A Distance Between Filtered Spaces Via Tripods

Understand

We present a simplified treatment of stability of filtrations on finite spaces.

• Interestingly, we can lift the stability result for combinatorial filtrations from [CSEM06] to the case when two filtrations live on different spaces without directly invoking the concept of interleaving.
• We then prove that this distance is intrinsic by constructing explicit geodesics between any pair of filtered spaces.
• Finally we use this construction to obtain a strengthening of the stability result.