We construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those model structures together, we construct a cofibrantly generated model structure on the category of all stratified spaces. In both model categories, weak equivalences are characterized by stratified homotopy groups.
"Homotopy theory of stratified spaces." Algebr. Geom. Topol. 21 (1) 507 - 541, 2021. https://doi.org/10.2140/agt.2021.21.507