Timeline for Why does passage to DG categories cure non-locality of derived categories?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 16, 2018 at 21:59 | comment | added | dhy | ...is to interpret $F$ as $\operatorname{Cone}(F|_U\oplus F|_V\rightarrow F|_{U\cap V})[1],$ and similarly for $G$. Then the data described above gives you a morphism between the two morphisms inside the cone. However, because you don't have functoriality of cones, you can't upgrade this to a morphism $F\rightarrow G$ without upgrading to some other (e.g. DG) setting. | |
| Jun 16, 2018 at 21:54 | comment | added | dhy | Here is my possibly incomplete understanding for why functorial cones (one big advantage of DG categories) are necessary for categorical descent. Assume you have an open cover of a variety $X$ by open sets $U,V$. To do categorical descent, you need to be able to reconstruct a morphism $f:F\rightarrow G$ from the data of the morphisms $f|_U$, $f|_V$, and a homotopy between their restrictions to $U\cap V.$ (In the derived world, you cannot keep track of this extra datum of a homotopy, which leads to non-functoriality of cones.) The natural way to do this... | |
| Jun 16, 2018 at 20:07 | vote | accept | CommunityBot | ||
| Jun 16, 2018 at 19:12 | answer | added | fosco | timeline score: 4 | |
| Jun 16, 2018 at 18:43 | comment | added | Mykola Pochekai | Because dg-category is almost the same as $k$-linear stable $(\infty,1)$-category so, it is something invariant and conceptual, and triangulated category framework has been introduced for computational reasons, rather then conceptual, so, it is difficult to expect that they will behave good. | |
| Jun 16, 2018 at 18:20 | history | asked | user74900 | CC BY-SA 4.0 |