Timeline for Understanding the definition of atlas of a stack over the category of manifolds
Current License: CC BY-SA 4.0
18 events
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| Jun 1, 2020 at 16:17 | history | rollback | Praphulla Koushik |
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| May 31, 2020 at 15:07 | comment | added | Praphulla Koushik | @Qfwfq About your first comment and my edit, I might have mixed up somethings.. Isn't it the case that, projection map $\underline{M}\times_{\mathcal{D}}\underline{N}\rightarrow \underline{N}$ induce a surjective submersion at the level of manifolds, irrespective of what Grothendieck topology I have choosen on $\text{Man}$.. This will make sure that, I get a Lie groupoid $(M\times_{\mathcal{D}M\rightrightarrows M)$.. source, target maps are assocaited projection maps.. so, proj.maps are surjective submersions at the level of manifolds, I dont know if we get source,target maps to be sur. subs. | |
| May 31, 2020 at 12:41 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| May 31, 2020 at 12:30 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Dec 4, 2018 at 22:37 | comment | added | Praphulla Koushik | With this definition we want $\{ U \times_Y V\rightarrow V \}$ to be a cover i.e., the map $U \times_Y V\rightarrow V$ is etale and surjective i.e., it is a surjective submersion... This is same thing as the condition with other Grothendieck topology.. Can other definition get affected with different Grothendieck topologies? What is the most widely used (if any) Grothendieck topology on the category of manifolds $\text{Man}$? | |
| Dec 4, 2018 at 22:28 | comment | added | Praphulla Koushik | In paper Differentiable stack and gerbes by Kai behrend and Ping Xu, they consider different Grothendieck topology on $\text{Man}$ (they denote this category by $\mathfrak{S}$)... It is in page $5$... "We endow $\mathfrak{S}$ with the Grothendieck topology given by the following notion of covering family. Call a family $\{U_i\rightarrow X\}$ of morphisms in $\mathfrak{S}$ with target $X$ a covering family of $X$, if all maps $U_i\rightarrow X$ are etale and the total map $\bigcup U_i\rightarrow X$ is surjective" | |
| Dec 4, 2018 at 22:25 | comment | added | Praphulla Koushik | @Qfwfq I just want to confirm one thing... So, here, the Grothendieck topology is given by surjective submersions... Given an object of $\text{Man}$ i.e., a manifold $M$, a cover for $M$ is simply given by one surjective submersion $V\rightarrow M$.Right? I have difficulty in understanding your definition of geometric stack..You said "[---] and the pullback $U\times_Y V\rightarrow V$ is a $J$-cover"... In an arbitrary Grothendieck topology it can happen that a cover may be of just one element... And here we want $\{U\times_Y V\rightarrow V\}$ to be that type of cover which has only one arrow. | |
| Jul 6, 2018 at 17:39 | comment | added | Qfwfq | See also the nLab page: ncatlab.org/nlab/show/geometric+stack | |
| Jul 6, 2018 at 17:39 | comment | added | Praphulla Koushik | Ok ok @Qfwfq I think I got it little.. | |
| Jul 6, 2018 at 17:35 | comment | added | Qfwfq | (...) You should think of $U\times_{Y} V$ as a disjoint union of patches of a manifold atlas (in th eclassical sense) and by $f:U\times_{Y} V\to V$ the "joint cover" i.e. the map that on each patch restricts to its open embedding into $V$. | |
| Jul 6, 2018 at 17:35 | comment | added | Praphulla Koushik | @Qfwfq ok ok. thanks. Can you give some reference where little more about geometric stacks is mentioned.. something al9ng the lines of what you said.. it looks very reasonable.. | |
| Jul 6, 2018 at 17:30 | comment | added | Qfwfq | Yeah, let's say we have a site $(C,J)$, and a stack $Y$ over it. Then we may call $Y$ "geometric" if it has an atlas $f:U\to Y$, i.e. a $1$-morphism of stacks where $U$ is in $C$ and "locally on $Y$" $f$ is in $J$. By the last condition I mean that for every morphism $V\to Y$ where $V$ is in $C$, $U\times_{Y} V$ is in $C$ and the pullback map $U\times_{Y} V \to V$ is a $J$-cover. | |
| Jul 6, 2018 at 17:19 | comment | added | Praphulla Koushik | @Qfwfq I think I understand something but not very sure.. you are saying condition on $f$ will vary depending on what Grothendieck topology I chose to put on Man.. I don’t think that article mentions anything about Grothendieck topology on Man, may be as they don’t want to discuss stack over arbitrary categories they did not care about grothendieck topology on Man.. | |
| Jul 6, 2018 at 16:55 | comment | added | Qfwfq | (...) Now, such a jointly surjective family of open embeddings $\{j_i : U_i\to M\}$ is almost by definition an atlas for the manifold $M$. The "union" map $f:X:=\amalg_i U_i\to M$ is no longer an open embedding but it's still open, surjective, and a local diffeo; and it's essentially the same as giving the covering family $\{j_i\}$ (if you admit non connected manifolds with a lot of components in Man, then $\{f\}$ is a honest (singleton) cover for a still equivalent Grothendieck topology, given by -if I'm not mistaken- surjective local diffeos). | |
| Jul 6, 2018 at 16:54 | comment | added | Qfwfq | Yes, $f$ should be a surjective submersion. But I think that, to understand the heuristics, the submersion thing is a bit of a red herring. The point is that you want $f$ to be a cover in the Grothendieck topology you chose to put on $\mathbf{Man}$. It happens that surjective submersions give a Grothendieck topology on Man, but an equally good (maybe even equivalent, considered that in the smooth category submersions have sections...) Grothendieck topology is simply given by families of jointly surjective open embeddings. (...) | |
| Jul 6, 2018 at 16:24 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Jul 6, 2018 at 6:12 | vote | accept | Praphulla Koushik | ||
| Jul 6, 2018 at 6:12 | history | answered | Praphulla Koushik | CC BY-SA 4.0 |