Universality of the Homotopy Interleaving Distance

Understand

As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces.

• These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability and inference theorems.
• Intuitively, whereas a strict interleaving between filtered spaces $X$ and $Y$ certifies that $X$ and $Y$ are approximately isomorphic, a homotopy interleaving between $X$ and $Y$ certifies that $X$ and $Y$ are approximately weakly equivalent.
• The main results of this paper are that homotopy interleavings induce an extended pseudometric $d_{HI}$ on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms.