Of late, a lot of people have been asking me about Tannakian categories, so I decided to post this short blurb (mainly to check the Latex-related features of wordpress and also to provide some bird’s-eye view of why we should care about Tannakian categories.)

To begin with:

Definition
A neutral Tannakian category $\mathcal{T}$ over a base field $k$ is a rigid abelian tensor category with an $k$-linear exact faithful tensor $\omega: \mathcal{T} \longrightarrow \textrm{Vect}_k$ ($\rm{Vect}_k$ being the category of vector spaces over $k$) which one calls the fiber functor.

This definition is stolen from a paper of Breen in the Motives volume (Proc. Symp. Pure Math. Vol 55, part I); As Breen notes, the idea is to imagine $\omega(X)$ as a fiber over $X \in \rm{Obj}(\mathcal{T})$; $\omega$ is, therefore, a sort of a vector bundle over $\mathcal{T}$. Incidentally, this is the original definition given by Saavedra-Rivano in his thesis with Grothendieck. Deligne in his Grothendieck Festschrift paper gives an equivalent definition where the “primacy” is on dual objects than internal homs (required for the “rigid” part in the above definition.)

The main reason why one should care about Tannakian categories is this:

Theorem
Let $X$ be an affine group scheme over $k$. Then the category of representations $\textrm{Rep}(X)$ of $X$ is a Tannakian category. Moreover the automorphisms of the fiber functor $\omega: \textrm{Rep}(X) \longrightarrow \rm{Vect}_k$ (simply the forgetful functor)

$\rm{Aut}^{\otimes} \omega \simeq X.$

The moral is simple: since the category of representations of an affine group scheme is Tannakian, just by studying the automorphisms of the fiber functor one recovers the affine group scheme itself (BTW, that by itself is a commutative but not necessarily co-commutative Hopf algebra.) One should compare the theorem above with the classical Pontryagin duality.

But for me (atleast), the real reason why Tannakian categories are interesting is because of the role they play in Grothendieck’s original conception of motives. Recall that the basic idea behind pure motives is this: let $\textrm{Smproj}$ be the category of smooth projective schemes over some base (field) and $\textrm{GrVect}_K$ be the category of graded vector spaces over a field $K$ of char 0 and $\mathcal{M}_{\sim}$ the category of pure motives (ie, motives of smooth projective schemes) with some adequate relation on the cycles $\sim$. In (the theory? fantasy? of) the original conception, we have three functors: the obvious contravariant one for good cohomology (Weil cohomology, for those in the know): $H: \textrm{Smproj} \longrightarrow \textrm{GrVect}_K$ and two other slightly mysterious functors: the functor of motivic cohomology that takes a sm. proj. scheme and associates to it a pure motive, and the realization functor $\omega: \mathcal{M}_{\sim} \longrightarrow \textrm{GrVect}_K$ which takes a motive and “realizes” it as a concrete cohomology theory such as deRham, etale with l-adic coeffs, …. Now the original idea was that if $\mathcal{M}_{\sim}$ was a Tannakian category then one should consider the realization functor as the fiber functor- in which case one has some very interesting Galois groups as groups of automorphisms at hand (he motivic Galois groups.) In practice though, all of this is very hard to construct explicitly- I will return to some of these issues in a less name-dropping and leisurely way in later posts.