TITLE: Tannakian Formalism
QUESTION [39 upvotes]: The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, why should this be enough information to recover the group? And does this work for other base fields (or rings?)?

REPLY [63 votes]: A very nice and very general version of Tannakian formalism is in Jacob Lurie's paper,
Tannaka duality for geometric stacks, arXiv:math/0412266. 
I like to think of Tannaka duality as recovering a scheme or stack from its category of coherent (or quasicoherent) sheaves, considered as a tensor category. 
From this POV the intuition is quite clear: having a faithful fiber functor to Vect (or more generally to R-modules) means your stack is covered (in the flat sense) by a point (or by Spec R). This is why you get (if you're over an alg closed field) that having a faithful fiber functor to Vect_k means you're sheaves on the quotient BG of a point by some group G, i.e. Rep G.. Over a more general base, you only locally look like a quotient of Spec R (or Spec k for k non-alg closed) by a group ---- ie you're a BG-bundle over Spec R, aka a G-gerbe. Even more generally, the kind of Tannakian theorem Jacob explains basically says that any stack with affine diagonal can be recovered from its tensor category of quasicoherent sheaves..
Actually the construction of the stack from the tensor category is just a version of the usual functor Spec from rings to schemes. Recall that as a functor, Spec R (k) = homomorphisms from R to k. So given a tensor category C let's define Spec C as the 
stack with functor of points Spec C(k) = tensor functors from C to k-modules (for any ring k, or algebra over the ground field etc). The Tannakian theorems then say for X reasonable
(ie a quasicompact stack with affine diagonal), we have X= Spec Quasicoh(X)  ---
so X is "affine in a quasicoherent-sheaf sense". Again, the usual Tannakian story is the case X=BG or more generally a G-gerbe.

REPLY [4 votes]: Some time ago I was also puzzled by this same question, and only now, after seeing yours, I start to think maybe I understand the idea. These are my own thoughts though, so you're encouraged to re-check them.
Consider for simplicity an affine algebraic group (on the opposite side would be a complex compact one). Then we know that the regular representation k[G] (respectively, L2(G)) should decompose as \sum R^*\otimes R.
Now what does knowing all the whole tensor category of representation imply? This means we can reconstruct the space k[G] as a G-representation by the above formula, so we know the functions on G! The multiplicatication of functions on G should be embedded in the product above, though I can't yet figure exactly how. 
E.g. for a linear group G there is some representation V which contains linear functions. Then functions of the form x^n can be obtained, up to conjugacy, as the highest vectors in V\otimes ... \otimes V, taken n times.
By the correspondence between functions and spaces, this would allow us to reconstruct the original space of group G. Now the action on the space of functions translates into the action of G.
So, I think if on this road one tries to define what the point of G is, one would arrive exactly at the defintion of it as the automorphisms of the fiber functor, the way it is in the Tannakian formalism.
I wonder if somebody could give a reference to a text explaining this point of view?