# nLab Tannakian category

Contents

category theory

## Applications

#### Algebra

higher algebra

universal algebra

duality

# Contents

## 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”).

## 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, Providence, R.I.: American Mathematical Society, 1994.

The relation to motives is disucssed in

• Pierre Deligne, James Milne, Tannakian categories (web)

• 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

Last revised on August 19, 2016 at 10:39:39. See the history of this page for a list of all contributions to it.