nLab automorphism 2-group

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 (delooping groupoids), then for $H \in Grp$ a group, its automorphism 2-group obtained this way is the strict 2-group

$$AUT(H) \;\coloneqq\; Aut_{Grp_2}(H) \;\coloneq\; Aut(\mathbf{B}H)$$

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

See inner automorphism 2-group.

Related concepts

• holomorph

• group, ∞-group,

• automorphism group, automorphism ∞-group,

• center, center of an ∞-group

• outer automorphism group, outer automorphism ∞-group