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Given a not too nasty topological space $X$, the category of sheaves of sets on $X$ remembers $X$.

Given a scheme $S$, the category of quasicoherent sheaves on $S$ remembers $S$.

Given a smooth manifold $M$, the category of ____ sheaves on $M$ remembers $M$.

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    $\begingroup$ 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.) $\endgroup$
    – Theo Johnson-Freyd
    Commented Dec 14, 2016 at 23:51
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    $\begingroup$ 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. $\endgroup$
    – Theo Johnson-Freyd
    Commented Dec 14, 2016 at 23:52
  • $\begingroup$ 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? $\endgroup$
    – Jules Lamers
    Commented Dec 15, 2016 at 15:35
  • $\begingroup$ 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... $\endgroup$
    – André Henriques
    Commented Dec 15, 2016 at 20:48

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Assuming the manifold is Hausdorff, the category of sheaves of modules over the sheaf of smooth functions do the trick.

This category is equivalent to the category of non-degenerate module over the ring of compactly supported smooth functions on the manifold, as it is a commutative ring, it remembers this ring and hence the manifold.

More simply, the ring of all smooth functions on the manifold can be reconstructed as the automorphism of the identity functor on this category. Moreover, this will also be true for any other subcategory of this that contains the unit module, so for example it also works for the category of vector bundle !

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  • $\begingroup$ 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. $\endgroup$
    – Bugs Bunny
    Commented Dec 15, 2016 at 9:09
  • $\begingroup$ 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. $\endgroup$
    – Simon Henry
    Commented Dec 15, 2016 at 11:17
  • $\begingroup$ After more thought: no you don't any assumption other than Hausdorff. $\endgroup$
    – Simon Henry
    Commented Dec 20, 2016 at 11:56
  • $\begingroup$ 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 $\operatorname{Mod}_{\mathbb R}$). $\endgroup$
    – Z. M
    Commented Jul 6 at 19:41

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