Higher quasi-categories vs higher Rezk spaces

We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category \Theta_n. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to be slightly modified to get a reasonable notion. We construct two Quillen equivalences between the model category of n-quasi-categories and the model category of Rezk \Theta_n-spaces showing that n-quasi-categories are a model for (\infty, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasi-categories and complete Segal spaces.Comment: 44 pages, v2: terminology changed (see Remark 5.27), Corollary 7.5 added, appendix A added, references added, v3: reorganization of Sections 5 and 6, more informal comments, new section characterizing strict n-categories whose nerve is an n-quasi-category, numbering has change

On the homotopy theory of stratified spaces

Let $P$ be a poset. We show that the $\infty$-category $\mathbf{Str}_P$ of $\infty$-categories with a conservative functor to $P$ can be obtained from the ordinary category of $P$-stratified topological spaces by inverting a class of weak equivalences. For suitably nice $P$-stratified topological spaces, the corresponding object of $\mathbf{Str}_P$ is the exit-path $\infty$-category of MacPherson, Treumann, and Lurie. In particular, the $\infty$-category of conically $P$-stratified spaces with equivalences on exit-path $\infty$-categories inverted embeds fully faithfully into $\mathbf{Str}_P$. This provides a stratified form of Grothendieck's homotopy hypothesis. We then define a combinatorial simplicial model structure on the category of simplicial sets over the nerve of $P$ whose underlying $\infty$-category is the $\infty$-category $\mathbf{Str}_P$. This model structure on $P$-stratified simplicial sets then allows us to easily compare other theories of $P$-stratified spaces to ours and deduce that they all embed into ours.Comment: v5: 41 pages. Minor edits. Added an additional argument on essential surjectivity. v4: 40 pages. Made some correction