nLab outer automorphism infinity-group

Definition

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

Write

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

for the internal automorphism ∞-group.

Then the n-truncation

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

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

Examples

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

Applications

• Outer automorphism $\infty$-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 $\infty$-group