Generalized persistence diagrams for persistence modules over posets

Abstract

When a category ${\mathscr {C}}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:{\mathbf {P}}\rightarrow {\mathscr {C}}$ from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel’s recent extension. Specifically, the barcode of any interval decomposable persistence modules $F:{\mathbf {P}}\rightarrow \mathbf {vec}$ of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset ${\mathbf {P}}$ of $F:{\mathbf {P}}\rightarrow {\mathscr {C}}$ in defining Patel’s generalized persistence diagram of F. Of particular importance is the fact that the generalized persistence diagram of F is defined regardless of whether F is interval decomposable or not. By specializing our idea to zigzag persistence modules, we also show that the zeroth level set barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel’s semicontinuity theorem about type ${\mathscr {A}}$ persistence diagram to Lipschitz continuity theorem for the category of sets.

Appendices

Limits and Colimits

We recall the notions of limit and colimit (Mac Lane 2013). Throughout this section I will stand for a small category.

Definition A.1

(Cone) Let $F:I\rightarrow {\mathscr {C}}$ be a functor. A cone over F is a pair $\left( L,(\pi _x)_{x\in \mathrm {ob}(I)}\right)$ consisting of an object L in ${\mathscr {C}}$ and a collection $(\pi _x)_{x\in \mathrm {ob}(I)}$ of morphisms $\pi _x:L \rightarrow F(x)$ that commute with the arrows in the diagram of F, i.e. if $g:x\rightarrow y$ is a morphism in I, then $\pi _y= F(g)\circ \pi _x$ in ${\mathscr {C}}$, i.e. the diagram below commutes.

In Definition A.1, the cone $\left( L,(\pi _x)_{x\in \mathrm {ob}(I)}\right)$ over F will sometimes be denoted simply by L, suppressing the collection $(\pi _x)_{x\in \mathrm {ob}(I)}$ of morphisms if no confusion can arise. A limit of a diagram $F:I\rightarrow {\mathscr {C}}$ is a terminal object in the collection of all cones:

Definition A.2

(Limit) Let $F:I\rightarrow {\mathscr {C}}$ be a functor. A limit of F is a cone over F, denoted by $\left( \varprojlim F,\ (\pi _x)_{x\in \mathrm {ob}(I)} \right)$ or simply $\varprojlim F$, with the following terminal property: If there is another cone $\left( L',(\pi '_x)_{x\in \mathrm {ob}(I)} \right)$ of F, then there is a unique morphism $u:L'\rightarrow \varprojlim F$ such that $\pi _x'=\pi _x\circ u$ for all $x\in \mathrm {ob}(I)$.

Remark A.3

It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then the terminal property of the limit guarantees its uniqueness up to isomorphism. For this reason, we will sometimes refer to a limit as the limit of a diagram.

Cocones and colimits are defined in a dual manner:

Definition A.4

(Cocone) Let $F:I\rightarrow {\mathscr {C}}$ be a functor. A cocone over F is a pair $\left( C,(i_x)_{x\in \mathrm {ob}(I)}\right)$ consisting of an object C in ${\mathscr {C}}$ and a collection $(i_x)_{x\in \mathrm {ob}(I)}$ of morphisms $i_x:F(x)\rightarrow C$ that commute with the arrows in the diagram of F, i.e. if $g:x\rightarrow y$ is a morphism in I, then $i_x= i_y\circ F(g)$ in ${\mathscr {C}}$, i.e. the diagram below commutes.

In Definition A.4, a cocone $\left( C,(i_x)_{x\in \mathrm {ob}(I)}\right)$ over F will sometimes be denoted simply by C, suppressing the collection $(i_x)_{x\in \mathrm {ob}(I)}$ of morphisms. A colimit of a diagram $F:I\rightarrow {\mathscr {C}}$ is an initial object in the collection of cocones over F:

Definition A.5

(Colimit) Let $F:I\rightarrow {\mathscr {C}}$ be a functor. A colimit of F is a cocone, denoted by $\left( \varinjlim F,\ (i_x)_{x\in \mathrm {ob}(I)}\right)$ or simply $\varinjlim F$, with the following initial property: If there is another cocone $\left( C', (i'_x)_{x\in \mathrm {ob}(I)}\right)$ of F, then there is a unique morphism $u:\varinjlim F\rightarrow C'$ such that $i'_x=u\circ i_x$ for all $x\in \mathrm {ob}(I)$.

Remark A.6

It is possible that a diagram does not have a colimit at all. However, if a diagram does have a colimit then the initial property of the colimit guarantees its uniqueness up to isomorphism. For this reason, we will sometimes refer to a colimit as the colimit of a diagram.

Remark A.7

(Restriction of an indexing poset) Let ${\mathbf {P}}$ be any poset and let ${\mathbf {Q}}$ be a subposet of ${\mathbf {P}}$. In categorical language, ${\mathbf {Q}}$ is a full subcategory of ${\mathbf {P}}$. Let $F:{\mathbf {P}}\rightarrow {\mathscr {C}}$ be a functor.

• (i) Assume that the limit of the restriction $F|_{{\mathbf {Q}}}$ exists. For any cone $\left( L', (\pi _p')_{p\in {\mathbf {P}}}\right)$ over F, its restriction $\left( L', (\pi _p')_{p\in {\mathbf {Q}}}\right)$ is a cone over the restriction $F|_{\mathbf {Q}}:{\mathbf {Q}}\rightarrow {\mathscr {C}}$. Therefore, by the terminal property of the limit $\left( \varprojlim F|_{{\mathbf {Q}}}, (\pi _q)_{q\in {\mathbf {Q}}}\right)$, there exists the unique morphism $u:L' \rightarrow \varprojlim F|_{{\mathbf {Q}}}$ such that $\pi _q'=\pi _q\circ u$ for all $q\in {\mathbf {Q}}$.

• (ii) Assume that the colimit of the restriction $F|_{{\mathbf {Q}}}$ exists. For any cocone $\left( C', (i_p')_{p\in {\mathbf {P}}}\right)$ over F, its restriction $\left( C', (i_p')_{p\in {\mathbf {Q}}}\right)$ is a cocone over the restriction $F|_{\mathbf {Q}}:{\mathbf {Q}}\rightarrow {\mathscr {C}}'$. Therefore, by the initial property of $\varinjlim F|_{{\mathbf {Q}}}$, there exists the unique morphism $u:\varinjlim F|_{{\mathbf {Q}}} \rightarrow C'$ such that such that $i'_q=u\circ i_q$ for all $q\in {\mathbf {Q}}$.

The rank invariant of a standard persistence module

In this section we review some important (standard) results about the rank invariant of one-dimensional or multidimensional persistence modules. Recall the poset ${\mathbf {U}}$ from Definition 2.10.

Definition B.1

(Rank invariant of a persistence module) Let $F:{\mathbf {R}}\rightarrow \mathbf {vec}$ be any persistence module. The rank invariant of F is defined as the map $\mathrm {rk}(F):{\mathbf {U}}\rightarrow {\mathbf {Z}}_+$ which sends each ${\mathbf {u}}=(u_1,u_2)\in {\mathbf {U}}$ to $\mathrm {rank}\left( F(u_1\le u_2)\right)$.

Remark B.2

(Category theoretical interpretation of the rank invariant) Let $F:{\mathbf {R}}\rightarrow \mathbf {vec}$ be a persistence module. For any ${\mathbf {u}}=(u_1,u_2)\in {\mathbf {U}}$, it is not difficult to check that

$$\begin{aligned} \left( F_{u_1},\left( \varphi _F(u_1,t)\right) _{t\in [u_1,u_2]}\right) \ \text{ and }\ \left( F_{u_2},\left( \varphi _F(t,u_2)\right) _{t\in [u_1,u_2]}\right) \end{aligned}$$

are a limit and a colimit of $F|_{[u_1,u_2]}$, respectively. The canonical LC map from $F_{u_1}$ to $F_{u_2}$ is definitely $\varphi _F(u_1,u_2)$, which is identical to $\varphi _F(t,u_2)\circ \varphi _F(u_1,t)$, for any $t\in [u_1,u_2]$. Therefore, $\mathrm {rk}(F)({\mathbf {u}})$ can be regarded as the rank of the canonical LC map of $F|_{[u_1,u_2]}$.

Remark B.3

In Definition B.1, $\mathrm {rk}(F)({\mathbf {u}})$ for ${\mathbf {u}}=(u_1,u_2)$ counts all the persistence features of the persistence module F which are born before or at $u_1$ and die after $u_2$. Also, when $u_1=u_2$, $\mathrm {rk}(M)({\mathbf {u}})$ is the dimension of $F_{u_1}$.

Remark B.4

(Rank invariant is order-reversing) In Definition B.1, for any pair ${\mathbf {u}}=(u_1,u_2)\le {\mathbf {u}}'=\left( u_1',u_2'\right)$ in ${\mathbf {U}}$, since

$$\begin{aligned} \varphi _F\left( u_1', u_2'\right) =\varphi _F\left( u_2, u_2'\right) \circ \varphi _F(u_1, u_2)\circ \varphi _F\left( u_1', u_1\right) , \end{aligned}$$

it holds that $\mathrm {rk}(F)\left( {\mathbf {u}}'\right) \le \mathrm {rk}(F)({\mathbf {u}}).$ Therefore, the map $\mathrm {rk}(F):{\mathbf {U}}\rightarrow {\mathbf {Z}}_+$ is an order-reversing map. This result generalizes to McCleary and Patel (2020, Proposition 4.4). Also, see Puuska (2020, Proposition 3.7).

Theorem B.5

(Completeness of the rank invariant for one-dimensional modules Carlsson and Zomorodian 2009) The rank invariant defined in Definition B.1 is a complete invariant for one-dimensional persistence modules, i.e. if there are two constructible persistence modules $F,G:{\mathbf {R}}\rightarrow \mathbf {vec}$ such that $\mathrm {rk}(F)=\mathrm {rk}(G)$, then F and G are isomorphic (see Definition F.1 for the meaning of constructible).

The rank invariant can also be defined for multidimensional modules $F:{\mathbf {R}}^n \rightarrow \mathbf {vec}$, $n>1$ : For any pair ${\mathbf {a}}\le {\mathbf {b}}$ in ${\mathbf {R}}^n$, let $\mathrm {rk}(F)({\mathbf {a}},{\mathbf {b}}):=\mathrm {rank}\left( \varphi _F({\mathbf {a}}, {\mathbf {b}})\right)$. This defines a function from the set $\{({\mathbf {a}},{\mathbf {b}})\in {\mathbf {R}}^n\times {\mathbf {R}}^n: {\mathbf {a}}\le {\mathbf {b}}\}$ to ${\mathbf {Z}}_+$. However, the map $\mathrm {rk}(F)$ is not a complete invariant for multidimensional modules, i.e. for any $n>1$, there exists a pair of persistence modules $F,G:{\mathbf {R}}^n\rightarrow \mathbf {vec}$ that are not isomorphic but $\mathrm {rk}(M)=\mathrm {rk}(N)$ (Carlsson and Zomorodian 2009).

Comparison with Ville Puuska’s rank invariant

In (Puuska 2020), Ville Puuska considers the set

$$\begin{aligned} \mathrm {Dgm}_{\mathbf {P}}:=\{(a,b)\in {\mathbf {P}}\times {\mathbf {P}}: a<b\} \end{aligned}$$

and defines the rank invariant of a functor $F:{\mathbf {P}}\rightarrow {\mathscr {C}}$ as the map $dF:\mathrm {Dgm}_{{\mathbf {P}}}\rightarrow {\mathscr {C}}$ sending (a, b) to $\mathrm {im}\left( \varphi _F(a,b)\right) \in \mathrm {ob}({\mathscr {C}})$. Even though this definition is a straightforward generalization of the rank invariant of Carlsson and Zomorodian (2009), when ${\mathbf {P}}=\mathbf {ZZ}$ and ${\mathscr {C}}=\mathbf {vec}$, this definition is not anywhere near a complete invariant of $F:{\mathbf {P}}\rightarrow {\mathscr {C}}$. Namely, there exists a pair of zigzag modules M, N such that $d_{\mathrm {I}}(M,N)=+\infty$ (Definition D.4) whereas $dM\cong dN$:

Example C.1

Consider the two zigzag modules $M,N:\mathbf {ZZ}\rightarrow \mathbf {vec}$ defined as follows: $M:=I^{(-\infty ,\infty )_\mathbf {ZZ}}$ and N is defined as

$$\begin{aligned}&N_{(i,i)}={\mathbb {F}}, N_{(i+1,i)}={\mathbb {F}}^2,\\&\varphi _N((i,i-1),(i,i))=\pi _1,\\&\varphi _N((i+1,i),(i,i))=\pi _2, \end{aligned}$$

where $\pi _1,\pi _2:{\mathbb {F}}^2\rightarrow {\mathbb {F}}$ are the canonical projections to the first and the second coordinate, respectively. Note that $dM(a,b)\cong dN(a,b)\cong {\mathbb {F}}$ for all $(a,b)\in \mathrm {Dgm}_{\mathbf {P}}$. However, it is not difficult to check that $\mathrm {barc}^{\mathbf {ZZ}}(M)=\left\{ \!\!\left\{ (-\infty ,\infty )_{\mathbf {ZZ}}\right\} \!\!\right\}$ and $\mathrm {barc}^{\mathbf {ZZ}}(N)=\left\{ \!\!\left\{ (i,i+2)_{\mathbf {ZZ}}:i\in {\mathbf {Z}}\right\} \!\!\right\}$. This implies that $d_\mathrm {B}\left( \mathrm {barc}^{\mathbf {ZZ}}(M),\mathrm {barc}^{\mathbf {ZZ}}(N)\right) =+\infty$, and in turn $d_{\mathrm {I}}(M,N)=+\infty$ by Theorem D.8.

Interleaving distance and existing stability theorems

1.1 Interleaving distance

We review the interleaving distance between ${\mathbf {R}}^d$ (or ${\mathbf {Z}}^d$)-indexed functors and between $\mathbf {ZZ}$-indexed functors (Botnan and Lesnick 2018; Chazal et al. 2009; Lesnick 2015).

1.1.1 Natural transformations

We recall the notion of natural transformations from category theory (Mac Lane 2013): Let ${\mathscr {C}}$ and ${\mathscr {D}}$ be any categories and let $F,G:{\mathscr {C}}\rightarrow {\mathscr {D}}$ be any two functors. A natural transformation $\psi :F\Rightarrow G$ is a collection of morphisms $\psi _c: F_c\rightarrow G_c$ in ${\mathscr {D}}$ for all objects $c\in {\mathscr {C}}$ such that for any morphism $f:c\rightarrow c'$ in ${\mathscr {C}}$, the following diagram commutes:

Natural transformations $\psi :F\rightarrow G$ are considered as morphisms in the category ${\mathscr {D}}^{\mathscr {C}}$ of all functors from ${\mathscr {C}}$ to ${\mathscr {D}}.$

1.1.2 The interleaving distance between ${\mathbf {R}}^d$ (or ${\mathbf {Z}}^d$)-indexed functors

In what follows, for any $\varepsilon \in [0,\infty )$, we will denote the vector $\varepsilon (1,\ldots ,1)\in {\mathbf {R}}^d$ by $\varvec{\varepsilon }$. The dimension d will be clearly specified in context.

Definition D.1

(${\mathbf {v}}$-shift functor) Let ${\mathscr {C}}$ be any category. For each ${\mathbf {v}}\in [0,\infty )^n$, the ${\mathbf {v}}$-shift functor $(-)({\mathbf {v}}):{\mathscr {C}}^{{\mathbf {R}}^d}\rightarrow {\mathscr {C}}^{{\mathbf {R}}^d}$ is defined as follows:

• (i) (On objects) Let $F:{\mathbf {R}}^d\rightarrow {\mathscr {C}}$ be any functor. Then the functor $F({\mathbf {v}}):{\mathbf {R}}^d\rightarrow {\mathscr {C}}$ is defined as follows: For any ${\mathbf {a}}\in {\mathbf {R}}^d$,

  $$\begin{aligned} F({\mathbf {v}})_{\mathbf {a}}:=F_{{\mathbf {a}}+{\mathbf {v}}}. \end{aligned}$$

  Also, for another ${\mathbf {a}}'\in {\mathbf {R}}^d$ such that ${\mathbf {a}}\le {\mathbf {a}}'$ we define

  $$\begin{aligned} \varphi _{F({\mathbf {v}})}({\mathbf {a}},{\mathbf {a}}'):= \varphi _{F}\left( {\mathbf {a}}+{\mathbf {v}}, {\mathbf {a}}'+{\mathbf {v}}\right) . \end{aligned}$$

  In particular, if ${\mathbf {v}}=\varvec{\varepsilon }\in [0,\infty )^d$, then we simply write $F(\varepsilon )$ in lieu of $F(\varvec{\varepsilon })$.

• (ii) (On morphisms) Given any natural transformation $\psi :F\Rightarrow G$, the natural transformation $\psi ({\mathbf {v}}):F({\mathbf {v}})\Rightarrow G({\mathbf {v}})$ is defined as $\psi ({\mathbf {v}})_{\mathbf {a}}=\psi _{{\mathbf {a}}+{\mathbf {v}}}:F({\mathbf {v}})_{\mathbf {a}}\rightarrow G({\mathbf {v}})_{\mathbf {a}}$ for each ${\mathbf {a}}\in {\mathbf {R}}^d$.

For any ${\mathbf {v}}\in [0,\infty )^d$, let $\psi _F^{\mathbf {v}}: F \Rightarrow F({\mathbf {v}})$ be the natural transformation whose restriction to each $F_{\mathbf {a}}$ is the morphism $\varphi _F({\mathbf {a}}, {\mathbf {a}}+{\mathbf {v}})$ in ${\mathscr {C}}$. When ${\mathbf {v}}=\varvec{\varepsilon }$, we denote $\psi _F^{\mathbf {v}}$ simply by $\psi _F^\varepsilon$.

Definition D.2

(${\mathbf {v}}$-interleaving between ${\mathbf {R}}^d$-indexed functors) Let ${\mathscr {C}}$ be any category. Given any two functors $F,G:{\mathbf {R}}^d\rightarrow {\mathscr {C}},$ we say that they are ${\mathbf {v}}$-interleaved if there are natural transformations $f:F\Rightarrow G({\mathbf {v}})$ and $g:G\Rightarrow F({\mathbf {v}})$ such that

• (i) $g({\mathbf {v}})\circ f = \psi _F^{2{\mathbf {v}}}$,

• (ii) $f({\mathbf {v}})\circ g = \psi _G^{2{\mathbf {v}}}$.

In this case, we call (f, g) a ${\mathbf {v}}$-interleaving pair. When ${\mathbf {v}}=\varepsilon (1,\ldots ,1)$, we simply call (f, g) $\varepsilon$-interleaving pair. The interleaving distance between $d_{\mathrm {I}}^{\mathscr {C}}$ is defined as

$$\begin{aligned} d_{\mathrm {I}}^{\mathscr {C}}(F,G):=\inf \{\varepsilon \in [0,\infty ):F,G\ \text{ are } \varvec{\varepsilon }\text{-interleaved }\}, \end{aligned}$$

(9)

where we set $d_{\mathrm {I}}^{\mathscr {C}}(F,G)=\infty$ if there is no $\varepsilon$-interleaving pair between F and G for any $\varepsilon \in [0,\infty )$. Then $d_{\mathrm {I}}^{\mathscr {C}}$ is an extended pseudo-metric for ${\mathscr {C}}$-valued ${\mathbf {R}}^d$-indexed functors. By replacing ${\mathbf {R}}^d$ by ${\mathbf {Z}}^d$ in Definitions D.1and D.2, we similarly obtain the interleaving distance between ${\mathbf {Z}}^d$-indexed functors.

Definition D.3

(Poset ${\mathbf {U}}$) The poset ${\mathbf {U}}:=\left\{ (u_1,u_2)\in {\mathbf {R}}^2: u_1\le u_2\right\}$ is equipped with the partial order inherited from ${\mathbf {R}}^{\mathrm {op}}\times {\mathbf {R}}$ (see Fig. 8 (A)) , i.e. $(u_1,u_2)\le (u_1',u_2')$ in ${\mathbf {U}}$ if and only if the interval $(u_1,u_2)\subset {\mathbf {R}}$ is contained in $(u_1',u_2')\subset {\mathbf {R}}$.

The reflection map ${\mathbf {r}}:{\mathbf {R}}^{\mathrm {op}}\rightarrow {\mathbf {R}}$ defined by $t\mapsto -t$ induces a poset isomorphism ${\mathbf {R}}^\mathrm {op}\times {\mathbf {R}}\rightarrow {\mathbf {R}}^2$ and this in turn induces an isomorphism ${\mathscr {C}}^{{\mathbf {R}}^2}\rightarrow {\mathscr {C}}^{{\mathbf {R}}^{\mathrm {op}}\times {\mathbf {R}}}.$ Therefore, the notions of $\varepsilon$-interleaving pair and the interleaving distance $d_{\mathrm {I}}^{\mathscr {C}}$ on $\mathrm {ob}({\mathscr {C}}^{{\mathbf {R}}^2})$ carry over to ${\mathbf {R}}^\mathrm {op}\times {\mathbf {R}}$-indexed or ${\mathbf {U}}$-indexed functors.

1.1.3 Extension functor and the interleaving for $\mathbf {ZZ}$-indexed functors

From Definition 2.10, recall the poset $\mathbf {ZZ}=\{(i,j)\in {\mathbf {Z}}^2: j=i\ \text{ or }\ j=i-1\}\subset {\mathbf {R}}^{\mathrm {op}}\times {\mathbf {R}}$. Let $\iota :\mathbf {ZZ}\hookrightarrow {\mathbf {R}}^\mathrm {op}\times {\mathbf {R}}$ be the canonical inclusion map. For any ${\mathbf {u}}=(u_1,u_2)\in {\mathbf {U}}$, let

$$\begin{aligned} \mathbf {ZZ}[\iota \le {\mathbf {u}}]:=\{a\in \mathbf {ZZ}: \iota (a)\le {\mathbf {u}}\} \end{aligned}$$

which is a subposet of $\mathbf {ZZ}$. Observe that $\mathbf {ZZ}[\iota \le {\mathbf {u}}]$ cannot be empty for any choice of ${\mathbf {u}}\in {\mathbf {U}}$. See Fig. 8b.

Fig. 8

a The shaded region stands for the poset ${\mathbf {U}}$. For ${\mathbf {u}},{\mathbf {u}}'\in {\mathbf {U}}$ as marked in the figure, we have ${\mathbf {u}}\le {\mathbf {u}}'$. b Fixing ${\mathbf {u}}\in {\mathbf {U}}$ as shown, the subposet $\mathbf {ZZ}[\iota \le {\mathbf {u}}]$ is indicated by the red points and the red arrows

We review the definition of extension functor $E:{\mathscr {C}}^{\mathbf {ZZ}}\rightarrow {\mathscr {C}}^{{\mathbf {U}}}$ of Botnan and Lesnick (2018) for a cocomplete category ${\mathscr {C}}$. For any $M: \mathbf {ZZ}\rightarrow {\mathscr {C}}$, define the functor ${\tilde{E}}(M):{\mathbf {R}}^\mathrm {op}\times {\mathbf {R}}\rightarrow {\mathscr {C}}$ as follows: For ${\mathbf {a}}\in {\mathbf {R}}^\mathrm {op}\times {\mathbf {R}}$,

$$\begin{aligned} {\tilde{E}}(M)({\mathbf {a}}):=\varinjlim M|_{\mathbf {ZZ}[(\iota \le {\mathbf {a}})]}. \end{aligned}$$

Also, for any pair ${\mathbf {a}}\le {\mathbf {b}}$ in ${\mathbf {R}}^{\mathrm {op}}\times {\mathbf {R}}$, since $\mathbf {ZZ}[\iota \le {\mathbf {a}}]$ is a subposet of $\mathbf {ZZ}[\iota \le {\mathbf {b}}]$, the linear map $\varinjlim M|_{\mathbf {ZZ}[(\iota \le {\mathbf {a}})]}\rightarrow \varinjlim M|_{\mathbf {ZZ}[(\iota \le {\mathbf {b}})]}$, is uniquely specified by the initial property of the colimit $\varinjlim M|_{\mathbf {ZZ}[(\iota \le {\mathbf {a}})]}$ (Definition A.5 and Remark A.7). This ${\tilde{E}}(M)$ is called the left Kan extension of M along $\iota :\mathbf {ZZ}\hookrightarrow {\mathbf {R}}^{\mathrm {op}}\times {\mathbf {R}}$. Given $M,N:\mathbf {ZZ}\rightarrow {\mathscr {C}}$ and a natural transformation $\Gamma :M\rightarrow N$, universality of colimits also yields an induced morphism ${\tilde{\Gamma }}:{\tilde{E}}(M)\rightarrow {\tilde{E}}(N)$. Given any $M:\mathbf {ZZ}\rightarrow {\mathscr {C}}$, the functor $E(M):{\mathbf {U}}\rightarrow {\mathscr {C}}$ is defined as the restriction ${\tilde{E}}(M)|_{\mathbf {U}}: {\mathbf {U}}\rightarrow {\mathscr {C}}$.

Definition D.4

(Interleaving distance between zigzag modulesBotnan and Lesnick 2018) Let ${\mathscr {C}}$ be a cocomplete category. For any $M,N:\mathbf {ZZ}\rightarrow {\mathscr {C}}$,

$$\begin{aligned} d_{\mathrm {I}}(M,N):=d_{\mathrm {I}}^{{\mathscr {C}}}\left( E(M),E(N)\right) . \end{aligned}$$

1.2 Existing stability theorems

In this section we review the existing stability results from Botnan and Lesnick (2018), McCleary and Patel (2020).

1.2.1 Stability theorem of Patel

We recall McCleary and Patel (2020, Theorem 5.6):

Theorem D.5

(Stability for constructible persistence modules) Let ${\mathscr {C}}$ be an essentially small, abelian¹² category. Let $F,G:{\mathbf {R}}\rightarrow {\mathscr {C}}$ be constructible (Definition F.1). Then,

$$\begin{aligned} d_\mathrm {B}\left( \mathrm {dgm}_{{\mathscr {C}}}(F),\mathrm {dgm}_{{\mathscr {C}}}(G)\right) \le d_{\mathrm {I}}(F,G). \end{aligned}$$

We remark that the persistence diagram $\mathrm {dgm}_{{\mathscr {C}}}(F)$ can be computed via the following process: (1) Re-indexing of $F:{\mathbf {R}}\rightarrow {\mathscr {C}}$ to obtain $D(F):{\mathbf {Z}}\rightarrow {\mathscr {C}}$ (Sect. F), (2) computing the persistence diagram of D(F) by Definition 3.13, and (3) the rescaling of the persistence diagram (Sect. F).

1.2.2 Bottleneck distance

We regard the extended real line $\overline{{\mathbf {R}}}:={\mathbf {R}}\cup \{-\infty ,\infty \}$ as a poset with the canonical order $\le$. Let $\overline{{\mathbf {U}}}:=\left\{ (u_1,u_2)\in \overline{{\mathbf {R}}}^2: u_1\le u_2\right\}$ be equipped with the partial order inherited from $\overline{{\mathbf {R}}}^{\mathrm {op}}\times \overline{{\mathbf {R}}}$. For ${\mathbf {u}}=(u_1,u_2),\ {\mathbf {v}}=(v_1,v_2)\in \overline{{\mathbf {U}}}$, let

$$\begin{aligned} \left\Vert {\mathbf {u}}-{\mathbf {v}}\right\Vert _\infty :=\max \left( \left|u_1-v_1\right|, \left|u_2-v_2\right|\right) . \end{aligned}$$

We can quantify the difference between two persistence diagrams or two barcodes of real intervals, using the bottleneck distance (Cohen-Steiner et al. 2007):

Definition D.6

(The bottleneck distance) Let $X_1,X_2$ be multisets of points in $\overline{{\mathbf {U}}}$. Let $\alpha :X_1\nrightarrow X_2$ be a matching, i.e. a partial injection. We call $\alpha$ an $\varepsilon$-matching if

• (i) for all ${\mathbf {u}}\in \mathrm {dom}(\alpha )$, $\left\Vert {\mathbf {u}}-\alpha ({\mathbf {u}})\right\Vert _\infty \le \varepsilon$,

• (ii) for all ${\mathbf {u}}=(u_1,u_2)\in X_1\setminus \mathrm {dom}(\alpha )$, $u_2-u_1 \le 2\varepsilon$,

• (iii) for all ${\mathbf {v}}=(v_1,v_2)\in X_2 \setminus \mathrm {im}(\alpha )$, $v_2-v_1 \le 2\varepsilon$.

Their bottleneck distance $d_\mathrm {B}(X_1,X_2)$ is defined as the infimum of $\varepsilon \in [0,\infty )$ for which there exists an $\varepsilon$-matching $\alpha :X_1\nrightarrow X_2$.

Recall that $\langle b,d \rangle _{\mathbf {ZZ}}$ for $b,d\in {\mathbf {Z}}$ denotes intervals of $\mathbf {ZZ}$ and that $\langle b,d \rangle$ for $b,d\in {\mathbf {R}}$ denotes intervals of ${\mathbf {R}}$.

Remark D.7

(1) Given a pair of multisets of intervals $\langle b,d \rangle _{\mathbf {ZZ}}$ of $\mathbf {ZZ}$, their bottleneck distance is defined by converting those multisets into the multisets of $\overline{{\mathbf {U}}}$ via the identification $\langle b,d \rangle _{\mathbf {ZZ}}\leftrightarrow (b,d)\in \overline{{\mathbf {U}}}$. (2) A map $\mathbf {Int}(\mathbf {ZZ})\rightarrow {\mathbf {Z}}_+$ can be considered as a multiset of $\mathbf {Int}(\mathbf {ZZ})$ in an obvious way, and thus such maps can also be compared in $d_\mathrm {B}$. (3) The bottleneck distance between multisets of real intervals $\langle b,d \rangle$ is also defined via the identification $\langle b,d \rangle \leftrightarrow (b,d)\in \overline{{\mathbf {U}}}$.

1.2.3 Algebraic stability for zigzag modules

Theorem D.8

(Bottleneck stability for zigzag modules Bjerkevik 2021; Botnan and Lesnick 2018) For any $M,N:\mathbf {ZZ}\rightarrow \mathbf {vec}$,

$$\begin{aligned} d_\mathrm {B}\left( \mathrm {barc}^{\mathbf {ZZ}}(M),\mathrm {barc}^{\mathbf {ZZ}}(N)\right) \le 2\cdot d_{\mathrm {I}}(M,N). \end{aligned}$$

Let M be any zigzag module. By $\mathrm {barc}^{\mathbf {ZZ}}_{{\mathbf {o}}}(M),\mathrm {barc}^{\mathbf {ZZ}}_{\mathbf {co}}(M),\mathrm {barc}^{\mathbf {ZZ}}_{\mathbf {oc}}(M)$ and $\mathrm {barc}^{\mathbf {ZZ}}_{{\mathbf {c}}}(M)$, we mean the subcollection of $\mathrm {barc}^{\mathbf {ZZ}}(M)$, consisting solely of the points of the form $(b,d)_{\mathbf {ZZ}}, [b,d)_{\mathbf {ZZ}}, (b,d]_{\mathbf {ZZ}},$ and $[b,d]_{\mathbf {ZZ}}$, respectively.

Remark D.9

Suppose that the two zigzag modules M, N in Theorem D.8 are the levelset zigzag persistent homology of any two Morse type functions $f,g:X\rightarrow {\mathbf {R}}$ (Definition G.1). Then, the inequality in Theorem D.8 can be extended and strengthened as follows (Botnan and Lesnick 2018; Carlsson et al. 2019, Theorem 4.11): For each $\star \in \{{\mathbf {o}},\mathbf {co},\mathbf {oc},{\mathbf {c}}\}$,

$$\begin{aligned} d_\mathrm {B}\left( \mathrm {barc}^{\mathbf {ZZ}}_\star (M),\mathrm {barc}^{\mathbf {ZZ}}_\star (N)\right)\le & {} d_{\mathrm {I}}(M,N)\\\le & {} \left\Vert f-g\right\Vert _\infty . \end{aligned}$$

Therefore, we also have

$$\begin{aligned}&d_\mathrm {B}\left( \mathrm {barc}^{\mathbf {ZZ}}(M),\mathrm {barc}^{\mathbf {ZZ}}(N)\right) \le \max _{\star }d_\mathrm {B}\left( \mathrm {barc}^{\mathbf {ZZ}}_\star (M),\mathrm {barc}^{\mathbf {ZZ}}_\star (N)\right) \\&\le d_{\mathrm {I}}(M,N)\le \left\Vert f-g\right\Vert _\infty , \end{aligned}$$

where the maximum is taken over all $\star \in \{{\mathbf {o}},\mathbf {co},\mathbf {oc},{\mathbf {c}}\}$.

Proof of Proposition 3.17

In this section we prove Proposition 3.17. For notational simplicity, we set ${\mathbf {P}}=\mathbf {ZZ}$. We remark that the proof extends to arbitrary connected locally finite posets.

1.1 Canonical bases and l-type intervals

Suppose that $M:\mathbf {ZZ}\rightarrow \mathbf {vec}$ is given as $M=\bigoplus _{c\in C} I^{J_c}$ for an index set C. Then for each $(i,j)\in \mathbf {ZZ}$, the dimension of $M_{(i,j)}$ is the total multiplicity of intervals containing (i, j) in $\mathrm {barc}^{\mathbf {ZZ}}(M)=\left\{ \!\!\left\{ J_c: c\in C\right\} \!\!\right\}$. Given any $c\in C$, and $(i,j)\in J_c$, let $e^{(i,j)}_c$ denote the 1 in the field ${\mathbb {F}}$ which corresponds to the (i, j)-component of $I^{J_c}$. Then for each $(i,j)\in \mathbf {ZZ}$, the vector space $M_{(i,j)}$ admits the canonical basis $B_{(i,j)}=\left\{ e^{(i,j)}_c: J_c \ \text{ contains } (i,j) \right\}$ and hence every $v\in M_{(i,j)}$ can be uniquely expressed as a linear combination of the elements in $B_{(i,j)}$, i.e.

$$\begin{aligned} v=\sum _{J_c\ni (i,j)} a_c e_c^{(i,j)}, a_c\in {\mathbb {F}}. \end{aligned}$$

(10)

This expression is called the canonical expression of v. The collection ${\mathscr {B}}=(B_{(i,j)})_{(i,j)\in \mathbf {ZZ}}$ is called the canonical basis of $M=\bigoplus _{c\in C} I^{J_c}$.

In order to prove Proposition 3.17, we identify a certain type of intervals of the poset $\mathbf {ZZ}$:

Definition E.1

(l-type intervals) Any interval I of $\mathbf {ZZ}$ is called l-type¹³ if I is of the form $(b,d)_\mathbf {ZZ}$ for some $b\in {\mathbf {Z}}\cup \{-\infty \}$ and $d\in {\mathbf {Z}}\cup \{\infty \}$ (see Fig. 2).

Notation E.2

Let $M:\mathbf {ZZ}\rightarrow \mathbf {vec}$ be any zigzag module and let $(i,j),(i',j')\in \mathbf {ZZ}$. For $v_{(i,j)}\in M_{(i,j)}$ and $v_{(i',j')}\in M_{(i',j')}$, we write $v_{(i,j)}\sim v_{(i',j')}$ if $(i,j),(i',j')$ are comparable and either

$$\begin{aligned} v_{(i,j)}&=\varphi _M((i',j'), (i,j))(v_{(i',j')})\ \text{(when } (i',j')\le (i,j)) \text{ or } \\ v_{(i',j')}&=\varphi _M((i,j), (i',j'))(v_{(i,j)})\ \text{(when } (i,j)\le (i',j')). \end{aligned}$$

Proof of Proposition 3.17

For simplicity of notation, we prove the proposition when ${\mathbf {P}}=\mathbf {ZZ}$ and $J=(-\infty ,\infty )=\mathbf {ZZ}$. The proof straightforwardly extends to any connected, locally finite poset ${\mathbf {P}}$ and $J\in \mathbf {Con}({\mathbf {P}})$. By Theorem 2.12, we may assume that $M=\bigoplus _{c\in C}I^{J_c}$, where $\mathrm {barc}^{\mathbf {ZZ}}(M)=\left\{ \!\!\left\{ J_c:c\in C\right\} \!\!\right\}$. In what follows, we identify $\mathbf {ZZ}$ with the integers ${\mathbf {Z}}$ via the bijection $(i,j)\mapsto i+j$. Therefore, by $\bigoplus _{k\in {\mathbf {Z}}} M_{k}$ and $\prod _{k\in {\mathbf {Z}}} M_{k}$, we will denote $\bigoplus _{(i,j)\in \mathbf {ZZ}} M_{(i,j)}$ and $\prod _{(i,j)\in \mathbf {ZZ}} M_{(i,j)}$ respectively.

• (i) The limit of M is (isomorphic to) the pair $\left( V,(\pi _k)_{k\in {\mathbf {Z}}}\right)$ described as follows:

  $$\begin{aligned} V:=\left\{ (v_k)_{k\in {\mathbf {Z}}}\in \prod _{k\in {\mathbf {Z}}} M_k:\ \forall k\in {\mathbf {Z}},\ v_{k}\sim v_{k+1} \right\} . \end{aligned}$$

  (11)

  For each $k\in {\mathbf {Z}}$, the map $\pi _k:V\rightarrow M_k$ is the canonical projection.

• (ii) The colimit of M is (isomorphic to) the pair $\left( U, (i_k)_{k\in {\mathbf {Z}}}\right)$ described as follows: U is the quotient vector space $\left( \bigoplus _{k\in {\mathbf {Z}}} M_k\right) /W$, where W is the subspace of the direct sum $\bigoplus _{k\in {\mathbf {Z}}} M_k$ which is generated by the vectors of the form $(\cdots ,0,\ldots ,0, v_k, -v_{k+1},0,\ldots ,0,\cdots )$ with $v_k\sim v_{k+1}$ (Notation E.2). Let q be the quotient map from $\bigoplus _{k\in {\mathbf {Z}}} M_k$ to $U=\left( \bigoplus _{k\in {\mathbf {Z}}}M_k\right) /W$. For $k\in {\mathbf {Z}}$, let the map $\bar{i_k}:M_k\rightarrow \bigoplus _{k\in {\mathbf {Z}}} M_k$ be the canonical injection. Then $i_k:M_k\rightarrow U$ is the composition $q\circ \bar{i_k}$.

Let $0\in {\mathbf {Z}}$. Since the canonical map $\psi _M:\varprojlim M \rightarrow \varinjlim M$ is equal to $i_0 \circ \pi _0$, it suffices to show that the dimension of the vector space $\mathrm {im}(i_0\circ \pi _0)$ is equal to the cardinality of $(-\infty ,\infty )_\mathbf {ZZ}$ in $\mathrm {barc}^{\mathbf {ZZ}}(M)$.

If $M_0$ is the zero space, then it is clear that there is no $(-\infty ,\infty )_\mathbf {ZZ}$ in $\mathrm {barc}^{\mathbf {ZZ}}(M)$ and that $\mathrm {im}(\pi _0)=0$. Therefore, $\mathrm {im}(i_0\circ \pi _0)=0$, and the statement directly follows.

Assume that $M_0$ is not trivial. Define

$$\begin{aligned} C_0:=\{c\in C:\ 0\in J_c\}. \end{aligned}$$

Since $M_0$ is a finite dimensional vector space, $\left|C_0\right|=\dim (M_0)$ is finite and hence we can write $C_0=\{c_1,\ldots ,c_m\}$, where $\dim (M_0)=m$. Also, we can write $M_0\cong \bigoplus _{j=1}^m{\mathbb {F}}_j$, where each ${\mathbb {F}}_j={\mathbb {F}}$ is the component of the interval module $I^{J_{c_j}}$ at $0\in \mathbf {ZZ}$. We identify $M_0$ with $\bigoplus _{j=1}^m{\mathbb {F}}_j$. Then, we claim that

$$\begin{aligned} \mathrm {im}(\pi _0)=\left\{ (a_1,\ldots ,a_m)\in \bigoplus _{j=1}^m{\mathbb {F}}_j:\ a_j=0\ \text{ if } J_{c_j} \text{ is } \text{ not } l\text{-type }\right\} . \end{aligned}$$

(12)

We prove this equality at the end of the proof.

For $j=1,\ldots ,m$, let $e_j:=(0,\ldots ,0,\underset{j-\text{ th }}{1},0,\ldots ,0)\in \bigoplus _{j=1}^m{\mathbb {F}}_j$. By equation (12), the set $B_0=\{e_j: J_{c_j} \text{ is } l\text{-type }\}$ is a basis of $\mathrm {im}(\pi _0)$. Therefore, the dimension of $\mathrm {im}(i_0\circ \pi _0)$ is equal to the dimension of the space that is spanned by the image of $B_0$ under the map $i_0:M_0\rightarrow U$. By invoking item (ii) above, if $J_{c_j}\ne (-\infty ,\infty )_\mathbf {ZZ}$, it follows that $i_0(e_j)=0\in U$.

Let $C_0^{\text{ full }}:=\{c\in C: J_c=(-\infty ,\infty )_{\mathbf {ZZ}}\}$, which is a subset of $C_0=\{c_1,\ldots ,c_m\}$. Assuming that $C_0^{\text{ full }}\ne \emptyset$, suppose that $C_0^{\text{ full }}=\{c_1,\ldots ,c_n\}$ for some $n\le m$ without loss of generality. Invoking item (ii) above, the set $\{i_0(e_{1}),\ldots , i_0(e_{n})\}$ is linearly independent in U. Therefore, we have that

$$\begin{aligned} \mathrm {rank}(i_0\circ \pi _0)=n=(\text{ the } \text{ multiplicity } \text{ of } (-\infty ,\infty )_\mathbf {ZZ} \text{ in } \mathrm {barc}^{\mathbf {ZZ}}(M)), \end{aligned}$$

as desired.

Finally we prove equation (12). First we prove “$\subset$”. Recall item (i) above and pick any $v=(v_k)_{k\in {\mathbf {Z}}}\in V$. Then $\pi _0(v)=v_0=(a_1,\ldots ,a_m)$. Suppose that $c_j\in C_0$ is such that $J_{c_j}$ is not l-type. This implies that the interval $J_{c_j}$ has an endpoint $r=(r_1,r_2)\in \mathbf {ZZ}$ where $r_1=r_2\in {\mathbf {Z}}$ and then either $(r_1+1,r_1)$ or $(r_1,r_1-1)$ is not in $J_{c_1}$. Without loss of generality, assume that $s=(r_1+1,r_1)\in \mathbf {ZZ}$ does not belong to $J_{c_1}$. By the choice of $v=(v_k)_{k\in {\mathbf {Z}}}$, we have $v_0\sim v_1\sim \ldots \sim v_{2r_1}\sim v_{2r_1+1}$, and this leads to that $a_j=0$.

Next we show “$\supset$”. Pick any $(a_1,\ldots ,a_m)$ in the RHS of equation (12). For each $k\in {\mathbf {Z}}$, define $v_k\in M_k$ using the canonical expression:

$$\begin{aligned} v_k:=\sum _{J_c\ni k} b_c e^{k}_{c}, b_c\in {\mathbb {F}}, \end{aligned}$$

where $b_c=0$ if $c\not \in C_0$ and $b_c=a_j$ if $c=c_j\in C_0=\{c_1,\ldots ,c_m\}$. Let $v:=(v_k)_{k\in {\mathbf {Z}}}$. Then, one can check that $\pi _0(v)=(a_1,\ldots ,a_m)$, completing the proof. $\square$

Constructible persistence modules and re-indexing

Patel generalizes the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible ${\mathbf {R}}$-indexed diagrams valued in a symmetric monoidal category (Patel 2018).

Definition F.1

(Constructible ${\mathbf {R}}$-indexed diagrams) Let $S=\{s_1<s_2<\ldots <s_n\}$ be a finite set of ${\mathbf {R}}$. A diagram $F:{\mathbf {R}}\rightarrow {\mathscr {C}}$ is S-constructible if

• (i) for $p\le q<s_1$, $\varphi _F(p, q)$ is the identity on e,

• (ii) for $s_i\le p\le q< s_{i+1}$, $\varphi _F(p, q)$ is an isomorphism,

• (iii) for $s_n\le p\le q$, $\varphi _F(p, q)$ is an isomorphism.

If $G:{\mathbf {R}}\rightarrow {\mathscr {C}}$ is T-constructible for some finite set $T\subset {\mathbf {R}}$, then we call G constructible.

1.1 Re-indexing an ${\mathbf {R}}$-indexed diagram by ${\mathbf {Z}}$

Let $F:{\mathbf {R}}\rightarrow {\mathscr {C}}$ be S-constructible with $S=\{s_1<s_2<\ldots <s_n\}$. A functor $D(F):{\mathbf {Z}}\rightarrow {\mathscr {C}}$ that contains all the algebraic information of F would be defined as follows:

$$\begin{aligned} D(F)_{i}&={\left\{ \begin{array}{ll} e,&{}\quad \text{ for } i\le 0,\\ F_{s_i},&{}\quad \text{ for } i=1,\ldots ,n,\\ F_{s_n},&{}\quad \text{ for } i\ge n+1. \end{array}\right. }\\ \varphi _{D(F)}\left( i,i+1\right)&= {\left\{ \begin{array}{ll} \mathrm {id}_e,&{}\text{ for } i\le 0,\\ \varphi _F(s_i,s_{i+1}), &{}\text{ for } i=0,1,\ldots ,n-1\text{, } \text{ where } \\ &{}s_0 \text{ is } \text{ arbitrarily } \text{ chosen } \text{ in } (-\infty ,s_1),\\ \mathrm {id}_{F_{s_n}},&{}\text{ for } i\ge n. \end{array}\right. } \end{aligned}$$

When ${\mathscr {C}}=\mathbf {vec}$, there exists a bijection from the barcode of F to that of D(F) via $[s_i,s_j)\mapsto [i,j-1]$ for $1\le i<j\le n$, and $[s_i,\infty ) \mapsto [i,\infty )$ for $1\le i\le n$.

1.2 Re-indexing a ${\mathbf {Z}}$-indexed diagram by $\mathbf {ZZ}$

Let $F:{\mathbf {Z}}\rightarrow {\mathscr {C}}$. Let us define $L(F):\mathbf {ZZ}\rightarrow {\mathscr {C}}$ as follows (Botnan and Lesnick 2018, Remark 4.5):

$$\begin{aligned} L(F)_{(i,i)}=L(F)_{(i+1,i)}&=F_i\\ \varphi _{L(F)}\left( (i+1,i),(i,i)\right)&=\mathrm {id}_{F_{i}}\\ \varphi _{L(F)}\left( (i-1,i),(i,i)\right)&=\varphi _F(i-1, i). \end{aligned}$$

When ${\mathscr {C}}=\mathbf {vec}$ and $F_i=0$ for $i\le 0$, there exists a bijection from the barcode of F to that of L(F) via $[a,b]\mapsto [a,b+1)_{\mathbf {ZZ}}$ for $a,b\in {\mathbf {Z}}$ with $a\le b$, and $[a,\infty )\mapsto [a,\infty )_{\mathbf {ZZ}}$ for $a\in {\mathbf {Z}}$.

Rigorous definition of Reeb graphs

In order to introduce the definition of Reeb graphs, we begin by introducing the notion of Morse type functions from Botnan and Lesnick (2018), Carlsson et al. (2009).

Definition G.1

(Morse type functions) Let X be a topological space. We say that a continuous function $p:X\rightarrow {\mathbf {R}}$ is of Morse type if

• (i) There exists a strictly increasing function ${\mathscr {G}}:{\mathbf {Z}}\rightarrow {\mathbf {R}}$ such that $\lim _{i\rightarrow +\infty }{\mathscr {G}}(i)=\infty$, $\lim _{i\rightarrow -\infty }{\mathscr {G}}(i)=-\infty$ and such that for each open interval $I_i=({\mathscr {G}}(i),{\mathscr {G}}(i+1))$ there exist a topological space $Y_i$ and a homeomorphism $h_i:I_i\times Y_i\rightarrow p^{-1}(I_i)$ with $f\circ h_i$ being the projection $I_i\times Y_i \rightarrow I_i$.

• (ii) Each homeomorphism $h_i:I_i\times Y_i \rightarrow p^{-1}(I_i)$ extends to a continuous function

  $$\begin{aligned} {\bar{h}}_i:\bar{I_i}\times Y_i \rightarrow p^{-1}(\bar{I_i}), \end{aligned}$$

  where $\bar{I_i}$ denotes the closure of $I_i$.

• (iii) For all $t\in {\mathbf {R}}$ and $k\in {\mathbf {Z}}_+$, $\dim \mathrm {H}_k\left( p^{-1}(t)\right) <\infty$.

We introduce the definition of Reeb graphs (De Silva et al. 2016).

Definition G.2

(Reeb graphs) Let X be a topological space and let $p:X\rightarrow {\mathbf {R}}$ be of Morse type. If the topological spaces $Y_i$ as in Definition G.1 (i) are finite sets of points with the discrete topology, then the pair (X, p) is said to be a Reeb graph.¹⁴ See Fig. 5 (A) for an illustrative example.