Inmathematics, aTannakian category is a particular kind ofmonoidal categoryC, equipped with some extra structure relative to a givenfieldK. The role of suchcategoriesC is to generalise the category oflinear representations of analgebraic groupG defined overK. A number of major applications of the theory have been made, or might be made in pursuit of some of the centralconjectures of contemporaryalgebraic geometry andnumber theory.
The name is taken fromTadao Tannaka andTannaka–Krein duality, a theory aboutcompact groupsG and theirrepresentation theory. The theory was developed first in the school ofAlexander Grothendieck. It was later reconsidered byPierre Deligne, and some simplifications made. The pattern of the theory is that ofGrothendieck's Galois theory, which is a theory about finitepermutation representations of groupsG which areprofinite groups.
The gist of the theory is that thefiber functor Φ of the Galois theory is replaced by an exact and faithfultensor functorF fromC to the category offinite-dimensionalvector spaces overK. The group ofnatural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the groupG of natural transformations ofF into itself, that respect the tensor structure. This is in general not an algebraic group but a more generalgroup scheme that is aninverse limit of algebraic groups (pro-algebraic group), andC is then found to beequivalent to the category of finite-dimensional linear representations ofG.
More generally, it may be that fiber functorsF as above only exists to categories of finite-dimensional vector spaces over non-trivialextension fieldsL/K. In such cases the group schemeG is replaced by agerbe on thefpqc site of Spec(K), andC is then equivalent to the category of (finite-dimensional) representations of.
LetK be a field andC aK-linearabelianrigid tensor (i.e., asymmetric monoidal) category such that. ThenC is aTannakian category (overK) if there is an extension fieldL ofK such that there exists aK-linearexact andfaithful tensor functor (i.e., astrong monoidal functor)F fromC to thecategory of finite dimensionalL-vector spaces. A Tannakian category overK isneutral if such exact faithful tensor functorF exists withL=K.[1]
The tannakian construction is used in relations betweenHodge structure andl-adic representation. Morally[clarification needed], the philosophy of motivestells us that the Hodge structure and the Galois representation associated to an algebraic variety are related to each other. The closely-related algebraic groupsMumford–Tate group andmotivic 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 categoriesare isomorphic to one another up to connected components.
Those areas of application are closely connected to the theory ofmotives. Another place in which Tannakian categories have been used is in connection with theGrothendieck–Katz p-curvature conjecture; in other words, in boundingmonodromy groups.
TheGeometric Satake equivalence establishes an equivalence between representations of theLanglands dual group of areductive groupG and certain equivariantperverse sheaves on theaffine Grassmannian associated toG. 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.
Wedhorn (2004) has established partial Tannaka duality results in the situation where the category isR-linear, whereR is no longer a field (as in classical Tannakian duality), but certainvaluation rings.Iwanari (2018) has initiated and developed Tannaka duality in the context ofinfinity-categories.
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