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  1. At crossed module it seems we are missing what i think should be the prototypical example: the relative second homotopy group π 2(X,A)\pi_2(X,A) together with the bundary map δ:π 2(X,A)π 1(A)\delta:\pi_2(X,A)\to \pi_1(A) and the π 1(A)\pi_1(A)-action on π 2(X,A)\pi_2(X,A). As someone confirms this example is correct I’ll add it to crossed module.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeMay 26th 2012

    I have added something about this as it is really a special case of the previous last example.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2017
    • (edited Oct 26th 2017)

    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.

  2. 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

    diff, v49, current

  3. second edit: misprints

    Ronald Brown

    diff, v50, current

  4. added another reference

    Ronald Brown

    diff, v51, current

  5. Correccted a reference.

    Ronald Brown

    diff, v51, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2020

    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.

    diff, v52, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2020

    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




    diff, v52, current

    • CommentRowNumber10.
    • CommentAuthorRichard Williamson
    • CommentTimeDec 29th 2020
    • (edited Dec 29th 2020)

    [Administrative note: comments 1-3 were originally from a different thread entitled ’[crossed module]’, I have merged that thread with the present one.]

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2021

    for no good reason I went and streamlined wording and formatting of the “diagrammatic definition”.

    One day we’ll bring this poor entry into shape.

    diff, v54, current

    • CommentRowNumber12.
    • CommentAuthorTim_Porter
    • CommentTimeMar 11th 2021


    diff, v55, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v57, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2023
    • (edited May 25th 2023)

    fixed a trivial typo in the Definition-section

    while at it, I have added more hyperlinks to technical terms and touched the formatting of the diagrams

    diff, v58, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2023

    added pointer to:

    diff, v59, current

    • CommentRowNumber16.
    • CommentAuthorTim_Porter
    • CommentTimeJun 7th 2023

    Typo fixed.

    diff, v61, current