Timeline for What is modern algebraic topology(homotopy theory) about?
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12 events
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| Jan 4, 2023 at 11:30 | comment | added | David White | A long time ago, Dmitri Pavlov left a comment asking for examples of localizations of model categories that do not present presentable infinity categories. Boris Chorny's work provides many examples, e.g., categories of functors between model categories. For this reason, he (and Rosicky) invented the notion of class-locally presentable. Other examples include Pro categories, e.g., in work of Isaksen. | |
| Jan 14, 2016 at 18:34 | comment | added | Dmitri Pavlov | @crystalline: “Localization” is used in Lurie's sense here, which really means “reflective localization”, not just inverting sets of arrows. | |
| Jan 14, 2016 at 18:33 | comment | added | Dmitri Pavlov | @DavidWhite: Localization in Lurie's book mean reflective localization, which corresponds to left Bousfield localizations in the language of model categories. Can you give an example of a model category that admits left Bousfield localizations at sets of morphisms and whose underlying ∞-category is not presentable? If not, I don't really see what your point is in the first place: the assumption of presentability is nontrivial, but the analogous assumptions for model categories are just as strong. | |
| Jan 14, 2016 at 17:25 | comment | added | David White | @Dmitri: When I say infinity categories are "best when presentable", I mean that many of the results in Lurie's work require presentability (e.g. localization). I still think this is a highly non-trivial assumption, and one that gets thrown around a lot. I asked a MO question a while back and we identified plenty of non-presentable infinity categories that people care about. With cellular model categories (or more generally cofibrantly generated), I can still do a fair number of the things I want, and can avoid that assumption. | |
| Jan 14, 2016 at 17:23 | comment | added | David White | @crystalline: You said "Ironically, it is impossible to talk about non-presentable things in the language of model categories though." This is entirely false. There are plenty of model categories whose underlying infinity category is not presentable, and the model structure is very important for working in these settings. Not every model category is Quillen equivalent to a combinatorial one. Also, I'm not thrilled about being told I have misconceptions by an anonymous user, and I'm inclined not to interact on this thread. Lastly, in the comment just above, that's not what I said or meant. | |
| Jan 14, 2016 at 15:14 | comment | added | Dmitri Pavlov | Saying that “(∞,1)-categories best when they are presentable” is no different from saying that ordinary categories work best when they are locally presentable. Although both statements are in a certain sense true, it's not like we have any alternatives when we are faced with a non-locally presentable category or ∞-category. And model categories certainly do not provide any additional advantages if we use them to present a non-locally presentable ∞-category. Not to mention that the ∞-category of spaces with weak homotopy equivalences is presentable. | |
| Jan 14, 2016 at 15:13 | comment | added | Zhen Lin | @crystalline Quillen's original definition of model category allows for non-trivial (essentially) small examples, such as the category of bounded chain complexes of finitely generated abelian groups. And if one is so inclined, there are ways to modify the definition of model category so that we can get every small $(\infty, 1)$-category – of course, one then starts to wonder what the point is... | |
| Jan 14, 2016 at 15:07 | comment | added | Dmitri Pavlov | Working with (non-Δ-generated) topological spaces in the language of model categories requires a lot of additional bookkeeping. One must constantly keep track of which objects are small with respect to which morphisms. While this is possible (and not terribly difficult per se), it is very painful to do, especially when working with nontrivial structures that incorporate topological space (e.g., topological rings etc.), where the required technicalities increase quickly. All of this is totally irrelevant to the underlying topological problem. | |
| Jan 14, 2016 at 14:25 | comment | added | Zhen Lin | @crystalline The $(\infty, 1)$-category corresponding to the Hurewicz model structure on $\mathbf{Top}$ is not locally presentable. | |
| Jan 14, 2016 at 13:34 | comment | added | David White | Since you asked in particular about spaces, let me remark that one of the frustrating things about $(\infty,1)$-categories is that they work best when they are presentable, but spaces as you're used to thinking about them are not. Thus, you have to use simplicial sets or $\Delta$-generated spaces. This is not a huge barrier, but I felt it was worth mentioning. It's another reason I have stuck with model categories in my own work. | |
| S Jan 14, 2016 at 13:33 | history | answered | David White | CC BY-SA 3.0 | |
| S Jan 14, 2016 at 13:33 | history | made wiki | Post Made Community Wiki by David White |