Stratifications on the Ran space (Q2141004)

The Ran space \(\operatorname{Ran}(X)\) of a space \(X\) is the space of all finite nonempty subsets \(\emptyset \neq S \subseteq X\), equipped with a topology that allows points to merge or split. Introduced in the context of algebraic geometry, it now has applications in algebraic topology and quantum algebra. The space \(\operatorname{Ran}(X)\) is naturally stratified by the unordered configuration spaces \(\operatorname{UConf}_k(X) = \{ S \subseteq X \mid \# S = k \}\) for \(k \geq 1\). While this stratification is useful, it can be coarser than needed, as it does not use any knowledge of \(X\). In this paper, given a metric space \(M\), the author introduces a stratification on \(\operatorname{Ran}(M) \times \mathbb{R}_{\geq 0}\) that is inspired by the study of persistent homology. This stratification is indexed by \([\mathsf{SC}]\), the poset of isomorphism classes of simplicial complexes and maps surjective on vertices. A point \((P, r) \in \operatorname{Ran}(M) \times \mathbb{R}_{\geq 0}\) lives in the stratum indexed by the Čech complex \([Č(P, r)] \in [\mathsf{SC}]\), whose vertices are the elements of \(P\) and whose simplices are subsets \(P' \subseteq P\) of configurations such that \(\bigcap_{x \in P'} \bar{B}(x, r) \neq \emptyset\), i.e., there exists a point at distance at most \(r\) to all the points of \(P'\). The main results of this paper are that: \begin{itemize} \item This is a stratification on \(\operatorname{Ran}(M) \times \mathbb{R}_{\geq 0}\); \item If \(M\) is semi-algebraic, then this defines a conical stratification. \end{itemize}