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A pair $(X,tX)$, with $X$ a simplicial set and $tX$ a collection of simplices of $X$, is said to be stratified if no $0$-simplex is in $X$ and all degenerate simplices of $X$ are in $tX$. Recall a stratified simplicial set $X$ is $n$-trivial if $X=|X|_n$, the collection of $n$-simplices of $X$. This definition generalizes the notion of marked and scaled simplicial sets, see for example Verity's paper on complicial sets, so that this obviously seems like a generalization of the notion of $\infty$ and $(\infty,2)$-categories. My questions are:

  • Is there a (left proper, combinatorial) Cartesian model structure on the category $\mathfrak{Strat}_n$ of $n$-trivial simplicial sets (see Example 102 of another paper by Verity) such that the fibrant objects are precisely the $(\infty,n)$-categories, i.e., such that they satisfy the Barwick-Schommer-Pries axioms?

  • With this model structure, is there a suitable notion of straightening?

I can take a reasonable guess for the first question - let the cofibrations be the monomorphisms of the underlying simplicial sets, the weak equivalences the maps that are equivalences under a left adjoint to a suitable generalization of the $\omega$-categorical nerve to these $n$-trivial simplicial sets, and the fibrations the maps with the RLP with respect to the acyclic cofibrations.

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  • $\begingroup$ If there was such a model structure, then I guess we could use Lurie's chunks (in particular Proposition T.4.2.4.4) to study straightening. $\endgroup$ – user62675 May 12 '15 at 22:27

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