2.1 Sheaf-function correspondence

Throughout this paper, $\mathbf {k}$ denotes a field. For a topological space X, we denote by $\mathrm {Mod}(\mathbf {k}_X)$ the category of sheaves of $\mathbf {k}$ -vector spaces on X, and $\mathrm {D}^b(\mathbf {k}_X)$ its bounded derived category. Let M be a real analytic manifold. The definitions and results of this section are exposed in detail in [Reference Kashiwara, Schapira and Houzel25, Chapters 8 & 9.7].

We denote by $\mathrm {Mod}_{\mathbb {R} c}(\mathbf {k}_M)$ the full subcategory of $\mathrm {Mod}(\mathbf {k}_M)$ consisting of constructible sheaves and by $\mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_M)$ the full subcategory of $\mathrm {D}^b(\mathbf {k}_M)$ whose objects are sheaves $F \in \mathrm {D}^b(\mathbf {k}_M)$ such that $\mathrm {H}^j(F) \in \mathrm {Mod}_{\mathbb {R} c}(\mathbf {k}_M)$ for all $j\in \mathbb {Z}$ . It is well known [Reference Kashiwara, Schapira and Houzel25, Th. 8.4.5] that the functor $ \mathrm {D}^b(\mathrm {Mod}_{\mathbb {R} c}(\mathbf {k}_M))\longrightarrow \mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_M) $ is an equivalence. The objects of $\mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_M)$ are still called constructible sheaves.

We denote by $\mathrm {CF}(M)$ the group of constructible functions on M. All the remaining results of this section are contained in [Reference Kashiwara, Schapira and Houzel25, Chapter 9.7].

To any constructible sheaf $F \in \mathrm {D}^b_{\mathbb {R} c }(\mathbf {k}_M)$ , it is possible to associate a constructible function $\chi (F) \in \mathrm {CF}(M)$ , called the local Euler characteristic of F, and defined by:

$$\begin{align*}\chi(F) (x) = \chi(F_x) = \sum_{i\in \mathbb{Z}} (-1)^i \mathrm{dim}_{\mathbf{k}}(\mathrm{H}^i(F)_x).\end{align*}$$

It is clear that for any distinguished triangle $F' \longrightarrow F \longrightarrow F" \stackrel {+1}{\longrightarrow }$ in $\mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_M)$ , one has $\chi (F) = \chi (F') + \chi (F")$ . Therefore, $\chi $ factorizes through the Grothendieck group $\mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_M))$ and there is a well-defined morphism of groups $\mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_M)) \longrightarrow \mathrm {CF}(M)$ mapping $[F]$ to $\chi (F)$ .

Lemma 2.8. Let $F \in \mathrm {D}^b_{\mathbb {R} c }(\mathbf {k}_M)$ with compact support, then

$$\begin{align*}\int \chi(F) ~\mathrm{d} \chi = \chi\left(\mathrm{R} \Gamma(M; F)\right) = \sum_{i\in \mathbb{Z}}(-1)^i \mathrm{dim}_{\mathbf{k}} \left(\mathrm{H}^i(M;F)\right). \end{align*}$$

We briefly review the construction of the direct image operation for constructible functions. Let $f : X \longrightarrow Y$ be a morphism of real analytic manifolds and $\varphi \in \mathrm {CF}(X)$ such that f is proper on $\mathrm {supp}(\varphi )$ . Then, for each $y\in Y$ , $\varphi \cdot 1_{f^{-1}(y)}$ is constructible and has compact support.

Definition 2.9. Keeping the above notations, one defines the function $f_\ast \varphi : Y \longrightarrow \mathbb {Z}$ by

$$\begin{align*}(f_\ast \varphi)(y) := \int \varphi \cdot 1_{f^{-1}(y)} ~\mathrm{d} \chi.\end{align*}$$

2.2 Convolution distance

We consider a finite-dimensional real vector space $\mathbb {V}$ endowed with a norm $\|\cdot \|$ . We equip $\mathbb {V}$ with the usual topology. Following [Reference Kashiwara and Schapira27], we briefly present the convolution distance, which is inspired from the interleaving distance between persistence modules [Reference Chazal, Cohen-Steiner, Glisse, Guibas and Oudot17]. We introduce the following notations:

$$ \begin{align*} s : \mathbb{V} \times \mathbb{V} \longrightarrow \mathbb{V}, ~~~s(x,y) = x + y \end{align*} $$

$$ \begin{align*} p_i : \mathbb{V} \times \mathbb{V} \longrightarrow \mathbb{V} ~~(i=1,2), ~~~p_1(x,y) = x,~p_2(x,y) = y. \end{align*} $$

The convolution bifunctor $- \star - \colon \mathrm {D}^b(\mathbf {k}_{\mathbb {V}})\times \mathrm {D}^b(\mathbf {k}_{\mathbb {V}}) \longrightarrow \mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ is defined as follows. For $F, \; G \in \mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ , we set

$$ \begin{align*} F \star G := \mathrm{R} s_! (F \boxtimes G). \end{align*} $$

For $r \geq 0$ and $x\in \mathbb {V}$ , let $B(x,r) = \{v \in \mathbb {V} \mid \|x-v\| \leq r\}$ . For $\varepsilon \in \mathbb {R}$ , we set

$$\begin{align*}K_{\varepsilon} := \begin{cases} \mathbf{k}_{B(0,\varepsilon)} \mathrm{~~if~~} \varepsilon \geq 0, \\ \mathbf{k}_{\mathrm{Int}(B(0,-\varepsilon))}[\mathrm{dim}(\mathbb{V})] \mathrm{~~if~~} \varepsilon < 0 \end{cases} \in \mathrm{D}^b(\mathbf{k}_{\mathbb{V}}).\end{align*}$$

The following proposition is proved in [Reference Kashiwara and Schapira27].

Proposition 2.12. Let $\varepsilon , \varepsilon '\in \mathbb {R}$ and $F \in \mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ . There are isomorphisms, functorial in F:

$$ \begin{align*} K_{\varepsilon} \star (K_{\varepsilon'} \star F) \simeq ( K_{\varepsilon} \star K_{\varepsilon'}) \star F \simeq K_{\varepsilon + \varepsilon'} \star F ~~~ and ~~~ K_0 \star F\simeq F. \end{align*} $$

If $\varepsilon \geq \varepsilon ' \geq 0 $ , there is a canonical morphism $\chi _{\varepsilon , \varepsilon '} \colon K_{\varepsilon }\longrightarrow K_{\varepsilon '}$ in $\mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ . It induces a canonical morphism $\chi _{\varepsilon , \varepsilon '} \star F \colon K_{\varepsilon } \star F \longrightarrow K_{\varepsilon '} \star F $ . In particular when $\varepsilon ' = 0$ , we get

(2.1) $$ \begin{align} \chi_{\varepsilon,0} \star F \colon K_{\varepsilon} \star F \longrightarrow F. \end{align} $$

Following [Reference Kashiwara and Schapira27], we recall the notion of $\varepsilon $ -isomorphic sheaves.

Definition 2.13. Let $F,G \in \mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ , and let $\varepsilon \geq 0$ . The sheaves F and G are $\varepsilon $ -isomorphic if there are morphisms $f : K_{\varepsilon } \star F \longrightarrow G$ and $g : K_{\varepsilon } \star G \longrightarrow F$ such that the diagrams

are commutative. The pair of morphisms $(f,g)$ is called a pair of $\varepsilon $ -isomorphisms.

Definition 2.14. For $F,G \in \mathrm {D}^b(\mathbf {k}_{\mathbb {V}}) $ , their convolution distance is

$$ \begin{align*} d_C(F,G) := \inf(\{\varepsilon \geq 0 \mid F ~\mathrm{and}~G~\mathrm{are}~\varepsilon -\mathrm{isomorphic}\} \cup \{ \infty \}). \end{align*} $$

It is proved in [Reference Kashiwara and Schapira27] that the convolution is, indeed, a pseudo-extended metric, that is, it satisfies the triangular inequality. Having isomorphic global sections is a necessary condition for two sheaves to be at finite convolution distance, as expressed in the following proposition, which can be found as [Reference Kashiwara and Schapira27, Remark 2.5 (i)].

Proposition 2.16. Let $F,G \in \mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ such that $d_C(F,G) < +\infty $ . Then

$$\begin{align*}\mathrm{R} \Gamma (\mathbb{V}; F) \simeq \mathrm{R} \Gamma (\mathbb{V};G).\end{align*}$$

Moreover, it satisfies the following important stability property.

Theorem 2.17. Let $u,v : X \longrightarrow \mathbb {V}$ be continuous maps, and let $F \in \mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ . Then,

$$\begin{align*}d_C(\mathrm{R} u_\ast F, \mathrm{R} v_\ast F) \leq \sup_{x \in X} \|u(x) - v(x)\|.\end{align*}$$

We will often make use of the following result, that we call the additivity of interleavings, which is a direct consequence of the additivity of the convolution functor.

Proposition 2.18 (Additivity of interleavings).

Let $(F_i)_{i \in I}$ and $(G_j)_{j\in J}$ be two finite families of $\mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ . For all $I' \subseteq I$ and $J' \subseteq J$ of the same cardinality (eventually empty) and for all bijections $\sigma : I' \longrightarrow J'$ , one has

$$\begin{align*}d_C(\oplus_{i\in i} F_i, \oplus_{j\in J} G_j) \leq \max \left ( \max_{i \in I'} d_C(F_i, G_{\sigma(i)}), \max_{i\in I\backslash I'}d_C(F_i,0), \max_{j\in J\backslash J'}d_C(G_j,0)) \right).\end{align*}$$

Proof. Let $I' \subseteq I$ and $J' \subseteq J$ of the same cardinality (eventually empty), and $\sigma : I' \longrightarrow J'$ a bijection. We set

$$\begin{align*}M = \max \left ( \max_{i \in I'} d_C(F_i, G_{\sigma(i)}), \max_{i\in I\backslash I'}d_C(F_i,0), \max_{j\in J\backslash J'}d_C(G_j,0)) \right).\end{align*}$$

If $M = + \infty $ , the inequality is true. Let us now assume that $M < + \infty $ . Let $\varepsilon> M$ . Then for all $i \in I \backslash I'$ , $F_i$ is $\varepsilon $ -interleaved with $0$ , so the canonical map $F_i \star K_{2\varepsilon } \longrightarrow F_i$ is zero. Similarly, for all $j \in J\backslash J'$ , $G_j$ is $\varepsilon $ -interleaved with $0$ , so the canonical map $G_j \star K_{2\varepsilon } \longrightarrow G_j$ is zero. Moreover, for all $i \in I'$ , there exists a pair of $\varepsilon $ -interleavings morphisms $f_i : F_i \star K_{\varepsilon } \longrightarrow G_{\sigma (i)}$ and $g_i : G_{\sigma (i)} \star K_{\varepsilon } \longrightarrow F_i$ .

Since $(\oplus _{i \in I} F_i) \star K_{\varepsilon } \simeq \oplus _{i \in I} (F_i \star K_{\varepsilon })$ and $(\oplus _{j \in J} G_j) \star K_{\varepsilon } \simeq \oplus _{j \in J} (G_j \star K_{\varepsilon })$ , and these direct sums are finite, we can define $f : \oplus _{i \in I} F_i \star K_{\varepsilon } \longrightarrow \oplus _{j \in J} G_{j}$ and $g : \oplus _{j \in J} G_j \star K_{\varepsilon } \longrightarrow \oplus _{i \in I} F_i$ uniquely by specifying $f_{ij} \in \mathrm {Hom}(F_i \star K_{\varepsilon }, G_j)$ and $g_{ji} \in \mathrm {Hom}(G_j \star K_{\varepsilon }, F_i)$ . We set, for all $(i,j) \in I \times J$

$$\begin{align*}f_{ij} = \begin{cases} f_i ~ \mathrm{if} ~ i \in I' ~ \mathrm{and} ~ j = \sigma(i) \\ 0 ~ \mathrm{else} \end{cases} \mathrm{and} \quad g_{ji} = \begin{cases} g_i ~ \mathrm{if} ~ i \in I' ~ \mathrm{and} ~ j = \sigma(i) \\ 0 ~ \mathrm{else} \end{cases}. \end{align*}$$

Let us verify that $(f,g)$ is an $\varepsilon $ -interleaving pair between $\oplus _{i\in i} F_i$ and $\oplus _{j\in J} G_j$ . For all $(i,l) \in I^2$ , one has

$$\begin{align*}\left (g \circ \left (f \star K_{\varepsilon} \right ) \right)_{il} = \sum_{j \in J} g_{jl} \circ (f_{ij} \star K_{\varepsilon}) = \begin{cases} \chi_{2\varepsilon,0}\star F_i ~\mathrm{if}~ i = l \\ 0 ~ \mathrm{else} \end{cases}. \end{align*}$$

Therefore, $g \circ \left (f \star K_{\varepsilon } \right ) = \chi _{2\varepsilon ,0}\star F $ . A similar computation yields $f \circ \left (g \star K_{\varepsilon } \right ) = \chi _{2\varepsilon ,0}\star G$ . Thus, $(f,g)$ is indeed an $\varepsilon $ -interleaving pair.

By taking the infimum over $\varepsilon> M$ , we get the desired inequality.

2.3 PL-sheaves and functions

We consider a finite-dimensional real vector space $\mathbb {V}$ endowed with a norm $\|\cdot \|$ . We equip $\mathbb {V}$ with the topology induced by the norm $\|\cdot \|$ , and $\mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ with the convolution distance $d_C$ associated to $\|\cdot \|$ . The notion of piecewise-linear(PL) sheaves was introduced by Kashiwara–Schapira in [Reference Kashiwara and Schapira27].

We shall denote by $\mathrm {D}^b_{\mathrm {PL}}(\mathbf {k}_{\mathbb {V}})$ the full subcategory of $\mathrm {D}^b(\mathbf {k}_{\mathbb {V}})$ consisting of PL sheaves. The two first points of the following approximation theorem are proved in [Reference Kashiwara and Schapira27], we provide a proof for three.

Following [Reference Lebovici29], we introduce the PL counterpart of constructible functions.

We denote by $\mathrm {CF}_{\mathrm {PL}}(\mathbb {V})$ the group of PL-constructible functions on $\mathbb {V}$ .

3.1 Convolution distance of the difference of compact convex subsets

Recall that for $x\in \mathbb {V}$ and $\varepsilon \geq 0$ , we denote by $B(x,\varepsilon )$ the closed ball of radius $\varepsilon $ centered at x.

Proof. Let F and $\varepsilon $ be as in the statement. By definition of interleavings, it is sufficient to prove that the canonical map $F \star K_{\varepsilon } \longrightarrow F$ is zero. Let $x \in \mathbb {V}$ . If $x \not \in \mathrm {supp}(F)$ , it is clear that the induced morphism $(F \star K_{\varepsilon })_x \longrightarrow F_x$ is zero. Let us assume that $x \in \mathrm {supp}(F)$ . By Equation (2.12) in [Reference Petit and Schapira37], one has

$$\begin{align*}(F \star K_{\varepsilon})_x \simeq \mathrm{R} \Gamma(B(x,\varepsilon);F) \simeq 0.\end{align*}$$

Therefore, the morphism $(F \star K_{\varepsilon })_x \longrightarrow F_x$ is zero in every case, which implies that $F \star K_{\varepsilon } \longrightarrow F$ is also zero.

Definition 3.4. Given $X \subset \mathbb {V}$ and $\varepsilon \geq 0$ , the $\varepsilon $ -thickening of X is defined by

$$\begin{align*}T_{\varepsilon}(X) := \{v \in \mathbb{V} \mid d(v,X) \leq \varepsilon\}.\end{align*}$$

Proof. For $y\in Y$ and $\varepsilon '> \varepsilon $ , one has the following distinguished triangle:

$$\begin{align*}\mathrm{R} \Gamma (B(y,\varepsilon'); \mathbf{k}_{Y\backslash X}) \longrightarrow \mathrm{R} \Gamma (B(y,\varepsilon'); \mathbf{k}_Y) \longrightarrow \mathrm{R} \Gamma (B(y,\varepsilon'); \mathbf{k}_X) \stackrel{+1}{\longrightarrow}.\end{align*}$$

By hypothesis, $B(y, \varepsilon ')\cap Y \cap X$ is nonempty and convex. Since X and Y are closed convex subsets, we deduce that the map $\mathrm {R} \Gamma (B(y,\varepsilon '); \mathbf {k}_Y) \longrightarrow \mathrm {R} \Gamma (B(y,\varepsilon '); \mathbf {k}_X)$ is an isomorphism. Therefore, $\mathrm {R} \Gamma (B(y,\varepsilon '); \mathbf {k}_{Y\backslash X}) \simeq 0$ , for all $y \in \mathrm {supp}(\mathbf {k}_{Y\backslash X}) \subset Y$ and $\varepsilon '> \varepsilon $ . Lemma 3.3 implies that $d_C(\mathbf {k}_{Y\backslash X},0) \leq \frac {\varepsilon }{2}$ .

Given an $\varepsilon $ -flag $X^\bullet = (X^i)_{i = 0\ldots n}$ , and , we define the spaces $\mathrm {Gr}_i(X^\bullet )$ by

$$\begin{align*}\mathrm{Gr}_0(X^\bullet) := X^0, ~~~\mathrm{and for all~} 1 \leq i\leq n, \mathrm{Gr}_i(X^\bullet) := X^i \backslash X^{i-1}. \end{align*}$$

It is immediate to verify that $\mathrm {Gr}_i(X^\bullet )$ is locally closed for all and that one has $X^n = \sqcup _i \mathrm {Gr}_i(X^\bullet ).$ Moreover, we set

$$\begin{align*}S(X^\bullet) := \bigoplus_{i = 0}^n \mathbf{k}_{\mathrm{Gr}_i(X^\bullet)} \in \mathrm{D}^b_{\mathbb{R} c}(\mathbf{k}_{\mathbb{V}}).\end{align*}$$

3.2 PL-case

The first step of our proof is the following concentration lemma in the PL case, that we will extend later on to arbitrary stratification by density of PL-sheaves with respect to the convolution distance.

Lemma 3.8. Let $\delta $ be a $d_C$ -dominated pseudo-extended metric on $\mathrm {CF}(\mathbb {V})$ . Let $\varphi \in \mathrm {CF}_{\mathrm {PL}}(\mathbb {V})$ with compact support. Therefore, there exists a finite set A, and for all $\lambda \in A$ , a nonempty compact subset $X_{\lambda } \subset \mathbb {V}$ which is a convex polytope such that $\varphi = \sum _{\lambda \in A} C_{\lambda } \cdot 1_{X_{\lambda }}$ , with $C_{\lambda } \in \mathbb {Z}$ . For $\lambda \in A$ , let $x_{\lambda } \in X_{\lambda }$ . Then one has

$$\begin{align*}\delta \left (\varphi,\sum_{\lambda \in A} C_{\lambda} \cdot 1_{\{x_{\lambda}\}} \right) = 0.\end{align*}$$

Proof. We consider the linear deformation retraction $H_{\lambda } : X_{\lambda } \times [0,1] \longrightarrow X_{\lambda }$ from $\{x_{\lambda }\}$ to $X_{\lambda }$ defined by

$$\begin{align*}H_{\lambda}(x,t) = (1-t)\cdot x_{\lambda} + t\cdot x. \end{align*}$$

We set $\ell _{\lambda } = \max \{\|x - x_{\lambda }\| \mid x \in X_{\lambda } \}$ and $\ell = \max _{\lambda } \ell _{\lambda }$ . Let $\varepsilon>0$ and $n = \lceil \frac {\ell }{\varepsilon }\rceil $ . We define for the sequence of subsets $X_{\lambda }^i := H_{\lambda } (X_{\lambda } \times {[0, \frac {i}{n}]})$ . By construction, $X_{\lambda }^\bullet = (X_{\lambda }^i)_{i=0\ldots n}$ is an $\varepsilon $ -flag. We depict an illustration of $X_{\lambda }^\bullet $ in Figure 1.

Let us define the following sheaves:

$$\begin{align*}F_{\varepsilon} &= \bigoplus_{\lambda \in A} S(X^{\bullet}_{\lambda})^{|C_{\lambda}|}[(1-\mathrm{sgn}(C_{\lambda}))/2],\\F &= \bigoplus_{\lambda \in A} \mathbf{k}_{\{x_{\lambda}\}}^{|C_{\lambda}|}[(1-\mathrm{sgn}(C_{\lambda}))/2].\end{align*}$$

Then one has

$$ \begin{align*} \chi(F_{\varepsilon}) &= \chi \left (\bigoplus_{\lambda \in A}S(X^{\bullet}_{\lambda})^{|C_{\lambda}|}[(1-\mathrm{sgn}(C_{\lambda}))/2] \right) \\ &= \sum_{\lambda\in A} C_{\lambda} \cdot \chi(S(X^{\bullet}_{\lambda}))\\ &= \sum_{\lambda\in A} C_{\lambda} \cdot 1_{X_{\lambda}} \quad \mathrm{(Proposition}\ 3.7\mathrm{-(1))}. \\ &= \varphi. \end{align*} $$

Figure 1 Illustration of the $\varepsilon $ -flag $X_{\lambda }^\bullet $ .

Similarly,

$$\begin{align*}\chi(F) = \sum_{\lambda \in A} C_{\lambda} \cdot 1_{\{x_{\lambda}\}}.\end{align*}$$

Moreover, one has by additivity of interleavings (Proposition 2.18)

$$ \begin{align*} d_C(F_{\varepsilon},F) &\leq \max_{\lambda \in A} d_C\left (S(X^{\bullet}_{\lambda})^{|C_{\lambda}|}[(1-\mathrm{sgn}(C_{\lambda}))/2] , \mathbf{k}_{\{x_{\lambda}\}}^{|C_{\lambda}|}[(1-\mathrm{sgn}(C_{\lambda}))/2] \right) \\ &= \max_{\lambda \in A} d_C\left (S(X^{\bullet}_{\lambda}), \mathbf{k}_{\{x_{\lambda}\}} \right ) \\ &\leq \frac{\varepsilon}{2} \leq \varepsilon \quad \mathrm{(Proposition}\ 3.7\mathrm{-(2)).} \end{align*} $$

Therefore, one has for all $k\in \mathbb {Z}_{>0}$

$$ \begin{align*} \delta\left (\varphi, \sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x_{\lambda}\}}\right ) &= \delta\left ( \chi(F), \chi(F_{\frac{1}{k}}) \right). \end{align*} $$

Since $\delta $ is $d_C$ -dominated, $(F_{\frac {1}{k}})_{k> 0}$ is a cohomologically bounded sequence of compactly supported constructible sheaves, and $d_C(F, F_{\frac {1}{k}}) \underset {k \longrightarrow +\infty }{\longrightarrow } 0$ , we conclude that

$$\begin{align*}\delta \left (\varphi, \sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x_{\lambda}\}} \right) =0.\end{align*}$$

Proposition 3.9. Let $\delta $ be a $d_C$ -dominated pseudo-extended metric on $\mathrm {CF}(\mathbb {V})$ . Let $\varphi \in \mathrm {CF}_{\mathrm {PL}}(\mathbb {V})$ with compact support, and let $x \in \mathbb {V}$ . Then one has

$$\begin{align*}\delta \left (\varphi,\left (\int \varphi ~\mathrm{d} \chi \right ) \cdot 1_{\{x\}} \right) = 0. \end{align*}$$

Proof. Given $u,v \in \mathbb {V}$ , we set $[u,v] = \{t\cdot u + (1-t)\cdot v \mid t \in [0,1]\}$ . Let us write $\varphi = \sum _{\lambda \in A} C_{\lambda } \cdot 1_{X_{\lambda }}$ , with A finite, $C_{\lambda } \in \mathbb {Z}$ and $X_{\lambda }$ compact convex polytopes. For $\lambda \in A$ , let $x_{\lambda } \in X_{\lambda }$ . Then by Lemma 3.8 applied to $\psi = \sum _{\lambda \in A} C_{\lambda } \cdot 1_{[x_{\lambda },x]}$ , one has

$$ \begin{align*} \delta \left (\psi, \sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x_{\lambda}\}}\right ) = 0 = \delta \left (\psi, \sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x\}}\right ). \end{align*} $$

Therefore,

$$\begin{align*}\delta \left (\sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x_{\lambda}\}}, \sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x\}} \right ) = 0.\end{align*}$$

We now apply Lemma 3.8 to $\varphi $ :

$$ \begin{align*} \delta \left(\varphi, \left (\int \varphi ~\mathrm{d} \chi \right ) \cdot 1_{\{x\}} \right) &= \delta \left(\varphi, \sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x\}} \right) \\ &\leq \delta \left (\varphi, \sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x_{\lambda}\}} \right) + \delta \left (\sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x_{\lambda}\}}, \sum_{\lambda\in A} C_{\lambda} \cdot 1_{\{x\}} \right) \\ &= 0.\\[-34pt] \end{align*} $$

3.3 General case

In this final section, we generalize the previous results to arbitrary stratifications, by PL approximation (Theorem 2.21).

Lemma 3.10. Let $\delta $ be a $d_C$ -dominated pseudo-extended metric on $\mathrm {CF}(\mathbb {V})$ . Let $\varphi \in \mathrm {CF}(\mathbb {V})$ with compact support, and let $x \in \mathbb {V}$ . Then one has

$$\begin{align*}\delta \left ( \varphi,\left ( \int \varphi ~\mathrm{d} \chi \right ) \cdot 1_{\{x\}} \right ) = 0. \end{align*}$$

Proof. Let $F \in \mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_{\mathbb {V}})$ with compact support such that $\varphi = \chi (F)$ . According to Theorem 2.21, for all $n\in \mathbb {Z}_{> 0}$ , there exists $F_n \in \mathrm {D}^b_{\mathrm {PL}}(\mathbf {k}_{\mathbb {V}})$ such that $d_C(F,F_n) \leq \frac {1}{n}$ , $\mathrm {supp}(F_n) \subset T_{\frac {1}{n}}(\mathrm {supp}(F))$ , and the sequence $(F_n)$ is cohomologically bounded. In particular, $F_n$ has compact support for all $n \geq 1$ . Moreover, by Proposition 2.16, one has for all $n \geq 1$ ,

$$\begin{align*}\mathrm{R} \Gamma (\mathbb{V}; F) \simeq \mathrm{R} \Gamma (\mathbb{V}; F_n). \end{align*}$$

Therefore, $\int \chi (F_n) ~\mathrm {d} \chi = \int \varphi ~\mathrm {d} \chi $ according to Lemma 2.8. Consequently, for all $n>0$

$$ \begin{align*}\delta \left (\varphi, \left (\int \varphi ~\mathrm{d} \chi \right ) \cdot 1_{\{x\}} \right ) &\leq \delta \left (\varphi, \chi(F_n) \right ) + \delta\left (\chi(F_n), \left (\int \varphi ~\mathrm{d} \chi \right ) \cdot 1_{\{x\}}\right) \\ &= \delta \left (\varphi, \chi(F_n) \right ) + \delta\left (\chi(F_n), \left ( \int \chi(F_n) ~\mathrm{d} \chi \right) \cdot 1_{\{x\}}\right) \\ &= \delta \left (\varphi, \chi(F_n) \right ) \quad \mathrm{(Proposition}\ 3.9) \\ &= \delta \left (\chi(F), \chi(F_n) \right ). \end{align*} $$

Since $\delta $ is $d_C$ -dominated, $(F_n)$ is a cohomologically bounded sequence of constructible compactly supported sheaves, and $d_C(F,F_n) \underset {n \longrightarrow +\infty }{\longrightarrow } 0$ , we conclude that

$$\begin{align*}\delta \left (\varphi, \left (\int \varphi ~\mathrm{d} \chi \right ) \cdot 1_{\{x\}} \right ) = 0.\\[-40pt] \end{align*}$$

Theorem 3.11. Let $\delta $ be a $d_C$ -dominated pseudo-extended metric on $\mathrm {CF}(\mathbb {V})$ , and let $\varphi ,\psi \in \mathrm {CF}(\mathbb {V})$ with compact supports be such that $\int \varphi ~\mathrm {d} \chi = \int \psi ~\mathrm {d} \chi $ . Then

$$\begin{align*}\delta(\varphi, \psi) = 0. \end{align*}$$

Proof. By the above lemma,

$$ \begin{align*} \delta(\varphi,\psi) &\leq \delta\left (\varphi, \left (\int \varphi ~\mathrm{d} \chi \right ) \cdot 1_{\{0\}}\right) + \delta\left (\left (\int \psi ~\mathrm{d} \chi \right ) \cdot 1_{\{0\}}, \psi\right ) \\ &= 0 \quad \mathrm{(Lemma}\ 3.10).\\[-37pt] \end{align*} $$

Corollary 3.12. Let $F,G \in \mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_{\mathbb {V}})$ with compact support such that $d_C(F,G) < +\infty $ . Then

$$\begin{align*}\delta(\chi(F), \chi(G)) = 0.\end{align*}$$

Corollary 3.13. Let X be a real analytic manifold, and let $\varphi \in \mathrm {CF}(X)$ with compact support. Also, consider $f,g : X \longrightarrow \mathbb {V}$ some morphisms of real analytic manifolds proper on $\mathrm {supp}(\varphi )$ . Then

$$\begin{align*}\delta (f_\ast \varphi, g_\ast \varphi ) = 0.\end{align*}$$

Proof. By [Reference Schapira39, Theorem 2.3], $f_\ast \varphi $ and $g_\ast \varphi $ are indeed constructible and have compact support by the hypothesis. Let $a_X : X \longrightarrow \{\mathrm {pt}\}$ and $a_{\mathbb {V}} : \mathbb {V} \longrightarrow \{\mathrm {pt}\}$ be the constant maps. Then by [Reference Schapira39, Section 2], under the identification $\mathrm {CF}(\{pt\}) \simeq \mathbb {Z}$ , one has

$$ \begin{align*} \int f_\ast \varphi ~\mathrm{d} \chi &= a_{\mathbb{V} \ast}(f_\ast \varphi) \\ &= (a_{\mathbb{V}}\circ f)_{\ast} \varphi \quad \mathrm{(Theorem}\ 2.11\mathrm{-3)} \\ &= a_{X \ast} \varphi \\ &= \int \varphi ~\mathrm{d} \chi. \end{align*} $$

Similarly, $\int g_\ast \varphi ~\mathrm {d} \chi = \int \varphi ~\mathrm {d} \chi = \int f_\ast \varphi ~\mathrm {d} \chi $ . Since both $f_\ast \varphi $ and $g_\ast \varphi $ have compact support, we conclude the proof by applying Theorem 3.11.

4.1 Persistence and sheaves

For a general introduction to multiparameter persistence, we refer the reader to [Reference Bakke Botnan and Lesnick11]. Let $d \geq 0$ . We equip $\mathbb {R}^d$ with the partial order $\leq $ , defined by $(x_1,\ldots ,x_d) \leq (y_1,\ldots ,y_d)$ iff $x_i \leq y_i$ for all i. We denote $(\mathbb {R}^d, \leq )$ for the associated poset category. The category of persistence modules with d-parameters, denoted by $\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)$ , is the category of functors $(\mathbb {R}^d,\leq ) \longrightarrow \operatorname {\mathrm {Mod}}(\mathbf {k})$ and natural transformations.

Persistence modules are usually compared using the interleaving distance, which is defined as follows. Let $\varepsilon \geq 0$ and $M \in \mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)$ . The $\varepsilon $ -shift of M is the persistence module $M[\varepsilon ]$ defined, for $x \leq y \in \mathbb {R}^d$ , by

$$\begin{align*}M[\varepsilon](x) = M(x + (\varepsilon,\ldots,\varepsilon)), ~~~M[\varepsilon](x\leq y ) = M(x + (\varepsilon,\ldots,\varepsilon) \leq y + (\varepsilon,\ldots,\varepsilon)).\end{align*}$$

This objectwise construction readily extends to an additive exact autoequivalence $\cdot [\varepsilon ] : \mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d) \to \mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)$ . The collection of linear maps $(M(x \leq x + (\varepsilon ,\ldots , \varepsilon )))_{x\in \mathbb {R}^d}$ induces a natural transformation $M \longrightarrow M[\varepsilon ]$ , denoted $\tau _{\varepsilon }^M$ . An $\varepsilon $ -interleaving between two persistence modules M and N in $\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)$ is the data of two morphisms $f : M \longrightarrow N[\varepsilon ]$ and $g : N \longrightarrow M[\varepsilon ]$ such that $g[\varepsilon ] \circ f = \tau _{2 \varepsilon }^M$ and $f[\varepsilon ] \circ g = \tau _{2 \varepsilon }^N$ . If there exists an $\varepsilon $ -interleaving between M and N, we say that they are $\varepsilon $ -interleaved and write $M \sim _{\varepsilon } N$ .

Definition 4.1. The interleaving distance between the persistence modules M and N in $\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)$ is the possibly infinite quantity

$$\begin{align*}d_I (M,N) := \inf \{\varepsilon \geq 0 \mid M \sim_{\varepsilon} N\}.\end{align*}$$

We now introduce the $\gamma $ -topology after Kashiwara–Schapira [Reference Kashiwara, Schapira and Houzel25, Section 3.5], as an intermediate between the Euclidean topology and the downset (or Alexandrov) topology. Let $\gamma = (\mathbb {R}_{\geq 0})^d$ . An open set $U \subset \mathbb {R}^d$ is $\gamma $ -open if it satisfies $U + \gamma = U$ . The set of $\gamma $ -open subsets of $\mathbb {R}^d$ indeed forms a topology of $\mathbb {R}^d$ , named the $\gamma $ -topology. We denote the associated topological space by $\mathbb {R}^d_\gamma $ . The identity map $\varphi _\gamma : \mathbb {R}^d \longrightarrow \mathbb {R}^d_\gamma $ , $x \mapsto x$ , is continuous, and induces an adjunction

Following [Reference Berkouk and Petit7, Section 4.2], it is possible to define an interleaving distance on $\mathrm {D}^b(\mathbf {k}_{\mathbb {R}^d_\gamma })$ , that we write $d_I^\gamma $ . Also, we endow $\mathbb {R}^d$ with the norm $\|\cdot \|_\infty $ defined by $\|x\|_\infty := \max _i |x_i|$ and denote by $d_C$ the associated convolution distance on $\mathrm {D}^b(\mathbf {k}_{\mathbb {R}^d})$ .

Moreover, in [Reference Berkouk and Petit7], the authors introduce a pair of adjoint functors

and prove the following.

Combining the above results, we obtain the following adjunction:

where the left adjoint functor is objectwise distance preserving, and the right adjoint is objectwise $1$ -Lipschitz.

We will also need the following lemma, that was not included in [Reference Berkouk and Petit7].

Proof. By [Reference Berkouk and Petit7, Fact 2.10] and [Reference Berkouk and Petit7, Proposition 2.11-(i)], one has

$$\begin{align*}\alpha^{-1}\circ \mathrm{R} \alpha_\ast \simeq \mathrm{id}_{\mathrm{D}^b(\mathbf{k}_{\mathbb{R}^d_\gamma})}.\end{align*}$$

Therefore, for any $M\in \mathrm {D}^b(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ , by Theorem 4.4- $(1)$ , one has

$$ \begin{align*} d_I(M, \mathrm{R} \alpha_\ast\alpha^{-1} M) &= d_I^\gamma(\alpha^{-1} M, \alpha^{-1} (\mathrm{R} \alpha_\ast\alpha^{-1} M)) \\ &= d_I^\gamma (\alpha^{-1} M, (\alpha^{-1} \mathrm{R} \alpha_\ast)\alpha^{-1} M)) \\ &= d_I^\gamma (\alpha^{-1} M, \alpha^{-1} M) \\ &= 0.\\[-37pt] \end{align*} $$

We denote by $\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ the full subcategory of $\mathrm {D}^b(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ whose objects are constructible persistence modules. Also, we denote $\mathrm {Pers}_{\mathbf {k},\mathbb {R} c}(\mathbb {R}^d)$ the intersection $\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)) \cap \mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)$ .

4.2 nonexistence of additive stable invariants of persistence modules

In this section, we identify $\mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c}(\mathbf {k}_{\mathbb {R}^d}))$ with $\mathrm {CF}(\mathbb {R}^d)$ , according to the sheaf-function correspondence (Theorem 2.6). Since $\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ is triangulated, its Grothendieck group is well defined. We let $\kappa $ be the map $\mathrm {ob}(\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))) \longrightarrow \mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)))$ sending a constructible persistence module to its $\mathrm {K}_0$ -class.

Definition 4.10. A pseudo-extended metric $\delta $ on $\mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)))$ is said to be $d_I$ -dominated if for all cohomologically bounded sequences $(M_n) \in \mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)) $ of compactly generated persistence modules, and $M \in \mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ compactly generated, one has

$$\begin{align*}d_I(M,M_n) \underset{n \longrightarrow +\infty}{\longrightarrow} 0 ~~\Longrightarrow \delta(\kappa(M),\kappa(M_n)) ~~ \underset{n \longrightarrow +\infty}{\longrightarrow} 0. \end{align*}$$

Any triangulated functor $T : \mathscr {C} \longrightarrow \mathscr {C}'$ between triangulated categories, induces a group morphism $\mathrm {K}_0(\mathscr {C}) \longrightarrow \mathrm {K}_0(\mathscr {C}')$ , that, for simplicity, we keep denoting by T. Given $\delta $ a pseudo-extended metric on $\mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)))$ , we let $\delta _\ast $ be the pseudo-extended metric defined on $\mathrm {CF}(\mathbb {R}^d)$ by

$$\begin{align*}\delta_\ast (\varphi, \psi) := \delta(\mathrm{R} (\alpha\circ\varphi_\gamma)_\ast \varphi, \mathrm{R} (\alpha\circ\varphi_\gamma)_\ast \psi). \end{align*}$$

Proof. Assume that $\delta $ is $d_I$ dominated. Let $(F_n)$ and F be compactly supported in $\mathrm {D}^b_{\mathbb {R} c }(\mathbf {k}_{\mathbb {R}^d})$ such that, $(F_n)$ is cohomologically bounded, and

$$\begin{align*}d_C(F,F_n) \underset{n \longrightarrow +\infty}{\longrightarrow} 0 .\end{align*}$$

Thus, by Theorems 4.3 and 4.4,

$$\begin{align*}d_I(\mathrm{R} (\alpha\circ\varphi_\gamma)_\ast F,\mathrm{R} (\alpha\circ\varphi_\gamma)_\ast F_n) \underset{n \longrightarrow +\infty}{\longrightarrow} 0.\end{align*}$$

The functor $\mathrm {R} (\alpha \circ \varphi _\gamma )_\ast $ has finite cohomological dimension [Reference Berkouk and Petit7, Proposition 3.11], thus, the sequence $(\mathrm {R} (\alpha \circ \varphi _\gamma )_\ast F_n)$ is cohomologically bounded and compactly generated by definition. Since $\delta $ is $d_I$ -dominated, we deduce that

$$ \begin{align*} \delta(\kappa(\mathrm{R} (\alpha\circ\varphi_\gamma)_\ast F),\kappa(\mathrm{R} (\alpha\circ\varphi_\gamma)_\ast F_n)) &= \delta (\mathrm{R} (\alpha\circ\varphi_\gamma)_\ast \chi(F),\mathrm{R} (\alpha\circ\varphi_\gamma)_\ast \chi(F_n)) \\ &= \delta_\ast(\chi(F), \chi(F_n)) \underset{n \longrightarrow +\infty}{\longrightarrow} 0. \end{align*} $$

Therefore, $\delta _\ast $ is $d_C$ -dominated. The proof of the converse works similarly.

Proof. According to Theorem 4.3, one has $d_I(M,N) = d_C ( (\alpha \circ \varphi _\gamma )^{-1} M, (\alpha \circ \varphi _\gamma )^{-1} N) < +\infty $ . Also, since $\delta $ is $d_I$ -dominated, $\delta _\ast $ is $d_C$ -dominated. Therefore, by Corollary 3.12,

(4.1) $$ \begin{align} & \delta_\ast(\chi((\alpha\circ\varphi_\gamma)^{-1} M), \chi((\alpha\circ\varphi_\gamma)^{-1} N)) \end{align} $$

(4.2) $$ \begin{align} &= \delta(\kappa(\mathrm{R}(\alpha\circ\varphi_\gamma)_\ast(\alpha\circ\varphi_\gamma)^{-1} M), \kappa(\mathrm{R}(\alpha\circ\varphi_\gamma)_\ast(\alpha\circ\varphi_\gamma)^{-1} N)) \end{align} $$

(4.3) $$ \begin{align} &= 0. \end{align} $$

Note that, by [Reference Kashiwara and Schapira27, Corollary 1.6], $\mathrm {R} \varphi _{\gamma \ast } \circ \varphi _\gamma ^{-1} \simeq \mathrm {id}_{\mathrm {D}^b(\mathbf {k}_{\mathbb {V}_\gamma })}$ . Therefore,

$$\begin{align*}\mathrm{R}(\alpha\circ\varphi_\gamma)_\ast(\alpha\circ\varphi_\gamma)^{-1} M \simeq \mathrm{R} \alpha_\ast \alpha^{-1} M.\end{align*}$$

By Lemma 4.5, $d_I(M,\mathrm {R} \alpha _\ast \alpha ^{-1} M) = 0$ . Since $\delta $ is $d_I$ -dominated, $\delta (\kappa (M), \kappa (\mathrm {R} \alpha _\ast \alpha ^{-1} M)) = 0$ . Similarly, $\delta (\kappa (N), \kappa (\mathrm {R} \alpha _\ast \alpha ^{-1} M)) = 0$ . From Equation $(4.3)$ , we deduce that

$$ \begin{align*} &\delta(\kappa(M), \kappa(N)) \\ &\leq \delta(\kappa(M), \kappa(\mathrm{R} \alpha_\ast \alpha^{-1} M)) + \delta(\kappa(\mathrm{R} \alpha_\ast \alpha^{-1} M), \kappa(\mathrm{R} \alpha_\ast \alpha^{-1} N)) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \delta(\kappa(\mathrm{R} \alpha_\ast \alpha^{-1} N), \kappa(N)) \\ &= 0 + 0 + 0. \\[-37pt]\end{align*} $$

Let $(G,+)$ be an abelian group, endowed with a pseudo-extended metric $\delta $ . We think of G as a group of invariants of persistence modules and of $\delta $ as a way of measuring dissimilarity between invariants.

Theorem 4.14. Let $\lambda : \mathrm {ob}(\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))) \longrightarrow G$ be an additive map with respect to exact triangles such that for all cohomologically bounded sequence $(M_n)$ , and M, compactly generated constructible persistence modules in $\mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ ,

$$\begin{align*}d_I(M,M_n)\underset{n \longrightarrow +\infty}{\longrightarrow} 0 \Longrightarrow \delta(\lambda(M), \lambda(M_n)) \underset{n \longrightarrow +\infty}{\longrightarrow} 0.\end{align*}$$

Then for all compactly generated constructible persistence modules $M,N \in \mathrm {D}^b_{\mathbb {R} c }(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ :

$$\begin{align*}d_I(M,N) < +\infty \Longrightarrow \delta (\lambda(M), \lambda(N)) = 0.\end{align*}$$

Proof. By the universal property of the Grothendieck group, there exists a unique group morphism $\Phi : \mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c}(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))) \longrightarrow G$ such that the following diagram of maps

is commutative. For $\varphi ,\psi \in \mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c}(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))) $ , define $\delta ^{-1}(\varphi , \psi ) := \delta (\Phi (\varphi ), \Phi (\psi )).$ It is a pseudo-extended metric on $\mathrm {K}_0(\mathrm {D}^b_{\mathbb {R} c}(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d)))$ . Moreover, $\delta ^{-1}$ is $d_I$ -dominated by commutativity of the above diagram. Therefore, by Corollary 4.12, for all $M,N \in \mathrm {D}^b_{\mathbb {R} c}(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ compactly generated such that $d_I(M,N) < +\infty $ , one has:

$$\begin{align*}\delta^{-1}(\kappa(M), \kappa(N)) = \delta(\Phi(\kappa(M)),\Phi(\kappa(N))) = \delta(\lambda(M), \lambda(N)) = 0.\\[-37pt] \end{align*}$$

Theorem 4.15. Let $\lambda : \mathrm {ob}(\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)) \longrightarrow G$ be an additive map with respect to short exact sequences such that for all $(N_n)$ and N, compactly generated constructible persistence modules in $\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ ,

$$\begin{align*}d_I(N,N_n)\underset{n \longrightarrow +\infty}{\longrightarrow} 0\Longrightarrow \delta(\lambda(N), \lambda(N_n)) \underset{n \longrightarrow +\infty}{\longrightarrow} 0. \end{align*}$$

Also, assume that the sum map $G \times G \longrightarrow G$ and the symmetric map $G \longrightarrow G$ are continuous with respect to $\delta $ . Then, for all compactly generated constructible persistence modules $M,N \in \mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ ,

$$\begin{align*}d_I(M,N) < +\infty \Longrightarrow \delta (\lambda(M), \lambda(N)) = 0.\end{align*}$$

Proof. Let $\lambda $ be as in the statement of the theorem. We extend it as a map $\overline {\lambda } : \mathrm {ob}(\mathrm {D}^b_{\mathbb {R} c}(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))) \longrightarrow G$ , by $\overline {\lambda }(M) := \sum _i (-1)^i \lambda (\mathrm {H}^i(M))$ (the sum is always finite by boundedness assumption, hence well defined in G). One checks easily that $\overline {\lambda }$ is additive with respect to exact triangles. Moreover, since the $\varepsilon $ -shift functor is exact, the cohomological functors $\mathrm {H}^i$ preserve interleavings. Therefore, for $(N_n)$ and N compactly generated persistence modules in $\mathrm {D}^b_{\mathbb {R} c}(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ , if $d_I(N,N_n)\underset {n \longrightarrow +\infty }{\longrightarrow } 0$ , then for all $i\in \mathbb {Z}$ , $d_I(\mathrm {H}^i(N),\mathrm {H}^i(N_n))\underset {n \longrightarrow +\infty }{\longrightarrow } 0$ . Therefore, by assumptions,

$$\begin{align*}\delta(\lambda(\mathrm{H}^i(N)),\lambda(\mathrm{H}^i(N_n))) \underset{n \longrightarrow +\infty}{\longrightarrow} 0, \quad \mathrm{for all~} i \in \mathbb{Z}.\end{align*}$$

Assume in addition that $(N_n)$ is cohomologically bounded, that is, there exists $C \in \mathbb {Z}_{\geq 0}$ such that for all $n \geq 0$ , $\mathrm {H}^i(N_n) \simeq 0$ whenever $|i |> C$ . We also assume without loss of generality that $\mathrm {H}^i(N) \simeq 0$ for all $|i |> C$ . Therefore, $\overline {\lambda }(N) = \sum _{i = -C}^C (-1)^i \lambda (\mathrm {H}^i(N))$ , and for all $n \geq 0$ , $\overline {\lambda }(N_n) = \sum _{i = -C}^C (-1)^i \lambda (\mathrm {H}^i(N_n))$ . By continuity of the group operations with respect to $\delta $ , one has

$$\begin{align*}\delta(\overline{\lambda}(N),\overline{\lambda}(N_n)) \underset{n \longrightarrow +\infty}{\longrightarrow} 0.\end{align*}$$

Therefore, $\overline {\lambda }$ satisfies the hypothesis of Theorem 4.14, and we can conclude for all compactly generated persistence modules $M,N \in \mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ , if $d_I(M,N)< +\infty $ , then

$$\begin{align*}\delta(\lambda(M),\lambda(N)) = \delta (\overline{\lambda}(M), \overline{\lambda}(N)) = 0.\\[-36pt] \end{align*}$$

Our effort to formulate the consequences of our main result in a purely persistent and nonderived setting, allows using the universality result proved by Lesnick in [Reference Lesnick30, Corollary 5.6], that we recall here. Given $f : X \longrightarrow \mathbb {R}^d$ , its sublevelsets filtration is the functor $\mathcal {S}(f) : (\mathbb {R}^d, \leq ) \longrightarrow \textbf {Top}$ defined by $\mathcal {S}(f)(x) := f^{-1} \{s \in \mathbb {R}^d \mid s \leq x\}$ , and its i-th persistence module is the functor $\mathcal {S}_i(f) :=\mathrm {H}_i(-; \mathbf {k}) \circ \mathcal {S}(f)$ , with $\mathrm {H}_i(-; \mathbf {k})$ the i-th singular homology with coefficients in $\mathbf {k}$ functor.

For $f : X \longrightarrow \mathbb {R}^d$ and $g : X \longrightarrow \mathbb {R}^d$ two continuous maps of topological space, one sets

$$\begin{align*}d_\infty (f,g) := \inf_{h : X \stackrel{\sim}{\longrightarrow} Y} \sup_{x\in X} \|f(x) - g \circ h (x)\|_\infty,\end{align*}$$

where h ranges over all homeomorphisms from X to Y.

Recall that a field $\mathbf {k}$ is said to be prime if it does not contain any proper subfields; therefore, if $\mathbf {k} = \mathbb {Q} $ or $\mathbb {F}_p$ , for p a prime number.

Theorem 4.16 [Reference Lesnick30].

Let $\mathbf {k}$ be a prime field and d be a pseudo-extended metric on $\mathrm {ob}(\mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d))$ such that for all maps of topological spaces $f : X \longrightarrow \mathbb {R}^d$ and $g : Y \longrightarrow \mathbb {R}^d$ , one has

$$\begin{align*}d(\mathcal{S}_i(f), \mathcal{S}_i(g)) \leq d_\infty(f,g).\end{align*}$$

Then $d \leq d_I$ .

Note that Lesnick’s proof relies on the existence of geometric lift for interleavings of persistence modules [Reference Lesnick30, Proposition 5.8], which holds without any assumption on the persistence modules. Therefore, Theorem 4.16 restricts to constructible persistence modules in the following way.

Theorem 4.17 (Universality, constructible version).

Let $\mathbf {k}$ be a prime field and d be a pseudo-extended metric on $\mathrm {ob}(\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d))$ such that for all maps of topological spaces $f : X \longrightarrow \mathbb {R}^d$ and $g : Y \longrightarrow \mathbb {R}^d$ such that $\mathcal {S}_i(f)$ and $\mathcal {S}_i(g)$ are constructible, one has

$$\begin{align*}d(\mathcal{S}_i(f), \mathcal{S}_i(g)) \leq d_\infty(f,g).\end{align*}$$

Then $d \leq d_I$ .

Combining Lesnick’s universality theorem with our Theorem 4.15, we obtain the following corollary.

Corollary 4.18. Let $\lambda : \mathrm {ob}(\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)) \longrightarrow G$ be an additive map with respect to short exact sequences such that for all maps of topological spaces $f : X \longrightarrow \mathbb {R}^d$ and $g : Y \longrightarrow \mathbb {R}^d$ such that $\mathcal {S}_i(f)$ and $\mathcal {S}_i(g)$ are constructible, one has

$$\begin{align*}\delta(\lambda(\mathcal{S}_i(f)), \lambda(\mathcal{S}_i(g))) \leq d_\infty(f,g).\end{align*}$$

Also, assume that the operations of G are continuous with respect to $\delta $ . Then for all compactly generated persistence modules $M,N \in \mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ , one has

$$\begin{align*}d_I(M,N) < +\infty \Longrightarrow \delta(\lambda(M), \lambda(N)) = 0.\end{align*}$$

4.3 Examples

In this section, we apply our results to well-known constructions that are common to TDA. We still denote by $\mathbb {V}$ a finite-dimensional real vector space endowed with a norm $\|\cdot \|$ .

4.3.1 Radon transforms

Radon transforms are a general class of transformations on constructible functions, for which there exists a well-formulated criterion of invertibility [Reference Schapira, Cohen, Giusti and Mora38]. The ECT is a particular instance of invertible Radon transform, which has found numerous applications [Reference Baryshnikov and Ghrist4, Reference Turner, Mukherjee and Boyer42, Reference Curry, Mukherjee and Turner20, Reference Crawford, Monod, Chen, Mukherjee and Rabadán18].

Let X and Y be two real analytic manifolds, and let $S \subset X \times Y$ be a locally closed subanalytic subset. Let $q_1$ and $q_2$ be the first and second projection defined on $X \times Y$ . We shall assume the following hypothesis:

(4.4) $$ \begin{align} q_2 \mathrm{~is proper on the closure of~} S \mathrm{~in}~ X \times Y. \end{align} $$

Definition 4.19. The Radon transform associated to S, is the group homomorphism $\mathscr {R}_S : \mathrm {CF}(X) \longrightarrow \mathrm {CF}(Y)$ defined by

$$\begin{align*}\mathscr{R}_S(\varphi) := q_{2 \ast} \left [ (\varphi \circ q_1) \cdot 1_S \right ]. \end{align*}$$

Remark 4.20. When $X = \mathbb {V}$ , $Y = \mathbb {S}^\ast \times \mathbb {R}$ , and $S_{ECT} = \{(v, (\xi , t)) \in X \times Y \mid \xi (v) \leq t\}$ , the transform $\mathscr {R}_{S_{ECT}}$ is the usual ECT. We have denoted $\mathbb {S}^\ast $ the unit dual sphere of $\mathbb {V}^\ast $ .

Proposition 4.21. Let $X = \mathbb {V}$ , and Y be real analytic manifold, and let $S \subset X \times Y$ satisfying hypothesis (4.4). Let $\delta $ be a pseudo-extended distance on $\mathrm {CF}(Y)$ such that $\delta \circ (\mathscr {R}_S \times \mathscr {R}_S)$ is $d_C$ -dominated. Then for all $\varphi ,\psi \in \mathrm {CF}(\mathbb {V})$ with compact support such that $\int \varphi \mathrm {d}\chi = \int \psi \mathrm {d}\chi $ , one has

$$\begin{align*}\delta\left(\mathscr{R}_S(\varphi), \mathscr{R}_S(\psi) \right) = 0.\end{align*}$$

Proof. This is a straightforward consequence of Theorem 3.11.

4.3.2 Amplitudes

In [Reference Giunti, Nolan, Otter and Waas22], the authors introduce the notion of amplitude on an abelian category $\mathscr {A}$ , as a notion of measurement of the size of objects of $\mathscr {A}$ , compatible with exact sequences.

Definition 4.22. Let $\mathscr {A}$ be an abelian category. An amplitude on $\mathscr {A}$ is a class function $\lambda : \mathrm {ob}(\mathscr {A}) \longrightarrow [0, + \infty ]$ satisfying $\lambda (0) = 0$ , and for all short exact sequence $0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0$

1. 1. $\lambda (A) \leq \lambda (B)$ ;

2. 2. $\lambda (C) \leq \lambda (B)$ ;

3. 3. $\lambda (B) \leq \lambda (A) + \lambda (C)$ .

The amplitude $\lambda $ will be said to be additive if $(3)$ is an equality.

Proposition 4.23. Let $\lambda $ be an additive amplitude on $\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ , such that for all $(M_n)$ and M compactly generated in $\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ , one has

$$\begin{align*}d_I(M_n,M) \underset{n \longrightarrow +\infty}{\longrightarrow} 0 \Longrightarrow |\lambda(M_n) - \lambda(M)| \underset{n \longrightarrow +\infty}{\longrightarrow} 0. \end{align*}$$

Then for all compactly generated persistence module $M, N \in \mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ such that $d_I(M,N) < +\infty $ , one has $\lambda (M) = \lambda (N)$ .

Proof. We apply Theorem 4.15 to the additive amplitude $\lambda $ , where $G = (\mathbb {R},+)$ is endowed with the standard metric. Thus, for all $M,N \in \mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ compactly generated, $|\lambda (M) - \lambda (N)| = 0$ .

Corollary 4.24. Assume that $\mathbf {k}$ is a prime field, and let $\lambda : \mathrm {ob}(\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)) \longrightarrow [0,+\infty ]$ be an additive amplitude on persistence modules such that for all maps of topological spaces $f : X \longrightarrow \mathbb {R}^d$ and $g : Y \longrightarrow \mathbb {R}^d$ , and all $i \in \mathbb {Z}_{\geq 0}$ , if $\mathcal {S}_i(f)$ and $\mathcal {S}_i(g)$ are constructible, then

$$\begin{align*}|\lambda(\mathcal{S}_i(f)) - \lambda(\mathcal{S}_i(g))| \leq d_\infty(f,g). \end{align*}$$

Then for all compactly generated and constructible $M,N \in \mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ , one has

$$\begin{align*}d_I(M,N) < +\infty \Longrightarrow \lambda(M) = \lambda(N).\end{align*}$$

4.3.3 Additive vectorizations

It is well known that persistence modules endowed with the interleaving distance do not embed isometrically into any Hilbert space [Reference Mitra and Virk34, Theorem 4.3]. Nevertheless, since most machine learning techniques take as input elements of a vector space, it is a very common strategy to define a so-called vectorization of persistence modules, that is, a map $\Phi : \mathrm {ob}(\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)) \longrightarrow \mathbb {W}$ , where $\mathbb {W}$ is a real vector space, usually endowed with a norm $\|\cdot \|$ . To name a few, see for instance persistence landscapes [Reference Bubenik13], persistent images [Reference Adams, Emerson, Kirby, Neville, Peterson, Shipman, Chepushtanova, Hanson, Motta and Ziegelmeier1] or multiparameter persistence landscapes [Reference Vipond43].

Proposition 4.25. Let $\Phi : \mathrm {ob}(\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)) \longrightarrow (\mathbb {W}, \| \cdot \|)$ be an additive vectorization of persistence modules satisfying for all $(M_n)$ and M in $\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)$ :

$$\begin{align*}d_I(M_n,M) \underset{n \longrightarrow +\infty}{\longrightarrow} 0 \Longrightarrow \|\Phi(M_n) - \Phi(M)\| \underset{n \longrightarrow +\infty}{\longrightarrow} 0. \end{align*}$$

Then for all compactly generated and constructible persistence modules $M,N \in \mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d) $ , if $d_I(M,N) < +\infty $ , then $\Phi (M) = \Phi (N)$ .

Proof. The proof is similar to Proposition 4.23.

Corollary 4.26. Assume that $\mathbf {k}$ is a prime field, and let $\Phi : \mathrm {ob}(\mathrm {Pers}_{\mathbf {k}, \mathbb {R} c}(\mathbb {R}^d)) \longrightarrow (\mathbb {W}, \| \cdot \|)$ be an additive vectorization of persistence modules such that for all maps of topological spaces $f : X \longrightarrow \mathbb {R}^d$ and $g : Y \longrightarrow \mathbb {R}^d$ , and all $i \in \mathbb {Z}_{\geq 0}$ such that $\mathcal {S}_i(f)$ and $\mathcal {S}_i(g)$ are constructible and compactly generated, one has

$$\begin{align*}\|\Phi(\mathcal{S}_i(f)) - \Phi(\mathcal{S}_i(g))\| \leq d_\infty(f,g). \end{align*}$$

Then for all compactly generated and constructible persistence modules $M,N \in \mathrm {Pers}_{\mathbf {k}}(\mathbb {R}^d) $ , if $d_I(M,N) < +\infty $ , then $\Phi (M) = \Phi (N)$ .

Remark 4.27. The construction $\|\Phi (\cdot )\|$ , where $\Phi $ is an additive vectorization of persistence modules, provides a very general mean of defining not necessarily additive amplitudes of persistence modules. One interpretation of Proposition 4.25 and Corollary 4.26 is that such amplitudes can never be reasonably controlled, either by the interleaving distance $d_I$ on persistence modules nor by the infinite distance $d_\infty $ on functions.