  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.
