Cohomology and Extensions
Special and general types
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
Let moreover be a braided ∞-group, with second delooping .
A central extension of by is a fiber sequence in of the form
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 .
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