Tannakian formalism information

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 generalise 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 an exact and faithful tensor functor F from C to the category of finite-dimensional vector spaces over K. 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 G of natural transformations of F into itself, that respect the tensor structure. This is in general not an algebraic group but a more general group scheme that is an inverse limit of algebraic groups (pro-algebraic group), and C is then found to be equivalent to the category of finite-dimensional linear representations of G.

More generally, it may be that fiber functors F as above only exists to categories of finite dimensional vector spaces over non-trivial extension fields L/K. In such cases the group scheme G is replaced by a gerbe ${\displaystyle {\mathcal {G}}}$ on the fpqc site of Spec(K), and C is then equivalent to the category of (finite-dimensional) representations of ${\displaystyle {\mathcal {G}}}$.