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The Formal Theory of Tannaka Duality
Daniel Schäppi, University of Chicago, Illinois
A publication of the Société Mathématique de France.
 Astérisque 2013; 140 pp; softcover Number: 357 ISBN-10: 2-85629-773-0 ISBN-13: 978-2-85629-773-5 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 2-categories 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