nLab
center of an infinity-group

Idea

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

Definition

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

$$\mathrm{AUT}(G)\to \mathrm{Out}(G)$$AUT(G) \to Out(G)

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

The homotopy fiber of this morphism

$$BZ(G)\to \mathrm{Aut}(G)\to \mathrm{Out}(G)$$\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 $𝒳=$ ∞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 $[G\stackrel{\mathrm{Ad}}{\to }\mathrm{Aut}(G)]$. The morphism $\mathrm{AUT}(G)\to \mathrm{Out}(G)$ is a fibration hence its homotopy fiber is, up to equivalence, the ordinary fiber, which is the crossed module $(G\stackrel{\mathrm{Ad}}{\to }\mathrm{Inn}(G))$, where $\mathrm{Inn}(G)\subset \mathrm{Aut}(G)$ 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 $BZ(G)$.

Related concepts

• group, 2-group, ∞-group,

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

• center, center of an $\mathrm{\infty }$-group,

• inner automorphism group

• outer automorphism group, outer automorphism ∞-group