Timeline for Sections of gerbes that can "vanish"
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 31, 2017 at 9:01 | comment | added | David Roberts♦ | @Dmitry well, I guess one mathematician's geometry is another's sheaf theory. There was more, applications to differential geometry, which is why I really want the video as well. | |
| Jul 31, 2017 at 8:56 | comment | added | Dmitri Pavlov | @DavidRoberts: What does "geometric" mean then? Coherent sheaves of rank at most 1 categorify functions in the same way as line bundles categorify invertible functions, so it seems like a natural answer to your question. Furthermore, they fully match the criteria given by you: at the open subset where they have rank 1 they give a trivialization, and at the closed subset where they have rank 0 they are degenerate (i.e., isomorphic to 0). | |
| Jul 30, 2017 at 22:05 | comment | added | David Roberts♦ | @DmitriPavlov that's more like it, but it was even more geometric in the talk. | |
| Jul 30, 2017 at 10:06 | comment | added | Dmitri Pavlov | @DavidRoberts: I see. Start with coherent sheaves of rank at most 1, suspend, and sheafify. The resulting stack contains bundle gerbes and their isomorphisms (because line bundles are coherent sheaves of rank at most 1), but also contains morphisms with nontrivial vanishing locus, corresponding to coherent sheaves of rank at most 1 that are not line bundles. | |
| Jul 29, 2017 at 23:27 | comment | added | David Roberts♦ | @allen no, it was nothing like that. More like a circle bundle where the circles degenerate. | |
| Jul 29, 2017 at 23:26 | comment | added | David Roberts♦ | @dmitri no, it was a rank one thing. As I said, outside the vanishing locus, it's a trivialisation. | |
| Jul 29, 2017 at 17:44 | comment | added | Allen Knutson | If you think that a line bundle is the associated bundle construction $\mathbb C \times^{U(1} M \to M$, then can you think of a gerbe as a principal $BU(1)$ bundle and use some representation of $BU(1)$? (I don't know how to think about those.) | |
| Jul 29, 2017 at 16:37 | comment | added | Dmitri Pavlov | The notion of noninvertible morphisms between bundle gerbes is due to Konrad Waldorf, see “More morphisms between bundle gerbes”. | |
| Jul 29, 2017 at 0:47 | history | asked | David Roberts♦ | CC BY-SA 3.0 |