nLab infinity-group extension

Idea

The notion of $\infty$-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, $\infty$-group extensions are the special case of principal ∞-bundles whose base space is the moduli ∞-stack of the group being extended.

Definition

Let $\mathcal{H}$ an (∞,1)-topos and $G, A, \hat G \in Grp(\mathbf{H})$ be ∞-groups with deloopings $\mathbf{B}G$, $\mathbf{B}A$ and $\mathbf{B}\hat G$, respectively.

Definition

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

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

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 $\mathbf{B}^2 A \in \mathbf{H}$.

Definition

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

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

Remark

Def. equivalently says that

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

• the extension is classified by the group cohomology class

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

Remark

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

$$H_{Grp}(G, \mathbf{B}^n\mathcal{A}) = \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^{n+1}\mathcal{A}) = H_{Grp}^{n+1}(G, \mathcal{A}) \,.$$

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 $\infty$-group extensions is equivalent to

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

In particular they are classified by the intrinsic $n+1$st $A$-cohomology of $\mathbf{B}G$.

Examples

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

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

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

Related concepts

• ∞-group cohomology

• central extension, higher 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)