aa r X i v : . [ m a t h . A T ] O c t Persistent homotopy theory
J.F. Jardine ∗ Department of MathematicsUniversity of Western OntarioLondon, Ontario, [email protected]
October 28, 2020
Abstract
Vietoris-Rips and degree Rips complexes are represented as homotopytypes by their underlying posets of simplices, and basic homotopy stabilitytheorems are recast in these terms. These homotopy types are viewedas systems (or functors), which are defined on a parameter space. Thecategory of systems of spaces admits a partial homotopy theory that isbased on controlled equivalences, suitably defined, that are the output ofhomotopy stability results.
Introduction
A prototypical homotopy stability result asserts that, if one adds points to adata set X that are close in a suitable sense to form a new data set Y , thenthe corresponding inclusion V ∗ ( X ) → V ∗ ( Y ) of Vietoris-Rips systems is a strongdeformation retract up to a bounded shift, where the bound depends linearlyon how close the points of Y are to points of X .The language in this last paragraph is a bit colloquial, and it involves newterms that need to be explained. In particular, a system of spaces X is a functor s X s where s is a member of the real parameter poset [0 , ∞ ) and each X s isa “space” or simplicial set, while a map of systems is a natural transformationof functors.For a finite metric space X (a data set), the Vietoris-Rips complexes s V s ( X ) form such a system, and an inclusion of finite metric spaces X ⊂ Y induces a transformation V s ( X ) → V s ( Y ) that is natural in the distance param-eter s . Recall that V s ( X ) is the finite simplicial complex whose simplices aresubsets σ of X such that the distance d ( x, y ) ≤ s for all x, y ∈ σ . ∗ Supported by NSERC. V s ( X ) is defined as an abstract simplicial com-plex, and one usually makes it into a space by constructing its realization. Analternative is to put a total order on the vertices (which is consistent with listingthe data set X ), and then form an associated simplicial set as a subobject ofa simplex that is determined by the order on X . This simplicial set also has arealization, which is homeomorphic to the realization of the abstract simplicialcomplex. Both routes lead to the same space, and hence represent the samehomotopy type.There is a different approach. The basic method of this paper is to treatthe poset P s ( X ) of simplices of V s ( X ) as a homotopy theoretic object in itsown right by using the nerve BP s ( X ) of P s ( X ). The space BP s ( X ) is thebarycentric subdivision of the Vietoris-Rips complex V s ( X ), and therefore hasthe same homotopy type.This construction may seem fraught with complexity, but one can restrict tolow dimensional simplices as necessary. The advantage of the poset approach isthat the nerves of the posets P s ( X ) can be employed to great theoretical effect,by using basic features of Quillen’s theory of homotopy types of posets [8].For example, suppose that Y is a finite metric space, and that X is a subsetof Y . Suppose that r ≥ y ∈ Y there is an x ∈ X such that d ( x, y ) < r , where d is the metric on Y . Then one constructsa retraction function θ : Y → X by insisting that θ ( y ) is a point of X such that d ( y, θ ( y )) < r . It follows from the triangle identity shows the function θ inducesa poset morphism θ : P s ( Y ) → P s +2 r ( X ), and there is a diagram of morphisms P s ( X ) / / i (cid:15) (cid:15) P s +2 r ( X ) i (cid:15) (cid:15) P s ( Y ) / / θ ssssssssss P s +2 r ( Y ) (1)in which the upper triangle commutes on the nose, and the bottom trianglecommutes up to a natural transformation that fixes P s ( X ). The horizontal andvertical morphisms are the natural inclusions.This construction translates directly to a proof of the Rips stability theoremafter applying the nerve functor — this is Theorem 4 below. There is a cor-responding construction and result for the degree Rips filtration (Theorem 6),where one uses a more interesting distance criterion that involves configurationspaces. We also present, in Theorem 5, a quick proof of the version of the Ripsstability theorem given by Blumberg-Lesnick [1] that uses only poset techniques.These results are proved in Section 2. The basic terminology appears inSection 1, along with a relatively simple model for the fundmental groupoid ofthe space BP s ( X ).The diagram (1) is a “homotopy interleaving”, and is a strong deformationretraction up to a shift — in this case the shift is 2 r . Its existence implies that2here is a commutative diagram π n ( BP s ( X ) , x ) / / i ∗ (cid:15) (cid:15) π n ( BP s +2 r ( X ) , x ) i ∗ (cid:15) (cid:15) π n ( BP s ( Y ) , x ) / / θ ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ π n ( BP s +2 r ( Y ) , x )of maps between homotopy groups for each choice of base point x ∈ X . Thereare similar induced diagrams in path components and in homology groups.It follows that if α ∈ π n ( BP s ( X ) , x ) maps to 0 ∈ π n ( BP s ( Y ) , x ), then α maps to 0 in π n ( BP s +2 r ( X ) , x ), so that the vertical maps i ∗ are 2 r -mono-morphisms, suitably defined. Similarly, the maps i ∗ are 2 r -epimorphisms, inthat every β ∈ π n ( BP s ( Y ) , x ) maps to an element of π n ( BP s +2 r ( Y ) , x ) whichis in the image of the homomorphism i ∗ .The maps i ∗ : π n ( BP s ( X ) , x ) → π n ( BP ( Y ) , x ) are 2 r -isomorphisms in thesense that they are 2 r -monomorphisms and a 2 r -epimorphisms. A similar ob-servation holds for path components, and one says that the 2 r -interleaving pro-duced by the Rips stability theorem is a 2 r -equivalence.More generally, one defines families of r -equivalences of systems of spacesfor all r ≥
0, and a map X → Y of systems of spaces is a controlled equivalence if it is an r -equivalence for some r ≥ f : X → Y is a sectionwise fibration if all of its constituentmaps f : X s → Y s are fibrations of simplicial sets. Sectionwise cofibrations andsectionwise weak equivalences are defined analogously.Quillen’s triangle axiom CM2 does not hold for the class of r -equivalences,but a modification is possible: Lemma 12 implies, for example, that if f : X → Y is an s -equivalence and g : Y → Z is an r -equivalence, then the composite g · f : X → Z is an ( r + s )-equivalence.It is shown, in a sequence of lemmas leading to Theorem 15, that maps p : X → Y which are both sectionwise fibrations and r -equivalences pull backto maps which are sectionwise fibrations and 2 r -equivalences. The doubling ofthe parameter from r to 2 r reflects the usual two obstructions in the argumentfor the corresponding classical result for simplicial sets.It is tempting to think that Theorem 15 has a dual formulation that holdsfor sectionwise cofibrations, but such a result has not been proved.We still have partial glueing results. Perhaps most usefully, if there is apushout diagram of systems A / / i (cid:15) (cid:15) C i ∗ (cid:15) (cid:15) B / / D i is a sectionwise cofibration, then i ∗ is a sectionwise cofibration, and thefollowing statements hold:1) If the map π A → π B is an r -isomorphism then the map i ∗ : π C → π D is an r -isomorphism.2) If the maps H k ( A ) → H k ( B ) are r -isomorphisms in homology (arbi-trary coefficients) for k ≥
0, then the maps H k ( C ) → H k ( D ) are 2 r -isomorphisms.These statements are proved in Lemma 18 of this paper — the arguments arenot difficult.We also show, in Lemma 17, that if i : A → B is an r -interleaving, or astrong deformation retraction up to shift r , then the same holds for the map i ∗ : C → D .This applies in particular to cofibrations that arise from stability theorems.Thus, if i is a map BP ∗ ( X ) → BP ∗ ( Y ) that is associated to an inclusion X ⊂ Y of finite metric spaces that satisfies d H ( X, Y ) < r , then the map i ∗ : C → D isa strong deformation retraction up to shift 2 r . Contents A data set X is a finite subset of a metric space Z . The collection of data setsin Z with inclusions between them forms a poset, which is denoted by D ( Z ).Suppose that s ≥
0, and that X is a data set in Z . Write P s ( X ) for theposet of all subsets σ ⊂ X such that d ( x, y ) ≤ s for all x, y ∈ σ .The poset P s ( X ) is the poset of simplices of the Vietoris-Rips complex V s ( X )of X . The members σ ⊂ X of P s ( X ) are simplices of dimension n −
1, where n = | σ | is the number of elements of σ .Each poset P s ( X ) is a finite category. Other examples of finite posets aregiven by the finite ordinal numbers n = { , , . . . , n } , with the obvious ordering.There is a functorial method of assigning a simplicial set BC to a smallcategory C , where the n -simplices of BC are the functors α : n → C , or stringsof composable morphisms in C of length n . The simplicial structure maps of4 C are defined by composition with the functors (poset maps) m → n betweenfinite ordinal numbers. The simplicial set BC is variously called the nerve orthe classifying space of C .A group G is a category (groupoid) with one object, and BG is a model forthe classifying space of G .The nerve construction can also be applied to the ordinal number posets n ,and there is a natural isomorphism B n ∼ = ∆ n , where ∆ n is the standard n -simplex in simplicial sets.The nerve functor C BC also preserves products, so that there is anisomorphism B ( C × ) ∼ = BC × ∆ . The existence of this isomorphism implies that the nerve functor takes naturaltransformations to simplicial homotopies.It is standard to identify natural transformations with homotopies in thisform of categorical homotopy theory.We have poset inclusions σ : P s ( X ) ⊂ P t ( X ) , s ≤ t, for the data set X .Observe that P ( X ) is the discrete poset (category) whose objects are theelements of X , and that P t ( X ) is the poset P ( X ) of all subsets of X for t sufficiently large.There is an isomorphism of posets P ( X ) ∼ = × m , where is the poset { , } and m is the cardinality of the set X . The isomor-phism sends a subset A of X to the m -tuple ( ǫ x ) x ∈ X , where ǫ x = ( x ∈ A , and0 if x / ∈ A .It follows that there is an isomorphism of simplicial sets B P ( X ) ∼ = (∆ ) × m . In particular, the simplicial set (or space) BP t ( X ) is contractible if t is suffi-ciently large.The Vietoris-Rips complex V s ( X ) is a finite abstract simplicial complex, and P s ( X ) is its poset of simplices.The realization | V s ( X ) | of V s ( X ) is constructed, as a space, by glueing affinesimplices together along face relationships, and it is standard to identify thesimpilicial complex V s ( X ) with its realization.5he nerve BP s ( X ) of the poset P s ( X ) is a simplicial set whose realizationis the barycentric subdivision sd( V s ( X )) of V s ( X ). The subdivision sd( V s ( X ))is naturally weakly equivalent to V s ( X ) [3, III.4], [4].There is a non-canonical method of associating a simplicial set structure to V s ( X ) that arises from a total ordering, or listing φ : N ∼ = −→ X of the elements of the data set X . In the presence of such a listing, the set X has N + 1 elements, and the simplicial set V s ( X ) is the subcomplex of thestandard simplex ∆ N whose non-degenerate simplices are the members of theoriginal abstract simplicial complex. In this case, the simplicial set V s ( X ) isoriented by the total ordering φ on X . Its realization, as a simplicial set, ishomeomorphic to the realization of the underlying abstract simplicial complex,so that its homotopy type is independent of the ordering.The method of this paper is to identify the poset P s ( X ) with the homotopytype BP s ( X ) directly, without either constructing a realization or assuming aparticular list structure on X . This is consistent with the general identificationof small categories with homotopy types, which was pioneered by Quillen [7],[8] during the early development of algebraic K -theory.Suppose that k is a non-negative integer. The poset P s ( X ) has a subobject P s,k ( X ) ⊂ P s ( X ), which is the subposet of simplices σ such that each element x ∈ σ has at least k distinct “neighbours” y in X (not necessarily in σ ) suchthat d ( x, y ) ≤ s .The poset P s,k ( X ) is the poset of simplices of the degree Rips complex (orLesnick complex) L s,k ( X ).For s ≤ t we have a diagram of poset inclusions P s ( X ) σ / / P t ( X ) P s,k ( X ) O O σ / / P t,k ( X ) O O P s.k +1 ( X ) σ / / O O P t.k +1 ( X ) O O The notation σ will always be used for poset inclusions associated to changes ofdistance parameter.Observe also that1) P s, ( X ) = P s ( X ) for all s , and2) P s,k ( X ) = ∅ for k sufficiently large.The objects P ∗ ,k ( X ) form the degree Rips filtration of the Vietoris-Rips systemof posets P ∗ ( X ). 6he simplicial set BP s ( X ) is model for V s ( X ) in the homotopy category,but it may seem intractably large since all simplices of V s ( X ) are vertices of BP s ( X ), while one can only practically recover the low dimensional part of thesimplicial structure of V s ( X ) in concrete examples. That said, one only needslow dimensional simplices to compute low dimensional homotopy or homologygroups of BP s ( X ). This is illustrated as follows.Suppose, generally, that the poset P is a subobject of a power set P ( X ) isa subposet that is closed under taking non-empty subsets, so that P defines anabstract simplicial complex.Suppose given a list x , . . . , x k of elements of X such that d ( x i , x j ) ≤ s . Thislist may have repeats, and can be viewed as a function x : { , , . . . , k } → X which may not be injective. Write[ x , . . . , x k ] = { x } ∪ · · · ∪ { x k } . in X . This set can be identified with the image of the function x .There is a graph Gr ( P ) whose vertices are the singleton elements (vertices) { x } of P , and there is an edge x → y if [ x, y ] is an object of P .Observe that there is an edge [ x, y ] : x → y if and only if there is an edge[ y, x ] : y → x , and there is an edge [ x, x ] : x → x .Write Γ( P ) for the category generated by Gr ( P ), subject to relations definedby the simplices [ x , x , x ]. Then we have the following: Proposition 1.
The category Γ( P ) is a groupoid, and there are equivalences Γ( P ) ≃ G ( P ) ≃ π ( BP ) , where π ( BP ) is the fundamental groupoid of the space BP , and G ( P ) is the freegroupoid on the poset P . The groupoid G ( P ) can be identified up to natural equivalence with thefundamental groupoid π ( BP ) by [3, III.2.1]. In more detail, πBP is isomr-phic to G ( P ∗ ( BP )) where P ∗ ( BP ) is the path category of BP , and there is anisomorphism P ∗ ( BP ) ∼ = P , essentially by inspection (see also [5]). Corollary 2.
The category Γ( P s ( X )) is a groupoid, and there are equivalences Γ( P s ( X )) ≃ G ( P s ( X )) ≃ π ( BP s ( X )) . There is an equivalence of groupoids π ( BP s ( X )) ≃ π ( V s ( X )), since thespaces BP s ( X ) and V s ( X ) are weakly equivalent, and so the fundamental groupoid π ( V s ( X )) is weakly equivalent to Γ( P s ( X )). Proof of Proposition 1.
We show that1) The category Γ( P ) is a groupoid.2) There is an equivalence of groupoids Γ( P ) ≃ G ( P ).7or the first claim, the edges [ x, x ] represent 2-sided identities, on account ofthe existence of the simplices [ x, x, y ] and [ x, y, y ]. Then the simplices [ x, y, x ]and [ y, x, y ] are used to show that each edge [ x, y ] represents an invertible mor-phism of Γ s ( X ).For the second claim, pick an element x σ ∈ σ for each simplex σ ∈ P .For σ ⊂ τ in P , the list x σ , x τ consists of elements of τ , so that [ x σ , x τ ] ⊂ τ is a simplex of P . If σ ⊂ τ ⊂ γ are morphisms of P , then [ x σ , x τ , x γ ] ⊂ γ is asimplex of P , and so there is a commutative diagram x σ [ x σ ,x τ ] / / [ x σ ,x γ ] ! ! ❈❈❈❈❈❈❈❈ x τ [ x τ ,x γ ] (cid:15) (cid:15) x γ in Γ( P ). It follows that sending the morphism σ ⊂ τ to [ x σ , x τ ] : x σ → x τ defines a functor P → Γ( P ), which induces a functor φ : G ( P ) → Γ( P ) . Suppose that [ x, y ] : x → y is an edge of the graph Gr ( P ). Then theassociated inclusions { x } → [ x, y ] ← { y } in P define a morphism [ x, y ] ∗ : { x } → { y } of G ( P ).If [ x, y, z ] is a simplex of P then the diagram of inclusions { y } z z ✈✈✈✈✈✈✈✈✈ (cid:15) (cid:15) ❍❍❍❍❍❍❍❍❍ [ x, y ] / / [ x, y, z ] [ y, z ] o o { x } < < ②②②②②②②② ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ / / [ x, z ] O O { z } o o i i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ b b ❉❉❉❉❉❉❉❉ is used to show that [ y, z ] ∗ · [ x, y ] ∗ = [ x, z ] ∗ in the groupoid G ( P ).It follows that the assignment that takes an edge x → y of Gr ( P ) to themorphism [ x, y ] ∗ : { x } → { y } defines a functor ψ : Γ( P ) → G ( P ) . The inclusions { x σ } ⊂ σ define a natural isomorphism ψ · φ ∼ = −→ G ( P ) , and the composite φ · ψ is the identity on Γ( P ).8 Stability
Suppose that Z is a metric space, and let D ( Z ) be the poset of finite subsets(data sets) in Z .The poset D ( Z ) has the Hausdorff metric d H , which can be described heuris-tically, relative to a fixed r ≥
0, as follows:1) Suppose that X ⊂ Y in D ( Z ). Then d H ( X, Y ) < r if for all y ∈ Y thereis an x ∈ X such that d ( y, x ) < r .2) For arbitrary X, Y ∈ D ( Z ): d H ( X, Y ) < r if and only if (equivalently)a) d H ( X, X ∪ Y ) < r and d H ( Y, X ∪ Y ) < r .b) for all x ∈ X there is a y ∈ Y such that d ( x, y ) < r , and for all y ∈ Y there is an x ∈ X such that d ( y, x ) < r .We also have the following: Lemma 3.
Suppose that X and Y are data sets in a metric space Z , andsuppose that d H ( X ∩ Y, X ) < r .Then d H ( Y, X ∪ Y ) < r . Lemma 3 is easily proved. The statement can be visualized by the followingdiagram of labelled inclusions: X ∩ Y / / r (cid:15) (cid:15) Y r (cid:15) (cid:15) X / / X ∪ Y Now suppose that X ⊂ Y are data sets in Z , and supppose that d H ( X, Y ) Suppose X ⊂ Y in D ( Z ) such that d H ( X, Y ) < r . Then there isa homotopy commutative diagram of poset morphisms P s ( X ) σ / / i (cid:15) (cid:15) P s +2 r ( X ) i (cid:15) (cid:15) P s ( Y ) σ / / θ ssssssssss P s +2 r ( Y ) (3) in which the upper triangle commutes, and the lower triangle commutes up to ahomotopy that fixes the subobject P s ( X ) . The diagram (3) in the statement of Theorem 4 is a homotopy interleaving .Theorem 4 is a form of the Rips stability theorem. The form of this result thatappears in the Blumberg-Lesnick paper [1] is the following: Theorem 5. Suppose given X, Y ⊂ Z are data sets with d H ( X, Y ) < r .Then there are maps φ : P s ( X ) → P s +2 r ( Y ) and ψ : P s ( Y ) → P s +2 r ( X ) such that ψ · φ ≃ σ : P s ( X ) → P s +4 r ( X ) and φ · ψ ≃ σ : P s ( Y ) → P s +4 r ( Y ) . Theorem 5 is a consequence of Theorem 4, but it also has a poset-theoreticproof, given below, that follows the outline given by Blumberg-Lesnick [1], anduses Quillen’s Theorem A [7], [3, IV.5.6]. The use of Theorem A for proofs ofstability results was introduced by Memoli [6]. Proof of Theorem 5. Set U = { ( x, y ) | x ∈ X, y ∈ Y, d ( x, y ) < r } . The poset P s,X ( U ) ⊂ P ( U ) consists of all subsets σ ⊂ U such that d ( x, x ′ ) ≤ s for all ( x, y ) , ( x ′ , y ′ ) ∈ σ . Define the poset P s,Y ( U ) similarly, by constrainingdistances between coordinates in Y . 10rojection on the X -factor defines a poset map p X : P s,X ( U ) → P s ( X ), andprojection on the Y -factor defines p Y : P s,Y ( U ) → P s ( Y ). The maps p X and p Y are weak equivalences, by Quillen’s Theorem A. In effect, the slice category p X /σ can be identified with the power set of the collection of all pairs ( x, y )such that x ∈ σ , and power sets are contractible posets.There are inclusions P s,X ( U ) ⊂ P s +2 r,Y ( U ) , P s,Y ( U ) ⊂ P s +2 r,X ( U ) , by the triangle identity, and these maps define the maps φ and ψ , respectvely,via the weak equivalences p X and p Y .Suppose that X is a finite subset of a metric space Z . Write X k +1 dis for theset of k + 1 distinct points of X , and think of it as a subobject of Z k +1 . Theproduct Z k +1 has a product metric space structure, and so we have a Hausdorffmetric on its poset D ( Z k +1 ) of finite subsets.We have the following analogue (and generalization) of Theorem 4: Theorem 6. Suppose X ⊂ Y in D ( Z ) such that d H ( X k +1 dis , Y k +1 dis ) < r . Thenthere is a homotopy commutative diagram of poset diagrams P s,k ( X ) σ / / i (cid:15) (cid:15) P s +2 r,k ( X ) i (cid:15) (cid:15) P s,k ( Y ) σ / / θ qqqqqqqqqq P s +2 r,k ( Y ) in which the upper triangle commutes, and the lower triangle commutes up to ahomotopy which fixes the image of P s,k ( X ) .Proof. Write P s,k ( X ) for the set of one-point members (vertices) of P s,k ( X ).Suppose that y ∈ P s,k ( Y ) − P s,k ( X ) . Then there are k points y , . . . , y k of Y , distinct from y such that d ( y, y i ) < s . There is a ( k +1)-tuple ( x , x , . . . , x k )such that d (( x , . . . , x k ) , ( y, y , . . . , y k )) < r, by assumption. Then d ( y, x ) < r , d ( y i , x i ) < r , and so d ( x , x i ) < s + 2 r , and x is a vertex of P s +2 r,k ( X ). Set θ ( y ) = x , and observe that d ( y, θ ( y )) < r .If [ y , . . . , y p ] is a simplex of P s,k ( Y ) then [ θ ( y ) , . . . , θ ( y p )] is a simplex of P s +2 r,k ( Y ), as is the subset[ y , . . . , y p , θ ( y ) , . . . , θ ( y p )] . Finish according to the method of proof for Theorem 4.A data set Y ∈ D ( Z ) is finite, so there is a finite string of parameter values0 = s < s < · · · < s r , consisting of the distances between elements of Y . I say that the s i are the phase-change numbers for Y . 11 orollary 7. Suppose that X ⊂ Y in D ( Z ) and that d H ( X k +1 dis , Y k +1 dis ) < r .Suppose that r < s i +1 − s i . Then the inclusion i : P s i ,k ( X ) → P s i ,k ( Y ) is aweak homotopy equivalence. Lemma 8. Suppose that X ⊂ Y in D ( Z ) and that d H ( X k +1 dis , Y k +1 dis ) < r .Suppose that that Y k +1 dis = ∅ . Then d H ( X kdis , Y kdis ) < r .Proof. Suppose that { y , . . . , y k − } is a set of k distinct points of Y . Thenthere is a y k ∈ Y which is distinct from the y i , for otherwise Y has only k elements. Then ( y , y , . . . , y k ) is a ( k + 1)-tuple of distinct points of Y . Thereis a ( k + 1)-tuple ( x , . . . , x k ) of distinct points of X such that d (( y , . . . , y k − , y k ) , ( x , . . . , x k − , x k )) < r. It follows that d (( y , . . . , y k − ) , ( x , . . . , x k − )) < r. Corollary 9. Suppose X ⊂ Y in D ( Z ) such that d H ( X k +1 dis , Y k +1 dis ) < r for somenon-negative number r . Then for ≤ j ≤ k there is a homotopy commutativediagram of poset diagrams P s,j ( X ) σ / / i (cid:15) (cid:15) P s +2 r,j ( X ) i (cid:15) (cid:15) P s,j ( Y ) σ / / θ qqqqqqqqqq P s +2 r,j ( Y ) in which the upper triangle commutes, and the lower triangle commutes up to ahomotopy that fixes the image of P s,j ( X ) .Proof. Use Theorem 6 and Lemma 8. A system of simplicial sets (or spaces) is a functor X : [0 , ∞ ) → s Set whichtakes values in the category of simplicial sets. One also says that such a functoris a diagram of simplicial sets with index category [0 , ∞ ). A map of systems X → Y is a natural transformation of functors that are defined on [0 , ∞ ).We shall also discuss systems of sets, groups and chain complexes as functorsdefined on the poset [0 , ∞ ), which take values in the respective categories. Examples 1) The functors s V s ( X ) , BP s ( X ) are systems of spaces, for a data set X ⊂ Z .The functor s P s ( X ) is a system of posets.2) If X ⊂ Y ⊂ Z are data sets, the induced maps P s ( X ) → P s ( Y ) and V s ( X ) → V s ( Y ) define maps of systems P ∗ ( X ) → P ∗ ( Y ) and V ∗ ( X ) → V ∗ ( Y ).12here are various ways to discuss homotopy theories of systems. The oldestof these is the projective model structure of Bousfield and Kan [2], althoughthey do not use the term “projective” — this term arose much later in motivichomotopy theory.In the projective structure, a map f : X → Y is a weak equivalence (re-spectively fibration) if each map X s → Y s is a weak equivalence (respectivelyfibration) of simplicial sets. The maps which are both weak equivalences andfibrations are called trivial fibrations.A map A → B of systems is a projective cofibration if it has the “left liftingproperty” with respect all maps which are weak equivalences and fibrations.Projective cofibrations are intensely studied and important, but will not beused here.A map of systems f : X → Y such that each map f : X s → Y s is aweak equivalence (respectively fibration) is often said to be a sectionwise weakequivalence (respectively sectionwise fibration). A sectionwise cofibration is amap of systems A → B such the each map A s → B s is a monomorphism(or cofibration) of simplicial sets. Every projective cofibration is a sectionwisecofibration, but the converse is not true.Suppose that X ⊂ Y in D ( Z ) such that d H ( X, Y ) < r . The Rips stabilitytheorem (Theorem 4) says that we have a homotopy interleaving BP s ( X ) σ / / i (cid:15) (cid:15) BP s +2 r ( X ) i (cid:15) (cid:15) BP s ( Y ) σ / / θ qqqqqqqqqqq BP s +2 r ( Y )where the upper triangle commutes and the lower triangle commutes up tohomotopy which is constant on the space BP s ( X ).Then we have the following:1) The natural transformation i : π BP ∗ ( X ) → π BP ∗ ( Y ) is a 2 r -mono-morphism : if i ([ x ]) = i ([ y ]) in π BP s ( Y ) then σ [ x ] = σ [ y ] in π BP s +2 r ( X ).2) The transformation i : π BP ∗ ( X ) → π BP ∗ ( Y ) is a 2 r -epimorphism :given [ y ] ∈ π BP s ( Y ), σ [ y ] = i [ x ] for some [ x ] ∈ π BP s +2 r ( X ).3) All natural transformations i : π n ( BP ∗ ( X ) , x ) → π n ( BP ∗ ( Y ) , i ( x )) ofhomotopy group functors are 2 r -isomorphisms in the sense that they areboth 2 r -monomorphisms and 2 r -epimorphisms.The statements 1)–3) are “derived”, and depend on having a way to talkabout higher homotopy groups.There is a functorial weak equivalence γ : X → Ex ∞ X of simplicial sets,where Ex ∞ X is a system of Kan complexes, and therefore have combinatoriallydefined homotopy groups [3]. Thus, for example, the notation π n ( BP s ( X ) , x )can refer to the combinatorial homotopy group π n (Ex ∞ BP s ( X ) , x ).13here is an alternative, in that one could use the adjunction weak equivalence η : X → S ( | X | ), where S is the singular functor and | X | is the topological real-ization of X . The combinatorial homotopy groups of the Kan complex S ( | X | )coincide up to natural isomorphism with the standard homotopy groups of thespace | X | . In this case, we would write π n ( BP s ( X ) , x ) to mean π n ( | BP s ( X ) | , x ).There is a natural isomorphism π n (Ex ∞ Y, y ) ∼ = π n ( | Y | , y )for all simplicial sets Y and vertices y of Y , so the combinatorial and topologicalconstructions produce isomorphic homotopy groups.The Kan Ex ∞ functor is combinatorial and therefore plays well with alge-braic constructions, while the realization functor is familiar but transcendental.The homotopy groups π n ( BP s ( X ) , x ) coincide with the homotopy groups π n ( V s ( X ) , x ) of the Vietoris-Rips complex V s ( X ) up to natural isomorphism.A similar observation holds for the extant constructions of the degree Ripscomplexes.The natural maps γ : Y → Ex ∞ Y and η : Y → S ( | Y | ) are fibrant mod-els for simplicial sets Y , in that the maps are weak equivalences which takevalues in fibrant simplicial sets (Kan complexes). Both constructions preservemonomorphisms.We shall write Y → F Y for an arbitrary fibrant model construction thatpreserves monomorphisms. A formal nonsense argument implies that the choiceof fibrant model does not matter.Suppose f : X → Y is a map of systems. Say that f is an r -equivalence if1) the map f : π ( X ) → π ( Y ) is an r -isomorphism of systems of sets2) the maps f : π k ( X t , x ) → π k ( Y t .f ( x )) are r -isomorphisms of systems ofgroups for t ≥ s , for all s ≥ x ∈ X s . Remark 10. Note the variation of condition 2) from the Vietoris-Rips example.In the general definition, we do not assume that all simplicial sets X s of thesystem X have the same vertices, so the base points of condition 2) have tobe chosen section by section. This is relevant for comparisons of degree Ripssystems BP ∗ ,k ( X ) → BP ∗ ,k ( Y ). Examples : 1) A map f : X → Y is a sectionwise equivalence if and only if itis a 0-equivalence.2) If r ≤ s and f : X → Y is an r -equivalence, then f is an s -equivalence.3) If X ⊂ Y are (finite) data sets in a metric space Z , then there is a homotopy r -interleaving BP s ( X ) σ / / i (cid:15) (cid:15) BP s + r ( X ) i (cid:15) (cid:15) BP s ( Y ) σ / / θ qqqqqqqqqq BP s + r ( Y )14or sufficiently large r : take r/ > d H ( X, Y ). This means that the map ofsystems BP s ( X ) → BP s ( Y ) is an r -equivalence for large r .A similar observation holds for the comparison BP s,k ( X ) → BP s,k ( Y ) ofdegree Rips systems.Say that a map of systems X → Y is a controlled equivalence if it is an r -equivalence for some r ≥ Lemma 11. Suppose given a diagram of systems X f / / ≃ (cid:15) (cid:15) Y ≃ (cid:15) (cid:15) X f / / Y in which the vertical maps are sectionwise weak equivalences. Then f is an r -equivalence if and only if f is an r -equivalence. Lemma 12. Suppose given a commutative triangle X f / / h ❅❅❅❅❅❅❅❅ Y g (cid:15) (cid:15) Z of maps of systems,’Then if one of the maps is an r -equivalence, a second is an s -equivalence,then the third map is a ( r + s ) -equivalence.Proof. The arguments are set theoretic. We present an example.Suppose X, Y, Z are systems of sets, h is an r -isomorphism and g is an s -isomorphism. Given z ∈ Y t , g ( z ) = h ( w ) for some w ∈ X t + s . Then g ( z ) = g ( f ( w )) in Z t + s so z = f ( w ) in Y t + s + r . It follows that f is an ( r + s )-epimorphism.Lemma 12 is an approximation of the triangle axiom for weak equivalencesin the definition of a Quillen model structure.There is a calculus of controlled equivalences and sectionwise fibrations,which starts with the following result and concludes with Theorem 15. Lemma 13. Suppose that p : X → Y is a sectionwise fibration of systems ofKan complexes, and that p is an r -equivalence. hen each lifting problem ∂ ∆ n α / / (cid:15) (cid:15) X s (cid:15) (cid:15) σ / / X s +2 rp (cid:15) (cid:15) ∆ n β / / θ Y s σ / / Y s +2 r can be solved up to shift r in the sense that the indicated dotted arrow liftingexists. Lemma 13 is the analogue of a classical result of simplicial homotopy the-ory [3, I.7.10], and its proof is a variant of the standard obstruction theoreticargument for that result. Proof of Lemma 13. The original diagram can be replaced up to homotopy bya diagram ∂ ∆ n ( α , ∗ ,..., ∗ ) / / (cid:15) (cid:15) X sp (cid:15) (cid:15) σ / / X s + rp (cid:15) (cid:15) ∆ n β / / Y s σ / / Y s + r (4)Then p ∗ ([ α ]) = 0 in π n − ( Y s , ∗ ), so σ ∗ ([ α ]) = 0 in π n − ( X s + r , ∗ ).The trivializing homotopy for σ ( α ) in X s + r defines a homotopy from theouter square of (4) to the diagram ∂ ∆ n ∗ / / (cid:15) (cid:15) X s + rp (cid:15) (cid:15) ∆ n ω / / Y s + r The element [ ω ] ∈ π n ( Y s +2 r , ∗ ) lifts to an element of π n ( X s +2 r , ∗ ) up tohomotopy, giving the desired lifting. Lemma 14. Suppose that p : X → Y is a sectionwise fibration of systems ofKan complexes, and that all lifting problems ∂ ∆ n / / (cid:15) (cid:15) X s (cid:15) (cid:15) σ / / X s + rp (cid:15) (cid:15) ∆ n / / θ Y s σ / / Y s + r have solutions up to shift r , in the sense that the dotted arrow exists making thediagram commute. Then the map p : X → Y is an r -equivalence. roof. If p ∗ ([ α ]) = 0 for [ α ] ∈ π n − ( X s , ∗ ), then there is a diagram on the leftabove. The existence of θ gives σ ∗ ([ α ]) = 0 in π n − ( X s + r , ∗ ).The argument for r -surjectivity is similar.Lemma 13 and Lemma 14 together imply the following: Theorem 15. Suppose given a pullback diagram X ′ / / p ′ (cid:15) (cid:15) X p (cid:15) (cid:15) Y ′ / / Y where p is a sectionwise fibration and an r -equivalence.Then the map p ′ is a sectionwise fibration and a r -equivalence.Proof. All lifting problems ∂ ∆ n α / / (cid:15) (cid:15) X ′ s (cid:15) (cid:15) σ / / X ′ s +2 rp ′ (cid:15) (cid:15) ∆ n β / / θ Y ′ s σ / / Y ′ s +2 r for p ′ have solutions up to shift 2 r , since it is a pullback of a map p that has thatproperty by Lemma 13. Then Lemma 14 implies that p ′ is a 2 r -equivalence.Suppose that i : A → B is a sectionwise cofibration of projective cofibrantsystems (i.e. systems of monomorphisms), and form the diagram A η / / i (cid:15) (cid:15) F A i ∗ (cid:15) (cid:15) B η / / F B in which the horizontal maps are fibrant models, and in particular sectionwiseequivalences. Typically, one sets F A = Ex ∞ A .Say that the map i admits an r -interleaving if, for all s , there are maps θ : B s → F A s + r such that the diagram A s σ / / i (cid:15) (cid:15) A s + r η / / F A s + r B s θ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ F A s + ri ∗ (cid:15) (cid:15) B s θ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ σ / / B s + r η / / F B s + r commutes up to a homotopy which restricts to the constant homotopy on A s .An r -interleaving is effectively a strong deformation retraction up to shift r . Lemma 16. Suppose that the map i : A → B admits an r -interleaving. Thenthe map i : A → B is an r -equivalence.Proof. The map η : A s + r → F A s + r is a weak equivalence, so that θ : B s → F A s + r induces functions θ ∗ : π B s → π A s + r and homomorphisms θ ∗ : π n ( B s , i ( x )) → π n ( A s + r , σ ( x )) . The diagram π A s σ / / i (cid:15) (cid:15) π A s + ri (cid:15) (cid:15) π B s σ / / θ : : ✉✉✉✉✉✉✉✉✉ π B s + r commutes, so that the map of systems of sets π A → π B is an r -isomorphism.The diagram π n ( A s , x ) σ / / i (cid:15) (cid:15) π n ( A s + r , σ ( x )) i (cid:15) (cid:15) π n ( B s , i ( x )) θ ♠♠♠♠♠♠♠♠♠♠♠♠ σ / / π n ( B s + r , σi ( x ))also commutes. In effect, σ ( i ( x )) = i ( θ ( i ( x )), and the homotopy σ ≃ i · θ restricts to the identity on i ( x ), so that [ σ ( α )] = [ i ( θ ( α )] in π n ( B s + r , σi ( x ))for any representing simplex α : ∆ n → F B s of a homotopy group element[ α ] ∈ π n ( B s , i ( x )). Lemma 17. Suppose that i : A → B is a sectionwise cofibration between sys-tems, such that i admits an r -interleaving. Suppose also that the diagram A α / / i (cid:15) (cid:15) C i ′ (cid:15) (cid:15) B β / / D (5) is a pushout. Then the map i ′ admits an r -interleaving. roof. The composites B s θ −→ F A s + r α ∗ −−→ F C s + r ,C s σ −→ C s + r η −→ F C s + r together determine a unique map θ ′ : D s → F C s + r .The homotopy i ∗ · θ ≃ η · σ is defined by a map h : B s × ∆ → F B s + r . This homotopy restricts to a constant homotopy on A s , which means that thediagram A s × ∆ pr / / i × ∆ (cid:15) (cid:15) A s σ / / A s + r η / / F A s + ri ∗ (cid:15) (cid:15) B s × ∆ h / / F B s + r commutes, where pr is a projection.The diagram A s × ∆ α × ∆ / / i × ∆ (cid:15) (cid:15) C s × ∆ i ∗ × ∆ (cid:15) (cid:15) B s × ∆ β × ∆ / / D s × ∆ is a pushout, and the composites B s × ∆ h −→ F B s + r β ∗ −→ F D s + r C s × ∆ pr −→ C s σ −→ C s + r η −→ F C s + r i ∗ −→ F D s + r together determine a homotopy h ′ : D s × ∆ → F D s + r from i ∗ · θ ′ to η · σ . The homotopy h ′ restricts to the constant homotopy on C s ,by construction.Theorem 15 says that the pullback of a map which is a sectionwise fibrationand an r -equivalence is a sectionwise fibration and a 2 r -equivalence. The “dual”statement, namely that a pushout of a map which is a cofibration and an r -equivalence is a cofibration and a 2 r -equivalence, has not been proved for anyrelevant class of cofibrations.The practical examples of cofibrations which are r -weak equivalences arecofibrations which admit r -interleavings. These arise from stability results, andLemma 17 says that the class of cofibrations which admit r -interleavings isclosed under pushout. 19hus, if one has a pushout diagram such as (5) for which the cofibration i admits an r -interleaving, then the induced map i ∗ also admits an r -interleaving.The map i ∗ induces r -isomorphisms H k ( C ) → H k ( D ) and π C → π D .We have more general statements for homology and path components, asfollows. Lemma 18. Supppose given a pushout diagram A α / / i (cid:15) (cid:15) C (cid:15) (cid:15) B β / / D of systems in which i is a cofibration. Then the following hold:1) If the map π A → π B is an r -isomorphism, then the π C → π D is an r -isomorphism.2) If the maps H k ( A ) → H k ( B ) are r -isomorphisms for k ≥ (arbitrarycoefficients, then the map H k ( C ) → H k ( D ) is a r -isomorphism, for k ≥ .Proof. Statement 1) follows from Lemma 19 below, since the path componentfunctor preserves pushouts.For statement 2) there is a system of exact sequences · · · → H k ( A ) → H k ( B ) → H k ( B/A ) ∂ −→ H k − ( A ) → H k − ( B ) → . . . An element chase within this system shows that the map 0 → H k ( B/A ) is a2 r -isomorphism for all k ≥ 0. One uses the system of exact sequences · · · → H k ( C ) → H k ( D ) → H k ( B/A ) ∂ −→ H k − ( C ) → H k − ( DB ) → . . . to show that the map H k ( C ) → H k ( D ) is a 2 r -isomorphism for all k ≥ Lemma 19. Suppose that the diagram A α / / f (cid:15) (cid:15) C f ′ (cid:15) (cid:15) B / / D is a pushout of systems of sets, and that f is an r -bijection. Then f ′ is an r -bijection.Proof. The map f has an epi-monic factorization A p −→ Z j −→ B A / / p (cid:15) (cid:15) C p ′ (cid:15) (cid:15) Z j (cid:15) (cid:15) / / Z ′ j ′ (cid:15) (cid:15) B / / D The map j ′ is a sectionwise monomorphism, and j ′ is an r -epimorphism since j is an r -epimorphism.The sectionwise epimorphism p ′ is constructed by collapsing images of fi-bres of p to points, and it follows that p ′ is an r -monomorphism as well as asectionwise epimorphism.The conclusion follows: the composite j ′ · p ′ is an r -monomorphism and an r -epimorphism. References [1] Andrew J. Blumberg and Michael Lesnick. Universality of the homotopyinterleaving distance. CoRR , abs/1705.01690, 2017.[2] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and local-izations . Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol.304.[3] P. G. Goerss and J. F. Jardine. Simplicial Homotopy Theory , volume 174 of Progress in Mathematics . Birkh¨auser Verlag, Basel, 1999.[4] J. F. Jardine. Simplicial approximation. Theory Appl. Categ. , 12:No. 2,34–72 (electronic), 2004.[5] J. F. Jardine. Path categories and resolutions.