Astérisque 2013; 140 pp; softcover Number: 357 ISBN10: 2856297730 ISBN13: 9782856297735 List Price: US$63 Member Price: US$50.40 Order Code: AST/357
 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. The author provides 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 he develops a "formal theory of Tannaka duality" inspired by Ross Street's "formal theory of monads." He applies his results to certain categories of filtered modules which are used to study \(p\)adic Galois representations. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in Tannaka duality, pseudomonoids, and Hopf monoidal comonads. Table of Contents  Introduction
 The category of filtered modules
 Outline of the Tannakian biadjunction
 The Tannakian biadjunction for general 2categories
 Details for the Tannakian biadjunction in \(\mathbf{Mod}(\mathcal{V})\)
 The recognition theorem in \(\mathbf{Mod}(\mathcal{V})\)
 Cosmoi with dense autonomous generator
 Further simplifications when \(\mathcal{V}\) is abelian
 Tannakian duality for bialgebras and Hopf algebras
 Affine groupoids over commutative rings
 The Tannakian biadjunction for Gray monoids
 Base change
 Appendix A. Density in cosmoi with dense autonomous generator
 Appendix B. Monoidal biadjunctions
 Appendix C. A technical lemma
 Appendix D. Tannaka duality for pseudomonoidal comonoids
 Bibliography
