# nLab automorphism 2-group

### Context

#### Higher category theory

higher category theory

# Contents

## Automorphism 2-groups

For $C$ any 2-category and $c \in C$ any object of it, the category $Aut_C(c) \subset Hom_C(c,c)$ of auto-equivalences of $c$ and invertible 2-morphisms between these is naturally a 2-group, whose group product comes from the horizontal composition in $C$.

If $C$ is a strict 2-category there is the notion of strict automorphism 2-group. See there for more details on that case.

For instance if $C = Grp_2 \subset Grpd$ is the 2-category of group obtained by regarding groups as one-object groupoids, then for $H \in Grp$ a group, its automorphism 2-group obtained this way is the strict 2-group

$AUT(H) := Aut_{Grp_2}(H)$

corresponding to the crossed module $(H \stackrel{Ad}{\to} Aut(H))$, where $Aut(H)$ is the ordinary automorphism group of $H$.

## Inner automorphism 2-groups

Revised on September 7, 2011 21:03:52 by Urs Schreiber (82.93.78.115)