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At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group $\pi_2(X,A)$ together with the bundary map $\delta:\pi_2(X,A)\to \pi_1(A)$ and the $\pi_1(A)$-action on $\pi_2(X,A)$. As someone confirms this example is correct I’ll add it to crossed module.
I have added something about this as it is really a special case of the previous last example.
At crossed module – References I moved the pointer to Henry Whitehead’s book from bottom to top and made explicit that this is the origin of the concept. Also added a pointer to “crossed module” from the entry J. H. C. Whitehead.
The point of these changes is to give due credit to Whitehead for the originality of his work both prewar and post war, and to clarify the relations between the papers. In particular, my 1980 paper was not claimed to be anything but derivative: every step was in one or other of the papers of W I referred to. However, the fact that W’s papers gave a universal property in 2-dimensional relative homotopy theory was a key clu8e to consider relative situations in the 1978 paper with Higgins, and so filtered spaces in the later papers. This of course suggested my joint work with Loday, and so the nonabelian tensor product of groups which act o each other.
Peiffer was a student of Reidemeister, but of course communication between the English and German groups was not possible during the war.
Ronald Brown
I have touched the formatting and the wording of the first few lines. In particular I fixed the broken hyperlinks to J.H.C. Whitehead.
also fixed the formatting of a few references. Notice that
``...''
does not work as intended here (the parser reads the first symbol as the instruction to print what follows as raw code).
Use instead
"..."
if you do want quotation marks. But for article titles we usually use, instead, italics via
_..._
or
*...*
[Administrative note: comments 1-3 were originally from a different thread entitled ’[crossed module]’, I have merged that thread with the present one.]
added pointer to:
added pointer to:
Saunders MacLane, §XII.8 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
John Baez, Aaron Lauda, pp. 45 of: HDA V: 2-Groups, Theory and Applications of Categories 12 (2004) 423-491. [arXiv:math.QA/0307200]
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