Timeline for What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$?
Current License: CC BY-SA 4.0
30 events
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| May 1, 2020 at 14:30 | comment | added | Praphulla Koushik | @WillSawin Yes. That is the second exercise (I think) of Hartshorne's book :P | |
| May 1, 2020 at 14:26 | comment | added | Will Sawin | @PraphullaKoushik I just mean that open sets on $X \times Y$ in the Zariski topology don't arise from the products of open sets on $X$ and open sets on $Y$, which you can find explained in any introduction to the Zariski topology. | |
| May 1, 2020 at 13:45 | comment | added | Praphulla Koushik | @WillSawin It looks like I need to read a little more.. Can you suggest some reference for "(at least in AG) products of sites are not so well-behaved."... It is expected to be not so well-behaved :D Just wanted to read some more... | |
| May 1, 2020 at 12:52 | comment | added | Will Sawin | @PraphullaKoushik The sheaf $\mathcal G$ is what I mean to call the first definition. I agree the first definition is very appropriate for defining torsors. The second definition is the site of $G$, which involves considering open subsets of the group $G$, or $G \times X$. I think this is not so helpful because (at least in AG) products of sites are not so well-behaved. | |
| May 1, 2020 at 10:20 | vote | accept | Adittya Chaudhuri | ||
| May 1, 2020 at 9:37 | answer | added | dorebell | timeline score: 3 | |
| May 1, 2020 at 9:05 | comment | added | Adittya Chaudhuri | @Asvin. Ok. I have asked the later part of this question as a different question here mathoverflow.net/questions/359056/… | |
| May 1, 2020 at 9:04 | history | edited | Adittya Chaudhuri | CC BY-SA 4.0 |
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| May 1, 2020 at 5:32 | comment | added | Asvin | I am just using geometric category roughly but it generalizes sets/topological spaces/schemes with zariski or etale topology /manifolds/infty spaces/whatever. The functor just assigns to a map T to X, the set of G torsors of T. | |
| May 1, 2020 at 5:27 | comment | added | Adittya Chaudhuri | @Asvin Also you mentioned "In general in any geometric category, you can ask if there is some object that represents the functor in your defn (1)". I did not get what did you mean by the "functor" in my defn (1). | |
| May 1, 2020 at 5:23 | comment | added | Adittya Chaudhuri | @Asvin I am trying to understand your comment. But can you explain in little detail what do you mean by geometric category? Is it the same thing as mentioned in ncatlab.org/nlab/show/geometric+category? Also is there any precise definition of geometric object in a category? | |
| May 1, 2020 at 4:55 | comment | added | Asvin | I guess you are really asking if there is some "geometric" object $\mathcal C/G$ for any site $\mathcal C$? In this generality, I don't think the question even makes sense. The paper you link to are working with $\infty$-topos and there the question makes sense. In general in any geometric category, you can ask if there is some object that represents the functor in your defn (1). | |
| May 1, 2020 at 4:35 | comment | added | Praphulla Koushik | What do they mean when they say site? site of what? If on the site $(\mathcal{C},\mathcal{A})$ (here $\mathcal{C}$ is a category and $\mathcal{A}$ is a Grothendieck topology on $\mathcal{C}$) does not have any extra structure, why would they call a sheaf of groups on $(\mathcal{C},\mathcal{A})$ to be a "geometric" group? | |
| May 1, 2020 at 4:20 | history | edited | Adittya Chaudhuri | CC BY-SA 4.0 |
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| May 1, 2020 at 4:20 | comment | added | Praphulla Koushik | "But the second definition seems inappropriate for defining groups and torsors." I do not completely understand this... Given a group $G$, one can consider the sheaf $\mathcal{G}$ on $X$, defined as $\mathcal{G}(U)=\{\text{smooth maps } U\rightarrow G\}$. Then, there is a notion of $H^1(X,\mathcal{G})$.. The paper "Introduction to Language of stacks and gerbes" by Moerdijk says this set $H^1(X,\mathcal{G})$ is in one-one correspondence with the the set of isomorphism classes of $G$-torsors on $X$.. So, in what sense (2) is inappropriate here? @WillSawin | |
| May 1, 2020 at 4:15 | comment | added | Adittya Chaudhuri | @WillSawin Sir, I have edited my question by writing some extra paragraphs at the end. It would be very helpful if you kindly go through my edits at your leisure. | |
| May 1, 2020 at 4:13 | history | edited | Adittya Chaudhuri | CC BY-SA 4.0 |
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| May 1, 2020 at 3:32 | comment | added | Will Sawin | @PraphullaKoushik I don't know what the second one is so I guess I mean the first. | |
| May 1, 2020 at 2:42 | comment | added | Praphulla Koushik | @WillSawin when you say site of $X$ you mean the category $\mathcal{O}(X)$ of open subsets of $X$ equipped with some Grothendieck topology or the topological stack $\underline{X}\rightarrow \text{Top}$ associated to the topological space $X$? | |
| Apr 30, 2020 at 19:16 | comment | added | Will Sawin | @PraphullaKoushik I just mean that there is not so much data about a map of spaces $Y \to X$ that can be expressed naturally in the language of the site of $X$ (rather than in the language of the space of $X$). The main data that would be preserved would be the sheaf of sections of $Y$, as well as the site of $Y$ and its map to $X$. But the second definition seems inappropriate for defining groups and torsors. | |
| Apr 30, 2020 at 19:03 | comment | added | Adittya Chaudhuri | @WillSawin Sir, can you please give me little details about why do you think when we pass to site from topological space, (1) and (3) becomes indistinguishable? Also what do you mean by (3) when we pass to site? Unless I understand what is meant by (3) (in the context of site) I am not able to understand (3)=(1) in the context of site. | |
| Apr 30, 2020 at 18:59 | comment | added | Praphulla Koushik | @WillSawin Can you expand on what do you mean by “ forget the topological space and pass to the site”? | |
| Apr 30, 2020 at 18:55 | comment | added | Adittya Chaudhuri | @WillSawin Yes I agree that (1) is the sheaf of section of (3). Actually this is my question " When we pass to site "Is indeed (1) and (3) are indistiguishable or there exist some notion of the form $\pi':P' \rightarrow X'$ where $P'$, $X'$ may be some categories and $\pi'$ is a functor. Also we may replace $G$ by Higher Groups like 2 groups or something of that sort. | |
| Apr 30, 2020 at 18:50 | comment | added | Will Sawin | Sure, but (1) is just the sheaf of sections of (3). If you forget the topological space and pass to the site, these definitions are pretty much indistinguishable. | |
| Apr 30, 2020 at 18:45 | comment | added | Adittya Chaudhuri | @WillSawin Sir, by (1) I mean a $\tilde{G}$-torsor i.e sheaf $S$ over a topological space $X$ with an action of a group valued sheaf $\tilde{G} $over $X$ and an existence of an open cover $\cup U_{\alpha}$ of $X$ such that each $S(U_{\alpha})$ is non empty. Now from a $\tilde{G}$ torsor we can produce an element of $H^1(X, \tilde{G})$ and from that element we can produce a principal $G$ bundle $\pi:P \rightarrow X$ where $P$ is a topological space obtained from the data given by the element of $H^1(X,G)$ and $\pi$ is a continuous function from $P$ to $X$. This $\pi:P \rightarrow X$ is my (3). | |
| Apr 30, 2020 at 18:17 | comment | added | Will Sawin | What distinction is being drawn between definitions (1) and (3)? | |
| Apr 30, 2020 at 17:42 | comment | added | Adittya Chaudhuri | @PraphullaKoushik I am actually not sure about what $G$ is when $C$ is a site. My reference for 1,2 and 3 is arxiv.org/abs/math/0212266 | |
| Apr 30, 2020 at 17:30 | answer | added | Praphulla Koushik | timeline score: 0 | |
| Apr 30, 2020 at 16:47 | comment | added | Praphulla Koushik | What is your $G$ when you want to define notion of a $\widetilde{G}$-torsor over $\mathcal{C}$ when $\mathcal{C}$ is a site? Is it just a group object in the category $\mathcal{C}$? Do you want to give a reference from where you are borrowing your definitions 1,2 and 3? | |
| Apr 30, 2020 at 15:33 | history | asked | Adittya Chaudhuri | CC BY-SA 4.0 |