Stratifications on the Ran Space

We describe a partial order on finite simplicial complexes. This partial order provides a poset stratification of the product of the Ran space of a metric space and the nonnegative real numbers, through the Čech simplicial complex. We show that paths in this product space respecting its stratification induce simplicial maps between the endpoints of the path.


Motivation
Persistent homology uses filtrations of algebraic objects [7], most often simplicial complexes, to produce persistence diagrams. Simplicial complexes have been used [1,14] to probe the topology of the underlying space. A key aim of this exposition is to combine the filtration of a particular simplicial complex with the different choices of simplicial complexes that can be made by sampling a space. With such a combination, we are motivated to answer the following questions: -If there is a path in from one finite sample of M to another: -can the changes in filtrations between these samples be precisely described? -can the persistent homology computation of a new sample be simplified by using the results of a different sample?
-Can we construct a space of all possible persistence diagrams by keeping track of homological changes of the simplicial complex coming from a sample of M and a distance r, as both change?
These questions, some of which have been already considered [5], would be greatly This is the first introduction, to our knowledge, of this poset.

Overview
The scope of the present work is to stratify Ran(M) × R 0 . Fix M to be a metric space, and to a finite subset P ⊆ M, build theČech complex. This simplicial complex has P ⊆ P defining a (|P |−1)-simplex whenever the intersection of (closed) r-balls around the points of P is non-empty. We denote byČ this assignment of a simplicial complex to a pair (P , r) ∈ Ran(M) × R 0 of a sample P of M and a nonnegative distance r.
Our contributions are first in the introduction of a partial order on isomorphism classes of simplicial complexes [SC] in Section 3.2. We also introduce the concept of a "frontier simplicial complex" that refines the notion of a simplicial complex in Section 4. . Finally, we show in Construction 1 that every path γ in Ran(M)×R 0 that respects its stratification induces a unique simplicial mapČ(γ (0)) →Č(γ (1)).

Background
Let SC be the set of finite, abstract simplicial complexes. 1 A simplicial complex C is defined by its vertices and simplices, that is, a pair of sets (V (C), S(C)) with S(C) ⊆ P (V (C)) closed under taking subsets.

Topological Spaces
Let X be a topological space.
for any X, Y ⊆ M. On the product space Ran(M) × R 0 we use the sup-norm Definition 1 Given a pair (P , r) ∈ Ran(M) × R 0 , theČech complex on P with radius r is the simplicial complex with vertices P , and P ⊆ P a simplex whenever p∈P B(p, r) = ∅. This assignmentČ : Some of the spaces we are interested in are semialgebraic. Recall that a set in R N is semialgebraic if it can be expressed as a finite union of sets of the form for polynomial functions f i , g i : R N → R.

Stratifications
Let (A, ) be a poset, or simply A when is clear from context.

Example 1
The set of simplices of a simplicial complex C forms a poset under inclusion. This is called the face poset of C.
Remark 1 A poset (A, ) may be interpreted as a category, whose objects are A and Hom(a, b) = * if a b and ∅ otherwise. A poset may also be interpreted as a topological space endowed with the Alexandrov topology, whose basis contains open sets of the form U a := {b ∈ A : a b}, for all a ∈ A.
When A is clear from context, f is simply called a stratification, and X is called Astratified by f , or just A-stratified, or even stratified. We write X a := {x ∈ X : f (x) = a} for the strata of X and A >a := {b ∈ A : b > a} for the subposet based at a particular element a ∈ A.
Given two stratifications f : X → A and g : Y → B, a stratified map from f to g is a pair of continuous maps φ 0 : X → Y and φ 1 : Definition 4 Let f : X → A be an A-stratification of X. Then X is conically stratified at x ∈ X by f if there exist -a topological space Z, -an A >f (x) -stratified topological space L, and -an stratified map Z × C(L) → X that is an open embedding whose image contains x.
The space X is conically stratified by f if it is conically stratified at every x ∈ X by f , in which case we call f a conical stratification of X.
The product Z × C(L) is canonically stratified by projection to the cone factor, that is, by the map (z, c) → g(c) for g a stratification of C(L). Fig. 1, the spaces C 2 , S 2 , S 3 are conically stratified, while C 1 , C 3 , S 1 are not. The spaces C 1 , S 1 fail to be conically stratified at every point on the equator, while C 3 fails to be conically stratified at the complex number 1 (see Example 3).

Supporting Results
In this section we develop ideas that support the main statements of Section 4. First we explore the implications for conical stratifications. Fig. 1 for examples of spaces satisfying the frontier condition.

Conical Stratifications
Lemma 1 Let f be an A-stratification of a topological space X whose strata are pathconnected. If f is a conical stratification, then f satisfies the frontier condition.
First note that L does not depend on x, as the image of ϕ contains an open neighborhood has the same associated L (up to a stratified homeomorphism). Indeed, suppose that x ∈ X b exists with an open embedding ϕ x : Z x × C(L x ) and L = L x . Given a path γ : I → X b from x to x , letting L γ (t) be the A >f (γ (t)) -stratified space guaranteed to exist by Definition 4, at t = sup t∈I {L γ (s) = L ∀ s t} we will arrive at a contradiction to the previous observation.
Next, suppose that (X a \ X a ) ∩ X b = ∅, and let x ∈ (X a \ X a ) ∩ X b . Given the stratified cone g : C(L) → A b from the embedding ϕ, it follows that b a, since every open neighborhood of x in X intersects X a . Hence C(L) b ⊆ C(L) a , as the stratum C(L) b of the cone point b is adjacent to all other strata of the cone, and a is in the Fig. 1 Three stratifications of the circle and the sphere, with higher vertical position indicating higher order in the poset. The spaces C 1 and C 2 are great circles through the poles of S 1 and S 2 , respectively. See Examples 2, 3, 4 for observations about these stratifications image of g by assumption. Hence Z × C(L) b ⊆ Z × C(L) a , both viewed as subsets of Z × C(L). By continuity of the embedding ϕ, it follows that Since Finally, since L is the same for all elements of X b , a must be in the image of the associated cone map, and this is enough to conclude that every element of X b has a neighborhood within the closure of X a . Hence X b ⊆ X a .
The converse of Lemma 1 is false, as Example 3 shows.
Example 3 Consider the circle C 3 from Fig. 1, embedded as the unit circle in the complex numbers C. This circle is stratified by the poset with relations x j y j and x j +1 y j for all j ∈ Z >0 . To ensure continuity of the stratifying map at the complex number 1, we add the relations x 1 x j for all j ∈ Z 2 . The stratifying map f : C 3 → A is given by That is, the black dot in C 3 in Fig. 1 corresponds to x 1 , each red dot corresponds to an x j 2 , and each blue interval corresponds to a y j . The frontier condition is satisfied trivially for strata (C 3 ) x j , as they are already closed in hence the frontier condition is also satisfied here.
However, C 3 is not conically stratified at 1 = e i2π . Indeed, following Definition 4, we note that Z must be { * }, as {1} = (C 3 ) x 1 is 0-dimensional. So if C 3 were conically stratified at 1, there must be some open neighborhood of 1 that is the homeomorphic image of a cone To have an open embedding C(L) → C 3 , the cone C(L) must have strata that correspond to strata in the open neighborhood of 1. Since every neighborhood of 1 contains elements of the form e iθ where θ ∈ (0, ε), for every ε > 0, such a construction would imply that there are distinct 0-dimensional strata in C(L) corresponding to (C 3 ) x = {e i2π/ }, for every integer > 2π/ε. This is a contradiction, as the only 0-dimensional stratum in C(L) is the cone point.
Example 4 In Fig. 1, C 3 is compatible with C 2 , and C 2 is compatible with C 1 . Similarly, S 3 is compatible with S 2 , and S 2 is compatible with S 1 .
A stratification is semialgebraic if all its strata are semialgebraic sets.
Lemma 2 Let f be a semialgebraic stratification of a closed semialgebraic set X. Then there exists a conical semialgebraic stratification of X compatible with f . Proof Let f : X → A be as in the statement. By [15,Theorem II.4.2], there exists a simplicial complex K whose geometric realization |K| is homeomorphic to X, and a stratification g : |K| → (S(K), ⊆) that refines f . We recall briefly that the geometric realization |K| is a topological space embedded in Euclidean space, with n-simplices represented by n-dimensional subspaces.
This stratification of the geometric realization of a simplicial complex [13, Definition A.6.7] is the canonical one, identifying interiors of simplices with their corresponding simplices in the face poset (S(K), ⊆). This map is conical by [13, Proposition A.6.8].
The simplicial complex K is unique (up to simplicial complex isomorphism) only if X is bounded [15,Remark II.4.3]. Next we develop a new structure on simplicial complexes.

Simplicial Complexes
For C, C ∈ SC, a simplicial map is a function V (C) → V (C ) such that the induced map on S(C) has image in S(C ). In other words, a simplicial map sends simplices of C to simplices of C .
For C ∈ SC, we denote by [C] the set of simplicial complexes isomorphic to C. In a similar fashion, we write [SC] for the set of isomorphism classes of simplicial complexes.

Definition 5 Let be the relation on [SC] given by [C]
[C ] whenever there is a simplicial map C → C that is surjective on vertices. [C ] must be injective on vertices, and so injective on simplices. Similarly, the same properties hold any map C → C inducing [C ] [C]. Hence we have a map C → C that is bijective on simplices, so C ∼ = C , and For transitivity, suppose that [C] [C ] and [C ] [C ]. Then there exists a simplicial map C → C that is surjective on V (C ), as well a simplicial map C → C that is surjective on V (C ). The composition of these two simplicial maps is a simplicial map C → C , and The same arguments show that defines a preorder on SC. Figure 3 shows the Hasse diagram of ([SC], ) for all simplicial complexes up to 3 vertices.

Remark 2
The assumption that all simplicial complexes in SC are finite is key to proving Lemma 3, as anti-symmetry needs to compare sizes of sets. Figure 4 gives one such example where anti-symmetry is violated in the non-finite case.

Main Results
There is a natural point-counting map Ran(M) → Z >0 , which is a stratification by Briefly, havingr r 1 guarantees that points will not merge in the open ball, and having r r 2 guarantees that simplices among the P i are neither lost nor gained in the open ball.
In other words, there is a well-defined and surjective map φ : Q → P for which φ(q) = P i whenever q ∈ B(P i ,r/4). Next, we claim φ is a simplicial map. Take Q ⊆ Q and suppose thatČ(Q , s) is a (|Q | − 1)-simplex. Let P = {P 0 , . . . , P } ⊆ P be such that Q ⊆ i=1 B(P i ,r/4) and Q ∩ B(P i ,r/4) = ∅, for 1 i . Suppose, for contradiction, thatČ(P , r) is not a (|P | − 1)-simplex, or equivalently, thatčr(P , r) < 0. Then 0 čr(P , r) +r/2 (by Eq. 8 and thatr r 2 ) = r − d M (P ,čs(P )) +r/2 (by definition ofČech radius) Case 2: There is some P ⊆ P with |P | > 1 andčr(P , r) = 0. Then r 2 = 0 from Eq. 8, so let and letr be the smallest of the two values r 1 and r 2 . As in Case 1, we claim the open neighborhood B ∞ ((P , r),r/4) of (P , r) is contained within [Č] −1 (U [C] ). The proof of this claim proceeds as in the first case: the only place that r 2 was used was to state that 0 čr(P , r) +r/2, in showing thatČ(P , r) is indeed a (|P | − 1)-simplex. Ifčr(P , r) = 0, then we already have this conclusion, and it is unnecessary to get to the contradiction. That is, φ still extends to a simplicial map, and [Č] −1 (U [C] ) is open in this case as well. However, Lemma 1 implies that [Č] is not a conical stratification.

It follows that
Example 5 Consider the space of at most 2 points Ran 2 (I ) on the unit interval I , and the space X = Ran 2 (I ) × R 0 , as shown in Fig. 6. Take x = ({p 1 , p 2 }, r) ∈ X, with p 1 = 0, p 2 = 1 2 and r = 1 4 . For y = ({p 1 , p 2 }, r < r), note that (11) Moreover, y is in the closure of of both X b and X a , that is, (X a \X a )∩X b = ∅. However, for z = ({p 1 , p 2 }, r > r) we see that z ∈ X b and z ∈ X a , meaning that X b ⊆ X a . Hence [Č] does not satisfy the frontier condition, and so by Lemma 1 cannot be a conical stratification.
One solution is to make a new stratum for points similar to x in Example 5. That is, for every  By "closed under taking supersets" we mean σ ∈ F (C) implies τ ∈ F (C) whenever σ ⊆ τ and τ ∈ S(C). A map of frontier simplicial complexes (V , S, F ) → (V , S , F ) is defined analogously to a map of simplicial complexes. That is, we require it to be a map on the vertices V → V which must induce a map on simplices S → S and on frontier simplices F → F . Figure 7 shows maps among all non-empty frontier simplicial complexes with at most 3 vertices.
Given a pair (P , r) ∈ Ran(M) × R 0 , augmentingČ(P , r) with the set F such that P ∈ F whenever P ∈ S(Č(P , r)) andčr(P , r) = 0 defines a frontier simplicial complex. This follows as theČech radius is 0 when the intersection of closed r-balls around the elements of P is non-empty but does not contain an open set.  The last statement follows as the proof of Theorem 1 was split up into two cases where thě Cech radius is and is not zero, so all that remains is to keep track of the frontier simplices throughout the proof. For a clearer result, we restrict to semialgebraic sets and fix an upper bound n ∈ Z >0 . We also employ some results about the algebra of semialgebraic sets, specifically that products [ to itsČech setčs(P ). TheČech set, from Eq. 5, is a semialgebraic set, as it is the intersection of balls, and the function that measures distance to a semialgebraic set is also semialgebraic, by [15, I.2.9.11]. Finally, a subset of a semialgebraic set defined by semialgebraic functions on the first set is itself semialgebraic in R N , by [

Stratifying Paths
For X, a topological space, recall Sing(X) is the simplicial set of continuous maps |Δ k | → X, where Δ k is the standard k-simplex. Let A be a poset and f : X → A a stratification.

Definition 7
An entrance path in X is a continuous map σ : |Δ k | → X for which there exists a chain a 0 · · · a k in A such that f (σ (0, . . . , 0, t i , . . . , t k )) = a k−i and t i = 0, for all i.
Contrast this with the more common definition of an exit path, as in [18], which is the same, but with f (σ (t 0 , . . . , t i , 0, . . . , 0)) = a i and t i = 0, for all i. The choice of "entrance" instead of "entry" comes from interpreting "exit" as a noun rather than a verb. Some examples of entrance paths are given in Fig. 8.
The subsimplicial set of Sing(X) of all entrance paths is denoted Sing A (X). In this context, a very roundabout way of defining theČech mapČ from Definition 1 would be as an assignmentČ is the nerve. This description is useful, however, when generalizing from points (0-simplices) to paths (1-simplices), in which case we only have to change the subscripts from 0 to 1.
Remark 5 Recall from Section 3.2 that a simplicial map C → C is a map V (C) → V (C ) which, when applied to elements of S(C), gives elements of S(C ). The claim in Fig. 9 The class [Č](γ i (t, 1 − t)) for i = 1, 2 is constant for all t, and for i = 3, 4 is constant only for t ∈ (0, 1]. The simplicial maps associated to γ 1 and γ 2 are both the identity, while the map associated to γ 3 is different from the one associated to γ 4 (and neither are the identity) Construction 1 that the γ i satisfy the conditions of a simplicial map follows by several observations: -Any γ i (t, 1 − t) may coincide only for t = 0, that is, at the end of the path.
-An intersection i B(γ i (t, 1 − t), r t ) that is non-empty for t = 1 must be non-empty for all t ∈ (0, 1], else γ would not be an entrance path. -The only possibility of i B(γ i (t, 1 − t), r t ) being non-empty for all t ∈ (0, 1] and empty for t = 0 is if r 0 = 0, in which case all the γ i (0, 1) coincide, which describes a surjective map from a simplex to a single vertex. -Since the balls B are closed, it is impossible to preserve simplicial complex isomorphism class by making one intersection empty at the same instant t ∈ (0, 1) as another is made non-empty.
Here we have used r t for the R 0 component of γ (t, 1 − t).
ConsideringČ asČech 0 and with the construction above ofČech 1 , we are tempted to generalize the result further. Examples abound of C, C ∈ SC with different simplicial maps C → C that are surjective on vertices, but it is not immediate that it is possible to construct an entrance path into some [SC]-stratified Ran(M) × R 0 joining such simplicial maps. That is, we do not immediately find counterexamples to Conjecture 2, so we hope it is true.
We conclude this section with some observations about paths.

Discussion
We have presented a thorough description of the space Ran(M) × R 0 , motivated by its interpretation as the space of all simplicial complexes on a metric space M. Our description gives a stratification [Č] based on theČech construction of a simplicial complex on M. This stratification may be refined into a structurally cleaner but more opaque conical stratification (Theorem 2), as well as a combinatorially motivated stratification, though it is unclear if the latter is conical (Remark 3). We use [Č] to associate paths with simplicial maps in Section 4.2, relating them to existing constructions in persistent homology (Remark 6) and conjecturing that the association extends to continuous maps of higher-dimensional simplices (Conjecture 2).
This approach prompts questions about the new concepts we introduced: -What does the poset of frontier simplicial complexes look like? -Is the [FSC]-stratification of Ran(M) × R 0 conical?
We are also motivated to push further the inquiry into interpreting paths: -Does theČech map and its generalizationČech 1 to paths extend to higher-dimensional simplices?
The choice of working with isomorphism classes of simplicial complexes, in which the vertices have no order, and simplicial sets (for the entrance paths and the nerve), in which the 0-simplices are ordered, does not make our work easier. An alternative approach would have been to take the nerve of the face poset of a simplicial complex, which is a simplicial set, instead of the simplicial complex itself. Part of the appeal of using isomorphism classes is that less information is remembered, hence it is not immediate that using simplicial sets would help.
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