On Tannaka Duality

TL;DR

This paper explores the connections between Tannaka duality and Galois theory, extending the duality to abstract tensor categories and providing detailed proofs of the equivalence in the vector space case.

Contribution

It develops a detailed proof of Tannaka duality for vector spaces and relates different generalizations of Tannaka and Galois theories within tensor categories.

Findings

Proves the Tannaka equivalence of categories for vector spaces.

Establishes the correspondence between different Tannaka duality generalizations.

Provides detailed proofs and clarifications of existing theories.

Abstract

The purpose of this work is twofold: to expose the existing similarities between the generalizations of the Tannaka and Galois theories, and on the other hand, to develop in detail our own treatment of part of the content of Joyal and Street [1] paper, generalizing from vector spaces to an abstract tensor category. We also develop in detail the proof of the Tannaka equivalence of categories in the case of vector spaces. Saavedra Rivano [2], Deligne and Milne [3] generalize classical Tannaka theory to the context of K-linear tensor (or monoidal) categories. They obtain a lifting-equivalence into a category of \group representations" for a finite-dimensional vector space valued monoidal functor. This lifting theorem is similar to the one of Grothendieck Galois theory [4] for a finite sets valued functor. On the other hand, Joyal and Street [1] work on the algebraic side of the duality between algebra and geometry, and also obtain a lifting-equivalence, but now to the category of finite-dimensional comodules over a Hopf algebra. In this work we follow the ideas of Joyal and Street and prove their lifting-equivalence, and then we show how it corresponds, by dualizing, to the ones of Saavedra Rivano [2] and Deligne and Milne [3]. Text is in spanish, but a brief english introduction is provided.