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

TL;DR

This paper extends the concept of inner automorphism groups from ordinary groups to strict 2-groups, defining associated 3-groups and exploring their structure and relation to universal 2-bundles.

Contribution

It introduces the 3-group of inner automorphisms for strict 2-groups and relates it to mapping cones and exact sequences, advancing the understanding of higher automorphism structures.

Findings

Defined the 3-group of inner automorphisms for strict 2-groups

Connected the structure to the mapping cone of the identity on G_{(2)}

Showed the exact sequence involving G_{(2)}, INN_0(G_{(2)}), and Σ G_{(2)}

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.