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nLab > nLab General Discussions: Alexander-Whitney and Eilenberg-Zilber in augmented grading notation

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    • CommentRowNumber1.
    • CommentAuthorMirco Richter
    • CommentTimeFeb 17th 2012
    • (edited Feb 17th 2012)
    The definition of the Eilenberg-Zilber and the Alexander-Whitney map in the nLab entries

    (http://ncatlab.org/nlab/show/Eilenberg-Zilber+map)

    (http://ncatlab.org/nlab/show/Alexander-Whitney+map)

    are given in the "standard simplicial dimension notation". However in the setting of abelian simplicial groups
    and chain complexes we have frequently the situation where we work with augmented simplicial sets
    and in that scenario there is the 'upshifted dimension counting' where we define the dimension of the augmented
    simplex as zero instead of $(-1)$. (Explained for example in

    http://ncatlab.org/nlab/show/simplex+category

    My question is now how this affects the definition of the above maps and since I can't find anything on the web
    I suggest to add such a augmented definition to the nLab entries on those topics.

    If someone can post a link or something where this is worked out, I will change the entry if you people agree.
    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 17th 2012

    I am not sure that it helps but Andy Tonks worked out the Eilenberg-Whitney stuff for crossed complexes in his thesis. As this is ’many object’ it does have the aspect of an augmented situation and is worth looking at anyhow. The result is discussed in Brown-Higgins-Sivera: Nonabelian Algebraic Topology. (p. 360 I think).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2012

    My question is now how this affects the definition of the above maps

    What beyond the evident shift in their integer labels do you have in mind? I am not sure if I understand your question.

    • CommentRowNumber4.
    • CommentAuthorMirco Richter
    • CommentTimeFeb 17th 2012
    • (edited Feb 17th 2012)
    I guess that is all I have in mind + how they work on the augmented parts.
    (The point is that the shift looks not complete trivial, as for example in the AW map
    the sum is now only up to (n-1), isn't it? ... And as far as I see the shuffle sum must
    be written different,too)

    Do you have an explicit term?
    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 17th 2012

    Mirco: Andy Tonks thesis discusses a lot of things relevant to this. It is on line here. (I forgot to paste the link last time.)

    • CommentRowNumber6.
    • CommentAuthorMirco Richter
    • CommentTimeFeb 17th 2012
    I had a short look on it, but there it is only implicit since he is working on crossed complexes and I think its better
    to wait and see if there is already an explicit expression somewhere. (Or have I overlooked in your reference?)

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