Timeline for When do two topoi have the same cohomology of constant sheaves
Current License: CC BY-SA 4.0
11 events
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| May 7, 2021 at 2:43 | comment | added | user106561 | @Denis-CharlesCisinski Thanks for your comment. The book you mentioned is helpful to me. I know where I can find the answers I need based on these very kind comments. | |
| May 6, 2021 at 16:23 | comment | added | D.-C. Cisinski | @yulincai Although it is true that this kind of questions is subsumed by shape theory of $\infty$-topoi, with Marc Hoyois' paper on higher Galois theory essentially saying everything we need to know, there is a little book of Moerdijk called "Classifying spaces and classifying topoi" which studies this kind of things for $1$-topoi, and manipulates explicit homotopies and geometric realizations, with rather explicit (and nice) examples related to the theory of fofliations. | |
| May 6, 2021 at 15:36 | answer | added | Zhen Lin | timeline score: 6 | |
| May 6, 2021 at 13:03 | comment | added | user106561 | @მამუკაჯიბლაძე Yes, I agree with what you said. Do we have similar sufficient condition, such "homotopic" for two topoi to have cohomology with any constant coefficients? | |
| May 6, 2021 at 13:01 | comment | added | user106561 | @DenisNardin Thanks for your answer. I will check the related theory. In my case, what I care the most is, when do two topoi have the same cohomology for constant sheaves. Based on your answer, I think the answer should be "if their $\infty$-topoi have the same shapes" (of course, it is a sufficient condition). Is it right? Thanks. | |
| May 6, 2021 at 12:17 | comment | added | მამუკა ჯიბლაძე | For toposes of the form $\mathrm{Set}^G$, for a group $G$, you get just cohomology of $G$ with constant coefficients. There are many acyclic groups, having trivial cohomology with any constant coefficients. They will have different fundamental groups however, which is just $G$ in this case. | |
| May 6, 2021 at 8:10 | comment | added | Denis Nardin | @yulincai Precisely from any topos you can construct an ∞-topos (and viceversa - it's an adjunction) and every ∞-topos has a shape (which is something like a pro-homotopy type). In the case of the small étale topos of a scheme this is what's usually known as the étale homotopy type of the scheme. The connection with locally constant sheaves is explored in this paper, and it does require some hypotheses on your topos to be well behaved. | |
| May 6, 2021 at 7:54 | comment | added | user106561 | @Tim Campion Thanks for your answer. I know nothing about $\infty$-topoi, but I think what you are saying is that, a topoi is a special case of $\infty$-topos, and in the theory of $\infty$-topoi, we have a similar theory of homotopy? | |
| May 6, 2021 at 5:56 | comment | added | Tim Campion | For $\infty$-topoi, one has the shape of the $\infty$-topos. | |
| May 6, 2021 at 4:54 | review | First posts | |||
| May 6, 2021 at 5:53 | |||||
| May 6, 2021 at 4:48 | history | asked | user106561 | CC BY-SA 4.0 |