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

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

2007-08-13

arxiv.org/abs/0708.1741v2

Journal of Homotopy and Related Structures, vol. 3(1), 2008, pp.193--245

Mathematics

Category Theory

references added, relation to simplicial constructions expanded, version to appear in JHRS

Scientific paper

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.

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