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 Čech simplicial complex. We show that paths in this product space respecting its stratification induce simplicial maps between the endpoints of the path.
Bibliographical noteAcknowledgements: The author was partly sponsored by EP/P025072/1. Thanks to Ben Antieau and Shmuel Weinberger for insightful discussions and guidance. Thanks to Justin Curry for suggesting the counterexample of Fig. 4. Thanks to the reviewers for helpful comments and corrections.
Data Availability StatementData Availability: Data sharing not applicable to this article as no datasets were generated or analysed during
the current study.
- Simplicial complexes
- Configuration space
- Entrance path