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The theorem states that, for a field $k$ of characteristic 0, any $k$-linear tensor category with $End(1)=k$ satisfying a condition that each object is annihilated by a Schur functor, is equivalent to a category of parity preserving representations of a supergroup $G$.

The statement is independent of the fiber functor, yet the proof seems to construct one by itself, by constructing a functor to $R$-modules for some large $R$ first and modifying it. I would like to understand how the proof proceeds from here.

Is this supposed to be analogous to the proof of Deligne for the non-super case? I found that the approach in Joyal & Street which realize $G$ as a spectrum of coend of the fiber functor is particularly clear to grasp. Is there a similar way of obtaining a supergroup out of a super fiber functor ?

One last question is: what aspects of super vector spaces make it possible to capture every tannakian categories?

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  • $\begingroup$ I find that the categorical dimensions are helpful in understanding it. If I'm not mistaken, then the categories you're talking about are precisely those where all objects have integer dimensions. That basically defines the fiber functor, because if you have a simple object $X$ with dimension $\pm n$, it must be sent to a vector space of dimension $n$, with grading $\pm 1$. So super vector spaces are in a sense the simplest category where all objects have integer dimensions. See also arxiv.org/abs/0906.0620. $\endgroup$
    – Manuel Bärenz
    May 2, 2015 at 16:34

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