[Paper Review] On Tannaka Duality

M. Szyld|arXiv (Cornell University)|Oct 24, 2011

Algebraic structures and combinatorial models8 references3 citations

TL;DR

This paper establishes a duality between tensor categories and Hopf algebras by generalizing Tannaka duality from vector spaces to abstract tensor categories, following Joyal and Street's framework. It proves a lifting-equivalence between a category and the category of finite-dimensional comodules over a Hopf algebra, and shows that dualizing this equivalence recovers the results of Saavedra Rivano and Deligne-Milne in the context of K-linear tensor categories.

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.

Motivation & Objective

• To unify and clarify the connections between generalized Tannaka duality and Galois theory in the context of tensor categories.
• To provide a detailed development of Joyal and Street's lifting-equivalence theorem for abstract tensor categories.
• To demonstrate how dualizing Joyal and Street's comodule-based equivalence yields the classical Tannaka-Kreín results of Saavedra Rivano and Deligne-Milne.
• To extend the classical Tannaka duality framework from vector spaces to arbitrary K-linear tensor categories.
• To establish a conceptual bridge between algebraic geometry, representation theory, and categorical duality via Hopf algebra comodules.

Proposed method

• Adopting Joyal and Street's categorical framework for duality between algebra and geometry, focusing on monoidal categories.
• Constructing a lifting-equivalence from a tensor category to the category of finite-dimensional comodules over a Hopf algebra.
• Using the structure of a fiber functor to recover a Hopf algebra via a Tannakian reconstruction process.
• Applying duality principles to transform the comodule category back into a representation-theoretic setting.
• Comparing the resulting equivalence with the classical Tannaka-Kreín theorems in the context of K-linear tensor categories.
• Leveraging the formalism of monoidal functors and natural transformations to prove the equivalence in full generality.

Experimental results

Research questions

• RQ1 How does Joyal and Street's lifting-equivalence in the context of comodules over a Hopf algebra relate to classical Tannaka duality?
• RQ2 What is the precise categorical dual of Joyal and Street's comodule equivalence, and how does it recover the results of Saavedra Rivano and Deligne-Milne?
• RQ3 In what way does the generalization of Tannaka duality from vector spaces to abstract tensor categories preserve the essential structure of representation theory?
• RQ4 How can the lifting-equivalence in the comodule setting be systematically dualized to recover the classical Tannaka-Kreín reconstruction theorems?
• RQ5 What conditions ensure that a monoidal functor to finite-dimensional vector spaces induces a Hopf algebra whose comodules recover the original category?

Key findings

• The paper establishes a lifting-equivalence between a rigid tensor category and the category of finite-dimensional comodules over a Hopf algebra, generalizing Joyal and Street's result.
• Dualizing this equivalence yields the classical Tannaka-Kreín reconstruction theorems of Saavedra Rivano and Deligne-Milne in the context of K-linear tensor categories.
• The construction demonstrates that the category of comodules over the reconstructed Hopf algebra is equivalent to the original category, under the given fiber functor.
• The proof of the lifting-equivalence is developed in detail, providing a categorical foundation for Tannaka duality in abstract tensor categories.
• The framework shows that the duality between algebra and geometry, as formalized by Joyal and Street, generalizes naturally to the setting of K-linear tensor categories.
• The results confirm that the classical Tannaka duality for vector spaces is a special case of a broader categorical duality involving Hopf algebras and comodules.