Homotopy theory of stratified spaces

TL;DR

This paper develops a new homotopy-theoretic framework for stratified spaces by constructing model structures that relate stratified spaces to diagram categories, enabling algebraic analysis of their homotopy properties.

Contribution

It introduces a cofibrantly generated model structure on stratified spaces and establishes a Quillen equivalence to diagram categories of simplicial sets, advancing the algebraic topology of stratified spaces.

Findings

Model structure on stratified spaces constructed

Quillen equivalence with diagram categories established

Weak equivalences characterized by stratified homotopy groups

Abstract

In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those model structures together, we construct a cofibrantly generated model structure on the category of all stratified spaces. In both model categories, weak-equivalences are characterized by stratified homotopy groups.