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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 25th 2010
   • PermaLink
   Author: Urs
   Format: Markdownadded two more properties to the <a href="http://ncatlab.org/nlab/show/nerve#properties_of_the_nerve_of_a_category_6">list of properties of nerves of categories</a> at [[nerve]]

   added two more properties to the list of properties of nerves of categories at nerve

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 2nd 2015
   • (edited Apr 2nd 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat we had at _[[nerve]]_ on the ordinary nerve of categories was not very helpful to lay people. I have expanded this now [here](http://ncatlab.org/nlab/show/nerve#NerveOfACategory) by inserting (before the abstract definition as the the restriction of $Cat(-,\mathcal{C})$ to $\Delta$) the low-brow definition, spelling it all out in components.

   What we had at nerve on the ordinary nerve of categories was not very helpful to lay people. I have expanded this now here by inserting (before the abstract definition as the the restriction of $Cat(-,\mathcal{C})$ to $\Delta$) the low-brow definition, spelling it all out in components.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeAug 30th 2015
   • (edited Aug 30th 2015)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThe description of the face maps from dimension 1 to dimension 0 was a bit strangely worded. I have tried out a different wording. (I think I noticed that in the entry on W-bar of a simplicially enriched groupoid, the corresponding faces are not defined and this is also true for most sources on that construction, (including some of my own papers, :-(), which is bizarre!)

   The description of the face maps from dimension 1 to dimension 0 was a bit strangely worded. I have tried out a different wording.

   (I think I noticed that in the entry on W-bar of a simplicially enriched groupoid, the corresponding faces are not defined and this is also true for most sources on that construction, (including some of my own papers, :-(), which is bizarre!)

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeAug 30th 2015
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexFor future readers, the permanent link to ‘list of properties of nerves of categories’ in Urs #1 is now <http://ncatlab.org/nlab/show/nerve#PropNerveCat> (and has been for a while too).

   For future readers, the permanent link to ‘list of properties of nerves of categories’ in Urs #1 is now http://ncatlab.org/nlab/show/nerve#PropNerveCat (and has been for a while too).

5. • CommentRowNumber5.
   • CommentAuthorJohn Baez
   • CommentTimeNov 27th 2018
   • PermaLink
   Author: John Baez
   Format: MarkdownItexThe page [[nerve]] has a proposition saying > A [[simplicial set]] is the nerve of a [[groupoid]] precisely if _all_ [[horns]] have _unique_ fillers. but I think this is wrong: it seems the (1,0)-horns and (1,1)-horns don't have unique fillers. In other words, there needn't be a unique morphism with a given source, or a given target. It seems uniqueness kicks in for the 2-dimensional horns. Am I confused?

   The page nerve has a proposition saying

   A simplicial set is the nerve of a groupoid precisely if all horns have unique fillers.

   but I think this is wrong: it seems the (1,0)-horns and (1,1)-horns don’t have unique fillers. In other words, there needn’t be a unique morphism with a given source, or a given target. It seems uniqueness kicks in for the 2-dimensional horns. Am I confused?

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 27th 2018
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexHmm, so what is a horn for $\Delta^1$? You remove the interior and one face, right? So a horn is just an object. And so of course if every object had a unique morphism to it or out of it, then the simplicial set would be the point. So you are right: it's an odd mismatch with the quasi-category definition, where _inner_ horns are only nontrivial starting in dimension 2, so nothing needs to be said to restrict attention to the relevant dimensions.

   Hmm, so what is a horn for $\Delta^1$? You remove the interior and one face, right? So a horn is just an object. And so of course if every object had a unique morphism to it or out of it, then the simplicial set would be the point. So you are right: it’s an odd mismatch with the quasi-category definition, where inner horns are only nontrivial starting in dimension 2, so nothing needs to be said to restrict attention to the relevant dimensions.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeNov 28th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexPresumably whoever wrote that was thinking only about $n$-horns for $n\gt 1$.

   Presumably whoever wrote that was thinking only about $n$-horns for $n\gt 1$.

8. • CommentRowNumber8.
   • CommentAuthorJohn Baez
   • CommentTimeNov 29th 2018
   • (edited Nov 29th 2018)
   • PermaLink
   Author: John Baez
   Format: MarkdownItexOkay, so if nobody has already done it, I'll add a note saying that the nerve of a groupoid has unique horn-fillers for $n$-horns with $n > 1$. If we dropped the inequality we'd only get nerves of _discrete_ groupoids.

   Okay, so if nobody has already done it, I’ll add a note saying that the nerve of a groupoid has unique horn-fillers for $n$-horns with $n > 1$. If we dropped the inequality we’d only get nerves of discrete groupoids.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 29th 2018
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex@John, >If we dropped the inequality we’d only get nerves of discrete groupoids. Ah, yes. So my claim (simplicial set = point) is just slightly too strong. Thanks for the correct version.

   @John,

   If we dropped the inequality we’d only get nerves of discrete groupoids.

   Ah, yes. So my claim (simplicial set = point) is just slightly too strong. Thanks for the correct version.