A Parallel Section Functor for 2-Vector Bundles

We associate to a 2-vector bundle over an essentially finite groupoid a 2-vector space of parallel sections, or, in representation theoretic terms, of higher invariants, which can be described as homotopy fixed points. Our main result is the extension of this assignment to a symmetric monoidal 2-functor $\mathrm{Par}:\mathbf{2}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{B}\mathbf{u}\mathbf{n}\mathbf{G}\mathbf{r}\mathbf{p}\mathbf{d}\to \mathbf{2}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}$. It is defined on the symmetric monoidal bicategory $\mathbf{2}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{B}\mathbf{u}\mathbf{n}\mathbf{G}\mathbf{r}\mathbf{p}\mathbf{d}$ whose morphisms arise from spans of groupoids in such a way that the functor $\mathrm{Par}$ provides pull-push maps between 2-vector spaces of parallel sections of 2-vector bundles. The direct motivation for our construction comes from the orbifoldization of extended equivariant topological field theories.