symmetric monoidal (∞,1)-category of spectra
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
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 – playing the role of a category of representations/actions of some algebraic structure over some base ring – and equipped with a monoidal functor to -modules or similar – playing the role of the forgetful functor which forgets the action (the “fiber functor”).
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, 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
Last revised on October 3, 2022 at 06:50:54. See the history of this page for a list of all contributions to it.