Tannakian category
Encyclopedia
In mathematics
, a tannakian category is a particular kind of monoidal category
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 Tannaka-Krein duality
, a theory about compact group
s 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, which is rather elaborate in detail in the exposition of Saavedra Rivano, 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 transformation
s 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).
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 motive
s. 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.
tensor category, together with a K-tensor functor to the category of K-vector spaces that is exact
and faithful.http://www.math.purdue.edu/~jinhyun/note/tannaka/tannaka.pdf
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a tannakian category is a particular kind of monoidal category
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
C, equipped with some extra structure relative to a given field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
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
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
The name is taken from Tannaka-Krein duality
Tannaka-Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. Its natural extension to the non-Abelian case is the Grothendieck duality theory....
, a theory about compact group
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
s G and their representation theory. The theory was developed first in the school of Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
. It was later reconsidered by Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
, and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory
Grothendieck's Galois theory
In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry...
, which is a theory about finite permutation representations of groups G which are profinite groups.
The gist of the theory, which is rather elaborate in detail in the exposition of Saavedra Rivano, 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 transformation
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
s 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
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
) 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).
Applications
The construction is used in cases where a Hodge structureHodge structure
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...
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
Galois module
In mathematics, a Galois module is a G-module where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module...
, by means of a mediating tannakian category it generates.
Those areas of application are closely connected to the theory of motive
Motive (algebraic geometry)
In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
s. Another place in which tannakian categories have been used is in connection with the Grothendieck–Katz p-curvature conjecture
Grothendieck–Katz p-curvature conjecture
In mathematics, the Grothendieck–Katz p-curvature conjecture is a problem on linear ordinary differential equations, related to differential Galois theory and in a loose sense analogous to the result in the Chebotarev density theorem considered as the polynomial case...
; in other words, in bounding monodromy groups.
Formal definition
A neutral tannakian category is a rigid abelianAbelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
tensor category, together with a K-tensor functor to the category of K-vector spaces that is exact
Exact functor
In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
and faithful.http://www.math.purdue.edu/~jinhyun/note/tannaka/tannaka.pdf