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

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.