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Classical Tannakian reconstruction recovers an affine group scheme $G$ over $k$ from the category of its linear representations over a field $k$ (as the automorphism group of the forgetful functor to the category $Vect_k$ of finite-dimensional vector spaces over $k$.)

This has many generalizations in the literature as I am slowly learning. One that I have not yet found is for other kinds of "representations" of $G$. For example, what if we consider homomorphisms of $G$ into algebraic groups different from $GL_n$'s, but here it is not even clear to me what the target category of the fiber functor would be.

Are there natural examples of a group's recovery from tensor abelian categories other than its linear reprsentations?

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  • $\begingroup$ If the representations aren't linear, where would the abelian structure come from? $\endgroup$
    – Jon Pridham
    Commented Jul 29, 2022 at 16:47
  • $\begingroup$ One sometimes defines a homomorphism of $G$ into an algebraic group as a tensor functor from the category of representations of that algebraic group into the category of representations of $G$. This is convenient when representations of $G$ are more conveniently defined as a particular category of sheaves. This suggests to me that there is nothing to say about Tannakian reconstruction of $G$ from representations into other algebraic groups - those are just another way of looking at the ordinary category of representations. $\endgroup$
    – Will Sawin
    Commented Jul 29, 2022 at 18:05

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