nLab
outer automorphism infinity-group

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Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

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Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Definition

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

Write

AUT(G):=Aut̲(BG)[BG,BG]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

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

is the outer automorphism \infty-group of GG.

Examples

Applications

Revised on September 7, 2011 21:04:30 by Urs Schreiber (82.93.78.115)