Tannakian formalism

Let K be a field and C a K-linear abelian rigid tensor (i.e., a symmetric monoidal) category such that \mathrm{End}(\mathbf{1})\cong K. Then C is a Tannakian category (over K) if there is an extension field L of K such that there exists a K-linear exact and faithful tensor functor (i.e., a strong monoidal functor) F from C to the category of finite dimensional L-vector spaces. A Tannakian category over K is neutral if such exact faithful tensor functor F exists with L=K. ==Applications==

The tannakian construction is used in relations between Hodge structure and l-adic representation. Morally, the philosophy of motives tells us that the Hodge structure and the Galois representation associated to an algebraic variety are related to each other. The closely related algebraic groups Mumford–Tate group and motivic Galois group arise from categories of Hodge structures, category of Galois representations and motives through Tannakian categories. Mumford-Tate conjecture proposes that the algebraic groups arising from the Hodge strucuture and the Galois representation by means of Tannakian categories are isomorphic to one another up to connected components. Those areas of application are closely connected to the theory of motives. Another place in which Tannakian categories have been used is in connection with the Grothendieck–Katz p-curvature conjecture; in other words, in bounding monodromy groups. The Geometric Satake equivalence establishes an equivalence between representations of the Langlands dual group {}^L G of a reductive group G and certain equivariant perverse sheaves on the affine Grassmannian associated to G. This equivalence provides a non-combinatorial construction of the Langlands dual group. It is proved by showing that the mentioned category of perverse sheaves is a Tannakian category and identifying its Tannaka dual group with {}^L G. ==Extensions==

has established partial Tannaka duality results in the situation where the category is R-linear, where R is no longer a field (as in classical Tannakian duality), but certain valuation rings. has initiated and developed Tannaka duality in the context of infinity-categories. ==References==