nLab
center of an infinity-group

Contents

Idea

The generalization of the notion of center from groups to ∞-groups.

Definition

For GGrp(𝒳)G \in \infty Grp(\mathcal{X}) an ∞-group, there is a canonical morphism

AUT(G)Out(G) AUT(G) \to Out(G)

from its automorphism ∞-group AUT(G):=Aut̲ 𝒳(BG)AUT(G) := \underline{Aut}_{\mathcal{X}}(\mathbf{B}G) to its outer automorphism ∞-group.

The homotopy fiber of this morphism

BZ(G)Aut(G)Out(G) \mathbf{B} Z(G) \to Aut(G) \to Out(G)

is the delooping of an ∞-group Z(G)Z(G). This is the center of GG.

Examples

Centers of ordinary groups

For 𝒳=\mathcal{X} = ∞Grpd and GG 0-truncated, it is an ordinary discrete group. Its automorphism 2-group is the strict 2-group coming from the crossed module [GAdAut(G)][G \stackrel{Ad}{\to} Aut(G)]. The morphism AUT(G)Out(G)AUT(G) \to Out(G) is a fibration hence its homotopy fiber is, up to equivalence, the ordinary fiber, which is the crossed module (GAdInn(G))(G \stackrel{Ad}{\to} Inn(G)), where Inn(G)Aut(G)Inn(G) \subset Aut(G) is the group of inner automorphisms. This is equivalent to (Z(G)1)(Z(G) \to 1), where Z(G)Z(G) is the ordinary center of GG, and this is the crossed module corresponding to BZ(G)\mathbf{B}Z(G).

Created on September 7, 2011 15:12:57 by Urs Schreiber (82.113.99.42)