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1. • CommentRowNumber1.
   • CommentAuthorMirco Richter
   • CommentTimeMar 21st 2012
   • (edited Mar 21st 2012)
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   Author: Mirco Richter
   Format: MarkdownItexJust to be sure because I can't find the following on the nLab: If $A$ is a simplicial abelian group, that in addition is $n$-coskeletal for some $n \in \mathbb{N}$, then the normalized Moore complex is concentrated in degree $\leq n$, right? That is because for an $n$-coskeletal simplicial set, we can say that any horn map $\partial_j : A_m \to \partial_j A_m$ is bijective for all $m \geq (n+1)$. (Here $\partial_j A_m$ means the $j$-horn set in dim $m$) Hence in particular the kernel of $\partial_m$ consist only of the identity in degree $m$ and so the same holds for the kernels of the $d_j$'s for all $0 \leq j \leq (m-1)$. Consequently the normalization, i.e. the intersection $\cap_{j=0}^{m-1} d^m_j$ is the trivial group in any dimension $m \geq (n+1)$ and that is what is called "concentrated in degree $\leq n$". Is that correct?

   Just to be sure because I can’t find the following on the nLab:

   If $A$ is a simplicial abelian group, that in addition is $n$-coskeletal for some $n \in \mathbb{N}$, then the normalized Moore complex is concentrated in degree $\leq n$, right?

   That is because for an $n$-coskeletal simplicial set, we can say that any horn map $\partial_j : A_m \to \partial_j A_m$ is bijective for all $m \geq (n+1)$. (Here $\partial_j A_m$ means the $j$-horn set in dim $m$) Hence in particular the kernel of $\partial_m$ consist only of the identity in degree $m$ and so the same holds for the kernels of the $d_j$’s for all $0 \leq j \leq (m-1)$. Consequently the normalization, i.e. the intersection $\cap_{j=0}^{m-1} d^m_j$ is the trivial group in any dimension $m \geq (n+1)$ and that is what is called “concentrated in degree $\leq n$”.

   Is that correct?

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeMar 21st 2012
   • (edited Mar 21st 2012)
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   Author: Tim_Porter
   Format: MarkdownItexMirco: your change to [[Moore complex]] does not make sense. You suggested an alternating last face, but the premise is that the group is a simplcial group and not a simplicial abelian group. The process of truncations and its relationship with coskeleta was explored by Conduché in the simplcial group case, and probably in Illusie's thesis for the abelian one. I think there is a discussion in the Menagerie (see the nLab entry on that). The answer to your question is, I think, yes.

   Mirco: your change to Moore complex does not make sense. You suggested an alternating last face, but the premise is that the group is a simplcial group and not a simplicial abelian group.

   The process of truncations and its relationship with coskeleta was explored by Conduché in the simplcial group case, and probably in Illusie’s thesis for the abelian one. I think there is a discussion in the Menagerie (see the nLab entry on that). The answer to your question is, I think, yes.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2012
   • (edited Mar 21st 2012)
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   Author: Urs
   Format: MarkdownItexThere are different models of "the" Moore complex, all [[quasi-isomorphism|quasi-isomorphic]] but not all truncated, if one is. What matters is not so much that the complex is truncated, but that its homology groups are concentrated in a given range. And this is the the case if the original simplicial group was $n$-truncated (aka $n+1$-coskeltal): by one of the statements of the [[Dold-Kan correspondence]] it in particular identifies the homotopy groups of a simplicial group with the homology groups of its corresponding chain complex.

   There are different models of “the” Moore complex, all quasi-isomorphic but not all truncated, if one is.

   What matters is not so much that the complex is truncated, but that its homology groups are concentrated in a given range. And this is the the case if the original simplicial group was $n$-truncated (aka $n+1$-coskeltal): by one of the statements of the Dold-Kan correspondence it in particular identifies the homotopy groups of a simplicial group with the homology groups of its corresponding chain complex.