at 15:40 | comment | added | Lao-tzu | I had also a sketch of reading the book of Dwyer-Hirschhorn-Kan-Smith time ago and know a little about relative categories. And I think Clark Barwick works a lot on this. | |
| Dec 26, 2018 at 15:39 | comment | added | Harry Gindi | Fair comment =]! | |
| Dec 26, 2018 at 15:38 | comment | added | Dmitri Pavlov | @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion. | |
| Dec 26, 2018 at 15:31 | comment | added | Harry Gindi | @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram. | |
| Dec 26, 2018 at 15:24 | comment | added | Dmitri Pavlov | @HarryGindi: I think it is better to use the term “(∞,1)-category” for this purpose, since “∞-category” is by now all but synonymous with “quasicategory”, due to Lurie's books. I disagree with your claim about definitions of homotopy (co)limits. In my opinion, the simplest definition of homotopy (co)limits is (for the functorial case) as the homotopy left adjoint of the constant diagram functor, which works equally well for any of the 5 models listed above. For the nonfunctorial case one simply unfolds the adjunction property, which also works for any of the 5 models. | |
| Dec 26, 2018 at 15:17 | comment | added | Harry Gindi | @DimitriPavlov By $\infty$-category, I did mean any model, not specifically quasicategories. In fact the model I had in mind was indeed relative categories. Quasicategories do happen to have the simplest definition of homotopy limit and colimit, however. | |
| Dec 26, 2018 at 14:49 | comment | added | Dmitri Pavlov | @Lao-tzu: You do know relative categories (i.e., categories with weak equivalences), which is what remains once you remove (co)fibrations from a model category. A relative category is all what you need to define all (∞,1)-categorical constructions, including homotopy (co)limits. The additional data of (co)fibrations is only useful for guiding computations, such as commuting left and right adjoint functors, e.g., commuting homotopy limits past homotopy colimits, see my answer mathoverflow.net/questions/287091/… | |
| Dec 26, 2018 at 14:41 | comment | added | Lao-tzu | @Dmitri Pavlov Thanks! I have only rather vague impression about $\infty$-categories, I don't read a large part of Lurie's HTT, instead only some short notes about that hence not systematic. The other four models you mentioned I completely don't know, only saw the name somewhere from time to time. | |
| Dec 26, 2018 at 14:26 | comment | added | Dmitri Pavlov | @Lao-tzu: One should remember that there are five commonly used models for (∞,1)-categories (not to be confused with ∞-categories), namely relative categories, simplicial categories, Segal categories, Rezk categories, and quasicategories (alias ∞-categories). None of these models is fully invariant, and quasicategories require just as much noninvariant manipulations as the other models (e.g., think of (co)Cartesian fibrations, left/right fibrations, etc.). Quasicategories are mentioned more often these days due to Lurie's books, which are based on them. | |
| Dec 26, 2018 at 14:19 | comment | added | Dmitri Pavlov | @Lao-tzu: My definition of (functorial) homotopy colimits is that the homotopy colimit functor is homotopy left adjoint to the constant diagram functor. In the case under discussion, the homotopy left adjoint can be computed as the left derived functor of the colimit functor, which is a left Quillen functor. This definition is fully invariant under weak equivalences, as it should be. Quasicategories (or ∞-categories) do not provide any additional advantages over relative categories in terms of defining homotopy (co)limits. | |
| Dec 26, 2018 at 13:34 | comment | added | Lao-tzu | @Harry Gindi Jardine's local homotopy doesn't mention homotopy (co)limits if I remember correctly. Anyway, thanks a lot for clarifying. | |
| Dec 26, 2018 at 13:28 | comment | added | Harry Gindi | @Lao-Tzu Yes, this is true. Homotopy limits are invariant under weak equivalence of injectively fibrant diagrams (because the limit functor is right-Quillen) | |
| Dec 26, 2018 at 13:24 | comment | added | Lao-tzu | Since in $\infty$-category, the limits are fully homotopy invariant, this will imply homotopy limits are homotopy invariance for arbitrary model category (I'm assuming homotopy limits in a model category is either take functorial fibrant replacement then use the formulas in Hirschhorn's book, or the right derived functor of $\lim$, but not the formulas in Hirschhorn's book only). If one uses the formulas in Hirschhorn's book only, then the homotopy limits only depend on the simplicial structure by the formulas, and not on the model structure (if we are in a simplicial model category). | |
| Dec 26, 2018 at 13:20 | comment | added | Harry Gindi | First question: Yes, the homotopy limit of a diagram in a model category is one way to compute the limit of the diagram in the underlying $\infty$-category. Second question: Maybe have a look at Jardine's book on the homotopy theory of sheaves on a site. There is something nontrivial going on. | |
| Dec 26, 2018 at 13:15 | comment | added | Lao-tzu | @Harry Gindi I find in Morel-Voevodsky, A1-homotopy theory of schemes, at the begining of section 2.1.4, they use a non-categorical definition of homotopy (co)limits for simplicial sheaves—just take section-wise homotopy (co)limits (of simplicial sets) then sheafify, for me this is weird and I don't think it coincides with any of the above things I mentioned in my first comment. | |
| Dec 26, 2018 at 13:11 | comment | added | Lao-tzu | @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $\infty$-categories (when taking the $\infty$-category associated to the given model category) or there is some relation you can explain? | |
| Dec 26, 2018 at 12:32 | comment | added | Harry Gindi | The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $\infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations. | |
| Dec 26, 2018 at 11:38 | comment | added | Lao-tzu | What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $\lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def? | |
| Dec 26, 2018 at 11:30 | vote | accept | Lao-tzu | ||
| Dec 26, 2018 at 4:37 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |