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 $ℋ$ an (∞,1)-topos and $G,A,\stackrel{^}{G}\in \mathrm{Grp}\left(H\right)$ be ∞-groups with deloopings $BG$, $BA$ and $\stackrel{^}{G}$, respectively.

###### Definition

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

$BA\stackrel{i}{\to }B\stackrel{^}{G}\stackrel{p}{\to }BG\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{B}A \stackrel{i}{\to} \mathbf{B}\hat G \stackrel{p}{\to} \mathbf{B}G \,.
###### Remark

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

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

###### Definition

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

$\begin{array}{ccc}BA& \stackrel{\Omega c}{\to }& B\stackrel{^}{G}\\ & & ↓\\ & & BG& \stackrel{c}{\to }& {B}^{2}A\phantom{\rule{thinmathspace}{0ex}}.\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

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

• the extension is classified by the group cohomology class

$\left[c\right]\in {\pi }_{0}H\left(BG,{B}^{2}A\right)={H}_{\mathrm{Grp}}^{2}\left(G,A\right)\phantom{\rule{thinmathspace}{0ex}}.$[\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 $A={B}^{n}𝒜$, then the above says that extension of $G$ by the $n$-fold delooping/suspension ${B}^{n}𝒜$ is classified by degree-$n$ group cohomology

${H}_{\mathrm{Grp}}\left(G,{B}^{n}𝒜\right)={\pi }_{0}H\left(BG,{B}^{n+1}𝒜\right)={H}_{\mathrm{Grp}}^{n+1}\left(G,𝒜\right)\phantom{\rule{thinmathspace}{0ex}}.$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

${\mathrm{Ext}}^{n}\left(G,A\right)≔H\left(BG,{B}^{n+1}A\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $BG$.

## References

The general concept is discussed in section 4.3 of

Extensions by braided 2-groups are discussed in

Revised on October 10, 2013 03:29:51 by Urs Schreiber (89.204.135.176)