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 Čech simplicial complex. We show that paths in this product space respecting its stratification induce simplicial maps between the endpoints of the path.
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Acknowledgements
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. ??. Thanks to the reviewers for helpful comments and corrections.
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This paper is based on the doctoral thesis of Jānis Lazovskis, defended May 2019 at the University of Illinois at Chicago.
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Lazovskis, J. Stratifications on the Ran Space. Order 39, 143–158 (2022). https://doi.org/10.1007/s11083-021-09568-1
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DOI: https://doi.org/10.1007/s11083-021-09568-1