center of an infinity-group
The generalization of the notion of center from groups to ∞-groups.
For an ∞-group, there is a canonical morphism
from its automorphism ∞-group to its outer automorphism ∞-group.
The homotopy fiber of this morphism
is the delooping of an ∞-group . This is the center of .
Centers of ordinary groups
For ∞Grpd and 0-truncated, it is an ordinary discrete group. Its automorphism 2-group is the strict 2-group coming from the crossed module . The morphism is a fibration hence its homotopy fiber is, up to equivalence, the ordinary fiber, which is the crossed module , where is the group of inner automorphisms. This is equivalent to , where is the ordinary center of , and this is the crossed module corresponding to .
Created on September 7, 2011 15:12:57
by Urs Schreiber