Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
  
  
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMirco Richter
   • CommentTimeMar 21st 2012
   • (edited Mar 21st 2012)
   • PermaLink
   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 is a simplicial abelian group, that in addition is -coskeletal for some , then the normalized Moore complex is concentrated in degree , right?

   That is because for an -coskeletal simplicial set, we can say that any horn map is bijective for all . (Here means the -horn set in dim ) Hence in particular the kernel of consist only of the identity in degree and so the same holds for the kernels of the ’s for all . Consequently the normalization, i.e. the intersection is the trivial group in any dimension and that is what is called “concentrated in degree ”.

   Is that correct?

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeMar 21st 2012
   • (edited Mar 21st 2012)
   • PermaLink
   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)
   • PermaLink
   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 -truncated (aka -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.