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ON THE STRUCTURAL THEOREM OF PERSISTENT HOMOLOGY 7 1 KILLIANMEEHAN,ANDREIPAVLICHENKO,ANDJANSEGERT 0 2 r Abstract. This note addresses the categorical framework for the computa- a tion of persistent homology, without reliance on a particular computational M algorithm. Thecomputationofpersistenthomologyiscommonlysummarized as a matrix theorem, which we call the Matrix Structural Theorem. Any of 7 the various algorithms for computing persistent homology constitutes a constructive proof of the Matrix Structural Theorem. We show that the Matrix ] Structural Theorem is equivalent to the Krull-Schmidt property of the cate- T goryoffilteredchaincomplexes. TheKrull-Schmidtpropertyisestablishedby A abstract categorical methods, yielding a nonconstructive proof of the Matrix Structural Theorem. . h These results provide the foundation for an alternate categorical frame- t work for persitent homology, bypassing the usual persistence vector spaces a andquiverrepresentations. m [ 3 But the power of homology is seldom of much efficacy, except v in those happy dispositions where it is almost superfluous. 5 5 with apologies to Edward Gibbon 0 2 0 . Contents 1 0 1. Two Manifestations of the Structural Theorem 2 7 1.1. Matrix Structural Theorem 2 1 1.2. Categorical Structural Theorem and Structural Equivalence 5 : v 2. Proving the Categorical Structural Theorem 6 i X 2.1. Additive and Krull-Schmidt Categories 6 2.2. Persistence Objects and Filtered Objects 7 r a 2.3. Chain Complexes and Filtered Chain Complexes 8 3. Two CategoricalFrameworks for Persistent Homology 10 3.1. Standard Framework using Persistence Vector Spaces 10 3.2. Alternate Framework using Quotient Categories 11 4. Proving Structural Equivalence 13 4.1. Forward Structural Equivalence 13 4.2. Reverse Structural Equivalence 19 Appendix A. Bruhat Uniqueness Lemma 21 Appendix B. Constructively Proving the Matrix Structural Theorem 22 B.1. Linear Algebra of Reduction 22 B.2. Matrix Structural Theorem via Reduction 24 References 27 Date:March6,2017. 1 2 KILLIANMEEHAN,ANDREIPAVLICHENKO,ANDJANSEGERT 1. Two Manifestations of the Structural Theorem 1.1. MatrixStructuralTheorem. Persistenthomology[10,6]hasachievedgreat success as a powerful and versatile tool, particularly for topological analysis of data. A number ofexcellentsurveysandintroductions is availablein the literature [18, 9, 22, 12, 21]. The characterization of decompositions in terms of barcode in- variants plays a central role in the theory and applications of persistent homology. Thecomputationofpersistenthomologyiscommonlysummarizedasamatrixthe- orem,whichwecallthe MatrixStructuralTheoreminthis paper. Inthis paperwe study the categoricalsetting for the Matrix Structural Theorem, and introduce an alternate categoricalframework for understanding the decomposition properties of persistent homology. For simplicity, we start with the ungraded version of the structural theorem. A differential matrix is a square matrix D satisfying D2 = 0. We’ll say a differential matrix is Jordan if it is in Jordan normalform, meaning it decomposes as a block- diagonal matrix built from copies of the two differential Jordan block matrices 0 1 J = [0] , K = . (cid:20)0 0(cid:21) We’ll say a differential matrix D is almost-Jordan if there exists a permutation matrix P such that the differential matrix P−1DP is Jordan. Given an almost- JordandifferentialmatrixD,itistrivialtoconstructsuchapermutationmatrixP. We will say a square matrix B is triangular if it is upper-triangularand invertible. The standard algorithm for computing persistent homology is based on the pa- pers [10, 23, 24]. The result of a persistent homology computation, not depending on a choice of algorithm, is conveniently summarized [8, 18] as a matrix factoriza- tion: Theorem 1.1. (Ungraded Matrix Structural Theorem) Any differential matrix D factors as D =BDB−1 where D is an almost-Jordan differential matrix and B is a triangular matrix. It is the triangular condition that makes this interesting: without the triangular condition, this would follow immediately from the ordinary Jordan normal form. Furthermore,thematrixDisuniquebyastandardresultfromBruhatfactorization, as reviewed in Appendix A. We’ll call D the persistence canonical form of the differential matrix D. A column of the triangular matrix B is in kerD iff the corresponding column of D is zero. We will say that B is normalized if each such column has diagonal entry equal to 1. It is always possible to normalize B by scalar multiplication of columns, but even with normalization B is not unique in general. AconstructiveproofofTheorem1.1followsfromanyofthealgorithmsfor computing persistent homology. In Appendix B we discuss a simple version of the standardalgorithm,withcompleteproofsusingordinarylinearalgebraofmatrices. Example 1.2. Consider the filtered simplicial complex: PERSISTENT HOMOLOGY 3 b b b b b b a a a c a c a c a c p≤0 p=1 p=2 p=3 p=4 p=5 p≥6 With the usual convention for an adapted basis, the ordering of basis elements pri- oritizes filtration level over degree. The initial basis of simplices is then ordered so thefiltration level(denotedbyprescript) is nondecreasing, andwithin each filtration level the degree (denoted by postscript) is nondecreasing. Using lexicographic order tobreakanyremainingties,theinitialadaptedbasisis a , b , ab , c , bc , ac , abc , 1 0 1 0 2 1 3 0 4 1 5 1 6 2 and the boundary operator over the field F = Q of rationals is represented by the differential matrix a b ab c bc ac abc 1 0 1 0 2 1 3 0 4 1 5 1 6 2 a 0 0 −1 0 0 −1 0 1 0 b 0 0 1 0 −1 0 0 1 0   ab 0 0 0 0 0 0 1 2 1 D = c  0 0 0 0 1 1 0 . 3 0   bc  0 0 0 0 0 0 1  4 1   ac  0 0 0 0 0 0 −1  5 1   abc  0 0 0 0 0 0 0  6 2   The persistence canonical form is a b ab c bc ac abc 1 0 1 0 2 1 3 0 4 1 5 1 6 2 a 0 0 0 0 0 0 0 1 0 b 0 0 1 0 0 0 0 1 0   ab 0 0 0 0 0 0 0 2 1 D = c  0 0 0 0 1 0 0 , 3 0   bc  0 0 0 0 0 0 0  4 1   ac  0 0 0 0 0 0 1  5 1   abc  0 0 0 0 0 0 0  6 2   asverifiedbycheckingthatD =B−1DB for thetriangular (andnormalized) matrix a b ab c bc ac abc 1 0 1 0 2 1 3 0 4 1 5 1 6 2 a 1 −1 0 −1 0 0 0 1 0 b 0 1 0 0 0 0 0 1 0   ab 0 0 1 0 1 −1 0 2 1 B = c  0 0 0 1 0 0 0 . 3 0   bc  0 0 0 0 1 −1 0  4 1   ac  0 0 0 0 0 1 0  5 1   abc  0 0 0 0 0 0 −1  6 2   4 KILLIANMEEHAN,ANDREIPAVLICHENKO,ANDJANSEGERT The persistence canonical form D is almost-Jordan in general, and in this example it happens to be actually Jordan. The matrix B represents the basis change to the new adapted basis a , b , ab , c , bc , ac , abc . The filtration level remains 1 0 1 0 2 1 3 0 4 1 5 1 6 2 nondecreasing because B is triangular. Each basis element retains pure degree, although Theorem 1.1 does not explicitly address issues of degree. The matrix D represents the boundary operator relative to the new adapted basis. We prefer to prioritize degree over filtration level in ordering the elements of an adapted basis. This has the advantage of encoding the degree in the matrix block structure. The following versionof the structuraltheorem is then manifestly compatible with the grading by degree: Theorem 1.3. (Matrix Structural Theorem) Any block-superdiagonal differential matrix D factors as D =BDB−1 where D is a block-superdiagonal almost-Jordan differential matrix and B is a block-diagonal triangular matrix. The block-diagonal structure of B ensures that the transformed basis elements re- tainpuredegree. ThepersistencecanonicalformDinheritstheblock-superdiagonal structure of the differential D. It is always possible to normalize B by scalar multiplication of columns as in the ungraded case. Any of the algorithmic proofs of Theorem 1.1 [10, 23, 24] can be used to prove Theorem 1.3 by keeping track of degrees. We discuss this point for the standard algorithm in Appendix B.2. Example 1.4. We again consider the filtered chain complex of Example 1.2, but with basis order prioritizing degree over filtration level. Now the degree of basis elements (denoted by postscript) is nondecreasing, and within a degree the filtration level (denoted by prescript) of basis elements is nondecreasing. Using lexicographic ordertobreakanyremainingties,theinitialadaptedbasisisnow a , b , c , ab , bc , ac , abc , 1 0 1 0 3 0 2 1 4 1 5 1 6 2 and the boundary operator over the field F = Q of rationals is now represented by the block-superdiagonal differential matrix a b c ab bc ac abc 1 0 1 0 3 0 2 1 4 1 5 1 6 2 a 0 0 0 −1 0 −1 0 1 0 b 0 0 0 1 −1 0 0 1 0   c 0 0 0 0 1 1 0 3 0 D = ab  0 0 0 0 0 0 1 . 2 1   bc  0 0 0 0 0 0 1  4 1   ac  0 0 0 0 0 0 −1  5 1   abc  0 0 0 0 0 0 0  6 2   The persistence canonical form inherits the block-superdiagonal structure PERSISTENT HOMOLOGY 5 a b c ab bc ac abc 1 0 1 0 3 0 2 1 4 1 5 1 6 2 a 0 0 0 0 0 0 0 1 0 b 0 0 0 1 0 0 0 1 0   c 0 0 0 0 1 0 0 3 0 D = ab  0 0 0 0 0 0 0 , 2 1   bc  0 0 0 0 0 0 0  4 1   ac  0 0 0 0 0 0 1  5 1   abc  0 0 0 0 0 0 0  6 2   as verified by checking that D = B−1DB for the block-diagonal triangular (and normalized) matrix a b c ab bc ac abc 1 0 1 0 3 0 2 1 4 1 5 1 6 2 a 1 −1 −1 0 0 0 0 1 0 b 0 1 0 0 0 0 0 1 0   c 0 0 1 0 0 0 0 3 0 B = ab  0 0 0 1 1 −1 0 . 2 1   bc  0 0 0 0 1 −1 0  4 1   ac  0 0 0 0 0 1 0  5 1   abc  0 0 0 0 0 0 −1  6 2   The persistence canonical form D is almost-Jordan, but not actually Jordan in this example. The matrix B represents the basis change to the new adapted basis a , b , c , ab , bc , ac , abc . Since B is block-diagonal, each basis element 1 0 1 0 3 0 2 1 4 1 5 1 6 2 retains pure degree, and the degree remains nondecreasing. Since B is triangular, the filtration level remains nondecreasing within each degree. The computation of this matrix B via the standard algorithm is worked out in Appendix B.2. 1.2. CategoricalStructuralTheoremandStructuralEquivalence. AKrull- Schmidt categoryis an additive category where objects decompose nicely as direct sums of indecomposable objects. In chapter 2, we study the additive category of filtered chain complexes in the setting of Krull-Schmidt categories, starting with a review of Krull-Schmidt categories in section 2.1. A filtered complex will be called basic if its boundary operator can be represented by differential matrix consisting ofa singleJordanblock. We will use nonconstructivecategoricalmethods to prove the following structural theorem for the category of filtered complexes: Theorem1.5. (Categorical StructuralTheorem)Thecategoryoffilteredcomplexes is Krull-Schmidt. A filtered complex is indecomposable iff it is basic. In chapter 3 we will prove the equivalence of the matrix and the categorical versions of the structural theorem. One direction is proved in section 3.1: Proposition 1.6. (Forward Structural Equivalence) The Matrix Structural Theo- rem implies the Categorical Structural Theorem. ThisisfollowedbyadetailedexampleofaKrull-Schmidtdecomposition. Theother direction is proved in section 3.2: 6 KILLIANMEEHAN,ANDREIPAVLICHENKO,ANDJANSEGERT Proposition 1.7. (Reverse Structural Equivalence) The Categorical Structural Theorem implies the Matrix Structural Theorem. Combining the Categorical Structural Theorem 1.5 and the Reverse Structural EquivalenceProposition1.7yieldsanonconstructivecategoricalproofoftheMatrix Structural Theorem 1.3. This contrasts with the various constructive algorithmic proofs of Theorem 1.3, one of which is reviewed in Appendix B.2. The construc- tivealgorithmicproofsexplainhowpersistenthomologyworks,thenonconstructive proof explains why persistent homology works. 2. Proving the Categorical Structural Theorem 2.1. Additive and Krull-Schmidt Categories. This section reviews the rel- evant background from category theory. General references for category theory include [16, 4, 1]. Additive categories are discussed in [16, 20]. Krull-Schmidt categories are discussed in [13, 17, 3]. Definition 2.1. A category is additive if: (1) Each Hom(X,Y) is an abelian group, and the morphism composition map Hom(Y,Z)×Hom(X,Y)→Hom(X,Z) is biadditive/bilinear. (2) There exists a zero object 0. (3) Any finite collection of objects X ,X ,...,X has a direct sum 1 2 n X ⊕X ⊕···⊕X . 1 2 n An additive category is linear over the field F if each Hom(X,Y) is a finite- dimensionalF-vectorspace,andeachmapHom(Y,Z)×Hom(X,Y)→Hom(X,Z) describing composition of morphisms (g,f) 7→ g◦f is F-bilinear. All of the categories we will be studying are linear. The endomorphism ring of an object X in an additive category is the Abelian group Hom(X,X) of endomorphisms, with multiplicative structure given by composition of endomorphisms. In a linear category, the endomorphism ring is an F-algebra. Definition 2.2. A ring is local if: (1) 16=0. (2) If an element f does not have a multiplicative inverse, then the element 1−f has a multiplicative inverse. The local property is important, because a finite direct sum decomposition into summands with a local endomorphism rings is essentially unique: Theorem 2.3. (e.g. [13] Theorem 4.2) Let X be an object in an additive category, and suppose there are two finite decompositions X ⊕···⊕X =X =Y ⊕···⊕Y 1 m 1 n into(nonzero) objects with local endomorphism rings. Then m=n and there exists a permutation π such that X ≃Y for each 1≤i≤m. i π(i) An object X in an additive category is decomposable if it is the direct sum X = Y ⊕Z of two nonzero objects Y and Z. An indecomposable object, often abbreviated as an indecomposable, is a nonzero object that is not decomposable. Lemma 2.4. An object is indecomposable if it has a local endomorphism ring. PERSISTENT HOMOLOGY 7 Proof. We will show that the endomorphism ring of a decomposable object X is not local. We may assume that X = Y ⊕Z with Y and Z nonzero. Then neither f =1 ⊕ 0 nor 1 −f =1 ⊕ 1 −f =0 ⊕ 1 has a multiplicative inverse in Y Z X Y Z Y Z the ring Hom(X,X)=Hom(Y ⊕Z,Y ⊕Z). (cid:3) A Krull-Schmidt category has properties that guarantee both the existence and essential uniqueness of finite direct sum decompositions of any object, see e.g. [13, 17] for more details: Definition 2.5. An additive category is Krull-Schmidt if: (1) Every object admits a finite decomposition as a sum of indecomposables. (2) Every indecomposable has a local endomorphism ring. Recall that an additive category is Abelian if every morphism has a kernel and a cokernel,everymonic morphismis normal(is the kernelofsome morphism), and every epic morphism is conormal (is the cokernel of some morphism). Note that Definition2.5ofKrull-Schmidtcategorydoesnotassumethattheadditivecategory is Abelian, or even the existence of kernels and cokernels. We are primarily inter- ested in the the linear category of filtered chain complexes, which is not Abelian. But we will use Abelian categories and their subcategories to show that this cate- goryisnonethelessKrull-Schmidt. Atiyah’sCriterion[3,13]providesaverygeneral sufficient condition for an Abelian category to be Krull-Schmidt. Since all of our categories are linear, we will only need the following special case: Theorem 2.6. (Atiyah’s Criterion) A linear Abelian category is Krull-Schmidt. TheproofofAtiyah’sCriterionisnonconstructive. Itneitherprovidesanalgorithm todecomposeagivenobjectasadirectsumofindecomposables,noraclassification of indecomposables. 2.2. Persistence Objects and Filtered Objects. Persistence objects [6] and filtered objects [20] are described by categorical diagrams. Suppose that X is a linear Abelian category (and therefore Krull-Schmidt by Theorem 2.6). We will study persistence indexedby the integersZ with their standardpartialorder≤. A persistence object in X is a diagram X in the category X of type • ··· X X X ··· . p−1 p p+1 ′ A morphism of persistence objects f : X → X is a commutative diagram of • • • “ladder” type ··· X X X ··· p−1 p p+1 p−1f pf p+1f ′ ′ ′ ··· X X X ··· p−1 p p+1 The category of persistence objects in X is Abelian, with pointwise kernels, cok- ′ ernels, and direct sums. The set of morphisms X → X between two persis- • • tence objects is a vector space, but not finite-dimensional in general. We will say a categorical diagram is tempered if all but finitely many of its morphisms are ′ isomorphisms. Since X is linear, the set of morphisms X → X between two • • tempered persistence objects is a finite-dimensional vector space. The tempered 8 KILLIANMEEHAN,ANDREIPAVLICHENKO,ANDJANSEGERT persistence objects comprise a strictly full Abelian subcategory of the persistence objects. Theorem 2.6 now yields: Proposition 2.7. Let X be a linear Abelian category. The category of tempered persistence objects in X is Krull-Schmidt. We next discuss subobjects in a linear Abelian category X. We make the addi- tionalassumptionthatthecategoryX isconcrete,meaningthatanobjectinX isa set with some additional structure, and a morphism in X is a map of sets compatible with the additional structure. For example, the linear Abelian category V of (finite-dimensional)vectorspacesisaconcretelinearAbeliancategory. Asubobject ′ morphism X ֒→X in X is a morphism that is an inclusion of the underlying sets. ′ We say X is a subobject of X iff such a subobject morphism exists. A subobject morphism is monic, and the composition of subobject morphisms is a subobject ′ ′ morphism. Any object X in X has a zero subobject 0 ֒→ X , and is its own sub- ′ ′ ′ ′ object X ֒→ X . A subobject X ֒→ X is proper if X 6= X . A nonzero object is said to be simple if it does not have a proper nonzero subobject. A simple object is obviously indecomposable, but an indecomposable object need not be simple. A filtered object in a concrete linear Abelian category X is a special type of tempered persistence object in X. We say a tempered persistence object X in X • is bounded below if there exists an integer j such that X =0 whenever p≤j. We j say X is a filtered object if it is bounded below and if every arrow is a subobject • morphism: ··· X X X ··· . p−1 p p+1 The filtered objects in X comprise a strictly full subcategory of the tempered persistence objects. The properties of monic morphisms have several consequences. A filtered object diagram has a limit and a colimit in the categorical sense (see e.g. [4] for limits and colimits). The limit is 0 since the diagram is bounded below. The colimit X is X for k sufficiently large (satisfying X = X whenever k p k ≤p). Finally,anysummandofafilteredobjectis isomorphicto afilteredobject. Combining these facts yields: Lemma 2.8. Let X be a linear Abelian category. The category of filtered objects in X is Krull-Schmidt. A filtered object X in X is indecomposable iff its colimit • X is an indecomposable object in X. We notethatthefilteredobjectscompriseasubcategoryofthetemperedpersis- tence objects, but this subcategory is not Abelian because a morphism of filtered objects may have a kernel and/or cokernel that is not a filtered object. So Lemma 2.8 is not merely a corollary of Theorem 2.6. Finally we observe that the category of persistence objects in a (concrete) linear Abelian category is itself a (concrete) linear Abelian category, to which Lemma 2.8 applies. 2.3. Chain Complexes and Filtered Chain Complexes. A persistence vector space is a persistence object in the (concrete) linear Abelian category X = V of (finite-dimensional) F-vector spaces. Tempered persistence vector spaces are well- understoodviathetheoryofquiverrepresentations. AnonemptysubsetI ⊆Zwill becalledanintervalifc∈I whenevera≤c≤bwitha∈I andb∈I. Weassociate to aninterval I ⊆Z the interval persistence vector space I constructed as follows: • PERSISTENT HOMOLOGY 9 I = F whenever p ∈ I, I = 0 whenever p ∈/ I, and every arrow F → F is the p p identity morphism 1. We will often omit the bullet prescript when context allows. For example, the interval persistence vector space [1,4)= [1,4) is the diagram of • vector spaces ··· 0 F 1 F 1 F 0 ··· p=0 p=1 p=2 p=3 p=4 associated to the interval [1,4) = {1,2,3} ⊆ Z. Proposition 2.7 applies to the linear Abelian category of tempered persistence vector spaces. Furthermore, the well-studied representation theory of A quivers (see e.g. [19]) carries over by a n limiting argument to prove the following structural theorem for the category of tempered persistence vector spaces: Theorem2.9. ThecategoryoftemperedpersistencevectorspacesisKrull-Schmidt. A tempered persistence vector space is indecomposable iff it is isomorphic to an interval. Theorem 2.9 can be applied to cochain complexes. A cochain complex, or cocomplex for short, is a tempered persistence vector space V • ··· V ∂p V ∂p+1 V ··· p−1 p p+1 with the property that the composition of successive morphisms ∂p+1◦∂p is zero. The kernel of a morphism between cocomplexes is a cocomplex, as is the cokernel, so the cocomplexes comprise a strictly full Abelian subcategory of the tempered persistence vector spaces. Theorem 2.6 and Theorem 2.9 now yield the structural result: Proposition 2.10. The category Copof cocomplexes is linear and Abelian, and therefore Krull-Schmidt. A cocomplex is indecomposable iff it is isomorphic to an interval cocomplex. Chain complexes are dual to cochain complexes. A complex (short for chain complex) is a tempered diagram V• in V of type ··· Vn−2 ∂n−1 Vn−1 ∂n Vn ··· with the property that the composition of successive morphisms ∂n−1◦∂n is zero. ′ A morphism of complexes f• : V• → V• is a commutative ladder diagram. The category V of vector spaces is isomorphic to its opposite category Vop via the duality functor that takes a vector space to its dual and a linear map to its trans- pose/adjoint. Duality takes the category Cop of cocomplexes to the category of complexes C. A complex is called an interval complex if its dual is an interval cocomplex, and Proposition 2.10 becomes: Proposition 2.11. The category C of complexes is linear and Abelian, and therefore Krull-Schmidt. A complex is indecomposable iff it is isomorphic to an interval complex. The interval complexes are easily classified. An interval complex I• is associated to an interval I ⊆ Z as follows: n ∈ I whenever I = F, and n ∈/ I whenever n I =0. Sinceacomplexcannothaveadjacentisomorphisms,theintervalcomplexes n are in bijective correspondence with the intervals I ⊆ Z of cardinality at most 10 KILLIANMEEHAN,ANDREIPAVLICHENKO,ANDJANSEGERT two. We will often omit the bullet postscript when the context allows. We denote by J[n] = {n} ⊆ Z the intervals of cardinality one. For example, the complex J[1]=J[1]• is the diagram of vector spaces ··· 0 F 0 0 ··· n=0 n=1 n=2 n=3 The indecomposable complex J[n] is simple. We denote by K[n] = [n,n+1] ⊆ Z the intervals of cardinality two. For example, the complex K[1] = K[1]• is the diagram of vector spaces ··· 0 F F 0 ··· 1 n=0 n=1 n=2 n=3 The indecomposable complex K[n] has exactly one nonzero proper subobject J[n] ֒→ K[n]. For example, the subobject morphism of complexes J[1] ֒→ K[1] is the commutative ladder diagram ··· 0 F 0 0 ··· 1 ··· 0 F F 0 ··· 1 n=0 n=1 n=3 n=4 WenowreturntothetheCategoricalStructuralTheorem1.5. Afilteredcomplex is a diagram in the category C of complexes ··· V V V ··· . p−1 • p • p+1 • We will say a filtered complex is basic if its colimit V• is isomorphic to an interval complex. The first statement of Proposition 2.11 tells us that the category C of complexes is linear and Abelian. Then Lemma 2.8 tells us that the category of filtered complexes is Krull-Schmidt. The second statement of Proposition 2.11 classifies the indecomposable filtered complexes, completing the proof of: Theorem1.5. (Categorical StructuralTheorem)Thecategoryoffilteredcomplexes is Krull-Schmidt. A filtered complex is indecomposable iff it is basic. Thebasicfilteredcomplexesareeasilyclassifiedsinceweknowallpropersubobjects of interval complexes, namely 0֒→J[n], 0֒→K[n], and J[n]֒→K[n]. Details and examples of basic filtered complexes appear in Chapter 4. 3. Two Categorical Frameworks for Persistent Homology 3.1. Standard Framework using Persistence Vector Spaces. The structural theoremforthecategoryoftemperedpersistencevectorspaces,Theorem2.9,isthe foundation for the standard framework for persistent homology. For each integer n, the homology of degree n is a functor H : C → V from the n category C of complexes to the category V of vector spaces. An object C in C is a diagram of vector spaces ··· Vn−1 ∂n Vn ∂n+1 Vn+1 ···