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automorphism 2-group

Automorphism 2-groups

For $C$ any 2-category and $c\in C$ any object of it, the category ${\mathrm{Aut}}_C(c)\subset {\mathrm{Hom}}_C(c,c)$ 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 $C={\mathrm{Grp}}_2\subset \mathrm{Grpd}$ is the 2-category of group obtained by regarding groups as one-object groupoids, then for $H\in \mathrm{Grp}$ a group, its automorphism 2-group obtained this way is the strict 2-group

$$\mathrm{AUT}(H):={\mathrm{Aut}}_{{\mathrm{Grp}}_2}(H)$$AUT(H) := Aut_{Grp_2}(H)

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

Inner automorphism 2-groups

See inner automorphism 2-group.

Related concepts

• group, ∞-group,

• automorphism group, automorphism ∞-group,

• center, center of an ∞-group

• outer automorphism group, outer automorphism ∞-group