#### 1. What is a Tannakian category?

##### 1.1 Abstracting the categorical properties of Rep(G)

The notion of neutral Tannakian category is modeled on the categories Rep(G) of finite-dimensional representations of an (affine) __algebraic group__ G (or more generally an __affine group scheme__) over a fixed field k. Two simple examples to keep in mind with k=C=complex numbers are: (i) G a finite group, (ii) G a connected, reductive algebraic group, e.g., G=GL(n) or SO(n) or Sp(2n). Another classical setting, (iii) G a connected compact Lie group, can be reduced to (ii), so it also serves as a motivating example.

A rough analogy is given by the way abelian categories are modeled on the category of abelian groups. Some basic operations on abelian groups and homomorphisms between are: subgroup, quotient, kernel, cokernel, direct sum etc. An __abelian category__ is then roughly a category in which these operations continue to make sense.

Since the above operations on abelian groups apply also to representations and morphisms between them, the categories Rep(G) are abelian categories. However, they possess two additional key operations: tensor product and dual. Moreover, if we consider representations over the fixed field k, we have a forgetful functor from Rep(G) to the category Vec_k of k-vector space which associates to a representation its underlying vector space. Thus, a __neutral Tannakian category__ C is roughly an abelian category C with a tensor product, duals and a functor to Vec_k which is supposed to generalize the forgetful functor in the case of Rep(G); such a functor (which should respect the tensor product and dual) is called a __fiber functor__. The more general notion of Tannakian category (not necessarily neutral) is more complicated; roughly it does not assume the existence of a fiber functor over k, but requires the weaker hypothesis that there exists a fiber functor over some field extension k' of k (possibly of infinite degree. [It is important and nontrivial to make all of the above precise.]

1.2 Tannaka Duality: Reconstructing G from Rep(G)

Return to the analogy between the category Ab of abelian groups and general abelian categories. While a general abelian category A is an abstraction of the category Ab, there are several embedding theorems which give us a way of "going back" from A to Ab. For example, one has (cf. [Freyd, Th. 7.14]) : Every (small) abelian category is equivalent to an exact subcategory of Ab.

If we cross over from abelian to Tannakian categories, the analogy leads us to question to what extent we can go back from a general Tannakian category C to a category Rep(G) of representations. For a compact topological group G, Tannaka showed that the category Rep(G) of continuous finite-dimensional representations uniquely determines G (up to isomorphism); in fact one can reconstruct G from Rep(G). An exposition of Tannaka's result is given in [Chevalley, Chap. VI, Th. 5]. A generalization of this result to neutral Tannakian categories was given by Saavedra (based on ideas of Grothendieck): Every neutral Tannakian category is equivalent to Rep(G) for some affine group scheme G. Saavedra also claimed a generalization to general Tannakian categories (where the group G has to be replaced by a fancier object called a __gerb__), but his argument in the general case was found to have an important error [Deligne-Milne, 3.15]. A correct statement and proof for general Tannakian categories was finally given by Deligne [Deligne].

#### 2. Applications of Tannakian categories

##### 2.1. The Tannakian viewpoint

If G is either a compact Lie group or a connected, reductive, complex algebraic group, the categories Rep(G) have been studied for roughly a century and they are ubiquitous throughout mathematics. Even in this very classical setting, the Tannakian viewpoint has been extremely powerful. This includes: (i) Studying the representations of G from the point of view of the category-theoretic properties of Rep(G) (ii) A duality between properties of G which are defined intrinsically "inside G" and those defined by means of any representation of G and independent of the choice of representation.

An important example of (i) is the notion of a __tensor generator__: A representation V such that every other representation W is obtained from V by successive application of the Tannakian operations (tensor product, dual and the abelian category operations of sub, quotient, direct sums). In the two classical cases above (and more generally for any affine (linear) algebraic group), every faithful representation is a tensor generator (and for compact Lie groups, the existence of a faithful representation follows from the Peter-Weyl theorem).

Given a tensor generator V, one may hope to prove that every representation -- i.e. every element of Rep(G) -- satisfies some property P as follows: (a) Show that V has property P, (b) Show that the property P is stable under the Tannakian operations e.g. if V_1 and V_2 have property P then so does the tensor product of V_1 and V_2. If (a) and (b) hold, the property P holds for all elements of Rep(G).

Here is a classical example of (ii): For G=GL(n) (over an algebraically closed field), an element is __semisimple__ if it is diagonalizable; at the other extreme it is called unipotent if its only eigenvalue is 1. Then a general element x admits a unique __Jordan decomposition__ x=su, with s semisimple, u unipotent and su=us. The notions of semisimple and unipotent elements, and that of Jordan decomposition generalize to an arbitrary algebraic group [Springer, Th. 2.4.8]. These notions can be defined intrinsically, without appealing to a representation of G, but also an element of G is semismple (resp. unipotent) if and only if its image has the property under every representation of G. These two complementary ways to think of semismple and unipotent elements may be viewed as the "Tannakian viewpoint" in this special case.

##### 2.2. New groups from Tannakian categories

In the previous section, we considered groups that had been discovered long ago and applied the observation that the associated representation category Rep(G) is Tannakian. So the direction was roughly "from groups to Tannakian categories". It turns out that the other direction is at least as natural. In fact, for most pairs (G, Rep(G)), it is perhaps more likely that you will first encounter Rep(G) and then deduce the existence of G by means of the Saavedra-Deligne Theorems mentioned above. Stronger yet, there are several important groups for which there is no known direct definition (as solutions to equations, or by generators and relations etc.) and the only known definition is to first define a neutral Tannakian category C with fiber functor and then define our group as "the" group G for which Rep(G) is equivalent to C. Some key examples are discussed in the next section.

#### 3. The ubiquity and universality of Tannakian categories

Since their introduction in the late 60'/early 70's, Tannakian categories have turned out to be ubiquitous throughout mathematics. Furthermore, they have often been used to unify different fields. Some examples:

##### 3.1. Hodge theory

The category R-HS of __real Hodge structures__ is neutral Tannakian over R=real numbers; it is equivalent to Rep(G) where G is the multiplicative group of nonzero complex numbers C* viewed as a real group (in other words Weil restriction of G_m from C to R) and we consider representations defined over R.

The category Q-HS^{pol} of polarizable __Q-Hodge structures__ is neutral Tannakian over Q=rational numbers. This is an example where there is no direct description of the associated group G such that Rep(G) is equivalent to Q-HS^{pol}

The categories of __variation of Hodge structure__ (over R or Q) also give neutral Tannakian categories.

##### 3.2. Fundamental group

Several categories of local systems on a topological space (resp. scheme) are Tannakian and their associated group is closely related to (completions) of the fundamental group (resp. etale fundamental group). See [Saavedra, Chap. 6, sect. 1.1].

##### 3.3. Motives

Grothendieck defined a category of (Grothendieck) motives and conjectured that it is Tannakian. This conjecture remains open; Grothendieck observed that it would follow from the "Standard Conjectures" about algebraic cycles on algebraic varieties. Assuming that Grothendieck's category is Tannakian, the associated group would be the "motivic Galois group", a generalization of the Galois group that should control all algebraic varieties (over a field which embeds in C). Unconditional definitions of a candidate for the motivic Galois group have been given by Nori and Ayoub.

Deligne constructed a category of "motives with absolute Hodge cycles" which is Tannakian and provides a substitute for Grothendieck's category which does not require knowing the standard conjectures [Deligne-Milne, sect. 5].

##### 3.4 The Langlands Program -- automorphic representations

Langlands conjectured that the collection of all automorphic representations of all GL(n) (for all n>0) should form a Tannakian category. The associated group would be the conjectural "Langlands group".

##### 3.5 Conjectures

Several of the most outstanding open conjectures in mathematics admit formulations in terms of Tannakian categories, or follow from statements about Tannakian categories. We have already mentioned the relationship between the Standard Conjectures and Grothendieck's category of motives.

Two additional monumental conjectures in algebraic geometry, in fact more striking than the Standard Conjectures, are the Hodge and Tate conjectures. The Hodge conjecture is equivalent to the Tannakian statement that the functor "Betti realization" is fully faithful from the category of motives to the category of Q-Hodge structures. Similarly, the Tate conjecture is equivalent to the statement that the functor "l-adic realization" is fully faithful from the category of motives to the category of Q_l vector spaces with continuous Galois action.

Langlands "Principle of Functoriality" conjectures about the relationship between automorphic representations of different groups is closely related to his his conjecture (3.4) that automorphic representations form a Tannakian category.

#### References

[Chevalley] C. Chevalley "Theory of Lie groups" Vol 1

[Deligne] P. Deligne "Categories Tannakiennes" In "The Grothendieck Festschrift" Vol 2

[Deligne-Milne] P. Deligne and J. Milne "Tannakian Categories" In "Hodge cycles, motives, and Shimura varieties" Lect. Notes Math. 900

[Freyd] P. Freyd "Abelian categories"

[Saavedra] N. Saavedra-Rivano "Categories Tannakiennes" Lect. Notes Math. 265

[Springer] T. Springer "Linear Algebraic Groups"

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