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

Abstract

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.

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

References in corpus (3)

• Higher-Dimensional Algebra V: 2-Groups
• Parallel Transport and Functors
• Smooth Functors vs. Differential Forms

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• 3-form Yang-Mills based on 2-crossed modules
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