On the representations of 2-groups in {Baez-Crans} 2-vector spaces

TL;DR

This paper shows that the representation theory of finite 2-groups in Baez-Crans 2-vector spaces simplifies to classical group representations, with higher homotopy invariants not affecting the core theory.

Contribution

It demonstrates that the representation theory of finite 2-groups in Baez-Crans 2-vector spaces reduces to linear representations of their isomorphism class groups, simplifying the understanding of such representations.

Findings

Representation theory reduces to classical group representations.

Higher homotopy invariants do not influence the representation theory.

Expected similar results for topological 2-groups.

Abstract

We prove that the theory of representations of a finite 2-group $\mathbb{G}$ in Baez-Crans 2-vector spaces over a field $k$ of characteristic zero essentially reduces to the theory of $k$-linear representations of the group of isomorphism classes of objects of $\mathbb{G}$, the remaining homotopy invariants of $\mathbb{G}$ playing no role. It is also argued that a similar result is expected to hold for topological representations of compact topological 2-groups in suitable topological Baez-Crans 2-vector spaces.