Non-Abelian Class Field Theory for Riemann Surfaces

Let T be a Tannakian category with a fiber functor ω : T → VerC, where VerC denotes the category of finite dimensional C-vector spaces. An object t ∈ T is called reducible if there exist non-zero objects x, y ∈ T such that t = x ⊕ y. An object is called irreducible if it is not reducible. If moreover every object x of T can be written uniquely as a sum of irreducible objects x = x1 ⊕ x2 ⊕ . . . ⊕ xn, then T is called a unique factorization Tannakian category. Usually, we call xi’s the irreducible components of x. A Tannakian subcategory S of a unique factorization Tannakian category T is called completed if for x ∈ S, all its irreducible components xi’s in T are also in S. S is called finitely generated if as a Tannakian category, it is generated by finitely many objects. Moreover, S is called finitely completed if (a) S is finitely generated; (b) S is completed; and (c) Autω ∣