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

Abstract

The regular representation of an essentially finite 2-group G in the 2-category 2Vect k of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it are computed. It is next shown that all hom-categories in Rep 2Vect k ( 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 ω : Rep 2Vect k ( G ) → 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 V ect k G of functors from the underlying groupoid of G to V ect k , on the one hand, and the k-linear category E nd ( ω ) of pseudonatural endomorphisms of ω , on the other hand. We conclude that E nd ( ω ) is a 2-vector space, and we (partially) describe a basis of it.