In mathematics, a Tannakian category is a particular kind of C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory.
The name is taken from Tadao Tannaka and Tannaka–Krein duality, a theory about compact groups G and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups G which are profinite groups.
The gist of the theory is that the fiber functor Φ of the Galois theory is replaced by a tensor functor T from C to K-Vect. The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group (a priori only a monoid) of natural transformations of T into itself, that respect the tensor structure. This is by nature not an algebraic group, but an inverse limit of algebraic groups (pro-algebraic group).
A neutral Tannakian category is a , such that there exists a K-tensor functor to the that is exact and faithful.
The construction is used in cases where a Hodge structure or l-adic representation is to be considered in the light of group representation theory. For example, the Mumford–Tate group and motivic Galois group are potentially to be recovered from one cohomology group or Galois module, by means of a mediating Tannakian category it generates.
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.
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In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968).
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Pierre René, Viscount Deligne (dəliɲ; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled Théorème de Lefschetz et critères de dégénérescence de suites spectrales (Theorem of Lefschetz and criteria of degeneration of spectral sequences).
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