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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.
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.
An extension $\hat G$ of $G$ by $A$ is a fiber sequence of the form
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 $\mathbf{B}^2 A \in \mathbf{H}$.
A higher central extension $\hat G$ of $G$ by $A$ is a fiber sequence in $\mathbf{H}$ of the form
$\mathbf{B}\hat G \to \mathbf{B}G$ is an $\mathbf{B} A$-principal ∞-bundle over $\mathbf{B}G$;
the extension is classified by the group cohomology class
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
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.
Notably for abelian $A$, by the main classification result at principal ∞-bundles, the ∞-groupoid of $\infty$-group extensions is equivalent to
In particular they are classified by the intrinsic $n+1$st $A$-cohomology of $\mathbf{B}G$.
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.
The general concept is discussed in section 4.3 of
Extensions by braided 2-groups are discussed in
Last revised on September 14, 2020 at 10:43:20. See the history of this page for a list of all contributions to it.