2K-Theory and 2-Vector Bundles

• 2K-theory is a categorified extension of classical K-theory that classifies 2-vector bundles using Grothendieck group constructions in symmetric monoidal bicategories.
• It deploys a 2-stack framework to organize coherent structures on super-algebra bundles over Lie groupoids with equivariant and orbifold adaptations.
• Its spectrum-level realization and link to higher algebraic K-theory offer new insights into twisted cohomological invariants in geometry and mathematical physics.

2K-theory generalizes algebraic topological K-theory into the framework of 2-categories, organizing the classification of “2-vector bundles” and their higher-categorical symmetries. At its core, 2K-theory captures the Grothendieck group of equivalence classes of 2-vector bundles, which are categorified vector bundles whose fibers are not vector spaces but objects in a symmetric monoidal bicategory (specifically, finite-dimensional ℤ/2-graded super-algebras, invertible bimodules, and intertwiners). The theory admits further generalization to equivariant, orbifold, and higher groupoid contexts, providing a foundation for “twisted” and “higher” K-theoretic invariants in geometry and mathematical physics (Huan, 22 Jan 2026).

1. 2-Vector Bundles over Lie Groupoids

A 2-vector bundle over a Lie groupoid $X_1 \rightrightarrows X_0$ is a categorified bundle whose fibers are objects of a symmetric monoidal bicategory $s2$:

• Objects: finite-dimensional $\mathbb{Z}/2$-graded super-algebras $A = A_0 \oplus A_1$.
• 1-Morphisms: finite-dimensional graded $(B, A)$-bimodules $M$.
• 2-Morphisms: even $B$-$A$-intertwiners between bimodules.
• Composition: via relative tensor product $N \otimes_B M$ with canonical associators and unitors.

Given a hypercover $$f_\bullet: \Gamma_\bullet \twoheadrightarrow X_\bullet$$, a 2-vector bundle $V$ consists of:

• Super-algebra bundle $A \to \Gamma_0$.
• Invertible even bimodule bundle $M \to \Gamma_1$, with $M_\gamma$ an $A_{t(\gamma)}$-$A_{s(\gamma)}$-bimodule for $\gamma \in \Gamma_1$.
• Coherent associator invertible intertwiner (the “multiplication” $\mu$) over $\Gamma_2$:

$$\mu: M_{\gamma_2} \otimes_{A_{s(\gamma_2)}} M_{\gamma_1} \xRightarrow{\cong} M_{\gamma_2 \gamma_1}$$

satisfying the pentagon identity on $\Gamma_3$.

• Unitor intertwiner $u_x: A_x \xRightarrow{\cong} M_{\mathrm{id}_x}$, satisfying triangle identities.

For the trivial groupoid, this data reduces to a classical super-algebra bundle with compatible bimodule isomorphisms (Huan, 22 Jan 2026).

2. The Symmetric Monoidal 2-Stack Structure

The assignment $X_\bullet \mapsto 2\mathrm{Vect}_k(X_\bullet)$ defines a 2-prestack over Lie groupoids, upgraded via the plus-construction to a symmetric monoidal 2-stack $2\mathrm{Vect}_k$. The monoidal structure is induced by direct sum “$\oplus$” both for super-algebras and their bimodules:

• $\oplus$ is performed fiberwise.
• All coherence (associativity, commutativity) is strict in this context.
• Internal equivalence classes are stable under $\oplus$.

This categorical sum enables the formation of a commutative monoid of internal equivalence classes of 2-vector bundles, as required for Grothendieck completion.

3. Definition and Formulation of 2K-Theory

2K-theory for a Lie groupoid $X$ is defined as the Grothendieck group of internal-equivalence classes of invertible 2-vector bundles:

$$K^{(2)}(X) = \mathrm{Groth} \left( \pi_0(2\mathrm{Vect}_k(X)) \right)$$

Here, $\pi_0$ is the set of internal-equivalence classes with respect to equivalences in the bicategory of 2-vector bundles. The group operation is induced from $\oplus$:

$[V \oplus W] = [V] + [W]$

For example, for $X = pt/G$ (the “orbifold point”), $2K(X)$ is the Grothendieck group of super-algebra bundles on $pt$ modulo the action of $G$, providing a height-2 analogue of twisted equivariant K-theory (Huan, 22 Jan 2026).

4. The 2K-Theory Spectrum and Homotopical Realization

The bicategory $M(s2)$ comprises super-algebras as objects, invertible bimodules as 1-morphisms, and invertible intertwiners as 2-morphisms. $M(s2)$ admits a strict symmetric monoidal structure under $\oplus$:

• Its 2-nerve $2\mathrm{Nerve}(M(s2))$ is a permutative 2-category.
• The group completion $K(s2) = \Omega B|2\mathrm{Nerve}(M(s2))|$ is a connective infinite loop space.
• There is a canonical connective spectrum $\mathbb{K}(s2) = \mathbb{A}(M(s2))$, whose zeroth space is $K(s2)$.

The homotopy groups satisfy:

$$\pi_0 K(s2) \cong \mathrm{Groth}(\pi_0 M(s2)), \quad \pi_1 K(s2) \cong \pi_0(\mathrm{Aut}(\mathbf{1})), \quad \pi_n K(s2) = \pi_{n-1}(B(M(s2)))\, (n>1)$$

A classification theorem asserts that, for any $X$, internal equivalence classes of 2-vector bundles correspond bijectively to based homotopy classes of maps $|\mathrm{Nerve}\,X| \to |2\mathrm{Nerve}(M(s2))|$.

5. Equivariant, 2-Equivariant, and Orbifold 2K-Theory

The 2K-theory formalism extends to equivariant and higher groupoid contexts:

• Coherent Lie 2-Groups: A coherent Lie 2-group is a group object in the bicategory Bibun of Lie groupoids, bibundles, and bibundle-maps, with multiplication $m$, identity $e$, and higher invertible coherence 2-cells.
• $G_\bullet$-Equivariant 2-Vector Bundles: For a $G_\bullet$-action on $X_\bullet$, a $G_\bullet$-equivariant 2-vector bundle is a pseudofunctor $BG_\bullet \to 2\mathrm{Vect}_k(X_\bullet)$ with additional compatible data (e.g., bibundle maps and modifications $\phi_T$, $l_T$ satisfying pentagon and triangle identities).
• The bicategory $2\mathrm{Vect}_k^G(X)$ collects $G_\bullet$-equivariant 2-vector bundles and their morphisms.

2-Equivariant 2K-theory is defined by passing to invertible objects and forming the Grothendieck group:

$$2K_{G_\bullet}(X_\bullet) = \mathrm{Groth} \left( \pi_0\, 2\mathrm{Vect}_k^G(X_\bullet) \right)$$

This specializes to the classification of $G_\bullet$-equivariant 2-vector bundles modulo equivariant internal equivalence.

For more general groupoid objects (“2-orbifolds”), the same construction applies, yielding a 2K-theory of orbifolds, which refines ordinary orbifold K-theory by allowing super-algebra twists in the fibers.

6. Relationship with Higher Algebraic K-Theory and Applications

2K-theory encodes higher-categorical twists and symmetries in geometric, representation-theoretic, and topological settings:

• It elevates the classical notion of vector bundle K-theory to a level in which bundles themselves are categorified (with fibers modeled as 2-vector spaces).
• It provides a setting for height-2 analogues of twisted equivariant K-theories, relevant for generalized cohomology, representation theory of 2-groups, and approaches to quantum symmetries.
• The spectrum-level presentation realizes 2K-theory as a generalized cohomology theory with a specific universal property with respect to symmetric monoidal bicategories.

A plausible implication is that further exploration of higher $n$K-theories, where fibers are $n$-categories of modules over superalgebras, may parallel the extension from vector bundles to 2-vector bundles, and potentially lead to a full hierarchy of “higher twisted” cohomology theories.

7. Outlook and Open Problems

• The current construction ensures a well-behaved theory for 2-vector bundles over Lie groupoids with possible higher group actions, and yields a robust spectrum representing the theory.
• An open problem is the explicit characterization of 2K-theory for more general classes of higher stacks and the direct computation for specific geometric examples, such as compact orbifolds or moduli spaces arising in mathematical physics.
• The extension of the improved convergent-Gaussian constructions used in related fields (e.g., Borel-Leroy summability in $\phi^{2k}$ theories) to the context of 2K-theory remains unexplored.
• The relation of 2K-theory to categorified versions of KK-theory or to other bivariant K-theoretic frameworks is open for investigation.

References: (Huan, 22 Jan 2026)