On the homotopy theory of stratified spaces

TL;DR

This paper develops a new homotopy theory for stratified spaces that aligns with existing theories and confirms a stratified version of Grothendieck's homotopy hypothesis, encompassing many geometric examples.

Contribution

It introduces a novel homotopy theory for P-stratified spaces and proves its equivalence to an ∞-category framework, confirming a conjecture and unifying various stratified space theories.

Findings

Establishes an equivalence between stratified spaces and ∞-categories with a conservative functor to P.

Proves a stratified version of Grothendieck's homotopy hypothesis.

Shows that conically stratified spaces and smooth stratified spaces embed into the new theory.

Abstract

Let $P$ be a poset. We define a new homotopy theory of suitably nice $P$-stratified topological spaces with equivalences on strata and links inverted. We show that the exit-path construction of MacPherson, Treumann, and Lurie defines an equivalence from our homotopy theory of $P$-stratified topological spaces to the $\infty$-category of $\infty$-categories with a conservative functor to $P$. This proves a stratified form of Grothendieck's homotopy hypothesis, verifying a conjecture of Ayala-Francis-Rozenblyum. Our homotopy theory of stratified spaces has the added benefit of capturing all examples of geometric interest: conically stratified spaces fit into our theory, and the Ayala-Francis-Tanaka-Rozenblyum homotopy theory of conically smooth stratified spaces embeds into ours.