Any group G gives rise to a 2-group of inner automorphisms,
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)β
there is a 3-group INN(G(2)β) and a slightly smaller 3-group
INN0β(G(2)β) of inner automorphisms. We describe these for
G(2)β any strict 2-group, discuss how INN0β(G(2)β) can be
understood as arising from the mapping cone of the identity on G(2)β and
show that its underlying 2-groupoid structure fits into a short exact sequence
G(2)ββINN0β(G(2)β)βΞ£G(2)β.
As a consequence, INN0β(G(2)β) encodes the properties of the
universal G(2)β 2-bundle.Comment: references added, relation to simplicial constructions expanded,
version to appear in JHR