On stratifications and poset-stratified spaces

TL;DR

This paper explores the mathematical relationship between traditional stratified spaces and poset-stratified spaces, which are both important in various areas of mathematics such as algebraic geometry and higher category theory.

Contribution

It provides a detailed analysis of how these two notions of stratified spaces are related, clarifying their connections and differences.

Findings

Established formal relations between stratified and poset-stratified spaces

Clarified the conditions under which these notions are equivalent or distinct

Enhanced understanding of stratification in different mathematical contexts

Abstract

A stratified space is a topological space equipped with a \emph{stratification}, which is a decomposition or partition of the topological space satisfying certain extra conditions. More recently, the notion of poset-stratified space, i.e., topological space endowed with a continuous map to a poset with its Alexandrov topology, has been popularized. Both notions of stratified spaces are ubiquitous in mathematics, ranging from investigations of singular structures in algebraic geometry to extensions of the homotopy hypothesis in higher category theory. In this article we study the precise mathematical relation between these different approaches to stratified spaces.