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quick entry for infinity-group extension, just so that I can complete links at related entries.
I don’t have a good internet access in the near future but I wonder about the connection between the definition in the stub above and the following:
An extension of $\infty$-group is a sequence $N\to G\to Q$ of $\infty$ groups and $\infty$ group maps such that $N\to G$ is $\infty$ normal and Q is the $\infty$ qoutient
Sorry for the slow reply. For some reason I see this only now.
The answer is: it’s equivalent.
As indicated in the entry, this is discussed in Principal ∞-bundles – theory, presentations and applications (schreiber).
I see there’s a talk soon that will fit here.
David Jaz Myers, Higher Schreier Theory
In a 1926 article, Otto Schreier gave a classification of all extensions of a group G by a (non-abelian) group K. This classification of extensions has come to be known as Schreier theory, and has been reformulated many times by many authors since. Just as central extensions by an abelian group are classified by group cohomology in degree 2, Schreier theory can be seen as an example of a classification by non-abelian group cohomology.
Higher Schreier theory concerns the classification of extensions of higher groups. Breen has generalized Schreier theory to sheaves of 2-groups. In this talk, we will give a proof of Schreier theory for oo-groups in homotopy type theory - and therefore for sheaves of ∞-groups by interpreting in various oo-toposes. Our main theorem is: Let G and K be ∞-groups. Then the type of extensions of G by K is equivalent to the type of actions of G on the delooping BK.
One can immediately see the resemblence of this formulation of higher Schreier theory to the classification of split extensions of G by K by the homomorphic actions of G on K. We can derive this classification, and some others, as an immediate corollary.
We will also discuss the notion of central extensions, and navigate some subtleties concerning the notion of centrality for higher groups.
But we should link also to nonabelian group cohomology.
So this talk by Myers is available now. Time to compare with infinity-group extension, higher central extension and center of an infinity group.
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