On the Regular Representation of an (essentially) Finite 2-group

The regular representation of an essentially finite 2-group G in the 2-category 2Vectk of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in Rep 2Vectk pGq 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 ω : Rep k pGq Ñ 2Vectk is representable with the regular representation as representing object. As a consequence we obtain a k-linear equivalence between the 2-vector space VectGk of functors from the underlying groupoid of G to Vectk, on the one hand, and the k-linear category Endpωq of pseudonatural endomorphisms of ω, on the other hand. We conclude that Endpωq is a 2-vector space, and we (partially) describe a basis of it.