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added to group extension a section on how group extensions are torsors and on how they are deloopings of principal 2-bundles, see group extension – torsors
am working on group extension. Have
restructured the general section-outline a bit, separating the Properties- section into
General
Central extensions
Abelian extensions
Nonabelian extensions (Schreier theory)
added the basics to the Definition section
added statement and detailed proof of the equivalence $CentrExt(G,A) \simeq H^2_{Grp}(G,A)$ in Properties - Central extensions - Classification by group cohomology (only just finished typing a first round, deserves more polishing)
added at Properties - Abelian group extensions for the moment at least pointers to other entries which contain more details;
tried to add numbered Definition/Proposition-environments and Proof-environments to the existing section Properties - Nonabelian extensions (Schreier theory) but I ran out of steam here, this needs another go of polishing.
have expanded and polished the section
Properties – Central group extensions – Formulation in homotopy theory .
Started a section Split exensions and semidirect product groups.
Wanted to do much more. But the lab is mostly down, so it’s tedious. Have to quit now.
Did you mean ‘an fiber sequence’ in the ideas section? I tried a fix but could not be sure that I had captured what you might have intended so undid it. (Fixed a typo later on). I found the difficulty was to capture the use of the lax action of $G$ on $A$ in simple enough terms for an ideas section. The use of fiber sequence does sort of do that, but then it was not clear to me that the entry on fiber sequences really was specific enough on the case of groups…. so I left it alone! In any case it is very nicely explained further down the entry.
There seemed to be some hats missing. I think I fixed them all but may have missed some.
Ah, I didn’t look at that first sentence in the Idea-section for a while…
The statement of fiber sequence there is okay, but it should in addition say that the following equivalent statements hold (which in this form also work for $\infty$-groups)
$G \simeq \hat G \sslash K$;
the fiber sequence exhibits a normal inclusion.
I’ll edit this as soon as the you have unlocked the entry.
I have added a paragraph pointing out the link between the splittings of a split extension and non-abelian 1-cocycles / derivations
at group extension in the section Central extensions classified by group cohomology there was a two places silently assumed that the given 2-cocycle is normalized. I have now made this explicit.
I have been trying to find universal central extension in the Lab (as there was a non-active link on another page). Has anyone seen an entry that handles this and its link with other areas?
There is this notion here: universal central extension
But this needs disambiguation, one also speaks of “universal central extensions” for loop groups etc.
Thanks. When I have time I will add in something on Milnor’s K-theory.
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