nLab
infinity-group extension

Context

Group Theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

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,G^Grp(H)G, A, \hat G \in Grp(\mathbf{H}) be ∞-groups with deloopings BG\mathbf{B}G, BA\mathbf{B}A and G^\hat \mathbf{G}, respectively.

Definition

An extension G^\hat G of GG by AA is a fiber sequence of the form

BAiBG^pBG. \mathbf{B}A \stackrel{i}{\to} \mathbf{B}\hat G \stackrel{p}{\to} \mathbf{B}G \,.
Remark

Equivalently this says that AG^A \to \hat G is a normal morphism of ∞-groups and that GG^GG \simeq \hat G \to G is its quotient.

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

Definition

A central extension G^\hat G of GG by AA is a fiber sequence in H\mathbf{H} of the form

BA Ωc BG^ BG c B 2A.. \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

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

  • the extension is classified by the group cohomology class

    [c]π 0H(BG,B 2A)=H 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 AA is an Eilenberg-MacLane object A=B n𝒜A = \mathbf{B}^{n}\mathcal{A}, then the above says that extension of GG by the nn-fold delooping/suspension B n𝒜\mathbf{B}^n\mathcal{A} is classified by degree-nn group cohomology

H Grp(G,B n𝒜)=π 0H(BG,B n+1𝒜)=H 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 GG 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 AA, by the main classification result at principal ∞-bundles, the ∞-groupoid of \infty-group extensions is equivalent to

Ext n(G,A)H(BG,B n+1A). \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+1n+1st AA-cohomology of BG\mathbf{B}G.

Examples

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)