nLab center of an infinity-group

Idea

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

Definition

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

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

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

The homotopy fiber of this morphism

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

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

Examples

Centers of ordinary groups

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

Related concepts

• higher central extension

• group, 2-group, ∞-group,

• automorphism group, automorphism 2-group, automorphism ∞-group,

• center, center of an $\infty$-group,

• inner automorphism group

• outer automorphism group, outer automorphism ∞-group