nLab Tannakian category

Idea

The notion of Tannakian category is an abstraction of the kind of data that enters the Tannaka reconstruction theorem: it is a suitable monoidal category $\mathcal{C}$ – playing the role of a category of representations/actions of some algebraic structure over some base ring $R$ – and equipped with a monoidal functor $\mathcal{C} \to R Mod$ to $R$-modules or similar – playing the role of the forgetful functor which forgets the action (the “fiber functor”).

Related concepts

• Tannaka duality

• Deligne's theorem on tensor categories

References

• N. Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics 265, Springer-Verlag, Berlin-New York, (1972).

• Pierre Deligne, Catégories Tannakiennes, The Grothendieck Festschrift, Volume 2, 111–195. Birkhäuser, 1990.

• Larry Breen, Tannakian categories, in Motives, Proceedings of Symposia in Pure Mathematics, 55, part 1, Providence, R.I.: American Mathematical Society, 1994, pp. 337-376.

The relation to motives is discussed in

• Pierre Deligne, James Milne: Tannakian categories, in: Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900, Springer (1982) [doi:10.1007/978-3-540-38955-2_4, webpage]

• Pierre Deligne, James Milne, A. Ogus, K-y Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, 900, Springer-Verlag, Berlin-New York, (1982). Annotated and corrected version (2012) (PDF).

• Yves André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, 17, Paris: Société Mathématique de France. (2004)

• R. Sujatha, Motives from a categorical point of view, notes from Franco-Asian Conference on Motives, IHES, 2006. PDF