The formal theory of Tannaka duality
By: Daniel Schäppi
Series: Astérisque -- 357; Astérisque -- 357.
Publisher: Société mathématique de France
Published: 2013
Place of publication: Paris
Language: eng
Number of pages: 140
ISBN-13: 9782856297735
Copyright date: 2013
Subjects: Categories (Mathematics); Duality theory (Mathematics)
Other titles: Tannaka duality
Description: A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a field extension. If we are working over an arbitrary commutative ring rather than a field, the categories of representations cease to be abelian. We provide a list of sufficient conditions which ensure that an additive tensor category is equivalent to the category of representations of an affine groupoid scheme acting on an affine scheme, or, more generally, to the category of representations of a Hopf algebroid in a symmetric monoidal category. In order to do this we develop a "formal theory of Tannaka duality" inspired by Ross Street's "formal theory of monads." We apply our results to certain categories of filtered modules which are used to study p-adic Galois representations.
Notes: Includes bibliographical references (pages 137-140).

Text in English; abstract in both English and French.
Table of contents: title: 1. Introduction; title: 2. The category of filtered modules; title: 3. Outline of the Tannakian biadjunction; title: 4. The Tannakian biadjunction for general 2-categories; title: 5. Details for the Tannakian biadjunction in Mod (V); title: 6. The recognition theorem in Mod (V); title: 7. Cosmoi with dense autonomous generator; title: 8. Further simplifications when V is abelian; title: 9. Tannakian duality for bialgebras and Hopf algebras; title: 10. Affine groupoids over commutative rings; title: 11. The Tannakian biadjunction for Gray monoids; title: 12. Base change; title: Appendices.