Topology Seminar: Stratified homotopy theory

Stratified spaces appear as natural objects in singularity theory. Goresky and MacPherson introduced intersection cohomology to extend cohomological properties of closed manifolds to stratified spaces, and it proved to be a powerful tool to study those objects. However, intersection cohomology is not homotopy invariant, rather it is invariant with respect to homotopies that "preserve" the stratification. This begs the question: does there exist a model category of stratified spaces which reflects this stratified notion of homotopy, and if so, is intersection cohomology representable in it?

We answer the first part of this question using a simplicial model category of filtered simplicial sets. As a category, it is only the category of simplicial sets over the classifying space of some fixed poset, but as a presheaf category, it inherits a model structure constructed using a natural cylinder object. We show that this category is simplicial, then we get stratified versions of Kan complexes and of homotopy groups that characterise fibrations and weak equivalences.