Tannakian category - Knowlpedia

Let $k$ be a field. A neutral Tannakian category over $k$ is an essentially small (isomorphism classes form a set), $k$ -linear abelian rigid symmetric monoidal category $\mathcal C$ with $\operatorname{End}(\mathbf 1)=k$ , equipped with an exact faithful $k$ -linear strong symmetric monoidal 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: maps $e:X^\vee\otimes X\to\mathbf1$ and $i:\mathbf1\to X\otimes X^\vee$ satisfy

(\operatorname{id}_X\otimes e)\circ(i\otimes\operatorname{id}_X)=\operatorname{id}_X, \qquad (e\otimes\operatorname{id}_{X^\vee})\circ(\operatorname{id}_{X^\vee}\otimes i)=\operatorname{id}_{X^\vee},

with the canonical associators and unitors inserted. Strong symmetric monoidality means specified natural isomorphisms $\omega(X)\otimes\omega(Y)\cong\omega(X\otimes Y)$ and $k\cong\omega(\mathbf1)$ , compatible with associators, both unitors, and symmetries. The hom-spaces and composition are $k$ -linear, and tensor product is $k$ -bilinear on morphisms.

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).

Here $\operatorname{Rep}_k(G)$ denotes finite-dimensional algebraic representations. 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.