Timeline for 2-limits of triangulated categories
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 6, 2021 at 15:11 | comment | added | D.-C. Cisinski | In fact, in the case where the indexing category is $\mathbb{N}^{op}$, there is a $lim^1$ obstruction against this equivalence, which vanishes for constructible $\ell$-adic sheaves in the context considered by Deligne (we do not only need to work with $\mathbb{Z}/\ell^n$-linear coefficients, but also with schemes of finite type over a finite field or over a separably closed field to ensure that all cohomology groups are finite). | |
| Nov 6, 2021 at 15:11 | comment | added | D.-C. Cisinski | Even if the diagram can be lifted to $\infty$-categories, it is not true in general that the homotopy category of the limit of $\infty$-categories is equivalent to the limit of homotopy categories. | |
| Dec 30, 2020 at 14:15 | comment | added | Dan Petersen | Note though that Deligne takes a 2-limit and you're taking a 2-colimit! And his argument uses that the rings $\mathbf Z/\ell^n$ are finite... | |
| Dec 30, 2020 at 12:19 | comment | added | ThomasZ | Thanks Dan that's very useful- I was trying to see if I could get away without using stable $\infty$-categories. I think in my setup (finitely presented modules over a commutative ring), the argument of Deligne also goes through- but I'll have a think if I can enhance that to the stable $\infty$-case. | |
| Dec 30, 2020 at 12:17 | vote | accept | ThomasZ | ||
| Dec 29, 2020 at 21:54 | history | answered | Dan Petersen | CC BY-SA 4.0 |