Fundamental pushouts of n-complicial sets
aa r X i v : . [ m a t h . A T ] M a y FUNDAMENTAL PUSHOUTS OF n -COMPLICIAL SETS VIKTORIYA OZORNOVA AND MARTINA ROVELLI
Abstract.
The paper focuses on investigating how certain relationsbetween strict n -categories are preserved in a particular implementa-tion of ( ∞ , n ) -categories, given by saturated n -complicial sets. In thismodel, we show that the ( ∞ , n ) -categorical nerve of n -categories is ho-motopically compatible with -categorical suspension and wedge. Asan application, we show that certain pushouts encoding composition in n -categories are homotopy pushouts of saturated n -complicial sets. Introduction
Since 1950s, category theory has established itself as a language to phrasemathematical phenomena in a uniform way. Recent developments in thestudy of the cobordism hypothesis, in derived algebraic geometry and inbrave new algebra, highlighted the presence and role played by higher mor-phisms, as well as the fact that axioms defining a categorical structure shouldbe weakened, replacing equalities with higher isomorphisms. This perspec-tive sparked new interest in the study of generalizations of the notion ofan ordinary category, in the form of an n -category and then of an ( ∞ , n ) -category.While it is still unfolding its significance in algebraic topology, higher cate-gory theory arose in 1960s with the original purpose of encoding non-abeliancohomology into the language of n -categories . The notion of an n -categoryencapsulates the idea that beyond objects and morphisms between them,there are also morphisms between morphisms, called -morphisms, mor-phisms between those, called -morphisms, and so on up to level n . All thesemorphisms compose associatively along morphisms of lower dimensions.Composition of morphisms was traditionally requested to satisfy strictequational conditions, such as strict associativity, and this led to a rich theoryof strict enriched category theory. However, many examples of interest thatnaturally present a higher categorical structure, such as several categories ofcobordisms, derived categories, or the categorical structure given by points,paths and higher homotopies in a topological space, fail to satisfy theseaxioms. Mathematics Subject Classification.
Key words and phrases. n -categories, ( ∞ , n ) -categories, complicial sets, suspension -category, pushout of n -categories. eemingly very different in nature, the notion of an n -category had to thenbe weakened in order to accommodate homotopical phenomena, becoming it-self a homotopical notion, and in the 1990’s the notion of an ( ∞ , n ) -category started making its way. An ( ∞ , n ) -category should consist of objects, re-garded as -morphisms, and k -morphisms between ( k − -morphisms forany k ; these morphisms must moreover compose weakly associatively alongmorphisms of a lower dimension and are all weakly invertible for k > n .While the theory of strict n -categories is unambiguous, the defining guide-lines for the notion of ( ∞ , n ) -category have been given a precise meaningin different models. All models are conjecturally equivalent, although somecomparisons showing equivalences of the corresponding homotopy theoriesare still missing.Regardless of the model, the collection of ( ∞ , n ) -categories should as-semble at least into an ( ∞ , -category ( ∞ , n ) C at , enlarging the ( ∞ , -category of strict n -categories n C at , and the inclusion of ( ∞ , -categories N : n C at ֒ → ( ∞ , n ) C at has been realized in many models, often implementedby a type of nerve construction. It is interesting to understand how thisembedding behaves with natural constructions of a categorical flavour, giventhat nerve constructions typically behave poorly with respect to construc-tions involving left adjoint functors and colimits.The goals of this article is to show that in a specific model of ( ∞ , n ) -categories, Verity’s saturated n -complicial sets [Ver08a, Rie18, OR18], twotypes of constructions, suspension and wedge , are compatible with the em-bedding. As a motivating application, we show that the nerve embeddingpreserves certain fundamental relations between n -categories, that encodecomposition and invertibility of morphisms.In other models, such as Barwick’s n -fold complete Segal spaces [Bar05,Lur09b], Rezk’s Θ n -spaces [Rez10] and Ara’s n -quasicategories [Ara14], theanalogous statements are essentially part of the axioms. However, given thelack of model comparisons with saturated n -complicial sets for n ≥ andthe unexplored compatibility of existing model comparisons with the nerveembedding for n = 2 , the result could not be imported at no cost.In Section 2 we introduce the suspension of - and ( ∞ , -categories, whichcan be seen as a left adjoint to taking the hom - or ( ∞ , -category betweentwo objects of a - or ( ∞ , -category. Roughly speaking, the suspension ofa - or ( ∞ , -category P is a - or ( ∞ , -category with two objects and aunique interesting hom-category given by P . Then, we show in Section 3 asTheorem 2.9 the following compatibility of nerve and suspension. Theorem A.
In the model of saturated -complicial sets, for any -category P there is an equivalence of ( ∞ , -categories N (Σ P ) ≃ Σ( N P ) between the nerve of the suspension and the suspension of the nerve. n Section 4 we introduce the wedge of two n - or ( ∞ , n ) -categories, a par-ticular way of gluing along an object, and we show in Section 5 the followingcompatibility of nerve and wedge, which will appear as Theorem 4.9. Theorem B.
In the model of saturated n -complicial sets, for any n -categories A and A ′ there is an equivalence of ( ∞ , n ) -categories N ( A ∨ A ′ ) ≃ N A ∨ N A ′ between the nerve of their wedge and the wedge of their nerves. As anticipated, we now elaborate on how Theorems A and B can be usedto then show that the nerve embedding preserves certain valuable pushouts.For ≤ m ≤ n , an m -morphism of an n -category D is represented bya functor C m → D , where C m is the free m -cell, so one can regard all freecells as the building blocks of n -categories. For instance, the free -, - and -cells can be depicted as C = C = ⇓ C = . Composition operations are governed by pasting diagrams, which can berealized as certain pushouts of n -categories, which are instances of Barwick–Schommer-Pries’ “fundamental pushouts” from [BSP11]. For instance, com-position of -morphisms along objects and along -morphisms are encodedin the pushouts in n C at ( ∗ ) ⇓ ⇓⇓ ⇓ and ⇓ ⇓⇓⇓ In the new setup ( ∞ , n ) C at , one can make sense of cells and shapes ob-tained as the fundamental pushouts from ( ∗ ) as ( ∞ , n ) -categories. Cellsshould still detect morphisms, and the pushouts should still encode compo-sition of morphisms. However, for this to be meaningful, the fundamentalpushouts regarded as ( ∞ , n ) -categories must be also the resulting pushoutin ( ∞ , n ) -categories. It is therefore expected, and included in the axioms fora model of ( ∞ , n ) -categories in the sense of [BSP11], that the fundamentalpushouts are preserved by the embedding.Using Theorems A and B we can show that the fundamental pushouts from( ∗ ) are preserved in the model of n -complicial sets, providing in particulara first step towards proving the equivalence of saturated n -complicial setswith other models. ore precisely, as an instance of Theorem B, we obtain the followingcorollary, asserting the preservation of the first fundamental pushout from( ∗ ), which will appear as Corollary 4.10. Corollary A.
In the model of saturated n -complicial sets, there is an equiv-alence of ( ∞ , -categories N ⇓ ⇓ ! ≃ N ⇓ ! ∐ N ( ) N ⇓ ! . Secondly, using Theorem A, we prove the following corollary, which assertsthe preservation of the second fundamental pushout from ( ∗ ), and will appearas Corollary 2.11. Corollary B.
In the model of saturated -complicial sets, there is an equiv-alence of ( ∞ , -categories N ⇓⇓ ! ≃ N ⇓ ! ∐ N ( ) N ⇓ . Finally, Theorem A also yields the following corollary, which will appear asCorollary 2.10. It asserts that the nerve embedding preserves the equivalencebetween the free -cell C and the free living -isomorphism. This is anothercondition that in other models is encoded into a completeness axiom, and isinstead combinatorially involved in the model of saturated complicial sets. Corollary C.
In the model of saturated -complicial sets, there is an equiv-alence of ( ∞ , -categories N ⇓ ∼ = ! ≃ N ( ) . Acknowledgements.
We would like to thank Clark Barwick and ChrisSchommer-Pries for explanations concerning fundamental pushouts, and Len-nart Meier and Emily Riehl for helpful conversations on this project. Thismaterial is based upon work supported by the National Science Founda-tion under Grant No. DMS-1440140 while the authors were in residence atthe Mathematical Sciences Research Institute in Berkeley, California, dur-ing the Spring 2020 semester. The first-named author thankfully acknowl-edges the financial support by the DFG grant OZ 91/2-1 with the projectnr. 442418934. ontents Introduction 11. Background on n -complicial sets 52. Nerve vs suspension - The results 123. Nerve vs suspension - The proofs 204. Nerve vs wedge - The results 265. Nerve vs wedge - The proofs 32References 371. Background on n -complicial sets We assume the reader to be familiar with the basics of strict higher cat-egory theory (see e.g. [Lei04]) and with the model categorical language (seee.g. [Hir03, Hov99]), and we recall the preliminary material that will be usedin the paper.The category n C at of n -categories is defined recursively as the categoryof categories enriched over the category of ( n − -categories, assuming thatthe category of -categories is the category S et of sets with the cartesianproduct. In particular, an n -category D consists of a set of objects and forany objects x, x ′ an ( n − -category Map D ( x, x ′ ) , together with a horizontalcomposition that defines a functor of hom- ( n − -categories ◦ : Map D ( x, x ′ ) × Map D ( x ′ , x ′′ ) → Map D ( x, x ′′ ) . For n = ∞ , the convention above specializesto an ω -category , as in [Str87, Ver08a].The following model structure models the standard homotopy theory of n -categories. It recovers the canonical model structure for -categories aswell as Lack’s model structure for -categories from [Lac02]. Theorem 1.1 ([LMW10, Thm 5]) . Let n ∈ N ∪ {∞} . The category n C at supports a cofibrantly generated model structure in which • all n -categories are fibrant; • the weak equivalences are precisely the n -categorical equivalences. In this paper, we will consider a model of ( ∞ , n ) -categories due to Veritybased on the following mathematical object. Definition 1.2. A simplicial set with marking is a simplicial set endowedwith a subset of simplices of strictly positive dimensions that contain all de-generate simplices, called thin or marked . We denote by m s S et the categoryof simplicial sets with marking and marking preserving simplicial maps. Originally referred to as simplicial set with hollowness in [Str87] and later as stratifiedsimplicial set e.g. in [Ver08a]. emark . The underlying simplicial set functor m s S et → s S et respectslimits and colimits, since it is both a left and a right adjoint (see e.g. [Ver08a,Obs. 97]), and it preserves and reflects monomorphisms, since it is a faithfulright adjoint. Moreover, as explained in [Ver08a, Obs. 109], • a simplex is marked in a limit of simplicial sets with marking lim i ∈ I X i ifand only if it is marked in each component X i for i ∈ I , and • a simplex is marked in a colimit of simplicial sets with marking colim i ∈ I X i if and only if it admits a marked representative in X i for some i ∈ I .The following model structure provides a model for the homotopy theoryof ( ∞ , n ) -categories. It is obtained applying Verity’s machinery [Ver08a,§6.3] to a special set of anodyne extensions, described in [Rie18] and recalledin Definition 1.7. Theorem 1.4 ([OR18, Thm 1.28]) . Let n ∈ N ∪ {∞} . The category m s S et supports a cofibrantly generated left proper model cartesian structure where • the fibrant objects are precisely the saturated n -complicial sets , i.e., thosewith the right lifting property with respect to the elementary anodyne ex-tensions, recalled in Definition 1.7; • the cofibrations are precisely the monomorphisms (of underlying simplicialsets).We call this model structure the model structure for ( ∞ , n ) -categories , andwe call the weak equivalences the ( ∞ , n ) -weak equivalences. The interpretation is that, in a saturated n -complicial sets, the marked k -simplices are precisely the k -equivalences. We refer the reader e.g. to [Rie18]for further elaboration on this viewpoint.In order to recall the elementary anodyne extensions, we need also thefollowing preliminary terminology and notation. Definition 1.5.
A sub-simplicial set with marking X of a simplicial set withmarking Y is regular if a simplex of X is marked in X if and only if it ismarked in Y . Notation 1.6.
We denote • by ∆[ m ] the standard m -simplex in which exactly the degenerate simplicesare marked; • by ∆[ m ] t the standard m -simplex in which the only marked non-degeneratesimplex is the top-dimensional one; • by ∆ k [ m ] , for ≤ k ≤ m , the standard m -simplex in which a non-degenerate simplex is marked if and only if it contains the vertices { k − , k, k + 1 } ∩ [ m ] ; • by ∆ k [ m ] ′ , for ≤ k ≤ m , the standard m -simplex with marking obtainedfrom ∆ k [ m ] by additionally marking the ( k − -st and ( k + 1) -st face of ∆[ m ] ; by ∆ k [ m ] ′′ , for ≤ k ≤ m , the standard m -simplex with obtained from ∆ k [ m ] ′ by additionally marking the k -th face of ∆[ m ] ; • by Λ k [ m ] , for ≤ k ≤ m , the regular sub-simplicial set of ∆ k [ m ] withmarking whose simplicial set is the k -horn Λ k [ m ] ; • by ∆[3] eq the -simplex in which the non-degenerate marked simplicesconsist of all - and -simplices, as well as -simplices [02] and [13] ; • by ∆[3] ♯ the -simplex in which all simplices in positive dimensions aremarked. Definition 1.7.
Let n ∈ N ∪ {∞} . An ( ∞ , n ) -elementary anodyne extension is one of the following maps of simplicial sets with marking.(1) The complicial horn extension , i.e., the canonical map Λ k [ m ] → ∆ k [ m ] for m ≥ and ≤ k ≤ m, which is an ordinary horn inclusion on the underlying simplicial sets.(2) The thinness extension , i.e., the canonical map ∆ k [ m ] ′ → ∆ k [ m ] ′′ for m ≥ and ≤ k ≤ m, which is an identity on the underlying simplicial set.(3) The triviality extension map, i.e., the canonical map ∆[ l ] → ∆[ l ] t for l > n, which is an identity on the underlying simplicial set.(4) The saturation extension , i.e., the canonical map ∆[3] eq ⋆ ∆[ l ] → ∆[3] ♯ ⋆ ∆[ l ] for l ≥ − which is an identity on the underlying simplicial set. Here, the con-struction ⋆ denotes the join construction of simplicial sets with marking,which is recalled in Definition 2.4.Although there is no explicit description of generating acyclic cofibrationsfor this model structure, the elementary anodyne extensions provide a goodapproximation, in the sense of the following lemma. Lemma 1.8.
A functor F : m s S et → M is left Quillen when m s S et isendowed with the model category for ( ∞ , n ) -categories and M is any modelcategory if and only if F is a left adjoint, it respects cofibrations and sendsall elementary anodyne extensions from Definition 1.7 to weak equivalencesof M . Note that the last condition was phrased slightly different in [OR18], namely we used aselementary saturation anodyne extensions the maps ∆[ l ] ⋆ ∆[3] eq → ∆[ l ] ⋆ ∆[3] ♯ for l ≥ − .As a consequence of the discussion following [RV20a, Def. D.7.9], the model structuresresulting from both conditions are equal (in the presence of the remaining elementaryanodyne extensions). We chose to work with this convention to simplify the proof ofProposition 2.5. roof. By Cisinski–Olschok theory (see e.g. [Ols09, Theorem 3.16, Lemma3.30]), one can show that the fibrations between fibrant objects in the modelstructure for ( ∞ , n ) -categories are precisely the maps having the right lift-ing property with respect to the elementary anodyne extensions from Defi-nition 1.7. By adjointness, if F is a left adjoint functor that respects cofibra-tions, it sends elementary anodyne extensions to weak equivalences if andonly if the right adjoint preserves fibrations between fibrant objects. By[JT07, Proposition 7.15], this is equivalent to saying that F is a left Quillenfunctor, as desired. (cid:3) As a special case of the slice model structures, constructed e.g. in [Hir15],we also obtain model structure on the category m s S et ∗ of pointed simplicialsets with marking and on the category m s S et ∗ , ∗ of bi-pointed simplicial setswith marking. Proposition 1.9.
The category m s S et ∗ , resp. m s S et ∗ , ∗ , supports a cofi-brantly generated left proper model structure where • the fibrant objects are precisely the pointed, resp. bipointed, simplicial setswith marking whose underlying simplicial sets with marking are saturated n -complicial sets. • the cofibrations are precisely the monomorphisms (on underlying simplicialsets).We call this model structure the model structure for pointed ( ∞ , n ) -categories ,resp. the model structure for bi-pointed ( ∞ , n ) -categories . We fix the following terminology.
Definition 1.10.
A map of simplicial sets with marking X → Y is a compli-cial inner anodyne extension if it can be written as a retract of a transfinitecomposition of pushouts of maps of the following form:(1) inner complicial horn extensions Λ k [ m ] → ∆ k [ m ] for m > and < k < m, (2) complicial thinness extensions ∆ k [ m ] ′ → ∆ k [ m ] ′′ for m ≥ and ≤ k ≤ m. Remark . One can prove with standard model categorical techniques thefollowing formal properties of complicial inner anodyne extensions.(1) Any complicial inner anodyne extension is an ( ∞ , n ) -acyclic cofibration.(2) The underlying simplicial map of a complicial inner anodyne extensionis an inner anodyne extension of simplicial sets.(3) The class of complicial inner anodyne extensions is closed under transfi-nite composition and pushouts. e will produce several complicial inner anodyne extensions using thefollowing one. Lemma 1.12.
For m ≥ and < k < m , let Λ k [ m ] ′ denote the regularsubset of ∆ k [ m ] ′ whose underlying simplicial set is given by the k -horn Λ k [ m ] .The inclusion Λ k [ m ] ′ → ∆ k [ m ] ′′ for m ≥ , < k < m is a complicial inner anodyne extension.Proof. The desired inclusion can be written as a composite Λ k [ m ] ′ ֒ → ∆ k [ m ] ′ ֒ → ∆ k [ m ] ′′ where the second arrow is a thinness anodyne extension and the first arrowis a pushout of an elementary inner complicial inner horn extension Λ k [ m ] Λ k [ m ] ′ ∆ k [ m ] ∆ k [ m ] ′ . This proves the claim. (cid:3)
For any n -category D , Street [Str87] defined a simplicial nerve N D interms of the n -truncated orientals O n [ m ] . The n -category O n [ m ] should bethought as the free n -category over an m -simplex. For a precise account onorientals we refer the reader to [Str87] or [AM16, §7].When n = 2 , we will make use of the following explicit description of the -truncated oriental. Definition 1.13.
Let m ≥ . The -truncated m -oriental is the -category O [ m ] in which(0) there are m + 1 objects x , . . . , x m ;(1) the -morphisms are freely generated under composition by the -morphisms f ij : x i → x j for i ≤ j ;(2) the -morphisms are generated under composition by the -morphisms α ijk : f ik ⇒ f jk ◦ f ij for i < j < k , subject to the relations that for any i < j < k < s (id f ks ◦ h α ijk ) ◦ v α iks = ( α jks ◦ h id f ij ) ◦ v α ijs . Remark . When regarded as a simplicial category, O [ m ] is isomorphic to C [∆[ m ]] , the homotopy coherent realization of the standard simplex, as stud-ied in [Lur09a, Def. 1.1.5.1]. In particular, there we find the following alter-native description. For any ≤ i, j ≤ m the hom-category Map O [ m ] ( x i , x j ) s given by Map O [ m ] ( x i , x j ) := [1] j − i − j > i [0] j = i ∅ j < i. This can be reformulated further saying that each -morphism of O [ m ] from x i to x j is uniquely represented as a subset of { i, i +1 , . . . , j − , j } containing i and j , and each -morphism is uniquely represented as an inclusion of suchsubsets.The geometry of orientals is such that one can define the following nerve. Definition 1.15.
Let n ∈ N ∪ {∞} . The Street nerve N D of an n -category D is the simplicial set in which • an m -simplex is an n -functor O n [ m ] → D . • the simplicial structure is induced by the geometry of orientals.For n = 2 , the Street nerve was studied in detail by Duskin in [Dus02],and can be described explicitly as follows. Definition 1.16.
The nerve N D of a -category D is the -coskeletal sim-plicial set in which(0) a -simplex consists of an object of D : x ; (1) a -simplex consists of a -morphism of D : x y ; a (2) a -simplex consists of a -cell of D of the form c ⇒ b ◦ a : yx z ; ba c (3) a -simplex consists of four -cells of D that satisfy the following relation. w z w z = x y x y e eda b c a d cf and in which the simplicial structure is as indicated in the pictures.The Street nerve can be endowed with the following marking, originallyconsidered by Roberts in unpublished work and Street in [Str87], furtherstudied by Verity in [Ver08a], and later discussed by Riehl in [Rie18]. efinition 1.17. Let n ∈ N ∪ {∞} . The Roberts–Street nerve is the simpli-cial set with marking N RS D , in which • the underlying simplicial set is the Street nerve N D , and • an m -simplex of N D is marked in N RS D if and only if the corresponding n -functor O n [ m ] → D sends the top-dimensional m -cell of O n [ m ] to anidentity of D . In particular, all simplices in dimension at least n + 1 aremarked.We will use the following pointset and homotopical properties of N RS . Proposition 1.18.
The Roberts–Street nerve N RS : n C at → m s S et • is a right adjoint functor, and in particular respects all limits; • is a homotopical functor between the model structure for n -categories andthe model structure for ( ∞ , n ) -categories if n ≤ .Proof. The fact that N RS is a right adjoint can be found in [Ver08a, §10.3].We now argue that if n ≤ the functor N RS is a homotopical functor, usingthe following auxiliary construction, considered e.g. in [Rie18, §3.2].Given any n -category for n ≤ , one can consider the simplicial set withmarking N ♮ D in which the simplicial set is N D and in which(1) a -simplex is marked in N ♮ D if and only if the representing -morphismin D is an equivalence.(2) a -simplex is marked in N ♮ D if and only if the representing -morphismin D is an isomorphism.(3) all simplices of N ♮ D in dimension or higher are marked.There is a natural inclusion of simplicial sets with marking N RS D → N ♮ D ,which can be seen to be an ( ∞ , n ) -weak equivalence combining [OR19a,Thm 5.2] and [OR18, Prop. 1.31]. The construction extends to a functor N ♮ : n C at → m s S et , which can be seen to be homotopical combining [OR19a,Thm 4.12] and [OR18, Prop. 1.31].Now, suppose we are given a weak equivalence of n -categories F : D → D ′ for n ≤ . It fits into the following commutative diagram N RS D N RS D ′ N ♮ D N ♮ D ′ . N RS FN ♮ F By previous considerations, the vertical maps and the bottom map are equiv-alences of ( ∞ , n ) -categories, so the top map must also be one. (cid:3) . Nerve vs suspension - The results
In this section, we illustrate the results and applications related to thecompatibility of nerve and suspension constructions.We recall the -categorical suspension . Definition 2.1.
Let D be a -category D . The suspension of D is the -category Σ D in which(a) there are two objects x ⊥ and x ⊤ (b) the hom- -categories given by Map Σ D ( a, b ) := D if a = x ⊥ , b = x ⊤ [0] if a = b, ∅ if a = x ⊤ , b = x ⊥ (c) there is no nontrivial horizontal composition. Example . Let k, l ≥ . • The suspension Σ[ k ] of the poset [ k ] is the free k -tuple of vertically com-posable -morphisms, namely the -category [1 | k ] belonging to Joyal’s cellcategory Θ . • The suspension
Σ([ k ] × [ l ] op ) of the poset [ k ] × [ l ] op can be understood asa quotient of the -truncated oriental O [ k + 1 + l ] as explained by thefollowing proposition. • The suspension Σ I of the free isomorphism I is the walking -isomorphism. Proposition 2.3.
For any k, l ≥ − there is a natural isomorphism of -categories Σ([ k ] × [ l ] op ) ∼ = O [ k ] \O [ k + 1 + l ] / O [ l ] between the suspension of the poset [ k ] × [ l ] op and the quotient O [ k ] \O [ k +1 + l ] / O [ l ] of the -truncated ( k + 1 + l ) -oriental O [ k + 1 + l ] obtained bycollapsing O [ k ] ∼ = O [ { , . . . , k } ] ֒ → O [ k + 1 + l ] to one point and O [ l ] ∼ = O [ { k + 1 , . . . , k + 1 + l } ] ֒ → O [ k + 1 + l ] to a different point.Proof. We define a -functor ϕ : O [ k + 1 + l ] → Σ([ k ] × [ l ] op ) that is natural in k and l using the description of orientals in terms of objects,generating - and -morphisms as discussed in Definition 1.13 and the de-scription of - and -morphisms of Σ([ k ] × [ l ] op ) as objects and -morphismsof [ k ] × [ l ] op . The -categorical suspension Σ D appears in [BSP11] as σ ( D ) . It also often appears inthe literature as a special case of a simplicial suspension. For instance, the homwise nerve N ∗ (Σ D ) of the suspension Σ D is a simplicial category that agrees with what would bedenoted as U ( N D ) in [Ber07], as S ( N D ) in [Joy07], as [1] N D in [Lur09a], and as [ N D ] in [RV20b]. Part of the arguments are inspired by [Ver07, §4] and [AM14]. a) On objects, we set for any ≤ i ≤ k + 1 + lϕ ( x i ) := (cid:26) x ⊥ if ≤ i ≤ kx ⊤ if k + 1 ≤ i .(b) On generating -morphisms, we set for any ≤ i < j ≤ k + 1 + lϕ ( f ij ) := id x ⊥ if ≤ i < j ≤ k, ( i, j − k − if ≤ i ≤ k < j, id x ⊤ if k < i < j. (c) On generating -morphisms, we set for any ≤ i < j < s ≤ k + 1 + lϕ ( α ijs ) := id id x ⊥ if ≤ i < j < s ≤ k, ( i, s − k − < ( j, s − k − if ≤ i < j ≤ k < s, ( i, s − k − < ( i, j − k − if ≤ i ≤ k < j < s, id id x ⊤ if k < i < j < s. To see that ϕ is well-defined on -morphisms and functorial, it is enoughto observe that Σ([ k ] × [ l ] op ) is a category enriched in posets, and any two -morphisms with the same source and target must coincide. By inspection,the -functor ϕ is also natural in both k and l .The -functor ϕ induces a -functor e ϕ : O [ k ] \O [ k + 1 + l ] / O [ l ] → Σ([ k ] × [ l ] op ) , and we argue that it is the desired isomorphism of -categories.(0) The -functor e ϕ is bijective on objects by construction.(1) The -functor e ϕ is bijective on -morphisms. Indeed, a careful inspectionshows that the non-identity -morphisms of O [ k ] \O [ k + 1 + l ] / O [ l ] arerepresented uniquely by f i i for i ≤ k < i , and essentially by definitionthe -morphisms of Σ([ k ] × [ l ] op ) are uniquely described as ( i , i − k − for i ≤ k < i .(2) The -functor e ϕ is bijective on -morphisms. To see this, recall fromRemark 1.14 that each -morphism of O [ k + 1 + l ] from to k + 1 + l is uniquely represented as a -morphism of the poset P ( { , , . . . , k + l, k + 1 + l } ) between subsets containing and k + 1 + l . Followingthis viewpoint, each non-identity -morphism of O [ k ] \O [ k + 1 + l ] / O [ l ] from x ⊥ to x ⊤ is uniquely represented as a -morphism of the poset P ( { , , . . . , k + l, k + 1 + l } ) of the form (cid:8) , , . . . , i − , i , i , i + 1 , . . . k + l, k + 1 + l }{ , , . . . , i ′ − , i ′ , i ′ , i ′ + 1 , . . . , k + l, k + 1 + l (cid:9) with i ≤ k < i and i ′ ≤ k < i ′ . In particular, i ≤ i ′ and i ′ ≤ i . By inspection, such -morphism is sent by e ϕ to the -morphism of ([ k ] × [ l ] op ) represented by the -morphism of [ k ] × [ l ] op ( i , i − k − i ′ , i ′ − k − which is the generic -morphism in Σ([ k ] × [ l ] op ) from x ⊥ to x ⊤ . (cid:3) We recall the join of simplicial sets with marking, which extends the or-dinary join for simplicial sets. Definition 2.4.
The join
X ⋆ X ′ of simplicial sets with marking is thesimplicial set defined as follows. • The set of m -simplices is given by ( X ⋆ X ′ ) m = a k + l = m − ,k,l ≥− X k × X ′ l where both X − and X ′− are singletons by definition. • The faces and degeneracies of a simplex ( σ, σ ′ ) ∈ X k × X ′ l ⊂ ( X ⋆ X ′ ) m are given by d i ( σ, σ ′ ) = (cid:26) ( d i σ, σ ′ ) if ≤ i ≤ k, ( σ, d i − k − σ ′ ) if k + 1 ≤ i ≤ m = k + 1 + l, and s i ( σ, σ ′ ) = (cid:26) ( s i σ, σ ′ ) if ≤ i ≤ k, ( σ, s i − k − σ ′ ) if k + 1 ≤ i ≤ m = k + 1 + l. • A simplex ( σ, σ ′ ) is marked if either σ is marked in X or σ ′ is marked in X ′ (or both). Proposition 2.5.
Regarding
X ⋆ ∆[0] as pointed on the -simplex x ⊤ comingfrom ∆[0] , the marked join with a -simplex defines a functor ( − ) ⋆ ∆[0] : m s S et → m s S et ∗ that is a left Quillen functor when m s S et is endowed with the model structurefor ( ∞ , n ) -categories and m s S et ∗ is endowed with the pointed model structurefor ( ∞ , n ) -categories. In particular, it is homotopical.Proof. The fact that the marked join with a point ( − ) ⋆ ∆[0] : m s S et → m s S et ∗ defines a left adjoint functor is addressed in [Ver08b, Def. 33]. ByLemma 1.8, in order to prove that it is left Quillen we only need to showit respects cofibrations and it sends all types of elementary ( ∞ , n ) -anodyneextensions to ( ∞ , n ) -weak equivalences. The unmarked version of the join construction appears in [EP00], [Joy08, §3], [Lur09a,§1.2.8] and [RV15, §2.4]. The marked version is in [Ver08b, Obs. 34] or [Rie18, Def. 3.2.5].
0) The functor ( − ) ⋆ ∆[0] takes cofibrations to cofibrations, as it can be seenwith a routine verification using the explicit description of simplices inthe suspension.(1) The functor ( − ) ⋆ ∆[0] takes any complicial horn extension to an ( ∞ , n ) -weak equivalence, as shown in [Ver08b, Lemma 39].(2) The functor ( − ) ⋆ ∆[0] takes any complicial thinness extension to an ( ∞ , n ) -weak equivalence, as shown in [Ver08b, Lemma 39].(3) The functor ( − ) ⋆ ∆[0] takes each saturation extension to a saturationanodyne extension, using the isomorphism ∆[ l ] ⋆ ∆[0] ∼ = ∆[ l + 1] . Indeed,this is a consequence of [OR18, Rmk 1.20], discussed in more detail in[RV20a, App. D].(4) The functor ( − ) ⋆ ∆[0] takes each triviality extension to an ( ∞ , n ) -weakequivalence. To see this, consider a triviality anodyne extension ∆[ m ] → ∆[ m ] t for m > n . The map ∆[ m ] ⋆ ∆[0] → ∆[ m ] t ⋆ ∆[0] is then anidentity on the underlying simplicial sets, with marking only differing indimensions m, m + 1 > n . In particular, the map of simplicial set withmarking can be seen as a pushout along a certain coproduct of trivialityextensions ∆[ p ] → ∆[ p ] t for p > n , and is in particular an ( ∞ , n ) -weakequivalence. (cid:3) We now define the suspension of simplicial sets with marking. We denoteby ∆[ − the empty simplicial set. Definition 2.6.
The suspension Σ X of a simplicial set with marking X isthe simplicial set with marking defined by the pushout of simplicial sets withmarking X ⋆ ∆[ −
1] ∆[0] ⋆ ∆[ − X ⋆ ∆[0] Σ X. Equivalently, Σ X can be understood as the quotient Σ X ∼ = ( X ⋆ ∆[0]) / X of X ⋆ ∆[0] modulo
X ⋆ ∆[ − ∼ = X . In particular, • there are two -simplices, one represented by any -simplex of X and onerepresented by the -simplex of ∆[0] , which we call x ⊥ and x ⊤ respectively. • the set of m -simplices for m > is given by all k -simplices of X for ≤ k ≤ m − as well as the m -fold degeneracies of the two -simplices x ⊥ and x ⊤ , namely (Σ X ) m ∼ = { s m x ⊥ } ∐ X m − ∐ . . . ∐ X ∐ { s m x ⊤ } . A suspension for simplicial sets (without marking) due to Kan appears in [Kan63,KW65], and is also mentioned in [GJ09, §III.5]. We refer the reader to [Ste15] for a surveyon classical simplicial suspension constructions. the set of non-degenerate m -simplices for m > is given by the non-degenerate ( m − -simplices of X . • a non-degenerate m -simplex σ is marked in Σ X if and only if it is markedas an ( m − -simplex of X . Lemma 2.7.
Regarding Σ X as a simplicial set with marking bipointed on x ⊥ and x ⊤ , the marked suspension defines a functor Σ : m s S et → m s S et ∗ , ∗ that is a left Quillen functor between the model structure for ( ∞ , n ) -categoriesand the model structure for bipointed ( ∞ , n + 1) -categories. In particular, itis homotopical and it respects connected colimits as a functor Σ : m s S et → m s S et .Proof. The fact that the suspension Σ defines a functor is a straightforwardverification, and we now describe its right adjoint functor, which we denote hom : m s S et ∗ , ∗ → m s S et , in terms of the right adjoint of ( − ) ⋆ ∆[0] fromProposition 2.5, which we denote P ⊲ : m s S et ∗ → m s S et .On objects, the right adjoint is given by ( Z, a, b ) hom Z ( a, b ) , where hom Z ( a, b ) is defined by the pullback of simplicial sets with marking hom Z ( a, b ) P ⊲b Z ∆[0] Z, a and the construction extends to a functor. To see that this functor is theright adjoint to the suspension, and observe that a map Σ X → Z under a, b corresponds to a commutative diagram simplicial sets with marking X ∆[0] X ⋆ ∆[0] Z ∆[0] ab which corresponds to a commutative diagram of simplicial sets with marking X P ⊲b Z ∆[0] Z, a which corresponds to Z → hom X ( a, b ) , as desired. e now show that Σ : m s S et → m s S et ∗ , ∗ is a left Quillen functor betweenthe model structure for ( ∞ , n + 1) -categories and the model structure forbipointed ( ∞ , n + 1) -categories. • The functor Σ respects cofibrations, as it can be seen with a routine veri-fication using the explicit description of simplices in the suspension. • The functor Σ respects ( ∞ , n + 1) -weak equivalences. To this end, sup-pose that f : X → Y is a weak equivalence of marked simplicial sets, andconsider the commutative diagram ∆[0] X ∼ = X ⋆ ∆[ − X ⋆ ∆[0]∆[0] Y ∼ = Y ⋆ ∆[ − Y ⋆ ∆[0] . = f⋆ ∆[ − f f⋆ ∆[0] We observe that all vertical arrows are weak equivalences (the first is anidentity, the second is a weak equivalence by assumption, and the third isa weak equivalence as a consequence of Proposition 2.5). Since the modelstructure for ( ∞ , n + 1) -categories is left proper and the right horizontalarrows can be seen to be cofibrations by direct inspection, we can apply thegluing lemma (obtained combining the dual of [Hir03, Cor. 13.3.8, Prop.13.3.4]) to conclude that the map induced on the pushout diagrams Σ X Σ Y Σ f is an ( ∞ , n + 1) -weak equivalence.We now show that Σ : m s S et → m s S et ∗ , ∗ is a left Quillen functor be-tween the model structure for ( ∞ , n ) -categories and the model structurefor bipointed ( ∞ , n + 1) -categories. Thanks to Lemma 1.8 and previousconsiderations, it is enough to show that Σ∆[ n + 1] → Σ∆[ n + 1] t is aweak equivalence in the model structure for ( ∞ , n + 1) -categories. Thismap is an isomorphism on the underlying simplicial sets (both isomorphicto ∆[ n + 2] / ∆[ n + 1] ), and the only difference in marking is that in theright-hand side the top-dimensional ( n + 2) -simplex is marked. This meansthat the map Σ∆[ n + 1] → Σ∆[ n + 1] t is a pushout ∆[ n + 2] ∆[ n + 2] t Σ∆[ n + 1] Σ∆[ n + 1] t of a triviality extension ∆[ n + 2] → ∆[ n + 2] t , and is therefore an ( ∞ , n + 1) -acyclic cofibration, as desired. (cid:3) We now compare the nerve of a suspension and the suspension of a nerve. emark . Let P be a -category. Recall from Proposition 2.3 that for any m ≥ there is an isomorphism of -categories O [ k ] \O [ m + 1] / O [ m − k ] ∼ =Σ([ k ] × [ m − k ] op ) .(1) We have a canonical map of simplicial sets Σ( N P ) → N (Σ P ) , • that is identity on -simplices, namely sends x ⊥ to x ⊥ and x ⊤ to x ⊤ ,and • that sends an ( m + 1) -simplex f : [ k ] → P with ≤ k ≤ m of Σ( N P ) for m ≥ to the ( m + 1) -simplex of N (Σ P ) O [ m + 1] O [ k ] \O [ m + 1] / O [ m − k ] ∼ = Σ([ k ] × [ m − k ] op ) Σ( P × [0] op ) ∼ = Σ P . Σ( f × !) The resulting map Σ( N P ) → N (Σ P ) of simplicial sets is an inclusion.(2) The map can be enhanced to a map of simplicial sets with marking Σ( N RS P ) → N RS (Σ P ) , which is a regular inclusion.The following theorem was anticipated as Theorem A, and will be provenin the next section. Theorem 2.9.
Let P be a -category.(1) The canonical inclusion Σ( N P ) → N (Σ P ) is an inner anodyne extension, and in particular a categorical equiva-lence.(2) The canonical inclusion Σ( N RS P ) → N RS (Σ P ) is a complicial inner anodyne extension, and in particular an ( ∞ , -weakequivalence. As applications of the theorem, we obtain the following two corollaries,which were anticipated as Corollary B and Corollary C.
Corollary 2.10.
Let I denote the free-living isomorphism category.(1) The canonical map of simplicial sets N [1] ֒ → N (Σ I ) is categorical equivalence.
2) The canonical map of simplicial sets with marking N RS [1] ֒ → N RS (Σ I ) is an ( ∞ , -weak equivalence.Proof. We prove Part (2); Part (1) is similar, observing that the unmarkedversion of Lemma 2.7 also holds (by adapting the proof to the unmarkedcontext using [Lur09a, Lem. 2.1.2.3]). We have an equivalence of (discrete) -categories [0] ֒ → I . Since N RS is homotopical by Proposition 1.18, we obtain an ( ∞ , -acycliccofibration N RS [0] ֒ → N RS I . Since the suspension is homotopical by Lemma 2.7, we obtain an ( ∞ , -acyclic cofibration Σ( N RS [0]) ֒ → Σ( N RS I ) . Since we can commute nerve and suspension up to equivalence by Theo-rem 3.11, we then obtain an ( ∞ , -acyclic cofibration N RS (Σ[0]) ֒ → N RS (Σ I ) , as desired. (cid:3) Corollary 2.11.
Let m ≥ .(1) The canonical map of simplicial sets N [1 | ∐ N [1 | . . . ∐ N [1 | N [1 | | {z } m ֒ → N [1 | m ] is a categorical equivalence.(2) The canonical map of simplicial sets with marking N RS [1 | ∐ N RS [1 | . . . ∐ N RS [1 | N RS [1 | | {z } m ֒ → N RS [1 | m ] is an ( ∞ , -weak equivalence.Proof. We prove part (2); Part (1) is similar, observing that the unmarkedversion of Lemma 2.7 also holds (by adapting the proof to the unmarkedcontext using [Lur09a, Lem. 2.1.2.3]).We know by [Joy08, Prop.2.13] that the spine inclusion ∆[1] ∐ ∆[0] . . . ∐ ∆[0] ∆[1] | {z } m ֒ → ∆[ m ] s an inner anodyne extension of simplicial sets. In fact, it can be upgradedto a complicial inner anodyne extension N RS [1] ∐ N RS [0] . . . ∐ N RS [0] N RS [1] | {z } m ֒ → N RS [ m ] . This can be seen by either enhancing the original argument to a markedcontext, or by recognizing it as an instance of Corollary 4.10, in which k i = 0 for all i . Since the suspension is homotopical by Lemma 2.7, we obtain an ( ∞ , -acyclic cofibration Σ( N RS [1] ∐ N RS [0] . . . ∐ N RS [0] N RS [1]) | {z } m ֒ → Σ N RS [ m ] . Since the suspension commutes with connected colimits by Lemma 2.7, weobtain an ( ∞ , -acyclic cofibration Σ N RS [1] ∐ Σ N RS [0] . . . ∐ Σ N RS [0] Σ N RS [1] | {z } m ֒ → Σ N RS [ m ] and using Theorem 3.11, we obtain an ( ∞ , -acyclic cofibration N RS Σ[1] ∐ N RS Σ[0] . . . ∐ N RS Σ[0] N RS Σ[1] | {z } m ֒ → N RS Σ[ m ] , as desired. (cid:3) Nerve vs suspension - The proofs
The aim of this section is to prove Theorem 2.9. We will prove (2) andobtain (1) as a corollary.In order to do a detailed analysis of N (Σ P ) , we will use the an explicitdescription of the nerve of suspension -categories, that involves the followingsimplicial set. Lemma 3.1 ([OR19b, Lemma 1.3]) . Let P be a category. The collection of P -matrices Mat m P := a k,l ≥− ,k + l = m − (cid:8) σ : [ k ] × [ l ] op → P (cid:9) for m ≥ defines a simplicial set Mat P in which (1) faces are given by removing precisely one row or one column;(2) degeneracies are given by doubling precisely one row or one column andinserting identities;(3) the non-degenerate simplices are the ones where no two consecutive rowsand no two consecutive columns coincide. See [OR19b, Lemma 1.3] for a precise description of the simplicial structure of
Mat P . e have the following identification. Theorem 3.2 ([OR19b, Theorem 1.4]) . Let P be a -category. There is anisomorphism of simplicial sets N (Σ P ) ∼ = Mat P . In particular, an m -simplex of the Duskin nerve of the suspension Σ P canbe described as a functor [ k ] × [ l ] op → P , together with k, l ≥ − such that k + l = m − .Remark . Let P be a -category. Under the isomorphism from Theo-rem 3.2, N m (Σ P ) ∼ = a k,l ≥− ,k + l = m − (cid:8) σ : [ k ] × [ l ] op → P (cid:9) , each m -simplex of N Σ P can be uniquely described as a functor σ : [ k ] × [ l ] op → P , which can be pictured as a “matrix” valued in P p l p l − · · · p p l p l − · · · p ... ... . . . ...p kl p k ( l − · · · p k . In particular, for any k there is a unique k -simplex of the form [ k ] × [ − op ∼ = ∅ → P , which can be imagined as a column of length k and empty widthand corresponds to the k -fold degeneracy of x ⊥ . Similarly, for any l ≥ there is a unique l -simplex of the form [ − × [ l ] op ∼ = ∅ → P , which can beimagined as a row of length l and empty width and corresponds to the l -folddegeneracy of x ⊤ . Remark . Let P be a -category.(1) Under the identification from Theorem 3.2, we see that the canonicalmap from Remark 2.8 Σ( N P ) → N (Σ P ) • is the identity on -simplices, namely sends x ⊥ to x ⊥ and x ⊤ to x ⊤ ,and • sends an ( m +1) -simplex σ : [ m ] → P of Σ( N P ) to the ( m +1) -simplex σ : [ m ] ∼ = [ m ] × [0] op → P .(2) Furthermore, a non-degenerate ( m + 1) -simplex of Σ( N P ) is marked in Σ( N RS P ) if and only if and only if the corresponding m -simplex of N P is marked in N RS P . emark 3.3 suggests that the number of rows k is a relevant feature ofsimplices of N (Σ P ) : the “type”. This notion was already considered andwidely discussed in [OR19b, §2]. Definition 3.5.
Let P be a -category. Let σ : [ k ] × [ m − k − op → P bean m -simplex of N (Σ P ) . The type of σ is the integer k . Remark . Let P be a -category. The type k of an m -simplex σ of N (Σ P ) given in the form σ : O [ m ] → Σ P can also be recognized as k = − if σ = s m x ⊤ , max { ≤ s ≤ m | σ ( s ) = x ⊥ } else. m if σ = s m x ⊥ This means that the n -functor σ : O [ m ] → Σ P sends the first k + 1 objectsof O [ m ] to x ⊥ and the remaining objects to x ⊤ : σ ( s ) = (cid:26) x ⊥ for any vertex ≤ s ≤ k of O [ m ] x ⊤ for any vertex k + 1 ≤ s ≤ m of O [ m ] . We will also make use of another useful feature of simplices: the “suspectindex”, and of a class of simplices of N (Σ P ) : the “suspect simplices”. Definition 3.7.
Let P be a -category. Let σ : [ k ] × [ d − k ] op → P be a ( d + 1) -simplex of N (Σ P ) of type k . • The suspect index of σ is the minimal ≤ r ≤ k such that for all r ≤ i ≤ k each row { i } × [ d − k ] op → P is constant. If there is no such integer, wedefine the suspect index to be k + 1 . • A simplex σ is called suspect if it is degenerate or if it is non-degenerateof type k and suspect index r ≤ k and σ (cid:0) ( r − , < σ ( r, (cid:1) = id σ ( r − , . Example . Let P be a -category, and g a non-identity morphism. Con-sider the following two -simplices of N (Σ P ) p p p p p p p p p p p p f = f f = = p p p p p p p p p p p p g =id= f = f f = = They both have type , and have suspect index . However, given that g isnot an identity, only the first one is a suspect simplex.We record for future reference the following features of the faces of asuspect simplex. These properties, whose proof we omit, can be deducedfrom a careful case distinction for the types. emma 3.9. Let P be -category. Let σ be a non-degenerate suspect ( d + 1) -simplex of N (Σ P ) of type k and suspect index r ≤ k . The a -th face of σd a ( σ ) is a suspect simplex if ≤ a ≤ r − of suspect index at most ( r − if a = r − of type k − and suspect index r if a = r a suspect simplex if r + 1 ≤ a ≤ k of type k if k + 1 ≤ a ≤ k + 1 + l. For the sake of intuition, one can verify the validity of the lemma in theexample below.
Example . The following pictures display the a -th face of the suspect -simplex of index and type considered in Example 3.8. p p p p p p p p p f = f f = = p p p p p p p p p f = f f = = p p p p p p p p p g =id= = a = 0 ≤ r − a = 1 = r − a = 2 = r p p p p p p p p p p p p p p p p p f f = r + 1 ≤ a = 3 ≤ k k + 1 ≤ a = 4 ≤ k + 1 + l We now use the explicit description of simplices of N (Σ P ) to give anexplicit description of the comparison map from Remark 2.8.We can now prove (2) of Theorem 2.9. Theorem 3.11.
For any category P , the canonical inclusion Σ( N RS P ) → N RS (Σ P ) is a complicial inner anodyne extension, and in particular an ( ∞ , -weakequivalence. In order to prove the theorem, we will add all simplices of N RS (Σ P ) missing from Σ( N RS P ) inductively on their (ascending) dimension d , their(descending) type k , and their (ascending) suspect index r . Proof.
In order to show that the inclusion Σ( N RS P ) → N RS (Σ P ) is a com-plicial inner anodyne extension, we will realize it as a transfinite compositeof intermediate complicial inner anodyne extensions Σ( N RS P ) =: X ֒ → X ֒ → · · · ֒ → X d − ֒ → X d ֒ → · · · ֒ → N RS (Σ P ) . or d ≥ , we let X d be the smallest regular subsimplicial set of N (Σ P ) containing X d − , all d -simplices of N (Σ P ) , as well as the suspect ( d + 1) -simplices of N (Σ P ) . Note that X already contains all non-degenerate -simplices of N RS (Σ P ) and that there are no non-degenerate suspect -simplices. We see that the difference between X d − and X d are the non-degenerate non-suspect d -simplices and the non-degenerate suspect ( d + 1) -simplices.In order to show that the inclusion X d − ֒ → X d is a complicial inneranodyne extension for all d ≥ , we realize it as a composite of intermediatecomplicial inner anodyne extensions X d − =: Y d ֒ → Y d − ֒ → . . . ֒ → Y k +1 ֒ → Y k ֒ → . . . ֒ → Y = X d . For ≤ k < d , let Y k be the smallest regular subset of X d containing Y k +1 as well as all non-degenerate suspect ( d + 1) -simplices e τ of N (Σ P ) of type k and all non-degenerate non-suspect d -simplices of type k − . Note that Y d already contains all non-degenerate d -simplices of type d − and that anysuspect ( d + 1) -simplex of type d is necessarily degenerate and thus can bechecked to be also already in Y d . We see using Lemma 3.9 that the differencebetween Y k and Y k +1 are the non-degenerate suspect ( d +1) -simplices of type k and possibly some of their faces (precisely those that are not suspect andthose that are not of a higher type).In order to show that the inclusion Y k +1 ֒ → Y k is a complicial inner an-odyne extension for ≤ k ≤ d − , we realize it as a filtration made byintermediate complicial inner anodyne extensions Y k +1 =: W ֒ → W ֒ → . . . ֒ → W r − ֒ → W r ֒ → . . . ֒ → W k = Y k . For < r ≤ k , we let W r be the smallest regular simplicial subset of Y k containing W r − as well as all suspect ( d + 1) -simplices of N RS (Σ P ) of type k and suspect index r , namely those e τ for which each i -th row constant for r ≤ i ≤ k . Note that any simplex of suspect index is degenerate andcan be checked to be already in X ⊂ W . We see using Lemma 3.9 thatthe difference between W r − and W r are the non-degenerate suspect ( d + 1) -simplices e τ of type k and suspect index r and the non-degenerate non-suspect d -simplices τ of type k − and suspect index r .There is a bijective correspondence between the ( d + 1) - and d -simplicesmentioned above, as follows. On the one hand, given any such τ one canbuild the suspect ( d + 1) -simplex e τ : [ k ] × [ d − k ] op → P of N RS (Σ P ) of suspect index r obtained from τ by adding as r -row theconstant map { r } × [ d − k ] op → P with value τ ( r − , ; on the other hand,given any suspect ( d + 1) -simplex e τ , one obtains τ as τ = d r ( e τ ) .We now record some relevant properties the ( d + 1) -simplices e τ as above. We argue that by induction and using Lemma 3.9 the r -horn of e τ belongsto W r − ; in particular, the r -horn defines a map of (underlying) simplicialsets Λ r [ d + 1] → W r − . Indeed, using Lemma 3.9 we see that the a -th face of e τ is already in W r − for a = r since: ♦ if ≤ a ≤ r − , the face d a ( e τ ) is a suspect d -simplex, and in particularit belongs to X d − ⊂ W r − . ♦ if a = r − , the face d a ( e τ ) has suspect index at most ( r − , and inparticular it belongs to W r − (even in X d − if r = 1 ). ♦ if r + 1 ≤ a ≤ k , the face d a ( e τ ) is a suspect d -simplex, and in particularit belongs to X d − ⊂ W r − . ♦ if k + 1 ≤ a ≤ k + 1 + l , the face d a ( e τ ) is of type k , and in particular itbelongs to Y k +1 ⊂ W r − . • We argue that the r -th horn of e τ defines a map of simplicial sets Λ r [ d + 1] → W r − with marking. To this end, we observe that that all simplices are markedin dimensions at least in W r − , no non-degenerate simplices are markedin dimension in Λ r [ d +1] (because < r < d +1 ), and the only marked -simplex of Λ r [ d + 1] is the -dimensional face { r − , r, r + 1 } . In particular,it is enough to show that now that this face is mapped to a degenerate -simplex of W r − . If r < k , then all the vertices of the -dimensionalface { r − , r, r + 1 } are mapped to x ⊥ , and the -simplex is mapped tothe degenerate -simplex at x ⊥ . If r = k , then the -dimensional face { r − , r, r + 1 } is mapped to the -simplex of N (Σ P ) e τ ( r − , e τ ( r, , = which is degenerate because e τ is a suspect simplex of suspect index r . • If furthermore τ is marked, we argue that the r -th horn of e τ defines a mapof simplicial sets with marking Λ r [ d + 1] ′ → W r − , with the simplicial set with marking Λ r [ d + 1] ′ defined in Lemma 1.12. Tothis end, we need to show the ( r − -st and ( r + 1) -st faces are mapped toa marked simplex of W r − . This is true when d > because all simplicesin dimension at least are marked in W r − , and we now address the case d = 2 . In this case, the non-degenerate suspect -simplex e τ is necessarily f the form e τ (1) e τ (0) e τ (0) e τ (0) . e τ (10) e τ (10) == and in particular k = 1 = r . The zeroth face of e τ is degenerate and thusmarked, and the second face of e τ must be marked because it is inhabitedby the same -morphism of Σ P (so -morphism of P ) as τ , which is markedby assumption.By filling all r -horns of suspect ( d + 1) -simplices e τ of W r , we then obtaintheir r -th face τ , which was missing in W r − , as well as the suspect ( d + 1) -simplex e τ itself. This can be rephrased by saying that there is a pushoutsquare ` τ non-marked Λ r [ d + 1] ∐ ` τ marked Λ r [ d + 1] ′ ` τ non-marked ∆ r [ d + 1] ∐ ` τ marked ∆ r [ d + 1] ′′ W r − W r . Since the involved horn inclusions are in fact inner horn inclusions, the inclu-sions of simplicial sets with marking Λ r [ d + 1] ֒ → ∆ r [ d + 1] and Λ r [ d + 1] ′ ֒ → ∆ r [ d + 1] ′′ are complicial inner anodyne extensions by Lemma 1.12.It follows that W r − ֒ → W r is an anodyne for any ≤ r ≤ d − j , that theinclusion Y j − ֒ → Y j for any ≤ j ≤ d , the inclusion Y j − ֒ → Y j for any ≤ j ≤ d , the inclusion X d − ֒ → X d for any d ≥ , and Σ( N RS P ) → N RS (Σ P ) are complicial inner anodyne extensions, as desired. (cid:3) As an instance of Remark 1.11 (or by reading the previous proof ignoringthe marking), we obtain the following corollary, which is (1) of Theorem A.
Corollary 3.12.
For any category P , the canonical inclusion Σ( N P ) → N (Σ P ) is an inner anodyne extension, and in particular a categorical equivalence. Nerve vs wedge - The results
In this section, we illustrate the results and applications related to thecompatibility of nerve and certain gluing construction that we call “wedge”.
Definition 4.1.
Let n ∈ N ∪ {∞} . Let A be an n -category, and a ⊤ (resp. a ⊥ ) an object of A . The object a ⊤ (resp. a ⊥ ) is a cosieve object (resp. sieveobject ) if the following equivalent conditions are met. The equivalence of the conditions can be seen as a special case of the argument from[AM14, §2.3]. Given any object b ∈ A , the hom ( n − -category Map A ( a ⊤ , b ) (resp. Map A ( b, a ⊥ ) )is given by Map A ( a ⊤ , b ) = (cid:26) { id a ⊤ } b = a ⊤ ∅ b = a ⊤ (cid:18) resp. Map A ( b, a ⊥ ) = (cid:26) { id a ⊥ } b = a ⊥ ∅ b = a ⊥ (cid:19) • The inclusion a : [0] ֒ → A is a cosieve (resp. sieve), as defined in [AM14,§2.3] under the name of cocrible (resp. crible ), i.e., there is an n -functor χ : A → [1] that restricts to an isomorphism of n -categories χ − { } ∼ = { a ⊤ } (resp. χ − { } ∼ = { a ⊥ } ) . Example . Let P be a -category (e.g. P = [ k ] ). The suspension -category Σ P (e.g. Σ P = [1 | k ] ) has a (unique) cosieve object, given by thelast object, and a (unique) sieve object, given by the first object.We consider the following type of pushout of n -categories along (co)sieveobjects. Definition 4.3.
Let n ∈ N ∪{∞} . The wedge of two n -categories A endowedwith a cosieve object a ⊤ and A ′ with a sieve object a ′⊥ is the pushout [0] AA ′ A ∨ A ′ . a ⊤ a ′⊥ As a motivating example, the wedge construction is useful to expressrelation between the n -categories belonging to Joyal’s categories Θ n (seee.g. [Joy97]). Example . For any k, k ′ ≥ (or even more generally k, k ′ ∈ Θ n − ), thewedge of [1 | k ] and [1 | k ′ ] is isomorphic to the -category belonging to Θ (resp. n -category belonging to Θ n ) denoted [1 | k ] ∨ [1 | k ′ ] ∼ = [2 | k, k ′ ] . More generally, for any m, m ′ ≥ , k i , k ′ i ′ ≥ (resp. k i , k ′ i ′ ∈ Θ n − ) for i =1 , . . . , m and i ′ = 1 , . . . , m ′ , the wedge of [ m | k , . . . k m ] and [ m ′ | k ′ , . . . , k ′ m ′ ] is isomorphic to [ m + m ′ | k , . . . , k m , k ′ , . . . , k ′ m ′ ] . A wedge of -categories maps to their product, as explained by the fol-lowing. Remark . Let n ∈ N ∪ {∞} , and A and A ′ two n -categories as in Defini-tion 4.3, in particular endowed with functors χ : A → [1] and χ ′ : A ′ → [1] .The inclusions A ∼ = A × ∗ id × a ′⊥ −−−−→ A × A ′ a ⊤ × id ←−−−− ∗ × A ′ ∼ = A ′ induce a canonical map A ∨ A ′ → A × A ′ , hich fits into a commutative diagram of n -categories A ∨ A ′ A × A ′ [2] [1] × [1] . χ × χ ′ → → In particular, get map
A ∨ A ′ → A × A ′ . The maps above turns out to be an inclusion as a consequence of thefollowing theorem. In particular, a wedge of n -categories can be understoodas a sub- n -category of the product. Theorem 4.6.
Let n ∈ N ∪ {∞} , and A and A ′ two n -categories as inDefinition 4.3. There is a pullback square of n -categories A ∨ A ′ A × A ′ [2] [1] × [1] . χ × χ ′ → → In particular,(a) the objects of
A ∨ A ′ are of the form ( a, a ′⊥ ) or ( a ⊤ , a ′ ) for some object a ∈ A or a ′ ∈ A ′ .(b) the mapping ( n − -categories are as follows Map
A∨A ′ (( a, a ′ ) , ( b, b ′ )) ∼ = Map A ( a, b ) if a ′ = b ′ = a ′⊥ Map A ′ ( a ′ , b ′ ) if a = b = a ⊤ , Map A ( a, a ⊤ ) × Map A ′ ( a ′⊥ , b ′ ) if b = a ⊤ and b = a ′⊥ , ∅ else.(c) A and A ′ are full subcategories of A ∨ A ′ .Proof. Let Q be the pullback of the map [2] → [1] × [1] along χ × χ ′ Q A × A ′ [2] [1] × [1] . By inspection we see that(a) the objects of Q are of the form ( a, a ′⊥ ) or ( a ⊤ , a ′ ) for some object a ∈ A or b ∈ A ′ . The case n = 2 of the theorem could be treated more directly with techniques from[AM14, §7.2], the case n = 3 could be treated more directly with techniques from [Gag19,§4.3], and the case in which A and A ′ are suspension -categories is treated in the proofof [OR19b, Thm 4.4]. b) the mapping ( n − -categories are as follows are given by Map Q (( a, a ′ ) , ( b, b ′ )) ∼ = Map A ( a, b ) if a ′ = b ′ = a ′⊥ Map A ′ ( a ′ , b ′ ) if a = b = a ⊤ , Map A ( a, a ⊤ ) × Map A ′ ( a ′⊥ , b ′ ) if b = a ⊤ and b = a ′⊥ , ∅ else.(c) the composition ( n − -functors in the first two cases is induced by thecomposition in A and A ′ . Moreover, the composition ( n − -functorsinvolving the third case are determined by composition in A and in A ′ .Consider the n -functors i A : A → Q and i A ′ : A ′ → Q defined on objects by i A ( a ) = ( a, a ′⊥ ) and i A ′ ( a ′ ) = ( a ⊤ , a ′ ) , and inducedby the isomorphisms above on hom- ( n − -categories. We argue that thecommutative diagram of n -categories [0] A ′ A Q a ′⊥ a ⊤ i A′ i A is a pushout of n -categories, proving the desired statement.In order to prove that Q satisfies the universal property of pushouts, wesuppose to be given a commutative diagram of n -categories formed by thesolid arrows [0] A ′ A Q D . a ′⊥ a ⊤ i A′ α ′ i A α F We show how to construct an n -functor F : Q → D so that the diagramcommutes, and we leave the verification of the uniqueness to the reader.(0) We define F on objects by F ( a, a ′⊥ ) = α ( a ) and F ( a ⊤ , a ′ ) = α ′ ( a ′ ) . (1) We define F on hom- ( n − -categories F : Map Q (( a, a ′ ) , ( b, b ′ )) → Map D ( F ( a, a ′ ) , F ( b, b ′ )) • if b = b ′ = a ′⊥ as the functor α : Map A ( a, b ) −→ Map D ( α ( a ) , α ( b )) . • if a = a ′ = a ⊤ as the functor α ′ : Map A ′ ( a ′ , b ′ ) −→ Map D ( α ′ ( a ′ ) , α ′ ( b ′ )) . if b = a ⊤ and a ′ = a ′⊥ as the functor α ′ ( − ) ◦ α ( − ) : Map A ( a, a ⊤ ) × Map A ′ ( a ′⊥ , b ′ ) −→ Map D ( α ( a ) , α ′ ( b ′ )) . • otherwise as the functor ∅ ! −→ Map D ( F ( a, b ) , F ( a ′ , b ′ )) . The fact that F is compatible with identities and with most instances ofcomposition is straightforward, and we verify compatibility with compositionin one of the two interesting cases (the other one is analog).To this end, we need to check the commutativity of the following diagramof ( n − -categories: Map Q (( a, a ′⊥ ) , ( b, a ′⊥ )) × Map Q (( b, a ′⊥ ) , ( a ⊤ , a ′ )) Map Q (( a, a ′⊥ ) , ( a ⊤ , a ′ ))Map D ( F ( a, a ′⊥ ) , F ( b, a ′⊥ )) × Map D ( F ( b, a ′⊥ ) , F ( a ⊤ , a ′ ))Map D ( F ( a, a ′⊥ ) , F ( a ⊤ , a ′ )) . ◦ Q F × F F ◦ D Inserting the definitions and identifications above, we can identify this dia-gram with the following one:
Map A ( a, b ) × Map A ( b, a ⊤ ) × Map A ′ ( a ′⊥ , a ′ )Map A ( a, a ⊤ ) × Map A ′ ( a ′⊥ , a ′ )Map D ( α ( a ) , α ( b )) × Map D ( α ( b ) , α ( a ⊤ )) × Map D ( α ′ ( a ′⊥ ) , α ′ ( a ′ ))Map D ( α ( a ) , α ( a ⊤ )) × Map D ( α ′ ( a ′⊥ ) , α ′ ( a ′ ))Map D ( α ( a ) , α ( b )) × Map D ( α ( b ) , α ′ ( a ′ )) Map D ( α ( a ) , α ′ ( a ′ )) . ◦ A × id α × α × α ′ α × α ′ id ×◦ D ◦ D ◦ D This diagram commutes since α is a functor and ◦ D is associative. (cid:3) We can define an analog wedge for simplicial sets along -simplices. efinition 4.7. The wedge of two simplicial sets with marking X with aspecified -simplex x ⊥ and X ′ with a specified -simplex x ′ is the pushoutof simplicial sets with marking ∆[0] XX ′ X ∨ X ′ . xx ′ We can now compare nerve of wedge with wedge of nerve as follows.
Remark . Let n ∈ N ∪ {∞} , and A and A ′ two n -categories as in Defini-tion 4.1.(1) There is a commutative diagram ∆[0] N A N A ′ N ( A ∨ A ′ ) . a ⊤ a ′⊥ By the universal property of pushouts we obtain a canonical map ofsimplicial sets N A ∨ N A ′ → N ( A ∨ A ′ ) , which is an inclusion. Under the identification from Theorem 4.6, thismap sends an m -simplex of N A ∨ N A ′ of the form σ : O [ m ] → A (resp. σ ′ : O [ m ] → A ) to the m -simplex of N ( A ∨ A ′ ) given by ( σ, s m a ′⊥ ) : O [ m ] → A ∨ A ′ (resp. ( s m a ⊤ , σ ′ ) : O [ m ] → A ∨ A ′ ).Moreover, a pair of n -functors ( σ, σ ′ ) , where σ : O [ m ] → A and σ ′ : O [ m ] →A ′ , defines an m -simplex of N ( A ∨ A ′ ) if and only if χσ ( s ) ≥ χ ′ σ ′ ( s ) for any vertex ≤ s ≤ m of O [ m ] . (2) The map of simplicial sets can be enhanced to a map of simplicial setswith marking N RS A ∨ N RS A ′ → N RS ( A ∨ A ′ ) , which is a regular inclusion, given that A and A ′ are full sub- n -categoriesof A ∨ A ′ .The main result of this section is that the nerve construction commuteswith the wedge construction up to a suitable notion of weak equivalence. Theorem 4.9.
Let n ∈ N ∪ {∞} , and A and A ′ two n -categories as inDefinition 4.3.(1) The canonical map of simplicial sets N A ∨ N A ′ → N ( A ∨ A ′ ) s an inner anodyne extension, and in particular a categorical equivalenceand a weak homotopy equivalence.(2) The canonical map of simplicial sets with marking N RS A ∨ N RS A ′ → N RS ( A ∨ A ′ ) is a complicial inner anodyne extension, and in particular an ( ∞ , n ) -weak equivalence. The theorem will be proven in the next section.Recall from Example 4.4 that -categories of the form [ m | k , . . . , k m ] arethe objects of Θ (more generally, that n -categories of the form [ m | k , . . . , k m ] are the objects of Θ n for k , . . . , k m ∈ Θ n − ). As an application of Theo-rem A, we obtain the following corollary, which was anticipated as Corol-lary A. Corollary 4.10.
Let m ∈ N and k , . . . , k m ∈ N (or k , . . . , k m ∈ Θ n − ).(1) The canonical map of simplicial sets N [1 | k ] ∨ · · · ∨ N [1 | k m ] ֒ → N [ m | k , . . . , k m ] is an inner anodyne extension, and in particular a categorical equivalenceand a weak homotopy equivalence.(2) The canonical map of simplicial sets with marking N RS [1 | k ] ∨ · · · ∨ N RS [1 | k m ] ֒ → N RS [ m | k , . . . , k m ] is complicial inner anodyne extension, and in particular an ( ∞ , n ) -weakequivalence.Proof. We observe that the object of any ( m + 1) -point suspension asdefined in [OR19b, §4] is a sieve object, and the object m of any ( m + 1) -point suspension is a cosieve object. Each of the two claims is proven usingthe corresponding statement of Theorem 4.9 by induction on m , specializingto A = [ m | k , . . . , k m ] and A ′ = [1 | k m +1 ] . (cid:3) Nerve vs wedge - The proofs
The aim of this section is to prove Theorem 4.9. We will show (2), andobtain (1) as a corollary.
Remark . Let n ∈ N ∪ {∞} , and A and A ′ two n -categories as in Defini-tion 4.3, in particular endowed with functors χ : A → [1] and χ ′ : A ′ → [1] .(1) As a consequence of Theorem 4.6, there is a canonical inclusion of sim-plicial sets N ( A ∨ A ′ ) ֒ → N ( A × A ′ ) ∼ = N A × N A ′ . oreover, a pair of n -functors ( ρ, ρ ′ ) , where ρ : O [ m ] → A and ρ ′ : O [ m ] →A ′ , defines an m -simplex of N ( A ∨ A ′ ) if and only if χρ ( s ) ≥ χ ′ ρ ′ ( s ) for any vertex ≤ s ≤ m of O [ m ] . (2) Furthermore, a simplex ( ρ, ρ ′ ) of N ( A ∨ A ′ ) is marked in N RS ( A ∨ A ′ ) ifand only if both components ρ and ρ ′ are marked in N A and N A ′ . Thismeans that we obtain a regular inclusion of simplicial sets with marking N RS ( A ∨ A ′ ) ֒ → N RS A × N RS A ′ . We will make use of the following features of simplices of N ( A ∨ A ′ ) . Definition 5.2.
Let n ∈ N ∪ {∞} and A and A ′ two n -categories as inDefinition 4.3. Let ( ρ, ρ ′ ) be an m -simplex of N ( A ∨ A ′ ) . The type of ( ρ, ρ ′ ) is the pair of integers ( k ρ , k ρ ′ ) defined by k ρ ( ′ ) = (cid:26) − if χ ( ′ ) ρ ( ′ ) = 1 , max { ≤ s ≤ m | χ ( ′ ) ρ ( ′ ) ( s ) = 0 } else.In particular, since χρ ( s ) ≥ χ ′ ρ ′ ( s ) for any vertex ≤ s ≤ m , we have that k ρ ′ ≥ k ρ . Remark . The definition can be rephrased by saying that any n -functor ρ : O [ m ] → A ∨ A ′ sends • the first k ρ + 1 vertices of O [ m ] to A \ { a ⊤ } , • the next k ρ ′ − k ρ vertices of O [ m ] to a ⊤ = a ′⊥ ∈ A ∨ A ′ , • and the final m − k ρ ′ vertices of O [ m ] to A ′ \ { a ′⊥ } .We will also make use of another useful feature of simplices of N ( A ∨ A ′ ) :the “suspect index”, and of a class of simplices of N ( A ∨ A ′ ) : the “suspectsimplices”. We chose the same terminology as in Section 3 because thesenotions play similar roles as those in the argument from Theorem 3.11. Definition 5.4.
Let n ∈ N ∪ {∞} and A and A ′ two n -categories as inDefinition 4.3. Let ( ρ, ρ ′ ) be a ( d + 1) -simplex of N ( A ∨ A ′ ) . • The suspect index of ( ρ, ρ ′ ) is the maximal r with k ρ + 1 ≤ r ≤ k ρ ′ forwhich there exists a simplex α ′ of N A ′ such that ρ ′ = s r − . . . s k ρ α ′ , and k ρ if such α ′ does not exist. • The simplex ( ρ, ρ ′ ) is called suspect if it is degenerate or in N A ∨ N A ′ orif it is of suspect index k ρ + 1 ≤ r ≤ k ρ ′ and k ρ ′ ≥ k ρ + 1 and in addition ρ = s r α for some simplex α of N A .We record for future reference the faces of a suspect simplex, as well astheir types. emma 5.5. Let n ∈ N ∪ {∞} and A and A ′ two n -categories as in Defi-nition 4.3. Let ( ρ, ρ ′ ) = ( s r α, s r − . . . s k ρ α ′ ) be a suspect ( d + 1) -simplex of N ( A ∨ A ′ ) of suspect index k ρ + 1 ≤ r ≤ k ρ ′ which is not in N A ∨ N A ′ . The a -th face of ( ρ, ρ ′ ) d a ( ρ, ρ ′ ) is a suspect simplex if ≤ a ≤ r − of type ( k ρ , k ρ ′ − and suspect index r − if a = r of type ( k ρ , k ρ ′ − and suspect index r if r + 1 = a ≤ k ρ ′ of type ( k ρ , k ρ ′ ) if r + 1 = a = k ρ ′ + 1 a suspect simplex if r + 2 ≤ a ≤ d + 1 . Proof.
From the simplicial identities, we obtain the formulas for the a -thface of ( ρ, ρ ′ ) is d a ( ρ, ρ ′ ) = ( s r − d a α, s r − . . . s k ρ − d a α ′ ) if ≤ a ≤ k ρ ( s r − d a α, s r − . . . s k ρ α ′ ) if k ρ + 1 ≤ a < r ( α, s r − . . . s k ρ α ′ ) if a = r ( α, s r − . . . s k ρ d k ρ +1 α ′ ) if r + 1 = a ≤ k ρ ′ ( α, s r − . . . s k ρ d k ρ +1 α ′ ) if r + 1 = a = k ρ ′ + 1( s r d a − α, s r − . . . s k ρ d a − r + k ρ α ′ ) if r + 1 < a ≤ k ρ ′ ( s r d a − α, s r − . . . s k ρ d a − r + k ρ α ′ ) if k ρ ′ < a ≤ d + 1 . From a careful case distinction, we obtain that the type of the a -th face of ( ρ, ρ ′ ) is k d a ( ρ,ρ ′ ) = ( k ρ − , k ρ ′ − if ≤ a ≤ k ρ ( k ρ , k ρ ′ − if k ρ + 1 ≤ a < r ( k ρ , k ρ ′ − if a = r ( k ρ , k ρ ′ − if r + 1 = a ≤ k ρ ′ ( k ρ , k ρ ′ ) if r + 1 = a = k ρ ′ + 1( k ρ , k ρ ′ − if r + 1 < a ≤ k ρ ′ ( k ρ , k ρ ′ ) if k ρ ′ < a ≤ d + 1 . as desired. (cid:3) We can now prove (2) of Theorem B.
Theorem 5.6.
Let n ∈ N ∪ {∞} and A and A ′ two n -categories as inDefinition 4.3. The canonical map of simplicial sets with marking N RS A ∨ N RS A ′ → N RS ( A ∨ A ′ ) is a complicial inner anodyne extension, and in particular an ( ∞ , n ) -weakequivalence. In order to prove the theorem, we will add all simplices of N RS (Σ P ) missing from Σ( N RS P ) inductively on their (ascending) dimension d , the(descending) difference of types b := k ρ ′ − k ρ , the (descending) type of thesecond component k ρ ′ , and their (descending) suspect index r . roof. In order to show that the inclusion N A ∨ N A ′ ֒ → N ( A ∨ A ′ ) is a com-plicial inner anodyne extension, we will realize it as a transfinite compositeof intermediate complicial inner anodyne extensions N A ∨ N A ′ =: X ֒ → X ֒ → . . . ֒ → X d − ֒ → X d ֒ → . . . ֒ → N ( A ∨ A ′ ) . For d ≥ , we let X d be the smallest regular subsimplicial set of N ( A ∨ A ′ ) containing X d − , all d -simplices of N ( A ∨ A ′ ) as well as the suspect ( d + 1) -simplices of N ( A∨A ′ ) . Note that X contains all -simplices of N ( A∨A ′ ) aswell as all that all suspect -simplices of N ( A ∨ A ′ ) are in X by definition.We see using Lemma 5.5 that the difference between X d and X d − are thenon-degenerate non-suspect d -simplices and the non-degenerate suspect ( d +1) -simplices.In order to show that the inclusion X d − ֒ → X d is a complicial inneranodyne extension for all d ≥ , we realize it as a composite of intermediatecomplicial inner anodyne extensions X d − =: Y d ֒ → Y d − ֒ → . . . ֒ → Y b +1 ֒ → Y b ֒ → . . . ֒ → Y = X d . For d − ≥ b ≥ , let Y b be the smallest regular subset of X d containing Y b +1 as well as all suspect ( d + 1) -simplices ( e σ, e σ ′ ) of N ( A ∨ A ′ ) of type ( k e σ , k e σ ′ ) for which k e σ ′ − k e σ = b + 1 . Note that any ( d + 1) -simplex of type difference d + 1 and any d -simplex of type difference d is in X ⊂ Y d . The differencebetween Y b and Y b +1 is given by the non-degenerate suspect ( d + 1) -simplices ( e σ, e σ ′ ) with type difference k e σ ′ − k e σ = b + 1 and their d -dimensional faces notalready present in Y b +1 . These are exactly the non-degenerate, non-suspect d -simplices ( σ, σ ′ ) of N ( A ∨ A ′ ) of type difference k σ ′ − k σ = b . Indeed, onthe one hand one can use Lemma 5.5 to check that all faces of ( e σ, e σ ′ ) thatare not already present in Y b +1 are non-degenerate non-suspect simplices oftype difference b ; on the other hand, any such d -simplex ( σ, σ ′ ) occurs as aface of the ( d + 1) -suspect simplex ( s r σ, s r − σ ′ ) , with r − being the suspectindex of ( σ, σ ′ ) . In particular, we have that Y = X d .In order to show that the inclusion Y b +1 ֒ → Y b is a complicial inner an-odyne extension for d − ≥ b ≥ , we realize it as a filtration made byintermediate complicial anodyne extensions Y b +1 =: Z d ֒ → Z d − ֒ → . . . ֒ → Z k +1 ֒ → Z k ֒ → . . . ֒ → Z b = Y b . For d > k ≥ b , we let Z k be the smallest regular subset of Y b containing Z k +1 as well as all ( d + 1) -simplices ( e σ, e σ ′ ) of Y b of type ( k e σ , k e σ ′ ) = ( k − b, k + 1) .Note that any ( d + 1) -simplex of type ( d − b, d + 1) is already in X ⊂ Z d .The difference between Z k and Z k +1 are the non-degenerate suspect ( d + 1) -simplices of N ( A∨A ′ ) of type ( k e σ , k e σ ′ ) = ( k − b, k +1) and their d -dimensionalfaces not already present in Z k +1 , which can be seen (using Lemma 5.5) tobe exactly all non-degenerate, non-suspect d -simplices ( σ, σ ′ ) of N ( A ∨ A ′ ) of type ( k − b, k ) . In particular by definition we have that Z b = Y b .In order to show that the inclusion Z k +1 ֒ → Z k is a complicial inneranodyne extension for d − ≥ k ≥ b , we realize it as a filtration made by ntermediate complicial inner anodyne extensions. Z k +1 =: W k +2 ֒ → W k +1 ֒ → . . . ֒ → W r +1 ֒ → W r ֒ → . . . ֒ → W k − b +1 = Z k . For k + 1 ≥ r ≥ k − b + 1 , we let W r be the smallest regular simplicialsubset of Z k containing W r +1 as well as the ( d + 1) -suspect simplices of Z k of suspect index r . In particular by definition we have that W k − b +1 = Z k .This means that the difference between W r +1 and W r are the non-degeneratesuspect ( d +1) -simplices ( e σ, e σ ′ ) of N ( A∨A ′ ) of type ( k − b, k +1) and suspectindex r , and their d -dimensional faces not already present in W r +1 , whichcan be seen (again using Lemma 5.5) to be exactly and the non-degeneratenon-suspect d -simplices ( σ, σ ′ ) of type ( k − b, k ) and suspect index r − .There is a bijective correspondence between the ( d + 1) - and d -simplicesmentioned above, as follows. On the one hand, given any such d -simplex ( σ, σ ′ ) one can build the suspect ( d + 1) -simplex ( e σ, e σ ′ ) := ( s r σ, s r − σ ′ ) of N RS ( A∨A ′ ) of suspect index r and type ( k − b, k +1) ; vice versa, given anysuch suspect ( d + 1) -simplex ( e σ, e σ ′ ) , one obtains ( σ, σ ′ ) as ( σ, σ ′ ) = d r ( e σ, e σ ′ ) .Let ( e σ, e σ ′ ) be a ( d +1) -suspect simplex in W r not in W r +1 , and let’s recordthe following relevant properties. • We argue that the r -horn of ( e σ, e σ ′ ) belongs to W r +1 ; in particular, the r -horn defines a map of (underlying) simplicial sets Λ r [ d + 1] → W r +1 . Indeed, using Lemma 5.5 we see that the a -th face of ( e σ, e σ ′ ) is already in W r +1 for a = r since: ♦ if ≤ a ≤ r − , the face d a ( e σ, e σ ′ ) is a suspect d -simplex, and inparticular it belongs to X d − ⊂ W r +1 . ♦ if a = r + 1 ≤ k + 1 , the face d a ( e σ, e σ ′ ) is a d -simplex of type ( k − b, k ) and suspect index r , and in particular it belongs to W r +1 . ♦ if a = r + 1 = k + 2 , the face d a ( e σ, e σ ′ ) is of type ( k − b, k + 1) and inparticular it belongs to Y b +1 ⊂ W r +1 . ♦ if r + 2 ≤ a ≤ d + 1 , the face d a ( e σ, e σ ′ ) is a suspect d -simplex, and inparticular it belongs to X d − ⊂ W r +1 . • We argue that the r -th horn of ( e σ, e σ ′ ) defines a map of simplicial sets Λ r [ d + 1] → W r +1 with marking. To this end, we need to show that any face containing { r − , r, r + 1 } is mapped to a marked simplex of W r − . This is truebecause a (not necessarily top-dimensional) face of ( e σ, e σ ′ ) that containsthe vertices { r − , r, r + 1 } is necessarily degenerate in both coordinates,given that ( e σ, e σ ′ ) = ( s r σ, s r − σ ′ ) . If furthermore ( σ, σ ′ ) is marked, the r -th horn of ( e σ, e σ ′ ) defines a map ofsimplicial sets with marking Λ r [ d + 1] ′ → W r +1 , with the simplicial set with marking Λ r [ d + 1] ′ defined in Lemma 1.12. Tothis end, we need to show that the top r -dimensional simplex, as well asthe ( r − st and ( r + 1) -st faces are mapped to a marked simplex of W r − .The top-dimensional r -simplex, and by Lemma 5.5 its ( r − -st face, aremapped to suspect simplices of W r − , so in particular degenerate in bothcomponents and marked. By direct computation, or using the explicitformulas given in the proof of Lemma 5.5, one finds that the ( r + 1) -st faceis degenerate in the second component and that the first component is thesimplex σ , which is marked by assumption, and it is therefore mapped toa pair of marked simplices.We can thus fill all r -horns of suspect ( d + 1) -simplices of W r to obtaintheir ( r + 1) -th face, which was missing in W r +1 , as well as the suspect ( d + 1) -simplex itself.In particular, the discussion shows that there is a pushout square ` ( σ,σ ′ ) non-marked Λ r [ d + 1] ∐ ` ( σ,σ ′ ) marked Λ r [ d + 1] ′ ` ( σ,σ ′ ) non-marked ∆ r [ d + 1] ∐ ` ( σ,σ ′ ) marked ∆ r [ d + 1] ′′ W r +1 W r . The involved horn inclusions are in fact inner horn inclusions, so the inclu-sions of simplicial sets with marking Λ r [ d + 1] ֒ → ∆ r [ d + 1] and Λ r [ d + 1] ′ ֒ → ∆ r [ d + 1] ′′ are complicial inner anodyne extensions by Lemma 1.12.It follows that W r +1 ֒ → W r is an anodyne for any k +1 ≥ r ≥ k − b +1 , thatthe inclusion Z k +1 ֒ → Z k for any d − ≥ k ≥ b , the inclusion Y b +1 ֒ → Y b forany d − ≤ b ≤ , the inclusion X d − ֒ → X d for any d ≥ , and N A∨ N A ′ ֒ → N ( A ∨ A ′ ) are complicial inner anodyne extensions, as desired. (cid:3) As an instance of Remark 1.11 (or by reading the previous proof ignoringthe marking), we obtain the following corollary, which is (1) of Theorem B.
Corollary 5.7.
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Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany
E-mail address : [email protected] Mathematical Sciences Institute, The Australian National University,Canberra, Australia
E-mail address : [email protected]@anu.edu.au