# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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?)