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The answer to my question is almost certainly "not much" — at least, I've asked a few people, and that was their answer. But I'd like to refine this answer, and MathOverflow seems like the best place.

I learned from David Ben-Zvi in an answer to this question the following theorem:

Let $\mathcal C$ be a 1-category, and consider the category $\operatorname{Rep}(\mathcal C)$ of 1-functors $\mathcal C \to \operatorname{1Vect}$. It is a (symmetric) monoidal category by "pointwise tensor product", i.e. pulling back along the diagonal map $\mathcal C \to \mathcal C^{\times 2}$. Conversely, we can consider some sort of "spec" of $\operatorname{Rep}(\mathcal C)$, namely the category of monoidal functors (and monoidal natural transformations) $\operatorname{Rep}(\mathcal C) \to \operatorname{1Vect}$. In fact, this "spec" is equivalent as a category to $\mathcal C$.

Given this, it is natural to ask the following three questions (or combinations thereof):

  • Recognition: which monoidal categories are of the form $\operatorname{Rep}(\mathcal C)$ for some $\mathcal C$?
  • Bump up $n$: modulo definitions, it is clear what the statement is with "$1$" replaced by "$n$". For example, the "$0$" version of the above says that a set is recoverable up to isomorphism from its algebra of all functions (the 0-category $\operatorname{0Vect}$ is precisely the ground field).
  • Internalize: is there a similar statement for "topological categories" and "continuous functors", for example? A version of in algebrogeometric land is in these questions (see also the answer here).

I'm not asking for definite answers to any of these directions, because I expect that telling the complete story is hard. But I am hoping for references to the existing literature. Hence: "What's already known (in the literature) about higher-categorical reconstruction theorems?"

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I don't know about monoidal categories, but many dg categories are of the form Mod(A) where A is a dg algebra... See mathoverflow.net/questions/33877/… –  Kevin H. Lin Aug 9 '10 at 21:32
    
Incidentally, I lied a bit in the statement of the "theorem" above. When defining "spec", or at least this "spec with residue field $\rm Vect$", of a monoidal cocomplete linear category like ${\rm Rep}C$, I think that one should insist that the functors to $\rm Vect$ be not just ("strongly") monoidal but also cocontinuous. This is like insisting that an algebra homomorphism respect not just multiplication but also addition. –  Theo Johnson-Freyd Aug 10 '10 at 2:29
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1 Answer 1

In his PhD, Giorgio Trentinaglia (see http://arxiv.org/abs/0809.3394) considers the reconstruction of an orbifold groupoid from its category of "representations".

At first, one might guess that "representation" means vector bundle. But Trentinaglia works with a larger category of non-locally-trivial vector bundles: the dimension is allowed to jump.

Wether or not the same reconstruction result holds true with actual vector bundles depends on the global quotient conjecture that claims that every orbifold is a global quotient by a compact Lie group.

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