Stratified Homotopy Theory (1908.01366v1)

Abstract: A stratified space is a topological space together with a decomposition into strata corresponding to different types of singularities. Examples of such spaces appear everywhere in topology and geometry. The study of stratified spaces involves invariants such as intersection cohomology which are only invariants under stratification-preserving homotopies. In this thesis, we study the homotopy theory of stratified spaces with respect to those stratified homotopies. To do so, we construct model categories for stratified spaces and we introduce new invariants to characterize them, the filtered homotopy groups. A stratified space can be seen as a topological space $X$ together with a continuous map to a poset of strata $X\to P$. We begin our study by restricting ourselves to the filtered case, where the poset of strata is fixed. We define the model category of filtered simplicial sets, show that it admits a description "`a la Kan", and characterize its weak-equivalences using the filtered homotopy groups. We deduce a proof of a filtered version of Whitehead theorem. We then construct a model category of filtered spaces. Its weak-equivalences are the morphisms that induce isomorphisms on all filtered homotopy groups, and its fibrations satisfy a filtered version of Serre's lifting conditions. We show that it is Quillen-equivalent to a category of diagrams of simplicial sets. We then work toward a comparison between the model categories of filtered simplicial sets and of filtered spaces. They are connected by a Quillen-adjunction, similar to the classical Kan-Quillen adjunction. We conjecture that it is in fact a Quillen equivalence. Lastly, working with the notion of Quillen bifibration, we show that there are model categories of stratified spaces and of stratified simplicial sets. We show that the two are related by an adjunction that preserve weak equivalences.