Representation theory of 2-groups on finite dimensional 2-vector spaces

In this paper we unfold the 2-category structure of the representations of a (strict) 2-group on (a suitable version of) Kapranov and Voevodsky’s 2-category of finite dimensional 2-vector spaces and we discuss the relationship with classical representation theory of groups on finite dimensional vector spaces. In particular, we prove that the monoidal category of representations of any group G appears as a full subcategory of the category of endomorphisms of a particular object in the 2-category of representations of G when G is thought of as a 2-group with only identity arrows. As an easy consequence of the unfolding process, we also see that every 2-group with a compact Lie group as base group has a rank one representation faithful with respect to the base group, contrary to a claim by Barrett and Mackaay (unpublished work).