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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 â€™many objectâ€™ 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).
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.
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.)
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