aa r X i v : . [ m a t h . A T ] F e b A STRATIFIED KAN-QUILLEN EQUIVALENCE
SYLVAIN DOUTEAU
Abstract.
In this paper, we exhibit a Quillen equivalence between two model categories encodingthe homotopy theory of stratified spaces : the model category of filtered simplicial sets, and thatof filtered spaces. Additionally, we introduce a new class of filtered spaces, that of vertical filteredCW-complexes, providing a nice model for the homotopy category of stratified spaces
Contents
1. Preliminaries and sketch of proof 31.1. Filtered simplicial sets and diagrams 31.2. Filtered and strongly filtered spaces 51.3. Proof of the main theorem and outline of the paper 72. Filtered simplicial sets and subdivisions 72.1. Equivalence between diagrams and filtered simplicial sets 72.2. factoring the subdivision 123. Vertical filtered CW-complexes and the homotopy theory of filtered spaces 173.1. Some results on vertical filtered CW-complexes 173.2. Proving the Quillen equivalence between Top P and Top N ( P ) C be a category, in which we want to invert some class of morphisms W ⊂ C . One can tryto define directly the localized category C [ W − ], but even if it exists, this "homotopy category" isnot the proper setting to do homotopical algebra. Indeed, many of the constructions that mightbe available in C , such as colimits and limits, will not have well-behaved counterparts in C [ W − ].Instead, one would want some notion of derived (co)-limits, invariant with respect to maps in W .For this, one needs more than just the datum of the category C [ W − ].This is the problem that Quillen’s model categories fix [Qui67]. In a model category, in addition topicking a class W of weak-equivalences, one also picks two extra classes of maps, the cofibrations andfibrations. Those classes satisfy axioms that are inspired by the properties of cellular maps, Serrefibrations and weak homotopy equivalences. This leads to a natural construction of a homotopyrelation between morphisms, and the realization of the localized category C [ W − ] as an explicitcategory, usually written Ho( C ), whose objects are a subclass of the objects of C and whose mapsare homotopy classes of maps in C . In this setting, one can define derived functors, homotopy(co)-limits, or further localize with respect to additional morphisms.Moreover, Quillen defined equivalences between model categories, which are now known asQuillen equivalences. All previously mentioned constructions are invariant under Quillen equiv-alences. In particular, Quillen equivalent model categories have equivalent homotopy categories. For this reason, given a particular homotopy category Ho( C ), it is very useful to find a model cat-egory D such that Ho( D ) is equivalent to Ho( C ), but it is even more important to understand theQuillen equivalence class of D . Abstractly, a homotopy theory is a class of Quillen equivalent modelcategories.Among model categories, simplicial model categories play an important role because they grantaccess to simplicial sets of morphisms. These enriched sets of morphisms are derived versions ofthe classical sets of morphisms. They are the main invariants of a homotopy theory, and provide anatural source for many homotopical invariants [DK80].The homotopy category corresponding to the classical homotopy theory of spaces can be de-scribed as the category of CW-complexes and homotopy classes of continuous maps. There are twoimportant model categories corresponding to this homotopy category : the category of simplicialsets, sSet, together with the Kan model structure, and the category of topological spaces, Top,together with the Quillen model structure. Those two model categories are related by a classicaladjunction,(1) || − || : sSet ↔ Top : Singand Quillen showed that it was a Quillen equivalence [Qui67]. This means that both model cate-gories encode the same homotopy theory, that of spaces. In particular, any computation of homo-topy limits and colimits, of homotopy localizations or of simplicial mapping spaces can be done ineither model category. This is especially important since the combinatorial nature of the categoryof simplicial sets grants access to a lot of natural constructions. For example, through the use ofsimplicial (abelian) groups, one can naturally define classifying spaces for groups, or loop spaces.Nowadays, there is a growing interest in understanding the homotopy theory of stratified spaces,see [AFR15, Hai18, NL19, Waa20]. This is in part motivated by the observation that classicaltopological objects can be better understood through the additional data of a stratification (see[Dou19c, Ør20] for some recent work on the subject). Recall that stratified spaces are topologicalspaces, together with the data of some decomposition into strata, usually given as a continuousmap toward a poset, X → P . Historically, they were introduced to deal with manifolds withsingularities, with the goal of extending invariants of manifolds to those objects. This is whatis accomplished by Goresky and MacPherson’s intersection homology [GM80]. Nowadays, finerinvariants of those stratified spaces have been introduced, such as the ( ∞− )category of exit-paths[Tre09, Woo09, Lur]. Those invariants are only well-behaved with respect to stratified maps andstratified homotopy equivalences, which leads to the study of the associated homotopy theory.Following Quillen’s strategy, the author of this paper defined two model categories for spacesstratified over some partially ordered set, P . One for the simplicial side, the model categoryof filtered simplicial sets sSet P [Dou18], which is a simplicial model category, and one for thetopological side, the model category of filtered spaces Top P [Dou19b]. The former is similar to theKan model structure on the category of simplicial sets : its cofibrations are the monomorphisms,its fibrations are characterized by lifting conditions against horns, and its weak-equivalences aredetected by suitably defined homotopy groups. The latter is similar to the Quillen model structureon topological spaces. All objects are fibrant, weak-equivalences are defined as maps inducingisomorphisms on all (filtered) homotopy groups and fibrations are characterized by lifting conditionssimilar to those defining Serre fibrations. Furthermore, the adjunction (1) has an equivalent in thestratified setting :(2) || − || P : sSet P ↔ Top P : Sing P STRATIFIED KAN-QUILLEN EQUIVALENCE 3
Given the similarity between the classical and the stratified context, one might expect this adjunc-tion to be a Quillen equivalence, but it is in fact not even a Quillen adjunction since Sing P doesnot preserve fibrant objects (see [Dou18, Example 4.14]). Nevertheless, it is known that using asuitably defined filtered subdivision, one recovers a Quillen adjunction [Dou19a, Corollaire 8.2.4],(3) || sd P ( − ) || P : sSet P ↔ Top P : Ex P Sing P In this paper, we prove that the Quillen adjunction (3) is a Quillen equivalence.
Thisresult fits into a larger corpus of work, investigating the homotopy theory of stratified spaces anda stratified version of the homotopy hypothesis (see [AFR15, Hai18, NL19, Waa20]). This theoremunites two different points of view on the homotopy theory of stratified spaces. A topological pointof view, carried by the category of filtered spaces, Top P , in which pseudo-manifolds and mapsbetween them live. And a combinatorial point of view, given by the simplicial model categorysSet P . The latter is the natural context in which to interpret the higher invariants of stratifiedspaces, such as the ∞ -category of exit-paths. It should also be noted that this results does notfollow from a straight-forward adaptation of Quillen’s original proof of the equivalence between Topand sSet. Instead, it relies on a comparison with a third model category of diagrams of simplicialsets.The proof of this result is broken down into two parts. One on the simplicial side, the otheron the topological side. Along the way, we prove that several useful adjunctions are also Quillenequivalences. Notably, we fill in a gap in the proof of [Dou19b, Theorem 2.15]. In addition, we alsointroduce vertical filtered CW-complexes, a nice class of stratified spaces, which provides a modelfor the homotopy category of stratified spaces (see Corollary 3.14).1. Preliminaries and sketch of proof
In this section, we recall the needed definitions and theorems from [Dou18] and [Dou19b], andgive a sketch of the proof of the main theorem.1.1.
Filtered simplicial sets and diagrams.Definition 1.1.
Let P be a partially ordered set. A filtered simplicial set over P is the data of asimplicial set X , together with a simplicial map toward the nerve of P , ϕ X : X → N ( P ). A filteredmap between filtered simplicial sets f : ( X, ϕ X ) → ( Y, ϕ Y ) is a simplicial map f : X → Y such thatthe following triangle commutes X YN ( P ) fϕ X ϕ Y The category of filtered simplicial sets over P , sSet P , is the category of filtered simplicial sets over P and filtered maps. Definition 1.2.
A filtered simplex is the data of a simplicial map ϕ : ∆ n → N ( P ). Such a filteredsimplex will usually be denoted ∆ ϕ instead of (∆ n , ϕ ). Alternatively, by identifying ∆ ϕ withthe image of ϕ in N ( P ), we will write ∆ ϕ = [ p , . . . , p n ]. A non-degenerate filtered simplex isthe data of an injective simplicial map ϕ : ∆ n → N ( P ). Alternatively, filtered simplices can beseen as simplices of the simplicial set N ( P ). From this point of view, one can associate to eachfiltered simplex ϕ : ∆ n → N ( P ), the unique non-degenerate simplex of N ( P ), ∆ ¯ ϕ such that ∆ ϕ is SYLVAIN DOUTEAU a degeneracy of ∆ ¯ ϕ . We write ∆( P ) and R ( P ) for the full sub-categories of sSet P , generated bythe filtered simplices and the non-degenerate filtered simplices, respectively. Remark . There is an equivalence of categories sSet P ≃ Fun(∆( P ) op , Set).
Definition 1.4.
Let Diag P be the category of functors :Diag P = Fun( R ( P ) op , sSet)Furthermore, we consider Diag P with its projective model structure, in which weak-equivalencesand fibrations are determined level-wise. It is a cofibrantly generated model category (see [Dou19b,proposition 2.2] for the generating sets of (trivial) cofibrations). Definition 1.5.
Define the bi-functor − ⊗ − : sSet × sSet P → sSet P as K ⊗ ( X, ϕ X ) ( K × X, ϕ X ◦ pr X )If ( X, ϕ X ) ∈ sSet P is fixed, the functor − ⊗ ( X, ϕ X ) : sSet → sSet P admits a right adjointMap(( X, ϕ X ) , − ) : sSet P → sSet defined on objects asMap(( X, ϕ X ) , ( Y, ϕ Y )) n = Hom sSet P (∆ n ⊗ ( X, ϕ X ) , ( Y, ϕ Y ))Recall that sSet P equipped with the functors : −⊗− : sSet × sSet P → sSet P , and Map( − , − ) : sSet opP × sSet P → sSet, is a simplicial category (see [Dou18, section 3]). Definition 1.6.
Define D as the functor D : sSet P → Diag P ( X, ϕ X ) (cid:26) R ( P ) op → sSet∆ ϕ Map(∆ ϕ , ( X, ϕ X ))where Map is the simplicial Hom in sSet P . The functor D admits a left adjoint, C : Diag P → sSet P ,defined as follows. Let C be the full subcategory of R ( P ) op × R ( P ) whose objects are the pairs(∆ ϕ , ∆ ψ ) such that ∆ ψ ⊂ ∆ ϕ . To any functor in Diag P , F : R ( P ) op → sSet, associate the followingfunctor F ⊗ R ( P ) : C → sSet P (∆ ϕ , ∆ ψ ) F (∆ ϕ ) ⊗ ∆ ψ Then, define C ( F ) as the colimit : C ( F ) = colim C F ⊗ R ( P )One can then define a model structure on sSet P ([Dou18, Theorem 3.19]) Theorem 1.7.
The category sSet P admits a simplicial combinatorial model structure where : • the cofibrations are the monomorphisms, • the fibrations are defined by lifting conditions against "admissible horn" • the weak-equivalence are the maps f : X → Y such that D ( f fib ) : D ( X fib ) → D ( Y fib ) is aweak-equivalence in Diag P , where ( − ) fib is any fibrant replacement functor. We will also make heavy use of two endofunctors of sSet P , sd P and Ex P , for which we quicklyrecall the definitions. See [Dou18, section 2.1] for complete definitions. STRATIFIED KAN-QUILLEN EQUIVALENCE 5
Definition 1.8.
Let P be a partially ordered set. Let sd P ( N ( P )) ⊂ sd( N ( P )) × N ( P ) be thesub-simplicial set containing the following simplices :sd P ( N ( P )) = { ((∆ ϕ , . . . , ∆ ϕ k ) , ∆ ψ ) | ∆ ϕ ⊂ · · · ⊂ ∆ ϕ k ⊂ N ( P ) , ψ : ∆ k → N ( P ) , ψ (∆ k ) ⊂ ∆ ϕ } Alternatively, under the identification ∆ ψ = [ p , . . . , p k ] we can describe sd P ( N ( P )) as { ((∆ ϕ , p ) , . . . , (∆ ϕ k , p k )) | ∆ ϕ ⊂ · · · ⊂ ∆ ϕ k ⊂ N ( P ) , p ≤ · · · ≤ p k , p i ∈ ∆ ϕ , ∀ ≤ i ≤ k } This simplicial set comes equiped with the two projections : pr : sd P ( N ( P )) → sd( N ( P )) and pr : sd P ( N ( P )) → N ( P ).Let ( X, ϕ X ) be a filtered simplicial set. Its filtered subdivision sd P ( X, ϕ X ) has for underlyingsimplicial set the following pull-back : sd P ( X, ϕ X ) sd( X )sd P ( N ( P )) sd( N ( P )) sd P ( ϕ X ) sd( ϕ X ) pr The filtration on sd P ( X, ϕ X ) is then given by the compositionsd P ( X, ϕ X ) sd P ( N ( P )) N ( P ) sd P ( ϕ X ) pr This definition extends to a well-defined functorsd P : sSet P → sSet P Furthermore, there is a natural transformation l.v P : sd P → Id . The functor sd P admits a rightadjoint, Ex P , which comes with a natural transformation β : Id → Ex P .1.2. Filtered and strongly filtered spaces.
Throughout this paper, Top denotes the categoryof ∆-generated spaces (see [Dug03], and [FR08]). All topological spaces are assumed to be ∆-generated.
Definition 1.9.
Let P be a partially ordered set. A filtered space over P is the data of • a space, X , • a continuous map ϕ X : X → P , where P is given the Alexandrov topology.A filtered map f : ( X, ϕ X ) → ( Y, ϕ Y ) is a continuous map f : X → Y , such that the followingtriangle commutes X YP fϕ X ϕ Y The category of filtered spaces and filtered maps is denoted Top P . Definition 1.10.
A strongly filtered space over P is the data of • A topological space X , • A continuous map ϕ X : X → || N ( P ) || , where || N ( P ) || is the realization of the simplicial set N ( P ). SYLVAIN DOUTEAU
A strongly filtered map f : ( X, ϕ X ) → ( Y, ϕ Y ) is a continuous map f : X → Y such that thefollowing commutative triangle commutes : X Y || N ( P ) || ϕ X f ϕ Y The category of strongly filtered spaces and strongly filtered maps is denoted Top N ( P ) . Definition 1.11.
Let ϕ P : || N ( P ) || → P be the continuous map, defined by, if t is a pointin the interior of ∆ ψ = [ p , . . . , p k ] ⊂ N ( P ), ϕ P ( t ) = p k . Composition of the filtration with ϕ P : || N ( P ) || → P induces a functor ϕ P ◦ − : Top N ( P ) → Top P ( X, ϕ X : X → || N ( P ) || ) ( X, ϕ P ◦ ϕ X : X → P )A right adjoint to this functor, − × P || N ( P ) || : Top P → Top N ( P ) , is defined on objects as thefollowing pull-back : ( Y, ϕ Y ) × P || N ( P ) || || N ( P ) || ( Y, ϕ Y ) P The strong filtration is then given by projection on the second factor.We then have the expected adjunctions relating (strongly) filtered spaces and filtered simplicialsets.
Definition 1.12.
Let (
X, ϕ X ) be a filtered simplicial set. Define || ( X, ϕ X ) || N ( P ) ∈ Top N ( P ) as || ( X, ϕ X ) || N ( P ) = ( || X || , || ϕ X || : || X || → || N ( P ) || )By defining || − || N ( P ) on maps as the usual realisation, we get a functor || − || N ( P ) : sSet P → Top N ( P ) Define || − || P : sSet P → Top P as the composition ( ϕ P ◦ − ) || − || N ( P ) Proposition 1.13.
The functor || − || N ( P ) admits a right adjoint, Sing N ( P ) : Top N ( P ) → sSet P ,such that, for any strongly filtered space, ( X, ϕ X ), the following diagram is a pullback squareSing N ( P ) ( X, ϕ X ) Sing( X ) N ( P ) Sing( || N ( P ) || ) Sing( ϕ X ) The composition Sing P = Sing N ( P ) ( − × P || N ( P ) || ) is then a right adjoint to || − || P . Definition 1.14.
Define the functors C as the composition C = || C ( − ) || N ( P ) : Diag P → Top N ( P ) ,and D as the composition D = D Sing N ( P ) : Top N ( P ) → Diag P . STRATIFIED KAN-QUILLEN EQUIVALENCE 7
Proof of the main theorem and outline of the paper.
In this paper, we prove thefollowing theorem :
Theorem 1.15.
The adjunction || sd P ( − ) || P : sSet P ↔ Top P : Ex P Sing P is a Quillen-equivalence.Proof. The proof of the main theorem can be summarized by the following diagram.sSet P sSet P Diag P Top N ( P ) Top P sd P Sd P Ex P D ||−|| P ex P C C D ϕ P ◦− −× P || N ( P ) || Sing P In section 2.2, we will construct a pair of adjoint functor Sd P : sSet P ↔ Diag P : ex P such that C ◦ Sd P ≃ sd P , and ex P ◦ D ≃ Ex P (Proposition 2.12). This implies that the functor Ex P Sing P is isomorphic to ex P D Sing P . By [Dou18, Remark 4.16], the latter is isomorphic to ex P ( D ( − × P || N ( P ) || )). In particular, it suffices to show that all three of the adjunctions : (Sd P , ex P ), ( C , D ),and ( ϕ P ◦ − , − × P || N ( P ) || ) are Quillen equivalences. • The adjunction (Sd P , ex P ) is defined and studied in section 2, and by Theorem 2.16, itis a Quillen equivalence. We also show in this section that the adjoint pairs ( C, D ) and(sd P , Ex P ) are Quillen equivalences (Theorem 2.1 and Proposition 2.17) • The adjunction ( C , D ) is a Quillen equivalence by [Dou19b, Theorem 2.12]. • The adjunction ( ϕ P ◦ − , − × P || N ( P ) || ) is studied in section 3. The proof of Theorem 2.15in [Dou19b], stating that it is a Quillen equivalence was missing arguments. To fill the gap,we introduce the notion of vertical filtered CW-complexes in subsection 3.1, and use themto prove that it is indeed a Quillen equivalence (Theorem 3.15). (cid:3) Filtered simplicial sets and subdivisions
In this section, we investigate the adjunctions between the category of filtered simplicial setssSet P and the category of diagrams, Diag P . In subsection 2.1, we first prove that the adjoint pair( C, D ) is a Quillen equivalence (Theorem 2.1). Then, in subsection 2.2, we describe a pair of adjointfunctors (Sd P , ex P ), factoring the adjoint pair (sd P , Ex P ) through ( C, D ) (Propositions 2.11 and2.12), and we show that it is a Quillen equivalence (Theorem 2.16).2.1.
Equivalence between diagrams and filtered simplicial sets.
In this subsection, we provethe following theorem.
Theorem 2.1.
The adjoint pair ( C, D ) is a Quillen equivalence. We will make use the following lemma, from [Dou18].
SYLVAIN DOUTEAU
Lemma 2.2 ([Dou18, Proposition 3.6, Proposition 3.8]) . The pair (
C, D ) is a Quillen adjunction.Furthermore, D reflects weak-equivalences between fibrant objects. Proof of Theorem 2.1.
By [Hov99, Corollary 1.3.16(c)], since D reflects weak-equivalences betweenfibrant objects, it is enough to show that for any cofibrant diagram F , the map F → D (( C ( F )) fib )is a weak-equivalence, where ( − ) fib is any fibrant replacement functor. Since weak-equivalences inDiag P are defined levelwise, one needs to check for all ∆ ϕ ∈ R ( P ) that the map F (∆ ϕ ) → Map(∆ ϕ , ( C ( F )) fib )is a weak-equivalence of simplicial sets. Since F is cofibrant, by lemma 2.3, it is enough to checkthat the map F (∆ ϕ ) → Map(∆ ϕ , ( F (∆ ϕ ) ⊗ ∆ ϕ ) fib )Is a weak-equivalence. Now, let F (∆ ϕ ) → K be some fibrant replacement of F (∆ ϕ ) (say, K =Ex ∞ ( F (∆ ϕ ))), and consider the following commutative diagram F (∆ ϕ ) Map(∆ ϕ , ( F (∆ ϕ ) ⊗ ∆ ϕ ) fib ) K Map(∆ ϕ , ( K ⊗ ∆ ϕ ) fib )By construction, the left arrow is a trivial cofibration. Furthermore, since − ⊗ ∆ ϕ preserves trivialcofibrations, so is F (∆ ϕ ) ⊗ ∆ ϕ → K ⊗ ∆ ϕ . In addition, ( − ) fib also preserves weak-equivalences,and takes value in fibrant objects. Since D preserves weak-equivalences between fibrant objects,the right arrow is also a weak-equivalence. By two out of three, it is then sufficient to show that thebottom arrow is a weak-equivalence. First, notice that K ⊗ ∆ ϕ → ( K ⊗ ∆ ϕ ) fib is a weak-equivalencebetween fibrant objects. Which means that the mapMap(∆ ϕ , K ⊗ ∆ ϕ ) → Map(∆ ϕ , ( K ⊗ ∆ ϕ ) fib )is a weak-equivalence. Since it also factors the bottom arrow, it is sufficient to show that thefollowing map is a weak-equivalence. K → Map(∆ ϕ , K ⊗ ∆ ϕ )But the codomain of this map is isomorphic to Map(∆ | ϕ | , K ) which is homotopy equivalent to K . (cid:3) Lemma 2.3.
Let F be a cofibrant object in Diag P , and ∆ ϕ be a non-degenerated simplex in R ( P ).The map F (∆ ϕ ) ⊗ ∆ ϕ → C ( F ) induces a weak-equivalenceMap(∆ ϕ , ( F (∆ ϕ ) ⊗ ∆ ϕ ) fib ) → Map(∆ ϕ , ( C ( F )) fib )for any fibrant replacement functor ( − ) fib .The proof of Lemma 2.3 will rely on the following definition and lemmas. Definition 2.4.
Let (
X, ϕ X ) be a filtered simplicial set, and ∆ ϕ ∈ R ( P ) a non-degenerate filteredsimplex. We say that a filtered simplex σ : ∆ ψ → ( X, ϕ X ) is of ϕ -type, if ∆ ψ is a degeneracy of∆ ϕ . We define the ϕ -part of X , X ϕ ⊂ X , as the subsimplicial set of X generated by the simplicesof ϕ -type. ( X ϕ , ( ϕ X ) | X ϕ ) is a filtered sub-simplicial set of ( X, ϕ X ), that we will denote ( X, ϕ X ) ϕ .This defines a functor ( − ) ϕ : sSet P → sSet P STRATIFIED KAN-QUILLEN EQUIVALENCE 9
Lemma 2.5.
Let (
X, ϕ X ) be a filtered simplicial set and ∆ ϕ a non-degenerate filtered simplex.Then, the inclusion ( X, ϕ X ) ϕ → ( X, ϕ X ) induces an isomorphismMap(∆ ϕ , ( X, ϕ X ) ϕ ) → Map(∆ ϕ , ( X, ϕ X )) Proof.
Let (
X, ϕ X ) be a filtered simplicial set and ∆ ϕ ∈ R ( P ) a non-degenerate filtered simplex.A simplex in Map(∆ ϕ , ( X, ϕ X )) is a filtered simplicial map σ : ∆ n ⊗ ∆ ϕ → ( X, ϕ X ). Passing to the ϕ -part, we get ( σ ) ϕ : (∆ n ⊗ ∆ ϕ ) ϕ → ( X, ϕ X ) ϕ But any simplex in ∆ n ⊗ ∆ ϕ is a face of a simplex of ϕ -type, which means that (∆ n ⊗ ∆ ϕ ) ϕ =∆ n ⊗ ∆ ϕ . In particular, all simplices of Map(∆ ϕ , ( X, ϕ X )) factor through ( X, ϕ X ) ϕ . (cid:3) Lemma 2.6.
The functor ( − ) ϕ : sSet P → sSet P preserves weak-equivalences and fibrant objects. Proof.
We will prove both assertion separately. First, note that ( − ) ϕ preserves all colimits, whichmeans it is a left adjoint. We will show that it is a left Quillen functor. This will prove the firstclaim, since a left Quillen functor preserves weak-equivalences between cofibrant objects, and allobjects are cofibrant in sSet P . Since sSet P is cofibrantly generated, it is enough to show thatthe generating (trivial) cofibrations are sent to (trivial) cofibrations. Since, ( − ) ϕ clearly preservesmonomorphism, it is true for cofibration. Let Λ ψk → ∆ ψ be an admissible horn inclusion. ∆ ψ mustbe a degenerate simplex of N ( P ), let ∆ ¯ ψ ∈ R ( P ) be the non-degenerate simplex such that ∆ ψ is adegeneracy of ∆ ¯ ψ . Consider the following cases : • If ∆ ϕ ∆ ¯ ψ then no simplex in ∆ ψ is of ϕ -type. In particular, we have (cid:16) Λ ψk → ∆ ψ (cid:17) ϕ = ∅ → ∅ which is a weak-equivalence. • If ∆ ϕ ( ∆ ¯ ψ , then, the top dimensional simplex of ∆ ψ is not of ϕ -type. Furthermore,since Λ ψk → ∆ ψ is admissible, d k (∆ ψ ) is of ¯ ψ -type, which means it is also not of ϕ -type. Inparticular, (Λ ψk ) ϕ = (∆ ψ ) ϕ , and the generating trivial cofibration is sent to an isomorphism. • If ∆ ϕ = ∆ ¯ ψ , then (∆ ψ ) ϕ = ∆ ψ . Furthermore, one has(Λ ψk ) ϕ = a j = kd j ∆ ψ of ϕ -type d j ∆ ψ This is a generalized horn in the sense of definition 2.7, and since Λ ψk is an admissible horn,there exists ǫ = ± ψ ( k ) = ψ ( k + ǫ ). We conclude by lemma 2.8.For the second part of Lemma 2.6, let ( X, ϕ X ) be a fibrant object in sSet P . We need to show that( X, ϕ X ) ϕ is fibrant. By [Dou18, Lemma 1.17], it is enough to show that the map ( X, ϕ X ) ϕ → N ( P )admits the right lifting property against all maps of the form∆ ⊗ ∂ (∆ ψ ) ∪ { ǫ } ⊗ ∆ ψ → ∆ ⊗ ∆ ψ , Where ∆ ψ is any filtered simplex, and { ǫ } is one of the two faces of ∆ . Consider such a liftingproblem :(4) ∆ ⊗ ∂ (∆ ψ ) ∪ { ǫ } ⊗ ∆ ψ ( X, ϕ X ) ϕ ∆ ⊗ ∆ ψ N ( P ) fg Once again, we distinguish three cases, depending on which type of simplex ∆ ψ is. We will write∆ ¯ ψ for the non-degenerate simplex corresponding to ∆ ψ . • If ∆ ¯ ψ ∆ ϕ , then there exists no lifting problem of the form (4). Indeed, by definition of( X, ϕ X ) ϕ , Hom(∆ ψ , ( X, ϕ X ) ϕ ) = ∅ . • If ∆ ¯ ψ = ∆ ϕ , consider the composition of the top map in (4) with the inclusion ( X, ϕ X ) ϕ → ( X, ϕ X ). Since, ( X, ϕ X ) is fibrant, there exists a lift ∆ ⊗ ∆ ψ → ( X, ϕ X ). But ∆ ψ is of ϕ -type, and so are the top dimensional simplices of ∆ ⊗ ∆ ψ , so this lift lands in ( X, ϕ X ) ϕ ,providing a lift in (4). • If ∆ ¯ ψ ( ∆ ϕ , consider the restriction of f to { ǫ } ⊗ ∆ ψ , f |{ ǫ }⊗ ∆ ψ : ∆ ψ → ( X, ϕ X ) ϕ . Bydefinition of ( X, ϕ X ) ϕ , there must exist some simplex σ : ∆ µ → ( X, ϕ X ) such that ∆ µ is of ϕ -type, and the restriction of f is a face of σ . Furthermore, σ lands in ( X, ϕ X ) ϕ . Considerthe modified lifting problem :∆ ⊗ ∂ (∆ ψ ) ∪ { ǫ } ⊗ ∆ µ ( X, ϕ X ) ϕ ( X, ϕ X )∆ ⊗ ∆ µ N ( P ) f ∪ ( { ǫ }⊗ σ ) e g The leftmost arrow is a cofibration, and since its domain and codomain retract to { ǫ } ⊗ ∆ µ ,it is also a stratified homotopy equivalence, which means it is a trivial cofibration. Since( X, ϕ X ) is fibrant, there must exist a lift, e g : ∆ ⊗ ∆ µ → ( X, ϕ X ). Since ∆ µ is of ϕ -type, soare the top dimensional simplices of ∆ ⊗ ∆ µ , which means e g lands in ( X, ϕ X ) ϕ . Restricting e g to ∆ ⊗ ∆ ψ ⊂ ∆ ⊗ ∆ µ gives a lift of (4), g : ∆ ⊗ ∆ ψ → ( X, ϕ X ) ϕ . (cid:3) Definition 2.7.
Let ψ : ∆ n → N ( P ) be a filtered simplex, and S a non-empty, proper subsetof { , . . . , n } . Define the generalized horn Λ ψS as the sub-simplicial set of ∆ ψ generated by thefollowing faces Λ ψS = a j S d j (∆ ψ ) Lemma 2.8.
Let Λ ψS be a generalized horn. If there exists j and ǫ = ± j ∈ S , j + ǫ S and ψ ( j ) = ψ ( j + ǫ ), then the inclusion Λ ψS → ∆ ψ is a trivial cofibration. Proof.
We will prove this statement by induction on the cardinality of S . Let ψ : ∆ n → N ( P ) be afiltered simplex and S ⊂ { , . . . , n } be a non-empty proper subset. Since S is non-empty, we have | S | ≥
1. If | S | = 1, S = { s } and the generalized horn, Λ ψS , is just the horn Λ ψS . By hypothesis,there exists ǫ = ± ψ ( s ) = ψ ( s + ǫ ), which means the horn Λ ψs is an admissible horn, andthe maps Λ ψs → ∆ ψ is a trivial cofibration.Now, assume | S | >
1. By hypothesis, there exists j ∈ S satisfying the hypothesis. Pick a = j ∈ S ,and let A = S \ { a } . The map Λ ψS → ∆ ψ factors through the inclusion Λ ψS → Λ ψA which fits intothe following pushout square d a (∆ ψ ) ∩ Λ ψS Λ ψS d a (∆ ψ ) Λ ψA STRATIFIED KAN-QUILLEN EQUIVALENCE 11
Since | A | = | S | −
1, it is enough to show that Λ ψS → Λ ψA is a trivial cofibration. Since trivialcofibrations are stable under pushout, we need to show that the leftmost map is a trivial cofibration.Notice that d a (∆ ψ ) is isomorphic to the following filtered simplex ψ ′ : ∆ n − → N ( P ) i (cid:26) ψ ( i ) if i < aψ ( i + 1) if i ≥ a Furthermore, under this isomorphism, d a (∆ ψ ) ∩ Λ ψS is isomorphic to Λ ψ ′ B , where B = ( D a ) − ( A )(with D a : { , . . . , n − } → { , . . . , n } the strictly increasing map omitting a ). By construction | B | = | A | < | S | . Furthermore, by setting j ′ = ( D a ) − ( j ) ∈ B , one has j ′ + ǫ = ( D a ) − ( j + ǫ ) ∈ B and ψ ′ ( j ′ ) = ψ ( j ) = ψ ( j + ǫ ) = ψ ′ ( j ′ + ǫ ). In particular, B satisfy the hypothesis of Lemma 2.8, soby the induction hypothesis Λ ψ ′ B → ∆ ψ ′ is a trivial cofibration. (cid:3) Lemma 2.9.
Let F be a cofibrant object in Diag P , and ∆ ϕ ∈ R ( P ) a non-degenerate filteredsimplex. The map, F (∆ ϕ ) ⊗ ∆ ϕ → C ( F )is injective, and its image is ( C ( F )) ϕ . Proof.
Since F is a cofibrant object in Diag P , by [Dou19b, Lemma 2.13], for any non-degeneratefiltered simplex ∆ ϕ ∈ R ( P ), the map i ϕ : F (∆ ϕ ⊗ ∆ ϕ ) → C ( F ) is a monomorphism. Furthermore,by construction, C ( F ) = [ ∆ ϕ ∈ R ( P ) i ϕ ( F (∆ ϕ ) ⊗ ∆ ϕ )Now, given a simplex of ϕ -type, σ : ∆ ψ → C ( F ), this simplex must land in some i µ ( F (∆ µ ) ⊗ ∆ µ ).Furthermore, for such a map to exist, one must have ∆ ϕ ⊂ ∆ µ . But then, if ∆ µ = ∆ ϕ , σ mustland in i µ ( F (∆ µ ) ⊗ ∆ ϕ ), which is identified in C ( F ) with a subsimplicial set of i ϕ ( F (∆ ϕ ) ⊗ ∆ ϕ ). Inparticular, all simplices of ϕ -type in C ( F ) are in i ϕ ( F (∆ ϕ ) ⊗ ∆ ϕ ). Since, conversely, F (∆ ϕ ) ⊗ ∆ ϕ is generated by simplices of ϕ -type, we have ( C ( F )) ϕ = i ϕ ( F (∆ ϕ ) ⊗ ∆ ϕ ), which concludes theproof. (cid:3) Proof of Lemma 2.3.
Let F be a cofibrant object in Diag P , and ∆ ϕ ∈ R ( P ) a non-degeneratefiltered simplex. Consider the following commutative diagram( C ( F )) ϕ C ( F )(( C ( F )) ϕ ) fib ( C ( F )) fib The vertical map are weak-equivalences. Since ( − ) ϕ preserve weak-equivalences by lemma 2.6, thisis also true in the following diagram( C ( F )) ϕ ( C ( F )) ϕ ((( C ( F )) ϕ ) fib ) ϕ (( C ( F )) fib ) ϕ Since the top map is the identity, by two out of three, the bottom map must also be a weak-equivalence. Furthermore, by lemma 2.6, its domain and codomain are fibrant object. Since D isright Quillen, the map Map(∆ ϕ , ((( C ( F )) ϕ ) fib ) ϕ ) → Map(∆ ϕ , (( C ( F )) fib ) ϕ )must be a weak-equivalence. But then, by lemma 2.5, one can rewrite the domain and codomainas follows : Map(∆ ϕ , (( C ( F )) ϕ ) fib ) → Map(∆ ϕ , ( C ( F )) fib )Finally, by lemma 2.9, we have a weak-equivalenceMap(∆ ϕ , ( F (∆ ϕ ⊗ ∆ ϕ )) fib ) → Map(∆ ϕ , ( C ( F )) fib ) (cid:3) factoring the subdivision.Definition 2.10. Let ∆ ϕ ∈ R ( P ) be a non-degenerate filtered simplex. Define the simplicial setSd P ( N ( P ))(∆ ϕ ) as the image of the map(sd P ( N ( P ))) ϕ → sd( N ( P ))Let ( X, ϕ X ) be a filtered simplicial set. Define the simplicial set Sd P ( X, ϕ X )(∆ ϕ ) as the followingpullback Sd P ( X, ϕ X )(∆ ϕ ) sd( X )Sd P ( N ( P ))(∆ ϕ ) sd( N ( P )) sd( ϕ X ) Proposition 2.11.
Let ∆ ψ ⊂ ∆ ϕ be two non-degenerate filtered simplices and ( X, ϕ X ) a filteredsimplicial set. There is an inclusion Sd P ( X, ϕ X )(∆ ϕ ) ⊂ Sd P ( X, ϕ X )(∆ ψ ). In particular, Sd P is afunctor Sd P : sSet P → Diag P furthermore, Sd P has a right adjoint ex P : Diag P → sSet P . Proof.
First, note that Sd P ( N ( P ))(∆ ϕ ) = { ( σ , . . . , σ n ) | ∆ ϕ ⊂ σ } ⊂ sd( N ( P )). Indeed, givensuch a simplex of sd( N ( P )), if ∆ ϕ = [ p , . . . , p k ], then (( σ , p ) , . . . , ( σ , p k ) , ( σ , p k ) , . . . , ( σ n , p k ))is a simplex in (sd P ( N ( P ))) ϕ whose image in sd( N ( P )) is ( σ , . . . , σ n ). In particular, if ∆ ψ ⊂ ∆ ϕ are two non-degenerate filtered simplices, there is an inclusion of subsimplicial sets of sd( N ( P )) :Sd P ( N ( P ))(∆ ϕ ) ⊂ Sd P ( N ( P ))(∆ ψ ) STRATIFIED KAN-QUILLEN EQUIVALENCE 13
Next, let (
X, ϕ X ) be a filtered simplicial set, and consider the following commutative diagram :Sd P ( X, ϕ X )(∆ ψ ) sd( X )Sd P ( X, ϕ X )(∆ ϕ ) Sd P ( N ( P ))(∆ ψ ) sd( N ( P ))Sd P ( N ( P ))(∆ ϕ )By construction, the squares on the front and on the back are pullback squares. This meansthat there is an unique map Sd P ( X, ϕ X )(∆ ϕ ) → Sd P ( X, ϕ X )(∆ ψ ) making the diagram commute.Furthermore, this map make the leftmost square into a pullback square. In particular, it must bea monomorphism. We have shown that Sd P ( X, ϕ X ) : R ( P ) op → sSet is a functor. Furthermore,the functoriality of sd, and of pullbacks, implies that Sd P : sSet P → Diag P is a functor. Sincepullback commute with colimits, Sd P is a colimit-preserving functor between presheaf category,and it admits a right adjoint, that we will write ex P : Diag P → sSet P . (cid:3) Proposition 2.12.
There is an isomorphism of functor C ◦ Sd P ≃ sd P Proof.
Let (
X, ϕ X ) be a filtered simplicial set. By lemma 2.13, sd P ( X, ϕ X ) is naturally isomorphicto the colimit of the functor G : C → sSet P (∆ ϕ , ∆ ψ ) ((sd P ( X, ϕ X )) ϕ ) ψ By lemma 2.15, the functor G is isomorphic to Sd P ( X, ϕ X ) ⊗ R ( P ). Since by definition, C (Sd P ( X, ϕ X )) =colim Sd P ( X, ϕ X ) ⊗ R ( P ), we have a natural isomorphism C (Sd P ( X, ϕ X )) ≃ sd P ( X, ϕ X ) (cid:3) Lemma 2.13.
Let (
X, ϕ X ) be a filtered simplicial set. Define the functor G : C → sSet P (∆ ϕ , ∆ ψ ) (( X, ϕ X ) ϕ ) ψ Then, (
X, ϕ X ) ≃ colim C G . Proof.
Let us first prove that G : C → sSet P is well-defined. If (∆ ϕ , ∆ ψ ) , (∆ ϕ , ∆ ψ ) ∈ C are suchthat ∆ ϕ ⊂ ∆ ϕ and ∆ ψ ⊂ ∆ ψ (that is, if there is a map in C (∆ ϕ , ∆ ψ ) → (∆ ϕ , ∆ ψ )), we have (( X, ϕ X ) ϕ ) ψ ⊂ (( X, ϕ X ) ϕ ) ψ . Indeed, we have the following sequence of inclusion (usinglemma 2.14) ( X, ϕ X ) ϕ ⊂ ( X, ϕ X )(( X, ϕ X ) ϕ ) ϕ ⊂ ( X, ϕ X ) ϕ (( X, ϕ X ) ϕ ) ψ ⊂ (( X, ϕ X ) ϕ ) ψ ( X, ϕ X ) ϕ ) ψ ⊂ (( X, ϕ X ) ϕ ) ψ So the functor G is well-defined. Then, by lemma 2.14, if ∆ ψ ⊂ ∆ ϕ are non-degenerate simplices,(( X, ϕ X ) ϕ ) ψ = ( X, ϕ X ) ϕ ∩ ( X, ϕ X ) ψ . In particular, colim G corresponds to the gluing of the( X, ϕ X ) ϕ along their intersection. In particular, it is a subspace of ( X, ϕ X ), and it is enough toshow that any simplex in X is in some ( X, ϕ X ) ϕ . But if σ : ∆ ψ → ( X, ϕ X ) is a filtered simplex,then σ lands in the ¯ ψ -part of ( X, ϕ X ), where ∆ ψ is a degeneracy of the non-degenerate simplex∆ ¯ ψ , which concludes the proof. (cid:3) Lemma 2.14.
Let (
X, ϕ X ) be a filtered simplicial set, and ∆ ϕ , ∆ ϕ and ∆ ϕ non-degeneratefiltered simplices. We have the following equalities and inclusion.(( X, ϕ X ) ϕ ) ϕ = ( X, ϕ X ) ϕ (( X, ϕ X ) ϕ ) ϕ = ∅ if ∆ ϕ ∆ ϕ (( X, ϕ X ) ϕ ) ϕ = ( X, ϕ X ) ϕ ∩ ( X, ϕ X ) ϕ if ∆ ϕ ⊂ ∆ ϕ (( X, ϕ X ) ϕ ) ϕ ⊂ (( X, ϕ X ) ϕ ) ϕ if ∆ ϕ ⊂ ∆ ϕ ((( X, ϕ X ) ϕ ) ϕ ) ϕ = (( X, ϕ X ) ϕ ) ϕ if ∆ ϕ ⊂ ∆ ϕ ⊂ ∆ ϕ Proof.
The first equality follows from the definition of (
X, ϕ X ) ϕ . Then, if ∆ ϕ ∆ ϕ , ( X, ϕ X ) ϕ contains no filtered simplices of type ϕ , which implies the second equality. Now, let ∆ ϕ ⊂ ∆ ϕ betwo non-degenerate filtered simplices. There are two obvious inclusions (( X, ϕ X ) ϕ ) ϕ ⊂ ( X, ϕ X ) ϕ and ( X, ϕ X ) ϕ , which gives half of the third equality. Conversely, let σ be a simplex in ( X, ϕ X ) ϕ ∩ ( X, ϕ X ) ϕ . Since σ ∈ ( X, ϕ X ) ϕ , there must be a simplex of type ϕ , τ : ∆ ψ → ( X, ϕ X ), such that σ is some face of τ . But since σ is also in ( X, ϕ X ) ϕ , σ must be equal to some face of τ of type ϕ , with ∆ ϕ ⊂ ∆ ϕ ⊂ ∆ ϕ . In particular, σ must be a subface of a face of τ of type ϕ , andso σ ∈ (( X, ϕ X ) ϕ ) ϕ . This same reasoning implies the fourth assertion. The fifth follows fromapplying the third equality to ( X, ϕ X ) ϕ , then using the fourth assertion.((( X, ϕ X ) ϕ ) ϕ ) ϕ = (( X, ϕ X ) ϕ ) ϕ ∩ (( X, ϕ X ) ϕ ) ϕ = (( X, ϕ X ) ϕ ) ϕ (cid:3) Lemma 2.15.
Let (
X, ϕ X ) be a filtered simplicial set and ∆ ϕ a non-degenerate simplex. There isa natural isomorphism, Sd P ( X, ϕ X )(∆ ϕ ) ⊗ ∆ ϕ ≃ (sd P ( X, ϕ X )) ϕ Proof.
Let us first prove the claim for (
X, ϕ X ) = N ( P ). The filtered simplicial set (sd P ( N ( P ))) ϕ contains exactly the simplices of the form(5) (( σ , q ) , . . . , ( σ k , q k )) , ∆ ϕ ⊂ σ , { q , . . . , q k } ⊂ { p , . . . , p n } with ∆ ϕ = [ p , . . . , p n ]. Indeed, let (( σ , q ) , . . . , ( σ k , q k )) be such a simplex. It is enough toconstruct a simplex of sd P ( N ( P )), (( τ , r ) , . . . , ( τ l , r l )) such that for all 0 ≤ i ≤ k there exists STRATIFIED KAN-QUILLEN EQUIVALENCE 15 some 0 ≤ j ≤ l such that ( σ i , q i ) = ( τ j , r j ), and such that { r , . . . , r l } = { p , . . . , p n } . But onecan obtain such a simplex by adding vertices of the form ( σ k , p i ) for every k such that q k < p i and q k +1 > p i , so all simplices of the form (5) are faces of simplices of ϕ -type. Conversely, a simplex of ϕ -type of sd P ( N ( P )) must be of the form (( σ , q ) , . . . , ( σ k , q k )), with { q , . . . , q k } = { p , . . . , p n } .Since it is a simplex of sd P ( N ( P )) one must also have q i ∈ σ for all 0 ≤ i ≤ k , and so ∆ ϕ ⊂ σ .In particular, all faces of a simplex of ϕ -type of sd P ( N ( P )) satisfy the conditions of (5). But (5)also describes the simplices of the product Sd P ( N ( P )) ⊗ ∆ ϕ (see the description of Sd P ( N ( P )) inthe proof of Proposition 2.11), which concludes the proof for the case ( X, ϕ X ) = N ( P ).Let ( X, ϕ X ) be a filtered simplicial set, consider the following commutative diagram(sd P ( X, ϕ X )) ϕ Sd P ( X, ϕ X )(∆ ϕ ) sd( X )(sd P ( N ( P ))) ϕ Sd P ( N ( P ))(∆ ϕ ) sd( N ( P ))By definition of Sd P , the rightmost square is a pullback square. Furthermore, it is enough to showthat the leftmost square is a pullback square since the isomorphism (sd P ( N ( P ))) ϕ ≃ Sd P ( N ( P ))(∆ ϕ ) ⊗ ∆ ϕ will then induce the desired isomorphism. We will show that the outer square is a pullbacksquare. This is true by definition of sd P if one omits the exponent ( − ) ϕ , so it is enough to show thatsd P ( ϕ X ) − ((sd P ( N ( P ))) ϕ ) = (sd P ( X, ϕ X )) ϕ . The inclusion (sd P ( X, ϕ X )) ϕ ⊂ sd P ( ϕ X ) − ((sd P ( N ( P ))) ϕ )follows from the fonctoriality of ( − ) ϕ . On the other hand, fix some simplex in sd P ( ϕ X ) − ((sd P ( N ( P ))) ϕ ).It must be of the form( σ, (( σ , q ) , . . . , ( σ k , q k ))) , σ : ∆ ψ → ( X, ϕ X ) , σ ⊂ · · · ⊂ σ k ⊂ ∆ ψ , q i ∈ σ , ≤ i ≤ k with σ of type µ , with ∆ ϕ ⊂ ∆ µ , and { q , . . . , q k } ⊂ { p , . . . , p n } . Proceeding as before, we canobtain this simplex as the face of some simplex of ϕ -type by adding vertices of the form ( σ k , p i ) forthe p i not equal to any of the q j . In particular, any simplex in sd P ( ϕ X ) − ((sd P ( N ( P ))) ϕ ) is theface of a simplex of ϕ -type of sd P ( X, ϕ X ). This prove that all square in the previous commutativediagram are pullback squares, which completes the proof. (cid:3) Theorem 2.16.
The adjoint pair ( Sd P , ex P ) is a Quillen equivalence.Proof. Let us first prove that it is a Quillen adjunction. It is enough to show that Sd P sends thegenerating (trivial) cofibrations of sSet P to (trivial) cofibrations in Diag P . We will start with thecofibrations. Let ∂ (∆ ψ ) → ∆ ψ be the inclusion of the boundary of some filtered simplex ∆ ψ . Forall ∆ ϕ ∈ R ( P ), define F ϕ to be the diagram defined by R ( P ) op → sSet∆ µ (cid:26) Sd P (∆ ψ )(∆ ϕ ) if ∆ µ ⊂ ∆ ϕ ∅ if ∆ µ ∆ ϕ We will decompose the map Sd P ( ∂ (∆ ψ )) → Sd P (∆ ψ ) as a sequence of cofibrations as follows. First,define F n +1 = Sd P ( ∂ (∆ ψ )), where n is the dimension of the image of ∆ ψ in N ( P ). Then, for any ∆ ϕ ∈ R ( P ), write | ϕ | for its dimension. For 0 ≤ k ≤ n , define F k as the following pushout S | ϕ | = k F ϕ ∩ F k +1 F k +1 S | ϕ | = k F ϕ F k We will show that F ≃ Sd P (∆ ψ ). Since by construction, the latter is covered by the F ϕ , it isenough to show that for any ∆ ϕ = ∆ µ such that | ϕ | = | µ | = k , F ϕ ∩ F µ ⊂ F k +1 . First, if F ϕ ∩ F µ = ∅ , one must have ∆ ϕ ∩ ∆ µ = ∅ , write this simplex ∆ ν . Furthermore, the intersectionmust be generated by F ϕ (∆ ν ) ∩ F µ (∆ ν ). By the proof of proposition 2.11, one sees that allsimplices of this intersection must be of the form ( σ , . . . , σ l ), with ∆ ϕ ⊂ ψ ( σ ) and ∆ µ ⊂ ψ ( σ )Since ∆ ϕ = ∆ µ , there must exist some non-degenerate filtered simplex, ∆ ǫ , containing both ∆ ϕ and ∆ µ , and such that ∆ ǫ ⊂ σ . But then, | ǫ | > k , which means the simplex is in F k +1 . Inparticular, F ≃ Sd P (∆ ψ ), and the map Sd P ( ∂ (∆ ψ )) → Sd P (∆ ψ ) can be obtained by successivepushout along maps of the form F ϕ ∩ F k +1 → F ϕ . In particular, it is enough to show that thoseare cofibrations. But the former diagram is of the form R ( P ) op → sSet∆ µ (cid:26) F ϕ ∩ F k +1 (∆ ϕ ) if ∆ µ ⊂ ∆ ϕ ∅ if ∆ µ ∆ ϕ where F ϕ ∩ F k +1 (∆ ϕ ) is the subset of F ϕ (∆ ϕ ) = Sd P (∆ ψ )(∆ ϕ ), defined by F ϕ ∩ F k +1 (∆ ϕ ) = { ( σ , . . . , σ l ) | (∆ ϕ ′ ⊂ σ , and ∆ ϕ ( ∆ ϕ ′ ) or (∆ ϕ ⊂ σ and σ l = ∆ ψ ) } . In particular, the maps F ϕ ∩ F k +1 → F ϕ are cofibrations (see [Dou19b, Proposition 2.2]), whichconcludes the proof in the case of cofibrations.Now, consider a generating trivial cofibration in sSet P , an admissible horn inclusion Λ ψk → ∆ ψ . We already know that its image by Sd P is a cofibration, (between cofibrant objects) so it isenough to show that it is also a weak equivalence. From [Dou18, Proposition 2.9], we know thatsd P (Λ ψk ) → sd P (∆ ψ ) is a weak-equivalence in sSet P . Furthermore, by Proposition 2.12 it is theimage of the map Sd P (Λ ψk ) → Sd P (∆ ψ ) by the functor C . By Theorem 2.1, C is a left adjoint in aQuillen equivalence, so it reflects weak-equivalences between cofibrant objects, which implies thatSd P (Λ ψk ) → Sd P (∆ ψ ) is a weak-equivalence.It then follows from Theorem 2.1 and Proposition 2.17, and the two out of three properties ofQuillen equivalences among Quillen adjunction that (Sd P , ex P ) is a Quillen equivalence. (cid:3) Proposition 2.17.
The adjoint pair (sd P , Ex P ) is a Quillen equivalence. Proof.
By definition, sd P preserve cofibrations. Furthermore, by [Dou18, Proposition 2.9], it alsopreserves trivial cofibrations, so it is a left Quillen functor. Now, let f : ( X, ϕ X ) → ( Y, ϕ Y ) be some STRATIFIED KAN-QUILLEN EQUIVALENCE 17 map in sSet P . Consider the commutative diagram( X, ϕ X ) ( Y, ϕ Y )sd P ( X, ϕ X ) sd P ( Y, ϕ Y ) f l.v P l.v P sd P ( f ) By [Dou18, Lemma A.3], the vertical maps are weak-equivalences, which means by the two out ofthree property that f is a weak-equivalence if and only if sd P ( f ) is a weak-equivalence. By [Hov99,Corollary 1.3.16(b)], we only need to prove that for any fibrant object in sSet P , ( X, ϕ X ), the co-unit ǫ : sd P (Ex P ( X, ϕ X )) → ( X, ϕ X ) is a weak-equivalence. Consider the commutative diagramsd P ( X, ϕ X ) ( X, ϕ X )sd P (Ex P ( X, ϕ X )) l.v P sd P ( β ) ǫ As we have seen, the top map is a weak-equivalence. Furthermore, since (
X, ϕ X ) is fibrant, by[Dou18, Lemma A.5], the map β : ( X, ϕ X ) → Ex P ( X, ϕ X ) is a weak-equivalence. Since sd P preserve weak-equivalences, sd P ( β ) is also a weak-equivalence, and by two out of three, so is ǫ : sd P (Ex P ( X, ϕ X )) → ( X, ϕ X ). (cid:3) Vertical filtered CW-complexes and the homotopy theory of filtered spaces
The goal of this section is to prove that the Quillen adjunction ϕ P ◦ − : Top N ( P ) ↔ Top P : −× P || N ( P ) || is a Quillen equivalence, thereby fixing a gap in the proof of [Dou19b, Theorem 2.15].In [Dou19b], we constructed model categories Top P and Top N ( P ) related by a Quillen adjunctionand showed that it was a Quillen equivalence. The proof of that last claim was incomplete, andso we give a full proof here. By [Hov99, Corollary 1.3.16(c)], it suffices to show that the unit ofthis adjunction, evaluated at a cofibrant object ( X, ϕ X ) ∈ Top N ( P ) is a weak-equivalence. This isthe content of Theorem 3.15, which is proven in subsection 3.2. In order to prove this theorem,in subsection 3.1, we introduce the notion of vertical filtered CW-complexes, (Definition 3.1). Weshow that any cofibrant object in Top N ( P ) is (strongly) homotopy equivalent to a vertical filteredCW-complex (Proposition 3.13), allowing us to reduce the proof of Theorem 3.15 to such objects.3.1. Some results on vertical filtered CW-complexes.Definition 3.1.
For n ≥
0, and ∆ ϕ ⊂ N ( P ) a non-degenerate filtered simplex, the filtered cell ofdimension ( n, ϕ ) is the strongly filtered space ( B n ⊗ ∆ ϕ , pr ∆ ϕ ), where B n is the ball in dimension n . Its boundary is the subspace S n − ⊗ ∆ ϕ , with the induced filtration. A filtered CW-complex isa strongly filtered space which can be obtained by inductively gluing cells along their boundaries.More precisely. ( X, ϕ X ) is a filtered CW -complex if a skeletal decomposition of X has been chosen: X ⊂ X ⊂ · · · ⊂ X n ⊂ · · · ⊂ ( X, ϕ X )such that : • X = S n ∈ N X n . • X is a disjoint union of 0-cells. (That is, cells of the form ∆ ϕ , for ∆ ϕ ∈ N ( P ) of anydimension). • For all n ≥ X n +1 can be obtained as the following pushout : ` e α S n ⊗ ∆ ϕ α X n ` e α B n +1 ⊗ ∆ ϕ α X n +1 ` χ α where all maps are strongly filtered.A filtered CW-complex is vertical if in addition, for any cell e α , its attaching map χ α : S n α ⊗ ∆ ϕ α → X n α satisfy the following condition : If ( x, t ) ∈ S n α ⊗ ∆ ϕ α , and χ α ( x, t ) lies in the interior of a cell e β of dimension ( m, ∆ ψ ), and has coordinates (( χ α ) ( x, t ) , ( χ α ) ( x, t )) in that cell. then : • ∆ ϕ α ⊂ ∆ ψ • For all s ∈ ∆ ϕ α , χ α ( x, s ) lies in the interior of the cell e β and has coordinates (( χ α ) ( x, t ) , s ).A filtered map f : ( X, ϕ X ) → ( Y, ϕ Y ) between vertical filtered CW-complexes is vertical if forany cell of X , e α , and any cell of Y , e β whose interior intersects f ( e α ), the following conditions aresatisfied : • ∆ ϕ α ⊂ ∆ ϕ β • For ( x, t ) ∈ B n α ⊗ ∆ ϕ α , if f α ( x, t ) = ( y, t ) ∈ e β , then for all s ∈ ∆ ϕ α , f α ( x, s ) = ( y, s ).We write Vert P for the category of vertical filtered CW-complexes over P and vertical filtered maps. Construction 3.2.
To any vertical filtered CW-complex, (
X, ϕ X ), we associate a (non-filtered)CW-complex, L ( X ), in the following way. Consider some attaching map χ α : S n α ⊗ ∆ ϕ α → X n α .Pick a cell, e β of X whose interior intersects the image of χ α , and consider the restriction χ α : χ − α ( e β ) → B n β ⊗ ∆ ϕ β . The verticality hypothesis guarantees that on this restriction, χ α can be decomposed as the productof two maps : • a continuous map L ( χ βα ) from a subset of : S n α to B n β • the strongly filtered inclusion ∆ ϕ α ֒ → ∆ ϕ β We construct a non-filtered CW-complex, L ( X ), by giving it a n -cell, L ( e α ), for each ( n, ∆ ϕ )-cell, e α , of X , and building the attaching maps from the L ( χ βα ). More explicitly, we define L ( X ) byinduction. Set L ( X ) = a e α , dim( e α )=(0 , ∆ ϕα ) B Then, assume that L ( X ) n has been built, and let e α be a cell of dimension ( n + 1 , ∆ ϕ α ). As wehave seen there are well defined maps L ( χ βα ) from subsets of S n to the cells, L ( e β ), of L ( X ) n . Since χ α is an attaching map, the pre-images χ − α ( e β ), for all cells e β ∈ X n , must cover S n ⊗ ∆ ϕ α . Since( X, ϕ X ) is a vertical filtered CW-complex, this implies that the codomains of the L ( χ βα ), must cover S n , and that they coincide on the intersection of the codomains. In turn, they can be glued toproduce an attaching map for L ( e α ), L ( χ α ) : S n → L ( X ) n . We can now define L ( X ) n +1 as the STRATIFIED KAN-QUILLEN EQUIVALENCE 19 following pushout : ` e α , dim( e α )=( n +1 , ∆ ϕα ) S n L ( X ) n ` e α , dim( e α )=( n +1 , ∆ ϕα ) B n +1 L ( X ) n +1 ` L ( χ α ) Furthermore, assume f : ( X, ϕ X ) → ( Y, ϕ Y ) is a vertical filtered map between vertical filtered CW-complexes. Just as earlier, if e α is a cell of X , and e β is a cell of Y whose interior intersects f ( e α ),the verticality condition allows us to decompose a suitable restriction of f as the product of twomaps : • a continuous map L ( f βα ) from a subset of B n α to B n β , • the strongly filtered inclusion ∆ ϕ α ֒ → ∆ ϕ β .First, one can glue the L ( f βα ) for all the cells e β intersecting f ( e α ). This gives a map L ( f α ) : B n α → L ( Y ). One can then glue those L ( f α ) together, to get a map L ( f ) : L ( X ) → L ( Y ).We have just defined a functor from the category of vertical filtered CW-complexes to the categoryof CW-complexes. We will see that, provided we keep the information about the filtered dimensionsof the cells, this functor retains all the information about vertical filtered CW-complexes. To makethis statement precise, we need the following definition. Definition 3.3.
Let K be a CW-complex. A P -labeling of K is the data of a map λ K from theset of cells of K to the set of non-degenerate simplices of N ( P ) such that, if e α intersects theclosure of e β , then λ K ( e β ) ⊂ λ K ( e α ). A label-preserving map between P -labeled CW-complexes f : ( K, λ K ) → ( L, λ L ) is a continuous map f : K → L such that for any cell of K , e α and any cellof L , e β , if f ( e α ) intersects e β , then λ K ( e α ) ⊂ λ L ( e β ). Let CW P be the category of P -labeledCW-complexes and label-preserving maps. Proposition 3.4.
The categories Vert P and CW P are equivalent. Proof.
We will construct both functors in this equivalence. In construction 3.2, we constructeda functor L : Vert P → CW. We need to define a P -labeling on its image. Let ( X, ϕ X ) be avertical filtered CW-complex. The CW-complex L ( X ) contains a cell L ( e α ) for each cell e α of X .If e α is a cell of dimension ( n α , ∆ ϕ α ), we define the labeling on L ( e α ) as λ L ( X ) ( L ( e α )) = ∆ ϕ α .We need to check that the map λ L ( X ) satisfies the condition of being a P -labeling. Assume that L ( e α ) intersects the closure of L ( e β ). This means that L ( e α ) intersects the image of the attachingmap L ( χ β ) : S n β − → L ( X ) n β − . By construction of L ( X ), this implies that e α intersects theimage of the attaching map of e β , χ β : S n β − ⊗ ∆ ϕ β → X n β − . But, by the verticality condition,this implies that ∆ ϕ β ⊂ ∆ ϕ α . The same check at the level of maps shows that the association( X, ϕ X ) → ( L ( X ) , λ L ( X ) ) induces a well-defined functor Vert P → CW P . By abuse of notation,we will still write L for this functor. From now on, L ( X, ϕ X ) denotes the P -labeled CW-complex,while L ( X ) denotes the underlying CW-complex (without labels).We still need to construct the inverse functor. Let ( K, λ K ) be a P -labeled CW-complex. Wewill define a vertical filtered CW-complex ( V ( K ) , ϕ V ( K ) ) with one ( n, ∆ ϕ )-cell for every n -cell of K with label ∆ ϕ . Define the 0-skeleton of V ( K ) as V ( K ) = a e α , dim( e α )=0 λ K ( e α ) . Then, assume that V ( K ) n has been built, and let e α be a ( n + 1)-cell of K . The attaching map χ α : S n → K n lands in cells e β all satisfying λ K ( e α ) ⊂ λ K ( e β ). In addition, for any such e β , thereis a well defined restriction χ βα : χ − α ( e β ) → e β . Since the cell V ( e β ) is in V ( K ) n , and of dimension( n β , λ K ( e β )), we can define a map V ( χ βα ) : χ − α ( e β ) ⊗ λ K ( e α ) → V ( e β ) as the product of the map χ βα , and the inclusion λ K ( e α ) ֒ → λ K ( e β ). Gluing the V ( χ βα ) together, we get a vertical filtered map V ( χ α ) : S n ⊗ λ K ( e α ) → V ( K ) n . We can now define V ( K ) n +1 as the following push-out : ` e α , dim( e α )= n +1 S n ⊗ λ K ( e α ) V ( K ) n ` e α , dim( e α )= n +1 B n +1 ⊗ λ K ( e α ) V ( K ) n +1 ` V ( χ α ) The proof that V sends label-preserving maps to vertical maps is similar. This imply that V : CW P → Vert P is a well-defined functor. But by construction, V ( L ( X, ϕ X )) ≃ ( X, ϕ X ) and L ( V ( K, λ K )) ≃ ( K, λ K ), which concludes the proof. (cid:3) Remark . The equivalence of categories of proposition 3.4 is compatible with homotopies inthe following sense : two vertical maps f, g : (
X, ϕ X ) → ( Y, ϕ Y ) are homotopic through a verticalhomotopy if and only if V ( f ) and V ( g ) are homotopic through a homotopy preserving labels. Thisfollows from the fact that V and L preserve cylinders. Remark . Let (
K, λ K ) be a P -labeled CW-complex, and U ⊂ K a subspace. One can define asubspace V ( U, λ K ) ⊂ V ( K, λ K ) in the following way. For any cell e α ∈ K , consider the subspace( e α ∩ U ) ⊗ λ K ( e α ) ⊂ V ( e α ). The union of those subspaces is the subspace V ( U, λ K ). Furthermore,if we assume U ⊂ K to be open, then V ( U, λ K ) ⊂ V ( K, λ K ) is an open subset. Definition 3.7.
A vertical map between vertical filtered CW-complexes f : ( X, ϕ X ) → ( Y, ϕ Y ) issaid to be cellular if for all cell of X , e α , f ( e α ) ⊂ Y n α .We will make use of the following few lemmas Lemma 3.8.
Let (
X, ϕ X ) be a vertical filtered CW-complex, and χ : S n ⊗ ∆ ϕ → ( X, ϕ X ) be acellular vertical map. Define ( Y, ϕ Y ) = ( X, ϕ X ) ∪ χ ( B n +1 ⊗ ∆ ϕ ) as the following pushout. S n ⊗ ∆ ϕ ( X, ϕ X ) B n +1 ⊗ ∆ ϕ ( Y, ϕ Y ) χ Then, (
Y, ϕ Y ) is a vertical filtered CW-complex, and ( X, ϕ X ) is a subcomplex of ( Y, ϕ Y ). Proof.
By assumption, (
X, ϕ X ) is a vertical filtered CW-complex, which means it admits a skeletaldecomposition : X ⊂ X ⊂ · · · ⊂ X n ⊂ · · · ⊂ X We define the following skeletal decomposition for Y . For k ≤ n , let Y k = X k . Then, for k > n ,let Y k = X k ∪ χ B n +1 ⊗ ∆ ϕ . Since χ is supposed to be cellular, we have Im( χ ) ⊂ X n , and the Y k STRATIFIED KAN-QUILLEN EQUIVALENCE 21 are all well-defined, and contain X k . Furthermore, for k = n , Y k +1 is obtained as the followingpushout ` e α ∈ X dim( e α )=( k +1 , ∆ ϕα ) S k ⊗ ∆ ϕ α Y k ` e α ∈ X dim( e α )=( k +1 , ∆ ϕα ) B k +1 ⊗ ∆ ϕ α Y k +1 ` χ α And, for k = n , Y n +1 can be obtained as the following pushout : S n ⊗ ∆ ϕ ` ` e α ∈ X dim( e α )=( n +1 , ∆ ϕα ) S n ⊗ ∆ ϕ α Y n B n +1 ⊗ ∆ ϕ ` ` e α ∈ X dim( e α )=( n +1 , ∆ ϕα ) B n +1 ⊗ ∆ ϕ α Y n +1 χ ` ` eα χ α This proves that (
Y, ϕ Y ) is a vertical filtered CW-complex. It clearly contains ( X, ϕ X ) as a subspace,and by construction, cells of ( X, ϕ X ) are also cells of ( Y, ϕ Y ), which means that ( X, ϕ X ) is asubcomplex. (cid:3) Lemma 3.9.
Let (
A, ϕ A ) , ( X, ϕ X ) and ( Z, ϕ Z ) be three vertical filtered CW-complexes, and let f : ( A, ϕ A ) → ( Z, ϕ Z ) be a cellular vertical map, and i : ( A, ϕ A ) → ( X, ϕ X ) the inclusion of asubcomplex (that is, all cells of A are cells of X ). Assume ( W, ϕ W ) is the following pushout( A, ϕ A ) ( Z, ϕ Z )( X, ϕ X ) ( W, ϕ W ) fi jg Then, (
W, ϕ W ) admits a structure of vertical filtered CW-complex for which j is the inclusion of asub-complex, and g is a cellular vertical map. Proof.
Consider the filtration of X , A ∪ X ⊂ A ∪ X · · · ⊂ X . For all n ≥
0, we have the push-out ` e α ∈ X \ A dim( e α )=( n +1 , ∆ ϕα ) S n ⊗ ∆ ϕ α A ∪ X n ` e α ∈ X \ A dim( e α )=( n +1 , ∆ ϕα ) B n +1 ⊗ ∆ ϕ α A ∪ X n +1 ` χ α Now define Z ∪ A X as Z ∪ ` e α ∈ X \ A dim( e α )=(0 , ∆ ϕα ) ∆ ϕ α . By construction, there exists a map f : A ∪ X → Z ∪ A X extending f . Now assume that Z ∪ A X n has been constructed, together with a cellularvertical map f n : A ∪ X n → Z ∪ A X n and define Z ∪ A X n +1 as the following push-out ` e α ∈ X \ A dim( e α )=( n +1 , ∆ ϕα ) S n ⊗ ∆ ϕ α A ∪ X n Z ∪ A X n ` e α ∈ X \ A dim( e α )=( n +1 , ∆ ϕα ) B n +1 ⊗ ∆ ϕ α A ∪ X n +1 Z ∪ A X n +1 ` χ α f n f n +1 Since, (
X, ϕ X ) is a filtered vertical CW-complex, the attaching maps χ α must land in the n -skeletonof X . But since f n is cellular, this implies that for any cell of dimension n + 1, the attaching maps f ◦ χ α must land in the n -skeleton of Z ∪ A X n . In particular, by lemma 3.8, Z ∪ A X n +1 is a filteredvertical CW-complex. Furthermore, since by assumption f n was vertical and cellular, and since f n +1 is the identity when restricted to the interior of the newly added cells, the map f n +1 is stillvertical and cellular. We then conclude by lemma 3.10. (cid:3) Lemma 3.10.
Let ( Y n ) n ∈ N be a family of vertical filtered CW-complexes together with maps i n : Y n ֒ → Y n +1 that are inclusions of sub-complexes. Then Y = colim Y n is a vertical filteredCW-complex, and the maps Y n → Y are inclusions of sub-complexes for all n ≥ Proof.
The filtered space Y admits a vertical CW-structure where it has a cell for each cell eventuallyappearing in the sequence Y ⊂ Y ⊂ ... . This structure turns all the Y n into sub-complexes. (cid:3) Proposition 3.11.
Let (
X, ϕ X ) be a filtered simplicial set. The realisation of its subdivision, || sd P ( X, ϕ X ) || N ( P ) admits the structure of a vertical filtered CW-complex. Proof.
Given a filtered simplicial set X , One can consider the filtration by sub-simplicial sets X n ⊂ X generated by simplices of dimension ≤ n . This gives rise to the description of || sd P ( X, ϕ X ) || N ( P ) as the colimit colim n || sd P ( X n , ϕ X n ) || N ( P ) . By lemma 3.10, it suffices to show that || sd P ( X , ϕ X ) || N ( P ) is a vertical filtered CW-complex (this is clear, since it is a disjoint union of points), and that forall n ≥
0, the map || sd P ( X n , ϕ X n ) || N ( P ) → || sd P ( X n +1 , ϕ X n +1 ) || N ( P ) is the inclusion of a sub-complex. Note that for all n ≥ || sd P ( X n +1 , ϕ X n +1 ) || N ( P ) can be described as the followingpush-out : ` σ ∈ X n.d. n +1 || sd P ( ∂ ∆ ϕ σ ) || N ( P ) || sd P ( X n , ϕ X n ) || N ( P ) ` σ ∈ X n.d. n +1 || sd P (∆ ϕ σ ) || N ( P ) || sd P ( X n +1 , ϕ X n +1 ) || N ( P ) ` ( || sd P ( σ ) || N ( P ) ) | ∂ ∆ ϕσ Working by induction and using lemma 3.9, it is enough to show that the top map is cellular andvertical, and that the leftmost map is the inclusion of a sub-complex. Since those properties arepreserved by taking disjoint unions, we have to show that :
STRATIFIED KAN-QUILLEN EQUIVALENCE 23 • For any filtered simplex, ∆ ϕ , || sd P (∆ ϕ ) || N ( P ) is a vertical filtered CW-complex, and || sd P ( ∂ ∆ ϕ ) || N ( P ) →|| sd P (∆ ϕ ) || N ( P ) is the inclusion of a sub-complex. • for any filtered simplex of X , σ : ∆ ϕ σ → X , of dimension n + 1, the map(6) || sd P ( ∂ (∆ ϕ σ )) || N ( P ) → || sd P ( X n , ϕ X n ) || N ( P ) is cellular and vertical.The former is covered in the first part of Lemma 3.12. For the latter, consider a filtered simplexof X , σ : ∆ ϕ σ → X of dimension n + 1, and consider its restriction to a face d i ∆ ϕ σ → X for some0 ≤ i ≤ n + 1. By definition, this restriction is a filtered simplex of dimension n of X , τ : ∆ ϕ τ → X .In particular, it realizes to a map || sd P ( d i ∆ ϕ σ ) || N ( P ) = || sd P (∆ ϕ τ ) || N ( P ) → || sd P ( X n , ϕ X n ) || N ( P ) Which can be assumed to be cellular and vertical, by the induction hypothesis. Since cellular-ity and verticality are local conditions, it suffices to check that they are satisified for all cells of || sd P ( ∂ (∆ ϕ σ )) || N ( P ) to conclude that (6) is cellular and vertical. But all those cells appear in at leastone of the || sd P ( d i ∆ ϕ σ ) || N ( P ) . We deduce that the map (6) is vertical and cellular. Furthermore,the map induced by σ : || sd P (∆ ϕ σ ) || N ( P ) → || sd P ( X n +1 , ϕ X n +1 ) || N ( P ) is the push-out of the vertical and cellular map (6) along a sub-complex inclusion. By lemma 3.9,this implies that it is a vertical and cellular map, concluding the proof. (cid:3) Lemma 3.12.
Let ∆ ϕ be a filtered simplex. || sd P (∆ ϕ ) || N ( P ) admits a structure of vertical filteredCW-complex, such that || sd P ( ∂ ∆ ϕ ) || N ( P ) and || sd P ( d i ∆ ϕ ) || N ( P ) are subcomplexes. Proof.
Throughout this proof, we will identify S n with || ∂ ∆ n +1 || and B n +1 with || ∆ n +1 || . Let ∆ ϕ be a filtered simplex. We will define a vertical filtered CW-complex X , and show that it is filteredhomeomorphic to || sd P (∆ ϕ ) || N ( P ) . Recall that if ∆ ψ is a possibly degenerate filtered simplex, ∆ ψ is the unique non-degenerate simplex of which ∆ ψ is a degeneracy. X will have a cell of dimension( k, ∆ ψ ), e ψ ,...,ψ k , for each strictly increasing chain ∆ ψ ( ∆ ψ ( · · · ( ∆ ψ k ⊂ ∆ ϕ . Define X as X = a ∆ ψ ⊂ ∆ ϕ ∆ ψ Now, let n ≥
0, and assume that X n has been built, with a ( k, ∆ ψ )-cell for every chain ∆ ψ ( ∆ ψ ( · · · ( ∆ ψ k ⊂ ∆ ϕ , for k ≤ n . Consider some chain ∆ ψ ( ∆ ψ ( · · · ( ∆ ψ k +1 ⊂ ∆ ϕ . Weneed an attaching map χ ψ ,...,ψ k +1 : S n ⊗ ∆ ψ → X n to glue the corresponding cell. Identify S n with || ∂ ∆ n +1 || . We can then cut the boundary of the cell e ψ ,...,ψ k +1 into the subspaces || d i ∆ n +1 || ⊗ ∆ ψ , for 0 ≤ i ≤ n + 1. For i = 0, the cells e ψ ,..., b ψ i ,...,ψ k +1 are of dimension ( n, ∆ ψ ). Inparticular, consider the homeomorphism ( D i ) − : || d i ∆ n +1 || → || ∆ n || . Taking the product of thishomeomorphism, with the identity Id : ∆ ψ → ∆ ψ gives part of the attaching map χ iψ ,...,ψ k +1 : || d i ∆ n +1 || ⊗ ∆ ψ ≃ || ∆ n || ⊗ ∆ ψ e ψ ,..., b ψi,...,ψn −−−−−−−−−→ X n In the case i = 0, the cell e ψ ,...,ψ n +1 is of dimension ( n, ψ ), with ∆ ψ ⊆ ∆ ψ . This meansthat we can similarly define a vertical map from || d ∆ n +1 || ⊗ ∆ ψ to the closure of e ψ ,...,ψ n +1 , bytaking the product of ( D ) − with the inclusion ∆ ψ ֒ → ∆ ψ . By construction, on any intersection d i ∆ n +1 ∩ d j ∆ n +1 , the partial attaching maps χ i and χ j coincide, meaning they can be glued to yieldan attaching map χ ψ ,...,ψ n +1 : || ∂ ∆ n +1 || ⊗ ∆ ψ → X n . We can then define X n +1 as the followingpush-out: ` ∆ ψ ( ··· ( ∆ ψn +1 ⊂ ∆ ϕ S n ⊗ ∆ ψ X n ` ∆ ψ ( ··· ( ∆ ψn +1 ⊂ ∆ ϕ B n +1 ⊗ ∆ ψ X n +1 ` χ ψ ,...,ψn +1 It remains to be shown that (
X, ϕ X ) is filtered homeomorphic to || sd P (∆ ϕ ) || N ( P ) . Recall thatsd P (∆ ϕ ) can be defined as the following pullbacksd P (∆ ϕ ) sd(∆ ϕ )sd P ( N ( P )) sd( N ( P )) sd P ( ϕ ) sd( ϕ ) Since the functor || − || N ( P ) preserves pullback, it suffices to show that X fits in a pullback square(7) X || sd(∆ ϕ ) |||| sd P ( N ( P )) || || sd( N ( P )) || fg and that the stratification on X is given by the composition X → || sd P ( N ( P )) || → || N ( P ) || . First,note that by construction, || sd(∆ ϕ ) || is homeomorphic to L ( X ) (as defined in remark 3.2). Inparticular, f will send the cells e ψ ,...,ψ n to the corresponding cells L ( e ψ ,...,ψ n ). And, for any pointin this cell, ( x, t ) ∈ B n ⊗ ∆ ψ , set f ψ ,...,ψ k ( x, t ) = x ∈ L ( e ψ ,...,ψ n ).Now to define g , recall that sd P ( N ( P )) is a simplicial complex with a vertex for each pair (∆ ψ , p ),where ∆ ψ is a non-degenerate filtered simplex and p ∈ ∆ ψ . The higher dimensional simplices ofsd P ( N ( P )) are then given by the tuples { (∆ ψ , p ) , . . . , (∆ ψ n , p n ) } , where ∆ ψ ⊂ · · · ⊂ ∆ ψ n isan increasing chain of non-degenerate simplices, and p i ∈ ∆ ψ , for all 0 ≤ i ≤ n . Considera cell in ( X, ϕ X ), e ψ ,...,ψ n . Topologically, it is a product of two simplices, ∆ n and ∆ ψ . Assuch, one can triangulate it in a canonical way. Label the vertices of ∆ n by the ∆ ψ i and write[ p , . . . , p k ] = ∆ ψ . The vertices of the triangulation of ∆ n ⊗ ∆ ψ will then be given by the pairs(∆ ψ i , p j ), for 0 ≤ i ≤ n , 0 ≤ j ≤ k . Furthermore, simplices of this triangulation will be given bychains ((∆ ψ i , p j ) , . . . , (∆ ψ il , p j l )), where ∆ ψ im ⊂ ∆ ψ im +1 , and p j m ≤ p j m +1 for all 0 ≤ m ≤ l − e ψ ,...,ψ n is then given by sending the vertices ofthe simplicial sets to the corresponding points in the cell, and then extending linearly. One checksthat this also turns the map f defined earlier into a simplicial map. On the other hand, we cannow define g ψ ,...,ψ n : || ∆ n || ⊗ ∆ ψ → || sd P ( N ( P )) || by setting g (∆ ψ i , p j ) = (∆ ψ i , p j ) and extendinglinearly.It remains to be shown that the square (7) is a pullback square. Let us first prove that it iscommutative. Let e ψ ,...,ψ n ∈ X be a cell. With the previous triangulation of the cell the restrictionof f and g to e ψ ,...,ψ n are simplicial. In particular, it is enough to check the commutativity at thelevel of vertices. Given (∆ ψ i , p j ) a vertex of the triangulation, we have f (∆ ψ i , p j ) = ∆ ψ i , and its STRATIFIED KAN-QUILLEN EQUIVALENCE 25 image in sd( N ( P )) is ∆ ψ i . On the other hand, we have g (∆ ψ i , p j ) = (∆ ψ i , p j ), whose image insd( N ( P )) is ∆ ψ i . We conclude that, (7) is commutative. Now, consider a simplex in sd( N ( P )),(∆ µ , . . . , ∆ µ n ), with ∆ µ ( · · · ( ∆ µ n ⊂ N ( P ). Its preimage in sd P ( N ( P )) is isomorphic as asimplicial complex to ∆ n ⊗ ∆ µ . On the other hand, its preimage in sd(∆ ϕ ) is the full sub-simplicialcomplex spanned by the vertices ∆ ψ such that ∆ ψ = ∆ µ i for some 0 ≤ i ≤ n . Finally, its preimagein X is the subspace spanned by all the cells of the form e ψ ,...,ψ k , where for all 0 ≤ i ≤ k , ∆ ψ i = ∆ µ j for some 0 ≤ j ≤ n . But one notes that such a sell e ψ ,...,ψ k ≃ ∆ k ⊗ ∆ ψ is precisely the productof (∆ ψ , . . . , ∆ ψ k ) ⊗ ∆ ψ ⊂ sd P ( N ( P )) with (∆ ψ , . . . , ∆ ψ k ) ⊂ sd(∆ ϕ ), over (∆ µ , . . . , ∆ µ n ). Inparticular, the square (7) is indeed a pull-back square. Furthermore, the stratification on ( X, ϕ X ),which is given on cells of the form || ∆ n || ⊗ ∆ µ as the projection on ∆ µ , clearly factors through g . In particular, ( X, ϕ X ) is a vertical filtered CW structure on || sd P (∆ ϕ ) || N ( P ) . Furthermore,the subspace || sd P ( ∂ ∆ ϕ ) || N ( P ) is spanned by the cells of the form e ψ ,...,ψ n where ∆ ψ n = ∆ ϕ . Inparticular, it is indeed a sub-complex. Similarily, || sd P ( d i ∆ ϕ ) || N ( P ) is the sub-complex containingall cells e ψ ,...,ψ n such that ∆ ψ n ⊂ d i ∆ ϕ . (cid:3) Proposition 3.13.
Let (
X, ϕ X ) be a cofibrant object in Top N ( P ) . There exists a vertical filteredCW-complex, ( Y, ϕ Y ), such that ( X, ϕ X ) and ( Y, ϕ Y ) are filtered homotopy equivalent in Top N ( P ) . Proof.
By proposition 2.12, the adjunction || sd P ( − ) || N ( P ) : sSet P ↔ Top N ( P ) : Ex P Sing N ( P ) is isomorphic to the adjunction ( || C (Sd P ( − )) || N ( P ) , ex P D Sing N ( P ) ), which can be rewritten as( C Sd P , ex P D ). By [Dou19b, Theorem 2.12] and Theorem 2.1, this is a composition of Quillenequivalences, so it must be a Quillen equivalence. But then, since all objects of Top N ( P ) are fibrantand all objects of sSet P are cofibrant, the co-unit || sd P (Ex P Sing N ( P ) ( X, ϕ X )) || N ( P ) → ( X, ϕ X )is a weak-equivalence in Top N ( P ) . By Proposition 3.11 the domain of this map admits the structureof a vertical filtered CW-complex. Furthermore, since both objects are fibrant and cofibrant, themap must be a filtered homotopy equivalence. (cid:3) Corollary 3.14.
Let Vert P / ∼ be the category whose objects are vertical filtered CW-complexesand whose maps are classes of strongly filtered maps up to filtered homotopy. Then there is anequivalence of categories Vert P / ∼ ≃ Ho ( T op N ( P ) ) Proof.
Since Top N ( P ) is a model category and all of its objects are fibrant, there is an equivalenceof categories Cof / ∼ ≃ Ho(Top N ( P ) ), where Cof ⊂ Top N ( P ) is the full subcategory of cofibrantobjects. But, since by proposition 3.13, all cofibrant objects of Top N ( P ) are filtered homotopyequivalent to vertical filtered CW-complexes, one can replace Cof by Vert P . (cid:3) Proving the Quillen equivalence between Top P and Top N ( P ) . In this subsection, weprove the following.
Theorem 3.15.
Let ( X, ϕ X ) be a cofibrant object in Top N ( P ) , the unit (8) ( X, ϕ X ) → ( X, ϕ P ◦ ϕ X ) × P || N ( P ) || is a weak-equivalence in Top N ( P ) . Since the adjunction between Top P and Top N ( P ) preserves filtered homotopy equivalences, byproposition 3.13, it is enough to show the result for vertical filtered CW-complexes. We first maketwo observations : Lemma 3.16.
The map || N ( P ) || N ( P ) → || N ( P ) || P × P || N ( P ) || is a weak-equivalence in Top N ( P ) . Proof. by [Dou19b, Lemma 2.16] the map || N ( P ) || P → P is a trivial fibration in Top P . Since thefunctor − × P ×|| N ( P ) || preserves trivial fibrations, the map || N ( P ) || P × P || N ( P ) || → P × P || N ( P ) || ≃ || N ( P ) || is a trivial fibration in Top N ( P ) . Furhermore, the composition || N ( P ) || → || N ( P ) || P × P || N ( P ) || → || N ( P ) || is the identity. By two out of three, the first map is a weak-equivalence in Top N ( P ) . (cid:3) Lemma 3.17.
Let (
X, ϕ X ) be an object in Top N ( P ) , the following commutative diagram is apull-back square : ( X, ϕ X ) ( X, ϕ P ◦ ϕ X ) × P || N ( P ) |||| N ( P ) || N ( P ) || N ( P ) || P × P || N ( P ) || ( Id X ,ϕ X ) ϕ X ( ϕ X ,Id )( Id,Id ) Proof.
This follows from a direct calculation of the pull-back. (cid:3)
Those two observations imply the following lemma.
Lemma 3.18.
Let X be some topological space and ∆ ϕ a non-degenerate filtered simplex. Then,the map X ⊗ || ∆ ϕ || N ( P ) → ( X ⊗ || ∆ ϕ || P ) × P || N ( P ) || is a weak-equivalence in Top N ( P ) . Proof.
Write ∆ ϕ = [ p , . . . , p n ], and write Q = { p < · · · < p n } ⊂ P for the corresponding linearsub-poset. Then || ∆ ϕ || = || N ( Q ) || . Furthermore, by lemma 3.17 we have the following pullbacksquare in the category Top N ( Q ) X ⊗ || ∆ ϕ || N ( Q ) ( X ⊗ || ∆ ϕ || Q ) × Q || N ( Q ) |||| N ( Q ) || N ( Q ) || N ( Q ) || Q × Q || N ( Q ) || pr ∆ ϕ ( Id,Id ) We know by lemma 3.16 that the bottom map is a weak-equivalence in Top N ( Q ) . Since Top N ( Q ) is a right proper model category, it is enough to show that the right map is a fibration in Top N ( Q ) to deduce that the top map is a weak-equivalence in Top N ( Q ) . But the map pr ∆ ϕ : X ⊗ || ∆ ϕ || Q →|| N ( Q ) || Q is clearly a fibration in Top Q , since it is a projection. In turn, its image by the rightQuillen functor −× Q || N ( Q ) || is a fibration in Top N ( Q ) . Now, the inclusion Q ⊂ P induces a functorTop N ( Q ) → Top N ( P ) that preserves all weak-equivalences, and that sends the map X ⊗|| ∆ ϕ || N ( Q ) → ( X ⊗ || ∆ ϕ || Q ) × Q || N ( Q ) || to the map X ⊗ || ∆ ϕ || N ( P ) → ( X ⊗ || ∆ ϕ || P ) × P || ∆ ϕ || . So it remains tobe shown that the inclusion( X ⊗ || ∆ ϕ || P ) × P || ∆ ϕ || → ( X ⊗ || ∆ ϕ || P ) × P || N ( P ) || STRATIFIED KAN-QUILLEN EQUIVALENCE 27 is a weak-equivalence in Top N ( P ) . Let ∆ ψ be a non-degenerate filtered simplex and consider anelement, f , in Map( || ∆ ψ || N ( P ) , ( X ⊗ || ∆ ϕ || P ) × P || N ( P ) || ), of dimension n . It can be decomposedas the product of a map in Top P , f P : ∆ n ⊗ ∆ ψ → X ⊗ || ∆ ϕ || P , and the inclusion || ∆ ψ || N ( P ) →|| N ( P ) || N ( P ) . In particular, for f P to exist, we must have ∆ ψ ⊂ ∆ ϕ . We conclude that f must liein Map( || ∆ ψ || N ( P ) , ( X ⊗ || ∆ ϕ || P ) × P || ∆ ϕ || ), which conclude the proof. (cid:3) Before moving on to the proof of Theorem 3.15, we need a few preliminary results and definitions
Lemma 3.19.
Let α : ( X, ϕ X ) → ( Y, ϕ Y ) be a map in Top N ( P ) . It is a weak-equivalence if andonly if, for all commutative square of the form(9) ( X, ϕ X ) ( Y, ϕ Y ) S n − ⊗ ∆ ϕ B n ⊗ ∆ ϕαf gh where n ≥ ϕ is a non-degenerate filtered simplex, there exists h : B n ⊗ ∆ ϕ → ( X, ϕ X ) suchthat h | S n − ⊗ ∆ ϕ = f and α ◦ h is filtered homotopic to g relative to S n − ⊗ ∆ ϕ .The result also holds in Top P . Proof.
To prove the direct implication, consider a factorisation of α as follows( X, ϕ X ) ( Z, ϕ Z ) ( Y, ϕ Y ) jr p such that, α = p ◦ j , with p a fibration, r ◦ j = Id X and j ◦ r is filtered homotopic to Id Z , relativeto j ( X ). Such a decomposition can be produced by considering the path-space associated to α (see[Dou19b, Lemma 2.11]). We now have the following lifting problem( X, ϕ X ) ( Z, ϕ Z ) ( Y, ϕ Y ) S n − ⊗ ∆ ϕ B n ⊗ ∆ ϕj pf gh ′ Since α is a weak-equivalence by hypothesis, and since j is a filtered homotopy equivalence, p mustbe a trivial fibration. This means that there exists some lift h ′ : ∆ n ⊗ ∆ ϕ → ( Z, ϕ Z ). Taking h = r ◦ h ′ gives the desired lift. Indeed, we have r ◦ h ′| S n − ⊗ ∆ ϕ = r ◦ j ◦ f = f . On the other hand, α ◦ h = α ◦ r ◦ h ′ = p ◦ j ◦ r ◦ h ′ , which is filtered homotopic to p ◦ h ′ = g . The homotopy between j ◦ r and Id Z is constant on j ( X ), in which lies j ◦ f ( S n − ⊗ ∆ ϕ ), this implies that the homotopybetween α ◦ h and g is relative to S n − ⊗ ∆ ϕ .To prove the converse, we need to prove that f induces isomorphisms on all filtered homotopygroups. Let φ : ∆ ϕ → ( X, ϕ X ) be some pointing of ( X, ϕ X ). Any element in π n (Map(∆ ϕ , ( X, ϕ X )) , φ )can be represented by a map f : S n ⊗ ∆ ϕ → ( X, ϕ X ), sending {∗} ⊗ ∆ ϕ to the chosen pointing.Assume that the element represented by α ◦ f is trivial in π n (Map(∆ ϕ , ( Y, ϕ Y )) , α ◦ φ ). This meansthat α ◦ f extends to a map g : B n +1 ⊗ ∆ ϕ → ( Y, ϕ Y ). Now, by the lifting property (9), wededuce that there exists a map h : B n +1 ⊗ ∆ ϕ → ( X, ϕ X ) extending f , which means that f rep-resents the trivial element in π n (Map(∆ ϕ , ( X, ϕ X )) , φ ). This implies that α induces an injectivemap on filtered homotopy groups. To show the surjectivity, assume that g : S n ⊗ ∆ ϕ → ( Y, ϕ Y ) isa representant of some element in π n (Map(∆ ϕ , ( Y, ϕ Y )) , α ◦ φ ). The map g can be seen as a map e g : B n ⊗ ∆ ϕ → ( Y, ϕ Y ) sending S n − ⊗ ∆ ϕ to Im( α ◦ φ ) ≃ ∆ ϕ . In particular, the restriction of e g to S n − ⊗ ∆ ϕ lifts to ( X, ϕ X ), where it lands in Im( φ ). The lifting property (9) then guarantees thatthere exists some map h : B n ⊗ ∆ ϕ → ( X, ϕ X ), sending S n − ⊗ ∆ ϕ to the pointing of X , whoseimage by α is filtered homotopic to e g relative to S n − ⊗ ∆ ϕ . In particular, the lift h represents anelement of π n (Map(∆ ϕ , ( X, ϕ X )) , φ ) in the preimage of the element represented by g . This impliesthat α induces a surjection on filtered homotopy groups. The same proof works in the case ofTop P . (cid:3) Construction 3.20.
Let (
X, ϕ X ) be a vertical filtered CW-complex, and ∆ ϕ a non-degeneratefiltered simplex. Let L ( X ) ϕ be the subcomplex of L ( X ) containing all cells with label ∆ ψ suchthat ∆ ϕ ⊂ ∆ ψ . Let λ ϕ be the constant labeling map with value ∆ ϕ . We define ( X, ϕ X ) ϕ = V ( L ( X ) ϕ , λ ϕ ). Note that the we have a well defined composition of label-preserving maps( L ( X ) ϕ , λ ϕ ) ( L ( X ) ϕ , λ L ( X ) ) ( L ( X ) , λ L ( X ) ) Id L ( X ) ϕ Applying V to this map gives a well-defined monomorphism ( X, ϕ X ) ϕ ֒ → ( X, ϕ X ), though it isnot the inclusion of a subcomplex. By abuse of notation, we will write X ϕ for the subspace of X underlying ( X, ϕ X ) ϕ . Lemma 3.21.
Let f : K ⊗ ∆ ϕ → ( X, ϕ X ) be a map in Top N ( P ) with K some topological space, ∆ ϕ a non-degenerate filtered simplex and ( X, ϕ X ) a vertical filtered CW-complex. Then, Im( f ) ⊂ X ϕ . Proof.
By assumption, f is strongly filtered. This implies that for any x ∈ K , and any t in theinterior of ∆ ϕ , f ( x, t ) must lie in a cell of dimension ( n, ∆ ψ ), with ∆ ϕ ⊂ ∆ ψ . By continuity of f , this implies that Im( f ) must lie in the closure of the union of all cells of dimension ( n, ∆ ψ )with ∆ ϕ ⊂ ∆ ψ . Any cell (say of dimension ( m, ∆ µ )) intersecting this closure must intersect theboundary of some cell of dimension ( n, ∆ ψ ) with ∆ ϕ ⊂ ∆ ψ . But this implies that ∆ ψ ⊂ ∆ µ .Now, if f ( x, t ) lies in some cell of dimension ( n, ∆ ψ ), the fact that f is strongly filtered impliesthat f ( x, t ) = ( y, t ) ∈ ∆ n ⊗ ∆ ϕ ⊂ ∆ n ⊗ ∆ ψ for some y ∈ ∆ n . In particular, f ( x, t ) must lie in( X, ϕ X ) ϕ . (cid:3) Remark . Lemma 3.21 implies that the stratified homotopy groups of a vertical filtered CW-complex can be computed from its associated P-labeled CW-complex. Indeed, let (
X, ϕ X ) be somevertical filtered CW-complex, ∆ ϕ be a non-degenerate filtered simplex, and φ : || V || → ( X, ϕ X ) apointing of ( X, ϕ X ). For simplicity, we will assume that φ is the inclusion of some 0-cell. Then,any element in sπ n (( X, ϕ X ) , φ )(∆ ϕ ) can be represented by a map f : S n ⊗ ∆ ϕ → ( X, ϕ X ). But,by lemma 3.21, such a map must land in X ϕ ≃ L ( X ) ϕ ⊗ ∆ ϕ . In particular, one can define e f : S n ⊗ ∆ ϕ → L ( X ) ϕ ⊗ ∆ ϕ by e f ( x, ( t , . . . , t n )) = ( pr ( f ( x, (1 , , . . . , , ( t , . . . , t n )). The map e f is filtered homotopic to f , by construction, and is of the form V ( g : S n → L ( X ) ϕ ) for some map g between P -labeled CW-complexes. Doing the same for maps of the form B n +1 ⊗ ∆ ϕ → ( X, ϕ X ),we get an isomorphism sπ n (( X, ϕ X ) , φ )(∆ ϕ ) ≃ π n ( L ( X ) ϕ , φ ) . Construction 3.23.
Let (
X, ϕ X ) be a vertical filtered CW-complex, and let p ∈ P . Let L ( X ) ≤ p bethe subcomplex of L ( X ) containing all cells with label ∆ ψ such that p ∈ ∆ ψ . For ∆ ψ = [ q , . . . , q m ] anon-degenerate filtered simplex, such that p = q i for some 0 ≤ i ≤ m , define tr ≤ p (∆ ψ ) = [ q , . . . , q i ].We define ( X, ϕ X ) ≤ p = V ( L ( X ) ≤ p , tr ≤ p ◦ λ L ( X ) ). As before, we will write X ≤ p to denote thesubspace of X underlying ( X, ϕ X ) ≤ p . STRATIFIED KAN-QUILLEN EQUIVALENCE 29
Now, let ∆ ϕ = [ p , . . . , p n ] be a non-degenerate filtered simplex. Define the subspace X ≤ ϕ ⊂ X ≤ p n to be the following union : X ≤ ϕ = [ p i ∈ ∆ ϕ ( ϕ P ◦ ϕ X ) − ( p i ) ∩ X ≤ p n In other word X ≤ ϕ is the union of the p i -strata of X ≤ p n . Lemma 3.24.
Let g : K ⊗ ∆ ϕ → ( X, ϕ P ◦ ϕ X ) be a map in Top P , with K some topological space,∆ ϕ a non-degenerate filtered simplex and ( X, ϕ X ) a vertical filtered CW-complex. Then Im( g ) liesin X ≤ ϕ . Proof.
Consider the following subset of || ∆ ϕ || = || [ p , . . . , p n ] || A = { ( t , . . . , t n ) | X t i = 1 , ≤ t i ≤ , ∀ ≤ i ≤ n − , < t n ≤ } . it is dense in || ∆ ϕ || , and all points in A are mapped to p n by the stratification ϕ P . Since g is amap in Top P , it preserves the stratifications over P , and so must send K ⊗ A to the p n -stratum of( X, ϕ P ◦ ϕ X ). But the p n -stratum of ( X, ϕ P ◦ ϕ X ) is contained in X ≤ p n , which is a closed subsetof X . This implies that Im( g ) lies in X ≤ p n . Furthermore, any point in Im( g ) lies in the p i -stratumof ( X, ϕ P ◦ ϕ X ) for some 0 ≤ i ≤ n , which implies that Im( g ) lies in X ≤ ϕ . (cid:3) Lemma 3.25.
Let (
X, ϕ X ) be a vertical filtered CW-complex, and ∆ ϕ be a non-degenerate filteredsimplex. There exists some open U ϕ ⊂ X ≤ ϕ such that X ϕ ⊂ U ϕ , and ( U ϕ , ϕ P ◦ ϕ X ) deformationretracts to ( X ϕ , ϕ P ◦ ϕ X ) in the category Top P . Proof.
Consider the inclusion of CW-complexes L ( X ) ϕ ⊂ L ( X ) ≤ p . By classical results, [Hat02,Proposition A.5] There exists an open neighborhood of L ( X ) ϕ into L ( X ) ≤ p , W ϕ , that deformationretracts to L ( X ) ϕ . Let us write i : L ( X ) ϕ → W ϕ for the inclusion, r : W ϕ → L ( X ) ϕ for theretraction and H : W ϕ × [0 , → W ϕ for the homotopy. Note that H can be chosen such that for apoint x in the interior of a cell e α , H ( x, t ) lies in the closure of e α for all t . In particular, those mapsare label-preserving. Now define a P -labeled CW-complex ( Z, λ Z ) as follows. Set Z = L ( X ) ≤ p , and λ Z ( e α ) = λ L ( X ) ( e α ) ∩ ∆ ϕ . This is well defined since for any cell e α ∈ L ( X ) ≤ p , p ∈ λ L ( X ) ( e α ) ∩ ∆ ϕ .Now, define ( Y, ϕ Y ) = V ( Z, λ Z ). Clearly, there is a sequence of inclusions of subspaces( X, ϕ X ) ϕ ⊂ ( Y, ϕ Y ) ⊂ ( X, ϕ X ) ≤ ϕ Note that since (
Y, ϕ Y ) is a filtered subspace of ( X, ϕ X ) ≤ p , ϕ Y is simply the restriction of ϕ X to Y . Furthermore, the open subset, ( W ϕ , λ L ( X ) ) ⊂ ( L ( X ) ≤ p , λ L ( X ) ) , lifts to some open subset V ( W ϕ ) ⊂ X ≤ p . Define U ϕ = V ( W ϕ ) ∩ X ≤ ϕ . We then have the sequence of inclusions( X ϕ , ϕ P ◦ ϕ X ) ( Y ∩ U ϕ , ϕ P ◦ ϕ X ) ( U ϕ , ϕ P ◦ ϕ X ) j j We will show that both admit deformation retracts. First, consider some cell e α of Y such that e α intersects U ϕ , and e α is not a cell of X ϕ . By construction of Y , there must be some cell e ′ α ∈ Z such that e α = V ( e ′ α ) ≃ e ′ α ⊗ λ Z ( e ′ α ). But then, the intersection e α ∩ U ϕ is filteredhomeomorphic to ( e ′ α ∩ W ϕ ) ⊗ λ Z ( e ′ α ). In particular, using this homeomorphism, we can define r ,α : e α ∩ U ϕ → X ϕ as a product of r α : e ′ α ∩ W ϕ → L ( X ) ϕ and the inclusion λ Z ( e ′ α ) ⊂ ∆ ϕ , andthe homotopy H ,α : ( e α ∩ U ϕ ) × [0 , → ( e α ∩ U ϕ ) as the product of H and the identity of λ Z ( e ′ α ).The continuity of the maps H and r then guarantees that the r ,α and H ,α glue together to forma retraction r : Y ∩ U ϕ → X ϕ and a homotopy H : Y ∩ U ϕ × [0 , → Y ∩ U ϕ between the identityof Y ∩ U ϕ and r ◦ j . One note that all the maps involved are in fact vertical. In particular, j
10 SYLVAIN DOUTEAU admits a deformation retraction in the category Top N ( P ) , which means it also does in the categoryTop P .Now, let e α = V ( e ′ α ) be a cell in X ≤ p of dimension ( k, ∆ ψ α ). The intersection e α ∩ U ϕ can befurther decomposed as ( e α ∩ V ( W ϕ )) ∩ X ≤ ϕ . In particular, there is a filtered homeomorphism inTop P e α ∩ U ϕ ≃ ( e ′ α ∩ W ϕ ) ⊗ || ∆ ψ α || ≤ ϕP Where, || ∆ ψ α || ≤ ϕP is the union of the p i -strata of || ∆ ψ α || P . Alternatively, if we write ∆ ψ α =[ q , . . . , q m ], we can describe the subset || ∆ ψ α || ≤ ϕP ⊂ || ∆ ψ α || P as follows : || ∆ ψ α || ≤ ϕP = { ( t , . . . , t m ) | max { q i | t i = 0 } ∈ ∆ ϕ } One can then define a filtered retract || ∆ ψ α || ≤ ϕP → || ∆ ψ α ∩ ∆ ϕ || P as follows. If ( t , . . . , t m ) ∈|| ∆ ψ α || ≤ ϕP , define ( t , . . . , t m ) ϕ = ( s , . . . , s m ) with s i = t i if q i ∈ ∆ ϕ , and s i = 0 else, and write | ( s , . . . , s m ) | = P i s i . Then, one can define the retraction as follows r ,ψ α : || ∆ ψ α || ≤ ϕP → || ∆ ψ α ∩ ∆ ϕ || P ( t , . . . , t m ) ( t , . . . , t m ) ϕ | ( t , . . . , t m ) ϕ | We can then take the linear homotopy, H ,α , between the identity of || ∆ ψ α || ≤ ϕP and the composition || ∆ ψ α || ≤ ϕP || ∆ ψ α ∩ ∆ ϕ || P || ∆ ψ α || ≤ ϕPr ,α j ,α Note that by construction, the maps j ,α , r ,α and H ,α are filtered over P , and are compatiblewith inclusions ∆ ψ α ֒ → ∆ ψ β . Furthermore, taking the product of j ,α with the identity of e ′ α , weget a map (that we will still write j ,α ) j ,α : ( e ′ α ∩ W ϕ ) ⊗ || ∆ ψ α ∩ ∆ ϕ || P → ( e ′ α ∩ W ϕ ) ⊗ || ∆ ψ α || ≤ ϕP In particular, the j ,α can be glued together to produce the inclusion j : Y ∩ U ϕ → U ϕ . Doing thesame for the r ,α produces a retract r : U ϕ → Y ∩ U ϕ and similarly for the H ,α giving a homotopybetween j ◦ r and Id U ϕ . Since the r ,α and H ,α are filtered maps over P , so are r and H . Inparticular, j admits a deformation retraction in Top P . (cid:3) Proof of Theorem 3.15.
Let (
X, ϕ X ) be a filtered CW-complexes. By Lemma 3.19, it is enough toshow that for any n ≥
0, any non-degenerate filtered simplex ∆ ϕ , and any commutative diagramin Top N ( P ) of the form(10) ( X, ϕ X ) ( X, ϕ P ◦ ϕ X ) × P || N ( P ) || S n − ⊗ ∆ ϕ B n ⊗ ∆ ϕαf gh there exists a lift up to homotopy h : B n ⊗ ∆ ϕ → ( X, ϕ X ) in Top N ( P ) . By lemma 3.21, we can replace( X, ϕ X ) with the subspace ( X, ϕ X ) ϕ . By lemma 3.24, we can replace ( X, ϕ P ◦ ϕ X ) × P || N ( P ) || by STRATIFIED KAN-QUILLEN EQUIVALENCE 31 ( X ≤ ϕ , ϕ P ◦ ϕ X ) × P || N ( P ) || . The lifting problem (10) then turns into( X, ϕ X ) ϕ ( X ≤ ϕ , ϕ P ◦ ϕ X ) × P || N ( P ) || S n − ⊗ ∆ ϕ B n ⊗ ∆ ϕf gh We then decompose the lifting problem as follows (note that in future diagrams, we omit thestratifications). X ϕ X ϕ × P || N ( P ) || U ϕ × P || N ( P ) || X ≤ ϕ × P || N ( P ) || S n − ⊗ ∆ ϕ B n ⊗ ∆ ϕαf gh h h We will construct the h i sequentially. Note that while working on the right side of the liftingproblem, we can omit the " × P || N ( P ) || " and consider that all spaces and maps live in Top P . Considerthe following lifting problem. U ϕ X ≤ ϕ S n − ⊗ ∆ ϕ B n ⊗ ∆ ϕα ◦ f gh Write ∆ ϕ = [ p , . . . , p n ]. Let us show that g ( B n ⊗ { p } ) lies in U ϕ . If ∆ ϕ = { p } , this followsfrom the definition of U ϕ , and tautologically implies the existence of a lift h . For the other cases,let ( x, (1 , , . . . , B n ⊗ { p } , the point g ( x, (1 , , . . . , e α of X ≤ p , of dimension ( k, ∆ ψ α ). But for any p i ∈ ∆ ϕ , g ( x, (1 , , . . . , p i -stratum of X ≤ p . Indeed, consider the sequence g ( x, (1 − l , , . . . , , l , , . . . , l isin the ( i + 1)-th position. It lies in the p i -stratum of X but converges to g ( x, (1 , , . . . , l tends to infinity. This implies that p i ∈ ∆ ψ α for all p i ∈ ∆ ϕ , which means that ∆ ϕ ⊂ ∆ ψ α . Thisimplies that W ϕ contains the (non-filtered) cell e ′ α which satisfies V ( e ′ α ) = e α . In turn, V ( W ϕ )contains the cell e α , and so the p -stratum of e α is contained in U ϕ . In particular, U ϕ must containthe image of B n ⊗ { p } under g . Now, consider the following subsets of B n ⊗ ∆ ϕ , for ǫ > ǫ ( B n ⊗ ∆ ϕ ) = { ( x, ( t , . . . , t n )) | t ≥ − ǫ } ⊂ B n ⊗ ∆ ϕ For all ǫ >
0, tr ǫ ( B n ⊗ ∆ ϕ ) is compact, which means that ( g (tr ǫ ( B n ⊗ ∆ ϕ ))) ǫ> is a family ofnested compact in X ≤ ϕ . As we have shown, their intersection lies in the open set U ϕ , which impliesthat their exists some ǫ > g (tr ǫ ( B n ⊗ ∆ ϕ )) lies in U ϕ . On the other hand, fix somehomeomorphism between S n − × [0 ,
1[ and B n \ { } . The family ( g ( S n − × [0 , δ ] ⊗ ∆ ϕ )) <δ< , is afamily of nested compacts in X ≤ ϕ . Their intersection is g ( S n − × { } ⊗ ∆ ϕ ) = α ◦ f ( S n − ⊗ ∆ ϕ ) ⊂ U ϕ . This implies that their exist 0 < δ < g ( S n − × [0 , δ ] ⊗ ∆ ϕ ) lies in U ϕ . Let A n,ϕ ( ǫ, δ ) = tr ǫ ( B n ⊗ ∆ ϕ ) ∪ S n − × [0 , δ ] ⊗ ∆ ϕ ⊂ B n ⊗ ∆ ϕ , and write j for the correspondinginclusion. We have shown that g ( A n,ϕ ( ǫ, δ )) lies in U ϕ . By lemma 3.26, there exists a retraction r : B n ⊗ ∆ ϕ → A n,ϕ ( ǫ, δ ), and a filtered homotopy H : ( B n ⊗ ∆ ϕ ) × [0 , → B n ⊗ ∆ ϕ between Id B n ⊗ ∆ ϕ and j ◦ r , relative to A n,ϕ ( ǫ, δ ) in Top P . We then obtain the lift h by taking h = g ◦ r .Now that h has been constructed, consider the second commutative diagram in Top P : X ϕ U ϕ S n − ⊗ ∆ ϕ B n ⊗ ∆ ϕα ◦ f h h By lemma 3.25, the top map is a filtered homotopy equivalence in Top P . In particular, it is aweak-equivalence in Top P . By lemma 3.19, this implies that a lift up to homotopy, h , must exist.Now turn to the last lifting problem :( X, ϕ X ) ϕ ( X ϕ , ϕ P ◦ ϕ X ) × P || N ( P ) || S n − ⊗ ∆ ϕ B n ⊗ ∆ ϕf h h By construction, (
X, ϕ X ) ϕ ≃ L ( X ) ϕ ⊗ ∆ ϕ . By Lemma 3.18, this means that the top map is aweak-equivalence in Top N ( P ) , which implies by lemma 3.19 that the desired lift must exist. (cid:3) Lemma 3.26.
The inclusion j : A n,ϕ ( ǫ, δ ) = tr ǫ ( B n ⊗ ∆ ϕ ) ∪ ( S n − × [0 , δ ]) ⊗ ∆ ϕ → B n ⊗ ∆ ϕ admit a deformation retraction r : B n ⊗ ∆ ϕ → A n,ϕ ( ǫ, δ ) in Top P . Proof.
Write ∆ ϕ = [ p , . . . , p k ], and ∆ b ϕ = [ p , . . . , p k ]. And fix an identification of B n with { ( x , . . . , x n ) | | ( x , . . . , x n ) | ≤ } ⊂ R n , such that the subspace S n − × [0 , δ ] is identified with { ( x , . . . , x n ) | − δ ≤ | ( x , . . . , x n ) | ≤ } . Define the following subspaces of B n ⊗ ∆ ϕ : B n,ϕ ( ǫ, δ ) = { (( x , . . . , x n ) , ( t , . . . , t k )) | | ( x , . . . , x n ) | ≤ − δ, t ∈ [0 , − ǫ ] } C n,ϕ ( ǫ, δ ) = { (( x , . . . , x n ) , ( t , . . . , t k )) | ( | ( x , . . . , x n ) | = 1 − δ, and t ∈ [0 , − ǫ ])or (( x , . . . , x n ) | ≤ − δ, and t = 1 − ǫ ) } By construction, we have B n ⊗ ∆ ϕ = A n,ϕ ( ǫ, δ ) ∪ C n,ϕ ( ǫ,δ ) B n,ϕ ( ǫ, δ )In particular, it is enough to show that the inclusion C n,ϕ ( ǫ, δ ) ֒ → B n,ϕ ( ǫ, δ ) admits a deformationretraction. Now, consider the following map : { ( t , . . . , t k ) | t ∈ [0 , − ǫ ] } 7→ [0 , × || ∆ b ϕ || P ( t , . . . , t k ) ( t − ǫ , ( t , . . . , t k )1 − t )it is a filtered homeomorphism in Top P , between the subspace { ( t , . . . , t k ) | t ∈ [0 , − ǫ ] } ⊂ || ∆ ϕ || P and [0 , × || ∆ b ϕ || P . In particular, it induces a filtered homeomorphism B n,ϕ ( ǫ, δ ) ≃ B n ⊗ ([0 , × || ∆ b ϕ || P )under this identification, the map C n,ϕ ( ǫ, δ ) → B n,ϕ ( ǫ, δ ) becomes S n − ⊗ ([0 , × || ∆ b ϕ || P ) ∪ B n ⊗ ( { } × || ∆ b ϕ || P ) → B n ⊗ ([0 , × || ∆ b ϕ || P ) STRATIFIED KAN-QUILLEN EQUIVALENCE 33
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