Timeline for Why do we need enriched model categories?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 25, 2023 at 17:15 | vote | accept | Arshak Aivazian | ||
| May 22, 2023 at 20:42 | comment | added | Emily | @AivazianArshak Hey, thank you, that's very sweet of you to say :) I'm really happy the things I asked/answered/wrote were interesting to someone else. Again, thank you so much :) | |
| May 22, 2023 at 18:50 | comment | added | Arshak Aivazian | @Emily thank you, indeed (regarding the second question) it would have been better written that way. By the way, I've been noticing you for a long time and I was very interested to read many of your questions, answers and your github page. | |
| May 22, 2023 at 18:15 | comment | added | Emily | Just to reiterate: this is a good question in my opinion. | |
| May 22, 2023 at 18:13 | comment | added | Emily | Maybe a useful way to rephrase it (which I think would help with the downvotes as well) would be to ask how enriched model categories fit today in current research in homotopy theory, like asking for examples of papers where enriched model categories (besides simplicially enriched ones) play a role in proving or formulating some of the crucial results in those papers, how using enriched $\infty$-categories instead in those papers would compare, whether or not one could use just unenriched model categories instead to achieve the same thing, and so on. | |
| May 22, 2023 at 18:13 | comment | added | Emily | @AivazianArshak I think this is a good question. Maybe part of the reason for the downvotes is that there have been a bunch of controversial questions on MO of this form ("do we still need model categories", "why are non-$\infty$-categorical models of spectra needed", etc.) | |
| May 22, 2023 at 15:51 | comment | added | Arshak Aivazian | I don't quite understand why there are three negative votes. The view of model categories as a computational tool (which I write in the first paragraph) is shared by some significant portion of people in the field (for example, 1, 2). But even if you don't support it, I still ask two specific interesting questions (I think). | |
| May 22, 2023 at 15:49 | comment | added | Arshak Aivazian | @FernandoMuro Yes, I have indeed read literature where enriched model categories are studied per se and this motivated my question. For example, Emily Riehl - Categorical homotopy theory, Guillou, May - Enriched model categories and presheaf categories. | |
| May 14, 2023 at 8:06 | comment | added | Fernando Muro | @AivazianArshak it looks like you’ve just found references where model categories are studies per se, otherwise you would already have an answer. You could maybe share what you already know about the topic. Nevertheless I think you won’t find any motivation to use them unless they show up in your own research. | |
| May 14, 2023 at 2:20 | answer | added | David White | timeline score: 9 | |
| May 13, 2023 at 1:41 | comment | added | Arshak Aivazian | @FernandoMuro Yes, of course! I was hoping that I managed to put the question this way: if I (as I write in the first paragraph) have no interest in (enriched) model categories per se, what other reasons could I have to care about them? Motivated by this, I ask two more specific questions. I thought about explicitly mentioning that, of course, someone might be interested in them per se, but I considered it superfluous in the final text :) | |
| May 12, 2023 at 19:47 | comment | added | Dmitri Pavlov | As far as I know, the particular question of (fully homotopy coherent) comparison between enriched model categories and enriched quasicategories has not been settled yet. Only recently, the equivalence between combinatorial model categories and presentable quasicategories was established in arxiv.org/abs/2110.04679, and as far as know, there were no attempts so far to prove similar results in the enriched or monoidal case. (Since model categories are homotopy (co)complete and we also want to avoid size issues, the combinatoriality assumption is completely natural here.) | |
| May 12, 2023 at 15:03 | comment | added | Fernando Muro | Maybe you just don’t need them, the same way as you’re not interested in model categories. Each person has their own interests, that’s fine. | |
| May 12, 2023 at 8:56 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 12, 2023 at 8:51 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| May 12, 2023 at 8:43 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |