Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The notion of -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, -group extensions are the special case of principal ∞-bundles whose base space is the moduli ∞-stack of the group being extended.
Let an (∞,1)-topos and be ∞-groups with deloopings , and , respectively.
An extension of by is a fiber sequence of the form
Equivalently this says that is a normal morphism of ∞-groups and that is its quotient.
Let moreover be a braided ∞-group, with second delooping .
A higher central extension of by is a fiber sequence in of the form
is an -principal ∞-bundle over ;
the extension is classified by the group cohomology class
If here is an Eilenberg-MacLane object , then the above says that extension of by the -fold delooping/suspension is classified by degree- group cohomology
In particular if 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 , by the main classification result at principal ∞-bundles, the ∞-groupoid of -group extensions is equivalent to
In particular they are classified by the intrinsic st -cohomology of .
For the (∞,1)-category of (∞,1)-sheaves on some site the Dold-Kan correspondence embeds chain complexes of abelian sheaves over into . Under this embedding ordinary Ext-groups and the shifted extensions that they classify (see here) identify with -group extensions in the above sense.
The string 2-group is an extension in Smooth∞Grpd of the spin group by the circle 2-group.
The fivebrane 6-group is an extension in 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.