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https://hdl.handle.net/2440/54076
Type: | Journal article |
Title: | The inner automorphism 3-group of a strict 2-group |
Author: | Roberts, D. Schreiber, U. |
Citation: | The Journal of Homotopy and Related Structures, 2008; 3(1):193-245 |
Publisher: | Tbilisi Centre for Mathematical Sciences |
Issue Date: | 2008 |
ISSN: | 1512-2891 |
Statement of Responsibility: | David Michael Roberts and Urs Schreiber |
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. |
Published version: | http://jhrs.rmi.acnet.ge/volumes/2008/volume3-1.htm |
Appears in Collections: | Aurora harvest 5 Mathematical Sciences publications |
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