Timeline for Cohomology of Grothendieck topology
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Jun 22, 2021 at 13:42 | vote | accept | Stefan Witzel | ||
| Jun 22, 2021 at 6:27 | comment | added | Agustí Roig | @ZhenLin Okay. You're right. In our papers, the Godement resolution for a Grothendieck site with enough points depends on the chosen set of points. | |
| Jun 22, 2021 at 1:38 | comment | added | Zhen Lin | Yes, of course there is a problem with sites without enough points, but actually what I was thinking about is sites with too many points. The Godement resolution in my understanding is supposed to be a canonical functorial resolution, but if you have to arbitrarily choose a sufficiently large set of points then you lose canonicity. This problem does not arise for topological spaces (or locales) because there is only a set of points (up to isomorphism). | |
| Jun 21, 2021 at 6:36 | comment | added | Agustí Roig | @DavidRoberts Thank you. | |
| Jun 21, 2021 at 0:39 | comment | added | David Roberts♦ | @AgustíRoig it's better to give the doi links, since Springer's links have been known to change in the past: Godement resolutions and sheaf homotopy theory, Godement resolution and operad sheaf homotopy theory | |
| Jun 20, 2021 at 19:29 | answer | added | Dan Petersen | timeline score: 12 | |
| Jun 19, 2021 at 8:52 | comment | added | Dan Petersen | @Agustí Presumably Zhen Lin means that the Godement resolution only works on a site with enough points, as in your papers. | |
| Jun 19, 2021 at 7:33 | comment | added | David Roberts♦ | Also, McLarty's work (arxiv.org/abs/1102.1773) showing derived functor cohomology works in a weak set theory is relevant here, since he makes sure the various resolution techniques go through. (published ref: Review of Symbolic Logic 13 Issue 2 (2020) pp. 296–325 doi.org/10.1017/S1755020319000340) | |
| Jun 19, 2021 at 6:31 | comment | added | Agustí Roig | @ZhenLin Maybe I'm not understanding what you mean, but I think there is indeed a Godement resolution for Grothendieck sites: link.springer.com/article/10.1007/s13348-014-0123-x and link.springer.com/article/10.1007/s13348-016-0171-5 | |
| Jun 18, 2021 at 15:10 | comment | added | Zhen Lin | The category of abelian group objects in a Grothendieck topos is a Grothendieck abelian category – you get local presentability by the usual yoga and you inherit the required exactness properties because sheafification is exact – so the Tôhoku paper is indeed applicable. It would be nice if the OP could indicate whether this is the level of answer sought. | |
| Jun 17, 2021 at 18:16 | history | edited | LSpice | CC BY-SA 4.0 |
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| Jun 17, 2021 at 16:51 | history | became hot network question | |||
| Jun 17, 2021 at 13:55 | comment | added | Benjamin Steinberg | Johnstone’s book on topos theory from way back has a section on topos cohomology. Not the elephant book but his original book. The book is not exactly reader friendly for non category theorists but it is shorter than SGA. I think he users Barr's covering theorem to get that there are enough injectives but once you have that you are good to go. There are no schemes in the book | |
| Jun 17, 2021 at 12:59 | comment | added | David Roberts♦ | One takes the topos of sheaves on the site, then considers abelian group objects in it, then maybe there are some assumptions that ensure you have an abelian category with the right properties to then use Grothendieck's Tohoku paper? | |
| Jun 17, 2021 at 11:32 | comment | added | Zhen Lin | The words used are the same, and indeed one is a generalisation of the other, but a sheaf on a Grothendieck site is a different thing from a sheaf on a topological space so you need different tools to compute cohomology. (For example, there is no analogue of the Godement resolution.) | |
| Jun 17, 2021 at 10:48 | history | edited | Asaf Karagila♦ |
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| S Jun 17, 2021 at 9:47 | history | suggested | user122276 |
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| Jun 17, 2021 at 9:46 | review | Suggested edits | |||
| S Jun 17, 2021 at 9:47 | |||||
| Jun 17, 2021 at 8:51 | history | asked | Stefan Witzel | CC BY-SA 4.0 |