A Persistent Version of Latschev’s Theorem

Oudot, Steve; Waas, Lukas

doi:10.4230/LIPIcs.SoCG.2026.82

A Persistent Version of Latschev’s Theorem

Steve Oudot

Inria Saclay, and École Polytechnique, Palaiseau, France Lukas Waas

University of Oxford, UK

Abstract

Latschev’s theorem provides sufficient conditions on a metric space $M$ and $\delta>0$ for the homotopy type of $M$ to agree with that of the Vietoris-Rips complex $\mathcal{R}^{\delta}(\mathbb{M})$ of any nearby space $\mathbb{M}$ in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair $(M,f\colon M\to\mathbb{R}^{N})$ and $\delta>0$ for the persistent homotopy type of the sublevel set filtration of $(M,f)$ to be interleaved with that of the function-Rips complex ${\normalsize\mathcal{R}}^{\delta}(\mathbb{M}^{\bullet})$ of any nearby pair $(\mathbb{M},\mathbb{f})$ . In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.

Keywords and phrases:

Topological data analysis (TDA), metric geometry, Vietoris-Rips complex, homotopy theory, multi-parameter persistent homology

Funding:

Lukas Waas: Lukas Waas acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 – 390900948 (the Heidelberg STRUCTURES Excellence Cluster). Furthermore, Lukas Waas is a member of the Oxford/Max Planck collaboration and this research was funded in in part by EPSRC international centre to centre collaboration grant EP/Z531224/1.

Copyright and License:

2012 ACM Subject Classification:

Mathematics of computing

\rightarrow

Algebraic topology

Related Version:

Full Version: “Function-Rips complexes in persistent homotopy theory: Stability and Latschev theorems”: https://doi.org/10.48550/arXiv.2603.23460 [31]

Acknowledgements:

This work started when the two authors attended the TDA week at Kyoto University in June 2025. It then continued as the two authors participated in the thematic program Topological Data Analysis, Persistence And Representation Theory Intertwined (TP25TD) at the Okinawa Institute of Science and Technology, from June to August 2025.

DOI:

10.4230/LIPIcs.SoCG.2026.82

Event:

42nd International Symposium on Computational Geometry (SoCG 2026)

Editors:

Hee-Kap Ahn, Michael Hoffmann, and Amir Nayyeri

Series and Publisher:

Leibniz International Proceedings in Informatics, Schloss Dagstuhl – Leibniz-Zentrum für Informatik

1 Introduction

This work connects three lines of research in TDA and metric topology. The first – and main – one aims at generalizing Latschev’s result about the homotopy type of Vietoris-Rips complexes built on Gromov-Hausdorff approximations of compact Riemannian manifolds.

Theorem (Latschev [25]).

Let $M$ be a closed Riemannian manifold. Then there exists $\delta_{0}>0$ such that, for every $0<\delta\leq\delta_{0}$ , there exists an $\varepsilon>0$ such that, for any metric space $\mathbb{M}$ with Gromov-Hausdorff distance to $M$ less than $\varepsilon$ , $M$ is homotopy equivalent to the geometric realization $|\mathcal{R}^{\delta}(\mathbb{M})|$ of the Vietoris-Rips complex $\mathcal{R}^{\delta}(\mathbb{M})$ of parameter $\delta$ .

To our knowledge, this result has been extended in the following directions:

$\blacksquare$

Replacing Riemannian manifolds with spaces of curvature bounded above [23];
$\blacksquare$

Quantifying the upper bounds on $\varepsilon$ and $\delta$ [27, 29];
$\blacksquare$

Letting $\delta$ go past its upper bound on specific spaces, for instance the circle [1] or ellipses [3];
$\blacksquare$

Considering variants of the Vietoris-Rips complex, such as the selective Rips complex [26].

Here, we extend Latschev’s result in another direction: To spaces equipped with filtration functions (Theorem 1 below). In this setting, the role of the homotopy type of $M$ is taken by the so-called persistent homotopy type of the sublevel set filtration of a function $f\colon M\to\mathbb{R}^{N}$ , denoted $M^{\bullet}$ (Notation 19). The role of the Vietoris-Rips complex of $\mathbb{M}$ is now taken by the filtered simplicial complex $\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})$ , called the function-Rips complex. It is obtained by filtering the ordinary Rips complex of fixed parameter $\delta$ , $\mathcal{R}^{\delta}(\mathbb{M})$ , through the Rips-complexes (at scale $\delta$ ) of the sublevel sets of a function $\mathbb{f}\colon\mathbb{M}\to\mathbb{R}^{N}$ (Example 12). Finally, the role of correspondences (for the Gromov-Hausdorff distance) is taken by so-called filtered correspondences (Definition 32), and the role of homotopy equivalences is taken by a notion of interleavings in the persistent homotopy category (Definition 28, see also [24]). In the following, $M$ denotes a compact metric space with curvature bounded above by some $\kappa\in\mathbb{R}$ (Recollection 5) and $f\colon M\to\mathbb{R}^{N}$ a componentwise $1$ -Lipschitz map. Finally, $\rho^{\kappa}_{M}$ denotes the minimum of the convexity radius $\varrho_{M}$ of $M$ (see Recollection 4) and of either $\frac{\pi}{4\sqrt{\kappa}}$ if $\kappa>0$ or $\infty$ if $\kappa\leq 0$ .

Theorem 1 (Persistent Latschev’s theorem).

Let $0<\delta\leq\rho^{\kappa}_{M}$ and $\varepsilon\leq\min\{\rho^{\kappa}_{M}-\delta,\frac{1}{7}\delta\}.$ Then, for any filtered $\varepsilon$ -correspondence $(M,f)\approx_{\varepsilon}(\mathbb{M},\mathbb{f})$ , there is a $\frac{7}{4}\delta$ -interleaving $M^{\bullet}\simeq_{\frac{7}{4}\delta}\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})$ in the persistent homotopy category.

Note that, in contrast to the setup of Latschev’s theorem, we cannot hope to get a homotopy equivalence between filtered spaces here. This is because the individual sublevel sets of $f$ may be highly pathological even though $M$ itself satisfies strict regularity conditions. Note also that we cannot expect interleavings at the level of topological spaces: any such interleaving would induce homeomorphisms at infinity, which generally do not exist.

In the special case $N=0$ (i.e., $\mathbb{R}^{N}$ is a singleton), the homotopy theoretic interleaving $M^{\bullet}\simeq_{\frac{7}{4}\delta}\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})$ is just an isomorphism in the homotopy category, i.e., a zig-zag of weak homotopy equivalences. So, by an application of Whitehead’s theorem, Theorem 1 recovers the quantitative version of Latschev’s theorem given in [29], with the same bounds on $\delta$ and $\varepsilon$ . As Theorem 1 is phrased at the persistent homotopy level, it also induces an interleaving at the persistent homology level. This connects our result to another line of research, the aim of which is to estimate the persistent homology of functions from finite point samples. In [14], the authors proposed an estimator for the persistent homology of $L^{\bullet}$ , for a Lipschitz map $g\colon L\to\mathbb{R}$ , from a finite sampling $\mathbb{L}\subset L$ of its domain $L$ . The proof technique leveraged in this line of work made use of the interleaving between Vietoris-Rips and Čech complexes, and ultimately led to an estimator given by the image of morphisms of persistence modules $\textnormal{im}(H_{\ast}\mathcal{R}^{\delta}(\mathbb{L}^{\bullet})\to H_{\ast}% \mathcal{R}^{2\delta}(\mathbb{L}^{\bullet}))$ , where $\mathbb{L}$ is filtered by $g|_{\mathbb{L}}$ . This estimator was then specialized to the case where $g$ is a density estimator in the context of unsupervised learning, yielding the clustering algorithm ToMATo [15]. It was later extended to more general noise models [10] and, more recently, to $\mathbb{R}^{N}$ -valued maps:

Theorem ([4]).

Let $(L,g)$ be such that $L$ is a compact metric space with $\varrho_{L}>0$ and $g$ is $1$ -Lipschitz. Let $\delta<\frac{1}{2}\varrho_{L}$ and $\varepsilon\leq\frac{1}{2}\delta.$ Then, for any subset $\mathbb{L}\subset L$ that is $\varepsilon$ -dense in $L$ , there is a $2\delta$ -interleaving $H_{\ast}(L^{\bullet})\simeq_{2\delta}\textnormal{im}(H_{\ast}\mathcal{R}^{% \delta}(\mathbb{L}^{\bullet})\to H_{\ast}\mathcal{R}^{2\delta}(\mathbb{L}^{% \bullet}))$ of persistence modules.

Theorem 1 complements this result by showing that, under a more restrictive regularity condition on $M$ (namely: that $M$ has curvature bounded above), a single filtered Rips complex is enough to estimate the persistent homology of $f$ , via a different proof approach that does not proceed through the interleaving between Vietoris-Rips and Čech complexes. The question of whether this is possible had been open since the beginning of this line of work, and it has important implications, including algorithmic ones. For instance, computing a free presentation of $\textnormal{im}(H_{\ast}\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})\to H_{\ast}% \mathcal{R}^{2\delta}(\mathbb{M}^{\bullet}))$ involves computing the free cover of a certain pullback for which efficient specialized algorithms exist only when $N=1$ or $2$ (see [4]); otherwise, one must resort to Schreyer’s algorithm [33] with doubly exponential complexity in $N$ . By contrast, free presentations of $H_{\ast}\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})$ can be computed efficiently for any $N\geq 1$ via specialized algorithms for the computation of Gröbner bases [18].

Similarly to the proof of the classical Latschev theorem, the proof of our persistent version (Theorem 1) is a two-step process. Firstly, one needs a persistent version of Hausmann’s theorem allowing us to relate $M^{\bullet}$ to $\mathcal{R}^{\delta}(M^{\bullet})$ .

Theorem 2 (Persistent Hausmann’s Theorem).

Let $0<\delta\leq\rho^{\kappa}_{M}$ . Then there is a $\delta$ -interleaving $M^{\bullet}\simeq_{\frac{3}{4}\delta}\mathcal{R}^{\delta}(M^{\bullet})$ in the persistent homotopy category.

This result, proven in the full version [31], is a rather straightforward consequence of a powerful new technique called metric thickening, which provides a more geometrical model for the Vietoris-Rips complex [2, 19]. The crucial and technically more difficult part of proving Theorem 1 is to relate $\mathcal{R}^{\delta}(M^{\bullet})$ and $\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})$ . Here, we prove the following result:

Theorem 3 (Approximation of function-Rips persistent homotopy type).

Let $0<\delta\leq\rho^{\kappa}_{M}$ and $\varepsilon\leq\min\{\rho^{\kappa}_{M}-\delta,\frac{1}{7}\delta\}$ . Then, for any filtered $\varepsilon$ -correspondence $(\mathbb{M},\mathbb{f})\approx_{\varepsilon}(M,f)$ , there is a $\delta$ -interleaving $\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})\simeq_{\delta}\mathcal{R}^{\delta}(% M^{\bullet})$ in the persistent homotopy category.

Figure 1: Shrinking transformation arrows (colored) and the object’s structure morphisms (black).

To derive this result, we first prove a bivariate interleaving $\mathcal{R}^{\bullet}(M^{\bullet})\simeq_{\varepsilon}\mathcal{R}^{\bullet}(% \mathbb{M}^{\bullet})$ from the filtered correspondence $(\mathbb{M},\mathbb{f})\approx_{\varepsilon}(M,f)$ in Section 3 (Proposition 34). Then, to leverage this interleaving, in Section 4, we introduce and study the general notion of a shrinking transformation: a transformation of the form $F^{\bullet_{1},\bullet_{2}}\to F^{S\bullet_{1},\bullet_{2}+C\bullet_{1}}$ , with $C\geq 0$ and $0\leq S<1$ , allowing one to move backwards in one parameter at the cost of increasing the other in a bivariate persistent object $F^{\bullet,\bullet}$ , as illustrated in Figure 1. Shrinking transformations can be used to turn interleavings in two variables $F^{\bullet,\bullet}\simeq_{\varepsilon}G^{\bullet,\bullet}$ into interleavings of the form $F^{\delta,\bullet}\simeq_{C\delta+(1+C)\varepsilon}G^{\delta,\bullet}$ (Theorem 44). Then, in Section 5 we work out conditions on $M$ that guarantee the existence of such a shrinking transformation on $\mathcal{R}^{\bullet}(M^{\bullet})$ (Theorem 51). One incarnation of these conditions is verified in the quantitative version of Latschev’s theorem [29] and produces a shrinking transformation with $C=S=\frac{3}{4}$ . From these conditions, combined with Propositions 34 and 44, we get Theorem 3. Together with Theorem 2, this gives the $\frac{7}{4}\delta$ -interleaving $M^{\bullet}\simeq_{\frac{7}{4}\delta}\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})$ of Theorem 1.
We note that most of our techniques, particularly Theorems 44 and 51, are quite general and seem widely applicable. Indeed, in the full version of this article [31], we will use them to prove local stability theorems for the persistent homotopy types of function-Rips complexes.

2 Background

2.1 Spaces of curvature bounded above

Recollection 4 (Geodesics and convexity).

Let $M$ be a metric space with metric $d$ . By a (constant speed) geodesic, we mean a map $\gamma\colon[0,1]\to M$ such that $d(\gamma(s),\gamma(t))=|s-t|\,d(\gamma(0),\gamma(1))$ for all $s\leq t\in[0,1]$ . We say that a subset $A\subset M$ is convex, if every pair of points $x,y\in A$ is connected by a unique constant speed geodesic $\gamma$ from $x$ to $y$ in $M$ , and furthermore every such geodesic is contained in $A$ . We denote

\varrho_{M}(x):=\sup\{r>0\mid B_{s}(x)\textnormal{ is convex, for all }s<r\}

the convexity radius at $x\in M$ . The convexity radius of $M$ is defined as $\varrho_{M}:=\inf_{x\in M}\varrho_{M}(x).$

Recollection 5 (Curvature bounded above).

Given $\kappa\in\mathbb{R}$ , we call $M_{\kappa}$ the following space:

1.

for $\kappa>0$ , the sphere of radius $\frac{1}{\sqrt{\kappa}}$ in $\mathbb{R}^{3}$ , equipped with the geodesic distance;
2.

for $\kappa=0$ , the euclidean plane $\mathbb{R}^{2}$ ;
3.

for $\kappa<0$ , the rescaling of the hyperbolic plane $\mathbb{H}$ by $\frac{1}{\sqrt{-\kappa}}$ .

We denote by $\varpi_{\kappa}$ the diameter of $M_{\kappa}$ ( $\varpi_{\kappa}=\frac{\pi}{\sqrt{\kappa}}$ if $\kappa>0$ and $\varpi_{\kappa}=\infty$ if $\kappa\leq 0$ ). We say that $M$ is a $\textnormal{CAT}(\kappa)$ space if any two points with distance at most $\varpi_{\kappa}$ are connected by a geodesic, and furthermore, every geodesic triangle of perimeter smaller than $2\varpi_{\kappa}$ is no thicker than a comparison triangle in $M_{\kappa}$ (see Figure 2 and [31, 9] for a rigorous definition). A metric space $M$ is called of curvature bounded above by $\kappa$ - or a $\textnormal{CBA}(\kappa)$ space for short - if every point $x\in M$ admits a neighborhood that is a $\textnormal{CAT}(\kappa)$ space. For example, every Riemannian manifold with sectional curvature bounded above by $\kappa$ is a $\textnormal{CBA}(\kappa)$ space.

Notation 6.

Let $M$ be a compact $\textnormal{CBA}(\kappa)$ space. We denote $\rho^{\kappa}_{M}:=\min\{\varrho_{M},\frac{\varpi_{\kappa}}{4}\}>0$ .

Figure 2: Illustration of equilateral triangles in curvatures

\kappa<0

,

\kappa=0

, and

\kappa>0

, respectively.

2.2 Models for persistent homotopy types

We will use three models for persistent homotopy types in the following, arising from spaces, simplicial complexes, and simplicial sets, the last of which we discuss in Section 2.4.

Notation 7.

We denote by sCplx the category of simplicial complexes and simplicial maps, and by Top the category of topological spaces and continuous maps. We denote by $K$ the topological realization of a simplicial complex $K$ . Often, we will just treat a complex as a space, and leave the realization implicit. The standard $n$ -simplex, given by the set of non-empty subsets of $\{0,\dots,n\}$ , will be denoted $\Delta^{n}_{c}$ .

Definition 8.

In the following, $\mathbb{P}$ will always denote a partially ordered set. When we treat $\mathbb{P}$ as a category, we mean the category whose objects are given by the elements of $\mathbb{P}$ , and where there is a unique morphism $x\to y$ whenever $x\leq y$ .
Let C be a category. A $\mathbb{P}$ -persistent object in C is a functor $F\colon\mathbb{P}\to\textnormal{{C}}$ . A morphism of $\mathbb{P}$ -persistent objects is a natural transformation between such functors. We denote by $\textnormal{{C}}^{\mathbb{P}}$ the category of $\mathbb{P}$ -persistent objects in C and their morphisms.

Notation 9.

We will usually denote persistent objects $F\colon\mathbb{P}\to\textnormal{{C}}$ in the form $F^{\bullet}$ , to indicate their functoriality in $\mathbb{P}$ . When the objects in a category C have a specific name, such as simplicial complexes, topological spaces or metric spaces, we will often refer to $\mathbb{P}$ -persistent objects in C by adding the prefix persistent to that name.

Notation 10.

$N\geq 0$ will always be a non-negative integer and $\mathbb{R}^{N}$ will be considered as a partially ordered set via componentwise comparison. By a space function pair (often just pair), we will mean a topological space $X$ together with a function $f\colon X\to\mathbb{R}^{N}$ . When we speak of a metric pair, this will mean that we equip $X$ with a metric, inducing its topology.

Example 11.

Given a pair $(X,f)$ , the sublevel sets $X^{u}:=f^{-1}\{v\in\mathbb{R}^{N}\mid v\leq u\}$ , together with functoriality given by inclusions, gives rise to a persistent space $u\mapsto X^{u}$ , which we denote by $X^{\bullet}$ . We abuse notation here insofar as the construction evidently depends on $f$ . As all structure maps are given by inclusions $X^{\bullet}$ is often also referred to as a filtered space.

Example 12 (function-Rips complex).

Given a metric space $\mathbb{M}$ and $\delta\geq 0$ , the open Vietoris-Rips complex of $\mathbb{M}$ (only Rips complex henceforth), denoted $\mathcal{R}^{\delta}(\mathbb{M})$ , is the simplicial complex whose set of simplices is given by

\{\{x_{0},\dots,x_{n}\}\mid x_{i}\in\mathbb{M},d(x_{i},x_{j})<\delta% \textnormal{, for all }0\leq i,j\leq n\}.

Varying the parameter $\delta\geq 0$ , one obtains a persistent simplicial complex $\mathcal{R}^{\bullet}(\mathbb{M})\colon\mathbb{R}_{\geq 0}\to\textnormal{{% sCplx}}$ , with functoriality on relations given by inclusions. Suppose now that $\mathbb{M}$ is additionally equipped with a (not-necessarily continuous) function $\mathbb{f}\colon\mathbb{M}\to\mathbb{R}^{N}$ . Then, for every $u\in\mathbb{R}^{N}$ , one can consider the Vietoris-Rips complex of $\mathbb{M}^{u}:=\{x\in\mathbb{M}\mid\mathbb{f}(x)\leq u\}$ . Varying $u$ with $\delta$ fixed gives rise to a persistent simplicial complex $\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})\colon\mathbb{R}^{N}\to\textnormal{{% sCplx}}$ , $u\mapsto\mathcal{R}^{\delta}(\mathbb{M}^{u})$ , called the function-Rips complex. Varying both $u$ and $\delta$ gives rise to the bivariate variant $\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})\colon\mathbb{R}_{\geq 0}\times% \mathbb{R}^{N}\to\textnormal{{sCplx}}$ , $(u,\delta)\mapsto\mathcal{R}^{\delta}(\mathbb{M}^{u})$ , called the bivariate function-Rips complex.

2.3 Persistent homotopy theory

Conceptually speaking, a $\mathbb{P}$ -persistent homotopy type should be a $\mathbb{P}$ -indexed functor valued in the homotopy theory or $\infty$ -category of spaces, not just the ordinary $1$ -category of spaces (see also [21, 8]). As we do not expect familiarity with $\infty$ -categorical language, we will instead work with the more elementary notion of relative categories, which we refer to as homotopy theories here.¹¹1This is justified insofar as relative categories provide a model for $(\infty,1)$ -categories [28, 6, 7].

Definition 13.

By a homotopy theory we will mean a relative category; that is, a pair consisting of a category C and a wide subcategory $W\subset\textnormal{{C}}$ . The morphisms in $W$ are called weak equivalences. A morphism in C that is in $W$ will often be denoted by the symbol $\xrightarrow{\simeq}$ .

Example 14.

Recall that a weak homotopy equivalence between topological spaces is a continuous map that induces isomorphisms on the sets of path components and on all homotopy groups. The category Top of topological spaces equipped with the subcategory of weak homotopy equivalences forms a homotopy theory which we denote by $\mathcal{S}\textnormal{{paces}}$ .²²2It follows as a consequence of Whitehead’s theorem that this homotopy theory is equivalent to the one given by CW-complexes and homotopy equivalences.

Notation 15.

Given a homotopy theory $\mathcal{C}=(\textnormal{{C}},W)$ and another category $\mathbb{I}$ , we denote by $\mathcal{C}^{\mathbb{I}}$ the homotopy theory $(\textnormal{{C}}^{\mathbb{I}},W^{\mathbb{I}})$ , where $W^{\mathbb{I}}$ is the wide subcategory of $\textnormal{{C}}^{\mathbb{I}}$ consisting of those natural transformations $\varphi\colon F\Rightarrow G$ such that for every object $i\in\mathbb{I}$ , the morphism $\varphi_{i}\colon F(i)\to G(i)$ is in $W$ . ³³3Note that this will generally only produce the $\infty$ -categorical functor category when $\mathcal{C}$ is sufficiently well-behaved, for example when it extends to a model category, as is the case for our examples ([28]). To indicate that we study functors (persistent objects) $F\colon\mathbb{I}\to\textnormal{{C}}$ in the context of the whole homotopy theory $\mathcal{C}$ we will often use the notation $F\colon\mathbb{I}\to\mathcal{C}$ .

Example 16.

When $\mathcal{C}=\mathcal{S}\textnormal{{paces}}$ is the homotopy theory of spaces, then $\mathcal{S}\textnormal{{paces}}^{\mathbb{P}}$ will be referred to as the $\mathbb{P}$ -persistent homotopy theory of spaces.

Notation 17.

Given a homotopy theory $\mathcal{C}=(\textnormal{{C}},W)$ , one can associate to it its homotopy category $\textnormal{ho}(\mathcal{C})$ , defined as the localization $\textnormal{{C}}[W^{-1}]$ of C at the weak equivalences $W$ .

Recollection 18.

Together with the canonical localization functor, $\textnormal{{C}}\to\textnormal{{C}}[W^{-1}]=\textnormal{ho}(\mathcal{C})$ , $\textnormal{ho}(\mathcal{C})$ is characterized by the universal property that any functor $\textnormal{{C}}\to\textnormal{{D}}$ that sends weak equivalences to isomorphisms factors uniquely through $\textnormal{ho}(\mathcal{C})$ . Explicitly, objects of $\textnormal{ho}(\mathcal{C})$ are the same as those of C, and morphisms are given by equivalence classes of zig-zags of morphisms in C, where backward-pointing arrows are weak equivalences. In this sense, $\textnormal{ho}(\mathcal{C})$ is the $1$ -category obtained by formally inverting the weak equivalences.

Notation 19.

We abuse notation insofar as we use the same symbols for objects and morphism in C and their images in $\textnormal{ho}(\mathcal{C})$ . In the context of the persistent homotopy theory $\mathcal{S}\textnormal{{paces}}^{\mathbb{P}}$ , a persistent space or persistent simplicial complex $X^{\bullet}$ will often be referred to as a persistent homotopy type. Note that while, set theoretically speaking, the persistent homotopy type $X^{\bullet}$ is the same as its underlying persistent space, in the context of the homotopy category $\textnormal{ho}(\mathcal{S}\textnormal{{paces}}^{\mathbb{U}})$ , two persistent homotopy types are isomorphic when they are connected by a zig-zag of weak equivalences.

2.4 Persistent simplicial sets

At least half of the persistent homotopy types we are studying in this article arise from purely combinatorial data provided in the form of persistent simplicial complexes. It is thus convenient to have a model for the persistent homotopy theory of spaces that is more combinatorial in nature. This can be achieved by working with simplicial sets⁴⁴4From a categorical perspective, simplicial sets are a much preferable combinatorial model than simplicial complexes. Note, for example, that simplicial sets come with a natural weak equivalence $\textnormal{sd}_{b}X\xrightarrow{\simeq}X$ between the barycentric subdivision $\textnormal{sd}_{b}X$ of $X$ and $X$ , which we use in Section 5.. The reader not familiar with the theory of simplicial sets can treat them as a black-box that extends the category of simplicial complexes in a convenient way (see [20, 21] for an introduction).

Recollection 20.

Recall that, conceptually speaking, a simplicial set is like a simplicial complex, where the simplices are ordered, and one allows for faces of simplices to collapse to lower dimensions. Categorically, this idea can be formalized as follows: Denote by $\Delta$ the category of finite linear posets $[n]=\{0\leq\dots\leq n\}$ , for $n\geq 0$ , with order-preserving maps. The category of simplicial sets sSet is the category of functors from $\Delta^{\textnormal{op}}$ into Set, i.e., $\textnormal{{Set}}^{\Delta^{\textnormal{op}}}$ . Given a simplicial set $X\in\textnormal{{sSet}}$ and $n\geq 0$ , the set $X([n])$ is denoted $X_{n}$ , and called the set of $n$ -simplices of $X$ . A simplex $\sigma\in X_{n}$ that is not in the image of a structure map $X_{k}\to X_{n}$ , for some $k<n$ , is called non-degenerate.

Example 21.

As a simplicial set, the function-Rips complex at $u\in\mathbb{R}^{N}$ and $\delta\geq 0$ can be modeled by a simplicial set with $n$ -simplices

\mathcal{R}^{u,\delta}(M,f)_{n}=\{(x_{0},\dots,x_{n})\mid x_{i}\in M;d(x_{i},x% _{j})<\delta;f(x_{i})\leq u\textnormal{ for all }i,j\in[n]\},

and functoriality on $\Delta^{\textnormal{op}}$ given by precomposition.

Notation 22.

Given $n\in\mathbb{N}$ , the image of $[n]$ under the Yoneda embedding $\Delta\hookrightarrow\textnormal{{Set}}^{\Delta^{\textnormal{op}}}$ , $[k]\mapsto\Delta([k],[n])$ , is denoted by $\Delta^{n}$ and referred to as the $n$ -simplex. The functor $[n]\mapsto\Delta^{n}$ defines a fully faithful embedding $\Delta\hookrightarrow\textnormal{{sSet}}$ , by which we treat $\Delta$ as a subcategory of sSet.

Recollection 23.

Simplicial sets admit a topological realization functor. Observe that $\Delta$ embeds into sCplx by sending $[n]$ to the standard simplex $\Delta^{n}_{c}$ . The composition $\Delta\to\textnormal{{sCplx}}\xrightarrow{-}\textnormal{{Top}}$ defines a topological realization functor on $\Delta\subset\textnormal{{sSet}}$ , which extends canonically to a colimit-preserving functor $-\colon\textnormal{{sSet}}\to\textnormal{{Top}}$ , so that the realization of a simplicial set $X$ , $X$ , is glued from realizations of its simplices.

The most important reason why simplicial sets are so useful is that they can be used to define a homotopy theory equivalent to that of topological spaces.

Recollection 24.

A simplicial map $\varphi\colon X\to Y$ in sSet is called a weak homotopy equivalence if its topological realization $\varphi\colon X\to Y$ is a homotopy equivalence. Denote the wide subcategory of weak homotopy equivalences by $W_{\textnormal{Kan}}$ , and the resulting relative category $(\textnormal{{sSet}},W_{\textnormal{Kan}})$ by $\textnormal{{sSet}}_{\textnormal{Kan}}$ . It is a fundamental fact of homotopy theory, that the topological realization functor $-\colon\textnormal{{sSet}}\to\textnormal{{Top}}$ then defines a so-called equivalence of homotopy theories ( $\infty$ -categories) $\textnormal{{sSet}}_{\textnormal{Kan}}\to\mathcal{S}\textnormal{{paces}}$ (see [32]). Conceptually speaking, this means that any homotopy theoretic construction or argument concerning $\mathcal{S}\textnormal{{paces}}$ can equivalently be performed in $\textnormal{{sSet}}_{\textnormal{Kan}}$ . For our purposes, however, it suffices to observe that the functor $-\colon\textnormal{{sSet}}^{\mathbb{P}}\to\textnormal{{Top}}^{\mathbb{P}}$ descends to an equivalence of categories $-\colon\textnormal{ho}\big((\textnormal{{sSet}},W_{\textnormal{Kan}})^{\mathbb% {P}}\big)\to\textnormal{ho}(\mathcal{S}\textnormal{{paces}}^{\mathbb{P}})$ , compatible with any reparametrization of the indexing poset $\mathbb{P}$ .

Recollection 25.

In the same way as homotopies induce identifications of maps in the homotopy category $\textnormal{ho}(\mathcal{S}\textnormal{{paces}})$ , morphisms of persistent simplicial sets of the form $X^{\bullet}\times\Delta^{1}\to Y^{\bullet}$ (where $X^{\bullet}\times\Delta^{1}$ denotes the indexwise product) – so-called elementary homotopies – induce identifications of persistent simplicial maps in $\textnormal{ho}(\textnormal{{sSet}}_{\textnormal{Kan}}^{\mathbb{P}})$ .

Recollection 26.

We frequently want to treat simplicial complexes as simplicial sets. There is a canonical fully faithful embedding $N_{s}\colon\textnormal{{sCplx}}\hookrightarrow\textnormal{{sSet}}$ , mapping a simplicial complex, $K$ , to the simplicial set given by $N_{s}(K)_{n}=\textnormal{{sCplx}}(\Delta_{c}^{n},K)$ , functorial in the obvious way in $n$ and $K$ (see [16, § 1.4]). For example, the simplicial set obtained by applying $N_{s}$ to a function-Rips complex is precisely the one described in Example 21. We will usually omit $N_{s}$ from the notation. Observe that from a homotopy-theoretic perspective, this is justified by the fact that, given $K\in\textnormal{{sCplx}}$ , there is a canonical homotopy equivalence $N_{s}{(K)}\xrightarrow{\simeq}K$ (see [30, 5]) even though the equality $K\cong N_{s}(K)$ does not hold on the homeomorphism level. Hence, for our purposes, we can freely identify the two realizations.

2.5 Interleavings in the homotopy category

One of the core advantages of persistent settings is that they allow for approximate notions of equivalence, so called interleavings (see [17] for a more general setting). In the following, $\mathcal{C}$ will denote some homotopy theory. To define interleavings, we need the following notation.

Notation 27.

Henceforth, $\mathbb{U}\subset\mathbb{R}^{N}$ , $\mathbb{U}^{\prime}\subset\mathbb{R}^{N^{\prime}}$ will always denote upsets of $\mathbb{R}^{N}$ , i.e., subsets fulfilling $x\in\mathbb{U},x\leq y\implies y\in\mathbb{U}$ .
Suppose we are given a map of posets $S\colon\mathbb{U}^{\prime}\to\mathbb{U}$ and a persistent object $F\colon\mathbb{U}\to\mathcal{C}$ . We write $F^{S(\bullet)}$ to denote the persistent object obtained by precomposing $F$ with $S$ . In the special case where $S$ is an inclusion of posets, we use the notation $F^{\bullet}|_{\mathbb{U}^{\prime}}$ to denote the restriction of $F$ to $\mathbb{U}^{\prime}$ . Suppose we are given another such map $S^{\prime}\colon\mathbb{U}^{\prime}\to\mathbb{U}$ such that $S(x)\leq S^{\prime}(x)$ , for all $x\in\mathbb{U}^{\prime}$ . In this case, we will use the notation $s\colon F^{S(\bullet)}\to F^{S^{\prime}(\bullet)}$ to denote the natural transformation induced by the relations $S(x)\leq S^{\prime}(x)$ . Given $\varepsilon\geq 0$ write $(-)+\varepsilon\colon\mathbb{U}\to\mathbb{U}$ for the map of posets given by $x\mapsto x+\varepsilon(1,\dots,1)$ .

Definition 28.

An $\varepsilon$ -interleaving in the homotopy category between $F^{\bullet},G^{\bullet}\colon\mathbb{U}\to\mathcal{C}$ consists of morphisms $\varphi\colon F^{\bullet}\to G^{\bullet+{\varepsilon}}\quad\text{ and }\quad% \psi\colon G^{\bullet}\to F^{\bullet+{\varepsilon}}$ in $\textnormal{ho}(\mathcal{C}^{\mathbb{U}})$ such that the diagrams

(1)

in $\textnormal{ho}(\mathcal{C}^{\mathbb{U}})$ commute. We will denote such interleavings in the form $\varphi\colon F\simeq_{\varepsilon}G\colon\psi.$

$\blacktriangleright$ Remark 29.

Two interleavings $\varphi\colon F^{\bullet}\simeq_{\varepsilon}G^{\bullet}\colon\psi$ and $\varphi^{\prime}\colon G^{\bullet}\simeq_{\varepsilon^{\prime}}J^{\bullet}% \colon\psi^{\prime}$ compose to an interleaving $(\varphi^{\prime})^{+\varepsilon}\circ\varphi\colon F^{\bullet}\simeq_{% \varepsilon+\varepsilon^{\prime}}J^{\bullet}\colon\psi^{+\varepsilon^{\prime}}% \circ\psi^{\prime}$ (see, for example, [17]). It follows that $d_{IH}(F^{\bullet},G^{\bullet}):=\inf\{\varepsilon\geq 0\mid F^{\bullet}\simeq% _{\varepsilon}G^{\bullet}\},$ defines an extended pseudo distance on $\mathcal{C}^{\mathbb{U}}$ called the interleaving distance in the homotopy category (see [24]). Observe, also, that any interleaving $\varphi\colon F^{\bullet}\simeq_{\varepsilon}G^{\bullet}\colon\psi$ gives rise to an interleaving $s\circ\varphi\colon F^{\bullet}\simeq_{\varepsilon^{\prime\prime}}G^{\bullet}% \colon s\circ\psi$ , for any $\varepsilon^{\prime\prime}\geq\varepsilon$ .

Example 30.

When $\mathbb{U}=\mathbb{R}^{0}=\{0\}$ , we can identify $\mathcal{C}^{\mathbb{U}}=\mathcal{C}$ , and $(-)+\varepsilon\colon\mathbb{U}\to\mathbb{U}$ is the identity. Then, an $\varepsilon$ -interleaving in $\textnormal{ho}(\mathcal{C})$ is the same as an isomorphism in $\textnormal{ho}(\mathcal{C})$ .

3 Stability of the bivariate function-Rips persistent homotopy type

To study the stability properties of the assignment $(\mathbb{M},\mathbb{f})\mapsto\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})$ , it is useful to consider the latter as a bivariate construction by also varying $\delta$ .

Notation 31.

Given a persistent object of the form $F\colon\mathbb{R}_{\geq 0}\times\mathbb{U}\to\textnormal{{C}}$ , we will use the notation $F^{\bullet,\bullet}$ to indicate the fact that it is a functor in two variables.

Definition 32 (See also [11]).

Let $(\mathbb{M}_{0},\mathbb{f}_{0})$ and $(\mathbb{M}_{1},\mathbb{f}_{1})$ be metric pairs over $\mathbb{R}^{N}$ , and let $\varepsilon\geq 0$ . A filtered $\varepsilon$ -correspondence between $(\mathbb{M}_{0},\mathbb{f}_{0})$ and $(\mathbb{M}_{1},\mathbb{f}_{1})$ is a subset $\mathfrak{C}\subset\mathbb{M}_{0}\times\mathbb{M}_{1}$ such that $\pi_{\mathbb{M}_{0}}(\mathfrak{C})=\mathbb{M}_{0}$ and $\pi_{\mathbb{M}_{1}}(\mathfrak{C})=\mathbb{M}_{1}$ , and such that, for all $(x,y),(x^{\prime},y^{\prime})\in\mathfrak{C}$ , it holds that

\displaystyle|d_{\mathbb{M}_{0}}(x,x^{\prime})-d_{\mathbb{M}_{1}}(y,y^{\prime}% )|\leq\varepsilon\textnormal{ and }|\mathbb{f}_{0}(x)-\mathbb{f}_{1}(y)|_{% \infty}\leq\varepsilon.

We will denote filtered $\varepsilon$ -correspondences in the form $\mathfrak{C}\colon(\mathbb{M}_{0},\mathbb{f}_{0})\approx_{\varepsilon}(\mathbb% {M}_{1},\mathbb{f}_{1})$ .

$\blacktriangleright$ Remark 33.

Filtered correspondences give rise to a distance on metric pairs $(\mathbb{M}_{0},\mathbb{f}_{0})$ and $(\mathbb{M}_{1},\mathbb{f}_{1})$ , defined as the infimum of the values $\varepsilon$ such that there exists a filtered $\varepsilon$ -correspondence $(\mathbb{M}_{0},\mathbb{f}_{0})\approx_{\varepsilon}(\mathbb{M}_{1},\mathbb{f}% _{1})$ . When $N=0$ , this distance is exactly twice the ordinary Gromov-Hausdorff distance.

The additional flexibility of varying the Rips parameter $\delta$ allows for a proof of the following stability result, homological variants of which can be found in several places ([12, 8]).

Proposition 34.

A filtered $\varepsilon$ -correspondence $\mathfrak{C}\colon(\mathbb{M}_{0},\mathbb{f}_{0})\approx_{\varepsilon}(\mathbb% {M}_{1},\mathbb{f}_{1})$ gives rise to an interleaving $\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet}_{0})\simeq_{\varepsilon}\mathcal{R}% ^{\bullet}(\mathbb{M}^{\bullet}_{1})$ in $\textnormal{ho}(\mathcal{S}\textnormal{{paces}}^{\mathbb{R}_{\geq 0}\times% \mathbb{R}^{N}})$ .

We use Recollections 26 and 24 and construct the interleaving using the simplicial set model for the bivariate Rips complex. We will leverage a universal property of the Rips simplicial set here, which we state for the case of filtered simplicial sets $X^{\bullet,\bullet}\in\textnormal{{sSet}}^{\mathbb{R}_{\geq 0}\times\mathbb{R}% ^{N}}$ , i.e., where all structure morphisms are inclusions (see Proposition 36 below.) It allows one to construct morphisms $X^{\bullet,\bullet}\to\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})$ by defining a map with target $\mathbb{M}$ on the vertices of $X^{\bullet,\bullet}$ (see [13] which uses the simplicial complex analog). We write $X^{\infty}=\bigcup_{(\delta,u)\in\mathbb{R}_{\geq 0}\times\mathbb{R}^{N}}X^{% \delta,u}$ .

Construction 35.

Let $X\colon\mathbb{R}_{\geq 0}\times\mathbb{R}^{N}\to\textnormal{{sSet}}$ be a filtered simplicial set and $(\mathbb{M},\mathbb{f})$ be a metric pair over $\mathbb{R}^{N}$ . We denote by $S(X,(\mathbb{M},\mathbb{f}))\subset\textnormal{{Set}}(X^{\infty}_{0},\mathbb{M})$ the set of such maps $\varphi\colon X^{\infty}_{0}\to\mathbb{M}$ that fulfill the condition that, for all $(\delta,u)\in\mathbb{R}_{\geq 0}\times\mathbb{R}^{N}$ , it holds that:

1.

for all $x\in X^{\delta,u}_{0}$ , we have $\mathbb{f}(\varphi(x))\leq u$ ;
2.

for all $1$ -simplices $\sigma\in X^{\delta,u}_{1}$ , with vertices $x_{0}$ and $x_{1}$ , we have $d(\varphi(x_{0}),\varphi(x_{1}))<\delta$ .

Associating to a morphism $X^{\bullet,\bullet}\to\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})$ the induced simplicial map $X^{\infty}\to\mathcal{R}^{\infty}(\mathbb{M},\mathbb{f})=\mathcal{R}^{\infty}(% \mathbb{M})$ and evaluating on vertices to obtain a map $X^{\infty}_{0}\to\mathcal{R}^{\infty}(\mathbb{M})_{0}=\mathbb{M}$ , defines a map

\eta\colon\textnormal{{sSet}}^{\mathbb{R}_{\geq 0}\times\mathbb{R}^{N}}(X^{% \bullet,\bullet},\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet}))\to S(X,(\mathbb{% M},\mathbb{f})).

Proposition 36 (See full version [31]).

$\eta$ , as in Construction 35, defines a natural bijection.

$\blacktriangleright$ Remark 37.

Conceptually speaking, the injectivity part of Proposition 36 means that a morphism $X^{\bullet,\bullet}\to\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})$ is uniquely determined by its values on vertices (the values can be identified with elements of $\mathbb{M}$ ). The surjectivity and well-definedness state that a map $X^{\infty}_{0}\to\mathbb{M}$ extends if and only if the criteria of Construction 35 are verified.

$\blacktriangleright$ Remark 38.

We will often need to use a modified version of Proposition 36, where instead of $\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})$ , we consider a reparametrized version in which $\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})\colon\mathbb{R}_{\geq 0}\times% \mathbb{R}^{N}\to\textnormal{{sSet}}$ is replaced by a composition $\mathbb{R}_{\geq 0}\times\mathbb{R}^{N}\xrightarrow{(\delta^{\prime},u^{\prime% })}\mathbb{R}_{\geq 0}\times\mathbb{R}^{N}\xrightarrow{\mathcal{R}^{\bullet}(% \mathbb{M}^{\bullet})}\textnormal{{sSet}}$ for poset maps $\delta^{\prime},u^{\prime}$ . Then, the analogous statement holds, replacing in Construction 35 $u$ with $u^{\prime}(\delta,u)$ and $\delta$ with $\delta^{\prime}(\delta,u)$ in the defining inequalities for $S(X^{\bullet,\bullet},(\mathbb{M},\mathbb{f}))$ . Furthermore, we will often encounter the modified case where we work with filtered simplicial sets $X^{\bullet,\bullet}$ over $\mathbb{I}\times\mathbb{R}^{N}$ , with $\mathbb{I}=[0,\rho]$ or $\mathbb{I}=\{\rho\}$ , and we restrict $\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})$ to $\mathbb{I}\times\mathbb{R}^{N}$ . In this case, again the essentially same statement holds, with the only change being the replacement of $X^{\infty}$ by $X^{\rho,\infty}=\bigcup_{u\in\mathbb{R}^{N}}X^{\rho,u}$ and the defining condition for $S(X^{\bullet,\bullet},(\mathbb{M},\mathbb{f}))$ needing to be verified only for pairs $(\delta,u)\in\mathbb{I}\times\mathbb{R}^{N}$ .

Next, let us construct the simplicial maps that define the interleaving of Proposition 34:

Construction 39.

Given a filtered $\varepsilon$ -correspondence $\mathfrak{C}$ , we define maps

\displaystyle\varphi\colon\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet}_{0})\to% \mathcal{R}^{\bullet+\varepsilon}(\mathbb{M}^{\bullet+\varepsilon}_{1})\quad{% \textnormal{ and }}\quad\psi\colon\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet}_{% 1})\to\mathcal{R}^{\bullet+\varepsilon}(\mathbb{M}^{\bullet+\varepsilon}_{0})

by applying Proposition 36 as follows. Observe that for $X^{\bullet,\bullet}=\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet}_{0})$ , we have $X^{\infty}_{0}=\mathbb{M}_{0}$ . Then $\varphi$ is defined under Proposition 36 by choosing for each $x\in\mathbb{M}_{0}$ an element $y\in\mathbb{M}_{1}$ with $(x,y)\in\mathfrak{C}$ and defining $\varphi(x):=y$ . The map $\psi$ is defined analogously.

Proof of Proposition 34.

The conditions on $\mathfrak{C}$ ensure exactly that the (shifted) requirements of Propositions 36 and 38 are fulfilled. It remains to see that $\varphi$ and $\psi$ define an $\varepsilon$ -interleaving in the homotopy category. We show that $\psi^{+\varepsilon}\circ\varphi\simeq s$ , the other equality being shown analogously. To this end, we need to construct a homotopy $H\colon\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet}_{0})\times\Delta^{1}\to% \mathcal{R}^{\bullet+2\varepsilon}(\mathbb{M}^{\bullet+2\varepsilon}_{0})$ between $\psi^{+\varepsilon}\circ\varphi$ and $s$ . We again use Proposition 36. In this case, it implies that the persistent simplicial maps $\psi^{+\varepsilon}\circ\varphi$ and $s$ are elementarily homotopic, if for every pair $x,y\in\mathbb{M}_{0}$ it holds that $d(\psi^{+\varepsilon}(\varphi(x)),s(y))\leq d(x,y)+2\varepsilon.$ Indeed, the inequality

d(\psi^{+\varepsilon}(\varphi(x)),s(y))=d(\psi^{+\varepsilon}(\varphi(x)),y)% \leq d(\varphi(x),\varphi(y))+\varepsilon\leq d(x,y)+2\varepsilon,

holds by definition of $\varphi$ and $\psi$ in terms of $\mathfrak{C}$ . $\hfill\blacktriangleleft$

4 The shrinking trick

For the remainder of this section, we will fix some homotopy theory $\mathcal{C}$ . By an interleaving, we will always mean interleaving in the homotopy category.

Notation 40.

When fixing either parameter of a persistent object $F^{\bullet,\bullet}\colon\mathbb{R}_{\geq 0}\times\mathbb{U}\to\mathcal{C}$ , we will use the notation $F^{\delta,\bullet}$ or $F^{\bullet,u}$ , to denote the resulting persistent objects on $\mathbb{U}$ , or $\mathbb{R}_{\geq 0}$ respectively. At times, we will also use the notation $\bullet_{1}$ and $\bullet_{2}$ to indicate the first and second variable, respectively. For example $F^{\bullet_{1},\bullet_{2}+{\bullet_{1}}}$ indicates the persistence module given by precomposing $F$ with the endofunctor $(\delta,u)\mapsto(\delta,u+\delta)$ of $\mathbb{R}_{\geq 0}\times\mathbb{U}$ .

Suppose we are given $\delta\geq 0$ , as well as two persistent objects $F^{\bullet,\bullet},G^{\bullet,\bullet}\colon\mathbb{R}_{\geq 0}\times\mathbb{% U}\to\mathcal{C}$ , together with an $\varepsilon$ -interleaving $\varphi\colon F^{\bullet,\bullet}\simeq_{\varepsilon}G^{\bullet,\bullet}\colon\psi$ . We want to deduce an interleaving $F^{\delta,\bullet}\simeq_{\varepsilon^{\prime}}G^{\delta,\bullet},$ for some $\varepsilon^{\prime}$ depending on $\varepsilon$ and $\delta$ . The issue at hand is, of course, that the interleaving morphisms $\varphi$ and $\psi$ only procure morphisms $F^{\delta,\bullet}\to G^{\delta+\varepsilon,\bullet+{\varepsilon}}\quad\text{ % and }\quad G^{\delta,\bullet}\to F^{\delta+\varepsilon,\bullet+{\varepsilon}}.$ To amend this, we introduce an additional structure on $G$ , which we call a shrinking transformation. This transformation allows us to decrease the $\delta$ -part of the persistence parameter of $G$ , at the cost of increasing the $u$ -part.

Notation 41.

We fix constants $0<S<1,C\geq 0$ and $\rho>0$ in $\mathbb{R}$ .

Definition 42.

A shrinking transformation for $G\colon\mathbb{R}_{\geq 0}\times\mathbb{U}\to\mathcal{C}$ is a morphism

\tau\colon G^{\bullet_{1},\bullet_{2}}|_{[0,\rho]\times\mathbb{U}}\to G^{S% \bullet_{1},\bullet_{2}+C\bullet_{1}}|_{[0,\rho]\times\mathbb{U}}

in $\textnormal{ho}(\mathcal{C}^{[0,\rho]\times\mathbb{U}})$ such that the following diagram in $\textnormal{ho}(\mathcal{C}^{[0,\rho]\times\mathbb{U}})$ commutes:

(2)

Definition 43.

Let $\delta\in[0,\rho]$ and $0\leq\varepsilon$ be such that $\varepsilon\leq\min\{\rho-\delta,\frac{1-S}{1+S}\delta\}.$ As we assumed $\varepsilon+\delta\leq\rho$ , the morphism $\tau$ from Definition 42 restricts to a well-defined morphism $\tau\colon G^{\delta+\varepsilon,\bullet+{\varepsilon}}\to G^{S(\delta+% \varepsilon),\bullet+{\varepsilon}+C(\delta+{\varepsilon})}$ , called by the same name by abuse of notation. We say that an $\varepsilon$ -interleaving $\varphi\colon F^{\bullet,\bullet}\simeq_{\varepsilon}G^{\bullet,\bullet}\colon\psi$ is $\delta$ -compatible with $\tau$ if the following diagram in $\textnormal{ho}(\mathcal{C}^{\mathbb{U}})$ commutes:

(3)

Theorem 44.

Let $G\colon\mathbb{R}_{\geq 0}\times\mathbb{U}\to\mathcal{C}$ be a persistent object in $\mathcal{C}$ , equipped with a shrinking transformation $\tau$ on $G$ . Let $\varepsilon,\delta\geq 0$ be such that $\varepsilon\leq\min\{\rho-\delta,\frac{1-S}{1+S}\delta\}$ .
Furthermore, let $F\colon\mathbb{R}_{\geq 0}\times\mathbb{U}\to\mathcal{C}$ be another persistent object, together with an $\varepsilon$ -interleaving $\varphi\colon F\simeq_{\varepsilon}G\colon\psi$ that is $\delta$ -compatible with $\tau$ . Then there is an interleaving

\varphi^{\prime}\colon F^{\delta,\bullet}\simeq_{(1+C)\varepsilon+C\delta}G^{% \delta,\bullet}\colon\psi^{\prime},

explicitly given by the compositions

	$\displaystyle\varphi^{\prime}\colon F^{\delta,\bullet}\xrightarrow{\varphi}G^{% \delta+\varepsilon,\,\bullet+{\varepsilon}}\xrightarrow{\tau}G^{S(\delta+% \varepsilon),\,\bullet+(1+C)\varepsilon+C\delta}\xrightarrow{s}G^{\delta,\,% \bullet+(1+C)\varepsilon+C\delta},$
	$\displaystyle\psi^{\prime}\colon G^{\delta,\bullet}\xrightarrow{\tau}G^{S% \delta,\,\bullet+C\delta}\xrightarrow{\psi}F^{S\delta+\varepsilon,\,\bullet+C% \delta+\varepsilon}\xrightarrow{s}F^{\delta,\,\bullet+(1+C)\varepsilon+C\delta}.$

Proof.

The fact that all morphisms $s$ and $\tau$ in the definition of $\varphi^{\prime}$ and $\psi^{\prime}$ are well-defined follows from the inequalities $\delta+\varepsilon\leq\rho$ and $S(\delta+\varepsilon)\leq S\delta+\varepsilon\leq S(\delta+\varepsilon)+% \varepsilon\leq\delta$ , which hold by the assumption on $\varepsilon$ and $\delta$ . We now need to verify the defining commutativity conditions of an interleaving. To simplify notation, we define $\varepsilon^{\prime}:=(1+C)\varepsilon+C\delta$ and $\bullet_{\gamma}:=\bullet_{2}+\varepsilon+C(\delta+\varepsilon)+CS(\delta+% \varepsilon).$ To simplify notation even further, we will omit $F$ and $G$ from the notation and only spell out the superscript. Whether $F$ or $G$ is meant will be uniquely determined by the specified morphisms. Now, consider the first composition $\psi^{\prime}\circ\varphi^{\prime}$ . To verify that $\psi^{\prime}\circ\varphi^{\prime}=s$ it suffices to verify the commutativity of the following diagram.

Observe that for every $s$ arrow to be well-defined, we require the inequality $S(\delta+\varepsilon)+\varepsilon\leq\delta$ , which we have already seen above. Observe that the canonical shift morphisms $s$ commute with essentially every other morphism in sight (in the appropriate contextual sense). The commutativities of the cells in the diagram follow either from the universal commutativity of $s$ or from Equations 2 and 3. This proves the first interleaving equality.
The second interleaving equality follows by chasing the following diagram.

Note that none of the cells require the $\delta$ -compatibility condition. Instead, one only uses the naturality properties of $s$ , one of the two interleaving equalities, and the defining property of the shrinking transformation. $\hfill\blacktriangleleft$

5 Shrinking transformations for the bivariate function-Rips complex

We now want to apply Theorem 44 together with Proposition 34 to prove Theorem 3.

Notation 45.

For the remainder of this section, let $M$ be a $\textnormal{CBA}(\kappa)$ space, for some $\kappa\in\mathbb{R}$ , equipped with a $1$ -Lipschitz function $f\colon M\to\mathbb{R}^{N}$ (with respect to the $|-|_{\infty}$ -norm on $\mathbb{R}^{N}$ ), and let $S, C$ and $\rho$ be as in Notation 41.

To establish the existence of a shrinking morphism, we use a classical technique in simplicial homotopy theory: Leveraging subdivisions to construct maps in the homotopy category.

Notation 46.

For the remainder of this section, we fix what we call a subdivision functor, by which we mean a pair consisting of a functor $\textnormal{sd}\colon\textnormal{{sSet}}_{\textnormal{Kan}}\to\textnormal{{% sSet}}_{\textnormal{Kan}}$ that preserves inclusions and colimits, together with a natural weak equivalence $\lambda\colon\textnormal{sd}\xRightarrow{\simeq}1_{\textnormal{{sSet}}}.$

Figure 3: Illustration of the last vertex map

\lambda_{b}

on

\textnormal{sd}_{b}\Delta^{2}

. The purple arrows indicate where the vertices are mapped by

\lambda_{b}

. All but the green simplex are collapsed to lower-dimensional simplices.

Example 47.

For the purpose of this article, the only relevant example is the barycentric subdivision functor $\textnormal{sd}_{b}\colon\textnormal{{sSet}}\to\textnormal{{sSet}}$ together with the last vertex map $\lambda_{b}\colon\textnormal{sd}_{b}X\xrightarrow{\simeq}X$ (see [22]). Let us recall the special case $X=\mathcal{R}^{\delta}(M)$ , which is the only one we will need. When $X=\mathcal{R}^{\delta}(M)$ , each vertex of $\textnormal{sd}_{b}\mathcal{R}^{\delta}(M)$ corresponds to a sequence $(x_{0},\dots,x_{n})$ of elements of $M$ that are less than $\delta$ apart from one another, and such that $x_{i}\neq x_{i+1}$ for all $i\in[n-1]$ . Then. $\lambda_{b}\colon\textnormal{sd}_{b}\mathcal{R}^{\delta}(M)\xrightarrow{\simeq% }\mathcal{R}^{\delta}(M)$ is given on vertices by mapping $(x_{0},\dots,x_{n})\in\textnormal{sd}_{b}(\mathcal{R}^{\delta}(M))_{0}$ to $x_{n}\in M=\mathcal{R}^{\delta}(M)_{0}$ .

Shrinking transformations will then be obtained from the following types of maps:

Definition 48.

By a pseudo-barycenter map, we mean a map $\Theta\colon\textnormal{sd}\mathcal{R}^{\rho}(M)_{0}\to M$ such that for all $0<\delta\leq\rho$ :

(a)

For every $x\in\textnormal{sd}\mathcal{R}^{\delta}(M)_{0}$ , we have $d(\lambda(x),\Theta(x))\leq C\delta$ ;
(b)

For $x,y\in\textnormal{sd}\mathcal{R}^{\delta}(M)_{0}$ , if there is a $1$ -simplex from $x$ to $y$ in $\textnormal{sd}\mathcal{R}^{\delta}(M)$ , then
$d(\Theta(x),\Theta(y))<S\delta$ and $d(\lambda(x),\Theta(y))<S\delta$ .

Figure 4: Illustration of the pseudo-barycenter map

\Theta

.

Example 49.

Let $\rho=\rho^{\kappa}_{M}$ . Consider the map $\Theta$ that maps a vertex $\sigma=(x_{0},\dots,x_{n})\in\textnormal{sd}_{b}(\mathcal{R}^{\rho}(M))$ to the center of a minimal enclosing ball $B$ of $\{x_{0},\dots,x_{n}\}\subset M$ - as illustrated in Figure 4. It follows from a geodesic version of Jung’s theorem (see [23, Lemma 3.4], and also [34] for a related construction) that for any two finite sets $\{x_{0},\dots,x_{n}\}\subset\{y_{0},\dots,y_{m}\}$ of diameter $\delta<\rho^{\kappa}_{M}$ , the associated centers $\Theta(x_{0},\dots,x_{n})$ and $\Theta(y_{0},\dots,y_{m})$ are well-defined and unique, and fulfill $d(\Theta(x_{0},\dots,x_{n}),\Theta(y_{0},\dots,y_{m}))<\frac{3}{4}\delta$ . As an immediate consequence, it follows that $\Theta$ defines a pseudo-barycenter map with constants $S=C=\frac{3}{4}$ .

Construction 50.

We use Recollections 24 and 26 and construct the shrinking transformation in the homotopy category of persistent simplicial sets. Since sd preserves colimits, we have $\textnormal{sd}(\mathcal{R}^{\bullet}(M^{\bullet}))^{\infty,\rho}_{0}=% \textnormal{sd}(\mathcal{R}^{\rho}(M))_{0}$ . Now, under this identification, we can apply Propositions 36 and 38 to uniquely extend $\Theta$ to a morphism $\tau^{\prime}\colon\textnormal{sd}(\mathcal{R}^{\bullet_{1}}(M^{\bullet_{2}}))% |_{[0,\rho]\times\mathbb{R}^{N}}\to\mathcal{R}^{S\bullet_{1}}(M^{\bullet_{2}+C% \bullet_{1}})|_{[0,\rho]\times\mathbb{R}^{N}}$ (see the proof of Theorem 51). Finally, in the persistent homotopy category $\textnormal{ho}(\textnormal{{sSet}}_{\textnormal{Kan}}^{[0,\rho]\times\mathbb{% R}^{N}})$ , we can invert the weak equivalence $\lambda\colon\textnormal{sd}\mathcal{R}^{\bullet}(M^{\bullet})\simeq\mathcal{R% }^{\bullet}(M^{\bullet})$ and define $\tau_{\Theta}$ as the composition:

\tau_{\Theta}\colon\mathcal{R}^{\bullet_{1}}(M^{\bullet_{2}})|_{[0,\rho]\times% \mathbb{R}^{N}}\xrightarrow{\lambda^{-1}}\textnormal{sd}(\mathcal{R}^{\bullet_% {1}}(M^{\bullet_{2}})|_{[0,\rho]\times\mathbb{R}^{N}})\xrightarrow{\tau^{% \prime}}\mathcal{R}^{S\bullet_{1}}(M^{\bullet_{2}+C\bullet_{1}})|_{[0,\rho]% \times\mathbb{R}^{N}}.

Using this construction, we can state the main result of this section.

Theorem 51.

The morphism $\tau_{\Theta}\colon\mathcal{R}^{\bullet_{1}}(M^{\bullet_{2}})|_{[0,\rho]\times% \mathbb{R}^{N}}\to\mathcal{R}^{S\bullet_{1}}(M^{\bullet_{2}+C\bullet_{1}})|_{[% 0,\rho]\times\mathbb{R}^{N}}$ is a shrinking transformation and $\delta$ -compatible with every $\varepsilon$ -interleaving $\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})\simeq_{\varepsilon}\mathcal{R}^{% \bullet}(M^{\bullet})$ arising from a correspondence $\mathfrak{C}\colon(\mathbb{M},\mathbb{f})\approx_{\varepsilon}(M,f)$ , for $\delta\leq\rho$ and $\varepsilon\leq\min\{\rho-\delta,\frac{1-S}{1+S}\delta\}$ .

Proof.

Throughout the proof, we are going to make use of the modified version of Proposition 36 in Remark 38. To simplify notation, we will write $\mathcal{R}^{\bullet,\bullet}$ for $\mathcal{R}^{\bullet}(M^{\bullet})|_{[0,\rho]\times\mathbb{R}^{N}}$ . We first show that $\tau^{\prime}$ , as defined in the previous construction, is indeed a well-defined morphism of persistent simplicial sets. Observe that compared to Proposition 36 the target persistent simplicial set has transformed under the parameter change $\bullet_{1}^{\prime}=S\bullet_{1}$ , $\bullet_{2}^{\prime}=\bullet_{2}+C\bullet_{1}.$ The conditions that we need to verify for $\tau^{\prime}$ to define a well-defined morphism (Remark 38) are explicitly given as follows. Let $\delta\in[0,\rho]$ and $u\in\mathbb{R}^{N}$ . We need to verify the following:

1.

Let $\sigma\in\textnormal{sd}\mathcal{R}^{\delta}(M)_{1}$ be a $1$ -simplex with vertices $x,y\in\textnormal{sd}\mathcal{R}^{\delta}(M)_{0}$ . Then the inequality $d(\Theta(x),\Theta(y))<S\delta$ holds. This condition is assumed by Property b.
2.

For $x\in\textnormal{sd}\mathcal{R}^{u,\delta}_{0}$ , it holds that $f(\Theta(x))\leq u+C\delta.$ This is a consequence of the inequalities

$f(\lambda(x))\leq u\quad\textnormal{and}\quad|f(\lambda(x))-f(\Theta(x))|_{% \infty}\leq d(\lambda(x),\Theta(x))\leq C\delta.$

The first of these holds as $\lambda(x)\in\mathcal{R}^{u,\delta}_{0}$ . The second holds by Property a, which implies that $d(\lambda(x),\Theta(x))\leq C\delta$ , together with the assumption that $f\colon M\to\mathbb{R}^{N}$ is $1$ -Lipschitz.

Next, let us verify the shrinking transformation condition. We need to show that the diagram

(4)

commutes. By precomposing with $\lambda$ , this is equivalent to showing that the diagram

(5)

commutes. In fact, it turns out that the two morphisms of persistent simplicial sets $s\circ\tau^{\prime}$ and $s\circ\lambda$ are elementarily homotopic. To see this, consider the map

\displaystyle H\colon(\textnormal{sd}\mathcal{R}^{\bullet_{1},\bullet_{2}}% \times\Delta^{1})_{0}=(\textnormal{sd}\mathcal{R}^{\bullet_{1},\bullet_{2}})_{% 0}\times\{0,1\}

\displaystyle\to M;\quad(x,i)\mapsto\begin{cases}\lambda(x),&i=0,\\ \Theta(x),&i=1,\end{cases}

and again apply Propositions 36 and 38. As we already know that $H_{0}$ and $H_{1}$ fulfill the conditions of Proposition 36, it only remains to verify the condition on $1$ -simplices $\sigma\in(\textnormal{sd}\mathcal{R}^{\bullet_{1},\bullet_{2}}\times\Delta^{1}% )_{1}$ in Proposition 36 in the case where the vertices $x$ and $y$ are in $\textnormal{sd}\mathcal{R}^{\delta}(M)_{0}\times\{0\}$ and $\textnormal{sd}\mathcal{R}^{\delta}(M)_{0}\times\{1\}$ respectively. Observe that, by the definition of the simplicial product, we can identify $\sigma$ with a simplex $\sigma^{\prime}\in\textnormal{sd}\mathcal{R}^{\delta}(M)_{1}$ , whose vertices we will also denote by $x$ and $y$ by abuse of notation. Then, the condition explicitly states that $d(\lambda(x),\Theta(y))<\delta$ which follows from Property b in the definition of a pseudo-barycenter map. This shows that $s\circ\tau^{\prime}$ and $s\circ\lambda$ are elementarily homotopic; thus, the shrinking transformation condition holds. It remains to show the compatibility with interleavings arising from a filtered correspondence. To this end, let $\mathfrak{C}$ be a filtered $\varepsilon$ -correspondence between $(M,f)$ and $(\mathbb{M},\mathbb{f})$ , and let $\varphi$ and $\psi$ be the induced $\varepsilon$ -interleaving maps. Furthermore, let $\varepsilon\leq\min{\frac{1-S}{1+S}\delta,\rho-\delta}$ . We write $\mathcal{S}^{\bullet_{1},\bullet_{2}}$ for $\mathcal{R}^{\bullet_{1}}(\mathbb{M}^{\bullet_{2}})|_{[0,\rho]\times\mathbb{R}% ^{N}}$ and $\mathcal{S}^{\bullet_{1}}$ for $\mathcal{R}^{\bullet_{1}}(\mathbb{M})$ . We now need to verify the commutativity of the outer diagram in

which we have marked in red. Observe that the left square commutes by the naturality of $\lambda$ , and that the lower left triangle commutes by definition. It thus suffices to show that the remaining cell to the right commutes. To this end, we show that the two morphisms of persistent simplicial sets $s\circ\psi\circ\tau^{\prime}\circ\textnormal{sd}\varphi$ and $s\circ\lambda$ are elementarily homotopic. To see this, we again use Propositions 36 and 38, but this time applied to $\textnormal{sd}\mathcal{S}^{\delta,\bullet_{2}}\times\Delta^{1}$ . Arguing exactly as above, we define a homotopy by extending the map

\displaystyle H^{\prime}\colon(\textnormal{sd}\mathcal{S}^{\delta,\infty}% \times\Delta^{1})_{0}=(\textnormal{sd}\mathcal{S}^{\delta})_{0}\times\{0,1\}

\displaystyle\to\mathbb{M};\quad(x,i)\mapsto\begin{cases}\lambda(x),&i=0.\\ (\psi\circ\tau^{\prime}\circ\textnormal{sd}\varphi)(x),&i=1;\end{cases}

By the same arguments as above, we can reduce to proving the following condition on $1$ -simplices $\sigma\in(\textnormal{sd}\mathcal{S}^{\delta,\bullet_{2}}\times\Delta^{1})_{1}$ in Proposition 36 in the case where the vertices $x$ and $y$ are in $\textnormal{sd}(\mathcal{S}^{\delta,\bullet_{2}})_{0}\times\{0\}$ and $\textnormal{sd}(\mathcal{S}^{\delta,\bullet_{2}})_{0}\times\{1\}$ , respectively: Given such a $\sigma$ , it holds that

d(\lambda(x),(\psi\circ\tau^{\prime}\circ\textnormal{sd}\varphi)(y))<\delta.

To see this, we first apply the defining property of a correspondence to obtain

d(\lambda(x),(\psi\circ\tau^{\prime}\circ\textnormal{sd}\varphi)(y))\leq% \varepsilon+d(\varphi(\lambda(x)),(\tau^{\prime}\circ\textnormal{sd}\varphi)(y% ))=\varepsilon+d(\varphi(\lambda(x)),\Theta((\textnormal{sd}\varphi)(y))).

By naturality of $\lambda$ , we have $\varphi(\lambda(x))=\lambda((\textnormal{sd}\varphi)(x))$ . Hence, we only need to show that $d(\lambda(\textnormal{sd}\varphi(x)),\Theta((\textnormal{sd}\varphi)(y)))<% \delta-\varepsilon.$ By assumption, there exists a $1$ -simplex $\sigma^{\prime}\in\textnormal{sd}\mathcal{S}^{\delta,\bullet_{2}}_{1}$ from $x$ to $y$ . Consequently, $\textnormal{sd}\varphi(\sigma^{\prime})\in\textnormal{sd}\mathcal{R}^{\delta+% \varepsilon,\bullet_{2}+\varepsilon}_{1}$ is a $1$ -simplex from $(\textnormal{sd}\varphi)(x)$ to $(\textnormal{sd}\varphi)(y)$ with $\delta+\varepsilon\leq\rho$ . By Property b, we thus have

d(\lambda(\textnormal{sd}(\varphi)(x)),\Theta((\textnormal{sd}\varphi)(y)))<S(% \delta+\varepsilon).

Finally, the assumption $\varepsilon\leq\frac{1-S}{1+S}\delta$ implies that $S(\delta+\varepsilon)\leq\delta-\varepsilon,$ which concludes the proof. $\hfill\blacktriangleleft$ We can now finally combine the main results of this article to prove Theorem 3.

Proof of Theorem 3.

By Proposition 34, the filtered correspondence $(\mathbb{M},\mathbb{f})\approx_{\varepsilon}(M,f)$ gives rise to an interleaving $\varphi\colon\mathcal{R}^{\bullet}(\mathbb{M}^{\bullet})\simeq_{\varepsilon}% \mathcal{R}^{\bullet}(M^{\bullet})\colon\psi$ in the homotopy category. By Example 49, $(M,f)$ admits a pseudo-barycenter map $\Theta$ with respect to $\rho=\rho^{\kappa}_{M}$ and $C=S=\frac{3}{4}$ . Observe that $\frac{1-S}{1+S}=\frac{1}{7}$ . By Theorem 51, $\Theta$ induces a shrinking transformation $\tau$ compatible with $\varphi$ and $\psi$ . Applying Theorem 44, we obtain an interleaving $\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})\simeq_{\frac{7}{4}\varepsilon+\frac% {3}{4}\delta}\mathcal{R}^{\delta}(M^{\bullet})$ . Finally, since $\varepsilon\leq\frac{1}{7}\delta$ we have $\frac{7}{4}\varepsilon+\frac{3}{4}\delta\leq\delta$ , and we obtain an interleaving $\mathcal{R}^{\delta}(\mathbb{M}^{\bullet})\simeq_{\delta}\mathcal{R}^{\delta}(% M^{\bullet})$ . $\hfill\blacktriangleleft$

References

[1] Michał Adamaszek and Henry Adams. The Vietoris–Rips complexes of a circle. Pacific Journal of Mathematics, 290(1):1–40, 2017.
[2] Michal Adamaszek, Henry Adams, and Florian Frick. Metric reconstruction via optimal transport. SIAM Journal on Applied Algebra and Geometry, 2(4):597–619, 2018. doi:10.1137/17M1148025.
[3] Michał Adamaszek, Henry Adams, and Samadwara Reddy. On Vietoris–Rips complexes of ellipses. Journal of Topology and Analysis, 11(03):661–690, 2019.
[4] Ethan André, Jingyi Li, David Loiseaux, and Steve Y. Oudot. Estimating the persistent homology of $\mathbb{r}^{n}$ -valued functions using function-geometric multifiltrations. arXiv preprint, 2024. arXiv:2412.04162.
[5] Omar Antolín-Camarena. Turning simplicial complexes into simplicial sets. unpublished notes, available at https://www.matem.unam.mx/˜omar/notes/ssets-from-complexes.html. accessed 26 July 2021.
[6] Clark Barwick and Daniel M. Kan. Relative categories: another model for the homotopy theory of homotopy theories. Indagationes Mathematicae, 23(1–2):42–68, 2012. doi:10.1016/j.indag.2011.10.002.
[7] Julia E. Bergner. A survey of $(\infty,1)$ -categories. In John C. Baez and J. Peter May, editors, Towards Higher Categories, volume 152 of IMA Volumes in Mathematics and its Applications, pages 69–83. Springer, 2009. doi:10.1007/978-1-4419-1524-5_2.
[8] Andrew J. Blumberg and Michael Lesnick. Stability of 2-parameter persistent homology. Foundations of Computational Mathematics, 24(2):385–427, 2024. doi:10.1007/S10208-022-09576-6.
[9] Martin R. Bridson and André Haefliger. Metric Spaces of Non-Positive Curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 1999. doi:10.1007/978-3-662-12494-9.
[10] Mickaël Buchet, Frédéric Chazal, Tamal K. Dey, Fengtao Fan, Steve Y. Oudot, and Yusu Wang. Topological analysis of scalar fields with outliers. In 31st International Symposium on Computational Geometry (SoCG 2015), volume 34 of Leibniz International Proceedings in Informatics (LIPIcs), pages 827–841. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2015. doi:10.4230/LIPIcs.SOCG.2015.827.
[11] Gunnar Carlsson and Facundo Mémoli. Multiparameter hierarchical clustering methods. In Hermann Locarek-Junge and Claus Weihs, editors, Classification as a Tool for Research, Studies in Classification, Data Analysis, and Knowledge Organization, pages 63–70. Springer, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-10745-0_6.
[12] Frédéric Chazal, David Cohen-Steiner, Leonidas J. Guibas, Facundo Mémoli, and Steve Y. Oudot. Gromov-hausdorff stable signatures for shapes using persistence. Computer Graphics Forum, 28(5):1393–1403, 2009. doi:10.1111/j.1467-8659.2009.01516.x.
[13] Frédéric Chazal, Vin de Silva, and Steve Y. Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193–214, 2014. doi:10.1007/s10711-013-9937-z.
[14] Frédéric Chazal, Leonidas J. Guibas, Steve Y. Oudot, and Primož Skraba. Scalar field analysis over point cloud data. Discrete & Computational Geometry, 46(4):743–775, 2011. doi:10.1007/s00454-011-9360-x.
[15] Frédéric Chazal, Leonidas J. Guibas, Steve Y. Oudot, and Primoz Skraba. Persistence-based clustering in Riemannian manifolds. Journal of the ACM, 60(6):41:1–41:38, 2013. doi:10.1145/2535927.
[16] Denis-Charles Cisinski. Higher Categories and Homotopical Algebra. Number 180 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2019.
[17] Vin de Silva, Elizabeth Munch, and Anastasios Stefanou. Theory of interleavings on categories with a flow. Theory and Applications of Categories, 33(21):583–607, 2018.
[18] Oliver Gäfvert. Topological and geometrical methods in data analysis. PhD thesis, KTH Royal Institute of Technology, 2021.
[19] Patrick Gillespie. Vietoris thickenings and complexes are weakly homotopy equivalent. Journal of Applied and Computational Topology, 8(1):35–53, 2024. doi:10.1007/s41468-023-00135-8.
[20] Paul G. Goerss and John F. Jardine. Simplicial Homotopy Theory, volume 174 of Progress in Mathematics. Birkhäuser, Basel, reprint of the 1999 original edition, 2009. doi:10.1007/978-3-0346-0189-4.
[21] John F. Jardine. Persistent homotopy theory. arXiv preprint arXiv:2002.10013, 2020. Revised version dated October 26, 2020. arXiv:2002.10013.
[22] Daniel M. Kan. On c.s.s. complexes. American Journal of Mathematics, 79(3):449–476, 1957. doi:10.2307/2372450.
[23] Rafal Komendarczyk, Sushovan Majhi, and Will Tran. Topological stability and Latschev-type reconstruction theorems for spaces of curvature bounded above. arXiv preprint arXiv:2406.04259, 2024. arXiv:2406.04259.
[24] Edoardo Lanari and Luis Scoccola. Rectification of interleavings and a persistent Whitehead theorem. Algebraic & Geometric Topology, 23(2):803–832, 2023. doi:10.2140/agt.2023.23.803.
[25] Janko Latschev. Vietoris–Rips complexes of metric spaces near a closed Riemannian manifold. Archiv der Mathematik, 77(6):522–528, 2001. doi:10.1007/PL00000526.
[26] Boštjan Lemež and Žiga Virk. Finite reconstruction with selective Rips complexes. arXiv preprint arXiv:2205.05525, 2022. arXiv:2205.05525.
[27] Sunhyuk Lim, Facundo Mémoli, and Osman Berat Okutan. Vietoris–Rips persistent homology, injective metric spaces, and the filling radius. Algebraic & Geometric Topology, 24(2):1019–1100, 2024. doi:10.2140/agt.2024.24.1019.
[28] Jacob Lurie. Higher Topos Theory, volume 170 of Annals of Mathematics Studies. Princeton University Press, 2009. doi:10.1515/9781400830558.
[29] Sushovan Majhi. Demystifying Latschev’s theorem: Manifold reconstruction from noisy data. Discrete & Computational Geometry, 74(2):544–568, 2025. doi:10.1007/s00454-024-00655-9.
[30] Nina Otter. Magnitude meets persistence. Homology theories for filtered simplicial sets. Homology, Homotopy and Applications, 24(2):365–387, 2022. doi:10.4310/HHA.2022.v24.n2.a12.
[31] Steve Oudot and Lukas Waas. Function-Rips complexes in persistent homotopy theory: Stability and Latschev theorems. arXiv preprint arXiv:2603.23460, 2026. arXiv:2603.23460.
[32] Daniel G. Quillen. Homotopical Algebra, volume 43 of Lecture Notes in Mathematics. Springer-Verlag, Berlin – New York, 1967. doi:10.1007/BFb0097438.
[33] Frank Olaf Schreyer. A standard basis approach to syzygies of canonical curves. Journal für die reine und angewandte Mathematik, 421:83–124, 1991.
[34] Matthew C. B. Zaremsky. Contractible Vietoris–Rips complexes of $\mathbb{z}^{n}$ . Proceedings of the American Mathematical Society, 154(2):503–508, 2026.

[bib.bib1] [1] Michał Adamaszek and Henry Adams. The Vietoris–Rips complexes of a circle. Pacific Journal of Mathematics, 290(1):1–40, 2017.

[bib.bib2] [2] Michal Adamaszek, Henry Adams, and Florian Frick. Metric reconstruction via optimal transport. SIAM Journal on Applied Algebra and Geometry, 2(4):597–619, 2018. doi:10.1137/17M1148025.

[bib.bib3] [3] Michał Adamaszek, Henry Adams, and Samadwara Reddy. On Vietoris–Rips complexes of ellipses. Journal of Topology and Analysis, 11(03):661–690, 2019.

[bib.bib4] [4] Ethan André, Jingyi Li, David Loiseaux, and Steve Y. Oudot. Estimating the persistent homology of $\mathbb{r}^{n}$ -valued functions using function-geometric multifiltrations. arXiv preprint, 2024. arXiv:2412.04162.

[bib.bib5] [5] Omar Antolín-Camarena. Turning simplicial complexes into simplicial sets. unpublished notes, available at https://www.matem.unam.mx/˜omar/notes/ssets-from-complexes.html. accessed 26 July 2021.

[bib.bib6] [6] Clark Barwick and Daniel M. Kan. Relative categories: another model for the homotopy theory of homotopy theories. Indagationes Mathematicae, 23(1–2):42–68, 2012. doi:10.1016/j.indag.2011.10.002.

[bib.bib7] [7] Julia E. Bergner. A survey of $(\infty,1)$ -categories. In John C. Baez and J. Peter May, editors, Towards Higher Categories, volume 152 of IMA Volumes in Mathematics and its Applications, pages 69–83. Springer, 2009. doi:10.1007/978-1-4419-1524-5_2.

[bib.bib8] [8] Andrew J. Blumberg and Michael Lesnick. Stability of 2-parameter persistent homology. Foundations of Computational Mathematics, 24(2):385–427, 2024. doi:10.1007/S10208-022-09576-6.

[bib.bib9] [9] Martin R. Bridson and André Haefliger. Metric Spaces of Non-Positive Curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin, 1999. doi:10.1007/978-3-662-12494-9.

[bib.bib10] [10] Mickaël Buchet, Frédéric Chazal, Tamal K. Dey, Fengtao Fan, Steve Y. Oudot, and Yusu Wang. Topological analysis of scalar fields with outliers. In 31st International Symposium on Computational Geometry (SoCG 2015), volume 34 of Leibniz International Proceedings in Informatics (LIPIcs), pages 827–841. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2015. doi:10.4230/LIPIcs.SOCG.2015.827.

[bib.bib11] [11] Gunnar Carlsson and Facundo Mémoli. Multiparameter hierarchical clustering methods. In Hermann Locarek-Junge and Claus Weihs, editors, Classification as a Tool for Research, Studies in Classification, Data Analysis, and Knowledge Organization, pages 63–70. Springer, Berlin, Heidelberg, 2010. doi:10.1007/978-3-642-10745-0_6.

[bib.bib12] [12] Frédéric Chazal, David Cohen-Steiner, Leonidas J. Guibas, Facundo Mémoli, and Steve Y. Oudot. Gromov-hausdorff stable signatures for shapes using persistence. Computer Graphics Forum, 28(5):1393–1403, 2009. doi:10.1111/j.1467-8659.2009.01516.x.

[bib.bib13] [13] Frédéric Chazal, Vin de Silva, and Steve Y. Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193–214, 2014. doi:10.1007/s10711-013-9937-z.

[bib.bib14] [14] Frédéric Chazal, Leonidas J. Guibas, Steve Y. Oudot, and Primož Skraba. Scalar field analysis over point cloud data. Discrete & Computational Geometry, 46(4):743–775, 2011. doi:10.1007/s00454-011-9360-x.

[bib.bib15] [15] Frédéric Chazal, Leonidas J. Guibas, Steve Y. Oudot, and Primoz Skraba. Persistence-based clustering in Riemannian manifolds. Journal of the ACM, 60(6):41:1–41:38, 2013. doi:10.1145/2535927.

[bib.bib16] [16] Denis-Charles Cisinski. Higher Categories and Homotopical Algebra. Number 180 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2019.

[bib.bib17] [17] Vin de Silva, Elizabeth Munch, and Anastasios Stefanou. Theory of interleavings on categories with a flow. Theory and Applications of Categories, 33(21):583–607, 2018.

[bib.bib18] [18] Oliver Gäfvert. Topological and geometrical methods in data analysis. PhD thesis, KTH Royal Institute of Technology, 2021.

[bib.bib19] [19] Patrick Gillespie. Vietoris thickenings and complexes are weakly homotopy equivalent. Journal of Applied and Computational Topology, 8(1):35–53, 2024. doi:10.1007/s41468-023-00135-8.

[bib.bib20] [20] Paul G. Goerss and John F. Jardine. Simplicial Homotopy Theory, volume 174 of Progress in Mathematics. Birkhäuser, Basel, reprint of the 1999 original edition, 2009. doi:10.1007/978-3-0346-0189-4.

[bib.bib21] [21] John F. Jardine. Persistent homotopy theory. arXiv preprint arXiv:2002.10013, 2020. Revised version dated October 26, 2020. arXiv:2002.10013.

[bib.bib22] [22] Daniel M. Kan. On c.s.s. complexes. American Journal of Mathematics, 79(3):449–476, 1957. doi:10.2307/2372450.

[bib.bib23] [23] Rafal Komendarczyk, Sushovan Majhi, and Will Tran. Topological stability and Latschev-type reconstruction theorems for spaces of curvature bounded above. arXiv preprint arXiv:2406.04259, 2024. arXiv:2406.04259.

[bib.bib24] [24] Edoardo Lanari and Luis Scoccola. Rectification of interleavings and a persistent Whitehead theorem. Algebraic & Geometric Topology, 23(2):803–832, 2023. doi:10.2140/agt.2023.23.803.

[bib.bib25] [25] Janko Latschev. Vietoris–Rips complexes of metric spaces near a closed Riemannian manifold. Archiv der Mathematik, 77(6):522–528, 2001. doi:10.1007/PL00000526.

[bib.bib26] [26] Boštjan Lemež and Žiga Virk. Finite reconstruction with selective Rips complexes. arXiv preprint arXiv:2205.05525, 2022. arXiv:2205.05525.

[bib.bib27] [27] Sunhyuk Lim, Facundo Mémoli, and Osman Berat Okutan. Vietoris–Rips persistent homology, injective metric spaces, and the filling radius. Algebraic & Geometric Topology, 24(2):1019–1100, 2024. doi:10.2140/agt.2024.24.1019.

[bib.bib28] [28] Jacob Lurie. Higher Topos Theory, volume 170 of Annals of Mathematics Studies. Princeton University Press, 2009. doi:10.1515/9781400830558.

[bib.bib29] [29] Sushovan Majhi. Demystifying Latschev’s theorem: Manifold reconstruction from noisy data. Discrete & Computational Geometry, 74(2):544–568, 2025. doi:10.1007/s00454-024-00655-9.

[bib.bib30] [30] Nina Otter. Magnitude meets persistence. Homology theories for filtered simplicial sets. Homology, Homotopy and Applications, 24(2):365–387, 2022. doi:10.4310/HHA.2022.v24.n2.a12.

[bib.bib31] [31] Steve Oudot and Lukas Waas. Function-Rips complexes in persistent homotopy theory: Stability and Latschev theorems. arXiv preprint arXiv:2603.23460, 2026. arXiv:2603.23460.

[bib.bib32] [32] Daniel G. Quillen. Homotopical Algebra, volume 43 of Lecture Notes in Mathematics. Springer-Verlag, Berlin – New York, 1967. doi:10.1007/BFb0097438.

[bib.bib33] [33] Frank Olaf Schreyer. A standard basis approach to syzygies of canonical curves. Journal für die reine und angewandte Mathematik, 421:83–124, 1991.

[bib.bib34] [34] Matthew C. B. Zaremsky. Contractible Vietoris–Rips complexes of $\mathbb{z}^{n}$ . Proceedings of the American Mathematical Society, 154(2):503–508, 2026.

A Persistent Version of Latschev’s Theorem

Abstract

Keywords and phrases:

Funding:

Copyright and License:

2012 ACM Subject Classification:

Related Version:

Acknowledgements:

DOI:

Event:

Editors:

Series and Publisher:

1 Introduction

Theorem (Latschev [25]).

Theorem 1 (Persistent Latschev’s theorem).

Theorem ([4]).

Theorem 2 (Persistent Hausmann’s Theorem).

Theorem 3 (Approximation of function-Rips persistent homotopy type).

2 Background

2.1 Spaces of curvature bounded above

Recollection 4 (Geodesics and convexity).

Recollection 5 (Curvature bounded above).

Notation 6.

2.2 Models for persistent homotopy types

Notation 7.

Definition 8.

Notation 9.

Notation 10.

Example 11.

Example 12 (function-Rips complex).

2.3 Persistent homotopy theory

Definition 13.

Example 14.

Notation 15.

Example 16.

Notation 17.

Recollection 18.

Notation 19.

2.4 Persistent simplicial sets

Recollection 20.

Example 21.

Notation 22.

Recollection 23.

Recollection 24.

Recollection 25.

Recollection 26.

2.5 Interleavings in the homotopy category

Notation 27.

Definition 28.

▶ Remark 29.

Example 30.

3 Stability of the bivariate function-Rips persistent homotopy type

Notation 31.

Definition 32 (See also [11]).

▶ Remark 33.

Proposition 34.

Construction 35.

Proposition 36 (See full version [31]).

▶ Remark 37.

▶ Remark 38.

Construction 39.

Proof of Proposition 34.

4 The shrinking trick

Notation 40.

Notation 41.

Definition 42.

Definition 43.

Theorem 44.

Proof.

5 Shrinking transformations for the bivariate function-Rips complex

Notation 45.

Notation 46.

Example 47.

Definition 48.

Example 49.

Construction 50.

Theorem 51.

Proof.

Proof of Theorem 3.

References

$\blacktriangleright$ Remark 29.

$\blacktriangleright$ Remark 33.

$\blacktriangleright$ Remark 37.

$\blacktriangleright$ Remark 38.