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PB-groupoids vs VB-groupoids

Alfonso Garmendia, Francesco Cattafi

TL;DR

The paper extends the classic VB–principal bundle correspondence to a 2-categorical setting by introducing PB-groupoids: diagrams of Lie groupoids and principal 2-groupoid actions that generalise principal bundles over groupoids. It builds a bridge via two main constructions: (i) PB-groupoids from VB-groupoids using 2-representations to form associated VB-groupoids, and (ii) adapted frame PB-groupoids $Fr^{sbis}(E_G)$ that recover PB-groupoids from VB-groupoids, establishing a canonical 1–1 VB⇄PB correspondence. The development hinges on the general linear 2-groupoid $GL(l,k)$ and two notions of 2-representation, tying together frames, 2-actions, and associated bundles. The framework is illustrated with examples including tangent and dual VB-groupoids, and it suggests further directions toward multiplicative structures on groupoids, PB-algebroids, and differentiable stacks. Overall, the work provides a robust higher-categorical generalisation of the VB–PB correspondence with concrete geometric constructions and duality compatibilities.

Abstract

In this paper we establish the principal bundle counterpart of the well-known and widely applied notion of vector bundle groupoid (VB-groupoid). In particular, we provide a general notion of principal bundle groupoid (PB-groupoid) as a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid. This recovers as particular cases various constructions involving structural Lie 2-groups, Lie groupoids and Lie groups. Moreover, given any VB-groupoid, we detect which frames are compatible with the underlying structure and show that the set of such frames has a natural PB-groupoid structure, with the action of the general linear 2-groupoid of the appropriate ranks. We use this construction, as well as that of the associated bundle induced by a 2-representation, to extend the classical correspondence between vector bundles and principal bundles to a new correspondence between VB-groupoids and PB-groupoids. We conclude with several examples and applications.

PB-groupoids vs VB-groupoids

TL;DR

The paper extends the classic VB–principal bundle correspondence to a 2-categorical setting by introducing PB-groupoids: diagrams of Lie groupoids and principal 2-groupoid actions that generalise principal bundles over groupoids. It builds a bridge via two main constructions: (i) PB-groupoids from VB-groupoids using 2-representations to form associated VB-groupoids, and (ii) adapted frame PB-groupoids that recover PB-groupoids from VB-groupoids, establishing a canonical 1–1 VB⇄PB correspondence. The development hinges on the general linear 2-groupoid and two notions of 2-representation, tying together frames, 2-actions, and associated bundles. The framework is illustrated with examples including tangent and dual VB-groupoids, and it suggests further directions toward multiplicative structures on groupoids, PB-algebroids, and differentiable stacks. Overall, the work provides a robust higher-categorical generalisation of the VB–PB correspondence with concrete geometric constructions and duality compatibilities.

Abstract

In this paper we establish the principal bundle counterpart of the well-known and widely applied notion of vector bundle groupoid (VB-groupoid). In particular, we provide a general notion of principal bundle groupoid (PB-groupoid) as a diagram of Lie groupoids and principal bundles, together with the action of a (strict) Lie 2-groupoid. This recovers as particular cases various constructions involving structural Lie 2-groups, Lie groupoids and Lie groups. Moreover, given any VB-groupoid, we detect which frames are compatible with the underlying structure and show that the set of such frames has a natural PB-groupoid structure, with the action of the general linear 2-groupoid of the appropriate ranks. We use this construction, as well as that of the associated bundle induced by a 2-representation, to extend the classical correspondence between vector bundles and principal bundles to a new correspondence between VB-groupoids and PB-groupoids. We conclude with several examples and applications.
Paper Structure (25 sections, 22 theorems, 167 equations)

This paper contains 25 sections, 22 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Main results and structure of the paper
  3. Further motivations and future applications
  4. Notations and conventions
  5. Acknowledgements
  6. Background on 2-objects
  7. VB-groupoids
  8. 2-vector bundles
  9. Lie 2-groupoids
  10. From PB-groupoids to VB-groupoids
  11. 2-Actions
  12. 2-Representations
  13. Principal 2-actions
  14. PB-groupoids
  15. First half of the correspondence
  16. ...and 10 more sections

Key Result

Lemma 2.1

Any anchored 2-vector bundle $E_1 \xrightarrow{\delta} E_0 \to M$ induces a VB-groupoid over the unit groupoid $M \rightrightarrows M$: \begin{tikzcd} {E_1 \times_M E_0} & M \\ {E_0} & M \arrow[from=1-1, to=2-1, shift left=.5ex, "\tis"] \arrow[from=1-1, to=2-1, shift right=.5ex, "\tit"'] \arro Conversely, given a VB-groupoid $E_\mathcal{G}\rightrightarrows E_M$ over the unit groupoid $\mathc

Theorems & Definitions (42)

  • Lemma 2.1: anchored 2-vector bundles and VB-groupoids over the unit groupoid
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • ...and 32 more