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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 23rd 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex[Question](http://mathoverflow.net/questions/56339/artin-mazur-codiagonal-preserves-kan-objects) at MO. Reproduced below: >The Artin-Mazur codiagonal $\nabla:ssSet \to sSet$ is right adjoint to the total decalage functor $Dec:sSet \to ssSet$. The total decalage functor is defined to be precomposition with the ordinal sum functor $+:\Delta\times \Delta \to \Delta$, $+([n],[m]) = [n+m+1]$. >If we agree to call a bisimplicial set $X= ([n]\times [m] \mapsto X_{nm})$ a _bi-Kan complex_ if each simplicial set $[k] \mapsto X_{km}$ and $[k] \mapsto X_{nk}$ is a Kan complex (for all $n$ and $m$), then I'm wondering if it is known in the literature whether $\nabla X$ is a Kan complex. Perhaps in Artin-Mazur's monograph? >Note that a bi-Kan complex is not necessarily the same as an internal Kan complex in the category of Kan complexes (I haven't checked this - just a warning) In particular, this is relevant to showing that $\bar W G$ is globally Kan for a topological simplicial group $G$.

   Question at MO. Reproduced below:

   The Artin-Mazur codiagonal $\nabla:ssSet \to sSet$ is right adjoint to the total decalage functor $Dec:sSet \to ssSet$. The total decalage functor is defined to be precomposition with the ordinal sum functor $+:\Delta\times \Delta \to \Delta$, $+([n],[m]) = [n+m+1]$.

   If we agree to call a bisimplicial set $X= ([n]\times [m] \mapsto X_{nm})$ a bi-Kan complex if each simplicial set $[k] \mapsto X_{km}$ and $[k] \mapsto X_{nk}$ is a Kan complex (for all $n$ and $m$), then I’m wondering if it is known in the literature whether $\nabla X$ is a Kan complex. Perhaps in Artin-Mazur’s monograph?

   Note that a bi-Kan complex is not necessarily the same as an internal Kan complex in the category of Kan complexes (I haven’t checked this - just a warning)

   In particular, this is relevant to showing that $\bar W G$ is globally Kan for a topological simplicial group $G$.

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 23rd 2011
   • (edited Feb 23rd 2011)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI am! but often do not look at MO. (but have answered it there as well now!) I assume you have checked the various papers by Cegarra, e.g. <i>The behaviour of the W-construction on the homotopy theory of bisimplicial sets.</i> Manuscripta Math. 124 (2007), 427–457 (With J. Remedios). I do not have a copy in front of me so cannot cross check. His webpage is at http://www.ugr.es/~acegarra/Trabajos.html. Another very relevant paper to check is http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.3820v1.pdf This seems to say that the answer is YES. see Fact 2.8 page 9. (Possibly also http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.4103v1.pdf might be useful.)

   I am! but often do not look at MO. (but have answered it there as well now!)

   I assume you have checked the various papers by Cegarra, e.g.
   The behaviour of the W-construction on the homotopy theory of bisimplicial sets. Manuscripta Math. 124 (2007), 427–457 (With J. Remedios).

   I do not have a copy in front of me so cannot cross check. His webpage is at http://www.ugr.es/~acegarra/Trabajos.html. Another very relevant paper to check is http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.3820v1.pdf

   This seems to say that the answer is YES. see Fact 2.8 page 9.

   (Possibly also http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.4103v1.pdf might be useful.)

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 23rd 2011
   • (edited Feb 23rd 2011)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexIf you check the MO question now, you will find the same answer (and a good remark by Harry!) My reason for re-replying here is to emphasise that the Granada group working with Cegarra, etc. have done a lot of very good stuff in this area and it may be a good idea to double check in some of the other papers for results that are useful for nLab threads. I must admit to not knowing enough of their work well enough. (It should also be said that the city of Granada has some wonderful things in it... not just the Alhambra.)

   If you check the MO question now, you will find the same answer (and a good remark by Harry!) My reason for re-replying here is to emphasise that the Granada group working with Cegarra, etc. have done a lot of very good stuff in this area and it may be a good idea to double check in some of the other papers for results that are useful for nLab threads. I must admit to not knowing enough of their work well enough. (It should also be said that the city of Granada has some wonderful things in it… not just the Alhambra.)