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.