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• CommentRowNumber1.
• CommentAuthorDG
• CommentTimeSep 19th 2015
• (edited Sep 19th 2015)

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.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeSep 20th 2015

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!

• CommentRowNumber3.
• CommentAuthorDG
• CommentTimeSep 20th 2015
Done.
• CommentRowNumber4.
• CommentAuthorZhen Lin
• CommentTimeSep 20th 2015

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.

• CommentRowNumber5.
• CommentAuthorKarol Szumiło
• CommentTimeSep 20th 2015

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeSep 21st 2015

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

• CommentRowNumber7.
• CommentAuthorDG
• CommentTimeSep 21st 2015
Oh, yes, of course ...
Done.