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

Abstract

We construct a theory of 2-vector bundles over a Lie groupoid, with fibers modeled by the bicategory of finite-dimensional super algebras, bimodules, and intertwiners. These 2-vector bundles form a symmetric monoidal 2-stack. We define the 2K-theory as the Grothendieck completion of the homotopy category of 2-vector bundles; yielding a category that naturally contains ordinary K-theory as the endomorphism ring of the trivial 2-vector bundle and contains twisted K-theories as morphisms between different twistings. We prove a classification theorem establishing an equivalence between the homotopy category of 2-vector bundles over a Lie groupoid and the homotopy category of simplicial maps from the nerve of the Lie groupoid to a certain classifying space. Moreover, we extend the framework to the 2-equivariant setting. For a Lie groupoid acted by a coherent Lie 2-group, we introduce the bicategory of 2-equivariant 2-vector bundles and define the 2-equivariant 2K-theory. An equivariant classification theorem is established. Explicit computations of 2-equivariant 2K-theories are carried out for the 2-groups BA, with A an abelian group, and Lie groups G. For BA, the classification recovers the representation rings $Z[t,t^{-1}]$ for $A = U(1)$ and $Z[t]/(t^n-1)$ for $A = Z/n$, providing a direct verification of Lurie's prediction about 2-equivariant elliptic cohomology. For a Lie group G, we show that the 1-morphisms in $2K_G(pt)$ correspond to (projective) super representations of G, with ordinary equivariant K-theory $K_G(pt)$ appearing as the endomorphism ring of the trivial object. Finally, we introduce weak groupoid objects internal to a bicategory, and define 2-orbifold 2-vector bundles and their 2K-theory.