Contents

group theory

cohomology

# Contents

## 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

The general concept is discussed in section 4.3 of

Extensions by braided 2-groups are discussed in