A Parallel Section Functor for 2-Vector Bundles

TL;DR

This paper develops a symmetric monoidal 2-functor that assigns 2-vector spaces of parallel sections to 2-vector bundles over groupoids, extending the concept to a bicategory with applications in topological field theories.

Contribution

It introduces a 2-functor from a bicategory of 2-vector bundles over groupoids to 2-vector spaces, generalizing the notion of parallel sections in a higher categorical setting.

Findings

Defines a 2-vector space of parallel sections as homotopy fixed points.

Constructs a symmetric monoidal 2-functor extending the assignment to a bicategory.

Provides pull-push maps between 2-vector spaces of sections in the bicategory.

Abstract

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 $\operatorname{Par} : \mathbf{2VecBunGrpd} \to \mathbf{2Vect}$. It is defined on the symmetric monoidal bicategory $\mathbf{2VecBunGrpd}$ whose morphisms arise from spans of groupoids in such a way that the functor $\operatorname{Par}$ 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.