Correspondence Modules and Persistence Sheaves: A Unifying Perspective on One-Parameter Persistent Homology
aa r X i v : . [ m a t h . A T ] J un CORRESPONDENCE MODULES AND PERSISTENCESHEAVES: A UNIFYING FRAMEWORK FORONE-PARAMETER PERSISTENT HOMOLOGY
HAIBIN HANG AND WASHINGTON MIO
Abstract.
We develop a unifying framework for the treatment of var-ious persistent homology architectures using the notion of correspon-dence modules. In this formulation, morphisms between vector spacesare given by partial linear relations, as opposed to linear mappings. Inthe one-dimensional case, among other things, this allows us to: (i) treatpersistence modules and zigzag modules as algebraic objects of the sametype; (ii) give a categorical formulation of zigzag structures over a con-tinuous parameter; and (iii) construct barcodes associated with spacesand mappings that are richer in geometric information. A structuralanalysis of one-parameter persistence is carried out at the level of sec-tions of correspondence modules that yield sheaf-like structures, termedpersistence sheaves. Under some tameness hypotheses, we prove inter-val decomposition theorems for persistence sheaves and correspondencemodules, as well as an isometry theorem for persistence diagrams ob-tained from interval decompositions of persistence sheaves. Applicationsinclude: (a) a Mayer-Vietoris sequence that relates the persistent homol-ogy of sublevelset filtrations and superlevelset filtrations to the levelsethomology module of a real-valued function and (b) the construction ofslices of 2-parameter persistence modules along negatively sloped lines. Introduction
The primary goal of this paper is to develop a framework for the unifiedtreatment of various one-parameter persistent homology architectures thathave been formulated and studied over the years by many authors (cf. [23,9, 17]). In the process, we broaden the landscape to include novel persistenthomology structures that can become valuable tools for summarizing andgaining additional insight on the organization and shape of complex dataobjects and datasets. The focus is on persistent homology parameterizedover the real line, but there are many connections with multi-parameterpersistence.
Mathematics Subject Classification.
Primary: 55N31; Secondary: 18F20.
Key words and phrases.
Persistent homology, correspondence modules, persistencesheaves.This research was partially supported by NSF grant DMS-1722995.
Our constructs are based on two main concepts: correspondence modules( c -modules) and persistence sheaves ( p -sheaves). In short, in a correspon-dence module, a morphism between two vectors spaces U and V is given bya partial linear relation; that is, a linear subspace of U × V . Letting CVec bethe category whose objects are the vector spaces over a (fixed) field k andthe morphisms from U to V are the partial linear relations from U to V , acorrespondence module over a poset ( P, ) is a functor F : ( P, ) → CVec.Partial linear relations previously have been used in topological data analysisby Burghelea and Haller to study circle-valued mappings [8].Recall that a persistence module ( p -module) over ( P, ) is a functor from( P, ) to Vec, the category of vector spaces (over k ) and linear mappings[30, 17]. A linear map T : U → V may be viewed as a CVec-morphismby replacing it with its graph G T . Thus, via graphs, any p -module maybe thought of as a c -module. Note that a “backward” mapping S : V → U also may be viewed as a CVec-morphism from U to V by replacing itwith G ∗ S , where the operation ∗ swaps the coordinates of G S . From thisviewpoint, persistence structures in which morphisms are given by linearmappings, regardless of whether the mappings are forward or backward,can all be formulated under the same category theory framework. Thus,zigzag modules [10, 27] also may be treated as c -modules. In particular,correspondence modules lead to a category theory formulation of zigzagpersistence over a continuous parameter t ∈ R . Correspondence modulesalso lead to new persistent homology invariants associated with structuraland functional data, as further detailed below.This manuscript focuses on c -modules over ( R , ≤ ). Persistence sheaves areintroduced mainly because it is difficult to directly analyze persistent struc-tures in the CVec category; however, they also are of interest in their ownright. We note that, in the context of persistent homology, sheaf-theoreticalapproaches were first investigated by Curry [22] and, more recently, by otherauthors [29, 1, 2]. The sections of a c -module V over any interval I ⊆ R form a vector space F ( I ). Moreover, if I ⊆ J , there is a restriction homo-morphism F JI : F ( J ) → F ( I ). In addition, sections satisfy certain localityand gluing properties, thus yielding a sheaf-like structure that we term p -sheaf. Analysis of the structure of p -sheaves has the advantage of placing usback in the Vec category, at the expense of replacing the domain category( R , ≤ ) with the category (Int , ⊆ ), whose objects are the intervals of the realline with morphisms given by inclusions. The central results of the paper,obtained through the study of general p -sheaves, are as follows:(I) Under some tameness hypotheses, we prove interval decompositiontheorems for c -modules and p -sheaves that lead to barcode or persis-tence diagram representations of their structures;(II) An isometry theorem that states that, for any two interval decompos-able p -sheaves, the bottleneck distance between their persistence dia-grams is the same as an interleaving distance between the p -sheaves. ORRESPONDENCE MODULES 3 (This interleaving distance extends the usual interleaving distance be-tween p -modules, cf. [15, 17]).One of the applications discussed in the paper reflects well the fact that c -modules unify different types of 1-parameter persistence in a single categorytheory framework: for a function f : X → R , we construct a Mayer-Vietorissequence that ties together the persistent homology modules obtained fromthe sublevelset and superlevelset filtrations of X , induced by f , and thelevelset homology c -module of ( X, f ).Rooted in the works of Frosini [25] and Robins [35], over the years, per-sistent homology has experienced a vigorous development on many fronts,including theory, computation, and applications. In another seminal piece,in which the expression persistent homology was introduced, Edelsbrunner etal. developed an algorithm to compute the persistent homology of a filteredsimplicial complex and introduced an early form of persistence diagrams[24]. Another landmark is the work of Zomorodian and Carlsson [37], wherethe persistent homology of a discrete simplicial filtration is formulated as agraded module over the polynomial ring k [ x ] from which one obtains inter-val decompositions under appropriate finiteness hypotheses. Barcodes wereintroduced in [14] as an alternative representation of an interval decomposi-tion. This work led to the insight that the fundamental structure underlyingpersistent homology is that of an inductive system of vector spaces referredto as a persistence module. In this algebraic setting, Crawley-Boevey showedthat any pointwise finite-dimensional p -module over ( R , ≤ ) admits an inter-val decomposition [21, 6]. As such, these p -modules may be represented bybarcodes, persistence diagrams, or Bubenik’s persistence landscapes [7]. Wedevelop sheaf-theoretical analogues of the techniques of [21] to prove thedecomposition theorems for c -modules and p -sheaves. This involves a hier-archy of decompositions that ultimately leads to an interval decomposition.Zigzag modules, introduced in [10], are a variant of p -modules parameter-ized over discrete subsets of R . In a zigzag module, a morphism may point ineither direction. The prototypical example is levelset persistence of a func-tion f : X → R . For t ∈ R , let X [ t ] := f − ( t ). Although there is no naturalmapping from X [ s ] to X [ t ], for s < t , the interlevel set X ts := f − [ s, t ] canact as an interpolant, as there are inclusions X [ s ] ֒ → X ts ← ֓ X [ t ]. Thus, solong as we restrict the parameter values to a discrete set, we get a sequenceof spaces connected by morphisms alternating from forward to backward.Passing to homology, we obtain a levelset zigzag module, for which an in-terval decomposition theorem holds [27, 5]. This formulation is adequateto study the level sets of Morse-type functions, but insufficient for moregeneral continuous functions, as there seems to be something inherentlydiscrete about such structures. The question of how to give a category the-ory formulation of zigzag structures over R has been asked by many andposed explicitly in [33]. For levelset persistence, using a series of zigzagstructures over different discrete subsets, Carlsson et al. showed that one H. HANG AND W. MIO still can define persistence diagrams associated with a continuous param-eter t ∈ R [11]. However, the question remained at the level of modules.Correspondence modules provide a solution to the problem, as detailed inSection 7.1. Our approach via c -modules removes the homology of inter-level sets as objects in the sequence, treating them as CVec-morphisms viathe graphs of the homomorphisms induced on homology by the inclusionmaps X [ s ] ֒ → X ts ← ֓ X [ t ], as explained above. Connections between levelsetpersistence and 2-parameter persistence have been studied in [4, 18]. Asexplained in Section 7, p -sheaves allow us to formulate interlevel persistenceas an exact persistence module over two continuous parameters.The stability of persistence diagrams is another theme of central inter-est, as it is important to ensure that persistent homology can be used re-liably in data analysis. A breakthrough result in this direction is the cele-brated stability theorem of Cohen-Steiner, Edelsbrunner and Harer [19]. If f, g : X → R satisfy some regularity conditions, then d b ( D f , D g ) ≤ k f − g k ∞ ,where D f and D g are the persistence diagrams associated with the sub-levelset filtration of X induced by f and g , respectively, and d b denotesbottleneck distance. As one often is interested in comparing functional datadefined on different domains, extensions to this setting have been studiedin [26, 28]. For structural data (finite metric spaces), Chazal et al. showedthat the persistence diagram of the Vietoris-Rips filtration is stable withrespect to the Gromov-Hausdorff distance [16]. As the transition from func-tions, or other filtrations, to persistence diagrams goes through persistencemodules, Chazal et al. [15, 17] introduced an interleaving distance d I be-tween p -modules to analyze stability at an algebraic level. The IsometryTheorem, proven by Lesnick [30] and Chazal et al. [15, 17], states that d b ( D ( U ) , D ( V )) = d I ( U , V ), where U and V are p -modules over R and D ( U )denotes the persistence diagram of U . We prove an isometry theorem for p -sheaves as a further extension of such stability results.The paper is organized as follows. Section 2 introduces some basic termi-nology and the notion of correspondence modules. Section 3 is devoted to thebasic properties of persistence sheaves. Tameness properties of c -modulesand p -sheaves that imply interval decomposability are discussed in Section4. The decomposition theorems for c -modules and p -sheaves are proven inSection 5 and the Isometry Theorem in Section 6. Section 7 discusses appli-cations to: (i) levelset persistence; (ii) the construction of a Mayer-Vietorissequence for c -modules; and (iii) 1-dimensional slices of 2-parameter persis-tence modules along lines of negative slope. We close the paper with somediscussion in Section 8.2. Correspondence Modules
Preliminaries.
We denote by Vec the category whose objects are thevector spaces over a fixed field k with linear mappings as morphisms. Aposet ( P, ) is treated as a category whose objects are the elements of P . ORRESPONDENCE MODULES 5
For s, t ∈ P , there is a single morphism from s to t if s t , and noneotherwise. Abusing notation, we also denote the morphism by s t .A persistence module ( p -module) over P is a functor V : P → Vec. Cor-respondence modules, defined next, generalize p -modules by allowing moregeneral morphisms between vector spaces. Definition 2.1.
Let U and V be vector spaces. A (linear, partial) corre-spondence from U to V is a linear subspace C ⊆ U × V . We denote the setof all such linear correspondences by C ( U, V ). If C ∈ C ( U, V ), then:(i) The domain of C is defined as Dom( C ) = π U ( C ) and the image of C as Im( C ) = π V ( C ), where π U and π V denote the projections onto U and V , respectively. The kernel of C is defined as ker( C ) = { u ∈ U | ( u, ∈ C } .(ii) The reverse correspondence C ∗ ∈ C ( V, U ) is defined as C ∗ = { ( v, u ) : ( u, v ) ∈ C } . Example . Let T : U → V be a linear map and G T the graph of T .Then, G T ∈ C ( U, V ) and G ∗ T ∈ C ( V, U ). The graph of the identity map I V : V → V gives the diagonal correspondence ∆ V := G I V . Definition 2.3.
Let C ∈ C ( U, V ) and C ∈ C ( V, W ). The composition C ◦ C ∈ C ( U, W ) is defined as C ◦ C = { ( u, w ) ∈ U × W : ∃ v ∈ V with ( u, v ) ∈ C and ( v, w ) ∈ C } .With this composition operation, we form a category CVec having vectorspaces over k as objects and correspondences C ∈ C ( U, V ) as morphismsfrom U to V . In this category, the identity morphism of an object V is ∆ V ,the graph of the identity map. Lemma 2.4.
Let C ⊆ U × V be a correspondence. If Dom( C ) = U and ker( C ∗ ) = 0 , then C is the graph of a linear mapping T : U → V . In partic-ular, isomorphisms in CVec are given by linear mappings. More precisely,let C ⊆ U × V and D ⊆ V × U be correspondences such that D ◦ C = ∆ U and C ◦ D = ∆ V . Then, there is an isomorphism T : U → V such that G T = C and G ∗ T = D .Proof. The assumptions on C imply that, for each u ∈ U , there is a unique v ∈ V such that ( u, v ) ∈ C . Define T by T ( u ) = v , which is linear withthe desired properties. For the statement about isomorphisms, D ◦ C =∆ U implies that Dom( C ) = U . Thus, to show that C is the graph of alinear mapping T : U → V , it suffices to verify that ker ( C ∗ ) = 0, whichfollows from the fact the C and D are inverse morphisms. Similarly, thereis S : V → U such that G S = D . Note that D ◦ C = ∆ U and C ◦ D = ∆ V imply that S ◦ T = I U and T ◦ S = I V . (cid:3) Definition 2.5. A correspondence module (abbreviated c -module) over aposet ( P, ) is a functor V : P → CVec.We adopt the following notation:
H. HANG AND W. MIO (i) V t for the vector space associated with t ∈ P ; that is, V t := V ( t );(ii) v ts for the morphism associated with s t ; that is, v ts := V ( s t ). Definition 2.6.
Let U and V be c -modules over P .(i) A morphism F : U → V is a natural transformation from the functor U to the functor V . In other words, a collection of compatible morphisms f t ∈ C ( U t , V t ), t ∈ P , meaning that f t ◦ u ts = v ts ◦ f s , for any s t . Amorphism F is an isomorphism if f t is a CVec isomorphism, ∀ t ∈ P .(ii) U is a submodule of V if U t is a subspace of V t , ∀ t ∈ P , and u ts =( U s × U t ) ∩ v ts , for any s ≤ t . Definition 2.7.
The graph functor G : Vec → CVec is defined by G ( V ) = V ,for any object V , and G ( T ) = G T , for any morphism T . Example . Persistence Modules and Zigzag Modules as c -Modules(i) Let U : P → Vec be a persistence module over P . The composition V = G ◦ U yields a correspondence module. Thus, any persistencemodule may be viewed as a c -module via the graphs of its morphisms.(ii) Consider the poset P n = { , , , . . . , n } with the usual ordering ≤ . Azigzag module V over P n is a sequence V p ←→ V p ←→ V ←→ . . . ←→ V n − p n ←→ V n , (1)where each V i is a k -vector space and p i ←→ denotes either a forwardhomomorphism f i : V i − → V i or a backward homomorphism g i : V i → V i − [10]. Let C i ⊆ V i − × V i be defined by:(a) C i = G f i , the graph of f i , if p i is a forward homomorphism;(b) C i = G ∗ g i , the reverse of the graph of g i , if p i is a backward homo-morphism.Set C ii = ∆ V i ⊆ V i × V i , for 0 ≤ i ≤ n , and C ij = C j ◦ . . . ◦ C i +1 ⊆ V i × V j ,for 0 ≤ i < j ≤ n . These correspondences induce a c -module structureon V .We denote by CMod ( P ), or simply CMod, the category whose objectsare the c -modules over P with natural transformations as morphisms. Next,we show that morphisms in CMod have images in the category theory sense.Zero morphisms, kernels and cokernels are not well defined in CMod. How-ever, in some special situations, we can associate a kernel or a cokernel c -module to a morphism. Definition 2.9.
Let U and V be c -modules over P and F : U → V a CModmorphism given by compatible CVec morphisms f t : U t → V t .(i) Letting Im t = Im( f t ) and im ts := v ts ∩ (Im s × Im t ), define Im( F ) :=(Im t , im ts ), s, t ∈ P , s t .(ii) Letting K t = ker( f t ) and k ts = u ts ∩ ( K s × K t ), define ker( F ) := ( K t , k ts ), s, t ∈ P , s t . ORRESPONDENCE MODULES 7 (iii) Let Q t = coker( f t ) and q ts ⊆ Q s × Q t be the subspace given by([ v s ] , [ v t ]) ∈ q ts if and only if there exist a s ∈ Im( f s ) and a t ∈ Im( f t )such that ( v s + a s , v t + a t ) ∈ v ts . Define coker( F ) := ( Q t , q ts ), s, t ∈ P , s t . Proposition 2.10 (Images and Kernels) . If F : U → V is a CMod mor-phism, then (i) Im( F ) is a submodule of V and the inclusion Im( F ) ֒ → V is an imageof the morphism F in the CMod category; (ii) If G : W → U is another CMod morphism and the sequence W t U t V tg t f t is exact in CVec, ∀ t ∈ P , then ker( F ) is a submodule of U ; (iii) If V is a persistence module, then coker ( F ) is a c -module.Proof. (i) To verify that Im( F ) is a submodule of V , it suffices to check thecomposition rule im tr = im ts ◦ im sr , for any r s t . The inclusion im tr ⊇ im ts ◦ im sr is straightforward. For the reverse inclusion, let ( v r , v t ) ∈ im tr .Then, there exist v s ∈ V s , u r ∈ U r and u t ∈ U t such that ( v r , v s ) ∈ v sr ,( v s , v t ) ∈ v ts , ( u r , v r ) ∈ f r and ( u t , v t ) ∈ f t . Our goal is to show that v s ∈ Im( f s ), as this implies that ( v r , v t ) ∈ im ts ◦ im sr . Since ( u r , v s ) ∈ v sr ◦ f r = f s ◦ u sr , there exists u s ∈ U s such that ( u r , u s ) ∈ u sr and ( u s , v s ) ∈ f s , showingthat v s ∈ Im f s , as desired. The universal property for images in CMod iseasily verified for Im( F ).(ii) The assumption that the sequences are exact implies that ker( F ) =Im( G ). Hence, (i) implies that ker( F ) is a submodule of U .(iii) Since V is a p -module, the correspondences v ts are given by graphs oflinear mappings; that is, v ts = G φ ts , where φ ts : V s → V t are linear mappings.Let coker( F ) := ( Q t , q ts ). For r s t , we first show that q tr ⊆ q ts ◦ q sr .Let ([ v r ] , [ v t ]) ∈ q tr . Then, there exist a r ∈ Im( f r ) and a t ∈ Im( f t ) suchthat ( v r + a r , v t + a t ) ∈ v tr , which means that φ tr ( v r + a r ) = v t + a t . Let v s = φ sr ( v r ) and a s = φ sr ( a r ). A diagram chase shows that a s ∈ Im( f s ).Clearly, φ sr ( v r + a r ) = v s + a s and φ ts ( v s + a s ) = v t + a t , showing that([ v r ] , [ v s ]) ∈ q sr and ([ v s ] , [ v t ]) ∈ q ts ; that is, ([ v r ] , [ v t ]) ∈ q ts ◦ q sr . For theconverse inclusion, suppose that ([ v r ] , [ v s ]) ∈ q sr and ([ v s ] , [ v t ]) ∈ q ts . Then,there exist a r ∈ Im( f r ), a s , b s ∈ Im( f s ) and a t ∈ Im( f t ) such that φ sr ( v r + a r ) = v s + a s and φ ts ( v s + b s ) = v t + a t . The fact that F is a CMod morphismimplies that c t = φ ts ( a s − b s ) ∈ Im( f t ). Then, φ tr ( v r + a r ) = φ ts ( v s + a s ) = φ ts ( v s + b s + a s − b s )= v t + a t + φ ts ( a s − b s ) = v t + ( a t + c t ) , (2)which implies that ([ v r ] , [ v t ]) ∈ q tr , concluding the proof. (cid:3) We close this section with a discussion of direct sums of c -modules. H. HANG AND W. MIO
Definition 2.11.
Let { V λ , λ ∈ Λ } , be an indexed collection of c -modules.Define the direct sum V = ⊕ λ ∈ Λ V λ by V t = ⊕ λ ∈ Λ V λt , with correspondences v ts = ⊕ λ ∈ Λ V λ ( s t ) ⊆ V s × V t , for any s t . Proposition 2.12. If { V λ , λ ∈ Λ } is an indexed collection of c -modules,then the direct sum V = ⊕ λ ∈ Λ V λ also is a c -module.Proof. We only need to show that v ts ◦ v sr = v tr , for any r s t . We firstverify that v ts ◦ v sr ⊇ v tr . Given ( v r , v t ) ∈ v tr , write ( v r , v t ) = P λ ∈ Λ c λ ( v λr , v λt ),where ( v λr , v λt ) ∈ V λ ( r t ) and all but finitely many coefficients c λ ∈ k vanish. For each λ ∈ Λ, there exists v λs ∈ V λs , such that ( v λr , v λs ) ∈ V λ ( r s )and ( v λs , v λt ) ∈ V λ ( s t ). Letting v s = ⊕ λ ∈ Λ c λ v λs , it follows that ( v r , v s ) ∈ v sr and ( v s , v t ) ∈ v ts . Thus, ( v r , v t ) ∈ v ts ◦ v sr , as claimed.Conversely, let ( v r , v t ) ∈ v ts ◦ v sr . Then, there exists v s ∈ ⊕ λ ∈ Λ V λs suchthat ( v r , v s ) ∈ v sr and ( v s , v t ) ∈ v ts . Write( v r , v s ) = X λ ∈ Λ c λ ( v λr , v λs ) and ( v s , v t ) = X λ ∈ Λ d λ ( v λs , v λt ) , (3)where ( v λr , v λs ) ∈ V λ ( r s ), ( v λs , v λt ) ∈ V λ ( s t ) and all but finitely manyscalars c λ , d λ ∈ k vanish. Then, v s = P λ ∈ Λ c λ v λs and v s = P λ ∈ Λ d λ v λs ,which implies c λ = d λ , ∀ λ ∈ Λ. Hence, ( v r , v t ) = P λ ∈ Λ c λ ( v λr , v λt ) ∈ v tr . (cid:3) Definition 2.13.
A correspondence module V is indecomposable if V ∼ = V ⊕ V implies that either V = 0 or V = 0.2.2. Interval Correspondence Modules.
We now specialize to corre-spondence modules over ( R , ≤ ). We introduce interval c -modules associatedwith each interval I ⊆ R . Unlike interval p -modules (cf. [17]), there may beup to four non-isomorphic interval c -modules associated with I .Let E be the set of extended, decorated real numbers defined as E = R × { + , −} ∪ {−∞ , + ∞} . (4)For t ∈ R , we use the abbreviations t + := ( t, +), t − := ( t, − ), and t ∗ foreither t + or t − . Throughout the paper, E is equipped with the total ordering ≤ given by:(i) t ∗ ≤ t ∗ , for any t , t ∈ R satisfying t < t ;(ii) t − ≤ t + , for any t ∈ R ;(iii) −∞ ≤ t ∗ ≤ + ∞ , ∀ t ∈ R , and −∞ ≤ + ∞ ;(iv) p ≤ p , ∀ p ∈ E .For p, q ∈ E , we write p < q to mean that p ≤ q and p = q . Decoratednumbers give a uniform notation for intervals in R , whether open, closed, orhalf-open. We adopt the following identification between objects in Int andelements of { ( p, q ) ∈ E | p < q } (cf. [17]):(a) For t , t ∈ R with t < t , ( t , t ) ↔ ( t +1 , t − ), [ t , t ) ↔ ( t − , t − ),( t , t ] ↔ ( t +1 , t +2 ), and [ t , t ] ↔ ( t − , t +2 );(b) For t ∈ R , the one-point interval [ t, t ] corresponds to ( t − , t + ) ∈ E ; ORRESPONDENCE MODULES 9 (c) For t ∈ R , ( −∞ , t ) ↔ ( −∞ , t − ), ( −∞ , t ] ↔ ( −∞ , t + ), ( t, + ∞ ) ↔ ( t + , + ∞ ) and [ t, + ∞ ) ↔ ( t − , + ∞ );(d) The entire real line R corresponds to ( −∞ , + ∞ ) ∈ E . Definition 2.14.
Let ( p, q ) ∈ E , p < q , represent an interval in R . Wedefine c -modules I [ p, q ], I [ p, q i , I h p, q ] and I h p, q i associated with ( p, q ) asfollows (see Fig. 1):(i) Let I denote any of the above c -modules and t ∈ R . Define I ( t ) = k , ∀ t ∈ ( p, q ), and I ( t ) = 0, otherwise. For s, t ∈ R and s ≤ t , set I ( s ≤ t ) = ∆ k if s, t ∈ ( p, q ), and I ( s ≤ t ) = 0 × s, t / ∈ ( p, q );(ii) I [ p, q ]( s ≤ t ) = 0 × s / ∈ ( p, q ) or t / ∈ ( p, q );(iii) I [ p, q i ( s ≤ t ) = k × s ∈ ( p, q ) and t ∈ ( q, + ∞ ); I [ p, q i ( s ≤ t ) = 0 × s ∈ ( −∞ , p );(iv) I h p, q ]( s ≤ t ) = 0 × k if s ∈ ( −∞ , p ) and t ∈ ( p, q ); I h p, q ]( s ≤ t ) = 0 × t ∈ ( q, + ∞ );(v) I h p, q i ( s ≤ t ) = k × s ∈ ( p, q ) and t ∈ ( q, + ∞ ); I h p, q i ( s ≤ t ) = 0 × k if s ∈ ( −∞ , p ) and t ∈ ( p, q ). Remark . (a) If ( p, q ) ∈ E is a finite interval, the four modules in Definition 2.14 fallin different isomorphism classes. If p = −∞ , then I [ p, q i = I h p, q i and I [ p, q ] = I h p, q ]. Similarly, if q = + ∞ , I [ p, q i = I [ p, q ] and I h p, q ] = I h p, q i . If ( p, q ) = ( −∞ , + ∞ ), all four c -modules coincide.(b) We adopt the pictorial representation of these four types of intervalmodules indicated in Fig. 1. Lemma 2.16 (Indecomposability) . Interval c -modules are indecomposable.Proof. The argument is standard. Let I be an interval c -module of any ofthe types described in Definition 2.14. The corresponding interval in R isdenoted I . Suppose η is a c -module isomorphism from I to U ⊕ V . ByLemma 2.4, ∀ t ∈ I , η t is the graph of a linear isomorphism φ t : k → U t ⊕ V t .Let π : U ⊕ V → U ⊕ V denote projection onto U followed by inclusion into U ⊕ V . Then, ψ t = φ − t ◦ π ◦ φ t : k → k is an idempotent of k . Thus, ψ t is given by multiplication by 0 or 1. Since φ t induce a c -module morphism,one can verify that this scalar is independent of t ∈ I . Hence, either U = 0or V = 0. (cid:3) Persistence Sheaves
Henceforth, all c -modules will be over ( R , ≤ ). In this section, we developa framework for the study of the structure of c -modules over R based on theconcept of persistence sheaves.Let 2 R be the category whose objects are the subsets of R with inclusionof sets as morphisms. We may think of the objects of 2 R as the open sets of All types:
Interval ( p, q )0 × × × I [ p, q ]: × × I [ p, q i : k × × I h p, q ]: × × k I h p, q i : k × × k Figure 1.
Interval c -modules associated with the interval ( p, q ). R in the discrete topology. We denote by Int the subcategory of 2 R whoseobjects are the intervals I ⊆ R . Definition 3.1. (Presheaves)(i) A discrete presheaf is a contravariant functor from 2 R to Vec; that is,a functor G : (2 R ) op → Vec.(ii) A persistence presheaf is a functor F : Int op → Vec.Here, the superscript op denotes the opposite category. Clearly, any dis-crete presheaf defines a persistence presheaf via restriction to Int. For apersistence presheaf F , we adopt the following terminology:(a) We refer to an element of the vector space F ( I ) as a section of F over I .(b) For I ⊆ J , we refer to the linear map F JI := F ( I ⊆ J ) : F ( J ) → F ( I )as a restriction homomorphism . If s ∈ F ( J ), we sometimes use thenotation s | I for F JI ( s ).(c) If I = { a } is a singleton, we simplify the notation for F ( I ), F JI and s | I to F a , F Ja and s | a , respectively.(d) If s ∈ F ( I ), the support of s is defined as supp[ s ] := { a ∈ I : F Ia ( s ) = 0 } .(e) If s and s ′ are sections of F , we write s s ′ to indicate that s is arestriction of s ′ .Similar notation and terminology are adopted for discrete presheaves. Remark . Let F be the restriction of a discrete presheaf G to Int. If J ⊆ R is an interval and A ⊆ J , we abuse notation and write F JA for the ORRESPONDENCE MODULES 11 restriction homomorphism “inherited” from G . Similarly, if s ∈ F ( J ), wewrite s | A for F JA ( s ). More generally, if V is a subspace of F ( J ), we set V | A = { s | A : s ∈ V } . Definition 3.3. (Morphisms)(i) Let F and G be persistence presheaves. A morphism from F to G isa natural transformation Φ : F → G ; that is, a collection Φ := { φ I : F ( I ) → G ( I ) : I ∈ Int } of linear mappings such that G JI ◦ φ J = φ I ◦ F JI ,for any I ⊆ J .(ii) Φ : F → G is an isomorphism if there is a morphism Ψ : G → F suchthat Ψ ◦ Φ and Φ ◦ Ψ are the identity morphisms of F and G , respectively. Definition 3.4.
Let V be a c -module and A ⊆ R .(i) A section of V over A is an indexed family s = ( v t ), v t ∈ V t and t ∈ A ,satisfying ( v r , v t ) ∈ v ts , for every r, t ∈ A with r ≤ t . The domain of s is the set A and we denote it Dom( s ). The support of s is definedas supp[ s ] = { t ∈ Dom( s ) : v t = 0 } . Pointwise addition and scalarmultiplication induce a k -vector space structure on the collection of allsections over A . If s is a section over B and A ⊆ B , the restriction of s to A is denoted s | A .(ii) The discrete presheaf of sections of V , denoted D V , is the contravariantfunctor that associates to each A ⊆ R the vector space D V ( A ) of allsections of V over A . If A ⊆ B , then D V ( A ⊆ B ) is the linear mappinggiven by s s | A , for any section s over B .(iii) The persistence presheaf of sections of V is the restriction of D V to Int.To define persistence sheaves, we introduce the notion of connected cov-ering. Definition 3.5.
Let X be a set and X = ∪ X λ , λ ∈ Λ, a covering of X by subsets X λ . The covering is connected if for any non-trivial partitionΛ = Λ ⊔ Λ , the intersection of the sets X = ∪ λ ∈ Λ X λ and X = ∪ λ ∈ Λ X λ is non-empty. Lemma 3.6.
Let X = ∪ X λ , λ ∈ Λ , be a covering of X with X λ = ∅ , ∀ λ ∈ Λ .The covering is connected if and only if, for any λ, µ ∈ Λ , there is a finitesequence λ , . . . , λ n ∈ Λ such that λ = λ , λ n = µ and X λ i − ∩ X λ i = ∅ , for ≤ i ≤ n .Proof. Consider the equivalence relation on Λ generated by λ ∼ µ if X λ ∩ X µ = ∅ . Then, λ ∼ µ if and only if a sequence as above exists. It is simpleto verify that the covering is connected if and only if [ λ ] = Λ, ∀ λ ∈ Λ. (cid:3) Lemma 3.7.
Let { X λ , λ ∈ Λ } be a connected covering of a set X . If V ⊆ X is such that, for each λ ∈ Λ , X λ ⊆ V or X λ ∩ V = ∅ , then V = X or V = ∅ .Proof. Let Λ = { λ ∈ Λ | X λ ⊆ V } and Λ = { λ ∈ Λ | X λ ∩ V = ∅} . Thisgives a partition of Λ with the property that X ∩ X = ∅ , with X and X as in Definition 3.5. Since the covering is connected, either Λ = ∅ orΛ = ∅ . This implies that V = X or V = ∅ . (cid:3) Definition 3.8.
Let F be a persistence presheaf. F is a persistence sheaf (abbreviated p -sheaf) if the following conditions are satisfied:(i) (Locality) For any covering I = ∪ I λ , λ ∈ Λ, of I by non-empty intervals I λ , if s ∈ F ( I ) is such that s | I λ = 0, ∀ λ ∈ Λ, then s = 0;(ii) (Connective Gluing) For any connected covering I = ∪ I λ , λ ∈ Λ, of I by intervals I λ , if s λ ∈ F ( I λ ), λ ∈ Λ, are sections of F such that s λ | I λ ∩ I λ ′ = s λ ′ | I λ ∩ I λ ′ , ∀ λ, λ ′ ∈ Λ, then there is a section s ∈ F ( I ) suchthat s | I λ = s λ , ∀ λ ∈ Λ.Note that property (ii) differs from the usual gluing property for sheavesbecause of the connectivity condition on coverings.
Proposition 3.9.
The persistence presheaf of sections of a c -module V is a p -sheaf.Proof. Locality is clearly satisfied, so we verify the connective gluing prop-erty. Let I = ∪ I λ , λ ∈ Λ, be a connected covering of an interval I byintervals I λ , and let s λ ∈ V ( I λ ), λ ∈ Λ, be sections of V that agree onoverlaps. There is a well-defined family s = ( v t ), t ∈ I , such that s | I λ = s λ , ∀ λ ∈ Λ. We need to show that s is a section. Let S be the collection ofall sections r of V satisfying r s ; that is, sections r that coincide withthe restriction of s to some subinterval of I . S is non-empty because therestriction of s to any I λ gives a section. Moreover, ( S, ) is a partiallyordered set such that each chain in S has an upper bound in S . By Zorn’slemma, S contains a maximal element ˆ s defined on an interval J ⊆ I . Byconstruction, for each λ ∈ Λ, I λ ∩ J = ∅ or I λ ⊆ J , for otherwise, we wouldbe able to extend ˆ s to a larger interval. Since J = ∅ , Lemma 3.7 ensuresthat J = I , proving that s = ˆ s . (cid:3) Definition 3.10.
Let F λ , λ ∈ Λ, be p -sheaves. The direct sum ⊕ F λ is the p -sheaf defined by:(i) ( ⊕ F λ )( I ) = ⊕ F λ ( I ), for any interval I ;(ii) ( ⊕ F λ )( I ⊆ J ) = ⊕ F λ ( I ⊆ J ). Definition 3.11.
Let F and G be p -sheaves. F is a subsheaf of G if (i)for any interval I , F ( I ) ⊆ G ( I ) and (ii) for any I ⊆ J and s ∈ F ( J ), F JI ( s ) = G JI ( s ).It is easy to verify that the intersection of a collection of subsheaves of a p -sheaf F is also a subsheaf of F . Definition 3.12.
Let S be a set of sections of a p -sheaf F . The p -sheafgenerated by S is the intersection of all subsheaves G of F with the propertythat if s ∈ S , then s is a section of G . This subsheaf of F is denoted F h S i . Definition 3.13.
Let ( p, q ) ∈ E be an interval. We define the interval p -sheaves k [ p, q ], k [ p, q i , k h p, q ] and k h p, q i associated with ( p, q ), as follows: ORRESPONDENCE MODULES 13 (i) k [ p, q ]( I ) = k if I ⊆ ( p, q ) and k [ p, q ]( I ) = 0 otherwise. The element1 ∈ k [ p, q ]( p, q ) is called the unit section of k [ p, q ].(ii) k [ p, q i ( I ) = k if I ⊆ ( p, + ∞ ) and I ∩ ( p, q ) = ∅ , and k [ p, q i ( I ) = 0otherwise. The element 1 ∈ k [ p, q i ( p, + ∞ ) is called the unit section of k [ p, q i .(iii) k h p, q ]( I ) = k if I ⊆ ( −∞ , q ) and I ∩ ( p, q ) = ∅ , and k h p, q ]( I ) = 0otherwise. The element of 1 ∈ k h p, q ]( −∞ , q ) is called the unit section of k h p, q ].(iv) k h p, q i ( I ) = k if I ∩ ( p, q ) = ∅ and k h p, q i ( I ) = 0 otherwise. Theelement 1 ∈ k h p, q i ( −∞ , + ∞ ) is called the unit section of k h p, q i .(v) If F is any of the above p -sheaves and I ⊆ J , then F JI is the identitymap if F ( J ) = F ( I ) = k , and F JI is the trivial map, otherwise.We refer to each of these four possibilities as the type of the interval p -sheafassociated with ( p, q ). Proposition 3.14.
For any interval ( p, q ) ∈ E , p < q , the following holds: (i) k [ p, q ] is isomorphic to the p -sheaf of sections of I [ p, q ] ; (ii) k [ p, q i is isomorphic to the p -sheaf of sections of I [ p, q i ; (iii) k h p, q ] is isomorphic to the p -sheaf of sections of I h p, q ] ; (iv) k h p, q i is isomorphic to the p -sheaf of sections of I h p, q i .Proof. The proof is straightforward. (cid:3)
Proposition 3.15.
Let s be a section of a p -sheaf F with Dom( s ) = ( x, y ) ∈ E , x < y , and let F h s i be the subsheaf generated by s . Suppose that supp[ s ] is an interval ( p, q ) ∈ E , where x ≤ p < q ≤ y . Then, the followingstatements hold: (i) If x = p and q < y , then F h s i is isomorphic to k [ p, q i . (ii) If x < p and q = y , then F h s i is isomorphic to k h p, q ] . (iii) If x < p and q < y , then F h s i is isomorphic to k h p, q i . (iv) If x = p and q = y , then F h s i is isomorphic to k [ p, q ] .Proof. For statement (i), by the connective gluing property, there is a section s over ( p, + ∞ ) such that s | ( p,y ) = s and s | ( q, + ∞ ) = 0. Note that s must bea section of any subsheaf of F having s as a section. For an interval I ⊆ ( p, + ∞ ), we denote by h s | I i the subspace of F ( I ) spanned by s | I . Let Int p,q be the collection of all intervals satisfying I ⊆ ( p, + ∞ ) and I ∩ ( p, q ) = ∅ .Define a persistence presheaf G ⊆ F by G ( I ) = h s | I i if I ∈ Int p,q , and G ( I ) = 0 otherwise. Note that s | I = 0, ∀ I ∈ Int p,q , and s | I = 0 if I / ∈ Int p,q .By construction, G is a sub-presheaf of F h s i .Let Φ : G → k [ p, q i be the homomorphism defined as follows: φ I ( s | I ) = 1if I ∈ Int p,q , and φ I = 0, otherwise. Similarly, define Ψ : k [ p, q i → G by ψ I (1) = s | I if I ∈ Int p,q and ψ II = 0, otherwise. Then, for any interval I , φ I and ψ I are mutual inverses, so Φ is a presheaf isomorphism. Since k [ p, q i is a sheaf, G is not just a presheaf, but a subsheaf of F . Thus, G = F h s i ,showing that F h s i is isomorphic to k [ p, q i . The proofs for the other casesare similar. (cid:3) Tameness
This section discusses tameness conditions for p -sheaves and c -modulesunder which we prove interval decomposition theorems in Section 5. Definition 4.1.
Let F be a p -sheaf.(i) F satisfies the descending chain condition (DCC) on images if for anyascending sequence I ⊆ I ⊆ I ⊆ . . . of intervals and any interval I ⊆ ∩ I n , the chain Im( F I I ) ⊇ Im( F I I ) ⊇ . . . is stable, that is, iteventually becomes constant;(ii) F satisfies the descending chain condition (DCC) on kernels if for anyascending sequence I ⊆ I ⊆ I ⊆ . . . of intervals and any interval I ⊇ ∪ I n , the chain ker( F II ) ⊇ ker( F II ) ⊇ . . . is stable. Definition 4.2.
Let V be a c -module and t ∈ R .(i) V satisfies the descending chain condition (DCC) on images at t if forany sequence . . . ≤ ℓ ≤ ℓ ≤ t , the chain Im( v tℓ ) ⊇ Im( v tℓ ) ⊇ . . . stabilizes in finitely many steps, and for any sequence t ≤ u ≤ u ≤ . . . , the chain Dom( v u t ) ⊇ Dom( v u t ) ⊇ . . . is stable.(ii) V satisfies the descending chain condition (DCC) on kernels at t if forany sequence t ≤ . . . ≤ u ≤ u , the chain ker( v u t ) ⊇ ker( v u t ) ⊇ . . . isstable, and for any sequence ℓ ≤ ℓ ≤ . . . ≤ t , the chain ker ( v tℓ ) ∗ ⊇ ker ( v tℓ ) ∗ ⊇ . . . is stable. Here, ∗ denotes the operation of reversingcorrespondences (see Definition 2.1). Definition 4.3. (Tameness)(i) A p -sheaf F is tame if it satisfies the DCC on both images and kernels.(ii) A c -module is virtually tame if it satisfies the DCC on both images andkernels at each t ∈ R .(iii) A c -module is tame if its p -sheaf of sections is tame. Remark . If a c -module V has the property that all correspondences v ts , s < t , are finite-dimensional, then V is virtually tame. In particular, apointwise finite-dimensional c -module is virtually tame.The next examples show that the sheaf of sections of a virtually tame c -module is not necessarily tame. Example . Let k be a field. Consider the c -module V given by V t = k , ∀ t ∈ R , with connecting morphisms v ts = k × k , for any s < t , and v tt =∆ k , the graph of the identity map, ∀ t ∈ R . V is virtually tame becauseeach V t is 1-dimensional. Furthermore, it admits the interval decomposition V ∼ = ⊕ x ∈ R h x i , where h x i denotes the interval module of type h i supportedon the singleton { x } . Note, however, that F ≇ ⊕ x ∈ R F x , where F x is the p -sheaf of sections of h x i . Indeed, for any interval I ⊆ R , F ( I ) comprise allsequences ( v t ) with v t ∈ k and t ∈ I . On the other hand, the sections ofthe direct sum p -sheaf are the sections over I whose supports are finite sets.Although F satisfies the DCC on images, F does not satisfy the DCC on ORRESPONDENCE MODULES 15 kernels and therefore is not tame. Indeed, consider the chain I n = [0 , n ] andlet I = [0 , + ∞ ). Then, the chain ker( F II n ) is not stable. Example . This example shows that the virtual tameness of V does notimply the tameness of F even for persistence modules. Let I n = (0 , /n ], I n := [0 + , /n + i be the associated interval p -module of type [ i , and V = ⊕ n I n . V is pointwise finite dimensional and thus virtually tame. However,its sheaf of sections is not tame, as the DCC on kernels is not satisfied.Indeed, let I = (0 , + ∞ ) and consider the chain J n = (1 /n, + ∞ ). Then, thechain ker( F IJ n ) is strictly decreasing, thus not stable. Remark . In spite of the above examples, in applications, we are primar-ily interested in pointwise finite-dimensional c -modules V whose structurechanges only at a finite set of points. More precisely, there is a finite set T = { t , · · · , t n } ⊆ R with t i < t i +1 , 1 ≤ i < n , such that if s, t ∈ R satisfy s ≤ t < t , t n < s ≤ t , or t i < s ≤ t < t i +1 , for some i , then the relation v ts is an isomorphism. Under this assumption, dim F ( I ) < ∞ , for any interval I , implying that F is tame. Proposition 4.8.
Let F be a p -sheaf and I ⊆ I ⊆ I ⊆ . . . an ascendingsequence of intervals. Then, the following holds: (i) If F satisfies the DCC on images and I ⊆ ∩ I n , then Im( F ∪ I n I ) =Im( F I m I ) for m large enough; (ii) If F satisfies the DCC on kernels and ∪ I n ⊆ I , then ker( F I ∪ I n ) =ker( F II m ) for m large enough.Proof. (i) Set I = I . For any n ≥
0, the DCC on images, applied tothe interval I n and the chain I n +1 ⊆ I n +2 ⊆ . . . , ensures that there ex-ists N ( n ) > n such that Im( F I m I n ) = Im( F I N ( n ) I n ), ∀ m ≥ N ( n ). Set V n =Im( F I N ( n ) I n ) ⊆ F ( I n ). Then, F I m I n ( V m ) = V n , for any m ≥ n . By construc-tion, for any s ∈ V , there is a sequence of sections s n ∈ V n , n ≥
1, suchthat F I m I n ( s m ) = s n , for m ≥ n . By the connective gluing property, thereexists s ∈ F ( ∪ I n ) such that F ∪ I n I n ( s ) = s n , ∀ n ≥
0. Thus, F ∪ I n I ( s ) = s .This implies that Im( F ∪ I n I ) ⊇ V = Im( F I m I ), ∀ m ≥ N (0). The inclusionIm( F ∪ I n I ) ⊆ Im( F I m I ) is clearly satisfied ∀ m ≥ N (0), so this proves theclaim.(ii) By the DCC on kernels, we can choose n > F II n ) =ker( F II n ), for any n ≥ n . Clearly, ker( F I ∪ I n ) ⊆ ker( F II n ). For the oppositeinclusion, let s ∈ ker( F II n ), so that s | I n = 0, ∀ n > n . By locality, s | ∪ I n = 0,which implies s ∈ ker( F I ∪ I n ). Hence, ker( F II n ) ⊆ ker( F I ∪ I n ), concluding theproof. (cid:3) Our next goal is to prove a version of Proposition 4.8 for virtually tame c -modules. We begin with an extension result for sections of a c -module. Proposition 4.9.
Let V be a c -module and A ⊆ R . If V is virtually tame,then any section of V over A can be extended to a section over an intervalcontaining A .Proof. Let f be a section over A and let S be the set of all sections thatextend f over some set containing A . For g, h ∈ S , write g ≤ h to mean that h extends g . Note that ( S, ≤ ) is a poset in which each ascending chain has anupper bound. By Zorn’s lemma, there exists a maximal element ¯ f ∈ S . Weclaim that the domain of ¯ f is an interval. Suppose not, then there exists r / ∈ Dom( ¯ f ) such that L = { t ∈ Dom( ¯ f ) | t < r } and U = { t ∈ Dom( ¯ f ) | r < t } are not empty. Choose an increasing sequence { ℓ n } ⊆ L with lim ℓ n = sup L and a decreasing sequence { u n } ⊆ U with lim u n = inf U . If sup L ∈ L , weassume that ℓ n = sup L , ∀ n . Similarly, u n = inf U , for every n , if inf U ∈ U .For n ≥
1, setIm ℓ n ,u n r = { v ∈ V r | ( ¯ f ( ℓ n ) , v ) ∈ v rℓ n and ( v, ¯ f ( u n )) ∈ v u n r } , (5)a non-empty affine subspace of V r . Note that these affine subspaces form anested sequence Im ℓ ,u r ⊇ Im ℓ ,u r ⊇ Im ℓ ,u r ⊇ . . . (6)We show that this sequence is stable. To this end, letker rℓ n ,u n = { v ∈ V r | (0 , v ) ∈ v rℓ n and ( v, ∈ v u n r } , (7)which also form a nested sequenceker rℓ ,u ⊇ ker rℓ ,u ⊇ ker rℓ ,u ⊇ . . . (8)Note that the stability of the sequence ker rℓ n ,u n implies the stability ofIm ℓ n ,u n r . Indeed, suppose ∃ N such that ker rℓ n ,u n = ker rℓ N ,u N , ∀ n ≥ N . Forany v n ∈ Im ℓ n ,u n r , we may writeIm ℓ n ,u n r = v n + ker rℓ n ,u n . (9)Since the choice of v N ∈ Im ℓ N ,u N r in (9) is arbitrary, using (6) and thestability of kernels, we haveIm ℓ N ,u N r = v N + ker rℓ N ,u N = v n + ker rℓ N ,u N = v n + ker rℓ n ,u n = Im ℓ n ,u n r , (10) ∀ n ≥ N , as claimed. The stability of (8) follows from ker rℓ n ,u n = ker( v rℓ n ) ∗ ∩ ker( v u n r ) and the fact that the right-hand side of this equation stabilizes byvirtual tameness. To conclude, pick v ∈ ∩ Im ℓ n ,u n r and extend ¯ f to r via theassignment ¯ f ( r ) = v . This contradicts the maximality of ¯ f . (cid:3) Proposition 4.10. If F is the p -sheaf of sections of a virtually tame c -module V , then for any ascending sequence of intervals I ⊆ I ⊆ I ⊆ . . . the following holds: (i) If A ⊆ ∩ I n is finite, then Im( F ∪ I n A ) = Im( F I m A ) for m large enough; (ii) If A ⊆ I ⊇ ∪ I n , where I is an interval and A is a finite set, then ker( F I ∪ I n ) | A = ker( F II m ) | A for m large enough. ORRESPONDENCE MODULES 17
Proof. (i) Since F ∪ I n A = F I m A ◦ F ∪ I n I m , it follows that Im( F ∪ I n A ) ⊆ Im( F I m A ), ∀ m . For the reverse inclusion, let f ∈ Im( F I m A ). We show that, for m sufficiently large, there is a section g defined over ∪ I n such that g | A = f .Let s = min A and t = max A . We construct g by gluing sections overthe following three intervals: I = [ s , t ], I − = { x ∈ ∪ I n : x ≤ s } , and I + = { x ∈ ∪ I n : x ≥ t } .Let [ s n , t n ] ⊆ I n , n ≥
1, be a sequence of intervals such that [ s n , t n ] ⊆ [ s n +1 , t n +1 ], A ⊆ ∩ [ s n , t n ], and ∪ [ s n , t n ] = ∪ I n , ∀ n ≥
1. Note that we alsohave A ⊆ [ s , t ] ⊆ [ s , t ]. For each n ≥
0, the virtual tameness of V impliesthat the descending chainsIm( v s n s n +1 ) ⊇ Im( v s n s n +2 ) ⊇ . . . and Dom( v t n +1 t n ) ⊇ Dom( v t n +2 t n ) ⊇ . . . are stable. Thus, ∃ N n ≥ n such that Im( v s n s m ) and Dom( v t m t n ) are constantfor m ≥ N n . Set L n := Im( v s n s m ) and U n := Dom( v t m t n ), for m ≥ N n . Byconstruction, for any x n ∈ L n and y n ∈ U n , n ≥
0, there exist x n +1 ∈ L n +1 and y n +1 ∈ U n +1 such that( x n +1 , x n ) ∈ v s n s n +1 and ( y n , y n +1 ) ∈ v t n +1 t n . (11)If f ∈ Im( F I m A ) and m ≥ N , then x := f s ∈ L and y = f t ∈ U .Iteratively, as described above, construct x n ∈ L n and y n ∈ U n satisfying(11). The sequences { x n } and { y n } , n ≥
0, together with f , yield a sectionof V over the set { s n : n ≥ } ∪ A ∪ { t n : n ≥ } . By Proposition 4.9, thissection can be extended to a section over ∪ I n with the desired properties.(ii) F II m = F ∪ I n I m ◦ F I ∪ I n implies that ker( F I ∪ I n ) ⊆ ker( F II m ). Therefore,ker( F I ∪ I n ) | A ⊆ ker( F II m ) | A , for every m . For the reverse inclusion, let A = A ∩ ( ∪ I n ), A − = { a ∈ A : a < t, ∀ t ∈ ∪ I n } and A + = { a ∈ A : a > t, ∀ t ∈∪ I n } . Set r − = max A − , r + = min A + , I − = I \ ( r − , + ∞ ) and I + = I \ ( −∞ , r + ) . Let f ∈ ker( F II m ) | A . Since A is finite, ∃ N > A ⊆ I m , for m ≥ N . Hence, f | A ≡ m ≥ N . Pick a section ˆ f ∈ ker( F II m ) suchthat ˆ f | A = f and set g = ˆ f | I − ∪ I + . Next, we show that we can extend g to a section h over I − ∪ I + ∪ n I n such that h | ∪ I n ≡
0, provided that m issufficiently large. Note that any such h will have the property that h | A = f .Let [ s n , t n ], n ≥
1, be as in the proof of (i). By construction, r − < . . . ≤ s ≤ s and t ≤ t ≤ . . . < r + . By virtual tameness, there exists n ≥ N such that the chainsker v s r − ⊇ ker v s r − ⊇ . . . and ker( v r + t ) ∗ ⊇ ker( v r + t ) ∗ ⊇ . . . (12)are stable at s n and t n , n ≥ n , respectively. Hence, if f ∈ ker( F II m ) | A , m ≥ n , we have that g r − ∈ ker( v s n r − ) and g r + ∈ ker( v r + t n ) ∗ , for any n ≥ n ,which implies that the section g can be extended, as claimed. By Proposition4.9, we can further extend g to a section over I . This concludes the proof. (cid:3) Decomposition Theorems
In this section, we prove interval decomposition theorems for tame p -sheaves and virtually tame c -modules that lead to representations of theirstructures by barcodes or persistence diagrams. We develop a sheaf-theoreticalanalogue of the techniques employed by Crawley-Boevey to obtain such de-compositions for pointwise finite-dimensional persistence modules [21].5.1. Coverings.
The arguments and constructions in [21] use the notion of sections of a vector space whose definition we recall next. To avoid confusionwith sections of c -modules and p -sheaves, we rename them splittings. Definition 5.1.
Splittings of Vector Spaces (cf. [21])(i) A splitting of a vector space V is a pair ( F − , F + ) of subspaces F − ⊆ F + ⊆ V .(ii) A collection { ( F − λ , F + λ ) : λ ∈ Λ } of splittings of V is disjoint if for all λ = µ , either F + λ ⊆ F − µ or F + µ ⊆ F − λ ;(iii) A collection of splittings { ( F − λ , F + λ ) : λ ∈ Λ } covers V if for any sub-space U ⊆ V , with U = V , ∃ λ ∈ Λ such that U + F − λ = U + F + λ ; andit strongly covers V provided that for all subspaces U, W ⊆ V with W * U , ∃ λ ∈ Λ such that U + ( F − λ ∩ W ) = U + ( F + λ ∩ W ). Proposition 5.2 (Crawley-Boevey [21]) . Let { ( F − λ , F + λ ) : λ ∈ Λ } be a setof splittings that is disjoint and covers V . (i) If W λ is a complement of F − λ in F + λ , ∀ λ ∈ Λ , then the inclusions W λ ⊆ V induce a direct sum decomposition V = ⊕ λ ∈ Λ W λ . (ii) If { ( G − σ , G + σ ) : σ ∈ Σ } is another set of splittings that is disjoint andstrongly covers V , then the family { ( F − λ + G − σ ∩ F + λ , F − λ + G + σ ∩ F + λ ) : ( λ, σ ) ∈ Λ × Σ } of splittings is disjoint and covers V . Corollary 5.3.
Suppose that { ( F − λ , F + λ ) : λ ∈ Λ } is disjoint and covers V and { ( G − σ , G + σ ) : σ ∈ Σ } is disjoint and strongly covers V . If W σ,λ is acomplement of ( F − λ ∩ G + σ ) + ( G − σ ∩ F + λ ) in G + σ ∩ F + λ , then: (i) The inclusions W σ,λ ⊆ V induce a decomposition V = ⊕ σ,λ W σ,λ ; (ii) For any λ ∈ Λ , the inclusions F − λ , W σ,λ ⊆ F + λ induce a direct sumdecomposition F + λ = F − λ ⊕ ( ⊕ σ W σ,λ ) .Proof. (i) Note that G + σ ∩ F + λ F − λ ∩ G + σ + G − σ ∩ F + λ ≃ F − λ + G + σ ∩ F + λ F − λ + G − σ ∩ F + λ . (13)Since the isomorphism in (13) is induced by inclusion, a complement of( F − λ ∩ G + σ ) + ( G − σ ∩ F + λ ) in G + σ ∩ F + λ is also a complement of F − λ + G − σ ∩ F + λ in F − λ + G + σ ∩ F + λ . Thus, the claim follows from Proposition 5.2. ORRESPONDENCE MODULES 19 (ii) Since W σ,λ is a complement of F − λ + G − σ ∩ F + λ in F − λ + G + σ ∩ F + λ , we have W σ,λ ∩ F − λ = 0 and W σ,λ ⊆ F + λ , ∀ σ ∈ Σ. For a fixed λ , to simplify notation,set H − σ := F − λ + G − σ ∩ F + λ and H + σ := F − λ + G + σ ∩ F + λ . Now, we show that ⊕ σ W σ,λ ∩ F − λ = 0. Let v λ + v σ + · · · + v σ n = 0 be a relation with v λ ∈ F − λ and v σ i ∈ W σ i ,λ . Since the splittings { ( G − σ , G + σ ) : σ ∈ Σ } of V are disjoint, so arethe splittings { ( H − σ , H + σ ) : σ ∈ Σ } of F + λ . By disjointness, we may assumethat H + σ i ⊆ H − σ i +1 , for all i < n . Then, v σ n = − v λ − v σ − · · · − v σ n − ∈ H − σ n because v λ ∈ F − λ ⊆ H − σ n and v σ i ∈ H + σ i ⊆ H − σ n . Hence, v σ n ∈ H − σ n ∩ W σ n ,λ =0. Similarly, we show that all other terms in the relation vanish. Therefore, ⊕ σ W σ,λ ∩ F − λ = 0 and ⊕ σ W σ,λ ⊆ F + λ .Choose W λ ⊆ F + λ such that W λ ⊕ (cid:0) ⊕ σ W σ,λ (cid:1) is a complement of F − λ in F + λ . Since { ( F − λ , F + λ ) : λ ∈ Λ } is disjoint and covers V , it follows that (cid:16) M λ W λ (cid:17) ⊕ (cid:16) M σ,λ W σ,λ (cid:17) = V. (14)On the other hand, by (i), ⊕ σ,λ W σ,λ = V . Thus, W λ = 0, ∀ λ ∈ Λ. Thisproves that F + λ = F − λ ⊕ ( ⊕ σ W σ,λ ). (cid:3) Let F be a p -sheaf and x, y, p, q ∈ E satisfy x ≤ p < q ≤ y . We use thefollowing abbreviations:Im ( x,y )( p,q ) = Im (cid:0) F ( x,y )( p,q ) (cid:1) ker ( x,y )( p,q ) = ker (cid:0) F ( x,y )( p,q ) (cid:1) Im (¯ x,y )( p,q ) = ∪ z
Decomposition of Tame p -Sheaves. Our next goal is to decomposea tame p -sheaf F as a direct sum of “atomic” subsheaves that are sheaf-theoretical analogues of interval c -modules. The building blocks of thisdecomposition are described next. Definition 5.6.
For any p -sheaf F and x, y ∈ E with x < y , we let:(i) F [ x, y ] be a complement of Im (¯ x,y )( x,y ) + Im ( x, ¯ y )( x,y ) in Im ( x,y )( x,y ) = F ( x, y );(ii) F [ x, y i be a complement of ker (¯ x, + ∞ )( y, + ∞ ) + ker ( x, + ∞ )(¯ y, + ∞ ) in ker ( x, + ∞ )( y, + ∞ ) ;(iii) F h x, y ] be a complement of ker ( −∞ , ¯ y )( −∞ ,x ) + ker ( −∞ ,y )( −∞ , ¯ x ) in ker ( −∞ ,y )( −∞ ,x ) ;(iv) F h x, y i be a complement of ker ( −∞ , + ∞ ))¯ x,y ( + ker ( −∞ , + ∞ )) x, ¯ y ( in ker ( −∞ , + ∞ )) x,y ( . Proposition 5.7.
For any p -sheaf F , the following statements hold: (i) If x, y ∈ E , x < y , and I ⊆ ( x, y ) is an interval, then Im ( x,y )( x,y ) (cid:12)(cid:12) I = F [ x, y ] (cid:12)(cid:12) I ⊕ (cid:16) Im (¯ x,y )( x,y ) (cid:12)(cid:12) I + Im ( x, ¯ y )( x,y ) (cid:12)(cid:12) I (cid:17) ;(ii) If x, y ∈ E , x < y , and I ⊆ ( x, + ∞ ) is an interval, then ker ( x, + ∞ )( y, + ∞ ) (cid:12)(cid:12) I = F [ x, y i (cid:12)(cid:12) I ⊕ (cid:16) ker (¯ x, + ∞ )( y, + ∞ ) (cid:12)(cid:12) I + ker ( x, + ∞ )(¯ y, + ∞ ) (cid:12)(cid:12) I (cid:17) ;(iii) If x, y ∈ E , x < y , and I ⊆ ( −∞ , y ) is an interval, then ker ( −∞ ,y )( −∞ ,x ) (cid:12)(cid:12) I = F h x, y ] (cid:12)(cid:12) I ⊕ (cid:16) ker ( −∞ , ¯ y )( −∞ ,x ) (cid:12)(cid:12) I + ker ( −∞ ,y )( −∞ , ¯ x ) (cid:12)(cid:12) I (cid:17) ;(iv) If x, y ∈ E , x < y , and I is any interval, then ker ( −∞ , + ∞ )) x,y ( (cid:12)(cid:12) I = F h x, y i (cid:12)(cid:12) I ⊕ (cid:16) ker ( −∞ , + ∞ ))¯ x,y ( (cid:12)(cid:12) I + ker ( −∞ , + ∞ )) x, ¯ y ( (cid:12)(cid:12) I (cid:17) . Proof.
Here we just prove (ii), the proofs of the other statements being sim-ilar. Since, by definition, ker ( x, + ∞ )( y, + ∞ ) = F [ x, y i ⊕ (cid:16) ker (¯ x, + ∞ )( y, + ∞ ) + ker ( x, + ∞ )(¯ y, + ∞ ) (cid:17) , wejust need to show that the two summands in this direct sum decompositionremain independent after restriction to I ; that is, F [ x, y i (cid:12)(cid:12) I ∩ (cid:16) ker (¯ x, + ∞ )( y, + ∞ ) (cid:12)(cid:12) I + ker ( x, + ∞ )(¯ y, + ∞ ) (cid:12)(cid:12) I (cid:17) = 0 . (19)Given s ∈ F [ x, y i and s ∈ (cid:16) ker (¯ x, + ∞ )( y, + ∞ ) + ker ( x, + ∞ )(¯ y, + ∞ ) (cid:17) with s | I = s | I , let s = s − s . Clearly, s | I = 0. If I ∩ ( x, y ) = ∅ , the left-hand side of (19) equals0, so the proof is trivial. If I ∩ ( x, y ) = ∅ , by the gluing property, s is the sumof two sections from ker (¯ x, + ∞ )( y, + ∞ ) + ker ( x, + ∞ )(¯ y, + ∞ ) . Hence, s ∈ ker (¯ x, + ∞ )( y, + ∞ ) + ker ( x, + ∞ )(¯ y, + ∞ ) ,which implies that s = s + s ∈ ker (¯ x, + ∞ )( y, + ∞ ) + ker ( x, + ∞ )(¯ y, + ∞ ) , so that s = s = 0.The result follows. (cid:3) For x, y ∈ E , x < y , we may view F [ x, y i as a persistence presheaf with F [ x, y i ( I ) = F [ x, y i| I , if I ⊆ ( x, + ∞ ), and F [ x, y i ( I ) = 0, otherwise. Mor-phisms are induced by restriction of sections of F . Similarly, we may treat F h x, y ], F h x, y i and F [ x, y ] as persistence presheaves, where F h x, y ]( I ) = 0 if I * ( −∞ , y ) and F [ x, y ]( I ) = 0 if I * ( x, y ). Henceforth, we refer to theseinterchangeably as p -sheafs or spaces of sections, the meaning determinedby the context.Let m [ x, y ] = dim( F [ x, y ]), m [ x, y i = dim( F [ x, y i ), m h x, y ] = dim( F h x, y ]),and m h x, y i = dim( F h x, y i ). Proposition 5.8. If F is a p -sheaf and x, y ∈ E , then: (i) F [ x, y ] ∼ = m [ x, y ] k [ x, y ] , if −∞ < x < y < + ∞ ; (ii) F [ x, y i ∼ = m [ x, y i k [ x, y i , if −∞ < x < y ≤ + ∞ ; (iii) F h x, y ] ∼ = m h x, y ] k h x, y ] , if −∞ ≤ x < y < + ∞ ; (iv) F h x, y i ∼ = m h x, y i k h x, y i , if −∞ ≤ x < y ≤ + ∞ .Proof. We prove (ii), the proofs of the other statements being similar. Choosea basis B = { s λ : λ ∈ Λ } of the space of sections F [ x, y i . By definition, foreach λ ∈ Λ, Dom( s λ ) = ( x, + ∞ ). Note that supp[ s λ ] = ( x, y ). Indeed,suppose ∃ t ∈ ( x, y ) such that s λ ( t ) = 0. Then, we may write s λ as thesum of sections in ker (¯ x, + ∞ )( y, + ∞ ) and ker ( x, + ∞ )(¯ y, + ∞ ) , which implies that s λ = 0, acontradiction. By Proposition 3.15, the p -sheaf generated by s λ satisfies F h s λ i ∼ = k [ x, y i . Since B is a basis, F [ x, y i ∼ = m [ x, y i k [ x, y i . (cid:3) Lemma 5.9 (Decomposition Lemma) . Let F be a tame p -sheaf. If p, q ∈ E , p < q , then the space of sections F ( p, q ) may be decomposed as F ( p, q ) = M −∞ Since F satisfies the DCC on images, by Lemma 5.4 the families ofsplittings ( F − x , F + x ) = (cid:0) Im (¯ x,q )( p,q ) , Im ( x,q )( p,q ) (cid:1) , (20) −∞ ≤ x ≤ p , and ( G − y , G + y ) = (cid:0) Im ( p, ¯ y )( p,q ) , Im ( p,y )( p,q ) (cid:1) , (21) q ≤ y ≤ + ∞ , are disjoint and strongly cover F ( p, q ). By the connectivegluing property for sections, we have: F + x ∩ G + y = Im ( x,q )( p,q ) ∩ Im ( p,y )( p,q ) = Im ( x,y )( p,q ) ,F − x ∩ G + y = Im (¯ x,q )( p,q ) ∩ Im ( p,y )( p,q ) = Im (¯ x,y )( p,q ) ,F + x ∩ G − y = Im ( x,q )( p,q ) ∩ Im ( p, ¯ y )( p,q ) = Im ( x, ¯ y )( p,q ) . (22) ORRESPONDENCE MODULES 23 Therefore, F + x ∩ G + y ( F − x ∩ G + y ) + ( F + x ∩ G − y ) = Im ( x,y )( p,q ) Im (¯ x,y )( p,q ) + Im ( x, ¯ y )( p,q ) = Im ( x,y )( x,y ) (cid:12)(cid:12) I Im (¯ x,y )( x,y ) (cid:12)(cid:12) I + Im ( x, ¯ y )( x,y ) (cid:12)(cid:12) I , (23)where the last equality follows from the fact that I = ( p, q ) ⊆ ( x, y ). ByProposition 5.7(i) and (23), F [ x, y ] | I is a complement of ( F − x ∩ G + y ) + ( F + x ∩ G − y ) in F + x ∩ G + y . Thus, Corollary 5.3(i) implies that F ( p, q ) = M −∞ Now we show that, in (24), the summands F [ x, + ∞ ] (cid:12)(cid:12) I , −∞ < x ≤ p ,may be replaced with L p To complete the proof, we decompose the last summand in (29).Consider the families of splittings( F − x , F + x ) = (cid:0) ker ( −∞ , + ∞ )( −∞ , ¯ x ) (cid:12)(cid:12) I , ker ( −∞ , + ∞ )( −∞ ,x ) (cid:12)(cid:12) I (cid:1) , (30) −∞ ≤ x < + ∞ , and( G − y , G + y ) = (cid:0) ker ( −∞ , + ∞ )(¯ y, + ∞ ) (cid:12)(cid:12) I , ker ( −∞ , + ∞ )( y, + ∞ ) (cid:12)(cid:12) I (cid:1) , (31) −∞ < y ≤ + ∞ , that are disjoint and strongly cover F ( −∞ , + ∞ ) (cid:12)(cid:12) I . Argu-ing as in Step 2 and using the fact that F + x ∩ G + y ( F − x ∩ G + y ) + ( F + x ∩ G − y ) = ker ( −∞ , + ∞ )) x,y ( (cid:12)(cid:12) I ker ( −∞ , + ∞ ))¯ x,y ( (cid:12)(cid:12) I + ker ( −∞ , + ∞ )) x, ¯ y ( (cid:12)(cid:12) I , (32)we obtain the decomposition F [ −∞ , + ∞ ] (cid:12)(cid:12) I = M −∞≤ x Moreover, F ∼ = M −∞ Let V be a virtually tame c -module and F its p -sheaf ofsections. If p, q ∈ E , p < q , then the space of sections F ( p, q ) satisfies F ( p, q ) | A = M −∞ In this section, we prove one of the main results of this paper, the stabil-ity of persistence diagrams associated with interval decomposable p -sheaves.We define the interleaving distance d I ( F, G ) between any two p -sheaves F and G and, assuming that the sheaves are decomposable, we also definethe bottleneck distance d b ( dgm ( F ) , dgm ( G )) between their persistence dia-grams. The Isometry Theorem states that d b ( dgm ( F ) , dgm ( G )) = d I ( F, G ) , (37)for any decomposable p -sheaves F and G . The inequality d b ( dgm ( F ) , dgm ( G )) ≥ d I ( F, G ) (38)follows directly from the definition of d I and d b . However, the proof of thealgebraic stability statement d b ( dgm ( F ) , dgm ( G )) ≤ d I ( F, G ) (39)involves rather delicate arguments. ORRESPONDENCE MODULES 27 Interleavings. This section introduces the notions of interleaving andinterleaving distance for p -sheaves, extending the corresponding concepts forpersistent modules [15, 17] to our setting, as needed in the formulation ofthe Isometry Theorem.Given a decorated number p = t ∗ ∈ E , t ∈ R , and ǫ ∈ R , let p + ǫ := ( t + ǫ ) ∗ ,with the additional convention that −∞ + ǫ = −∞ and + ∞ + ǫ = + ∞ . Definition 6.1. (Dilations and Erosions)(i) If I = ( p, q ) ∈ E is an interval, the ǫ -dilation of I , ǫ ≥ 0, is defined as I ǫ := ( p − ǫ, q + ǫ ).(ii) The ǫ -erosion of I = ( p, q ), ǫ > 0, is defined as I − ǫ := ( p + ǫ, q − ǫ ), if p + ǫ < q − ǫ . Otherwise, I − ǫ = ∅ .(iii) For ǫ > Z ⊆ R , if Z = ⊔ λ ∈ Λ I λ is its representation as the disjointunion of its connected components, define Z − ǫ = ⊔ λ ∈ Λ I − ǫλ . Definition 6.2. Let F and G be p -sheaves and ǫ ≥ ǫ -homomorphism Φ : F → G is a collection { φ II − ǫ : F ( I ) → G ( I − ǫ ) : I ∈ Int } of linear maps such that G J − ǫ I − ǫ ◦ φ JJ − ǫ = φ II − ǫ ◦ F JI , for any I ⊆ J , with theconvention that G ( I − ǫ ) = 0 if I − ǫ = ∅ . We refer to a 0-homomorphismsimply as a homomorphism.(ii) The ǫ -erosion of F is the ǫ -homomorphism e ǫF : F → F given by { F II − ǫ : I ∈ Int } . (Note that e F is the identity.)To motivate the definition of interleaving and explain how it generalizesthe notion of interleaving of persistence modules (cf. [15, 17]), let V be a p -module and F be its sheaf of sections. Note that vectors in V s , s ∈ R ,are in one-to-one correspondence with sections of F over [ s, + ∞ ), where v s ∈ V s corresponds to the section ( v t ) t ≥ s given by v t = v ts ( v s ). Under thiscorrespondence, the transition map v ts becomes a ( t − s )-erosion map. Thedifference for more general p -sheaves is that we need to consider sectionsover all intervals, not just those of the form [ s, + ∞ ). Definition 6.3 (Interleaving) . Let F and G be p -sheaves.(i) An ǫ - interleaving between F and G , ǫ ≥ 0, is a pair of ǫ -homomorphismsΦ ǫ : F → G and Ψ ǫ : G → F such that Ψ ǫ ◦ Φ ǫ = e ǫF and Φ ǫ ◦ Ψ ǫ = e ǫG .(Note that a 0-interleaving is an isomorphism.)(ii) Given ǫ ≥ F and G are ǫ + - interleaved if they are ( ǫ + δ )-interleavedfor every δ > interleaving distance between F and G is defined as d I ( F, G ) = inf { ǫ > F and G are ǫ -interleaved } = min { ǫ ≥ F and G are ǫ + -interleaved } , with the convention that d I ( F, G ) = ∞ if no interleaving exists. Remark . The two forms of Definition 6.3(iii) are equivalent because if F and G are ǫ -interleaved, then they are ǫ + -interleaved. Note, however, thatthe converse is not necessarily true. Moreover, d I is an extended pseudo-metric on the space of isomorphism classes of p -sheaves. Remark . Given s, t ∈ R , s ≤ t , there are up to four intervals ( x, y ) ∈ E ,defined by s and t , given by the different combinations of decorations for s and t . More precisely, let A s,t = { ( x, y ) ∈ { s − , s + } × { t − , t + } : x < y } , (40)with the convention that −∞ ∗ = −∞ and + ∞ ∗ = + ∞ . Then, the intervalsassociated with s and t are ( x, y ) ∈ A s,t . For a fixed interval-sheaf type,say k [ , ], any two p -sheaves in the collection { k [ x, y ] : ( x, y ) ∈ A s,t } are 0 + -interleaved. The same applies to the other three types: k h x, y ], k [ x, y i ,and k h x, y i . Hence, the interleaving distance is blind to changes in thedecorations of the endpoints of an interval.6.2. Persistence Diagrams. To motivate the definition of persistence di-agrams, suppose that a p -sheaf F is interval decomposable; that is, F ∼ = M −∞ Given any p -sheaf F , define its decorated persistence di-agram Dgm( F ) as the quadruple of functions ([ m ] , h m ] , [ m i , h m i ) with do-mains(i) Dom([ m ]) = { ( x, y ) ∈ E : −∞ < x < y < + ∞} ,(ii) Dom([ m i ) = { ( x, y ) ∈ E : −∞ < x < y ≤ + ∞} ,(iii) Dom( h m ]) = { ( x, y ) ∈ E : −∞ ≤ x < y < + ∞} ,(iv) Dom( h m i ) = { ( x, y ) ∈ E : −∞ ≤ x < y ≤ + ∞} ,given by [ m ]( x, y ) = dim F [ x, y ] , [ m i ( x, y ) = dim F [ x, y i , h m ]( x, y ) = dim F h x, y ] , h m i ( x, y ) = dim F h x, y i . ORRESPONDENCE MODULES 29 As pointed out in Remark 6.5, for a given interval p -sheaf type, the inter-leaving distance is not sensitive to changes in the decoration of the endpointsof an interval. Thus, to obtain an isometry theorem, we should not distin-guish persistence diagrams associated with those intervals. To this end, asin [17], we introduce “undecorated” versions of persistence diagrams. Definition 6.7. Given any p -sheaf F , we define its (undecorated) persis-tence diagram dgm ( F ) as the quadruple of functions ([ m ] , [ m i , h m ] , h m i )with domains(i) Dom([ m ]) = { ( s, t ) ∈ R × R : s < t } ,(ii) Dom([ m i ) = { ( s, t ) ∈ R × R : s < t } ,(iii) Dom( h m ]) = { ( s, t ) ∈ R × R : s < t } ,(iv) Dom( h m i ) = { ( s, t ) ∈ R × R : s ≤ t } ,given by [ m ]( s, t ) = P ( x,y ) ∈ A s,t [ m ]( x, y ), with A s,t as in (40). The functionvalues [ m i ( s, t ), h m ]( s, t ), and h m i ( s, t ) are defined similarly. Remark . We often refer to the domain of each of the four functions ina p -diagram as a multiset and to the value of the function at ( s, t ) as themultiplicity of ( s, t ). We also frequently treat the multiplicity functions asdefined on R × R by extending them to be zero outside their original domains.Notice that for singletons ( s − , s + ) ∈ E , s ∈ R , forgetting decorationsreturns points on the diagonal ∆ = { ( s, s ) : s ∈ R } . Nonetheless, in Defini-tion 6.7(i)-(iii), ∆ is not included in the domain of the multiplicity functionbecause k [ s − , s + ], k h s − , s + ] and k [ s − , s + i are all 0 + -interleaved with thetrivial p -sheaf. This is not the case, however, for k h s − , s + i . Indeed, the δ -erosion map for global sections of this p -sheaf is the identity, for any δ > k h s − , s + i is not δ -interleaved with the trivial p -sheaf, for any δ > p -diagrams. In order to define the bottleneck distancebetween p -diagrams, we first discuss a variant of the notion of matching ofmultisets that is suited to our goals. Abusing terminology, we often refer toa multiset ( A, f ), with multiplicity function f , simply as A . The support of A is defined as supp( A ) = { a ∈ A : f ( a ) = 0 } .A partial matching between the multisets ( A, f ) and ( B, g ) is a multisetinjection from a subset of A to B . More formally, a resolution of A is amapping π A : ˆ A → A such that | π − A ( a ) | = f ( a ), ∀ a ∈ A . A partial matching between A and B is an injection σ : ¯ A → ˆ B , where ¯ A ⊆ ˆ A . The partialmatching σ is surjective if Im( σ ) = ˆ B . We abuse terminology and refer toelements of ˆ A as elements of the multiset A , and to σ ( a ) as an element ofthe multiset B .As usual, the bottleneck distance will be based on the ℓ ∞ -distance on theextended plane, denoted d ∞ : R × R → R and given by d ∞ (( s , t ) , ( s , t )) = max {| s − s | , | t − t |} , (42) with the convention that |∞ − ∞| = 0. Note that d ∞ (cid:0) ( s, t ) , ∆ (cid:1) = | s − t | / Definition 6.9. Let A and B be multisets with domain R × R , ǫ > L > 0. A partial matching σ between A and B is an ( ǫ, L )- matching provided that:(i) if σ ( a ) = b , then d ∞ ( a, b ) ≤ ǫ ;(ii) if a ∈ supp( A ) \ Dom( σ ), then d ∞ ( a, ∆) ≤ Lǫ ;(iii) if b ∈ supp( B ) \ Im( σ ), then d ∞ ( b, ∆) ≤ Lǫ .A partial matching σ is a full ǫ -matching if it is a multiset bijection between A and B that satisfies condition (i) above. Definition 6.10. The persistence diagrams dgm i = ( f i , f i , f i , f i ), i ∈{ , } , are ǫ -matched if the following holds:(i) f and f are ( ǫ, f and f are ( ǫ, f and f are ( ǫ, f and f are fully ǫ -matched. Definition 6.11. The bottleneck distance between persistence diagrams isthe extended pseudo-metric defined as d b ( dgm , dgm ) = inf { ǫ > dgm and dgm are ǫ -matched } . The next proposition shows that a converse to the stability of persistencediagrams holds. We adopt the following notation:[ F ] = M −∞ Let F and G be decomposable p -sheaves. If dgm ( F ) and dgm ( G ) are ǫ -matched, ǫ > , then F and G are ǫ + -interleaved. Thus, d I ( F, G ) ≤ d b ( dgm ( F ) , dgm ( G )) . Proof. Let ǫ ≥ 0. If dgm ( F ) = ( f , f , f , f ) and dgm ( G ) = ( g , g , g , g )are ǫ -matched, there exist:(a) an ( ǫ, f and g ;(b) an ( ǫ, f and g ;(c) an ( ǫ, f and g ;(d) a full ǫ -matching between f and g .We show that the subsheaves [ F ] and [ G ] are ǫ + -interleaved. The argu-ment for the components of type h ], [ i , and h i of F and G are similar. ORRESPONDENCE MODULES 31 As the diagrams disregard decorations of the endpoints of an interval, sup-pose that for s i , t i ∈ R , s i < t i , i = 1 , 2, the interval modules k [ s ∗ , t ∗ ] and k [ s ∗ , t ∗ ] are ( ǫ, ǫ -matching of diagrams. Then, d ∞ (( s , t ) , ( s , t )) ≤ ǫ , which implies that k [ s ∗ , t ∗ ] and k [ s ∗ , t ∗ ] are ( ǫ + δ )-interleaved for any δ > 0. Assembling all of these pairwise interleavings,yields the desired ( ǫ + δ )-interleaving. (cid:3) Algebraic Stability. This section is devoted to the proof of the alge-braic stability of persistence diagrams; that is, the statement that d b ( dgm ( F ) , dgm ( G )) ≤ d I ( F, G ) . The general strategy for the proof of stability, particularly in Lemma 6.27and Theorem 6.28 below, is similar to that used by Bjerkevik [3] to studyalgebraic stability in the context of multi-parameter persistence modules.The arguments needed for p -sheaves, however, are quite distinct. Definition 6.13. Let F be a p -sheaf. A set B F of non-trivial sections of F is a basis of F if for any interval I ⊆ R , the set B F ( I ) := { f | I : f ∈ B F , I ⊆ Dom( f ) and f | I } is a basis of F ( I ). Lemma 6.14. If B F is a basis of a p -sheaf F , then each section in B F hasconnected support. Moreover, if f ∈ B F , then the subsheaf F h f i spanned by f (see Definition 3.12) is isomorphic to an interval p -sheaf.Proof. Suppose that f ∈ B F has disconnected support. Then, there exist r < s < t such that f | r = 0, f | s = 0 and f | t = 0. By the gluing property, f may be decomposed as f = a + b , where a x = f x , for x ∈ L := Dom( f ) ∩ ( −∞ , s ], and a x = 0, for x ∈ R := Dom( f ) ∩ [ s, + ∞ ). This implies that b x = f x , ∀ x ∈ R and b | L ≡ 0. Note that both a and b are non-trivialsections. By the definition of basis, a and b can be uniquely expressed aslinear combinations a = c f + X f λ ∈ B F ( I ) \{ f } c λ f λ and b = d f + X f λ ∈ B F \{ f } d λ f λ . (45)Since f = a + b , it follows that c + d = 1. On the other hand, a | L = c f | L + X c λ f λ | L and b | R = d f | R + X d λ f λ | R . (46)Since both a | L and b | R are non-trivial and coincide with f | L and f | R , respec-tively, it follows that c = d = 1. This contradicts the fact that c + d = 1.The fact that, for any f ∈ B F , the subsheaf F h f i is isomorphic to aninterval sheaf now follows from Proposition 3.15. (cid:3) Let F be interval decomposable. If Φ : P λ ∈ Λ F λ → F is a p -sheaf isomor-phism, where each F λ is an interval p -sheaf, then the images under Φ of theunit sections s λ of F λ (see Definition 3.13) form a basis of F . The followingproposition provides a converse statement. Hence, we may view an intervaldecomposition of a p -sheaf as a choice of basis. Proposition 6.15. If B F is a basis of F , then there are interval p -sheaves F λ , λ ∈ Λ , and an isomorphism Φ : P λ ∈ Λ F λ → F that maps the unitsections s λ of F λ bijectively onto B F .Proof. Set G λ := F h f λ i . Lemma 6.14 implies that, for each λ , there is anisomorphism φ λ : F λ → G λ , where F λ is an interval p -sheaf. Moreover, φ λ may be chosen to map the unit section s λ of F λ to f λ . The fact that B F isa basis implies that the inclusions G λ ⊆ F , λ ∈ Λ, induce an isomorphism ı : ⊕ G λ → F . Then, the composite Φ := ı ◦ ( ⊕ λ φ λ ) is an isomorphism withthe desired properties. (cid:3) Let f and g be sections of F and G , respectively, with connected support.By Proposition 3.15, F h f i and G h g i are (isomorphic to) interval p -sheaves. Definition 6.16. Under the above assumptions, define f and g to be ǫ - matched if:(i) F h f i and G h g i are interval p -sheaves of the same type;(ii) d H (supp( f ) , supp( g )) ≤ ǫ , where d H denotes (extended) Hausdorff dis-tance. Lemma 6.17. Under the assumptions of Definition 6.16, if the sections f and g are ǫ -matched, then F h f i and G h g i are ǫ -interleaved.Proof. Define ǫ -homomorphisms Φ ǫ : F h f i → G h g i and Ψ ǫ : G h g i → F h f i by Φ ǫ ( f ) = g | Dom( f ) − ǫ and Ψ ǫ ( g ) = f | Dom( g ) − ǫ . This completes characterizethese homomorphisms because the p -sheaves are spanned by f and g . Then,(Φ ǫ , Ψ ǫ ) is an ǫ -interleaving. (cid:3) Definition 6.18. Given δ > 0, a section is said to be δ - trivial if it getsmapped to the zero section under the δ -erosion map. Otherwise, the sectionis called δ - significant . We denote by X ǫF ⊆ B F the set of all ǫ -significantsections in B F . Definition 6.19. ( ǫ -Mappings and ǫ -Matchings)(i) A mapping σ : Dom( σ ) ⊆ B F → B G is an ǫ - mapping if f and σ ( f ) are ǫ -matched, ∀ f ∈ Dom( σ ).(ii) B F and B G are said to be ǫ - matched if there exists an injective ǫ -map σ : Dom( σ ) ⊆ B F → B G such that X ǫF ⊆ Dom( σ ) and X ǫG ⊆ Im( σ ). Proposition 6.20. If B F and B G are ǫ -matched, then dgm ( F ) and dgm ( G ) are ǫ -matched.Proof. This follows directly from the definitions of ǫ -matchings for sectionsand for persistence diagrams. (cid:3) By Proposition 6.20, we can translate a diagram-matching problem intoa question of matching bases and this is how we approach the proof of thestability theorem. Lemma 6.21. Let ǫ > . If there exist injective ǫ -mappings σ : X ǫF → B G and τ : X ǫG → B F , then there is an ǫ -matching between B F and B G . ORRESPONDENCE MODULES 33 Proof. Let G be the bipartite graph with vertex set partitioned as B F ⊔ B G and whose edge set is the union of the graphs of the maps σ and τ . Wedenote an edge { u, v } , where σ ( u ) = v , by u σ −→ v . Similarly, u τ −→ v if v = τ ( u ). Since σ and τ are injections, each vertex of G has degree ≤ ǫ -matching over each connected component of G andtake the union of these to obtain the desired matching. Let C be anyconnected component of G and V C its vertex set. Let C ǫF := X ǫF ∩ V C and C ǫG := X ǫG ∩ V C , the sets of vertices in C comprised of ǫ -significant sectionsof F and G , respectively. Then, at least one of the following propertiesis satisfied: (a) C ǫG ⊆ Im σ | C ǫF or (b) C ǫF ⊆ Im τ | C ǫG . Indeed, there is asequence . . . τ −→ v − σ −→ v τ −→ v σ −→ . . . (47)such that V C = ∪ i { v i } . This sequence may be infinite or finite on eitherend. Note that if it is finite on the right, then the last element is a sectionthat is 2 ǫ -trivial. If the sequence is infinite on the left or finite startingwith an edge of type σ −→ , then the construction implies that C ǫG ⊆ Im σ | C ǫF ,regardless of the behavior of the sequence on the right end. Similarly, if itis finite on the left, starting with a type τ −→ edge, then C ǫF ⊆ Im τ | C ǫG .If property (a) above is satisfied, then σ | C ǫF : C ǫF → B G ∩ C is an ǫ -matching between B F ∩ V C and B G ∩ V C . Let ζ := τ | C ǫG : C ǫG → B F ∩ V C . Ifproperty (b) is satisfied, then τ − | Im( ζ ) : Im( ζ ) ⊇ C ǫF → C ǫG is an ǫ -matchingbetween B F ∩ V C and B G ∩ V C . Assembling the matchings over all connectedcomponents of G yields an ǫ -matching between B F and B G . (cid:3) We now proceed to the core argument in the proof of the algebraic stabilityof persistence diagrams, namely, the construction of the injective ǫ -mappingscalled for in the hypothesis of Lemma 6.21. In the remainder of this section,we index the bases of F and G as B F = { f λ : λ ∈ Λ F } and B G = { g µ : µ ∈ Λ G } , respectively. We also adopt the following abbreviations: D λ := Dom( f λ ) , D µ := Dom( g µ ) , (48) Z λ := D λ \ supp( f λ ) , Z µ := D µ \ supp( g µ ) . (49)For ǫ ≥ 0, Φ ǫ : F → G and Ψ ǫ : G → F denote ǫ -homomorphisms. Using B F and B G , an ǫ -morphism Φ ǫ may be represented by a “matrix” (cid:0) φ µλ (cid:1) ,where Φ ǫ ( f λ ) = X µ ∈ Λ G φ µλ g µ | D − ǫλ , (50)for any λ ∈ Λ F . Here, we make the convention that φ µλ = 0 if g µ | D − ǫλ = 0;that is, D − ǫλ ⊆ Z µ . Thus, for a fixed λ , only finitely many entries φ µλ can benon-zero. Similarly, Ψ ǫ may be represented by a matrix (cid:0) ψ λµ (cid:1) whose entries,for a fixed µ , vanish for all but finitely many values of λ . Lemma 6.22. If φ µλ = 0 , then D − ǫλ ⊆ D µ and Z − ǫλ ⊆ Z µ . Moreover, theinterval p -sheaves F h f λ i and G h g µ i are of the same type. Proof. D − ǫλ ⊆ D µ follows directly from the definition of φ µλ . Suppose that Z λ = ∅ and let I ⊆ R be a connected component of Z λ . Then,0 = Φ ǫ ( f λ | I ) = X µ ∈ Λ G φ µλ g µ | I − ǫ . (51)Since non-trivial sections in { g λ | I − ǫ } are independent, we have φ µλ g µ | I − ǫ = 0for any µ . Hence, φ µλ = 0 implies that g µ | I − ǫ = 0. Thus, Z − ǫλ ⊆ Z µ .Lemma 6.14 shows that both F h f λ i and G h g µ i are isomorphic to interval p -sheaves. The fact that these interval p -sheaves are of the same type followsdirectly from D − ǫλ ⊆ D µ and Z − ǫλ ⊆ Z µ . (cid:3) To state the next lemma, we introduce some terminology. Let α : E → R × N be given by α ( t + ) = ( t, 1) and α ( t − ) = ( t, − ∀ t ∈ R , α (+ ∞ ) = (+ ∞ , α ( −∞ ) = ( −∞ , − − ” operations in R × N thatperform coordinate-wise addition or subtraction. If we order the elementsof R × N by ( a, n ) < ( b, m ) if a < b , or if a = b and n < m , then α is orderpreserving. Lemma 6.23. Let λ, λ ′ ∈ Λ F and suppose that φ µλ ψ λ ′ µ = 0 for some µ ∈ Λ G .If any one of the conditions (i) F h f λ i ≃ k [ p, q ] , F h f λ ′ i ≃ k [ p ′ , q ′ ] and α ( q ) − α ( p ) ≥ α ( q ′ ) − α ( p ′ ) , (ii) F h f λ i ≃ k [ p, q i , F h f λ ′ i ≃ k [ p ′ , q ′ i and α ( p ) + α ( q ) ≤ α ( p ′ ) + α ( q ′ ) , (iii) F h f λ i ≃ k h p, q ] , F h f λ ′ i ≃ k h p ′ , q ′ ] and α ( p ) + α ( q ) ≥ α ( p ′ ) + α ( q ′ ) , (iv) F h f λ i ≃ k h p, q i , F h f λ ′ i ≃ k h p ′ , q ′ i and α ( q ) − α ( p ) ≤ α ( q ′ ) − α ( p ′ ) .is satisfied, then g µ is ǫ -matched with either f λ or f λ ′ .Proof. By Lemma 6.22, the interval p -sheaf G h g µ i has the same type as theintervals p -sheaves F h f λ i and F h f ′ λ i and(1) D − ǫλ ⊆ D µ and Z − ǫλ ⊆ Z µ ;(2) D − ǫµ ⊆ D λ ′ and Z − ǫµ ⊆ Z λ ′ .Thus, to prove the lemma, it suffices to show that the assumptions that(3) d H (supp [ f λ ] , supp [ g µ )] > ǫ and(4) d H (supp [ f λ ′ ] , supp [ g µ )] > ǫ lead to a contradiction. We divide the argument into four cases. Case 1. Suppose that (i) is satisfied, so that we may write G h g µ i ≃ k [ x, y ].By (1), we have x ≤ p + ǫ and y ≥ q − ǫ , whereas (3) implies that x < p − ǫ or y > q + ǫ . If x < p − ǫ , the inequality y ≥ q − ǫ ensures that α ( y ) − α ( x ) > α ( q ) − α ( p ) . (52)If y > q + ǫ , the inequality x ≤ p + ǫ also implies (52). Similarly, (2) and(4) yield α ( q ′ ) − α ( p ′ ) > α ( y ) − α ( x ) . (53)Then, (52) and (53) imply that α ( q ) − α ( p ) < α ( q ′ ) − α ( p ′ ), which contradictsthe hypothesis. ORRESPONDENCE MODULES 35 Case 2. Suppose that (ii) is satisfied and write g µ ≃ k [ x, y i . By (1), wehave x ≤ p + ǫ and y ≤ q + ǫ , whereas (3) implies that x < p − ǫ or y < q − ǫ .Either possibility yields α ( x ) + α ( y ) < α ( p ) + α ( q ). Similarly, by (2) and(4), we have α ( p ′ ) + α ( q ′ ) < α ( x ) + α ( y ). Then, α ( p ′ ) + α ( q ′ ) < α ( p ) + α ( q ),which is a contradiction. Case 3. Suppose that (iii) is satisfied and let g µ ≃ k h x, y ]. By (1) we have x ≥ p − ǫ and y ≥ q − ǫ , whereas (3) implies that x > p + ǫ or y > q + ǫ ,either giving α ( x ) + α ( y ) > α ( p ) + α ( q ). Similarly, by (2) and (4), we have α ( p ′ ) + α ( q ′ ) > α ( x ) + α ( y ). Then, α ( p ′ ) + α ( q ′ ) > α ( p ) + α ( q ), which is acontradiction. Case 4. Suppose that (iv) holds and write g µ ≃ k h x, y i . By (1) we have x ≥ p − ǫ and y ≤ q + ǫ , whereas (3) implies x > p + ǫ or y < q − ǫ ,either yielding α ( y ) − α ( x ) < α ( q ) − α ( p ). Similarly, by (2) and (4), we have α ( q ′ ) − α ( p ′ ) < α ( y ) − α ( x ). Hence, α ( q ′ ) − α ( p ′ ) < α ( q ) − α ( p ), contradictingthe hypothesis. (cid:3) Definition 6.24. Given f ∈ B F , define m ǫG ( f ) ⊆ B G as the the subset ofall sections g ∈ B G such that f and g are ǫ -matched.Let A = { f λ , . . . , f λ m } ⊆ B F be a finite, indexed sub-collection of basiselements with the property that all interval subsheaves F h f λ i i , 1 ≤ i ≤ m ,are of the same type and the inequalities specified in the hypotheses ofLemma 6.23 are satisfied for any i ≤ j . More precisely, if F h f λ i i ≃ k [ p i , q i ]and F h f λ j i ≃ k [ p j , q j ], i ≤ j , then α ( q i ) − α ( p i ) ≥ α ( q j ) − α ( p j ), andsimilarly for other interval sheaf types. Define ν ( A ) := { g µ ∈ B G : φ µλ i ψ λ j µ = 0 for some pair i ≤ j } . (54)Note that | ν ( A ) | < ∞ because, for each fixed pair i ≤ j , only finitely manyentries φ µλ i and φ λ j µ can be non-zero. Moreover, Lemma 6.23 implies thateach section in ν ( A ) is ǫ -matched with at least one section in A . Thus, wehave ν ( A ) ⊆ ∪ f ∈ A m ǫG ( f ) . (55)We remark that the special ordering of the elements of A is essential in theargument that shows that (55) holds. Proposition 6.25. Let A be as above. If (Φ ǫ , Ψ ǫ ) is an ǫ -interleaving and A ⊆ X ǫF (that is, all sections in A are ǫ -significant), then | A | ≤ | ν ( A ) | < ∞ . Proof. Let ν ( A ) = { g µ , . . . , g µ n } . Since Ψ ǫ ◦ Φ ǫ = e ǫF and all sections in A are 2 ǫ -significant, we have that P nl =1 φ µ l λ i ψ λ j µ l = δ ij , for any i ≤ j , where δ ij is Kronecker’s delta. Thus, the product of the matrices K := ( φ µ l λ i ) m × n and L := ( ψ λ j µ l ) n × m is a triangular matrix with diagonal entries all equal to1. Hence, the product KL has rank m . This implies that | ν ( A ) | = n ≥ rank ( K ) ≥ rank ( KL ) = m = | A | . (cid:3) As pointed out earlier, the general strategy for the proof of stability issimilar to that used by Bjerkevik in the study of multi-parameter persis-tence [3], although the arguments for p -sheaves differ substantially. A keymatching result needed in the proof of the stability theorem, we employ thefollowing classic result in combinatorics and graph theory. Theorem 6.26 (Hall’s Matching Theorem) . Let { Z x : x ∈ X } be a collectionof non-empty subsets of a set Z such that | A | ≤ | ∪ x ∈ A Z x | < ∞ , for anyfinite set A ⊆ X . Then, there exists an injective map ζ : X → Z such that ζ ( x ) ∈ Z x , ∀ x ∈ X . Lemma 6.27 (The Matching Lemma) . Suppose that F and G are decom-posable p -sheaves with bases B F and B G , respectively, and let X := { f ∈ X ǫF : | m ǫG ( f ) | < ∞} . If F and G are ǫ -interleaved, then there exists an injective ǫ -map ζ : X → B G .Proof. We construct ζ by applying Hall’s Matching Theorem to Z := B G , Z f := m ǫF ( f ), for any f ∈ X . The fact that m ǫF ( f ) = ∅ follows from (55)and Proposition 6.25 applied to the singleton { f } . Given any non-emptyfinite set A ⊆ X , write it as the disjoint union A = [ A ] ⊔ [ A i ⊔ h A ] ⊔ h A i , (56)where [ A ] only contains sections f such that F h f i ∼ = k [ p, q ], the subset [ A i only contains sections f such that F h f i ∼ = k [ p, q i , and similarly for theother two terms. If any of these subsets is empty, we simply discard it.We can index the elements of [ A ] appropriately, so that [ A ] satisfies theassumptions of Proposition 6.25. Therefore, for this labeling of [ A ], we have | [ A ] | ≤ | ∪ f ∈ [ A ] m ǫG ( f ) | . Similar inequalities can be obtained for the otherthree subsets of A after appropriate labeling of their elements. Note thatthe sets ∪ f ∈ [ A ] m ǫG ( f ), ∪ f ∈ [ A i m ǫG ( f ), ∪ f ∈h A ] m ǫG ( f ), and ∪ f ∈h A i m ǫG ( f ) arepairwise disjoint because they contain sections that span interval sheaves ofdifferent types. Therefore, | A | = | [ A ] | + | [ A i| + |h A ] | + |h A i| ≤ | ∪ f ∈ A m ǫG ( f ) | . (57)By Theorem 6.26, there is an injection ζ : X → B G such that ζ ( f ) ∈ m ǫG ( f ), ∀ f ∈ X . In other words, ζ is an injective ǫ -map. (cid:3) Theorem 6.28 (The Algebraic Stability Theorem) . Let F and G be decom-posable p -sheaves and ǫ > . If F and G are ǫ -interleaved, then B F and B G are ( ǫ + δ ) -matched, for any δ > . Therefore, d b ( dgm ( F ) , dgm ( G )) ≤ d I ( F, G ) . Proof. We begin by reducing the argument to the case in which both B F and B G are countable sets. Let (Φ ǫ , Ψ ǫ ) be an ǫ -interleaving between F and G represented in the bases B F and B G by the matrices ( φ µλ ) and ( ψ λµ ),respectively. Let G be the bipartite graph with vertex set partitioned as ORRESPONDENCE MODULES 37 B F ⊔ B G , and with an edge between f λ and g µ if φ µλ = 0 or ψ λµ = 0. Then,any connected component C of G is a bipartite subgraph whose vertex set V is countable. This can be seen as follows. If we fix v ∈ V , then for anyinteger n ≥ 0, the set B n := { v ∈ C : d ( v , v ) ≤ n } is finite, where d ( v , v ) isthe distance in the graph C . Hence, V = ∪ n ≥ B n is countable. Moreover,it is a straightforward consequence of Definition 6.3 that if F and G are ǫ -interleaved, so are F h C i and G h C i , the subsheaves of F and G spannedby C and C , respectively.If there is an ( ǫ + δ )-matching between C and C , for any connectedcomponent C of G , then assembling the matchings over all connected com-ponents of G , we obtain an ( ǫ + δ )-matching between B F and B G . Thus,without loss of generality, we assume that B F and B G are countable.By Lemma 6.21, to construct an ( ǫ + δ )-matching between B F and B G , itsuffices to construct injective ( ǫ + δ )-maps σ : X ǫF → B G and τ : X ǫG → B F .To construct σ , we filter the countable set X ǫF ⊆ B F and define it inductivelyover the filtration. Let X := { f ∈ X ǫF : | m ǫG ( f ) | < ∞} (58)and ζ : X → B G be an injective ǫ -map whose existence is guaranteed by theMatching Lemma (Lemma 6.27).Before proceeding with the construction, we recall a standard fact: if N , N , · · · ⊆ Z are infinite subsets of a set Z , there exist infinite subsets N ′ i ⊆ N i , i ≥ 1, such that N ′ i ∩ N ′ j = ∅ , for any i = j .List the sections in X ǫF \ X as { f i : i ≥ } . For any fixed δ > 0, since | m ǫG ( f i ) | = ∞ , by Lemma 6.29 below, there exists a sequence S i := { g ni : n ≥ } ⊆ m ǫG ( f i ) (59)such that any two sections in S i are δ -matched. Furthermore, by the remarkin the previous paragraph, we can assume that S i ∩ S j = ∅ , for any i = j . If X = ∅ , we simply define σ ( f i ), i ≥ 1, to be any element of S i . This concludesthe construction of σ because it is an injective ǫ -mapping, in particular, aninjective ( ǫ + δ )-map. If X = ∅ , let X := X \ ∪ i ≥ ζ − ( S i ) (60)and define σ : X → B G as σ = ζ | X , which is an injective ǫ -map. For i ≥ 1, inductively define X i := X i − ∪ ζ − ( S i ) ∪ { f i } . (61)Assuming that an injective ( ǫ + δ )-map σ i − : X i − → B G has been con-structed, we extend it to σ i : X i → B G , as follows:(i) σ i ( f i ) = g i ;(ii) if f ∈ ζ − ( S i ) ⊆ X and ζ ( f ) = g ji , set σ i ( f ) = g j +1 i .Since f and g ji are ǫ -matched and g ji and g j +1 i are δ -matched, we have that f and σ i ( f ) = g j +1 i are ( ǫ + δ )-matched. Thus, σ i is an injective ( ǫ + δ )-map. As X ǫF = ∪ i ≥ X i , the mapping σ : X ǫF → B G given by σ | X i = σ i has thedesired properties.Similarly, we construct an injective ( ǫ + δ )-map τ : X ǫG → B F . By Lemma6.21, there exists an ( ǫ + δ )-matching between B F and B G .For the stability statement, let ǫ > d I ( F, G ). Then, for any δ > 0, thereis an ( ǫ + δ )-matching between B F and B G . Taking the infimum over δ > d b ( dgm ( F ) , dgm ( G )) ≤ ǫ . Taking the infimum over ǫ > 0, weobtain d b ( dgm ( F ) , dgm ( G )) ≤ d I ( F, G ). (cid:3) Lemma 6.29. Let f ∈ B F be a ǫ -significant section and S ⊆ B G aninfinite subset with the property that each section g ∈ S is ǫ -interleaved with f . Then, for any δ > , there exists an infinite sub-collection S δ ⊆ S suchthat any two sections in S δ are δ -interleaved.Proof. Without loss of generality, we may assume that S is countable. Sup-pose that supp( f ) = ( s ∗ , t ∗ ) and let S = { g n : n ≥ } with supp [ g n ] =( s ∗ n , t ∗ n ), ∀ n ≥ 1. Then, we have s − ǫ ≤ s n ≤ s + ǫ and t − ǫ ≤ t n ≤ t + ǫ forall n . Hence, { ( s n , t n ) } is a bounded set under the d ∞ metric and thereforehas at least one accumulation point, say, ( s , t ). This implies that, for any δ > 0, we can choose a sub-collection of intervals { ( s ∗ n i , t ∗ n i ) } ∞ i =1 , each con-tained in the δ/ s − , t +0 ), with s n i → s and t n i → t . Byconstruction, any pair of sections in S δ = { g n i : i ≥ } are δ -interleaved. (cid:3) Theorem 6.30 (The Isometry Theorem) . If F and G are decomposable p -sheaves, then d b ( dgm ( F ) , dgm ( G )) = d I ( F, G ) . Proof. This follows from Proposition 6.12 and Theorem 6.28. (cid:3) Applications The formulation of persistent structures developed in this paper largelyhas been motivated by applications. This section explores some of these ap-plications, explaining how correspondence modules relate to levelset zigzagpersistence, to slices of 2-D persistence modules, as well as how to obtain ho-mological barcodes richer in geometric information than those obtained fromsublevel (or superlevel) set filtrations or extended persistence [20]. Amongother things, we establish a Mayer-Vietoris sequence relating sublevel andsuperlevel set homology modules to levelset homology modules.7.1. Levelset Persistence. Let f : X → R be a continuous function de-fined on a topological space X . It is of great interest to summarize thetopological changes in the level sets of f across function values, as suchsummaries can provide valuable insights on f . However, unlike sublevelsets, there are no natural mappings relating different level sets, so a com-mon practice is to use interlevel sets to interpolate level sets in a zigzagstructure [10, 12, 11]. To be more precise, for s ≤ t , denote the interlevelset between s and t by X ts := f − ([ s, t ]). To further simplify notation, write ORRESPONDENCE MODULES 39 X [ t ] for the level sets X tt . At the topological space level, we have inclusions X [ s ] ֒ → X ts ← ֓ X [ t ] that induce homomorphisms H ∗ ( X [ s ]) H ∗ ( X ts ) H ∗ ( X [ t ]) φ ts ψ ts (62)on homology (with field coefficients). Thus, for an increasing sequence ( t n ),we obtain a zigzag module . . . H ∗ ( X [ t n − ]) H ∗ ( X t n t n − ) H ∗ ( X [ t n ]) . . . φ nn − ψ nn − (63)Using correspondences, we eliminate the homology of interlevel sets from(63), treating the triple ( H ∗ ( X t n t n − ) , φ nn − , ψ nn − ) as a CVec-morphism from H ∗ ( X [ t n − ]) to H ∗ ( X [ t n ]). This has the virtue of leaving only the homol-ogy of the level sets as objects in the sequence, also leading to a categoricalformulation of level set persistence that easily extends to a continuous pa-rameter t ∈ R .To state the next theorem, recall that we denote the graph of a mapping T by G T and the operator that reverses correspondences by ∗ . For s, t ∈ R , s ≤ t , define a correspondence h ts ⊆ H ∗ ( X [ s ]) × H ∗ ( X [ t ]) by h ts = G ∗ ψ ts ◦ G φ ts ,and let H −∗ ( f ) := ( H ∗ ( X [ t ]) , h ts ), s, t ∈ R , s ≤ t , which we refer to as thelevelset c -module associated with f . As in [11], the homology theory usedis Steenrod-Sitnikov homology [32]. Theorem 7.1. Let X be a locally compact polyhedron. If f : X → R is aproper continuous function and H ∗ is Steenrod-Sitnikov homology with fieldcoefficients, then H −∗ ( f ) is a c -module.Proof. To prove that H −∗ ( f ) is a c -module, it suffices to verify the validity ofthe composition rule h tr = h ts ◦ h sr , for any r ≤ s ≤ t . For the argument wepresent, it will be useful to consider the inclusions of interlevel sets X sr ֒ → X tr ← ֓ X ts , for r ≤ s ≤ t , and the induced homomorphisms H ∗ ( X sr ) H ∗ ( X tr ) H ∗ ( X ts ) . ρ s,tr σ tr,s (64)The proof amounts to a chase in the commutative diagram H ∗ ( X tr ) H ∗ ( X sr ) H ∗ ( X ts ) H ∗ ( X [ r ]) H ∗ ( X [ s ]) H ∗ ( X [ t ]) , ρ s,tr σ tr,s φ sr φ tr ψ sr φ ts ψ ts ψ ts (65)noting that the assumptions on X and f along with the fact that H ∗ isSteenrod-Sitnikov homology imply that the center diamond in (65) is exact (Proposition 3.7 of [11]). This means that the sequence H ∗ ( X [ s ]) H ∗ ( X sr ) ⊕ H ∗ ( X ts ) H ∗ ( X tr ) α β (66)is exact, where α ( a ) = φ sr ( a ) ⊕ ψ ts ( a ) and β ( a, b ) = ρ s,tr ( a ) − σ tr,s ( b ).To check that h ts ◦ h sr ⊆ h tr , let ( v r , v s ) ∈ h sr and ( v s , v t ) ∈ h ts . Set v = ρ s,tr ◦ φ sr ( v s ) = φ tr ( v s ). From (65), it follows that φ tr ( v r ) = ψ ts ( v t ) = v , whichimplies that ( v r , v t ) ∈ h tr . For the reverse inclusion, let ( v r , v t ) ∈ h tr , whichmeans that φ tr ( v r ) = ψ ts ( v t ). The commutativity of the diagram impliesthat w = φ sr ( v r ) and w = ψ ts ( v t ) satisfy ρ s,tr ( w ) = σ tr,s ( w ). By exactness,there exists v s ∈ H ∗ ( X [ s ]) such that ψ sr ( v s ) = w and φ ts ( v s ) = w . Thus,( v r , v s ) ∈ h sr and ( v s , v t ) ∈ h ts , showing that ( v r , v t ) ∈ h ts ◦ h sr . (cid:3) Remark . If dim H ∗ ( X [ t ]) < ∞ , ∀ t ∈ R , then the levelset c -module H −∗ ( f ) is virtually tame, thus admitting an interval decomposition. Thisis the case, for example, if X is a locally compact polyhedron and f is aproper piecewise-linear map. As in [11], using rectangle measures, one maydefine persistence diagrams under the more general setting of Theorem 7.1,without requiring an interval decomposition of H −∗ ( f ). However, we refrainfrom exploring this point of view in this paper. Example . Let f : X → R be as indicated in Fig. 2. The interval decom-position of the levelset c -module H − ( f ) contains all four types of intervalmodules, as indicated in the barcode. The bar types follow the conventionmade in Remark 2.15. X R f − − Figure 2. Persistence diagram for H − ( f ). Remark . There is an interesting connection between the present formu-lation of levelset homology using c -modules and 2-D persistence modules.Consider R with the partial ordering in which ( s , t ) ( s , t ) if s ≥ s and t ≤ t . As a category, this ordering corresponds to R op × R . Let∆ + := { ( s, t ) ∈ R : s ≤ t } and ∆ − := { ( s, t ) ∈ R : s ≥ t } . Similar tothe extension of zigzag modules to an exact 2-D persistence module [4, 18],it is possible to extend a c -module to a persistent module over R . Placethe c -module H −∗ ( f ) along the diagonal ∆ ⊆ R . Under the assumptions of ORRESPONDENCE MODULES 41 Theorem 7.1, one can define an exact p -module over ∆ + , extending H −∗ ( f ),whose vector space at ( s, t ) ∈ ∆ + is V ( s,t ) := H ∗ ( X ts ) and whose morphisms v ( s ,t )( s ,t ) are the mappings on homology induced by the inclusions X t s ֒ → X t s .Letting F denote the p -sheaf of sections of the c -module H −∗ ( f ), define a per-sistence module over ∆ − by V ( s,t ) := F ([ t, s ]) and v ( s ,t )( s ,t ) := F [ t ,s ][ t ,s ] . Onecan verify that these two structures combine to yield a single exact 2-Dpersistence module V over R op × R .7.2. A Mayer-Vietoris Sequence. Let f : X → R be a continuous func-tion. Under the assumptions of Theorem 7.1, in this section, we constructa Mayer-Vietoris (M-V) sequence of c -modules for covers of X given bysublevel and superlevel sets of f . The level set c -modules H − i ( f ), i ≥ 0, con-structed in Section 7.1, represents the homology of the intersections of theelements of these covers. Recall that CMod is the category whose objectsare the c -modules (over R ) with natural transformations as morphisms. Ingeneral, morphisms in CMod do not have kernels. However, in this partic-ular case, there is sufficient structure to formulate an exact M-V sequence.We begin with the construction of the objects in the sequence.Denote the sublevel sets of f by X t := f − (( −∞ , t ]) and the superlevelsets of f by X t := f − ([ t, + ∞ )). For s ≤ t and i ≥ 0, let ı ts : H i ( X s ) → H i ( X t ) and ts : H i ( X t ) → H i ( X s ) be the morphisms induced by the inclu-sions X s ⊆ X t and X t ⊆ X s , respectively. We denote the graphs of ı ts and ts by I ts and J ts , respectively. Then, H ∨ i ( f ) := ( X t , I ts ) and H ∧ i ( f ) := ( X t , J ts )are the homology c -modules associated to the sublevel and superlevel filtra-tions of X , respectively. The direct sum H ∨ i ( f ) ⊕ H ∧ i ( f ) is the homology c -module associated with the covers X = X t ∪ X t , t ∈ R . Note that, for s ≤ t ,the elements ( a s , b s ) ∈ H i ( X s ) ⊕ H i ( X s ) and ( a t , b t ) ∈ H i ( X t ) ⊕ H i ( X t ) arein correspondence in H ∨ i ( f ) ⊕ H ∧ i ( f ) if and only if ı ts ( a s ) = a t and ts ( b t ) = b s .We also define the “constant” c -module H i ( X ) in which the vector space overany t ∈ R is H i ( X ) with the diagonal subspace ∆ H i ( X ) as correspondence,for any s ≤ t .Now we define the relevant c -module morphisms for the M-V sequence.Let p ti : H i ( X t ) → H i ( X ) and q ti : H i ( X t ) → H i ( X ) be the mappings inducedon homology by the inclusions X t ⊆ X and X t ⊆ X . The commutativity ofthe diagrams H ∗ ( X s ) H ∗ ( X t ) H ∗ ( X ) ı ts p si p ti H ∗ ( X s ) H ∗ ( X t ) H ∗ ( X ) q si q ti ts (67)for any s ≤ t , implies that the mappings p ti − q ti : H i ( X t ) ⊕ H i ( X t ) → H i ( X ),given by ( a t , b t ) p ti ( a t ) − q ti ( b t ), induce a c -module morphism p i − q i : H ∨ i ( f ) ⊕ H ∧ i ( f ) → H i ( X ) . (68) Similarly, the inclusions X [ t ] ⊆ X t and X [ t ] ⊆ X t , t ∈ R , induce mappings ψ ti : H i ( X [ t ]) → H i ( X t ) and φ ti : H i ( X [ t ]) → H i ( X t ) on homology, which inturn induce a c -module morphism ψ i ⊕ φ i : H − i ( f ) → H ∨ i ( f ) ⊕ H ∧ i ( f ) . (69)To define the connecting c -module morphisms, we work under the as-sumptions of Theorem 7.1. For each t ∈ R , consider the cover X = X t ∪ X t ,as well as the coarser cover X = X t ∪ X s , for s ≤ t . Naturality of the Mayer-Vietoris sequence implies that inclusions yield a commutative diagram H i ( X ) H i − ( X [ s ]) H i − ( X s ) ⊕ H i − ( X s ) H ( X ) H i ( X ) H i − ( X ts ) H i − ( X t ) ⊕ H i − ( X s ) H ( X ) H i ( X ) H i − ( X [ t ]) H i − ( X t ) ⊕ H i − ( X t ) H ( X ) . ∂ i id φ ts id∂ i ∂ i id ψ ts id (70)In particular, for any i ≥ s ≤ t , the diagram H i − ( X [ s ]) H i ( X ) H i − ( X ts ) H i − ( X [ t ]) φ ts ∂ i ∂ i ψ ts (71)commutes, showing that the mappings ∂ i : H i ( X ) → H i − ( X [ t ]), t ∈ R ,induce a c -module connecting morphism∆ i : H i ( X ) → H − i − ( X ) . (72) Theorem 7.5. If f : X → R is a proper, continuous function defined on alocally compact polyhedron X and H ∗ is Steenrod-Sitnikov homology, thenthe Mayer-Vietoris sequence . . . H i +1 ( X ) H − i ( f ) H ∨ i ( f ) ⊕ H ∧ i ( f ) . . . ∆ i +1 ψ i ⊕ φ i p i − q i is exact in the CMod category.Proof. If W U V G F are any two consecutive morphisms in the sequence,by construction, W t U t V tg t f t is exact, ∀ t ∈ R . By Proposition 2.10(i)and (ii), all morphisms in the sequence have well-defined image and kernel c -modules. CMod exactness follows from CVec exactness at each t ∈ R . (cid:3) Example . Let f : X → R be projection of the space depicted in Fig. 3to a horizontal axis. By Proposition 2.10(iii), coker( p − q ) is a c -module.The barcode for coker( p − q ), also shown in the figure, exactly capturesthe horizontal spread of the 1-dimensional cycles of X . ORRESPONDENCE MODULES 43 − . − − . . . X R f Figure 3. Barcode for the cokernel of p − q : H ∨ ( f ) ⊕ H ∧ ( f ) → H ( X ).In this example, the barcode for extended persistence [20] also encodes thesame geometric properties, but c -modules present the information in a cat-egory theory framework that naturally integrates various different types ofpersistence architectures.7.3. Slicing 2-D Persistence Modules. Multidimensional persistent ho-mology is of great practical interest, as topological analysis of complex datafrequently gives rise to topological or simplicial filtrations that depend onmultiple parameters. However, unlike the one-dimensional case in whichthe barcode of a sufficiently tame p -module yields a complete invariant, itis impossible to obtain a discrete, complete representation of the structureof multidimensional p -modules [13]. As such, it is of interest to define, al-beit incomplete, computable and informative invariants for these modules,such as rank invariants [13, 34] and some numeric invariants [36], to sum-marize their structural properties. Lesnick and Wright slice 2-D persistencemodules along affine lines of non-negative slopes to obtain a family of one-dimensional p -modules whose structures may be described by persistencediagrams [31]. Here, we show how to define a c -module structure alongaffine lines of negative slope. If the original 2-D persistence module is point-wise finite dimensional, then these negatively sloped slices are virtually tamethus admitting an interval decomposition.We view ( R , ≤ ) as a poset, where ( x , y ) ≤ ( x , y ) if and only if x ≤ x and y ≤ y . Let ℓ ⊆ R be an affine line of negative slope. We fix anorientation for ℓ via the unit vector u = e iθ , − π/ < θ < 0, parallel to ℓ . Thisinduces a linear ordering on ℓ given by s t if and only if ( t − s ) · u ≥ 0. Givena 2-D persistence module U : R → Vec, we define a c -module V : R → CVec,termed the slice of U along ℓ , as follows. The vector space at t ∈ ℓ is V t := U t .To define the correspondences, we introduce some notation. For a, b ∈ R ,let a ∨ b ∈ R be the (unique) element that is initial with respect to theproperty that a ≤ a ∨ b and b ≤ a ∨ b . If a = ( x , y ) and b = ( x , y ) satisfy x ≤ x and y ≥ y , then a ∨ b := ( x , y ). For s, t ∈ ℓ with s t , let P ( s, t ) be collection of all finite sequences T = ( t i ) ni =0 of points in ℓ satisfying s = t r t r t i r i t = t n r n t n − r n − v s = w z w z w i z i v t = w n z n w n − z n − (i) a staircase (ii) staircase interpolation Figure 4. Interpolating v s ∈ V s and v t ∈ V t along a staircase. t . . . t n , t = s and t n = t . Letting r i = t i − ∨ t i , define the staircaseΓ T associated to T ∈ P ( s, t ) as the sequence in R given by t ≤ r ≥ t ≤ . . . ≥ t n − ≤ r n ≥ t n , (73)as depicted in Fig. 4(i). Using staircases, define a correspondence v ts , asfollows. Definition 7.7. A pair ( v s , v t ) ∈ v ts if and only if for any staircase Γ T , T ∈ P ( s, t ), there are vectors that interpolate v s and v t along Γ T , as illustratedin Fig. 4(ii). More precisely, there are vectors w i ∈ U t i , 0 ≤ i ≤ n , and z i ∈ U r i , 1 ≤ i ≤ n , such that(i) w = v s and w n = v t ;(ii) u r i t i − ( w i − ) = z i and u r i t i ( w i ) = z i , for 1 ≤ i ≤ n . Lemma 7.8. Let V be a slice of a 2-D persistence module U along a lineof negative slope, and let S, T ∈ P ( s, t ) with S ⊆ T . If v s ∈ V s and v t ∈ V t may be interpolated along Γ T , then v s and v t also may be interpolated alongthe coarser staircase Γ S .Proof. Using an iterative argument, it suffices to consider the case where S and T differ by a single element, say, T = { t , . . . , t j , . . . , t n } and S = { t , . . . , t j − , ˆ t j , t j +1 , . . . , t n } , (74)where ˆ t j indicates deletion of t j . Using the notation of Definition 7.7, sup-pose w i ∈ U t i and z i ∈ U r i interpolate v s and v t along Γ T . Let ¯ r j = t j − ∨ t j +1 and ¯ z j = u ¯ r j t j − ( w j − ). Then,¯ z j = u ¯ r j t j − ( w j − ) = u ¯ r j r j ◦ u r j t j ( w j ) = u ¯ r j r j +1 ◦ u r j +1 t j ( w j )= u ¯ r j r j +1 ( z j +1 ) = u ¯ r j r j +1 ◦ u r j +1 t j +1 ( w j +1 ) = u ¯ r j t j +1 ( w j +1 ) . (75)Thus, ¯ z j = u ¯ r j t j − ( w j − ) = u ¯ r j t j +1 ( w j +1 ), showing that the vectors w , . . . , w j − , w j +1 , . . . , w n and r , . . . , r j − , ¯ r j , r j +2 , . . . , r n (76) ORRESPONDENCE MODULES 45 interpolate v s and v t along the staircase Γ S . (cid:3) Theorem 7.9. Let U : R → Vec be a pointwise finite-dimensional persis-tence module. If ℓ ⊆ R is a negatively sloped line, then the slice of U along ℓ is a virtually tame c -module.Proof. Let V be the slice of U along ℓ . To show that V is a c -module, itsuffices to verify the composition rule v tr = v ts ◦ v sr for morphisms. We beginwith the inclusion v ts ◦ v sr ⊆ v tr . Let ( v r , v s ) ∈ v sr and ( v s , v t ) ∈ v ts . Given T ∈ P ( r, t ), let T = T ∪ { s } . Write T as a union T = T ∪ T , T ∈ P ( r, s ),and T ∈ P ( s, t ). By assumption, we may interpolate v r and v s along Γ T , aswell as v s and v t along Γ T . Concatenating these interpolations, we obtainan interpolation of v r and v t along Γ T . By Lemma 7.8, v r and v t also can beinterpolated along the coarser staircase Γ T . This proves that ( v r , v t ) ∈ v tr .For the reverse inclusion, let ( v r , v t ) ∈ v tr and s ∈ ℓ be such that r s t .Our goal is to show that there exists v s ∈ V s = U s such that ( v r , v s ) ∈ v sr and ( v s , v t ) ∈ v ts . Given T ∈ P ( r, s ) and T ∈ P ( s, t ), let V T ,T be the affinesubspace of V s comprising those vectors v s ∈ V s such that:(i) v r and v s may be interpolated along Γ T ;(ii) v s and v t may be interpolated along Γ T .Note that V T ,T is non-empty because ( v s , v t ) ∈ v ts implies that v r and v t may be interpolated along the staircase Γ T ∪ T . To conclude the argument,we show that W s := \ T ∈ P ( r,s ) T ∈ P ( s,t ) V T ,T = ∅ , (77)as this implies that any v s ∈ W s satisfies ( v r , v s ) ∈ v sr and ( v s , v t ) ∈ v ts , asdesired.Let A = { V T ,T : T ∈ P ( r, s ) and T ∈ P ( s, t ) } , partially ordered viainclusion. Since V s is finite dimensional, each descending chain in A stabilizesin finitely many steps, thus having a lower bound. By Zorn’s Lemma, A has aminimal element V R ,R . We show that V R ,R ⊆ V T ,T , for any T ∈ P ( r, s )and T ∈ P ( s, t ). Let S = T ∪ R and S = T ∪ R . By Lemma 7.8, V S ,S ⊆ V T ,T and V S ,S ⊆ V R ,R . By minimality, V R ,R = V S ,S ⊆ V T ,T . Hence, W s = V R ,R = ∅ , as claimed.The virtual tameness of V follows from the assumption that U is pointwisefinite dimensional. (cid:3) Closing Remarks This paper introduced and developed two main concepts: (i) correspon-dence modules as a generalization of such structures as persistence modulesand zigzag modules that are of interest in topological data analysis and(ii) persistence sheaves that provide a pathway to the structural analysis of c -modules. Using sheaf-theoretical arguments, we proved interval decompo-sition theorems for sufficiently tame c -modules and p -sheaves parameterized over R , as well as a stability theorem for persistence-diagram representa-tions of p -sheaves. Applications discussed in the paper include: (1) a newformulation of continuously parameterized level-set persistence in a cate-gory theory framework; (2) a Mayer-Vietoris sequence that brings togetherlevelset, sublevelset and superlevelset homology modules of a real-valuedfunction; and (3) 1-dimensional slices of 2-dimensional persistence modulesalong lines of negative slope.This study of persistent homology from the viewpoint of c -modules and p -sheaves opens up several avenues for further investigation. Here, we discusssome of the questions that the results of this paper raise.(a) The Isometry Theorem was proven in the context of decomposable p -sheaves. On the other hand, Theorem 5.14 shows that any virtually tame c -module V admits a (unique) interval decomposition and therefore maybe represented by a persistence diagram. Is there an isometry theoremfor decomposable c -modules under a suitable notion of interleaving?(b) As pointed out in Remark 7.4, the levelset homology c -module of afunction, viewed as parameterized over the diagonal ∆ ⊆ R , can beextended to a 2-D exact persistence module over the entire plane. Abovethe diagonal, the extension is given by the homology of interlevel sets andbelow the diagonal by sections of the c -module over closed intervals. Canthis construction be extended to arbitrary c -modules? One of our plansis to construct and investigate such extensions using limits and colimitsin the CVec category. Such extensions will provide an additional tool toanalyze the structure of c -modules.(c) This paper has focused primarily on structural and stability questions as-sociated with c -modules. However, the practical relevance of c -modulesdepends heavily on the computability of interval decompositions. Thus,a basic problem is that of developing and implementing an algorithm tocalculate the persistence diagram of a sufficiently tame c -module. References [1] N. Berkouk and G. Ginot. A derived isometry theorem for constructible sheaves on R . arXiv:1805.09694, 2018.[2] N. Berkouk, G. Ginot, and S. Oudot. Level-sets persistence and sheaf theory.arXiv:1907.09759, 2019.[3] H. B. Bjerkevik. Stability of higher-dimensional interval decomposable persistencemodules. arXiv: 1609.02086v2, 2016.[4] M. Botnan and M. Lesnick. Algebraic stability of zigzag persistence modules. Alge-braic Geom. Topol. , 18(6):3133–3204, 2018.[5] M. B. Botnan. Interval decomposition of infinite zigzag persistence modules. Proc.Am. Math. Soc. , 145(8):3571–3577, 2017.[6] M. B. Botnan and W. Crawley-Boevey. Decomposition of persistence modules. Proc.Am. Math. Soc. , doi: 10.1090/proc/14790, in press.[7] P. Bubenik. Statistical topological data analysis using persistence landscapes. J.Mach. Learn. Res. , 16(1):77–102, 2015.[8] D. Burghelea and S. Haller. Topology of angle valued maps, barcodes and Jordanblocks. J Appl. and Comput. Topology , 1(1):121–197, 2017. ORRESPONDENCE MODULES 47 [9] G. Carlsson. Topology and data. Bull. Am. Math. Soc. , 46(2):255–308, 2009.[10] G. Carlsson and V. de Silva. Zigzag persistence. Found. Comput. Math. , 10(4):367–405, 2010.[11] G. Carlsson, V. de Silva, S. Kaliˇsnik, and D. Morozov. Parametrized homology viazigzag persistence. Algebraic Geom. Topol. , 19(2):657–700, 2019.[12] G. Carlsson, V. de Silva, and D. Morozov. Zigzag persistent homology and real-valuedfunctions. In Proceedings of the Twenty-Fifth Annual Symposium on ComputationalGeometry , SCG 09, pages 247–256, New York, NY, USA, 2009. Association for Com-puting Machinery.[13] G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. DiscreteComput. Geom. , 42(1):71–93, 2009.[14] G. Carlsson, A. Zomorodian, A. Collins, and L. J. Guibas. Persistence barcodes forshapes. Int. J. Shape Model. , 11(02):149–187, 2005.[15] F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Y. Oudot. Proximity ofpersistence modules and their diagrams. In Proceedings of the Twenty-Fifth AnnualSymposium on Computational Geometry , SCG 09, pages 237–246, New York, NY,USA, 2009. Association for Computing Machinery.[16] F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. M´emoli, and S. Y. Oudot. Gromov-Hausdorff stable signatures for shapes using persistence. Comput. Graph. Forum ,28(5):1393–1403, 2009.[17] F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The Structure and Stability of Per-sistence Modules . Springer Briefs in Mathematics. Springer International Publishing,2016.[18] J. Cochoy and S. Oudot. Decomposition of exact pfd persistence bimodules. DiscreteComput. Geom. , 63(2):255–293, 2020.[19] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete Comput. Geom. , 37(1):103–120, 2007.[20] D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Extending persistence usingPoincar´e and Lefschetz duality. Found. Comput. Math. , 9:133–134, 2009.[21] W. Crawley-Boevey. Decomposition of pointwise finite-dimensional persistence mod-ules. J. Algebra Its Appl. , 14(05):1550066, 2015.[22] J. Curry. Sheaves, cosheaves and applications. arXiv:1303.3255, 2013.[23] H. Edelsbrunner and J. Harer. Persistent homology – a survey. Contemporary Math-ematics , 453:257–282, 2008.[24] H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and sim-plification. Discrete. Comput. Geom. , 28(4):511–533, 2002.[25] P. Frosini. A distance for similarity classes of submanifolds of a euclidean space. Bull.Aust. Math. Soc. , 42(3):407–415, 1990.[26] P. Frosini, C. Landi, and F. M´emoli. The persistent homotopy type distance. Homol.Homotopy Appl. , 21(2):231–259, 2019.[27] P. Gabriel. Unzerlegbare darstellungen I. Manuscripta Math. , 6(1):71–103, 1972.[28] H. Hang, F. M´emoli, and W. Mio. A topological study of functional data and Fr´echetfunctions of metric measure spaces. J Appl. and Comput. Topology , 3(4):359–380,2019.[29] M. Kashiwara and P. Schapira. Persistent homology and microlocal sheaf theory. J.Appl. Comput. Topol. , 2(1–2):83–113, 2018.[30] M. Lesnick. The theory of the interleaving distance on multidimensional persistencemodules. Found. Comput. Math. , 15(3):613–650, 2015.[31] M. Lesnick and M. Wright. Interactive visualization of 2-d persistence modules.arXiv:1512.00180, 2015.[32] J. Milnor. On the Steenrod homology theory (first distributed 1961). In S. C. Ferry,A. Ranicki, and J. M. Rosenberg, editors, Novikov Conjectures, Index Theorems, and Rigidity: Oberwolfach 1993 , volume 1 of London Mathematical Society Lecture NoteSeries , pages 79–96. Cambridge University Press, 1995.[33] S. Y. Oudot. Persistence theory: from quiver representations to data analysis , volume209. American Mathematical Society Providence, 2015.[34] A. Patel. Generalized persistence diagrams. J. Appl. Comput. Topol. , 1(3-4):397–419,2018.[35] V. Robins. Towards computing homology from finite approximations. In TopologyProceedings , volume 24, pages 503–532, 1999.[36] J. Skryzalin and G. Carlsson. Numeric invariants from multidimensional persistence. J. Appl. Comput. Topol. , 1(1):89–119, 2017.[37] A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput.Geom. , 33(2):249–274, 2005. Department of Mathematics, Florida State University, Tallahassee, FL32306-4510 USA E-mail address : [email protected] Department of Mathematics, Florida State University, Tallahassee, FL32306-4510 USA E-mail address ::