AFFINE GROUP SCHEMES // REP CATEGORIES // TANNAKA DUALITY // FIBER FUNCTORS
Tannaka duality is a far-reaching generalization of Pontryagin duality: an affine group scheme G over a field k is completely determined by its category of representations Rep(G) together with the forgetful (fiber) functor omega: Rep(G) -> Vec_k.
The group is RECOVERED from its shadow — the category of things it acts on. This is reconstruction: knowing the representations is knowing the group. Deligne proved the converse: every rigid abelian tensor category with a fiber functor IS Rep(G) for some G.
A fiber functor omega: C -> Vec_k is an exact faithful tensor functor from a Tannakian category C to finite-dimensional vector spaces. It "forgets the symmetry" — like evaluating a local system at a point to get a vector space. Different fiber functors correspond to different base points, connected by torsors.
The space of fiber functors is a G-torsor (principal homogeneous space). Choosing a fiber functor is like choosing coordinates — the group G acts transitively on all possible choices. This is the geometric content of Tannaka duality.
A Tannakian category is a rigid abelian k-linear tensor category (C, tensor, 1) where every object has a dual. "Rigid" means internal Hom exists and behaves like duals of representations. The key axioms: tensor product is exact in each variable, End(1) ~ k, and every object has finite length.
Deligne's theorem: If C is a Tannakian category over a field k of characteristic 0 and admits a fiber functor to Vec_K for some extension K/k, then C ~ Rep(G) for an affine group scheme G over k. The group scheme G = Aut^tensor(omega) is pro-algebraic.
Fundamental examples: (1) Rep(G) for algebraic group G is Tannakian with omega = forgetful. (2) Local systems on (X,x) form a Tannakian category with omega = stalk at x, giving G = pi_1^alg(X,x). (3) Hodge structures form a Tannakian category, giving the Mumford-Tate group. (4) Motives form a (conjectural) Tannakian category, giving the motivic Galois group.
In characteristic p, or when the category has non-trivial parity, one must allow super group schemes. Deligne proved: every tensor category of moderate growth over an algebraically closed field of char 0 is super-Tannakian — it is Rep(G) for an affine super group scheme G. The super condition detects signs in the symmetric monoidal structure.