I've recently realized that there is a gap in my understanding of the Tannakian formalism for reconstructing an (algebraic) group from its category of (finite-dimensional) representations. To warm up, here is a version I do understand. Let (C,⊗,flip) be a symmetric monoidal category C-linear category satisfying some technical conditions (the details of which depend on exactly what you're trying to do, but e.g. "abelian and every object is dualizable" should suffice), and suppose that F:C→VectC is a faithful symmetric monoidal C-linear (exact, ...) functor. Then G=End⊗(F), the monoid of monoidal natural transformations of F, is a (affine algebraic over C) group, and C is equivalent to the category of G-representations.
Versions of the above statement are due to Deligne, and some are older and some are newer. A very important theorem of Deligne's is that (depending on the precise set-up of the problem) the fiber functor F isn't needed. Indeed, a result that I continue to find amazing is that there exists a unique-up-to-(necessarily nonunique!)-isomorphism fiber functor C→VectC iff every object in C has nonnegative-integer dimension. (In case it's not clear, everywhere I write "Vect" and so on, I mean to imply the category of finite dimensional vector spaces.)
I'm not so worried about the above version of Tannakian theory (although I'm sure I missed some details), as about the "super" version. Let me start with the generalization of the second paragraph. Recall that the category SVectC of super vector spaces is the free (technical conditions) monoidal category generated by VectC and an object X that ⊗-squares to the unit object, with the symmetric structure uniquely determined by the request that flip:X⊗X→X⊗X is minus the identity. (As a monoidal category, but not as a symmetric monoidal category, SVectC is equivalent to the category of representations of Z/2.) The generalization of the second paragraph, also due to Deligne, is that a rigid symmetric monoidal (technical conditions) category has a fiber functor to SVectC iff every object has (possibly negative) integer dimension is annihilated by some Schur functor.
So let's take some category C, like C=SVectC itself. My problem is to understand the generalization of the first statement. See, the identity functor SVectC→SVectC has a nontrivial symmetric monoidal automorphism, which acts by −1 on the "fermionic" generating object X. So the naive theorem fails: it is not true that SVectC is equivalent to the category of End⊗(id)-representations in SVectC.
One potential fix is to strengthen the condition that C be C-linear to the condition that it be tensored over VectC or over SVectC (depending on where the fiber functor lands). Then the identity functor id:SVectC→SVectC has only the identity automorphism when thought of as a functor of SVectC-tensored categories.
But my actual interest is in the category DGVect of chain complexes of (non-super) vector spaces. This category is not tensored over SVect, but it does have a fiber functor to it. When you calculate the endomorphism supergroup group of this fiber functor, you get the group G=Gm⋉G0|1a, which is the group of affine transformations of the odd line. But the category of G-representations in SVect is the category of chain complexes of supervector spaces, not the category of chain complexes of regular vector spaces. Not that this latter thing is a bad category — it's just that I'd like to describe the other one too. And on the other hand, G isn't defined over Vect, so I can't say that DGVect is "the category of G-modules in Vect".
So my question is: how does Tannakianism treat the category of chain complexes? More generally, how does it treat a category with a fiber functor to SVect? I guess I could have also asked an even easier example (although it's not my main motivation): how can I describe Vect as a Tannakian category over SVect (with the unique inclusion)? In all cases, I have a fiber functor F:C→SVect and I can construct the supergroup G=End⊗(F), but to get C back I seem to need some way to mandate how G interacts with the canonical Z/2 that acts on SVect — how, and what is this generalization of supergroup?
This post imported from StackExchange MathOverflow at 2015-03-30 11:50 (UTC), posted by SE-user Theo Johnson-Freyd