Stratifications on the Ran space

TL;DR

This paper introduces a stratification of the product space involving the Ran space and nonnegative reals using a partial order on simplicial complexes, revealing how paths induce simplicial maps.

Contribution

It defines a new partial order on simplicial complexes that stratifies the product of the Ran space and nonnegative reals, linking paths to simplicial maps.

Findings

Stratification of the product space via a partial order on simplicial complexes.

Paths respecting the stratification induce simplicial maps.

Provides a new framework for understanding the topology of the Ran space.

Abstract

We describe a partial order on finite simplicial complexes. This partial order provides a poset stratification of the product of the Ran space of a metric space and the nonnegative real numbers, through the \v Cech simplicial complex. We show that paths in this product space respecting its stratification induce simplicial maps between the endpoints of the path.