Invariants for tame parametrised chain complexes
aa r X i v : . [ m a t h . A T ] N ov INVARIANTS FOR TAME PARAMETRISED CHAIN COMPLEXES
WOJCIECH CHACH ´OLSKI, BARBARA GIUNTI and CLAUDIA LANDI
Abstract
We set the foundations for a new approach to TopologicalData Analysis (TDA) based on homotopical methods at chaincomplexes level. We present the category of tame parametrisedchain complexes as a comprehensive environment that includesseveral cases that usually TDA handles separately, such as per-sistence modules, zigzag modules, and commutative ladders. Weextract new invariants in this category using a model structureand various minimal cofibrant approximations. Such approxi-mations and their invariants retain some of the topological, andnot just homological, aspects of the objects they approximate.
Introduction
Data analysis is often about simplifying, ignoring most of the information availableand extracting what might be meaningful for the task at hand. The same strategyof extracting summaries is also at the core of topology. In recent years, these twobranches have merged, giving rise to Topological Data Analysis (TDA) [ ].TDA can benefit from a broad spectrum of existing homotopical tools for extractingsuch summaries. Currently, the most popular is persistent homology. The first step inpersistence theory is to transform data into spatial information via, for example, theVietoris-Rips construction. The second step is typically the extraction of the homologyof the obtained spaces, resulting in a so-called persistence module, effectively studiedby enumerating its indecomposables [ ].Despite its success, TDA based on persistent homology has some limitations. Firstly,one is limited to objects for which it is possible to list their indecomposable summands[ ], and whose decompositions can be computed algorithmically. For example, theclass of commutative ladders cannot be analysed using its indecomposables because itis of wild representation type [ , ]. On the other hand, the indecomposables of theclass of zigzag modules are fully described, but so far there is no efficient software toanalyse them [ , ]. Secondly, applying homology might be too drastic, disregardinga large amount of geometric information.The main goal of this paper is to show how to use homotopy theory not only toovercome the previous issues but also to open the way towards new invariants. Thesame strategy of disregarding some information and focusing on aspects that might berelevant is also at the core of homotopy theory, with colocalization being an exampleof such a process. In colocalization, the simplification is achieved by approximatingarbitrary objects by other objects that are simpler and more manageable, such as the lass of cofibrant objects in a model category. Our work is based on the realisationthat the category tame ([0 , ∞ ) , ch) of tame [0 , ∞ )-parametrised chain complexes overa field admits a model structure for which there is a surprisingly simple decompositiontheorem describing all indecomposable cofibrant objects (see Theorem 4.2). This is soeven though the entire category tame ([0 , ∞ ) , ch) is of wild representation type. Thestructure theorem 4.2 identifies cofibrant objects in tame ([0 , ∞ ) , ch) with sequencesof persistence diagrams augmented with points on the diagonal (see Section 5) whichwe call Betti diagrams.The cofibrant objects can be then used to approximate arbitrary objects in tame ([0 , ∞ ) , ch).Proving that such minimal approximations exist (see Theorem 2.7) has been essentialin this work.Model categories are convenient for ensuring the existence of certain morphisms orapproximations. A common difficulty in working with model categories, however, isthe lack of algorithmic constructions producing such morphisms and approximations.Approximations in model categories are often constructed using universal proper-ties and require performing large limits. Extracting calculable invariants from suchapproximations, which is essential in TDA, is often not feasible. In this article, wemake a great effort to describe all the constructions, factorisations, and approxima-tions explicitly. All the steps we perform for the tame [0 , ∞ )-parametrised chaincomplexes in perspective can be implemented.Considering tame [0 , ∞ )-parametrised chain complexes instead of vector spaces hasanother advantage. Persistence modules, zigzag modules, and commutative laddersare special objects in tame ([0 , ∞ ) , ch). Thus, this category allows for a comprehensivetheory in which different objects that are handled separately by standard persistencetheory can be studied and compared together. Furthermore, for persistence modules,cofibrant minimal approximations provide complete invariants (see 5.4).In conclusion, we propose a refined approach to the persistence pipeline: first, con-vert the input into a parametrised simplicial complex. Second, extract a parametrisedchain complex. Third, form a minimal cofibrant approximation of the extractedparametrised chain complex. Finally, represent the minimal cofibrant approximationby its Betti diagrams. Related works.
The model structure described in Section 2 is a special case of aprojective model structures on a tame subcategory of functor categories [ , ].The structure theorem 4.2 describing cofibrant objects in tame ([0 , ∞ ) , ch) appearsalso in, for example, [ , , ], although the language of model categories is notused there. An interpretation from the point of view of Morse theory was givenin [ ]. In [ ], Meehan, Pavlichenko and Segert show that the category of filteredchain complexes is a Krull-Schmidt category. In [ ], Usher and Zang generalise thetheory of barcodes to filtered Floer-type complexes, considering chain complexes ofinfinite dimension whose parametrisation is not tame. They prove a singular valuedecomposition theorem for such complexes and identify two types of barcodes of them:the verbose and the concise . In the finite case, such barcodes correspond respectivelyto the Betti diagrams and the minimal Betti diagrams of cofibrant objects in oursetting (see Section 5).The point of view of homotopy theory is entering the TDA subject also for purposesdifferent from ours. For example, [ , , , ] are about lifting the stability theorem2f persistence to homotopy stability theorems, to make it applicable to a wider classof datasets.
1. Minimality
Let M be a model category [ , ]. This means that three classes of morphismsin M are chosen: weak equivalences ( ∼ −→ ), fibrations ( ։ ), and cofibrations ( ֒ → ).These classes and M are required to satisfy the following axioms: MC1.
Finite limits and colimits exist in M . MC2. If f and g are morphisms in M such that gf is defined and if two of the threemorphisms f , g , gf are weak equivalences, then so is the third. MC3. If f is a retract of g and g is a fibration, a cofibration, or a weak equivalence,then so is f . MC4.
Consider a commutative square in M consisting of the solid morphisms: X EY B α β
Then a morphism, depicted by the dotted arrow and making this diagram com-mutative, exists under either of the following two assumptions: (i) α is a cofi-bration and a weak equivalence and β is a fibration, or (ii) α is a cofibrationand β is a fibration and a weak equivalence. MC5.
Any morphism g can be factored in two ways: (i) g = βα , where α is a cofibrationand β is both a fibration and a weak equivalence, and (ii) g = βα , where α isboth a cofibration and a weak equivalence and β is a fibration.In particular, MC1 guarantees the existence of the initial object, denoted by ∅ ,and of the terminal object, denoted by ∗ . An object X in a model category M iscalled cofibrant if the morphism ∅ → X is a cofibration. If the morphism X → ∗ isa fibration, then X is called fibrant .Axiom MC5 above guarantees existence of certain factorisations of morphisms. Itdoes not specify any uniqueness. Typically, a morphism in a model category admitsmany such factorisations. There are however model categories in which among allthese factorisations there is a canonical one called minimal [ , ]: Definition 1.1.
Let g : X → Y be a morphism in M . A factorisation g = βα , where α is cofibration and β is a fibration and a weak equivalence, is called minimal if everymorphism φ which makes the following diagram commutative is an isomorphism: AX YA β ∼ αα g β ∼ φ A minimal factorisation of ∅ → X is called a minimal cover of X .According to the above definition, we can think about a minimal cover of X as amorphism β : cov( X ) → X such that: (i) cov( X ) is cofibrant, (ii) β is both a fibration3nd a weak equivalence, and (iii) any morphism φ which makes the following diagramcommutative is an isomorphism: cov( X )cov( X ) X β ∼ β ∼ φ Minimal factorisations are unique:
Proposition 1.2.
Let g : X → Y be a morphism in M . Assume βα = g = β ′ α ′ areminimal factorisations. Then there is an isomorphism φ making the following diagramcommutative: X A ′ A Y α ′ α β ′ ∼ β ∼ φ Proof.
Let φ and ψ be any morphisms making the following diagram commute, whichexist by the lifting axiom MC4: X A ′ A Y α ′ α β ′ ∼ ψβ ∼ φ Then by the definition of minimal factorisations, the compositions φψ and ψφ areisomorphisms. Consequently, so are φ and ψ .Two objects X and Y in M are called weakly equivalent if there is a sequenceof weak equivalences of the form: X A A · · · A k Y ∼ ∼ ∼ ∼ ∼ Similarly to factorisations of morphisms in a model category, the collection ofobjects weakly equivalent to a given object is large. There are model categories, how-ever, where this collection contains a canonical object called a minimal representative:
Definition 1.3.
An object X in M is called minimal if it is cofibrant, fibrant, andany weak equivalence φ : X → X is an isomorphism. A minimal representative ofan object X in M is a minimal object in M which is weakly equivalent to X .Minimal representatives are unique up to isomorphisms: Proposition 1.4.
Let X ′ and Y ′ be minimal representatives of respectively X and Y . Then X and Y are weakly equivalent if and only if X ′ and Y ′ are isomorphic.Proof. If X ′ and Y ′ are isomorphic, then X and Y are weakly equivalent. Assume X and Y are weakly equivalent. Then X ′ and Y ′ are also weakly equivalent. Sincethey are both cofibrant and fibrant there are weak equivalences φ : X ′ ∼ −→ Y ′ and ψ : Y ′ ∼ −→ X ′ . By the definition of the minimality, the compositions φψ and ψφ areisomorphisms. Consequently, so are φ and ψ and hence X ′ and Y ′ are isomorphic.4roposition 1.2 and Proposition 1.4 ensure the uniqueness of minimal factorisa-tions, minimal covers and minimal representatives. These propositions however donot imply their existence, which has to be proven separately and it does depend onthe considered model category. Definition 1.5.
A model category satisfies the minimal factorisation axiom if allminimal factorisations exist in this category. It satisfies the minimal representativeaxiom if all minimal representatives exist in this category.Many model categories, particularly of combinatorial flavour, satisfy the minimalfactorisation axiom. However the standard model structure on topological spaces [ ]does not.
2. Tame [0 , ∞ ) -parametrised objects Let M be a category. The symbol [0 , ∞ ) denotes the poset of non-negative real num-bers. Functors of the form X : [0 , ∞ ) → M are also referred to as [0 , ∞ )-parametrisedobjects. The value of X at t in [0 , ∞ ) is denoted by X t and X s t : X s → X t denotesthe morphism in M that X assigns to s t . The morphism X s t is also referred toas the transition morphism in X from s to t . Definition 2.1.
A sequence τ < · · · < τ k in [0 , ∞ ) discretises X : [0 , ∞ ) → M if X s t : X s → X t may fail to be an isomorphism only when there is a in [ k ] such that s < τ a t . A functor X : [0 , ∞ ) → M is called tame if there is a sequence that dis-cretises it. The symbol tame ([0 , ∞ ) , M ) denotes the category whose objects are tamefunctors X : [0 , ∞ ) → M and whose morphisms are the natural transformations.If τ < · · · < τ k discretises X : [0 , ∞ ) → M , then the transitions of the restrictionsof X to the intervals [0 , τ ),. . . , [ τ k − , τ k ), and [ τ k , ∞ ) are isomorphisms. Note thatif τ < · · · < τ k discretises X : [0 , ∞ ) → M , then so does any of its refinements (asequence µ < · · · < µ n is a refinement of τ < · · · < τ k if { τ . . . , τ k } is a subset of { µ . . . , µ n } ).2.2. Kan extensions.
Consider a sequence of k composable morphisms in M : X · · · X kX < X k − Let M admit all finite colimits. Let g : X → Y be a morphism intame ([0 , ∞ ) , M ) and 0 = τ < · · · < τ k a sequence discretising both X and Y . Byinduction on a in [ k ], define morphisms ¯ g τ a : X τ a → Q τ a and ˆ g τ a : Q τ a → Y τ a in M as follows:For a = 0 : (¯ g : X → Q ) := (1 : X → X ) (ˆ g : Q → Y ) := ( g : X → Y )For a = 1 . . . , k : Q τ a := colim( Y τ a − g τa − ←−−−− X τ a − X τa − <τa −−−−−−−→ X τ a )¯ g τ a : X τ a → Q τ a and ˆ g τ a : Q τ a → X τ a are the unique morphisms making the followingdiagram commutative, where the inside square is pushout: X τ a − X τ a Y τ a − Q τ a Y τ a X τa − <τa g τa − g τa ¯ g τa Y τa − <τa ˆ g τa For a = 1 , . . . , k , define Q τ a − <τ a : Q τ a − → Q τ a to be the composition of the mor-phism Y τ a − → Q τ a represented by the bottom horizontal arrow in the above dia-gram and ˆ g τ a − : Q τ a − → Y τ a − . Let Q : [0 , ∞ ) → M be the tame functor given bythe Kan extension of the sequence of morphisms { Q τ a − <τ a } along 0 = τ < · · · < τ k g : X → Q and ˆ g : Q → Y the natural transformationgiven by { ¯ g τ a } a =0 ,...,k and { ˆ g τ a } a =0 ,...,k (see 2.2). Note that g = ˆ g ¯ g .The isomorphism type of the functor Q and the factorisation g = ˆ g ¯ g do not dependon the choice of the sequence that discretises X and Y . If ¯ f : X → P and ˆ f : P → Y are natural transformations constructed with respect to another such a sequence, thenthere is a unique isomorphism φ : Q → P for which the following diagram commutes: QX YP ˆ g ¯ g ¯ f g ˆ fφ Theorem 2.4. Let M be a model category. The following choices of weak equiva-lences, fibrations and cofibrations form a model structure on tame ([0 , ∞ ) , M ) . Amorphism g : X → Y in tame ([0 , ∞ ) , M ) is a • weak equivalence if g t : X t → Y t is a weak equivalence for all t . • fibration if g t : X t → Y t is a fibration for all t . • cofibration if ˆ g t : Q t → Y t (see 2.3) is a cofibration for all t . Due to tameness, to prove g : X → Y in tame ([0 , ∞ ) , M ) is a weak equivalence,or a fibration, or a cofibration, only finitely many verifications need to be performed.If 0 = τ < · · · < τ k discretises both X and Y , then g is a weak equivalence (respec-tively, a fibration) if and only if g τ a : X τ a → Y τ a is a weak equivalence (respectively,a fibration) in M for any a = 0 , . . . , k . Similarly, g is a cofibration if and only ifˆ g τ a : Q τ a → Y τ a is a cofibration in M for any a = 0 , . . . , k . It is important to realisehowever that for g to be a cofibration is it not enough for g t to be a cofibration forall t . Proposition 2.5. Let M be a model category.1. If g : X → Y is a cofibration in tame ([0 , ∞ ) , M ) , then g t : X t → Y t is a cofi-bration in M for any t in [0 , ∞ ) .2. An object X in tame ([0 , ∞ ) , M ) is cofibrant if and only if X is cofibrant and,for any s < t in [0 , ∞ ) , the transition morphism X s 0. Furthermore, ˆ g = ( ∅ → X ) and ˆ g τ a : Q τ a = X τ a − → X τ a is the transition morphism in X for a > 0. The statement is then a direct conse-quence of the definition of a cofibration in tame ([0 , ∞ ) , M ). Proof of Theorem 2.4. MC1: This is a consequence of the fact that there is a sequencethat discretises all elements in a finite collection of tame functors.MC2 and MC3: These follows from the fact that M satisfies these axioms, and fromthe functoriality of the mediating morphism ˆ g .MC4: Consider a commutative square in tame ([0 , ∞ ) , M ): X EY B α β where either α is a cofibration and β is a fibration and a weak equivalence, or α is acofibration and a weak equivalence and β is a fibration. We need to show that there isa morphism φ : Y → E which if added to the above square would make the obtaineddiagram commutative. Let us choose a sequence 0 = τ < · · · < τ k that discretises allfunctors in this square. We are going to define by induction on a in [ k ] morphisms φ τ a : Y τ a → E τ a . We then use this sequence to get the desired φ : Y → E .Set φ : Y → E to be any morphism in M that makes the following squarecommutative. It exists by the axiom MC4 in M . X E Y B α β φ Assume a > φ τ b : Y τ b → E τ b for b < a . We can then formthe following commutative diagram, where the indicated arrows are cofibrations byProposition 2.5.1: X τ a − X τ a E τ a − E τ a Q τ a Y τ a − Y τ a B τ a − B τ a α τa − α τa ¯ α τa β τa − ˆ α τa φ ′ φ τa − φ τa β τa All the horizontal arrows represent the transition morphisms, φ ′ : Q τ a → E τ a is induced8y the universal property of a pushout, and φ τ a : Y τ a → E τ a is any morphism thatmakes the following diagram commute, whose existence is guaranteed by axiom MC4: E τ a Q τ a B τ a Y τ a β τa ˆ α τa φ ′ φ τa MC5: Consider a morphism g : A → X in tame ([0 , ∞ ) , M ). Let us choose a sequence0 = τ < · · · < τ k that discretises both A and X . By induction on a = 0 , . . . , k , weare going to construct the appropriate factorisations g τ a = β τ a α τ a . Set α : A ֒ → Y and β : Y ։ X to be the factorisation of g : A → X , where one of α , β is also aweak equivalence. Such a factorisation exists by MC5 in M . Assume a > α τ b : A τ b ֒ → Y τ b and β τ b : Y τ b ։ X τ b for b < a . We can then define: Q τ a := colim( A τ a A τ a − Y τ a − A τa − <τa α τa − )and form the following commutative diagram: A τ a − Y τ a − X τ a − Y τ a A τ a Q τ a X τ a g τa − α τa − A τa − <τa β τa − X τa − <τa β τa α ′ g τa α τa β ′ α ′′ where the left square is pushout and β ′ : Q τ a → X τ a is induced by the universalproperty of the pushout. The morphisms α ′′ : Q τ a → Y τ a and β τ a : Y τ a → X τ a formthe appropriate factorisation of β ′ into the composition of either a cofibration whichis a weak equivalence and a fibration, or a cofibration and a fibration which is a weakequivalence. Set α τ a : A τ a → Y τ a to be the composition α ′′ α ′ , and Y τ a − <τ a : Y τ a − → Y τ a to be the composition of Y τ a − → Q τ a and α ′′ : P τ a → Y τ a . Define Y to bethe Kan extension along 0 = τ < · · · < τ k of the sequence { Y τ a − <τ a } a =1 ,...,k (see2.2). Let α : A → Y and β : Y → X be the natural transformations induced by thesequences of morphisms { α τ a : A τ a → Y τ a } a =0 ,...,k and { β τ a : Y τ a → X τ a } a =0 ...,k . Byconstruction, α is a cofibration and β is a fibration. Furthermore, depending on thechoice of the factorisations of β ′ : Q τ a → X τ a , either α or β is a weak equivalence.2.6. Minimal factorisations . Assume M satisfies the minimality axiom. Consider amorphism g : A → X in tame ([0 , ∞ ) , M ). Perform the same constructions as in theproof of MC5 but instead of taking arbitrary factorisations consider the minimalones. In step zero, we take morphisms α : X ֒ → Y and β : Y ∼ −→→ X that forma minimal factorisation of g : A → X . Analogously, in the a -th step we take mor-phisms α ′′ : Q τ a ֒ → Y τ a and β τ a : Y τ a ∼ −→→ X τ a which form a minimal factorisation of β ′ : Q τ a → X τ a . We claim that the obtained morphisms α : A ֒ → Y and β : Y ∼ −→→ X form a minimal factorisation of g : A → X . We just proved:9 heorem 2.7. If the model category M satisfies the minimal factorisation axiom,then so does tame ([0 , ∞ ) , M ) . Corollary 2.8. If the model category M satisfies the minimal factorisation axiom,then so does tame([0 , ∞ ) k , M ) for any k = 1 , , . . . . 3. Chain complexes of vector spaces Let K be a field and N = { , , . . . } the set of natural numbers. A (non-negativelygraded) chain complex of K -vector spaces is a sequence of linear functions X = { δ n : X n +1 → X n } n ∈ N of K -vector spaces, called differentials , such that δ n δ n +1 = 0for all n in N . In the notation of the differentials we often ignore their indexes andsimply denote them by δ , or δ X to indicate which chain complex is considered.A chain complex X is called compact if L n ∈ N X n is finite dimensional [ ]. Thishappens if and only if X n is finite dimensional for all n and X n is trivial for n ≫ Homology. The following vector spaces are called respectively the n -th cycles andthe n -th boundaries of X : Z n X := ( X if n = 0ker( δ n − : X n → X n − ) if n > B n X := im( δ n : X n +1 → X n )Since δ n δ n +1 = 0, the n -th boundaries B n X is a vector subspace of the n -th cycles Z n X . The quotient Z n X/B n X is called the n -th homology of X and is denoted by H n X . We write ZX , BX and HX to denote the non-negatively graded vector spaces { Z n X } n ∈ N , { B n X } n ∈ N , and { H n X } n ∈ N .3.2. Model structure. A morphism of chain complexes g : X → Y is a sequence oflinear functions { g n : X n → Y n } n ∈ N such that g n δ X = δ Y g n +1 for all n . Such a mor-phism maps boundaries and cycles in X to boundaries and cycles in Y . The inducedmap on homologies is denoted by Hg : HX → HY . If Hg : HX → HY is an iso-morphism, then g is a weak equivalence. If g n : X n → Y n is an epimorphism for all n > n = 0), then g is a fibration. If g n : X n → Y n isa monomorphism for all n > 0, then g is a cofibration. This choice of weak equiva-lences, fibrations and cofibrations defines a model structure on the category of chaincomplexes, denoted by Ch (see [ , ]). Consider the full subcategory of Ch givenby compact chain complexes. The same choices of weak equivalences, fibrations, andcofibrations, as for Ch, define a model structure on such a subcategory, denoted bych.3.3. Suspension. The suspension of a chain complex X , denoted by SX is a chaincomplex such that: δ n : ( SX ) n +1 → ( SX ) n = ( X → n = 0 − δ n − : X n → X n − if n > g : X → Y of chain complex is a morphism10 g : SX → SX such that:( Sg ) n : ( SX ) n → ( SY ) n = ( → n = 0 g n − : X n − → Y n − if n > g Sg is a functor denoted by S : Ch → Ch.Note that H ( SX ) = 0 and H n SX is isomorphic to H n − X for all n > 0. Further-more, if f is a cofibration or a weak equivalence, then so is Sf , and if f is a fibration,then Sf is a fibration if and only if f is an epimorphism.The desuspension of a chain complex X , denoted by S − X , is a chain complexsuch that: δ n : ( S − X ) n +1 → ( S − X ) n = ( − δ : X → Z X if n = 0 − δ n +1 : X n +2 → X n +1 if n > g : X → Y of chain complex is amorphism S − g : S − X → S − X such that:( S − g ) n : ( S − X ) n → ( S − Y ) n = ( g : Z X → Z Y if n = 0 g n +1 : X n +1 → Y n +1 if n > g S − g is a functor denoted by S − : Ch → Ch.Note that H n S − X is isomorphic to H n +1 X . If f is a fibration, cofibration or aweak equivalence, then so is S − f . Furthermore, S − SX is isomorphic to X , and SS − X is isomorphic to X if and only if X = 0.We now provide some explicit constructions of chain complexes used essentiallyin 3.8 to compute the standard decomposition and the minimal representative in ch.3.4. Cofiber sequences. Let f : X → Y be a morphism of chain complexes. Define achain complex Cf , called the cofiber of f , a cofibration i : Y ֒ → Cf , and a fibration p : Cf ։ SX , as follows: Y · · · Y Y Y Y Cf · · · Y ⊕ X Y ⊕ X Y ⊕ X Y SX · · · X X X i δ Y δY [ ] δY [ ] δY [ ] p h δY f − δX i [ ] h δY f − δX i [ ] h δY f − δX i [ ] [ δY f ] − δX − δX − δX The cofibration i and the fibration p form an exact sequence, called the cofibersequence of f : 0 Y Cf SX i p Consider two maps of chain complexes f : X → Y and g : W → Z . Each of themleads to a cofiber sequence. A natural transformation between these exact sequencesis by definition a triple of morphisms of chain complexes Sα : SX → SW , β : Y → Z γ : Cf → Cg which make the following diagram commute:0 Y Cf SX Z Cg SW iβ pγ Sαi p Commutativity of this diagram has two consequences. First, γ is of the form: γ n = ( β : Y → Z if n = 0 h β n h n − α n − i : Y n ⊕ X n − → Z n ⊕ W n − if n > h = { h n : X n → Z n +1 } n > satisfies the equa-tion βf − gα = δ Z h + hδ X , which means that h is a homotopy between βf and gα . Itfollows that the set of natural transformations between the two cofiber sequences isin bijection with the set of triples consisting of morphisms α : X → W and β : Y → Z ,and a homotopy h between βf and gα . We illustrate such a triple in form of a diagram: X YW Z fα h βg The symbol C ( α, β, h ) : Cf → Cg denotes the morphism γ : Cf → Cg , correspondingto this triple ( α, β, h ).In the case h = 0, such diagrams corresponds to a commutative squares: X YW Z fα βg = X YW Z fα βg In this case, the corresponding morphism between the cofibers is denoted simply by C ( α, β ) : Cf → Cg .In the case the differentials δ Z and δ X are trivial (in all degrees), the followingimplication holds (homotopy commutative square is commutative): X YW Z fα h βg implies X YW Z fα βg Comparison morphism. Let f : X → Y be a morphism of chain complexes. Con-sider the quotient morphism q : Y → Y /f ( X ) and define the comparison morphism Cf → Y /f ( X ) to be: Cf · · · Y ⊕ X Y ⊕ X Y Y /f ( X ) · · · ( Y /f ( X )) ( Y /f ( X )) ( Y /f ( X )) h δY f − δX i [ q ] h δY f − δX i [ q ] [ δY f ] q δ δ δ If f is a monomorphism, then the comparison morphism Cf ։ Y /f ( X ) is a weakequivalence. 12.6. Factorisation. The complex C X is also denoted by CX and called the cone on X . Explicitly, the cofibration i : X ֒ → CX is given by: X · · · X X X X CX · · · X ⊕ X X ⊕ X X ⊕ X X i δ X δX [ ] δX [ ] δX [ ] h δX − δX i h δX − δX i h δX − δX i [ δX ] Note that HCX = 0.The complex S − CX is also denoted by P X and called the path complex on X .We also use the symbol p : P X → X to denote the fibration given by the desuspension S − p : S − CX → S − SX = X . Explicitly: P X · · · X ⊕ X X ⊕ X X ⊕ X X X · · · X X X X p h − δX δX i [ ] h − δX δX i [ ] h − δX δX i [ ] [ − δX ] δXδX δX δX δ X Note that HP X = 0.Since HP X = 0 = HCX , the fibration p : P X ։ X and the cofibration i : X ֒ → CX fit into the following factorisations of the morphisms 0 → X → P X CX X p ∼∼ i These morphisms i and p can be used to construct explicit factorisations of arbi-trary morphisms in Ch, whose existence is guaranteed by axiom MC5: any g : X → Y fits into a commutative diagram: X ⊕ P YX YCX ⊕ Y [ g p ] g [ ] ∼ h ig i [ 0 1 ] ∼ These factorisations are natural, however in general not minimal (see Definition 1.1).To obtain minimal factorisations we cannot perform natural constructions and wewill be forced to make some choices.3.7. Graded vector spaces. A (non-negatively) graded K -vector space is by definition asequence of K -vector spaces V = { V n } n ∈ N . Such a graded vector space is concentratedin degree k if V n = 0 for all n = k . Graded vector spaces concentrated in degree 0 areidentified with vector spaces.Let V = { V n } n ∈ N be a graded K -vector space. The same symbol V is also usedto denote the chain complex { V n +1 → V n } n ∈ N with the trivial differentials. Inthis case, HV = V and hence any weak equivalence φ : V → V is an isomorphism. Infact, an arbitrary chain complex X is minimal (see Definition 1.3) if and only if allits differentials are trivial. More generally any cofibration α : X ֒ → Y for which thechain complex Y /α ( X ) has all trivial differentials satisfies the following minimalitycondition: any weak equivalence φ : Y → Y for which αφ = α is an isomorphism. To13ee this consider a commutative diagram with exact rows:0 X Y Y /α ( X ) 00 X Y Y /α ( X ) 0 α φα Using the long sequences of homologies for each row, we can conclude the morphism Y /α ( X ) → Y /α ( X ) is a weak equivalence and hence an isomorphism as Y /α ( X ) isassumed to have all differentials trivial. We can then use the exactness of the rows toget that φ is also an isomorphism.To denote the n -fold suspension of K we use the symbol S n . Explicitly, S n is thechain complex concentrated in degree n such that ( S n ) n = K . For example S = K .The complex S n is called the n -th sphere . The cone CS n is denoted by D n +1 andcalled the ( n + 1)-st disk . Explicitly:( D n +1 ) k = ( K if k = n or k = n + 10 otherwise , δ k = ( k = n Standard decomposition and minimal representative. Let X be a chain complex.Consider the morphisms p : CBX ։ SBX ← X : δ X (see 3.4). Axiom MC4 guaran-tees existence of a morphism φ : CBX → X making the following diagram commuta-tive: 0 XCBX SBX ∼ δpφ The restriction of any such φ to i : BX ֒ → CBX is the standard inclusion BX ֒ → ZX ֒ → X . This can be seen by looking at the long exact sequences of homologiesapplied to the rows in the following commutative diagram:0 BX CBX SBX ZX X SBX i pφ δ The morphism φ leads therefore to a pushout square (in particular φ is a cofibration): BX CBXZX X i φ Since considered coefficients are in a field and all the differentials in BX , ZX and HX are trivial, there is a morphism s : HX → ZX , whose composition with thequotient ZX ։ HX is 1 HX . For any such s , the morphism (cid:2) i s (cid:3) : BX ⊕ HX → ZX is an isomorphism. It follows that so is the morphism (cid:2) φ s (cid:3) : CBX ⊕ HX → X ,where the symbol s also denotes the composition of s : HX → ZX and the inclusion ZX ֒ → X . We call CBX ⊕ HX the standard decomposition of the chain complex X . Since CBX has trivial homology, the morphism s : HX → X is a weak equivalenceand hence HX is the minimal representative (see Definition 1.3) of X .14.9. Minimal factorisations. Let g : X → Y be a morphism of chain complexes. Toconstruct its minimal factorisation (see Definition 1.1) we perform the following steps:1. Take the kernel j : W ֒ → X of g : X → Y ;2. Choose an isomorphism W ≃ −→ CBW ⊕ HW (see 3.8);3. Consider the composition: W CBW ⊕ HW CBW ⊕ CHW ≃ α [ i ]4. Use axiom MC4 to construct a morphism φ : X → CBW ⊕ CHW , which fitsinto the following commutative diagram: W CBW ⊕ CHWX j α ∼ φ 5. The morphism (cid:2) φg (cid:3) : X → ( CBW ⊕ CHW ) ⊕ Y is then a cofibration.We are now ready to state: Proposition 3.10. The following factorisation is minimal: ( CBW ⊕ CHW ) ⊕ YX Y [ 0 1 ] ∼ g h φg i Proof. Let ψ = h ψ ψ ψ ψ i : ( CBW ⊕ CHW ) ⊕ Y → ( CBW ⊕ CHW ) ⊕ Y be a mor-phism making the following diagram commutative: X ( CBW ⊕ CHW ) ⊕ Y ( CBW ⊕ CHW ) ⊕ Y Y h φg ih φg i [ 0 1 ] ∼ [ 0 1 ] ∼ ψ Commutativity of the bottom triangle implies ψ = 0 and ψ = 1. Since W is thekernel of g , commutativity of the top triangle implies commutativity of: W CBW ⊕ CHWCBW ⊕ CHW αα ψ The quotient ( CBW ⊕ CHW ) /α ( W ) = SHW has all differentials trivial. The mor-phism ψ : CBW ⊕ CHW → CBW ⊕ CHW is therefore an isomorphism (see 3.7).It follows that so is ψ = (cid:2) ψ ψ (cid:3) We just proved that ch satisfies the minimal factorisation axiom. By Theorem 2.7,it follows that tame ([0 , ∞ ) , ch) also satisfies the minimal factorisation axiom, andthus any tame parametrised chain complex admits a minimal cover. In Section 4, weprovide a characterisation of such minimal covers.15 . Cofibrations in tame ([0 , ∞ ) , ch) In this section we discuss cofibrations in the model category tame ([0 , ∞ ) , ch) asdescribed in Theorem 2.4. According to Proposition 2.5, an object X in tame ([0 , ∞ ) , ch)is cofibrant if and only if the transition X s The Kan extensions (see 2.2) given by the data described in thefollowing table are called interval spheres :Index n , s < e = ∞ n , s = e < ∞ n , s < e < ∞ Name I n [ s, ∞ ) I n [ s, s ) I n [ s, e ) k k ] → ch S n D n +1 i : S n ֒ → D n +1 Inclusion [ k ] ⊂ [0 , ∞ ) s s s < e For example, I [5 , ∞ ) : [0 , ∞ ) → ch is a functor whose value at t < t is S . Similarly, I [5 , 5) : [0 , ∞ ) → ch has value 0 if t < 5, and D if 5 t . Thefunctor I [5 , 7) : [0 , ∞ ) → ch has three values: 0 if t < S if 5 t < 7, and D if7 t . The transition morphisms in I [5 , 7) are either the identities, or the inclusion0 ֒ → S or the inclusion i : S ֒ → D .A morphism I n [ s, e ) → s = e . Thus, intervalspheres of type I n [ s, s ) are the only interval spheres for which the chain complex I n [ s, s ) t has trivial homology for every parameter t in [0 , ∞ ).The main result of this section is the structure theorem (compare with [ , , ]): Theorem 4.2. (1) Any cofibrant object in tame ([0 , ∞ ) , ch) is isomorphic to a directsum ⊕ li =1 I n i [ s i , e i ) , where l could possibly be .(2) If ⊕ li =1 I n i [ s i , e i ) ∼ = ⊕ l ′ j =1 I n ′ j (cid:2) s ′ j , e ′ j (cid:1) , then l = l ′ and there is a permutation σ of the set { , . . . , l } such that n i = n ′ σ ( i ) , s i = s ′ σ ( i ) , and e i = e ′ σ ( i ) for any i . To prove Theorem 4.2, we first need to characterise cofibrations in tame ([0 , ∞ ) , ch)and explain how to enumerate morphisms out of I n [ s, e ). We start with cofibrations: Proposition 4.3. For every morphism g : X → Y in tame ([0 , ∞ ) , ch) , the followingstatements are equivalent:1. g is a cofibration;2. g t : X t → Y t is a monomorphism for every t in [0 , ∞ ) , and Y /g ( X ) is cofibrant;3. g t : X t → Y t is a monomorphism for every t in [0 , ∞ ) and, for all s < t in [0 , ∞ ) , the following is a pullback square: X s X t Y s Y tX s V W W U α α β β It leads to two vector spaces: P := lim( W U W β β ) Q := colim( W V W α α )and two linear functions α : V → P and β : Q → U that make the following diagramscommutative, where the inside squares are respectively a pullback and a pushout: V P W W U α αα β β V W W Q U α α β β β Then α : V → P is surjective if and only if β : Q → U is injective. ⇒ : The first part of follows from Proposition 2.5.1, and the second from thefact that in a model category cofibrations are preserved by pushouts. ⇒ : For all s < t in [0 , ∞ ), we have the following commutative diagram, where theindicated arrows are cofibrations in ch: X s Y s Y s /g ( X ) s X t Y t Y t /g ( X ) tX s Let X be an object in tame ([0 , ∞ ) , ch) .1. Let n be a natural number, s be in [0 , ∞ ) , and x be in Z n X s . Then the morphism I ( x ) : I n [ s, ∞ ) → X (see 4.4) is a cofibration if and only if X is cofibrant and x is not in the image of X t Proof of Theorem 4.2. (2) This is a consequence the fact that I n [ s, e ) is indecom-posable, for all n and s e (see [ , ]). (1) Let X in tame ([0 , ∞ ) , ch) be cofibrant. Choose a sequence 0 = τ < · · · < τ k discretising X . The morphism X τ a − <τ a : X τ a − → X τ a is then a cofibration in ch for every a = 1 , . . . , k .Assume first all the differentials in X t are trivial for all t . In this case, X is isomor-phic to L n > X n . Let l n := dim X n and l τ a n := dim coker( X τ a − <τ a n : X τ a − n → X τ a n )for a = 1 , . . . , k . Then X n is isomorphic to L ka =0 L l τan j =1 I n [ τ a , ∞ ) and consequently X is isomorphic to: M n > k M a =0 l τan M j =1 I n [ τ a , ∞ )Assume now there is a non-trivial differential in X and set:(a) n to be the smallest natural number for which δ : X tn +1 → X tn is non trivial forsome t . This assumption implies X tn = Z n X t for any t .(b) e to be the smallest τ a for which δ : X τ a n +1 → X τ a n is non trivial.(c) s to be the smallest τ a such that τ a e and for which the following intersectioncontains a non zero element:im (cid:0) X τ a en : Z n X τ a = X τ a n ֒ → X en (cid:1) ∩ im (cid:0) δ : X en +1 → X en (cid:1) = 0We claim that these choices imply X is isomorphic to I n [ s, e ) ⊕ X ′ . We can thenapply the same strategy to X ′ . If X ′ has a non-trivial differential, we split out of X ′ another direct summand of the form I n ′ [ s ′ , e ′ ) for s ′ e ′ in [0 , ∞ ). Tamenessguarantees that this process eventually terminates and we end up with an objectwith all the differentials being trivial, which we can decompose as described aboveand the theorem would be proven.It remains to show our claim that X is isomorphic to I n [ s, e ) ⊕ X ′ . For that, wemake some choices:1. Choose a non zero vector v in the intersection from step (c) above.2. Choose x in X sn = Z n X s and y in X en +1 such that: X s en ( x ) = v = δ ( y ).3. Consider the morphism I ( x, y ) : I n [ s, e ) → X (see 4.5).The reason why we made all these choices is to ensure I ( x, y ) : I n [ s, e ) → X is acofibration (see Proposition 4.7.2).Let φ : X → I n [ s, s ) be a morphism that fits into the following commutative dia-gram, where the top horizontal morphism is the standard inclusion (see 4.5). Existence19f such a φ is guaranteed by axiom MC4: I n [ s, e ) I n [ s, s ) X I ( x,y ) ∼ φ If t < e , then the differential δ : X tn +1 → X tn is trivial. Thus, for any s t < e , thelinear function φ tn +1 : X tn +1 → I n [ s, e ) tn +1 has to be trivial. It follows that φ : X → I n [ s, s ) factors through the standard inclusion: I n [ s, e ) X I n [ s, s ) iφψ The composition I n [ s, e ) X I n [ s, e ) I ( x,y ) ψ is therefore the identity and con-sequently X is isomorphic to a direct sum I n [ s, e ) ⊕ X ′ . 5. Betti diagrams of objects Let X be a cofibrant object in tame ([0 , ∞ ) , ch). According to Theorem 4.2 thereare unique functions { β n X : Ω → { , , . . . }} n =0 , ,... , called Betti diagrams of X ,such that X is isomorphic to: M n M ( s,e ) ∈ Ω ( I n [ s, e )) β n X ( s,e ) Betti diagrams have finite support: the set supp( β n X ) := { ( s, e ) ∈ Ω | β n X ( s, e ) = 0 } is finite for every n . Thus, to describe an isomorphism type of a cofibrant object X intame ([0 , ∞ ) , ch), a sequence of functions { β n X : Ω → { , , . . . }} n =0 , ,... with finitesupports needs to be specified. Such functions are also called persistence diagrams (see[ ]). Betti diagrams are complete invariants of cofibrant objects in tame ([0 , ∞ ) , ch)and they play a fundamental role in persistence and TDA.In this section, we explain various ways of assigning Betti diagrams to arbitraryobjects (not only cofibrant) in tame ([0 , ∞ ) , ch). For such general objects one shouldnot expect these invariants to be complete. Our strategy is to approximate arbitraryobjects by cofibrant objects and use Theorem 4.2 to extract Betti diagrams from theobtained approximations.Minimal representatives (Definition 1.3) and minimal covers (Definition 1.1) are themost fundamental constructions that convert an arbitrary object in tame ([0 , ∞ ) , ch)into a cofibrant one. This leads to two invariants of an isomorphism class of X intame ([0 , ∞ ) , ch) which are called minimal Betti diagrams and Betti diagrams : X { β n X ′ : Ω → { , , . . . }} n =0 , ,... { β n cov( X ) : Ω → { , , . . . }} n =0 , ,... minimal Betti diagrams Betti diagrams where X ′ is a minimal representative of X and cov( X ) → X is a minimal cover of X .20e also use the symbols β min n X and β n X to denote respectively β n X ′ and β n cov( X ).Moreover, if X and Y are weakly equivalent, then β min n X = β min n Y .To understand relationship between these invariants, we first characterise minimalobjects (see Definition 1.3) in tame ([0 , ∞ ) , ch). Let X be cofibrant in tame ([0 , ∞ ) , ch).Consider its decomposition into a direct sum of interval spheres (Theorem 4.2). If thisdecomposition contains a component of the form I n [ s, s ), then by projecting it away,we obtain a self weak equivalence of X which is not an isomorphism. Thus, if X isminimal, its decomposition cannot contain such components, which is equivalent tohaving β n X ( s, s ) = 0 for all n and all s . This implication can be reversed: Proposition 5.1. 1. An object X in tame ([0 , ∞ ) , ch) is minimal if and only if itis cofibrant and β n X ( s, s ) = 0 for all natural numbers n and all s in [0 , ∞ ) .2. A morphism c : X → Y is a minimal cover if and only if X is cofibrant, c isa weak equivalence and a fibration, and no direct summand of X of the form I n [ s, s ) , for some n and s , is in the kernel of c (is mapped via c to ).Proof. We start with describing how a self weak equivalence of an object X intame ([0 , ∞ ) , ch) leads to its decomposition as a direct sum. Let f : X → X be aweak equivalence. Since all objects in tame ([0 , ∞ ) , ch) are compact, there is a nat-ural number l for which f l = f l + k for all k > 0. We can then form the followingcommutative diagram where the top horizontal morphism is an isomorphism:im( f l ) im( f l ) X X X f l f l Commutativity of this diagram, and the facts that the top horizontal morphism is anisomorphism and f l is a weak equivalence, have two consequences: X is isomorphic toa direct sum im( f l ) ⊕ ker( f l ), and the morphisms X ։ im( f l ) and ker( f l ) → Assume X is cofibrant and β n X ( s, s ) = 0 for all n and s . Let f : X → X be aself weak equivalence and l be such that X is isomorphic to im( f l ) ⊕ ker( f l ) andthe morphism ker( f l ) → f l )is a direct sum of interval spheres of the form I n [ s, s ). Since by the assumption X does not have such components, ker( f l ) = 0 and consequently f l and hence f areisomorphisms. That proves statement . Consider a morphism c : X → Y such that X is cofibrant, c is a weak equivalenceand a fibration. If the kernel of c contains a direct summand of X of the form I n [ s, s ),then by projecting it away, we would obtain a weak equivalence f : X → X such that cf = c and which is not an isomorphism, preventing c to be a minimal cover.Assume now that the kernel of c does not contain any direct summand of X of theform I n [ s, s ). Consider a weak equivalence f : X → X such that cf = c . Choose l forwhich X is isomorphic to im( f l ) ⊕ ker( f l ) and the morphism ker( f l ) → f l ) is a direct sum of intervalspheres of the form I n [ s, s ). Since ker( f l ) is in ker( c ) and it is a direct summand of X , the assumption implies ker( f l ) is trivial, and as before f is an isomorphism. Corollary 5.2. Let X be an object in tame ([0 , ∞ ) , ch) (not necessarily cofibrant). . A cofibrant object X ′ in tame ([0 , ∞ ) , ch) is a minimal representative of X if andonly if it is weakly equivalent to X and β n X ′ ( s, s ) = 0 for all natural numbers n and all s in [0 , ∞ ) .2. Let X ′ be the minimal representative of X and cov( X ) its minimal cover. Then β n X ′ ( s, e ) = β n cov( X )( s, e ) for all s < e . According to the above corollary, the minimal Betti diagrams and the Betti dia-grams may differ only on the diagonal ∆ := { ( s, s ) ∈ Ω } ⊂ Ω. The minimal Bettidiagrams ignore the diagonal by assigning 0 to all its elements (see Proposition 5.1),reflecting the fact that minimal representative does not contain any component withtrivial homology. The Betti diagrams on the other hand do not ignore the diago-nal and retain information about components of the cover cov( X ) that have trivialhomology.For certain objects in tame ([0 , ∞ ) , ch), the minimal Betti diagrams and the Bettidiagrams coincide and provide a complete set of invariants: Proposition 5.3. Assume X and Y in tame ([0 , ∞ ) , ch) are such that all the differ-entials of X t and Y t are trivial for all t in [0 , ∞ ) . Then:1. X and Y are isomorphic if and only if they are weakly equivalent.2. X and Y are isomorphic if and only if their minimal Betti diagrams are equal.3. The minimal cover and the minimal representative of X are isomorphic.4. The minimal Betti diagrams and Betti diagrams of X coincide.Proof. Statement is a consequence of the fact that X and Y are isomorphic to theirrespective homologies. Statement follows from statement . To show statement ,choose a minimal representative X ′ of X and a weak equivalence f : X ′ → X . Since X is isomorphic to its homology, f is a fibration and hence it is also a minimal coverof X . Finally statement is a consequence of statement .5.4. Tame [0 , ∞ ) -parametrised vector spaces. We regard tame [0 , ∞ ) -parametrisedvector spaces , also known as persistence modules, as objects in tame ([0 , ∞ ) , ch)whose values are concentrated only in degree 0 for all parameters t in [0 , ∞ ) (see 3.7).Such objects in tame ([0 , ∞ ) , ch) satisfy the assumption of Proposition 5.3 and hencetheir isomorphism types are uniquely determined by their Betti diagrams. Further-more, minimal representatives and minimal covers of such objects coincide. It followsthat for a tame [0 , ∞ )-parametrised vector space X , we have β n X ( s, s ) = 0 for all n and s (see Proposition 5.1). Since homology in positive degrees of X is trivial, we alsoget β n X = 0 for all n > 0. Thus, the isomorphism type of X is uniquely determinedby its 0-th Betti diagram β X : Ω → { , , . . . } . In this case β X coincides with theusual persistence diagram of X [ ]. 6. Betti diagrams of morphisms In this section we explain various ways of assigning Betti diagrams to a morphism g : X → Y in tame ([0 , ∞ ) , ch). 22.1. Minimal factorisations. If g : X → Y is a cofibration, then the quotient Y /g ( X )is cofibrant and we can take its Betti diagrams β n ( Y /g ( X )). If g : X → Y is not acofibration, we can consider its minimal factorisation (see Definition 1.1): AX Y β ∼ gα and assign to g the Betti diagrams β n ( A/α ( X )) of the quotient A/α ( X ). These Bettidiagrams are invariants of the isomorphism type of g , and in the case X = 0 recoverthe Betti diagrams of Y discussed in Section 5.6.2. The cover of the cofiber. Instead of taking the minimal factorisation of g , we canapply the cofiber construction (see 3.4) parameterwise, to obtain an exact sequencein tame ([0 , ∞ ) , ch): 0 Y Cg SX i p We can then assign to g the Betti diagrams β n cov( Cg ) of the minimal cover cov( Cg )of the cofiber Cg . The diagrams β n cov( Cg ) depend on the isomorphism type of g , andas before, in the case X = 0, recover the Betti diagrams of Y discussed in Section 5.6.3. The cofiber of the covers. We can extract a cofibrant object out of g : X → Y yetin another way. Use axiom MC4 to choose a morphism g ′ that fits into the followingcommutative square, where the vertical morphisms denote the minimal covers:cov( X ) cov( Y ) X Y c X ∼ g ′ c Y ∼ g Since cov( X ) and cov( Y ) are cofibrant, then so is the cofiber Cg ′ , and hence we cantake its Betti diagrams β n Cg ′ . Although in this construction we made a choice of g ′ , the obtained Betti diagrams do not depend on it and hence provide invariantsof the isomorphism type of g . To prove this independence, consider the morphism C ( c X , c Y ) : Cg ′ → Cg , induced by the commutativity of the square above (see 3.4).It fits into the following commutative diagram with exact rows, where the indicatedmorphisms are weak equivalences, cofibrations, and fibrations:0 cov( Y ) Cg ′ S cov( X ) 00 Y Cg SX c Y ∼ i C ( c X ,c Y ) ∼ p Sc X ∼ i p Proposition 6.4. The following factorisation is minimal: Cg ′ cov( Y ) Cg C ( c X ,c Y ) ∼ ic Y i Furthermore, if X is cofibrant, then C ( c X , c Y ) : Cg ′ → Cg is a minimal cover. ic Y : cov( Y ) → Cg does not depend on g ′ , neither does itsminimal factorisation. Therefore, Proposition 6.4 implies that the isomorphism typeof Cg ′ does not depend on the choice of g ′ and consequently neither β n Cg ′ . Proof of Proposition 6.4. Consider a self equivalence f : Cg ′ → Cg ′ of the factorisa-tion. It fits into the following commutative diagram:cov( Y ) Cg ′ S cov( X )cov( Y ) Cg ′ S cov( X ) Y Cg SX c Y ∼ i C ( c X ,c Y ) ∼ f p Sc X ∼∼ c Y i ∼ C ( c X ,c Y ) p ∼ Sc X i i The induced morphism S cov( X ) → S cov( X ) is then a weak equivalence and hencean isomorphism as Sc X : S cov( X ) → SX is the minimal cover. The morphism f istherefore an isomorphism as well.Assume X is cofibrant. We are going to use the criteria from Proposition 5.1 toargue that in this case C ( c X , c Y ) is a minimal cover. Since cov( X ) = X , the followingsquare is a pullback: cov( Y ) Cg ′ Y Cg ic Y ∼ C ( c X ,c Y ) ∼ i It follows that the kernel of C ( c X , c Y ) coincide with the kernel of c Y . Consider a directsummand of Cg ′ of the form I n [ s, s ), for some n and s , that belongs to the kernelof C ( c X , c Y ). Then this summand belongs also to the kernel of c Y . In particular,it is included in cov( Y ) and hence it is also a direct summand of cov( Y ). Thatcontradicts the criteria from Proposition 5.1 applied to c Y . We can conclude thatsuch summands do not exist and hence, according to the same criteria, C ( c X , c Y ) isa minimal cover.With a morphism g : X → Y in tame ([0 , ∞ ) , ch), we have associated four cofibrantobjects: A/α ( X ) (see 6.1), cov( Cg ) (see 6.2), Cg ′ (see 6.3), and the minimal repre-sentative of cov( Cg ). These cofibrant objects lead to Betti diagrams β n ( A/α ( X )), β n cov( Cg ), β n Cg ′ , and β min n cov( Cg ). Since all these cofibrant objects are weaklyequivalent to each other, all these Betti diagrams agree for all ( s, e ) in Ω such that s < e . They may have different values only on the diagonal ∆ ⊂ Ω.6.5. Commutative ladders. A commutative ladder is by definition an object intame ([0 , ∞ ) , ch) whose values at all parameters are chain complexes which are nontrivial only in degrees zero and one. For example, I [ s, s ) is a commutative ladder.Similarly, so is the Kan extension of D → s < e of ele-ments in [0 , ∞ ) (see 2.2). A tame [0 , ∞ )-parametrised vector space is also an exampleof a commutative ladder. In general, however, in contrast to tame [0 , ∞ )-parametrisedvector spaces, the minimal Betti diagrams of a commutative ladder can fail to be equalto its Betti diagrams. For example, the minimal representative of I [ s, s ) is trivial,however its minimal cover is I [ s, s ). Furthermore, again in contrast to tame vector24paces, Betti diagrams are not complete invariants of commutative ladders. For exam-ple, I [ s, s ) and the Kan extension of D → s < e ofelements in [0 , ∞ ) have isomorphic minimal covers and hence same Betti diagrams.If f : X → Y is a morphism of tame [0 , ∞ )-parametrised vector spaces, then itscofiber Cf is a commutative ladder. Any commutative ladder Z is the cofiber ofits differential δ : Z → Z , where we regard Z and Z as [0 , ∞ )-parametrised vectorspaces. Thus, we can extract from Z various Betti diagrams assigned to the morphism δ : Z → Z . For example, we can consider the Betti diagrams of the quotient of thecofibration in the minimal factorisation of the differential δ : Z → Z (see 6.1). Wecan also apply to δ the procedure described in 6.3 to obtain another sequence of Bettidiagrams. Thus, a commutative ladder leads to four sequences of Betti diagrams.These Betti diagrams are not arbitrary. Let β n : Ω → { , , . . . } be any of these Bettidiagrams extracted from a commutative ladder. Since for n > 1, there are no non-trivial morphisms from the interval sphere I n [ s, s ) into any commutative ladder, β n ( s, s ) = 0 for n > s in [0 , ∞ ). As values of commutative ladders have nohomology in degrees strictly bigger than 1, we then also get β n = 0, for n > 7. Zigzags Discrete zigzags. Throughout this section, k is assumed to be a positive naturalnumber. Elements of the set { r, l } are called directions, r stands for right and l for left.Let c = ( c , . . . , c k ) be a sequence of directions i.e., elements of { r, l } . Such a sequencedetermines a poset structure ” → ” on { , , . . . , k } where, for a < b in { , , . . . , k } , a ← b if c a +1 = · · · = c b = l , and a → b if c a +1 = · · · = c b = r . This poset is denotedby [ k ] c and the sequence c is called its profile . A profile c consisting of only r ’s iscalled standard and the induced poset structure on { , , . . . , k } is denoted by [ k ].Here are graphical illustrations of [4] c for 3 different profiles:0 ← ← → → → → → → ← → ← → X : [ k ] c → Ch, the following needs to be specified: • k + 1 chain complexes X a for every a in { , , . . . , k } , • k chain morphisms: X a − → a : X a − → X a for every a such that c a = r , and X a → a − : X a → X a − for every a such that c a = l .A discrete zigzag is by definition a functor of the form X : [ k ] c → Ch for some k and some profile c .7.2. Straightening zigzags.