# What is known about higher-categorical reconstruction theorems? (reference request)

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