nLab
outer automorphism infinity-group

Definition

Let $𝒳$ be an (∞,1)-topos and $G\in \mathrm{\infty }\mathrm{Grpd}(𝒳)$ an n-truncated ∞-group object, for some $n\in \mathbb{N}$ (an n-group in $𝒳$).

Write

$$\mathrm{AUT}(G):=\underline{\mathrm{Aut}}(BG)\hookrightarrow [BG,BG]\in 𝒳$$AUT(G) := \underline{Aut}(\mathbf{B}G) \hookrightarrow [\mathbf{B}G, \mathbf{B}G] \in \mathcal{X}

for the internal automorphism ∞-group.

Then the n-truncation

$$\mathrm{Out}(G):={\tau }_n\mathrm{AUT}(G)\in \mathrm{\infty }\mathrm{Grp}(𝒳)$$Out(G) := \tau_n AUT(G) \in \infty Grp(\mathcal{X})

is the outer automorphism $\mathrm{\infty }$-group of $G$.

Examples

• For $𝒳=$ ∞Grpd and $n=0$, $G$ is an ordinary discrete group, and $\mathrm{AUT}(G)$ is its automorphism 2-group. Then $\mathrm{Out}(G)$ is the ordinary group of ordinary outer automorphisms.

Applications

• Outer automorphism $\mathrm{\infty }$-groups control part of the nonabelian cohomology of ∞-gerbes. See there for more details.

Related concepts

• group, ∞-group,

• automorphism group, automorphism ∞-group,

• center, center of an ∞-group

• outer automorphism group, outer automorphism $\mathrm{\infty }$-group