Tannakian category - Knowlpedia

Definition

Let $k$ be a field. A neutral Tannakian category over $k$ is a $k$ -linear abelian rigid symmetric monoidal category $\mathcal C$ with $\operatorname{End}(\mathbf 1)=k$ , equipped with an exact faithful $k$ -linear tensor functor

\omega:\mathcal C\longrightarrow\operatorname{Vect}^{\mathrm{fd}}_k.

The functor $\omega$ is a fiber functor. “Rigid” means every object has a tensor dual.

Tannaka reconstruction

The tensor automorphisms of the fiber functor form an affine group scheme

G=\operatorname{Aut}^{\otimes}(\omega),

and Tannaka duality gives a tensor equivalence

\mathcal C\simeq\operatorname{Rep}_k(G).

Thus the group is reconstructed from its category of representations together with the forgetful fiber functor.

If a fiber functor exists only after extending scalars, the category is Tannakian but not necessarily neutral over $k$ ; its fiber functors form a gerbe rather than selecting one group scheme over $k$ .

Langlands examples

The geometric Satake category is Tannakian, and its Tannaka group is the Langlands dual group. Categories of local systems or motives, when equipped with a suitable fiber functor, similarly produce Galois or motivic groups.

References

Pierre Deligne and James S. Milne, “Tannakian Categories,” in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900, Springer, 1982, 101–228. Milne.
James S. Milne, “Tannakian categories: origins and summary,” 2025. arXiv.