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John Baez, Danny Stevenson, The Classifying Space of a Topological 2-Group, In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) )Algebraic Topology_. Abel Symposia, vol 4. Springer, Berlin, Heidelberg. (arXiv:0801.3843, doi:10.1007/978-3-642-01200-6_1)
Danny Stevenson, Classifying theory for simplicial parametrized groups (arXiv:1203.2461)
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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?
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(since unstable operations on stable cohomology is really operations on their image in non-abelian cohomology)
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finally an original reference:
and:
I can’t find this item online:
but searching for it brings up the following references, which had been missing here and which I have added to the entry now:
Paul Dedecker, Cohomologie à coefficients non abéliens et espaces fibrés, Bulletins de l’Académie Royale de Belgique 41 (1955) 1132-1146 [persee:barb_0001-4141_1955_num_41_1_69497]
Paul Dedecker, Sur La Cohomologie Non Abelienne I (Dimension Deux), Canadian Journal of Mathematics 12 (1960) 231-251 [doi:10.4153/CJM-1960-019-7]
have also re-organized the list of references around this (here) to be more systematic
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.
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…
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?
No, part I is abelian.
Part II goes nonabelian.
(7.2) for 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.
This is Schreier:
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added pointer to Lurie 2014 for the understanding of non-abelian cohomology as homotopy classes of maps into any (pointed) space
took the liberty of also pointing to my 2009 Oberwolfach talk (here) where I had started using this perspective
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The description of a sequential Omega-spectrum looks like it is missing an Omega!
Have also !include-ed (here) the table “cohomlogy and gauge fieds” (see there).
(This and the surrounding paragraphs are following the new introduction to The Character Map in Equivariant Twistorial Cohomotopy.)
This is looking very accessible.
Typo: you have ’realtive homotopy’ towards the end of the ’Twisted non-abelian cohomology’ subsection in an image.
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