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2-Equivariant 2-Vector bundles and 2K-theories

Zhen Huan

TL;DR

The paper develops a higher-categorical analogue of vector bundles by constructing $2$-vector bundles over Lie groupoids using the bicategory of finite-dimensional super algebras, bimodules, and intertwiners, and defines $2K$-theory via a classifying-space construction. It then extends this framework to the $2$-equivariant setting under coherent $2$-group actions and to orbifold contexts through weak groupoid objects, yielding $2$-equivariant and $2$-orbifold $2K$-theories and corresponding spectra. Central techniques include descent via hypercovers, the plus construction to obtain $2$-stacks, and Osorno’s spectrum construction adapted to the super-algebra/bimodule model. The results provide a robust higher-categorical generalization of (equivariant) K-theory, with potential connections to higher elliptic cohomology and twisted theories in geometry and physics.

Abstract

We construct a theory of 2-vector bundles over a Lie groupoid, with fibers modeled by the bicategory of super algebras, bimodules and intertwiners. We demonstrate that these 2-vector bundles form a symmetric monoidal 2-stack. From this structure, we define the 2K-theory as the Grothendieck group of the internal equivalence classes of the 2-vector bundle over the given Lie groupoid, and we construct the spectra representing this theory. We then extend this framework to the equivariant setting. For any Lie groupoid equipped with an action by a coherent 2-group, we introduce the bicategory of 2-equivariant 2-vector bundles over it. This leads to the definition of 2-equivariant 2K-theory as the Grothendieck group of the internal equivalence classes in the bicategory. Furthermore, we define a higher analogue of orbifold, which generalizes Lie groupoids with a 2-group action, and construct the bicategory of 2-orbifold 2-vector bundles. Finally, we can define the 2-orbifold 2K-theory.

2-Equivariant 2-Vector bundles and 2K-theories

TL;DR

The paper develops a higher-categorical analogue of vector bundles by constructing -vector bundles over Lie groupoids using the bicategory of finite-dimensional super algebras, bimodules, and intertwiners, and defines -theory via a classifying-space construction. It then extends this framework to the -equivariant setting under coherent -group actions and to orbifold contexts through weak groupoid objects, yielding -equivariant and -orbifold -theories and corresponding spectra. Central techniques include descent via hypercovers, the plus construction to obtain -stacks, and Osorno’s spectrum construction adapted to the super-algebra/bimodule model. The results provide a robust higher-categorical generalization of (equivariant) K-theory, with potential connections to higher elliptic cohomology and twisted theories in geometry and physics.

Abstract

We construct a theory of 2-vector bundles over a Lie groupoid, with fibers modeled by the bicategory of super algebras, bimodules and intertwiners. We demonstrate that these 2-vector bundles form a symmetric monoidal 2-stack. From this structure, we define the 2K-theory as the Grothendieck group of the internal equivalence classes of the 2-vector bundle over the given Lie groupoid, and we construct the spectra representing this theory. We then extend this framework to the equivariant setting. For any Lie groupoid equipped with an action by a coherent 2-group, we introduce the bicategory of 2-equivariant 2-vector bundles over it. This leads to the definition of 2-equivariant 2K-theory as the Grothendieck group of the internal equivalence classes in the bicategory. Furthermore, we define a higher analogue of orbifold, which generalizes Lie groupoids with a 2-group action, and construct the bicategory of 2-orbifold 2-vector bundles. Finally, we can define the 2-orbifold 2K-theory.
Paper Structure (21 sections, 6 theorems, 127 equations)

This paper contains 21 sections, 6 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. 2-vector space
  4. Bilimit and bi-pullback
  5. Bilimits and adjunction
  6. Bi-pullbacks and some symbols
  7. Some symbols used in the paper
  8. 2-Vector bundles and 2K-theory
  9. 2-Vector Bundles over a Lie groupoid
  10. 2-prestack of 2-vector bundles
  11. 2-stack of 2-vector bundles
  12. 2K-theory and its spectra
  13. 2-equivariant 2-Vector bundles and 2-equivariant $2K$-theory
  14. 2-equivariant 2-Vector bundles over a Lie groupoid
  15. 2-equivariant $2K$-theory and its spectra
  16. ...and 6 more sections

Key Result

Lemma 3.9

Let $\mathcal{A}$ and $\mathcal{B}$ be super algebra bundles over $\mathcal{X}_0$. Let $\mathcal{M}$ and $\mathcal{M}'$ be two $\mathcal{A}-\mathcal{B}$-bimodule bundle over $\mathcal{X}_1$.

Theorems & Definitions (81)

  • Definition 2.1
  • Example 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Remark 3.7
  • Remark 3.8
  • ...and 71 more