Optimising the topological information of the A_∞-persistence groups

06/19/2017 ∙ by Francisco Belchí, et al. ∙ 0

Persistent homology typically studies the evolution of homology groups H_p(X) (with coefficients in a field) along a filtration of topological spaces. A_∞-persistence extends this theory by analysing the evolution of subspaces such as V := Ker Δ_n_| H_p(X)⊆ H_p(X), where {Δ_m}_m≥1 denotes a structure of A_∞-coalgebra on H_*(X). In this paper we illustrate how A_∞-persistence can be useful beyond persistent homology by discussing the topological meaning of V, which is the most basic form of A_∞-persistence group. In addition, we explore how to choose A_∞-coalgebras along a filtration to make the A_∞-persistence groups carry more faithful information.

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Introduction

Persistent homology [5, 6] computes the (persistent) Betti numbers of a sequence of topological spaces and continuous maps

created by varying a parameter such as time, thickness, intensity, height, etc. Depending on the context, this can allow us to discover highly non-linear structure in data or to compute novel geometric descriptors of shapes. For instance, G. Carlsson et al. considered a point cloud dataset built from 3 by 3 high-contrast patches from grey-scale natural images and studied an unknown space

on which the points accumulated with high density. They used persistent homology to estimate some Betti numbers of

, but went further to find that the 2-skeleton of formed a Klein bottle [4]. This extra knowledge was then used as starting point to develop a dictionary for texture representation [18]. This is a great example of how important it can be to find persistent topological information beyond the level of Betti numbers. -persistence [2] aims at doing so in a semi-automated way by studying persistent topological information at the level of -structures – algebraic constructions that can encode attributes related to cup product and higher order Massey products (see Fig. 1).

Figure 1. Persistence applied to point cloud datasets: (a) Betti numbers contribute valuable topological information about solid objects, able to distinguish e.g., a circumference from a disc. Persistent homology adapts these invariants to the study of point cloud datasets, so that a sample from a circumference can be told apart from that of a disc. (b) -structures provide a much more detailed structural description than that of Betti numbers. They contain all the information of the cup product, and hence can tell apart e.g., a torus from a wedge of spheres . On top of that, they contain information related to (higher order) Massey products, allowing us to also distinguish e.g., 3 unlinked rings from the Borromean rings (see Fig. 3). (c) In this context, the aim of -persistence is to use the strengths of -(co)algebras to sharpen the tool of persistent homology, e.g., to allow the distinction of more involved point cloud datasets.

We plan to work with -coalgebras on the homology of a space and with -algebras on its cohomology . These structures codify a good deal of topological information of , as this paper will illustrate. For all concepts below related to -structures, see §1. Given two homotopy equivalent spaces, the transferred -coalgebras they induce coincide up to isomorphism. Hence, ideally, we would like to study the persistence of this whole isomorphism class of -structures. Unfortunately, this class is too large to compute in general, so we end up having to sacrifice some of this structure in return for computability. More concretely, after choosing a field of coefficients, we fix a transferred -coalgebra for each space in

and think of defining a vector subspace

for every , so that the following two conditions hold:

  1. contains enough information from the -coalgebra to be useful.

  2. is simple enough to allow a feasible persistence computation.

Persistent homology encodes in a barcode the evolution along of the vector space . By (2) we mean that, similarly, we should be able to encode in a barcode the evolution along of the subspace . In [2], we used zigzag persistence [3] to prove that satisfies (2), where here and henceforth denotes the restriction of the map to the subspace .

In this article, we do the following. On the one hand, we draw a more comprehensive picture of ’s trade-off between simplicity and retained information from the -structure. To do so, we show both what we lose by not using the whole isomorphism class of -structures, and what we still gain with respect to Betti numbers. Specifically, we show that is not a homotopy invariant in the most general setting (Ex. 3.1), but this dimension can still recover in some cases information not readily available in the Betti numbers or even the cup product (Thm. 3.2, Cor. 3.3 , Prop. 3.5 and 3.6). On the other hand, recall that the presence of a homological feature in the sequence may sometimes be represented in a zigzag barcode by two or more different, totally independent intervals. This can mislead us to believe that they may be detecting more than one feature, as exemplified in [23, §5.6.6]. As -persistence uses a tweaked version of a zigzag barcode (see [2, Def. 2.8]), in §4 we will show how this ambiguity affects -persistence and discuss a way to bypass this issue.

This work is organised as follows: in §1 we provide some background on -structures. In §2 we recall, from an algebraic point of view, the ordinary and versions of barcodes. These are tools used to encode persistence information. In addition, we give an alternative proof of Lemma 2.4, which has as corollary the barcode decomposition theorem of persistent homology (Thm. 2.3). In §3 we illustrate when -persistence can be useful beyond persistent homology by discussing the topological meaning of the measurements and their duals in cohomology (Ex. 3.1, Prop. 3.5 and 3.6, Thm. 3.2 and Cor. 3.3 ). The results in this section can be used to choose the right to study the persistence of and they also suggest a new persistence approach to links. In §4 we exhibit how the aforementioned zigzag ambiguity problem affects -persistence and propose a way to deal with it. This involves showing that we can sometimes make coherent choices of -structures on the sequence so that what -persistence has to decompose is a persistence module (Thm. 4.2). We then prove the barcode decomposition theorem of -persistence for the case of such coherent choices of -structure (Thm. 4.3) to stress how much things simplify by bypassing the need for zigzag machinery. We leave some of the proofs for §5 and finish with some conclusions in §6.

The results in §3 and §4 should be of interest both to algebraic topologists and to more applied scientists involved in data analysis in some way.

Let us fix some notation and assumptions for the rest of the paper. For the basic notions of algebraic topology used (such as cup product) we refer the reader to classics such as [9]. Throughout this paper, we will always work over a field of coefficients which we will usually omit from the notation. For instance, will denote the homology of with coefficients in . Every topological space considered will be assumed to have finite dimensional homology groups in all degrees, . We will denote by

a finite sequence of topological spaces and continuous maps and, for any integers and , we will denote by

the linear maps induced in homology by the composition

1. -structures

Looking for algebraic structures that include all the information of the homology groups and even of the standard cohomology algebra given by the cup product, and provide still more detailed homological information, we naturally bump into -structures.

Definition 1.1.

An -coalgebra structure on a graded vector space is a family of maps

of degree such that, for all , the following Stasheff identity holds:

The identities for state that if is an -coalgebra, then is a differential on and the comultiplication is coassociative up to the chain homotopy . Moreover, any differential graded coalgebra (DGC henceforth) can be viewed as an -coalgebra by setting and for all An -coalgebra is called minimal if

Definition 1.2.

A morphism of -coalgebras

is a family of maps

of degree , such that for each , the following identity holds:

We say that the morphism of -coalgebras is:

  • an isomorphism if is an isomorphism,

  • a quasi-isomorphism if induces an isomorphism in homology.

There are some trivial -coalgebra structures one can always endow a graded vector space with, such as the one given by for all . This structure does not give in general any new information, so instead, we will focus our attention in a special kind of -coalgebras we will refer to as the transferred ones. As we will see between this section and §3, out of all the -coalgebra structures one could endow with, the transferred ones can give particularly meaningful information about the homotopy type of . For instance, we mentioned that measures the degree to which the coassociativity of can be relaxed in an -coalgebra . However, even if is strictly coassociative, may be non-trivial, and in a transferred -coalgebra, this non-triviality can give crucial information (see §3).

Definition 1.3.

We will say that an -coalgebra on the homology of a space is a transferred -coalgebra (induced by ) if it is minimal and quasi-isomorphic to the -coalgebra

where denotes the singular chain complex of and denotes an approximation to the diagonal. We will drop the ’induced by from the notation when no confusion is possible.

Analogously, in a reduced homology setting, we will refer to an -coalgebra as a transferred one if it is minimal and quasi-isomorphic to the induced -coalgebra at the reduced chains level

An immediate consequence of Definition 1.3 is that all transferred -coalgebras on induced by are isomorphic.

In the proof of Prop. 3.5 we will use the following remark.

Remark 1.4.

For every space , there always exist transferred -coalgebras on induced by , which can be computed in several ways, most of them amounting to an application of the Homotopy Transfer Theorem [11, 12] (hence the name transferred -coalgebras). This can be used as an algorithm that takes as input a diagram of the form

(1.1)

in which is a chain complex, is a DGC with comultiplication , and the degree 0 chain maps and and the degree 1 chain homotopy make the following hold: , and is a chain homotopy between and , i.e., . The output of the algorithm includes an explicit minimal -coalgebra structure on with and such that, if , then for all .

The dual notion of an -coalgebra is that of an -algebra. Depending on the tools more suitable to making the computations, it can be more convenient to work with one notion or the other.

Definition 1.5.

An -algebra structure on a graded vector space is a family of maps

of degree such that, for all , the following Stasheff identity holds:

Everything said for -coalgebras dualizes to -algebras. In an -algebra , measures the associativity of but even if is strictly associative, may be non-trivial and give useful information (see §3). The Homotopy Transfer Theorem also works for differential graded algebras (DGA henceforth) and -algebras, and we can define transferred -algebras on the cohomology of a space, , just as in Definition 1.3, using the cup product instead of an approximation to the diagonal.

In particular, for any transferred -algebra on , coincides with the cup product. Hence, transferred -structures encode all the information in the homology groups of and in its cohomology algebra as well, but there is more. For instance, in [2, Thm. 1.3] we exhibited a way to build pairs of spaces with isomorphic homology groups and isomorphic cohomology algebras but non-isomorphic transferred -coalgebras (on their homology), and T. Kadeishvili proved in [10, Prop. 2] that under mild conditions on a topological space , any transferred -algebra on its cohomology determines the cohomology of its loop space, , whereas the cohomology ring of alone does not.

2. Barcodes

In persistence, we are interested in finding out how the topology evolves along the sequence of topological spaces and continuous maps created by varying a parameter such as time, thickness, intensity, height, etc. This is done by applying algebraic-topology constructions to and decomposing the result into the smallest pieces possible in a certain sense, obtaining a graphical representation called a barcode. In this section we recall, from an algebraic point of view, the notions of barcode in persistent homology [5, 6] and -persistence [2]. In addition, we give an alternative proof of the decomposition theorem of persistent homology.

Definition 2.1.

For every and , the persistent homology group of the sequence between and is defined as the vector space

and its corresponding persistent Betti number is defined as

Definition 2.2.

A persistence module is a finite sequence of finite dimensional vector spaces and linear maps between them of the form

The Fundamental Theorem of Persistent Homology can be stated as follows:

Theorem 2.3.

[5, §3] (Fundamental Theorem of Persistent Homology) For any integer , there exists a unique multiset of intervals of the form for , and intervals of the form for , such that for all  , the dimension equals the number of intervals in (counted with multiplicities) which contain the interval .

In particular, equals the number of intervals in (counted with multiplicities) which contain the integer .

The multiset in Thm. 2.3 is known as the barcode of . Thm. 2.3 is a consequence of the following result:

Lemma 2.4.

[5, §3] Let be a persistence module and set

Then there exists a unique multiset of intervals of the form for , and intervals of the form for , such that for all  , the dimension can be computed as the sum of the multiplicity of each interval in that contains .

G. Carlsson and A. Zomorodian first proved Lemma 2.4 and therefore Thm. 2.3 in its full generality in [5, §3] with a slightly different notation and vocabulary. They made use of the structure theorem of finitely generated modules over Principal Ideal Domains. Next we give an alternative proof of Lemma 2.4 (and therefore of Thm. 2.3) making use, instead, of the following well-known fact in linear algebra:

Lemma 2.5.

(Frobenius inequality on matrix ranks) Let and be matrices such that the products and are defined. Then

Here is an alternative proof of Lemma 2.4:

Proof.

Let us set

Let us assume that there existed a multiset consisting of:

  • a number of intervals of the form for all , and

  • a number of intervals of the form for all ,

such that, for all  ,

Then, an inclusion-exclusion-type argument shows that, for all , is forced to be

(2.1)

Hence, if exists, it must be unique. On the other hand, the existence of amounts to holding for every . Finally, notice that Lemma 2.5 implies that these inequalities hold, proving the existence of . ∎

Thm. 2.3 then follows from applying Lemma 2.4 to the persistence module

For the rest of this section, choose a transferred -coalgebra structure on the homology of each space in .

Definition 2.6.

For every , and , the -persistent group between and is the vector space

These are the -persistence groups in terms of transferred -coalgebras in homology.

In -persistence [2], we study the evolution of vector subspaces such as . The problem is that the maps do not restrict, in general, to maps (see for instance [2, Thm. 3.1]), therefore not producing a persistence module Despite that, we can still completely understand the persistent groups in Def. 2.6 by using zigzag techniques [3]. Indeed, here is the counterpart of Thm. 2.3:

Theorem 2.7.

[2, Cor. 2.9, Def. 2.8], For any pair of integers and , there exists a unique multiset of intervals of the form for , such that for all  , the dimension equals the number of intervals in (counted with multiplicities) which contain the interval .

In particular, equals the number of intervals in (counted with multiplicities) which contain the integer .

The multiset in Thm. 2.7 is called the -barcode of . The construction of this barcode involves zigzag decompositions and in §4 we will deal with a hazard this presents.

3. Topological meaning of -persistence groups

It is well known that a torus, a Klein bottle and something as simple as a wedge of spheres have isomorphic homology groups with coefficients, but that the cup product tells these 3 spaces apart. These objects are not only mathematical amusements – they can actually model datasets. For instance, a quasi-periodic signal can produce a sliding window embedding which is dense in a torus [17] and high-contrast patches in grey-scale images can produce a point cloud which is dense in a space whose 2-skeleton is a Klein bottle [4]. Since persistence can be used to study these datasets (as recalled in Fig. 1), it is therefore relevant to have a persistence approach to cup product. A particular case of -persistence yields one such approach, since the multiplication in a transferred -algebra is precisely the cup product. Inasmuch as this product is well understood, in this section we will focus on understanding what -persistence can achieve with the higher operations and , for .

The key to understanding the persistent homology groups is to know what means. Similarly, the key to understanding -persistence groups as in Definition 2.6, , is to know what means. The meaning of the former is easy to grasp, since reduces to the Betti number . The meaning of the later is a little more involved, so we now explore some of the topological information encoded in the dimension of (and dually the dimension of the cokernel of the map ). We will start by showing that, in the worst case, different -structures transferred by the same space can lead to different values of if (Ex. 3.1). However, we identify situations in which this does cannot happen. In such situations, these values provide information on the homotopy type of topological spaces, beyond that of the standard cohomology algebra (Thm. 3.2, Cor. 3.3 and Prop. 3.5 and 3.6).

Some results in this section involve folklore notions of either rational homotopy theory or homological perturbation theory, but to the best of our knowledge, there is no proof in the literature of any of the results we will prove here.

Let us start by stating that numbers like can give different values depending on the chosen transferred -structure.

Example 3.1.

Let be wedge of the complex projective plane and a 7-sphere

Then, the reduced rational homology of , , admits two (isomorphic) transferred -coalgebra structures and such that

This example (which we explain in detail in §5) could lead some to think that there is no use in using quantities such as to distinguish topological spaces, but there is no need to throw the baby out with the bath water. The rest of this section is devoted to prove that in some circumstances these numbers can still tell apart non-homotopically-equivalent topological spaces whose homology groups and even cohomology algebras are isomorphic.

We continue by stating in Thm. 3.2 (which we prove in §5) that isomorphic -coalgebras induce the same numbers for several values of .

Theorem 3.2.

Let and be two isomorphic minimal -coalgebra structures on a graded vector space such that for all . Let us set

and

Then and

for all integers and .

Applying Thm. 3.2 to transferred -structures on yields sufficient conditions for the numbers to be homotopy invariants. We state this as the following corollary:

Corollary 3.3.

Let be a transferred -coalgebra structure on the reduced homology of a space , and let us set

Then, and the numbers (for integers and ) are independent of the choice of transferred -coalgebra structure on . Moreover, since homotopy equivalent spaces induce isomorphic transferred -coalgebras, and every such are invariants of the homotopy type of .

Using Thm. 3.2, one can infer similar statements to Cor. 3.3 for transferred -coalgebras in (non-reduced) homology and dualize them to transferred -algebras in cohomology (reduced or not). Cor. 3.3 shows -operations that vanish up to a certain so that the possible non-vanishing at level becomes important. Following this idea, we now show how the numbers

can sometimes find extra topological information. We do this by using the connection between -structures, Massey products and higher linking numbers. We will consider links in (i.e., collections of embeddings of a circumference into which do not intersect) and embeddings of arbitrary spheres in a sphere of a higher dimension. Let us start with 2-component links: the unlinked of Fig. 1(a) and the Hopf link of Fig. 1(b).

(a)
(b)
Figure 2. Betti numbers cannot distinguish the two unlinked annuli of (a), from the Hopf link (two disjoint annuli linked as in (b)). However, the standard cohomology algebra on and can tell them apart. In particular, since all the information in the stardard cohomology algebra is contained in any transferred -algebra, we conclude that we can distinguish these two links by using -algebras.

It is easy to see that for all and, using Alexander Duality, holds as well. Recall that the cup product in the complement in of two knots can detect their linking number [20, §5.D]. In this case, the fact that has a 0 linking number but does not, implies that

whereas

Hence, the cup product, which we recall is the operation in any transferred -algebra , can tell apart these two links.

Let us look now at two 3-component links: the unlinked of Fig. 2(a) and the Borromean rings of Fig. 2(b). The latter satisfy that if we remove any of the rings, the other two become unlinked, but the three of them considered at once cannot be pulled apart. It is easy to see that and holds for all . Moreover, for both and , if we remove one of the circumferences, the other two become unlinked. Hence, since the remaining two circumferences are unlinked, their linking number is 0 and the cup product of the Alexander duals of those fundamental classes vanishes. The fact that this pairwise unlinkedness holds for the three circumferences implies that the cup product

is 0 for both and . Indeed, we need a 3-fold operation such as the triple Massey product to detect the 3-fold linkedness of the Borromean rings.

The triple Massey product [24] on the cohomology algebra of a space, , is defined only for those triples of classes such that

for which such triple Massey product is a specific subset

(a)
(b)
Figure 3. Betti numbers and the cup product cannot distinguish the three unlinked annuli of (a), from the Borromean rings (three disjoint annuli linked as in (b)). However, using -algebras on and can tell them apart (see Prop. 3.5).

The second ingredient we need to prove Thm. 3.5 is an interesting relation between -structures and Massey products that tells us that, given a topological space , the higher -multiplication on any transferred -algebra on computes Massey products up to a sign:

Theorem 3.4.

[21, §12], [13, Thm 3.1, Cor A.5]. For any transferred -algebra on the cohomology of a space , if are cohomology classes for which the triple Massey product is defined, then

where

Despite any two transferred -algebras on being isomorphic, their higher multiplications may behave slightly differently (as seen in the dual scenario in Ex. 3.1), but this result gives us ways to take advantage of some particular choices of as in the following proposition.

Proposition 3.5.

Let   and be as above, the 3-component trivial unlinked link and the Borromean rings, respectively. Then, admits a transferred -algebra structure such that

whereas does not.

Proof.

Let us start with the unlinked . Using stereographic projection, it is easy to see that is homotopy equivalent to minus two circumferences and an infinitely long straight line, where these 3 objects sit in 3 disjoint regions of . The space we just described is, in turn, homotopy equivalent to the wedge of spheres

As any wedge of spheres, admits a CW decomposition that yields a cellular cochain complex with trivial coboundary operator . Thus, for all and we can therefore choose and to be isomorphisms that make to be the zero map in a diagram dual to (1.1),

Applying Remark 1.4 in terms of -algebras produces a transferred -algebra such that for all . Therefore, admits a transferred -algebra such that

Let us look now at the Borromean rings . On the one hand, W. S. Massey proved in [15] that the Alexander dual of the fundamental classes of the Borromean rings, which form a basis of , satisfy

On the other hand, Thm. 3.4 tells us that for any transferred -algebra on , the following holds:

Both things together yield the conclusion

Let us end this section by proving one last result, extending Prop. 3.5, and mentioning that other examples using for other values of can be analogously obtained.

Let , and . Consider three spaces in (homeomorphic to three spheres of different dimensions) determined by the pairs of equations

-dimensional sphere
-dimensional sphere
-dimensional sphere

Notice that the case yields Borromean rings.

Recall that the Jordan-Brower separation theorem tells us that for any space homeomorphic to , the complement consists of two path components – the inside and the outside of . Any two of the spheres are separated by a space homeomorphic to , which plays an analogous role to pairwise unlinkedness for knots. For instance, lies outside the sphere

while lies inside it. Now we can use Alexander duality in the exact same way as before. Letting be the disjoint union of and , consider the cohomology classes and
corresponding to the dual of the fundamental classes of and . The fact that these spheres can be separated as explained makes the pairwise cup product of these cohomology classes vanish, so that their triple Massey product is defined. Furthermore, W. S. Massey also proved in [15, §4] that this Massey product cannot contain the 0 cohomology class. Hence, the following result follows from using the proof of Prop. 3.5 mutatis mutandis:

Proposition 3.6.

With the notation above, any transferred -algebra on will have

where denotes .

As we explained, the triple Massey product is defined whenever certain 2-fold products vanish (namely, when the cup products are 0). Similarly, for every there is also an -fold Massey product [14] defined whenever the -fold Massey products vanish (in the sense that those sets contain the 0 cohomology class). On the other hand, the operation of a transferred -algebra computes -fold Massey product similarly to the case shown in Thm. 3.4 (see [13, Thm 3.1, Cor A.5] and more recently [16]). Hence, one can think of results similar to Prop. 3.5 and 3.6 for bodies instead of just 3, which will involve information about the -operation . There is also a relation between -algebras and higher order Whitehead products of a similar flavour to that of -algebras and higher order Massey products. The interested reader can explore this in [22, Thm. V.7(7)] and more recently in [1].

Finally, notice that results such as Prop. 3.5 and 3.6 can be exploited by -persistence to study datasets with an underlying (low or high-dimensional) link structure. Also, if we know in which particular context we are working on, we can use the results in this section to choose the right to focus on when using -persistence.

4. Compatible -structures to avoid zigzag ambiguities

Given the sequence of topological spaces and continuous maps

let denote a transferred -coalgebra on , for each , and let and be two integers. Once all this is fixed, to simplify notation, write to represent the restriction of to

. Let us look at a particular example for a moment.

Example 4.1.

Let us assume that , and that there exists a homology class such that

Notice that a homology class can indeed behave like this, as seen in [2, §3]. It is straightforward to check that the existence of the class , whose image by falls into at of the terms forming , results in the creation, in the -barcode of , of two (apparently unrelated) intervals and , each containing at most 500 integers, which amounts to only of the terms in .

Ex. 4.1 shows that the -persistent groups and their corresponding barcodes are not as faithful as we would like in describing the evolution of subspaces such as along . This is due to the need of using zigzag persistence [3] in the construction of the -barcodes. One way to avoid this issue is to choose sequences and transferred -structures along in a coherent manner to allow us to compute -persistence via persistence modules rather than via more general zigzag modules. In terms of the -barcode in homology, this boils down to being able to compute a transferred -coalgebra on , for each in , such that the following will hold for every

(4.1)

In this section we show a way to construct filtrations and the corresponding transferred -coalgebras so that assumption (4.1) holds (see Thm. 4.2), and prove the barcode decomposition theorem of -persistence for the case of such compatible choices of -structures (see Thm. 4.3) to stress how much things simplify, even at a theoretical level, by bypassing the need for zigzag machinery.

Here is a construction exhibiting an example of a choice of -structures that make assumption (4.1) hold.

Theorem 4.2.

Let be a 1-connected CW complex and let be an integer. For all , let denote the wedge , for some . Then there exists a transferred -coalgebra on the reduced rational homology of , for each , such that the following inclusion holds for every , and :

where denotes the restriction of to and denotes the map induced by the inclusion .

We save the proof of Thm. 4.2 for §5 and proceed to prove the following particular case of the fundamental decomposition theorem of -persistence [2, Thm. 2.7], to show how much the assumption (4.1) simplifies things; compare this to the proof in [2, Thm. 2.7].

Theorem 4.3.

Fix some integers and . If denotes the restriction of to and holds for all , then there exists a unique multiset of intervals of the form for , and intervals of the form for , such that for all  , the dimension equals the number of intervals in (counted with multiplicities) which contain the interval .

In particular, equals the number of intervals in (counted with multiplicities) which contain the integer .

Proof.

Fix a transferred -coalgebra structure on , for all and fix two integers and . If the inclusion holds for all , then the maps in the persistence module

restrict to maps that form the persistence (sub)module

(4.2)

Since , applying Lemma 2.4 to (4.2) finishes the proof. ∎

Remark 4.4.

In the terminology introduced in [2, Def. 2.5], Ex. 4.1 shows that a class that -falls asleep and -wakes up again would be represented in the -barcode as apparently independent intervals, each corresponding to a different period in which the class has been -awake, and assumption (4.1) would amount to making sure no class can -wake up once it has -fallen asleep.

5. Appendix

We relegated to this section the proof of Ex. 3.1 and Thm. 3.2 and 4.2. We will start by recalling some results involving rational homotopy theory, whose standard reference is [7], mostly to explain the relation between Quillen minimal models and transferred -coalgebras.

Remark 5.1.

Let us start by noting that -coalgebra structures on a graded vector space

are in one-to-one correspondence with differentials in the complete tensor algebra

with . Here, denotes the desuspension of , i.e., , the grading in is given by

where

and the product is given by concatenation.

Giving a differential in this algebra is equivalent to giving its restriction