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## One parameter persistence

One-parameter persistence is the persistence theory associated to the poset (P; ) = (R; ). Hence, a one-parameter persistence module is an object of Pers(kR).

The main example of one-parameter persistence module is: given a function u : X ! R (not necessarily continous), one defines its sub-level sets filtration as the functor S(u) from the poset category (R; 1 ) to the category of topological spaces Top,1 u)(s) = u ( ; s) defined by S( 1 ( , and

S(u)(s t) is the inclusion of u ( ; s) into u ; t). For n 0, the n-th sublevel set persistence modules associated to u is the functor Sn(u) := Hsingn S(u); where Hsingn denotes the n-th singular homology functor with coeﬃcients in k.

It is easy to verify that the functor T : ([0; +1); ) ! End((R; )) defined by T (« ) : s 7!s + » is a flow on (R; ). This flow induces an interleaving distance on Pers(kR) which allows us to express the stability properties of the mapping u 7! nS(u).

**Theorem 2.1.19** ( [CdSGO16])

Let u; v : X ! R be two maps from a topological space X to R, and n 0. Then: dTI (Sn(u); Sn(v)) sup ju(x) v(x)j: x2X

Moreover, dTI is universal amongst all other metrics satisfying this sta-bility property.

**Theorem 2.1.20** ( [Les12] – Thm. 5.5)

Let k be a prime field, ie. k = Z=pZ with p a prime number or k = Q. Let d : Obj(Pers(kR)) Obj(Pers(kR)) ! R which satisfies axioms (M1) (M4) (see prop. 2.1.13). Assume also that for any u; v : X ! R and n 0, d satisfies:

d(Sn(u); Sn(v)) sup ju(x) v(x)j: x2X

Then d dTI .

In addition to satisfying this stability property, pfd one-parameter per-sistence modules (def. 2.1.4) enjoy a combinatorial description:

Theorem 2.1.21 ( [CB12] – Thm. 1.1)

For any pointwise finite dimensional one-parameter persistence module M, there exists a unique multi-set of intervals of R (that is, an interval can appear several times in the list) noted B(M) , such that : I2B(M) B(M) is the barcode of M.

Therefore, the barcode of M, which consists of a list (with repetition) of intervals of R, completely determines the isomorphism class of M. If u : X ! R is such that Sn(u) is pfd, one defines its n-th sublevel sets barcode as the barcode of Sn(u). There is no reason a priori that one can compute the interleaving distance between two persistence modules in a combinatorial way, from the knowledge of their barcodes. Nevertheless, the following result asserts that it is possible in the very specific context of one-parameter persistence. Usually, it is referred to as the isometry theorem.

**Theorem 2.1.22** ( [Les15] – Thm. 3.4)

Let dTB be the bottleneck distance associated to dTI . Then for M and N any pointwise finite dimensional persistence modules over R: dTI (M; N) = dTB(M; N):

This theorem is at the core of all the applications of persistence, both in machine learning and in theoretical areas of mathematics such as symplectic geometry.

### Level-sets persistence

Level-sets persistence is the persistence theory associated to a certain class of persistence modules over the poset + = f(x; y) 2 R2 j x + y > 0g, endowed with the following order: (x; y) (x0; y0) () x x0 and y y0:

Level-sets persistence take its name from the level-set filtration asso-ciated to a real-valued function. Consider u : X ! R a map from a topological space X to R. Then, one defines the level-sets filtration of u as the functor L(u) from the poset category ( +; ) 1to the category of topological spaces Top, defined by L(u)((x; y)) = u (] x; y[), and L(u)((x; y) (x0; y0)) is the inclusion of u 1(] x; y[) into u 1(] x0; y0[).

For n 0, the n-th level-sets persistence module associated to u is the functor Ln(u) := Hsingn L(u); where Hsingn denotes the n-th singular homology functor with coeﬃcients in k.

It is easy to verify that the functor T : ([0; +1); ) ! End(( +; )) defined by T (« ) : s 7!s + (« ; « ) is a flow on ( +; ). This flow induces an interleaving distance on Pers(k + ) which allows us to express the stability properties of the mapping u 7! L(u).

**Theorem 2.1.23** ( [BL17]) Let u; v : X ! R be two maps from a topological space X to R, and n 0.

Then: dTI (Ln(u); Ln(v)) sup ju(x) v(x)j: x2X

We shall extend the definition of the shift functor to vectors of (R 0)2. Let M 2 Obj(Pers(k + )) and s = (s1; s1) 2 (R 0)2. We shall denote in the sequel sx = (s1; 0) and sy = (0; s2). The persistence module M[s] is defined by: M[s](v) = M(v + s) for any v 2 +, and M[s](v w) = M(v + s w + s).

It is immediate to check that there is a canonical s-smoothing morphism: sM : M ! M[s]:

When the context is clear, we shall not make explicit the smoothing morphisms in our diagrams. To a persistence module M over + and any s 2 (R 0)2, one can associate a commutative diagram in Pers(k + ) :

M[sy] M[s]

/M[sx]

This induces the short complex:

Mfsg := M ! M[sx] M[sy]! M[s] (2.6)

where the first map is sMx M[sx] M[sy]

sMy and the second one is ( sy ; sx )

in matrix notations.

**Definition 2.1.24**

An object M 2 Pers(k + ) is a middle-exact persistence module if the com-plexes Mfsg are exact for every s 2 R2>0.

**Remark 2.1.25**

We think of middle-exact persistence modules as being the analogue for the poset + of half the terms of the Mayer-Vietoris long exact sequence relating the various homology groups of two open subsets of a space, their reunion and intersection. What is missing to have a long exact sequence are precisely the connecting homomorphisms relating homology groups of diﬀerent degrees. In Section 4.2.1, we will precisely introduce an additional data on a (graded) middle-exact object of Pers(k + ) to obtain such long exact sequences.

Middle-exact persistence modules have a barcode decomposition similar to persistence modules over R that we now describe. First we specify the various geometric types, called blocks, of the barcode.

**Notation.** Given a < b in R [ f 1g, we shall denote by ha; bi any of the four intervals ]a; b[; [a; b]; ]a; b]; [a; b[.(x; x)

Figure 2.1 – On the left a block of type bb pictured in blue. On the middle a block of type vb pictured in green. On the right a block of type db in red and its dual block of type bb+ in yellow. The various coordinates refers to the intersection points of the boundaries of the blocks with the anti-diagonal as well as the extremum of the birth or death blocks. The dashed boundary lines means that the boundary line is not part of the block.

**Definition 2.1.26**

A block B is a subset of R2 of the following type:

1. A birthblock (bb for short) if there exists (a; b) 2 R2 such that B =

ha; 1i hb; 1i, where a and b can both equal simultaneously. Moreover, we will write that B is of type bb+ if a + b > 0, and of type bb if a + b 0.

2. A deathblock (db for short) if there exists (a; b) 2 R2 such that B = ; ai h ; bi. Moreover, we will write that B is of type db+ if a + b > 0 and of type db if not.

A horizontalblock (hb for short) if there exists a 2 R and b 2 R [ f+1g such that B = R ha; bi.

A verticalblock (vb for short) if there exists a 2 R [ f+1g and b 2 R such that B = ha; bi R.

**Remark 2.1.27**

Blocks are defined over the whole plane R2 and not just +.

**Remark 2.1.28**

Note that a deathblock B is characterized by its supremum 2, that is supfs 2 Bg together with the data of whether its two boundary lines are in the block or not (note that the supremum is inside B if and only if both boundaries lines are). Similarly a birth block B0 is characterized by its infimum inffs

2. which is easily seen to be, if B = h ; xi h ; yi, the point (x; y) 2 R2

B0g and the data of whether its boundary lines are in B or not. Note also that the vertical and horizontal blocks never have finite extrema.

**Definition 2.1.29** (Duality between death and birth blocks)

The dual of a deathblock B is the birthblock By whose infimum is the supremum of B and whose vertical (resp. horizontal) boundary line is in By if and only if the vertical (resp. horizontal) boundary line of B is not.

Dually we define the dual Cy of a birthblock C as the death block whose supremum is the infimum of C and whose vertical (resp. horizontal) boundary line is in Cy if and only if the the vertical (resp. horizontal) boundary line of C are not.

**Remark 2.1.30**

The rule B 7!By is involutive: (By)y = B, thereby it exhibits a perfect duality between death and birth blocks. Furthermore, note that the dual of a deathblock is of type bb+ if and only if the deathblock has a non-trivial intersection with + i.e. is in db+.

One easily observe that for a block B, the set B \ + is an interval of the poset +. Therefore the persistence module kB\ + is well defined according to definition 2.1.2. For simplicity, we will omit + and write kB for kB\ + . This leads to the following remark:

**Remark 2.1.31**

If B is in db , then kB = 0. Therefore we will usually not consider the block modules associated to such negative deathblocks. In what follows, the reader can safely assume that when we speak about a deathblock we mean an element of db+, unless otherwise stated.

Let us denote, for s 2 +, B s = ft s; t 2 Bg; this is a block of the same type as B (up to sign ).

**Lemma 2.1.32**

Let B be a block and s 2 +. There is a canonical isomorphism kB[s] kB s

**Theorem 2.1.33 ( [CO17], [BCB18])**

Let M 2 Pers(k + ) be middle exact and pointwise finite dimensional (pfd).

Then there exists a unique multiset of blocks B(M) such that: M ’ M kB: B2B(M) B(M) is the barcode of M.

**Proposition 2.1.34**

Let u : X ! R be a continuous map of topological spaces. Then Ln(u) is a middle-exact persistence module for all n 0.

**Proof**

Let s = (s1; s2) 2 (R>0)2 and n 0. The complex Ln(u)fsg is exact if and only if it is exact when evaluated on all points (x; y) 2 +. The sequence Ln(u)(x; y)! L n(u)[sx](x; y) Ln(u)[sy](x; y)! L n(u)[s](x; y) is exact, since it is the middle part of the Mayer-Vietoris sequence asso-ciated to the open covering:

u 1(] x s1; y + s2[) = u 1 (] x s1; y[) [ u 1 (] x; y + s2[) ;

u 1 (] x s1; y[) \ u 1 (] x; y + s2[) = u 1 (] x; y[) :

Therefore, if u is such that Ln(u) is pfd, it admits a decomposition as a direct sum of block modules. Therefore, we can define the n-th level-sets barcode of u as the multiset of blocks B(Ln(u)). Similarly to the case of one-parameter persistence, one can ask whether it is possible to find a combinatorial expression for the interleaving distance between two middle-exact persistence modules. More precisely, if the interleaving distance is equal to its associated bottleneck distance, noted dTB. Bjerkevik gave a positive answer to this question.

**Theorem 2.1.35 ( [Bje16] – Thm 3.3)**

Let M and N be two pfd middle-exact persistence modules over +, then: dTI (M; N) = dTB(M; N):

Note that this result is not true without the middle-exacteness assump-tion. In [Bje16, Ex. 5.2], the author gives an example of two persistence modules over +, M and N, which are finite direct sums of interval modules and verify:

dTI (M; N) = 1; dTB(M; N) = 3:

Figure 2.2 – The topological space X in R2 and its subset X0 in red.

A natural question to ask is to compare the diﬀerence in topological information given by the sublevel-sets and the level-sets persistence modules construction. As one could expect, the level-sets persistence modules of a function contains strictly more information than its sub-levelsets module. This statement will be made precise in chapter 5. For now, we only present an example illustrating this phenomenon.

**Table of contents :**

**1 Introduction **

**2 Background & first results **

2.1 Persistence theory

2.2 Persistence and sheaves

2.3 Stable resolutions of persistence modules

**3 The derived isometry theorem **

3.1 Introduction and preliminaries

3.2 Homomorphisms in Db Rc(kR)

3.3 Structure of « -interleavings

3.4 Isometry theorem and graded barcodes

3.5 Applications

**4 Level-sets persistence and sheaves **

4.1 Introduction

4.2 The category MV(R)

4.3 Stable sheaf theoretic interpretation of persistence

4.4 Isometric pseudo-equivalence between MV(R)sf and Db Rc(kR) 126

4.5 Applications

**5 Ephemeral persistence modules and distance comparison **

5.1 Introduction

5.2 Sheaves on and Alexandrov topology

5.3 Ephemeral persistence modules

5.4 Distances on categories of sheaves

5.5 Comparison of the convolution and the interleaving distance

**6 Conclusion **

A Brief introduction to abelian sheaves

A.1 Abelian categories and their derived category

A.2 Sheaves of vector-spaces

**Bibliography**