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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2009
    • (edited Dec 15th 2009)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <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> <blockquote> What is going on at <a href="https://ncatlab.org/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>? </blockquote> <p>The second part of the answer is the statement:</p> <blockquote> The same. </blockquote> <p>;-)</p> <p>I'll expand on that eventually.</p> </div>
    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeDec 15th 2009
    • (edited Dec 15th 2009)

    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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2009
    • (edited Dec 16th 2009)

    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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2009
    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeDec 17th 2009
    • (edited Dec 17th 2009)

    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.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeDec 18th 2009

    You plan to write something else at connection on a comodule, which is why the redirect is only temporary?

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeDec 18th 2009

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021

    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.)

    diff, v13, current

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 2nd 2021
    • (edited Sep 2nd 2021)

    Gille and Szamuely’s Central Simple Algebras and Galois Cohomology, p.40 mentions noncommutative H 1H^1, but not in terms of crossed homomorphisms, I think.

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 2nd 2021

    How about Milne’s Algebraic Groups, p. 76.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021

    Thanks! The definition is on p. 25. Crossed homomorphisms is equation (1). Great, am adding this item now.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021

    Regarding Milne, I don’t see it yet, could you point me to the intended page number in the pdf here? Thanks!

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021

    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.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021

    added pointer to:

    diff, v14, current

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 2nd 2021

    My Internet’s down too.

    It’s in section 3k of the published book,p. 76.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021

    Ah, thanks! Okay, will adjust section pointers yet once more.

    (My internet connection was fine, but the nLab server just took its daily downtime.)

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021
    • (edited Sep 2nd 2021)

    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.

    • CommentRowNumber18.
    • CommentAuthorjesuslop
    • CommentTimeAug 18th 2023

    In “idea”, changed abelian cohomology degree letter typo, it is of degree nn \in \mathbb{N} instead of kk \in \mathbb{N}.

    diff, v17, current