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Automorphism 2-groups

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

If CC 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 2GrpdC = Grp_2 \subset Grpd is the 2-category of group obtained by regarding groups as one-object groupoids, then for HGrpH \in Grp a group, its automorphism 2-group obtained this way is the strict 2-group

AUT(H):=Aut Grp 2(H) AUT(H) := Aut_{Grp_2}(H)

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

Inner automorphism 2-groups

See inner automorphism 2-group.

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