Representation theory of 2-groups on Kapranov and Voevodsky’s 2-vector spaces

In this paper the 2-category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2-group G on (some version of) Kapranov and Voevodsky’s 2-category of (complex) 2-vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invariants π0(G), π1(G) and [α] ∈ H 3(π0(G), π1(G)) classifying G. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain that the monoidal category of linear representations (more generally, the category of [z]-projective representations, for any given cohomology class [z] ∈ H2(π0(G),C ∗)) of the first homotopy group π0(G) as well as its category of representations on finite sets both live in Rep2MatC (G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of morphisms between suitable 1-dimensional representations) and the second as a subcategory of the homotopy category of Rep 2MatC (G).