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

In this paper, the 2-category $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$ of (weak) representations of an arbitrary (weak) 2-group $\mathbb{G}$ 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 $\pi_0(\mathbb{G})$, $\pi_1(\mathbb{G})$ and $[\alpha]\in H^3(\pi_0(\mathbb{G}),\pi_1(\mathbb{G}))$ classifying $\mathbb{G}$. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain the the {\it monoidal} category of linear representations (more generally, the category of $[z]$-projective representations, for any given cohomology class $[z]\in H^2(\pi_0(\mathbb{G}),\mathbb{C}^*)) of the first homotopy group$\pi_0(\mathbb{G})$as well as its category of representations on finite sets both live in$\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$, the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of morphisms between suitable 1-dimensional representations) and the second as a subcategory of the homotopy category of$\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$.