aa r X i v : . [ m a t h . A T ] N ov PERSISTENCE AND SHEAVES
KARTHIK YEGNESH
Abstract.
In this note, we descibe a mild generalization of Carlsson and Zomorodian’s per-sistent homology of filtered topological spaces, namely persistent sheaf cohomology . Givena sheaf of abelian groups on a filtered topological space, we obtain a global description ofthe sheaf cohomology classes present across the space by studying the persistence of thecohomology classes. As an application, we study the persistent cellular sheaf cohomology ofnetwork coding sheaves developed by Ghrist and Hiraoka in [2]. The persistence of networkcoding sheaf cohomology classes across a filtered digraph (which represents network deterio-ration phenomena) provides insight into the stability of certain information flows across thenetwork. Introduction
Let X be a topological space and φ : X → R a continuous real-valued function. Traditionalpersistent homology theory seeks to obtain a global description of the homological propertiesof X via examining the singular homology of subspaces of X induced by φ . That is, persistenthomology studies the “persistence” of the homology groups of each subspace of the filteredspace φ − ( −∞ , i ] = X ֒ → φ − ( −∞ , i ] = X ֒ → . . . ֒ → φ − ( −∞ , i n ] = X n = X , where i j < i k for j < k and n < ∞ . The persistence of homology classes in the filtration yieldsuseful information regarding their significance in the global picture of X . In this paper, westudy the data of an abelian sheaf on a filtered topological space via studying the persistenceof the sheaf cohomology functor restricted to sub-spaces in the filtration. As an application,we study the persistent cellular sheaf cohomology of a network coding sheaves developed in [2].The persistence of network coding sheaf cohomology classes provide insight into the stabilityof certain information flows across the network.2. Background
We will recall some basic definitions regarding persistent homology and (co)sheaf (co)homology.We will assume some background with basic algebraic topology, including singular homologyand some category theory. For more background, the read is encouraged to look at [1] and [3].2.1.
Persistent Homology.
Persistent homology is a tool which one uses to study the birthand death of topological features in a filtered space.
Definition 2.1.
Let X • be a filtered topological space, i.e a space X equipped with a sequenceof nested subspaces X ⊂ X ⊂ . . . ⊂ X n = X . Let H n ( X ) denote the n th singular homology(with coefficients in a field k ) vector space of X . Fix indices i and j with i ≤ j . The ( i, j )-persistent n th homology vector space H i,jn ( X • ) is defined as H i,jn ( X • ) = im( H n ( X i ) → H n ( X j )),where H n ( X i ) → H n ( X j ) is induced by the inclusion X i ⊂ X j . Remark 2.2.
The dimension of the k -vector space H i,jn ( X • ) is the number of n -dimensionalholes present in subspace X i that are also present in X j . Example 2.3. If i = j , then it is clear that H i,jn ( X • ) = H n ( X i ) since the k -linear map inducedby id : X i → X i must be the identity map on H n ( X i ). Sheaves and Cech cohomology.
We recall some relevant facts about sheaves and co-homology. In this paper, we will use both sheaves and cosheaves, but the information thissubsection is easily dualized for the case of cosheaves. An excellent survey of cosheaf theorycan be found in Justin Curry’s thesis [3].Basically, a sheaf is an association of “data” (what exactly that means depends on thesituation) to open sets of a topological space that is compatible with inclusions
U ֒ → V of opensets of the space. Formally: Definition 2.4.
Let X be a topological space and C an abelian category (the reader cansafely imagine C to be Ab, the category of abelian groups). A C -valued presheaf F on X isa contravariant functor F : Open( X ) op → C from the category of open subsets of X to J . If U ⊂ X , an element x ∈ F ( U ) is a section of F over U . For a pair of embedded open subsets V ⊂ U ⊂ X , the induced map on the inclusion F ( U ) → F ( V ) is called the restriction map. Apresheaf F on X is a sheaf if for any open U ⊂ X and any open cover { U i } of U , the sequence0 → F ( U ) → L i F ( U i ) → L i F ( T i U i ) is exact. Note that if F is a sheaf, then F ( ∅ ) = 0,where 0 denotes the zero object of C (e.g the trivial abelian group, the trivial k -vector space,etc.). Example 2.5.
The presheaf which associates to each open set U ⊂ X its singular n th coho-mology H n ( U ; G ) with coefficients in some abelian group G is a sheaf. The sheaf condition inthis case is satisfied because the functor H n satisfies the Mayer-Vietoris property. Non-Example 2.6.
The constant presheaf F : Open( X ) → Ab which sends each open set tothe abelian group Z and each morphism U → V to id Z is not a sheaf. Definition 2.7.
Let X be a topological space and U = { U i } an open cover of X , and F apresheaf of abelian groups on X . The group of ˇCech k -chains associated to U is the group C k ( U , F ) = L i F ( U , ,...,k ), where U , ,...,k = T ki =0 U i .Equipped with differentials ∂ k : C k ( U ; F ) → C k +1 ( U ; F ), we obtain a ˇCech complex C • ( U ; F ) =0 → C ( U ; F ) → C ( U ; F ) → . . . → C k ( U ; F ) → . . . . We denote the k th ˇCech cohomologygroup associated to F and covering U by ˇ H n ( U ; F ) = ker( ∂ k ) / im( ∂ k − ). Remark 2.8. ˇCech cohomology in general does not coincide with sheaf cohomology (definedvia derived functors), but for our purposes and eventual applications, ˇCech cohomology suffices.2.3.
Cellular Sheaves.
In order to make (co)sheaf (co)homology computable in practical sce-narios, one often restrict attention to sheaves over cell complexes which are valued in the cate-gory of vector spaces over a (usually finite) field. In this paper, we will develop out techniquesin the generality of arbritrary topological spaces. However, our main application of studyingthe persistence of network coding sheaf cohomology classes will involve cellular language.
Definition 2.9.
Let X be a cell complex (see [3]). Let Cell( X ) denote the poset of cells in X ,regarded as a category in which there exists a single arrow x → y if and only if x is a face of y .A cellular sheaf F on X is a functor F : Cell( X ) → Vect k , where Vect k denotes the categoryof vector spaces over the field k . Definition 2.10.
Given a cellular sheaf F on X , one can define cellular sheaf cohomology . Itis a computationally feasible adaptation of the Cech/Sheaf cohomology of sheaves on generaltopological spaces or Grothendieck sites. Cell( X ) k denote the set of k -dimensional cells of X . Write x ≤ y if x is a face of y . For λ and θ two cells in X , denote by [ λ, θ ] the signedincidence relation [3, Definition 6.1.9]. Let C ( X ; F ) k = L c ∈ Cell( X ) k F ( c ) . Define the differential ∂ k : C ( X ; F ) k → C ( X ; F ) k +1 by ∂ k ( c ) = P θ ≤ c [ θ : c ] α c,θ for c ∈ C ( X ; F ) k and α c,θ being the ERSISTENCE AND SHEAVES 3 restriction. Since ∂ = 0, define the k th cellular sheaf cohomolgy k -vector space H kc ( X ; F ) =ker( ∂ k ) / im( ∂ k − ).Cellular sheaf cohomology (particularly in the contex of network coding sheaves [1]) will beused as an application of our tools.3. Persistent Sheaf Cohomology
We can adjust the definition of persistent homology slightly to obtain persistent sheaf coho-mology . We will place the restriction that the spaces in consideration are cell complexes, so weare dealing with cellular sheaf cohomology.
Notation 3.1.
Let X be a finite topological space. Denote by X • a filtration X ⊂ X ⊂ . . . ⊂ X n = X . We will assume that the X i are indexed over an ordered set. Each X i is endowedwith the subspace topology induced by the inclusion i : X i ֒ → X . Let F : X op → Vect k bea Vect k -valued sheaf over X (which restricts to a sheaf over all the X i ). Fix a covering U on X . This restricts to coverings U i on each of the X i . Let C k ( U , F ) = L i F ( U , ,...,k ) (seeDefinition 2.4). Let Ω i denote the inclusion Ω i : X i ֒ → X i +1 .We obtain the following commutative diagram, called the sheaf persistence complex . Construction 3.2. ... (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) → . . . → C kk − ( U i − ; F ) Ω i − (cid:15) (cid:15) ∂ i − k − / / C kk ( U i − ; F ) Ω i − (cid:15) (cid:15) ∂ i − k / / C kk +1 ( U i − ; F ) → . . . → Ω i − (cid:15) (cid:15) → . . . → C kk − ( U i ; F ) Ω i (cid:15) (cid:15) ∂ ik − / / C kk ( U i ; F ) Ω i (cid:15) (cid:15) ∂ ik / / C kk +1 ( U i ; F ) → . . . → Ω i (cid:15) (cid:15) → . . . → C kk − ( U i +1 ; F ) (cid:15) (cid:15) ∂ i +1 k − / / C kk ( U i +1 ; F ) (cid:15) (cid:15) ∂ i +1 k / / C kk +1 ( U i +1 ; F ) → . . . → (cid:15) (cid:15) ... ... ...The cohomology vector spaces of this complex will be used in the definition of co-persistentsheaf cohomology , which we now provide. Definition 3.3.
Let H k ( X i ; F ) denote the k th sheaf cohomology vector space of F restrictedto X i , i.e H k ( X ; F ) = ker( ∂ ik ) / im( ∂ ik − ) as dictated by the above diagram. Let X • be a filteredtopological space and F a cellular cosheaf on it. The k th ( i, j ) co-persistent sheaf cohomol-ogy vector space H ki,j ( X ; F ) is defined as im( H k ( X j ; F ) → H k ( X i ; F )), where H k ( X j ; F ) → H k ( X i ; F ) is induced by the inclusion X i ֒ → X j .Notice that instead of defining the persistent cosheaf homology, we choose to define co-persistent sheaf cohomology. The vector space H ki,j encodes the Cech cohomology classes ofthe restriction F | X j that are not destroyed when passing to the subspace X i . The analogue ofthis in the singular homology world would be studying which homology classes are present ina subspace that are not disrupted when passing to a smaller subspace. KARTHIK YEGNESH
Remark 3.4.
Let e F be the (sheafification of) the constant k -valued presheaf F : Open( X ) → Vect k . Then there is an isomorphism of k -vector spaces H ki,j ( X ; F ) ≃ im( H k ( X j , k ) → H k ( X i , k )). In other words, we can obtain “persistent singular cohomology” as a special caseof persistent sheaf cohomology in the same way that singular cohomology is a special case ofsheaf cohomology. Remark 3.5.
This is very easily dualizable to obtain persistent cosheaf homology .In the context of this paper, co-persistent cosheaf cohomology enables us to study which NCsheaf cohomology classes persist through a deteriorating network. We will switch to “persistentcellular sheaf cohomology,” but this is defined in the exact same way as the more general case.3.1.
Sheaf Cohomological Persistence Modules and Diagrams.
The theory of quiverrepresentations and persistence diagrams plays a large role in the development of persistenthomology in the sense of Carlsson and Zomorodian. We will describe a similar scenario on thecontext of persistent (co)sheaf homology. We recall a definition first.
Definition 3.6. A persistence module of length n is a sequence of vector spaces V i over a field k indexed over { i ∈ N | i ≤ n } equipped with k linear maps ϕ i : V i → V i +1 . Equivalently, this isa functor from the small category • → . . . → • (with n objects) to Vect k .It is a classical result that every persistence module admits a unique decomposition intodirect sums of so called interval modules . An interval module is a persistence module of theform 0 → . . . → k → . . . → k → H n : TopSpaces → Vect k to a filtered topological space X • = X ֒ → X ֒ → . . . . The lengths of the intervals in the canonical interval decomposition (which represent thelifetimes of homology classes) are recorded in a multiset called a persistence diagram . Definition 3.7.
Let X • be a filtered topological space. Let Z ∞ denote the set Z ∪ {∞} . Adegree n persistence diagram is roughly a multiset over Z ∞ × Z ∞ , where the multiplicity µ ( i, j ) ofa point ( i, j ) ∈ Z ∞ is dim( H i,jn ( X • )). The points parametrize the lifetimes of homology classesin X • . The points of a persistence diagram therefore correspond to ”intervals” in the intervaldecomposition of the persistence module obtain via the homology of X • . The multiplicityfunction records the number of intervals of a particular length and position exist.The notion of a persistence diagram thus can be generalized to any situation in which thereis a suitable interpretation of a persistence module. Construction 3.8.
Let F be a sheaf of vector spaces on filtered topological space X • = X ֒ → X ֒ → . . . . Let U be a covering of X and denote by U i the restriction of U to X i . Byapplying the Cech cohomology functor H k ( U − ; F ) to each X i , we obtain a persistence module H k ( U ; F ) → H k ( U ; F ) → . . . . This admits a direct-sum decomposition into interval modules.Let Int k ( i, j ) (for i ≤ j ≤ ∞ ) denote the set of interval modules of length j − i such that thefirst non-trivial vector space in each of the interval modules is at position i and last at position j . Remark 3.9.
The interval modules in the last construction represent the lifetimes of sheafcohomology classes in the X i as one passes to increasingly smaller subspaces of X . The longintervals describe sheaf cohomology classes which “persist” through the “deteriorating space,”while the short intervals indicate that certain classes do not. Remark 3.10.
If instead we use the cosheaf homology functor where the cosheaf is the constantfunctor taking values in the field k , then the persistence module obtained is precisely the oneobtained by taking persistent homology with coefficients in k . ERSISTENCE AND SHEAVES 5
We now define degree n sheaf persistence diagrams based on the persistence modules associ-ated to a sheaf on a filtered space described previously. Construction 3.11.
Fix a sheaf F : X • → Vect k . For each n ∈ N , construct a multisetDgm n ( X • ; F ) over Z ∞ × Z ∞ called the degree n sheaf persistence diagram associated to F as follows. If Int n ( i, j ) = ∅ , add a point ( i, j ) ∈ Dgm n ( X • ; F ). The multiplicity function µ : Dgm n ( X • ; F ) → N is given by µ ( i, j ) = | Int n ( i, j ) | .The multiset Dgm n ( X • ; F ) provides a global description of the lifetimes of sheaf cohomol-ogy classes across the filtered space X i . For example, clustering of points along the diagonalindicates an instability of sheaf cohomology data when passing to smaller subspaces as dictatedby the nature of the filtration.4. Application to Information Flow in Deteriorating Networks
We will now present an application of our tools to studying information flow across unstablenetworks. We first recall the basics of network coding sheaves found in [2].4.1.
Network Coding Sheaves.
Let F be a network viewed as a directed graph. Denoteby V ( G ) and E ( G ) its sets of nodes and edges, respectively. Assume that there exist sets S, T ⊂ V ( G ) called sources and targets . Define a function cap : E ( G ) → N which assigns toeach edge e in F , its capacity cap( e ) ∈ N . Let Vect k denote the category of finite vector spacesand k -linear maps for some Galois field k . We can construct a cellular sheaf F : G → Vect k called a network coding sheaf according to the definition below. Definition 4.1. A network coding sheaf F : G → Vect k is a cellular sheaf constructed asfollows. To each edge e ∈ E ( G ), F ( e ) = k cap( e ) . For a node v , denote by In( v ) the setof directed edges that are directed towards v . For a node v , F ( v ) = L e ∈ In( v ) cap( e ). Therestriction maps are given by the canonical projections.The main result of [2] is the following. Theorem 4.2.
Let F be a NC sheaf on directed graph F . Then the th cellular sheaf cohomologyvector space H ( G ; F ) is equivalent to the total information flows on F . Construction 4.3.
Let F be a finite directed graph (regarded as a 1-dimensional cell complex)with the structure necessary to construct a NC sheaf F : G → Vect k . Let st : E ( G ) → R bea function on the edge set of F assigning to each edge e in F its strength st( e ) ∈ R ≥ . Wemay constrain the domain to the positive reals, but this is not entirely necessary. Denote by F i the subgraph of F consisting of all nodes in addition to edges e such that st( e ) ≤ e . Callpositive real r critical if F r = G r − ǫ for some ǫ >
0, i.e the subcomplex F i is strictly larger that F r − ǫ . Denote by { c , c , . . . } the set of critical values in increasing order. We have a filtration G st • = G c ⊂ G c ⊂ . . . ⊂ G c ∞ . Remark 4.4.
The filtration G st • is solely indicative of network link strength. The smallestsubcomplexes contain the edges with the weakest strength functions. Notation 4.5.
Fox X a finite directed graph with a strength function f , X f • will denote thefiltered cell complex in the sense of Construction 4.3. Proposition 4.6.
The co-persistent NC sheaf cohomology vector spaces H i,j ( X • ; F ) are equiv-alent to the information flows on network F which survive with the removal of edges E ( X j \ X i ) .Proof. By [2, Theorem 8], the vector spaces H ( X i ) and H ( X j ) are equivalent to the informa-tion flows across sub-networks F i and F j , respectively. The image of the map on vector spaces H k ( X j ; F ) → H k ( X i ; F ) induced by the inclusion F i ֒ → X j is generated by precisely the NC KARTHIK YEGNESH sheaf cohomology classes that are present in F j but are also present in the sub-network F i .This is the definition of H i,j ( X • ; F ), so the proof is completed. (cid:3) One can also gain useful information from the degree 0 persistence diagram Dgm ( X • ; F )associated to the NC sheaf F and the filtered network F • . Construction 4.7.
Applying the degree 0 NC sheaf cohomology functor H ( − ; F ) to thefiltered network X • , we obtain a persistence module (and therefore a persistence diagram).The decomposition into interval modules indicates which information flows on the networkpersist through the network’s deterioration based on their length. The longest intervals survivethrough the most edge deterioration and vice versa. Remark 4.8.