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<p>created <a href="https://ncatlab.org/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a></p>
<p>the secret title of this entry is "Schreier theory done right". (where "right" is right from the <a href="https://ncatlab.org/nlab/show/nPOV">nPOV</a>)</p>
<p>this is the first part of the answer to</p>
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What is going on at <a href="https://ncatlab.org/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>?
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<p>The second part of the answer is the statement:</p>
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The same.
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<p>;-)</p>
<p>I'll expand on that eventually.</p>
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Accompanying article group extension focusing on Schreier's theory for nonabelian group extensions traditional way. Added also an entry within my personal lab about Paul Dedecker (zoranskoda), one of the pioneers.
went again through nonabelian group cohmology and tried to carefully match all formulas (concerning precise order of the terms and points where the automorphism actions apply) to the diagrams, following the "convention L B" as detailed at strict 2-group - in terms of crossed modules.
Should be right now. Mistakes here are like sign errors. Not terribly important in general, but sometimes very important, and usually annoying to track down.
added at nonabelian group cohomology a section Homotopies between 2-cocycles.
To be compared with Zoran's Comparing different extensions: 2-coboundaries at group extension
I created a bit by an error connection on a comodule thinking on comodules over additive comonads. But for now the material still fits into the connection for a coring so I left only a redirect FOR THE MOMENT at connection on a comodule and expanded entry connection for a coring, having also few words on comonad case. Today I largely expanded group extension with a 2-coboundary section and more, on the basis of my LaTeX notex from 1997 and being stimulated by discussions with Urs. I now also created monadic descent and symmetric comonad.
You plan to write something else at connection on a comodule, which is why the redirect is only temporary?
I hope so, but I am yet not clear about how the grand scheme will work. Once the material is built up, this will be easier to do then at this point.
I have added pointer to
which – weirdly – remains the only reference I have found so far that makes explicit non-abelian group cohomology in degree 1, as the set of crossed-conjugation equivalence classes of crossed homomorphisms.
Other references I have seen (for instance the otherwise nice account by Milne 2017) which do highlight the full non-abelian generality of crossed homomorphisms all switch back to abelian coefficients when it comes to defining group cohomology.
Coming up with the non-abelian generalization is of course immediate, but it would still nice to see a citable reference that admits this. (Groupprops is a great resource, but it feels less citable than a textbook.)
Gille and Szamuely’s Central Simple Algebras and Galois Cohomology, p.40 mentions noncommutative H1, but not in terms of crossed homomorphisms, I think.
How about Milne’s Algebraic Groups, p. 76.
Thanks! The definition is on p. 25. Crossed homomorphisms is equation (1). Great, am adding this item now.
Regarding Milne, I don’t see it yet, could you point me to the intended page number in the pdf here? Thanks!
Now that we are back online, I see that Milne’s pdf has the definition in section 27.a (which didn’t make it into the published version, apparently). Am adding these pointers now to the entry.
added pointer to:
My Internet’s down too.
It’s in section 3k of the published book,p. 76.
Ah, thanks! Okay, will adjust section pointers yet once more.
(My internet connection was fine, but the nLab server just took its daily downtime.)
Wow, so I see that they broke his manuscript a fair bit in the publication process, making him use the definition of crossed homomorphisms 12 chapters before they are being introduced.
The following is stated:
“Notice that when K has nontrivial automorphisms, this differs in general from the ordinary degree 2 abelian group cohomology even if K is abelian.”
Is there a concrete example where an extension has a nontrivial class in H2ab(G,K) but is trivial in H2nonab(G,K)?
and I guess as a more basic question, what is the action of K on Aut(K) to construct Aut(K)//K? Is it just left-multiplication by the corresponding adjoint automorphism? i.e. k:ϕ↦Ad(k)⋅ϕ?
Yes.
Given any crossed module Ht→Gα→Aut(H), the corresponding group internal to Cat has
group of objects G
group of morphisms H⋊G
where a morphism (h,g)∈H⋊G has
domain g
codomain t(h)g.
(I remember these relations by thinking of the graphics on pp. 37 here or p. 10 here. This kind of explanation is also at strict 2-group – In terms of crossed modules, but this was written in the early days of the nLab when we didn’t have good graphics support yet, so the rendering is quite clunky.)
In other words, the underlying groupoid is the homotopy quotient G⫽H where H acts on G by left multiplication under t.
Now for the case at hand of the automorphism 2-group, the crossed module is
Ht≔Ad→G≔Aut(H)α≔id→Aut(H)and the corresponding homotopy quotient is Aut(H)⫽H with H acting on Aut(H) by left multiplication under Ad.
Got it, thanks. Then Aut(K)⫽ˆG is similarly defined by now using the crossed module (K↪ˆG), correct? And also the ξ in Degree 2 cocycles is a typo?
The “ξ” was indeed a typo for “χ”, have fixed it now, thanks for catching this. Have also re-rendered with TikZ the first displayed diagram in the proof of Prop. 2.1.
The group ˆG and its action on elements α∈Aut(K) appears in this diagram. The corresponding crossed module is of the form ˆG→Aut(K), I’d say.
If I have a central extension 1→K→ˆG→G→1 for K abelian, the 2-cocycle is a map f:BG→B2K, and the homotopy fiber F can be confirmed to be homotopy equivalent to BˆG by using the long exact sequence of homotopy groups.
If, on the other hand, I allow G to act nontrivially on K, then the extension information is captured more precisely by f:BG→BAUT(K) for AUT(K) the automorphism 2-group of K. Even though the article states that Aut(K)//ˆG is the homotopy fiber F of this map, I thought the long exact sequence would give that π1(F)=ˆG, but it instead gives 1→K→π1(F)→ker(G→Aut(K))→1, so in particular π1(F)=K if G acts faithfully. Am I missing something?
[I am technically on a short Easter vacation with family. But now I am sitting on a park bench in the sun while family is checking out some boutiques. So briefly:]
The long exact sequence of homotopy groups induced by a homotopy fiber sequence of the form F⟶BG⟶BAUT(K) starts out as
⋯⟶1⟶Z(K)⟶π1(F)⟶G⟶Out(K)⟶π0(F)→*Is that what you are looking at, though?
Yes, that’s what I am looking at (I should have made clear that I kept assuming that K is abelian, so Z(K)=K and Out(K)=Aut(K)). So if at least some element of G acts nontrivially on K then π1(F), since it fits in the sequence 1→K→π1(F)→ker(G→Aut(G))→1, “shrinks”, so to speak, instead of being isomorphic to the group extension the extension data determines, which is confusing me.
And happy Easter !
Okay, I see.
The resolution should be that in computing π1(F) you need to beware that F is not a delooping and has Aut(K) worth of objects.
Fixing the basepoint at which to compute π1 to be any α∈Aut(K), then (under your assumption that G→Out(K) is injective) its automorphisms in F are, according to the diagrams shown in the entry, 2-morphisms in BAUT(K) from α to itself, and these form the group Z(K) — which is in accord with the long exact sequence that we are looking at.
I see, that makes sense though it becomes rather inconvenient for what I had in mind. I was thinking of working with ˆG bundles over a space M starting from lifting a G bundle, and that the G bundles that could be lifted to a ˆG bundle are those for which the composition M→BG→BAUT(K) is homotopically trivial. But it seems you actually don’t get a lift to a ˆG bundle unless G acts trivially on K… which morphism, then, out of BG, has BˆG as the homotopy fiber so that I can talk about ˆG bundles as lifts of (certain) G bundles?
This will still work, just for a group different from the ˆG in the entry. Namely F is equivalent to a delooping B˜G and that ˜G takes the place of ˆG in your construction.
So I’m getting from what you are saying that, generally, the fibration BK→BˆG→BG really does not have an additional map from BG that extends the fiber sequence, it usually just ends there with BG, right?
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