Universality of the homotopy interleaving distance

As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability inference theorems. Intuitively, whereas strict interleaving spaces X X Y"> Y encoding="application/x-tex">Y certifies that approximately isomorphic, homotopy weakly equivalent. The main results this paper interleavings induce an extended pseudometric alttext="d Subscript upper H I"> d H I encoding="application/x-tex">d_{HI} on spaces, is the universal satisfying natural invariance axioms. To motivate these axioms, also observe (or more generally, any two axioms additional “homology bounding” axiom) can be formulate lifts several fundamental theorems from algebraic (homological) level Finally, consider problem persistent Whitehead theorem terms interleavings. We provide counterexample naive formulation result.