  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. To motivate these axioms, we also
observe that $d_{HI}$ (or more generally, any pseudometric satisfying these two
axioms and an additional "homology bounding" axiom) can be used to formulate
lifts of several fundamental TDA theorems from the algebraic (homological)
level to the level of filtered spaces. Finally, we consider the problem of
establishing a persistent Whitehead theorem in terms of homotopy interleavings.
We provide a counterexample to a naive formulation of the result.
