Stratified Homotopy Theory and a Whitehead Group for Stratified Spaces
MMaster Thesis
Stratified Homotopy Theory and a
Whitehead Group for Stratified Spaces
Lukas WaasSupervisor: Prof. Dr. Markus BanaglFaculty of Mathematics and Computer ScienceUniversity of HeidelbergOctober 21, 2020Last Updated on February 13, 2021 a r X i v : . [ m a t h . A T ] F e b bstract Simple homotopy theory deals with the question of when a homotopy equivalence f between sufficiently combinatorial topological spaces X and Y can be representedthrough a sequence of elementary combinatorial moves, called elementary expan-sions and collapses. It turns out, this question is answered completely by an ob-struction element, the Whitehead torsion of f , inside of an algebraic group, theWhitehead group of X . In this master thesis, we extend this classical perspective tothe world of stratified homotopy theory. To obtain a well established framework towork in, we prove a series of results on two model categories of simplicial sets andtopological spaces, both equipped with a notion of filtration, introduced by SylvainDouteau in his PHD thesis. Weak equivalences in these categories are essentiallystratum preserving maps that induce weak homotopy equivalences on strata andhigher homotopy links. Making use of a filtered version of the simplicial approxima-tion theorem, we show that as long as one restricts to finite filtered simplicial sets,their homotopy theory provides a good model for the filtered topological one. Inparticular, we show that there is a fully faithful embedding of homotopy categoriesfrom the (finite) simplicial into the topological setting. We also use these results tocharacterize the morphisms in the topological filtered homotopy category betweenfiltered spaces that are triangulable and stratified in some very general sense - whichincludes most definitions of stratified spaces - as stratified homotopy classes of stra-tum preserving maps. Having then gained a good theoretical understanding of howstratified homotopy theory can be modeled in the world of simplicial sets, we pro-pose a class of combinatorial elementary expansions for filtered simplicial sets thatgeneralize both the classical ones as well as a class of stratified expansions suggestedby Banagl et al. We prove that they fulfill a series of axioms, suggested by Eckmannand Siebenmann for the construction of simple homotopy theory. In doing so, weobtain a combinatorially defined Whitehead group and torsion for filtered simplicialsets (and hence also for triangulable stratified spaces). We then begin a detailedinvestigation of their formal properties, proving for example that a Mayer-Vietorisformula holds. Using these results we show that every filtered simplicial set hasthe simple homotopy type (in the sense induced by these expansions) of a filteredsimplicial complex of the same dimensions. We then apply the results we obtainedon the connection between filtered simplicial sets and filtered topological spaces toobtain a more topological description of the Whitehead group and to generalizethe Whitehead torsion to arbitrary stratum preserving maps of triangulated filteredspaces. Finally, we prove that our simple homotopy theory is a generalization of theclassical one, in the sense that it agrees with the latter when one considers triviallyfiltered simplicial sets as CW-complexes. usammenfassung Einfache Homotopietheorie besch¨aftigt sicht mit der Frage, wann eine Homotopie-¨aquivalenz f zwischen hinreichend kombinatorisch gearteten R¨aumen X und Y durch eine Abfolge von elementaren kombinatorischen Operationen, genannt ele-mentare Erweiterungen und Kollapse, dargestellt werden kann. Eine vollst¨andigeAntwort auf diese Frage liefert ein Obstruktionselement in der sogenannten White-headgruppe von X , die Whitehead torsion von f . In dieser Masterarbeit verallge-meinern wir diese klassischen Ergebnisse auf stratifizierte Homotopietheorie. Unserhomotopietheoretisches Framework hierf¨ur sind zwei Modellkategorien von filtri-erten Objekten - eine von simplizialen Mengen, eine von topologischen R¨aumen- eingef¨uhrt von Sylvain Douteau in seiner Promotion. In beiden Kategorien sindschwache ¨Aquivalenzen dadurch charakterisiert, dass sie schwache Homotopie¨aqui-valenzen auf Strata und (h¨oheren) Homotopielinks induzieren. Mit Hilfe einer strat-ifizierten Version des simplizialen Approximationssatzes beweisen wir, dass sich diesimpliziale Homotopiekategorie (unter leichten Endlichkeitsannahmen) volltreu indie topologische einbettet, also die simplizial filtrierte Welt ein gutes Homotopiemod-ell f¨ur die topologische bietet. Weiterhin charakterisieren wir die Morphismen in dentopologischen Homotopiekategorie zwischen einer Klasse von endlich triangulier-baren stratifizierten R¨aumen, die beinahe alle klassischen Beispiele stratifizierterR¨aume beinhaltet. Sie sind durch Homotopieklassen strata erhaltender Abbildun-gen unter stratifizierter homotopie gegeben. Mit diesen Ergebnissen ausgestattet,beginnen wir dann unsere Untersuchung einfacher stratifizierter Homotopietheorie.Wir schlagen eine Klasse elementarer Erweiterungen vor, welche sowohl die klas-sischen elementaren Erweiterungen als auch eine Klasse stratifizierter Erweiterun-gen die k¨urzlichen von Banagl et al. vorgeschlagen wurden, verallgemeinert. F¨urdiese Beweisen wir, dass sie einen Axiomensatz erf¨ullen, der von Eckmann undSiebenmann f¨ur die Konstruktion einer einfachen Homotopietheorie vorgeschlagenwurde. Inbesondere erhalten wir eine kombinatorische Whiteheadgruppe und White-head torsion f¨ur filtrierte simpliziale Mengen, und damit auch f¨ur triangulierbarestratifizierte R¨aume. Als n¨achstes folgt eine detailierte Untersuchung von formalenEigenschaften dieser Whiteheadgruppe und der zu ihr korrespondierenden einfachen¨Aquivalenzen. Inbesondere beweisen wir eine Mayer-Vietoris Formel und zeigenmit ihrer Hilfe, dass jede filtrierte simpliziale Menge den einfachen Homotopietypeines filtrierten Simplizialkomplexes hat. Dann nutzen wir die im ersten Teil derArbeit gewonnen Resultate, um die Whitehead torsion auf triangulierte filtrierteR¨aume zu verallgemeinern und erhalten so eine Reihe ¨aquivalenter Interpretationender stratifizierten Whiteheadgruppe. Zu guter Letzt beweisen wir mit Hilfe dieserCharakterisierungen, dass unsere Konstruktion der Whiteheadgruppe im Falle triv-ialer Filtration nat¨urlich isomorph zu der klassischen ist. cknowledgements First and foremost I want to thank my advisor, Professor Markus Banagl. He wasthe one who awakened my interest in topology during my first calculus course as afreshman and has accompanied my studies of the subject ever since. Next, I wantto thank the Studienstiftung des Deutschen Volkes for their continued financial andintellectual support. Finally, big thanks go out to my friends Ricardo, Jonas, Dario,Tim and Tim and my girlfriend Rike for their help in proofreading and huntingdown the commas I tend to spice up every sentence with. ontents P -filtered topologicalspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.1.2 The category of P -filtered simplicial sets . . . . . . . . . . . . . . . 271.1.3 The Douteau model structure on P -filtered simplicial sets . . . . . 311.1.4 A representation result for homotopy classes . . . . . . . . . . . . 371.2 Realizations of weak equivalences . . . . . . . . . . . . . . . . . . . . . . . 411.2.1 Strongly filtered topological spaces . . . . . . . . . . . . . . . . . . 421.2.2 Weak equivalence of homotopy links . . . . . . . . . . . . . . . . . 451.2.3 The realization theorem . . . . . . . . . . . . . . . . . . . . . . . . 581.3 The filtered simplicial approximation theorem . . . . . . . . . . . . . . . . 591.3.1 Standard constructions for filtered simplicial complexes . . . . . . 601.3.2 Subdivisions and relative subdivisions . . . . . . . . . . . . . . . . 651.3.3 Proof of the filtered simplicial approximation theorem . . . . . . . 691.3.4 Filtered ordered simplicial complexes . . . . . . . . . . . . . . . . . 791.3.5 Last vertex maps of relative subdivisions . . . . . . . . . . . . . . . 821.3.6 Filtered simplicial approximation, part two . . . . . . . . . . . . . 861.4 A more detailed analysis of H Top P . . . . . . . . . . . . . . . . . . . . . 891.4.1 | − | P : H s Set finP → H
Top P is fully faithful . . . . . . . . . . . . . 891.4.2 Homotopy classes into fibrant, finitely triangulable filtered spaces . 92 W h P , A P , τ P and of simple equivalences . . . . . . . 1372.3.3 Filtered ordered simplicial complexes and simple homotopy type . 1482.4 The Whitehead group in a topological filtered setting . . . . . . . . . . . . 1522.4.1 Comparison to the classical Whitehead group . . . . . . . . . . . . 156 Appendices 166A Appendix 167
A.0.1 A result on pullbacks of colimits . . . . . . . . . . . . . . . . . . . 167A.0.2 A basic set theoretic manipulation . . . . . . . . . . . . . . . . . . 168 hapter 0
Introduction
The goal of this work is the extension of methods of a rather classical field of algebraictopology, simple homotopy theory, to one that has recently seen a lot of development,stratified homotopy theory. While our investigation is purely theoretical, we want tomotivate it by an example originating from topological data analysis (TDA).The pipeline of TDA is often described as follows. Given some dataset, for examplea point cloud in euclidean space, one wants to identify some of its topological features.As a point cloud on its own is discrete, the first step to do so is to replace it by anappropriate, more geometrically interesting object. This is usually done by connectingits points through simplices, thus obtaining a simplicial complex (or a family of simpli-cial complexes, depending on certain parameters), following certain rules that consider,for example, the distance between points. Finally, one applies a topological invariant tothis resulting simplicial complex, obtaining some algebraic data, that one can interpret.This invariant is typically (co)homology with coefficients in some field. Homology ofcourse is a homotopy invariant. Recently, however, there has been a lot of interest in re-placing homology in this final step by some other (not necessarily topological) invariant(see for example [BH11]). Doing so, one hopes to pick up on different, and potentiallyfiner features than homology can identify (see for example [RBSL19] and [BH11]). Onesuch invariant is intersection homology, as suggested in [BH11]. Intersection homologyis an invariant of spaces equipped that are equipped with a filtration that is particularlyadapted to working with spaces with singularities. In this setting, the classical pipelinebegins with a point cloud in which some points have been marked as belonging to asingularity. The simplicial complex K obtained from this point cloud (for examplethrough the Vietoris-Rips construction) is then naturally equipped with a subcomplex K given by the full subcomplex spanned by the points marked as singular. Such a pairis called a filtered simplicial complex (filtered over { , } ). For this filtered simplicialcomplex K = ( K ⊂ K ), one then computes the intersection homology (with respectto some perversity and formal codimension, in the sense of [Fri20]).1 HAPTER 0. INTRODUCTION R sampled from a pinched torus. The rightshows a Vietoris-Rips complex built from it, up to dimension 3. Three-dimensionalsimplices are marked in a lighter green. K is taken to be the pinched point.In practice, the filtered complex K can be rather large and high dimensional even ifthe point cloud is located in low dimensional euclidean space. For computational reasons,one may therefore ask the question whether it is possible to replace K by a smaller filteredcomplex, without changing its intersection homology. This is essentially the approachof [BMS20]. There, the authors characterize certain elementary combinatorial movesthat do not change the intersection homology - the stratified homotopy type, to be moreprecise - of a filtered simplicial complex. These moves are a generalization of elementarycollapses of simplicial complexes, due to Whitehead. One says that a simplicial complex K collapses elementarily to L or equivalently, L expands elementarily to K if K isobtained from L through filling a horn. That is, K is obtained from L by adding asimplex that has all but one of its proper faces present in L together with the missingface.Figure 2: To the left is an elementary collapse in the classical setting. The addedsimplex and its face are marked in blue. The picture on the right shows a reducedcomplex obtained from the filtered complex in Figure 1 through the collapses describedin [BMS20].The realization of the inclusion L (cid:44) → K is then a homotopy equivalence. A map HAPTER 0. INTRODUCTION f : | K | → | K | homotopic to the composition of the realizations of such elementaryexpansions and their inverses is called a simple homotopy equivalence. The field con-cerned with the question of the relationship between simple homotopy equivalences andgeneral homotopy equivalences is simple homotopy theory. At the same time, the fieldconcerned with the question of finding a homotopy theoretical setting to formulate inter-section homology in is stratified homotopy theory. Thus, the study of such combinatorialoperations in the stratified (filtered) setting lies in the intersection of these two fields.If one wants to obtain a deeper understanding of how to use such operations in prac-tice, what other possible combinatorial manipulations might be useful, and what theirtheoretical limitations are - that is, which homotopy equivalences (in some appropriatestratified sense) can be represented through such operations - this intersection is whatone needs to study. Let us call it simple stratified homotopy theory.A filtered space is (classically) a topological space, X , together with a filtration byclosed subspaces X ⊂ ... ⊂ X n = X . Such a space X then decomposes (on a setlevel) into the disjoint union of the subspaces X k = X k \ X k − . These spaces are calledstrata. Filtered spaces arise naturally in numerous fields of mathematics. Usually, theycan be thought of as specifying singularities in a space. For example, a filtration canbe obtained by repeatedly taking the singular locus in a complex algebraic variety. Inthese types of settings, the filtrations are often particularly ”nice” when it comes to theshape of their strata (for a complex variety they are smooth complex manifolds) andthe interactions between the strata. Depending on what kind of niceness conditions onefocuses on, one arrives at various different notions of a stratified spaces (for a good intro-duction, we recommend for example [Ban07] and [Fri20]). Stratified homotopy theoryis concerned with the question of finding a good homotopy theoretical setting for thesekind of objects. Intersection homology for example is generally not a homotopy invari-ant, however, it is invariant under homotopy equivalences that respect the stratificationin a appropriate way, called stratified homotopy equivalences (see [Fri20, Prop. 4.1.10.]).In the non-stratified setting, i.e. the classical one, the question whether a homotopyequivalence f : | K | → | L | between two realized (finite) simplicial complexes (or CW-complexes, with the geometrically analogous notions of expansions and collapses, see[Coh73]) is a simple homotopy equivalence is entirely answered by a certain element ofthe so called Whitehead group, denoted W h ( K ), of K . This element is called the White-head torsion of f , τ ( f ). It turns out that τ ( f ) is 0 if and only if f is a simple homotopyequivalence. One might think of this result as the fundamental theorem of simple homo-topy theory. The Whitehead group admits numerous equivalent constructions rangingfrom the purely geometric-combinatorial to algebraic constructions, entirely dependenton the fundamental group (groupoid to be more precise) of K (see for ex. [Coh73]). Sim-ilarly, there are several different interpretations of what τ ( f ) = 0 entails geometrically.In case where f is an h -cobordism - that is, K is (a CW-structure on) a closed, connectedmanifold M , | L | is a compact manifold with boundary M (cid:116) N , and f is the inclusioninto the boundary, in addition to being a homotopy equivalence - then if dim( M ) ≥ HAPTER 0. INTRODUCTION τ ( f ) essentially measures whether f is given by a boundary inclusion M (cid:44) → M × I .This result is called the s -cobordism theorem, and it is widely regarded to be one of thegreatest contributions of simple homotopy theory (see for example [WV00] for a detailedstudy of this phenomenon in the topological, smooth and PL setting). Much of the workon simple stratified homotopy theory has been done with this kind of result in mind.See for example [Wei94] for a good overview. However, this is not the approach we aregoing to take. The goal of this master thesis is to develop the combinatorial perspective- that is, the one induced by collapses and expansions - for the stratified setting. Tobe more precise, we identify a certain class of elementary expansions that preserve the(weak) stratified homotopy type (in the sense of [Dou19a]). We then prove the existenceof a Whitehead group and Whitehead torsion for this setting, which behave analogouslyto the classical perspective, that is, they measure whether an equivalence can be builtfrom these simple combinatorial operations. This is be the content of Chapter 2. To thebest of our knowledge, no such work has been done by previous authors. Comparingour formulation of simple stratified homotopy theory to the one of previous authors willcertainly prove an interesting topic for future work.To formulate our combinatorial perspective, we make use of the category of simpli-cial sets filtered over a poset P , s Set P , introduced in [Dou18]. Much of the work ofthis master thesis consists of showing what they model (homotopically speaking) onthe topological side. This is necessary if one wants to understand what the combinato-rial operations and the induced notions of Whitehead group and Torsion measure froma more topological perspective. In [Dou19b], the author has begun this comparisonprocess. He has introduced model structures on s Set P and on Top P , the category oftopological spaces filtered over P , and studied their interaction. We expand on thiswork. In fact, we show that the realization functor between these categories preservesweak equivalences and, as long as one restricts to a sufficiently compact setting, inducesan equivalence of categories on the homotopy categories. These types of investigationsare the content of Chapter 1. While a priori not necessary for the construction of theWhitehead invariants, this allows us to interpret them from the topological perspective.Independently from this particular application, we think that they might add to a deeperunderstanding of stratified homotopy theory, in particular when it comes to a homotopytheoretical formulation of intersection homology. As we already mentioned in the previous paragraph, the content of this thesis is splitup into two partially independent investigations. In this subsection, we give a roughoverview of our main results. They are marked in bold font. When stating the contenttheorems, we often state them in a more conceptual form to avoid too many definitionsin this introductory statement. The precise statements are of course found under therespective references.
HAPTER 0. INTRODUCTION Chapter 1:
This chapter is an investigation into two suggested model categories forstratified homotopy theory, suggested in [Dou19a] and [Dou18]. It can be read entirelyindependently from any questions on simple homotopy theory, however, the choice ofsetting is strongly motivated by such questions. For example, we restrict to the case of afixed underlying type of stratification. This is sufficient for the study of most notions ofequivalence, but of course not for a homotopy theory of more general maps of stratifiedspaces. In [Dou18] and [Dou19a], Douteau introduced a model category of simplicialsets, s
Set P , and one of topological spaces, Top P , both filtered over some fixed partiallyordered set P . The objects in Top P are (certain) topological spaces, filtered by a familyof closed subspaces indexed over P . This induces a notion of strata, indexed over P ,as in the case where P is linear. The morphisms are then such maps that retain thestratification index. The two categories are connected through a realization, singularsimplicial set adjunction, | − | P (cid:97) Sing P . The notion of weak equivalences in both casescan be expressed by inducing isomorphisms on analogues of homotopy groups, adaptedto the stratified setting. Passing to these weak equivalences that are slightly more gen-eral than general stratified homotopy equivalences allows for the extra degree of freedomnecessary for our combinatorial formulation of simple stratified homotopy theory. Chap-ter 1 is mainly concerned with connecting these two model categories. To be precise,our main results here are the following.First off, it is the content of Theorem 1.2.23 that the realization functor | − | P :s Set P → Top P preserves weak equivalences. We show this result by using results of[Dou19b] in a more rigid filtered setting and translating these to the usual filtered settingby a comparisson of homotopy links argument, Theorem 1.2.5 .In particular, as an immediate consequence of this, we have that | − | P induces a functorof the respective homotopy categories, denoted by adding an H . Under the additionalassumption that we restrict to the full subcategory of H s Set P , H s Set finP , given by simpli-cial sets with finitely many non-degenerate simplices, we further show
Theorem 1.4.1 :The induced functor of homotopy categories H s Set finP → H
Top P is fully faithful.To show the latter result, we make use of a filtered analogue of the simplicial approxima-tion theorem. This result is not entirely new. The proof and the theorem are based onmethods and results found in [Sch71]. However, there are some technical problems in thesource which we were able to circumvent. Furthermore, we keep explicit track of sub-divisions, which is be necessary for our purposes. The proof is decidedly more involvedthan the non-filtered setting. We finally obtain Theorem 1.3.44 , a filtered analogueto the simplicial approximation theorem, which also states that stratification respectinghomotopies between realizations of morphisms in s
Set P , between appropriately filteredordered simplicial complexes, are witnessed by a sufficiently high degree of subdivision.This result, independently from the remainder of this work, should be interesting whenit comes to comparing the filtered PL-setting to the topological one. We finally reducethe question of general filtered simplicial sets to that of filtered simplicial complexes, by HAPTER 0. INTRODUCTION P -filtered simplicial set can be replaced bya P -filtered ordered simplicial complex, up to simple equivalence. This is a consequenceof our result Theorem 2.3.23 , which states that this even holds up to an appropriatenotion of simple equivalence.As a side effect of the simplicial approximation theorem, we are also able to actually char-acterize the morphisms in H Top P between objects that are not cofibrant with respectto the model structure. At least between compact, piecewise linear filtered spaces thatare stratified in some appropriate sense, we show that they are actually given by stratumrespecting homotopy classes between maps of P -filtered spaces ( Theorem 1.4.7 ). Thisresult hints at the fact that there might be an additional model structure on
Top P ,having the same equivalences, where the fibrant objects are actually spaces that are”nicely” stratified. We hope to prove such a result in later work. Chapter 2:
Having gained a better understanding of our stratified homotopy cate-gories, we then move on to Chapter 2, the simple stratified setting. We do so in thesetting of s
Set P . Hence, most of the chapter only requires Section 1.0.1 and Section 1.1.2- Section 1.1.4 to obtain an abstract understanding. Only at Section 2.4, do the resultsof Chapter 1 come into play. However, they are of course useful to have in mind as theyjustify that our combinatorial musings actually have relevance to the topological realm.Our approach to simple homotopy theory is the usage of a general category theoret-ical framework for simple homotopy theory developed independently by Eckmann andSiebenmann ([eck] and [Sie70]). For this framework to be applicable, one needs to spec-ify a class of morphisms (taking the role of expansions), that fulfill a certain set ofaxioms (see Theorem 2.1.3). Our analogue to the elementary expansions, described inthe classical setting in our introduction before this subsection, are given by pushouts ofwhat Douteau calls “admissible horn inclusions”. These admissible horn inclusions area generating set for the trivial cofibrations in s Set P . The induced notion of expansionis called (finite) filtered strong anodyne extensions, FSAE, for short. They are a moregeneral class of operations than the ones introduced in [BMS20] and generally only pro-vide weak equivalences in s Set P and Top P , not stratified homotopy equivalences (i.e.homotopy equivalences in the sense in the sense of stratum preserving homotopies ofstratum preserving maps). However, whenever two realizations of filtered simplicial setsthat are ”nicely” stratified, not just filtered, are connected by a zigzag of such expan-sions, they are actually stratified homotopy equivalent, by an analogue to the Whiteheadtheorem (Theorem 1.1.15) or an appeal to Theorem 1.4.7. At the same time, this ad-ditional degree of freedom makes them behave more similar to the classical expansions.We begin illustrating these advantages in Section 2.2.1. We then translate methodsdeveloped in the non-stratified setting by Moss ([Mos19]) to the stratified one to ob-tain a series of equivalent characterizations of FSAEs which are often easier to verify( Proposition 2.2.22 ). HAPTER 0. INTRODUCTION
Proposition 2.2.35 .As a consequence of these types of results, we are then able to show that the axiomsof Theorem 2.1.3 are fulfilled in the setting of s
Set finP and FSAEs (
Theorem 2.3.4 ).In particular, this gives us a of Whitehead group and Whitehead torsion that measureswhether a morphism in the homotopy category H s Set finP is given by a zigzag of FSAEs,i.e. is a simple equivalence in this setting. We show that these behave much like theclassical torsion and Whitehead group, thus laying a solid theoretical groundwork for fur-ther, more geometrically minded analysis. This work does not yet contain any attemptsat computing our Whitehead group. However, a first step in this direction, allowing for apossible reduction to the classical case, is that we have shown a Mayer-Vietoris formula(
Proposition 2.3.16 ).Next, we obtain a P -filtered simplicial set analogue to the well known result that everyfinite CW-complex has the weak homotopy type of a finite simplicial complex. This isthe content of Theorem 2.3.23 , which can be proven independently from any of ourresults in Chapter 1, and in fact was the deciding argument there for the reduction fromfiltered simplicial sets to filtered ordered simplicial complexes.Then, we move on to obtaining an interpretation of the Whitehead torsion for morphismsin the topological filtered homotopy category
Top P . This is of course through the useof methods from Chapter 1. In Proposition 2.4.4, we then summarize all the equivalentdescriptions of our Whitehead group and Whitehead torsion, that we obtained. Amongthem is that the Whitehead torsion of a weak equivalence between realizations of finite P -filtered simplicial sets, f : | X | P → | Y | P , disappears if and only if it comes from azigzag of (realizations of) elementary expansions in H Top P . The Whitehead group of | X | P (as before) is then described by the class of isomorphism | X | P → | Y | P in H s Set P modulo post composition with such simple equivalences.Finally, we prove that our theory is an extension of the classical one. That is, forthe special case where P = (cid:63) is a one-point set we show that our Whitehead group isnaturally isomorphic to the non filtered one, under the embedding of simplicial sets intoCW-complexes. This is the content of Theorem 2.4.5 . We use this subsection to fix some notation and conventions. We freely make use ofstandard language of category theory, trying to add a reference when we deviate fromthe most basic notions. • Throughout all of this thesis, we denote by P some fixed, possibly infinite, partiallyordered set. By a flag in P , we mean a finite, linearly ordered subset of P . Flagsare denoted in the form { p ≤ ... ≤ p n } , for p i ∈ P . We also need flags that allow HAPTER 0. INTRODUCTION d-flags , where the d stands for degenerate. Theyare denoted in the form ( p ≤ ... ≤ p n ), for p i ∈ P . We avoid the more standard[ − ] notation as it interferes with the standard notation of setting [ n ] := { , ..., n } . • By Set , Ab and AbMon we denote the categories of sets, abelian groups andabelian monoids respectively. • When we speak of topological spaces, we do not mean arbitrary ones, but suchtopological spaces that are ∆-generated (see [nLa20e]). This means that theyhave the final topology with respect to all continuous maps coming from real-izations of standard simplices (or equivalently from finite dimensional euclideanspace). The category of such spaces, together with (continuous) maps as mor-phisms is denoted by
Top . This is a convenient category of topological spaces(in the sense of [nLa20c]). The reason we do not go with the more standardcompactly-generated weakly Hausdorff spaces, is that we are constantly be work-ing with partially ordered sets equipped with the Alexandroff topology (the closedsets are the down-sets). These are only T when they are discrete. Limits in the∆-generated category are obtained, by taking limits in Top and then taking thefinal topology with respect to all maps from simplices into this space. Colimits arecomputed just as in the naive topological category. We will take care to mentionpotential issues arising from this, when they occur. However, most constructionswe use are purely categorical here and the differences in topoology are basicallyirrelevant from a homotopy theoretical perspective. This is so, as there is no dif-ference to these settings as long as one “probes” with ∆-generated objects suchas spheres. The reader having any remaining doubts is recommended a look at[Dug20] and at [nLa20e]. • We freely make use of much of the standard language of algebraic topology. Stan-dard notation of spaces includes I for the unit interval and S n and D n for the n -dimensional sphere and disk respectively. • Besides standard results and notions of algebraic topology, we freely make use ofthe language of simplicial sets and of simplicial homotopy theory. The category ofsimplicial sets is denotes by s
Set . We use most of the standard notation, foundfor example in [GJ12]. • Further, combinatorial objects of interest are (abstract) ordered and nonorderedsimplicial complexes. By a simplicial complex K we mean a set K (0) (of vertices),together with a subset of its power set that only contains finite sets and is closedunder the subset relation. We have allowed the empty simplex here as it makessome notation more canonical, for example when it comes to joins, but of coursethis is purely a matter of taste. We freely use most of the language associatedto simplicial complexes (simplices, subdivisions, faces etc.), found for example in[Spa89, Ch. 3]. By sCplx , we denote the category of simplicial complexes, withmorphisms given by simplicial maps (maps of simplicial complexes), i.e. maps on HAPTER 0. INTRODUCTION sCplx o . • Finally, we make free use of the language of model categories, in particular simpli-cial ones. Most of this can be found for example in [Hir09], but we give referenceswhen we first use a new terminology deviating from the most basic definitions. Thehomotopy category of a model category, obtained by localizing the weak equiva-lences, is denoted by adding a H to the name of the category. We sometimes usethis notation for types of homotopy categories, that do not necessarily originatefrom a notion of model category as well. Morphisms in these homotopy categoriesthat come from ones in the original category are denoted in the form [ f ]. If themodel category is simplicial we denote by [ X, Y ] the set of simplicial homotopyclasses between X and Y . For the classical simplicial structure on topologicalspaces, for example, these are just the homotopy classes of continuous maps. hapter 1 Stratified Homotopy Theory
In this chapter, we are going to set and describe the framework, in which our stratifiedsimple homotopy theory is formulated. As already mentioned, this is based on workof Douteau in [Dou19a, Dou18, Dou19b]. Since we have just given a rather detailedoverview of the structure and content of this chapter in the overview of results, werefrain from giving another such summary here. The respective sections and subsectionsare each equipped with such a short summary anyway.
Throughout this work, we are constantly be working with objects that are in some sensefiltered over a partially ordered set. As it would be quite a hassle to introduce the analo-gous notation and language over and over again, we summarize some generalities in thissubsection. We follow the choice of language in [Dou19b] here. This is in no way intendedto be a category theoretical theory of what filtrations are, and purely intended as fixingsome language and general constructions. In fact, the language is somewhat conflictingwith what is usually defined to be a category of filtered objects (see Remark 1.0.10).While writing out such a general theory might be an interesting undertaking, we didnot find that it really adds anything mathematically here. In particular, if one deviatestoo far from the examples we give here, this language can turn out to be somewhatnonsensical. It is not intended in this way. In other words: Is this construction phrasedappropriately for a general theory of filtered objects? Definitely not. Is it sufficient toquickly introduce language for a multitude of scenarios we care about? Yes, it is.Denote by
Poset the category of partially ordered sets, with morphisms given by orderpreserving maps.
Construction 1.0.1.
Let C be some category. (The reader should have the categoriesof (∆-generated) topological spaces, simplicial sets, and simplicial complexes in mind).Furthermore, let F : Poset −→ C be some functor. Then, for each P ∈ Poset , considerthe over category C /F ( P ) , over F ( P ). That is, the category with objects given by arrows p C : C → F ( P ) in C and morphisms given by commutative diagrams:10 HAPTER 1. STRATIFIED HOMOTOPY THEORY C C (cid:48) F ( P ) p C p C (cid:48) .We call C /F ( P ) the category of P -filtered objects with stratum preserving morphisms in C (with respect to F ) . We mostly omit the ”with respect to F ” from here on out,but one should note that it is important to keep track of it if one wants to obtain anotion of strata. If the objects and morphisms in C have particular names, let us sayspaces and maps, the objects and morphism in C /F ( P ) are called P -filtered spaces andstratum preserving maps respectively. We usually omit explicitly mentionining that themorphisms are to be stratum preserving, but one should keep in mind that there areother notions of morphisms of filtered objects (see remark 1.0.10). By a slight abuse ofnotation, we often refer to a P -filtered object by its underlying object in C . The arrow p C is then referred to as the filtration of C .This construction is functorial in a number of ways, summarized in Construction 1.0.4.We start, however, with a series of examples of such categories so that the reader hassome examples in mind. In our cases, the generated filtered categories have somewhatmore concrete descriptions than just being some arrow category. We give such descrip-tions in the respective subsections. Example 1.0.2.
Our main examples for F as in Construction 1.0.1 and hence categoriesof filtered objects are the following. • Filtered topological spaces:
Given a partially ordered set P we can equip itwith its Alexandroff topology; that is, the topology with closed sets given by thesets that are closed below under the partial order on P . By abuse of notation, wealso refer to this space as P , as it contains all the data of the original partiallyordered set. Then P is in fact a ∆-generated space. To see this, we need to showthat a set whose inverse image under every map from the realization of a n -simplex | ∆ n | → P is closed is a downset. Let A be such a set. In fact, it suffices to probewith intervals instead of arbitrary simplices. For p ∈ A , p (cid:48) ≤ p consider the map σ : I −→ P [0 , . (cid:51) t (cid:55)−→ p (cid:48) (0 . , (cid:51) t (cid:55)−→ p. This is clearly continuous. But as (0 . ,
1] is not closed, we have p ∈ A = ⇒ p (cid:48) ∈ A. Thus, the Alexandroff topology construction induces a functor
Poset −→ Top , HAPTER 1. STRATIFIED HOMOTOPY THEORY P , we denote this category by Top P . Top P is studied in detailin the next subsection. • Filtered simplicial complexes:
Given a partially ordered set, the subset ofits power set, given by all finite linearly ordered subset (flags) of P , sd( P ) is asimplicial complex, called the nerve of P . Even more, it is an ordered simplicialcomplex (see Section 1.3.4) in the obvious way. The induced covariant map onpower sets makes this construction functorial, inducing functorssd( − ) : Poset −→ sCplx sd( − ) : Poset −→ sCplx o . We denote the induced categories of (ordered) filtered simplicial complexes by sCplx P , ( sCplx o P ). They are studied in detail in Section 1.3. • Filtered simplicial sets:
Consider the fully faithful embedding of the simplexcategory ∆ into
Poset . Using this embedding, one obtains the nerve functor N ( − ) : Poset → s Set , by sending P to ∆ (cid:44) → Poset
Hom
Poset ( − ,P ) −−−−−−−−−−→ Set . Explicitly, the n -simplices of N ( P ) are given by (not necessarily strictly) increasingsequences in P of length n . In other words, they are given by the d-flags in P .The length n of such a d-flag J is denoted by J . Note that N ( P ) can bealternatively thought of as the image of sd( P ) under a Yoneda-style embedding sCplx o (cid:44) → s Set . We do this in detail in Section 1.3.4. The induced category of P -filtered simplicial sets is be denoted by s Set P . If ∆ n → N ( P ) is the filtrationof an n -simplex, then by definition of N ( P ), it is uniquely determined by a d -flag J = ( p ≤ ... ≤ p n ). We denote such a filtered simplex by ∆ J . We begin with amore detailed description of s Set P in Section 1.1.2. Example 1.0.3.
It can also be interesting to consider different cases of P , for theConstruction 1.0.1. For example if P = (cid:63) , the one-point set, and F is a functor thatpreserves terminal objects, then the category C /F ( (cid:63) ) is the arrow category over a terminalobject, which is isomorphic to the original category C under the forgetful functor. Hence,in Example 1.0.2, we just obtain the categories Top , sCplx , sCplx o and s Set .Construction 1.0.1 is functorial in the following sense. The reader will of coursenotice that the following construction can be decomposed into a few varieties of smallerfunctorialities, but as the examples relevant to us are mostly of the following particularshape this summarized one makes it easier to refer to this construction.
HAPTER 1. STRATIFIED HOMOTOPY THEORY Construction 1.0.4.
Let C and D be two categories and P, Q ∈ Poset . Consider adiagram of functors
C D
Poset
LF G together with a morphism f : LF ( P ) → G ( Q ). Then, for each P ∈ Poset , we obtain afunctor C /F ( P ) → D /G ( Q ) , by applying L to arrows, and then post composing them with f . Dual to this, if we aregiven a diagram of functors C D
Poset
RF G together with a morphism g : F ( P ) → RG ( Q ) which C has pullbacks along, then weobtain a functor D /G ( Q ) → C /F ( P ) by first applying R to each arrow, and then changing base (i.e. pulling back) along g .This is particularly interesting in the case, where L (cid:97) R are an adjoint pair and f and g correspond to each other under this adjunction. Then, as base change and postcompo-sition are also adjoint, one obtains that the induced functors also form an adjoint pair(with the same left-right positions).The first main example one should consider is the case where Q ⊂ P is a subposetand R is given by the identity functor. In this case this leads to an abstract definitionof strata. Definition 1.0.5.
In the setting of Construction 1.0.1, fix some partially ordered sets Q ⊂ F . Then we denote by ( − ) Q : C /P → C /Q the functor obtained by basechanging along F ( Q ) → F ( P ) (as in Construction 1.0.4). If Q = { p } is a one point set, and F preserves terminal objects we can also think of ( − ) { p } as having target in C /F ( { p } ) ∼ = C . Then, for C ∈ C , we denote this by C p and call it the p -th stratum of C . Furthermore, also for p ∈ P , we denote C ≤ p := C { q ≤ p } . Example 1.0.6. If C = Top and P = [ n ] is some finite linearly ordered set, then afiltered space is a filtered space in the sense of Friedman (see for [Fri20] (up to beingHausdorff)). Then the strata in our sense are the (∆-fication of) the strata in the usualsense. The p -th filtration space, denoted T p in Friedmans book, is then given by (theunderlying space of) T ≤ p . HAPTER 1. STRATIFIED HOMOTOPY THEORY Remark 1.0.7.
We should make another remark on notation at this point. The reasonwe chose not to go with the classical notation - of using lower indices for strata andsuperscript indices for filtration degrees - is that in any sort of inductive proof onequickly runs out of places where one can put their indices. In fact, our situation is alreadyconfusing enough, as it is standard to use the lower index for the set of n -simplices in asimplicial set X . We always denote this set by X ([ n ]) to avoid any possible confusion.The superscript is reserved for enumeration of objects.Other frequently occurring examples of functors arising through Construction 1.0.4are the following. Example 1.0.8.
We consider a few examples of functors induced in the filtered setting. • Subdivisions : Recall the barycentric subdivision functors, constructed as follows.On a simplex, take its power set as a partially ordered set, then apply sd( − ) or N ( − ) respectively, inducing a functor ∆ op → C , where C is either s Set , sCplx o or sCplx . This is then left Kan extended to the category of (ordered) simplicialcomplexes and simplicial sets respectively. More explicitly for simplicial sets oneextends via sd( X ) = lim −→ ∆ n → X,n ∈ N sd(∆ n ) and for a simplicial complex K , one justtakes sd( K ) to be the functor of Example 1.0.2 applied to the set of simplices of K ordered by inclusion. For ordered simplicial complexes and more generally orderedsimplicial sets, sd comes together with a last vertex mapl.v. : sd(∆ n ) −→ ∆ n , induced on the simplex level by sending a vertex in the subdivision σ = ( x ≤ ... ≤ x k ) to x k , the maximal vertex with respect to the linear ordering on σ . Thisinduces a natural transformationl.v. : sd( − ) → C , where C is either s Set or sCplx o . Now, in Construction 1.0.4, for C either ofthe three categories above, L = sd the respective subdivision functor and f thecorresponding last vertex map sd( N ( P )) → N ( P ) , sd ( P ) → sd( P )we obtain filtered barycentric subdivision functors. Explicitly, the filtration of asubdivision of a simplex of X is given, by assigning to each vertex σ the maximal p in p X ( σ ).Figure 1.1: Subdivision of the filtered simplex over { , } given by J = (0 ≤ ≤ HAPTER 1. STRATIFIED HOMOTOPY THEORY − ). This is also todistinguish them, from another type of subdivision functor that we introduce lateron in Construction 1.1.31. In the case where C is either the category of filteredsimplicial sets or of ordered simplicial complexes, the last vertex map is filteredand hence induces a natural transformationl.v. : sd( − ) −→ C /P . This particular type of subdivision is of course be central to our investigation ofthe filtered simplicial approximation theorem in Section 1.3. • Realizations : Let | − | be either of the classical realization functors from s
Set , sCplx o , sCplx into Top . Then the realizations of | sd( P ) | = | N ( P ) | are naturallyfiltered as follows. Each realized simplex in | sd( P ) | is given by the convex span (cid:110) (cid:88) p ∈J t p | p | ⊂ R J | t p ≥ , (cid:88) p ∈J t p = 1 (cid:111) , where J is some flag in P and we denote by | p | the unit vector in R J correspondingto p . Then, we send such an element (cid:80) p ∈J t p | p | to max { p ∈ J | t p > } . This de-fines a natural transformation between | sd( − ) | and the Alexandroff space functor.Taking L to be | − | and f to be this natural transformation in Construction 1.0.4then induces filtered realization functors | − | P : C P −→ Top P , again all denoted the same. Explicitly, the interior of a simplex σ ∈ X is assignedto the stratum corresponding to max { p X ( σ ) } .Figure 1.2: Filtration and induced strata of the realization of a standard simplex corre-sponding to the flag J = { ≤ ≤ } over P = { , , } .These functors are of course compatible with the various subdivision and embed-ding functors between different choices of C , described in more detail in Section 1.3. HAPTER 1. STRATIFIED HOMOTOPY THEORY • Singular simplicial sets: | − | is of particular interest when C is the categoryof simplicial sets, as then it admits a right adjoint given by the singular simplicialset functor Sing : Top −→ s Set (see for ex [GJ12]). Now, take g : N ( P ) → Sing( P ) to be the adjoint mapfrom the natural map | N ( P ) | → P , from the last bulletpoint and R = Sing( − ).Then Construction 1.0.4 induces a right adjoint Sing P ( − ) to | − | P . Explicitly,Sing P ( X ) ⊂ Sing( X ) is given only by such singular simplices, that are of theshape σ : | ∆ J | P → X , for d-flags J . The filtration is then given by mapping σ to J . | − | P and Sing P ( − ) are of course be the integral instruments of connecting homotopytheory in the simplicial and the topological setting.To finish this section, we should make a quick note on how limits and colimits arecomputed in these categories of filtered objects. Remark 1.0.9.
Let C be an arbitrary category and c ∈ C . It is an easily verified fact onover categories that if C is complete (cocomplete) (as Top and s
Set are), then so is C /c .Limits of a diagram F : J → C /c are computed by first composing with the forgetfulfunctor into C , then adding c and the arrows into it to the resulting diagram, and finallytaking the limit of this diagram in C and its structure morphism into c as the limit in C /c . For example, the product in s Set P of X and Y is given by the diagonal of thepullback square X × N ( P ) Y XY N ( P ) p X p Y .Colimits are computed, by again taking the composition with the forgetful functor, andtaking the colimit in C together with its induced arrow into c . That is, the forgetfulfunctor C /c → C preserves and reflect colimits. In particular, colimits in s Set P or Top P are computed on the underlying simplicial sets (spaces), and then naturally filtered.We should end this subsection with a quick remark on possible linguistic confusionthat may arise. Remark 1.0.10.
Sadly, the nomenclature around stratified and filtered objects is some-what inconsistent. This is in particular the case, when it comes to when something shouldbe called stratified as opposed to filtered. To just give a small outlook on the choice ofpossible nomenclature, here is a taste: In [Wei94], a map of spaces filtered over the same P is said to be stratified, if it is stratum preserving in our sense. In [Dou19b], such amap is called a filtered map. A filtered map, in [Wei94] however, is one that satisfies f ( X ≤ k ) ⊂ Y ≤ k instead of f ( X k ) ⊂ Y k . The term stratified, is reserved for maps betweenfiltered spaces over different posets, given by commutative diagrams HAPTER 1. STRATIFIED HOMOTOPY THEORY X YP Q. p X p Y In [Hai18], a stratified space is just what we defined to be a filtered space. In [CSAT18]and [Fri20], they are however defined to be filtered space that additionally satisfy the socalled Frontier Condition. Classically, stratified spaces are often expected to have stratathat at least homologically are manifold like. As to not add to the confusion, we onlyuse the term stratified space with additional qualifiers such as homotopically or locallyconelike. When it comes to maps, we decided to go with stratum preserving, simplybecause it seems to be the most descriptive. Sadly, this has the awkward side effect thatthe objects and morphisms in our categories are not named analogously. In particular,when we are citing from [Dou18] and [Dou19a], one should be careful to do the necessarytranslations.
Our study of stratified homotopy theory does heavily apply the language of model cat-egories. We do not start the undertaking of giving an introduction to this theory here,as it would go beyond the scope of this master thesis. We recommend [Hir09] for an in-troduction. There are several approaches that attempt to use model categories to studystratified homotopy theory out there at the moment. We have decided to go with theone recently published by Douteau in his thesis [Dou19b] which is partially summarizedin [Dou19a] and [Dou18]. Roughly speaking, the perspective of homotopy theory comedown to defining weak equivalences to be morphisms that induce weak equivalences(in the classical sense) on all strata and (generalized) links (the spaces connecting thestrata), as opposed to stratified homotopy equivalences. Throughout this thesis, we tryto motivate why we think this setting is appropriate in particular for our study of simplehomotopy theory. In this section, we give a quick introduction into the work done inthese publications. The intention however is mostly one of giving and motivating defini-tions, and for all proofs we refer to the respective sources. The reader familiar with theresults in [Dou19a] and [Dou18] can probably skip this section as our own work beginsfrom the section following it. P -filtered topologicalspaces It is a frequent phenomenon in mathematics that for the successful study of a restrictiveclass of objects it can be immensely helpful to pass to a larger class. Doing so onegains access to a greater range of operations without constantly having to worry aboutleaving the restrictive setting. In some sense, this is what filtered spaces are to stratifiedspaces. Intersection homology for example, while originally defined and intended onlyfor pseudomanifolds, extends to all spaces filtered over a finite linear set (see [Kin85]).
HAPTER 1. STRATIFIED HOMOTOPY THEORY P , Top P , introduced inExample 1.0.2. We start this subsection by giving a less abstract description of objectsand morphisms in Top P and introduce the various enrichments of it. We then startinvestigating notions of homotopy in this setting. In doing so, we also summarize themost important results on the model structure defined by Douteau on this category in[Dou19a] and [Dou19b]. We mostly follow [Dou19b] for this.In Example 1.0.2 we defined the category of P -filtered topological spaces to be the over-category of Top over P , where the latter is thought of as a topological space, equippedwith the Alexandroff topology. Remark 1.1.1. A P -filtered topological space X can alternatively be defined as a space X ∈ Top together with a family ( X ≤ p ) p ∈ P of closed subspaces such that X p ⊂ X p (cid:48) ⇐⇒ p ≤ p (cid:48) , for p, p (cid:48) ∈ P . The strata X p of X are then explicitly given by the subspace (in the∆-generated sense) of X given by X p = X ≤ p − (cid:91) p (cid:48)
Top P for appropriate P . Taking this point of view, a morphism in Top P f : X → Y alternatively described as a map of the underlying topological spaces suchthat f ( X p ) ⊂ Y p ,for all p ∈ P. This justifies calling them stratum preserving.Figure 1.3: Visualization of a filtered space over P = [2] and its strata, obtained bytaking the wedge sum of a S and a D . HAPTER 1. STRATIFIED HOMOTOPY THEORY
Top P is enriched and copowered (see [nLa20f] and [nLa20d] for defini-tions) over the closed monoidal category Top . This is accomplished as follows.
Construction 1.1.2.
The category
Top is a closed monoidal category enriched overitself. To be more precise, one obtains a monoidal structure through the product, − ×− . And an inner hom-functor by equipping Hom
Top ( T, T (cid:48) ) with the ∆-ification of thecompact open topology, for
T, T (cid:48) ∈ Top (see also [Dug20]). We denote the resultingspace by C ( T, T (cid:48) ). We now equip Hom
Top P ( X, Y ) with the (∆-generated) subspacetopology with respect to the inclusionHom
Top P ( X, Y ) ⊂ C ( X, Y ) , where we omitted the forgetful functor into Top in the right mapping space. We denotethese spaces by C P ( X, Y ). Furthermore,
Top P is copowered over Top , with the copower − ⊗ X (cid:97) C P ( X, − ) given by the P -filtered space T × X π X −−→ X p X −−→ P. for T ∈ Top and X ∈ Top P . For details see [Dou19b, Sec. 5.2].The copowering with Top naturally induces a notion of homotopy on
Top P . Definition 1.1.3.
Two stratum preserving maps of P -filtered spaces f , f : X → Y are said to be stratum preserving homotopic or also stratified homotopic if there exists astratum preserving map H : I ⊗ X → Y such that the diagram X (cid:116) X ∼ = ( (cid:63) (cid:116) (cid:63) ) ⊗ X YI ⊗ X f (cid:116) f i ⊗ H commutes. i : (cid:63) (cid:116) (cid:63) (cid:44) → I denotes the inclusion of the endpoints here. It is an easyverification that this notion of homotopy behaves much like the classical one, that is,being homotopic is symmetric, reflexive, transitive and compatible with compositionsetc. We use most of the standard nomenclature, also used in the non-filtered setting,adding the prefix stratum preserving or stratified . So by nomenclature such as stratifiedhomotopy classes, stratum preserving homotopy equivalence, etc. we mean the obviousthing. Stratum preserving homotopy equivalences are also often referred to as stratifiedhomotopy equivalences. For X, Y ∈ Top P , we denote by [ X, Y ] P the set of stratifiedhomotopy classes of stratum preserving maps X → Y . Furthermore, stratum preservinghomotopy between morphisms in Top P is denoted by (cid:39) P .It is a straighforward verification using standard properties of mapping spaces thatthis notion of homotopy is compatible with the enriched structure in the following sense. Lemma 1.1.4.
The enriched hom functor C P ( − , − ) : Top opP × Top P → Top sends stratified homotopic morphisms in
Top P into homotopic ones. In particular, strat-ified homotopy equivalence induce homotopy equivalences on C P ( − , − ) . HAPTER 1. STRATIFIED HOMOTOPY THEORY Remark 1.1.5.
For many intents and purposes, stratum preserving homotopy equiva-lence is a very useful notion of isomorphism. Singular intersection homology for example,descends to a functor on stratified homotopy classes, and hence is a stratified homotopyequivalence invariant [Fri20, Prop. 4.1.10.]. However, for many other intends and pur-poses, such as the simple homotopy theory (see also our comments in Section 2.2.1)we define in this work, stratified homotopy equivalence seems to be too discrete of anequivalence relation. Consider for example the map indicated in the following.Figure 1.4: Illustration of a stratum preserving map between two filtrations of S . It isgiven by collapsing the blue region to a point.This map clearly induces a homotopy equivalence of the underlying spaces. However,it is not a stratum preserving homotopy equivalence. Geometrically, this is since thereis no way of continuously mapping the cycle on the right to the one on the left, withoutmapping the red into the blue part. An easy way to see this more formally is to computethe intersection homology (see for ex. [Fri20]), I H ( − , Q ) (simplicially) for both sides.As the left hand side gives a filtration of S that turns it into CS set by invariance offiltration (see [Fri20, Ch. 2]) we obtain: I H ( lhs, Q ) = H ( S , Q ) = Q . For the right hand side we can not make a similar argument, as the 0-stratum is of thewrong dimension. But an easy computation, or the geometric argument that no 1-cyclecan intersect the 0-stratum transversally, shows that I H ( rhs, Q ) = 0 . Similarly to passing from homotopy equivalences to weak homotopy equivalences, it turnsout to be useful to broaden the notion of stratum preserving homotopy equivalence alittle, to a notion lying between stratum preserving homotopy equivalence and (weak)homotopy equivalence of the underlying spaces.To do so, we need the notion of homotopy links.
Construction 1.1.6.
Let J be a flag in P and X ∈ Top P . Consider the space C P ( | ∆ J | P , X ) ∈ Top . If J = { p } , then this gives the p -th stratum. If J = ( p < p ), HAPTER 1. STRATIFIED HOMOTOPY THEORY p -th stratum in the p -th. Following this nomenclature, we call C P ( | ∆ J | P , X ) the J -thhomotopy link of X and denote it by Hol P ( J , X ) . We also call such objects generalized homotopy links. Functoriality of C P and | − | P induces a functor Hol P : Top P −→ Top sd( P ) op into the category of space valued presheaves on sd( P ), where we think of sd( P ) as acategory by taking the category induced by the partial order of inclusion of simplices. Remark 1.1.7.
One should motivate a little bit why the space in Construction 1.1.6 arecalled links. Recall that, for a simplicial complex K with a full subcomplex S ⊂ K , thelink of S in K is the subcomplex given by all simplices that do not intersect K but arecontained in a simplex doing so. For an illustration, see Figure 1.5. In a sense, the linkcontains geometric information about how S sits in K . If one takes a first barycentricsubdivision first, then furthermore the link has a natural map into S , given by sendinga vertex σ ∈ sd( K ) given by a simplex σ of K to its intersection with S . Then the linkgives the boundary of a mapping cylinder neighbourhood of | S | in | K | , induced by thismap. For piecewise linear filtered spaces (that is, PL spaces filtered by closed PL spaces)one can always construct a regular neighbourhood in this way and the link is unique upto PL homeomorphism.Figure 1.5: An illustration of the link and of (the homotopy type of the) homotopy linksof a PL-model of the space in Figure 1.3. This is computed using Proposition 1.1.8. HAPTER 1. STRATIFIED HOMOTOPY THEORY J = ( p < p )) gives a functorialreplacement for this. It is in fact a good replacement in the following sense. Proposition 1.1.8.
Let X be a metric space filtered over P = { , } . Let N be aneighbourhood of X in X , such that there exists a deformation retraction r : N × I → N of N into X which is stratum preserving up to t = 1 . Then N (with the inducedfiltration) is filtered homotopy equivalent to the P -filtered space (cid:16) X ⊂ Cyl ( f ) (cid:17) , where Cyl ( f ) is the mapping cylinder (with the teardrop topology, see [Qui88]) of the startingpoint evaluation map f : Hol P (cid:0) (0 ≤ , X (cid:1) −→ Hol P (cid:0) (0) , X (cid:1) ∼ = X induced by the inclusion (0) (cid:44) → (0 ≤ . For a proof see [Qui88, Lem, 2.4] with a correction found in [Fri03, A. 1]. In par-ticular, the homotopy link is actually homotopy equivalent to the link in the PL sense,wherever that exists.It is an immediate consequence of Lemma 1.1.4 that Hol P ( J , − ) sends stratum pre-serving homotopy equivalences into (weak) homotopy equivalences. It turns out that,for most examples of stratified spaces, the converse holds. Such a result was first statedby Miller in [Mil13, Thm. 6.3] and it has been the starting point of a wealth of in-vestigations into stratified homotopy theory. We use the following Whitehead-Theoremstyle version of such a result, shown in [Dou18]. First however, we have need for a fewdefinitions in the simplicial setting. Definition 1.1.9.
Let J = ( p ≤ ... ≤ p n ) be a d -flag in P of length n . Consider a horninclusion Λ nk (cid:44) → ∆ n . Denote by Λ J k ⊂ ∆ J the P -filtered space given byΛ nk (cid:44) → ∆ n p ∆ J −−−→ N ( P ) . We call such a horn inclusion Λ J k (cid:44) → ∆ J admissible if p k = p k +1 or p k = p k − ; that is,if J is degenerate with the k -th vertex repeated. We also say that k is J - admissible . Example 1.1.10.
The following illustrates a few examples of admissible and not ad-missible horn inclusions.Figure 1.6: Examples of horn inclusions of 2-simplices, filtered over P = 0 , HAPTER 1. STRATIFIED HOMOTOPY THEORY J = (0 ≤ ≤
1) and k = 2 and 0 respectively.(2) and (4) show horn inclusions for J = (0 ≤ ≤
1) and k = 0 and 2 respectively. Outof these, only (3) and (4) are admissible.Later on, in our investigation of stratified simple homotopy theory, admissible horninclusions will be the elementary building blocks for simple equivalences. For now, thefollowing homotopical characterization should make the definition of admissible horninclusion a little more motivated. Lemma 1.1.11.
A horn inclusion i k : ∆ J k (cid:44) → ∆ J is admissible if and only if | i k | P is astratum preserving homotopy equivalence.Proof. Let J = ( p ≤ ... ≤ p n ). For the only if part, let without loss of generality p k = p k +1 . Now, consider the deformation retraction of | ∆ n | onto its ( k + 1)-th facegiven by R : I ⊗ | ∆ J | P −→ | ∆ J | P ( s, (cid:88) i ∈ [ n ] t i | p i | ) (cid:55)−→ (cid:88) i ∈ [ n ] \{ k,k +1 } t i | p i | + ( st k +1 + t k ) | p k | + (1 − s ) t k +1 | p k +1 | . This is easily checked to be stratum preserving (elementarily, as a quick consequenceof Lemma 1.3.11, or by using an underlying simplicial stratum preserving homotopy asin [Dou18, Prop 1.13.]). Furthermore, it restricts to a deformation retraction of | ∆ J k | P onto the ( k + 1)-th face. Now, for the converse, assume that | i k | P : | Λ J k | P (cid:44) → | ∆ J | P isa stratum preserving homotopy equivalence and p k is not repeated in J . Consider therestriction of | i k | P to Q = P \ { p k } ⊂ P .Figure 1.7: An illustration of ( | Λ J k | ) Q , for k = 0 and J = (0 ≤ ≤
1) and J = (0 ≤ ≤ Top copower struc-tures on
Top P and Top Q are clearly compatible under the base change functor). In HAPTER 1. STRATIFIED HOMOTOPY THEORY (cid:0) | ∆ J | P (cid:1) Q = { (cid:88) i ∈ [ n ] t i | p i | ∈ | ∆ J | P | t i = 0 for i < k = ⇒ t k = 0 } This set is clearly convex, hence contractible. For the horn we obtain (cid:0) | Λ J k | P (cid:1) Q = .. . (cid:110) (cid:88) i ∈ [ n ] t i | p i | ∈ | ∆ J | P | (cid:0) t i = 0, for i < k = ⇒ t k = 0 (cid:1) and (cid:0) t i = 0 for some i ∈ [ n ] , i (cid:54) = k (cid:1)(cid:111) which contracts linearly onto the boundary of the k -th face via the linear homotopybetween the identity and (cid:88) i ∈ [ n ] t i | p i | (cid:55)−→ (cid:88) i ∈ [ n ] \{ k } t i − t k | p i | . In particular, the underlying space of (Λ J k ) Q is homotopic to S n − (hence empty for n = 1), in contradiction to the contractibility of the one underlying (∆ J ) Q .If one now takes the perspective that realizations of simplicial subsets should givecofibrations (in the sense of model categories) in Top P , then in light of Lemma 1.1.11the following definition should not be surprising. Definition 1.1.12. A P -filtered space X ∈ Top P is called an f-stratified space (wherethe f stands for fibrant) if it has the right lifting property with respect to all realizationsof admissible horns inclusions; that is, if for each d -flag J in P , and k J -admissible andeach solid diagram in Top P as below, a dashed arrow making the diagram commuteexists. | Λ J k | P X | ∆ J | P Remark 1.1.13.
These spaces are often called fibrant spaces as their Sing P ( − ) is fibrantwith respect to a certain model structure, defined later on in Section 1.1.2. We find thisnomenclature a bit confusing, as it conflicts with the model structure defined on Top P we will be using. It seems to be an open question whether there exists a model structureon Top P with respect to which these spaces are actually the fibrant objects. We thusdecided to call them f-stratified. Example 1.1.14.
Most examples of stratified spaces encountered in practice are f-stratified. More explicitly all homotopically stratified metric spaces with finite stratifi-cation (see [NL19, Prop. 8.1.2.6.]) and all conically stratified spaces (see [Dou18, Prop.4.212]) are f-stratified. Hence, the class of all f-stratified spaces also includes variousof the classical definitions of stratified space, such as Whitney stratified, Thom-Matherstratified and topologically stratified spaces (see for example [NL19, Rem. 5.1.0.14]).
HAPTER 1. STRATIFIED HOMOTOPY THEORY
Theorem 1.1.15. [Dou18][Thm. 4.23] Let g : X → Y be a stratum preserving map off-stratified spaces over P that are stratum preserving homeomorphic to the realizationsof P -filtered simplicials sets. Then g is a stratum preserving homotopy equivalence ifand only if, for each flag J of P , Hol P ( J , g ) is a weak homotopy equivalence. Remark 1.1.16.
If one strengthens the requirements of Theorem 1.1.15 to X and Y being conically stratified, then it in fact suffices to check for flags of length ≤
2, i.e. forhomotopy links and strata. (See [Dou18, Cor. 5.12]).The approach to stratified homotopy theory we follow here is essentially to takethe equivalent characterization of stratum preserving homotopy equivalences in Theo-rem 1.1.15 for a definition. For most stratified spaces classically encountered, the newnotion of equivalence just coincides with stratum preserving (stratified) homotopy equiv-alences. However, for filtered spaces farther away from what is typically called stratified,this provides extra degrees of freedom allowing for the construction of a combinatorialsimple homotopy theory in Section 2.3.1 (see also Section 2.2.1 for reasons why this no-tion is preferable). In fact, this notion of weak equivalence fits into a (simplicial) modelstructure on
Top P . This was originally shown in [Dou19a], and is formulated in theform we are going to use in [Hai18, 1.3.7]. Essentially what one does is post-composethe Hol functor with Sing. One then right-transfers the projective model structure ons Set sd( P ) over to Top P along this functor (see [nLa20a] for a definition of combinatorialmodel categories). Theorem 1.1.17.
There exists a combinatorial model structure on
Top P , which isexplicitly described as follows. • f : X → Y is a weak equivalence if and only if Hol P ( J , f ) is a weak equivalencefor each flag J in sd( P ) . • f : X → Y is a fibration if and only if Hol P ( J , f ) is a serre fibration for each flag J in sd( P ) . • The acyclic cofibrations are generated by (cid:8) | Λ nk | ⊗ | ∆ J | P (cid:44) → | ∆ n | ⊗ | ∆ J | P | J ∈ sd( P ) , n ≥ , k ∈ [ n ] (cid:9) . • The cofibrations are generated by (cid:8) | ∂ ∆ n | ⊗ | ∆ J | P (cid:44) → | ∆ n | ⊗ | ∆ J | P | J ∈ sd( P ) , n ≥ (cid:9) . In particular, by a weak equivalence of P -stratified spaces we mean a P -stratifiedmap as in the first bullet point of Theorem 1.1.17. We denote by H Top P the homotopy HAPTER 1. STRATIFIED HOMOTOPY THEORY
Top P . We call this modelstructure the Henrique-Douteau model structure on Top P , following the nomenclaturein [Hai18]. The obvious question arises how the category H Top P differs from the oneobtained by only localizing stratum preserving homotopy equivalences. Essentially thedifference comes down to allowing certain thickenings of strata. Example 1.1.18.
Consider the stratum preserving map shown in Figure 1.4. We havealready seen that it is not a stratum preserving homotopy equivalence. However, it is aweak equivalence of filtered spaces. If X denotes the space on the left, and Y the spaceon the right, then Proposition 1.1.8 shows that (up to homotopy equivalences) the mapson homotopy links are given as in the following figure.Figure 1.8: Illustration of the induced maps on generalized homotopy links of Figure 1.4,up to homotopy equivalence.This of course is not in contradiction to Theorem 1.1.15 as X is not f-stratified.This strata thickening interpretation is made more precise by Proposition 1.1.39 lateron. However, as long as one is interested in most classical examples of stratified spaces(and willing to allow for some compactness restrictions), these two categories actuallyagree. We have shown this in Corollary 1.4.4. Remark 1.1.19.
The choice of calling such maps weak equivalences is not purely dueto consistency with model theoretic language. In fact, one can define filtered homotopygroups, which essentially come down to being the homotopy groups of the generalizedhomotopy links. Then a weak equivalence is precisely one that defines isomorphismson all of these homotopy groups (for appropriate choice of basepoints). For details see[Dou18].
Remark 1.1.20.
The astute reader might be somewhat surprised by the choice of fibra-tions and cofibrations in Theorem 1.1.17. Especially, after the definition of f-stratifiedspace, one would expect the acyclic cofibrations to be generated by the realizations ofadmissible horn inclusions, and the fibrations to be defined accordingly. In particular,this would make the f-stratified spaces the fibrant objects and justify this nomenclature.In fact, it seems highly likely that such a model structure does exist, and can be ob-tained by right-transfer from one defined on the category s
Set P (see Theorem 1.1.28). HAPTER 1. STRATIFIED HOMOTOPY THEORY
Top P is actually a simplicialone (see [GJ12, Ch.2, Sec. 2] for a definition). The additional simplicial structure, isconstructed as follows. Construction 1.1.21.
The enrichment and copower structure of
Top P over Top in-duces one over s
Set as follows. Let
X, Y ∈ Top P and S ∈ s Set . • The copowering is given by S ⊗ X = | S | ⊗ X. • The simplicial enrichment is then induced, by defining
M ap ( X, Y ) : ∆ op (cid:44) → s Set −⊗ X −−−→ Top P Hom
Top P ( − ,Y ) −−−−−−−−−−→ Set ;that is,
M ap ( X, Y )([ n ]) = Hom Top P (∆ n ⊗ X, Y ). • Finally, the power structure X S is induced by the inclusion (cid:71) p ∈ P Hom
Top ( | S | , X p ) (cid:44) → C ( | S | , X );that is, one equips the set of maps | S | → X whose image only intersects one stratumin X with the (∆-generated) subspace topology, with respect to this inclusion.All of these are functorial in both arguments and checked to define a simplicial structurein [Dou19b, 5.1.3].Then we have: Proposition 1.1.22. [Hai18] The model structure defined in Theorem 1.1.17 is combi-natorial and simplicial.
This ends our introduction into the category
Top P . P -filtered simplicial sets In some sense, the study of simple homotopy theory can be understood as asking thequestion: Can a specified ”weak equivalence” be witnessed in a purely combinatorialway and if yes, how can we tell? In particular, to even make this question well-defined,one needs a combinatorial model for the homotopy setting that one is working in. Orig-inally, Whitehead used simplicial complexes ([Whi39]) as such a model. However, the
HAPTER 1. STRATIFIED HOMOTOPY THEORY P , which we defined in Example 1.0.2. For a more detailed introductionincluding proofs we refer to [Dou18], [Dou19b] or [Hai18]. We should first make the no-tational remark that we denote the set of n simplices of a (filtered) simplicial set X by X ([ n ]) and not by X n , as is usually done, to avoid any possible confusion with the strata.To begin with, it can be helpful to obtain a somewhat more explicit and geometricaldescription of the category of P -filtered simplicial sets. Remark 1.1.23.
First note that N ( P ) ∈ s Set is not just any simplicial set; It is onethat comes from an ordered simplicial complex. We analyze this relationship in moredetail in Section 1.3.4, but for now it suffices to realize that a n -simplex in N ( P ) isentirely determined by its family of vertices, that is, its images under the face mapsinduced by the inclusions ∆ (cid:44) → ∆ n . We denote these vertices for σ ∈ S ([ n ]) by x i,σ , for i ∈ [ n ]. In particular, a simplicial map from an arbitrary simplicial set f : S → N ( P )is entirely determined by f [0] : S ([0]) → N ( P )([0]) = P . Conversely, a map on the0-skeletons g : S ([0]) → P extends to a simplicial map f : S → N ( P ) if and only if foreach simplex σ ∈ S ([ n ]), g ( x ,σ ) ≤ ... ≤ g ( x n,σ ) . HAPTER 1. STRATIFIED HOMOTOPY THEORY P -filteredsimplicial set can simply be thought of as a simplicial set X together with a map p X : X ([0]) → P , such that whenever two vertices x and x are connected by a 1-simplexfrom x to x in X , then p X ( x ) ≤ p X ( x ). In this formulation, morphism f : X → Y is simply a map of the underlying simplicial sets such that p X ( x ) = p Y ( f ( x )) , for all vertices x of X . Again, this justifies the term stratum preserving. Example 1.1.24.
The following figure shows two simplicial sets whose vertices havebeen assigned values in P = { , } . However, only the first one of the two is a filteredsimplicial set in the sense of Remark 1.1.23. In the second one, the upper left 1-simplexis attached with the wrong orientation.Figure 1.9: Examples of two simplicial sets, whose vertices have been assigned values in P = { , } .While this perspective is very useful for a visual understanding and for verificationthat a map stratum preserving, there is another perspective that turns out to be verypowerful from a model theoretic point of view. Construction 1.1.25.
We denote by ∆( P ), the category of simplices of N ( P ); thatis the subcategory of s Set P of filtered simplicial sets with underlying simplicial set astandard simplex. Less abstractly, by Remark 1.1.23, ∆( P ) is equivalently given by theset of d-flags of P , with a morphism of d-flags ( p ≤ ... ≤ p n ) → ( q ≤ ... ≤ q m ) given bya map of posets f : [ n ] → [ m ] such that p f ( i ) = q i . Geometrically this is just the categoryof filtered simplices with stratum preserving face inclusions and degeneracy maps. Now,consider the Yoneda-Style functors Set P (cid:44) → Set ∆( P ) op X (cid:55)→ (cid:8) ∆( P ) (cid:44) → s Set P Hom s Set P ( − ,X ) −−−−−−−−−−→ Set (cid:9) , doing the obvious thing on morphisms. This defines an equivalence of categories. Infact, the analogous statement is true for any category of presheaves, not just for s Set . HAPTER 1. STRATIFIED HOMOTOPY THEORY X ∈ Set ∆( P ) op to ˜ X defined by˜ X ([ n ]) = (cid:71) J ∈ ∆( P ) , J = n X ( J ) , with the obvious induced face and degeneracy maps and functoriality induced by func-toriality of (cid:70) . This is naturally filtered by just sending σ ∈ ˜ X to the d-flag J , denotingits component in the disjoint union. (For more details see [Dou18, Prop. 1.3])This presheaf perspective is be particularly useful as it allows for the usage of workof Cisinski on model structures of presheaves [Cis06].The category s Set P is a simplicial category (see [GJ12, Ch.2, Sec. 2]). The simpli-cial structure is constructed as follows: Construction 1.1.26.
Let S ∈ s Set and
X, Y ∈ s Set P . • We define their outer product S ⊗ X , as the composition S × X π X −−→ X p X −−→ N ( P ) . This is clearly functorial in both arguments and associative (up to natural isomor-phism) with respect to products, in both arguments. This defines the copowering. • The simplicial enrichment on s
Set P is then induced by defining M ap ( X, Y ) : ∆ op (cid:44) → s Set −⊗ X −−−→ s Set P Hom s Set P ( − ,Y ) −−−−−−−−−−→ Set . That is,
M ap ( X, Y )([ n ]) = Hom s Set P (∆ n ⊗ X, Y ). • Finally, the powering is given (in the presheaf perspective of Construction 1.1.25)by X S : ∆( P ) op (cid:44) → s Set
P S ⊗− −−−→ s Set P Hom s Set P ( − ,X ) −−−−−−−−−−→ Set . That is, X S ( J ) = Hom s Set P ( S ⊗ ∆ J , X ) . All of these are clearly functorial, in all arguments involved, in the obvious way. Theydefine a simplicial structure on s
Set P (see [Dou18, Sec. 3]).In particular, we have a notion of (strictly) simplicial homotopy on s Set P and theresulting notions of simplicial homotopy equivalence, simplicial homotopy class etc., in-duced by the cylinder functor given by ∆ ⊗ − (see [Hir09, Ch. 9.5] for definitions.).We will sometimes switch up the order of arguments in this functor, in particular whenwe use it to define homotopies. However, it should always be clear from context whichone the space inducing the filtration is. We denote by [ X, Y ] P the set of simplicial HAPTER 1. STRATIFIED HOMOTOPY THEORY s Set P ( X, Y ) by the equivalence relationgenerated by elementary simplicial homotopies, also called strictly simplicial homotopies.It is of course already clear for the case P = (cid:63) and the classical theory simplicial ho-motopy theory (see for ex. [GJ12]) that strictly simplicial homotopy does not define anequivalence relation for arbitrary targets. For this to work, one needs to pass to a class ofobjects that fulfill certain horn filler conditions. More generally, for simplicial homotopyclasses in a simplicial model category to properly describe the homotopy category, oneneeds the source object to be cofibrant and the target to be fibrant. This is made moreprecise in the following standard proposition of model category theory (see for example[DS95, Prop. 5.11.] together with [Hir09, Prop. 9.5.24.]). Proposition 1.1.27.
Let M be a simplicial model category. Let X, Y ∈ M . If X iscofibrant and Y is fibrant, then the natural map [ X, Y ] → Hom HM ( X, Y ) is a bijection. P -filtered simplicial sets In Construction 1.1.25 we noted that s
Set P can be thought of as the category ofpresheaves Set ∆( P ) op , where ∆( P ) is the category of d -flags in P . In particular, thisopens s Set P up to the usage of work of Cisinski on model structures on categoriesof presheaves ([Cis06]). In [Dou18], Douteau uses this perspective to define a modelstructure on s Set P that makes the simplicial analogue to f -stratified spaces the fibrantobjects. Theorem 1.1.28. [Dou18, Thm. 2.14 and Thm. 3.4] With respect to the simplicialstructure defined in Construction 1.1.26, s Set P can be given the structure of a combi-natorial, simplicial model category such that: • The cofibrations are the monomorphisms, that is, the inclusions of P -filtered sim-plicial sets X (cid:44) → Y . They are generated by the boundary inclusions ∂ ∆ J (cid:44) → ∆ J ,for d -flags J in P . • The acyclic cofibrations are generated by the class of admissible horn inclusions Λ J k (cid:44) → ∆ J , for d -flags J in P and k J -admissible. (Furthermore, they are alsothe saturated class (see Definition 2.2.9) generated by these inclusions). • The weak equivalences are the morphisms f : X → Y that induce bijections onsimplicial homotopy classes f ∗ : [ Y, Z ] P → [ X, Z ] P , for all fibrant Z (in the sense of the definition of the previous item). HAPTER 1. STRATIFIED HOMOTOPY THEORY H s Set P and call this model structurethe Douteau model structure on s Set P . One should directly note the following connec-tion to the topological setting, which is immediate from the adjunction | − | P (cid:97) Sing P . Lemma 1.1.29. [Dou18] X ∈ Top P is a f-stratified space if and only if Sing P ( X ) ∈ s Set P is fibrant, with respect to the Douteau model structure. While the definition of fibration is clearly motivated by the topological setting (seeDefinition 1.1.12), this characterization of weak equivalence is of course completely ab-stract. In fact, the characterization given here is simply the one holding in any simpli-cial, left proper model category, and is completely unspecific (see [Hir09, Thm 7.8.6.]together with [Hir09, Prop. 9.5.24.]). If one wants to get a better understanding of theweak equivalences and the filtered homotopy category, it can be very helpful to have anexplicit description of fibrant replacement functors. Note that in the case where P = (cid:63) this is of course already well understood as then the model structure described in Theo-rem 1.1.28 is just the classical Kan-Quillen model structure on s Set (cid:63) = s
Set P . Douteauexplicitly constructs such a fibrant replacement functor in [Dou18]. We try to motivateits description a little bit here. Recall first the classical setting of the Kan-Quillen modelstructure on the category of simplicial sets. One obtains fibrant replacements as follows(see [GJ12, Ch III. Sec. 4], for a detailed analysis of Ex ∞ ). Remark 1.1.30.
Fibrant replacements in the category of simplicial sets, with respect tothe Kan-Quillen model structure, are in a sense ”geometric up to adjoint”. To be moreprecise, the situation is the following. The subdivision functor s
Set → s Set admits a(simplicial) right adjoint Ex( − ). Explicitly it is defined by the compositionEx( X ) : ∆ op (cid:44) → s Set sd −→ s Set
Hom s Set ( − ,X ) −−−−−−−−−→ Set , i.e. Ex( X )([ n ]) = Hom s Set (sd(∆ n ) , X ), with functoriality induced in the obvious way.Pulling simplices back with the last vertex map, induces a monomorphism X (cid:44) → Ex( X ).A fibrant replacement in the Kan-Quillen model structure is then given by the structuremap into the following limit.1 s Set (cid:44) → Ex (cid:44) → Ex (cid:44) → ... (cid:44) → lim ←− Ex n =: Ex ∞ Geometrically, we should think of Ex ∞ as being the right adjoint to ”infinite barycentricsubdivision”. As every object in the Kan-Quillen model structure is cofibrant, we thenhave Hom H s Set ( X, Y ) ∼ = [ X, Ex ∞ ( Y )] , induced by the inclusion Y (cid:44) → Ex ∞ ( Y ), for X, Y ∈ s Set . In the case where X is finitehowever, a compactness argument shows that every arrow in H s Set is already witnessedby some finite n , and the same holds for every identification of arrows happening whenone passes to the homotopy category. Using the induced adjunction sd n (cid:97) Ex n , thisessentially means that every homotopy class as well as every homotopy is witnessed bysome degree of barycentric subdivision. (We do this argument in detail in the filtered HAPTER 1. STRATIFIED HOMOTOPY THEORY
X, Y ] ∼ = [ | X | , | Y | ], we may understand the fact that Y (cid:44) → Ex ∞ ( Y ) gives a fibrantreplacement as a simplicial approximation theorem. This is also the perspective we takefor the proof of one of our main results (Theorem 1.4.1).Douteau takes the analogous approach for his fibrant replacement. Construction 1.1.31.
Consider the subdivision functor sd : s
Set P → s Set P inducedby subdivision on s Set , as in Example 1.0.8. This is not be the subdivision functor weuse. Instead, define sd P ( N ( P )) as the P -filtered simplicial subset of sd( N ( P )) ⊗ N ( P )(filtered via the second component) defined bysd P ( N ( P ))([ k ]) := (cid:110)(cid:0) ( σ ⊂ ... ⊂ σ k ) , τ (cid:1) | ˆ τ ⊂ ˆ σ (cid:111) ⊂ (cid:0) sd( N ( P )) ⊗ N ( P ) (cid:1) ([ k ]) , where by the hat we denote the unique non-degenerate flags that the d -flags σ and τ degenerate from. Note that this actually defines a simplicial set, i.e. that the definingproperty for simplices is closed under face and degeneracy maps of sd( N ( P )) ⊗ N ( P ).The projection to the first component, defines a (non stratum preserving) simplicial mapsd P ( N ( P )) → sd( N ( P )). Now, for some X ∈ s Set P consider the following pullbackdiagram. X (cid:48) sd( X )sd P ( N ( P )) sd( N ( P )) sd( p X ) X (cid:48) is naturally filtered via the composition X (cid:48) → sd P ( N ( P )) p sd P ( N ( P )) −−−−−−−→ N ( P ). We setsd P ( X ) to this filtered space. This defines a functorsd P : s Set P −→ s Set P , through functoriality of all the constructions involved. We call this functor the ( P -)filtered subdivision functor. The subscript P is there to distinguish it from the subdivi-sion defined in Example 1.0.8. To the latter we refer to as the naive subdivision functor. For details see [Dou18, Subsec. 2.1].
Lemma 1.1.32. sd P preserves all small colimits.Proof. Later on, it will alternatively follow directly from the fact that sd P admits a rightadjoint. Alternatively, it follow from the fact that sd preserves colimits, together withthe fact that base change in Set and hence in any category of presheaves does.
Example 1.1.33.
As a consequence of Lemma 1.1.32 it really suffices to understandwhat sd P looks like when it is applied to filtered simplices ∆ J . In other words, we couldhave alternatively constructed sd P by left Kan-extension of a functor defined only forfiltered simplices. For these it can explicitly computed as:sd P (∆ J )[ k ] ∼ = (cid:110)(cid:0) ( σ , p ) ≤ ... ≤ ( σ k , p k ) (cid:1) | p i ∈ σ , ∀ i (cid:111) ⊂ sd(∆ J ⊗ N ( P )) , HAPTER 1. STRATIFIED HOMOTOPY THEORY ≤ is with respect to the product order induced by ⊂ and ≤ P . The face anddegeneracy maps are induced by leaving out and repeating entries ([Dou18, Subsec.2.1]). Below, we have illustrated two such examples of subdivisions of filtered simplices.Geometrically, this means that sd P can be understood as thickening each stratum intothe higher ones. This can also be seen in the following illustration of sd P for two filteredsimplices. Figure 1.10: Illustration of sd P (∆ J ) for P = { , , } .As a direct consequence of Lemma 1.1.32, we can construct a last vertex map forsd P . Construction 1.1.34.
By the left Kan extension perspective taken in Example 1.1.33,to construct a last vertex map sd P → s Set P it suffices to construct such a map for thefiltered simplices, and show that it is compatible with face and degeneracy maps. Let J be a d-flag in P . By the definition of sd(∆ J ) and the fact that the underlying simplicialset of ∆ J is ∆ n = N ([ n ]), with n = J , it suffices to construct a map on the verticesand show that it extends. Such a map is defined by (cid:0) ( x ≤ ... ≤ x i ) , p (cid:1) (cid:55)−→ max { i ∈ [ k ] | p ∆ J ( x i ) = p } . I.e. instead of taking the last vertex of a simplex, one takes the last vertex lying overthe correct p , thus making the map P -filtered. This, induces a natural transformationl.v. P : sd P −→ s Set P . For details see [Dou18, Def. 3.3.4]. We denote the n -th iteration of this functor by sd nP .Further, we denote by l.v. nP the natural transformation sd nP → s Set P induced by n -timescomposition of l.v. P . HAPTER 1. STRATIFIED HOMOTOPY THEORY sd P . Construction 1.1.35. [Dou18, Def 2.4] Let Ex P : s Set P → s Set P be the functordefined just as in Remark 1.1.30, replacing ∆ by ∆( P ), s Set by s
Set P (thought of thecategory of presheaves over ∆( P )) and sd by sd P . This defines a right adjoint to sd P .As in Remark 1.1.30, through pulling back simplices with l.v. P one obtains a naturalinclusion 1 s Set P (cid:44) → Ex P ( − ). Passing to the limit we obtain a functor Ex ∞ P : s Set P (cid:44) → Ex ∞ P together with a natural inclusion 1 s Set P (cid:44) → Ex ∞ P .This construction defines a fibrant replacement functor in the Douteau model struc-ture on s Set P . Proposition 1.1.36. [Dou18, Lem. 3.3.19 + App. B] The natural transformation s Set P (cid:44) → Ex ∞ P defines a cofibrant fibrant replacement (approximation). We will be making extensive use of this somewhat explicit description of fibrant re-placements. We start this process in Section 1.1.4. One should probably lose a few wordson why sd P is the ”correct” choice of subdivision here, and not the more straightfor-ward sd. First, note the following result, which probably makes the classical subdivisionseem like a quite intuitive choice at first. It is of course well known in the non filteredsetting. It uses Proposition 1.3.19, which essentially states that sd gives a subdivision,in the sense of simplicial complexes, in a filtered way. However, as we only use this forheuristic purposes up to Section 1.3.4, there is no risk of circularity here. Proposition 1.1.37.
For X ∈ s Set P there is an isomorphism | sd( X ) | P ∼ = | X | P . Furthermore, this isomorphism is stratum preserving homotopic to the realization of l.v. :sd → s Set P . If X is a filtered ordered simplicial complex (see Definition 1.3.35), it can betaken to be defined by the barycentric subdivision isomorphism (see proposition 1.3.19).Proof. First note, that in the case of a general filtered simplicial set the respectiveisomorphism is not induced by barycentric subdivision, as the latter is not natural withrespect to degeneracy maps. In fact, the homeomorphisms above can only be madenatural in the homotopy category. The proof is pretty much identical to the (non-trivial) proof found for example in [FP90, Thrm. 4.6.4] if one checks that all maps andhomotopies involved are stratum preserving. However, as all maps there are defined ona simplex level and all homotopies are given by straight line homotopies this is readilyverified.
Remark 1.1.38.
Our goal is to use the model structure on s
Set P for a study of H Top P .If, motivated by the filtered Whitehead theorem (Theorem 1.1.15), we agree that thenotion of fibrancy given by Theorem 1.1.28 is the correct one, then the correct notionof subdivision should be one that allows an adjoint inducing a fibrant replacement asin Remark 1.1.30. We will see in a bit that this works out for sd P . sd however, while HAPTER 1. STRATIFIED HOMOTOPY THEORY P is a consequence of our resultTheorem 1.4.1 together with the results in Section 1.1.4. sd however, can not witness allmorphisms in H Top P . By Proposition 1.1.37, sd witnesses at most the morphisms inthe category obtained by taking stratified homotopy classes in Top P . That it witnessesall of these (between realizations of filtered simplicial complexes) is precisely the contentof Theorem 1.3.44. The category obtained in this fashion, however, is not H Top P .We have already seen that there are countless examples of weak homotopy equivalencesin Top P that are not stratum preserving homotopy equivalences (see Example 1.1.18).The thickenings of strata, described in Example 1.1.33, allow precisely for the additionaldegree of freedom that is obtained when localizing all weak equivalences of P -filteredtopological spaces.The fact that sd P witnesses the passage to H s Set P in the sense of Remark 1.1.38 isalso reflected in the following statement which is integral to the proof of Theorem 1.4.1. Proposition 1.1.39. [Dou19b, Prop. 8.1.1] Let Y ∈ s Set P . Then | sd P ( Y ) | P ∈ Top P is a cofibrant object in the Henrique-Douteau model structure.Proof. The content of [Dou19b, Prop. 8.1.1] actually is a slightly stronger statement,using the strongly filtered category
Top N ( P ) (see Section 1.2.1) but the forgetful func-tor Top N ( P ) → Top P is a left Quillen functor, and hence preserves cofibrant objects([Dou19a, Subsec 2.4]).As a consequence of this, together with the fact that every object in Top P is fibrantand Proposition 1.1.27, we obtain that sd P sends morphisms that realize to weak equiv-alences, to such morphisms that realize to stratum preserving homotopy equivalences. Example 1.1.40.
Consider the following simplicial set model for the weak equivalencein Example 1.1.18. It is not hard to see that the realization of sd P applied to it isactually a stratum preserving homotopy equivalence. HAPTER 1. STRATIFIED HOMOTOPY THEORY P witnesses things happening in the homotopy category H s Set P ,at least as long as one restricts to simplicial sets with finitely many non-degeneratesimplices. So far, morphisms in H s Set P are rather abstract things, at least within a finite frame-work. That is due to the fact that fibrant replacements in s Set P are usually highlycomplicated infinite filtered simplicial sets, even if X is finite. In fact, Ex ∞ P ( X ) is clearlyinfinite whenever X is not a point or the empty simplicial set. However, if we restrictto filtered simplicial sets with finitely many non-degenerate simplices, we can use theadjunction sd P (cid:97) Ex P to give a more explicit description of morphisms in the homotopycategory. Definition 1.1.41.
We call a P -filtered simplicial set X finite if the underlying simplicialset has only finitely many non-degenerate vertices. We denote by s Set finP and H s Set finP the respective full subcategories of s
Set P and H s Set P given by taking finite filteredsimplicial sets as objects. Lemma 1.1.42.
The finite P -filtered simplicial sets are precisely the compact (small)objects in s Set P (see [nLa20b] for a definition).Proof. First off, as colimits are computed on the underlying simplicial sets (Remark 1.0.9)one easily reduces this statement to the respective one about simplicial sets. Now, let
HAPTER 1. STRATIFIED HOMOTOPY THEORY S ∈ s Set P be compact. Then, consider the filtered diagram given by the finite simplicialsubsets of S and inclusions of the latter, S . We have S = lim −→ S as every simplex iscontained in some finite sub simplicial set. By the definition of compactness, the iden-tity 1 S : S → S then factors through some finite S i in the diagram. Hence, S is a subsimplicial set of a finite simplicial set and hence finite. Conversely, let S be a finitesimplicial set, and S (cid:48) be a filtered colimit diagram in s Set . We show that the naturalmorphism lim −→ S (cid:48) i ∈S (cid:48) Hom s Set ( S, S (cid:48) i ) −→ Hom s Set ( S, lim −→ S )is a bijection. Denote by S the diagram given by the non-degenerate simplices of S .Then we have natural bijections:lim −→ S (cid:48) i ∈S (cid:48) Hom s Set ( S, S (cid:48) i ) ∼ = lim −→ S (cid:48) i ∈S (cid:48) Hom s Set (lim −→ S , S (cid:48) i ) ∼ = lim −→ S (cid:48) i ∈S (cid:48) (cid:0) lim ←− ∆ nj ∈S Hom s Set (∆ n j , S (cid:48) i ) (cid:1) ∼ = lim ←− ∆ ni ∈S (cid:0) lim −→ S (cid:48) i ∈S (cid:48) Hom s Set (∆ n j , S (cid:48) i ) (cid:1) ∼ = lim ←− ∆ nj ∈S (cid:0) lim −→ S (cid:48) i ∈S (cid:48) S (cid:48) i ([ n j ]) (cid:1) ∼ = lim ←− ∆ nj ∈S (cid:0) ( lim −→ S (cid:48) i ∈S (cid:48) S (cid:48) i )([ n j ]) (cid:1) ∼ = lim ←− ∆ nj ∈S (cid:0) Hom s Set (∆ n j , lim −→ S (cid:48) ) (cid:1) ∼ = Hom s Set ( S, lim −→ S (cid:48) )composing to the bijection we needed to verifiy. Note that while all of the other bijectionsinvolved hold for arbitrary S , we needed the fact that S is a finite diagram in the thirdbijection to commute finite limits with filtered colimits in the category Set .As the objects of simple homotopy theory are classicaly finite in some combinatorialsense, these are the types of filtered simplicial sets we mostly are concerned with. Inthe finite setting, sd P can serve as a more geometric placeholder for Ex ∞ P . We use thissection to elaborate on this. Most of what follows now is certainly well known in theunfiltered setting, but as we needed a proof for the filtered one anyway, we are just goingto go ahead and give them here. We start by illuminating the homotopy relationshipbetween Ex P and sd P a bit. Lemma 1.1.43.
Consider the diagram in s Set P sd P ( X ) X Ex P (sd P ( X )) l.v. P η , HAPTER 1. STRATIFIED HOMOTOPY THEORY where η is the unit of the adjunction sd P (cid:97) Ex P . It commutes in H s Set P .Proof. First, note that sd P ( X ) l.v. P −−−→ X is a weak equivalence (see [Dou18, A.3]). As X (cid:44) → Ex P ( X ) is an anodyne extension [Dou19b, Appendix 2], Ex P preserves weakequivalences. In particular,Ex P (sd P ( X )) Ex P (l.v. P ) −−−−−−→ Ex P ( X )is a weak equivalence. Hence, it suffices to show that the diagram commutes in s Set P after composing both η and sd P ( X ) (cid:44) → Ex P (sd P ( X )) with this weak equivalence. Weshow this elementarily. Let ∆ J σ −→ sd P ( X ) be a (filtered) simplex of sd( X ). Considerthe following commutative diagramsd P (∆ J ) sd P ( X ) sd P ( X )∆ J sd P ( X ) X sd P ( σ )l.v. P sd P (l.v. P )l.v. P l.v. P σ l.v. P .Now, the image of σ under the composition containing η is given by first postcomposingwith the bottom right arrow, then applying sd P (which corresponds to η ), and finallypostcomposing with the right vertical. Hence, by funtoriality of sd P ( X ) this is given bythe composition starting at the upper left, then going all the way to the right horizontallyand then down. On the other hand, the image of σ under the other composition isgiven by first pulling σ back with the left vertical and then postcomposing with thebottom right horizontal. By commutativity of the diagram, this agrees with the firstconstruction.Just as above (or by a simple inductive argument) one obtains the more generalversion: Lemma 1.1.44.
Let n ≥ . Consider the diagram in s Set P sd nP ( X ) X Ex nP (sd nP ( X )) l.v. nP η ,where η is the unit of the adjunction sd nP (cid:97) Ex nP . It commutes in H s Set P . As a consequence we obtain the following.
Lemma 1.1.45.
For n ∈ N , consider a diagram in s Set P sd nP ( X ) X Y Ex nP ( Y ) l.v. nP ˆ ff HAPTER 1. STRATIFIED HOMOTOPY THEORY where ˆ f is the adjoint to f under sd nP (cid:97) Ex nP . Then this diagram commutes in H s Set P .Proof. By the unit counit form of an adjunction, f is given by X η −→ Ex nP (sd nP ( X )) Ex nP ( ˆ f ) −−−−→ Ex nP ( Y ) . Hence, the diagram in the statement refines to a diagram:sd nP ( X ) X Ex nP (sd nP ( X )) Y Ex nP ( Y ) l.v. nP ˆ ffη Ex nP ( ˆ f ) .The large right triangle commutes by naturality of the inclusion into Ex nP . The lowerleft triangle commutes by assumption. The upper right triangle commutes in H s Set P ,by Lemma 1.1.44. Hence, the outer square also commutes in H s Set P .As a direct consequence of this result, we obtain the following useful representationresult for morphisms in H s Set finP . Proposition 1.1.46.
Let X α −→ Y be a morphism in H s Set finP . Then there is an n ∈ N and a morphism sd nP ( X ) f −→ Y such that α fits into a commutative diagram sd nP ( X ) X Y [l.v. nP ] [ f ] α .In particular, α = [ f ] ◦ [l.v. nP ] − . Proof. As Y (cid:44) → Ex ∞ P ( Y ) is a fibrant replacement of Y , α fits into a commutative diagram X Y Ex ∞ P ( Y ) α [ f ] ,for some morphisms f in s Set P . As X is finite (hence compact), and Ex nP ( Y ) (cid:44) → Ex ∞ P ( Y ) is a weak equivalence, this diagram splits off a commutative diagram HAPTER 1. STRATIFIED HOMOTOPY THEORY X Y Ex nP ( Y ) α [ f ] for some sufficiently large n in N and some morphism f : X → Ex nP ( Y ). Now, the resultfollows by Lemma 1.1.45. Now, that we have a notion of weak equivalence on s
Set P and one on Top P , the obviousquestion arises of what the relationship between the two is. In particular, one would liketo understand the relationship with respect to the realization functor | − | P : s Set P −→ Top P . In the case where P is a singleton, it is of course a well known classical result that | − | preserves weak equivalences and induces an equivalence of homotopy categories (whereon the right hand side one uses the Quillen-Model structure [GJ12, Thm. 11.4]).With the model structures given by Theorem 1.1.28 and Theorem 1.1.17 however, | − | P is not the left adjoint of a Quillen equivalence. In fact, its right adjoint Sing P does noteven preserve fibrant objects (see [Dou19b, Sec. 8.1] for an example) and | − | P does notpreserve cofibrant objects. This might of course not be due to the choice of weak equiv-alences on Top P , but due to the choice of fibrations (or cofibrations respectively.) So, itmight very well still be true that for some other choice of model structures, inducing thesame homotopy theory, the two form a Quillen equivalence (see also Remark 1.1.20).We will not take the path of trying to define different model structures here. How-ever, we are going to show that the realization functor does preserve weak equivalences.A first step to showing this result was taken in [Dou19b, Thm. 8.3.7]. There, it is shownthat the realization functor to the more rigid model category of topological spaces over | N ( P ) | , Top N ( P ) , called the category of strongly filtered topological spaces, preservesweak equivalences. The latter is Quillen equivalent to Top P [Dou19a, Thm. 215]. Theright adjoint functor of this Quillen equivalence fits into a commutative diagram:s Set P Top N ( P ) Top P |−| N ( P ) |−| P . So, one might hope that it preserves weak equivalences making the the same also holdfor the composition. We show however in Example 1.2.3 that this is fact not the case.Nevertheless, we show that it holds for all weak equivalences between objects of the
HAPTER 1. STRATIFIED HOMOTOPY THEORY | X | N ( P ) for X ∈ s Set P . This is the content of Theorem 1.2.23. It is an immediateconsequence of Theorem 1.2.5, which essentially says that for realizations of P -filteredsimplicial sets the (generalized) homotopy links in the N ( P ) and the P setting are weaklyhomotopy equivalent. We start by very roughly summarizing some results and definition of [Dou19a, Sec. 1].The category
Top N ( P ) of strongly ( P -)filtered topological spaces is obtained by takingthe composition Poset N (cid:44) −→ s Set |−| −−→
Top instead of the Alexandroff space functor, in Example 1.0.2. Objects and morphisms in
Top N ( P ) are called strongly ( P -)filtered spaces and strongly stratum preserving maps respectively. Strongly stratum preserving maps are called strongly filtered maps in[Dou19a]. Analogously to the case of Top P (just replace P by | N ( P ) | in Construc-tion 1.1.2) this category is enriched over Top . We denote the hom space by C N ( P ) ( X, Y ),for
X, Y ∈ Top N ( P ) . Also, just as in the P case we obtain a cotensoring with the pre-cisely analogous construction of T ⊗ X , for T ∈ Top and X ∈ Top N ( P ) . Again tensoringwith I induces a notion of strongly stratified homotopy and strongly stratum preservinghomotopy equivalence . Then C N ( P ) ( − , − ) is compatible with this notion of homotopy.Further, one also has a realization functor | − | N ( P ) : s Set P → Top N ( P ) together with a right adjointSing N ( P ) : Top N ( P ) −→ s Set P . They are constructed via Construction 1.0.4, by taking: L = | − | , R = Sing , f = 1 Top and finally g as the unit of the adjunction N ( P ) → Sing ◦ | N ( P ) | . Using, the realization | − | N ( P ) we can then define a (strong) homotopy link functor.Hol N ( P ) : Top N ( P ) −→ Top sd( P ) op X (cid:55)−→ C N ( P ) ( | − | N ( P ) , X ) , as in Construction 1.1.6. We call a morphism X f −→ Y in Top N ( P ) a ( weak equivalence ofstrongly ( P -)filtered spaces ) if for each flag J ∈ sd( P ) the induced map Hol N ( P ) ( J , f ) isa weak homotopy equivalence of topological spaces. As int he P -filtered setting one showsthat every strongly stratum preserving homotopy equivalence is also a weak equivalenceof strongly filtered spaces. Finally, as for Top P , one obtains from [Dou19a][Thm. 2.8]the analogue to Theorem 1.1.17: Theorem 1.2.1.
There exists a cofibrantly generated model structure on
Top N ( P ) suchthat a morphism f : X → Y is HAPTER 1. STRATIFIED HOMOTOPY THEORY • a fibration if Hol N ( P ) ( J , f ) is a Serre fibration, for each flag J ∈ sd( P ) ; • a weak equivalence if f is a weak equivalence of strongly filtered topological spacesover P , that is if for each flag J ∈ sd( P ) , Hol N ( P ) ( J , f ) is a weak homotopyequivalence; • a cofibration if and only if it has the left lifting property against all acyclic fibra-tions. With respect to this model structure and the Douteau-Henrique model structure ons
Set P one obtains: Theorem 1.2.2. [Dou19b, Thm. 8.3.7] Both functors of the adjunction | − | N ( P ) : s Set P ←→ Top N ( P ) : Sing N ( P ) preserve weak equivalences. The Quillen equivalence between
Top N ( P ) and Top P described in [Dou19a, Subsec.2.4] is then given by the following construction. There is a forgetful functor into thecategory Top P given by postcomposing with the filtration of | N ( P ) | P , | N ( P ) | → P . Wedenote this functor by F : Top N ( P ) −→ Top P .F has a (enriched) right adjoint G : Top P −→ Top N ( P ) , given by pulling back along | N ( P ) | → P . By the construction of | − | P , F fits into anon the nose commutative diagram:s Set P Top N ( P ) Top P |−| N ( P ) |−| P F . Due to this, one might expect that the composition of |−| N ( P ) and F , |−| P also preservesweak equivalences. One might hope to prove this, by showing that F preserves weakequivalences. In fact, it turns out that the pair ( F, G ) induces a Quillen equivalence,between the two model structures given in Theorem 1.1.17 and Theorem 1.2.1 (see[Dou19a, Thm. 2.15]). However, this of course does not mean that F preserves all weakequivalences, only that it preserves weak equivalences between cofibrant objects. Forgeneral weak equivalences, F does indeed not preserve them: Example 1.2.3.
Consider the strongly stratum preserving map shown in Figure 1.12.The strong filtration is given by the height map, defined by projection to the vertical.The map is defined by identifying the upper two and lower two line segments respectively.
HAPTER 1. STRATIFIED HOMOTOPY THEORY P = [1].Figure Figure 1.13 shows the induced maps of Figure 1.12 on generalized homotopylinks, where the latter are depicted up to homotopy equivalence, on the right hand side.Homotopy links were computed using Proposition 1.1.8.Figure 1.13: The induced maps on the generalized homotopy links of Figure 1.12. Ho-motopy links are shown up to homotopy equivalence.While on the left hand side all of the maps are easily seen to be homeomorphismseven, this is clearly not the case on the right. In particular, F does not sustain theproperty of being a weak equivalence here.The failure of F to preserve weak equivalences here comes from the fact that F didnot preserve the weak homotopy type of the strongly filtered spaces. As F is an enrichedfunctor, it induces a natural inclusion of homotopy links:Hol N ( P ) ( J , X ) = C N ( P ) ( | ∆ J | N ( P ) , X ) F (cid:44) −→ C P ( F ( | ∆ J | N ( P ) ) , F ( X )) = Hol P ( J , F ( X )) , for a flag J ∈ sd( P ) and X ∈ Top N ( P ) . This induces a natural transformation: α : Hol N ( P ) (cid:44) → Hol P ◦ F By naturality and the two out of three property for weak equivalences, one obtains:
HAPTER 1. STRATIFIED HOMOTOPY THEORY Lemma 1.2.4.
Let f : X → Y in Top N ( P ) be such that both α X and α Y are weakhomotopy equivalences, at each J ∈ sd( P ) . Then f is a weak equivalence of strongly P -filtered spaces if and only if F ( f ) is a weak equivalence of P -filtered spaces. It is the content of the next subsection to show that the requirements of F are fulfilledif X and Y are isomorphic (or even weaker, strongly stratum preserving homotopyequivalent) to realizations of filtered simplicial sets. The content of this subsection is the proof of the following theorem:
Theorem 1.2.5.
Let X ∈ s Set P . Then, for each flag J ∈ sd( P ) , the inclusion ofgeneralized homotopy links: α | X | N ( P ) , J : Hol N ( P ) ( J , | X | N ( P ) ) (cid:44) → Hol P ( J , | X | P ) is a weak equivalence of topological spaces. We are first going to reduce to the case where X is a finite filtered simplicial set.This allows us to assume for the homotopy links to be metrizable and in particularparacompact. Recall the following result on CW-complexes. Proposition 1.2.6. [Hat02][Proposition A.1] A compact subspace of a CW-complex iscontained in a finite subcomplex.
It is in an immediate consequence of this result that homotopy groups of a CW-complex K can be computed as the colimit over the homotopy groups of finite subcom-plexes of K . We obtain an analogous result in the filtered case: Proposition 1.2.7. X ∈ s Set P . Let X i be the filtered diagram given by the finite filteredsimplicial subsets of X . Then, for each J ∈ sd( P ) , n ≥ and x in some sufficientlylarge | X i | the morphisms: lim −→ ( π (Hol N ( P ) ( J , | X i | N ( P ) ))) −→ π (Hol N ( P ) ( J , | X | N ( P ) )) , lim −→ ( π n (Hol N ( P ) ( J , | X i | N ( P ) ) , x )) −→ π n (Hol N ( P ) ( J , | X | N ( P ) ) , x ) and lim −→ ( π (Hol P ( J , | X i | P ))) −→ π (Hol P ( J , | X | P )) , lim −→ ( π n (Hol P ( J , | X i | P ) , x )) −→ π n (Hol P ( J , | X | P ) , x ) , are isomorphisms (where for n ≥ the colimit is of course only taken over the finalsubdiagram given by X i such that x ∈ | X i | ). HAPTER 1. STRATIFIED HOMOTOPY THEORY Proof.
We are going to prove the result for n ≥ P . The proof of the N ( P )case is completely analogous. The same can be said about the 0 vs. the n ≥ Y ∈ Top P , by the copower structure on Top P , π n (Hol P ( J , Y ) , x ) canalternatively described as the subquotient of Hom Top P ( S n ⊗ | ∆ J | P , Y ) given by suchmaps that restrict to x on { (cid:63) } × | ∆ J | P modulo stratified homotopies relative to thissubspace, where (cid:63) denotes the basepoint of S n . The pointed filtered space S n ⊗ | ∆ J | is clearly compact and the realization of a simplicial set is a CW-complex. Hence, byProposition 1.2.6, the map coming from the colimit is onto. Applying the same argumentto homotopies gives injectivity.We are prove the following version of Theorem 1.2.5 which by Proposition 1.2.7implies the former. Theorem 1.2.8.
Let X ∈ s Set P be locally finite (i.e. such that | X | is locally compact).Then, for each flag J ∈ sd( P ) , the inclusion of generalized homotopy links: α | X | N ( P ) J : Hol N ( P ) ( J , | X | N ( P ) ) (cid:44) → Hol P ( J , | X | P ) is a homotopy equivalence. So, from here on out we assume X to be locally finite. Further, we drop the indicesfrom α . Next, we should make a quick remark on the topology of the mapping spaces. Remark 1.2.9.
Recall that we take
Top to be the category of ∆-generated spaces;i.e. such spaces that have the final topology with respect to simplices (for details see[Dug20]). Recall further that the topology on C ( T, T (cid:48) ), for
T, T (cid:48) ∈ Top , is the ”∆-fication”, denoted k ∆ , of the compact open topology. That is, one takes the final topologywith respect to all maps from simplices into the mapping space with the compact opentopology. However, for questions of homotopy equivalence, this distinguishment is reallynot that important. This is so, as k ∆ preserves products where one of the factors islocally compact (see [Dug20]) and hence homotopy equivalences. So, we instead takethe compact open topology on C , for the remainder of this section. This has theadvantage of making Hol P ( J , Y ) a metrizable space, when Y is metrizable. A metric isthen given by d ( f, g ) = sup x ∈| ∆ J | P ( f ( x ) , g ( x )). The same construction works for N ( P ).In particular, this is the case when Y = | X | P for X a finite filtered simplicial set (ascompact CW-complexes are metrizable [FP93][Thm. A]). We make use of this fact lateron. Note that it would of course suffice to assume that the filtered simplicial sets arelocally finite so that the resulting spaces are locally compact.As a first step in the proof of Theorem 1.2.8 we now reduce to the case, where J = P .In particular, we can just assume P to be a finite linearly ordered set P = [ q ], q ∈ N . Construction 1.2.10.
Recall the restriction functors described in Definition 1.0.5.Then, by construction of the generalized homotopy links, one immediately obtains nat-ural homeomorphisms (given by the postcomposition):Hol N ( J ) ( J , Y J ) ∼ = Hol N ( P ) ( J , Y ) , Hol J ( J , Y J ) ∼ = Hol P ( J , Y ) , HAPTER 1. STRATIFIED HOMOTOPY THEORY Y either in Top N ( P ) or in Top P . By construction, we have a natural inclusion F ( Y J ) (cid:44) → F ( Y ) J (induced by the universal property of the pullback). In general however, this inclusionwill not be a filtered homeomorphism. This is illustrated in Figure 1.14 for example.Figure 1.14: Comparison of F ( Y J ) and F ( Y ) J in the case, where P = [2], J = (0 ≤ Y = | ∆ J | N ( P ) . The right hand triangle is missing only the point in the 0-stratum.In the case where Y is the realization of a locally filtered simplicial set, one can givea more explicit construction of Y J . Before we do so, we introduce some notation thatcomes in handy during the technical parts of this section. Construction 1.2.11.
Given a d-flag I in P , and a point ξ in | ∆ I | P or in | ∆ P | wedenote by ξ i its p i -th coordinate in the standard realization in R I , where p i is the i -thentry of I . Even more, for a subset D ⊂ P we denote by ξ D the vector given by such ξ i where p i ∈ D . In case p i ∈ D holds for no p i we take this to be 0. By a slight abuseof notation, we will sometimes think of these vectors as embedded into R I so that wecan write expressions like ξ D + ξ D (cid:48) . For any such vector, we denote by || − || the sumof its values. Further, note that the vector ( || ξ { p } || ) p ∈ P is precisely the value of ξ under | ∆ I | P → | N ( P ) | , where we think of the right hand side as realized in R P . The pointsin the p -th stratum of | ∆ J | P , for p ∈ P , are then precisely those points ξ , with ξ { p } (cid:54) = 0and ξ { p (cid:48) } = 0 for p (cid:48) ∈ I with p (cid:48) > p . Remark 1.2.12.
Let Y ∈ s Set P and J ∈ sd( P ). One can think of Y J as the fullsimplicial subset spanned by such vertices that map into J under Y → N ( P ). Then( | Y | N ( P ) ) J ∼ = ( | Y J | ) N ( P ) . This follows immediately from the fact that the realizationfunctor into Top from simplicial sets, sustains finite limits (see [nLa20g]).However, ( | Y | P ) J is a strictly larger space than this (see Figure 1.14). It is given by theunion of the strata with index in J . We will be making use of another explicit descriptionquite frequently. Let Y be locally finite. Denote by ∆ i the diagram given by the non-degenerate simplices of Y . By Proposition A.0.1, ( | Y | P ) J is given by lim −→ (cid:0) ( | ∆ i | P ) J (cid:1) .Hence, for most intends and purposes, it suffices to know what ( | ∆ | P ) J looks like for a HAPTER 1. STRATIFIED HOMOTOPY THEORY I = ( p ≤ ... ≤ p k ) be a d-flag in P . We identify the vertices of ∆ I with the corresponding unit vectors in R I and | ∆ I | with their convex hull. Then( | ∆ I | P ) J = (cid:110) ξ ∈ | ∆ I | P (cid:12)(cid:12) ξ fulfills Condition 1.1 (cid:111) . with max( { p i | || ξ { p i } || > } ) ∈ J (1.1)Under this condition, we clearly have || ξ J || > Proposition 1.2.13.
Let Y = | ˆ Y | N ( P ) where ˆ Y is a locally finite filtered simplicial set.Then the inclusion F ( Y J ) (cid:44) → F ( Y ) J , from Construction 1.2.10, is a stratum preservinghomotopy equivalence.Proof. By Proposition A.0.1, we can construct a homotopy inverse on the simplex level,and check that the map as well as the homotopies are compatible with degeneracy andface maps, hence glue to a global one. For a filtered simplex ∆ I , with notation as inRemark 1.2.12, the inclusion is (cid:110) ξ ∈ | ∆ I | P (cid:12)(cid:12)(cid:12) || ξ J || = 1 (cid:111) ⊂ (cid:110) ξ ∈ | ∆ I | P (cid:12)(cid:12)(cid:12) fulfills Condition 1.1 (cid:111) . Under Condition 1.1, we clearly have || ξ J || >
0. Define a retract, r , of this inclusion, i ,via ξ (cid:55)→ || ξ J || ξ J . By Condition 1.1, this does not change the maximal indices for which ξ does not disap-pear and hence is stratum preserving. Clearly, r ◦ i = 1 ( | ∆ I | N ( P ) | ) J . Conversely, i ◦ r is stratified homotopic to the identity on ( | ∆ I | P ) J by the straight linehomotopy, which is easily checked to be well-defined. It is not hard to see that thestraight line homotopy of i ◦ r and the latter identity is compatible with degeneracyand face maps, hence induces a global stratum preserving deformation retraction of F ( Y J ) (cid:44) → F ( Y ) J .We have now reduced the proof of Theorem 1.2.8 to the case where J = P = [ q ],for some q ∈ N . We will still not write | N ( P ) | at every possible location, just because itmakes the notation look horribly convoluted. By using | N ( P ) | instead of | ∆ P | = | ∆ J | ,we usually indicate what role in the proof the space is taking at this moment. One shouldhowever keep in mind that on as P -filtered spaces we have | ∆ J | = | ∆ P | = | N ( P ) | . Wenow illustrate the proof of Theorem 1.2.8 on a π level, i.e. pointwise and for q = 1,before we give a rigorous proof in more generality. HAPTER 1. STRATIFIED HOMOTOPY THEORY Example 1.2.14.
It can be illustrative to see how every point in γ ∈ Hol P ( J , | X | P )lies in the path component of one in Hol N ( P ) ( J , | X | N ( P ) ), for J = [1] the linear setwith 2 elements. Then Hol P ( J , | X | P ) is the space of paths, starting in the 0-stratumand immediately leaving it. The proof of this statement serves as a model for the moregeneral case. Essentially, the idea is the followingUnder this condition, we clearly have || ξ J || > X red the subspace of | X | N ( J ) obtained by by pulling back along[0 , (cid:44) → I ∼ = | N ([1]) | . By Proposition A.0.1 this space is glued together from subspacesof simplices of the form | ∆ I | redN ( P ) = (cid:110) ξ ∈ | ∆ I | P (cid:12)(cid:12)(cid:12) || ξ { } || > (cid:111) . We have illustrated this space in Figure 1.15.Figure 1.15: Illustration of an example of X red .Now, X red admits a rescaling map: ρ : X red × P | N ( P ) | −→ | X | N ( P ) defined on a simplex level by( ξ, ( t , t )) (cid:55)→ t || ξ { } || ξ { } + t || ξ { } || ξ { } . It is not hard to see that this also makes sense for || ξ { j } || = 0 as then also t j = 0, andone can just set the corresponding summand to 0. This construction is compatible withdegeneracy and face maps, and hence extends to all of X red . Note that ρ fits into acommutative diagram X red × P | N ( P ) | | X | N ( P ) | N ( P ) | π | N ( P ) | ρ ; HAPTER 1. STRATIFIED HOMOTOPY THEORY ρ is stratified homotopic to X red × P | N ( P ) | π X red −−−−→ X red (cid:44) → | X | P , by a straight line homotopy R : (cid:0) X red × P | N ( P ) | (cid:1) × ∆ → | X | P constructed simplex-wise (as a morphism in Top P ). Now, X red is an open neighbourhoodof ( | X | P ) in | X | P . Thus, if we restrict γ to a sufficiently small neighbourhood of 0, ithas image in X red . So, by continuously scaling down the domain of definition, γ lies inthe same path component as some (stratified) path γ (cid:48) with image in X red . Next, oneuses R to continuously rescale γ (cid:48) to a map over | N ( P ) | . This is done by definingˆ γ : | ∆ J | P −→ X red × P | N ( P ) | ( t , t ) (cid:55)−→ (cid:0) γ (cid:48) ( t , t ) , ( t , t ) (cid:1) and then applying R to obtain a stratified homotopy from γ (cid:48) to a strongly stratumpreserving map. The whole process is illustrated in the following picture.Figure 1.16: Illustration of the process described in Example 1.2.14.We start the proof of the general case by replicating the “retracting- γ -part”of Example 1.2.14. To make notation a little bit more concise, we omit the index fromthe realization functors for simplices whenever it is clear from context what is meant. Construction 1.2.15.
For a d-flag I in P = [ q ] denote by | ∆ I | red the filtered subspaceof | ∆ J | given by | ∆ I | red := (cid:110) ξ ∈ | ∆ I | (cid:12)(cid:12)(cid:12) ξ { p } = 0 = ⇒ ξ { p (cid:48) } = 0 , for p (cid:48) ≥ p ∈ I (cid:111) . Further, denote by X red the space filtered over P obtained by pulling back | X | N ( P ) along | ∆ P | red (cid:44) → | ∆ P | = | N ( P ) | . X red is essentially defined as the subspace of points ξ ∈ | X | N ( P ) fulfilling the implication (cid:0) p | X | N ( P ) ( ξ ) (cid:1) { p } = 0 = ⇒ p | X | P ( ξ ) < p, HAPTER 1. STRATIFIED HOMOTOPY THEORY p ∈ P .Now, consider the following shrinking map λ : | ∆ P | −→ | ∆ P | induced by affinely extending p (cid:55)→ bar( | ∆ [ p ] | ) ∈ | ∆ [ p ] | ⊂ | ∆ P | , which sends a vertex to the barycenter of the simplex spanned by all vertices smaller thanit in the linear order on P . It is clearly stratum preserving. We denote these barycentersby bar([ p ]) for short. This map is stratified homotopic to the identity through a straightline homotopy. In particular, the inclusion Hol N ( P ) ( P, | X | N ( P ) ) (cid:44) → Hol P ( P, | X | P ) ishomotopic to the map given by σ (cid:55)→ σ ◦ λ. Furthermore, λ has image in | ∆ P | red . To see this, just note that any point in the image ξ is of the shape ξ = (cid:88) p ∈ P t p bar([ p ]) = (cid:88) p ∈ P t p p + 1 ( | | + ... + | p | ) , for t p ≥ , (cid:80) p ∈ P t p = 1 (using again the | − | notation to denote the respective unitvectors). Hence, if ξ { p } = (cid:88) p (cid:48) ≥ p t p (cid:48) p (cid:48) + 1 = 0 , so are all t (cid:48) p with p (cid:48) ≥ p and hence also ξ { p (cid:48) } = 0, for p (cid:48) ≥ p . Thus, for any σ ∈ Hol N ( P ) ( P, | X | N ( P ) ) we have σ ( λ ( ξ )) ∈ X red . As an immediate corollary of Construction 1.2.15 we obtain:
Corollary 1.2.16.
Let X ∈ s Set P be locally finite. We use the notation from Con-struction 1.2.15. Then up to homotopy the inclusion α : Hol N ( P ) ( P, | X | N ( P ) ) (cid:44) → Hol P ( P, | X | P ) factors as Hol N ( P ) ( P, | X | N ( P ) ) Hol P ( P, | X | P )Hol P ( P, X red ) , with the dashed arrow given by σ (cid:55)→ (cid:110) ξ (cid:55)→ σ ( λ ( ξ )) (cid:111) . HAPTER 1. STRATIFIED HOMOTOPY THEORY
Construction 1.2.17.
Note that, by the natural isomorphism X red × P | N ( P ) | = (cid:0) | X | N ( P ) × | N ( P ) | | ∆ P | red (cid:1) × P | N ( P ) |∼ = (cid:12)(cid:12) X | N ( P ) × | N ( P ) | (cid:0) | ∆ P | red × P | N ( P ) | (cid:1) , the left hand side is a pullback in the sense of Proposition A.0.1. Hence, it is given bya colimit over the diagram | ∆ I i | red × P | N ( P ) | , induced by pulling back the realizations of the non-degenerate simplices of X . The pro-jection to the second component makes this a colimit over | N ( P ) | , i.e. one in Top N ( P ) .For a d-flag I ∈ N ( P ), define the straightening map ρ I : | ∆ I | red × P | N ( P ) | −→ | ∆ I | ( ξ, ( t , ..., t q )) (cid:55)−→ (cid:88) p ∈ P t p || ξ { p } || ξ { p } , where the summands are taken to be 0, when || ξ { p } || = 0. We should make a few remarkson why this is well-defined and continuous. First off, note that for right hand side vectorin R I to actually lie in | ∆ J | , we have to have ξ { p } = 0 = ⇒ t p = 0 . Otherwise, the vector does not have 1 as the sum over its entries. To see this is the case,let p ∈ P and ξ { p } = 0. Note that as ξ ∈ | ∆ I | red we have that p | ∆ J | P ( ξ ) < p . Hence, aswe have taken the fiber product over P on the left hand side, this also means t p (cid:48) = 0,for p (cid:48) ≥ p .We can replicate the same argument, but replacing equality with convergence, to showthat the map defined in this fashion is also continuous. It is clearly strongly stratumpreserving, using the projection to the second component on the left hand side and | ∆ I | → | ∆ P | = | N ( P ) | on the right hand side. Furthermore, note that it fits into acommutative diagram | ∆ I | red | ∆ I | red × P | N ( P ) | | ∆ I | d ρ I , where d denotes the map induced by 1 | ∆ I | red and the composition with the strong filtra-tion of | ∆ I | red . Denote by R I : (cid:16) | ∆ I | red × P | N ( P ) | (cid:17) × I −→ | ∆ I | HAPTER 1. STRATIFIED HOMOTOPY THEORY ρ I and | ∆ I | red × P | N ( P ) | π | ∆ I| red −−−−−→ | ∆ I | red (cid:44) → | ∆ I | As both maps are stratum preserving, so is the straight line homotopy (as the strata ofa filtered simplex are convex, see also Lemma 1.3.11). One can easily check that R iscompatible with the face and degeneracy maps of X . Hence, this induces a stratifiedhomotopy R : (cid:16) X red × P | N ( P ) | (cid:17) ⊗ ∆ → | X | P between a strongly stratum preserving map ρ and the stratum preserving map X red × P | N ( P ) | π X red −−−−→ X red (cid:44) → | X | P . Finally, note that R maps into X red at every point in time, but t = 1. Remark 1.2.18. ρ can be slightly extended. Instead of pulling back to | ∆ P | red × P | N ( P ) | , we can pull back to (cid:16) | ∆ P | red × P | N ( P ) | (cid:17) ∪ (cid:16) | ∆ P | × | N ( P ) | | N ( P ) | (cid:17) ⊂ | ∆ P | × | N ( P ) | = | ∆ P | . We essentially add points on the diagonal. Denote this space by ˆ∆. One can easily checkthat the formula for the simplexwise definition of ρ I in Construction 1.2.17 extends to | ∆ I | × | N ( P ) | ˆ∆ = (cid:16) | ∆ I | red × P | N ( P ) | (cid:17) ∪ (cid:16) | ∆ I | × | N ( P ) | | N ( P ) | (cid:17) , where the union on the right hand side is to be understood in | ∆ I | × | N ( P ) | . On (cid:16) | ∆ I | × | N ( P ) | | N ( P ) | (cid:17) this is just given by the the projection to the first component. Bythe same argument as in Construction 1.2.15, we obtain an extension of ρ to | X | N ( P ) × | N ( P ) | ˆ∆ = (cid:16) X red × P | N ( P ) | (cid:17) ∪ (cid:16) | X | N ( P ) × | N ( P ) | | N ( P ) | (cid:17) ⊂ | X | N ( P ) × P | N ( P ) | . (1.2)ˆ ρ fits into a commutative diagram | X | N ( P ) | X | N ( P ) × | N ( P ) | ˆ∆ | X | N ( P )1 d ˆ ρ , (1.3)where d is just the identification with the second component of the union in (1.2). Proposition 1.2.19.
The dashed map from Corollary 1.2.16 is a homotopy equivalence.
HAPTER 1. STRATIFIED HOMOTOPY THEORY Proof.
First, note that under the enriched the adjunction F : Top N ( P ) ←→ Top P : G we have a natural homeomorphism:Hol P ( P, X red ) ∼ = Hol N ( P ) ( P, X red × P | N ( P ) | ) , where again the argument in the right hand Hol is strongly filtered by the projection to | ∆ P | = | N ( P ) | . Under this natural homeomorphism the dashed map corresponds to σ (cid:55)→ (cid:110) ξ (cid:55)→ ( σ ◦ λ ( ξ ) , ( ξ , ..., ξ q )) (cid:111) . Denote this map by η . Denote by ρ ∗ the mapHol N ( P ) ( P, X red × P | N ( P ) | ) −→ Hol N ( P ) ( P, | X | N ( P ) )induced by ρ . We claim that ρ ∗ is a homotopy inverse to η .First, consider the homotopy between the two mapsHol N ( P ) ( P, | X | N ( P ) ) −→ Hol P ( P, | X | P )given by σ (cid:55)→ σ ◦ λ and the inclusion that is induced by the straight line homotopy from λ to the indentity. Up to the last point in time, this maps into Hol P ( P, X red ), and at t = 1 it maps into Hol N ( P ) ( P, | X | N ( P ) ). Hence, if we post-compose this homotopy withHol P ( P, | X | P ) ∼ = Hol N ( P ) ( P, | X | P × P | N ( P ) | ) , (using(1.2)) it factors throughHol N ( P ) ( P, | X | N ( P ) × | N ( P ) | ˆ∆) (cid:44) → Hol N ( P ) ( P, | X | P × P | N ( P ) | ) . That is, we obtain a new homotopy: H : Hol N ( P ) ( P, | X | N ( P ) ) × ∆ −→ Hol N ( P ) ( P, | X | N ( P ) × N ( P ) ˆ∆) .H is given by σ (cid:55)−→ (cid:110) ξ (cid:55)→ ( σ ◦ λ ( ξ ) , ( ξ , ..., ξ )) (cid:111) , for t = 0; (1.4) σ (cid:55)−→ (cid:110) ξ (cid:55)→ ( σ ( ξ ) , ( ξ , ..., ξ )) (cid:111) , for t = 1. (1.5)Now, consider the compositionHol N ( P ) ( P, | X | N ( P ) ) Hol N ( P ) ( P, | X | N ( P ) )Hol N ( P ) ( P, | X | N ( P ) × | N ( P ) | ˆ∆) H t Hol N ( P ) ( P, ˆ ρ ) . HAPTER 1. STRATIFIED HOMOTOPY THEORY t = 0 the horizontal is ρ ∗ ◦ η . By(1.5) and the fact that σ is strongly stratumpreserving, H maps into Hol N ( P ) ( P, | X | N ( P ) × | N ( P ) | | N ( P ) | ). Therefore, by (1.3), whichessentially states that ˆ ρ is the projection to the first component on | X | N ( P ) × | N ( P ) | | N ( P ) | and again, by (1.5), we obtain that, for t = 1, the horizontal is the identity. In particular, ρ ∗ ◦ η (cid:39) η ◦ ρ ∗ is also homotopic to the identity. We first post-compose η ◦ ρ ∗ with the enriched adjunction homeomorphismHol N ( P ) ( P, X red × P | N ( P ) | ) ψ −→ ∼ Hol P ( P, X red )and then with the inclusion into Hol P ( P, | X | P ), i ∗ . Note that the adjunction homeo-morphism ψ is given by first including into Hol P ( P, F ( X red × P | N ( P ) | )) via α and thenpushing forward with the projection to X red , which we denote π ∗ . We then have acommutative diagramHol N ( P ) ( P, X red × P | N ( P ) | ) Hol P ( P, | X | P )Hol P ( P, F ( X red × P | N ( P ) | )) i ∗ ◦ ψ ◦ η ◦ ρ ∗ ˜ λ Hol P ( P,F ( ρ )) , where ˜ λ is given by first including into Hol P ( P, F ( X red × P | N ( P ) | )), via α , and thenpulling back with λ . Using the straight line homotopy between λ and the identity,˜ λ is homotopic to α . Denote this homotopy by L . R induces a homotopy R ∗ fromHol P ( P, F ( ρ )) to i ∗ ◦ π ∗ . Now, consider the diagonally composed homotopy of L and R ∗ , i.e. D t := R ∗ ,t ◦ L t . At t = 1, this is i ∗ ◦ ψ . At t = 0 it is i ∗ ◦ ψ ◦ η ◦ ρ ∗ , by the commutativity of the last diagram.For t ∈ (0 , R maps into X red and hence, D t maps into Hol P ( P, X red ). In particular, D factors through Hol P ( P, X red ). That is, there is a unique ˜ D with i ∗ ◦ ˜ D = D . Then at t = 0, ˜ D is ψ ◦ η ◦ ρ ∗ . As D is i ∗ ◦ ψ at t = 1, ˜ D is ψ at t = 1. As ψ is a homeomorphism,in particular we also obtain η ◦ ρ ∗ (cid:39) Proposition 1.2.20.
In the setting of Corollary 1.2.16,
Hol P ( P, X red ) (cid:44) → Hol P ( P, | X | P ) is a homotopy equivalence. One would hope that one can make a similar Argument as in Example 1.2.14. How-ever, note that the argument there involved the fact that for q = 1, a neighbordhood of | | ∈ | ∆ P | is always mapped into X red . This can not immediately be replicated in thegeneral case. The argument needs a little bit of refinement. HAPTER 1. STRATIFIED HOMOTOPY THEORY Construction 1.2.21.
Let k ∈ P . Define | ∆ P | red,k := { ξ ∈ | ∆ P | | ξ p = 0 = ⇒ ξ p (cid:48) = 0, for all p < k, p ≤ p (cid:48) } . Define the filtered space X k by pulling back | X | N ( P ) along | ∆ P | red,k (cid:44) → | ∆ P | . Note, howin case where X = ∆ I for a d-flag I in P this is equivalently described by | ∆ I | red,k = { ξ ∈ | ∆ I | | ξ { p } = 0 = ⇒ ξ { p (cid:48) } = 0, for all p < k, p ≤ p (cid:48) } . As these spaces are easily seen to be ∆-generated (in fact they can be triangulated),the pullback here, is actually the pullback in the naive topological category (by Propo-sition A.0.1). Thus, we can actually think of X k as actual topological subspaces of | X | P and not just in the ∆-generated subspace sense (i.e. as the ∆-fication of a subspace).They are then alternatively described as the inverse image under p | X | N ( P ) of | ∆ P | red,k .Clearly, we then have: X red = X q ⊂ X q − ⊂ ... ⊂ X = | X | P . Furthermore, X k +1 is a neighbourhood of the k -th stratum of X k in X k . To see this,note first that, by the pullback construction, we only need to show this for | ∆ P | . The k -th stratum is given by such ξ ∈ | ∆ P | red,k where ξ k > ξ p (cid:48) = 0, for p (cid:48) ≥ k + 1.Hence, the defining property of | ∆ P | red,k +1 is clearly fulfilled in a small open ball aroundsuch a point.Instead of directly showing that the inclusion in Proposition 1.2.20 is a homotopyequivalence, we show the analogous statement for the inclusions X k +1 (cid:44) → X k . Proposi-tion 1.2.20 is then an immediate consequence of this. Lemma 1.2.22.
In the setting of Construction 1.2.21, for k ∈ P , the map Hol P ( P, X k +1 ) (cid:44) → Hol P ( P, X k ) induced by X k +1 (cid:44) → X k admits a deformation retraction.Proof. First note that, by our local finiteness assumption, all spaces involved are metriz-able and the sources in all mapping spaces involved are compact. In particular, thecompact open topology on all mapping spaces involved can be thought of as comingfrom the supremum metric. This makes most continuity verifications very easy and theyare mostly omitted.Consider the map S : | ∆ P | × [0 , −→ | ∆ P | ;( ξ, s ) (cid:55)−→ ( ξ [ k − , ξ k + (1 − s ) || ξ { k +1 ,...,q } || , sξ { k +1 ,...,q } ) . Note that this is the identity at s = 1, stratum preserving outside of s = 0 and that itrestricts to the identity on | ∆ [ k ] | ⊂ | ∆ P | . We now claim the following. HAPTER 1. STRATIFIED HOMOTOPY THEORY Claim 1.2.22.1.
There exists a map of topological spaces: t : Hol P ( P, X k ) × ∆ −→ I such that(i) t ( σ, s ) = 0 if and only if s = 0.(ii) σ (cid:16) S (cid:0) ξ, t (cid:0) σ, || ξ { k +1 ,...,q } || (cid:1)(cid:1)(cid:17) ∈ X k +1 , for σ ∈ Hol P ( P, X k ) and ξ ∈ | ∆ P | .We are going to prove this claim at the end of the proof. We show first how it impliesthe statement of the lemma. Consider the map S : Hol P ( P, X k ) × ∆ −→ Hol P ( P, X k )( σ, s ) (cid:55)−→ (cid:110) ξ (cid:55)→ σ (cid:0) S (cid:0) ξ, (1 − s ) + st (cid:0) σ, || ξ { k +1 ,...,q } || ) (cid:1)(cid:17)(cid:111) . We need to verify that this is well-defined in the sense that the map described on theright hand side is actually stratum preserving. As S is stratum preserving at each timeoutside of 0, we only need to worry about the case where the second argument of S i.e.(1 − s ) + st ( σ, || ξ { k +1 ,...,q } ) || is 0. Then, in particular s = 1, hence t ( σ, || ξ { k +1 ,...,q } || ) = 0and thereby, using (i) we obtain that || ξ { k +1 ,...,q } || = 0. Thus, ξ ∈ | ∆ [ k ] | which by thedefinition of S implies S ( ξ, s ) = ξ and thereby preservation strata is also verified in thiscase. Now, note furthermore that H maps Hol P ( P, X k +1 ) into itself. At s = 0, it isgiven by pulling back with S ( − ,
1) which is the identity. At s = 1, by (ii), it mapsinto Hol P ( P, X k +1 ). Thus, we have verified that indeed Hol P ( P, X k +1 ) (cid:44) → Hol P ( P, X k )admits a deformation retraction.It remains to show Claim 1.2.22.1. We denote by d the metric on | ∆ P | induced bythe supremum norm on R P . Let ε ∈ (0 , ε := { ξ ∈ | ∆ P | | || ξ { k,...,q } || ≥ ε } , ∆ k,ε := | ∆ [ k ] | ∩ ∆ ε ⊂ | ∆ P | . Alternatively, the latter is given by such ξ ∈ | ∆ [ k ] | with ξ k ≥ ε . Clearly, ∆ k,ε is acompact subset of the k -th stratum of | ∆ P | and ∆ ε is compact. Furthermore, note thatsup inf ξ ∈ S (∆ ε × [0 ,t ]) ξ (cid:48) ∈ ∆ k,ε d ( ξ, ξ (cid:48) ) t → −−→ . (1.6)Now, let σ i ∈ Hol P ( P, X k ). As σ i is stratum preserving, it maps ∆ k,ε into the k -thstratum of X k , X kk . Hence, as we have seen in Construction 1.2.21 that X k +1 is aneighbourhood of X kk , a neighbourhood of ∆ k,ε maps into X k +1 under σ i . Hence, by(1.6), we also get σ ( S (∆ ε × [0 , t i,ε ]) ⊂ X k +1 (1.7) HAPTER 1. STRATIFIED HOMOTOPY THEORY t i,ε >
0. By the definition of the compact open topology, thisalso holds in a neighbourhood of σ i , U i ⊂ Hol P ( P, X k ). Now, cover Hol P ( P, X k ) by suchneighbourhoods to obtain a covering ( U i ) together with t i,ε as above. As Hol P ( P, X red )is metrizable, in particular, it is paracompact. Let ϕ i be a partition of unity subordinateto ( U i ). Next, define t ε : Hol P ( P, X k ) −→ (0 , σ (cid:55)−→ (cid:88) ϕ i ( σ ) t i,ε . Then, for each σ ∈ Hol P ( P, X k ), by (1.7), we obtain σ (cid:16) S (cid:0) ∆ ε × [0 , t ε ( σ )] (cid:1)(cid:17) ⊂ X k +1 . (1.8)Suppose we have done this construction for ε = n . Consider the covering of (0 ,
2) givenby the open intervals I n := (2 − ( n +1) , − ( n − ) for n ≥ . Take a family of bump functions ψ n on [0 ,
2] such that ψ n has support in ( I n ) and is 1 on [2 − n , − ( n − ]. Then, define: t : Hol P ( P, X k ) × ∆ −→ [0 , σ, s ) (cid:55)−→ s (cid:89) n ≥ (cid:16) − ψ n ( s ) (cid:0) − t − n ( σ ) (cid:1)(cid:17) . Note that the product on the right hand side is always bounded by 1, so the expressionactually makes sense even at s = 1. Further, note that locally in the s coordinate, for s >
0, this is actually just a finite product. Furthermore, as (cid:81) ... is always smallerthan one, this is easily seen to be continuous in s = 1 also. We need to see that thisconstruction fulfills Items (i) and (ii) of Claim 1.2.22.1. The first is immediate fromthe facts that clearly t ( σ,
0) = 0, that the product at any other time is finite, and that t − n ( σ ) >
0, for all σ ∈ Hol P ( P, X k ). For the second, note that for s ∈ [2 − n , − ( n − ]and σ ∈ Hol P ( P, X k ) we have: t ( σ, s ) ≤ t − n ( σ ) (1.9)Now, let ξ ∈ | ∆ P | . If || ξ k +1 ,...,q || = 0, then the expression in (ii) is just σ ( ξ ) and thus p | ∆ P | ( ξ ) = p | X | P ( σ ( ξ )) ≤ k . But for those p we have X kp = X k +1 p . If || ξ { k +1 ,...,q } || >
0, take n ≥ || ξ k +1 ,...,q || ∈ [2 − n , − ( n − ]. Then ξ ∈ ∆ − n and by (1.9) t ( σ, || ξ { k +1 ,...,q } || ) ≤ t − n ( σ ). Hence, by (1.8), the result follows.This finished the proof of Theorem 1.2.8 and hence of Theorem 1.2.5. The following realization theorem is now an immediate consequence of Theorem 1.2.5.
Theorem 1.2.23.
The realization functor: | − | P : s Set P −→ Top P preserves weak equivalences. HAPTER 1. STRATIFIED HOMOTOPY THEORY Proof.
We have already seen that | − | P factors as F ◦ | − | N ( P ) . By Theorem 1.2.2 | − | N ( P ) preserves weak equivalences. Furthermore, by Theorem 1.2.5, every stronglyfiltered space of the shape | X | N ( P ) fulfills the requirements of Lemma 1.2.4 for F topreserve weak equivalences. Hence, the composition, that is | − | P , preserves weakequivalences.This result, even despite the lack of a Quillen equivalence between the simplicial andthe topological model categories, gives strong justification, for the former being a goodcandidate to “model” the latter. By the universal property of the localization, | − | P induces a functor | − | P : H s Set P −→ H Top P , denoted the same by abuse of notation. We conjecture that this is in fact an equivalenceof categories. While we do not have a proof of this statement, we will at least seelater on (in Theorem 1.4.1) that this functor is fully faithful if one restricts to finitefiltered simplicial sets. In particular, one obtains an equivalence of categories between thehomotopy category of finite filtered simplicial sets and the category of filtered topologicalspaces that are weakly equivalent to the realization of a finite filtered simplicial set. To understand the connection between the simplicial and the topological categories offiltered objects, we take a detour via the category of simplicial complexes. This approachmight seem a bit unusual as the usage of simplicial sets and model categories tends tocircumvent any appeals to the more classical and rigid simplicial complexes. However,as we are mainly concerned with understanding the underlying homotopy categories, theusage of a simplicial approximation theorem allows for a (admittedly somewhat brute-force) connection of the two settings. This finally reflects in the fully faithful embeddingdescribed in Theorem 1.4.1. However, despite the path we are following maybe not be-ing the most elegant from a model category perspective, it has the nice side-effect ofshedding a lot more light on the piecewise linear world of filtered spaces. Much of thecontent of this section is a priori independent from the world of simplicial sets and canbe understood as an investigation of the purely piecewise linear setting.The simplicial approximation theorem ([Spa89, Ch. I, Sec. 4., Theorem 8]) is prob-ably one of the most powerful and most used theorems in classical algebraic topology.It states that for any map continuous map φ : | K | → | L | between the realizations offinite simplicial complexes K and L there exists a subdivision K (cid:48) of K and a simplicialmap f : K (cid:48) → L , such that | K (cid:48) | ∼ −→ | K | φ −→ | L | is homotopic to | f | (for most of thestandard language on simplicial complexes used here we refer to [Spa89, Ch. I]). Thus,it allows one to reduce many questions on maps of polyhedra to the setting of purelycombinatorial maps of simplicial complexes (see for example the proof of the Lefschetzfixed-point theorem in [Bre13, Ch.4, Sec. 23]). HAPTER 1. STRATIFIED HOMOTOPY THEORY L ⊂ K such that φ on | L | is already simplicial (see[Zee64]). In particular, applying this relative version to homotopies, one obtains thatevery continuous map of compact polyhedra is homotopic to a piecewise linear map,and that two piecewise linear maps are homotopic if and only if they are homotopicthrough a piecewise linear homotopy. In other words, one obtains an equivalence ofcategories between the p.l. homotopy category of compact polyhedra and the full sub-category of the homotopy category of topological spaces given by compact polyhedra.Thus, at least as long as one is interested mostly in spaces homotopy equivalent to thelatter, many questions of homotopy theory can be reduced to the piecewise linear setting.This of course begs the question whether a similar statement can be made in the filtered(stratified setting). Such a result was given by C.H. Schwartz in [Sch71]. However, theproof there has several flaws. (To our best understanding condition ¯ γ of [Sch71, Theorem2] is only fulfilled if the map is locally constant on the subcomplex ˜ L . In addition tothat, the proof of the existence of the homotopy, as it is of now, seems to be based on amistaken assumption in [Sch71, equation 1.51] which leads to the linear interpolation inthe homotopy not being well-defined.) Furthermore, the volume of Mathematica (Cluj)it is published in seems to be rather hard to access as of this moment. In the Englishlanguage, to the best of our knowledge, there is no correct proof of such a theoremavailable at all.The proof of the simplicial approximation theorem for the filtered (stratified) settingis loosely based on the one in [Sch71]. However, as we restrict our-self to the finite set-ting, our proofs are somewhat more concise. Furthermore, we give an explicit descriptionof the subdivision used. Lastly, we give a new proof of the existence of the homotopy,circumventing the difficulties mentioned above. Recall that in Example 1.0.2 we constructed a category of P -filtered simplicial complexesnamely sCplx P . Similarly to the setting of P -filtered simplicial sets this category admitsthe following more explicit description. Remark 1.3.1.
Note that, by the definition of maps of simplicial complexes, a filteredsimplicial complex can be equivalently characterized as a simplicial complex togetherwith a map p : K (0) → P such that whenever { x , ..., x k } is a simplex of K , then { p ( x ) , ..., p ( x k ) } is a flag in P . Furthermore, a morphism of P -filtered simplicial com-plexes f : K → L is equivalently a simplicial map of the underlying simplicial complexessuch that p L ( f ( x )) = p K ( x )), for x ∈ K (0) . This justifies calling such morphisms stratumpreserving simplicial maps.Recall that we equipped sCplx P with the realization functor | − | P into Top P inExample 1.0.8. For readabilities sake, we mostly omit the P index, for the remainder of HAPTER 1. STRATIFIED HOMOTOPY THEORY
Remark 1.3.2.
The realization functor is naturally isomorphic to the functors given bythe following construction. For a filtered simplicial complex K , consider R K (0) ≥ (wherein the infinite case we take the weak topology induced by finite dimensional subspaces).We identify x ∈ K (0) with the vector in R K (0) ≥ that has 1 at x and 0 for all other entries,and denote this vector by | x | . We send K to (cid:110) ξ = (cid:88) t x | x | ∈ R K (0) ≥ | (cid:88) t x = 1 , { x | t x (cid:54) = 0 } ∈ K (cid:111) ⊂ R K (0) ≥ , filtered by ξ = (cid:88) t x | x | (cid:55)−→ p K ( x m ) , where the latter is maximal with respect to t x m (cid:54) = 0. For σ ∈ K , we denote by ˚ σ theopen simplex in | K | P corresponding to σ . By construction, ˚ σ lies in the stratum of | K | P corresponding to the maximum of the flag p K ( σ ), i.e.˚ σ ⊂ | K | max( p K ( σ )) . We usually think of filtered simplicial complexes as being embedded in R K (0) ≥ in thisfashion, which makes the description of linear homotopies somewhat cleaner. Construction 1.3.3.
Given a A subcomplex of a P -filtered simplicial complex A , thefiltration of K naturally restricts to a filtration of A . Hence, we think of any fullsubcomplex of a P -filtered complex as also being P -filtered. For such a pair A ⊂ K offiltered complexes, we denote by K − A the ( P -filtered) subcomplex of K spanned byall vertices not in A . We use the ” − ” to distinguish this from the set of simplices not in A , denoted K \ A .We now quickly recap some standard constructions from p.l. topology. For moredetails see for example [RS12]. Construction 1.3.4.
Recall that for two simplicial complexes K and K the join of thetwo, K (cid:63) K , is given by the simplicial complex with vertices K (0)0 (cid:116) K (0)1 and simplicesgives by subsets σ of K (0)0 (cid:116) K (0)1 such that K (0)0 ∩ σ ∈ K and K (0)1 ∩ σ ∈ K . Thesimplex in K (cid:63) K given by σ (cid:116) τ for σ ∈ K and τ ∈ K is denoted by σ (cid:63) τ . Note thatif K and L are filtered simplicial complexes over P , the join is in general not naturallyfiltered. There might be vertices x and y connected by a 1-vertex in the join such thatneither p K ( x ) ≤ p L ( y ) nor p L ( y ) ≤ p K ( x ). Thus, one has to be a little bit careful withusing joins here and if we talk about the join of filtered complexes, we usually meanthe underlying simplicial complex. Further, recall that for two simplices σ and τ in K , σ (cid:63) K τ ⊂ K (0)0 denotes the union of σ and τ . In general of course, σ (cid:63) K τ will not be asimplex of K . We use the same notation for filtered simplicial complexes.As expected from the proof of a simplicial approximation theorem, we are be makingextensive use of the notion of stars. HAPTER 1. STRATIFIED HOMOTOPY THEORY Construction 1.3.5.
Recall (see [RS12, Ch. 3]) that the closed star of a simplex σ ina simplicial complex K , denoted star K ( σ ), is the subcomplex given by all simplices thatare contained in a simplex containing σ . Recall further that the open star of a simplex σ in K , denoted star K ( σ ), is the set of all simplices containing σ . We denote by | star K ( σ ) | the union of the open simplices in star K ( σ ), | star K ( σ ) | := (cid:91) τ ∈ star K ( σ ) ˚ τ ⊂ | K | . The choice of notation should not mislead one to think that star K ( σ ) is actually asimplicial complex. We use the same notation in the filtered setting.The following properties of the star constructions are immediate from the non-filteredcase. Remark 1.3.6.
Let K be a filtered simplicial complex and σ = { x , ..., x k } ∈ K . Thenthe following hold.(i) star K ( σ ) = (cid:84) x i ∈ σ star K ( x i )(ii) star K ( σ ) ⊂ (cid:84) x i ∈ σ star K ( x i )(iii) | star K ( σ ) | = { (cid:80) x ∈ K (0) t x x ∈ | K | | t x > x ∈ σ } (iv) | star K ( σ ) | is an open neighbourhood of ˚ σ in | K | .(v) | star K ( σ ) | = | star K ( σ ) | Now, let τ ∈ K be another simplex. Then the following are equivalent: • σ (cid:63) K τ in K . • star K ( σ ) ∩ star K ( τ ) (cid:54) = ∅ • | star K ( σ ) | ∩ | star K ( τ ) | (cid:54) = ∅ • star K ( σ ) ∩ star K ( τ ) (cid:54) = ∅ • | star K ( σ ) | ∩ | star K ( τ ) | (cid:54) = ∅ In particular, one obtains a geometrical condition for when a set of vertices { x (cid:48) , ..., x (cid:48) k } ⊂ K (0) forms a simplex in K ; i.e. that the intersection of their (realized) open stars is non-empty. Now, if f : K → L is a map of simplicial complexes, then:(a) | f | ( | star K ( σ ) | ) ⊂ | star L ( f ( σ )) | (b) | f | ( | star K ( σ ) | ) ⊂ | star L ( f ( σ )) | As an immediate consequence of this remark together with Remark 1.3.16, we obtainan equivalent characterization of when a map on the vertices of filtered complexes inducesone of filtered complexes.
HAPTER 1. STRATIFIED HOMOTOPY THEORY Lemma 1.3.7.
Let
K, L ∈ sCplx P and f : K (0) → L (0) be a map such that f ( p K ( x )) = p L ( f ( x )) . Then f extends to a morphism in sCplx P if and only if for every σ ∈ K , (cid:84) x ∈ σ star L ( f ( x )) or equivalently (cid:84) x ∈ σ | star L ( f ( x )) | is non-empty. Even when the filtration condition of Lemma 1.3.7 is not fulfilled, the open stars canbe used to obtain an upper boundary for the stratum of an open simplex.
Lemma 1.3.8.
Let K ∈ sCplx P and x ∈ K . Then for any ξ ∈ | star K ( x ) | , p K ( x ) ≤ p | K | ( ξ ) . Proof.
Let τ be a simplex such that ξ ∈ ˚ τ and x ∈ τ . Then p K ( x ) ≤ max( p K ( τ )) = p | K | ( ξ ) . We also need a few remarks about line segments.
Definition 1.3.9.
Let ξ and η in | K | be two points that lie in a common simplex | σ | .Denote by [ ξ, η ] := { (1 − t ) ξ + tη | t ∈ [0 , } ⊂ | σ | ⊂ | K | ⊂ R K (0) the affine line segment between them. We call this the closed line segment between ξ and η . We further denote by ( ξ, η ) := [ ξ, η ] \ { ξ, η } and call this the open line segment between ξ and η . Furthermore, we define half openline segments in the obvious way.By construction, every open line segment lies in a unique open simplex of | K | , cor-responding to the join of τ, τ (cid:48) ⊂ σ with ξ ∈ ˚ τ and η ∈ ˚ τ (cid:48) . As a direct consequencethere is the following lemma, which will be quite useful later on for the construction ofhomotopies. Lemma 1.3.10.
Let K ∈ sCplx P and σ ∈ K . Then, for any half open line segment [ ξ, η ) with ξ ∈ | star K ( σ ) | , we also have [ ξ, η ) ⊂ | star K ( σ ) | .Proof. As | star K ( σ ) | is open and ξ ∈ star K ( σ ), we have ( ξ, η ) ∩ | star K ( σ ) | (cid:54) = ∅ . Hence,as the open simplices of | K | are pairwise disjoint, and | star K ( σ ) | is the union of suchsimplices, we get τ ∈ star K ( σ ), where τ is the unique simplex with ( ξ, η ) ⊂ ˚ τ . Inparticular, [ ξ, η ) = { ξ } ∪ ( ξ, η ) ⊂ | star K ( σ ) | . Furthermore, line segments also interact rather nicely with the stratification.
Lemma 1.3.11.
Let | σ | be a simplex of a filtered simplicial complex | K | . Let ξ, η ∈ | σ | and p | K | ( ξ ) ≤ p | K | ( η ) . Then ( ξ, η ] lies in the p | K | ( η ) -th stratum of | K | . HAPTER 1. STRATIFIED HOMOTOPY THEORY Proof.
This is immediate from the characterization of | K | p as the set of elements (cid:110) (cid:88) x ∈ K (0) t x | x | ∈ | K | | p = max ( p K ( { x | t x > } )) (cid:111) ⊂ R K (0) . Finally, we need to make use of regular neighbourhoods (see [RS12, Ch. 3]).
Definition 1.3.12.
Recall that the simplicial neighbourhood of a subcomplex L ⊂ K , N ( K, L ), is the subcomplex of K given by N ( K, L ) := (cid:91) σ ∈ L star K ( σ ) . Denote by ∂N ( K, L ) the intersection of K − L and N ( K, L ). Then one has K = ( K − L ) ∪ N ( K, L ) . Furthermore, we denote by ˚ N ( K, L ) the set of simplices given by N ( K, L ) \ ∂N ( K, L ).Just as for stars, denote by | ˚ N ( K, L ) | the union of the open simplices in ˚ N ( K, L ) in | K | .An open line segment ( ξ, η ) ⊂ | N ( K, L ) | with ξ ∈ | L | and η ∈ | K − L | is called a ray of N ( K, L ). We use the same notation in the filtered setting.Similarly to the setting of stars one then has:
Remark 1.3.13.
Let K ∈ sCplx P and A be a full subcomplex of K . Then the followinghold:(i) N ( K, A ) = { σ (cid:63) K τ | σ ∈ A, τ ∈ ∂N ( K, A ) s.t. σ (cid:63) τ ∈ K } (ii) | ˚ N ( K, A ) | is an open (regular in the topological sense) neighbourhood of | A | in | K | .(iii) | ˚ N ( K, A ) | \ | A | is the disjoint union of the rays of N ( K, A ).(iv) | ˚ N ( K, A ) | \ | A | is covered by sets of the shape | star K ( { a, x } ) | , for a ∈ A (0) and x ∈ ∂N ( K, A ) (0) , { a, x } ∈ K .We will be making explicit use of the constructions that turn | ˚ N ( K, L ) | into a regularneighbourhood later on so we now go a little bit more into detail, describing them here.We use the notation from remark 1.3.2. Construction 1.3.14.
Let K ∈ sCplx P and A a be full subcomplex of K . Considerthe map s A : | K | −→ [0 , (cid:88) x ∈ ( K − A ) (0) t x | x | + (cid:88) a ∈ A (0) t a | a | (cid:55)−→ (cid:88) x ∈ ( K − A ) (0) t x . HAPTER 1. STRATIFIED HOMOTOPY THEORY s − A ( { } ) = | K − A | , s − A ( { } ) = | A | and s − A ((0 , | ˚ N ( K, A ) | \ | A | . s A restrictedto | N ( K, A ) | is called the radial parameter . Furthermore, we have the two maps: α A : | ˚ N ( K, A ) | −→ | A | (cid:88) x ∈ ∂N ( K,A ) (0) t x | x | + (cid:88) a ∈ A (0) t a | a | (cid:55)−→ (cid:88) a ∈ A (0) t a (cid:80) a ∈ A (0) t a | a | ,β A : | N ( K, A ) | \ | A | −→ | ∂N ( K, A ) | (cid:88) x ∈ ∂N ( K,A ) (0) t x | x | + (cid:88) a ∈ A (0) t a | a | (cid:55)−→ (cid:88) x ∈ ∂N ( K,A ) (0) t x (cid:80) x ∈ ∂N ( K,A ) (0) t x | x | . For ξ ∈ | ˚ N ( K, A ) | \ | A | , ( α A ( ξ ) , β A ( ξ )) is then the unique ray of N ( K, A ) containing ξ .Further, the three maps are then related via: ξ = (1 − s A ( ξ )) α A ( ξ ) + s A ( ξ ) β A ( ξ ) . (1.10)Furthermore, we obtain a deformation retraction of | ˚ N ( K, A ) | onto | A | via: H : | ˚ N ( K, A ) | × ∆ −→ | ˚ N ( K, A ) | ( ξ, t ) (cid:55)−→ (1 − t ) ξ + tα A ( ξ ) . Note that, by Lemma 1.3.11, this is a stratum preserving map up to t = 1. Such ahomotopy is called a nearly strict homotopy (see [Qui88]). Just as in the classical setting, the filtered simplicial setting has a notion of subdivision.This of course just comes down to subdividing the underlying simplicial complexes in away compatible with the filtration.
Definition 1.3.15.
Let K ∈ sCplx P . A subdivision ( K (cid:48) , s ) of K , written K (cid:48) (cid:67) K ,is a filtered simplicial complex K (cid:48) over P , together with an inclusion s : K (cid:48) (0) (cid:44) → | K | fulfilling the following conditions: • s respects strata, that is p | K | ( s ( x (cid:48) )) = p K (cid:48) ( x (cid:48) ). • For every σ ∈ K (cid:48) there exists a τ ∈ K such that s ( σ ) ⊂ | τ | . • The induced linear extension of s | K (cid:48) | → | K | with respect to | K | ⊂ R K (0) ≥ is ahomeomorphism. Remark 1.3.16.
Note that under the conditions of Definition 1.3.15 the induced map | K (cid:48) | → | K | is automatically a stratum preserving homeomorphism. To see this, let σ bea simplex in K (cid:48) and ξ ∈ ˚ σ . Let τ be the minimal simplex in K such that s ( σ ) ⊂ | τ | .Then s ( ξ ) ⊂ ˚ τ (compare [Spa89, Ch. I, Sec. 4]). Now, let x (cid:48) m be a maximal vertex in HAPTER 1. STRATIFIED HOMOTOPY THEORY σ with respect to p K (cid:48) , i.e. one such that ξ lies in the p K (cid:48) ( x (cid:48) m )-th stratum. Then, byminimality of τ , and the fact that s respects strata, we also know that for a maximalvertex x m of τ we get p K ( x m ) = p K (cid:48) ( x (cid:48) m ). In particular, ˚ τ and hence also the image of ξ lies in the p K (cid:48) ( x (cid:48) m )-th stratum.In the following, we will usually omit any mention of the map s and just write K (cid:48) (cid:67) K . Remark 1.3.17.
Clearly, just as in the classical setting (cid:67) behaves transitively i.e. sub-divisions can be composed. That is, for filtered simplicial complexes K (cid:48)(cid:48) , K (cid:48) , K we haveand subdivisions K (cid:48)(cid:48) (cid:67) K (cid:48) , K (cid:48) (cid:67) K one obtains a subdivision K (cid:48)(cid:48) (cid:67) K by composing themap K (cid:48)(cid:48) (0) → | K (cid:48) | with the induced stratum preserving homeomorphism | K (cid:48) | → | K | .Since we will frequently be subdividing simplicial complexes K together with a fullsubcomplex A , the following lemma will find a lot of - albeit sometimes implicit - usage. Lemma 1.3.18.
Let A ⊂ K be a full subcomplex of a (filtered) simplicial complex.Further, let K (cid:48) (cid:67) K be a subdivision of K and A (cid:48) (cid:67) A the induced subdivision of A .Then A (cid:48) ⊂ K (cid:48) is again a full subcomplex.Proof. Let σ (cid:48) ∈ K (cid:48) , such that all of its vertices lie in A (cid:48) . Let σ ∈ K be the minimalsimplex supporting σ (cid:48) . That is, σ is minimal with respect to | σ (cid:48) | ⊂ | σ | , where we haveidentitfied | K (cid:48) | and | K | under the induced p.l. isomorphism. It suffices to show σ ∈ A .By minimality, either | σ | = | σ (cid:48) | or some vertex of σ (cid:48) is contained in the interior of σ .In the latter case, this directly implies σ ∈ A , and hence σ (cid:48) in A (cid:48) . In the former, thismeans the vertices in σ agree with those in σ (cid:48) , and hence, by fullness of A , we obtain σ ∈ A .Recall that in Example 1.0.8, we constructed a barycentric subdivision functor sd : sCplx P → sCplx P . As the name suggests the filtered barycentric subdivision actuallygives a subdivision in the sense of Definition 1.3.15. Proposition 1.3.19.
For K ∈ sCplx P , consider the map s : sd( K ) (0) → | K | sending σ to the barycenter of | σ | ⊂ | K | - We denote this by bar( σ ) . This induces a subdivision sd( K ) (cid:67) K. Proof.
By the classical statement (see [Spa89, Ch I, Sec. 3]) all one needs to show is that s respects strata. This holds by definition of the filtration of | K | as the barycenter of | σ | always lies in ˚ σ , and thus in the stratum corresponding to max( p K ( σ )) = p sd( K ) ( σ ).Denote by sd n ( K ) the n -times application of sd. Then as an immediate consequenceof this proposition and Remark 1.3.17 we have for K ∈ sCplx P a subdivisionsd n ( K ) (cid:67) K. Remark 1.3.20.
As the barycentric subdivision of a filtered simplicial set is just givenby putting a filtration on one of the underlying simplicial set, one obtains in particular the
HAPTER 1. STRATIFIED HOMOTOPY THEORY σ ∈ sd n ( K ) (cid:67) K denote by d ( σ ) the diameter of the image of | σ | in | K | , with respect to the metric inducedby | K | ⊂ R K (0) . Denote by D n the supremum over the d ( σ ) of sd n ( K ). Then, if K isfinite, D n n →∞ −−−→ . In particular, by Lebesgues Lemma, for every open covering of | K | there exists an N such that for all n ≥ N in the pulled back covering on | sd n ( K ) | every closed simplex iscontained in some open set of the covering.For the proof of the filtered simplicial approximation theorem, we will need to makeuse of relative barycentric subdivisions, akin to what Zeeman used in [Zee64]. Construction 1.3.21.
Let K be a filtered simplicial complex with a full subcomplex A . Consider the simplicial complex given by { σ (cid:63) { τ ⊂ ... ⊂ τ k } | σ (cid:63) K τ k ∈ K } ⊂ A (cid:63) sd( K − A ) . For a simplex in the above complex, we have by definition that p K ( σ ) ∪ p K ( τ k ) is a flagin P . In particular, the same holds for p K ( σ ) ∪ { max( p K ( τ )) , ..., max( p K ( τ k )) } ⊂ p K ( σ ) ∪ p K ( τ k ) . Hence, by Remark 1.3.16 the above complex is filtered by the map induced by the disjointunion of p A and p sd( K − A ) on A (0) (cid:116) sd( K − A ). We denote the induced filtered simplicialcomplex by sd( K rel A ) and call it the barycentric subdivision of K relative to A . Thiscomes with inclusions of filtered simplicial complexes A (cid:44) → sd( K rel A )sd( K − A ) (cid:44) → sd( K rel A ) , and sd( K rel A ) (0) = A (0) (cid:116) sd( K − A ) (0) . Figure 1.17: Illustration of a relative subdivision.
HAPTER 1. STRATIFIED HOMOTOPY THEORY K ) (cid:67) sd( K rel A ) (cid:67) K. Proposition 1.3.22.
In the setting of Construction 1.3.21, consider the maps s : sd( K ) (0) −→ | sd( K rel A ) | sd( K − A ) (0) (cid:51) τ (cid:55)−→ | τ | , sd( A ) (0) (cid:51) σ (cid:55)−→ bar( σ ) ∈ | σ | , sd( N ( K, A )) (0) (cid:51) σ n (cid:63) K τ m (cid:55)−→ n + 1 m + n + 2 bar( σ ) + m + 1 m + n + 2 | τ | ∈ | σ (cid:63) K τ | , for σ and τ non-empty in the last case, and n and m indicating the dimensions, and s : sd( K rel A ) (0) −→ | K | sd( K − A ) (0) (cid:51) σ (cid:55)−→ bar( σ ) ∈ | σ | ,A (0) (cid:51) a (cid:55)−→ | a | . They induce filtered subdivisions sd( K ) (cid:67) sd( K rel A ) (cid:67) K. The composition of these two subdivisions is the barycentric one, namely sd( K ) (cid:67) K from Proposition 1.3.19.Proof. The first two conditions of Definition 1.3.15 are easily verified. Next we showthat the diagram of induced stratum preserving maps | sd( K ) | | K || sd( K rel A ) | ∼ commutes. As all of them are given by piecewise linear extension, it suffices to showthat this holds on the restriction to | sd( K ) | . For x in sd( K − A ) (0) and sd( A ) (0) this isobvious by definition. Now, in the case where x = σ n (cid:63) K τ m for τ, σ (cid:54) = ∅ we obtain: x (cid:55)−→ n + 1 m + n + 2 bar( σ ) + m + 1 m + n + 2 | τ |(cid:55)−→ n + 1 m + n + 2 bar( σ ) + m + 1 m + n + 2 bar( τ )= bar( σ (cid:63) K τ )= bar( x )by (local) linearity of the maps. Hence, we have shown commutativity. To see that allmaps involved are homeomorphisms, it suffices to restrict to the finite case as all spacesinvolved have the final topology with respect to finite subcomplexes. Then all the spacesinvolved are compact Hausdorff, hence it suffices to show bijectivity. We already knowthat the horizontal arrow is a bijection so it suffices to show that | sd( K rel A ) | → | K | is injective. This is immediate from the definition. HAPTER 1. STRATIFIED HOMOTOPY THEORY We now have all the necessary tools available for the statement and the proof of thefiltered simplicial approximation theorem. We first state and prove the theorem in thecase where P = [0 , ..., q ] is a finite linearly ordered set. We formally set K − := ∅ .For the remainder of this section, let K, L ∈ sCplx P be some fixed filtered simplicialcomplexes over P . Construction 1.3.23.
Let Σ = (Σ p ) p ∈ P be a family of natural numbers indexed over P . Define inductively subdivisions of K viasd Σ ≤− ( K ) := K and sd Σ ≤ p +1 ( K ) := sd Σ p (sd Σ ≤ p ( K ) rel sd Σ ≤ p ( K ≤ p )) . Further, set sd Σ ( K ) := sd Σ ≤ q ( K ) . Let A be a full subcomplex of K such that also, for all p ∈ P , K ≤ p ∪ A ⊂ K is a fullsubcomplex. Define analogously sd Σ ≤− ( K rel A ) := K sd Σ ≤ p +1 ( K rel A ) := sd Σ p +1 (sd Σ ≤ p ( K ) rel sd Σ ≤ p ( K ≤ p ∪ A )) . Then, set sd Σ ( K rel A ) := sd Σ ≤ q ( K rel A ) . Note that we have used Lemma 1.3.18 here to justify that in each step we again subdividerelative to a full subcomplex. By composability of subdivisions and Proposition 1.3.22we get a sequence of subdivisionssd Σ ( K ) = sd Σ ≤ q ( K ) (cid:67) ... (cid:67) sd Σ ≤− ( K ) = K ;sd Σ ( K rel A ) = sd Σ ≤ q ( K rel A ) (cid:67) ... (cid:67) sd Σ ≤− ( K rel A ) = K, where in each step, p + 1, there are no new vertices added to | K | ≤ p (the union of thelatter with | A | ).We then have: Theorem 1.3.24 (Filtered Simplicial Approximation A) . Let K be a finite filtered sim-plicial complex over a finite linearly ordered set P . Further, let L be another filteredsimplicial complex over P and φ : | K | P −→ | L | P be a stratum preserving map. Then there exists a Σ as in Construction 1.3.23 and astratum preserving simplicial map f : sd Σ ( K ) −→ L such that | f | P (cid:39) P (cid:0) | sd Σ ( K ) | P ∼ −→ | K | P φ −→ | L | P (cid:1) . HAPTER 1. STRATIFIED HOMOTOPY THEORY
Theorem 1.3.25 (Filtered Simplicial Extension) . Let K be a finite filtered simplicialcomplex over a finite linearly ordered set P . Let L be another filtered simplicial complexover P . Let A be a full subcomplex of K such that for all p ∈ P , the complex A ∪ K ≤ p ⊂ K is a full subcomplex. Let φ : | K | P −→ | L | P be a stratum preserving map such that(a) φ restricted to | A | is the realization of a stratum preserving simplicial map f A .(b) φ is such that for σ ∈ A , we have φ ( | star K ( σ ) | ) ⊂ | star L ( f A ( σ )) | .Then there is a Σ as in Construction 1.3.23 and a stratum preserving simplicial map sd Σ ( K rel A ) f −→ L such that f | A = f A . Remark 1.3.26.
First, it should be noted that the fullness conditions on the subcom-plexes A ∪ X p in Theorem 1.3.25 are really not too much of a restriction. They canalways be obtained, by simply subdividing both K and L once barycentrically.It is interesting to illuminate condition (b) of Theorem 1.3.25 a little bit. It arises, forexample, if φ is not only a simplicial map on A but also on the simplicial neighbourhood N ( K, A ) of A . In the non-filtered case, (at least up to subdivision and homotopy of φ rel | A | ) this can always be accomplished. Roughly speaking, to do this take a subdivision K (cid:48) (cid:67) K in a way that adds no new vertices to | A | , but vertices to the half way points(with respect to the radial parameter) to every open simplex in | ˚ N ( K, A ) | \ | A | . For amore detailed construction, see [Zee64]. Then the identity on | K | is homotopic relativeto | A | to a map that is given as follows. Take the identity on | K − A | and then mapthe new vertices in | N ( K, A ) | onto | A | along a last vertex map (for some appropriateordering). Then extend piecewise linearly. This map is given on N ( K (cid:48) , A ) by a simpli-cial map N ( K (cid:48) , A ) → A . In particular, φ is then homotopic to a map that is given on | N ( K (cid:48) , A ) | by the simplicial map N ( K (cid:48) , A ) −→ A f A −→ L. What we have done, is effectively constructing a particularly nice regular neighbour-hood. In the filtered case however, these homotopies can in general not be chosen to bestratum preserving. Take for example A = K ≤ p for some space with nontrivial filtra-tion. Clearly, if | K ≤ p | lies in the closure of | K | \ | K ≤ p | , there is no way to contract any HAPTER 1. STRATIFIED HOMOTOPY THEORY | K p | into | K p | in a stratum preserving way. Any such neighbourhoodcontains points outside of | K p | . In other words, only very rarely are subspaces of filteredspaces NDRs in a filtered way. However, for the application we are most interested in,i.e. for approximating homotopies, the situation is a lot more favourable as we can seein the next example. Example 1.3.27.
Let K be an ordered P -filtered simplicial complex and L another P -filtered simplicial complex. For another ordered simplicial complex M , denote by K ⊗ M the filtered (ordered) simplicial complex obtained by taking the product of orderedsimplicial complexes, and projecting to the first component, to obtain a filtration (seeSection 1.3.4 for details). Let H : | K ⊗ ∆ | ∼ = | K | ⊗ ∆ → | L | be any stratum preservinghomotopy, such that H restricted to | K |(cid:116)| K | comes from a stratum preserving simplicialmap f (cid:116) g : K (cid:116) K → L . Then, consider | K ⊗ sd ∆ | = | K | ⊗ [0 , , with the stratum preserving homeomorphism given by by thinking of sd ∆ as beingglued from 4 1-simplices. The homotopy:˜ H : | K ⊗ sd ∆ | = | K | ⊗ [0 , −→ | L | given by H from 1 to 3 and the constant homotopies on the remainder, fulfills all of therequirements of Theorem 1.3.25, but the fullness condition. To see this, just note thaton | N ( K ⊗ sd ∆ , K (cid:116) K ) | = | K ⊗ ∆ (cid:116) K ⊗ ∆ | ˜ H is given by the realization of K ⊗ ∆ (cid:116) K ⊗ ∆ −→ K (cid:116) K f (cid:116) g −−→ L, where the left map is just the disjoint union of the projections onto K . Thus, byRemark 1.3.26, it fulfills the requirements. We can then subdivide once barycentricallyto obtain the situation of Theorem 1.3.25.We proceed with the proof of these two theorems in several steps. Notationally westay in the setting of Theorem 1.3.25. First we construct the simplicial map in Theo-rem 1.3.25 and show it has several additional properties. In fact, we construct a mapon vertices in Proposition 1.3.28, fulfilling a set of conditions that imply that is extendsto a stratum preserving simplicial map as in Theorem 1.3.25. Finally, we show that for A = ∅ its realization is stratum preserving homotopic to φ , thus proving Theorem 1.3.24.For the remainder of this section, we denote B A,p := sd Σ ( K rel A ) − sd Σ ( K ≤ p − ∪ A rel A ) , for a string Σ as in Theorem 1.3.25 and p ∈ P . If A = ∅ we just write B p for this. Notethat in this case B p ∩ sd Σ ( K rel A ) ≤ p = sd Σ ( K rel A ) p . HAPTER 1. STRATIFIED HOMOTOPY THEORY | K (cid:48) | with | K | to make notation a little more concise. Proposition 1.3.28.
There exists a Σ and a map f : sd Σ ( K rel A ) (0) −→ L (0) such that the following conditions hold. Let K (cid:48) := sd Σ ( K rel A ) . Then: (i) For x ∈ K (cid:48) (0) , we have p K (cid:48) ( x ) ≤ p L ( f ( x )) . (ii) For p ∈ P and x ∈ K (cid:48) (0) ≤ p \ A (0) , we have φ ( | star K (cid:48) ( x ) ∩ B A,p | ) ⊂ | star L ( f ( x )) | . (iii) f agrees with f A on A (0) . Before we prove this proposition, we show that it implies Theorem 1.3.25 to motivatethe conditions a little more.
Corollary 1.3.29. f as in Proposition 1.3.28 extends to a morphism of filtered sim-plicial complexes f : K (cid:48) → L fulfilling: (i) For p ∈ P and σ ∈ ( K (cid:48) − A ) ≤ p , φ ( | star K (cid:48) ( σ ) ∩ B A,p | ) ⊂ | star L ( f ( σ )) | . (ii) f | A = f A .In particular, Theorem 1.3.25 holds.Proof of Corollary 1.3.29. What we have to show is that f extends to a map of sim-plicial complexes; that is, that for σ ∈ K (cid:48) f ( σ ) is a simplex in L and that furthermore(by Remark 1.3.16) for each x ∈ K (cid:48) (0) we also have p K (cid:48) ( x ) ≥ p L ( f ( x )) . For the first condition, note that, by the equivalent characterization of when a join ofsimplices is a simplex in K (Remark 1.3.6), it suffices to show that certain intersectionsof stars in L are non-empty. We start with the cases where σ ∈ A or σ ∈ K (cid:48) − A . For theformer, this is obvious as we already know that f extends to a simplicial map on A . Forthe latter case, let p = max( p K (cid:48) ( σ )). Then star K (cid:48) ( σ ) ∩ B A,p is non-empty. Hence, (usingfrom Remark 1.3.6 (i) and (ii)), together with (i) we also obtain that (cid:84) x i ∈ σ | star( f ( x i )) | is non-empty, hence by Remark 1.3.6 that f ( σ ) is a simplex of L . It remains to showthe case where the simplex lies in neither of the two. Since A is a full subcomplex of K (cid:48) ,such a simplex is of the shape σ (cid:63) K (cid:48) τ , with τ ∈ K (cid:48) − A and σ ∈ A . Now, let x ∈ τ be avertex, maximal with respect to p K (cid:48) and p := p K (cid:48) ( x ). As we have just seen before, φ ( | x | ) ∈ | star L ( f ( τ )) | . At the same time | x | ∈ | star K (cid:48) ( σ ) | = | star K ( σ ) | . HAPTER 1. STRATIFIED HOMOTOPY THEORY φ ( | x | ) ∈ | star L ( f ( σ )) | . In particular, | star L ( f ( σ )) | and | star L ( f ( τ )) | have non-empty intersection. Hence, f ( σ ) (cid:63) L f ( σ ) = f ( σ (cid:63) K (cid:48) τ ) is a simplex of L and thus f extends to a map of simplicialcomplexes.Now, preservation of strata is an immediate consequence of Lemma 1.3.8 and (ii) ofProposition 1.3.28 and the fact that φ preserves strata. Proof of Proposition 1.3.28.
We define Σ and f inductively. As only values of p (cid:48) lesseror equal to p occur in the definition of sd Σ ( − ) it makes sense to talk about the latter,even if Σ is only defined up to p . Assume inductively that we already have defined Σ upto p ∈ P and set K p := sd Σ ≤ q ( K rel A ). Denote by B pr,A the subcomplex of K p givenby K p − ( K p ≤ r ∪ A ), for r, p ∈ P . Further, assume we have already defined g r : K r, (0) −→ L (0) , for 0 ≤ r ≤ p such that g agrees with f A on A (0) , and that the following hold:(i) r For x ∈ (cid:0) K rr (cid:1) (0) , we have p K r ( x ) ≤ p L ( g p ( x )) . (ii) r For x ∈ (cid:0) K r − A (cid:1) (0) , we have φ ( | star K r ( x ) ∩ B rA,r | ) ⊂ | star L ( g r ( x )) | . (iii) r g r agrees with g r − on K r, (0) ≤ r − ∪ A (0) = K r − , (0) ≤ r − ∪ A (0) , for r ≥ x, y vs σ, τ respectively. For p = 0 consider the open covering of | K − A | given by pulling backthe covering of | L | by the open stars | star L ( y ) | , y ∈ L . By Remark 1.3.20, for somesufficiently large Σ we have that for each x ∈ sd Σ ( K − A ) there exists a y ∈ L suchthat φ ( | star sd Σ0 ( K − A ) ( x ) | ) ⊂ star L ( y ) . Set g ( x ) to such a y , for x ∈ sd Σ ( K − A ) and to f A ( x ) for x ∈ A . Then clearly theconditions (i) r , (ii) r and (iii) r , for r = 0, are satisfied.For the following p , no new vertices are added to | ˚ N ( K, A ) | and by (iii) r we keep g p = f A on A (0) . Hence, on A the condition (i) r is satisfied by assumption. As (ii) r does notdepend on what is happening on | A | , we may as well replace K by K − A and henceassume A = ∅ to make notation a little cleaner.For the inductive step p to p +1, let C be the subcomplex given by such simplices σ ∈ K p where φ ( | σ | ) ⊂ | ˚ N ( L, L ≤ p ) | − | L ≤ p | . Since φ is a stratum preserving map, C ⊂ B pp +1 .Then, by Remark 1.3.13, specifically (iv) of the latter, | C | is covered by the pulled HAPTER 1. STRATIFIED HOMOTOPY THEORY | star L ( { y p , y } ) | , for a 1-simplex { y p , y } ∈ L with y p ∈ L ≤ p and y ∈ ∂ ( N ( L, L ≤ p ). Hence, for some sufficiently large Σ p +1 , we have thatfor every x ∈ sd Σ p +1 ( C ), there are y p ∈ L p and a y ∈ ∂N ( L, L p ) such that { y p , y } ∈ L and φ ( | star K p +1 ( x ) ∩ sd Σ p +1 ( C ) | ) ⊂ star L ( { y p , y } ) ⊂ star L ( y ) . (1.11)This defines K p +1 . Now, for g p +1 , consider the following cases:(a) x ∈ K p +1 ≤ p = K p ≤ p : g p +1 ( x ) := g p ( x ) . (b) x ∈ B p +1 p +1 , | x | ∈ ˚ σ for σ ∈ K p , and σ has a least one vertex, x ∈ σ , such that p L (cid:0) g p ( x ) (cid:1) > p : g p +1 ( x ) := g p ( x ) , for such a x .(c) Neither are the case. Hence, x ∈ B p +1 p +1 . Take the σ ∈ K p such that | x | ∈ ˚ σ . Then | σ | ⊂ | B p +1 p +1 | . and g p ( σ ) ⊂ L ≤ p . As no new vertices were added to | K p | outside of | B p +1 p +1 | , we have that | B p +1 p +1 | = | B pp +1 | . In particular, σ ∈ B pp +1 ⊂ B pp . Hence, byRemark 1.3.6, specifically (i) and (ii) there together with (ii) r for r = p : φ ( | σ | ) ⊂ φ ( | star K p ( σ ) ∩ B pp | ) ⊂ (cid:92) x i ∈ σ | star L ( g p ( x i )) | ⊂ | ˚ N ( L, L p ) | . Here, the last inclusion comes from g p ( σ ) ⊂ L ≤ p . In other words, as φ preservesstrata, σ ∈ C . Now, set g p +1 ( x ) := y for a y given by (1.11).We need to see that this satisfies the inductive conditions ((i) r , (ii) r and (iii) r ), for r = p + 1. The first condition is immediate by construction. We have not changed g p +1 on vertices in K pp and outside of it we have systematically set g p +1 ( x ) to vertices instrata of index higher than p . The third condition is also obviously fulfilled.It remains to show (ii) r . For x ∈ K p +1 ≤ p , first note that as we have not added anynew vertices to | ˚ N ( K p , K pp ) | , | star K p ( x ) | = | star K p +1 ( x ) | and | B p +1 p +1 | = | B pp +1 | . Hence, inthis case (ii) r for r = p gives the result. We are left with the case x ∈ B p +1 p +1 . So, g p +1 ( x )is either given as in (b) on in (c).In case (b), let σ be the simplex with | x | ∈ | ˚ σ | and x ∈ σ with g p ( x ) / ∈ L ≤ p . Then | star K p +1 ( x ) | ⊂ | star K p ( x ) | and | B p +1 p +1 | = | B pp +1 | ⊂ | B pp | . Hence, φ ( | star K p +1 ( x ) ∩ B p +1 p +1 | ) ⊂ φ ( | star K p ( x ) ∩ B pp | ) ⊂ | star L ( g p ( x )) | = | star L ( g p +1 ( x )) | . HAPTER 1. STRATIFIED HOMOTOPY THEORY σ as in (c). For any vertex x i of σ we have g p ( x i ) ∈ L ≤ p . Inparticular, by (ii) r , for r = p , and again Remark 1.3.6 we obtain: φ ( | star K p ( σ ) ∩ B pp | ) ⊂ (cid:92) x i ∈ σ star L ( x i ) ⊂ | ˚ N ( L, L ≤ p ) | . Hence, as φ preserves strata, | star K p ( σ ) ∩ B pp +1 | is mapped into | ˚ N ( L, L ≤ p ) | \ | L ≤ p | . Inother words, star K p ( σ ) ∩ B pp +1 ⊂ C. In particular, | star K p +1 ( x ) ∩ B p +1 p +1 | ⊂ | star K p ( σ ) ∩ B pp +1 | ⊂ | C | and thereby | star K p +1 ( x ) ∩ B p +1 p +1 | = | star K p +1 ( x ) ∩ sd Σ ≤ p +1 C | . Hence, by construction of g p +1 ( K ), we have φ ( | star K p +1 ( x ) ∩ B p +1 p +1 | ) ⊂ star L ( { y p , y } ) ⊂ star L ( y ) = star L ( g p +1 ( x )) , for y and y p as in (c).Next we show that if we set f := g q the conditions in the statement of the propo-sition are fulfilled. (iii) is obvious by (iii) r and the start of the induction. For (i), notethat by construction for p ≤ q we have | B pA,p | = | B p +1 A,p | = ... = | B qA,p | = | B A,p | . and for x ∈ K (cid:48)≤ p = K p ≤ p : | star K p ( x ) | = | star K p +1 ( x ) | = ... = | star K q ( x ) | = | star K (cid:48) ( x ) | . Hence, (ii) follows by (iii) r together with (ii) r . We are lacking that f fulfills (i). Using(iii) r and (i) r we immediately obtain that, for each x ∈ K (cid:48) , p K (cid:48) ( x ) ≤ p L ( g q ( x )) = p L ( f ( x )) . This finishes the proof.To prove Theorem 1.3.24 we are left with showing the following.
Proposition 1.3.30.
The stratum preserving simplicial map K (cid:48) → L from Corol-lary 1.3.29 is stratified homotopic to φ . HAPTER 1. STRATIFIED HOMOTOPY THEORY Proof.
First, note that by Lemma 1.3.11, whenever the straight line homotopy betweento stratum preserving maps can be constructed it is automatically stratum preserving.For the straight line homotopy between a continuous map between realizations of sim-plicial complexes ψ : | ˜ K | → | ˜ L | and the realization of a simplicial map g : ˜ K → ˜ L toexist it suffices that, for any simplex σ ∈ ˜ K , the condition ψ (˚ σ ) ⊂ | star L ( g ( σ )) | is fulfilled. In particular, by (i) of Corollary 1.3.29, we can use the straight line homotopyon | K (cid:48) q (cid:116) ... (cid:116) K (cid:48) | . We now inductively extend this homotopy over K p := | B p (cid:116) K (cid:48) p − (cid:116) K (cid:48) p − (cid:116) ... (cid:116) K (cid:48) | , going from p = q to p = 0. We induce over the following assumption. There exists astratum preserving homotopy F p : K p ⊗ ∆ −→ | L | between the restrictions of φ and | f | , fulfilling additionally the following smallness con-dition. For r ≤ p , σ ∈ K (cid:48) r and ξ ∈ | star K (cid:48) ( σ ) ∩ B r | ∩ K p we also have F p ( { ξ } × [0 , ⊂ | star L ( f ( σ ) | . (1.12)Note how, by Lemma 1.3.10 and (i) of Corollary 1.3.29, this condition always holds on K q for any p as then F p ( { ξ } × [0 , φ ( ξ ) , | f | ( ξ )) , which is a half open line segment starting in | star L ( f ( σ ) | . This gives the start of theinduction. Now, from p + 1 to p , note that K p \ K p +1 = ( | ˚ N ( K (cid:48) , K (cid:48)≤ p ) | \ | K (cid:48)≤ p | ) ∩ | B p | = | ˚ N ( B p , K (cid:48) p ) | \ | K (cid:48) p | . In particular, by Remark 1.3.13, K p − K p +1 is given by the disjoint union of rays ( α, β )of N ( B p , K (cid:48) p ), with α ∈ | K (cid:48) p | and β ∈ | B p − K (cid:48) p | = | B p +1 | . Now, consider the maps α K (cid:48) p : | ˚ N ( B p , K (cid:48) p ) | −→ | K (cid:48) p | ; β K (cid:48) p : | N ( B p , K (cid:48) p ) | \ | K (cid:48) p | −→ | B p +1 | from Construction 1.3.14, sending a ξ ∈ K p − K p +1 to the respective end points of theray it is contained in. We omit the subscript K (cid:48) p from here on out. Now, define: λ : ( K p \ K p +1 ) × [0 , −→ | L | ( ξ, t ) (cid:55)−→ φ ((1 − t ) α ( ξ ) + tβ ( ξ )) for t ≤ , ( ξ, t ) (cid:55)−→ F p +1 ( β ( ξ ) , t −
1) for t ≥ . HAPTER 1. STRATIFIED HOMOTOPY THEORY s K (cid:48) p : | ˚ N ( B p , K (cid:48) p ) | −→ [0 , λ ( ξ, s ( ξ )) = φ ( ξ ) . Now, for a fixed ξ ∈ K p \ K p +1 , let r ≤ p and σ ∈ K (cid:48) p be such that ξ ∈ | star K (cid:48) ( σ ) ∩ B r | . Then, as ξ ∈ ( α ( ξ ) , β ( ξ )), we also have that( α ( ξ ) , β ( ξ )) ⊂ | star K (cid:48) ( σ ) ∩ B r | and hence that [ α ( ξ ) , β ( ξ )] ⊂ | star K (cid:48) ( σ ) ∩ B r | . Consequently, by (i) of Corollary 1.3.29, we obtain φ ([ α ( ξ ) , β ( ξ )]) ⊂ star L ( f ( σ )) . By the induction hypothesis, we also have F p +1 ( { β ( ξ ) } × [0 , ⊂ | star L ( f ( σ )) | . In particular, we obtain λ ( { ξ } × [0 , ⊂ | star L ( f ( σ )) | (1.13)and λ ( { ξ } × [0 , ⊂ | star L ( f ( σ )) | . Such a σ and r always exist as ξ ∈ | N ( B p , K (cid:48) p ) | . σ can be taken to be the simplex in K (cid:48) p such that α ( ξ ) ∈ ˚ σ . In this case, we also have | f | ( α ( ξ )) ∈ ˚ f ( σ ) . In particular, the following map defined by linearly interpolating between λ ( ξ, t ) and | f | ( α ( ξ )) is well-defined.Λ : ( K p \ K p +1 ) × [0 , × [0 , −→ | L | ( ξ, t, s ) (cid:55)−→ (1 − s ) | f | ( α ( ξ )) + sλ ( ξ, t )Then, by Lemma 1.3.10 and (1.13), we obtain that, for a σ as in (1.12), we always haveΛ( { ξ } × [0 , × (0 , ⊂ | star L ( f ( σ )) | . (1.14)Furthermore, by Lemma 1.3.11, for ξ ∈ K p \ K p +1 , ( α ( ξ ) , β ( ξ )] lies in the same stratumas β ( ξ ). Denote its index by p (cid:48) . Then, again by Lemma 1.3.11, the fact that Λ is HAPTER 1. STRATIFIED HOMOTOPY THEORY F p +1 is stratum preserving,we obtain that Λ( { ξ } × (0 , × (0 , ⊂ | L | p (cid:48) (1.15)Now, let κ : [0 , × [0 , / [0 , × { } −→ [0 , × [0 , , such that κ restricts to the linear homeomorphisms (where we indicate the orientationby the ordering of the interval):[0 , × { } −→ { } × [0 , , [1 , × { } −→ [0 , × { } , { } × [1 , −→ { } × [1 , , { } × [0 , −→ [1 , × { } . As Λ is constant along { ξ } × [0 , × { } and ( K p \ K p +1 ) is locally compact (hencequotients commute with products with the latter) Λ descends to a map¯Λ : ( K p \ K p +1 ) × (cid:0) [0 , × [0 , / [0 , × { } (cid:1) −→ | L | and then by composing with an inverse to 1 K p \K p +1 × κ induces a map:˜Λ : ( K p \ K p +1 ) × I −→ | L | such that: ˜Λ( ξ, t, s ) = φ ((1 − s ) α ( ξ ) + sβ ( ξ )) , if t = 0 F p +1 ( β ( ξ ) , t ) , if s = 1(1 − s ) | f | ( α ( ξ )) + s | f | ( β ( ξ )) , if t = 1 F p +1 ( α ( ξ ) , t ) , if s = 0 (1.16)Now, set F p on ( K p \ K p +1 ) × I to( K p \ K p +1 ) × ∆ ( K p \ K p +1 ) × I | L | ( ξ, t ) ( ξ, t, s ( ξ )) ˜Λ( ξ, t, s ( ξ )) . ˜Λ . By (1.10) of (1.3.14) and the piecewise linearity of | f | , we obtain that this in fact givesa homotopy between φ | K p \K p +1 and | f || K p \K p +1 . (For an illustration, see Figure 1.18).If ξ converges to ξ in the boundary of K p \ K p +1 , then s ( ξ ) converges either to 1 orto 0 and ξ to either β ( ξ ) or α ( ξ ) respectively. Hence, again by (1.16), we obtain acontinuous extension of F p +1 to K p . We need to check that this homotopy is in factstratum preserving, and that the inductive condition (1.12) holds. Let ξ ∈ K p \ K p +1 .Under the map ( K p \ K p +1 ) × ∆ −→ ( K p \ K p +1 ) × I ξ × [0 ,
1) maps into { ξ } × [0 , × (0 , . Under the inverse of 1 × κ the latter maps into { ξ } × (0 , × (0 , ∪ [0 , × { } . Hence, the inductive smallness condition follows by(1.14). Analogously, { ξ } × [0 ,
1] is mapped into (0 , × [0 ,
1] which in turn is mappedinto (0 , × (0 , F p is stratum preserving. HAPTER 1. STRATIFIED HOMOTOPY THEORY λ , Λ and F , marked respectively in different colours. Recall that an ordered simplicial complex is a simplicial complex K together with alinear ordering on every simplex σ ∈ K , compatible in the sense that if τ ⊂ σ , thenthe ordering on τ is the restriction of the one in σ . We usually write simplices in anordered simplicial complex in the shape { x ≤ ... ≤ x k } . Further, recall that a map ofordered simplicial complexes is a map of simplicial complexes restricting to a monotonousmap on each simplex. In Example 1.0.2 we defined the category of P -filtered orderedsimplicial complexes, sCplx o P . As we did for the several other categories of filteredobjects we introduced, we begin this subsection by giving some alternative descriptionsof this category. Remark 1.3.31.
Similarly to Remark 1.3.1, a P -filtered ordered simplicial complex isalternatively described as a simplicial complex K together with a map p K : K (0) → P fulfilling some condition. In the non-ordered case, this condition was that the imageof every simplex is a flag in P . In the ordered case, we need to reflect the fact that p K : K → sd( P ) is a map of ordered complexes. This is equivalently specified by:( x < y ) ∈ K = ⇒ p K ( x ) ≤ p K ( y ) . In other words, this requires that the ordering on K is such that vertices in lowerstrata always come before vertices in higher strata. A morphism of ordered filteredsimplicial complexes f : K → L is equivalently a map of the underlying ordered simplicialcomplexes, fulfilling p K ( x ) = p L (cid:0) f ( x ) (cid:1) . This justifies calling such simplicial maps stratumpreserving .The advantage of the ordered over the unordered category is that it embeds fullyfaithfully into s
Set P . We use this to apply the simplicial approximation theorems The-orem 1.3.24 and Theorem 1.3.25 in the setting of filtered simplicial sets. HAPTER 1. STRATIFIED HOMOTOPY THEORY Construction 1.3.32.
Consider the functor S o : sCplx o −→ s Set K (cid:55)−→ (cid:8) [ n ] (cid:55)→ Hom sCplx o (∆ n , K ) (cid:9) with the obvious functoriality on morphisms and ∆ n thought of as the ordered simplicialcomplex given by flags of [ n ].Simplicial sets of the form S o ( K ) have the property that every simplex σ ∈ X ([ k ]) isuniquely determined by its multiset of vertices, that is by ( x i ) i ∈ [ k ] allowing for permu-tations. This follows immediately from the definition of a map of (ordered) simplicialcomplexes. It turns out that this property completely describes simplicial sets that areisomorphic to S ( K ), for some K ∈ sCplx .To see this, denote by C the full subcategory of simplicial sets fulfilling the latter prop-erty. Consider the functor from s Set to sCplx constructed as follows. For X ∈ s Set define a simplicial complex with vertices X ([0]), by taking the simplices to be suchsubsets { x , ..., x k } ⊂ X that are the set of vertices of a simplex σ ∈ X ([ k ]). Thisconstruction becomes functorial by checking that for a map of simplicial set f : X → Y the induced map f ([0]) : X ([0]) → Y ([0]) induces a map of simplicial complexes. Weobtain a functor C : s Set −→ sCplx . In general, the image of this functor does not naturally carry the structure of an orderedsimplicial complex. This is due to the fact that two vertices in a simplicial set mightbe connected by two 1-simplices with different orientation. However, in the case where X ∈ C , the relation given on X ([0]) by x ≤ y ⇐⇒ x = d ( σ ) , y = d ( σ )for some σ ∈ X ([1]), turns C ( X ) into an ordered simplicial complex as then no twoconflicting orders occur on any simplex. For f : X → Y ∈ C , C ( f ) is then a map ofordered simplicial complexes, with respect to this additional structure. Hence, we obtaina functor: C o : s Set −→ sCplx o . We then have the following proposition.
Proposition 1.3.33.
In the setting of Construction 1.3.32, the functor S is a fullyfaithful embedding of categories. An inverse of its restriction in the image to C is givenby C o .Proof. By a slight abuse of notation, we denote the factorization of S into C also by S .It is a straightforward verification that 1 sCplx o ∼ = C ◦ S . For the other composition, weobtain a natural transformation to the identity by the mapHom sCplx o (∆ n , C o ( X )) → X ([ n ]) , HAPTER 1. STRATIFIED HOMOTOPY THEORY σ on the left to the unique simplex in X given by the multiset given by( σ ( { i } ) i ∈ [ n ] . It suffices to show, this is a bijection. This is immediate, from the fact that X ∈ C .Further, note that S o (sd( P )) is naturally isomorphic to N ( P ). In particular, as sCplx o P and s Set P are given by over-categories of these respectively, we obtain: Corollary 1.3.34. S o from Construction 1.3.32 induces a fully faithful embedding sCplx o P (cid:44) → s Set P . Filtered simplicial sets isomorphic to spaces in the image of this embedding are preciselythose X ∈ s Set P where every n -dimensional simplex σ : ∆ n → X is uniquely determinedby the multiset of its vertices, that is by the family ( σ ( { i } )) i ∈ [ n ] modulo permutation. Using this fact, we will often think of filtered ordered simplicial complexes as filteredsimplicial sets. In particular, the following definition makes sense:
Definition 1.3.35.
Let X ∈ s Set P be such that every n -dimensional simplex σ : ∆ n → X by its’ multiset of vertices. Then, we call X a P - filtered ordered simplicial complex ,or ( P -) FOS-complex for short. We nearly always omit the P .It is easy to see that this definition is also compatible with realization functors, i.e.that there are natural isomorphism between C C o −−→ sCplx o P → sCplx P |−| P −−−→ Top P and | − | P : s Set P → Top P . Where the functor sCplx o P → sCplx P is the forgetful functor forgetting about the or-derings on simplices. Hence, we do not distinguish between these two ways of realizingFOS-complexes, and denote them by |−| P , or just |−| , for the sake of notational brevity.It can also be useful to have the following alternative characterization of an FOS-complex. Lemma 1.3.36. A P -filtered simplicial set is an FOS-complex if and only if every non-degenerate simplex is uniquely determined by its set of vertices.Proof. This has nothing to do with the filtrations, so let us assume they are trivial.First, note that an FOS-complex has no non-degenerate simplices with degenerate faces.In particular, every non-degenerate simplex of dimension n -has precisely n differentvertices. Thus, being an FOS-complex implies the other property. Now, let X be asimplicial set satisfying the latter. Note that for such an X every nondegenerate simplexof dimension n has precisely n different vertices. We can always take a maximal face ofsuch a simplex without repetitions in the vertices. Such a face is non-degenerate and hasthe same underlying vertex set as the original simplex. Hence, it is the simplex. Next, let HAPTER 1. STRATIFIED HOMOTOPY THEORY σ and τ be two possibly degenerate simplices with the same multiset of vertices. Then, byassumption, the two simplices they degenerate from agree and have no repeating vertices.The respective degeneracy maps giving σ and τ are already completely specified by themultiplicities in their corresponding multisets. Hence, as the latter agree, σ and τ comefrom the same nondegenerate simplex and through the same degeneracy map, in otherwords σ = τ .We finish this subsection with a type of “dense up to homotopy equivalence” result,for the embedding S o . It is an immediate consequence of Theorem 2.3.23. The readerworried about circularity may be assured that we only use this result in Section 1.4. Thelargest part of the second chapter up to Section 2.4 is completely independent from thisresult. Proposition 1.3.37.
Every finite filtered simplicial set X ∈ s Set finP is isomorphic in H s Set P to a finite FOS-complex of the same dimension. In particular, many of the homotopy theoretic questions on filtered simplicial setscan be reduced to questions on FOS-complexes.
Another advantage, of FOS-complexes over nonordered ones, is that their subdivisionsadmit last vertex maps. This provides a way to represent subdivision homeomorphismsup to stratum preserving homotopy. We have already seen such constructions for filteredsimplicial sets in Section 1.1.2. The fully faithful inclusions of categories S o : sCplx o P (cid:44) → s Set P is compatible with subdivisions in the following sense. Construction 1.3.38.
Let K be any filtered simplicial complex. Then sd( K ) is natu-rally an FOS-complex, with the order on vertices given by inclusion. Hence, we can alsothink of sd as an endofunctor of sCplx o P , as well as as a functor sCplx P −→ sCplx o P . It is not hard to see that there is a natural isomorphismsd( S o ( K )) ∼ = S o (sd( K )) . Indeed, first note that the two definitions are clearly compatible on simplices and thenextend by the colimit definition of sd for simplicial sets. Hence, this definition is alsocompatible with simplicial set version of subdivision. Finally, consider the last vertexmap sd( K ) −→ K given on vertices by { x ≤ ... ≤ x k } (cid:55)→ x k . HAPTER 1. STRATIFIED HOMOTOPY THEORY −→ sCplx o P . Again, it is easy to see that this is compatible with the definition of the last vertextransformation sd −→ s Set P under the inclusion of categories given by S o .In the ordered setting, we can also equip the relative versions of subdivision, Con-struction 1.3.21, with last vertex maps. Construction 1.3.39.
Let K be an FOS-complex and A a full subcomplex such thatwhenever two vertices x ∈ K − A and a ∈ A lie in a common simplex we have a < x . Thenthe relative subdivision sd( K rel A ) becomes an FOS-complex (over P ), by taking theinduced ordering on sd( K − A ) (cid:116) A and setting a < σ , whenever two vertices a ∈ A (0) and σ ∈ sd( K − A ) (0) lie in a common simplex. Now, the last vertex map l.v. : sd( K ) → K factors into two maps of FOS-complexes,sd( K ) l −→ sd( K rel A ) l −→ K, fitting into the commutative diagram:sd( A ) A A sd( K ) sd( K rel A ) K sd( K − A ) sd( K − A ) K − A l.v. 1 l l The maps are constructed as follows. l is defined on vertices viasd( K − A ) (0) (cid:51) τ (cid:55)−→ τ (cid:0) sd( N ( X, A )) − sd( A ) (cid:1) (0) (cid:51) σ (cid:63) τ (cid:55)−→ τ sd( A ) (0) (cid:51) σ (cid:55)−→ l.v.( σ ) . This maps a simplex { σ ⊂ ... ⊂ σ k ⊂ σ k +1 (cid:63) τ ⊂ ... ⊂ σ k + l (cid:63) τ l } , with σ i ∈ A and τ j ∈ K − A , to { l.v.( σ ) ≤ ... ≤ l.v.( σ k ) ≤ τ ≤ ... ≤ τ l } = { l.v.( σ ) ≤ ... ≤ l.v.( σ k ) } (cid:63) { τ ≤ ... ≤ τ l } . As { l.v.( σ ) ≤ ... ≤ l.v.( σ k ) } ⊂ σ k + l (cid:63) τ k this actually defines an (ordered) simplex insd( K rel A ). It is clearly filtered, as we have assumed that vertices in A precede thosein K in the ordering. l is given on vertices by:sd( K − A ) (0) (cid:51) τ (cid:55)−→ l.v.( τ ) A (0) (cid:51) σ (cid:55)−→ σ, HAPTER 1. STRATIFIED HOMOTOPY THEORY σ (cid:63) { τ ≤ ... ≤ τ k } to σ (cid:63) { l.v.( τ ) ≤ ... ≤ l.v.( τ k ) } . Again, by construction of the FOS-complex sd( K rel A ) and the ordering assumptionon A ⊂ K , this defines an ordered simplex. Proposition 1.3.40.
In the setting of Construction 1.3.39, the realizations of the twoarrows l and l are stratified homotopic to the filtered subdivision homeomorphisms fromProposition 1.3.22.Proof. This is immediate through a use of straight line homotopies.We now want to iterate this construction. For this we need:
Lemma 1.3.41.
In the setting of Construction 1.3.39 let A (cid:48) ⊂ K be another fullsubcomplex fulfilling the ordering condition and containing A . Then the subcomplex sd( A (cid:48) rel A ) ⊂ sd( K rel A ) satisfies the requirements of Construction 1.3.39 as well.Proof. We have already seen in Lemma 1.3.18 that fullness is a property that is preservedunder subdivision. Therefore, it suffices to check the ordering condition. Let a (cid:48) ∈ sd( A (cid:48) rel A ) and x ∈ sd( K (cid:48) rel A ) − sd( A (cid:48) rel A ) be two vertices contained in a commonsimplex. In case a ∈ A , the result is immediate from the construction of the orderingon sd( K rel A ), as A ⊂ sd( A (cid:48) rel A ) . If not, then both lie in sd( K − A ). But here,the ordering relation holds for any, subcomplex sd( A (cid:48)(cid:48) ), A (cid:48)(cid:48) ⊂ K . This follows, bythe definition of the ordering on sd( K ) via inclusions of simplices, as no simplex notcontained in A (cid:48)(cid:48) can contain a simplex contained in A (cid:48)(cid:48) .We can now use this to study an iterative version of relative subdivisions of FOS-complexes. Construction 1.3.42.
Let K be an FOS-complex and A := ( A ⊂ ... ⊂ A n ) be a familyof full subcomplexes of K each satisfying the ordering condition of Construction . . . Now, define sd A ( K ) by induction over n as follows. For n = 1, just set sd A ( K ) :=sd( K rel A ). For n to n + 1, let ˜ A be the subfamily of A where A n +1 was removed. Byanother inductive use of Lemma 1.3.41 we set sd A ( K ) to the FOS-complex defined bysd A ( K ) := sd (cid:16) sd ˜ A ( K ) rel sd ˜ A ( A n +1 ) (cid:17) . Then, inductively by Proposition 1.3.22, we have subdivisionssd n ( K ) (cid:67) sd A ( K ) (cid:67) K, where the latter is so that no new vertices are added to | ˚ N ( K, A ) | . HAPTER 1. STRATIFIED HOMOTOPY THEORY Corollary 1.3.43.
Let K be an FOS-complex together with a family A as in Construc-tion 1.3.42 of length n . Then the ( n -times iterated) last vertex map l.v. n : sd n ( K ) → K factors into two maps of FOS-complees sd n ( K ) l A −→ sd A ( K ) l A −→ K fitting into the commutative diagram: sd n ( A ) A A sd A ( K ) sd A ( K ) K l.v. n l A l A .Furthermore, | l A | P and | l A | P are respectively stratified homotopic to the subdivisionhomeomorphisms | sd n ( K ) | P ∼ −→ | sd A ( K ) | P and | sd A ( K ) | P ∼ −→ | K | P .Proof. For l A , just take the iterative composition of the respective l from Construc-tion 1.3.39. This clearly make the right hand side of the diagram commute and isstratified homotopic to the subdivision homeomorphism by an inductive use of Proposi-tion 1.3.40. l n is constructed inductively as follows. For n = 1, take l from Construction 1.3.39 for A = A . For the inductive step from n to n + 1, set l n +10 to the composition:sd n +1 ( K ) = sd (cid:0) sd n ( K ) (cid:1) sd( l ˜ A ) −−−−→ sd (cid:0) sd ˜ A ( K ) (cid:1) l −→ sd (cid:0) sd ˜ A ( K ) rel sd ˜ A ( A n +1 ) (cid:1) = sd A ( K ) , with ˜ A as in Construction 1.3.42. The right hand side l is the one corresponding to theFOS-pair sd ˜ A ( A n +1 ) ⊂ sd ˜ A ( K ). Now, consider the diagram:sd n +1 ( A ) sd( A ) A sd n +1 ( K ) sd (cid:0) sd ˜ A ( K ) (cid:1) sd A ( K ) sd(l.v. n ) l.v.sd( l ˜ A ) l .This commutes by the induction hypothesis and functoriality of sd. Furthermore, bynaturality of l.v., the upper horizontal composition equals l.v. n +1 : sd n +1 ( A ) → A . Thisshows the commutativity assertion. We then have l A ◦ l A = l ˜ A ◦ l ◦ l ◦ sd( l ˜ A )= l ˜ A ◦ l.v. ◦ sd( l ˜ A )= l ˜ A ◦ l ˜ A ◦ l.v.= l.v. n +1 HAPTER 1. STRATIFIED HOMOTOPY THEORY | sd A ( K ) | P ∼ −→ | K | P as the latter is a stratum preserving homeomorphism.The composition | sd n ( K ) | P ∼ −→ | sd A ( K ) | P ∼ −→ | K | P is the subdivision homeomorphism corresponding to sd n ( K ) (cid:67) K , by Proposition 1.3.22.The latter is stratified homotopic to the realization of l.v. n by Proposition 1.1.37. On theother hand, we have already seen that | sd A ( K ) | P ∼ −→ | K | P . is homotopic to | l A | P . Hence,its composition with | l A | P is also homotopic to | l.v. n | P . This finishes the proof.In the next subsection, we use these relative last vertex maps to connect stratifiedtopological and stratified simplicial homotopy classes. In this subsection, we are going to prove the following alternative version of the filteredsimplicial approximation theorem.
Theorem 1.3.44 (Filtered simplicial approximation B) . Let
K, L be P -filtered orderedsimplicial complexes over P , where K is finite. Let | K | P φ −→ | L | P be a stratum preserving map. Then there exists an n ∈ N and a stratum preserving(ordered) simplicial map sd n ( K ) f −→ L such that | f | P (cid:39) P (cid:16) | sd n ( K ) | P | l.v. n | P −−−−→ | K | P φ −→ | L | P (cid:17) (cid:39) P (cid:16) | sd n ( K ) | P ∼ −→ | K | P φ −→ | L | P (cid:17) . Conversely, if two stratum preserving (ordered) simplicial maps f , f : K → L are suchthat | f | P (cid:39) P | f | P , then there exists an n ∈ N and a stratum preserving (ordered) simplicial map H : sd n ( K ⊗ ∆ ) → L such that the following diagrams commutes. sd n ( K ) K sd n ( K ⊗ ∆ ) L l.v. n sd n ( i j ) f j H .Here i j denotes inclusion of K into K ⊗ ∆ at the respective endpoint, for j = 0 , . HAPTER 1. STRATIFIED HOMOTOPY THEORY Proof.
We start by showing the results for a linearly ordered set P = [ q ]. Note that, fora sequence Σ = (Σ i ) i ∈ [ q ] in N , sd Σ ( K rel A ) from Section 1.3.3 is a special case of theiterative relative subdivision defined in Construction 1.3.42, where we take A := { A ⊂ ... ⊂ A (cid:124) (cid:123)(cid:122) (cid:125) Σ ⊂ A ∪ K ⊂ ... ⊂ A ∪ K (cid:124) (cid:123)(cid:122) (cid:125) Σ ⊂ A ∪ K ⊂ ... ⊂ A ∪ K q − ⊂ ... ⊂ A ∪ K q − (cid:124) (cid:123)(cid:122) (cid:125) Σ q } . Let Σ and f (cid:48) : sd Σ ( K ) → L be a filtered simplicial approximation of φ given by Theo-rem 1.3.24. Denote | Σ | := (cid:80) p ∈ [ q ] Σ p . Then note that as the filtration map is ordered,the subcomplexes K q ⊂ K fulfill the ordering condition of Corollary 1.3.43. Hence, bythe latter, we can pull back f (cid:48) with l A (using the notation of Corollary 1.3.43) to obtaina stratum preserving simplicial map f (cid:48)(cid:48) : sd | Σ | l A −→ sd Σ ( K ) → L such that | f (cid:48)(cid:48) | P (cid:39) P (cid:16) | sd | Σ | ( K ) | P | l.v. n | P −−−−→ | K | P φ −→ | L | P (cid:17) (cid:39) P (cid:16) | sd | Σ | ( K ) | P ∼ −→ | K | P φ −→ | L | P (cid:17) (1.17)Note however that this is not necessarily an ordered map. Now, set n = | Σ | + 1 and f : sd n ( K ) sd( f (cid:48)(cid:48) ) −−−−→ sd L l.v. −−→ L. Then all of the maps involved are maps of FOS-complexes and by naturality of l.v. and(1.17) we obtain: | f | P = | l.v. ◦ sd( f (cid:48)(cid:48) ) | P = | f (cid:48)(cid:48) ◦ l.v. | P (cid:39) P φ ◦ | l.v. ◦ l.v. | Σ | | P (cid:39) P φ ◦ | l.v. n | P finishing the first part of the proof.For the proof of the homotopy statement, let f , f : K → L be maps of FOS-complexesover P and H : | K ⊗ ∆ | P ∼ = | K | P ⊗ ∆ → | L | P be a stratum preserving homotopybetween their realizations. Then by Example 1.3.27, we can apply Theorem 1.2.8 to theinduced homotopy constructed in this example˜ H : | sd (cid:0) K ⊗ sd (∆ ) (cid:1) | P −→ | sd( L ) | P between | sd( f ) | P and | sd( f ) | P . Then, arguing just as we did for the previous statementbut using Theorem 1.3.25 instead, we obtain an n (cid:48) ∈ N and a morphism of filteredsimplicial complexes H (cid:48) : sd n (cid:48) (cid:16) sd (cid:0) K ⊗ sd (∆ ) (cid:1)(cid:17) −→ sd( L ) . HAPTER 1. STRATIFIED HOMOTOPY THEORY f j ) ◦ l.v. n (cid:48) on therespective inclusions of sd n (cid:48) +1 ( K ). Now, consider the map of FOS-complex l : sd ( K ⊗ ∆ ) (cid:0) l.v. ◦ sd ( π K ) , sd ( π ∆1 ) (cid:1) −−−−−−−−−−−−−−−−−→ K ⊗ sd (∆ ) .l restricts to l.v. followed by the inclusions i (cid:48) j : K (cid:44) → K ⊗ sd (∆ ) ,at the respectiveendpoints, j = 0 ,
1. Now, set n = n (cid:48) + 4 and H to the composition of maps of FOS-complexes H : sd n ( K ⊗ ∆ ) sd n (cid:48) +2 ( l ) −−−−−→ sd (cid:16) sd n (cid:48) +1 (cid:0) K ⊗ sd (∆ ) (cid:1)(cid:17) sd ( H (cid:48) ) −−−−→ sd ( L ) l.v. −−→ L. Pulling this back with sd n ( i j ) and using the restriction statements on H (cid:48) and l we obtainthe commutative diagram:sd n ( K ) sd n (cid:48) +2 ( K ) sd ( K ) K sd n ( K ⊗ ∆ ) sd n (cid:48) +2 (cid:0) K ⊗ sd (∆ ) (cid:1) sd ( L ) L sd n ( i j ) sd n (cid:48) +2 (l.v. ) sd n (cid:48) +2 ( i (cid:48) j ) sd(l.v. n (cid:48) ) sd ( f j ) l.v. f j sd n (cid:48) +2 ( l ) sd( H (cid:48) ) l.v. .By definition, the bottom horizontal composition is H . Furthermore, by naturality ofl.v., the top horizontal composition is l.v. n . Hence, this provides the diagram in thestatement of the theorem. We are left with showing the statements in the case where P is not a finite linear partially ordered set. Note that all of the simplicial complexesinvolved are finite. Thus, can without loss of generality restrict to the case where P isfinite. (Rigorously, this is done by pulling back to the finite subset (subcomplex spannedby) the elements of P in the image of the filtration, approximating, and then composingwith the inclusion into P again.) The finite, not linearly ordered case now is immediatelyreduced to the linear order one by refining the order on P to a linear one and using thefollowing lemma. Lemma 1.3.45.
Let P be a partially ordered set. Let P (cid:48) be another partially orderedset together with a bijective map of partially ordered set ρ : P → P (cid:48) . Denote by N ( ρ ) ∗ : s Set P −→ s Set P (cid:48) ρ ∗ : Top P −→ Top P (cid:48) the functors obtained by postcomposing filtrations with N ( ρ ) and (the map induced onthe Alexandroff spaces by) ρ respectively. Then these two functors are fully faithful, andfit into a diagram (commutative up to natural isomorphism): s Set P s Set P (cid:48) Top P Top P (cid:48) |−| P N ( ρ ) ∗ |−| P (cid:48) ρ ∗ . HAPTER 1. STRATIFIED HOMOTOPY THEORY Furthermore, N ( ρ ) ∗ commutes with ⊗ ( − ) , sd and l.v. .Proof. The commutativity statements are all immediate from the definitions of the re-spective functors and natural transformations. It remains to show that ρ ∗ and N ( ρ )are fully faithful. Both maps we post-compose with, ρ and N ( ρ ), are monomorphismsin their respective categories. Indeed, it is an elementary and easily verified fact onover-categories that postcomposing with a monomorphism is a fully faithful functor.This finishes our investigation into filtered simplicial complexes. We now return toinvestigating the relationship between Top P and s Set P . H Top P We now have several tools available to start a detailed analysis of H Top P . Our maintool is the filtered simplicial approximation theorem, Theorem 1.3.44. However, byProposition 1.3.37, this does not necessarily mean we need to restrict ourselfs only toFOS-complexes. | − | P : H s Set finP → H
Top P is fully faithful In the case where P = (cid:63) is a one-point set, the Douteau model structure on s Set P agrees with the classical Kan-Quillen one under the obvious isomorphism of categoriess Set P ∼ = s Set . In this setting, it is of course well known that the realization functordefines an equivalence of categories | − | : H s Set ∼ −→ H Top . By Theorem 1.2.23, the filtered realization functor also induces a functor | − | P : H s Set P → H Top P on homotopy categories. To the best of our knowledge it is notyet known whether this defines an equivalence of categories. However, we are going toshow in this section (Corollary 1.4.4) that if one restricts to the finite setting this functoris fully faithful. This shows that at least for filtered topological spaces that are weaklyequivalent to the realizations of finite filtered simplicial sets most question on homotopytheory can be reduced to the combinatorial setting. In particular, this allows us to de-fine the Whitehead group and Whitehead torsion in the topological setting in Section 2.4.As the title of the subsection states, we are now going to prove Theorem 1.4.1. Ourproof is somewhat unorthodox, at least from a model theoretical perspective, as we firstto reduce to FOS-complexes and then use simplicial approximation theorems. This iscertainly more of a classical than a modern approach to the problem. It comes at theprice of having to restrict to the finite setting. We conjecture that this is just due to theinsufficiency of our method of proof, and in fact | − | P : H s Set P −→ H Top P is an equiv-alence of categories (induced by a Quillen equivalence even, see also Remark 1.1.20).However, as this work was written with simple homotopy theory in mind, some com-pactness assumptions are really not that big of a price to pay. HAPTER 1. STRATIFIED HOMOTOPY THEORY Theorem 1.4.1.
The functor | − | P : H s Set finP −→ H
Top P induced by Theorem 1.2.23 is fully faithful.Proof. Let
X, Y ∈ s Set finP . We want to show thatHom H s Set P ( X, Y ) |−| P −−−→ Hom H Top P ( | X | P , | Y | P )is a bijection. First, note that, by Proposition 1.3.37, we may assume without loss ofgenerality that X and Y are finite FOS-complexes. We now want to reduce to the case,where the right hand side is given by actual stratified homotopy classes. By Proposi-tion 1.1.27 and the fact that by definition every object in Top P is fibrant, we have forcofibrant T and T (cid:48) arbitrary:Hom H Top P ( T, T (cid:48) ) ∼ = [ T, T (cid:48) ] P . (1.18)Now, consider the last vertex mapl.v. P : sd P ( X ) −→ X. By [Dou19b, A.3] or alternatively Proposition 2.3.19, the latter is a weak equivalence.In particular, the induced map | l.v. P | P : | sd P ( X ) | P −→ | X | P induces an isomorphism in H Top P . By Proposition 1.1.39, | sd P ( X ) | P is cofibrant.Furthermore, similarly to the case of sd one easily checks that sd P ( X ) is still an FOS-complex. Hence, we assume without loss of generality that both X and Y are finiteFOS-complex, and that | X | P is cofibrant. Thus, using Equation (1.18), we are left withshowing that | − | P : Hom H s Set P ( X, Y ) −→ Hom H Top P ( | X | P , | Y | P ) ∼ = [ | X | P , | Y | P ] P is a bijection. Now, let [ φ ] be any arrow on the right hand side, represented stratumpreserving map φ : | X | P → | Y | P . By the second version of the filtered simplicialapproximation theorem (Theorem 1.3.44), for some sufficiently large n and some map ofFOS-complexes f : sd n ( X ) f −→ Y we have[ φ ] ◦ [ | l.v. | nP ] = [ | f | P ] . [l.v. n ] is an isomorphism in H s Set P . In particular,[ φ ] = | [ f ] ◦ [l.v. n ] − | P showing surjectivity. Now, conversely let ϕ , ϕ be two arrows on the left hand side ofthe equation. By Proposition 1.1.46, both fit into a commutative diagram HAPTER 1. STRATIFIED HOMOTOPY THEORY nP ( X ) X Y [l.v. nP ] [ f i ] ϕ i For appropriate morphisms in s
Set finP f i . Without loss of generality, we can assume n = n . Furthermore, inverting [l.v. n i ], we can then assume that n i = 0. Hence, weare now in the setting where [ f i ] = ϕ i and | f | P (cid:39) P | f | P . By the homotopy part ofTheorem 1.3.44, there is a commutative diagram in s Set P :sd n ( X ) X sd n ( X ⊗ ∆ ) Y l.v. n sd n ( i j ) f j H .Corollary 2.3.20 here, states that l.v. : sd( − ) preserves weak equivalences (for finitearguments, but this is easily generalized). Again, we assure that there is no risk ofcircularity involved here. Thus, sd n ( X ) (cid:116) sd n ( X ) (cid:44) → sd n ( X ⊗ ∆ ) gives a cylinder objectfor sd n ( X ) that witnesses a left homotopy between f ◦ l.v. n and f ◦ l.v. n . Hence,[ f ◦ l.v. n ] = [ f ◦ l.v. n ]and, by the invertibility of [l.v.] in H s Set P , we obtain[ f ] = [ f ] . In particular, we obtain the following immediate corollary, using the fact that amorphism in the model category is a weak equivalence if and only if it descends to anisomorphism in the homotopy categoy.
Corollary 1.4.2.
A morphism f : X → Y in s Set finP is a weak equivalence with respectto the Douteau model structure if and only if | f | P is a weak equivalence with respect tothe Henrique-Douteau model structure. Hence, for spaces that are (stratified homotopy equivalent) stratum preserving home-omorphic to the realization of a finite simplicial set, questions on homotopies can oftenbe reduced to the combinatorial setting of filtered simplicial sets. It is useful to have aname for such spaces.
Definition 1.4.3.
Let T ∈ Top P . • A filtered simplicial set X ∈ s Set P together with a stratum preserving homeomor-phism | X | P ∼ = T is called a triangulation of T . If X is finite, we call it a finitetriangulation . • A filtered topological space admitting such a (finite) triangulation is called (finitely)triangulable
HAPTER 1. STRATIFIED HOMOTOPY THEORY
Corollary 1.4.4.
Denote by H Top t − finP the full subcategory of H Top P given by suchfiltered spaces that are weakly equivalent to a triangulable filtered space. Then | − | P : H s Set finP −→ H
Top P induces an equivalence of categories H s Set finP (cid:39) −→ H
Top t − finP . Corollary 1.4.5.
A morphism of finite filtered simplicial sets f : X → Y in s Set P isa weak equivalence, with respect to the Douteau model structure if and only if | f | P is aweak equivalence, with respect to the Henrique-Douteau model structure on Top P . As we have already shown in the proof of Theorem 1.4.1, the Hom-sets in H Top P , witha triangulable filtered space at the domain, admit fairly explicit descriptions. This isowed to the fact that cofibrant approximations are given by sd P ( − ). More precisely, inthis case we have a natural bijection:Hom H Top P ( | X | P , T ) ∼ = Hom H Top P ( | sd P ( X ) | P , T ) = (cid:2) | sd P ( X ) | P , T (cid:3) P . It would be nice to have such a result, not only for spaces cofibrant in the Douteau-Henrique model structure, but also for spaces arising more naturally in the study ofspaces with singularities. Such a result would make the homotopy category of filteredspaces a lot easier to understand. Recall (see Definition 1.1.12) that a f-stratified space(over P ) is a filtered space T that has the right lifting property with respect to admissiblehorn inclusions as in the diagram below. | Λ J k | P T | ∆ J | P As we already mentioned in Example 1.1.14, examples of such spaces include all conicallystratified spaces (in particular all PL pseudomanifolds and PL stratified spaces in theoriginal sense of Goresky and MacPherson) and all homotopically stratified spaces inthe sense of Quinn. One then has:
Lemma 1.4.6.
Let T ∈ Top P be f-stratified. Then T has the right lifting property withrespect to all realizations of (filtered) anodyne extensions A (cid:44) → B in s Set P .Proof. The class of morphisms having the left lifting property with respect to anotherclass of morphisms, is always saturated. That is, it is closed under coproducts, pushouts,transfinite compositions (see [nLa20h], for a definition) and retracts and contains all
HAPTER 1. STRATIFIED HOMOTOPY THEORY | − | P preserves colimits, and hence maps the saturated class of the latter into the saturatedclass generated by realizations of horn inclusions.In Remark 1.1.20, we conjectured that f-stratified spaces can in fact be taken to bethe fibrant objects in an appropriate model structure on Top P that has the same weakequivalence as the Henrique-Douteau model structure. A strong hint at such a resultis that it turns out that for f-stratified spaces, which are triangulable by finite FOS-complexes, the Hom sets in H Top P are actually given by sets of stratified homotopyclasses. Theorem 1.4.7.
Let
X, Y ∈ s Set finP such that X is an FOS-complex and Y such that | Y | P is an f-stratified space. Then the natural map [ | X | P , | Y | P ] P −→ Hom H Top P ( | X | P , | Y | P ) is a bijection.Proof. Let Y i (cid:44) −→ F ( Y ) be a cofibrant fibrant replacement for Y . Then, by Lemma 1.4.6 | i | P admits a retract r , which is a weak equivalence as it is a retract of a weak equivalenceby Theorem 1.2.23. Now, consider the following commutative diagram: (cid:2) X, F ( Y ) (cid:3) P Hom H s Set P ( X, F ( Y )) (cid:2) | X | P , | F ( Y ) | P (cid:3) P Hom H Top P ( | X | P , | F ( Y ) | P ) (cid:2) | X | P , | Y | P (cid:3) P Hom H Top P ( | X | P , | Y | P ) ∼ ∼ r ∗ ∼ , where the right lower vertical is the isomorphism induced by the weak equivalence r .The upper right vertical is an isomorphism by Theorem 1.4.1, which clearly extends toall P -filtered simplicial sets that are weakly equivalent to a finite one, i.e. also to F ( Y ).In particular, by commutativity, the bottom right horizontal is also onto. It remainsto show injectivity. Consider stratified homotopy classes of maps φ , φ : | X | P → | Y | P whose images in Hom H Top P ( | X | P , | Y | P ) agree. By pulling back with the subdivisionhomeomorphism | sd n ( X ) | P ∼ −→ | X | p and invoking Theorem 1.3.44, we may without lossof generality assume that both come from homotopy classes of morphisms of filteredsimplicial sets f , f : X → Y . They are mapped to by [ i ◦ f ] , [ i ◦ f ] ∈ [ X, F ( Y )] P respectively. Thus, by commutativity of the diagram and the fact that the composition”right, down, down” is a bijection, this already implies [ φ ] = [ φ ].In particular, this result also holds true for filtered spaces T X , T Y which are respec-tively stratified homotopy equivalent to realizations of such filtered simplicial sets. HAPTER 1. STRATIFIED HOMOTOPY THEORY Remark 1.4.8.
The additional assumption that X is an FOS-complex and not justa finite simplicial set should be omittable. To do this, one only needs to extend thefiltered simplicial approximation theorem to all finite P -filtered simplicial sets. One wayto approach this, might be to define a model structure on s Set P that uses sd instead ofsd P . However, one might just as well take the approach we illuminated in Remark 1.1.20,which should give a more complete understanding of the situation.This result gives a rather clear view of what the morphisms in H Top P look like formany classical examples of stratified spaces. To be more precise, we obtain: Corollary 1.4.9.
Let
Top f,fin,P LP be the full subcategory of f-stratified, compact polyhe-dra in
Top P , i.e. f-stratified spaces, triangulable by a finite FOS-complex. Let H naive Top f,fin,P LP be the category obtained from it, by identifying stratified homotopic maps. Then
Top f,fin,P LP (cid:44) → Top P induces a fully faithful embedding H naive Top f,fin,P LP (cid:44) → H
Top P . In particular, the analogous result holds, for the full subcategories of
Top f,fin,P LP given bysuch realizations of finite FOS-complexes that are pseudo-varieties, cs-stratified, conicallystratified or homotopically stratified spaces.
We should note that a similar statement has been conjectured by Douteau in [Dou19b,8.1.4]. However, there it is conjectured for the strongly filtered setting. hapter 2
Simple Stratified HomotopyTheory
Roughly speaking, the origin of simple homotopy theory is the question of which ho-motopy equivalences f : X → Y between appropriately combinatorial objects, say CW-complexes, can be represented through a sequence of certain combinatorial moves (ex-pansions and collapses). As there is already a lot of excellent introductory work intosimple homotopy theory, we refrain from giving an overview of the classical perspective.The reader completely new to the subject should find motivation and analogies for ourstratified perspective in [Coh73]. In the classical setting of CW-complexes (or simplicialcomplexes) and elementary expansions, it turns out that the answer to this questionis encoded by an element τ ( f ) - the Whitehead torsion - of a purely algebraic group W h ( X ) - the Whitehead group of X . In this chapter we ask and attempt to answerthe analogous question but for filtered (stratified) spaces. Before one can even dare toattempt giving an answer, one of course has to be a little bit more specific as to whatthe correct analogy should be. In other words:What are the “appropriately combinatorial objects”?What are the “homotopy equivalences”?What are the “combinatorial moves”?The first of these three questions we have already answered in Section 1.1.2 with the cat-egory s Set P of P -filtered simplicial sets. It simply seems the most well explored categoryof filtered objects of combinatorial nature that is not too restrictive when it comes topushout and mapping cylinder constructions. To answer the second question, we take theweak equivalences in the Douteau model structure on s Set P and the Henrique-Douteaumodel structure on Top P respectively. We have seen in various results of Chapter 1 (inparticular Theorem 1.2.23, Theorem 1.4.1 and Corollary 1.4.2) that they interact rathernicely. In particular, | − | P preserves and reflects weak equivalences as long as the sourceand target are finite filtered simplicial sets. The reason we chose weak equivalences overthe more rigid filtered homotopy equivalences partially lies in the answer to the third95 HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY P -filtered simplicial set has the simplehomotopy type of a P -filtered ordered simplicial complex (Theorem 2.3.23). Finally inSection 2.4, we use the results of Chapter 1 to give a series of equivalent descriptionsof the stratified Whitehead group and torsion, also generalizing the latter to continuousstratum preserving maps instead of purely simplicial ones (Proposition 2.4.4). We usethis to prove that the Whitehead group we defined agrees with the classical one in casewhere P = (cid:63) is a one point set (Theorem 2.4.5). Historically, several authors have been concerned with the question of how to general-ize Whitehead’s simple homotopy theory to other settings such as locally finite CW-complexes (see [Sie70], [eck]) or even more generally to more abstract homotopy theo-retical settings, for example categories of chaincomplexes (see [KP86]). Luckily, theseapproaches have all been formulated in a very general category theoretical setting, sothey are easily transferred to our question at hand. Here, we present the approach thatwas independently taken by Siebenmann as well as Eckmann and his student Bolthausen(see [Sie70], [eck]). It will turn out, later on in Section 2.1, that this approach can beused to formulate a simple homotopy theory for filtered simplicial sets, and ultimatelyfor finitely triangulated filtered spaces. However, we should note that the way the ax-ioms for this approach were formulated in [Sie70] and [eck] is slightly flawed. To be moreprecise, in the precise way they are phrased in it does not apply to any setting known tous and certainly not to the settings the authors were interested in (see Remark 2.1.4).
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY C be some category embedded in a larger cate-gory ˆ C that has the same objects as C but potentially more morphisms. Further, letΣ be a class of morphisms in C . Now, consider the localized category C (Σ − ) (see forexample [GZ12, Ch. I]) and denote by Q the structure functor Q : C → C (Σ − ). Definition 2.1.1.
A morphism in C (Σ − ) is called simple if it is given by a compositionof morphisms of the shape Q ( s ) , Q ( s ) − for s ∈ Σ. Two morphisms α, β in C − with thesame source X ∈ C are said to have the same simple morphism class if β = γ ◦ α forsome γ that is simple (clearly this construction gives an equivalence relation, and hencea well-defined notion of equivalence class). We denote the class of α by (cid:104) α (cid:105) . Denote by A ( X ) the class of simple morphism classes of morphisms with source X ∈ C . Denote by E ( X ) the subclass of A ( X ) given by classes (cid:104) α (cid:105) where some (and hence every) α (cid:48) ∈ (cid:104) α (cid:105) is an isomorphism in C (Σ − ). E ( X ) and hence A ( X ) has a distinguished element givenby (cid:104) X (cid:105) .Now, suppose we are in a sufficiently nice setting. That is, one where C (Σ − ) isequivalent to some homotopy category we are interested in studying and one where A ( X ) is sufficiently small, i.e. of set size. This is for example the case for ˆ C the categoryof finite CW-complexes, C the given by inclusions of subcomplexes and Σ the class ofcompositions of elementary expansions. We know that in these cases E in fact definesa functor on C (Σ − ) into monoids and E ( X ) defines a functor into abelian groups, theWhitehead group. ([eck],[Sie70]). One can ask the general question of what requirementsneed to be fulfilled for this to be the case. This is answered in the following definitionand theorem. Definition 2.1.2.
We say that a triple
C ⊂ ˆ C , Σ as above (and such that A ( X ) is of setsize for X ∈ C ) admits a Whitehead group if the following axioms are fulfilled.( A0 ) Σ contains all isomorphisms in ˆ C and is closed under composition.( A1 ) Let f : X → X (cid:48) and s : X → Y be morphisms in C . Then the the pushout diagram X YX (cid:48) Y (cid:48) fs s (cid:48) f (cid:48) , in ˆ C exists and furthermore f (cid:48) , s (cid:48) ∈ C . If further s ∈ Σ, then so is s (cid:48) . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY A2 ) Let f, g : X → Y be two morphisms in C such that Q ( f ) = Q ( g ). Then thereexists a commutative diagram in C X YY Z fg st , where s, t ∈ Σ. Theorem 2.1.3.
Let
C ⊂ ˆ C , Σ admit a Whitehead group.Then every morphism in C (Σ − ) is of the form Q ( s ) − Q ( f ) for some s ∈ Σ , f ∈ C .In particular, for each X ∈ C and each (cid:104) α (cid:105) ∈ A ( X ) there exists a a ∈ C such that (cid:104) α (cid:105) = (cid:104) Q ( a ) (cid:105) . Let X ∈ C . For (cid:104) α (cid:105) and (cid:104) β (cid:105) in A ( X ) define (cid:104) α (cid:105) + (cid:104) β (cid:105) by the simple isomorphismclass of the diagonal in a pushout diagram X YZ W ab in ˆ C , where Q ( a ) ∈ (cid:104) α (cid:105) and Q ( a ) ∈ (cid:104) β (cid:105) . Furthermore for (cid:104) α (cid:105) in A ( X ) and f : X → X (cid:48) ∈ C define f ∗ (cid:104) α (cid:105) ∈ E ( Y ) as (cid:104) Q ( a (cid:48) ) (cid:105) , where a (cid:48) is the pushout of a representative of α in C along f in ˆ C . Both of these constructions are well-defined.Then, C →
AbMon X (cid:55)→ ( A ( X ) , + , (cid:104) X (cid:105) ) { X f −→ X (cid:48) } (cid:55)→ {(cid:104) α (cid:105) (cid:55)→ f ∗ (cid:104) α (cid:105)} defines a functor into the category of abelian monoids. Furthermore, C → Ab X (cid:55)→ ( E ( X ) , + , (cid:104) X (cid:105) ) { X f −→ X (cid:48) } (cid:55)→ {(cid:104) α (cid:105) (cid:55)→ f ∗ (cid:104) α (cid:105)} defines a functor into the category of abelian groups. Both send s ∈ Σ to an isomorphism,i.e. induce functors: E : C (Σ − ) → Ab ,A : C (Σ − ) → AbMon . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY Remark 2.1.4.
Before we move on to a proof, it might be useful to shed a little bitof light on the usage of the two categories C and ˆ C here. Note that both the functo-riality as well as the group structure are defined by taking pushouts of representativesof equivalence classes. In general pushouts interact badly with homotopies and the in-duced equivalence relation (see for example [KP86, Ch.V]). For example, the pushoutof a homotopy equivalence might not be a homotopy equivalence anymore, which wouldinterfere with the functoriality of E above. Similarly glueings of homotopy equivalencesmight not be homotopy equivalences, interfering with the definition of addition. How-ever, there are of course classes of maps that interact more nicely, when it comes topushouts: Cofibrations (see for example [KP86, Ch. V Sec. 6, 7, Sec]). Note for exam-ple that the pushout of an acyclic cofibration of topological spaces is again an acycliccofibration ([KP86, Ch. I, Cor. 6.14]) and of course this also holds true in any modelcategory [Hir09]. Usually, the way to circumvent these difficulties is by replacing maps f : X → Y by the mapping cylinder inclusions X (cid:44) → M f , i.e. passing to homotopypushouts. This is for example how abstract simple homotopy theory is built purelycylinder based in [KP86, Ch. VI, Sec. 3]. Alternatively, one can restrict to whateverclass of morphisms interact nicely with homotopy and pushouts in this setting. This isessentially the step of passing from ˆ C to C . In many cases, one can then use an appropri-ate factorization system to represent morphisms in ˆ C by morphisms in C . Furthermore,one then hopes that C (Σ − ) is still equivalent as a category to the homotopy categoryone is interested in studying. This is for example also done implicitly in [Coh73], wherethe geometric Whitehead group is constructed from pairs of CW-complexes that are de-formation retractions, as any inclusion of a subcomplex of CW-complexes is a cofibration.In the original sources for this chapter ([eck],[Sie70]), however, no mention of a secondcategory ˆ C is made. In fact, the analogue to ( A1 ) in Definition 2.1.2 is just formulatedvia pushouts in C . While both authors are applying the construction precisely as wementioned in the above paragraph, they seem to be under the assumption that in thecategory with objects CW-complexes and with morphisms inclusions of subcomplexesthe pushout is given by the pushout in the category of CW-complexes ([eck, p.6], [Sie70,Paragraph 2]). This is false. Yes, the pushout diagram in CW-complexes does exist andall of its structure maps are still inclusions of subcomplexes, but it does not fulfil theexistence part of the universal property in the subcategory of inclusions of subcomplexes.Take for example the pushout along the empty complex of two points, (cid:63) . Trivially, theconstant map into another point makes the diagram (given by the maps from the emptycomplex into the points) commute. But there is no map (cid:63) (cid:116) (cid:63) → (cid:63) that is an inclusionof a subcomplex.Nevertheless, this is neither really hurtful to the results nor to the theoretical impact oftheir contributions as neither of them really uses the universal property of the pushoutanywhere in their proof. What the authors are really using the pushout for is to havea canonical way to produce commutative squares. To be more precise, they need a wayto produce commutative squares, unique up to isomorphism of diagrams and they want HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY A1 ).We now begin the proof of Theorem 2.1.3. All the following statements are to beunderstood in this setting. Lemma 2.1.5.
Every morphism in C (Σ − ) is of the form Q ( s ) − Q ( f ) , for some s ∈ Σ , f ∈ C .Proof. By definition of the localized category [GZ12, Ch. I], every morphism in C (Σ − )from X to Y is given by a zigzag X = X ↔ X ↔ ... ↔ X n = Y, where we allow all arrows in C in right direction and only arrows in Σ in left direction.We need to show that each such zigzag can be brought into the shape X f −→ Z s ←− Y, for s ∈ Σ, under the operations inducing the morphism structure in C (Σ − ) [GZ12, Ch.I]. Now, assume by induction that we have already proven this for zigzags up to length n −
1. The case n = 1 is obvious. Now, by the composition rules in a localized category,we can reduce a zigzag of length n to the shape X f −→ X s ←− X g ←→ Y, with f, g ∈ C and s ∈ Σ. If g points to the left, it is necessarily in Σ. Hence, as Σ isclosed under composition by ( A0 ), we are done. So let us assume g points to the right.Now, consider the diagram X X X YX (cid:48) s gf g (cid:48) s (cid:48) , where the right hand square is a pushout diagram in ˆ C . By ( A1 ), this square again liesin C and s (cid:48) ∈ Σ. By commutativity, the morphism in C (Σ − ) given by Q ( s (cid:48) ) − Q ( g (cid:48) ) Q ( f ) = Q ( s (cid:48) ) − Q ( g (cid:48) ◦ f )is then the same as the one induced by the zigzag we started with, concluding theproof.By the same argument as in the proof of Lemma 2.1.5, one obtains: HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Lemma 2.1.6.
Every simple morphism σ in C (Σ − ) is of shape Q ( s ) − Q ( s (cid:48) ) for some s, s (cid:48) ∈ Σ . We can now use this to obtain an alternative description of A ( X ) and E ( X ). Lemma 2.1.7.
For every (cid:104) α (cid:105) in A ( X ) there exists an a : X → Y in C such that (cid:104) α (cid:105) = (cid:104) Q ( a ) (cid:105) .Furthermore, for a : X → Y , b : X → Z in C : (cid:104) Q ( a ) (cid:105) = (cid:104) Q ( b ) (cid:105) if and only if there exists a commutative diagram in C : YX SZ sab t , with t, s ∈ Σ .Proof. The first statement is an immediate consequence of Lemma 2.1.6. For the secondstatement first note that, by Lemma 2.1.6, (cid:104) Q ( a ) (cid:105) = (cid:104) Q ( b ) (cid:105) if and only if there exists acommutative diagram in C (Σ − ) YX SZ Q ( s ) Q ( b ) Q ( a ) Q ( t ) , whith s, t ∈ Σ. Hence, we can apply ( A2 ) to s ◦ a and t ◦ b . Now, using that, by ( A0 ),Σ is closed under composition, we obtain the result.In particular, we can think of A ( X ) as the set of morphisms in C with source X modulo composition with morphisms in Σ. Using this, we write (cid:104) a (cid:105) instead of (cid:104) Q ( a ) (cid:105) for a ∈ C from now on. Note that this description of A ( X ) (the induced one of E ( X )to be more precise) is the one used by Cohen in [Coh73] if one takes ˆ C the categoryof finite CW-complexes with cellular maps, C the subcategory given by inclusions ofsubcomplexes and Σ the class of finite compositions of elementary expansions. Lemma 2.1.8.
The construction
C →
MonAb in Theorem 2.1.3 gives a well-definedfunctor into
MonAb . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proof.
Consider the following diagram of pushout squares in ˆ C : X X X X X X X X X f g f g g f g f g g f f , (2.1)with X = X . By composability of pushout squares, if all the small squares are pushoutsquares, then so are all rectangles in the diagram. Further, note that for any operationinvolving pushout, the choice of pushout diagram is up to natural isomorphism. In par-ticular, by ( A0 ), it is not relevant up to simple morphism class.We start by showing that “+” is well-defined on A ( X ). So let (cid:104) a (cid:105) , (cid:104) b (cid:105) be simple mor-phism classes in A ( X ), for a, b ∈ C . By Lemma 2.1.7, we know that in fact any simplemorphism class is of this shape and it suffices to show that the addition construction isinvariant under the composition of a and b by morphisms in Σ. So in (2.1) take f = a , g = b and f , g ∈ Σ. By ( A1 ), (2.1) then lies in C , and all the arrows in the lowerright square are in Σ. In particular, the diagonal of the upper left square differs from thediagonal of the large outer square by a composition with morphisms in Σ. As all squaresinvolved are cocartesian, this shows both the inner upper left and the outer pushoutdiagonals belong to the same simple morphism class.If one takes f = 1 X , then the upper left square with g = g ∈ C is cocarte-sian in ˆ C , showing that (cid:104) X (cid:105) in fact defines a neutral element with respect to addition.Addition clearly is commutative. To see associativity, consider a commutative cube X X X X X X X X f g h , (2.2)in ˆ C constructed by first taking the pushout of f and g and then pushing this di-agram forward along h . A quick diagram chase and the standard properties of thepushout show that all squares in the diagram are pushout squares. In particular, the HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY X to X is obtained both by pushing the diagonal of the upper face squarealong h and by pushing the diagonal of the left face square along f . If one takes f, g, h ∈ C , then by ( A1 ) the whole cube lies in C and the statement we have just shownis precisely the associativity of the addition on A ( X ). Summarizing, we have shown thatin fact ( A ( X ) , + , (cid:104) X (cid:105) ) defines an abelian monoid.We are now left with showing the functoriality of A ( − ). Again consider (2.1). Theargument for well-definedness of f ∗ (see definition in Theorem 2.1.3), for f : X → X (cid:48) ,works similarly to the well-definedness of “+”, by taking f = f , g to be g and g to be a morphism in Σ. One obtains f ∗ ◦ f ∗ = ( f ◦ f ) ∗ (wherever this is defined) bytaking f = f , f = f and g = a a representative of a simple morphism class andagain using composition of pushout diagrams. The proof that 1 X ∗ = 1 A ( X ) is prettymuch identical to the one that (cid:104) X (cid:105) gives a neutral element of addition and also the onethat f ∗ preserves the neutral element. To see that f ∗ is compatible with addition, againrefer to (2.2) and use the same argument as for associativity of ”+”.Now, that we have established that A ( − ) induces a functor C →
MonAb , we canalso check that it descends to a functor on C (Σ − ). We start by proving the followinguseful lemma about simple morphism classes of compositions. Lemma 2.1.9.
For f : X → Y in C and g : Y → Z in C the following equation holds: f ∗ (cid:104) g ◦ f (cid:105) = f ∗ (cid:104) f (cid:105) + (cid:104) g (cid:105) . Proof.
Just consider the following composition of pushout squares.
X Y ZY Y (cid:48) Z (cid:48) f f g . The sum to the right corresponds to the diagonal of the right square. f ∗ (cid:104) f (cid:105) correspondsboth to the middle vertical and the lower left horizontal. f ∗ (cid:104) g ◦ f (cid:105) corresponds to the lowerhorizontal composition (by composition of pushouts). But as the lower left horizontaland the middle vertical agree, the lower horizontal composition is the diagonal of theright hand square, showing the equation. Lemma 2.1.10.
Let s : X → X (cid:48) ∈ Σ . Then s induces an isomorphism of monoids s ∗ : A ( X ) → A ( X (cid:48) ) .Proof. We need to show that s ∗ is a bijection. Let (cid:104) a (cid:105) , (cid:104) a (cid:105) ∈ A ( X ) given by some f, g ∈C with source X . If s ∗ (cid:104) a (cid:105) = s ∗ (cid:104) a (cid:105) , then, by Lemma 2.1.7, we have two commutative HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY i = 0 ,
1) in C X Y i X (cid:48) Y (cid:48) i S s a i s (cid:48) i a (cid:48) i t i with all s and t simple by ( A1 ), such that t ◦ a (cid:48) = t ◦ a (cid:48) . Applying Q and chasing the diagrams, we obtain Q ( t ◦ s (cid:48) ) ◦ Q ( a ) = Q ( t ◦ s (cid:48) ) ◦ Q ( a ) . As, by ( A0 ), Σ is closed under compositions, we obtain (cid:104) a (cid:105) = (cid:104) a (cid:105) . Conversely, let (cid:104) b (cid:105) be in A ( Y ). Then by Lemma 2.1.9 we have an equation s ∗ (cid:104) b ◦ s (cid:105) = s ∗ (cid:104) s (cid:105) + (cid:104) b (cid:105) = s ∗ (cid:104) b (cid:105) = (cid:104) b (cid:105) . using the fact that s ∈ Σ and that s ∗ is a morphism of monoids. In particular, we haveshown surjectivity of s ∗ .By the universal property of the localization we obtain: Corollary 2.1.11. X (cid:55)→ A ( X ) induces a functor C (Σ − ) → MonAb as described inTheorem 2.1.3.
We are left with showing that the statements in Theorem 2.1.3 on E ( − ) hold. Infact, we are going to show that E ( X ) is the group of invertible elements of A ( X ). Inparticular, E is just given by postcomposing A : C (Σ − ) → MonAb with the functorthat sends an abelian monoid to the subgroup of its invertible elements. Again we startwith a series of lemmata.
Lemma 2.1.12.
Let f : X → X (cid:48) be a morphism in C . Q ( f ) has a left inverse if andonly if there exists a ¯ f : X (cid:48) → X (cid:48)(cid:48) in C such that ¯ f ◦ f ∈ Σ .Proof. This is an immediate consequence of Lemma 2.1.5 and ( A0 ) and ( A2 ). To bemore precise, if Q ( f ) has a left inverse, then, by Lemma 2.1.5, the inverse is of the shape Q ( s ) − ◦ Q ( ¯ f ), for some ¯ f with source X (cid:48) and s ∈ Σ. Hence Q ( ¯ f ◦ f ) = Q ( s ) . Now, apply ( A2 ) to obtain s (cid:48) and t in Σ such that t ◦ ¯ f ◦ f = s (cid:48) ◦ s. By ( A0 ) the right hand side of this equation lies in Σ. Now, set ¯ f = t ◦ ¯ f . This showsthe only if part. The if part is obvious. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Lemma 2.1.13. (cid:104) α (cid:105) ∈ A ( X ) is invertible (w.r.t. “+”) if and only if α is an isomorphismin C (Σ − ) i.e. (cid:104) α (cid:105) ∈ E ( X ) . In particular, E ( X ) is precisely the group of invertibleelements of A ( X ) .Proof. We start with the only if part. By Lemma 2.1.7, there exist a : X → Y and b : X → X (cid:48) in C representing (cid:104) α (cid:105) and its inverse. In particular, we have a commutativediagram in C X YX (cid:48) Y (cid:48) S ab b (cid:48) a (cid:48) t such that the composition of the diagonal of the square with t lies in Σ, and that thesquare is cocartesian. Inverting Q ( t ) we obtain that Q ( b (cid:48) ) has a right inverse. ByLemma 2.1.12, Q ( b ) has a left inverse. However, by ( A2 ) and Lemma 2.1.12, the prop-erty of Q ( f ) having a left inverse is stable under pushout. In particular, Q ( b (cid:48) ) also hasa left inverse, making it an isomorphism. But then Q ( a ) is such that composition withthe isomorphism Q ( t ) ◦ Q ( b (cid:48) ) makes it an isomorphism (as t ◦ b (cid:48) ◦ a ∈ Σ). In particular, Q ( a ) and hence also α is an isomorphism.Conversely, let (cid:104) α (cid:105) be such that α is invertible. Take a representative a : X → Y in C ,using Lemma 2.3.1) of (cid:104) α (cid:105) . Then Q ( a ) is invertible and in particular by Lemma 2.1.12we find some b : Y → Z such that b ◦ a ∈ Σ. Now, by Lemma 2.1.9 we then obtain anequation a ∗ (cid:104) b ◦ a (cid:105) = a ∗ (cid:104) a (cid:105) + (cid:104) b (cid:105) . But as b ◦ a ∈ Σ, the left hand side of this equation is 0. In particular, a ∗ (cid:104) a (cid:105) has an inverse.Since Q ( a ) is an isomorphism a ∗ is an isomorphism of monoids, by Corollary 2.1.11. Inparticular, (cid:104) a (cid:105) also has an inverse.This finishes the proof of Theorem 2.1.3. We now have a general theory at ourdisposal that allows us to construct Whitehead groups for a wide range of settings. Weapply this to filtered simplicial sets in Section 2.3.1. Now, that we have a general machinery for the construction of Whitehead groups avail-able, the next step is to specify what the combinatorial moves (expansions) - that is,the morphism in Σ (following the notation of Section 2.1) - should be. Similarly to theclassical theory, we take the perspective that these expansion should be generated insome sense by certain elementary moves. These moves are the (pushouts of) admissiblehorn inclusions (Definition 1.1.9). In Section 2.2.1, we compare these to another possiblechoice of candidate. We then begin studying the resulting class of expansions (calledfinite filtered strong anodyne extensions, or finite FSAEs for short) in detail through
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY P and sd are FSAEs(Proposition 2.2.24). We then show in Section 2.2.3 that FSAEs interact nicely withthe homotopy category H s Set P (Proposition 2.2.30), as they can be interpreted as rel-ative cell complexes in a small object argument. Finally, we prove that in most aspects(such as their interactions with mapping cylinders and products) they behave much likeclassical expansions, making them a good candidate for the construction of a simplehomotopy theory. This is the content of Section 2.2.4. All of these result are then beused in the next section, showing that the axioms of Definition 2.1.2 hold in this setting;that is, that we obtain a (combinatorial) stratified Whitehead group. While, by Theorem 2.1.3, any class of morphism in s
Set P compatible with the axiomsof Definition 2.1.2 serves for the construction of some sort of simple homotopy theory,the goal of this work is construct one that is - similarly to the classical setting - com-binatorial in nature. That is, we want the class Σ in Definition 2.1.2 to be given bycomposition of some type of elementary moves, which then ultimately define the wholesimple homotopy theory. In this subsection, we try to motivate our particular choiceof what an elementary expansion should be. Thus, this chapter is very example drivenand not particularly result heavy. The reader (who is in a hurry) will be fine with justreading the definitions and passing on to the next chapter.In their recent paper [BMS20], the authors have proposed a notion of elementary ex-pansion for filtered simplicial complexes over the poset { ≤ } . They used these toreduce the size of certain filtered simplicial complexes obtained in the pursuit of ap-plying intersection homology to a topological data analysis setting. They then went onto ask the question of whether there might be a larger class of expansion, allowing formore collapses. Translated to the language of filtered simplicial sets and generalized toarbitrary posets their definition essentially comes down to the following notion of “strictadmissibility”. For the sake of comparison, we also repeat the definition of an admissiblehorn inclusions (see Definition 2.2.1). Definition 2.2.1.
Let J = ( p ≤ ... ≤ p n ) be a d -flag in P . • A horn inclusion Λ J k (cid:44) → ∆ J , 0 ≤ k ≤ n , is called admissible if the k -th vertex isrepeated in ∆ J , i.e. either p k = p k − or p k = p k +1 . • A horn inclusion Λ J k (cid:44) → ∆ J , 0 ≤ k ≤ n , is called strictly admissible if it isadmissible and further p k is maximal in J .To be a little bit more concise, we sometimes just say k is admissible (strictly admissible)when a flag J is specified, to refer to the above. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Remark 2.2.2.
One should note that the first condition is equivalent to the inclusionΛ J k (cid:44) → ∆ J being a simplicial homotopy equivalence [Dou18, Proposition 1.13], and alsoto its realization being stratified homotopy equivalence (see Lemma 1.1.11). However ingeneral, only in the second case can this homotopy equivalence be made relative to thehorn inclusion, i.e. only then is the inclusion a stratum preserving strong deformationretract. For a construction of such a relative homotopy, see the orthogonal deformationretraction in [BMS20] and note that this works just as well for arbitrary P . Example 2.2.3.
Consider the admissible horn inclusions from Example 1.1.10, shownagain in Figure 2.1. While in each case, checking whether the conditions of Defini-tion 2.2.1 are fulfilled is of course very easy, it can be interesting to check why they arenot fulfilled from a more geometric perspective.(1) and (3) show the horn inclusions corresponding to the d-flag J = (0 ≤ ≤
1) for k = 2 and k = 0, respectively. While (3) is an admissible inclusion, (1) is not. A moregeometric way to see the latter is that the 0-stratum of the horn in (1) has two pathcomponents while the 0-stratum of the simplex has one. In particular, the inclusion cannot be a homotopy equivalence and thereby not admissible. (3) is not strictly admissible.Geometrically this is reflected in the fact that any map that collapses the simplex to thehorn relative to the horn has to map some of the red part into the blue part. Hence,such a map is not stratum preserving.(2) and (4) show the horn inclusions corresponding to the d-flag J = (0 ≤ ≤
1) for k = 0 and k = 2 respectively. Geometrically speaking, (2) can not be admissible sincein the horn the 1-stratum has two connected components but in the simplex, it only hasone. (4), however, even gives a strictly admissible inclusion. An inverse up to homotopyis given by projecting the lower face and the interior vertically upwards onto the hornand it is easy to check that this turns the horn inclusion into a stratum preserving strongdeformation retract. Figure 2.1: Examples of horn inclusions Definition 2.2.4.
An arrow X → Y in s Set P is called a (strict) elementary expansion HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY J k ∆ J X Y where the upper horizontal is a (strictly) admissible horn inclusion. The formal inverse(and the inverse in the homotopy category by an abuse of notation) of such an inclusionis called a (strict) elementary collapse . Remark 2.2.5.
As, by construction of the latter, every admissible horn inclusion is aacyclic cofibration in the Douteau model structure, and acyclic cofibrations are closedpushouts (cobase change), every elementary expansion is a weak homotopy equivalenceof P -filtered simplicial sets. In particular, by Corollary 1.4.2, they realize to weakequivalences of filtered spaces. However, these will in general not be stratified homotopyequivalences (see example below). In case of a strict elementary expansion however, | Λ J k (cid:44) → ∆ J | P is not only a homotopy equivalence, but also a homotopy equivalencerelative to | Λ J k | P by Remark 2.2.2. Hence, using the fact that | − | P preserves pushoutsand the universal property of the latter, one obtains a homotopy inverse of X (cid:44) → Y .Another way one could show this is by checking that in the case of strict horn inclusionsthe realization | Λ J k (cid:44) → ∆ J | P has the homotopy extension property with respect tostratum preserving homotopies. One can then use the standard argument found forexample in [KP86, Ch. 6] to show that stratified homotopy equivalences that have thisproperty are stable under pushouts. Example 2.2.6.
In Figure 2.2 we see a zigzag of elementary expansions. This givesa weak equivalence in s
Set P between combinatorials of the models of the spaces inFigure 1.8. Notice how out of these only (3) is strict. This is reflected in the fact thatthe realization of the filtered simplicial set on the left hand side is actually not stratifiedhomotopy equivalent to the one on the right hand side. We have already seen this inExample 1.1.18.Figure 2.2: Example of a sequence of elementary expansions between combinatorialmodels of the spaces in Example 1.1.18. Remark 2.2.7.
At first glance Example 2.2.6 might give the impression that for buildinga stratified simple homotopy theory ”up to elementary expansion” induces too coarseof an equivalence relation and one should only consider strict expansions. However, one
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Example 2.2.8.
Many geometric arguments in classical simple homotopy theory involvethe mapping cylinder in some shape or form (see for example the classical book by Cohen[Coh73] for a cellular or Whitehead’s original notes [Whi39] for a simplicial perspective).In particular, they often build upon the fact that for a (cellular) map f : X → Y in theclassical setting the inclusion into the mapping cylinder Y (cid:44) → M f is a composition of elementary expansions (or more generally a simple equivalence).However, in the filtered setting, this is hardly ever the case. Take for example P = { ≤ } and consider inclusion of ∆ J with J = (0 ≤ ⊗ ∆ J at 0 (seeFigure 2.3).Figure 2.3: Inclusion at 0 into the cylinder of ∆ J , for J = (0 ≤ . In fact, these two filtered simplicial sets can not even be transformed into eachother through strict elementary moves. To see this, note that the only way to removethe additional vertex in the 0-stratum is by also removing the lower 1-simplex of type J = (0 ≤
0) at the same or some earlier point in time. Denote this simplex by σ . Inboth cases, the simplex of with d-flag (0 ≤ ≤
1) attached to σ has to be removed first. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY J (cid:48) with J (cid:48) = (0 ≤ ≤ ≤
1) where the (0 ≤
0) part corresponds to σ . But even after such aremoval, σ is again attached to a simplex with d-flag (0 ≤ ≤ Set P . At the same time “up to elementary expansion” is not toocourse of an invariant to be of geometric interest as it models weak equivalences in theDouteau-Model structure on Top P . By Theorem 1.1.15, this means it models stratifiedhomotopy equivalences as long as source and target are stratified in some reasonablesense. One advantage of the definition of elementary expansions in Definition 2.2.4 is that inthe non-filtered case there is already a good amount of machinery available, when itcomes to the study of these objects. Much of this machinery generalizes to the filteredsetting. Recall, that in the classical Quillen model structure on simplicial sets the acycliccofibrations are also called anodyne extensions. (see for example [JT08, Ch. 3]). Recallfurther, that an equivalent characterization of this class of morphisms is given as follows.
Definition 2.2.9.
Let C be a category that has small colimits. Let Σ be a class ofmorphisms in C . Σ is called saturated if it fulfills the following properties.(i) Σ contains all isomorphisms.(ii) Σ is closed under pushouts. That is, if for a pushout square A BA (cid:48) B (cid:48) ss (cid:48) s ∈ Σ then s (cid:48) ∈ Σ.(iii) Σ is closed under arbitrary coproducts. That is, if A i → B i is a family of morphismsin Σ, then (cid:71) A i → (cid:71) B i ∈ Σ . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY ω -composites. That is, if A i → A i +1 , i ∈ ω , is a countable familyof morphisms in Σ, then so is A → lim −→ A i . Herem ω denotes the first countable ordinal.(v) Σ is closed under retracts. That is, if for a commutative diagram A A (cid:48)
AB B (cid:48) B s s (cid:48) s with both horizontal compositions the identity s (cid:48) ∈ Σ, then s ∈ Σ. Proposition 2.2.10. [JT08, Cor. 3.3.1.] Let H be the class of horn inclusions in s Set , Λ nk (cid:44) → ∆ n , for k ≤ n ∈ N . Let A be the smallest saturated class containing H . Then A is the class of anodyne extensions which is the class of acyclic cofibrations. The analogous statement holds for the Douteau model structure on s
Set P . Proposition 2.2.11. [Dou18, Thm. 2.14.] Let H be the class of admissible horn inclu-sions in s Set P . Let A be the smallest saturated class containing H . Then A is the classof acyclic cofibrations, with respect to the Douteau model structure. From here on out, when we refer to cofibrations we always mean cofibrations withrespect to the Douteau model structure on s
Set P if not stated otherwise. Out of theclosure properties in Definition 2.2.9 only the closure under retracts is non-constructive.It turns out, that a lot can still be said about the closure of the class of horn inclusionsunder property (i)-(iv) in Definition 2.2.9. Definition 2.2.12.
Let H be the class of admissible horn inclusionsΛ J k (cid:44) → ∆ J in s Set P . Let SA be the smallest class containing H that is closed underproperty (i)-(iv) of Definition 2.2.9. An element of SA is called a filtered strong anodyneextension , or FSAE for short. If both the target, as well as the source of an FSAE arefinite, it is called a finite FSAE.
By Proposition 2.2.11, every FSAE is an acyclic cofibration. In the context of smallobject arguments ([Hir09, Ch. 10.5]) the class constructed in Definition 2.2.12 is oftenreferred to as the relative cell complexes (with respect to the class of horn inclusions).We take this perspective in Section 2.2.3. The astute reader will immediately noticethat any composition of elementary expansions Definition 2.2.4 is an FSAE. It turns outthat the class of FSAEs is in fact the closure of elementary expansions under transfinitecomposition (see Proposition 2.2.22).We start with a series of examples of finite FSAEs. However, we will only show thatthey are in fact FSAEs at the end of this section.
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Example 2.2.13.
All of the following filtered simplicial sets involved are simplicialcomplexes. Thus, we only describe the morphisms on the vertices.(i) Let J = ( p ≤ ... ≤ p n ) and J (cid:48) be d-flags in P such that there is a degeneracymorphism ∆ J → ∆ J (cid:48) . Then any section of this morphism∆ J (cid:48) (cid:44) → ∆ J is a finite FSAE.(ii) Let J = ( p ≤ ... ≤ p n ) be a d-flag in P . Then the cofibration∆ J (cid:44) → sd(∆ J ) p k (cid:55)→ ( p ≤ ... ≤ p k )is a finite FSAE.(iii) Let J be a flag in P . Then the cofibration∆ J (cid:44) → sd P (∆ J ) p (cid:55)→ ( p, J )is a finite FSAE. See Figure 2.4.(iv) For any FSAE, f : X (cid:44) → Y , the induced mapsd P ( X ) sd P ( f ) (cid:44) −−−−→ sd P ( Y )is also an FSAE.Figure 2.4: Illustration of the FSAE in (iii). HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
X (cid:44) → Ex( X ) is a strong anodyne extension [Mos19, Thm. 22]. Douteau hastransferred some of this work to the setting of filtered simplicial sets in [Dou19b, Ch.3.3.2]. In this subsection, we expand on this generalization. We use this in Section 2.2.4to prove that the filtered analogon of [Mos19, Thm. 22], Proposition 2.2.35, still holds.This, together with the fact that FSAEs are just infinite compositions of elementaryexpansions, is one of the deciding steps in the development of a simple homotopy theoryfor filtered simplicial sets in section 2.3.1. Finally, at the end of this subsection we showa useful lemma (Lemma 2.2.25) that essentially allows us to replace arbitrary FSAEs byfinite compositions of elementary expansions in many situations.We start by translating some of the definition in [Mos19] into the filtered setting andshowing how they are related to the non-filtered on. This is pretty much a verbatimcopy of what is being done there, replacing simplicial sets by filtered simplicial sets andadding the admissibility conditions. Hence, we do not go into too much detail and referto [Mos19]. By a slight abuse of notation, we treat cofibrations A (cid:44) → B as inclusions ofsub-simplicial sets when it comes to notation such as B \ A , translating all the inclu-sions to general monomorphisms would be as tedious as it is trivial. For a ( P -filtered)simplicial set B , we denote by B n.d. the set of its non-degenerate simplices. For now, let A s (cid:44) −→ B be a cofibration in s Set P . Definition 2.2.14.
An anodyne presentation of s consists of an ordinal κ and a κ -indexed increasing family of sub-simplicial sets ( A α ) α ≤ κ of B satisfying: • A = A and A κ = B , • For every non-zero limit ordinal λ < κ , (cid:83) α<λ A α = A λ holds. • For every α < κ the inclusion A α (cid:44) → A α +1 is a pushout of a coproduct of admissiblehorn inclusions.In other words, A → B is the transfinite composition of pushouts of coproducts ofadmissible horn inclusions.Recall that a binary relation ≺ on a set S is called well founded if every non-empty S (cid:48) ⊂ S has a minimal element. That is, it exists an x ∈ S (cid:48) such that for no y ∈ S (cid:48) therelation y ≺ x holds. Definition 2.2.15. (i) A pairing T on s is a partition of B n.d. \ A n.d. into disjoint sets B I and B II togetherwith a bijection T : B II → B I . Simplices in B II or B I are called of type II or type I , respectively. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY proper if for each σ ∈ B II the following holds: There exists aunique k ≤ dim( T ( σ )) such that σ = d k ( T ( σ )) and k is such that Λ J k (cid:44) → ∆ J is anadmissible horn inclusion, where J is the d-flag corresponding to T ( σ )).(iii) A proper pairing T on s induces a relation on B II given by: σ ≺ τ if and only if σ (cid:54) = τ and σ is a face of T ( τ ). This is called the ancestral relation .(iv) The smallest transitive and reflexive relation on B n.d. \ A n.d. generated by T ( σ ) (cid:22) σ ,for σ ∈ B II , as well as d k ( σ ) (cid:22) σ , for σ ∈ B n.d. , is called the ancestral preorder and denoted by (cid:22) T .(v) A proper pairing T is called regular if its ancestral relation is well founded.It can be helpful to decode these definitions a bit. Remark 2.2.16.
Essentially, the point is that regular pairings are what arises from ananodyne presentation one does the following. For every horn Λ J k (cid:44) → ∆ J → B in B that is filled at some point of the presentation, if τ is the non-degenerate simplex of B corresponding to ∆ J → B , then τ is assigned to B II and σ := d k ( τ ) to B I . Onethen constructs T via T ( σ ) := τ . One easily verifies that this defines a proper pairing.The ancestral preorder for this pairing specifies in what order simplices appear in theanodyne presentation. The map B II → κx (cid:55)→ sup { α | σ / ∈ A α } is relation-preserving, with respect to the ancestral relation on the left. Any relationthat maps relation preserving into an ordinal is well founded (this is an easy exercise inelementary set theory). Thus, this means our pairing is regular.We should thus think of the condition of properness as specifying that the pairingactually pairs non-degenerate simplices with faces that correspond to admissible horninclusions. The condition of regularity on the other hand specifies that the adding ofsimplices happens in a well mannered order, i.e. that no simplex is added before all butone of its proper faces are present and that in the case of B n.d. \ A n.d being infinite, thiscan be done in a transfinitely iterative fashion. In practice, there is a useful alternativecharacterization of regularity that is often easy to verify. The proof is identical to thenon-filtered case in [Mos19]. Lemma 2.2.17. [Mos19, Lem. 14] Let T be a proper pairing on s . Then T is regularif and only if there exists a function Φ : B II −→ N such that, for each n and for all type II simplices σ and τ of dimension n , the implication σ ≺ τ = ⇒ Φ( σ ) < Φ( τ ) holds. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Example 2.2.18.
Figure 2.5 shows an anodyne presentation in s
Set as well as thecorresponding pairing and its ancestral relation. The paired simplices have been markedwith the same color, which should not be confused with a filtration.Figure 2.5: An anodyne presentation of an inclusion ∆ (cid:44) → ∆ .As there has already been quite a bit of work on pairings in the non-filtered settingin [Mos19], it can be useful to know when a proper pairing on the underlying simplicialsets also specifies a pairing of filtered simplicial sets. Lemma 2.2.19.
Let T be a pairing on s . Let s (cid:48) be the image of s under the forgetfulfunctor s Set P → s Set . Let T (cid:48) be the pairing induced by T on s (cid:48) . Then the followinghold. (i) T is proper if and only if T (cid:48) is proper and, for each x ∈ B II of type J , the unique k such that d k ( T ( x )) = x is such that Λ J k (cid:44) → ∆ J is admissible. (ii) T is regular if and only if T (cid:48) is regular.Proof. This is immediate from the definition of properness and the fact that the addi-tional regularity condition is clearly independent from the filtration maps into N ( P ).Next, we want to prove the converse of Remark 2.2.16, i.e. that every regular pairinggives rise to an anodyne presentation. Before we can prove this, we need a useful technicallemma. Lemma 2.2.20.
Let
A (cid:44) → B be an FSAE with proper pairing T and correspondingancestral preorder (cid:22) T . Let σ ∈ B n.d. \ A n.d. . Then the downset S generated by σ under (cid:22) T is still finite. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proof.
First note that σ (cid:48) (cid:23) T τ is equivalent to there being a finite sequence σ (cid:48) = σ (cid:23) T σ (cid:23) T ... (cid:23) T σ n = τ such that in every step either σ i +1 = d k ( σ i ) or σ i +1 = T ( σ i ). Hence, S = (cid:83) n ∈ ω S n where S n is inductively defined as the closure of S n − , first under the pairing T and then underthe face relation, and we set S = { σ } . In particular, as both closures preserve finiteness,each S n is finite. Now, assume S is infinite. Then, by finiteness of the S n , no S n cancontain all of S . In particular, (by refining a little if necessary) we obtain an infinitesequence σ = σ (cid:23) T σ (cid:23) T ... (cid:23) T σ n ... where in each step d k ( σ i ) = σ i +1 , for some k , or T ( σ i ) = σ i +1 . Let σ (cid:48) i be the subsequencegiven by such σ i where T ( σ i ) = σ i +1 . By definition of the ancestral preorder, we thenhave σ (cid:48) i (cid:31) σ (cid:48) i +1 . In particular, by well foundedness, there are only finitely many σ (cid:48) i . Butat the same time, these are precisely the elements of the sequence σ i where dim( σ i +1 ) > dim( σ i ). Everywhere else, dim( σ i +1 ) = dim( σ i ) −
1. In particular, the sequence σ i obtains negative dimension, which is a contradiction.The following remark corresponds to [Mos19, Prop. 12]. Remark 2.2.21.
Conversely to Remark 2.2.16, every regular pairing T gives rise to ananodyne presentation of length ω (i.e. of countable length) as follows. Define, usinginduction on the well foundedness of the ancestral relation: F : B II → Ord; σ (cid:55)→ sup { , F ( τ ) + 1 | τ ≺ σ } , where Ord denotes well founded class of all ordinals. As the ancestral relation is wellfounded and the set that we take the supremum over is necessarily finite by Lemma 2.2.20,one can inductively show that F ( τ ) is a natural number, i.e. the construction above mapsinto ω .One now defines A n , n ∈ ω , as the filtered simplicial subset of B with non-degeneratesimplices A nn.d. = A n.d. ∪ { σ, T ( σ ) ∈ B n.d. | F ( σ ) ≤ n } and checks that this fulfills the requirements of an anodyne presentation.These constructions are summarized in the following proposition: Proposition 2.2.22.
Let A s −→ B be a morphism in s Set P . Then the following areequivalent. (i) s is an FSAE. (ii) s is a cofibration and admits a regular pairing. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY s is a cofibration and admits a countable anodyne presentation. (iv) s is a cofibration and admits an anodyne presentation. (v) s is a transfinite composition of elementary expansions. (By a transfinite compo-sition over we mean an isomorphism here.) (vi) s lies in the smallest class of morphisms that contains all admissible horn inclu-sions and isomorphisms and is furthermore closed under pushout and transfinitecomposition.Proof. The equivalences (ii) ⇐⇒ (iii) ⇐⇒ (iv) are the content of Remark 2.2.21 andRemark 2.2.16.(v) implies (iv) by definition. Conversely, let ( A α ) α ≤ κ be an anodyne presentation of A (cid:44) → B . Each inclusion A α (cid:44) → A α +1 is given by a pushout (cid:70) i ∈ J Λ J i k i (cid:70) i ∈ J ∆ J i A α A α +1 with the vertical given by a disjoint union of admissible horn inclusions. Choose a wellfounded ordering of J , J ∼ ←− ψ κ α , for some ordinal κ α . Via transfinite induction define A α,β (cid:44) → A α,β +1 by the pushout:Λ J ψ ( β +1) k ψ ( β +1) ∆ J ψ ( β +1) A α,β A α,β +1 and via A α,β (cid:44) → A α,λ = lim −→ β (cid:48) <λ A α,β (cid:48) for limit ordinals λ ≤ κ α . Then (up to natural isomorphism) A α (cid:44) → A α +1 is given by thetransfinite composition of the diagram given by ( A α,β ) β<κ α All the maps to successorordinals are given by elementary expansions. Thus, the statement to be shown follows bythe fact that a transfinite composition of transfinite compositions is again a transfinitecomposition.Again, by definition, we have (i) = ⇒ (vi) and (iii) = ⇒ (i). So, by the equiva-lences we have already shown, it suffices to show (vi) = ⇒ (v). We need to showthat the class given by transfinite compositions of elementary expansions is closed underpushout and transfinite composition. For pushouts, this is clear by commutativity of HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Corollary 2.2.23.
Let A s (cid:44) −→ B be a cofibration such that B n.d. \ A n.d. is finite. Then,the following are equivalent. (i) s is a (finite) FSAE. (ii) s admits a regular pairing. (iii) s admits a finite anodyne presentation. (iv) s is a composition, possibly empty, of elementary expansions. (v) s lies in the smallest class of morphisms that contains all admissible horn inclusionsand isomorphisms and is closed under pushout and composition. Corollary 2.2.23 can now be used to finally show the following:
Proposition 2.2.24.
All cofibrations in Example 2.2.13 are finite FSAEs.Proof.
For (iv) note that as sd P is a left adjoint, it respects all colimits. Hence, by (v)of Corollary 2.2.23, it suffices to show that sd P sends admissible horn inclusions intofinite FSAEs. This in done in the proof of [Dou18, Prop. 2.9]. For the remaining threeexamples the proof is very similar and relies on (ii) of Corollary 2.2.23 and Lemma 2.2.17.We begin with (ii) of Example 2.2.13 as the proof is by far the most involved of theremaining three. Note that we may assume that J is non-degenerate, as the require-ments of a regular pairing are strictly stronger in the non-degenerate case. So, withoutloss of generality, let J = [ q ] for some q ∈ N . For p ∈ [ q ] ∪ {− } we define D p to be thefull filtered subcomplex of sd(∆ [ q ] ) spanned by the vertices σ such that for all p (cid:48) ∈ σ theimplication r ≤ min( p (cid:48) , p ) = ⇒ r ∈ σ holds. Thinking of ∆ [ q ] as a subcomplex of sd(∆ [ q ] ) via the cofibration in question, wethen have ∆ [ q ] = D q ⊂ D q − ⊂ ... ⊂ D − = sd(∆ [ q ] ) . Note that, due do a slight quirk in the indexing, which could certainly be fixed by ashift, we also have D q = D q − . However, this makes the remainder of the proof a littlemore clean when it comes to indexing. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY D p for q = 2.We construct a regular pairing for each of these inclusions (with the first one beingtrivial). Let p ∈ [ q ] and B := D p − , A := D p . For ( σ ≤ ... ≤ σ k ) ∈ D p − n.d. \ D pn.d. , bydefinition, there exists an i ∈ { , ..., k } such that σ i / ∈ D p . One easily deduces, p / ∈ σ i and there exists a p (cid:48) ≥ p such that p (cid:48) ∈ σ i . Let m ∈ { , ..., k } be maximal with respectto σ m not containing p . Therefore, as σ i ⊂ σ m we also have p (cid:48) ∈ σ m . Now, there aretwo cases: either m < k and σ m +1 = σ m ∪ { p } , or not. We set B I to the set of simpliceswith the former property, and B II to the set of simplices with the latter property. Now,define T : B II −→ B I ( σ ≤ ... ≤ σ k ) (cid:55)−→ ( σ ≤ ... ≤ σ m ≤ σ m ∪ { p } ≤ σ m +1 ≤ ... ≤ σ k ) . Note first that σ m ∪ { p } trivially still lies in D p . The map is clearly bijective. Further-more, as σ m contains a p (cid:48) > p , adding p to σ m does not change the stratum σ m it liesin. Hence, the admissibility condition of properness is fulfilled, making this a properpairing. Define Φ : B II −→ N ( σ ≤ ... ≤ σ k ) (cid:55)−→ m, where m is maximal with respect to σ m not containing p . Now, if ( σ ≤ ... ≤ σ n ) ≺ ( τ ≤ ... ≤ τ n ), then, by definition,( σ ≤ ... ≤ σ n ) ⊂ ( τ ≤ ... ≤ τ m ≤ τ m ∪ { p } ≤ τ m +1 ≤ ... ≤ τ n ) . As σ (cid:54) = τ and both lie in B II this means( σ ≤ ... ≤ σ n ) = ( τ ≤ ... ≤ τ m − ≤ τ m ∪ { p } ≤ τ m +1 ≤ ... ≤ τ n ) . In particular, Φ( σ ) = m − < m = Φ( τ ) . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY T .Figure 2.7: Illustration of the induced anodyne presentation of D (cid:44) → D − , for q = 2.The steps in this presentation are the ones corresponding to dim( σ ) = 0 , , σ ) =0 , , B I , B II , T and Φ and leave it to the reader to check that this actually definesa regular pairing. B and A here refer to the respective codomain and domain of thecofibration. For (i), first note that ,by composability of FSAEs, we may restrict to thecase J = ( x ≤ ... ≤ x k ≤ ... ≤ x n ) and J (cid:48) = ( x ≤ ... ≤ x k ≤ x k ... ≤ x n ). The sectionof the degeneracy map is the either given by the k -th or ( k + 1)-th face map. We do the k case, the other one is analogous. Denote the vertices of J (cid:48) by y , ..., y n +1 . A simplexin B n.d. \ A n.d. is given by a d-flag ( y i ≤ ... ≤ y k ≤ ... ≤ y i j ). We then set B I to thosesimplices where the vertex y k is followed by y k +1 , and B II to the set of those where it isnot. Further, set: T : B II −→ B I ( y i ≤ ... ≤ y k ≤ ...y i j ) (cid:55)−→ ( y i ≤ ... ≤ y k ≤ y k +1 ≤ ...y i j ) . Note that there is no relevant (in the sense of Lemma 2.2.17) ancestral relation to checkfor this pairing, so it is automatically regular.For (iii) a simplex in B n.d. \ A n.d. is given by a flag (( p , σ ) ≤ ... ≤ ( p n , σ n )) in P × sd(∆ J ),where p i ∈ p ∆ J ( σ ) and σ (cid:54) = J . Again, let m be maximal with respect to σ m (cid:54) = J .Further, let B I be the set of those flags where m < n and p m = p m +1 , and B II to theset of those where this does not hold. Define T : B II −→ B I (cid:0) ( p , σ ) ≤ ... ≤ ( p n , σ n ) (cid:1) (cid:55)−→ (cid:0) ( p , σ ) ≤ ... ≤ ( p m , σ m ) ≤ ( p m , J ) ≤ ... ≤ ( p n , J ) (cid:1) and Φ : B II −→ N (cid:0) ( p , σ ) ≤ ... ≤ ( p n , σ n ) (cid:1) (cid:55)−→ m, with m as above. We have illustrated the induced anodyne presentation in Figure 2.8. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY J = (0 ≤ ≤ . The steps in this presentation are theones corresponding to dim( σ ) = 0 , , σ ) = 0 , , Lemma 2.2.25.
Consider a diagram in s Set P as below C AB s , where A and C are finite simplicial sets and s is an FSAE. Then there exists a finitefiltered simplicial subset B (cid:48) ⊂ B and a factorization as below: C AB (cid:48) B s such that both A (cid:44) → B (cid:48) and B (cid:48) (cid:44) → B are FSAEs.Proof. By mono-epi factorization in sSet we may without loss of generality assume that
C, A ⊂ B . Now, choose a regular pairing for s , T . This exists by Proposition 2.2.22.Now, let S ⊂ B be the downset generated by C n.d. \ A n.d. in B n.d. \ A n.d. with respect to (cid:22) T . As C n.d. is finite, by Lemma 2.2.20, S is finite. Further, S ∪ A n.d. is closed under theface relation, by construction. In particular, there is a unique finite filtered sub-simplicialset B (cid:48) ⊂ B with B (cid:48) n.d. = S ∪ A n.d. . By definition of the ancestral preorder, (cid:22) T , and thefact that S = B (cid:48) n.d. \ A n.d. , we have σ ∈ B (cid:48) n.d. \ A n.d. if and only if T ( σ ) ∈ B (cid:48) n.d. \ A n.d. . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY T restricts to a pairing on A (cid:44) → B (cid:48) . This is clearly still regular and it is proper,as any subset of a well founded set is well founded. By the same argument, the sameholds for the restriction of T to B (cid:48) (cid:44) → B . In particular, we get a factorization diagramas in the claim. The reader familiar with Quillens small object argument (see for ex. [Hir09, Ch. 10.5])will have noticed that it applies to the class of admissible horn inclusions and that therelative cell complexes in this setting are just the FSAEs. The reader unfamiliar with theargument is advised a quick skim of [Hir09, Ch. 10.5] as we rely on some nomenclaturefrom this source.
Definition 2.2.26. [Hir09, Def. 10.5.15] Let C be a category and I a set of morphismsin C . We say that I permits the small object argument if the domains of the elements in I are small relative to I (see [Hir09, Def. 10.5.12]). Example 2.2.27.
The set of admissible horn inclusions H admits the small object argu-ment. This is immediate, as all objects involved are finite filtered simplicial sets. Hence,by Lemma 1.1.42, the sources of H are small with respect to any class of morphisms.The small object argument then states: Proposition 2.2.28. [Hir09, Prop. 10.5.16] If C is a cocomplete category and I isa set of maps in C that permits the small object argument, then there is a functorialfactorization of every map in C into a relative I -cell complex (see [Hir09, Def. 10.5.8])followed by a morphism that has the right lifting property with respect to all morphismsin I . Corollary 2.2.29.
There exists a functor F : s Set P → s Set P together with a naturaltransformation i −→ F such that for each X ∈ s Set P : (i) F ( X ) is fibrant. (ii) i X is an FSAE.Proof. By Example 2.2.27 we can apply Proposition 2.2.28 to the class of admissiblehorn inclusions H and terminal maps X → N ( P ). We obtain a functorial splitting XF ( X ) N ( P ) . where the vertical morphism lies in cell ( H ), i.e. is an FSAE and the horizontal morphismhas the right lifting property with respect to H , making it a fibration. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Set P , H s Set P . This is summarized in the fol-lowing proposition which we are using all throughout the remainder of this work. Proposition 2.2.30.
Let
X, Y ∈ s Set P . (i) Let X α −→ Y be a morphism in H s Set P . Then there exists a morphism f : X Z (cid:44) −→ in s Set P and an FSAE Y s −→ Z such that α = [ s ] − ◦ [ f ] . If X and Y are finite, then Z can be taken to be finite. (ii) Let X f,g −−→ Y be two morphisms in s Set P . Then [ f ] = [ g ] if and only if thereexists an FSAE Y s (cid:44) −→ Z such that s ◦ f and s ◦ g are strictly simplicially (stratumpreserving, elementary) homotopic. If X and Y are finite, then the same statementholds with Z finite.Proof. Denote by F, i −→ F the fibrant replacement functor from Corollary 2.2.29. Westart with the proof of ( i ). We only do the proof in the finite case, as the infinite one isa strict simplification of it. Let X, Y ∈ s Set fin P and X α −→ Y be a morphism in H s Set P .Then, as F ( Y ) is fibrant and X cofibrant, α fits into a commutative diagram X YF ( Y ) α [ f ] [ i Y ] , where f is a morphism X f −→ F ( Y ) in s Set P (see Proposition 1.1.27). Now, applyLemma 2.2.25 to get a factorization X YY (cid:48) F ( Y ) α [ f ] [ f (cid:48) ] [ i (cid:48) Y ] [ i Y ] , where we know of the outer diagram and all of the triangle diagrams but the most upperone that they are commutative, and where i (cid:48) Y is an FSAE and Y (cid:48) is finite. Using the factthat the morphism Y (cid:48) → F ( Y ) is an FSAE and hence an isomorphism in the homotopycategory and chasing the diagram shows that the most upper triangle also commutes.To summarize, we have shown [ i (cid:48) Y ] ◦ α = [ f (cid:48) ] HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY α = [ i (cid:48) Y ] − [ f (cid:48) ]for the FSAE i (cid:48) Y between finite filtered simplicial sets.The if part is immediate from the fact that every FSAE is a weak equivalence andagain Proposition 1.1.27. For the only if statement, consider the fibrant replacement Y i Y (cid:44) −→ F ( Y ). Then [ i Y ◦ f ] = [ i Y ◦ g ]. As X is cofibrant and F ( Y ) is fibrant, we have asimplicial homotopy H : X ⊗ ∆ → F ( Y ) between i Y ◦ f and i Y ◦ g (see for ex. [Hir09]).In other words there is a commutative diagram X (cid:116) X YX ⊗ ∆ F ( Y ) i (cid:116) i f (cid:116) g i Y H . If we apply Lemma 2.2.25 to this we obtain: X (cid:116) X YY (cid:48) X ⊗ ∆ F ( Y ) i (cid:116) i f (cid:116) g i Y i (cid:48) Y HH (cid:48) , where i (cid:48) Y is an FSAE of finite filtered simplicial sets. A quick diagram chase togetherwith the fact that Y (cid:48) → F ( Y ) is a monomorphism shows that the upper left part of thediagram commutes. In particular, i (cid:48) Y ◦ f and i (cid:48) Y ◦ g are homotopic through a simplicialhomotopy. Vital to many of the arguments we need for the construction of a Whitehead group forfiltered simplicial sets are the various interactions of strong anodyne extensions withmapping cylinders. In the classical cellular setting, this can be found in [Coh73, Ch. 2, §
5] and from a simplicial perspective in Whiteheads original publication in [Whi39, Sec.6]. We have already seen that the class of strict elementary expansions is rather limitingin this sense, c.f. Example 2.2.8. In this subsection we show that the same is not thecase for general elementary expansions.Recall that in a simplicial model category the simplicial mapping cylinder, M f of amorphism f : X → Y is defined via the pushout HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY X ∼ = X ⊗ ∆ YX ⊗ ∆ M ffi .Recall further that if X is cofibrant, then Y → M f is an acylic cofibration and X i (cid:44) −→ X ⊗ ∆ → M f is a cofibration. This allows to factor any such f into an acyclic cofi-bration followed by the induced weak equivalence p Y : M f → Y . For the category ofsimplicial sets with the Kan-Quillen model structure, the statement that Y (cid:44) → M f is aacyclic cofibration (i.e. an anodyne extension) can be slightly strengthened. This is aconsequence of the following result. Proposition 2.2.31. [Mos19, Prop. 16] Let A s (cid:44) −→ B be a strong anodyne extensionsand X i (cid:44) −→ Y be any cofibration (both in sSet ). Then B × X ∪ A × X A × Y (cid:44) → B × Y is also a strong anodyne extension. In particular, one obtains that the inclusions
X (cid:44) → X × ∆ at both endpoints is astrong anodyne extensions. Hence, for X f −→ Y a morphism in s Set P the inclusion of Y (cid:44) → M f , is a strong anodyne extension too, by stability under pushouts. This is ofcourse the simplicial analogue to the respective result for mapping cylinders of cellularmaps and cellular expansions (see for example [Coh73, Cor. 5.1A]). It turns out thisresult generalizes to the filtered setting. To obtain this in a fairly general form, we needthe following outer product construction. Definition 2.2.32.
First, note that for two partially ordered sets P and P (cid:48) there is anatural isomorphism N ( P ) × N ( P (cid:48) ) ∼ = N ( P × P (cid:48) ). Under this identification, we obtainan outer product functor − ⊗ − : s Set P × s Set P −→ s Set P × P (cid:48) , by simply taking the product of arrows into N ( P ) and N ( P (cid:48) ) respectively. In case where P (cid:48) is a point, this corresponds to the simplicial copower on s Set P under s Set P (cid:48) ∼ = s Set ,justifying the overload of notation. From the presheaf perspective of Construction 1.1.25this construction is given by: X ⊗ Y : ∆( P × P (cid:48) ) op (cid:44) → ∆( P ) op × ∆( P (cid:48) ) op X × Y −−−→ Set × Set −×− −−−→
Set . Proposition 2.2.33.
Let A s (cid:44) −→ B be a strong anodyne extension in s Set P and let X (cid:44) → Y be a cofibration in sSet P (cid:48) . Then B ⊗ X ∪ A ⊗ X A ⊗ Y (cid:44) → B ⊗ Y is a strong anodyne extension in sSet P × P (cid:48) . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proof.
This is really just a modification of the proof of [Mos19, Prop. 16]. The proofgoes consists of two steps. First we show that the class of X s (cid:44) −→ Y that have this propertyfor some fixed i is closed under:(i) arbitrary coproducts,(ii) pushouts,(iii) transfinite composition,and that the analogous statement holds with rolls of s and i swapped. For the secondstep, note that the class of cofibrations in sSet P (cid:48) is generated under these operationsby the boundary inclusions ∂ ∆ J (cid:48) (cid:44) → ∆ J (cid:48) , for d-flags J (cid:48) in P (cid:48) and the class of FSAEsis generated by the admissible horn inclusions Λ J k (cid:44) → ∆ J , for flags J in P . Hence, itsuffices to show that, for such J , J (cid:48) , k ,∆ J ⊗ ∂ ∆ J (cid:48) ∪ Λ J k ⊗ ∂ ∆ J (cid:48) Λ J k ⊗ ∆ J (cid:48) (cid:44) → ∆ J ⊗ ∆ J (cid:48) is an FSAE.The proof of the first step is analogous in both arguments, so we only do the one withfixed i . First note that the exterior product ⊗ does commute with colimits in both argu-ments (by the analogous statement in Set ). From this (i) immediately follows. For (ii),note that for an arrow A → A (cid:48) and B (cid:48) the pushout of s along this arrow, the followingdiagram is a pushout diagram. B ⊗ X ∪ A ⊗ X A ⊗ Y B ⊗ YB (cid:48) ⊗ X ∪ A (cid:48) ⊗ X A (cid:48) ⊗ Y B (cid:48) ⊗ Y An easy way to verify this statement is to note that by the alternative constructionof ⊗ via the perspective of categories of Set -valued presheaves (Construction 1.1.25)we may just as well check this in
Set , replacing ⊗ by × . This is an easy exerciseof manipulations of pushouts in a closed monoidal category. It can be found in theappendix (Lemma A.0.2). We are lacking (iii). So, let A : κ → sSet P × P (cid:48) be a transfinitecomposition diagram for some ordinal κ . By transfinite induction, let κ be minimal suchthat we have not shown closedness under transfinite composition (the induction start isobvious). If κ = κ (cid:48) + 1 is a successor ordinal then A κ = A κ (cid:48) . If κ (cid:48) is a successor ordinal,then the transfinite composition is just the one given by the restriction of A to κ (cid:48) . Sothe result follows by the induction hypothesis. If κ (cid:48) = κ (cid:48)(cid:48) + 1 is a successor ordinal, thenconsider the diagram below: A κ (cid:48)(cid:48) ⊗ X ∪ A ⊗ X A ⊗ Y A κ (cid:48)(cid:48) ⊗ YA κ (cid:48) ⊗ X ∪ A ⊗ X A ⊗ Y A κ (cid:48) ⊗ X ∪ A k (cid:48)(cid:48) ⊗ X A κ (cid:48)(cid:48) ⊗ Y A κ (cid:48) ⊗ Y = A κ ⊗ Y . (2.3)
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Set that the square in this diagram isa pushout diagram. By assumption, the upper vertical and the lower right morphismare FSAEs. By stability under pushouts and composition, the map on the bottom is anFSAE, but this is precisely the map, corresponding to the composition A → A κ (cid:48)(cid:48) → A κ (cid:48) = A κ i.e. to the transfinite composition we considered. Now, let κ be a limit ordinal.Just as before, consider the pushout square in (2.3) but replace (formally) κ (cid:48) with κ , κ (cid:48)(cid:48) with κ (cid:48) , and 0 with κ (cid:48)(cid:48) . We obtain a diagram κ → sSet P × P (cid:48) given by κ (cid:48)(cid:48) < κ (cid:48) (cid:55)→ { A κ ⊗ X ∪ A κ (cid:48)(cid:48) ⊗ X A κ (cid:48)(cid:48) ⊗ Y (cid:44) → A κ ⊗ X ∪ A k (cid:48) ⊗ X A κ (cid:48) ⊗ Y } . By the induction hypothesis and stability of FSAEs under pushout, all the arrows in thisdiagram are FSAEs. Further, by commutativity of colimits and the fact that ⊗ commuteswith colimits, this fulfils the requirements of a transfinite composition diagram, for eachlimit ordinal κ (cid:48) < κ . The colimit of this diagram is A κ ⊗ Y (again, this is easily checkedin Set by the same argument as before). In particular, A κ ⊗ X ∪ A ⊗ X A ⊗ Y (cid:44) → A κ ⊗ Y is a transfinite composition of FSAEs, making it an FSAE also.For the second part of the proof, consider a d-flag J = ( p ≤ ... ≤ p m ) in P , a d-flag J (cid:48) = ( p (cid:48) ≤ ... ≤ p (cid:48) m ) in P (cid:48) and some k ≤ n such that Λ J k (cid:44) → ∆ J is admissible.We use the pairing in the proof of [Mos19, Prop. 16] and Lemma 2.2.19 to show thatthe latter still works in the filtered setting. We give a sketch of this construction. Let A := ∆ J ⊗ ∂ ∆ J (cid:48) ∪ Λ J k ⊗ ∂ ∆ J (cid:48) Λ J k ⊗ ∆ J (cid:48) and B := ∆ J ⊗ ∆ J (cid:48) . We are now in the setting where all the simplicial sets involved come from simplicialcomplexes. On the underlying simplicial sets ⊗ is just given by the product of orderedsimplicial complexes. Every N -simplex in B is given by a finite sequence( µ , ν ) ≤ ( µ , ν ) ≤ ... ≤ ( µ N , ν N )in [ m ] × [ n ] with respect to the induced preorder on the product. The simplex is non-degenerate if at each step the µ - or the ν -entry strictly increases. The i -th face is ob-tained, by leaving out the i -th entry. Thus, i is admissible with respect to the product fil-tration of this simplex if and only if ( p µ i , p (cid:48) ν i ) = ( p µ i +1 , p (cid:48) ν i +1 ) or ( p µ i , p (cid:48) ν i ) = ( p µ i − , p (cid:48) ν i − ) . A non-degenerate simplex in B belongs to B n.d. \ A n.d. if and only if in the correspond-ing sequence the the ν i skip no value in [ n ] and the µ i skip no value in [ m ] other than k . Pictorially, we may think of such a sequence as a walk in the [ m ] × [ n ] grid as inFigure 2.9. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY m ] × [ n ] where P, P (cid:48) = [1] and J = (0 ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ J (cid:48) = (0 ≤ ≤ ≤ ≤ ≤ ≤
1) indicated by blue and red.At each time i in the walk, the next step is of type (+1 , +0) , (+0 , +1) , (+1 , +0), or inaddition to this possibly of type (+2 , +0) , (+2 , +1) if the next column is the k -th and itis skipped. For k < m , [Mos19] constructs a pairing on A (cid:44) → B by setting B I to consistof those simplices where the k -th column is not skipped and the step that leaves thecolumn is of type (+1 , k -th column is removed (see Figure 2.10 for an example).Figure 2.10: The walks corresponding to a pair of simplices in B n.d. \ A n.d. . The removedvertex is marked. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY < k , one obtains a pairing by taking B I to consist of those simplices wherethe corresponding walk does not skip the k -th column and the prior step is of type(+1 , +0). One then pairs such a simplex with the one given removing the first point onthe k -th column. In both cases Moss shows in [Mos19] that this gives a regular, properpairing on the underlying simplicial sets. Now, if p k = p k +1 we take the prior pairing, if p k = p k − we take the latter pairing on the underlying simplicial map A (cid:44) → B . To seethis gives a proper, regular pairing on A (cid:44) → B we just need to verify that in both casesthe requirements of Lemma 2.2.19 are met. That is, we need to check that, for each pairgiven by T , if i is the index of the removed point, then i is admissible. If p k = p k +1 ,then the removed point corresponds to a vertex of stratum ( p k , p (cid:48) j ), for some 0 ≤ j ≤ n .As the next point is reached through a (+1 , +0), it corresponds to a ( p k +1 , p (cid:48) j ) = ( p k , p (cid:48) j )vertex. In particular, i is admissible. The case p k = p k − works analogously.We are mostly interested in the case where either P or P (cid:48) is the partially orderedset with one element, i.e. the corresponding category of filtered simplicial sets is justthe category of simplicial sets. We obtain the following immediate corollary of Proposi-tion 2.2.33. Corollary 2.2.34.
Let X i (cid:44) −→ Y be a cofibration in sSet P and X (cid:48) i (cid:48) (cid:44) −→ Y (cid:48) be a cofibrationin sSet . Then if either of the two is an FSAE, so is the morphism in s Set P ∼ = s Set P × (cid:63) : Y ⊗ X (cid:48) ∪ X ⊗ X (cid:48) X ⊗ Y (cid:48) (cid:44) → Y ⊗ Y (cid:48) This result has a series of immediate consequences for the interactions of FSAEs withfiltered simplicial mapping cylinders (see Proposition 2.2.35). These can be thought ofas filtered, simplicial set analogues to the results found for example in ([Coh73, Ch. 2,]),[Whi39, Sec. 6], [KP86, Ch. 6.3]). They are the decisive arguments when it comesto verifying the Eckmann-Siebenmann axioms (Section 2.1) for the construction of aWhitehead group.
Proposition 2.2.35.
Let A s (cid:44) −→ B be an FSAE, B (cid:48) i (cid:44) −→ B be a cofibration and B f −→ Z be a morphism in s Set P . Then all of the following morphisms in s Set P (illustrated inFigure 2.11) are FSAEs. (i) i , i : Z (cid:44) → Z ⊗ ∆ (ii) Z (cid:44) → M f (iii) B ∪ A(cid:44) → M f ◦ s M f ◦ s (cid:44) → M f (iv) M f ◦ i (cid:44) → M f In particular, for any elementary stratum preserving simplicial homotopy Y ⊗ ∆ H −→ Z with f := H and g := H , denote by ˆ M H the pushout given by Y ⊗ ∆ M H Y ˆ M H . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Then the compositions (v) M f , M g (cid:44) → M H → ˆ M H are both FSAEs. Figure 2.11: An illustration of some examples of the FSAEs (i) to (v) in Proposi-tion 2.2.35.
Proof.
For (i) take Y = Z , X = ∅ , X (cid:48) = ∆ and i (cid:48) = i , i in Corollary 2.2.34. For (ii)note that Z (cid:44) → M f is given by the pushout diagram B B ⊗ ∆ Y M fi f . By (i), i is an FSAE, hence, so is Y (cid:44) → M f . For (iii) first consider the case of Corol-lary 2.2.34, where X = A, Y = B and X (cid:48) = ∆ (cid:116) ∆ , Y (cid:48) = ∆ , i (cid:48) = i (cid:116) i . We obtainthat B ∪ A i (cid:44) −→ A ⊗ ∆ A ⊗ ∆ ∪ A i (cid:44) −→ A ⊗ ∆ B (cid:44) → B ⊗ ∆ HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY B at 0 into the previous homotopy pushout j andconsider the following commutative diagram. We omit some of the pushout subscript,to simplify notation from here on out. B B ∪ A ⊗ ∆ ∪ B B ⊗ ∆ Z B ∪ A(cid:44) → M f ◦ s M f ◦ s M fj f f (cid:48) . The upper horizontal composition is just the inclusion of B into B ⊗ ∆ at 0 and thelower horizontal composition is Z (cid:44) → M f . In particular, the inner left and the outersquare are cocartesian. Hence, so is the inner right square. Thus, as we have just seenthat the upper right horizontal is an FSAE, the same holds for the morphism in (iii), bystability under pushouts. For (iv) take X = B (cid:48) , Y = B and X (cid:48) = ∆ , Y (cid:48) = ∆ , i (cid:48) = i .Then the result follows by a similar pushout argument as we used for (iii).We prove (v) for f . The case of g works analogously. First note that by (i), (iii)and (iv) the cofibrations M f (cid:44) → Y ⊗ ∆ ∪ M f (cid:44) → M H as well as their composition areFSAEs. The ∪ stands for the pushout of Y i (cid:44) −→ Y ⊗ ∆ along Y (cid:44) → M f here. Now,consider the commutative diagram Y ⊗ ∆ Y ⊗ ∆ ∪ ... M f M H Y M f ˆ M H . (2.4)The outer square is cocartesian by definition. The inner left square fits into Y Y ⊗ ∆ Y M f Y ⊗ ∆ ∪ ... M f M fi The horizontal compositions of 2.2.4 are the identity making the outer square cocartesian.The left inner square is cocartesian by definition. In particular, the right inner square,which is the left inner square in (2.4) is cocartesian. Hence, so is the inner right squarein (2.4). By stability under pushouts, this shows that M f (cid:44) → ˆ M H is an FSAE.In particular, Proposition 2.2.35 shows that with our specification of elementarycollapses we do not run into problems such as we illustrated in Example 2.2.8. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
In Section 2.1 we have reviewed a general setting that Whitehead groups can be con-structed in. Then, in in Section 2.2 we begun a detailed analysis of a class of expansionsof P -filtered simplicial sets, FSAEs, which work as a filtered analogue to the expansionin the classical simple homotopy theory. We now show that (finite) FSAEs fit into thisgeneral setting (Section 2.1) and thus obtain a Whitehead group and torsion for filteredsimplicial sets. We then study these invariants in detail, showing that they behave muchlike the classical ones (Section 2.3.2). Finally, we obtain the analogue to the result thatevery finite CW-complex is simple homotopy equivalent to a finite simplicial complexfor our setting (Section 2.3.3). Recall the setting of Section 2.1. Given a category C contained in a larger category ˆ C and a class of morphisms Σ in C fulfilling certain axioms, it allows us to construct aWhitehead group and a Whitehead torsion. In this subsection, we apply this theoryto the case where ˆ C is the category of finite filtered simplicial sets in s Set P , C is thesubcategory of ˆ C with the same objects morphisms taken to be only cofibrations in theDouteau model structure - i.e. inclusions of filtered simplicial subsets - and Σ is theclass of (finite) FSAEs.Recall that we denote by H s Set fin P the full subcategory of H s Set P given by finite filteredsimplicial sets. Further, let C ( C (cid:48) ) be the subcategory of s Set P given by finite (arbitrary)filtered simplicial sets and morphisms cofibrations and Σ (Σ (cid:48) ) the class of morphism in C ( C (cid:48) )that are FSAEs. As every FSAE is in particular an anodyne extension and theseare precisely the trivial cofibrations in the Douteau-Henrique model structure on s Set P [Dou18, Thm. 2.14], by the universal property of the localization of a category we obtaina commutative (on the nose) diagram of functors C (Σ − ) H s Set fin P C (cid:48) (Σ (cid:48)− ) H s Set P . All of the functors above are given by the identity on objects. It turns out, all of themare in fact fully faithful. Before we prove this, we start by proving two lemmata. Denoteby Q and Q (cid:48) the structure functors C → C (Σ − ) and C (cid:48) → C (cid:48) (Σ (cid:48)− respectively. Lemma 2.3.1.
Let X f,g (cid:44) −→ Y be two cofibrations in s Set P that are homotopic through a(elementary, stratum preserving) simplicial homotopy. Then Q (cid:48) ( f ) = Q (cid:48) ( g ) . If both X and Y are finite, then also Q ( f ) = Q ( g ) . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proof.
The argument both in the finite and in the infinite case are identical, so we provethe finite one. We fist show the case where the two are homotopic through a cofibration H : X ⊗ ∆ (cid:44) → Y . We then show an additional lemma to complete the proof of this one.Consider the two cofibrations X i ,i (cid:44) −−→ X ⊗ ∆ . If we can show that Q ( i ) = Q ( i ), weare done since then Q ( f ) = Q ( H ) ◦ Q ( i ) = Q ( H ) ◦ Q ( i ) = Q ( g ) . Now, note that sd(∆) = Λ . Consider the following diagram in s Set .∆ ∆ sd(∆ ) = Λ ∆ isd ( i ) d , for i the inclusion at 0 or 1. All of the maps in the diagram are FSAEs. In particular,if we apply X ⊗ ( − ) by Proposition 2.2.33, we obtain commutative diagrams of FSAEs X X ⊗ ∆ X ⊗ Λ X ⊗ ∆ i j i (cid:48) j ,j = 0 ,
1. By commutativity and the fact that all arrows involved become isomorphismsunder Q , it suffices to show that Q ( i (cid:48) ) = Q ( i (cid:48) ) . Let t : X ⊗ sd(∆ ) ∼ −→ X ⊗ sd(∆ ) be the isomorphism induced by the swap of endpointssd(∆ ) → sd(∆ ). Denote by i m : X (cid:44) → X ⊗ sd(∆ ) the inclusion induced by sending∆ to the middle vertex in sd(∆ ). The diagram in C X X ⊗ sd(∆ ) X ⊗ sd(∆ ) X ⊗ sd(∆ ) i m i m t commutes. By the same usage of Proposition 2.2.33 as before, one shows that i m is anFSAE. In particular, by commutativity of the last diagram, Q ( t ) = Q (1 X ⊗ sd(∆ ) )and it follows Q ( i (cid:48) ) = Q ( t ) Q ( i (cid:48) ) = Q (1) Q ( i (cid:48) ) = Q ( i (cid:48) )completing the first part of this proof. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Lemma 2.3.2.
Let X f (cid:44) −→ Y be a cofibration in s Set P . Then the mapping cylindersplitting X YM ff commutes in C (cid:48) (Σ (cid:48)− ) . If X and Y are finite, it also commutes in C (Σ − )) .Proof. Denote by H the structure map X ⊗ ∆ → M f . Since f is a cofibration, so is this(by stability under pushouts). X i (cid:44) −→ X ⊗ ∆ H −→ M f is the lower diagonal map in thestatement. X i (cid:44) −→ X ⊗ ∆ H −→ M f is the composition of the upper right with the vertical.In particular, by the first part of the proof of Lemma 2.3.1, the result follows. proof of Lemma 2.3.1; part two. We still have to show that Q ( f ) = Q ( g ) if the two arehomotopic through a simplicial homotopy H : X ⊗ ∆ → Y that is not necessarily acofibration. To show this, consider the following diagram in s Set P . M f X Y ˆ M H M ggf Here ˆ M H and the arrows into it are the ones constructed in Proposition 2.2.35 (i.e.ˆ M H is the mapping cylinder of H where the X ⊗ ∆ at 1 was collapsed to X .) ByProposition 2.2.35 Item (v), all arrows in the right part of the diagram are FSAEsand the upper right and lower right triangles are commutative by construction. ByLemma 2.3.2 the upper left and lower left triangle commute in C (Σ − ). Passing to C (Σ − ) we now have a diagram where: • The outer square commutes. • The upper left and lower left triangle commute. • The upper right and lower right triangle commute and consist only of isomor-phisms.In particular, a quick diagram chase shows that in C (Σ − ) we have Q ( f ) = Q ( g ). Proposition 2.3.3.
All of the functors in (2.3.1) are fully faithful. In particular, thehorizontals are isomorphisms of categories.
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proof.
We prove the finite case as the infinite one is basically just a simpler version ofthis one.We start by showing fullness. By Proposition 2.2.30 it suffices to show this that mor-phisms in H s Set fin P of shape [ f ] for some morphism f : X → Y in s Set fin P . Using mappingcylinder factorization we obtain the diagram X YM ff , which is commutative in H s Set P . As, M f is again finite, by Proposition 2.2.35, Y (cid:44) → M f is a (finite) FSAE, hence induces an isomorphism in C (Σ − ). So it suffices to show thatthe class of X (cid:44) → M f in H s Set P comes from an arrow in C (Σ − ). But this is obvious,as X (cid:44) → M f is a cofibration of finite filtered simplicial sets.It remains to show the faithfulness. First note that by the same argument as in Sec-tion 2.1 (i.e. the fact that we can always change direction in a zigzag by using pushouts)every morphism in C (Σ − ) is of the shape Q ( s ) − Q ( f ) where s is a (finite) FSAE and f amorphism in C . So suppose we are given two such morphisms Q ( s ) − Q ( f ) , Q ( s (cid:48) ) − Q ( g )with [ s ] − [ f ] = [ s (cid:48) ] − [ g ]. We need to show Q ( s ) − Q ( f ) = Q ( s (cid:48) ) − Q ( g ). By invertibilityof Q ( s ) , Q ( s (cid:48) ) we may without loss of generality assume that f, g have the same source X and target Y and that s = s (cid:48) = 1 Y . In particular, we then have[ f ] = [ g ] . By Proposition 2.2.30 there then exists an s ∈ Σ such that s ◦ f and s ◦ g are (stratumpreserving, elementarily) simplicially homotopic. By Lemma 2.3.1 this implies Q ( s ◦ f ) = Q ( s ◦ g ). As Q ( s ) is invertible, we obtain Q ( f ) = Q ( g ) . As a consequence of Proposition 2.3.3, our choices of C and Σ are promising can-didates for constructing a Whitehead group (with respect to FSAEs i.e. with respectto elementary expansions) on H s Set fin P ( H sSet). For X ∈ C , ( C (cid:48) ) and Σ, Σ (cid:48) as beforerecall the definitions of a simple morphism and of A ( X ) , E ( X ) (Definition 2.1.1). Forthe remainder of this section, we only be focus on the finite setting i.e. on C , Σ. Quiteclearly, all of the arguments below also work in the infinite one, with the exception that A ( X ) is in general probably not of set size in this case. If one is willing to work withgroups of class size, this might produce a fruitful theory as well.In the finite setting however, E ( X ) is clearly of set size, as the equivalence relationon E ( X ) is up to isomorphism. We now show that the triple C ⊂ ˆ C , Σ satisfies theaxioms in Definition 2.1.2, i.e. admits a Whitehead group.
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Theorem 2.3.4.
Let ˆ C be the full subcategory of s Set P given by finite filtered simplicialsets. Let C be the subcategory which has the same objects as C but only cofibrations in s Set P as morphisms. Let Σ be the class of FSAEs in ˆ C . Then C ⊂ ˆ C , Σ admits aWhitehead group (see Definition 2.1.2).Proof. ( A0 ) of Definition 2.1.2 is obvious as FSAEs contain all isomorphism and areclosed under composition by definition. Similiarly, as cofibrations are closed underpushouts and FSAEs are closed under pushouts by definition, we immediately obtain( A1 ). It remains to show ( A2 ). So let f, g : X → Y be two cofibrations of simplicialsets, such that Q ( f ) = Q ( g ). Then, in particular, [ f ] = [ g ]. By Proposition 2.2.30 andthe fact that (finite) FSAEs are stable under composition we may hence assume withoutloss of generality that f and g are elementary simplicially homotopic through a simplicialhomotopy H . By Proposition 2.2.35 (v), we then have a commutative diagram X M f M g ˆ M Hs f s g in C , with s f and s g FSAEs. Now, extend this diagram as follows:
XY M f , M g YY ⊗ ∆ ˆ M H Y ⊗ ∆ R f R g R f gi s f s (cid:48) f s (cid:48) g s g i All the squares ending in R s are obtained by pushing out. The upper left and upper rightsquare commute by definition. Now, as both f and g are cofibrations, we have that s (cid:48) f and s (cid:48) g are FSAEs, by Proposition 2.2.35 (iv). Furher i is an FSAE by Proposition 2.2.35(iv). In particular, by stability under pushouts, every arrow, but the ones in the first rowis an FSAE. Hence, by stability under composition and commutativity of the diagram,we have shown ( A2 ).Thus Theorem 2.3.4, we can apply Theorem 2.1.3 together with Proposition 2.3.3 toobtain two functors H s Set fin P ∼ = C (Σ − ) A −→ MonAb ; (2.5) H s Set fin P ∼ = C (Σ − ) E −→ Ab . (2.6) HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY A and E are listed. We are now in shape to finally define the Whitehead group of a finite filteredsimplicial set. Definition 2.3.5.
Let Let X ∈ s Set fin P . • Denote by A P the functor in (2.5). A P ( X ) is called the Whitehead monoid of the filtered simplicial set X . • Denote by
W h P the functor in (2.6). W h P ( X ) is called the Whitehead group of the filtered simplicial set X . • For a morphism α : X → Y in H s Set fin P , let τ P ( α ) ∈ A P ( X ) ( ... ∈ W h P ( X ) if α is an isomorphism) be the simple morphism class of the morphism in C (Σ − )corresponding to α . τ P ( α ) is called the Whitehead torsion of α . • For a morphism f : X → Y in s Set P , define τ P ( f ) := τ P ([ f ]) ∈ A P ( X ) ( ... ∈ W h P ( X ) if f is a weak equivalence). τ P ( f ) is called the Whitehead torsion of f .In the following subsection, we investigate some of the properties of A P , W h P and τ P . In particular, we give a more explicit description of their functoriality for morphismsin s Set P . It turns out, in many ways they behave just like the classical theory. This ofcourse stems from the fact, that most of what we have done here is a formalization ofthe arguments used there. More generally, it seems that if one keeps really good track ofwhat we are actually using, our argument should generalize to any cofibrantly generatedcombinatorial simplicial model category, fulfilling an analogue of Proposition 2.2.30 forcompact objects, and also the analogue of Proposition 2.2.33. W h P , A P , τ P and of simple equivalences The astute reader will have noticed that, while we have spent a great deal of time talkingabout FSAEs - the filtered simplicial analogue to expansions in the classical setting - wehave yet to define what a simple equivalence is, in the setting of finite filtered simplicialsets. This is because to really say anything interesting about these objects, it is helpfulto already have the theory of Theorem 2.1.3 in place. Showing that one can use FSAEsof finite filtered simplicial sets to construct a Whitehead group (a Whitehead monoid)and a Whitehead torsion has been the content of the last subsection, of Proposition 2.3.3,Theorem 2.3.4 and Definition 2.3.5 to be more precise. Now, that these are set up, wecan start our investigation of simple equivalences. It can be of interest to compare theresults in this section to the ones found for example in [Whi39] for a simplicial complexsetting, [Whi50], [Coh73] for a CW setting and [KP86] for an abstract setting, whichthey and their proofs are clearly inspired by.
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY C = s Set fin P , C the subcategory given by cofibrations in ˆ C and Σ the class of FSAEsin ˆ C and that we have a (canonical) isomorphism of categories C (Σ − ) ∼ = H s Set fin P , byProposition 2.3.3. Definition 2.3.6. • A morphism in α in H s Set fin P is called a simple equivalence if and only if themorphism in C (Σ − ) corresponding to α is a simple morphism. That is, if it isa finite composition of morphisms of shape [ s ] or [ s ] − , where s is an FSAE ins Set fin P . • Two objects in
X, Y ∈ H s Set P are said to have the same simple homotopy type or also to be simply equivalent if there is a simple equivalence X α −→ Y . • A morphism in s
Set fin P is called a simple equivalence if [ f ] is a simple equivalence. • We say two morphisms α, β in H s Set fin P with the same source have the same simplemorphism class if their corresponding morphism in C (Σ − ) have the same simplemorphism class. That is, if there is a simple equivalence σ in H s Set fin P such that α = σ ◦ β . The equivalence class generated by this relation is called the simplemorphism class of α , denoted by (cid:104) α (cid:105) . • The simple morphism class of a morphism f in s Set fin P is defined to be the simplemorphism class of [ f ], denoted by (cid:104) f (cid:105) . • A zigzag in s
Set fin P X = X ↔ ... ↔ X n = Y with all arrows FSAEs is called a deformation between X and Y . We also say X deforms into Y and vice versa. • A commutative diagram in s
Set fin P XY ... Y na a n , where the verticals are cofibrations, and the lower horizontal is a deformation from Y to Y n is called a deformation from a to a n . We also say a deforms into a n and vice versa.Clearly, every finite FSAE is a simple equivalence. Also, by definition, any deforma-tion induces a simple equivalence, by taking the zigzag as a morphism in H s Set fin P . Inparticular, we obtain a vast class of examples of simple equivalences, by using the resultsin Proposition 2.2.35 together with the fact that simple equivalences fulfill a two out ofthree rule, by definition. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Example 2.3.7.
Let X f −→ X (cid:48) be a morphism and X a (cid:44) −→ Y be a cofibration in H s Set fin P .Denote by i X , i X (cid:48) the respective inclusions into the mapping cylinder M f . Then thefollowing arrows are simple equivalences.(i) X ⊗ ∆ p X −−→ X (ii) M f p (cid:48) X −−→ X (cid:48) (iii) M f ∪ i X (cid:48) ,a Y p X (cid:48) ∪ a −−−−→ X (cid:48) ∪ f,a Y We have illustrated an example of the third in Figure 2.12.Figure 2.12: An example of the simple equivalence in (iii) for P = { , } with the 0-stratum marked in blue and the 1-stratum marked in red. The pushout square is to theleft, the simple equivalence to the right.The first two are an immediate consequence of the results in Proposition 2.2.35 andthe two out of three law. For the third one ... Proof. ..., consider first the pushout square:
Y X (cid:48) ∪ f,g YY ⊗ ∆ X (cid:48) ∪ i ◦ f,a Y ⊗ ∆ i φ . As the left vertical is an FSAE, so is the right. Secondly, consider the pushout square X ⊗ ∆ ∪ g,i Y M f ∪ i X (cid:48) ,a YY ⊗ ∆ X (cid:48) ∪ i ◦ f,a Y ⊗ ∆ ψ HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY g is a cofibration, by Proposition 2.2.33 the left vertical and hence also theright vertical is an FSAE. Now, consider the triangle diagram: M f ∪ i X (cid:48) ,a Y X (cid:48) ∪ f,a YX (cid:48) ∪ i ◦ f,a Y ⊗ ∆ p X (cid:48) ∪ aψ φ . (2.7)It is clearly not commutative in s Set P . However, a left inverse to φ and hence an inversein H s Set fin P is given by the collapsing the cylinder map, X (cid:48) ∪ i ◦ f,a Y ⊗ ∆ → X (cid:48) ∪ f,a Y .The latter fits into a commutative diagram: M f ∪ i X (cid:48) ,a Y X (cid:48) ∪ f,a YX (cid:48) ∪ i ◦ f,g Y ⊗ ∆ p X (cid:48) ∪ aψ . In particular, (2.7) commutes in H s Set fin P . Hence, by the two out of three property, weobtain that M f ∪ i X (cid:48) ,a Y p X (cid:48) ∪ a −−−−→ X (cid:48) ∪ f,a Y is a simple equivalence.We begin with a few statements that are immediate consequences of the isomorphismof categories C (Σ − ) ∼ = H s Set fin P and the results in Section 2.1. For the following, let X ∈ s Set fin P . Remark 2.3.8. τ P induces a one to one correspondence between the simple morphismclasses with source X and A P ( X ). In particular, α : X → Y in H s Set fin P is a simpleequivalence if and only if τ P ( α ) = 0. Due to this, we do not distinguish between elementsof A P ( X ) and simple morphism classes (of morphism with source X ) and for the sakeof notational brevity, we write (cid:104) f (cid:105) instead of τ P ( f ) for the remainder of this subsection.By Lemma 2.1.5 every morphism in H s Set fin P is of the shape [ s ] − ◦ [ a ] where a is acofibration and s is a FSAE. Hence, every simple morphism class in A P ( X ) is of the shape (cid:104) a (cid:105) for some cofibration a : X (cid:44) −→ Y in s Set fin P . By Lemma 2.1.7, two such cofibrationshave the same simple morphism class if and only if they differ up to a deformation. Inparticular, we can think also think of A P ( X ) as cofibrations in s Set fin P with source X modulo deformation.By the same argument as we used for the proof of Lemma 2.1.5 (i.e. stability ofFSAEs under pushout and composition), every deformation between object or morphismcan be replaced by one that is of the shape s : X → S ←− Y : t with s and t FSAEs.
Remark 2.3.9.
As a morphism in a model category is a weak equivalence if and only ifit induces an isomorphism in the homotopy category, every simple equivalence in s
Set fin P is a weak equivalence. In particular, f : X → Y in s Set fin P is a weak equivalence ifand only if τ P ( f ) ends up in W h P ( X ). Hence, we can think of W h P ( X ) as the simplemorphism classes of weak equivalences, with source X or alternatively by Remark 2.3.8as acyclic cofibrations with source X modulo deformation. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proposition 2.3.10.
Let f : X → Y be a morphism in s Set fin P . Then the following areequivalent: (i) f is a simple equivalence. (ii) [ f ] = [ t ] − ◦ [ s ] for s, t finite FSAEs. (iii) τ P ( f ) = 0(iv) X (cid:44) → M f is a simple equivalence.Proof. The first three statements are equivalent by Remark 2.3.8 and Lemma 2.1.6.Clearly, weak equivalences fulfill a two out of three property by definition. Hence, bymapping cylinder factorization and Proposition 2.2.35 (the fact that
Y (cid:44) → M f is anFSAE), we obtain that (i) is equivalent to (iv).Next up, we study the functoriality of A P and of W h P . By a slight abuse of no-tation, we denote any functoriality of the above by a ( − ) ∗ . That is, we use f ∗ for A P ([ f ]) , W h P ([ f ]), for f in s Set fin P and α ∗ for A P ( α ) , W h P ( α ) for α in H s Set fin P . Whatis meant, will usually be obvious from context. So far, we really only have an explicitdescription of α ∗ (cid:104) β (cid:105) after a choice of cofibrations a and b such that [ a ] = α ,[ b ] = β .By Theorem 2.1.3, α ∗ ( (cid:104) β (cid:105) ) is then given by (cid:104) b (cid:48) (cid:105) , where b (cid:48) is a pushout of b along a ins Set P . As is usually the case, for pushouts to interact well with homotopy properties,it suffices for one of the two maps to be a cofibration (and the objects to be cofibrant).This is reflected in Proposition 2.3.12. It is very reminiscent of the setting of homotopypushouts (see for ex. [KP86]), and also reflects in (iii) of Example 2.3.7. Before we provethis proposition, it is useful to know the following. Lemma 2.3.11.
Let X (cid:48) s −→ X be an FSAE in s Set fin P and (cid:104) α (cid:105) ∈ A P ( X ) . Then ( s ∗ ) − (cid:104) α (cid:105) = (cid:104) α ◦ [ s ] (cid:105) Proof.
By Remark 2.3.8, we may without loss of generality assume that α = [ a ], for acofibration a . Now, use Lemma 2.1.9 to obtain: s ∗ (cid:104) a ◦ s (cid:105) = s ∗ (cid:104) s (cid:105) + (cid:104) a (cid:105) = (cid:104) a (cid:105) . In particular, as s ∗ is a bijection this shows the result. Proposition 2.3.12.
Let X ∈ s Set fin P and f, g morphisms in s Set fin P with source f .Consider a pushout square in s Set fin P X X (cid:48)
Y Y (cid:48) fg g (cid:48)
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY and denote by the diagonal by d . If either f or g is a cofibration, then: f ∗ (cid:104) g (cid:105) = (cid:104) g (cid:48) (cid:105) , and (cid:104) f (cid:105) + (cid:104) g (cid:105) = (cid:104) d (cid:105) . Proof.
We start with the case where g is a cofibration. Denote by i X (cid:48) and i X therespective inclusions into M f . Then [ f ] = [ i (cid:48) X ] − ◦ [ i X ]. Now, consider the pushoutcomposition diagram X M f X (cid:48) Y M f ∪ i X ,g X (cid:48) Y (cid:48) i X g p X (cid:48) g (cid:48) (2.8)Then as [ p X (cid:48) ] = [ i X (cid:48) ] − , by Lemma 2.3.11 f ∗ (cid:104) g (cid:105) = (cid:104) X (cid:48) i X (cid:48) (cid:44) −−→ M f (cid:44) → M f ∪ i X ,g Y (cid:105) . In Example 2.3.7 we have shown that since g is a cofibration M f ∪ i X ,g Y → Y (cid:48) = X (cid:48) ∪ g,f Y is a simple morphism. Hence, by commutativity of (2.8), we obtain: f ∗ (cid:104) g (cid:105) = (cid:104) X (cid:48) i (cid:48) X (cid:44) −→ M f (cid:44) → M f ∪ i X ,g Y (cid:105) = (cid:104) X (cid:48) [ p X (cid:48) ] − −−−−→ M f → M f ∪ i X ,g Y → X (cid:48) ∪ g,f Y (cid:105) = (cid:104) g (cid:48) (cid:105) . For the proof of the case where f is a cofibration, again we refer to (2.8) but with theroles of f and g swapped. In this case, the result follows from composability of pushoutsand again the fact that if f is a cofibration, then M g ∪ i X ,f X (cid:48) → Y (cid:48) = Y ∪ f,g X (cid:48) is a simple equivalence (Example 2.3.7). By the same fact, we also have that the diagonalof the left hand square in (2.8) only differs from the diagonal of the rectangle by a simpleequivalence, showing the addition formula.As two immediate corollaries of this and Proposition 2.3.10 we obtain: Corollary 2.3.13.
Simple equivalences are stable under pushout along cofibrations in s Set fin P . Simple equivalences that are also cofibrations are stable under pushouts alongarbitrary morphisms in s Set fin P . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Corollary 2.3.14.
Let f : X → Y and g : Y → Z be morphisms in s Set fin P . Then f ∗ (cid:104) g ◦ f (cid:105) = f ∗ (cid:104) f (cid:105) + (cid:104) g (cid:105) . Proof.
Take a mapping cylinder factorization of f , f = p Y ◦ i X , and let i Y be theinclusion of Y into M f . Then set ˆ g := g ◦ p Y . Now, using Proposition 2.3.12, just as inLemma 2.1.9 one obtains: ( i X ) ∗ (cid:104) ˆ g ◦ i X (cid:105) = ( i X ) ∗ (cid:104) i X (cid:105) + (cid:104) ˆ g (cid:105) . Finally, applying ( p Y ) ∗ = ( i Y ) − ∗ to the last equation gives: f ∗ (cid:104) g ◦ f (cid:105) = f ∗ (cid:104) ˆ g ◦ i X (cid:105) = f ∗ (cid:104) i X (cid:105) + ( p Y ) ∗ (cid:104) ˆ g (cid:105) = f ∗ (cid:104) p Y ◦ i X (cid:105) + ( i Y ) − ∗ (cid:104) g ◦ p Y (cid:105) = f ∗ (cid:104) f (cid:105) + (cid:104) g ◦ p Y ◦ i Y (cid:105) = f ∗ (cid:104) f (cid:105) + (cid:104) g (cid:105) , where the second equality follows from the fact that p Y is a simple equivalence, and thethird equality follows from Lemma 2.3.11. Corollary 2.3.15.
Let X (cid:48) s −→ X be a simple equivalence in s Set fin P and (cid:104) α (cid:105) ∈ A P ( X ) .Then ( s ∗ ) − (cid:104) α (cid:105) = (cid:104) α ◦ [ s ] (cid:105) Proof.
The proof is identical to the one of Lemma 2.3.11, but this time using Corol-lary 2.3.14.The results in Corollary 2.3.13 is of course reminiscent of the interaction of cofi-brations and weak equivalences in a cofibrant model category. This makes on hope foranalogue to the cube lemma (see for example [Hov07, Lem. 5.2.6]). In the classicalsetting, such a result follows from the sum theorem [Coh73, Prop. 23.1]. In the filteredcase one can argue similarly. In fact, we now show the following result:
Proposition 2.3.16.
Consider a commutative cube X Y X Y X Y X Y h h h f hf , (2.9) HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY in s Set fin P , where the left and the right face are cocartesian, the upper morphisms inthese squares are cofibrations and h is a weak homotopy equivalence. Denote by f thediagonal in the left face. Then: (cid:104) h (cid:105) = f ∗ (cid:104) h (cid:105) + f ∗ (cid:104) h (cid:105) − f ∗ (cid:104) h (cid:105) . Proof.
We first reduce to the case where all maps in the left cocartesian face are cofi-brations. Using mapping cylinder factorization on the left vertical and factoring thispushout face we obtain a commutative diagram: X Y X Y M Y M (cid:48) YX Y X Y h h ˜ h s ˜ f =˜ h ˜ f = h f hs (cid:48) , (2.10)where ˜ f , ˜ f are cofibrations, as they are pushouts of cofibrations, and s is a simpleequivalence. By Corollary 2.3.13, s (cid:48) is also a simple equivalence. Denote by ˜ f thediagonal going from X to M (cid:48) . Assuming we have already shown the formula in the twocofibration case, we then obtain: (cid:104) ˜ h (cid:105) = ˜ f ∗ (cid:104) h (cid:105) + ˜ f ∗ (cid:104) ˜ h (cid:105) − ˜ f ∗ (cid:104) h (cid:105) (2.11)Now, by Corollary 2.3.15 and commutativity of the diagram, we have (cid:104) ˜ h (cid:105) = ( s (cid:48) ) − ∗ (cid:104) h (cid:105) ; (cid:104) ˜ h (cid:105) = ( s ) − ∗ (cid:104) h (cid:105) . Together with (2.11) we obtain: (cid:104) h (cid:105) = s (cid:48)∗ ( ˜ f ∗ (cid:104) h (cid:105) + ˜ f ∗ (cid:104) ˜ h (cid:105) − ˜ f ∗ (cid:104) h (cid:105) )= ( s (cid:48) ◦ ˜ f ) ∗ (cid:104) h (cid:105) + ( s (cid:48) ◦ ˜ f ) ∗ (cid:104) ˜ h (cid:105) − ( s (cid:48) ◦ ˜ f ) ∗ (cid:104) h (cid:105) = f ∗ (cid:104) h (cid:105) + f ∗ ( s ∗ (cid:104) ˜ h (cid:105) ) − f ∗ (cid:104) h (cid:105) = f ∗ (cid:104) h (cid:105) + f ∗ (cid:104) h (cid:105) − f ∗ (cid:104) h (cid:105) . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY φ the morphism X (cid:44) → X ∪ X ,i M h = X ∪ X ⊗ ∆ ∪ Y . φ is the bottomhorizontal composition in the commutative diagam: X Y X X ∪ X ⊗ ∆ X ∪ X ⊗ ∆ ∪ Y h i , where the right square is cocartesian and i is a cofibration, due to our assumption thatall the morphisms in the left square are cofibrations. By Proposition 2.2.33 the firstlower horizontal is an FSAE, in particular a weak equivalence. As weak equivalencesare stable under pushout along cofibrations in a cofibrant model category [Hir09, Prop.13.1.3], φ is also a weak equivalence. By Proposition 2.3.12, we have (cid:104) φ (cid:105) = f ∗ (cid:104) h (cid:105) . (2.12)Now, consider the pushout square X ∪ X M h X ∪ X M h X ∪ X M h X ∪ X M h = M hψ ψ , where we denote the diagonal by ψ . As Y → Y is a cofibration, so is ψ . In particular,by Proposition 2.3.12, we obtain (cid:104) ψ (cid:105) = (cid:104) ψ (cid:105) + (cid:104) ψ (cid:105) . (2.13)Denote by η , η , η the respective inclusions X (cid:44) → X ∪ M h , etc. By Proposition 2.3.12 (cid:104) η (cid:105) = (cid:104) h (cid:105) (2.14) (cid:104) η i (cid:105) = f i ∗ (cid:104) h i (cid:105) , for i = 1 , . (2.15)Furthermore, these factor as: X X (cid:44) → X ∪ M h X ∪ M h i φ η i ψ i . with i either 1 , φ ∗ (cid:104) η (cid:105) = φ ∗ (cid:104) φ (cid:105) + (cid:104) ψ (cid:105) (2.16) φ ∗ (cid:104) η i (cid:105) = φ ∗ (cid:104) φ (cid:105) + (cid:104) ψ i (cid:105) , for i = 1 , . (2.17) HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY φ is a weak equivalence, (cid:104) φ (cid:105) has a unique inverse. Combining (2.13), (2.16)and (2.17) we get: φ ∗ (cid:104) η (cid:105) = φ ∗ (cid:104) η (cid:105) + φ ∗ (cid:104) η (cid:105) − φ ∗ (cid:104) φ (cid:105) . As φ is a weak equivalence, φ ∗ is an isomorphism. Hence, (cid:104) η (cid:105) = (cid:104) η (cid:105) + (cid:104) η (cid:105) − (cid:104) φ (cid:105) . Now, insert (2.12), (2.14) and (2.15) into the last formula, to obtain : (cid:104) h (cid:105) = f ∗ (cid:104) h (cid:105) + f ∗ (cid:104) h (cid:105) − f ∗ (cid:104) h (cid:105) . As an immediate consequence of this, we have the classical cube lemma (see [Hov07,Lem. 5.2.6]) and a version of it for simple equivalences.
Corollary 2.3.17.
Consider a cube as in Proposition 2.3.16. Then if h , h and h areweak equivalences, so is h . Further, if h , h and h are simple equivalences then so is h .Proof. By Remark 2.3.9, being a weak equivalence is equivalent to having invertibleWhitehead torsion. As being invertible in a monoid is sustained under monoid mor-phisms and as sums of invertible elements are invertible the first statement follows byProposition 2.3.16. For the second statement, repeat the same argument replacing in-vertible by being 0.This nearly finishes our abstract investigation of the properties of the Whiteheadmonoid and torsion for filtered simplicial sets. We end the section with the followingcharacterization of simple equivalences, summarizing some of the results in this subsec-tion. This connects our results to the axiomatic approach taken in [KP86, Ch. 6, ] aswell as a series of examples instrumental to the remainder of this work.
Proposition 2.3.18.
Denote by S the class of simple equivalences in s Set fin P . Then S isgiven by the smallest class containing all isomorphisms and elementary expansions thatis closed under: (i) The two out of three law; (ii)
Finite coproducts; (iii)
Pushouts of cofibrations in S along arbitrary morphisms in s Set fin P .Proof. First we show that S is in fact closed under these operations. For the two out ofthree law, this is immediate by definition. For finite coproducts, apply Corollary 2.3.17to the case where X and Y are the empty filtered simplicial set. (iii) is part of thestatement of Corollary 2.3.13.To show the converse inclusion, first note that, by Corollary 2.2.23, the class generated in HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY M f → Y for X → Y a morphism in s Set fin P .Hence, again by two out of three, it suffices to show that this class contains all simpleequivalences that are also cofibrations. By Remark 2.3.9 and Remark 2.3.8 such a simpleequivalence a : X → Y fits into a commutative diagram X YS at s in s
Set fin P , where s and t are FSAEs. A final application of two out of three shows thatthen a is contained in the class, as we have already noted that all (finite) FSAEs are.Corollary 2.3.17 together with Proposition 2.3.18 can be used to easily provide auseful range of examples of simple equivalences. Proposition 2.3.19.
Let X ∈ s Set finP . Then both of the last vertex maps sd( X ) −→ X sd P ( X ) −→ X are simple equivalences.Proof. By the fact that both subdivision functors preserve cofibrations and an inductiveuse of Corollary 2.3.17 it suffices to show this for X = ∆ J , for J a d-flag in P . In thecase of sd, Example 2.2.13 (ii) provides a section to the morphism that is an FSAE, hencethe result follows by two out of three. In the case sd P , note that sd P preserves simpleequivalences. To see this, recall that as it has a right adjoint and sustains cofibrations,by Proposition 2.3.18, it suffices to show that it sends elementary expansions into simpleequivalences. But this was already shown in (iv) of Example 2.2.13. Let J (cid:48) be the non-degenerate flag in P that J degenerates from. By (i) of Example 2.2.13, the degeneracymap is a simple equivalence. By the naturality of the last vertex map, we then have acommutative diagram ∆ J ∆ J (cid:48) sd P (∆ J ) sd P (∆ J (cid:48) ) . So again by two out of three, it suffices to show that the last vertex map is a simpleequivalence for the case where J is non-degenerate. But, (iii) of Example 2.2.13 providesa section of the last vertex map that is given by a finite FSAE, completing the proofthrough a final appeal to two out of three.As an immediate corollary of this result we obtain by an application of two out ofthree: HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Corollary 2.3.20.
Both functors s Set finP sd −→ s Set finP ;s Set finP sd P −−→ s Set finP preserve simple equivalences.
If one is entirely concerned with showing the last two statements in the weaker settingof weak equivalences, this can of course be done, just as in the classical setting and thereis no restriction to finite simplicial sets necessary. Finally, we obtain an alternativedescription of the Whitehead torsion of a morphism in H s Set finP . Corollary 2.3.21.
Let α : X → Y be a morphism in H s Set finP and n ∈ N and f : sd nP ( X ) → Y such that α = [ f ] ◦ [l.v. nP ] − as in Proposition 1.1.46. Then τ P ( α ) = (l.v. nP ) ∗ (cid:0) τ P ( f ) (cid:1) . Proof.
This is immediate from Proposition 2.3.19 and Corollary 2.3.15.
Recall the following classical result.
Proposition 2.3.22. [Coh73, Prop. 7.2] Every finite CW-complex has the simple ho-motopy type of a finite simplicial complex (of the same dimension).
In this subsection, we prove a filtered analogue of this proposition in the setting ofs
Set . Recall that a filtered ordered simplicial complex or short FOS-complex is equiv-alently a filtered simplicial-set, such that every non-degenerate simplex is uniquely de-termined by the set of its vertices (see Lemma 1.3.36).
Theorem 2.3.23.
Every finite filtered simplicial set X ∈ s Set P has the simple homotopytype of a finite filtered ordered simplicial complex (over P ) of the same dimension. This will be particularly useful later on, when we are going to describe the connectionbetween the topological and the simplicial filtered homotopy categories. In practice, itallows us, to reduce to the setting of FOS-complexes and make use of a simplicial ap-proximation theorem. To show this result, we first begin with some generalities onthe relationship between FOS-complexes and filtered simplicial sets. Recall that a non-singular simplicial set is a simplicial set S such that each of its non-degenerate simplices∆ n → S is given by an inclusion of simplicial sets. A P -filtered simplicial set is called non-singular if its underlying simplicial set is non-singular.Clearly, every FOS-complex is a filtered non-singular filtered simplicial set. Further-more, in a certain sense, these properties improve under subdivision. Consider, forexample, the following simplicial model of a singular S . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY S filtered over P = { , } , with the standard colorscheme. The left hand side is a non-singular filtered simplicial set. After subdividingonce, we obtain an FOS-complex.This type of behavior is the content of the following lemma. Lemma 2.3.24.
Let K ∈ s Set P be a non-singular filtered simplicial set. Then sd( K ) is an FOS-complex.Proof. This has nothing to do with the filtrations, so we might just as well prove the non-filtered case. Every simplex of sd( K ) is given by a pair (∆ n τ −→ K, σ ) with σ ∈ sd(∆ n ).That is, it is given by ∆ k σ −→ sd(∆ n ) sd( τ ) −−−→ sd( K ) . Furthermore, by choosing n minimally, we may assume that σ = ( σ ⊂ ... ⊂ σ k ) fulfills σ k = [ n ], as otherwise, the simplex comes from the boundary of sd(∆ n ) contradictingminimality. Under the minimality assumption, a simplex is then uniquely determined bysuch a pair. In particular, the vertices of sd( K ) are then given by pairs (∆ n τ −→ K, [ n ]),where τ is non-degenerate. In other words, they correspond to the non-degeneratesimplices of K . The vertex set, for a simplex (∆ n τ −→ K, σ ) as above, is then given by { ∆ σ i σ i −→ ∆ n τ −→ K | i ∈ [ k ] } . This set is ordered by the containment relation of the σ i . By the minimality assumption,its maximal element always corresponds to τ . Hence, for two non-degenerate simplicesto have the same vertex set, they have to come from representing pairs ( τ, σ ), ( τ (cid:48) , σ (cid:48) )with τ = τ (cid:48) . Now, as τ is an inclusion by assumption, and thus pushing forward withit is injective, this means that the underlying set of the flags σ and σ (cid:48) in [ n ] agree. Asthey are non-degenerate, this implies that σ = σ (cid:48) . Hence, we have shown that everynon-degenerate simplex in sd( K ) is uniquely determined by its vertices, i.e. that K isan FOS-complex.The advantage of non-singular filtered simplicial sets over FOS-complexes is thatthey are stable under certain pushouts in s Set P . Lemma 2.3.25.
Consider a pushout diagram in s Set P HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
L KL (cid:48) K (cid:48) with the upper horizontal and the left vertical (and hence all arrows) cofibrations and K and L (cid:48) (and hence L ) non-singular filtered simplicial sets. Then K (cid:48) is also a non-singularfiltered simplicial set.Proof. This an easy exercise in composability of cofibrations.We now make use of a construction similar to the mapping cylinder of simplicialcomplexes, used in [Whi39]. The essential idea is that by using this mapping cylinderconstruction, the attaching maps of a simplex can be replaced by cofibrations, allowingan application of Lemma 2.3.25.
Construction 2.3.26.
Let K f −→ K (cid:48) be a morphism in s Set P with K and K (cid:48) FOS-complexes. For the remainder of this construction, we think of f as a morphism ofordered simplicial complexes in the classical sense as in Section 1.3.4, that is as a mapon the vertices preserving the ordering and filtration and fulfilling that the image of eversimplex (thought of a a set of vertices) is again a simplex. By the fully faithful embeddingfrom Corollary 1.3.34, a construction in this setting transfers into the setting of simplicialsets. Now, denote by M cxf the subcomplex of the join (recall Construction 1.3.4) of sd( K (cid:48) )and sd( K ) given by M cxf := (cid:110) { σ ⊂ ... ⊂ σ k } (cid:63) { τ k +1 ⊂ ... ⊂ τ k + l } | σ i ⊂ f ( τ k + j ) for all i, j (cid:111) . We have an induced filtration on the vertex set by the respective filtrations of sd( K ) andsd( K (cid:48) ). While such definition on the vertex set will generally not give a filtration for thejoin, since not any two vertices are contained in a common 1-simplex in sd( P ), it doesdefine one on M cxf , as f is stratum preserving. An ordering on M cxf is then given bytaking the two orderings on sd( K ) and sd( K (cid:48) ) and putting vertices in the latter beforeones in the former, whenever they lie in a common simplex. Clearly, M cxf comes withtwo inclusions sd( K (cid:48) ) (cid:44) → M cxf ← (cid:45) sd( K ) . Furthermore, the former inclusion comes with a retract M cxf −→ sd( K (cid:48) ) { σ ⊂ ... ⊂ σ k } (cid:63) { τ k +1 ⊂ ... ⊂ τ k + l } (cid:55)−→ { σ ⊂ ... ⊂ σ k ⊂ f ( τ k +1 ) , ... ⊂ f ( τ k + l ) } Hence, as for the usual mapping cylinder, one obtains a commutative mapping cylindersplitting diagram: sd( K ) M cxf sd( K (cid:48) ) sd( f ) . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY K and K (cid:48) are both of dimension lower than n , then M cxf is ofdimension lower than n + 1. Proposition 2.3.27.
In the setting of Construction 2.3.26, the inclusion K (cid:48) (cid:44) → M cxf isan FSAE. In particular, whenever K and K (cid:48) are finite, the retract M cxf −→ sd( K (cid:48) ) is asimple equivalence.Proof. Again we think of f also as a map of ordered filtered simplicial complexes. UsingProposition 2.2.22 we construct a regular pairing on ( M cxf ) n.d. \ ( K (cid:48) ) n.d. . Denote B := M cxf and A := K (cid:48) . We say that a non-degenerate simplex in B n.d. \ A n.d. , { σ (cid:40) ... (cid:40) σ k } (cid:63) { τ k +1 (cid:40) ... (cid:40) τ k + l ] is of type I if { σ , ..., σ k } (cid:54) = ∅ and σ k = f ( τ k +1 ). Note that aswe assume the simplex not to lie in A , such a τ k +1 always exists. Else, we say it is oftype II . Then T : B II −→ B I { σ (cid:40) ... (cid:40) σ k } (cid:63) { τ k +1 (cid:40) ... (cid:40) τ k + l } (cid:55)−→ { σ (cid:40) ... (cid:40) σ k (cid:40) f ( τ k +1 ) } (cid:63) { τ k +1 (cid:40) ... (cid:40) τ k + l } As f is stratum preserving, this in fact defines a proper pairing. To see that it is regular,we use Lemma 2.2.17 and set Φ : B II −→ N { σ (cid:40) ... (cid:40) σ k } (cid:63) { τ k +1 (cid:40) ... (cid:40) τ k + l } (cid:55)−→ l. It is an easy verification very much similar to the one in the proofs of Example 2.2.13that this gives a map such that σ ≺ σ (cid:48) = ⇒ f ( σ ) < f ( σ (cid:48) ) . Using the mapping cylinder splitting from Construction 2.3.26 we obtain from thisresult:
Corollary 2.3.28.
Let K f −→ K (cid:48) be a morphism of finite FOS-complexes in s Set P . Then sd( f ) factors into a cofibration of finite FOS-complexes followed by a simple equivalenceof FOS-complexes. We are now in shape to prove Theorem 2.3.23.
Proof of Theorem 2.3.23.
We prove the theorem via induction over the dimension of X .When X is 0-dimensional the result is trivial, as every 0-dimensional filtered simplicialset is also an FOS-complex. Now, if X is ( n + 1)-dimensional, then it it fits into apushout diagram (cid:70) ∂ ∆ J i (cid:70) ∆ J i ˆ X X (2.18)
HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY X is the n -skeleton of X with the induced filtration and the disjoint unions(together with their induced maps) are given by the non-degenerate n + 1-simplicesof X . By the induction hypothesis, ˆ X has the simple homotopy type of a finite n -dimensional FOS-complex K . By Proposition 1.1.46 this means that there is a k ∈ N and a simple equivalence sd kP ( ˆ X ) a −→ K . Now, as sd P sustains colimits and cofibrations,this gives us a composition of pushout diagrams (cid:70) sd kP (cid:0) ∂ ∆ J i (cid:1) (cid:70) sd kP (cid:0) ∆ J i (cid:1) sd kP ( ˆ X ) sd kP ( X ) K ˜ X. a ˜ a By Proposition 2.3.19, X is simple homotopy equivalent to sd kP ( X ). Furthermore, byCorollary 2.3.13, ˜ a is also a simple equivalence. Hence, it suffices to show that ˜ X hasthe simple homotopy type of a finite n + 1-dimensional FOS-complex. By applyingthe mapping cylinder factorization of Corollary 2.3.28 to (cid:70) sd kP (cid:0) ∂ ∆ J i (cid:1) → K , we againobtain a new pushout composition diagramsd (cid:16) (cid:70) sd kP (cid:0) ∂ ∆ J i (cid:1)(cid:17) sd (cid:16) (cid:70) sd kP (cid:0) ∆ J i (cid:1)(cid:17) ˜ K ˜ K (cid:48) sd( K ) sd( ˜ X ) , where ˜ K is a finite FOS-complex of dimension n + 1 (as it is given by the simplicialmapping cylinder from Construction 2.3.26), the first left vertical is a cofibration and˜ K → sd ( K ) is a simple equivalence. Thus, again by Corollary 2.3.13, ˜ K (cid:48) → sd( ˜ X ) is asimple equivalence. As the latter target is simply equivalent to ˜ X by Proposition 2.3.19,it suffices to show that ˜ K (cid:48) has the homotopy type of a finite ( n + 1)-dimensional FOS-complex. But by Lemma 2.3.25, ˜ K (cid:48) is a finite ( n + 1)-dimensional non-singular filteredsimplicial set. Hence, by one last appeal to Proposition 2.3.19, this is simply equivalentto the finite ( n + 1)-dimensional FOS-complex sd( ˜ K (cid:48) ), where we used Lemma 2.3.24 tosee that this is in fact an FOS-complex. So far our Whitehead group is defined as a weak equivalence invariant of finite filteredsimplicial sets and the Whitehead torsion as an invariant of morphisms in H s Set finP . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY H CW ,is equivalent to the homotopy category of topological spaces H Top (with respect to theQuillen Model Structure). In particular, one obtains an equivalence of categories fromthe full subcategory of H CW fin given by finite CW-complexes to the full subcategory of H Top given by topological spaces, weakly equivalent to a finite CW-complex. One canthen think of computing the Whitehead group and Whitehead torsion in the topologicalsetting, as choosing an inverse to this equivalence (i.e. for each space a weakly equivalentCW-complex) and then composing this equivalence with the Whitehead group definedon CW-complexes. Of course, while the isomorphism type of the Whitehead groups doesnot depend on this choice, the Whitehead torsion still does. That is, for this construc-tion to be well-defined, one needs to keep track of the choices of (weakly equivalent)CW-structure one makes.Now, to define simple homotopy theory in the filtered topological setting, we couldtake the approach of taking an inverse (up to isomorphism) to the restriction of the fullyfaithful functor | − | P : H s Set finP → H
Top P to its essential image (see Theorem 1.4.1),and then composing it with the Whitehead group functor. Note however that while thiswould make the Whitehead group well-defined up to natural isomorphism, the White-head torsion of a map would still depend on a choice of inverse. In fact, it would evendepend on the choice of inverse whether the Whitehead torsion is 0, i.e. whether a filteredmap is a simple equivalence, in some sense. This problem of course already occurs in theclassical setting. To make the Whitehead torsion well-defined, one needs to first makea choice of CW-structure (up to homotopy) for all spaces involved. This problem cansomewhat be amended using the fact that every homeomorphism is a simple equivalence(see [Coh73, App. Main Theorem]), thus giving a notion of simple equivalence, onlydepending on the homeomorphism type of the spaces involved. However, as we have nosuch result in the filtered setting at hand, a priori, the construction of Whitehead torsionalways depends on such choices. We thus restrict our-self to the setting, of realizationsof filtered simplicial sets. A generalization to the “up to weak equivalence” setting isstraightforward.Denote by | s Set finP | the category with objects given by the objects in s Set P and mor- HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY | s Set finP | ( X, Y ) = Hom
Top P ( | X | P , | Y | P ) . We call this the category of finitely triangulated spaces, filtered over P . Further, wedenote by H| s Set finP | the category with the same objects and morphisms given byHom H| s Set finP | ( X, Y ) = Hom H Top P ( | X | P , | Y | P ) . Both, by definition, embed fully faithfully into
Top P and H Top P respectively. ByTheorem 1.4.1, we immediately obtain: Corollary 2.4.1.
The realization functor | − | P : H s Set finP −→ H
Top P induces an isomorphism of categories H s Set finP ∼ −→ H| s Set P | fin . In particular, every morphism φ : | X | P → | Y | P in H| s Set P | is of the shape φ = [ | f | P ] ◦ [ | l.v. nP | P ] − , for sufficiently large n ∈ N and some f : sd nP ( X ) → Y . Such an f as in Corollary 2.4.1 is called a simplicial approximation to φ . Thus, wecan now think of the functors A P and W h P constructed in Section 2.3.1 as functorsbeing defined on H| s Set P | fin . Lemma 2.4.2.
Let φ be a morphism in H| s Set P | fin . Then the following are equivalent. (i) The morphism in H s Set finP corresponding to φ is a simple equivalence. (ii) f is a simple equivalence, where f is any simplicial approximation of φ .Proof. This is an immediate consequence of Theorem 2.3.23, Proposition 2.3.19 and thetwo out of three property for simple equivalences in H s Set finP .Using the isomorphism of categories H s Set finP ∼ = H| s Set finP | we can now transfermost of the nomenclature from the setting of stratum preserving simplicial maps to thesetting of stratum preserving maps of the underlying realizations. Definition 2.4.3. • A morphism φ in H| s Set fin P | is called a simple equivalence if any of the equivalentcharacterizations in Lemma 2.4.2 is fulfilled. • Two objects in
X, Y ∈ H| s Set finP | are said to have the same simple homotopy type or also to be simply equivalent if there is a simple equivalence φ : X → Y . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY • A morphism ϕ in | s Set fin P | is called a simple equivalence if [ ϕ ] is a simple equiva-lence. • We say two morphisms φ, φ (cid:48) in H| s Set fin P | with the same source have the samesimple morphism class if their corresponding morphism in H s Set finP have the samesimple morphism class. That is, if there is a simple equivalence σ in H| s Set fin P | such that φ = σ ◦ φ (cid:48) . The equivalence class generated by this relation is called the simple morphism class of φ , denoted by (cid:104) φ (cid:105) . • The simple morphism class of a morphism ϕ in | s Set fin P | is defined to be the simplemorphism class of [ ϕ ], denoted by (cid:104) ϕ (cid:105) . • The
Whitehead torsion , τ P ( φ ), of a morphism φ ∈ H| s Set P | , is defined as theWhitehead torsion of the corresponding morphism in s Set P . • The
Whitehead torsion τ P ( ϕ ) of a morphism ϕ ∈ | s Set P | is defined as τ P ([ ϕ ]).Summarizing results from the previous sections, we then have the following equivalentcharacterization of our Whitehead monoid, group and torsion. Proposition 2.4.4.
Let X ∈ s Set finP . Then there is a commutative diagram of bijections A P ( X ) (cid:110) a | X a (cid:44) −→ Y ∈ s Set finP a cofibration (cid:111) / deformation (cid:110) (cid:104) α (cid:105) | X α −→ Y ∈ H s Set finP (cid:111) (cid:110) (cid:104) f (cid:105) | X f −→ Y ∈ s Set finP (cid:111)(cid:110) (cid:104) φ (cid:105) | X φ −→ Y ∈ H| s Set finP | (cid:111) (cid:110) (cid:104) ϕ (cid:105) | X ϕ −→ Y ∈ | s Set finP | (cid:111) (cid:104)−(cid:105)|−| P |−| P .Furthermore, they restrict to bijections with W h P ( X ) if one adds the additional conditionof the respective arrows being an isomorphism/weak equivalence respectively. Under theseidentifications, the Whitehead torsion of an arrow ϕ : X → Y ∈ | s Set P | is equivalentlygiven by: τ P ( ϕ ) ... (cid:104) [ f ] ◦ [l.v. nP ] (cid:105) (l.v. nP ) − ∗ ( (cid:104) f (cid:105) ) (cid:104) [ ϕ ] (cid:105) (cid:104) ϕ (cid:105) ,where f : sd nP ( X ) → Y is a simplicial approximation of ϕ . HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proof.
The upper square was already described in detail in Remark 2.3.8. The lowersquare commutes by definition of simple morphism classes. Furthermore, the lower leftvertical is a bijection by Corollary 2.4.1 and the definition of simple equivalences in H| s Set P | . Hence, by commutativity, the lower left horizontal inclusion is onto, makingall the maps involved bijective. The statement on the Whitehead group follows analo-gously to Remark 2.3.9, from the fact that a morphism is a weak equivalence in a modelcategory if and only if its image in the homotopy category is an isomorphism. Finally,the statement on the Whitehead torsion is immediate from Corollary 2.3.21. The obvious question arises how our construction for the Whitehead group of a filteredsimplicial set relates to the classical construction of the Whitehead group of a CW-complex. In this section, we are going to show that our construction can be thoughtof as a generalization from the case where P is a one-point set to the case of arbitrarypartially ordered sets. To be more specific:In the case where P = (cid:63) , s Set P is isomorphic to s Set in the obvious way, and theDouteau model structure is the Kan-Quillen model structure on s
Set . The analogousstatement can be made for
Top (cid:63) , and the model structure on
Top (cid:63) ∼ = Top is just theclassical Kan-Quillen one (on ∆-generated spaces, to be precise). Note that we used“ (cid:63) ” to avoid any ambiguity with pointed spaces. Let H CW fin denote the homotopycategory of finite CW-complexes. By the cellular approximation theorem, H CW fin em-beds fully faithfully into H Top . The realization of a simplicial set naturally carries thestructure of a CW-complex. This induces a fully faithful embedding H s Set fin(cid:63) ∼ = H| s Set fin ∗ | = H| s Set fin | (cid:44) → H CW fin . As every finite CW-complex has the simple homotopy type of a finite simplicial complex(see [Coh73, Prop. 7.2]), this is even an equivalence of categories. Now, consider thefollowing diagram H s Set (cid:63) H CW fin Ab ∼ W h (cid:63)
W h , (2.19)where by W h we refer to the classical Whitehead group functor. In this subsection, weare going to prove the following result. (Note that for some reason, some authors definethe classical Whitehead torsion to live in the Whitehead group of the target space, notthe source. We always mean the corresponding element in the source).
Theorem 2.4.5.
The diagram (2.19) commutes up to a natural isomorphism, uniquelydescribed by the property that, for f ∈ | s Set fin(cid:63) | , τ (cid:63) ( f ) (cid:55)→ τ ( f ) , with τ ( f ) the classical Whitehead torsion. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY K (notnecessarily ordered) is a map e obtained by a pushout in simplicial complexes as belowΛ n ∆ n K K (cid:48) e ,where Λ n (cid:44) → K is an inclusion of a full subcomplex. Note that both conditions, injec-tivity and fullness, are necessary as colimits in the category of simplicial complexes aregenerally rather non-geometrical in their behavior. Remark 2.4.6.
Consider some arbitrary order of K (cid:48) . This induces compatible order-ings on the whole diagram. Then e maps to an elementary expansion in the sense ofDefinition 2.2.4, under the fully faithful inclusion of ordered simplicial complexes intosimplicial sets, S o from Corollary 1.3.34. Conversely, every finite (F)SAE of ordered sim-plicial complexes maps to a composition of elementary expansion of simplicial complexes,under the forgetful functor.For a fixed finite simplicial complex K , one then sets E scx ( K ) = (cid:8) K a (cid:44) −→ K (cid:48) | s.t. K (cid:48) finite and | a | is a homotopy equivalence (cid:9) / ∼ e , where ” ∼ e ” is the equivalence relation induced by composition with elementary expan-sions (and isomorphisms). The equivalence class of 1 K , (cid:104) K (cid:105) , turns this into a pointedset. Remark 2.4.7.
We should say a few words on the matter of why we are not justapplying the theory in Section 2.1. While at first sight it might be very tempting and infact the authors of [Sie70] seem to think it is a straightforward application, the theoryis not applicable. We have already noted in Remark 2.1.4 that pushouts in the categoryof inclusions of CW-complexes do not exits. The same argument holds for simplicialcomplexes. But even if one decides to work with the modified axioms that we adoptedin Section 2.1, one runs into the problem of pushouts, even along cofibrations in thelarger category of all simplicial complexes, being highly ungeometric. For example thepushout of two 1-simplices along their boundary is not a S as one might hope, but againa 1-simplex. To obtain the correct geometrical thing, one needs to either pass to a largercombinatorial category, such as simplicial sets or allow for some notion of subdivision inthe morphisms, i.e. pass to the piecewise linear setting (with the caveat, that even therenot all pushouts do exist). In particular, a priori E scx ( − ), as defined above, neithercarries a group structure, nor is it a functor. We circumvent these shortcoming, throughartificially introducing subdivisions in the next construction. It might be that if onevery carefully substitutes some mapping cylinder arguments through mapping cylindersof simplicial complexes, as used in [Whi39], such a step might not be necessary. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Construction 2.4.8.
Let K be a finite simplicial complex. In (iii) of Example 2.2.13,we have shown that, for any (F)SAE f : K (cid:44) → K (cid:48) , sd( f ) is again a (F)SAE. In particular,by Remark 2.4.6, the induced map of unordered simplicial complexes sd( f ) : sd( K ) (cid:44) → sd( K (cid:48) ) is again a composition of elementary expansions. Hence, sd induces a well-definedmap E scx ( K ) sd −→ E scx (sd( K )) . Clearly, this is a map of pointed sets. Now, we define ˜ E ( K ) as the colimit E scx ( K ) → E scx (sd( K )) → E scx (sd ( K )) → ... → ˜ E ( K ) . (2.20)We make the analogous construction for W h (cid:63) ( X ), for X some ordering of K . We canagain consider the colimit W h (cid:63) ( X ) sd −→ W h (cid:63) (sd( X )) sd −→ W h (cid:63) (sd ( X )) → ... → lim −→ W h (cid:63) (sd n ( X )) . However, as l.v. is a simple equivalence (Proposition 2.3.19) and by its naturality togetherwith the characterization of functoriality of the Whitehead group on simple morphisms(Corollary 2.3.15), the subdivision map is given by l.v. − ∗ . In particular, it is an isomor-phism. Hence, we may identify W h (cid:63) ( X ) with the colimit on the right. Now, under thisidentification, by the universal property of the colimit, we obtain a map of pointed setsΦ (cid:48) : ˜ E ( K ) → W h ∗ ( X ) by mapping (cid:10) sd n ( K ) a (cid:44) −→ K (cid:48) (cid:11) (cid:55)−→ (cid:10) sd n +1 ( K ) = sd n +1 ( X ) sd( a ) (cid:44) −−−→ sd( K (cid:48) ) (cid:11) . Note that as subdivisions of a simplicial complex are naturally ordered, the right handside is in fact a morphism of ordered simplicial complexes, and hence, by the fully faithfullembedding of the latter into simplicial sets, a morphism of simplicial sets. Furthermore,this naturally transfers a group structure onto ˜ E ( K ) as follows. For two inclusions a : sd n ( K ) (cid:44) → K , b : sd m ( K ) (cid:44) → K , first pass to a common degree of subdivision, i.e.for the sake of simplicity we assume n = m = 0. Then, subdivide once and take thepushout in simplicial sets sd( X ) = sd( K ) sd( K )sd( K ) L a b .Denote by d the diagonal. While L might not necessarily be a ordered simplicial complex,by Lemma 2.3.25 and Lemma 2.3.24, sd( L ) in fact is one. Thus, we set[ (cid:104) a (cid:105) ] + [ (cid:104) b (cid:105) ] := [ (cid:104) d (cid:48) (cid:105) ] , where d (cid:48) is the map of unordered simplicial complexes underlying sd( d ). It is easilyverified, using the fact that sd preserves pushouts in s Set and the same argument as inLemma 2.1.8, that this in fact defines an abelian monoid structure on ˜ E ( K ) and thatΦ (cid:48) is a monoid homomorphism. Showing that inverses exist is a little more subtle, butwill follow from the group structure on W h (cid:63) ( X ) in the end. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Lemma 2.4.9.
In the setting of Construction 2.4.8, Φ (cid:48) is onto.Proof. We start by considering
W h ∗ ( X ) from the simple morphism class perspective inProposition 2.4.4. Let (cid:104) α : X → Y (cid:105) be such a simple morphism class in W h (cid:63) ( X ). ByTheorem 2.3.23, we may without loss of generality assume that Y is a ordered simplicialcomplex. By Corollary 2.3.21, for some sufficiently large n and some map f of simplicialsset f : sd n ( X ) → Y , we have l.v. n ∗ (cid:104) f (cid:105) = (cid:104) α (cid:105) . Hence, under the colimit identification of Construction 2.4.8, the two classes agree. Now, f is a map of ordered simplicial complexes. Again, under the colimit identification, (cid:104) f (cid:105) = (cid:104) sd( f ) (cid:105) . The latter however, by Corollary 1.2.16, has the simple morphism class of some inclusionof ordered simplicial complexes (cid:104) f (cid:48) (cid:105) . Now, finally forgetting about the order structure and then subdividing, this clearly liesin the image of Φ (cid:48) .We now construct the natural transformation in Theorem 2.4.5.
Construction 2.4.10.
Clearly, the requirement on Whitehead torsions in Theorem 2.4.5already uniquely determines a map. Now, to see this is well-defined, first note that asin the filtered simplicial set setting, where the Whitehead group can be thought of assimple morphism classes of stratum preserving maps (Proposition 2.4.4), the classicalWhitehead group can also be thought of as homotopy classes of maps of finite CW-complexes with a fixed source, modulo postcomposition with simple equivalences. For asource, see for example [eck, Sec. 6], with the caveat that one needs to use the amendedaxioms of Section 2.1, for this to be correct. As clearly the realization of an elementaryexpansion of simplicial sets gives an elementary expansion of CW-complexes, passingfrom simple morphism classes in the s
Set -setting to simple morphism classes in theCW-setting by realization defines a map
W h (cid:63) ( X ) → W h ( | X | ) that fulfills the torsioncondition (for X ∈ s Set (cid:63) ). To see this in fact defines a natural transformation of groups,note that under the equivalence classes of inclusion of subsets (complexes) (see [Coh73,] for the CW-case), this map simply corresponds to (cid:104) a (cid:105) (cid:55)→ (cid:104)| a |(cid:105) . In particular, as functoriality and addition in both settings is defined via pushouts, and | − | preserves pushouts, this defines a natural transformation of abelian group valuedfunctors Φ :
W h ∗ ( X ) −→ W h ( | X | ) . Lemma 2.4.11. Φ as in Construction 2.4.10 is onto. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proof.
This time, we take the perspective of simple morphism clases of morphisms in H| s Set fin(cid:63) | and H CW fin respectively. For W h (cid:63) , this is given by Proposition 2.4.4.Now, with respect to these identifications, a simple morphism class (cid:104) φ (cid:105) is mapped tothe simple morphism class of the underlying map of topological spaces. In particular,as H| s Set fin(cid:63) | → H CW fin is fully faithful, the simple morphism class of every arrow | X | → T with target T = | Y | , the realization of a finite simplicial set, is met. But byProposition 2.3.22, the fact that every finite CW-complex has the simple homotopy typeof a simplicial complex (and taking some ordering on this complex), and the embeddingof ordered simplicial complexes into simplicial sets (Construction 1.3.32), this is the casefor every simple morphism class in the CW-setting. Lemma 2.4.12.
In the setting of Construction 2.4.8 and Construction 2.4.10, denoteby Φ (cid:48)(cid:48) the composition ˜ E ( K ) Φ (cid:48) −→ W h (cid:63) ( X ) Φ −→ W h ( | X | ) . Then this is explicitly given by (cid:10) sd n ( K ) a (cid:44) −→ K (cid:48) (cid:11) (cid:55)−→ (cid:10) | X | = | K | ∼ −→ | sd n ( K ) | | a | (cid:44) −→ | K (cid:48) | (cid:11) = | l.v. n | ∗ (cid:0) (cid:104)| a |(cid:105) (cid:1) . Proof.
By construction, Φ (cid:48) ( (cid:104) a (cid:105) ) for a as above, is given by(l.v. n +1 ) ∗ ( (cid:104) sd( a ) (cid:105) ) . Hence, under Φ this maps to (cid:0) | l.v. n +1 | (cid:1) ∗ ( | sd( a ) | ). Now, | l.v. n +1 | is the realization of asimple equivalence (Proposition 2.3.19), and in particular a simple equivalence of CW-complexes. Let l be a cellular homotopy inverse to it. Then as | l.v. n +1 | is homotopic tothe sudivision homeomorphism | sd n +1 ( K ) | ∼ −→ | K | , a homotopy inverse to l is homotopicto the inverse of the latter and hence, the latter is also a simple homotopy equivalence.Now, applying the analogue to Corollary 2.3.15 for the cellular Whitehead group (see[Coh73, Prop. 22.4]) we obtain:Φ (cid:48)(cid:48) (cid:0) l.v. n +1 ∗ ( (cid:104) sd( a ) (cid:105) ) (cid:1) = l − ∗ (cid:0) (cid:104)| sd( a ) |(cid:105) (cid:1) = (cid:68) | K | ∼ −→ | sd n +1 ( K ) | | sd( a ) | (cid:44) −−−−→ | sd( L ) | (cid:1)(cid:69) = (cid:68) | K | ∼ −→ | sd n +1 ( K ) | | sd( a ) | (cid:44) −−−−→ | sd( L ) | ∼ −→ | L | (cid:1)(cid:69) = (cid:68) | X | = | K | ∼ −→ | sd n ( K ) | | a | (cid:44) −→ | ( L ) | (cid:69) = | l.v. n | ∗ (cid:0) (cid:104)| a |(cid:105) (cid:1) where the third equality follows again from the fact that the subdivision isomorphismsare simple equivalences, the fourth from their naturality and the final one again from anappeal to the CW-analogue of Corollary 2.3.15.Next, we show Φ is injective. For this, we use a somewhat well known, but seeminglybadly documented result on the relationship between simple equivalences in the simplicialcomplex and the cellular sense. HAPTER 2. SIMPLE STRATIFIED HOMOTOPY THEORY
Proposition 2.4.13.
Let K be a finite simplicial complex. Using the notation fromConstruction 2.4.8 and Lemma 2.4.12, the kernel (in the sense of pointed sets) of themap E scx ( K ) → ˜ E ( K ) Φ (cid:48)(cid:48) −−→ W h ( | K | ) is trivial.Proof. A complete proof would essentially come down to replicating much of what White-head did in [Whi39], in the original simplicial setting. We instead refer there for most ofthe details. Without loss of generality, we may clearly assume | K | to be connected. Inthe proof of [Whi39, Thm. 20] it is effectively shown that an inclusion of finite simplicialcomplexes K (cid:44) → K (cid:48) belongs to (cid:104) K (cid:105) if a certain class element associated to it in thealgebraic Whitehead group of π ( | K | ), W h ( π ( | K | )), disappears. The proof back thenessentially used the same method as in the cellular setting, but without the availability ofthe language of CW-complexes. If one explicitly tracks down the construction there, anduses Lemma 2.4.12, one sees that the element is precisely the one given by first applyingΦ (cid:48)(cid:48) and then using the natural isomorphism W h ( | K | ) ∼ = W h ( π ( | K | )), constructed forexample in [Coh73, 1]. Hence, the statement.As a corollary we obtain (using that l.v. is an isomorphism): Corollary 2.4.14. Φ (cid:48)(cid:48) from Lemma 2.4.12 has trivial kernel. We can now comple the proof of Theorem 2.4.5.
Corollary 2.4.15.
Let Φ be as in Construction 2.4.10, Φ (cid:48) as in Construction 2.4.8 and Φ (cid:48)(cid:48) as in Lemma 2.4.12. Then all of them are isomorphisms of abelian monoids. Inparticular, ˜ E scx ( K ) is an abelian group and Theorem 2.4.5 holds.Proof. This is just putting together what we already know. We haveΦ (cid:48)(cid:48) = Φ ◦ Φ (cid:48) . By Lemma 2.4.9, Φ (cid:48) is onto. By Corollary 2.4.14, Φ (cid:48)(cid:48) and hence also Φ (cid:48) has trivial kernel.In particular, Φ (cid:48) is an map from an abelian monoid into an abelian group that is ontoand has trivial kernel. Such a map is always an isomorphism. Hence, Φ is injective. ByLemma 2.4.11 it is also onto, i.e. also an isomorphism. Thus, finally the same holds forΦ (cid:48)(cid:48) . ibliography [Ban07] Markus Banagl. Topological invariants of stratified spaces . Springer Science &Business Media, 2007.[BH11] Paul Bendich and John Harer. Persistent intersection homology.
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Appendix
A.0.1 A result on pullbacks of colimits
The following result on base changes is useful for local constructions on certain subspacesof realizations of filtered simplicial sets. Recall that limits in the category of ∆-generatedspaces are taken by taking limits in the naive category of topological spaces (i.e. arbitraryones) and then applying k ∆ , i.e. putting the final topology with respect to simplices onthe limit (see [Dug20]). Proposition A.0.1.
Let X ∈ s Set be a locally finite simplicial set, together with a map | X | ϕ X −−→ A in Top . Let B f −→ A be another map in Top . Let | ∆ i | be the diagram, givenby the realizations of the non-degenerate simplices of X . Further, denote by f ∗ ( | X | ) thetotal space of the base change of ϕ X along f and by f ∗ ( | ∆ i | ) the total space of the basechange of | ∆ i | → X → A along f . Consider the commutative diagram induced by theuniversal property of the pullback and colimit: lim −→ ( f ∗ ( | ∆ i | )) f ∗ ( | X | ) B .
Then the horizontal map is a homeomorphism. Furthermore, if f is such that, foreach non-degenerate simplex of X , ∆ i , the fiber product space in the naive categoryof topological space (that is topological spaces that are not necessarily ∆ -generated) of | ∆ i | → | X | → A and f is a ∆ -generated space, then so is the fiber product space inthe naive category of topological space of f and ϕ X , denoted f ∗ ( | X | ) naive . In particular,then, f ∗ ( | X | ) naive = f ∗ ( | X | ) . Proof.
To show that lim −→ ( f ∗ ( | ∆ i | )) −→ f ∗ ( | X | ) is a homeomorphism, it really suffices toshow that it is a quotient map. That it is bijective follows immediately from the fact thatin all of the categories involved, limits and colimits are given by limits and colimits in Set equipped with a topology and in
Set base change commutes with arbitrary colimits. We167
PPENDIX A. APPENDIX −→ ( f ∗ ( | ∆ i | ) naive ) −→ f ∗ ( | X | ) naive is a homeomorphism. Then, as colimits inthe ∆-generated category are computed in the naive topological one and as k ∆ preservescolimits, the result follows. It suffices to show that (cid:71) f ∗ ( | ∆ i | ) naive −→ f ∗ ( | X | ) naive is a quotient map. As pullbacks in naive topological spaces commute with arbitrarycoproducts, this map fits into a commutative diagram in the naive category of topologicalspaces (cid:70) f ∗ ( | ∆ i | ) naive (cid:70) | ∆ i | f ∗ ( | X | ) naive | X | B A ϕ X f with both squares cartesian. Hence, it suffices to show that the quotient map (cid:70) | ∆ i | →| X | stays a quotient map, under pulling back. However, it is a well known fact thatproper maps between locally compact Hausdorff spaces are universally closed (i.e. everypullback of such a map is closed). See for example [Sta20, Tag 005R] together with[Bou66, Ch.9 Prop. 7]. As X is locally finite, this is the case, i.e. the quotient map isproper, and | X | is locally compact Hausdorff. As a surjective, closed map is a quotientmap, this finishes the proof. A.0.2 A basic set theoretic manipulation
Lemma A.0.2.
For arrows in
Set A → B , X → Y , A → A (cid:48) and B (cid:48) = B ∪ A A (cid:48) theinduced diagram B × X ∪ A × X A × Y B × YB (cid:48) × X ∪ A (cid:48) × X A (cid:48) × Y B (cid:48) × Y is a pushout diagram.Proof. Really all we are going to need is the fact that in
Set the product preservescolimits as
Set is a close monoidal category. Now, consider the following composition ofcommutative diagrams. A × Y B × X ∪ A × X A × Y B × YA (cid:48) × Y B (cid:48) × X ∪ A (cid:48) × X A (cid:48) × Y B (cid:48) × Y PPENDIX A. APPENDIX − × Y preserves colimits, the larger outer commutativesquare is a pushout. So it suffices to show that the left hand square is a pushout square.This follows from the natural isomorphisms:( B × X ∪ A × X A × Y ) ∪ A × Y A (cid:48) × Y ∼ = B × X ∪ A × X A (cid:48) × Y ∼ = B × X ∪ A × X ( A (cid:48) × X ∪ A (cid:48) × X A (cid:48) × Y ) ∼ = ( B × X ∪ A × X A (cid:48) × X ) ∪ A (cid:48) × X A (cid:48) × Y ∼ = B (cid:48) × X ∪ A (cid:48) × X A (cid:48) × Y eclaration of AuthorshipEigenst¨andigkeitserkl¨arungeclaration of AuthorshipEigenst¨andigkeitserkl¨arung