Simplicial sets form a combinatorial model for the homotopy theory of topological spaces; more precisely geometric realisation and the total singular complex functor are adjoint functors providing a Quillen equivalence between the Quillen model structure on simplicial sets and the (standard) one on topological spaces. Indeed, the model structure on topological spaces is obtained by transferring the Quillen model structure on simplicial sets across this adjunction.

Fixing suitable stratifications of geometric simplices gives rise to stratified versions of geometric realisation and of the total singular complex functor, and hence to an adjunction between simplicial sets and stratified spaces. By transferring the Joyal (rather than the Quillen) model structure across this adjunction one obtains a model structure on stratified spaces. We claim that this model structure is a natural context in which to study the homotopy theory of stratified spaces. As evidence for this, its cofibrant-fibrant objects are closely related to Quinn's homotopically stratified spaces, and the resulting notion of homotopy equivalence between them is stratification-preserving homotopy.

The fibrant objects of Joyal's model structure are quasi-categories (simplicial models of infinity categories). Results about these can be transferred across the above adjunction to obtain homotopy-theoretic results about stratified spaces. As an example, one obtains a version of a recent theorem due to David Miller which characterises stratification-preserving homotopy equivalences (between cofibrant-fibrant stratified spaces) as those maps which induce an isomorphism on posets of strata, and weak homotopy equivalences between corresponding strata and homotopy-links.

Coauthor: Stephen Nand-Lal