A folk model structure on omega-cat

A folk model structure on omega-cat Yves Lafont ∗ , François Métayer † & Krzysztof W orytkiewicz ‡ September 1, 2018 Abstract The primary aim of this work is an intrinsic homo topy theory of strict ω -categories. W e establish a model structure on ω Cat , the category of strict ω -categ ories. The con structions leading to the model structure in que stion are expressed entirely wi thin the scope of ω Cat , building on a set of generating cofibrations and a class of weak equi v alences as basic items. All object are fibrant while free objects are cofibrant. W e further exhibit model structures of this type on n -categories for arbitrary n ∈ N , as specialisations of the ω -catego rical one along right adjoints. In particular , kno wn cases for n = 1 an d n = 2 nicely fit into the scheme. 1 Introd uction 1.1 Backgr ound and motivations The origin of the present work goes back to the following result [1, 24]: if a mon oid M can be presen ted by a finite, conflu ent and terminating rewriting system, then its third homolo gy group H 3 ( M ) is of finite type. The finiteness p roperty e xtends in f act to all d imensions [14], but th e above theorem may als o be refined in another direction: th e s ame hy pothesis implies that M h as finite derivation type [25], a property of homotopical nature. W e claim that these id eas are better expressed in terms of ω -categories (see [10, 1 1, 17]). Thus we work in the category ω Ca t , whose objects are the strict ω - categories an d the morphisms are ω -functor s (see Section 3 ). In fact, when consider ing the inter play b etween th e mo noid itself an d the space o f co mputations a ttached to any p resen- tation of it, on e readily obser ves that both o bjects suppor t a structu re of ω -category in a very direct way: this was the starting p oint o f [1 9 ], whic h intr oduces a notion of resolution f or ω -c ate gories, based o n co mputads [2 6 , 21] or poly graphs [6], the term inology we adopt here. Recall that a polygr aph S consists of sets o f cells of all dim en- sions, determining a freely generated ω -c ate gory S ∗ . A resolutio n of an ω - cate gory C by a polygr aph S is then an ω -functor p : S ∗ → C satisfying a certain lifting pr operty (see Section 5 below); [19] also defines a hom otopy re- lation between ω -functors an d sho ws th at an y two resolutio ns of the same ω -category are homotopically equi valent in this sense. This immediate ly s uggests lo oking for a ho motopy theory on ω Ca t in which the a bove resolutions becom e t rivial fibrations: the model stru cture we descr ibe here does exactly tha t. No tice, in add ition, that polyg raphs turn out to be th e cofibrant objec ts (see [20] a nd Sec tion 5 below). On the o ther hand, our mo del stru cture generalizes in a very precise sense the “folk” model st ructure on Cat (see [13]) as well a model s tructure on 2 Cat in a similar spirit (see [1 5, 16]). Incidentally , there is also a q uite different, Thomason -like, model structure o n 2 Cat ( see [27]). Its generalisation to ω Cat remains an open problem. Since [22], the notion of mod el structure has been grad ually recogn ized as the app ropriate abstract framework for developing homo topy th eory in a category C : it co nsists in three classes of morp hisms, weak equivalence s , fibrations , and cofibrations , subject to axioms whose e xact formulatio n h as some what ev olved in time. In practice, most model structures are cofib r antly generated . Th is means that there are sets I of generating cofibrations and J of generating trivial cofibrations which determine all the cofibrations and all the fibrations by lifting properties. Recall that, given a set I of morph isms, I - injectives are the morph isms which have the right lifting pr o perty with respect to I . They build a class denoted by I − inj . Likewise, I -cofi brations are th e morphisms having the left lifting pr o perty with respect to I − inj (see Section 2.2). The class of I -cofibration s is denoted by I − cof . Now , our ∗ Uni versi té de la Méditerranée (Aix-Marseille 2) - Institut de Mathémat iques de Luminy (UMR 6206 du CNRS) † Uni versi té Denis Diderot (UMR 7126 du CNRS) - L abor atoire Preuves, Programmes et Systèmes ‡ Akademia Górnicz o-Hutnic za Kraków - Kated ra Informatyki 1 construction is based on a theorem by J.Smith (see [3]): und er some fairly standard ass umption s o n the underlying category , conditions (S1) W has the 3 for 2 proper ty and is stable under retracts; (S2) I − inj ⊆ W ; (S3) I − cof ∩ W is closed under pushouts and transfinite composition s; (S4) W admits a solution set J ⊆ I − cof ∩ W at I . are sufficient to ob tain a model s tructure in which W , I and J are the weak equivalences, the generating cofib rations and the generatin g tri vial cofibrations, respectiv ely . 1.2 Organization of the paper Section 2 re views combinatorial model cate gories , with special emphasis on our v ersion of Smith’ s theorem (Sec- tion 2.4), while Section 3 recalls the basic definitions of globular s ets and ω -categories, and sets the notations. Section 4 is the core of the paper, that is the de ri v ation of our model structure by m eans of a set I of ge nerating cofibration s and a class W of weak equiv alences, satisfying conditions (S1) to (S4) . 1.2.1 Sketch of the main argument W e first defin e the set I of g enerating cofibration s, and establish clo sure pro perties we shall use later in the proof of condition (S3) . W e then define the class W of ω - weak eq ui valences, wh ich are at this stage our candidates for the rôle of weak equiv alences (Section 4.3). For this purp ose, we first need a no tion of ω -equiv alence between p arallel cells ( Sec- tion 4.2), together with crucial prop erties of this notion. W e then p rove condition (S2) , and par t of (S1) (Section 4 .3), as well as additio nal closure properties contributing to (S3) . At this stage, just one point of (S1) remains unpr ov ed, namely the assertion if f : X → Y and g ◦ f : X → Z belong to W , then so does g : Y → Z . This req uires an entirely n e w constru ction: we define an endo functor Γ of ω Cat , which to e ach ω -category X associates a n ω -category Γ( X ) of r ever sible cylinders in X . Section 4.4 summarizes th e main features of Γ , whereas the more technical p roofs ar e given in Appendix A. This e ventu ally leads to an alternati ve characte rization of weak equiv alences and to a complete proof of (S1) . As for conditio n (S3) , the difficult point is to prove the closure of I − cof ∩ W by pu shout, which does not follow from th e previously established pro perties. T he main o bstacle is that W itself is definitely n ot closed b y pushou t. What we need instead is a new class Z of immersions such that: i. Z is closed by pushou t; ii. I − cof ∩ W ⊆ Z ⊆ W , which completes th e p roof o f (S3) . I mmersions are defined in Section 4. 6, by u sing aga in the f unctor Γ in an essential way . Section 4.7 is de voted to the proof of the solution set condition (S4) . Precisely , we ha ve to b uild, for each i ∈ I , a set J i of ω -functo rs satisfy ing the follo wing property : for each commu tati ve sq uare X / / i   Z f   Y / / T (1) where i ∈ I and f ∈ W , there is a j ∈ J i such that (1) factors through j : X / / i   U j   / / Z f   Y / / V / / T (2) The whole solution set is then J def . = S i ∈ I J i . It turns out that in our case, the sets J i are just singletons. 2 1.2.2 Additional properties The end of the pape r is dev oted to two ad ditional po ints: Section 5 gi ves a char acterization o f cofibr ant o bjects as polygr aphs by interpr eting the results of [ 20 ] in terms o f ou r mod el structu re. Finally , Section 6 shows how the p resent m odel stru cture on ω Cat tran sfers to n Cat for any in te ger n : in particular, for n = 1 and n = 2 , we recover the abovementioned structures on Cat [13] and 2 Cat [15, 16]. 2 Combinatorial model categories W e recall some facts abou t model categories with locally-pr esentable underlying categories. 2.1 Locally pre sentable categories Let α be a regular cardinal. An α -filtered category F is a category s uch that i. for any set o f ob jects S with cardina lity | S | < α and fo r each A ∈ S there is an object T an d a mor phism f A : A → T ; ii. for any set of morp hisms M ⊆ F ( A, B ) with cardin ality | M | < α ther e is an object C and a m orphism m : B → C , such that m ◦ m ′ = m ◦ m ′′ for all m ′ , m ′′ ∈ M . W e say th at F is filtered in case α = ℵ 0 . In pa rticular , a dir ec ted (partially order ed) set is a filtered category . Recall that an α -filter ed colimit is a co limit of a functor D : I → C f rom a small α -filtered category I . L et C be a category . An object X ∈ C is α -presen table if the covariant representable functor C ( X , − ) : C → Sets preserves α -filtered colimits. This bo ils d o wn to the fact that a mo rphism from X to an α -filtered colimit factors throu gh some object o f the relevant α -filtered diagr am, in an essentially un ique way . If X is α -presen table, and β is a regular cardinal such that α < β , then X is also β -presentab le. W e say th at an object X ∈ C is pr esentable if ther e is a regular car dinal witnessing th is fact. If it is th e case, the smallest such cardinal, π ( X ) , is called X ’ s pr esentation r ank . Definition 1. Let α be a r e gular card inal. A cocomplete category C is locally α -pr esentable if th ere is a family G = ( G i ) i ∈ I of ob jects such that every o bject of C is an α -filtered colimit of a d iagram in the f ull subcategory spanned by the G i ’ s. W e say that a cocomplete category is locally finitely pr esentable if it is locally ℵ 0 -presentab le. Finally , we say that a cocomplete category is locally presentab le if there is a regular cardinal witnessing this f act. Definition 1 is equiv alent to th e orig inal one b y Gabriel an d Ulmer [ 7]. I t proves especially powerful to establish factorisation results, when combined with the small object ar g ument (Section 2.2) . L et β be a regular card inal. Recall that a β -colimit is a colim it of a functor D : I → C fro m a small category I such that | I 1 | < β . Proposition 1. Let β be a regular cardinal. A β -colimit of β -presentab le objects is β -presentab le. Remark 1. Let α be a regu lar cardinal an d C b e a locally α -p resentable category . By definition of local pr esentabil- ity , every object X ∈ C is an α -filtered colimit of a diagra m of α -presentable ob jects, so it is a β -colimit for a regular card inal β such th at α 6 β 6 | C 1 | + . Thus, by vir tue o f Prop osition 1, ev ery o bject o f C is p resentable (with a presentation rank possibly exceeding α ). ♦ 2.2 Small objects f or fr ee Let C be a ca te gory . Recall that its cate gory of morp hisms C → is defined as the functor ca te gory C ( ·→· ) , where ( · → · ) is the category g enerated b y the on e-arrow graph. L et f : X → Y and g : Z → T be morp hisms in C . W e say that f has the left-lifting property with respect to g , or equi valently tha t g has the right lif ting property with respect to f , if every co mmuting squar e ( u, v ) ∈ C → ( f , g ) admits a lift , that is a morphism h : Y → T makin g the following diagram commutativ e X u / / f   Z g   Y h > > v / / T 3 This relation is denoted by f ⋔ g . For any class of morphisms A , we define ⋔ A def . = { f | f ⋔ g , g ∈ A} A ⋔ def . = { g | f ⋔ g , f ∈ A} Proposition 2. Suppose f = f ′′ ◦ f ′ . Then − if f ′ ⋔ f then f is a retract of f ′′ ; − if f ⋔ f ′′ then f is a retract of f ′ . Proposition 2 is known as “the retract argument”. Let dom : C → → C and co d : C → → C be th e obvious functors picking the d omain and the cod omain of a morph ism, respecti vely . A fun ctorial factorisation in C is a triple F = ( F, λ, ρ ) where F : C → → C is a fu nctor while λ : do m → F an d ρ : F → cod are natural transfo rmations. Let L and R be classes of morphisms in C . W e say that the pair ( L , R ) admits a functorial factorisation ( F, λ, ρ ) provided that λ f ∈ L and ρ f ∈ R for all morphisms f ∈ C → . If ( F, λ, ρ ) is clear from the con te xt (or if it does not matter), we say by abuse of language that ( L , R ) is a functor ial factor isation. Let I be a set of m orphisms in a cocomp lete c ate gory C and I ∗ be th e closure of I und er pushout. T he class I -cell of r elative I -cell complexes is the closure of I ∗ under tran sfinite composition . Let I − inj def . = I ⋔ and I − cof def . = ⋔ ( I − inj) . Remark 2. If I ⊆ I ′ , then I − inj ⊇ I ′ − inj and J − inj = ( J − cof ) − inj . It is easy to see that I − cell ⊆ I − cof . ♦ The next proposition recalls standard formal properties of the classes just defined (see [8]). Proposition 3. I − inj as well as I − cof contain all iden tities. I − inj is closed un der composition and p ullback while I − cof is closed under retract, transfinite composition and pushout. W e may now state the crucial factorisation result we shall need: Proposition 4. Suppose that C is locally p resentable and let I be a set of mor phisms of C . Then ( I − cell , I − inj) is a functo rial f actorisation. Proof. The req uired factorisation is p roduced by the “small object argument”, d ue to Quillen (see also [9] for an extensi ve discussion): − For any f in C → , let S f be the set of morphisms of C → with domain in I and codo main f , that is S f = { s = ( u s , v s ) ∈ C → | dom( s ) = i s ∈ I , co d( s ) = f } . W e get a f unctor F : C → → C toge ther with natu ral tran sformations λ : dom → F an d ρ : F → co d determined by the inscribed pushout of the outer commutative square a s ∈ S f A s a s ∈ S f B s X Y F ( f ) a s ∈ S f i s   [( u s ) s ∈ S f ] / / [( v s ) s ∈ S f ] / / f   j 0 4 4 j j j j j j j j λ f             ρ f $ $ where, for each s ∈ S f , i s : A s → B s , and [( u s ) s ∈ S f ] , [( v s ) s ∈ S f ] are giv en by the universal p roperty of coprod ucts. 4 − By tra nsfinite iteratio n of the previous con struction, we get, for each o rdinal β , a triple ( F β , λ β , ρ β ) . Pr e- cisely , F 0 ( f ) def . = F ( f ) , λ 0 f def . = λ f , ρ 0 f def . = ρ f ; if β + 1 is a successor ordina l, then F β +1 ( f ) def . = F  ρ β f  , λ β +1 f def . = λ ρ β f ◦ λ β f , ρ β +1 f def . = ρ ρ β f , and if β be a limit ordin al, then F β ( f ) def . = colim γ <β F γ ( f ) while λ β f and ρ β f are giv en by transfinite composition and uni versal proper ty , respectiv ely . − Now notice th at, for ea ch ordinal β , λ β f belongs to I − cell , an d that ( λ β f , ρ β f ) is a fu nctorial factorisation. It remains to show that there is an o rdinal κ f or which ρ κ f belongs to I − inj . This is where loc al presentability helps: thus, let κ b e a regu lar card inal suc h that f or e ach i ∈ I , th e presentation rank π (dom i ) is strictly smaller than κ , and suppose that the outer square of the following diagram commutes: A B F κ ( f ) Y a s ∈ S ρ β f B s F β +1 ( f ) i   u / / v / / ρ κ f   in B ? ? j β +1 ? ? c β +1 ,κ ? ? Since A is κ -presentab le and F κ ( f ) is a κ - filtered colimit, there is a β < κ such that u factor s thr ough F β ( f ) as u = c β ,κ ◦ u ′ for some u ′ , with c β ,κ : F β ( f ) → F κ ( f ) the colimiting morp hism. I t follows then from the above construction that c β +1 ,κ ◦ j β +1 ◦ in B is a lift, whence ρ κ f ∈ I − inj , and we are don e. ⊳ 2.3 Model structur es a nd cofibrant g en eration W e say that a class A o f mor phisms has the 3 fo r 2 pr operty if whenever h = g ◦ f and a ny two ou t of the three morph isms f , g , h belong to A , then so does the third. W e now recall the basics of model structures, follo wing the presentation of [12]. Definition 2. A model structur e on a complete and cocomplete category C is given by three classes of morph isms, the class C of cofibrations , the class F of fibrations , an d the class W of weak equivalences , satisfy ing the fo llo wing condition s: (M1) W has the 3 for 2 proper ty; (M2) C , F and W are stable under retracts; (M3) C ∩ W ⊆ ⋔ F and F ∩ W ⊆ C ⋔ ; (M4) the pairs ( C ∩ W , F ) and ( C , F ∩ W ) are functor ial factorisations. 5 A complete and cocomplete category equipped with a model structure is called a model cate gory . T he members of F ∩ W are called trivial fibrations and the member s of C ∩ W are trivial cofibrations . Remark 3. There is a certain amo unt of redu ndancy in the definition of a model category as the class of fibrations is determined by the class of cofibrations and vice-versa: we have − F = ( C ∩ W ) − inj − F ∩ W = C − inj ; as well as − C = ⋔ ( F ∩ W ) ; − C ∩ W = ⋔ F . ♦ In most known model categories co fibrations and fibrations are gene rated by sets of mo rphisms. In the case of locally-pr esentable cate gories, we get the following definition: Definition 3. A locally-presentable mo del category is cofibrantly generated if there ar e two sets I , J of morphisms such that i. C = I − cof ; ii. C ∩ W = J − cof . The morphisms in I are called generating cofibrations while the morphisms in J are called generating tri vial cofi- brations . Locally -presentable, cofibrantly generated model categories are called combinato rial model categories . Notice that a locally-pr esentable model category is combinatorial if and only if F ∩ W = I − inj and F = J − inj . The wh ole point in the definition o f combinator ial mod el catego ries is the possibility to ap ply the small ob ject argument to arbitrary sets I and J . T he general case, ho we ver , requires extra conditions on those sets. 2.4 The solution set condition Let C be a category , i a morphism of C and W a class of mo rphisms of C . W e say that W adm its a solution set at i if the re is a set W i of morph isms such that any commutative sq uare • i   / / • w ∈ W   • / / • where w ∈ W factors throu gh some w ′ ∈ W i : • i   / / • / / w ′ ∈ W i   • w ∈ W   • / / • / / • If I is a set of morph isms, we say that W admits a solution set at I if it admits a solution set at any i ∈ I . W e now turn to Smith’ s theorem, on which our constru ction is based: Theorem 1. Let I be a set, and W a class of morphisms in a locally presentable category C . Suppose that (S1) W has the 3 for 2 proper ty and is stable under retracts; (S2) I − inj ⊆ W ; (S3) I − cof ∩ W is closed under pushou ts and transfinite compositions; (S4) W admits a solution set J ⊆ I − cof ∩ W at I . Then C is a combinatorial model category w here W is the class of weak equiv alences while I is a set of generating cofibration s and J is a set of genera ting tri vial cofibrations. 6 W e refer to [3] for a n e xtensive discussion of Theorem 1 . In the original statement, (S4) only requires the e xistence of a solution set, without any inclu sion c ondition. Th e present version brings a minor simplification in the treatment of our particular case. For the remaining of this section, we assume the hypotheses of Theorem 1. Lemma 1. (Smith) Suppose there is a class J ⊆ I − cof ∩ W such that each commutin g square • i   / / • w ∈ W   • / / • admits a factorisation • i   / / • / / j ∈J   • w ∈ W   • / / • / / • Then J − cof = I − cof ∩ W Lemma 1 is a ke y s tep in the proof of Theorem 1. This is Lemma 1.8 in [3 ], where a complete proof is giv en, based again on the small object argument combined with an inductio n step. Remark 4. W e have J − cof = I − cof ∩ W by Lemma 1, so in particular J − inj = ( I − cof ∩ W ) − inj by remark 2. ♦ Lemma 2. I − inj = J − inj ∩ W . Proof. “ ⊆ ” Since I − inj ⊆ W , by (S2) , we need to show that I − inj ⊆ J − inj . L et j ∈ J , f ∈ I − inj and suppose f ◦ u = v ◦ u for som e u and v . The small o bject argu ment produ ces a factorisation f = f ′′ ◦ f ′ with f ′ ∈ J − cof an d f ′′ ∈ J − inj , so th ere are p and q such that th e following diagram co mmutes (the existence o f q is a consequ ence of Remark 4): • • • • • • J ∋ j   u / / v / / J − c of ∋ f ′   J − inj ∋ f ′′   f ∈ I − inj                     p 6 6 q 7 7 “ ⊇ ” Let f ∈ J − inj ∩ W . The small object argument produ ces a f actorisation • • • I − cof ∋ f ′ | | y y y y y y y f ∈ ( J − inj ) ∩W   I − inj ∋ f ′′ " " E E E E E E E so f ′ ∈ I − cof ∩ W by (S1) . On the other hand f ∈ ( I − cof ∩ W ) − inj by Remark 4, so f ∈ I − inj by the retract argument (see Proposition 2). ⊳ 7 Proof of Theorem 1. Let C def . = I − cof and F def . = J − inj . It readily fo llo ws that W , C an d F are th e constituent classes of a model structure on C : − (M1) holds by hyp othesis; − (M2) holds by hyp othesis for W , by construction for C and F ; − as for (M3) , consider a comm utati ve square: • • • • I − cof ∋ c   f ∈ J − inj   / / / / If c ∈ W then this sq uare admits a lift by Rema rk 4. On the other h and, if f ∈ W then th is square admits a lift by Lemma 2; − (M4) holds because the f actorisations are constructed using the small object ar gumen t and hav e the required proper ties b y Lemma 2 and R emark 4, respectively . Therefo re C is a combinatorial model category by Remark 4. ⊳ 3 Higher dimensional categories This section is dev oted to a br ief review of hig her dimen sional categories, here defined as globular sets with structure. 3.1 Globular sets Let O be the small ca te gory whose ob jects ar e in te gers 0 , 1 , . . . , an d wh ose mo rphisms are generated by s n , t n : n → n +1 for n ∈ N , subject to the following equations: s n +1 ◦ s n = t n +1 ◦ s n , s n +1 ◦ t n = t n +1 ◦ t n . These e quations imply that th ere are exactly two mo rphisms from m to n if m < n , no ne if m > n , and on ly the identity if m = n . Definition 4. A globular set is a presheaf on O . In o ther words, a globular set is a functo r from O op to Sets . Globular sets and natural tra nsformations form a category Glo b . If X is a globular set, we denote b y X n the image o f n ∈ N by X ; members of X n are ca lled n -cells . By defin ing σ n = X (s n ) and τ n = X (t n ) , we get sour ce and tar get maps σ n , τ n : X n +1 → X n . More genera lly , whenev er m > n , on e defines σ n,m = σ n ◦ · · · ◦ σ m − 1 , τ n,m = τ n ◦ · · · ◦ τ m − 1 , so that σ n,m and τ n,m are m aps from X m to X n . Let u s call two n -ce lls x , y parallel when e ver n = 0 , o r n > 0 and σ n − 1 ( x ) = σ n − 1 ( y ) , τ n − 1 ( x ) = τ n − 1 ( y ) . W e write x k y w hene ver x , y , are parallel cells: • x $ $ y : : • W e will need a few additional notations about globular s ets: 8 − if u is an n +1 -cell, we write u : x → y wh ene ver σ n u = x and τ n u = y , in which case x k y ; − if m > n an d u is an m -ce ll, we write u : x → n y whenever σ n,m ( u ) = x and τ n,m ( u ) = y . Here a gain x , y ar e parallel n -cells; − we write u ⊲ n v if u : x → n y an d v : y → n z for some m -cells u , v and n -cells x , y , z ; − if n > 0 an d u is an n -cell, we write u ♭ for σ 0 ,n ( u ) and u ♯ for τ 0 ,n ( u ) , so that we get u : u ♭ → 0 u ♯ . 3.2 Strict ω -categories A strict ω -category is a g lob ular set C endowed with operations o f com position an d un its, satisfyin g the laws of associativity , units and interchan ge, as follows: − if u , v ar e m -cells such that u ⊲ n v , we write u ∗ n v for the n - composition of u with v (in diagram matic order) ; − if x is an n -cell, we write 1 x : x → x fo r the correspon ding n +1 - dimensional unit ; − if x is an n -cell and m > n , we write 1 m x for the corresponding m - dimensiona l unit . W e also write 1 n x for x ; − if m > n > p , we write u ∗ p v for 1 m u ∗ p v whenever u : x → p y is an n -cell and v : y → p z is an m -cell; − similarly , we write u ∗ p v for u ∗ p 1 m v whenever u : x → p y is an m -cell and v : y → p z is an n -ce ll. If m > n , the following identities hold for any m -cells u ⊲ n v ⊲ n w and for any m -cell u : x → n y : ( u ∗ n v ) ∗ n w = u ∗ n ( v ∗ n w ) , 1 m x ∗ n u = u = u ∗ n 1 m y . If m > n > p , the follo wing identities hold for any m -cells u ⊲ n u ′ and v ⊲ n v ′ such that u ⊲ p v (so that u ′ ⊲ p v ′ ), for any n -cells x ⊲ p y , and for any p -cell z : ( u ∗ n u ′ ) ∗ p ( v ∗ n v ′ ) = ( u ∗ p v ) ∗ n ( u ′ ∗ p v ′ ) , 1 m x ∗ p 1 m y = 1 m x ∗ p y , 1 m 1 n z = 1 m z . An ω - functor is a m orphism of glob ular sets preserving compositions an d u nits. Th us, ω - categories and ω -fu nctors build the cate gory ω Cat , which is our main object of study . The forgetf ul functor U : ω Cat → Glob is finitary m onadic [2] and Glob is a top os of presheaves on a small category: therefo re ω Cat is complete and cocomplete. On the other hand, the left adjoint to U takes a globular set to the fr ee ω -categor y it ge nerates. In particular , consider Y : O → Glob the Y oneda embedding: we get, for each n , a representab le glob ular set Y ( n ) = O ( − , n ) . Definition 5. For n ≥ 0 , the n -globe O n is the free ω -category generated by Y ( n ) . Notice that O n has exactly two non-id entity i -cells for i < n , exactly one non- identity n -cell, and no non-identity cells in dimensions i > n . Proposition 5. ω Cat is locally finitely presentable. Proof. It is a general fact that the representable objects Y ( n ) are finitely pr esentab le . Beca use U preserves filtered colimits, all n -globe s are finitely presentable objects in ω Ca t . ⊳ 3.3 Shift construction The following con struction will pr ov e essential in d efining the fun ctor Γ of Section 4 .4 below . Thus, given an ω -category C and two 0 -cells x , y in it, we define a new ω -catego ry [ x, y ] as follows: − ther e is an n -cell [ u ] in [ x, y ] for each n +1 -cell u : x → 0 y ; − fo r any n +1 -cells u, v : x → 0 y an d for an y n +2 -cell w : u → v , we have [ w ] : [ u ] → [ v ] in [ x, y ] ; − n -c omposition is defined by [ u ] ∗ n [ v ] = [ u ∗ n +1 v ] wh ene ver u ⊲ n +1 v ; − m -d imensional units are defined by 1 m [ u ] =  1 m +1 u  . 9 The verificatio n of the axiom s of ω -categor ies is straightfo rward. W e shall use some addition al operations de- scribed below . For any 0 -cells x, y , z , we get: − a precomposition ω -functor u · − : [ y , z ] → [ x, z ] for each 1-cell u : x → y , defined by u · [ v ] = [ u ∗ 0 v ] ; − a postcom position ω -functor − · v : [ x, y ] → [ x, z ] fo r each 1-cell v : y → z , defined by [ u ] · v = [ u ∗ 0 v ] ; − a compo sition ω -bifunctor − ⊛ − : [ x, y ] × [ y , z ] → [ x, z ] , defined by [ u ] ⊛ [ v ] = [ u ∗ 0 v ] . 4 The folk model structure The first step is to consider, f or each n , the globular set ∂ Y ( n ) ha ving the same cells as Y ( n ) except for removing the unique n -ce ll. T hus ∂ Y ( n ) generates an ω -category ∂ O n , the b oundary of the n -globe, and we get an inclusion ω -functor i n : ∂ O n → O n . Notice that, for each n , we get a pushout: ∂ O n O n O n ∂ O n +1 i n   i n / /   / / (3) The rest of this section is dev oted to the construction of a combinatorial model structure on ω Cat where I def . = { i n | n ∈ N } is a set of genera ting cofibrations. 4.1 I -injectives Notice that an ω -functo r f : X → Y in I − inj can eq ui valently be char acterised as verify ing the fo llo wing condition s: − fo r any 0-cell y in Y , there is a 0-cell x in X such that f x = y ; − fo r any n -cells x k x ′ in X and fo r any v : f x → f x ′ in Y , there is u : x → x ′ in X such that f u = v . Lemma 3. An ω -functo r f : X → Y in I − inj satisfies the following properties: − for any n -cell y in Y , there is an n -cell x in X such that f x = y ; − for any n -cells y k y ′ in Y , there are n -cells x k x ′ in X such that f x = y and f x ′ = y ′ . 4.2 Omega-equivalen ce Our defin ition of wea k equiv alences is b ased on two n otions: reversible cells an d ω -equivalence between pa rallel cells. These notions are defined by mutual coinduction. Definition 6. For any n -cells x k y in some ω -category: − we say that x and y are ω - equiv alent , and we write x ∼ y , if there is a reversible n +1 -cell u : x ∼ → y ; − we say that the n +1 -cell u : x → y is reversible , and we write u : x ∼ → y , if there is an n +1 -cell u : y → x such that u ∗ n u ∼ 1 x and u ∗ n u ∼ 1 y . Such a u is called a weak in verse of u . Notice that there is no base case in such a definition. Hence, we get infinite trees of cells of increasing dimension. W e now establish the first properties of re versible cells and ω -eq ui valence. 10 Lemma 4. For any ω -functo r f : X → Y and for any u : x ∼ → x ′ in X , we have f u : f x ∼ → f x ′ in Y . Hen ce, f preserves ∼ . Proof. Suppose that x , x ′ are n -cells with u : x ∼ → x ′ . By de finition, ther e is an n +1 - cell u : x ′ → x such that u ∗ n u ∼ 1 x and u ∗ n u ∼ 1 x ′ , whe nce reversible n +2 -c ells v : u ∗ n u ∼ → 1 x and v ′ : u ∗ n u ∼ → 1 x ′ . Now , by coinduc tion, f v : f u ∗ n f u ∼ → 1 f x and f v ′ : f u ∗ n f u ∼ → 1 f x ′ . Th erefore f u : f x ∼ → f x ′ . ⊳ Proposition 6. The relation ∼ is an ω -congr uence. Mo re precisely: i. For any n -cell x , we get 1 x : x ∼ → x . Hence, ∼ is reflexi ve. ii. For any reversible n +1 -cell u : x ∼ → y , we get u : y ∼ → x . Hence, ∼ is symmetric. iii. For any re versible n +1 -cells u : x ∼ → y and v : y ∼ → z , we get u ∗ n v : x ∼ → z . Hence, ∼ is transitiv e. i v . For any n -cells x, y , z , and f or any u : x → y , s, t : y → n z and v : s ∼ → t we ge t u ∗ n v : u ∗ n s ∼ → u ∗ n t . There is a similar property for postcompo sition. Hence, ∼ is compatible with comp ositions. Proof. For (i), the proof is by coin duction, whereas (ii) follows immediately from the definition. Let x , y , z , u , v , s and t as in (i v), and consider f , the precomp osition ω -functor u · − : [ y , z ] → [ x, z ] (Section 3.3 ). As v : s ∼ → t , we easily get [ v ] : [ s ] ∼ → [ t ] , so th at Lemma 4 app lies an d f [ v ] : [ s ] ∼ → [ t ] , whence u ∗ n v : u ∗ n s ∼ → u ∗ n t . The same h olds for po stcomposition. As for (iii) , suppose th at u : x ∼ → y and v : y ∼ → z . By d efinition, there ar e n +1 -cells u : y → u and v : z → y together with r e versible n +2 -cells w : u ∗ n u ∼ → 1 x and t : v ∗ n v ∼ → 1 y . By using th e compatibility property (iv) just established, we g et u ∗ n v ∗ n v ∗ n u ∼ u ∗ n 1 y ∗ n u = u ∗ n u . Also u ∗ n u ∼ 1 x . By coind uction, tr ansiti vity hold s in dim ension n +1 , whence u ∗ n v ∗ n v ∗ n u ∼ 1 x . Likewise v ∗ n u ∗ n u ∗ n v ∼ 1 z . Th erefore u ∗ n v : x ∼ → z . ⊳ There is a conv enient notion of weak uniquen ess, related to ω - equi valence. Definition 7. A co ndition C defines a weakly u nique cell u : x → y if w e have u ∼ u ′ for a ny other u ′ : x → y satisfying C . A less immediate, but crucial result is the following “weak di vision” property . Lemma 5. Any re versible 1 -cell u : x ∼ → y satisfies the left division property : − For any 1 -cell w : x → z , there is a weakly unique 1 -cell v : y → z such that u ∗ 0 v ∼ w . − For any 1 -cells s, t : y → z and for any 2 -cell w : u ∗ 0 s → u ∗ 0 t , there is a weakly unique 2 -cell v : s → t such that u ∗ 0 v ∼ w . − More generally , for all n > 0 , for any parallel n -cells s, t : y → 0 z and for an y n +1 -cell w : u ∗ 0 s → u ∗ 0 t , there is a weakly unique n +1 -cell v : s → t such that u ∗ 0 v ∼ w . Similarly , u : x ∼ → y satisfies the right division property . In fact, this also applies to any re versible 2 -cell u : x ∼ → y , seen as a reversible 1 -cell in the ω -category [ u ♭ , u ♯ ] . Proof. W e have a weak in verse u : y ∼ → x and some reversible 2-cell r : u ∗ 0 u ∼ → 1 y . − In the first case, we have u ∗ 0 v ∼ w if and only if v ∼ u ∗ 0 w . − In th e second ca se, u ∗ 0 v ∼ w imp lies ( r ∗ 0 s ) ∗ 1 v = ( u ∗ 0 u ∗ 0 v ) ∗ 1 ( r ∗ 0 t ) ∼ ( u ∗ 0 w ) ∗ 1 ( r ∗ 0 t ) by interchang e a nd c ompatibility . By left d i vision b y r ∗ 0 s (first case), this condition defin es a weakly unique v . Hence, we get weak uniqueness for l eft di vision by u . Moreover, th is condition implies u ∗ 0 u ∗ 0 v ∼ u ∗ 0 w by right division by r ∗ 0 t (first case), from which we get u ∗ 0 v ∼ w by weak uniq ueness applied to u . − Th e general case (for left and right division) is proved in the same way by induction on n . ⊳ 11 4.3 ω -W eak equi valences If we replace equality by ω -equiv alence in the definition of I -injectiv es, we get ω - weak equiv alences. Definition 8. An ω -functo r f : X → Y is an ω -weak equiv alence whenever it satisfies the following condition s: i. for any 0-cell y in Y , there is a 0-cell x in X such that f x ∼ y ; ii. for any n -cells x k x ′ in X and for any v : f x → f x ′ in Y , there is u : x → x ′ in X such that f u ∼ v . W e write W for the class of ω -weak equiv alences. Remark 5. As equality implies ω -equivalence (Pr oposition 6), we ha ve I − inj ⊆ W , which is exactly condition (S2) of Theorem 1. ♦ W e first rem ark that ω -equiv alences are weakly injective , in the sense of the following Lemma. Lemma 6. If f : X → Y is in W , then x ∼ x ′ for any x k x ′ in X such that f x ∼ f x ′ in Y . Proof. Let f ∈ W and x , x ′ parallel n -cells such that f x ∼ f x ′ . T here are n + 1 -cells u : f x → f x ′ and u : f x ′ → f x such that u ∗ n u ∼ 1 f x and u ∗ n u ∼ 1 f x ′ . Because f is a ω -weak equivalence, we get n +1 -cells v : x → x ′ and v : x ′ → x such that f v ∼ u an d f v ∼ u . By using Pro position 6,(iii) an d (iv), an d the preservation of compo sitions and units by f , f ( v ∗ n v ) ∼ u ∗ n u ∼ f (1 x ) By coindu ction, v ∗ n v ∼ 1 x and like wise v ∗ n v ∼ 1 x ′ , whence x ∼ x ′ . ⊳ The “3 f or 2 ” pr operty states that whenever two ω -fu nctors ou t o f f , g and h = g ◦ f ar e ω -weak eq ui valences, then so is the third. So the re are really three statements, that we shall address separately . Lemma 7. Let f : X → Y and g : Y → Z be ω -weak equivalences. Then g ◦ f : X → Z is in W . Proof. Suppose that f : X → Y , g : Y → Z are ω -w eak eq ui v alences and let h = g ◦ f . If z is a 0 - cell in Z , there is a 0 -cell y in Y suc h that g y ∼ z , and a 0 -cell x in X such that f x ∼ y . By Lemm a 4, h x ∼ g y , and by Prop osition 6,(iii), h x ∼ z . Now , let x , x ′ be two parallel n -cells in X and w : h x → h x ′ be an n +1 -cell in Z . The re is a v : f x → f x ′ such that g v ∼ w an d a u : x → x ′ such that f u ∼ v . By Lemma 4 an d Proposition 6,(iii) again, we get h u ∼ w and we are d one. ⊳ Lemma 8. Let f : X → Y , g : Y → Z be ω -functo rs and suppose tha t g and g ◦ f are ω -weak eq ui valences. Then f is in W . Proof. Let f , g an d h = g ◦ f such that g ∈ W and h ∈ W . Let y be a 0 - cell in Y , and z = g y . There is a 0 -cell x in X such that h x ∼ z . By L emma 6, f x ∼ y . Likewis e, let x , x ′ be p arallel n -cells in X and v : f x → f x ′ an n +1 -cell in Y . W e get g v : h x → h x ′ , therefo re there is an n +1 -cell u : x → x ′ in X such tha t h u ∼ g v . By Lemma 6, f u ∼ v and we are done. ⊳ The remain ing p art of the 3-f or -2 proper ty for W is significan tly har der to show and will be addr essed in Sec- tion 4.5. 12 Lemma 9. The class W is closed under retract and transfinite composition . Proof. The closure under retracts follows immediately from the definition, by using Lemma 4. As f or the closur e u nder transfinite compo sition, let α > 0 be an ord inal, v ie wed as a category with a u nique morph ism β → γ fo r each pair β ≤ γ of ordinals < α , and X : α → ω Cat a fu nctor , p reserving colimits. W e denote by w γ β the mo rphism X ( β → γ ) : X ( β ) → X ( γ ) , an d by ( X , w β ) the colimit of the d irected system ( X ( β ) , w γ β ) . Su ppose that each w β +1 β belongs to W . W e need to show that w 0 : X (0) → X is still a ω -weak equiv alence. W e first establish that f or each β < α , w β 0 ∈ W , by indu ction on β : − if β = 0 , w β 0 is the identity on X (0) , thus belong s to W ; − if β is a successor ordinal, β = γ + 1 and w β 0 = w γ +1 γ ◦ w γ 0 . By in duction, w γ 0 ∈ W , and by hypothesis w γ +1 γ ∈ W , h ence the result, by composition ; − if β is a limit ordinal, β = sup γ <β γ . L et n > 0 , ( x, y ) a pair of parallel n − 1 cells in X (0 ) and u : w β 0 ( x ) → w β 0 ( y ) an n - cell in X ( β ) . Because X p reserves colimits, there is alr eady a γ < β an d an n - cell v in X ( γ ) such that v : w γ 0 ( x ) → w γ 0 ( y ) an d w β γ ( v ) = u . By the in duction h ypothesis, w γ 0 is a ω -weak eq ui v alence, and there is a z : x → y in X (0) such tha t w γ 0 ( z ) ∼ v . By composing with w β γ , we g et w β 0 ( z ) ∼ u . The same argument applies to the case n = 0 , so that w β 0 ∈ W . Now we complete the proo f by in duction o n α itself: if α is a succe ssor ord inal, th en α = β + 1 and w 0 is w β 0 , hence belon gs to W , as we ju st proved. If α is a limit o rdinal, we reprod uce the argument of the limit case above, using again the fact that w β 0 is a ω -weak equiv alence for any β < α . ⊳ Corollary 1. I − cof ∩ W is closed under retract and transfinite composition . 4.4 Cylinders The proofs o f conditio n ( S2) , part of (S1) and (S3) were dir ectly based on our definitions of generating cofibr ations and ω -weak equivalences. As for the remaining poin ts, we sh all need a n e w co nstruction: to each ω -category X we associate an ω -ca te gory Γ( X ) whose cells are t he r eversible cylinders of X . The correspon dence Γ turns out to be f unctorial and end o wed with natu ral transfor mations from and to the id entity func tor . Rev ersible cylinders ar e in fact cylinders in the sen se of [ 19 ] a nd [18], satifying an add itional rev ersibility conditio n. In the present work , “cylinder” means “re versible c ylinder”, as the general case will not occur . Definition 9. By in duction on n , we define the n otion of n - cylinder U : x y y between n -cells x and y in som e ω -category: − a 0-cylinder U : x y y in X is giv en by a rev ersible 1 -cell U ♮ : x ∼ → y ; − if n > 0 , an n -cylinder U : x y y in X is given by tw o r e versible 1-cells U ♭ : x ♭ ∼ → y ♭ and U ♯ : x ♯ ∼ → y ♯ , together with some n − 1 -cylinder [ U ] : [ x ] · U ♯ y U ♭ · [ y ] in the ω -category [ x ♭ , y ♯ ] . If U : x y y is an n -cylinder, we write π 1 U and π 2 U for the n -cells x and y . x x ♭ x ♯ y y ♭ y ♯ U ♮   U ♭   x / / y / / U ♯   U ♮               W e also write π 1 X U and π 2 X U to emph asize the fact that U is an n -cylinder in the ω -category X . The next step is to show that n -cylinders in X a re the n -cells of a globular set. Definition 10. By induction on n , we define the source n -cylinder U : x y x ′ and the target n -cylinder V : y y y ′ of any n +1 -cylinder W : z y z ′ between n +1 -cells z : x → y and z ′ : x ′ → y ′ : 13 − if n = 0 , then U ♮ = W ♭ and V ♮ = W ♯ ; − if n > 0 , then U ♭ = V ♭ = W ♭ and U ♯ = V ♯ = W ♯ , wh ereas the two n − 1 -cylinders [ U ] and [ V ] are respectively defined as the s ource and the target of the n -cylinder [ W ] in the ω -category [ z ♭ , z ′ ♯ ] . In that case, we write W : U → V or also W : U → V | z y z ′ . x ♭ y ♭ x ♯ y ♯ W ♭   W ♯   x * * y : : x ′ * * y ′ : : U ♮   V ♮ p p z ) ) R R R R R R R z ′ ) ) W ♮ j j Lemma 10. W e have U k V for any n +1 -cylinder W : U → V . In other words, c ylinders form a globular s et. Proof. By induction on n . ⊳ Remark that the 0-source U and the 0-target V o f an n +1 -cylinder W are giv en by U ♮ = W ♭ and V ♮ = W ♯ . W e now define trivial cylinders . Definition 11. By induction on n , we define the trivial n -cylinder τ x : x y x for any n -cell x : − if n = 0 , then ( τ x ) ♮ = 1 x ; − if n > 0 , then ( τ x ) ♭ = 1 x ♭ and ( τ x ) ♯ = 1 x ♯ , whereas [ τ x ] is the tri vial cylinder τ [ x ] in [ x ♭ , x ♯ ] . W e also wr ite τ X x for τ x to emp hasize the fact that x is an n - cell of the ω -category X . The following result is a straightfor w ard consequence of the definition. Lemma 11. W e have τ x k τ y for any n -cells x k y , and τ z : τ x → τ y for any z : x → y . More genera lly , we get the following notion of de gene r ate cylinder : Definition 12. An n -cylinder between parallel cells is degenerate whenever n = 0 or n > 0 and its source and target are tri v ial. Remark that τ x k U k τ y f or any de generate n -cylinder U : x y y . The next easy lemma gives a more concrete description of degenerate cylinders: Lemma 12. i. For any de generate n -cylinder U : x y y , we get a re versible n +1 -cell U ♮ : x ∼ → y . ii. Conv ersely , any rev ersible n +1 -cell u : x ∼ → y correspo nds to a unique de generate n -cylinder U : x y y . In particular, the tri vial n -cylinder τ x : x y x is the degenerate n -cylinder gi ven by ( τ x ) ♮ = 1 x : x ∼ → x . Thus, fo r each ω -category X , we have defin ed a globular set Γ( X ) wh ose n -cells are n -cylind ers in X , to gether with globular morphisms π 1 X , π 2 X : Γ( X ) → X and τ X : X → Γ( X ) such that π 1 X ◦ τ X = id X = π 2 X ◦ τ X . X id X | | z z z z z z z z z τ X   id X " " D D D D D D D D D X Γ( X ) π 1 X o o π 2 X / / X Now we may define composition s of n -cylinder s in X , as well as u nits, in such a way th at the g lob ular set Γ( X ) becomes an ω -category: this is done in detail in appendix A (see also [19] and [18]). Th us, from no w on, Γ( X ) de- notes this ω -category . Likewise, π 1 X , π 2 X and τ X become ω -functor s. The following theorem, proved in appendix , summarizes the prop erties we actually use in the construction of our model st ructure . 14 Theorem 2. The cor respondence X 7→ Γ( X ) is the o bject part of an en dofunctor on ω Cat , an d π 1 , π 2 : Γ → id , τ : id → Γ are natural transformatio ns. In particular, we get f U : f x y f x ′ for any ω - functor f : X → Y and for any n -cylinder U : x y x ′ in X . W e end this p resentation of n -cylinders with the following important “transport” lemma. Lemma 13. For any parallel n -cylinders U : x y x ′ and V : y y y ′ , we have a topdown transport : i. For any z : x → y , there is z ′ : x ′ → y ′ together with a cylinder W : U → V | z y z ′ . ii. Such a z ′ is weakly unique: z ′ ∼ z ′′ for any z ′′ : x ′ → y ′ together with a cylinder W ′ : U → V | z y z ′′ . iii. Conversely , there is a cylinder W ′ : U → V | z y z ′′ for any z ′′ : x ′ → y ′ such that z ′ ∼ z ′′ . Similarly , we have a bottom up transpor t . Proof. W e pro ceed by induction on n . − If n = 0 , let U : x y x ′ and V : y y y ′ be parallel 0 -cylinders, and a 1 -cell z : x → y . By defin ition, there a re re versible 1 - cells u : x → x ′ and v : y → y ′ . Let u : x ′ → x a weak inv erse of u , and define z ′ = u ∗ 0 z ∗ 0 v . Now u ∗ 0 z ′ = u ∗ 0 u ∗ 0 z ∗ 0 v . As u ∗ 0 u ∼ 1 x , u ∗ 0 z ′ ∼ z ∗ 0 v , by using Proposition (6). When ce a reversible 2 - cell w : z ∗ 0 v ∼ → u ∗ 0 z ′ , that is a re versible 1 - cell, or 0 - cylinder , in the ω -category [ x, y ′ ] . Th us we get a 1 -cylinder W : U → V | z y z ′ , and (i) is proved. Suppose now that the re is a z ′′ : x ′ → y ′ together with a 1 -cylinde r W ′ : U → V | z y z ′′ . It f ollo ws th at u ∗ 0 z ′ ∼ z ∗ 0 v ∼ u ∗ 0 z ′′ , whence z ′ ∼ z ′′ by Lemma 5 . This p roves (ii). Suppo se finally th at z ′′ ∼ z ′ . W e get u ∗ 0 z ′′ ∼ u ∗ 0 z ′ ∼ z ∗ 0 v , and a cylinder W ′ : U → V | z y z ′′ as above, which proves (iii). − Sup pose that (i), (ii) and ( iii) hold in dimension n . Let U : x y x ′ , V : y y y ′ parallel n +1 -cylind ers and z : x → y an n +2 -cell. By definitio n, we ha ve re versible 1 -cells U ♭ = V ♭ : x ♭ ∼ → x ′ ♭ , U ♯ = V ♯ : x ♯ ∼ → x ′ ♯ , together with parallel n -cylind ers [ U ] : [ x ] · U ♯ y U ♭ · [ x ′ ] and [ V ] : [ y ] · y ♯ y V ♭ · [ y ′ ] in [ x ♭ , y ′ ♯ ] . Now [ z ] · U ♯ : [ x ] · U ♯ → [ y ] · V ♯ is an n +1 -cell [ w ] in [ x ♭ , y ′ ♯ ] . By the induc tion hypothesis, we get an n +1 -cell [ w ′ ] : U ♭ · [ x ′ ] → V ♭ · [ y ′ ] and an n +1 - c ylinder [ W 0 ] : [ U ] → [ V ] | [ w ] y [ w ′ ] in [ x ♭ , y ′ ♯ ] . By L emma 5, there is a [ z ′ ] : [ x ′ ] → [ y ′ ] such that [ w ′ ] ∼ U ♭ · [ z ′ ] . Th us, p art (iii) of the indu ction h ypothesis gives an n +1 -cylinder [ W ] : [ U ] → [ V ] | [ z ] · U ♯ y U ♭ · [ z ′ ] . But this defin es an n +2 -cylinder W : U → V | z y z ′ , and (i) holds in dimension n +1 . Moreover, by induction , the above ce ll [ w ′ ] is weakly uniqu e, and so is z ′ , by Lemma 5: this gives (ii) in dimensio n n +1 . Finally , if z ′′ ∼ z ′ , U ♭ · [ z ′′ ] ∼ U ♭ · [ z ′ ] in [ x ♭ , y ′ ♯ ] , and the induction hy pothesis g i ves an n +1 -cylinder [ W ′ ] : [ U ] → [ V ] | [ z ] · U ♯ y U ♭ · [ z ′′ ] , whence an n +2 -cylinder W ′ : U → V | z y z ′′ , so that (iii) holds in dimension n +1 . ⊳ Corollary 2. For each ω -category X , π 1 X , π 2 X are in I − inj and τ X is in W . Proof. Let U | x y x ′ and V | y y y ′ be p arallel n -cylind ers in X an d z : π 1 X U → π 1 X V an n +1 -cell. By Lemma 13, there is an n +1 -cylinder W : U → V such that π 1 X W = z . Th is p roves that π 1 X is in I − inj . Likewise, by bottom up transpo rt, π 2 X is in I − inj . But I − inj ⊆ W by (S2) so that π 1 X is a ω -weak eq ui valence. Now π 1 X ◦ τ X = id X , and by Lemma 8, τ X ∈ W . ⊳ 4.5 Gluing factorization For any ω -functor f : X → Y , we conside r the following pullback: Π( f ) Γ( Y ) X Y f ′   f ∗ π 1 Y / / f   π 1 Y / / W e write ˆ f : Π( f ) → Y for π 2 ◦ f ′ , so that the following diagram commute s: 15 Π( f ) Γ( Y ) X Y X Y f ′   f ∗ π 1 Y / / f   π 1 Y / / f   ˜ f / / τ Y / / id X % % id Y : : Since π 1 Y is in I − inj , so is its pullba ck f ∗ π 1 Y . By ( S2) , f ∗ π 1 Y is in W . As f ∗ π 1 Y ◦ ˜ f = id X , by Lemma 8, ˜ f is also a ω -weak equiv alence. Definition 13. The decomposition f = ˆ f ◦ ˜ f is called the gluing factorization of f . X ˜ f / / f 6 6 Π( f ) ˆ f / / Y The above constructions may be described more concretely as follows: − an n -ce ll in Π( f ) is a pair ( x, U ) where x is an n -ce ll in X and U : f x y y is an n -cylinder in Y ; − ˜ f x = ( x, τ f x ) fo r any n -cell x in X , and ˆ f ( x, U ) = π 2 U = y for any n -cylinder U : f x y y in Y . The gluing factorization leads to an extremely useful characterization of ω -weak equiv alences. Proposition 7. An ω -functo r f : X → Y is in W if and only if ˆ f : Π( f ) → Y is in I − inj . Proof. Suppose that ˆ f is in I − inj , then it is in W b y ( S2) ; as ˜ f is a ω - weak eq ui valence, so is the composition f = ˆ f ◦ ˜ f , by Lemma 7. Con versely , suppose that f is in W , and let us show that ˆ f is in I − inj : − For any 0-cell y in Y , there is a 0-cell x in X su ch that f x ∼ y . Hence, we ge t a re versible 1 - cell u : f x ∼ → y defining a 0-cylinder U : f x y y , so that ( x, U ) is a 0-c ell in Π( f ) and ˆ f ( x, U ) = y . − For any n - cells ( x, T ) k ( x ′ , T ′ ) in Π( f ) , we get parallel n -cylinders T : f x y y and T ′ : f x ′ y y ′ . For any n +1 -cell w : y → y ′ , Lemm a 13, bottom up d irection, g i ves v : f x → f x ′ together with V : T → T ′ | v y w . Since f is in W a nd x k x ′ , we get a n n +1 -cell u : x → x ′ such th at f u ∼ v . By Lemma 1 3, (iii), bo ttom up direc tion, we get U : T → T ′ | f u y w , so that ( u, U ) : ( x, T ) → ( x ′ , T ′ ) is an n +1 -cell in Π( f ) and ˆ f ( u, U ) = w . ⊳ Corollary 3. W is the smallest class contain ing I − inj which is closed under composition and right in verse. It is now possible to prove the remaining part of condition 3-for-2 for W . Lemma 14. If f : X → Y and h = g ◦ f : X → Z are in W , so is g : Y → Z . Proof. − For any 0-cell z in Z , there is a 0-cell x in X such that h x ∼ z . So we get g y ∼ z , where y = f x . − Let y k y ′ be n -cells in Y , and let w : g y → g y ′ be an n +1 -cell in Z . + By Prop osition 7 , ˆ f is in I − inj , so that Lemma 3 applies, an d we get x k x ′ in X an d parallel n - cylinders T : f x y y and T ′ : f x ′ y y ′ . + By Th eorem 2, we get parallel n -cylinders g T : h x y g y an d g T ′ : h x ′ y g y ′ . + By Pro position 7, ˆ h is in I − inj and we get u : x → x ′ together with U : g T → g T ′ | h u y w . + By Lem ma 13, (i) we get v : y → y ′ together with V : T → T ′ | f u y v . + By Th eorem 2, we get g V : g T → g T ′ | h u y g v . + By Lem ma 13, (ii), we get g v ∼ w . ⊳ 16 4.6 Immersions In order to complete the proo f of condition (S3) , we introduce a ne w class of ω - functors. Definition 14. An immersion is an ω -functo r f : X → Y satisfying the following three conditio ns: (Z1) there is a retraction g : Y → X such that g ◦ f = id X ; (Z2) there is an ω -functo r h : Y → Γ( Y ) such that π 1 Y ◦ h = f ◦ g and π 2 Y ◦ h = id Y ; (Z3) h ◦ f = τ Y ◦ f . I n other words, h is trivial on f ( X ) . X f / / id X 7 7 Y g / / X X f   Y g o o h   id Y   Y Γ( Y ) π 1 Y o o π 2 Y / / Y X f / / f   Y h   Y τ Y / / Γ( Y ) W e write Z for the class of immersions. Notice that, by naturality of τ , co ndition (Z3) can be replaced by the following one: (Z3’) h ◦ f = Γ( f ) ◦ τ X . The gluing constructio n of the pre vious section yields a characterization of immersions by a lifting property . Lemma 15. An ω -functo r f : X → Y is an imme rsion if an d on ly if ther e is an ω -functo r k : Y → Π( f ) such that k ◦ f = ˜ f and ˆ f ◦ k = id Y . X ˜ f / / f   Π( f ) ˆ f   Y k = = z z z z id Y / / Y Proof. Let f : X → Y , and suppose that there is a k : Y → Π( f ) satisfying the ab ov e lifting p roperty . Define g = f ∗ π 1 Y ◦ k and h = f ′ ◦ k . W e get g ◦ f = f ∗ π 1 Y ◦ k ◦ f = f ∗ π 1 Y ◦ ˜ f = id X , hen ce (Z1) . Als o π 1 Y ◦ h = π 1 Y ◦ f ′ ◦ k = f ◦ f ∗ π 1 Y ◦ k = f ◦ g an d π 2 Y ◦ h = π 2 Y ◦ f ′ ◦ k = ˆ f ◦ k = id Y , hence (Z2) . Finally h ◦ f = f ′ ◦ k ◦ f = f ′ ◦ ˜ f = τ Y ◦ f , hence (Z3) . Con versely , suppose that f : X → Y is an immersion, and l et g , h satify the conditions of Definition 14. By (Z2) , π 1 Y ◦ h = f ◦ g , so that the uni versal property of Π( f ) yields a unique k : Y → Π( f ) such that f ∗ π 1 Y ◦ k = g and f ′ ◦ k = h . Thus ˆ f ◦ k = π 2 Y ◦ f ′ ◦ k = π 2 Y ◦ h = id Y , by (Z2) . No w f ∗ π 1 Y ◦ k ◦ f = g ◦ f = id X = f ∗ π 1 Y ◦ ˜ f and f ′ ◦ k ◦ f = h ◦ f = τ Y ◦ f by (Z3) so that f ′ ◦ k ◦ f = f ′ ◦ ˜ f : by the univ ersal property of Π( f ) , this gives k ◦ f = ˜ f , an d we are done. ⊳ Corollary 4. I − cof ∩ W ⊆ Z . Proof. Suppose that f : X → Y belo ngs to I − cof ∩ W . As f ∈ W , by Proposition 7, ˆ f ∈ I − inj . Now f ∈ C of has the lef t lifting pro perty with re spect to ˆ f , so th at there is a k such that k ◦ f = ˜ f and ˆ f ◦ k = id Y . By Lemma 15, f is an imme rsion. ⊳ Lemma 16. Z ⊂ W . Proof. Suppose that f : X → Y is an immersion, and let g , h as in Definition 1 4: − For any 0-cell y in Y , we get h y : f x y y whe re x = g y . Henc e, we get ( h y ) ♮ : f x ∼ → y , so that f x ∼ y . − For any n -cells x k x ′ in X an d fo r any v : f x → f x ′ in Y , we have h v : f u y v where u = g v : x y x ′ . By (Z3) , the cylinder h v : τ f x → τ f x ′ is degener ate. Hence, we get ( h v ) ♮ : f u ∼ → v , so tha t f u ∼ v . ⊳ Lemma 17. Z is closed under pushout. 17 Proof. Let f : X → Y be an immersion, i : X → X ′ an ω -functor and f ′ : X ′ → Y ′ the pusho ut of f by i : X Y X ′ Y ′ f   i / / f ′   j / / Since f is an imm ersion, we h a ve g : Y → X and h : Y → Γ( Y ) satisfying co nditions (Z1) to (Z3) . By universality of the p ushout and by (Z3’) , we g et g ′ : Y ′ → X ′ and h ′ : Y ′ → Γ( Y ′ ) such that the following diagrams commute: X Y X ′ Y ′ X X ′ f   i / / f ′   j / / g   g ′      i / / id X   id X ′   X Y X ′ Y ′ Γ( Y ) Γ( Y ′ ) Γ( X ) Γ( X ′ ) f   i / / f ′   j / / h   h ′      Γ( j ) / / τ X   Γ( f ) " " τ X ′   Γ( f ′ ) | | Finally , conditions (Z1) to (Z3) for g ′ and h ′ follow from conditio ns (Z1) to (Z3) for g and h . ⊳ Corollary 5. I − cof ∩ W is closed under pushout. Proof. Let f ∈ I − cof ∩ W and f ′ a pushou t of f . By Corollary 4, f is an immersion , and so is f ′ by Lemma 17. By Lem ma 16, f ′ is a ω -weak equiv alence. No w I − cof is stable by pushou t, so that f ′ ∈ I − cof . Hence f ′ ∈ I − cof ∩ W and we are done. ⊳ 4.7 Generic square s By Y oned a’ s Lem ma, for each n , the functo r X 7→ X n , fro m ω Cat to Set s is represented by th e n - globe O n . Thus, to each n -cell x of X correspon ds a unique ω -fu nctor h x i : O n → X . Moreover , for an y pair x , x ′ of n -cells in X , the co ndition o f parallelism x k x ′ is equiv alent to h x i ◦ i n = h x ′ i ◦ i n . By the pushou t square (3) mentioned at the beginning of Section 4, we get a unique ω -functor h x, x ′ i : ∂ O n +1 → X . associated to any pair x , x ′ of parallel n -ce lls. This app lies in particular to the case wh ere x = x ′ = o , the u nique proper n -cell of O n . The cor responding ω -functor is den oted by o n = h o, o i : ∂ O n +1 → O n . Since ω Cat is locally presentab le, there is a factorization o n = p n ◦ k n with p n ∈ I − inj and k n ∈ I − cof . ∂ O n +1 k n / / o n 5 5 P n p n / / O n Now b y composition of k n with both ω -functors O n → ∂ O n +1 of the p ushout (3), we get j n , j ′ n : O n → P n such that the following diagram commu tes: ∂ O n O n O n ∂ O n +1 P n O n i n   i n / /   / / k n / / p n / / j n   id O n j ′ n 4 4 id O n 7 7 18 The following definition singles out an importan t part of the abov e diagram. Definition 15. The generic n -square is the following commutati ve square: ∂ O n i n   i n / / O n j n   O n j ′ n / / P n Remark 6. Notice that p n is in I − inj , hence in W , and th at p n ◦ j n = id O n . Therefor e j n ∈ W , by Lemma 8 . On the other hand i n ∈ I − cof . Sin ce I − cof is stable un der composition and pushou t, we ha ve j n ∈ I − cof ♦ The next result characterizes the relation of ω - equi valence in terms of suitable f actorization s. Lemma 18. For any n -cells x k x ′ in X , the following conditions are equiv alent: i. x ∼ x ′ ; ii. there is an ω -category Y and ω -functo rs k : ∂ O n +1 → Y , p : Y → O n and q : Y → X such that p ∈ I − inj and the following diagram commute s: ∂ O n +1 o n { { v v v v v v v v v k      h x,x ′ i # # G G G G G G G G G O n Y p o o _ _ _ _ q / / _ _ _ _ X ; iii. Th ere is an ω -functo r q : P n → X such that the following diagram commu tes: ∂ O n +1 o n { { v v v v v v v v v k n   h x,x ′ i # # G G G G G G G G G O n P n p n o o q / / _ _ _ _ X . Proof. If x ∼ x ′ , ther e is a reversible n +1 -ce ll u : x ∼ → x ′ which defines a degenerate n -cylind er U : x y x ′ . W e get τ X x k U , whereas π 1 X τ X x = π 2 X τ X x = π 1 X U = x an d π 2 X U = x ′ , so that the following diag rams commute: ∂ O n +1 o n / / h x,x i # # G G G G G G G G G h τ X x,U i   O n h x i   Γ( X ) π 1 X / / X ∂ O n +1 h x,x ′ i " " F F F F F F F F F h τ X x,U i   Γ( X ) π 2 X / / X Let f = h x i . By u ni versality of Π( f ) , we g et k : ∂ O n +1 → Π( f ) such that the following diagram commutes: ∂ O n +1 o n ) ) k / / _ _ _ h τ X x,U i , , Π( f ) f ∗ π 1 X / / f ′   O n f   Γ( X ) π 1 X / / X The desired factorizations are gi ven by Y = Π( f ) , p = f ∗ π 1 X and q = ˆ f = π 2 X ◦ f ′ . Hence, (i) implies (ii). Con versely , if we assume ( ii), then k g i ves us two n -cells y k y ′ in Y such that p y = p y ′ , q y = x a nd q y ′ = x ′ . Hence, we get y ∼ y ′ by Lemma 6 applied to p , and x ∼ x ′ by Lemma 4 applied to q . On the other h and, if we assume (ii), then k factors th rough k n by the lef t liftin g prop erty , an d so does h x, x ′ i . Hence (ii) implies (iii). Conversely , (iii) is just a special case of (ii). ⊳ 19 W e now turn to a ne w characterization of ω -weak equiv alences. Proposition 8. An ω -functo r f : X → Y is an ω -weak equivalence if and only if any comm utati ve square whose left arrow is i n and whose right arrow is f factors through the generic n -square. ∂ O n ( ( i n / / i n   O n j n   / / _ _ _ X f   O n j ′ n / / 6 6 P n / / _ _ _ Y Proof. Let f : X → Y be an ω -weak equi valence, and consider a commutative d iagram ∂ O n i n   / / X f   O n / / Y . W e show that it factors through the generic n -square: − If n = 0 , the commutative square is gi ven by some 0-cell y in Y : 0 / /   X f   1 h y i / / Y Since f is in W , ther e is a 0-cell x in X such that f x ∼ y , and by the previous lemma, we get q : P 0 → Y such that q ◦ k 0 = h f x, y i , wh ich means that the follo wing diagram commutes: 0 ' ' / /   1 j 0   h x i / / _ _ _ X f   1 j ′ 0 / / h y i 7 7 P 0 q / / _ _ _ Y − If n > 0 , the commutative square is gi ven by n − 1 -cells x k x ′ in X and some n -cell v : f x → f x ′ in Y : ∂ O n h x,x ′ i / / i n   X f   O n h v i / / Y Since f is in W , the re is u : x → x ′ in X such that f u ∼ v , and by Lemma 18 , we g et q : P n → Y such that q ◦ k n = h f u, v i , which means that the following diagram commutes: ∂ O n h x,x ′ i ( ( i n / / i n   O n j n   h u i / / _ _ _ X f   O n j ′ n / / h v i 6 6 P n q / / _ _ _ Y The conv erse is proved by the same argument. ⊳ 20 Corollary 6. The class W of ω -weak equivalences admits the solution set J = { j n | n ∈ N } . W e may finally state the cen tral result of this w ork: Theorem 3. ω Cat is a comb inatorial m odel category . Its class of weak equivalences is th e class W of ω -weak equiv alences while I and J are the sets of generating cofibration s and generating tri vial cofibrations, respectiv ely . Proof. ω Cat is locally presentable by proposition 5 while − con dition (S1) h olds by lemma 7, lemma 8, lemma 14 and lemma 9; − con dition (S2) h olds by remark 5; − con dition (S3) h olds by corollary 1 and corollary 5; − con dition (S4) h olds by corollary 6. ⊳ Remark 7. By corollar y 3, the model structure of theorem 3 is left-determined in the sense of [2 3 ]. 5 Fibrant and cofibrant objects Recall that, given a mode l category C , an object X of C is fibrant if the u nique mo rphism ! X : X → 1 is a fibration. Dually , X is cofibrant if the unique morphism 0 X : 0 → X is a cofibration. Now X is fibrant if and only if, f or any trivial cofibration f : Y → Z an d any u : Y → X , th ere is a v : Z → X such that v ◦ f = u : in fact, this implies that ! X : X → 1 has the righ t-lifting property with respect to trivial cofibration s. Y u / / f   X ! X   Z ! Z / / v > > ~ ~ ~ ~ 1 Like wise, X is cofibr ant if and only if for an y tri vial fibration p : Y → Z and any morphism u : X → Z ther e is a lift v : X → Y such that p ◦ v = u . 0 0 Y / / 0 X   Y p   X u / / v > > ~ ~ ~ ~ Z 5.1 Fibrant ω -categories In the f olk m odel stru cture o n ω Ca t , th e c haracterization of fibrant objects is th e simp lest p ossible, as shown by the following result. Proposition 9. All ω -categories are fibrant. Proof. Let X be an ω -category , f : Y → Z a trivial co fibration, an d u : Y → X an ω -f unctor . By Cor ollary 4, f is an immersion. In particu lar there is a r etraction g : Z → Y such that g ◦ f = id Y . Let v = u ◦ g . W e get v ◦ f = u ◦ g ◦ f = u . Hence X is fibrant. Y u / / f   X Z v > > ~ ~ ~ ~ g A A ⊳ 21 5.2 Cofibrant ω -categories Our un derstanding of the cofibr ant ob jects in ω Cat is based on a n appro priate notion of fr eely generated ω - category: no tice that the free ω -ca te gories in the s ense of the adjunction between ω Ca t an d Glob are not sufficient, as there are too few of the m. W e fir st describe a p rocess of genera ting free cells in eac h dimension . In dime nsion 0 , we just hav e a set S 0 and no opera tions, so that S 0 generates S ∗ 0 = S 0 . In dim ension 1 , gi ven a graph S ∗ 0 S 1 τ 0 o o σ 0 o o where S ∗ 0 is the set of vertices, S 1 the set of edg es, and σ 0 , τ 0 are th e source and target maps, there is a free category generated by it: S ∗ 0 S ∗ 1 τ 0 o o σ 0 o o . Now suppose that we add a new set S 2 together with a graph S ∗ 1 S 2 τ 1 o o σ 1 o o satisfying the bo undary conditions σ 0 ◦ σ 1 = σ 0 ◦ τ 1 and τ 0 ◦ σ 1 = τ 0 ◦ τ 1 . What we get is a com putad , a notion first introdu ced in [26], freely generating a 2 -category S ∗ 0 S ∗ 1 τ 0 o o σ 0 o o S ∗ 2 τ 1 o o σ 1 o o . This pattern h as been extended to a ll dimension s, giving rise to n -c omputads [21] or polygraph s [5, 6]. More precisely , let n G lob (resp. n Cat ) d enote the cate gory of n -globular sets (resp. n -categories), we g et a co mmutativ e diagram ( n +1) Cat / / U n   ( n +1) Glob   n Cat / / n Glob (4) where the hor izontal ar ro ws are the obvious fo rgetful fu nctors and the vertical a rro ws are truncation functors, removing all n +1 -cells. On the other han d, let n Cat + be the category defined by the following pullback square: n Cat + n Cat ( n +1) Glob n Glob V n   / /   / / (5) From (4) , we get a un ique func tor R n : ( n +1 ) Ca t → n Cat + such that V n R n = U n , whe re U n and V n are the tr uncation fun ctors appear ing in (4 ) and (5) repectively . N o w th e key to the constru ction of polygr aphs is the existence of a left-adjoint L n : n Cat + → ( n +1) Cat to this R n . Concretely , i f X is an n -categor y and S n +1 a set of n +1 -cells attached to X by X n S n +1 τ n o o σ n o o (6) satisfying the boundar y c onditions, then L n builds an ( n +1) -category whose explicit construction is giv en in [20]. Here we just mention the following proper ty of L n : let X + be an object of n Cat + giv en by an n -category X 0 · · · τ 0 o o σ 0 o o X n τ n − 1 o o σ n − 1 o o and a graph (6) then the n +1 -category L n X + has the same n -cells as V n X + . In other words, there is a set of n +1 -cells S ∗ n +1 such that L n X + has the form X 0 · · · τ 0 o o σ 0 o o X n τ n − 1 o o σ n − 1 o o S ∗ n +1 τ n o o σ n o o 22 Definition 16. n -po lygraphs are defined inductively by the follo wing conditions: − a 0 -poly graph is a set S (0) ; − an n +1 -po lygraph is an ob ject S ( n +1) of n Cat + such that V n S ( n +1) is of the form L n S ( n ) where S ( n ) is an n -polyg raph. Like wise, a polygr aph S is a sequence ( S ( n ) ) n ∈ N of n -polyg raphs such that, for each n , V n S ( n +1) = L n S ( n ) . The pu llback (5) g i ves a notion o f mo rphisms for n Ca t + , wh ich, b y inductio n, determines a notio n of m orphism between n -poly graphs, a nd p olygraphs. Thus we get a category Pol of polyg raphs and morph isms. By Defini- tion 16 and the abovementioned pro perty of L n , we may s ee a polyg raph S as an infinite diagram of the following shape: S 0   S 1   ~ ~ } } } } } } } ~ ~ } } } } } } } S 2   ~ ~ } } } } } } } ~ ~ } } } } } } } S 3   ~ ~ } } } } } } } ~ ~ } } } } } } } · · · ~ ~ } } } } } } } } ~ ~ } } } } } } } } S ∗ 0 S ∗ 1 o o o o S ∗ 2 o o o o S ∗ 3 o o o o · · · o o o o . (7) In (7), each S n is the set of ge nerators of the n -cells, the o blique d ouble arr o ws rep resent the attach ment of new n -cells on the previously defined n − 1 -category , thus defining an object X + of ( n − 1) Cat + , whereas S ∗ n is the set of n -cells in L n − 1 X + . The botto m line o f (7) d isplays the fr ee ω -c ate g ory generated by the po lygraph S . T his defines a functor Q : S 7→ S ∗ from Pol to ω Cat , wh ich is in fact a left-adjo int. A detailed description of the right-ad joint P : X 7→ P ( X ) from ω Cat to Pol is gi ven in [19]. It is now possible to state the main result of this section: Theorem 4. An ω -category is cofibrant if and only if it is freely generated by a polygraph. Suppose that X is freely g enerated by a p olygraph S , p : Y → Z is a trivial fibration an d u : X → Z is an ω -functor . It is easy to build a lift v : X → Y suc h that p ◦ v = u d imensionwise by using the un i versal property of the functor s L n . Y p   S ∗ u / / v > > } } } } Z Thus freely gen erated ω -categories are cofibrant. The proof o f the con verse is much hard er , and is the main purpose of [20]. The problem reduces to the fact that the full subcategory of ω Ca t whose objects are free on polygraph s is Cauchy complete, meaning that its idempotent morphisms split. The results of [1 9 ] may be revisited in the fr ame work of the folk mod el structure on ω Cat . In fact, a reso lution of an ω -category X by a po lygraph S is a trivial fibration S ∗ → X , henc e a cofibrant replacement of X . Notice tha t for each ω -category X , the cou nit of the adjunctio n between P o l and ω Ca t gi ves an ω -functor ǫ X : QP ( X ) → X which is a tri vial fibration, and defines the standard r esolu tion of X . 6 Model structure on n Cat In this section , w e show that the model structure on ω Cat we just describ ed yield s a mo del structure on the category n Cat of (strict, sma ll) n -categories for each integer n ≥ 1 . In p articular , we recover the known folk model structures on Cat [13] and 2 Cat [15, 16]. 23 Let n ≥ 1 be a fixed integer . There is an inclusion fun ctor F : n Cat → ω Cat which simply adds all necessary unit cells in dimension s k > n . This fun ctor F has a left adjoint G : ω Cat → n Cat . Precisely , if X is an ω -category and 0 ≤ k ≤ n , the k -cells of GX are exactly those of X f or k < n , whereas ( GX ) n is th e quotient of X n modulo the co ngruence generated by X n +1 . In oth er w ords, pa rallel n -cells x , y in X are congruent modulo X n +1 if and only if there is a sequence x 0 = x, x 1 , . . . , x p = y o f n -cells and a sequence z 1 , . . . , z p of n +1 -cells such that, for each i = 1 , . . . , p either z i : x i − 1 → n x i or z i : x i → n x i − 1 . Notice that the fu nctor F also has a r ight adjoint, namely the trunc ation functor U : ω Cat → n Cat which simply forgets all cells of dimension k > n . Theorem 5. The inclusion fu nctor F : n Cat → ω Cat creates a mo del structur e on n Cat , in which the weak equiv alences are the n -functo rs f such that F ( f ) ∈ W , and ( G ( i k )) k ∈ N is a family of generating cofibrations. The general situation is in vestigated in [4], whose proposition 2.3 states suf ficient condition s f or the transport of a model structure along an adjunction . In ou r particular case, these condition s boil down to the follo wing: (C1) the model structure on ω Ca t is cofibrantly generated ; (C2) n Cat is locally presentable; (C3) W is closed under filtered colimits in ω Cat ; (C4) F p reserves filtered colimits; (C5) If j ∈ J is a generating tri vial cofibration of ω Ca t , and g is a pushou t of G ( j ) in n Cat , then F ( g ) is a weak equiv alence in ω Cat . Conditions (C1) and (C2 ) are known already . Condition (C3) follows from the de finition of weak equiv alences and the fact that the ω -categor ies O n are finitely presen table objects in ω Cat . The functor F , being left adjo int to U , preserves all colimits, in particular filtered ones, hence (C4) . W e now tur n to th e pro of of the remaining co ndition (C5) . First remar k that GF is th e identity on n C at , so that the mon ad T = F G is idem potent and the monad multiplication µ : T 2 → T is th e identity . As a consequen ce, if η : 1 → T deno tes the unit of the monad, for each ω -category X T ( η X ) = 1 T ( X ) . (8) Also, for each ω -functo r of the form u : T ( X ) → T ( Y ) , T ( u ) = u. (9) Now let X be an ω -category . For each k > n , all k -cells o f T ( X ) ar e un its. Th erefore, by con struction of the connectio n functor Γ , all k -cells in Γ T ( X ) ar e also units, wh ich imp lies that Γ T ( X ) belongs to the imag e of F , whence T Γ T ( X ) = Γ T ( X ) . (10) W e successively get the natural transformations: η X : X → T ( X ) , Γ( η X ) : Γ( X ) → Γ T ( X ) , T Γ( η X ) : T Γ( X ) → T Γ T ( X ) = Γ T ( X ) , by (10). Thus λ X = T Γ( η X ) yields a natural transformation λ : T Γ → Γ T . 24 Lemma 19. The monad T on ω Cat preserves immersions. Proof. Let f : X → Y b e an immersion. W e want to sho w t hat f ′ = T ( f ) is still an immersion. By Definition 14, there are g : Y → X and h : Y → Γ( Y ) such that g ◦ f = id; π 1 Y ◦ h = f ◦ g ; π 2 Y ◦ h = id; h ◦ f = τ Y ◦ f . Let g ′ = T ( g ) : T ( Y ) → T ( X ) and h ′ = λ Y ◦ T ( h ) : T ( Y ) → Γ T ( Y ) , it is now sufficient to check the follo wing equations: g ′ ◦ f ′ = id; (11) π 1 T ( Y ) ◦ h ′ = f ′ ◦ g ′ ; ( 12) π 2 T ( Y ) ◦ h ′ = id; (13) h ′ ◦ f ′ = τ T ( Y ) ◦ f ′ . (14) Equation (11) is obviou s f rom f unctoriality . As for (1 2), we first notice that, by natu rality of π 1 , the following diagram commu tes: Γ( Y ) Γ( η Y )   π 1 Y / / Y η Y   Γ T ( Y ) π 1 T ( Y ) / / T ( Y ) . By applyin g T to the above diagram, we get T Γ( Y ) λ Y   T ( π 1 Y ) / / T ( Y ) T ( η Y )   Γ T ( Y ) T ( π 1 T ( Y ) ) / / T ( Y ) . Now , by (8), T ( η Y ) = 1 T ( Y ) and because Γ T ( Y ) = T Γ T ( Y ) , by (9), T ( π 1 T ( Y ) ) = π 1 T ( Y ) . Hence T ( π 1 Y ) = π 1 T ( Y ) ◦ λ Y . (15) Thus π 1 T ( Y ) ◦ h ′ = π 1 T ( Y ) ◦ λ Y ◦ T ( h ) , = T ( π 1 Y ) ◦ T ( h ) , = T ( π 1 Y ◦ h ) , = T ( f ◦ g ) , = T ( f ) ◦ T ( g ) , = f ′ ◦ g ′ . Equation s (13) an d (14) hold b y the same argum ents applied to th e natural transfo rmations π 2 and τ respectively . Hence T ( f ) is an imm ersion, and we are done. ⊳ Lemma 20. Let f : X → Y be an immersion, and suppo se the following square is a pushout in n Cat : G ( X ) G ( Y ) A B G ( f )   u / / g   v / / Then F ( g ) is an immersion. 25 Proof. As F is left adjoint to U , it preserves pushouts, and the follo wing square is a pushout in ω Cat : T ( X ) T ( Y ) F ( A ) F ( B ) T ( f )   F ( u ) / / F ( g )   F ( v ) / / Now f is an im mersion, and so is T ( f ) by Lem ma 19. As im mersions are closed by pu shouts (Lemma 17), F ( g ) is also an immersion. ⊳ Now let j b e a gen erating trivial cofibr ation in ω Cat , and g a pushou t of G ( j ) in n C at . By Coro llary 4, j is an imme rsion, so that L emma 2 0 applies, and F ( g ) is an immersion . By L emma 16, imm ersions are weak equiv alences, s o that F ( g ) ∈ W . Hence condition (C5) holds, and we are done. In case n = 1 , the weak equiv alences of n Cat are exactly th e equ i v alences of categories, wh ereas if n = 2 , they are th e bieq uivalences in the sense of [1 5]. Moreover , from the gene rating cofibr ations of ω Cat we immediately get a family of generating cofibrations in n Cat , namely the n -functor s G ( i k ) : G ( ∂ O k ) → G ( O k ) for all k ∈ N . By abuse of lan guage, let u s d enote G ( X ) = X whenever X is an ω -categor y of the form F ( Y ) , that is without non-identity cells in dimensions > n . Like wise, denote G ( f ) = f fo r each ω -functor f of the form F ( g ) . Wit h this conv ention − fo r each integer k ≤ n , G ( i k ) = i k ; − G ( i n +1 ) is the collapsing map i ′ n +1 : ∂ O n +1 → O n ; − fo r each k > n +1 , G ( i k ) is the identity on O n . Now the rig ht-lifting proper ty with respect to identities is clear ly v oid. Thus we only n eed a fin ite family of n +2 generating cofibrations i 0 , . . . , i n , i ′ n +1 . If n = 1 or n = 2 , these are precisely the gene rating cofibr ations of [1 3] an d [1 5] r especti vely . Th erefore the correspo nding model structures are particular cases of ours. A The functor Γ The aim of this section is to g i ve a com plete proo f of The orem 2. In o rder to d o tha t, we extend ω - functors to cylinders and we introduce the following operations: − left an d right action of cells on c ylinders, written u · V and U · v ; − con catenation of cylinders, written U ∗ V ; − multip lication of cylinders, written U ⊛ V ; − com positions of cylinders and the units , written U ∗ n V and 1 m U . W e must p rove the following p roperties: associativity an d un its fo r co mpositions, in terchange and iterated u nits, compatibility of Γ( f ) , π 1 , π 2 , τ with compo sitions and units, functoriality of Γ and natura lity of π 1 , π 2 , τ . Lemma 21. (functo riality) Any ω -functo r f : X → Y extends to c ylinders in a canonical way: i. for any n -cylinder U : x y x ′ in X , we get some n -cylinder f U : f x y f x ′ in Y ; 26 ii. we have f U k f V whenever U k V , and f W : f U → f V for any W : U → V ; iii. we have ( g ◦ f ) U = g f U for any ω -functo r g : Y → Z , and also id U = U . In other words, Γ defines a functor from ω Cat to Glob and the homom orphisms π 1 , π 2 are natural. Definition 17. (left and rig ht action) Precom position and postco mposition extend to cylinders. For any 0-cells x, y , z , we get: − the n -cylinder u · V in [ x, z ] , defined for any 1-cell u : x → y and for any n -cylinder V in [ y , z ] ; − the n -cylinder U · v in [ x, z ] , defined for any 1-cell v : y → z and for any n -cylinder U in [ x, y ] . Lemma 22. (bimod ularity) The following identities hold for any 0-cells x, y , z , t : − ( u ∗ 0 v ) · W = u · ( v · W ) for any 1-cells u : x → y and v : y → z , and for any n -cylinder W in [ z , t ] ; − ( U · v ) · w = U · ( v ∗ 0 w ) for any 1-cells v : y → z and w : z → t , and for any n -cylinder U in [ x, y ] ; − ( u · V ) · w = u · ( V · w ) for any 1-cells u : x → y and w : z → t , and for any n -cylinder V in [ y , z ] . Moreover , we hav e 1 x · U = U = U · 1 y for any 0-cells x, y and for any n -cylinder U in [ x, y ] . This is proved by functoriality . W e omit parentheses in such expressions: For instance, u · v · W stands for u · ( v · W ) , and U · v · w fo r ( U · v ) · w . Moreover , action will always ha ve precedence o ver oth er o perations: For instance, u · V ∗ W stands for ( u · V ) ∗ W . Definition 18. (concaten ation) By in duction o n n , we de fine th e n -cylinder U ∗ V : x y z for any n -cylinders U : x y y and V : y y z : − if n = 0 , then ( U ∗ V ) ♮ = U ♮ ∗ 0 V ♮ ; − if n > 0 , then ( U ∗ V ) ♭ = U ♭ ∗ 0 V ♭ and ( U ∗ V ) ♯ = U ♯ ∗ 0 V ♯ , whereas [ U ∗ V ] = [ U ] · V ♯ ∗ U ♭ · [ V ] . In both cases, we say that U and V are consecutive , and we write U ⊲ V . Lemma 23. (source and target of a c oncatenation) W e have U ∗ U ′ k V ∗ V ′ for any n -cylinders U k V and U ′ k V ′ such that U ⊲ U ′ and V ⊲ V ′ , an d W ∗ W ′ : U ∗ U ′ → V ∗ V ′ for any n +1 -cylinders W : U → V and W ′ : U ′ → V ′ such that W ⊲ W ′ . Lemma 24. (compatib ility of Γ( f ) with co ncatenation and τ ) The following identities h old any ω -functo r f : X → Y : − f ( U ∗ V ) = f U ∗ f V for any n -cylinders U ⊲ V in X ; − f τ x = τ f x for any n -cell x in X . In particular, the homomorp hism τ is natural. In the cases of precomp osition and postcomposition, we get the follo wing result: Lemma 25. (distributi vity over co ncatenation and τ ) The following id entities hold for any 0-cells x, y , z and f or any 1-cell u : x → y : − u · ( V ∗ W ) = u · V ∗ u · W for any n -cylinders V ⊲ W in [ y , z ] ; − u · τ [ v ] = τ [ u ∗ 0 v ] for any n +1 -cell v : y → 0 z . There are similar prope rties for right action. Lemma 26. (associativity and units for concaten ation) The following identities hold for any n -cylinders U ⊲ V ⊲ W and for any n -cylinder U : x y y : ( U ∗ V ) ∗ W = U ∗ ( V ∗ W ) , τ x ∗ U = U = U ∗ τ y . 27 Proof. W e pro ceed by induction on n . The case n = 0 is obvious. If n > 0 , the first iden tity is obtained as follows: [( U ∗ V ) ∗ W ] = [ U ∗ V ] · W ♯ ∗ ( U ∗ V ) ♭ · [ W ] (definition of ∗ ) = ([ U ] · V ♯ ∗ U ♭ · [ V ]) · W ♯ ∗ ( U ♭ ∗ 0 V ♭ ) · [ W ] (definition of ∗ ) = ([ U ] · V ♯ · W ♯ ∗ U ♭ · [ V ] · W ♯ ) ∗ U ♭ · V ♭ · [ W ] (distributi vity over ∗ ) = [ U ] · V ♯ · W ♯ ∗ ( U ♭ · [ V ] · W ♯ ∗ U ♭ · V ♭ · [ W ]) (inductio n hypothesis) = [ U ] · ( V ♯ ∗ 0 W ♯ ) ∗ U ♭ · ([ V ] · W ♯ ∗ V ♭ · [ W ]) (distributi vity over ∗ ) = [ U ] · ( V ∗ W ) ♯ ∗ U ♭ · [ V ∗ W ] (definition of ∗ ) = [ U ∗ ( V ∗ W )] . (definition of ∗ ) The second identity is obtained as follows, using distrib utivity over τ and the indu ction hypothesis: [ τ x ∗ U ] = [ τ x ] · U ♯ ∗ ( τ x ) ♭ · [ U ] = τ [ x ] · U ♯ ∗ 1 x ♭ · [ U ] = τ  x ∗ 0 U ♯  ∗ [ U ] = [ U ] , and similarly for the third one. ⊳ From now on, we shall omit parenth eses in concatenations. Lemma 27. (cylinders in a car tesian pr oduct) T here are natural isomor phisms of globular sets Γ( X × Y ) ≃ Γ( X ) × Γ( Y ) and Γ( 1 ) ≃ 1 , wh ich satisfy th e following cohere nce conditions with the can onical isomorp hisms ( X × Y ) × Z ≃ X × ( Y × Z ) and 1 × X ≃ X ≃ X × 1 : Γ(( X × Y ) × Z ) / /   Γ( X × ( Y × Z ))   Γ( X × Y ) × Γ( Z )   Γ( X ) × Γ( Y × Z )   (Γ( X ) × Γ( Y )) × Γ( Z ) / / Γ( X ) × (Γ( Y ) × Γ( Z )) Γ( 1 × X ) / /   Γ( X ) Γ( X × 1 ) o o   Γ( 1 ) × Γ( X )   Γ( X ) × Γ( 1 )   1 × Γ( X ) / / Γ( X ) Γ( X ) × 1 o o Remark 8. There is a coheren ce condition for the symmetry X × Y ≃ Y × X , but we shall not use it e xplicitly . ♦ Remark 9. By Lemmas 21 and 27, any ω -bifunctor f : X × Y → Z extends to cylinders in a canonical way . ♦ Definition 19. (multiplication ) Co mposition exten ds to cylinders: For any 0-cells x, y , z , we get the n -cylinder U ⊛ V in [ x, z ] , defined for any n -cylinders U in [ x, y ] and V in [ y , z ] . Lemma 28. (associativity of m ultiplication) Th e fo llo wing id entity hold s for any 0-c ells x, y , z , t , and for a ny n -cylinders U in [ x, y ] , V in [ y , z ] , W in [ z , t ] : ( U ⊛ V ) ⊛ W = U ⊛ ( V ⊛ W ) Proof. By functoriality , using coherence with the canonical isomorphism ( X × Y ) × Z ≃ X × ( Y × Z ) . ⊳ Remark 10. In Γ( X × Y ) ≃ Γ( X ) × Γ( Y ) , con catenation and τ can be defined comp onentwise. ♦ Using compatibility of Γ( f ) with concatena tion an d τ , we get the following result: Lemma 29. (compatib ility o f mu ltiplication with co ncatenation and τ ) The following id entities ho ld for any 0- cells x, y , z , for any n -cylinders U ⊲ U ′ in [ x, y ] and V ⊲ V ′ in [ y , z ] , and for any n +1 -cells u : x → 0 y and v : y → 0 z : ( U ∗ U ′ ) ⊛ ( V ∗ V ′ ) = ( U ⊛ V ) ∗ ( U ′ ⊛ V ′ ) , τ [ u ] ⊛ τ [ v ] = τ [ u ∗ 0 v ] . Remark 11. Any 0-cell x in X defines an ω -fu nctor h x i : 1 → X , from which we get Γ h x i : 1 ≃ Γ( 1 ) → Γ( X ) . It is easy to see that this homom orphism of globular sets correspond s to the sequence of tri vial n -cylinders τ 1 n x . ♦ Lemma 30. (represen tability) The follo wing identities hold for any 0-cells x, y , z : − u · V = τ 1 n [ u ] ⊛ V = τ  1 n +1 u  ⊛ V for any 1-cell u : x → y and for any n -cylinder V in [ y , z ] ; 28 − U · v = U ⊛ τ 1 n [ v ] = U ⊛ τ  1 n +1 v  for any 1-cell v : y → z and for any n -cylinder U in [ x, y ] . In other words, the (left and right) action of a 1-cell u is represented by the n -cylinder τ  1 n +1 u  . Proof. By functoriality , using coherence with the canonical isomorphisms 1 × X ≃ X ≃ X × 1 . ⊳ Definition 20. (extended action) For any 0- cells x, y , z , we e xtend left and right action to higher dimensional ce lls as follows: − u · V = τ [ u ] ⊛ V for any n +1 -cell u : x → y and for any n -cylinder V in [ y , z ] ; − U · v = U ⊛ τ [ v ] for any n +1 -cell v : y → z and for any n -cylinder U in [ x, y ] . Remark 12. In particular, we get u · V = 1 n +1 u · V fo r any 1-cell u : x → y and for an y n -c ylinder V in [ y , z ] , and similarly for the right action. This means that we ha ve i ndeed extended the action of 1-cells. ♦ Lemma 31. (extended bimodularity) The first three identities of lemma 22 e xtend to higher dimension al cells. Proof. By associati vity of multiplication and compatibility of multiplication with τ . ⊳ Lemma 32. (extended distrib utivity) The identities of lemma 25 extend to higher dimensional cells. Proof. The first identity is obtained as follows, using compatibility of multiplication with concatenation : u · ( V ∗ W ) = τ [ u ] ⊛ ( V ∗ W ) = ( τ [ u ] ∗ τ [ u ]) ⊛ ( V ∗ W ) = ( τ [ u ] ⊛ V ) ∗ ( τ [ u ] ⊛ W ) = u · V ∗ u · W. The second one follows from compatibility of multiplication with τ . ⊳ Lemma 33. (commu tation) The followi ng identities hold for any 0-cells x, y , z , for any n +1 -cells u, u ′ : x → 0 y and v , v ′ : y → 0 z , and for an y n -cylinders U : [ u ] y [ u ′ ] in [ x, y ] and V : [ v ] y [ v ′ ] in [ y , z ] : U · v ∗ u ′ · V = U ⊛ V = u · V ∗ U · v ′ . Proof. The first identity is obtained as follows, using compatibility of multiplication with concatenation : U · v ∗ u ′ · V = ( U ⊛ τ [ v ] ) ∗ ( τ [ u ′ ] ⊛ V ) = ( U ∗ τ [ u ′ ]) ⊛ ( τ [ v ] ∗ V ) = U ⊛ V , and similarly for the second one. ⊳ From now on, we shall always assume that m > n . Definition 21. (compo sitions) By induction on n , we define the m -cylinder U ∗ n V : R → n T | x ∗ n y y x ′ ∗ n y ′ for any m -cylinders U : R → n S | x y x ′ and V : S → n T | y y y ′ : − ( U ∗ 0 V ) ♭ = U ♭ = R ♮ and ( U ∗ 0 V ) ♯ = V ♯ = T ♮ , whereas [ U ∗ 0 V ] = x · [ V ] ∗ [ U ] · y ′ ; − if n > 0 , then ( U ∗ n V ) ♭ = U ♭ = V ♭ and ( U ∗ n V ) ♯ = U ♯ = V ♯ , whereas [ U ∗ n V ] = [ U ] ∗ n − 1 [ V ] . In both cases, we say that U and V are n - composab le , and we write U ⊲ n V . Lemma 34. (source and target of a composition) W e have U ∗ n U ′ k V ∗ n V ′ for any m -cylinders U k V and U ′ k V ′ such that U ⊲ n U ′ (so that V ⊲ n V ′ ), and W ∗ n W ′ : U ∗ n U ′ → V ∗ n V ′ for any m +1 -cylinders W : U → V and W ′ : U ′ → V ′ . Definition 22. (units) By indu ction on n , we define the m -cylinder 1 m U : U → n U | 1 m x y 1 m y for any n -cylinder U : x y y : − if n = 0 , then (1 m U ) ♭ = (1 m U ) ♯ = U ♮ , whereas [1 m U ] = τ  1 m U ♮  . In particular, we get  1 1 U  = τ  U ♮  ; − if n > 0 , then (1 m U ) ♭ = U ♭ and (1 m U ) ♯ = U ♯ , whereas [1 m U ] = 1 m − 1 [ U ] . Lemma 35. (source and target of a unit) W e have 1 m +1 U : 1 m U → 1 m U for any n -cylinder U . Remark 13. By constructio n, π 1 and π 2 are compatible with composition s and units. ♦ 29 Lemma 36. (associativity and units fo r compo sitions) T he following identities ho ld for any m -cylinders U ⊲ n V ⊲ n W and for any m -cylinder U : S → n T : ( U ∗ n V ) ∗ n W = U ∗ n ( V ∗ n W ) , 1 m S ∗ n U = U = U ∗ n 1 m T . Proof. W e pro ceed by induction on n . If n = 0 , the first iden tity is obtained as follows (with U : x y x ′ , V : y y y ′ and W : z y z ′ ): [( U ∗ 0 V ) ∗ 0 W ] = ( x ∗ 0 y ) · [ W ] ∗ [ U ∗ 0 V ] · z ′ (definition of ∗ 0 ) = x · y · [ W ] ∗ ( x · [ V ] ∗ [ U ] · y ′ ) · z ′ (definition of ∗ 0 ) = x · y · [ W ] ∗ x · [ V ] · z ′ ∗ [ U ] · y ′ · z ′ (distributi vity over ∗ ) = x · ( y · [ W ] ∗ [ V ] · z ′ ) ∗ [ U ] · y ′ · z ′ (distributi vity over ∗ ) = x · [ V ∗ W ] ∗ [ U ] · ( y ′ ∗ 0 z ′ ) (definition of ∗ 0 ) = [ U ∗ 0 ( V ∗ 0 W )] . (definition of ∗ 0 ) The second identity is obtained as follows (with U : x y y and S : x ♭ y y ♭ ), using distributi vity ov er τ : [1 m S ∗ 0 U ] = 1 m x ♭ · [ U ] ∗ [1 m S ] · y = 1 x ♭ · [ U ] ∗ τ [ 1 m S ♮ ] · y = [ U ] ∗ τ [1 m S ♮ ∗ 0 y ] = [ U ] , and similarly for the third one. If n > 0 , we apply the ind uction hypothesis. ⊳ Lemma 37. (compatib ility of τ with composition s and units) The following identities hold for any m -cells u ⊲ n v and for any n -cell x : τ ( u ∗ n v ) = τ u ∗ n τ v , τ 1 m x = 1 m τ x . Proof. By induction on n . If n = 0 , the first iden tity is obtained as follows, using distributi v ity over τ : [ τ ( u ∗ 0 v )] = τ [ u ∗ 0 v ] = τ [ u ∗ 0 v ] ∗ τ [ u ∗ 0 v ] = u · τ [ v ] ∗ τ [ u ] · v = u · [ τ v ] ∗ [ τ u ] · v = [ τ u ∗ 0 τ v ] . The second identity is obtained as follows: [ τ 1 m x ] = τ [1 m x ] = τ  1 m 1 x  = τ h 1 m ( τ x ) ♮ i = [1 m τ x ] . If n > 0 , we apply the ind uction hypothesis. ⊳ Lemma 38. (compatib ility of Γ( f ) with compo sitions and u nits) Th e following id entities ho ld any ω -functo r f : X → Y : − f ( U ∗ n V ) = f U ∗ n f V for any m -cylinders U ⊲ n V in X ; − f 1 m U = 1 m f U for any n -cylinder U in X . In the cases of precomp osition and postcomposition, we get the follo wing result: Lemma 39. (distributi vity o ver compositions and units) The following iden tities f or an y 0-cells x, y , z and for an y 1-cell u : x → y : − u · ( V ∗ n W ) = u · V ∗ n u · W for any m -cylinders V ⊲ n W in [ y , z ] ; − u · 1 m V = 1 m u · V for any n -cylinder V in [ y , z ] . There are similar prope rties for right action. Lemma 40. (compatib ility of conca tenation with com position and units) The following identities hold f or a ny m -cylinders U ⊲ n V and U ′ ⊲ n V ′ such that U ⊲ U ′ and V ⊲ V ′ , and for any n -cylinders S ⊲ T : ( U ∗ n V ) ∗ ( U ′ ∗ n V ′ ) = ( U ∗ U ′ ) ∗ n ( V ∗ V ′ ) , 1 m S ∗ 1 m T = 1 m S ∗ T . 30 Proof. W e pro ceed by induction on n . If n = 0 , the first iden tity is obtained as follows (with U : x y x ′ , U ′ : x ′ y x ′′ , V : y y y ′ and V ′ : y ′ y y ′′ ): [( U ∗ 0 V ) ∗ ( U ′ ∗ 0 V ′ )] = [ U ∗ 0 V ] · ( U ′ ∗ 0 V ′ ) ♯ ∗ ( U ∗ 0 V ) ♭ · [ U ′ ∗ 0 V ′ ] (definition of ∗ ) = ( x · [ V ] ∗ [ U ] · y ′ ) · V ′ ♯ ∗ U ♭ · ( x ′ · [ V ′ ] ∗ [ U ′ ] · y ′′ ) (definition of ∗ 0 ) = x · [ V ] · V ′ ♯ ∗ [ U ] · y ′ · V ′ ♯ ∗ U ♭ · x ′ · [ V ′ ] ∗ U ♭ · [ U ′ ] · y ′′ (distributi vity over ∗ ) = x · [ V ] · V ′ ♯ ∗ x · V ♭ · [ V ′ ] ∗ [ U ] · U ′ ♯ · y ′′ ∗ U ♭ · [ U ′ ] · y ′′ (commu tation) = x · ([ V ] · V ′ ♯ ∗ V ♭ · [ V ′ ]) ∗ ([ U ] · U ′ ♯ ∗ U ♭ · [ U ′ ]) · y ′′ (distributi vity over ∗ ) = x · [ V ∗ V ′ ] ∗ [ U ∗ U ′ ] · y ′′ (definition of ∗ ) = [( U ∗ U ′ ) ∗ 0 ( V ∗ V ′ )] . (definition of ∗ 0 ) In the commutatio n step, we use the fact that U ♯ = V ♭ and U ′ ♯ = V ′ ♭ since U ⊲ 0 V and U ′ ⊲ 0 V ′ . The second identity is obtained as follows, using distrib utivity over τ : [1 m S ∗ 1 m T ] = [1 m S ] · (1 m T ) ♯ ∗ (1 m S ) ♭ · [1 m T ] = τ  1 m S ♮  · T ♮ ∗ S ♮ · τ  1 m T ♮  = τ h 1 m S ♮ ∗ 0 T ♮ i ∗ τ h 1 m S ♮ ∗ 0 T ♮ i = τ h 1 m S ♮ ∗ 0 T ♮ i = τ h 1 m ( S ∗ T ) ♮ i = [1 m S ∗ T ] . If n > 0 , the first iden tity is obtained as follows: [( U ∗ n V ) ∗ ( U ′ ∗ n V ′ )] = [ U ∗ n V ] · ( U ′ ∗ n V ′ ) ♯ ∗ ( U ∗ n V ) ♭ · [ U ′ ∗ n V ′ ] (definition of ∗ ) = ([ U ] ∗ n − 1 [ V ]) · U ′ ♯ ∗ U ♭ · ([ U ′ ] ∗ n − 1 [ V ′ ]) (definition of ∗ n ) = ([ U ] · U ′ ♯ ∗ n − 1 [ V ] · U ′ ♯ ) ∗ ( U ♭ · [ U ′ ] ∗ n − 1 U ♭ · [ V ′ ]) (distributi vity o ver ∗ n − 1 ) = ([ U ] · U ′ ♯ ∗ U ♭ · [ U ′ ]) ∗ n − 1 ([ V ] · U ′ ♯ ∗ U ♭ · [ V ′ ]) (inductio n hypothesis) = [ U ∗ U ′ ] ∗ n − 1 [ V ∗ V ′ ] (definition of ∗ ) = [( U ∗ U ′ ) ∗ n ( V ∗ V ′ )] . (definition of ∗ n ) In the penultimate step, we use the fact that U ♭ = V ♭ and U ′ ♯ = V ′ ♯ since U ⊲ n V and U ′ ⊲ n V ′ . The second identity is obtained as follows, using distrib utivity over u nits and the induction hypothesis: [1 m S ∗ 1 m T ] = [1 m S ] · (1 m T ) ♯ ∗ (1 m S ) ♭ · [1 m T ] = 1 m − 1 [ S ] · T ♯ ∗ S ♭ · 1 m − 1 [ T ] = 1 m − 1 [ S ] · T ♯ ∗ 1 m − 1 S ♭ · [ T ] = 1 m − 1 [ S ] · T ♯ ∗ S ♭ · [ T ] = 1 m − 1 [ S ∗ T ] = [1 m S ∗ T ] . ⊳ Remark 14. In Γ( X × Y ) ≃ Γ( X ) × Γ( Y ) , com positions and units can be defined compon entwise. ♦ Using compatibility of Γ( f ) with composition s a nd units, we get the following result: Lemma 41. (compatib ility o f mu ltiplication with co mpositions and un its) The following iden tities hold for any 0-cells x, y , z , for any m -cylinders U ⊲ n U ′ in [ x, y ] and V ⊲ n V ′ in [ y , z ] , and for an y n -cylinders S in [ x, y ] and T in [ y , z ] : ( U ∗ n U ′ ) ⊛ ( V ∗ n V ′ ) = ( U ⊛ V ) ∗ n ( U ′ ⊛ V ′ ) , 1 m S ⊛ 1 m T = 1 m S ⊛ T . Lemma 42. (compatib ility of ac tion with comp ositions a nd u nits) The fo llo wing identities hold fo r any 0-cells x, y , z , for any m +1 -cells u, u ′ : x → 0 y such that u ⊲ n +1 u ′ , f or any m -cylinders V ⊲ n V ′ in [ y , z ] , for any n +1 -cell s : x → 0 y , and for any n -cylinder T in [ y , z ] : ( u ∗ n +1 u ′ ) · ( V ∗ n V ′ ) = u · V ∗ n u ′ · V ′ , 1 m +1 s · 1 m T = 1 m s · T . There are similar prope rties for right action. Proof. The first identity is obtained as follows, u sing compatibility of τ with compositions and the p re vious lemma: ( u ∗ n +1 u ′ ) · ( V ∗ n V ′ ) = τ [ u ∗ n +1 u ′ ] ⊛ ( V ∗ n V ′ ) = τ ([ u ] ∗ n [ u ′ ]) ⊛ ( V ∗ n V ′ ) = ( τ [ u ] ∗ n τ [ u ′ ]) ⊛ ( V ∗ n V ′ ) = ( τ [ u ] ⊛ V ) ∗ n ( τ [ u ′ ] ⊛ V ′ ) = u · V ∗ n u ′ · V ′ . The second identity is obtained as follows, using compatibility of τ with un its and the previous lemma: 1 m +1 s · 1 m T = τ  1 m +1 s  ⊛ 1 m T = τ 1 m [ s ] ⊛ 1 m T = 1 m τ [ s ] ⊛ 1 m T = 1 m τ [ s ] ⊛ T = 1 m s · T . ⊳ 31 Now we assume that m > n > p . Lemma 43. (interchan ge) Th e following identities hold for any m -cylinders U ⊲ n U ′ and V ⊲ n V ′ such that U ⊲ p V (so that U ′ ⊲ p V ′ ), for any n -cylinders S ⊲ p T , and for any p -cylinder R : ( U ∗ n U ′ ) ∗ p ( V ∗ n V ′ ) = ( U ∗ p V ) ∗ n ( U ′ ∗ p V ′ ) , 1 m S ∗ p 1 m T = 1 m S ∗ p T , 1 m 1 n R = 1 m R . Proof. W e pro ceed by induction on p . If p = 0 , the first iden tity is obtained as follows (with U : x y y , U ′ : x ′ y y ′ , V : z y t and V ′ : z ′ y t ′ ): [( U ∗ n U ′ ) ∗ 0 ( V ∗ n V ′ )] = ( x ∗ n x ′ ) · [ V ∗ n V ′ ] ∗ [ U ∗ n U ′ ] · ( t ∗ n t ′ ) (definition of ∗ 0 ) = ( x ∗ n x ′ ) · ([ V ] ∗ n − 1 [ V ′ ]) ∗ ([ U ] ∗ n − 1 [ U ′ ]) · ( t ∗ n t ′ ) (definition of ∗ n ) = ( x · [ V ] ∗ n − 1 x ′ · [ V ′ ]) ∗ ([ U ] · t ∗ n − 1 [ U ′ ] · t ′ ) (compatib ility of · with ∗ n − 1 ) = ( x · [ V ] ∗ [ U ] · t ) ∗ n − 1 ( x ′ · [ V ′ ] ∗ [ U ′ ] · t ′ ) (compatib ility of ∗ with ∗ n − 1 ) = [ U ∗ 0 V ] ∗ n − 1 [ U ′ ∗ 0 V ′ ] (definition of ∗ 0 ) = [( U ∗ 0 V ) ∗ n ( U ′ ∗ 0 V ′ )] . 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