  The so called \v{C}ech and Vietoris-Rips simplicial filtrations are designed
to capture information about the topological structure of metric datasets.
These filtrations are two of the workhorses in the field of topological data
analysis. They enjoy stability with respect to the Gromov-Hausdorff (GH)
distance, and this stability property allows us to estimate the GH distance
between finite metric space representations of the underlying datasets.
  Via the concept of Gromov's curvature sets we construct a rich theoretical
framework of valuation-induced stable filtration functors. This framework
includes the \v{C}ech and Vietoris-Rips filtration functors as well as many
novel filtration functors that capture diverse features present in datasets. We
further explore the concept of basepoint filtrations functors and use it to
provide a classification of the filtration functors that we identify.
