Any group $G$ gives rise to a 2-group of inner automorphisms, $\mathrm{ INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{ (2)}$ there is a 3-group $\mathrm{ INN}(G_{ (2)})$ and a slightly smaller 3-group $\mathrm{ INN}_0(G_{ (2)})$ of inner automorphisms. We describe these for $G_{ (2)}$ any strict 2-group, discuss how $\mathrm{ INN}_0(G_{ (2)})$ can be understood as arising from the mapping cone of the identity on $G_{ (2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{ (2)} \to \mathrm{ INN}_0(G_{ (2)}) \to \Sigma G_{ (2)}$. As a consequence, $\mathrm{ INN}_0(G_{ (2)})$ encodes the properties of the universal $G_{ (2)}$ 2-bundle.
Categories: math.CT math.GR