On directed homotopy equivalences and a notion of directed topological complexity

This short note introduces a notion of directed homotopy equivalence and of "directed" topological complexity (which elaborates on the notion that can be found in e.g. Farber's book) which have a number of desirable joint properties. In particular, being dihomotopically equivalent implies having bisimilar natural homologies (defined in Dubut et al. 2015). Also, under mild conditions, directed topological complexity is an invariant of our directed homotopy equivalence and having a directed topological complexity equal to one is (under these conditions) equivalent to being dihomotopy equivalent to a point (i.e., to being "dicontractible", as in the undirected case). It still remains to compare this notion with the notion introduced in Dubut et al. 2016, which has lots of good properties as well. For now, it seems that for reasonable spaces, this new proposal of directed homotopy equivalence identifies more spaces than the one of Dubut et al. 2016.

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1 Introduction

The aim of this note is to introduce another notion of directed homotopy equivalence than the one of [8], hoping to get other insights on directed topological spaces. The view we are taking here is that of topological complexity, as defined in [9], adapted to directed topological spaces.

Let us briefly motivate the interest of this “directed” topological complexity notion. In the very nice work of M. Farber, it is observed that the very important planification problem in robotics boils down to, mathematically speaking, finding a section to the path space fibration with . If this section is continuous, then the complexity is the lowest possible (equal to one), otherwise, the minimal number of discontinuities that would encode such a section would be what is called the topological complexity of . This topological complexity is both understandable algorithmically, and topologically, e.g. as having a continuous section is equivalent to being contractible. More generally speaking, the topological complexity is defined as the Schwartz genus of the path space fibration, i.e. is the minimal cardinal of partitions of into “nice” subspaces such that is continuous.

This definition perfectly fits the planification problem in robotics where there are no constraints on the actual control that can be applied to the physical apparatus that is supposed to be moved from point to point

. In many applications, a physical apparatus may have dynamics that can be described as an ordinary differential equation in the state variables

and in time , parameterized by control parameters , . These parameters are generally bounded within some set , and, not knowing the precise control law (i.e. parameters as a function of time ) to be applied, the way the controlled system can evolve is as one of the solutions of the differential inclusion where is the set of all with . Under some classical conditions, this differential inclusion can be proven to have solutions on at least a small interval of time, but we will not discuss this further here. Under the same conditions, the set of solutions of this differential inclusion naturally generates a dspace (a very general structure of directed space, where a preferred subset of paths is singled out, called directed paths, see e.g. [13]). Now, the planification problem in the presence of control constraints equates to finding sections to the analogues to the path space fibration222

That would most probably not qualify for being called a fibration in the directed setting.

taking a dipath to its end points. This is developed in next section, and we introduce a notion of directed homotopy equivalence that has precisely, and in a certain non technical sense, minimally, the right properties with respect to this directed version of topological complexity.

The development of the notion of directed topological complexity and of its properties, together with applications to optimal control is joint work with coauthors, and will be published separately.

Mathematical context :

The context is that of d-spaces [13].

Definition 1 ([13]).

A directed topological space, or d-space is a topological space equipped with a set of continuous maps (where is the unit segment with the usual topology inherited from ), called directed paths or d-paths, satisfying three axioms :

• every constant map is directed

• is closed under composition with non-decreasing maps from to

• is closed under concatenation

For a d-space, let us note by (resp. ) the topological space, with compact open topology, of dipaths (resp. the trace space, ie. modulo increasing homeomorphisms of the unit directed interval) in . (resp. ) is the sub-space of (resp. of ) containing only dipaths (resp. traces) from point to point . We write * for the concatenation map from to (resp. on trace spaces), which is continuous.

A dmap from d-space to d-space is a continuous map from to that also maps elements from to elements of (i.e. they preserve directed paths).

In what follows, we will be particularly concerned with the following map :

Definition 2.

We define the dipath space map333By analogy with the classical path space fibration - but fibration may be a bad term in that case in directed algebraic topology. of by for .

Because only contains directed paths, the image of is just a subset of , called . On the classical case, we do not need to force the restriction to the image of the path space fibration, since the notions of contractibility and path-connectedness are simple enough to be defined separately. In the directed setting, dicontractibility, and “directed connectedness” are not simple notions and will be defined here through the study of the dipath space map.

In order to study this map, in particular when looking at conditions under which there exists “nice” sections to it, we need a few concepts from directed topology.

2 Some useful directed topological constructs

Let be a d-space. We define as in [6], the preorder on , iff there exists a dipath from to . We define the category whose:

• objects are pairs of points of such that (i.e. objects are elements of )

• morphisms (called extensions) from to are pairs of dipaths of with going from to and going from to

We now define, for each d-space, the functor from to with :

• , where is a morphism from to and is a trace from to (i.e. and element of ).

Remark :

There is an obvious link to profunctors and to enriched category theory that we will not be contemplating here. Similar ideas from enriched category theory in directed algebraic topology have already been used in [15, 16] and [8].

A homotopy is a continuous function . We say that two maps are homotopic if there is a homotopy such that and . This is an equivalence relation, compatible with composition.

A d-homotopy equivalence is a dmap which is invertible up to homotopy, i.e., such that there is a dmap with and homotopic to identities. We say that two dspaces are d-homotopy equivalent if there is a d-homotopy equivalence between them.

In such a case, and being dmaps induce

 Pf : PX→PY

resp.

 Pg : PY→PX

which are continuously bigraded maps in the sense that and this bigrading is continuous in , (resp. , continuous in , ).

We write for the sub topological space of of dipaths from to in for some (resp. for the sub topological space of of dipaths in from to for some ).

3 Dihomotopy equivalences

3.1 Dihomotopy equivalence and dicontractibility

Definition 3.

Let and be two d-spaces. A dihomotopy equivalence between and is given by :

• A d-homotopy equivalence between and , and .

• A map continuously bigraded as such that is a homotopy equivalence between and

• A map , continuously bigraded as such that is a homotopy equivalence between and

• These homotopy equivalences are natural in the following sense

• For the two diagrams below (separately), for all , there exists (with domains and codomains induced by the diagrams below) such that they commute444Meaning both squares, one with and and the other with and ; and similarly for the diagram on the right hand side. up to homotopy

with and .

• For the two diagrams below (separately), for all there exists such that they commute up to homotopy

with and .

We sometimes write for the full data associated to the dihomotopy equivalence . Note that in the definition above, we always have the following diagrams that commute on the nose, so that the conditions above only consists of 6 commutative diagrams up to homotopy :

Remark :

This definition clearly bears a lot of similarities with Dwyer-Kan weak equivalences in simplicial categories (see e.g. [2]). The main ingredient of Dwyer-Kan weak equivalences being exactly that induces a homotopy equivalence. But our definition adds continuity and “extension” or “bisimulation-like” conditions to it, which are instrumental to our theorems and to the classification of the underlying directed geometry.

Remark :

There is an obvious notion of deformation diretract of , which is a dihomotopy equivalence such that the inclusion map from to is the left homotopy inverse of and . But there seems to be no reason that in general, two dihomotopically equivalent spaces are diretracts of a third one (the mapping cylinder in the classical case). It may be true with a zigzag of diretracts though, as in [8].

A dicontractible directed space is a space for which there exists a directed deformation retract onto one of its points. Particularizing once again the definition above, we get :

Definition 4.

Let be a d-space. is dicontractible if there is a continuous map , continuously bigraded, such that are homotopy equivalences (hence in particular, all path spaces of are contractible).

3.2 Strong dihomotopy equivalence

Algebraically speaking, there is a simple condition that enforces the bisimulation condition above (Lemma 1), that we call strong dihomotopy equivalence (see below Definition 5). We do not know if it is equivalent to dihomotopy equivalence for a large class of directed spaces.

Definition 5.

Let and be two d-spaces. A strong dihomotopy equivalence between and is given by :

• A d-homotopy equivalence between and , and .

• A map continuously bigraded as such that is a homotopy equivalence between and

• A map , continuously bigraded as such that is a homotopy equivalence between and

• (a) For all , and ,

• (b) For all , and ,

• (c) For all , and , there exists such that , and

• (d) For all , and , there exists such that , and

Lemma 1.

Strong dihomotopy equivalences are dihomotopy equivalences

Proof.

Let be a strong dihomotopy equivalence ; it comes with , and . Let , with and . In order to prove that is a dihomotopy equivalence, we must find such that the diagram below involving and commutes up to homotopy (the other diagram is commutative, for free)

This commutes indeed with and because of property (a) of strong dihomotopy equivalence .

Now, consider the diagram below, involving and

with and . Let and . Property implies that the diagram above commutes up to homotopy.

Similarly, let , property (c) of implies that the following diagram commutes up to homotopy by taking and

with and .

And finally, property (b) implies that the following diagram commutes up to homotopy, by taking and

3.3 Simple properties and examples of directed homotopy equivalences

The first obvious (but important) observation is that directed homotopy equivalence refines ordinary homotopy equivalence. Also, directed homotopy equivalence is an invariant of dihomeomorphic dspaces :

Lemma 2.

Let , be two directed spaces. Suppose there exists a dmap, which has an inverse, also a dmap. Then and are directed homotopy equivalent.

Proof.

Take , and . This data forms a directed homotopy equivalence. ∎

Now, natural homology [5] is going to be an invariant of dihomotopy equivalence, as it should be :

Lemma 3.

Let , be two directed spaces. Suppose and are directed homotopy equivalent. Then and have bisimilar natural homotopy and homology (in the sense of [8]).

Proof.

Call and the underlying dmaps, forming the homotopy equivalence which is a directed homotopy equivalence.

The bisimulation relation we are looking for is the relation :

 {((a,b),Pfa,b,(f(a),f(b))) | (a,b)∈ΓX}∪{((g(c),g(d)),Pgc,d,(c,d)) | (c,d)∈ΓY}

The diagrams defining the directed homotopy equivalence imply that is hereditary with respect to extension maps. ∎

Unfortunately, unlike Dwyer-Kan equivalences, or classical homotopy equivalences, our dihomotopy equivalences do not have the 2-out-of-3 property. We only have preservation by composition, as shown in next Lemma, but also, for surjective dihomotopy equivalences, two thirds of the 2-out-of-3 property, as shown in Proposition 1.

Lemma 4.

Compositions of dihomotopy equivalences are dihomotopy equivalences.

Proof.

Suppose and are dihomotopy equivalences. We have quadruples , as in Definition 3. Now, it is obvious to see that its composite is a dihomotopy equivalence from to . ∎

The problem for getting a general 2-out-of-3 property on dihomotopy equivalences can be exemplified as follows. Suppose that is a dihomotopy equivalence and that is a dihomotopy equivalence. In particular, by 2-out-of-3 on classical homotopy equivalence, we know that is a homotopy equivalence, with homotopy inverse . Consider now the following composites, for all

Because of 2-out-of-3 for classical homotopy equivalences, and as and in the diagram above are homotopy equivalences, is a homotopy equivalence. But we need it to be a homotopy equivalence for all , not only the ones in the image of . Similarly to Dwyer-Kan equivalences (see e.g. [2]), it is reasonable to add in the definition of dihomotopy equivalences the assumption that is surjective on points. Then we have

Proposition 1.

If and are such that and are surjective dihomotopy equivalences, then is a surjective dihomotopy equivalence.

Proof.

In that case, we get that induces homotopy equivalences from all (call the homotopy inverse ). Now, notice that, denoting by the homotopy inverse of , and by the homotopy inverse of , is a right homotopy inverse to , because

 Pf2f1(a),f1(b)∘(Pf1a,b∘F21a,b)=(Pf2f1(a),f1(b)∘Pf1a,b)∘F21a,b=(P(f2∘f1)a,b)∘F21a,b∼IdPZf2∘f1(a),f2∘f1(b)

We therefore have a right homotopy inverse and a homotopy inverse of . So,

 F2rc,d∼(F2c,d∘Pf2c,d)∘F2rc,d=F2c,d∘(Pf2c,d∘F2rc,d)∼F2c,d

and

 F2rc,d∘Pf2c,d∼F2c,d∘Pf2c,d∼IdPY(c,d)

therefore is a homotopy inverse of . Moreover, as a composition of maps with which are continuous in , it is a continuously bigraded map on .

We also have to look at the map induced by . In the following diagram

by 2-out-of-3 for classical homotopy equivalences, without any other assumptions, we get that is a homotopy equivalence (with homotopy inverse ) for all path spaces . We note as above that, denoting by the homotopy inverse of , is a left homotopy inverse of . As before, we easily get that and is a homotopy inverse of which forms a continuously bigraded map.

Now we have to check the extension diagrams of Definition 3. Let with , . As is surjective, and for some . As is a dihomotopy equivalence, we have the existence of as in the diagram below.

Now, we use the fact that is a dihomotopy equivalence, and we get a map such that the following diagram commutes up to homotopy

In the diagram above, (represented as a dashed arrow) is actually the composite as shown before ; similarly, is the composite . Therefore we have the extension property needed for . The five other diagrams can be proven in a similar manner, by pulling back or pushing forward the existence of maps using the diagrams for , , , (resp. , , and ).

Remark :

Similarly, if we have a surjective that is a dihomotopy equivalence, and such that is a surjective dihomotopy equivalence, first, there is no reason why should be surjective. Similarly as with before, we can prove that forms a continuous bigraded family of homotopy equivalences with a continuous bigraded family of homotopy inverses. The problem is with which we can only prove to have a continuous family of homotopy inverses on spaces . The only result we can have in general is dual to the one of Proposition 1. If and are such that and are dihomotopy equivalences with surjective homotopy inverses, then is a dihomotopy equivalence with surjective homotopy inverse.

Example 1.

The unit segment is dicontractible. The wedge of two segments is dicontractible (which makes shows the version of dicontractibility discussed here to be notably different from that used in the framework of [8]). Note that in view of applications to directed topological complexity, this is coherent with the fact that directed topological complexity should be invariant under directed homotopy equivalence. For any two pair of points in of directed segment or of a wedge of two segments, there is indeed a continuous map depending on this pair of points to the unique dipath going from one to the other.

Example 2.

The Swiss flag is not directed homotopy equivalent to the hollow square. This can be seen already using natural homology, that distinguishs the two, see e.g. [5].

More precisely, we consider the following d-spaces (SF on the left, HS on the right), coming from PV processes, which are subspaces of and whose points are within the white part in the square (the grey part represents the forbidden states of the program) and whose dipaths are non decreasing paths for the componentwise ordering on . They are homotopy equivalent using the two maps (and even dmaps), from to and from to , depicted below ( is the map on the left) :

The points in light grey are the points which do not belong to the image of those maps. The problem is that those two programs are quite different: has a dead-lock in and inaccessible states, while does not. Topologically, they do not have the same (directed) components in the sense of [11].

It is easy to see that although and are homotopy equivalences as well, does not induce a dihomotopy equivalence in our sense. Take a point in the lower convexity of the Swiss flag, and consider the constant dipath on in . maps the constant path on onto the constant path on but we can extend in this constant path to paths from to the image by of the upper right point, which is again the upper right point in . This extension makes the corresponding path space within homotopy equivalent to two points, whereas there are no path from to the right upper point in .

In natural homology, and do not have bisimilar natural homologies since, considering the pair of points in , all extensions of this pair of points will give 0th homology of there corresponding path space equal to , whereas there are extensions of any pair of points in which give 0th homology of there corresponding path space equivalent to .

Example 3.

Consider Figure 1 : this depicts a naive dihomotopy equivalence between and to its upper face (so to a point). More precisely, the dmap , which maps any point of to the point of just above of it, is a naive dihomotopy equivalence, whose inverse modulo dihomotopy is the embedding of into . Hence, and a dihomotopy from to is depicted in Figure 1.

Note that this naive dihomotopy equivalence, , does not induce a dihomotopy equivalence in our sense. As a matter of fact, consider points and  : is homotopy equivalent to two points whereas is homotopy equivalent to a point.

4 Dicontractibility and the dipath space map

Definition 6.

Let be a d-space. is said to be weakly dicontractible if its natural homotopy (equivalently, natural homology [5], by directed Hurewicz, all up to bisimulation) is the natural system - which we will denote by - on 1 (the final object in ) with value on the object, in dimension 0 and 0 in higher dimensions.

The following is a direct consequence of Lemma 3 but we give below a simple and direct proof of it :

Lemma 5.

Let be a dicontractible d-space. Then is weakly dicontractible in the sense of Definition 6.

Proof.

Suppose is dicontractible. Therefore, by Definition 4, we have a continuous map , continuously bigraded, which are are homotopy equivalences. All extension maps induce identities modulo homotopy, trivially. Now, these diagrams of spaces induce, in homology, a diagram which has as value on objects for dimension 0, and 0 for higher dimensions. It is a simple exercise to see that such diagrams are bisimilar to the one point diagram which has only as value in dimension 0, and 0 in higher dimension. This shows weak dicontractibility. ∎

Theorem 1.

Suppose is a contractible d-space. Then, the dipath space map has a continuous section if and only if is dicontractible.

Proof.

As is contractible, we have (the constant map) and (the inclusion) which form a (classical) homotopy equivalence. Trivially, and are dmaps.

Suppose that we have a continuous section of . There is an obvious inclusion map