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  1. I added a reference to a paper of mine

    Amnon Yekutieli

    diff, v48, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2020

    added pointer to

    • Jinpeng An. Zhengdong Wang, Nonabelian cohomology with coefficients in Lie groups,Trans. Amer. Math. Soc. 360 (2008), 3019-3040 (doi:10.1090/S0002-9947-08-04278-5)

    diff, v51, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2020

    added these pointers:

    diff, v51, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2020

    added pointer to

    diff, v51, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2020
    • (edited Aug 15th 2020)

    For citation purposes, what’s a good (textbook) account that makes explicit both the concept of non-abelian cohomology and its representation by classifying spaces (under suitable conditions)

    I am asking, because the literature on “non-abelian cohomology” tends to jump to arcane properties before ever saying clearly what the (simple) main structure of the general theory actually is. As a result, it is hard to tell a reader “see reference XYZ”, because if they don’t already know about the topic, they might not even recognize that XYZ is about this topic.

    The article Roberts-Stevenson 12 above stands out in this respect, as it does state these general principle on the first three pages, even if not quite in an expository way.

    But ignoring higher groups and stuff, just focusing on the ancient theory. What’s your preferred textbook(-style) reference that makes clear that non-abelian cohomology is a thing and that its ultimately about maps into classifying spaces?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2020

    I have tried to give some more logical order to the list of references.

    Now I have introduced 3 subsections under “References”, correspondoing to discussion in homotopical dimensioon 1, 2, \infty, respectively.

    diff, v52, current

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    • CommentAuthorUrs
    • CommentTimeAug 18th 2020

    added pointer to

    diff, v54, current

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    • CommentAuthorUrs
    • CommentTimeSep 2nd 2020
    • (edited Sep 2nd 2020)

    added pointer to

    diff, v57, current

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    • CommentAuthorUrs
    • CommentTimeSep 2nd 2020

    added pointer to

    diff, v57, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2020

    added publication data to:

    diff, v58, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2020

    added publication details to:

    • Carlos Simpson, Algebraic aspects of higher nonabelian Hodge theory, in: Fedor Bogomolov, Ludmil Katzarkov (eds.), Motives, polylogarithms and Hodge theory, Part II (Irvine, CA, 1998), Int. Press Lect. Ser., 3, II, Int. Press, 2002, 2016, 417-604. (arXiv:math/9902067, ISBN:9781571462909)

    diff, v59, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2021
    • (edited Jan 19th 2021)

    added pointer to

    (since unstable operations on stable cohomology is really operations on their image in non-abelian cohomology)

    diff, v61, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2021

    added pointer to:

    diff, v64, current

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    • CommentAuthorUrs
    • CommentTimeSep 2nd 2021

    added pointer to:

    diff, v65, current

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    • CommentAuthorUrs
    • CommentTimeJun 21st 2023

    added pointer to:

    diff, v68, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2023

    finally an original reference:

    • Alexander Grothendieck, Chapter V of: A General Theory of Fibre Spaces With Structure Sheaf, University of Kansas, Report No. 4 (1955, 1958) [pdf]

    diff, v68, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2023

    and:

    diff, v68, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2023

    I can’t find this item online:

    • Paul Dedecker, Cohomologie de dimension 2 à coefficients non abéliens, C. R. Acad. Sci. Paris 247 (1958) 1160-1163

    but searching for it brings up the following references, which had been missing here and which I have added to the entry now:

    have also re-organized the list of references around this (here) to be more systematic

    diff, v69, current

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 22nd 2023

    Added missing link to Dedecker’s 1958 article

    diff, v70, current

    • CommentRowNumber20.
    • CommentAuthorzskoda
    • CommentTimeJun 22nd 2023
    • Samuel Eilenberg, Saunders MacLane, Cohomology theory in abstract groups. II. Group extensions with a non-Abelian kernel, Ann. of Math. (2) 48, (1947). 326–341 jstor:1969174

    • Saunders MacLane, Cohomology theory in abstract groups. III. Operator homomorphisms of kernels. Ann. of Math. (2) 50, (1949). 736–761.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2023

    Thanks for highlighting. The actual definition of non-abelian group cohomology seem to be in part I (here). Incidentally, this seems to be the kind of original reference that, a while ago, I had been looking for to add to crossed homomorphism.

    We should add this also to nonabelian group cohomology. Will do so now…

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2023

    I just wrote:

    The actual definition of non-abelian group cohomology seem to be in part I

    Hm, or maybe not. I’ll need to have a closer look. Where do they actually define non-abelian group cohomology?

    • CommentRowNumber23.
    • CommentAuthorzskoda
    • CommentTimeJun 23rd 2023
    • (edited Jun 23rd 2023)

    No, part I is abelian.

    Part II goes nonabelian.

    (7.2) for f 3=1f_3=1 is a nonabelian 2-cocycle condition, while the equation before Lemma 7.2 expresses what is a nonabelian 3-cocycle and (9.3) explains when a 3-cocycle is a coboundary. Compare the section on “traditional approach” at group extension where this is extracted (using later expositions like Kurosh which are based on Schreier + Eilenberg-MacLane II).

    Nonabelian 1- 2- cocycle and coboundary conditions were stated beforehand By Schreier in 1928 or so, 3-cocycle in Teichmueller in 1940 (cited in Eilenberg MacLane II) and so on. A standard reference for “Galois” nonabelian cohomology (1-, 2-, 3-cocycles) is also Serre.

    • CommentRowNumber24.
    • CommentAuthorzskoda
    • CommentTimeJun 23rd 2023
    • (edited Jun 23rd 2023)

    This is Schreier:

    • Otto Schreier, Über die Erweiterung von Gruppen I., Monatsh. f. Mathematik und Physik 34, 165–180 (1926) doi
    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeAug 17th 2023
    • (edited Aug 17th 2023)