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1. • CommentRowNumber1.
   • CommentAuthorDG
   • CommentTimeSep 19th 2015
   • (edited Sep 19th 2015)
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   Author: DG
   Format: MarkdownItexAt [ncatlab.org/.../category+of+simplices](http://ncatlab.org/nlab/show/category+of+simplices), Proposition 2 says that for a simplicial set $X$, the category of non-degenerate simplices of $X$ is a *reflective* subcategory of the category of simplices of $X$. Is that true? Does someone have a reference for that statement? At the moment I can only see it in the case that $X$ has the property that every face of every non-degenerate simplex of $X$ is itself non-degenerate.

   At ncatlab.org/…/category+of+simplices, Proposition 2 says that for a simplicial set $X$, the category of non-degenerate simplices of $X$ is a reflective subcategory of the category of simplices of $X$. Is that true? Does someone have a reference for that statement? At the moment I can only see it in the case that $X$ has the property that every face of every non-degenerate simplex of $X$ is itself non-degenerate.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeSep 20th 2015
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   Author: Mike Shulman
   Format: MarkdownItexThis was a mistake of mine from several years ago (look at [revision 6](http://ncatlab.org/nlab/revision/category+of+simplices/6) for the incorrect argument), which Urs tried to fix in revision 7; but it looks like he removed the incorrect argument but left in the unjustified conclusion. In fact your condition (every face of a nondegenerate simplex is nondegenerate) is probably exactly what is needed to make the incorrect argument work. Please feel free to fix the page!

   This was a mistake of mine from several years ago (look at revision 6 for the incorrect argument), which Urs tried to fix in revision 7; but it looks like he removed the incorrect argument but left in the unjustified conclusion. In fact your condition (every face of a nondegenerate simplex is nondegenerate) is probably exactly what is needed to make the incorrect argument work. Please feel free to fix the page!

3. • CommentRowNumber3.
   • CommentAuthorDG
   • CommentTimeSep 20th 2015
   • PermaLink
   Author: DG
   Format: TextDone.

   Done.
4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeSep 20th 2015
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   Author: Zhen Lin
   Format: MarkdownItexIs there an established name for this condition? I feel like it should be called "having free degeneracies", but I get the feeling that means something else already.

   Is there an established name for this condition? I feel like it should be called “having free degeneracies”, but I get the feeling that means something else already.

5. • CommentRowNumber5.
   • CommentAuthorKarol Szumiło
   • CommentTimeSep 20th 2015
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   Author: Karol Szumiło
   Format: MarkdownItexZhen Lin, we had an inconclusive discussion about this [here](http://nforum.ncatlab.org/discussion/5534/terminology-for-certain-simplicial-sets/).

   Zhen Lin, we had an inconclusive discussion about this here.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeSep 21st 2015
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   Author: Mike Shulman
   Format: MarkdownItexThanks DG! But it would be even better to replace the incorrect statement with the correct one.

   Thanks DG! But it would be even better to replace the incorrect statement with the correct one.

7. • CommentRowNumber7.
   • CommentAuthorDG
   • CommentTimeSep 21st 2015
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   Author: DG
   Format: TextOh, yes, of course ... Done.

   Oh, yes, of course ...
   Done.