Homomorphisms in Db Rc(kR)

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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.

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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

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