nLab outer automorphism infinity-group

Contents

Context

Higher category theory

higher category theory

Contents

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.

Last revised on September 7, 2011 at 21:04:30. See the history of this page for a list of all contributions to it.