Pre-Talbot seminar, lecture 2
John Francis - Tannakian Formalism and the Barr-Beck Theorem
Tannakian formalism - gives a criterion for recognizing a tensor category C ⊗ as the representations of an affine algebraic group.
(1) Geometric motivation(2) Category theory, Barr-Beck(3) Tannaka-Krein duality
1. Say we have X some scheme, stack, etc. If you’re not comfortablewith this, think of some equations with coefficients in a field k, and theymight have symmetries. For any k-algebra R, you can say
R-valued solutionsX(R) =to these equations
With symmetries, you don’t just have a set. The symmetries give youa groupoid.
We want to study QCX , the category of quasi-coherent sheaves on
X. A quasi-coherent sheaf M on X consists of the data of an Amodule M (x) (= a quasi-coherent sheaf on Spec A) for every map
x : Spec A → X.
Unfortunately, geometry is hard - equations can be complicated, andso can symmetries. We’d like to reduce our problems to questions about
vector spaces.
Shape of a solution We want to say that the data of M ∈ QCX isequivalent to the data of a vector space + some extra structure on it.
p
Take # 1: Take global sections. We have a canonical map X → ∗,and pushing forward gives us a functor p∗ = Γ : QCX → QC∗ = k-mod.
We ask for the data of M to be given by the data of RΓ(M ) together
with some extra structure on it.
Problem: Is there a problem? [Nick says, “It’s not faithful.”]What? What does that have to do with anything? Okay, so Nick’spessimism says that RΓ can kill things - it’s not conservative. Actually, if X is affine, it’s just forgetful. It’s not a murderer. But if
X = P1 , then O(−1) has vanishing cohomology. We can’t put any
extra structure on zero, so Nick might be right.
Also, RΓ doesn’t preserve tensor structure, but we have no need togo there, since we’re already dead in the water.
Take # 2: Cover X. We pick an affine cover f : Spec A → X.
Then f ∗ : QCX → A-mods is conservative. This cover describes X bygluing, so we can form a simpicial object that maps to X:
1

2· · · X2 oπ1/
/X o ∆ X/ 0/ 1/o/Xπ2

where X1 = X0 × X0 , X0 = Spec A, and ∆ is the relative diagonal.
XX = colim Xi . This is called a geometric realization, and it is goodenough to describe QCX in terms of A-modules.
Unfortunately, this is more data than just k-modules. [Kobi mentions that A-modules are just k-modules with extra structure.] Well,the extra structure of an A-module is monadic, while the above iscomonadic. We don’t want to mix them. Bad idea. [Mixing makes it
difficult to understand the tensor structure.]
Geometrically, we are asking for X to be covered by Spec k. Asking
to be covered by a point is asking for equations to have a single solution
over k. We can try to build the simplicial object· · · oo//
/ Spec k × Spec k o/

/ Spec kX/X

Let’s suggestively write G = Spec k× Spec k, so X2 = ∗× ∗× ∗ = G× G.
XXX∗Then our simplicial structure looks likeo m //Go/···G × G o//∗/ X,with m denoting a multiplication map. In other words, covering X bya point gives an isomorphism X ∼= BG for G a monoid. In fact, G is agroup, because the symmetries in our stacks take values in groupoids.
If we took a more general notion of stack, allowing arbitrary categories
of symmetries, then we wouldn’t have a group. Anyway, describing
QCX in terms of vector spaces with extra data is the same as giving it
the structure of representations of a group.
This suggests the following: Given C ⊗ , if there exists a functor F :C ⊗ → k-mod⊗ that is conservative, and C ⊗ has duals, then C ⊗ ∼= Repk G
for G an affine group scheme. F is called a fiber functor - the motivation
for this terminology comes from a similar idea in fundamental groups
using sets instead of vector spaces.
There is an analogous idea in homotopy theory. A space that iscovered by a point is just a pointed connected space, which gives anequivalent theory to that of loop spaces, by the functors B (classifyingspace) and Ω (loop space).
2. Category theory. We’ll give a formal setup for describing the“extra structure.”

3F

Suppose we have some categories C  A. F and G are adjoint
Gfunctors if there exists a natural equivalence
HomA (F X, Y ) ∼= HomC (X, GY ).
A typical example is C = V ect, A = Comm − alg, F is the free algebra
functor, Sym∗ , and G forgets the algebra structure.
Objects of C in the essential image of F have some extra structure,
which we’d like to extract. First, we note that composition gives us
functors G ◦ F : C → C and F ◦ G : A → A. By using the above
natural equivalence on identity maps, we get natural transformations
idC → G ◦ F , called the unit, and F ◦ G → idA , called the counit.
Let C = F ◦ G. Then there is a natural map C → C ◦ C givenunit
by F G = F ◦ idC ◦ G −→ F GF G. This is a coassociative coalgebra
structure on C, called a comonad (or cotriple). We get a diagram
F̃COF



/ ComodC (A)GA
where ComodC (A) is the category of comodules over the comonad C.
F̃ is given by the unit map F (X) → F (GF )(X) = C ◦ F (X). It gives
us an approximation of C as “A + extra structure.” We’d like to know
how good this approximation is.
Barr-Beck Theorem If F is conservative + a modest additional hypothesis, then F̃ is an equivalence.
The hypothesis is that F preserves F -split equalizers, i.e., that itpreserves a few limits in addition to all colimits. There is an opposite
version, with modules over a monad, but we won’t use it.
F3. Tannakian formalism - says that if C ⊗  k-mod is a conservative
tensor functor, and if C has duals (i.e., is rigid), then C ⊗ ∼= Repk (G)for some G.
Asking for a conservative tensor functor to k-mod is a strong thing
to ask, just like asking a stack to be covered by a point.
Sketch of a proof
Our candidate G is given by tensor automorphisms of the fiber functor F . Aut⊗ (F ) is a group. In fact, for any k-algebra R, we can define
R-points by base change. There is a natural functor F ⊗ R : C ⊗ → Rmod, and we define Aut⊗ (F )(R) := Aut⊗ (F ⊗ R), so Aut⊗ (F ) is agroup scheme over k.

4Suppose that C ⊗ = Rep(G). Let’s check that we can recover Gthrough this precedure. There exists a homomorphism G → Aut⊗ (F ),where each g ∈ G gives us a commutative diagram
Vφg/Vg

φ/ V0V0
We can check that it is an equivalence by looking at the subcategory
in C generated by V . The two maps Aut⊗ (F ) ,→ GL(V ) ← GV havethe same image. We take a limit over all V ∈ C ⊗ and we are done.

That was the Tannaka part. This is the Krein part.
Given general C ⊗ with adjunction as above, C ⊗ ∼= ComodC (k-mod)
by Barr-Beck. C is colimit-preserving, so C(1) gets a coalgebra structure. We have an equivalence:
ComodC (k-mod) ∼= {comodules over the coalgebra C(1)}C(1) as a coalgebra gets an algebra structure via the marked arrow:
X ⊗ X 0P

PPPPPP
PPPPPP(/ X ⊗ X 0 ⊗ C(1)⊗2kkkkkkkkkkk ∗ku kkk(X ⊗ X 0 ) ⊗ C(1)Using this algebra structure (which is commutative), we get G =Spec C(1). Then, ComodC (k-mod) = ComodOG = Repk (G).