nLab
inner automorphism 2-group

Contents

Idea

The inner automorphism 2-group of a group is essentially the sub-2-group of the automorphism 2-group AUT(G)AUT(G) of GG of those automorphisms that are connected by a transformation to the identity: this makes the automorphism necessarily an inner automorphism.

In fact, more precisely the inner automorphism 2-group is the 2-group of these connecting transformations, i.e. it remembers the group element and the inner automorphism that it induces under conjugation.

Definition

Let GG be a group. Write BG\mathbf{B}G for its delooping.

The inner automorphism 2-group INN(G)INN(G) of GG is the strict 2-group

  • whose objects are diagrams

    Id BG η BG α \array{ & \nearrow \searrow^{\mathrlap{Id}} \\ \mathbf{B}G &\Downarrow^{\eta}& \mathbf{B}G \\ & \searrow \nearrow_{\mathrlap{\alpha}} }

    in Grpd.

  • morphisms κ:(η 1,α 1)(η 2,α 2)\kappa : (\eta_1, \alpha_1) \to (\eta_2, \alpha_2) are commuting triangles of transformations

    Id η 1 η 2 α 1 κ α 2 \array{ && Id \\ & {}^{\mathllap{\eta_1}}\swArrow && \seArrow^{\mathrlap{\eta_2}} \\ \alpha_1 &&\stackrel{\kappa}{\Rightarrow}&& \alpha_2 }

Equivalently, this is the action groupoid

INN(G)=G//GEG INN(G) = G//G \eqqcolon \mathbf{E}G

of GG acting on itself.

Equivalently , this is the strict 2-group corresponding to the crossed module

[INN(G)]=(GIdG) [INN(G)] = \left( G \stackrel{Id}{\to} G \right)

with action GAut(G)G \to Aut(G) given by right multiplication in GG.

This makes it evident that INN(G)INN(G) is contractible

BINN(G)*. \mathbf{B} INN(G) \stackrel{\simeq}{\to} {*} \,.

In fact, we may think of INN(G)INN(G) as the universal GG-principal bundle in its incarnation in Grpd (as opposed to the more tradition incarnation in Top, to which it is Quillen equivalent by the homotopy hypothesis theorem).

To emphasize this we also write

EGINN(G). \mathbf{E}G \coloneqq INN(G) \,.

We have a natural sequence of groupoids

GEGBG. G \to \mathbf{E}G \to \mathbf{B}G \,.

It is an old theorem by Graeme Segal that under nerve followed by geometric realization this maps to the sequence of topological spaces

GGG G \to \mathcal{E} G \to \mathcal{B}G

that is the universal GG-bundle over the classifying space G\mathcal{B}G in its incarnation in Top.

The 2-group structure on INN(G)INN(G) is evident, and hence makes the fact evident that the universal GG-bundle itself carries a group structure, which is compatibel with the group structure, in that the morphism GEG G \to \mathbf{E}G deloops to a morphism

BGBEG. \mathbf{B}G \to \mathbf{B} \mathbf{E}G \,.

This fact is useful in various applications in nonabelian cohomology.

Revised on December 19, 2009 17:41:06 by Toby Bartels (173.60.119.197)