On the Homotopy Theory of Stratified Spaces

Abstract

This thesis is broken into two parts. In the first part (Chapters 2 to 6) is dedicated to proving a 'homtopy hypothesis' for stratified spaces. Specifically, given a poset P, we show that the ∞-category Strₚ of ∞-categories with a conservative functor to P can be obtained from the ordinary category of P-stratified topological spaces by inverting a class of weak equivalences. For suitably nice P-stratified topological spaces, the corresponding object of Strₚ is the exit-path ∞-category of MacPherson, Treumann, and Lurie. To prove this stratified homotopy hypothesis, we define combinatorial simplicial model structure on the category of simplicial sets over the nerve of 𝑃 whose underlying ∞-category is the ∞-category Strₚ. This model structure on P-stratified simplicial sets allows us to easily compare other theories of P-stratified spaces to ours and deduce that they all embed into ours.

The second part (Chapters 7 to 9) explores a number of consequences of this stratified homotopy hypothesis, as well as related results on exit-path ∞-categories and constructible sheaves. This includes an overview of our joint work with Bariwck and Glasman on exit-path categories in algebraic geometry; this work uses as input the perspective on stratified spaces provided by our stratified homotopy hypothesis.