The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces

TL;DR

This paper develops a categorical framework for large-scale geometry, clarifies the metric Rips filtration, and applies it to hierarchically hyperbolic spaces, providing new models and universal properties for these geometric structures.

Contribution

It introduces a categorical approach to the metric Rips filtration, characterizes universal quasigeodesic cones, and constructs canonical models of hierarchically hyperbolic spaces using universal properties.

Findings

Characterized when the Rips colimit produces a canonical large-scale model.

Proved the quasigeodesic subcategory is closed under colimits.

Constructed a canonical model of HHS as a Rips graph of coarsely consistent tuples.

Abstract

We introduce a flexible, categorical framework for large-scale geometry that clarifies basic behaviour of the metric Rips filtration and streamlines some constructions in geometric group theory. The paper has two main parts. First, we develop the theory of the metric Rips filtration and its colimit in natural coarse categories: informally, we characterise when the Rips colimit produces a canonical large-scale model of a metric space and use this to prove that the quasigeodesic subcategory is closed under colimits in the coarsely Lipschitz category. We also establish adjointness properties of the Rips colimit and use them to characterise extremal metrics and universal morphisms from quasigeodesic sources. Second, we apply this machinery to characterise universal quasigeodesic cones via an explicit Rips-Tuple recipe. In the HHS setting this yields a concrete, canonical model of the total space: an HHS is quasi-isometric to a Rips graph of the space of coarsely consistent tuples in the product of its factor spaces. Moreover, we give a local-to-global criterion that promotes uniformly controlled, factorwise retractions to a canonical global hierarchical retraction. Because the approach is based on universal properties and uniformly controlled coarse data rather than inductive constructions, distance formulae, or hierarchy paths, it applies equally well to arbitrary families of metric spaces equipped with pairwise constraints.