Timeline for Why stable $\infty$-categories?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 27, 2023 at 13:10 | comment | added | user141099 | Could you please provide a reference for the fact that showing Zariski descent for $\mathscr{D}$ boils down to showing that this one diagram you wrote down is cartesian? | |
| May 29, 2021 at 18:41 | comment | added | Denis Nardin | @Wojowu I missed a word, corrected now. Thank you! | |
| May 29, 2021 at 18:41 | history | edited | Denis Nardin | CC BY-SA 4.0 |
Missed a word
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| May 29, 2021 at 18:09 | comment | added | Wojowu | "it's clear what should be a symmetric monoidal ∞-category: it's just a symmetric monoidal ∞-category which is stable and such that the tensor product is exact in each variable." I think this sentence doesn't say what you meant for it to say. | |
| May 29, 2021 at 18:00 | vote | accept | Gabriel | ||
| May 29, 2021 at 17:27 | comment | added | dhy | So if you want to define $\mathscr{D}(S)$ as a dg-category, you need to first define what it means to take a limit of dg-categories over an index $\infty$-category. It's pretty awkward to mix dg and $\infty$ in this way, and it's easier to just work $\infty$-categorically from the beginning. | |
| May 29, 2021 at 17:23 | comment | added | dhy | @DonuArapura Denis's response above is very good, but let me add one reason why you might choose to work with $\infty$-categories over dg-categories even if you cared only about characteristic zero (derived) objects. Consider the problem of defining the derived category of quasi-coherent sheaves on a derived scheme $S$. One way to do so is to take the limit of $\mathscr{D}(R)$ over all affine (derived) schemes $\operatorname{Spec} R$ mapping to $S$. This index category is an (unstable) $\infty$-category; in particular, it is not a dg category. (cont.) | |
| May 29, 2021 at 15:33 | comment | added | Denis Nardin | (cont.) is the setting of ∞-categories, where you can really do all these homotopical constructions "as if you were doing ordinary category theory" and everything "just works™". Therefore I find more convenient to work with stable ∞-categories and using $\mathbb{Z}$-linear (or $\mathbb{Q}$-linear, etc.) as needed. Especially since some important maps are not $\mathbb{Z}$-linear (among which there's the "Tate-valued Frobenius map" $R\to R^{tC_p}$, which has been important recently in work by Scholze et al) | |
| May 29, 2021 at 15:31 | comment | added | Denis Nardin | @DonuArapura I'm loathe to lengthen this already long answer, so I'll add a couple of things as a comment. From a modern perspective pretriangulated dg-categories are just $\mathbb{Z}$-linear stable ∞-categories (that is to say, stable ∞-categories $\mathcal{C}$ together with a "multiplication" $\operatorname{Perf}_{\mathbb{Z}}\times \mathcal{C}\to \mathcal{C}$). Therefore they are strictly less general than stable ∞-categories, although of course sometimes they are the right tool for the job. But I would argue that the proper home of both concepts (cont.) | |
| May 29, 2021 at 14:37 | comment | added | Donu Arapura | Thanks. OK, I'm convinced! Although, you touch on this briefly, could you say a bit more about the relation to dg-categories? I had the impression that they addressed some of the same issues. | |
| May 29, 2021 at 14:11 | history | answered | Denis Nardin | CC BY-SA 4.0 |