nLab
Tannakian category

Context

Category theory

Concepts

Universal constructions

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Theorems

Extensions

Applications

Algebra

Algebraic theories

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Algebras and modules

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Higher algebras

  • symmetric monoidal (∞,1)-category of spectra

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Model category presentations

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Geometry on formal duals of algebras

Theorems

Duality

  • abstract duality: ,

  • concrete duality: , , ,

Examples

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In QFT and String theory

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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 𝒞\mathcal{C} – playing the role of a category of representations/actions of some algebraic structure over some base ring RR – and equipped with a monoidal functor 𝒞RMod\mathcal{C} \to R Mod to RR-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.