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

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.