Timeline for Formality of a category of constructible sheaves
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 15, 2021 at 13:41 | comment | added | Laurent Cote | [Correction to my earlier comments: the sheafy-hom in question is indeed zero, and not the skyscraper sheaf as I was claiming earlier. I thank @BenG for correcting this misconception.] | |
| Oct 7, 2021 at 18:11 | comment | added | Aaron Mazel-Gee | Ah, thanks @BenG! So the category is not a direct sum, after all. | |
| Oct 6, 2021 at 20:39 | comment | added | Ben G | $hom(F_1,F_2)\simeq hom(\mathbb{C}\rightrightarrows 0, \mathbb{C}\rightrightarrows \mathbb{C})$ is correct, but that's not equivalent to $0$ -- there's a hom in degree 1. | |
| Oct 6, 2021 at 17:18 | comment | added | Laurent Cote | Maybe I've drastically misunderstood something, but if I have two constant sheaves $F_1, F_2$ , each supported on a different circle, then surely $\operatorname{Hom}(F_1, F_2)=\mathbb{C}$ (nothing derived here). The space of morphisms from $F_1$ to $F_2$ is global sections of the sheafy-hom, which in this case is a skyscraper. Did I misunderstand what you are saying? | |
| Oct 5, 2021 at 17:25 | comment | added | Aaron Mazel-Gee | Note that $F_2$ is a pushforward (equivalently a right Kan extension), so by adjunction you can take the pullback of $F_1$ to get $hom(F_1,F_2) \simeq hom( \mathbb{C} \rightrightarrows 0 , \mathbb{C} \rightrightarrows \mathbb{C} ) \simeq 0$. | |
| Oct 5, 2021 at 17:23 | comment | added | Aaron Mazel-Gee | I'm pretty sure that $hom(F_1,F_2) \simeq 0$ and reversely. It follows that this is the direct sum of the full subcategories on $F_1$ and $F_2$, each of which is formal (it's the one-object dg-category on cochains on the circle). I find it easiest to think about all of this in terms of exit-paths: S-constructible sheaves on $X$ are the same as representations of $Exit_S(X) \simeq (I_1 \leftleftarrows p \rightrightarrows I_2)$, a category with three objects and morphisms as indicated. [cont.] | |
| S Oct 3, 2021 at 13:58 | review | First questions | |||
| Oct 3, 2021 at 14:10 | |||||
| S Oct 3, 2021 at 13:58 | history | asked | Laurent Cote | CC BY-SA 4.0 |