nLab
infinity-group extension

Idea

The notion of $\mathrm{\infty }$-group extension generalizes the notion of group extension to homotopy theory/(∞,1)-category theory and from groups to ∞-groups. It is also a generalization to nonabelian cohomology of the shifted group extensions that are classified by Ext-groups.

Under forming loop space objects, $\mathrm{\infty }$-group extensions are the special case of principal ∞-bundles whose base space is the moduli ∞-stack of the group being extended.

Definition

Let $\mathscr{H}$ an (∞,1)-topos and $G,A,\hat{G}\in \mathrm{Grp}(H)$ be ∞-groups with deloopings $BG$, $BA$ and $\hat{G}$, respectively.

Definition

An extension $\hat{G}$ of $G$ by $A$ is a fiber sequence of the form

$$BA\stackrel{i}{\to }B\hat{G}\stackrel{p}{\to }BG.$$\mathbf{B}A \stackrel{i}{\to} \mathbf{B}\hat G \stackrel{p}{\to} \mathbf{B}G \,.

Remark

Equivalently this says that $A\to \hat{G}$ is a normal morphism of ∞-groups and that $G\simeq \hat{G}\to G$ is its quotient.

Let moreover $A$ be a braided ∞-group, with second delooping $B^{2}A\in H$.

Definition

A central extension $\hat{G}$ of $G$ by $A$ is a fiber sequence in $H$ of the form

$$\begin{array}{ccc}BA& \stackrel{\Omega c}{\to }& B\hat{G}\\ & & \downarrow \\ & & BG& \stackrel{c}{\to }& B^{2}A.\end{array}.$$\array{ \mathbf{B} A & \stackrel{\Omega \mathbf{c}}{\to} & \mathbf{B}\hat G \\ && \downarrow \\ && \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^2 A \,. } \,.

Remark

Def. 2 equivalently says that

• $B\hat{G}\to BG$ is an $BA$-principal ∞-bundle over $BG$;

• the extension is classified by the group cohomology class

  $$[c]\in {\pi }_{0}H(BG,B^{2}A)=H_{\mathrm{Grp}}^2(G,A).$$[\mathbf{c}] \in \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^2 A) = H^2_{Grp}(G,A) \,.

Remark

If here $A$ is an Eilenberg-MacLane object $A=B^n𝒜$, then the above says that extension of $G$ by the $n$-fold delooping/suspension $B^n𝒜$ is classified by degree-$n$ group cohomology

$$H_{\mathrm{Grp}}(G,B^n𝒜)={\pi }_{0}H(BG,B^{n+1}𝒜)=H_{\mathrm{Grp}}^{n+1}(G,𝒜).$$H_{Grp}(G, \mathbf{B}^n\mathcal{A}) = \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^{n+1}\mathcal{A}) = H_{Grp}^{n+1}(G, \mathcal{A}) \,.

In particular if $G$ here is 0-truncated (hence a plain group object in the underlying 1-topos) then this reproduces the traditional theory of group extensions of 1-groups by 1-groups.

Properties

Notably for abelian $A$, by the main classification result at principal ∞-bundles, the ∞-groupoid of $\mathrm{\infty }$-group extensions is equivalent to

$${\mathrm{Ext}}^n(G,A)≔H(BG,B^{n+1}A).$$\mathbf{Ext}^n(G, A) \coloneqq \mathbf{H}(\mathbf{B}G, \mathbf{B}^{n+1}A) \,.

In particular they are classified by the intrinsic $n+1$st $A$-cohomology of $BG$.

Examples

• For $H={\mathrm{Sh}}_{\mathrm{\infty }}(C)$ the (∞,1)-category of (∞,1)-sheaves on some site $C$ the Dold-Kan correspondence embeds chain complexes of abelian sheaves over $C$ into $H$. Under this embedding ordinary Ext-groups and the shifted extensions that they classify (see here) identify with $\mathrm{\infty }$-group extensions in the above sense.

• The string 2-group is an extension in $H=$ Smooth∞Grpd of the spin group by the circle 2-group.

• The fivebrane 6-group is an extension in $H=$ Smooth∞Grpd of the string 2-group by the circle 6-group.

Related concepts

• central extension

• centrally extended groupoid

References

The general concept is discussed in section 4.3 of

• Thomas Nikolaus, Danny Stevenson, Urs Schreiber, Principal ∞-bundles -- theory, presentations and applications

Extensions by braided 2-groups are discussed in

• Evan Jenkins, Extensions of groups by braided 2-groups (arXiv:1106.0772)