Gray tensor product and saturated N -complicial sets
aa r X i v : . [ m a t h . A T ] J u l GRAY TENSOR PRODUCT AND SATURATED N -COMPLICIAL SETS VIKTORIYA OZORNOVA, MARTINA ROVELLI, AND DOMINIC VERITY
Abstract.
We show that the pretensor and tensor products of sim-plicial sets with marking are compatible with the homotopy theory ofsaturated N -complicial sets (which are a proposed model of ( ∞ , N ) -categories), in the form of a Quillen bifunctor and a homotopical bi-functor, respectively. Overview
Higher category theory is becoming increasingly important as a unifyinglanguage for various areas of mathematics, most notably for algebraic topol-ogy and algebraic geometry, where many relevant structures occur naturallyas ( ∞ , N ) -categories , rather than strict N -categories. In this article, weare concerned with an ( ∞ , N ) -categorical version of the Crans–Gray tensorproduct [Gra74, Cra95], originally defined for strict N -categories in order toencode different flavors of lax natural transformations.The model of ( ∞ , N ) -categories that we consider, due to the third-namedauthor, is that of saturated N -complicial sets . A saturated N -complicial setis a simplicial set with marking satisfying extra conditions that guaranteethat the marked simplices behave as higher equivalences. In [Ver08a], heconstructed two pointset models of the Gray tensor product of simplicialsets with marking: the tensor ⊗ and the pretensor ⊠ , homotopically equiv-alent but each with different valuable properties, and showed that they arecompatible with the homotopy theory of (non-saturated) N -complicial sets.In this note, we provide the extra verification that enables us to concludethat the pretensor and the tensor products ⊠ and ⊗ are in fact also com-patible with the model structure for saturated N -complicial sets, in a sensethat will be made precise by Corollaries 2.3 and 2.6. Main Theorem.
For any N ∈ N , the bifunctors ⊠ and ⊗ are homotopi-cal with respect to the model structure on simplicial sets with marking forsaturated N -complicial sets, which model ( ∞ , N ) -categories. Mathematics Subject Classification.
Key words and phrases.
Gray tensor product, saturated n -complicial set, ( ∞ , n ) -category. he theorem was proven for N = 1 by Joyal [Joy08, Thm 6.1] in thecontext of quasi-categories and by Lurie [Lur09, Cor. 3.1.4.3] in the context ofmarked simplicial sets. During the final work on the completion of this paper,analogous result was shown for N = 2 by Gagna–Harpaz–Lanari [GHL20] inthe context of scaled simplicial sets. For general N , the result was previouslyobtained by the third-named author, and recently rediscovered by the firsttwo authors.Beside for its own interest, the result would play a role in work by Campion–Kapulkin–Maehara, in comparing cubical models of ( ∞ , N ) -categories tosaturated N -complicial sets, as indicated in [CKM20, Rmk 7.3, Conj. 7.4]. Acknowledgements.
We would like to thank Emily Riehl for bringing theproblem treated in this paper to the attention of the first two authors, andLennart Meier for helpful conversations on this project. This material isbased upon work supported by the National Science Foundation under GrantNo. DMS-1440140 while the authors were in residence at the MathematicalSciences Research Institute in Berkeley, California, during the Spring 2020semester. The first-named author thankfully acknowledges the financial sup-port by the DFG grant OZ 91/2-1 with the project nr. 442418934. Thethird-named author was supported by the Discovery Project DP190102432from the Australian Research Council.
Contents
1. Background on simplicial sets with marking 21.1. The model structures on simplicial sets with marking 31.2. Pretensor and tensor product of simplicial sets with marking 62. The main theorem 82.1. The formal part of the proof 102.2. Proof of Proposition 2.10 12References 231.
Background on simplicial sets with marking
We recall in this section the background material on simplicial sets withmarking, saturated complicial sets, and on the pretensor and tensor product, ⊠ and ⊗ . efinition 1.1. A simplicial set with marking is a simplicial set with a des-ignated subset of marked or thin positive-dimensional simplices that includesall degenerate simplices. A map of simplicial sets with marking is a simpli-cial map that preserves the marking. We denote by m s S et the category ofsimplicial sets with marking and maps of simplicial sets with marking.1.1. The model structures on simplicial sets with marking.
The fol-lowing notational conventions will be used to define saturated N -complicialsets and to describe the model structure for N -complicial sets on m s S et .The material is mostly drawn from [Ver08b, §§2.1-2.2], [Rie18] and [OR20b,§1], and we refer the reader to these references for a more detailed account. Notation 1.2.
We denote • by ∆[ − the empty simplicial set. • by ∆[ m ] the simplicial set with marking whose underlying simplicial setis ∆[ m ] and in which only degenerate simplices are marked. • by ∂ ∆[ m ] the simplicial set with marking whose underlying simplicial setis ∂ ∆[ m ] and in which only degenerate simplices are marked. • by ∆[ m ] t the simplicial set with marking whose underlying simplicial setis ∆[ m ] and in which only degenerate simplices and the top m -simplex aremarked. • by ∆ k [ m ] , for ≤ k ≤ m , the simplicial set with marking whose underlyingsimplicial set is ∆[ m ] and in which a non-degenerate simplex is marked ifand only if it contains the vertices { k − , k, k + 1 } ∩ [ m ] . • by ∆ k [ m ] ′ , for ≤ k ≤ m , the simplicial set with marking obtained from ∆ k [ m ] by additionally marking the ( k − -st and ( k + 1) -st face of ∆[ m ] . • by ∆ k [ m ] ′′ , for ≤ k ≤ m , the simplicial set with marking obtained from ∆ k [ m ] ′ by additionally marking the k -th face of ∆[ m ] . • by Λ k [ m ] , for ≤ k ≤ m , the simplicial set with marking whose underlyingsimplicial set is the k -horn Λ k [ m ] and whose simplex is marked if and onlyif it is marked in ∆ k [ m ] . • by ∆[3] eq the simplicial set with marking whose underlying simplicial setis ∆[3] , and the non-degenerate marked simplices consist of all - and -simplices, as well as -simplices [02] and [13] . • by ∆[3] ♯ the simplicial set with marking whose underlying simplicial set is ∆[3] , and all simplices in positive dimensions are marked. • by ∆[ ℓ ′ | eq | ℓ ] , for ℓ, ℓ ′ ≥ − , the simplicial set with marking ∆[ ℓ ′ ] ⋆ ∆[3] eq ⋆ ∆[ ℓ ] . • by ∆[ ℓ ′ | ♯ | ℓ ] , for ℓ, ℓ ′ ≥ − , the simplicial set with marking ∆[ ℓ ′ ] ⋆ ∆[3] ♯ ⋆ ∆[ ℓ ] . This notion is the same as stratified simplicial set in the sense of Verity [Ver08a], andis different from (but related to) marked simplicial set in the sense of Lurie [Lur09]. ere, ⋆ denotes the join of simplicial sets with marking, which can befound in [Ver08b, Observation 34] or [Rie18, Def. 3.2.5], and which we recallfor the reader’s convenience. Definition 1.3.
Given simplicial sets with marking X and Y , the join X ⋆Y is a simplicial set with marking whose underlying simplicial set is the join ofthe underlying simplicial sets, and in which an r -simplex x ⋆ y : ∆[ k ] ⋆ ∆[ r − k − → X ⋆ Y for − ≤ k ≤ r is marked if and only if the simplex x ismarked in X or the simplex y is marked in Y (or both). Definition 1.4.
For N ∈ N ∪{∞} , an elementary ( ∞ , N ) -anodyne extension is one of the following.(1) The complicial horn extension , i.e., the canonical map Λ k [ m ] → ∆ k [ m ] for m ≥ and ≤ k ≤ m, which is the ordinary horn inclusion on the underlying simplicial sets.(1’) The complicial thinness extension , i.e., the canonical map ∆ k [ m ] ′ → ∆ k [ m ] ′′ for m ≥ and ≤ k ≤ m, which is the identity on the underlying simplicial set.(2) The left saturation extension , i.e., the canonical map ∆[ ℓ | eq ] → ∆[ ℓ | ♯ ] for ℓ ≥ − , which is the identity on the underlying simplicial set.(3) The triviality extension map, i.e., the canonical map ∆[ p ] → ∆[ p ] t for p > N , which is the identity on the underlying simplicial set. Remark . We point out that the parameter N only plays a role in thetriviality anodyne extension in (3). In particular, complicial horn extensions,thinness extensions and saturation anodyne extensions are ( ∞ , N ) -anodynefor every N ∈ N ∪ {∞} . Definition 1.6.
Let X be a simplicial set with marking, and N ∈ N ∪ {∞} .(1) X is a complicial set , also called a weak complicial set , if it has the rightlifting property with respect to the complicial horn anodyne extensions Λ k [ m ] → ∆ k [ m ] and the thinness anodyne extensions ∆ k [ m ] ′ → ∆ k [ m ] ′′ for m ≥ and ≤ k ≤ m .(2) X is a saturated complicial set if it is a complicial set and it has the rightlifting property with respect to the left saturation anodyne extensions ∆[ ℓ | eq ] → ∆[ ℓ | ♯ ] for ℓ ≥ − .(3) X is a saturated N -complicial set if it is a saturated complicial set andit has the right lifting property with respect to the triviality anodyneextensions ∆[ p ] → ∆[ p ] t for p > N . or any N ∈ N , saturated N -complicial sets are a proposed model for ( ∞ , N ) -categories , and we refer the reader to [Ver08a, Rie18, OR20b] for adescription of the intuition behind this combinatorics.Roughly speaking, according to the intuition that the r -simplices of asimplicial set with marking represent r -morphisms and that the marked sim-plices represent r -equivalences, we can rephrase as follows.(1) In a complicial set r -morphisms can be composed, and composite of r -equivalences is an r -equivalence.(2) In a saturated complicial set r -equivalences satisfy the two-out-of-sixproperty.(3) In a saturated N -complicial set all r -morphisms are equivalences in di-mension r > N .There is a model structure on m s S et for saturated N -complicial sets. Theorem 1.7 ([Ver08a, Rie18, OR20b]) . Let N ∈ N ∪ {∞} . There is acofibrantly generated model structure on m s S et in which • the cofibrations are precisely the monomorphisms; • the fibrant objects are precisely the saturated N -complicial sets; • all elementary anodyne extensions are acyclic cofibrations.We call this model structure the model structure for ( ∞ , N ) -categories, or themodel structure for saturated N -complicial sets, we denote it by m s S et ( ∞ ,N ) ,and we call the acyclic cofibrations ( ∞ , N ) -acyclic cofibrations.Remark . As discussed in [Ver08b, Example 21], the generating cofibra-tions for the model structure for ( ∞ , N ) -categories are • the boundary inclusions ∂ ∆[ m ] → ∆[ m ] for m ≥ , • and the marking inclusions ∆[ m ] → ∆[ m ] t for m ≥ . We mentioned that, by construction, all left saturation extensions ∆[ ℓ | eq ] → ∆[ ℓ | ♯ ] for ℓ ≥ − are acyclic cofibrations. In fact, even the saturation ex-tensions of the more general form ∆[ ℓ ′ | eq | ℓ ] → ∆[ ℓ ′ | ♯ | ℓ ] for ℓ, ℓ ′ ≥ − areacyclic cofibrations. Lemma 1.9.
The saturation extension ∆[ ℓ ′ | eq | ℓ ] → ∆[ ℓ ′ | ♯ | ℓ ] for ℓ, ℓ ′ ≥ − is acyclic cofibration. The case N = ∞ is subtle, since there are at least two different viewpoints on whatan ( ∞ , ∞ ) -category should be. roof. The saturation extensions ∆[ ℓ ′ | eq | ℓ ] → ∆[ ℓ ′ | ♯ | ℓ ] have the left liftingproperty with respect to all saturated N -complicial sets, as shown [RV20,§D.7], and since they are isomorphisms on the underlying simplicial setsthey must also have the right lifting property with respect to all fibrationsbetween saturated N -complicial sets. We then conclude that they are acycliccofibrations as an instance of [JT07, Lemma 7.14]. (cid:3) Proposition 1.10 ([OR20a, Lemma 1.8]) . Let M be a model category. Aleft adjoint functor F : m s S et ( ∞ ,N ) → M is left Quillen if and only if itrespects cofibrations and sends all elementary anodyne extensions to weakequivalences of M . Pretensor and tensor product of simplicial sets with marking.
Inspired by the Crans–Gray tensor product of ω -categories from [Gra74,Cra95], which can be thought as strict ∞ -categories, Verity defined twomodels of Gray tensor products of simplicial sets with marking: the pretensor ⊠ and the tensor ⊗ . In this paper, we will work with the definition of thetensor product ⊗ , while the pretensor product ⊠ plays a more indirect role.For completeness, we recall both definitions. Notation 1.11 ([Ver08a, Notation 5]) . For any p, q ≥ , • the degeneracy partition operator is the map in ∆Π p,q : [ p + q ] → [ p ] and Π p,q : [ p + q ] → [ q ] defined by i (cid:26) i if i ≤ pp if i > p and i (cid:26) if i < pi − p if i ≥ p • the face partition operator is the map in ∆ ∐ p,q : [ p ] → [ p + q ] and ∐ p,q : [ q ] → [ p + q ] defined by i i and i p + i. Remark . As explained in [Ver08a, §1.6], any non-degenerate r -simplexof ∆[ r ] → ∆[ p ] × ∆[ q ] can be pictured as a path of length r in a rectangulargrid of size p × q . According to this interpretation, the ( p + q ) -simplex givenby (Π p,q , Π p,q ) : ∆[ p + q ] → ∆[ p ] × ∆[ q ] is the path with “first all to the right,then all up”, as shown in the following picture for p = 3 and q = 2 . In [Ver08b, Observation 62], Verity states the relationship between the Crans–Graytensor product of ω -categories and the tensor product of simplicial sets with marking,using the fact that ω -categories (in the form of strict complicial sets) form a reflectivesubcategory of simplicial sets with marking. Given two ω -categories, their Crans–Graytensor product can be obtained by reflecting their tensor product as simplicial sets withmarking. efinition 1.13 ([Ver08b, Def. 135]) . Given simplicial sets with marking X and Y , the pretensor X ⊠ Y is formed by taking the product of underlyingsimplicial sets and endowing it with a marking under which a non-degenerate r -simplex ( x, y ) : ∆[ r ] → X × Y is marked if either • it is a mediator , i.e., there exists < k < r and ( r − -simplices x ′ : ∆[ r − → X and y ′ : ∆[ r − → Y such that x = s k − x ′ = x ′ ◦ s k − and y = s k y ′ = y ′ ◦ s k . • it is a crushed cylinder , i.e., there exists a partition p, q of r = p + q andsimplices x ′ : ∆[ p ] → X and y ′ : ∆[ q ] → Y such that x = x ′ ◦ Π p,q and y = y ′ ◦ Π p,q , and either the simplex x ′ is marked in X or the simplex y ′ is marked in Y (or both).It is proven in [Ver08b, Lemma 142] that ⊠ is a bifunctor that preservescolimits in each variable. We then obtain the following adjunctions. Regard-ing the terminology of lax and oplax, we follow the same convention as e.g.[Lac10, AL19]. Proposition 1.14 ([Ver08a, Cor. 144]) . For any simplicial set with marking S there are adjunctions − ⊠ S : m s S et ⇄ m s S et : [ S, − ] oplax and S ⊠ − : m s S et ( ∞ ,N ) ⇄ m s S et ( ∞ ,N ) : [ S, − ] lax . However, the pretensor ⊠ is not associative, so it cannot be used to builda monoidal structure on m s S et . For this purpose, one can instead considerthe tensor product ⊗ (which however does not preserve colimits). Definition 1.15 ([Ver08a, Def. 128]) . Given simplicial sets with marking X and Y , the tensor X ⊗ Y is formed by taking the product of underlyingsimplicial sets and endowing it with a marking under which a non-degenerate r -simplex ( x, y ) : ∆[ r ] → X × Y is marked if for each p, q ≥ the partition r = p + q cleaves the simplex ( x, y ) , i.e., the p -simplex x ◦ ∐ p,q is marked in X or the q -simplex y ◦ ∐ p,q is marked in Y .Pretensor and tensor are equivalent in the following sense. Proposition 1.16 ([Ver08a, Lemma 149]) . For any simplicial sets withmarking X and Y the canonical inclusion X ⊠ Y ֒ → X ⊗ Y s an ( ∞ , N ) -acyclic cofibration for any N ∈ N ∪ {∞} . In particular thereis an objectwise weak equivalence − ⊠ − ≃ − ⊗ − : m s S et ( ∞ ,N ) × m s S et ( ∞ ,N ) → m s S et ( ∞ ,N ) . To highlight the difference between the pretensor and the tensor, we brieflydiscuss an example. We refer the reader to [Ver08a, §6.3] for a deeper treat-ment and for more details and examples.
Example . We consider the case of X = ∆[2] t and Y = ∆[1] . • the simplex ∆[2] → ∆[2] t × ∆[1] depicted asis a mediator, and is therefore marked in both ∆[2] t ⊠ ∆[1] and ∆[2] t ⊗ ∆[1] . • the simplex ∆[3] → ∆[2] t × ∆[1] depicted asis a crushed cylinder, and is therefore marked in both ∆[2] t ⊠ ∆[1] and ∆[2] t ⊗ ∆[1] . • the simplex ∆[2] → ∆[2] t × ∆[1] depicted asis cleaved by every partition, and is therefore marked in ∆[2] t ⊗ ∆[1] , butit is not marked in ∆[2] t ⊠ ∆[1] . • the simplex ∆[2] → ∆[2] t × ∆[1] depicted asis cleaved by the partitions (2 , and (0 , , but not by the partition (1 , ,and is therefore not marked neither in ∆[2] t ⊠ ∆[1] nor in ∆[2] t ⊗ ∆[1] .2. The main theorem
The main result is the following.
Theorem 2.1.
Let N ∈ N ∪ {∞} . For any simplicial set with marking S the adjunction − ⊠ S : m s S et ( ∞ ,N ) ⇄ m s S et ( ∞ ,N ) : [ S, − ] oplax is a Quillen pair. In particular, the functor − ⊠ S : m s S et ( ∞ ,N ) → m s S et ( ∞ ,N ) is homotopical. he theorem admits many essentially equivalent reformulations or directconsequences, which we collect as corollaries.Using Proposition 1.16 we obtain the following corollary. Corollary 2.2.
Let N ∈ N ∪ {∞} . For any simplicial set with marking S the functor − ⊗ S : m s S et ( ∞ ,N ) → m s S et ( ∞ ,N ) is homotopical. The statement can then be strengthened as follows.
Corollary 2.3.
Let N ∈ N ∪ {∞} . The functor − ⊗ − : m s S et ( ∞ ,N ) × m s S et ( ∞ ,N ) → m s S et ( ∞ ,N ) is homotopical. Lemma 2.4.
Let f : X → Y be a map of simplicial sets with marking. Then f is a weak equivalence in the model structure for saturated N -complicial setsif and only if f op is one.Proof of Lemma 2.4. We argue that ( − ) op is left Quillen, so in particularhomotopical, and hence respects weak equivalences. Given the canonicalisomorphism ( X op ) op ∼ = X from [Ver08b, Observation 38], we also obtainthat ( − ) op reflects weak equivalences, concluding the proof.To see that ( − ) op : m s S et ( ∞ ,N ) → m s S et ( ∞ ,N ) is a left Quillen functor,we observe the following.(0) Since ( − ) op is an isomorphism, if X → Y is a monomorphism, then X op → Y op is a monomorphism, so ( − ) op preserves cofibrations.(1) By [Ver08a, Observation 157], for for m ≥ and ≤ k ≤ m the map Λ k [ m ] op → ∆ k [ m ] op is the map Λ m − k [ m ] → ∆ m − k [ m ] , which is a weakequivalence in the model structure for saturated N -complicial sets. Inparticular, ( − ) op sends complicial horn extensions to weak equivalences.(2) By [Ver08a, Observation 125], for m ≥ and ≤ k ≤ m the map ∆ k [ m ] ′ op → ∆ k [ m ] ′′ op is the map ∆ m − k [ m ] ′ → ∆ m − k [ m ] ′′ , which is aweak equivalence in the model structure for saturated N -complicial sets.In particular, ( − ) op sends thinness extensions to weak equivalences.(3) By [Ver08a, Observation 107], for p > N the map ∆[ p ] op → ∆[ p ] op t is ∆[ p ] → ∆[ p ] t , which is a weak equivalence in the model structure for sat-urated N -complicial sets. In particular ( − ) op sends triviality extensionsfor p > N to weak equivalences.(4) For ℓ ≥ − , one can use [Ver08b, Observation 36] to show that themap ∆[ ℓ | eq ] op → ∆[ ℓ | ♯ ] op is the map ∆[3 eq | ℓ ] → ∆[3 ♯ | ℓ ] , which wasshown in Lemma 1.9 to be a weak equivalence in the model structure forsaturated N -complicial sets. In particular, ( − ) op sends left saturationextensions to weak equivalences. y Proposition 1.10, we then conclude that ( − ) op is a left Quillen functor,as desired. (cid:3) Proof of Corollary 2.3.
We already know from Theorem 2.1 that the functor ⊗ respects weak equivalences in the first variable, and we now check thatit respects weak equivalences in the second variable, too. If X → Y is aweak equivalence, by Lemma 2.4 the map X op → Y op is a weak equivalence.By Theorem 2.1 the map X op ⊠ S op → Y op ⊠ S op is a weak equivalence.By Proposition 1.16 the map X op ⊗ S op → Y op ⊗ S op , which is by [Ver08a,Lemma 131] the map ( S ⊗ X ) op → ( S ⊗ Y ) op , is a weak equivalence. UsingLemma 2.4, the map S ⊗ X → S ⊗ Y is then a weak equivalence, as desired. (cid:3) Using again Proposition 1.16 we obtain the following corollary.
Corollary 2.5.
Let N ∈ N ∪ {∞} . The functor − ⊠ − : m s S et ( ∞ ,N ) × m s S et ( ∞ ,N ) → m s S et ( ∞ ,N ) is homotopical. Since cofibrations in the model category m s S et ( ∞ ,N ) are checked on theunderlying simplicial set, we obtain the following corollary. Corollary 2.6.
Let N ∈ N ∪ {∞} . The functor − ⊠ − : m s S et ( ∞ ,N ) × m s S et ( ∞ ,N ) → m s S et ( ∞ ,N ) is a left Quillen bifunctor. In particular, for any simplicial set with marking S the adjunction S ⊠ − : m s S et ( ∞ ,N ) ⇄ m s S et ( ∞ ,N ) : [ S, − ] lax is a Quillen pair. The formal part of the proof.
In this subsection we prove Theo-rem 2.1 building on existing work of the third-named author and on a tech-nical fact (Proposition 2.10) whose proof will be postponed until the lastsubsection.
Proposition 2.7.
Let N ∈ N ∪ {∞} . For any m ≥ the pushout-pretensor ( J ⊠ ∆[ m ]) ∐ I ⊠ ∆[ m ] ( I ⊠ ∆[ m ] t ) → J ⊠ ∆[ m ] t of an ( ∞ , N ) -elementary anodyne extension I → J with the canonical map ∆[ m ] ֒ → ∆[ m ] t is an ( ∞ , ∞ ) -acyclic cofibration.Proof. By [Ver08a, Lemma 140] the pushout-pretensor of two entire maps inthe sense of [Ver08a, Notation 100], namely maps that are an isomorphismon the underying simplicial sets, is an isomorphism. Hence, in particularthe pushout-pretensor of a complicial thinness extension ∆ k [ m ] ′ ֒ → ∆ k [ m ] ′′ with the canonical map ∆[ m ] ֒ → ∆[ m ] t is an isomorphism. Moreover, it is xplained in the proof of [Ver08a, Lemma 169] that the pushout-pretensorof a complicial horn extension Λ k [ m ] ֒ → ∆ k [ m ] with the canonical map ∆[ m ] ֒ → ∆[ m ] t is an ( ∞ , ∞ ) -acyclic cofibration. (cid:3) Proposition 2.8.
Let N ∈ N ∪ {∞} . For any m ≥ the pushout-pretensor ( J ⊠ ∂ ∆[ m ]) ∐ I ⊠ ∂ ∆[ m ] ( I ⊠ ∆[ m ]) → J ⊠ ∆[ m ] of an elementary ( ∞ , N ) -anodyne extension I → J with a boundary inclusion ∂ ∆[ m ] ֒ → ∆[ m ] is an ( ∞ , N ) -acyclic cofibration.Proof. We treat each type of elementary anodyne extension.(1) It is explained in the proof of [Ver08a, Lemma 143] that the pushout-pretensor of an thinness elementary anodyne extension ∆ k [ m ] ′ ֒ → ∆ k [ m ] ′′ with a boundary inclusion is an ( ∞ , ∞ ) -acyclic cofibration.(2) It is explained in the proof of [Ver08a, Lemma 169] that the pushout-pretensor of a complicial horn Λ k [ m ] ֒ → ∆ k [ m ] extension with a bound-ary inclusion is an ( ∞ , ∞ ) -acyclic cofibration.(3) We will show in Proposition 2.10 that the pushout-tensor of a left sat-uration extension ∆[ ℓ | eq ] → ∆[ ℓ | ♯ ] with a boundary inclusion is an ( ∞ , ∞ ) -acyclic cofibration. By Proposition 1.16 (together with the factthat the pushout that needs to be analyzed is in fact a homotopy pushout),this implies that also that the pushout-pretensor of a left saturationextension ∆[ ℓ | eq ] ֒ → ∆[ ℓ | ♯ ] with a boundary inclusion is an ( ∞ , ∞ ) -acyclic cofibration.(4) We will show in Proposition 2.9 that the pushout-pretensor of a trivialityextension ∆[ p ] → ∆[ p ] t for p > N with a boundary inclusion is an ( ∞ , N ) -acyclic cofibration. (cid:3) The proof above made use of the following two propositions.
Proposition 2.9.
Let N ∈ N ∪ {∞} . For any m ≥ and p > N thepushout-pretensor (∆[ p ] t ⊗ ∂ ∆[ m ]) ∐ ∆[ p ] ⊗ ∂ ∆[ m ] (∆[ p ] ⊗ ∆[ m ]) → ∆[ p ] t ⊗ ∆[ m ] of an ( ∞ , N ) -triviality anodyne extension ∆[ p ] → ∆[ p ] t with a boundaryinclusion ∂ ∆[ m ] ֒ → ∆[ m ] is an ( ∞ , N ) -acyclic cofibration.Proof. The simplicial sets with marking (∆[ p ] t ⊗ ∂ ∆[ m ]) ∐ ∆[ p ] ⊗ ∂ ∆[ m ] (∆[ p ] ⊗ ∆[ m ]) and ∆[ p ] t ⊗ ∆[ m ] have the same underlying simplicial set, isomorphicto ∆[ p ] × ∆[ m ] . We observe that they also have the same set of marked r -simplices for r < p . Indeed, the set of marked simplices in dimension r < p is already contained in ∂ ∆[ p ] ⊗ ∆[ m ] . Moreover, for any r -simplex σ : ∆[ r ] → ∆[ p ] t ⊗ ∆[ m ] for r ≥ p we can consider the map of simplicial sets ∆[ r ] → (∆[ p ] t ⊗ ∂ ∆[ m ]) ∐ ∆[ p ] ⊗ ∂ ∆[ m ] (∆[ p ] ⊗ ∆[ m ]) , nd realize ∆[ p ] t ⊗ ∆[ m ] as the pushout along the union of many trivialityanodyne extensions: ` σ ∆[ r ] ` σ ∆[ r ] t (∆[ p ] t ⊗ ∂ ∆[ m ]) ∐ ∆[ p ] ⊗ ∂ ∆[ m ] (∆[ p ] ⊗ ∆[ m ]) ∆[ p ] t ⊗ ∆[ m ] In particular, the inclusion in question is an ( ∞ , N ) -acyclic cofibration, asdesired. (cid:3) Proposition 2.10.
Let N ∈ N ∪ {∞} . For any m ≥ and ℓ ≥ − thepushout-tensor (∆[ ℓ | ♯ ] ⊗ ∂ ∆[ m ]) ∐ ∆[ ℓ | eq ] ⊗ ∂ ∆[ m ] (∆[ ℓ | eq ] ⊗ ∆[ m ]) → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] of a saturation anodyne extension ∆[ ℓ | eq ] → ∆[ ℓ | ♯ ] with a boundary inclu-sion ∂ ∆[ m ] ֒ → ∆[ m ] is an ( ∞ , ∞ ) -acyclic cofibration. The proof of this proposition is postponed until the last section.We can now prove the theorem.
Proof of Theorem 2.1.
To see that − ⊠ S : m s S et ( ∞ ,N ) → m s S et ( ∞ ,N ) is aleft Quillen functor, we observe the following. • Since the underlyng simplicial set of the pretensor of simplicial sets withmarking is product of the underlying simplicial sets, if X → Y is amonomorphism, then X ⊠ S → Y ⊠ S is a monomorphism at the levelof underlying simplicial sets. In particular, − ⊠ S preserves cofibrations. • If I → J is an elementary anodyne extension, the map I ⊠ S → J ⊠ S canbe written as the pushout product J ⊠ ∆[ − ∐ I ⊠ ∆[ − I ⊠ S → J ⊠ S. It can then be deduced from Propositions 2.7 and 2.8 using the compat-ibility of pushouts and pretensor product with colimits that the functor − ⊠ S sends all elementary anodyne extensions to weak equivalences.By Proposition 1.10, we then conclude that the functor − ⊠ S is a left Quillenfunctor, as desired. (cid:3) Proof of Proposition 2.10.
In this subsection we provide the lastmissing verification.
Remark . A non-degenerate r -simplex σ : ∆[ r ] → ∆[ ℓ + 4] × ∆[ m ] ismarked in ∆[ ℓ | eq ] ⊗ ∆[ m ] (resp. ∆[ ℓ | ♯ ] ⊗ ∆[ m ] ) if and only if This reasoning is inspired by [Ver08a, Lemma 129]. the second projection pr σ is degenerate (in particular there exists a max-imal ≤ h ≤ r such that pr σ ( h −
1) = pr σ ( h ) and we call this h the degeneracy index of σ ), and • the partition face ∐ h,r − h of the first projection (pr σ ) ◦ ∐ h,r − h is markedin ∆[ ℓ | eq ] (resp. ∆[ ℓ | ♯ ] ).Informally speaking, the degeneracy index h of a simplex σ is the maximalvalue for which σ ( h ) is the final point of a horizontal piece in the path thatdescribes the simplex σ . Proof of Proposition 2.10.
For simplicity of notation, we write S := (∆[ ℓ | ♯ ] ⊗ ∂ ∆[ m ]) ∐ ∆[ ℓ | eq ] ⊗ ∂ ∆[ m ] (∆[ ℓ | eq ] ⊗ ∆[ m ]) . and we show by induction on l that the map S → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] is anacyclic cofibration for any m ≥ and any ℓ ≥ − .The simplicial sets with marking S and ∆[ ℓ | ♯ ] ⊗ ∆[ m ] have the sameunderlying simplicial set, isomorphic to ∆[ ℓ +4] × ∆[ m ] . By Remark 2.11, the r -simplices of ∆[ ℓ | ♯ ] ⊗ ∆[ m ] that are not marked in S are then characterizedas follows: An r -simplex is marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S if andonly if • the second projection pr σ is surjective, so in particular r ≥ m and (pr σ ) ◦ ∐ h,r − h = id ∆[ r − h ] : ∆[ r − h ] → ∆[ r − h ] , and • the partition face ∐ h,r − h of the first component pr σ is of the form (pr σ ) ◦ ∐ h,r − h = σ ′ ⋆ σ ′′ : ∆[ h − ⋆ ∆[1] → ∆[ ℓ ] ⋆ ∆[3] with σ ′′ ∈ { [01] , [03] , [12] , [23] } and σ ′ non-degenerate.We will now mark all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S by constructing a sequence of entire acyclic cofibrations S ֒ → S ֒ → S ֒ → S ֒ → S ֒ → S ֒ → S ∼ = ∆[ ℓ | ♯ ] ⊗ ∆[ m ] , which will prove the lemma. More precisely, we will mark(1) in S exactly all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S thatare contained in a copy of ∆[ ℓ − | ♯ ] ⊗ ∆[ m ] ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] by meansof induction hypothesis if ℓ > − .(2) in S all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ ∈{ [03] , [23] } (as well as other simplices) by means of saturation extensions.The generic simplex σ that is being marked in S can be depicted asfollows. . .. . .... ... . . . .... . . ℓ + 1 ℓ + 2 ℓ + 3 ℓ + 4 ... ... ... .... . .. . .... ... . . . ... ... ... ... .... . . σ ( h ) ℓ m (3) in S exactly all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [12] by means of thinness extensions. The generic simplex σ thatis being marked in S can be depicted as follows. . . .. . .... ... . . . .... . . ℓ + 1 ℓ + 2 ℓ + 3 ℓ + 4 ... ... ... .... . .. . .... ... . . . ... ... ... ... .... . . σ ( z ) σ ( h ) ℓ m (4) in S exactly all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [01] and pr σ hitting at most one of the values l + 3 and l + 4 by means of thinness extensions. The generic simplex σ that is beingmarked in S can be depicted as follows. . .. . .... ... . . . .... . . ℓ + 1 ℓ + 2 ℓ + 3 ℓ + 4 ... ... ... .... . .. . .... ... . . . ... ... ... ... .... . . σ ( z ) σ ( h ) ℓ m (5) in S exactly all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [01] and pr σ hitting both l + 3 and l + 4 , with last appearances of l + 2 and l + 3 in consecutive positions by means of thinness extensions.The generic simplex σ that is being marked in S can be depicted asfollows. . . .. . .... ... . . . .... . . ℓ + 1 ℓ + 2 ℓ + 3 ℓ + 4 ... ... ... .... . .. . .... ... . . . ... ... ... ... .... . .. . . σ ( z ) σ ( h ) ℓ m (6) in S exactly all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S (which in particular have σ ′′ = [01] and pr σ hitting both ℓ + 3 and ℓ + 4 ,with last appearances of ℓ + 2 and ℓ + 3 not in consecutive positions)by means of thinness extensions. The generic simplex σ that is beingmarked in S can be depicted as follows. . .. . .... ... . . . .... . . ℓ + 1 ℓ + 2 ℓ + 3 ℓ + 4 ... ... ... .... . .. . .... ... . . . ... ... ... ... .... . .. . .. . . σ ( z ) σ ( w ) σ ( h ) ℓ m We now proceed to explaining how to build the desired filtrations.(1) For ℓ = − we set S = S , and for ℓ > − we will obtain S from S by marking exactly all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S that are contained in a copy of ∆[ ℓ − | ♯ ] ⊗ ∆[ m ] ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] .For each ≤ i ≤ ℓ , we consider the map of simplicial sets with marking ∆[ ℓ − | eq ] ⊗ ∆[ m ] → S induced by the i -th face, and we can thenexpress the inclusions S ֒ → S as the pushout with a disjoint union ofthe inclusion ∆[ ℓ − | ♯ ] ⊗ ∂ ∆[ m ] ∐ ∆[ ℓ − | eq ] ⊗ ∂ ∆[ m ] ∆[ ℓ − | eq ] ⊗ ∆[ m ] ֒ → ∆[ ℓ − | ♯ ] ⊗ ∆[ m ] which are acyclic cofibrations given by the induction hypothesis: ` i ∈ [ l ] ∆[ ℓ − | ♯ ] ⊗ ∂ ∆[ m ] ∐ ∆[ ℓ − | eq ] ⊗ ∂ ∆[ m ] ∆[ ℓ − | eq ] ⊗ ∆[ m ] ` i ∈ [ l ] ∆[ ℓ − | ♯ ] ⊗ ∆[ m ] S S . In particular, S ֒ → S is an acyclic cofibration. Moreover, we have aninduced inclusion S ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] . (2) We obtain S from S by marking in particular for m ≤ r ≤ ℓ +4+ m all r -simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ ∈ { [03] , [23] } ,as well as some additional simplices. For any m ≤ r ≤ ℓ + 4 + m andany ~b := ( b ≤ . . . ≤ b ℓ ) an increasing sequence in [0 , ℓ + 4 + m − r ] , we argue that the simplicialmap ϕ : ∆[ l ] ⋆ ∆[3] ⋆ ∆[ r − ℓ − → ∆[ ℓ + 4] × ∆[ m ] efined by the formula i ( i, b i ) if ≤ i ≤ ℓ ( i, ℓ + 4 + m − r ) if ℓ + 1 ≤ i ≤ ℓ + 4( ℓ + 4 , m − r + i ) if ℓ + 5 ≤ i ≤ r is in particular a map of simplicial sets with marking ∆[ ℓ | eq | r − ℓ − → S .To see this, we suppose that γ ⋆ γ ⋆ γ : ∆[ r ] ⋆ ∆[ r ] ⋆ ∆[ r ] → ∆[ ℓ ] ⋆ ∆[3] ⋆ ∆[ r − ℓ − is a generic marked and non-degenerate ( r + 1 + r + 1 + r ) -simplex of ∆[ ℓ ] ⋆ ∆[3] ⋆ ∆[ r − ℓ − and we prove that the ( r +1+ r +1+ r ) -simplexof ∆[ ℓ + 4] × ∆[ m ] defined by the composite of maps of simplicial sets ∆[ r ] ⋆ ∆[ r ] ⋆ ∆[ r ] γ ⋆γ ⋆γ −−−−−→ ∆[ ℓ ] ⋆ ∆[3] ⋆ ∆[ r − ℓ − ϕ −→ ∆[ ℓ + 4] × ∆[ m ] is marked in ∆[ ℓ | eq ] ⊗ ∆[ m ] . Since γ ⋆γ ⋆γ is marked, one amongst the γ i ’s must be marked, and since moreover γ ⋆γ ⋆γ is non-degenerate thesimplex γ must be marked in ∆[3] eq . By construction, the degeneracyindex of the composite ϕ ◦ ( γ ⋆ γ ⋆ γ ) is r + 1 . Moreover, we see thatthe partition face ∐ r +1+ r ,r +11 of the first component of ϕ ◦ ( γ ⋆ γ ⋆ γ ) is of the form ∆[ r ] ⋆ ∆[ r ] γ ⋆γ −−−→ ∆[ ℓ ] ⋆ ∆[3] → ∆[ ℓ | eq ] and it is marked because it is the join of the marked simplex γ : ∆[ r ] → ∆[3] eq with another simplex of the form ∆[ r ] → ∆[ ℓ ] . This proves thatthe simplicial map ϕ does indeed preserve the marking.We then define the inclusion S ֒ → S as the pushout with the unionof a family of saturation extensions (which are acyclic cofibrations byLemma 1.9) of the form ∆[ ℓ | eq | r − ℓ − → ∆[ ℓ | ♯ | r − ℓ − : ` r ` ~b ∆[ ℓ | eq | r − ℓ − ` r ` ~b ∆[ ℓ | ♯ | r − ℓ − S S . In particular, S ֒ → S is an acyclic cofibration and we have addedall simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ ∈{ [03] , [23] } . Moreover, with a reasoning similar to the one producingthe map ∆[ ℓ | eq | r − ℓ − → S , one can show that there is an inducedmap S ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] . (3) We obtain S from S by marking for m ≤ r ≤ ℓ + 4 + m all r -simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [12] . For any such σ in question there is a degeneracy index h > and a unique maximal h ≤ z ≤ r so that pr σ ( z ) = ℓ + 3 . In particular, r − m ≤ z ≤ r . he new markings will be added by constructing a sequence of acycliccofibrations S =: S (0)2 ֒ → S (1)2 ֒ → · · · ֒ → S ( z )2 ֒ → S ( z +1)2 ֒ → · · · ֒ → S ( ℓ +4+ m )2 =: S such that S ( z )2 contains all missing markings for simplices of a given z .For any σ with a given z , the simplicial map ψ : ∆[ r + 1] → ∆[ ℓ + 4] × ∆[ m ] defined by the formula i σ ( i ) if ≤ i ≤ z ( ℓ + 4 , m − r + z ) if i = z + 1 σ ( i − if z + 1 < i ≤ r + 1 is in particular a map of simplicial sets with marking ∆ z +1 [ r + 1] ′ → S ( z − .To see this, we consider a non-degenerate marked s -simplex τ : ∆[ s ] → ∆[ r +1] of ∆ z +1 [ r +1] ′ , and we prove that the s -simplex of ∆[ ℓ +4] × ∆[ m ] defined by the composite of map of simplicial sets ψ ◦ τ : ∆[ s ] τ −→ ∆[ r + 1] ψ −→ ∆[ ℓ + 4] × ∆[ m ] is marked in S ( z − . • If τ contains { z, z + 1 , z + 2 } ∩ [ r + 1] , by construction the secondprojection of ψ ◦ τ is degenerate, with degeneracy index being thepreimage of z + 1 in ∆[ s ] . Moreover, the face partition of the firstcomponent of ψ ◦ τ contains the edge [( ℓ + 3)( ℓ + 4)] in ∆[ ℓ + 4] andso the simplex ψ ◦ τ is marked in S . • If τ = d z +2 , by construction the second projection of ψ ◦ τ is notsurjective (as it misses the value m − r + z +1 ) and moreover degenerate,with degeneracy index being the preimage of z + 1 in ∆[ r ] . Moreover,the face partition of the first component of ψ ◦ τ hits at least a -dimensional simplex of ∆[3] . In particular, ψ ◦ τ is marked already in ∆[ ℓ | ♯ ] ⊗ ∂ ∆[ m ] . • If τ = d z , we distinguish two cases. If z = h , by construction thesecond projection of ψ ◦ τ is degenerate, with degeneracy index z = h .Moreover, the face partition of the first component of ψ ◦ τ containsthe edge [( ℓ + 2)( ℓ + 4)] in ∆[ ℓ + 4] and so the simplex ψ ◦ τ is markedin S . If h < z , by construction the second projection of ψ ◦ τ isdegenerate, with degeneracy index h . Moreover, the face partition ofthe first component of ψ ◦ τ contains the edge [( ℓ + 2)( ℓ + 3)] in ∆[ ℓ + 4] and in fact the marking of the simplex ψ ◦ τ was added in S ( z − .This proves that the simplicial map ψ does indeed preserve the marking.We then define the inclusion S ( z − ֒ → S ( z )2 as the pushout with severalthinness extensions ∆ z +1 [ r + 1] ′ → ∆ z +1 [ r + 1] ′′ (as many as r -simplices as z varies): ` r ` z ` σ ∆ z +1 [ r + 1] ′ ` r ` z ` σ ∆ z +1 [ r + 1] ′′ S ( z − S ( z )2 . In particular S ( z − ֒ → S ( z )2 is an acyclic cofibration. Moreover, byconstruction there is an induced map S ( z )2 ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] . We then set S := S ( r )2 , so that in particular S ֒ → S is an acycliccofibration and we have marked all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [12] . Moreover, by construction we have aninduced map S ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] (4) We obtain S from S by marking for m ≤ r ≤ ℓ + 4 + m all r -simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [01] and pr σ hittingat most one of the values ℓ +3 and ℓ +4 . For any such σ in question thereis a unique maximal h ≤ z ≤ r so that pr σ ( z ) = ℓ + 2 . In particular, r − m ≤ z ≤ r . We will add the missing simplices by constructing asequence of anodyne extensions S =: S (0)3 ֒ → S (1)3 ֒ → · · · ֒ → S ( z − ֒ → S ( z )3 ֒ → · · · ֒ → S ( ℓ +4+ m )3 =: S such that S ( z )3 contains all missing simplices for a given z . For any σ with a given z , the simplicial map ψ : ∆[ r + 1] → ∆[ ℓ + 4] × ∆[ m ] defined by the formula i σ ( i ) if ≤ i ≤ z ( ℓ + 4 , z ) if i = z + 1 and pr σ ( z + 1) = ℓ + 4 or z = r, ( ℓ + 3 , z ) if i = z + 1 and pr σ ( z + 1) = ℓ + 3 ,σ ( i − if z + 1 < i ≤ r + 1 is in particular a map of simplicial sets with marking ∆ z +1 [ r + 1] ′ → S ( z − .To see this, we consider a non-degenerate marked s -simplex τ : ∆[ s ] → ∆[ r +1] of ∆ z +1 [ r +1] ′ , and we prove that the s -simplex of ∆[ ℓ +4] × ∆[ m ] defined by the composite of maps of simplicial sets ψ ◦ τ : ∆[ s ] τ −→ ∆[ r + 1] ψ −→ ∆[ ℓ + 4] × ∆[ m ] is marked in S ( z − . If τ contains { z, z + 1 , z + 2 } ∩ [ r + 1] , by construction the secondprojection of ψ ◦ τ is degenerate, with degeneracy index being thepreimage of z + 1 in ∆[ s ] . Moreover, the face partition of the firstcomponent of ψ ◦ τ contains the edge [( ℓ + 2)( ℓ + 3)] or [( ℓ + 2)( ℓ + 4)] in ∆[ ℓ + 4] and so the simplex ψ ◦ τ is marked in S . • If τ = d z +2 , by construction the second projection of ψ ◦ τ is notsurjective (as it misses the value m − r + z +1 ) and moreover degenerate,with degeneracy index being the preimage of z + 1 in ∆[ r ] . Moreover,the face partition of the first component of ψ ◦ τ hits at least a -dimensional simplex of ∆[3] . In particular, ψ ◦ τ is marked already in ∆[ ℓ | ♯ ] ⊗ ∂ ∆[ m ] . • If τ = d z , we distinguish two cases. If h = z , by construction thesecond projection of ψ ◦ τ is degenerate, with degeneracy index h = z .Moreover, the face partition of the first component of ψ ◦ τ containsthe edge [( ℓ + 1)( ℓ + 3)] or [( ℓ + 1)( ℓ + 4)] in ∆[ ℓ + 4] and so the simplex ψ ◦ τ is marked in S . If h < z , by construction the second projectionof ψ ◦ τ is degenerate, with degeneracy index h . Moreover, the facepartition of the first component of ψ ◦ τ contains the edge [( ℓ +1)( ℓ +2)] in ∆[ ℓ + 4] and in fact the marking of the simplex ψ ◦ τ was added in S ( z − .This proves that the simplicial map ψ does indeed preserve the marking.We then define the inclusion S ( z − ֒ → S ( z )3 as the pushout with manythinness anodyne extensions ∆ z +1 [ r + 1] ′ → ∆ z +1 [ r + 1] ′′ (as many as r -simplices σ as z varies): ` r ` z ` σ ∆ z +1 [ r + 1] ′ ` r ` z ` σ ∆ z +1 [ r + 1] ′′ S ( z − S ( z )3 . In particular S ( z − ֒ → S ( z )3 is an acyclic cofibration. Moreover, we havean induced map S ( z )4 ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] We then set S := S ( ℓ +4+ m )3 , so that in particular S ֒ → S is an acycliccofibration and we have marked all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [01] and pr σ hitting at most one of the values ℓ + 3 and ℓ + 4 . Moreover, we have an induced map S ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] . (5) We obtain S from S by marking for m ≤ r ≤ ℓ + 4 + m all r -simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [01] and pr σ hitting both ℓ + 3 and ℓ + 4 , with last appearances of ℓ + 2 and ℓ + 3 inconsecutive positions. More precisely, for any such σ in question thereis a unique maximal h + 1 ≤ z ≤ r − so that pr σ ( z ) = ℓ + 3 , and by ssumption pr σ ( z + 1) = ℓ + 4 and pr σ ( z −
1) = ℓ + 2 . In particular, r − m + 1 ≤ z ≤ ℓ + 3 + m . For any σ with a given z , the simplicial map ψ : ∆[ r + 1] → ∆[ ℓ + 4] × ∆[ m ] defined by the formula i σ ( i ) if ≤ i ≤ z ( m − r + z, ℓ + 4) if i = z + 1 σ ( i − if z + 1 < i ≤ r + 1 is in particular a map of simplicial sets with marking ∆ z +1 [ r + 1] ′ → S .To see this, we consider a non-degenerate marked s -simplex τ : ∆[ s ] → ∆[ r +1] of ∆ z +1 [ r +1] ′ , and we prove that the s -simplex of ∆[ ℓ +4] × ∆[ m ] defined by the composite of maps of simplicial sets ψ ◦ τ : ∆[ s ] τ −→ ∆[ r + 1] ψ −→ ∆[ ℓ + 4] × ∆[ m ] is marked in S . • If τ contains { z, z + 1 , z + 2 } ∩ [ r + 1] , by construction the secondprojection of ψ ◦ τ is degenerate, with degeneracy index being thepreimage of z + 1 in ∆[ s ] . Moreover, the face partition of the firstcomponent of ψ ◦ τ contains the edge [( ℓ + 3)( ℓ + 4)] in ∆[ ℓ + 4] andso the simplex ψ ◦ τ is marked in S . • If τ = d z +2 , by construction the second projection of ψ ◦ τ is notsurjective (as it misses the value m − r + z +1 ) and moreover degenerate,with degeneracy index being the preimage of z + 1 in ∆[ r ] . Moreover,the face partition of the second component of ψ ◦ τ hits at least a -dimensional simplex of ∆[3] . In particular, ψ ◦ τ is marked alreadyin ∆[ ℓ | ♯ ] ⊗ ∂ ∆[ m ] . • If τ = d z , by construction the second projection of ψ ◦ τ is degenerate,with degeneracy index h . Moreover, the face partition of the firstcomponent of ψ ◦ τ contains the edge [( ℓ + 1)( ℓ + 2)] in ∆[ ℓ + 4] anddoes not hit ℓ + 3 , so the marking of the simplex ψ ◦ τ was added in S .This proves that the simplicial map ψ does indeed preserve the marking.We define the inclusion S ֒ → S as the pushout with several thinnessextensions ∆ z +1 [ r + 1] ′ → ∆ z +1 [ r + 1] ′′ (as many as r -simplices σ as z varies): ` r ` z ` σ ∆ z +1 [ r + 1] ′ ` r ` z ` σ ∆ z +1 [ r + 1] ′′ S S . In particular S ֒ → S is an acyclic cofibration and we have marked allsimplices in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [01] and pr σ hitting oth the values ℓ + 3 and ℓ + 4 , with last appearances of ℓ + 2 and ℓ + 3 in consecutive positions. Moreover, we have an induced map S ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] . (6) We obtain S from S by marking for m ≤ r ≤ ℓ + 4 + m all missing r -simplices with σ ′′ = [01] and pr σ hitting both ℓ + 3 and ℓ + 4 , withlast appearances of ℓ + 2 and ℓ + 3 not in consecutive positions. Moreprecisely, for any such σ in question there is a unique maximal h < z < r so that pr σ ( z ) = ℓ + 3 . In particular, r − m ≤ z ≤ ℓ + 3 + m . We willadd them by constructing a sequence of acyclic cofibrations S =: S ( r − m )5 ֒ → S ( r − m +1)5 ֒ → · · · ֒ → S ( z − ֒ → S ( z )5 ֒ → · · · ֒ → S ( ℓ +3+ m )5 =: S such that S ( z )5 contains all missing simplices for a given z . For any σ with a given z , the simplicial map ψ : ∆[ r + 1] → ∆[ ℓ + 4] × ∆[ m ] defined by the formula i σ ( i ) if ≤ i ≤ z ( ℓ + 4 , m − r + z ) if i = z + 1 σ ( i − if z + 1 < i ≤ r + 1 is in particular a map of simplicial sets with marking ∆ z +1 [ r + 1] ′ → S ( z − .To see this, we consider a non-degenerate marked s -simplex τ : ∆[ s ] → ∆[ r +1] of ∆ z +1 [ r +1] ′ , and we prove that the s -simplex of ∆[ ℓ +4] × ∆[ m ] defined by the composite of maps of simplicial sets ψ ◦ τ : ∆[ s ] τ −→ ∆[ r + 1] ψ −→ ∆[ ℓ + 4] × ∆[ m ] is marked in S ( z − . • If τ contains { z, z + 1 , z + 2 } ∩ [ r + 1] , by construction the secondprojection of ψ ◦ τ is degenerate, with degeneracy index being thepreimage of z + 1 in ∆[ s ] . Moreover, the face partition of the firstcomponent of ψ ◦ τ contains the edge [( ℓ + 3)( ℓ + 4)] in ∆[ ℓ + 4] andso the simplex ψ ◦ τ is marked in S . • If τ = d z +2 , by construction the second projection of ψ ◦ τ is notsurjective (as it misses the value m − r + z +1 ) and moreover degenerate,with degeneracy index being the preimage of z + 1 in ∆[ r ] . Moreover,the face partition of the first component of ψ ◦ τ hits at least a -dimensional simplex of ∆[3] . In particular, ψ ◦ τ is marked already in ∆[ ℓ | ♯ ] ⊗ ∂ ∆[ m ] . • If τ = d z , we distinguish two cases depending on the value of w , beingthe maximal value for which pr σ ( w ) = ℓ + 2 . By assumption, h ≤ w < z − . If w = z − , by construction the second projection of ψ ◦ τ is degenerate, with degeneracy index h . Moreover, the face partition ofthe first component of ψ ◦ τ contains the edge [( ℓ + 1)( ℓ + 2)] in ∆[ ℓ + 4] nd hits ℓ + 2 and ℓ + 3 in consecutive positions for the last time andso the marking of ψ ◦ τ was added in S . If w < z − , by constructionthe second projection of ψ ◦ τ is degenerate, with degeneracy index h .Moreover, the face partition of the second component of ψ ◦ τ containsthe edge [( ℓ + 1)( ℓ + 2)] in ∆[ ℓ + 4] and in fact the marking of thesimplex ψ ◦ τ was added in S ( z − .This proves that the simplicial map ψ does indeed preserve the marking.We then define the inclusion S ( z − ֒ → S ( z )5 as the pushout with severalthinness extensions ∆ z +1 [ r + 1] ′ → ∆ z +1 [ r + 1] ′′ (as many as r -simplices σ as z varies): ` r ` z ` σ ∆ z +1 [ r + 1] ′ ` r ` z ` σ ∆ z +1 [ r + 1] ′′ S ( z − S ( z )5 . In particular S ( z − ֒ → S ( z )5 is an acyclic cofibration. Moreover, we havean induced map S ( z )5 ֒ → ∆[ ℓ | ♯ ] ⊗ ∆[ m ] . We then set S := S ( ℓ +3+ m )5 , so that in particular S ֒ → S is an acycliccofibration and we have marked all simplices σ marked in ∆[ ℓ | ♯ ] ⊗ ∆[ m ] and not in S with σ ′′ = [01] and pr σ hitting both the values ℓ + 3 and ℓ + 4 , with last appearances of l + 2 and l + 3 not in consecutivepositions. In particular, we have an isomorphism S ∼ = ∆[ ℓ | ♯ ] ⊗ ∆[ m ] . This concludes the proof. (cid:3)
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Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Ger-many
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