  Let $P$ be a poset. We define a new homotopy theory of suitably nice
$P$-stratified topological spaces with equivalences on strata and links
inverted. We show that the exit-path construction of MacPherson, Treumann, and
Lurie defines an equivalence from our homotopy theory of $P$-stratified
topological spaces to the $\infty$-category of $\infty$-categories with a
conservative functor to $P$. This proves a stratified form of Grothendieck's
homotopy hypothesis, verifying a conjecture of Ayala-Francis-Rozenblyum. Our
homotopy theory of stratified spaces has the added benefit of capturing all
examples of geometric interest: conically stratified spaces fit into our
theory, and the Ayala-Francis-Tanaka-Rozenblyum homotopy theory of conically
smooth stratified spaces embeds into ours.
