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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.
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