[Paper Review] Representative functions on discrete groupoids and duality with Hopf algebroids

Laiachi El Kaoutit|arXiv (Cornell University)|Nov 12, 2013

Homotopy and Cohomology in Algebraic Topology16 references3 citations

TL;DR

This paper establishes a duality between the category of discrete groupoids and geometrically transitive commutative Hopf algebroids via two contravariant functors: one assigning to each Hopf algebroid its characters groupoid, and the other assigning to each discrete groupoid its Hopf algebroid of representative functions. The key contribution is a generalization of the classical duality between discrete groups and commutative Hopf algebras to the groupoid setting, extending Tannaka-Krein duality principles.

ABSTRACT

The aim of this paper is to establish a duality between the category of discrete groupoids and the category of geometrically transitive commutative Hopf algebroids in the sense of P. Deligne and A. Bruguières. In one direction we have the usual contravariant functor which assigns to each Hopf algebroid its characters groupoid (the fiber groupoid at the ground field). In the other direction we construct the contravariant functor which associated to each discrete groupoid its Hopf algebroid of representative functions. This duality extends the well known duality between discrete groups and commutative Hopf algebras, and also sheds light on a new approach to Tannaka-Krein duality for compact topological groupoids. Our results are supported by several illustrative examples including topological ones.

Motivation & Objective

• To extend the classical duality between discrete groups and commutative Hopf algebras to the setting of groupoids.
• To define a contravariant functor from discrete groupoids to commutative Hopf algebroids using representative functions.
• To establish a duality between the category of discrete groupoids and the category of geometrically transitive commutative Hopf algebroids.
• To provide a new framework for understanding Tannaka-Krein duality in the context of topological groupoids.
• To generalize the role of representative functions from Lie and compact groups to discrete groupoids.

Proposed method

• Constructs a contravariant functor that assigns to each discrete groupoid its Hopf algebroid of representative functions over a field $\Bbbk$.
• Uses the fiber groupoid at the ground field to define the character groupoid functor from Hopf algebroids to discrete groupoids.
• Applies the theory of Tannakian categories and monoidal categories to define the universal object in the category of geometrically transitive commutative Hopf algebroids.
• Employs the construction $\mathscr{L}_{\Bbbk}(\omega) \cong \bigoplus_X \omega(X)^* \otimes_{\mathrm{T}_X} \omega(X)$ as the universal Hopf algebroid associated to a fiber functor.
• Relies on results from Deligne and Bruguiffres on Tannakian categories and the structure of Hopf algebroids with enough idempotents.
• Utilizes the category $\mathsf{Tanna}_{\Bbbk}$ of Tannakian categories with fiber functors to define the functor $\mathscr{L}_{\Bbb{k}}$ to $\mathsf{GTCHAlgd}_{\Bbb{k}}$.

Experimental results

Research questions

• RQ1 Can the duality between discrete groups and commutative Hopf algebras be extended to discrete groupoids and commutative Hopf algebroids?
• RQ2 How can representative functions on discrete groupoids be used to construct a Hopf algebroid structure?
• RQ3 What is the role of the character groupoid in the duality between Hopf algebroids and discrete groupoids?
• RQ4 How does this duality relate to Tannaka-Krein duality for compact topological groupoids?
• RQ5 What algebraic and categorical structures underlie the construction of representative functions on groupoids?

Key findings

• The paper establishes a contravariant duality between the category of discrete groupoids and the category of geometrically transitive commutative Hopf algebroids via the functors $\mathscr{R}_{\Bbbk}$ and $\chi_{\Bbb{k}}$.
• The Hopf algebroid of representative functions on a discrete groupoid is constructed as $\mathscr{L}_{\Bbb{k}}(\omega) \cong \bigoplus_X \omega(X)^* \otimes_{\mathrm{T}_X} \omega(X)$, forming a well-defined covariant functor to the category of geometrically transitive commutative Hopf algebroids.
• The duality generalizes the classical duality between discrete groups and commutative Hopf algebras, extending it to the groupoid setting.
• The construction of $\mathscr{L}_{\Bbb{k}}$ is shown to be a strict homomorphism of bicategories, enriching the categorical framework.
• The algebra of continuous representative functions on a topological groupoid with compact Hausdorff base space arises as the image of a Tannakian object under the generalized functor $\overline{\mathscr{L}}_{\Bbb{k}}$.
• The Hopf algebroid $\overline{\mathscr{L}}_{\Bbb{k}}(\omega)$ is projective as an $(A \otimes_{\Bbb{k}} A)$-module if the associated bimodule $\boldsymbol{\Sigma}$ is projective.