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 * over a base field is a rigid abelian tensor category with an -linear exact faithful tensor* * ( being the category of vector spaces over ) 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 as a fiber over ; is, therefore, a sort of a vector bundle over . 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 be an affine group scheme over . Then the category of representations of is a Tannakian category. Moreover the automorphisms of the fiber functor (simply the forgetful functor)

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 be the category of smooth projective schemes over some base (field) and be the category of graded vector spaces over a field of char 0 and the category of pure motives (ie, motives of smooth projective schemes) with some adequate relation on the cycles . 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): 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 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 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.

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July 10, 2007 at 9:06 pm

bbHooray for (finally!) some arithmetic geometry representation in the online world!

Now a slight “terminological” complaint: I don’t think the term “motivic cohomology” was ever used by Grothendieck to describe a universal cohomology theory (i.e: the functor you call motivic cohomology) . And the people who did coin it (Beilinson most likely) certainly don’t use it to mean a universal cohomology theory! In their theory/dream/fantasy, motivic cohomology groups are certain ext groups WITHIN an already existing derived category of mixed motives. In particular, motivic cohomology groups have “realisations” simply into abelian groups, and not into things like Galois representations/Hodge structures. A much better name, I think, would be absolute cohomology in analogy with the Galois side.

Sorry to submit an essentially contentless comment, but reading people call motivic cohomology a universal cohomology theory confused me for the longest time before I realised it wasn’t that.

July 11, 2007 at 1:45 pm

Abhijnan RejHi bb,

Thanks for the comment. You are right in saying that Grothendieck never used the term “motivic cohomology”. On the other hand, what I meant by (ab)using the term is that it is a functor that takes a smooth proj. scheme and associates to it a pure motive- now this pure motive does have (conjecturally atleast) a realization in terms of “known” cohomology theory. So in this picture, calling the functor motivic cohomology seemed appropriate.

Ofcourse, the term as is used today refers to Bellinson-Voevodsky, … which is that these are simply Ext groups in a (derived triangulated) category of mixed motives, but I guess thats a story for some other day.

Best,

Obi

November 26, 2007 at 2:10 am

What are Tannakian Categories? « The Lyceum Mathematikoi[...] in Algebra, Category Theory at 2:10 am by saij Read this and Find out: But for me (at least), the real reason why Tannakian categories are interesting is because of the [...]