On the regular representation of an (essentially) finite 2-group

The regular representation of an essentially finite 2-group $\mathbb{G}$ in the 2-category ${\mathbf{2}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_k$ of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in ${\mathbf{R}\mathbf{e}\mathbf{p}}_{{\mathbf{2}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_k}(\mathbb{G})$ are 2-vector spaces under quite standard assumptions on the field $k$, and a formula giving the corresponding “intertwining numbers” is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor {\boldmath\omega}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k is representable with the regular representation as representing object. As a consequence we obtain a $k$-linear equivalence between the 2-vector space ${\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_k^{\mathcal{G}}$ of functors from the underlying groupoid of $\mathbb{G}$ to ${\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_k$, on the one hand, and the $k$-linear category \mathcal{E} nd({\boldmath\omega}) of pseudonatural endomorphisms of {\boldmath\omega}, on the other hand. We conclude that $$\mathcal{E}$ nd({\boldmath$\omega$})$ is a 2-vector space, and we (partially) describe a basis of it.