During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for 2-categories, I also understand why $n$-fold Segal spaces provide a model for $(\infty, n)$-categories. What puzzles me is the completion of $n$-fold Segal spaces. There is a construction for $n=1$ in the case of bisimplicial sets by Rezk in his paper about the homotopy theory of homotopy theory. I can imagine that something similar can be done for Segal spaces which are simplicial spaces instead of bisimplicial sets. My questions are:

How does the completion generalize to $n$-fold Segal spaces?


Is the completion a fibrant replacement in the model category structure on simplicial spaces?


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