Timeline for Which category of sheaves on a manifold remembers the manifold?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 6 at 19:41 | comment | added | Z. M | Sorry, could you please explain how the category of modules over the ring of smooth functions is the same as the category of non-degenerate non-unital modules over the non-unital ring of compactly supported smooth functions? I am confused about non-unital rings and non-unital modules: they are subtly dependent on the base, so letting $A$ be a non-unital $\mathbb R$-algebra, then a non-unital $A$-module is different from a non-unital $\mathbb R$-linear $A$-module (i.e. a non-unital $A$-module in $\operatorname{Mod}_{\mathbb R}$). | |
| Dec 20, 2016 at 11:56 | comment | added | Simon Henry | After more thought: no you don't any assumption other than Hausdorff. | |
| Dec 15, 2016 at 15:16 | history | edited | Simon Henry | CC BY-SA 3.0 |
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| Dec 15, 2016 at 11:17 | comment | added | Simon Henry | Good question. For the first part (that this category is equivalent to the category of non-degenerate modules over the ring of compactly suported smooth function) one does not need any assumption other than hausdorffness. The fact that it recover the ring of smooth functions (or compactly supported smooth function) is trivial if you allow the monoidal structure as part of the data, a little less otherwise but I don't think it uses paracompactness. For the fact that the ring reconstruct the manifold itself, I haven't had much thought about it, but I'm sure someone else can answer. | |
| Dec 15, 2016 at 9:09 | comment | added | Bugs Bunny | Do you need paracompact or Lindelof? Or is it a general statement? I am just not so sure whether it will distinguish the long ray from the long line. | |
| Dec 14, 2016 at 20:36 | history | answered | Simon Henry | CC BY-SA 3.0 |