nLab
inner automorphism 2-group

Idea

The inner automorphism 2-group of a group is essentially the sub-2-group of the automorphism 2-group $\mathrm{AUT}(G)$ of $G$ 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 $G$ be a group. Write $BG$ for its delooping.

The inner automorphism 2-group $\mathrm{INN}(G)$ of $G$ is the strict 2-group

• whose objects are diagrams

  $$\begin{array}{cc}& \nearrow {\searrow }^{\mathrm{Id}}\\ BG& {\Downarrow }^{\eta }& BG\\ & \searrow {\nearrow }_{\alpha }\end{array}$$\array{ & \nearrow \searrow^{\mathrlap{Id}} \\ \mathbf{B}G &\Downarrow^{\eta}& \mathbf{B}G \\ & \searrow \nearrow_{\mathrlap{\alpha}} }

  in Grpd.

• morphisms $\kappa :({\eta }_1,{\alpha }_1)\to ({\eta }_2,{\alpha }_2)$ are commuting triangles of transformations

  $$\begin{array}{ccc}& & \mathrm{Id}\\ & {}^{{\eta }_1}⇙& & {⇘}^{{\eta }_2}\\ {\alpha }_1& & \stackrel{\kappa }{\Rightarrow }& & {\alpha }_2\end{array}$$\array{ && Id \\ & {}^{\mathllap{\eta_1}}\swArrow && \seArrow^{\mathrlap{\eta_2}} \\ \alpha_1 &&\stackrel{\kappa}{\Rightarrow}&& \alpha_2 }

Equivalently, this is the action groupoid

$$\mathrm{INN}(G)=G//G≕EG$$INN(G) = G//G \eqqcolon \mathbf{E}G

of $G$ acting on itself.

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

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

with action $G\to \mathrm{Aut}(G)$ given by right multiplication in $G$.

This makes it evident that $\mathrm{INN}(G)$ is contractible

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

In fact, we may think of $\mathrm{INN}(G)$ as the universal $G$-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

$$EG≔\mathrm{INN}(G).$$\mathbf{E}G \coloneqq INN(G) \,.

We have a natural sequence of groupoids

$$G\to EG\to BG.$$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

$$G\to ℰG\to ℬG$$G \to \mathcal{E} G \to \mathcal{B}G

that is the universal $G$-bundle over the classifying space $ℬG$ in its incarnation in Top.

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

$$BG\to BEG.$$\mathbf{B}G \to \mathbf{B} \mathbf{E}G \,.

This fact is useful in various applications in nonabelian cohomology.