2-Vector Bundles

• 2-vector bundles are a categorified generalization of vector bundles where fibers are objects in a 2-category (typically finite semisimple linear categories), enabling higher-categorical transition data.
• They employ bicategorical structures and higher cocycle conditions to achieve local-to-global gluing via functors and natural isomorphisms that satisfy coherence relations.
• Applications span extended topological field theory, twisted K-theory, and higher representation theory, bridging classical geometry with quantum phenomena.

A 2-vector bundle is a categorified generalization of the notion of a vector bundle, in which the fibers are not mere vector spaces but objects in a 2-category, typically finite semisimple linear categories (as in 2Vect), and the transition data involves not only linear maps but also functors and natural isomorphisms subject to higher coherence conditions. The theory of 2-vector bundles provides a bridge between classical bundle theory, higher representation theory, bundle gerbes, and extended topological field theory, and plays a foundational role in higher-categorical geometry, particularly via connections to K-theory, gerbes, and equivariant structures. Below is a detailed account of the key structures, classification, and applications of 2-vector bundles.

1. Foundational Structures: Bicategories, Fibers, and Morphisms

At the core of 2-vector bundle theory is the bicategory structure used to model the fibers and transition data. The principal model is the bicategory 2Vect, whose objects are finite-dimensional semisimple (super-)algebras, 1-morphisms are bimodules, and 2-morphisms are even bimodule intertwiners. Composition of 1-morphisms is given by the relative tensor product of bimodules, and all standard coherence isomorphisms, such as associators and unitors, are encoded as 2-morphisms. The symmetric monoidal structure is given by the (graded) tensor product of algebras and external tensor product of bimodules, with trivial grading as the unit object. Dualizability is also central: every finite-dimensional algebra is dualizable, with fully dualizability corresponding to semisimplicity and invertibility to central simplicity (the Brauer–Wall group providing the isomorphism classes) (Kristel et al., 2021).

A 2-vector bundle over a base (usually a manifold or groupoid) is described by specifying fibers as objects in 2Vect and transition data as functors and natural isomorphisms satisfying pentagon and triangle axioms on double and triple overlaps, respectively. This can be formulated as a 2-functor from a base groupoid (or manifold's open cover hypergroupoid) into a symmetric monoidal bicategory such as 2Vect or its super analog (Huan, 22 Jan 2026, Schweigert et al., 2017).

2. Cocycle Data, Gluing, and 2-Stack Structures

The local-to-global construction of 2-vector bundles requires a higher cocycle formalism. Over an open cover $\{U_i\}$ of a manifold, local triviality is encoded by providing equivalences to a fixed finite semisimple category (e.g., $\text{Vect}^n$), and transition data on $U_{ij}$ as 1-functors (auto-equivalences of the standard fiber), together with 2-isomorphisms on $U_{ijk}$ that satisfy higher cocycle conditions (pentagon identity on $U_{ijkl}$), reflecting the associativity of composition in the bicategory (Huan, 2022, Schweigert et al., 2017).

The "plus construction" of Nikolaus–Schweigert, a stackification process, upgrades such prestacks into 2-stacks, yielding descent bicategories and enabling global 2-vector bundles as objects in the 2-stack $2VectBd^+(M)$ on a manifold $M$. For equivariant or orbifold settings, the 2-stack can be defined on the site of Lie groupoids, banded by 2-groups, and organized via spans and homotopy pullbacks for morphisms (Huan, 22 Jan 2026, Schweigert et al., 2017).

3. Classification, Cohomology, and K-Theory

The classification of 2-vector bundles is governed by nonabelian (crossed module) cohomology and higher K-theory. For a fixed superalgebra $A$, the stack of $A$-Morita 2-vector bundles on $M$ is classified by \v{C}ech cohomology $\check H^1(M; (A_0^\times \to \mathrm{Aut}(A)))$, where $A_0^\times$ acts by conjugation and $\mathrm{Aut}(A)$ is the automorphism group (Kristel et al., 2021). In terms of descent data, this corresponds to 1-cocycles $g_{ij}$ in $\mathrm{Aut}(A)$ together with 2-cocycles $a_{ijk}$ in $A_0^\times$ satisfying compatibility (pentagon and triangle) relations up to coboundaries.

Over a Lie groupoid, the set of equivalence classes of 2-vector bundles is identified with homotopy classes $[\mathrm{Nerve}(G), \mathrm{Nerve}(M(s2))]$, where $M(s2)$ denotes the maximal subgroupoid of the bicategory of superalgebras with Morita equivalences. Taking the group completion of the classifying space gives a 2K-theory spectrum, and the Grothendieck group of the equivalence classes of 2-vector bundles is identified with $\pi_0$ of the mapping space into this spectrum (Huan, 22 Jan 2026).

4. Exemplary Models, Reductions, and Connections to Gerbes

The framework of 2-vector bundles encompasses and unifies various geometric and categorical objects:

• Line bundle and gerbe reduction: Taking the fiber category as $\mathbf{Vect}_k$ (ordinary vector spaces) yields classical vector bundles; choosing $BU(1)$ (the groupoid with one object and automorphism group $U(1)$) recovers bundle gerbes. The descent formalism for 2-vector bundles with $BU(1)$-fibers precisely matches the gerbe cocycle and 3-cocycle data (Huan, 2022, Kristel et al., 2021).
• Bundle gerbes and algebra bundles as sub-2-vector bundles: The 2-stack of $U(1)$-gerbes and the stack of bundles of central simple algebras embed as full sub-2-stacks of 2Vect, corresponding respectively to bundles classified by $H^2(M, U(1))$ and $H^1(M, \mathrm{PGL}_n)$ (Kristel et al., 2021).
• Examples: The Baas–Dundas–Rognes model for $\mathbb{C}$-linear 2-vector bundles on semisimple Frobenius manifolds exemplifies the local trivialization (in terms of module categories over direct sums of idempotents) and the gluing (permutation of idempotents and line bundle twists) as categorified transition functions (Amoreo et al., 2015).

5. Equivariant 2-Vector Bundles and Higher Structures

2-vector bundles can be refined to equivariant settings, where a Lie groupoid or a coherent Lie 2-group acts on the total 2-category. In the model for the string 2-group via free loop groups, a $G$-equivariant 2-vector bundle is equipped with an action functor and natural 2-isomorphisms reflecting the associator and unit of the 2-group action. The pentagon identity and higher coherence relations are enforced as required by higher group symmetry (Huan, 2022, Huan, 22 Jan 2026).

The theory further generalizes to 2-orbifolds (stacks presented by Lie groupoids with 2-group actions), with the symmetric monoidal 2-stack of 2-orbifold 2-vector bundles, and leads to invariants such as 2-orbifold 2K-theory, represented by associated spectra.

6. Parallel Section Functors, Homotopy Invariants, and Extended TQFT

A 2-vector bundle over a groupoid $I$ (i.e., a 2-functor $I\to 2Vect$) admits a "category of parallel sections," defined as the hom-category of 2-functors from the trivial bundle to the given bundle. This is concretely the category of sections together with coherent isomorphisms over groupoid morphisms, and abstractly is the homotopy limit or the space of homotopy fixed points of the 2-functor (Schweigert et al., 2017). The assignment extends to a symmetric monoidal 2-functor from the bicategory of 2-vector bundles on groupoids to 2Vect, and forms the basis for "orbifoldization" procedures in extended equivariant TQFT. When restricted, these functors recover classical parallel section functors or module categories for twisted group algebras.

7. Key Examples, Applications, and Future Directions

2-vector bundles appear in diverse contexts:

• Topological field theory: The canonical Baas–Dundas–Rognes 2-vector bundle over a semisimple Frobenius manifold encodes the maximal Calabi–Yau (Cardy) category of D-branes, resolving a conjecture of Segal regarding "elliptic objects" that should arise in 2D open/closed field theory (Amoreo et al., 2015).
• Twisted K-theory: The theory of 2-vector bundles is central to the construction of twisted K-theory, especially in relation to bundle gerbes and Morita bundle gerbes (Kristel et al., 2021).
• Representation theory and String geometry: Equivariant 2-vector bundles encode categories of representations for groupoids or 2-groups, as in the string 2-group and positive-energy loop group representations (Huan, 2022).
• Higher geometric quantization: The parallel section 2-functor approach provides a systematic framework for higher geometric quantization, particularly for groupoids and orbifolds with nontrivial higher symmetry (Schweigert et al., 2017).
• Double vector bundles and super-geometry: Double vector bundles (2-fold vector bundles) can be interpreted equivalently as $\mathbb{Z}^2$-graded manifolds of suitable type, showing the deep connections between higher vector bundle theory and graded supergeometry (Vishnyakova, 2016).

Ongoing research explores extensions to n-vector bundles, generalized cohomology classifications, applications in condensed matter (higher gauge theories), and new invariants in both pure mathematics and mathematical physics.