Rectification of interleavings and a persistent Whitehead theorem

The homotopy interleaving distance, a distance between persistent spaces, was introduced by Blumberg and Lesnick and shown to be universal, in the sense that it is the largest homotopy-invariant distance for which sublevel-set filtrations of close-by real-valued functions are close-by. There are other ways of constructing homotopy-invariant distances, but not much is known about the relationships between these choices. We show that other natural distances differ from the homotopy interleaving distance in at most a multiplicative constant, and prove versions of the persistent Whitehead theorem, a conjecture of Blumberg and Lesnick that relates morphisms that induce interleavings in persistent homotopy groups to stronger homotopy-invariant notions of interleaving.


Introduction
Context Many of the main theoretical tools of topological data analysis (TDA) come in the form of stability theorems.One of the best known stability theorems, due to Cohen-Steiner, Edelsbrunner and Harer [7], implies that if f; g W X !R are sufficiently tame functions, such as piecewise linear functions on the geometric realization of a finite simplicial complex, then d B .D n .f/; D n .g//Ä kf gk 1 : Here, D n .f/ denotes the n-dimensional persistence diagram of f .This consists of a multiset of points of the extended plane R 2 that captures the isomorphism type of the n th persistent homology of the sublevel sets of f , that is, of the functor R !Vec obtained by composing the sublevel-set filtration r 7 !f 1 .1; r W R !Top with the n th homology functor H n W Top !Vec, where R denotes the poset of real numbers and Vec denotes the category of vector spaces over some fixed field.The distance d B is the bottleneck distance, a combinatorial way of comparing persistence diagrams.This result was later refined by Chazal, Cohen-Steiner, Glisse, Guibas and Oudot in [4] to the algebraic stability theorem, which says that for F; G W R !Vec sufficiently tame functors, one has d B .D.F /; D.G// Ä d I .F; G/; where, as before, D.F / denotes the persistence diagram of F , which describes the isomorphism type of F , and d I denotes the interleaving distance, a distance between functors R !C for any fixed category C , which we recall below.
Stability theorems imply that pipelines like the following, popular in TDA, are robust to perturbations of the input data and can be used for inference purposes: data !persistent spaces For example, the algebraic stability theorem tells us that the last step is stable, if we endow persistent vector spaces (Vec R ) with the interleaving distance and persistence diagrams with the bottleneck distance, while functoriality implies that the second step is stable, if we also endow persistent spaces (Top R ) with the interleaving distance; see Bubenik and Scott [3].
Problem statement Although useful in some applications, the interleaving distance on Top R is often too fine; for instance, it is easy to see that Vietoris-Rips and other functors S W Met !Top R are not stable with respect to the Gromov-Hausdorff distance on metric spaces and the interleaving distance on Top R .However, when one composes these functors with a homotopy-invariant functor, such as homology H n W Top R !Vec R , the composite H n ı S W Met !Vec R turns out to be stable; see Chazal, Cohen-Steiner, Guibas, Mémoli and Oudot [5].So, in these cases, one way to make the first step in the pipeline above stable is to force the interleaving distance on Top R to be homotopyinvariant [2,Section 1.2].For this reason, many homotopy-invariant adaptations of the interleaving distance on Top R and related categories have been proposed; see eg Blumberg and Lesnick [2], Frosini, Landi and Mémoli [8] and Lesnick [15].In order to describe some of these adaptations, we recall the definition of the interleaving distance d I .
Let C be a category.Given ı 0 2 R and F W R !C , let F ı W R !C be given by F ı .r/ WD F.r Cı/, with the obvious structure morphisms.One says that F; G 2 C R are ı-interleaved if there exist natural transformations f W F ! G ı and g W G ! F ı such that g ı ıf W F ! F 2ı equals the natural transformation F ! F 2ı given by the structure morphisms of F , and such that f ı ı g W G ! G 2ı equals the natural transformation G ! G 2ı given by the structure morphisms of G. Then d I .F; G/ WD inf.fı 0 W F and G are ı-interleavedg [ f1g/: Blumberg and Lesnick [2] define X; Y 2 Top R to be ı-homotopy interleaved if there exist weakly equivalent persistent spaces X 0 ' X and Y 0 ' Y such that X 0 and Y 0 are ı-interleaved, and use homotopy interleavings to define the homotopy interleaving distance, denoted d HI .The homotopy interleaving distance is the (metric) quotient of the interleaving distance by the equivalence relation given by weak equivalence, in the sense that d HI is the largest homotopy-invariant distance that is bounded above by the interleaving distance.
Instead of taking a metric quotient, one can take the categorical quotient of Top R by weak equivalences, and define interleavings directly in the homotopy category, similar to what is done in eg Frosini, Landi and Mémoli [8] Kashiwara and Schapira [14] and Lesnick [15].In order to do this, one notes that the shift functors ./ ı W Top R !Top R preserve weak equivalences and thus induce functors ./ ı W Ho.Top R / !Ho.Top R /.This lets one copy the definition of interleaving, but in the homotopy category, which gives the notions of interleaving in the homotopy category and of interleaving distance in the homotopy category, denoted by d IHC .
A third option, also introduced in [2], is to compare objects of Top R using interleavings in Ho.Top/ R , called homotopy commutative interleavings, which give rise to the homotopy commutative interleaving distance, denoted by d HC .
We have described three homotopy-invariant notions of interleaving in decreasing order of coherence.On one end, homotopy interleavings can be equivalently described as homotopy coherent diagrams of spaces [2, Section 7].On the other end, homotopy commutative interleavings correspond to diagrams in the homotopy category of spaces.It is clear that d HI d IHC d HC , and that any of the homotopy-invariant interleavings induce interleavings in homotopy groups.
Two questions arise: Are the three distances in some sense equivalent or are they fundamentally different?If a map induces interleavings in homotopy groups, does it follow that the map is part of one of the homotopy-invariant notions of interleaving?A conjectural answer to the second question is given in [2,Conjecture 8.6], where it is conjectured that when X and Y are a kind of persistent CW-complex of finite dimension d 2 N, if there exists a morphism between them inducing a ı-interleaving in homotopy groups, then X and Y are cı-homotopy interleaved for a constant c that only depends on d.
Contributions Homotopy interleavings compose in any functor category of the form M R m for M a cofibrantly generated model category (Proposition 2.3).This allows us to state some of our results for any cofibrantly generated model category M, or for a category of spaces S, which can be instantiated to be any of the Quillen equivalent model categories of topological spaces or simplicial sets (Remark 2.1).Our first theorem is the following rectification result.
Theorem A Let M be a cofibrantly generated model category, let X; Y 2 M R , and let ı > 0 2 R. If X and Y are ı-homotopy commutative interleaved , then they are cı-homotopy interleaved for every c > 2.
It follows that we have 2d HC d HI d IHC d HC .The above rectification result is different from many such results in homotopy theory, where a diagram of a certain shape, in the homotopy category, is lifted to a strict diagram of the same shape.The difference lies in the fact that the shape of the strict diagram we construct is different from the shape of the diagram in the homotopy category.In fact, building on the suggestion in [2] of using Toda brackets to give a lower bound for the above rectification, we show (Proposition 3.12) that for M D Top, if cd HC d HI then c 3  2 , so that, in particular, d HC ¤ d HI .This means that rectification in the usual sense is not possible in general, and thus standard results are not directly applicable.We also show that Theorem A has no analogue for multipersistent spaces (Section 3.3).
Our second theorem relates morphisms inducing interleavings in homotopy groups to interleavings in the homotopy category.See Definition 5.7 for the notion of persistent CW-complex and Definition 5.2 for the notion of interleaving induced in persistent homotopy groups.Structure of the paper In Section 2, we recall and give references for the necessary background.In Section 3, we prove Theorem A, we provide a lower bound for the rectification of homotopy commutative interleavings between persistent spaces, and we show that Theorem A has no analogue for multipersistent spaces.In Section 4, we characterize projective cofibrant (multi)persistent simplicial sets as filtered simplicial sets.In Section 5, we prove Theorem B.
The main purpose of this section is to fix notation and to provide the reader with references.This section can be referred to as needed, but we do recommend going over Section 2.2 as it contains the notions of interleaving relevant to us.
We assume that the reader is comfortable with the language of category theory.Throughout the paper, we will use the term distance to refer to any extended pseudometric on a (possibly large) set X , that is, to any function d X W X X !OE0; 1 that is symmetric, satisfies the triangle inequality, and is 0 on the diagonal.

Spaces and model categories 2.1.1 Spaces
We work model-independently whenever possible.This means that whenever we say space we will mean either topological space or simplicial set.Results stated for spaces will hold for both possible models.The category of spaces will be denoted by S.
For a general introduction to simplicial sets, see eg [9] or [12,Chapter 3].We denote the geometric realization functor for simplicial sets by j jW sSet !Top.

Model categories
The theory of model categories was introduced in [18]; for a modern and thorough development of this theory we recommend [11] and [12].
We recall that two objects x; y 2 M of a model category M are said to be weakly equivalent if they are isomorphic in Ho.M/, which happens if and only if they are connected by a zigzag of weak equivalences in M.This is an equivalence relation, which we denote by x ' y.When there is risk of confusion, morphisms in Ho.M/ will be surrounded by square brackets OEf , to distinguish them from morphisms in M.  [12,Theorem 2.4.25].This model structure is, in some respects, better behaved than the Quillen model structure on topological spaces, and is in fact the model of space used in [2].We will not concern ourselves with these subtleties since, by the observations in Remark 2.1, there is no essential difference between using Top or Top CGWH when studying homotopy-invariant notions of interleaving.
We will make use of the notion of cofibrantly generated model category [12, Chapter 2, Section 2.1].Recall that the Kan-Quillen model structure on simplicial sets is cofibrantly generated, where a set of generating cofibrations consists of the boundary inclusions @ n ,! n for n 0 [12, Theorem 3.6.5].The Quillen model structure on topological spaces is also cofibrantly generated, with a set of generating cofibrations given by fS n 1 ,! D n g n 0 [12,Theorem 2.4.19].
We conclude by recalling the basic properties of projective model structures.Given a model category M and a small category I, the projective model structure on the functor category M I is, when it exists, the model structure whose fibrations (resp.weak equivalences) are those which are pointwise fibrations (resp.weak equivalences) of M.
The projective model structure on M I exists, and is cofibrantly generated, whenever M is cofibrantly generated.Moreover, if I and J are, respectively, generating cofibrations and generating trivial cofibration for the model structure of M, then fI.i; / ˇf W i 2 I; f 2 Ig and fI.i; / ˇg W i 2 I; g 2 J g are, respectively, generating cofibrations and generating trivial cofibrations for the projective model structure, where, given a functor F W I !Set and an object X 2 M, the functor F ˇX W I !M is defined by i 7 !`a2F.i/X [11,Section 11.6].For simplicity, we denote I.i; / ˇX by i ˇX .
We are especially interested in the projective model structure when the indexing category is a poset .P; Ä/.In this case, if r 2 P and X 2 M, then r ˇX is the functor that takes the value X on every s r , and has as value the initial object of M when sr .The nontrivial structure morphisms of this functor are the identity of X .
Note that we have a functor hW Ho.M I / !Ho.M/ I by the universal property of Ho.M I /.

Interleavings and interleavings up to homotopy 2.2.1 Strict interleavings
We denote the poset of real numbers with their standard order by R, and for m 2 N, we let R m be the set of m-tuples of real numbers with the product order.We set x m D fi W 1 Ä i Ä mg, so that ."i / i2 x m Ä .ıi / i2 x m 2 R m if and only if " i Ä ı i for all 1 Ä i Ä m.We denote the element .0;: : : ; 0/ 2 R m by 0.
Fix a category C and a natural number m 1.An m-persistent object of C is any functor of the form R m !C .We often refer to m-persistent objects simply as persistent objects or as multipersistent objects when we want to stress the fact that m is not necessarily 1. Fix persistent objects X; Y; Z 2 C R m , r; s 2 R m , and "; ı 0 2 R m .We use the following conventions.
Assume r Ä s.The structure morphism X.r / !X.s/ will be denoted by ' X r;s .The ı-shift to the left of X is the functor defined by mapping X to the persistent object ı X , with values given by .ıX /.r / D X.r ı/.
Natural transformations f W X !Y ı will be referred to as ı-morphisms, and will often be denoted by f W X !ı Y .Since we have natural bijections Note that an "-morphism f W X !" Y can be composed with a ı-morphism g W Y !ı Z, yielding an ."Cı/-morphismg " ıf W X !"Cı Y .This composition is associative and unital, and is natural with respect to shifts of morphisms.
Let " 1 ; " 2 ; ı 1 ; ı 2 0 2 R m .Note that an ." 1 ; " 2 /-interleaving between X and Y can be composed with any .ı 1 ; ı 2 /-interleaving between Y and Z, yielding an ." 1 Cı 1 ; " 2 Cı 2 /interleaving.The fact that interleavings compose implies that, when m D 1, the formula This is the interleaving distance on the class of objects of the category C R .This notion of distance can be extended to objects of the functor category C R m [15], but we will not make use of this extension.

Interleavings up to homotopy
If one is comparing objects of a category of functors of the form R m !M, for M a model category, it makes sense to want to find a homotopy-invariant notion of interleaving.In this paper, we consider the following three homotopy-invariant relaxations of the notion of interleaving.Let M be a cofibrantly generated model category and endow M R m with the projective model structure.Let X; Y 2 M R m and let "; ı 0 2 R m .
(2) Note that the shift functor ./ ı W M R m !M R m maps weak equivalences to weak equivalences.This implies that all the notions in Section 2.2.1 have analogues in the category Ho.M R m /.We say that X and Y are .";ı/-interleaved in the homotopy category if they are .";ı/-interleaved as objects of Ho.M R m /.
For each of the three homotopy-invariant notions of interleaving introduced above, we have a corresponding extended pseudometric on the collection of objects of the category M R .Let X; Y 2 M R .Following [2], we define the homotopy interleaving distance as The fact that the homotopy interleaving distance satisfies the triangle inequality follows from Proposition 2.3.The interleaving distance in the homotopy category is Again following [2], the homotopy commutative interleaving distance is defined as between the projective model structures respects interleavings, in the sense that shifts commute with both the left and right adjoints.This implies that, for any of the three homotopy-invariant notions of interleaving described above, we have that two functors on one side of the adjunction are .";ı/-interleaved if and only if their images (along the derived adjunction) on the other side are .";ı/-interleaved.In particular, if m D 1, the two adjoints give an isometry between M R and N R independently of whether we use d HI , d IHC or d HC .

Composability of homotopy interleavings
In this short section, we give a simplified proof of a generalization of the fact that homotopy interleavings can be composed, originally proved in [2,Section 4].This is generalized further in [20,Theorem 4.1.4].
Proof We start by depicting the pullback square in the statement: apply the functor ./ ı W C R m !C R m to the pullback square above, and use the uniqueness part of its universal property.Section 4]) Let M be cofibrantly generated , fix m 1, let X; Y; Z W R m !M, and let " 1 ; " 2 ; ı 1 ; ı 2 0 2 R m .If X and Y are ." 1 ; " 2 /homotopy interleaved and Y and Z are .ı 1 ; ı 2 /-homotopy interleaved , then X and Z are ." 1 Cı 1 ; " 2 Cı 2 /-homotopy interleaved.
Since M is cofibrantly generated, the projective model structure on M R m exists, and, by applying a functorial fibrant replacement M !M pointwise, we get a functorial fibrant replacement so, in particular, it preserves interleavings.With this in mind, we can assume that Y 0 and Y 00 are fibrant, which implies -and this is a general fact -that we have C 2 M R m and trivial fibrations C ! Y 0 and C ! Y 00 .Using Lemma 2.2, we can pull back the interleavings we were given along the trivial fibrations, as follows: Since trivial fibrations are stable under pullback, we have that X 00 ' X and Z 00 ' Z, and since interleavings compose, we have that X 00 and Z 00 are ." 1 Cı 1 ; " 2 Cı 2 /-interleaved, as required.
We remark that the idea of using pullbacks to prove a triangle inequality appears in [17].
3 Interleavings in M R and in Ho.M/ R This section is concerned with the rectification of homotopy commutative interleavings into homotopy interleavings.In Section 3.1, we prove Theorem A, which allows one to construct, for any c > 2, a cı-homotopy interleaving out of a ı-homotopy commutative interleaving, when working with 1-persistent objects of any cofibrantly generated model category M. We think of this result as giving a multiplicative upper bound of 2 for this rectification.In Section 3.2, we give a multiplicative lower bound of 3   2   for the rectification, when M is the category of spaces.In Section 3.3, we show that Theorem A has no analogue for multipersistent spaces.

Upper bound
Let Z Â R denote the posets of integers and real numbers respectively.The inclusion where br c is the largest integer bounded above by r .Note that, given m 0 2 Z, one has a notion of m-interleaving between functors A; B W Z !C , and that i W C Z !C R preserves these interleavings.
We start with a few simplifications.For ı > 0, let M ı W R !R be given by M ı .r/ D ı r .The following lemma allows us to work with integer-valued interleavings instead of ı-interleavings, and its proof is immediate.The following lemma allows us to work with Z-indexed persistent objects instead of R-indexed ones.Here, by homotopy interleaving between Z-indexed functors we mean the obvious adaptation of the notion of homotopy interleaving to Z-indexed functors with values in a model category.
Proof Note that X is 1-interleaved with i .i.X //, as, for all r 2 R, we have r 1 Ä br c Ä r Ä br c C 1. Since i preserves interleavings and weak equivalences, it is enough to show that homotopy interleavings between Z-indexed functors with values in a cofibrantly generated model category compose, which is a straightforward adaptation of Proposition 2.3 to Z-indexed functors.
The next straightforward lemma gives us a special replacement of an object of the category M Z , with M a model category, that will be useful when lifting structure from Ho.M/ Z to M Z .Lemma 3.3 Given a model category M and X 2 M Z , there exists X 2 M Z and a weak equivalence X !X such that X .i/ is cofibrant in M for every i 2 N; for every i 0, the structure morphism Dually, we can replace Y 2 M Z by a pointwise fibrant Y whose "negative" maps are fibrations.
The following lemma will allow us to lift interleavings in Ho.M/ Z to homotopy interleavings in M Z .Lemma 3.4 Let M be a model category.The functor hW Ho.M Z / !Ho.M/ Z is essentially surjective, conservative and full.In particular, if A; B 2 M Z become isomorphic in Ho.M/ Z , then they are weakly equivalent.
Proof It is clear that the functor is essentially surjective and full, so we only prove the last property.Assume we are given X; Y 2 Ho.M Z / together with a map f W hX !hY .Thanks to Lemma 3.3, we can assume that X (resp.Y ) is pointwise cofibrant (resp.fibrant) in M, and that all the nonnegative (resp.negative) structural maps in X (resp.Y ) are cofibrations (resp.fibrations).The map f can therefore be represented as a family fOEf i g i2Z of homotopy classes of maps of M. We construct a preimage of f under h inductively, starting with a choice of representatives f 0 i for the homotopy classes OEf i .The squares commute up to homotopy, and since x 0 and y 1 are, respectively, a cofibration and a fibration, we can deform f 0 1 and f 0 1 into homotopic maps Proof We start by giving the proof for the case m D 1, as in this case the main idea is more clear.We will use the following constructions.Let eW Z !Z be the functor that maps even numbers to themselves and an odd number n to n 1.Similarly, let oW Z !Z be the functor that maps odd numbers to themselves and an even number n to n 1.
Note that, for every C 2 M Z , we have that C is .1;0/-interleaved with e .C / and with o .C /, and that e .C / and o .C / are 1-interleaved.Now assume given a 1-interleaving between hA and hB in Ho.M/ Z , that is, assume that there are morphisms f i W hA.i / !hB.i C1/ and g i W hB.i / !hA.i C1/ in Ho.M/ rendering the following diagram commutative: Consider the object C 0 2 Ho.M/ Z given by one of the two diagonal zigzags of the diagram above; namely, let where n= =m denotes the largest integer l such that l m Ä n.
We are now ready to prove the main result of this section.
Theorem A Let M be a cofibrantly generated model category, let X; Y 2 M R , and let ı > 0 2 R. If X and Y are ı-homotopy commutative interleaved , then they are cı-homotopy interleaved for every c > 2.
Proof Let c > 2 and let m 1 2 Z be large enough so that .2mC 2/=m Ä c.By Lemma 3.1, we may assume that X; Y 2 M R are m-homotopy commutative interleaved and we must show that they are cm-homotopy interleaved.Since 2m C 2 Ä mc, Lemma 3.2 reduces the problem to showing that i .X / and i .Y / are 2m-homotopy interleaved in M Z , knowing that they are m-homotopy commutative interleaved.Proposition 3.5 now finishes the proof.

Lower bound
Theorem A implies that we have d HI Ä cd HC as distances on M R , for c D 2 and for every cofibrantly generated model category M. One could wonder if the constant c D 2 can be improved.In this section we show that, when M D S, we have c 3 2 .We do this by characterizing three-object persistent spaces which are 1-homotopy interleaved with a trivial persistent space in terms of the vanishing of a Toda bracket.The idea of using Toda brackets to prove that d HI ¤ d HC is suggested in [2, Example 7.3].
The Toda bracket is an operation on composable triples of homotopy classes of pointed maps, and was originally defined to compute homotopy groups of spheres [21].We are interested in the use of Toda brackets as an algebraic obstruction to the rectification of diagrams.We now describe the fundamental procedure involved in the definition of Toda brackets, and the few properties that we are interested in; see eg [1].
Let S denote the category of pointed spaces.For concreteness, in the arguments of this section we use S D Top.Let OE3 denote the category freely generated by the graph

! ! ! :
A diagram X 2 Ho.S / OE3 , which is given by X.0/; X.1/; X.2/; X.3/ 2 Ho.S / and homotopy classes of pointed maps OEf 0 W X.0/ !X.1/, OEf 1 W X.1/ !X.2/, and OEf 2 W X.2/ !X.3/, is a bracket sequence if OEf 1 ı OEf 0 and OEf 2 ı OEf 1 are equal to the null map, that is, to the homotopy class of the constant pointed map.Let X 0 2 Ho.S / OE3 be a bracket sequence and let X 2 S OE3 be such that hX Š X 0 , which exists by Lemma 3.4.We can, and do, assume that X takes values in CW-complexes.Consider the diagram of pointed spaces and pointed maps Since X 0 is a bracket sequence, we know that there exist (pointed) homotopies filling the squares in the diagram above.For Y a pointed space, let C Y denote its reduced cone.Each pair of such homotopies gives us pointed maps ˛W CX.0/ !X.2/ and ˇW CX.1/ !X.3/ such that ˛ı i D f 1 ı f 0 W X.0/ !X.2/ and ˇı i D f 2 ı f 1 , where i is the inclusion into the cone.In particular, we have a commutative square (3-1) ıw hich, by noticing that the pushout of the top and left morphisms is a model for the reduced suspension of X.0/, gives us an element of OE †X 0 .0/;X 0 .3/,where OE ; denotes homotopy classes of pointed maps.Definition 3.6 Let X 2 Ho.S / OE3 be a bracket sequence.Consider the subset of OE †X.0/; X.3/ consisting of all elements that can be obtained using the procedure above.This is the Toda bracket of X .We say that the Toda bracket vanishes if it contains the null map.
It is well known -see eg [1, Section 1] -that the nonvanishing of a Toda bracket is an obstruction to the rectification of the bracket sequence, in the following sense.Proposition 3.7 The Toda bracket of a bracket sequence X 0 2 Ho.S / OE3 vanishes if and only if there exists X 2 S OE3 with hX Š X 0 and with f 1 ı f 0 and f 2 ı f 1 equal to the null map.
Although Toda brackets are defined for diagrams of pointed spaces, one can extend them to unpointed spaces, provided the spaces are simply connected.This is what we do now.A simply connected space is a nonempty, connected space whose fundamental groupoid is trivial.Let S sc and S sc; denote the categories of simply connected spaces and of pointed, simply connected spaces, respectively.We have the following well-known fact and corollary.Lemma 3.8 The forgetful functor U W Ho.S sc; / !Ho.S sc / is an equivalence of categories.
Corollary 3.9 If X 2 Ho.S sc; / OE3 is such that the composite of consecutive maps of U .X / are null-homotopic, then X is a bracket sequence.
Let X; X 0 2 Ho.S sc; / OE3 be such that U .X / Š U .X 0 /.Then X is a bracket sequence if and only if X 0 is; in that case, the Toda bracket of X vanishes if and only if the Toda bracket of X 0 does.Corollary 3.9 implies that, for X 2 Ho.S sc / OE3 , there is a well-defined notion of X being a bracket sequence, namely that any lift X 0 2 Ho.S sc; / OE3 is a bracket sequence; in that case, we say that the Toda bracket of X vanishes if the Toda bracket of X 0 does.Let jW S OE3 !S Z be given by extending X 2 S OE3 to the right with the singleton space and to the left with the empty space.Let 2 S OE3 be the constant singleton space.Proposition 3.10 Let X 2 S OE3 sc .Then (1) h.j.X // 2 Ho.S/ Z is 1-interleaved with h.j.// if and only if hX is a bracket sequence; (2) if h.j.X // 2 Ho.S/ Z is 1-interleaved with h.j.//, then j.X / is 1-homotopy interleaved with j. / if and only if the Toda bracket of hX vanishes.
For the converse of (2), assume that j.X / and j. / are 1-homotopy interleaved.It follows that there exists a commutative diagram of pointed spaces and pointed maps with A and B contractible and X 0 2 S OE3 such that X 0 ' X , as diagrams of unpointed spaces.It suffices to show that the Toda bracket of X 0 vanishes.For this, note that, using the diagonal morphism A ! B, we can find maps ˛W CX 0 .0/! X0 .2/and In particular, in this case, there is a diagonal filler for the square (3-1) and thus the induced map †X 0 .0/! X0 .3/ is nullhomotopic, as required.
The following lemma is clear.
Edoardo Lanari and Luis Scoccola We are now ready to prove the lower bound.
Proposition 3.12 Let M D S. If d HI Ä cd HC then c 3 2 .
Proof By Lemma 3.11 and Proposition 3.10, it suffices to find a bracket sequence X 2 Ho.S / OE3 valued in simply connected spaces such that its Toda bracket does not vanish.Examples of this are given in [21].A classical example, referenced in [2], is S 4 !S 4 !S 3 !S 3 with the first and last maps degree 2 maps, and the middle map the suspension of the Hopf map.

Impossibility of rectification in higher dimensions
In this section, we show that Theorem A has no analogue for multipersistent spaces; we thank Alex Rolle for pointing this out to us.We prove this for m D 2 and remark that a similar argument works for m > 2.
Proposition 3.14 If m D 2, there is no constant c > 0 2 R such that for all ı > 0 2 R m , if X; Y 2 S R m are ı-homotopy commutative interleaved , then they are cı-homotopy interleaved.
Let sq denote the subposet of R 2 spanned by f.0; 0/; .0;1/; .1;0/; .1;1/g, so that a functor sq !C from sq to a category C corresponds to a commutative square in C .We will use the following well-known fact, which says that a homotopy commutative diagram can have different, nonequivalent lifts.For a specific instance see eg [10].
Lemma 3.15 There exist A; B W sq !S such that hA Š hB 2 Ho.S/ sq and such that A 6 ' B.
Proof of Proposition 3.14 Given a diagram AW sq !S, consider the bipersistent space A 0 W R 2 !S such that A 0 .r;s/ D ∅ whenever r or s are negative, A 0 .r;s/ D A.br c; bsc/ whenever 0 Ä r; s < 2, and A 0 .r;s/ is the singleton space whenever 0 Ä r; s and 2 Ä max.r; s/.Let A; B W sq ! S. Note that if .0;0/ Ä ı < 1 2 ; 1 2 2 R 2 and A 0 ; B 0 2 S R 2 are ı-homotopy interleaved, then we have A ' B. To prove the result, it is enough to show that there exist bipersistent spaces X; Y 2 S R 2 that are 0-homotopy commutative interleaved, ie such that hX Š hY , which are not ı-homotopy interleaved for any 0 Ä ı < 1 2 ; 1 2 2 R 2 .In order to do this, we can let A and B be as in Lemma 3.15 and take X D A 0 and Y D B 0 .

Projective cofibrant persistent simplicial sets
The purpose of this section is to characterize projective cofibrant persistent simplicial sets as filtered simplicial sets (Proposition 4.5).We work with simplicial sets indexed by an arbitrary poset .P; Ä/.Definition 4.1 A P -filtered simplicial set (filtered simplicial set when there is no risk of confusion) is a simplicial set X equipped with functions ˇn W X n !P , satisfying ˇn 1 .di .// Ä ˇn./ for every n 1, 2 X n , and boundary map ˇnC1 .si .// Ä ˇn./ for every n 0, 2 X n , and degeneracy map When there is no risk of ambiguity, we denote the filtered simplicial set .X; ˇ/ simply by X .Definition 4.2 Given a filtered simplicial set .X; ˇ/, define a persistent simplicial set 1 .X; ˇ/ 2 sSet P such that for r 2 P we have 1 .X; ˇ/.r / n D f 2 X n W ˇn. / Ä r g, with faces and degeneracies given by restricting the ones of X .By a standard abuse of language, We say that a persistent simplicial set is a filtered simplicial set if it is isomorphic to y Y for Y a filtered simplicial set.
The following result is a characterization of filtered simplicial sets among persistent simplicial sets by means of easily verified point-set conditions.
Lemma 4.3 A persistent simplicial set X 2 sSet P is a filtered simplicial set if and only if the following conditions are satisfied : (1) The structure morphism X.r / !X.r 0 / is a monomorphism for every r Ä r 0 in P .In particular, up to isomorphism, we may, and do, assume that X.r / is a subsimplicial set of X.r 0 /.
(2) For every simplex 2 S r 2P X.r /, the set ft 2 P W 2 X.t/g has a minimum.
Proof The only nontrivial part is that if X satisfies the two conditions in the statement then it is filtered.Set Y D S r 2P X.r /, which makes sense thanks to condition (1).Given 2 Y n , define ˇn./ WD minft 2 P W 2 X.t/g, which is well defined thanks to condition (2).We then have X Š y Y .The rest of the proof is clear.
The proof of the following lemma a straightforward application of Lemma 4.3.We use the term cell attachment to indicate any pushout of a generating cofibration r ˇ@ n !r ˇn .
Lemma 4.4 (1) A retract of a filtered simplicial set is filtered.
(2) If the domain of a cell attachment is a filtered simplicial set, then the codomain is too.
(3) Let be a limit ordinal and let X W ! sSet P be a diagram of persistent simplicial sets, where for each < we have that the map X !X C1 is a cell attachment.If X is a filtered simplicial set for every < , then X D colim < X is a filtered simplicial set.
The recognition principle for projective cofibrant persistent simplicial sets is now a consequence of Lemma 4.4 and the fact that the cofibrant objects in a cofibrantly generated model category are precisely the retracts of transfinite compositions of cell attachments [12, Proposition 2.1.18(b)].
Proposition 4.5 A persistent simplicial set is filtered if and only if it is projective cofibrant.
In practice, many of the persistent spaces relevant to topological data analysis are filtered simplicial sets.
Example 4.6 The Vietoris-Rips complex associated to a metric space .X; d X /, usually defined to be a persistent simplicial complex, can be turned into a persistent simplicial set by choosing a total order on X .It follows directly from its definition that this persistent simplicial set is filtered.Other examples of this form include the Čech complex and the filtrations of [6].
An example of a filtered multipersistent simplicial set is the following.Given a metric space .X; d X / together with a real-valued function f W X !R, one can construct a bifiltered simplicial set as follows.For each s 2 R, consider X s D f 1 .1; s and let F s;r be the Vietoris-Rips complex of X s at scale r .
We remark that persistent simplicial sets whose structure maps are monomorphisms are not necessarily filtered.This happens in practice when the same simplex "appears at different times", that is, when condition (2) in Lemma 4.3 is not satisfied.Examples of this include the degree-Rips bifiltration [16], and Vietoris-Rips applied to the kernel density filtration of [19].
5 Interleaving in Ho.S R m / and in homotopy groups In this section, we prove Theorem B. We start by defining the notions of persistent homotopy groups of a persistent space, and of morphism inducing an interleaving in persistent homotopy groups.The notion of persistent homotopy group we use is essentially the same as that of Jardine [13].
We model the n th homotopy group n .W; w/ of a pointed space .W; w/ by the set of pointed homotopy classes of pointed maps from the n-dimensional sphere S n into W .
The n th persistent homotopy group of X based at x is the persistent group n .X; x/W R m !Grp that is trivial at s r , and that is n .X.s/; ' X r;s .x// 2 Grp at s r .
Note that n is functorial for every n 2 N. Definition 5.2 Let "; ı 0 2 R m .Assume given a homotopy class of morphisms OEf W X 0 !Y 0" 2 Ho.S R m /.Let X 0 ' X be a cofibrant replacement, let Y 0 ' Y be a fibrant replacement, and let f W X !" Y be a representative of f .We say that OEf induces an .";ı/-interleaving in homotopy groups if the induced map 0 .f/W 0 .X / !" 0 .Y / is part of an .";ı/-interleaving of persistent sets, and if for every r 2 R m , every x 2 X.r /, and every n 1 2 N, the induced map n .f/W n .X; x/ !" n .Y; f .x// is part of an .";ı/-interleaving of persistent groups.
It is clear that the definition above is independent of the choices of representatives.
A standard result in classical homotopy theory is that a fibration of Kan complexes inducing an isomorphism in all homotopy groups has the right lifting property with respect to cofibrations [9,Theorem I.7.10].An analogous, persistent, result (Corollary 5.13), says that, for a fibration of fibrant objects inducing a ı-interleaving in homotopy groups, the lift exists up to a shift, which depends on both ı and on a certain "length" n 2 N associated to the cofibration.To make this precise, we introduce the notion of n-dimensional extension.
Definition 5.3 Let A; B 2 S R m and let n 2 N. A map j W A ! B is an n-dimensional extension (of A) if there exists a set I , a family of tuples of real numbers fr i 2 R m g i2I , and commutative squares of the form depicted on the left below, that together give rise to the pushout square on the right: Here, @D n ,! D n stands for S n 1 ,! D n if S D Top, and for @ n ,! n if S D sSet.
A single-dimensional extension is an n-dimensional extension for some n 2 N.
Definition 5.4 Let ÃW A ! B be a projective cofibration of S R m and let n 2 N. We say that Ã is an n-cofibration if it factors as the composite of n C 1 maps f 0 ; : : : ; f n , with f i an n i -dimensional extension for some n i 2 N. We say that A 2 S R m is n-cofibrant if the map ∅ !A is an n-cofibration.
The next lemma, which follows directly from Proposition 4.5, gives a rich family of examples of n-cofibrant persistent simplicial sets.Recall that a simplicial set is n-skeletal if all its simplices in dimensions above n are degenerate.
Lemma 5.5 Let A 2 sSet R m and let n 2 N.If A is projective cofibrant and pointwise n-skeletal , then it is n-cofibrant.
Example 5.6 The Vietoris-Rips complex VR.X / of a metric space X , as defined in Example 4.6, is n-cofibrant if the underlying set of X has finite cardinality jX j D nC1.
If one is interested in persistent (co)homology of some bounded degree n, then one can restrict computations to the .nC1/-skeleton of a Vietoris-Rips complex, which is .nC1/-cofibrant.
A result analogous to Lemma 5.5, but for persistent topological spaces, does not hold, as cells are not necessarily attached in order of dimension.This motivates the following definition.
Definition 5.7 Let n 2 N. A persistent topological space A 2 Top R m is an ndimensional persistent CW-complex if the map ∅ !X can be factored as a composite of maps f 0 ; : : : ; f n , with f i an i -dimensional extension.
Example 5.8 The geometric realization of any n-cofibrant persistent simplicial set is an n-dimensional persistent CW-complex.Lemma 5.9 Every n-dimensional persistent CW-complex is n-cofibrant.
We now make precise the notion of lifting property up to a shift.Definition 5.10 Let i W A ! B and p W Y !X be morphisms in S R m and let ı 0. We say that p has the right ı-lifting property with respect to i if for all morphisms A ! Y and B ! X making the square on the left below commute, there exists a diagonal ı-morphism f W B ! ı Y rendering the diagram commutative.The diagram on the left is shorthand for the one on the right: We now prove Lemma 5.12, an adaptation of a result of Jardine, which says that fibrations inducing interleavings in homotopy groups have a shifted right lifting property, as defined above.The main difference is that we work in the multipersistent setting.We use simplicial notation and observe that the corresponding statement for persistent topological spaces follows from the simplicial one by using the singular complexrealization adjunction.We recall a standard, technical lemma whose proof is given within that of eg [9, Theorem I.7.10].
Lemma 5.11 Consider a commutative square of simplicial sets (5-1) where p is a Kan fibration between Kan complexes.If there is commutative diagram like the one on the left below, for which the lifting problem on the right admits a solution, .id @ n f1g/ ˛h p hı.id @ n f0g/ p g gı.id n f0g/ .id n f1g/ ťhen the initial square (5-1) admits a solution.
Lemma 5.12 (cf [13, Lemma 14]) Let ı 0, and let f W X !Y 2 S R m induce a .0;ı/-interleaving in homotopy groups.If X and Y are projective fibrant and f is a projective fibration, then f has the right 2ı-lifting property with respect to boundary inclusions r ˇ@D n !r ˇDn for every r 2 R m and every n 2 N.
Proof Consider a commutative diagram as on the left below, which corresponds to the one on the right: (5-2) We must find a 2ı-lift for the diagram on the right.The proof strategy is to appeal to Lemma 5.11 to simplify ˛, then prove that at the cost of a ı-shift we can further reduce ˛to a constant map, and then show that the simplified lifting problem can be solved at the cost of another ı-shift.So we end up with a 2ı-lift, as in the statement.We proceed by proving the claims in opposite order.
We start by showing that (5-2) can be solved up to a ı-shift whenever ˛is constant.Let us assume that ˛is of the form ˛D for some 2 X.r / 0 .Since, then, ˇrepresents an element OEˇ 2 n .Y .r/; /, there exists a map ˛0 W n !X.r C ı/ whose restriction to @ n is constant on 2 X.r / 0 , and such that there is a homotopy hW ˇ' p˛0 relative to @ n .We can thus consider where H is a diagonal filler for the right-hand side square, which exists since the middle vertical map is a trivial cofibration of simplicial sets and p r Cı is a Kan fibration by assumption.The composite map H ı id n f1g is a lift for (5-2).
We now assume that ˛is of a specific, simplified form, and prove that, up to a ı-shift, we can reduce the lifting problem (5-2) to the case in which ˛is constant.Let us assume that d i .˛/D 2 X.r / 0 for every 0 < i Ä n, and set ˛0 D d 0 .˛/.Then ˛0 represents an element OE˛0 2 n 1 .X.r /; /, with the property that pOE˛0 D 0 2 n 1 .Y .r/; /.Since p induces a .0;ı/-interleaving in homotopy groups, witnessed by a homotopy h 0 W n 1 1 !X.r C ı/, constant on @ n 1 .If we set h 0 i D W n 1 !X.r C ı/ for every 0 < i Ä n 1 and h 0 0 D h 0 , we get a map h 0 W @ n 1 !X.r C ı/.We can now extend is such that h 0 ı .id@ n f0g/ D , so, thanks to Lemma 5.11, we have reduced this case to the case in which ˛is constant.
To conclude, we must show that we can reduce the original lifting problem (5-2) to one in which all but the 0 th faces of ˛are constant on a point 2 X.r / 0 .Let K W ƒ n 0 1 !ƒ n 0 be the homotopy that contracts the simplicial horn onto its vertex 0, which determines a diagram with k 1 D ˛ı j ı K, where j W ƒ n 0 !@ n is the inclusion of the horn into the boundary.We can now extend the map .˛;k 1 /W .@n f1g/ [ .ƒn 0 1 / !X.r / to a homotopy k W @ n 1 !X.r /.Similarly, we extend the map .id n f1g/ ǒbserving that ˛0 WD k j@ n f0g satisfies d i .˛0/D for 0 < i Ä n, and appeal to Lemma 5.11.
Corollary 5.13 Let ı 0 and let f W X !ı Y induce a ı-interleaving in homotopy groups.If X and Y are projective fibrant and f is a projective fibration, then f has the right .4.nC1/ı/-lifting property with respect to n-cofibrations for all n 2 N.
Proof By assumption, f W X !Y ı induces a .0;2ı/-interleaving in all homotopy groups.Now, an n-cofibration can be written as a composite of nC1 single-dimensional extensions, and any shift of a single-dimensional extension is again a single-dimensional extension, so it is enough to show that f has the right 4ı-lifting property with respect to single-dimensional extensions.
A single-dimensional extension is the pushout of a coproduct a i2I r i ˇ.@D n / ! a i2I r i ˇDn ; so it suffices that f has the right 4ı-lifting property with respect to coproducts of that form, which follows from Lemma 5.12 and the universal property of coproducts.
We are ready to prove Theorem B. Proof By Lemmas 5.5 and 5.9, X and Y are d-cofibrant.Let OEf W X !Y ı be as in the statement.Since OEf is a morphism in the homotopy category, we begin by choosing a convenient representative of it.We let p W X 0 !Y 0 be a projective fibration between projective fibrant objects such that there exist trivial cofibrations i W X !X 0 and j W Y !Y 0 with OEp ı OEi D OEj ı OEf , in Ho.S R m /.
On the one hand, note that, by construction, p .4dC3/ı ı g ı j D p .4dC3/ı ı g 0 D j , so, since OEj is an isomorphism, it follows that OEp .4dRemark 5.14 Together, Theorems A and B imply a version of the persistent Whitehead conjecture, which we recall as Conjecture 5.15.Our result is, in a sense, stronger than the one conjectured, since Theorem B, which addresses part (i) of the conjecture, applies to arbitrary multipersistent spaces.In another respect, our result is slightly weaker, as the conjecture is stated for cofibrant, pointwise CW-complexes, which does not necessarily imply being a persistent CW-complex in our sense.We believe that this is not an issue, as many of the cofibrant, pointwise CW-complexes persistent topological spaces that appear in applications are in fact persistent CW-complexes, as they are usually the geometric realization of a filtered simplicial complex.

Theorem B Fix m 1 2 N
and d 2 N. Let X; Y 2 S R m be (multi )persistent spaces that are assumed to be projective cofibrant and d-skeletal if S D sSet, or persistent CW-complexes of dimension d if S D Top.Let ı 0 2 R m .If there exists a morphism in the homotopy category X !Y ı 2 Ho.S R m / that induces ı-interleavings in all homotopy groups, then X and Y are .4.dC1/ı/-interleaved in the homotopy category.Together, Theorems A and B give a positive answer to a version of the persistent Whitehead conjecture [2, Conjecture 8.6] (see Remark 5.14 for a discussion and Conjecture 5.15 for a statement of the conjecture).
Quillen equivalence between cofibrantly generated model categories, then the induced Quillen equivalence[11, Theorem 11.6.5]M R m N R m
which render the above squares commutative.Inductively, we can iterate this procedure to find the desired preimage of f under h.The next result is the main rectification step involved in lifting interleavings in Ho.M/ Z to homotopy interleavings in M Z .Proposition 3.5 Let M be a model category and let A; B 2 M Z .Let m 1 2 Z.If hA and hB are m-interleaved in Ho.M/ Z , then A and B are 2m-homotopy interleaved in M Z .

Conjecture 5 . 15 [ 2 , Conjecture 8 . 6 ]
Suppose we are given connected , cofibrant X; Y W R !Top, with each X.r / and Y .r/ CW-complexes of dimension at most d, and f W X !Y ı inducing a ı-interleaving in all homotopy groups.Then there is a constant c, depending only on d, such that (i) f induces a cı-interleaving in the homotopy category Ho.Top R /;(ii) X and Y are cı-homotopy interleaved.Algebraic & Geometric Topology, Volume 23 (2023)The subscription price for 2023 is US $650/year for the electronic version, and $940/year ( C $70, if shipping outside the US) for print and electronic.Subscriptions, requests for back issues and changes of subscriber address should be sent to MSP.Algebraic & Geometric Topology is indexed by Mathematical Reviews, Zentralblatt MATH, Current Mathematical Publications and the Science Citation Index.Algebraic & Geometric Topology (ISSN 1472-2747 printed, 1472-2739 electronic) is published 9 times per year and continuously online, by Mathematical Sciences Publishers, c/o Department of Mathematics, University of California, 798 Evans Hall #3840, Berkeley, CA 94720-3840.Periodical rate postage paid at Oakland, CA 94615-9651, and additional mailing offices.POSTMASTER: send address changes to Mathematical Sciences Publishers, c/o Department of Mathematics, University of California, 798 Evans Hall #3840, Berkeley, CA 94720-3840.AGT peer review and production are managed by EditFlow ® from MSP. PUBLISHED BY mathematical sciences publishers nonprofit scientific publishing http://msp.org/© 2023 Mathematical Sciences Publishers disjointly supported Hamiltonians on symplectically aspherical manifolds YANIV GANOR and SHIRA TANNY 733 Realization of graded monomial ideal rings modulo torsion TSELEUNG SO and DONALD STANLEY 765 Nonslice linear combinations of iterated torus knots ANTHONY CONWAY, MIN HOON KIM and WOJCIECH POLITARCZYK 803 Rectification of interleavings and a persistent Whitehead theorem EDOARDO LANARI and LUIS SCOCCOLA 833 Operadic actions on long knots and 2-string links ETIENNE BATELIER and JULIEN DUCOULOMBIER 883 A short proof that the L p -diameter of Diff 0 .S; area/ is infinite MICHAŁ MARCINKOWSKI 895 Extension DGAs and topological Hochschild homology HALDUN ÖZGÜR BAYINDIR 933 Bounded cohomology of classifying spaces for families of subgroups KEVIN LI Two of the main model structures of interest to us are the Quillen model structure on Top, the category of topological spaces [12, Chapter 1, Section 2.4], and the Kan-Quillen model structure on sSet, the category of simplicial sets [12, Chapter 3].We recall that the geometric realization functor j jW sSet !Top is left adjoint to the singular functor SingW Top !sSet, and that, together, they form a Quillen equivalence [12, Chapter 1, Section 1.3, Theorem 3.6.7].For completeness, we mention that there is a subcategory Top CGWH Â Top, the category of compactly generated weakly Hausdorff topological spaces (called compactly generated spaces in [12, Definition 2.4.21]), that is often used instead of Top.The Quillen model structure on Top restricts to a model structure on Top CGWH , and the inclusion Top CGWH !Top is part of a Quillen equivalence Remark 3.13 Proposition 3.12 implies in particular that d HI ¤ d HC .As mentioned in the introduction, we know that we have d HI d IHC d HC , so it is natural to wonder whether we have d HI ¤ d IHC or d IHC ¤ d HC , or both.We leave these as open questions.
Theorem B Let X; Y 2 S R m be persistent spaces that are assumed to be projective cofibrant and d-skeletal if S D sSet, or persistent CW-complexes of dimension at most d if S D Top.Let ı 0 2 R m .If there exists a morphism in the homotopy category X !Y ı 2 Ho.S R m / that induces ı-interleavings in all homotopy groups, then X and Y are .4.dC1/ı/-interleaved in the homotopy category.