Tannakian Formalism

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.

This doesn't answer your question directly, but I think does explain why such theorems exist: if your notion of "representations" of some algebraic gadget are insufficient to reconstruct the gadget from the 'collection' of representations, then you better go think up better "representations"! I mean this rather loosely, so for example I'm claiming that thinking about representations of complex algebraic groups without also thinking about how they tensor would be lame.

This for example is why the "representation theory of a subfactor $N < M$" is all about the bimodules between $N$ and $M$.

The forgetful functor to $\text{Vect}$ I don't think of as so essential -- it lets you construct the original gadget in the category you expect, but if you don't have this function it shouldn't prevent you from believing that there's a "gadget object" in some other category.

I'm not sure how to answer the question about intuition. In regards to your second question, the Tannakian formalism works over any field and in fact works for more general categories than Rep(G). A category C is called a neutral Tannakian category over k (k a field) if it is a rigid abelian tensor category together with a k-linear tensor exact functor from C to k-Vect. This latter functor is known as the fiber functor (and is just the forgetful functor referenced in the original post when C = Rep(G)). In this case, there's a theorem that says any neutral Tannakian category is equivalent to the category of representations of an affine group scheme G, where G is given by the tensor automorphisms of the fiber functor.

One of the original references for this is a 1982 paper by Deligne and Milne titled 'Tannkian Categories.' Deligne also has an article in the more recent FGA explained.

My understanding is that sometimes k can even be replaced by an arbitrary commutative ring. I don't know general conditions under which this holds. However, a great example of the Tannakian formalism in action where k is a general commutative ring is the Mirkovic-Vilonen paper on the geometric Satake correspondence, which they prove in great generality (Ginzburg also has a nice paper on this topic, but only in characteristic zero). The Mirkovic-Vilonen paper can be found here http://arxiv.org/abs/math/0401222

In this paper, they prove that their fiber functor is represented by a k-module which is free over k (k any commutative ring here) and hence the Tannakian formalism still works. I don't know if this condition is also a necessary condition.

There's also a version of the Tannakian formalism over a scheme, so to speak. If we view a group G as a principal G-bundle on a point and the category of vector spaces as the category of vector bundles on a point, then we can try to generalize to a scheme. Namely, if we fix a G-bundle on a scheme X, then this induces an exact tensor functor from Rep(G) to the category of vector bundles on X using the associated bundle construction. It turns out that the converse holds: given an exact tensor functor from Rep(G) to vector bundles on X, this is equivalent to giving a G-bundle on X. There's a proof of this fact in a set of notes on the webpage for the seminar that Dennis Gaitsgory is currently running.

I think it's instructive to start by asking the easier question: "how do I recover a (non-commutative) ring $R$ from the category $Mod_R$ of $R$-modules"?

The first thing you might try is to identify the free $R$-module $R$ as an object of $Mod_R$. There is a particularly nice description of this object: it is the object which represents the forgetful functor $F: Mod_R \to Ab$ taking an $R$-module to its underlying abelian group. This is not anything subtle: all I'm saying here is that if $M$ is an $R$-module and $m\in M$, then there is a unique $R$-linear map $f\colon R \to M$ taking $1\in R$ to $m\in M$. Now because of abstract category theory, we know that the object representing a functor is unique up to isomorphism, so this really does characterise $R$ as an object of $Mod_R$. (More precisely, it identifies the pair $(R,1\in R)$ up to unique isomorphism.)

The above idea shows the importance of fibre/forgetful functors, but it doesn't yet allow us to recover $R$ as a ring, because it doesn't give us access to the ring structure. This is where auto/endomorphisms of the fibre functor enter the picture. Indeed, because $R$ represents the fibre functor, we know by Yoneda that $$End(F)=Hom(Hom(R,-),F)\cong F(R)=R$$ as abelian groups. When you trace through the Yoneda Lemma, this isomorphism is sending $r\in R$ to the endomorphism of the fibre functor which acts on an $R$-module $M$ by $m\mapsto rm$. (N.B. this map is not $R$-linear if $R$ is non-commutative, but it is an endomorphism of the fibre functor.) In particular, the isomorphism $End(F)\cong R$ is even an isomorphism of rings.

To summarise: we can recover any ring $R$ from the category $Mod_R$ and the forgetful functor $F: Mod_R \to Ab$, as the ring of endomorphisms of $F$.

I don't want to add too much more to an already lengthy answer, but I'll very briefly sketch how to recover the actual Tannakian reconstruction theorem from this viewpoint. If $G$ is an algebraic group over a field $k$, let $Rep_k(G)$ be the category of finite-dimensional $k$-linear representations of $G$. The affine ring $\mathcal O(G)$ of $G$ is the union of finite-dimensional $G$-subrepresentations, i.e. is an ind-object of $Rep_k(G)$. Dually, the dual $\mathcal O(G)^*$ is the inverse limit of a cofiltered system of finite-dimensional $G$-representations, so is a pro-object of $Rep_k(G)$. What one can show is that $\mathcal O(G)^*$ pro-represents the forgetful functor $Rep_k(G) \to Vec_k$, and so the same Yoneda argument as earlier establishes that $\mathcal O(G)^*$ can be recovered as the $k$-algebra of endomorphisms of $F$. You can then show that the tensor product and duals in $Rep_k(G)$ induce on $End(F)$ the structure of a cocommutative Hopf algebra, and that the Yoneda isomorphism $$End(F)\cong\mathcal O(G)^*$$ is an isomorphism of Hopf algebras. You can recover $G$ from $\mathcal O(G)^*$ as its scheme of grouplike elements, and so we get $$G \cong End(F)^{gplike} = Aut(F)$$ is the scheme of automorphisms of $F$.

(If you want to make this argument precise, you have to be careful about working in the category of pro-objects of $Rep_k(G)$, and interpreting words like "Hopf algebra" in that category.)

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, $L^2(G)$) 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 $V\otimes \cdots \otimes V$, 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 definition 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?