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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 20th 2014
    • (edited Feb 20th 2014)

    Somebody over lunch at the conference here said that the nn-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 nn-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 (n1)(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 nn-fold complete Segal space demands that the (n1)(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 (n1n-1)-fold complete Segal spaces is by adding lots of degeneracies, this means that the completeness condition on an nn-fold complete Segal space involves lots of degeneracy conditions in degree 0.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2019

    Egbert added pointer to

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

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

    diff, v20, current

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