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The inner automorphism 3-group of a strict 2-group

The inner automorphism 3-group of a strict 2-group

David Michael Roberts and Urs Schreiber

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 $$ \xymatrix{ G_{(2)} \ar[r] & \mathrm{INN}_0(G_{(2)}) \ar[r] & \mathbf{B} G_{(2)} } \,. $$ As a consequence, $\mathrm{INN}_0(G_{(2)})$ encodes the properties of the universal $G_{(2)}$ 2-bundle.

Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 193-244