## The context

Let $I=[0,∞)$ and consider the category of persistence modules $(V,π)$ indexed over $I$ satisfying:

- For all $t$ in $I$ but a closed discrete set of points $T$, there exists a neighborhood of $t$ on which $π$ is an isomorphism.
- For all $t$ in $I$ there exists some $ε$ such that $π$ is an isomorphism on $(t-ε,t]$.

For any continuous monotonous map $f:I→I$, we can consider the shifted module $(V[f],π[f])$ defined by: $$ V[f]_t := V_{ft}, \quad π[f]_{s,t} = π_{fs,ft}. $$ If $f:I→I$ also satisfies $f≥\mathrm{Id}$ (i.e. $fx≥x$ for all $x∈I$), then we get a natural morphism of persistence modules $V→V[f]$ (This can be more generally described as having a functor $\cdot[f]$ associated to $f$, and for any $f\leq g$ an associated natural transformation $\cdot[f] \to \cdot[g]$).

Let $O_I$ denote the monoid of continuous monotonous maps $f:I→I$ and $O_I⁺$ the submonoid of those with $f≥\mathrm{Id}$.

We can say that two persistence modules $V,W$ are $(f,g)$-interleaved, for $f,g∈O_I⁺$ if there exist morphisms $F:V→W[f]$ and $G:W→V[g]$ making the usual diagrams commute.

All this is for instance treated in this paper of Bubenik, de Silva, Scott in a way more general setting. The takeaway is that everything behaves very well. Note also that as soon as $V,W$ are interleaved, then $V_∞≅W_∞$.

The Rips complex filtration gives us the following **stability theorem**.

If $X,Y$ are finite metric spaces, then (maybe with a constant forgotten): $$ d_{\mathrm{int}}(V_X,V_Y) ≤ d_{\mathrm{GH}}(X,Y), $$

where $V_X$ (resp. $V_Y$) is the persistence module corresponding to homology of Rips complexes for $X$ (resp. $Y$), $d_{\mathrm{int}}$ is the interleaving distance and $d_{\mathrm{GH}}$ the Gromov-Hausdorff distance.

Following the proof of this fact (say in these notes of Polterovich, Rosen, Samvelyan, Zhang), one can generalize the above to:

If $X,Y$ are coarsely isometric metric spaces (maybe plus some tameness conditions) with "upper control functions" $ρ,θ$, then $V_X$ and $V_Y$ are, up to additive constants, $(ρ,θ)$-interleaved.

As an example, if we take not the standard homology in the Rips complexes, but simplicial uniformly finite homology (i.e. infinite sums of simplicies allowed, but with uniform bound on coefficients), we get that two coarsely isometric spaces $X,Y$ have interleaved persistence modules $V_X,V_Y$, thus $(V_X)_∞ ≅ (V_Y)_∞$. Since $(V_X)_∞$ is uniformly finite homology of $X$, this is a roundabout way to say that uniformly finite homology is invariant under coarse isometries.

We can specialize to quasi-isometries: If we consider the following "distance" on metric spaces: $$ d_{\mathrm{aff}}(X,Y) = \log \inf \{A ≥ 1\ :\ \text{$\exists$ a quasi-isometry between}$$ $$\text{$X$ and $Y$ with scaling constant $A$}\}, $$ and similarly on persistence modules: $$ d_{\mathrm{aff}}(V,W) = \log \inf \{A ≥ 1\ :\ \text{$\exists$ an affine interleaving between}$$ $$\text{ $V$ and $W$ with linear part $A$}\}, $$ we directly get that: $$ d_{\mathrm{aff}}(V_X,V_Y) ≤ d_{\mathrm{aff}}(X,Y), $$ which is a sort of “scaling” version of the stability theorem.

The same kind of path can presumably be taken with “exhaustion” functions and persistence homology w.r.t sublevel sets: If $X$ is some nice topological space and $f,g:X→I$ continuous such that $$ fx ≤ ρ(gx), gx ≤ θ(fx) \quad ∀x $$ for some $ρ,θ∈O_I⁺$, then the corresponding persistence modules will be $(ρ,θ)$-interleaved.

## The question

Can this line of thought (“coarsifying persistence”) be used/developped further? Has it been done somewhere already?