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In the Tannaka-Krein duality theory, the Krein theorem describes the conditions under which a given category $\varPi$ is a category of finite-dimensional representations of some compact group $G$:

  • $\varPi$ must be a subcategory in the monoidal category of finite-dimensional vector spaces (with the usual tensor product $\otimes$),

  • $\varPi$ must have a unit, i.e. an object $I$ such that $I\otimes X\cong X$ for all objects $X$ in $\varPi$,

  • every object $X$ of $\varPi$ can be decomposed into a sum of minimal objects, and

  • for any minimal objects $X$ and $Y$ the space of homomorphisms $\operatorname{Hom}_{\varPi}(X,Y)$ is either one-dimensional or equal to zero.

I believe there must be analogous results for different other Tannaka duality theories, which are constructed now for many situations, including the very general one where $\varPi$ is the category of representations of a monoid $A$ in a closed symmetric monoidal category $V$.

I think, I am reading wrong books, but I only know the Krein theorem for compact groups: even the formulation for algebras $A$ over $\mathbb C$ is not familiar to me. Can anybody enlighten me on what is known in this field?

EDIT. I apologize, I have poorly formulated this question. Apparently, it was necessary to directly criticize the article in nLab, but I underestimated the importance of this source. After Qiaochu Yuan's answer, the problem, as I see it, is that in the list of examples in this article there are no references to specific statements in the literature. I am sure that this is easily fixable, and I would be grateful if someone would provide these links (either here, or in the article itself).

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  • $\begingroup$ The ncat article you link includes a rather expansive list of what is known in this field, including categorical perspective using fiber functors and yoneda lemmas, which lead to generalizations of Tannaka duality to higher categories and a list of reconstruction theorms which are all, mutatis mutandis, the same. Its references are also fairly expansive, including Delign's Categories Tannakien. Do you have a specific question about the content of this article which is not clear? $\endgroup$ – Matthew Titsworth Sep 12 at 14:49
  • $\begingroup$ @MatthewTitsworth yes I don't understand details. For example, what is the description of the dual category for an algebra $A$ over $\mathbb C$? To say that this is just the category of $A$-modules is not an answer (this is as if in the Krein theorem it was written "the category of $G$-modules"). $\endgroup$ – Sergei Akbarov Sep 12 at 15:29
  • $\begingroup$ I thought the list in the nLab article was just a list of heuristic expectations, not exact statements. Are these indeed exact theorems? $\endgroup$ – Sergei Akbarov Sep 12 at 15:44
  • $\begingroup$ Matthew are you the author of this article? It would help if somebody give exact references to each line of this list (and correct the formulations). $\endgroup$ – Sergei Akbarov Sep 12 at 16:02
  • $\begingroup$ If you are going to assert that the actual answer is not an answer, then there's little which can be said to help you. Which details don't you understand? The objects of $Mod_A$ are modules over $A$. The morphisms are morphisms of modules. If you don't understand the definitions of Rings and Modules over them, these are standard, but can also be found on ncat or wikipedia. The ncat article also links to the definition of $Mod_A$ and includes a sketch of the special case where $A = k[G]$ to show that $Rep(G) \sim Mod_{k[G]}$ to link back to the representation theory of finite groups. $\endgroup$ – Matthew Titsworth Sep 12 at 17:44
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In the comments you ask:

The question is the following: if I have an "abstract" category (or even a subcategory in the category of vector spaces), how can I understand that this is the category of modules over an algebra $A$?

There is a sharp characterization of module categories as follows.

Gabriel's theorem: A category $C$ is the category of modules over a ring iff it is abelian, cocomplete, and has a compact projective generator $P$. The functor $\text{Hom}(P, -)$ then exhibits $C$ as the category of right modules over the ring $R = \text{End}(P)$.

For a proof see these two blog posts. "Generator" is a priori ambiguous but in this case all of the common definitions are equivalent.

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    $\begingroup$ The proof shows that we can replace "abelian" by "linear" (by which I mean $\mathbb{Z}$-enriched / preadditive), incidentally. $\endgroup$ – Qiaochu Yuan Sep 13 at 18:48

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