aa r X i v : . [ m a t h . C T ] F e b ORIENTALS AS FREE WEAK ω -CATEGORIES YUKI MAEHARA
Abstract.
The orientals are the free strict ω -categories on the simplices in-troduced by Street. The aim of this paper is to show that they are also thefree weak ω -categories on the same generating data. More precisely, we exhibitthe Street nerves of the orientals as fibrant replacements of the simplices inVerity’s model structure for complicial sets. Introduction
This paper concerns the simplicial approach to the theory of higher-dimensionalcategories. The fundamental notion in this approach is that of complicial set intro-duced by Roberts [Rob78]. Although Roberts conjectured the complicial sets to beprecisely the nerves of the (strict) ω -categories, even constructing the desired nervefunctor proved to be rather difficult. The construction was eventually achieved byStreet [Str87] who realised the simplices as ω -categories in the form of orientals .This led to a precise formulation of Roberts’ conjecture, which was subsequentlyproven by Verity [Ver08b, Theorem 266].Verity then began the study of weak complicial sets [Ver08a, Ver07], a homo-topical variant of the strict notion. They model weak ω -categories , i.e. structuresthat only satisfy the usual axioms for ω -category up to coherent, invertible highercells. Among many other results, Verity constructed a model structure capturingthe homotopy theory of weak complicial sets.The aim of this paper is to contribute to this theory by showing that the orientals,originally introduced as the free strict ω -categories on the simplices, are also thefree weak ω -categories on the same generating data. More precisely, we exhibit theStreet nerve of the n -th oriental O n as a fibrant replacement of ∆[ n ] in Verity’saforementioned model structure.In many ways, this paper draws insights from Steiner’s analysis of orientals in[Ste07]. In particular, we make use of his description of the ω -functors O m → O n as certain formal linear combinations of maps [ m ] → [ n ] in ∆, and much of ourproof is based on ideas that can be found in [Ste07, Propositions 5.9-11].1. Background
The combinatorics in this paper relies on Steiner’s description of ω -functorsbetween orientals as certain formal linear combinations of maps in ∆ [Ste07]. Thisdescription is reviewed in Section 1.1, and Section 1.2 discusses how Steiner recoversa notion of composition in this framework. On the other hand, the main result ofthis paper is formulated using the language of complicial sets. We present minimalbackground on complicial sets in Section 1.3, and refer the interested reader to[Ver08b, Ver08a, Ver07] for more on this subject. Section 1.4 discusses how thenotions of identity and composition in the two frameworks agree. O O O (0) (0) (1) (0 , (0) (2)(1) (0 , ,
1) (1 , , , Figure 1. O , O and O (0) (1) (2) (3) (0 ,
1) (2 , , , , (0 , ,
2) (0 , , (0 , , , (0) (1) (2) (3) (0 ,
1) (2 , , , , (1 , , , , Figure 2. O Notation.
Given a simplicial set X ∈ sSet def = (cid:2) ∆ op , Set (cid:3) , an element x ∈ X n and a simplicial operator α : [ m ] → [ n ], we write xα for the image of x under X ( α ) : X n → X m . Note that we have ( xα ) β = x ( αβ ) in this notation whenevereither side is defined.1.1. The category O . The n -th oriental O n , introduced by Street [Str87], is thefree ω -category on the n -simplex with atomic m -cells in bijective correspondencewith injective maps [ m ] → [ n ] in ∆. The following is a slightly more precise (butstill informal) description of O n . • The zeroth oriental O is the terminal ω -category, consisting of a single0-cell and no non-identity higher cells. • For n ≥
1, the “boundary” of O n may be constructed by gluing n + 1copies of O n − according to the structure of ∂ ∆[ n ]. The “interior” of O n is then filled with an n -cell that points from the composite of odd faces tothe composite of even ones.We have drawn (the atomic cells in) O n for n ≤ O , O and O can also be found. In thesefigures, and also throughout this paper, we denote a simplicial operator α : [ m ] → [ n ] by the sequence of its images (cid:0) α (0) , . . . , α ( m ) (cid:1) .We will not quote the precise definition of O n in this paper, and refer the inter-ested reader to [Str87]. Instead, we now recall Steiner’s description [Ste07] of thefull subcategory of ω -Cat spanned by the orientals.Let Z ∆ denote the free Ab-category on ∆. More explicitly, Z ∆ has the sameobjects as ∆ and its hom-abelian groups Z ∆ (cid:0) [ m ] , [ n ] (cid:1) are the free ones generatedby the corresponding hom-sets of ∆. The composition in Z ∆ is given by extendingthe one in ∆ linearly in each variable. We will write x α for the coefficient of RIENTALS AS FREE WEAK ω -CATEGORIES 3 α : [ m ] → [ n ] in x ∈ Z ∆ (cid:0) [ m ] , [ n ] (cid:1) so that x = X α ∈ ∆([ m ] , [ n ]) x α ∗ α. For given x ∈ Z ∆ (cid:0) [ m ] , [ n ] (cid:1) , we define its support bysupp( x ) def = (cid:8) α ∈ ∆ (cid:0) [ m ] , [ n ] (cid:1) : x α = 0 (cid:9) . Definition 1.1.
Let O ( m, n ) denote the subset of Z ∆ (cid:0) [ m ] , [ n ] (cid:1) consisting of those x such that:(O1) P α x α = 1; and(O2) for any injective maps β : [ p ] → [ m ] and γ : [ p ] → [ n ] in ∆, the coefficient( xβ ) γ is non-negative.Here xβ denotes the composite in Z ∆ where β is identified with its image underthe canonical functor ∆ → Z ∆.The ultimate (but unenlightening) justification for the rather mysterious condi-tion (O2) is Theorem 1.3 below. A more conceptual justification is given in theremark following that theorem. Proposition 1.2.
The sets O ( m, n ) determine a subcategory O of (the underlyingordinary category of ) Z ∆ .Proof. That the composition preserves (O1) is easy to check. To see that theanalogous statement holds for (O2), note that if a composite[ p ] [ k ] [ m ] [ n ] β χ φ is equal to an injective map γ , then in particular χβ must be injective. (cid:3) Theorem 1.3 ([Ste07, Theorem 4.2]) . The category O is isomorphic to the fullsubcategory of ω -Cat spanned by the orientals.Remark. Intuitively, the ω -functor O m → O n corresponding to x ∈ O ( m, n ) isobtained by pasting the positive terms in x (regarded as cells in O n ) along thenegative ones. For example, the element (0 , − (1 , ,
2) in O (1 ,
2) correspondsto the ω -functor O → O that picks out the composite of the 1-cells (0 ,
1) and(1 , β to be the identity at [ m ].Then (O2) requires all negative terms in x be degenerate. Interpreting using theintuition above, this is just the reasonable statement that we only paste m -cellsalong cells of dimension strictly smaller than m . The general case (with arbitraryinjective β ) asks the same condition to hold not just for x but also its faces.The canonical functor ∆ → Z ∆ can be easily checked to factor through O , andthe composite ∆ → O ֒ → ω -Cat is precisely the cosimplicial object Street usedto define the nerve functor. (See Section 1.3 below.) In particular, we obtain thesimplicial nerve of O n as follows. Definition 1.4.
We will write O ( − , n ) for the simplicial set obtained by restrictingthe obvious representable O op → Set along ∆ op → O op . The action of each β :[ p ] → [ m ] is thus given by linearly extending the precomposition∆ (cid:0) [ m ] , [ n ] (cid:1) → ∆ (cid:0) [ p ] , [ n ] (cid:1) : α αβ. YUKI MAEHARA
Note that x ∈ O ( m, n ) is degenerate at some k if and only if each α ∈ supp( x )is degenerate at k . Remark.
It is easy to deduce from the definition of O that we have O (0 , n ) = (cid:8) ( i ) : i ∈ [ n ] (cid:9) for any n ≥
0. This provides an obvious bijection [ n ] ∼ = O (0 , n ), and we will oftenidentify the two sets accordingly. In particular, given x ∈ O ( m, n ) and i ∈ [ m ], wewill treat x ( i ) (the image of x under the action of ( i ) : [0] → [ m ]) as if it were anatural number. Proposition 1.5 ([Ste07, Proposition 5.7]) . Let x ∈ O ( m, n ) . Then we have x (0) = min { α (0) : α ∈ supp( x ) } and x ( m ) = max { α ( m ) : α ∈ supp( x ) } . The operations ∨ k and ▽ k . Now we recall operations ∨ k and ▽ k on thesimplicial set Z ∆ (cid:0) − , [ n ] (cid:1) from [Ste07] which restrict to a sort of pasting and its witness on O ( − , n ) respectively. Definition 1.6.
Let x, y ∈ Z ∆ (cid:0) [ m ] , [ n ] (cid:1) with m ≥
1, and let 1 ≤ k ≤ m . Supposethat xδ k − = yδ k . Then we define x ▽ k y def = xσ k − xδ k − σ k − + yσ k − ,x ∨ k y def = x − xδ k − σ k − + y or equivalently, x ▽ k y def = xσ k − yδ k σ k − + yσ k − ,x ∨ k y def = x − yδ k σ k − − y Remark.
The assumption xδ k − = yδ k in Definition 1.6 should be interpreted asa composability condition. Then according to the intuition described after Theo-rem 1.3, x ∨ k y is precisely the pasting of x and y along their common face. Thesimplex x ▽ k y is to be thought of as a witness for this pasting (see the propositionbelow). See also [Ste12, Remark 6.4] for how ∨ k can indeed be seen as a compositionin a suitable category. Remark.
The operations we are denoting by ∨ k and ▽ k are what would be called ∨ k − and ▽ k − respectively in [Ste07]. We have shifted the index because, whileworking with the combinatorics of O ( − , n ), we found it less confusing for the k -thface (rather than the ( k + 1)-st one) of x ▽ k y to play a special role. We apologiseto the reader if they are already familiar with Steiner’s work and find our notationconfusing. In Steiner’s later papers such as [Ste12], the symbol ▽ is replaced by ∧ and the operations corresponding to ∨ k go unnamed.The following proposition is a direct consequence of Definition 1.6. Proposition 1.7.
For any x, y, k as in Definition 1.6, we have: • ( x ▽ k y ) δ k − = y ; • ( x ▽ k y ) δ k = x ∨ k y ; and • ( x ▽ k y ) δ k +1 = x . RIENTALS AS FREE WEAK ω -CATEGORIES 5 Proposition 1.8 ([Ste07, Proposition 5.4]) . O ( − , n ) is closed under the operations ▽ k and ∨ k . Proposition 1.9.
An element x ∈ Z ∆ (cid:0) [ m ] , [ n ] (cid:1) satisfies x = ( xδ k +1 ) ▽ k ( xδ k − ) ifand only if each simplicial operator in supp( x ) is degenerate at either k − or k .Proof. The “only if” direction is clear. For the “if” direction, suppose that eachsimplicial operator in supp( x ) is degenerate at either k − k . Since we have( xδ k +1 ) ▽ k ( xδ k − ) = xδ k +1 σ k − xδ k +1 δ k − σ k − + xδ k − σ k − , it suffice to check that α = αδ k +1 σ k − αδ k +1 δ k − σ k − + αδ k − σ k − holds for any α : [ m ] → [ n ] that is degenerate at either k − k . The latterstatement is straightforward to verify. (cid:3) The following proposition turns a sum ( x = y + z ) into a pasting ( x = u ∨ k v ),and particular instances of this fact are used in [Ste07, Propositions 5.9-11]. Itsproof is a straightforward manipulation of simplicial operators. Proposition 1.10.
Let x, y, z ∈ Z ∆ (cid:0) [ m ] , [ n ] (cid:1) be elements satisfying x = y + z , andlet ≤ k ≤ m . Define u = y + zδ k σ k − , v = yδ k − σ k − + z. Then we have uδ k − = vδ k , u ▽ k v = yσ k + zσ k − and u ∨ k v = x .Remark. Observe that most of the operations we consider only preserve the supportin a weak sense. For instance, we havesupp( xδ k ) ⊂ (cid:8) αδ k : α ∈ supp( x ) (cid:9) , supp( x ∨ k y ) ⊂ supp( x ) ∪ (cid:8) αδ k − σ k − : α ∈ supp( x ) (cid:9) ∪ supp( y ) . In general, one cannot replace the subset symbol by an equality; one has, for ex-ample, (cid:0) (0 , − (1 ,
1) + (1 , (cid:1) δ = (2) and (cid:0) (0 , , − (0 , , , , (cid:1) ∨ (cid:0) (0 , , − (2 , , , , (cid:1) = (0 , , − (0 , , , , O ( − , is strictly preserved in certain nice cases, andthis will be crucial in our arguments below.1.3. Street nerve and complicial sets.
The cosimplicial object ∆ → ω -Cat : n
7→ O n as described in the previous subsection induces a nerve functor ω -Cat → sSet. However, this functor is not full; for instance, the standard 2-simplex isthe nerve of the obvious 1-category [2], and we can consider the simplicial map∆[2] → O ( − ,
2) picking out the simplex (0 , , ω -functor [2] → O since we are sending the commutative triangle in [2] to anon-commutative one in O .To rectify this, Roberts proposed considering simplicial sets with distinguishedsimplices (to be thought of as “abstract commutative/identity simplices”). Theadjective to refer to these distinguished simplices has changed multiple times fromRoberts’ original neutral , to hollow , to thin , and then to marked . YUKI MAEHARA
Definition 1.11. A marked simplicial set ( X, tX ) is a simplicial set X togetherwith subsets tX n ⊂ X n of marked simplices for n ≥ morphism of marked simplicial sets f : ( X, tX ) → ( Y, tY ) is a simpli-cial map f : X → Y that preserves marked simplices. We denote the category ofmarked simplicial sets by msSet.We will often suppress tX and simply speak of a marked simplicial set X . Remark.
The reader is warned that this notion of marked simplicial set is differentfrom Lurie’s [Lur09, § Definition 1.12.
The
Street nerve of an ω -category C is the marked simplicial set ω -Cat( O ( − ) , C )in which an n -simplex F : O n → C is marked if and only if it sends the (unique)atomic n -cell in O n to an identity in C .With this definition, Street showed in [Str88] that the nerve of any ω -categoryis a complicial set in the sense we recall now. (In fact Street proved somethingstronger; see loc. cit .) Definition 1.13.
We say a morphism in msSet is regular if it reflects markedsimplices. In other words, f : ( X, tX ) → ( Y, tY ) is regular if f ( x ) ∈ tY n implies x ∈ tX n . By a regular subset ( A, tA ) of (
X, tX ), we mean a simplicial subset A ⊂ X equipped with the marking tA n = A n ∩ tX n . Definition 1.14.
For each n ≥
0, we will regard the standard n -simplex ∆[ n ] asa marked simplicial set equipped with the minimal marking ( i.e. only degeneratesimplices are marked). Also, for each 0 < k < n , we write: • ∆ k [ n ] for the object obtained from ∆[ n ] by further marking those simplices α : [ m ] → [ n ] with { k − , k, k + 1 } ⊂ im α ; • Λ k [ n ] for the regular subset of ∆ k [ n ] whose underlying simplicial set is the k -th horn; • ∆ k [ n ] ′ for the object obtained from ∆ k [ n ] by further marking δ k − and δ k +1 ; and • ∆ k [ n ] ′′ for the object obtained from ∆ k [ n ] ′ by further marking δ k .The class of ( inner ) complicial anodyne extensions is the closure of the union (cid:8) Λ k [ n ] ֒ → ∆ k [ n ] : 0 < k < n (cid:9) ∪ (cid:8) ∆ k [ n ] ′ ֒ → ∆ k [ n ] ′′ : 0 < k < n (cid:9) under coproducts, pushouts along arbitrary maps, and transfinite compositions. Definition 1.15.
A ( strict ) complicial set is a marked simplicial set X such that: • X has the unique right lifting property with respect to Λ k [ n ] ֒ → ∆ k [ n ] forall 0 < k < n ; • X has the unique right lifting property with respect to ∆ k [ n ] ′ ֒ → ∆ k [ n ] ′′ for all 0 < k < n ; and • the marked 1-simplices in X are precisely the degenerate ones.Now we can state what was conjectured by Roberts, made precise by Street, andproven by Verity. RIENTALS AS FREE WEAK ω -CATEGORIES 7 Theorem 1.16 ([Ver08b, Theorem 266]) . The Street nerve provides an equivalencebetween ω -Cat and the full subcategory of msSet spanned by the complicial sets.Remark. Since k − k + 1 have the same parity and k has the opposite one,we see from the informal description of O n that δ k − and δ k +1 lie on the same side(either source or target) of the interior n -cell, and δ k lie on the other. Observefurther that, aside from the degenerate ones, the marked simplices in ∆ k [ n ] areprecisely those that are not contained in δ k − , δ k , or δ k +1 . Thus the (marked)interior n -simplex of ∆ k [ n ] may be thought of as asserting an equality between δ k and the composite of δ k − and δ k +1 .In this sense, lifting against Λ k [ n ] ֒ → ∆ k [ n ] defines a sort of composition, andlifting against ∆ k [ n ] ′ ֒ → ∆ k [ n ] ′′ ensures that any composite of identities is itself anidentity. Remark.
Although the term complicial set originally meant Definition 1.15, todayit is more commonly used to refer to the weak variant where: • the unique right lifting property is replaced by a mere right lifting property; • the lifting is required for k = 0 and k = n too; and • the condition on marked 1-simplices is dropped.The homotopy theory of these weak complicial sets is captured by a model structureon msSet due to Verity [Ver08a, Theorem 100]. In particular, the weak complicialsets are precisely the fibrant objects therein. Since the (inner) complicial anodyneextensions of Definition 1.14 are examples of its trivial cofibrations, this modelstructure provides a suitable framework in which to interpret our main result (The-orem 2.1). Note, however, that it will not play any mathematical role in this paper.1.4. O ( − , n ) as complicial sets. From now on, we will regard O ( − , n ) as theStreet nerve of O n and in particular as a marked simplicial set. Lemma 1.17.
A simplex x ∈ O ( m, n ) is marked if and only if each simplicialoperator in supp( x ) is degenerate.Proof. Let ADC denote the category of augmented directed complexes and let K [ m ]and K [ n ] be such complexes suitably generated by ∆[ m ] and ∆[ n ] respectively (see[Ste07] for their precise definitions).Then we have a bijection ω -Cat( O m , O n ) ∼ = ADC (cid:0) K [ m ] , K [ n ] (cid:1) by [Ste07, The-orem 2.6]. (See [Ste04] for more on the relationship between ω -categories andaugmented directed complexes.) By examining the definition of ν (which can befound in [Ste07, Section 2]), one can check that the ω -functors O m → O n thatsend the atomic m -cell in O m to an identity correspond precisely to those maps K [ m ] → K [ n ] that send the basis element [0 , , . . . , m ] to 0.Steiner constructs a bijection ADC (cid:0) K [ m ] , K [ n ] (cid:1) ∼ = O ( m, n ) in [Ste07, Theorem4.1]. By examining its proof, one can check that such maps K [ m ] → K [ n ] in turncorrespond precisely to those x ∈ O ( m, n ) with all α ∈ supp( x ) degenerate. Thiscompletes the proof. (cid:3) With this characterisation of marked simplices, we can show the two notions ofcomposition to be equivalent.
Proposition 1.18.
Let x, y ∈ O ( m, n ) and suppose xδ k − = yδ k . Then thereexists a unique map ∆ k [ m + 1] → O ( − , n ) that sends δ k − and δ k +1 to y and x respectively, namely the one that picks out x ▽ k y . YUKI MAEHARA
Proof.
Let X ⊂ ∆ k [ m + 1] be the minimal regular subset containing δ k − and δ k +1 .We first show that the inclusion X ֒ → ∆ k [ m + 1] is a complicial anodyne extension.To see this, note that the simplices α in ∆ k [ m + 1] \ X are precisely those α : [ p ] → [ m + 1] whose images contain both k − k + 1. Thus we maypartition the set of non-degenerate simplices in ∆ k [ m + 1] \ X into pairs of the form { α, αδ i } where α : [ p ] → [ m + 1] sends i ∈ [ p ] to k . Moreover, it is straightforwardto see that α defines a regular map of marked simplicial sets α : ∆ i [ p ] → ∆ k [ m + 1].It follows that the inclusion X ֒ → ∆ k [ m + 1] may be obtained by filling Λ i [ p ] foreach such α in increasing order of p . This exhibits the inclusion as a complicialanodyne extension.By assumption, the simplices x, y specify a map X → O ( − , n ). Since O ( − , n ) isa complicial set, this map extends uniquely to ∆ k [ m + 1] by the above argument. Itnow remains to check that the map picking out x ▽ k y is indeed such an extension.We have ( x ▽ k y ) δ k − = y and ( x ▽ k y ) δ k +1 = x by Proposition 1.7, and we alsohave x ▽ k y ∈ O ( m + 1 , n ) by Proposition 1.8. That this map respects markingfollows from Proposition 1.9 and Lemma 1.17. This completes the proof. (cid:3) Main theorem
The purpose of this paper is to prove the following theorem.
Theorem 2.1.
The map ι n : ∆[ n ] → O ( − , n ) picking out the simplex (0 , . . . , n ) isa complicial anodyne extension for any n ≥ . We factorise the map ι n as follows. Let A ⊂ O ( − , n ) be the regular subsetconsisting of those x ∈ O ( m, n ) such that α − ( n ) has the same cardinality for all α ∈ supp( x ). Note that, since each x in the image of ι n has exactly one simplicialoperator in supp( x ), we have a chain of regular monomorphisms∆[ n ] → A ֒ → O ( − , n ) . (That ∆[ n ] → A is regular easily follows from Lemma 1.17.)The following lemma is the combinatorial heart of this paper. Lemma 2.2.
The inclusion
A ֒ → O ( − , n ) is a complicial anodyne extension.Proof. Observe that the simplices in O ( − , n ) \ A may be characterised as those m -simplices x for which there exist α, β ∈ supp( x ) and i ∈ [ m ] such that α ( i ) = n = β ( i ).Fix a non-degenerate m -simplex x in O ( − , n ) \ A . Let r be the rank of x , bywhich we mean rank( x ) def = min [ α ∈ supp( x ) α − ( n ) . For α ∈ supp( x ), let ˇ x α = ( x α , if α ( r ) < n, , if α ( r ) = n, ¯¯ x α = ( , if α ( r ) < n,x α , if α ( r ) = n. We also declare ˇ x α = ¯¯ x α = 0 for α / ∈ supp( x ) so that we have ˇ x, ¯¯ x ∈ Z ∆ (cid:0) [ m ] , [ n ] (cid:1) and x = ˇ x + ¯¯ x . (The decorations are supposed to remind the reader which one RIENTALS AS FREE WEAK ω -CATEGORIES 9 ˇ x ¯¯ x ? ? ⋆ ⋆ ? ?? ? n n ℓ r Figure 3. r , ˇ x , ¯¯ x and ℓ corresponds to the (in)equality.) Note that the first sentence of this proof impliesˇ x = 0 = ¯¯ x .Let ℓ be the level of x , by which we meanlevel( x ) def = min (cid:8) i ∈ [ m ] : α ( i ) = α ( r ) for all α ∈ supp(ˇ x ) (cid:9) . (See Fig. 3.) Claim 2.2.a.
We have ℓ ≥ .Proof of Claim. Suppose for contradiction that ℓ = 0.Let y = x (0 , . . . , r ) ∈ O ( r, n ). It follows from our assumption that any α ∈ supp( y ) satisfies either α ( r ) = n or α (0) = α ( r ) < n . Fix α of the latter sort, sayconstant at k . Then α is the unique element of supp( y ) that maps r to k . It followsthat y α = 1 if k = y ( r ) and y α = 0 otherwise. But since (cid:8) α (0 , . . . , r ) : α ∈ supp(¯¯ x ) (cid:9) is clearly contained in supp( y ), we have y ( r ) = n > k by Proposition 1.5. Thereforewe have X α ( r ) The condition ( † ) • rank( y ) = r − ; • level( y ) = ℓ ; and • y does not satisfy ( † ) .Moreover we have ˇ y = ˇ xδ ℓ and ¯¯ y = ¯¯ xδ ℓ .Proof of Claim. We first show that the support of y is given bysupp( y ) = (cid:8) αδ ℓ : α ∈ supp( x ) (cid:9) . Let α, β ∈ supp( x ) and suppose αδ ℓ = βδ ℓ . Since ℓ < r , this assumption inparticular implies α ( r ) = β ( r ), so we have either: • α ( r ) = β ( r ) < n , in which case α = αδ ℓ σ ℓ = βδ ℓ σ ℓ = β ; or • α ( r ) = β ( r ) = n , in which case α = αδ ℓ σ ℓ − = βδ ℓ σ ℓ − = β .In either case, we have α = β . This shows that supp( y ) may be described asasserted above.It is now easy to deduce that y is a non-degenerate simplex in O ( − , n ) \ A withrank( y ) = r − y ) = ℓ , and that we have ˇ y = ˇ xδ ℓ and ¯¯ y = ¯¯ xδ ℓ . Tosee that y does not satisfy ( † ), observe that if it did then ¯¯ x and hence x would bedegenerate at ℓ . (cid:3) Claim 2.2.c. If x does not satisfy ( † ) , then its parent w = ˇ xσ ℓ + ¯¯ xσ ℓ − is anon-degenerate ( m + 1) -simplex in O ( − , n ) \ A such that: • rank( w ) = r + 1 ; • level( w ) = ℓ ; and • w satisfies ( † ) .Moreover we have wδ ℓ = x .Proof of Claim. Let u = ˇ x + ¯¯ xδ ℓ σ ℓ − and v = ˇ xδ ℓ − σ ℓ − + ¯¯ x (see Fig. 5). Thenwe have w = u ▽ ℓ v and wδ ℓ = u ∨ ℓ v = x by Proposition 1.10. Thus to prove w ∈ O ( m + 1 , n ), it suffices (by Proposition 1.8) to show u, v ∈ O ( m, n ). Note thatboth u and v clearly satisfy (O1).Let β : [ p ] → [ m ] and γ : [ p ] → [ n ] be injective maps in ∆. We would like toshow that ( uβ ) γ ≥ • If ℓ / ∈ im( β ) then ( uβ ) γ = ( xβ ) γ ≥ • Suppose that β ( k ) = ℓ and β ( k − 1) = ℓ − ≤ k ≤ p . Notethat in this case ( uβ ) γ = (ˇ xβ ) γ since γ is injective and ¯¯ xβ is degenerate at k − 1. We may assume γ ( k ) < n for otherwise (ˇ xβ ) γ = 0. – If k ≤ p − β ( k + 1) ≤ r then ˇ xβ is degenerate at k , hence(ˇ xβ ) γ = 0. RIENTALS AS FREE WEAK ω -CATEGORIES 11 ˇ x ¯¯ x ? ? ♦ ⋆ ⋆ ? ?? ? † (cid:7) ? ? n n ˇ x ¯¯ xδ ℓ σ ℓ − ? ? ♦ ⋆ ⋆ ? ?? ? † † ? ? n n ˇ xδ ℓ − σ ℓ − ¯¯ x ? ? ⋆ ⋆ ⋆ ? ?? ? † (cid:7) ? ? n nxuv ℓ r Figure 5. x , u and v in Claim 2.2.c – Otherwise (ˇ xβ ) γ = ( xβ ′ ) γ ≥ β ′ : [ p ] → [ m ] is given by β ′ ( i ) = ( r, if i = k,β ( i ) , otherwise. • Suppose that β ( k ) = ℓ for some k ∈ [ p ] and ℓ − / ∈ im( β ). Then the sameargument as in the previous bullet point shows that (ˇ xβ ) γ is non-negative.So it suffices to prove (¯¯ xδ ℓ σ ℓ − β ) γ ≥ 0. Suppose (¯¯ xδ ℓ σ ℓ − β ) γ = 0. – If γ ( p ) = n then we have (¯¯ xδ ℓ σ ℓ − β ) γ = ( xβ ′ ) γ ≥ β ′ ( i ) = ℓ − i = k,r if i = p,β ( i ) otherwise. – If γ ( p ) < n then we have (¯¯ xδ ℓ σ ℓ − β ) γ = ( xβ ′ ) γ ′ ≥ β ′ :[ p + 1] → [ m ] and γ ′ : [ p + 1] → [ n ] are given by β ′ ( i ) = ℓ − i = k,r if i = p + 1 ,β ( i ) otherwise,and γ ′ ( i ) = ( n if i = p + 1 ,γ ( i ) otherwiserespectively.Now we show ( vβ ) γ ≥ • If ℓ − / ∈ im( β ) then ( vβ ) γ = ( xβ ) γ ≥ • Suppose that β ( k − 1) = ℓ − β ( k ) = ℓ for some 0 ≤ k ≤ p − 1. Notethat in this case ( vβ ) γ = (¯¯ xβ ) γ . Suppose further that (¯¯ xβ ) γ = 0 holds. – If γ ( p ) = n then we have (¯¯ xβ ) γ = ( xβ ′ ) γ where β ′ ( i ) = ( r if i = p,β ( i ) otherwise. – If β ( p ) < r then we have (¯¯ xβ ) γ = ( xβ ′ ) γ ′ ≥ β ′ : [ p + 1] → [ m ]and γ ′ : [ p + 1] → [ n ] are given by β ′ ( i ) = ( r if i = p + 1 ,β ( i ) otherwise,and γ ′ ( i ) = ( n if i = p + 1 ,γ ( i ) otherwiserespectively. • Suppose that β ( k ) = ℓ − k ∈ [ p ] and ℓ / ∈ im( β ). Then ( vβ ) γ =( uβ ′ ) γ where β ′ ( i ) = ( ℓ − i = k,β ( i ) otherwise.Thus we have already shown this coefficient to be non-negative.This proves that w ∈ O ( m + 1 , n ).Observe that, since σ ℓ and σ ℓ − have a common section δ ℓ , the support of w =ˇ xσ ℓ + ¯¯ xσ ℓ − is given bysupp( w ) = (cid:8) ασ ℓ : α ∈ supp(ˇ x ) (cid:9) ∪ (cid:8) ασ ℓ − : α ∈ supp(¯¯ x ) (cid:9) . It now easily follows that w is not contained in A , has rank r + 1 and level ℓ , andsatisfies ( † ). (cid:3) Claim 2.2.d. The set of non-degenerate simplices in O ( − , n ) \ A may be partitionedinto parent-child pairs.Proof. We must show that the parent and child constructions are inverse to eachother. Given x that does not satisfy ( † ), let w be its parent. Then it follows fromClaim 2.2.c that the child of w is indeed wδ level( w ) = wδ ℓ = x .Now suppose that x satisfies ( † ), and let y be its child. Then it follows fromClaim 2.2.b that the parent of y isˇ yσ level( y ) + ¯¯ yσ level( y ) − = ˇ xδ ℓ σ ℓ + ¯¯ xδ ℓ σ ℓ − . Since the condition ( † ) implies that ˇ x is degenerate at ℓ and ¯¯ x is degenerate at ℓ − x . (cid:3) The parent-child pairs correspond to the interior-face pairs of the horns to befilled in the final step of this proof. We will fill these horns in lexicographicallyincreasing order of their dimension, corank and level. Here by the corank of x wemean the difference dim( x ) − rank( x ); equivalently,corank( x ) def = max (cid:8) | α − ( n ) | : α ∈ supp( x ) (cid:9) . Observe that both the parent and child constructions preserve the corank (seeClaims 2.2.b and 2.2.c). RIENTALS AS FREE WEAK ω -CATEGORIES 13 Claim 2.2.e. If x satisfies ( † ) , then for any k ∈ [ m ] with k = ℓ , one of the followingholds: • xδ k is contained in A ; • xδ k is contained in O ( − , n ) \ A and satisfies ( † ) ; or • xδ k is contained in O ( − , n ) \ A and does not satisfy ( † ) , and we have (cid:0) corank( xδ k ) , level( xδ k ) (cid:1) < lex ( c, ℓ ) . Here < lex denotes the lexicographical order, so the assertion is that either wehave corank( xδ k ) < c , or we have corank( xδ k ) = c and level( xδ k ) < ℓ . Proof of Claim. If xδ k is contained in A or satisfies ( † ) then we are done. So supposethat xδ k is contained in O ( − , n ) \ A and does not satisfy ( † ). Note that the corankof xδ k is at most c , and if corank( xδ k ) = c then the level of xδ k is at most ℓ .The case k ≥ r is easy since xδ k necessarily has corank < c .Next consider the case k < ℓ . If xδ k has corank < c then we are done. Ifcorank( xδ k ) = c then xδ k necessarily has level < ℓ .The remaining case is when ℓ < k < r . In this case, observe that if xδ k hascorank c and level ℓ then it necessarily satisfies ( † ), contradicting our assumption.This completes the proof. (cid:3) Claim 2.2.f . Suppose that x is marked and does not satisfy ( † ) , and let w be itsparent. Then the faces wδ ℓ − and wδ ℓ +1 are also marked.Proof of Claim. By Lemma 1.17, the assumption that x is marked is equivalent toevery operator in supp( x ) being degenerate. Recall from the proof of Claim 2.2.cthat we have wδ ℓ +1 = ( u ▽ ℓ v ) δ ℓ +1 = u where u = ˇ x + ¯¯ xδ ℓ σ ℓ − and v = ˇ xδ ℓ − σ ℓ − + ¯¯ x . Since we havesupp( u ) ⊂ supp(ˇ x ) ∪ { αδ ℓ σ ℓ − : α ∈ supp(¯¯ x ) } and every simplicial operator in the larger set is degenerate, u is marked by Lemma 1.17.Similarly wδ ℓ − = v is marked. (cid:3) Finally we exhibit the inclusion A ֒ → O ( − , n ) as a complicial anodyne extension.For each m, c, ℓ ≥ 0, let P m,c,ℓ be the set of parent simplices of dimension m , corank c , and level ℓ . (Note that P m,c,ℓ = ∅ if any one of m , c or ℓ is 0.) For any ( m, c, ℓ ) ∈ ω × ω × ω , we will write B m,c,ℓ ⊂ O ( − , n ) for the minimal regular subset containing A and P m ′ ,c ′ ,ℓ ′ for ( m ′ , c ′ , ℓ ′ ) ≤ lex ( m, c, ℓ ). Then the assignation ( m, c, ℓ ) B m,c,ℓ defines a transfinite sequence in msSet (with each limit ordinal < ω × ω × ω beingmapped to the colimit of the previous objects) whose composite is precisely theinclusion A ֒ → O ( − , n ). Hence it suffices to show that each B m,c,ℓ − ֒ → B m,c,ℓ is a complicial anodyne extension. Note that, by Claim 2.2.e, the non-degeneratesimplices in B m,c,ℓ \ B m,c,ℓ − are precisely the simplices in P m,c,ℓ and their children.Moreover, any w ∈ P m,c,ℓ is marked in O ( − , n ) by Lemma 1.17, and furthermoreif wδ ℓ is marked in O ( − , n ) then so are wδ ℓ − and wδ ℓ +1 by Claim 2.2.f. Thus theinclusion B m,c,ℓ − ֒ → B m,c,ℓ may be factorised as a pushout of a w ∈ P m,c,ℓ (cid:0) Λ ℓ [ m ] ֒ → ∆ ℓ [ m ] (cid:1) followed by a pushout of a w (cid:0) ∆ ℓ [ m ] ′ ֒ → ∆ ℓ [ m ] ′′ (cid:1) where the second coproduct is taken over those w ∈ P m,c,ℓ with wδ ℓ marked. Thiscompletes the proof. (cid:3) Proof of Theorem 2.1. We prove the theorem by induction on n . The base case istrivial since ι : ∆[0] → O ( − , 0) is invertible. For the inductive step, fix n ≥ ι n − : ∆[ n − → O ( − , n − 1) is a complicial anodyne extension.Consider the factorisation ∆[ n ] → A ֒ → O ( − , n )of ι n . By Lemma 2.2, the second factor is a complicial anodyne extension. Observethat the first factor is isomorphic to ι n − ⊕ id : ∆[ n − ⊕ ∆[0] → O ( − , n − ⊕ ∆[0]where ⊕ is the join operation. It follows from the inductive hypothesis and [Ver08a,Lemma 39] (which states that the complicial anodyne extensions are suitably stableunder the join operation) that ∆[ n ] → A is a complicial anodyne extension. Thiscompletes the proof. (cid:3) Remark. Fix n ≥ 1, and let ∆[ n ] t denote the marked n -simplex (obtained from ∆[ n ]by marking the unique non-degenerate n -simplex). Since marked simplices are tobe thought of as identities, it is reasonable to expect the Street nerve of the ( n − O n to be a fibrant replacement of ∆[ n ] t . Unfortunately, thisresult does not seem to be accessible via our current approach. The “most fibrant”object we can prove to be weakly equivalent to ∆[ n ] t is N O n with all n -simplicesadditionally marked (which is not actually fibrant), though we will omit the proof. 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