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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 20th 2014
   • (edited Feb 20th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSomebody over lunch at the conference here said that the $n$-Lab somewhere leaves out a condition in the definition of [[n-fold complete Segal spaces]], namely "it's not just completeness, there is also a condition that many spaces are degenerate". We were offline and couldn't quite determine which entry was meant. Now I am online but alone, and I checked at _[[n-fold complete Segal space]]_, which doesn't really give any definition at all, but points to _[[(infinity,n)-category]]_ and _[[n-category object in an (infinity,1)-category]]_. I _think_ (am pretty sure) that there the correct definition is given, but I don't really have the leisure to check in detail right now. Instead, I suspect that everything on the nLab is correct but there is just a subtlety that maybe deserves to highligted more, namely for $n$-fold Segal spaces the completenss condition automatically involves more and more degeneracy condition due to the way that $\infty$-groupoids are regarded as degenerate cases of $(n-1)$-fold complete Segal spaces. To hint at that (don't have time for more right now), I have now added to _[[n-fold complete Segal space]]_ the following paragraph: > In analogy of how it works for [[complete Segal spaces]], the completness condition on an $n$-fold complete Segal space demands that the $(n-1)$-fold complete Segal space in degree zero is (under suitable identifications) the [[infinity-groupoid]] which is the [[core]] of the [[(infinity,n)-category]] which is being presented. Since the embedding of $\infty$-groupoids into ($n-1$)-fold complete Segal spaces is by adding lots of degeneracies, this means that the completeness condition on an $n$-fold complete Segal space involves lots of degeneracy conditions in degree 0.

   Somebody over lunch at the conference here said that the $n$-Lab somewhere leaves out a condition in the definition of n-fold complete Segal spaces, namely “it’s not just completeness, there is also a condition that many spaces are degenerate”.

   We were offline and couldn’t quite determine which entry was meant. Now I am online but alone, and I checked at n-fold complete Segal space, which doesn’t really give any definition at all, but points to (infinity,n)-category and n-category object in an (infinity,1)-category. I think (am pretty sure) that there the correct definition is given, but I don’t really have the leisure to check in detail right now.

   Instead, I suspect that everything on the nLab is correct but there is just a subtlety that maybe deserves to highligted more, namely for $n$-fold Segal spaces the completenss condition automatically involves more and more degeneracy condition due to the way that $\infty$-groupoids are regarded as degenerate cases of $(n-1)$-fold complete Segal spaces.

   To hint at that (don’t have time for more right now), I have now added to n-fold complete Segal space the following paragraph:

   In analogy of how it works for complete Segal spaces, the completness condition on an $n$-fold complete Segal space demands that the $(n-1)$-fold complete Segal space in degree zero is (under suitable identifications) the infinity-groupoid which is the core of the (infinity,n)-category which is being presented. Since the embedding of $\infty$-groupoids into ($n-1$)-fold complete Segal spaces is by adding lots of degeneracies, this means that the completeness condition on an $n$-fold complete Segal space involves lots of degeneracy conditions in degree 0.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 25th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexEgbert added pointer to * [[Simona Paoli]], _Simplicial Methods for Higher Categories -- Segal-type Models of Weak $n$-Categories_, Springer 2019 ([doi](https://www.springer.com/gb/book/9783030056735)) and I tweaked the formatting a little. Will add this to other entries, too <a href="https://ncatlab.org/nlab/revision/diff/n-fold+complete+Segal+space/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/n-fold+complete+Segal+space/20">v20</a>, <a href="https://ncatlab.org/nlab/show/n-fold+complete+Segal+space">current</a>

   Egbert added pointer to

   • Simona Paoli, Simplicial Methods for Higher Categories – Segal-type Models of Weak $n$-Categories, Springer 2019 (doi)

   and I tweaked the formatting a little. Will add this to other entries, too

   diff, v20, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 3rd 2022
   • (edited Oct 3rd 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexam deleting the following bibitem * [[Simona Paoli]], _Simplicial Methods for Higher Categories -- Segal-type Models of Weak $n$-Categories_, Springer (2019) &lbrack;[doi:10.1007/978-3-030-05674-2](https://doi.org/10.1007/978-3-030-05674-2), [toc pdf](https://link.springer.com/content/pdf/bfm%3A978-3-030-05674-2%2F1.pdf)&rbrack; and instead moving it to an appropriate entry *[[weakly globular n-fold category]]*, which I will create now (at least a stub for it) <a href="https://ncatlab.org/nlab/revision/diff/n-fold+complete+Segal+space/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/n-fold+complete+Segal+space/23">v23</a>, <a href="https://ncatlab.org/nlab/show/n-fold+complete+Segal+space">current</a>

   am deleting the following bibitem

   • Simona Paoli, Simplicial Methods for Higher Categories – Segal-type Models of Weak $n$-Categories, Springer (2019) [doi:10.1007/978-3-030-05674-2, toc pdf]

   and instead moving it to an appropriate entry weakly globular n-fold category, which I will create now (at least a stub for it)

   diff, v23, current