Timeline for Which category of sheaves on a manifold remembers the manifold?
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6 events
| when toggle format | what | by | license | comment | |
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| Dec 15, 2016 at 20:48 | comment | added | André Henriques | Theo: Yes, I'm looking for a Gabriel-type theorem (so no monoidal structure). Jules: the question I'm asking is arguably a little bit vague... I guess I want to get a picture of the kind of categories of sheaves that it is reasonable to consider on manifolds... | |
| Dec 15, 2016 at 15:35 | comment | added | Jules Lamers | Naive question to understand better what you're asking for: Are the two examples that you give "iff" statements? And if so, are you also looking for the minimal condition for the case of smooth manifolds? | |
| Dec 14, 2016 at 23:52 | comment | added | Theo Johnson-Freyd | So any category that can recover $C^\infty(M)$ can recover $M$. Any reasonable category of modules will have $C^\infty(M)$ as the endomorphism ring of the unit object. Do you allow your category to know its monoidal structure? Or are you looking for a Gabriel-type theorem that gets it from the category alone? In the Gabriel-type case, you cannot expect anything too functorial: there will typically be more category automorphisms than manifold automorphisms. | |
| Dec 14, 2016 at 23:51 | comment | added | Theo Johnson-Freyd | The Hausdorff $C^\infty$-manifold $M$ is remembered already by the ring $C^\infty(M)$, just as a commutative ring. This is something special about manifolds, and not obvious. I think if your set theory allows astronomically large sets, you should disallow your manifold from being too large. Second countable is more than enough to get your manifold small. (The question, as I understand it, boils down to recovering a set $X$ from the ring of all $\mathbb R$-valued functions on $X$. I know how to do this functorially if $X$ is smaller than the smallest measurable cardinal.) | |
| Dec 14, 2016 at 20:36 | answer | added | Simon Henry | timeline score: 6 | |
| Dec 14, 2016 at 20:12 | history | asked | André Henriques | CC BY-SA 3.0 |