Rigidification of quasi-categories

RIGIDIFICA TION OF QUASI-CA TEGORIES DANIEL DUGGER AND D A VID SPIV AK Abstract. W e giv e a new construction for rigi di fying a quasi-category in to a simplicial catego ry , and prov e that it is weakly equiv alent to the rigidification give n b y Lurie. Our construction comes f rom the use of nec klaces, which are simplicial sets obtained by stri nging simpli ces together. As an appli cation of these methods, we use our mo del to r epro v e some basic f acts from [L] ab out the rigidification pro cess. Contents 1. Int ro duction 1 2. Background on quasi-catego r ies 3 3. Necklaces 6 4. The categor ification functor 10 5. Homotopical mo dels for catego rification 16 6. Prop erties of c ategorifica tion 18 Appendix A. Leftov er pr o ofs 21 References 25 1. Intr oduction Quasi-ca tegories a re a certain genera lization o f categ ories, in which one has not only 1-mo rphisms but n -morphisms for every na tur al num ber n . They hav e be e n extensively studied by Cor dier and Porter [CP], by Joy a l [J1], [J 2], and by Lurie [L]. If K is a qua si-catego ry and x and y are tw o ob jects of K , then one may asso ciate a “ mapping space” K ( x, y ) w hich is a simplicial set. There are man y different cons tructions for these mapping spaces , but in [L] one par ticular mo del is g iven for which there are comp osition maps K ( y , z ) × K ( x, y ) → K ( x, z ) giving rise to a simplicial category . This simplicial ca tegory is denoted C ( K ), and it may be thought of as a rigidific ation o f the quasi-ca tegory K . It is proven in [L] that the homotopy theo ries of quasi- categorie s a nd simplicial categ ories are equiv a lent via this functor. In this pap er we int ro duce some new mode ls for the mapping spa ces K ( x, y ), which are particula rly easy to descr ibe and particularly easy to use—in fact they are just the nerves of ordinary categories (i.e., 1-ca tegories). Like Lur ie’s model, our mo dels admit compos ition maps giving rise to a s implicial category; so we ar e giving a new metho d for rigidifying qua si-catego ries. W e pr ov e that our co ns truction is homotopy equiv alen t (as a simplicia l categor y) to Lurie’s C ( K ). Moreov er, b ecaus e our mapping spaces are ner ves of categor ies there are ma ny standar d to ols av ailable 1 2 DANIEL DUGGER AND DA VID SPIV AK for analy zing their homotopy types. W e demonstrate the effectiveness of this by giving new pr o ofs of s ome basic fa cts ab o ut the functor C ( − ). One pay off of this appro ach is that it is p ossible to give a s treamlined pro o f of Lurie’s Quillen equiv alence betw een the homotopy theo ry of quasi- categor ie s and simplicial categor ies. This requir es, how ev er, a mor e detailed study o f the mo del category str ucture o n quasi-catego ries. W e will take this up in a sequel [DS ] and prov e the Quillen equiv a lence there. 1.1. Mapping s paces via si mplicial categories. Now we describe our res ults in more detail. A quas i- categor y is a simplicial s et that has the r ight-lifting-proper ty with r e sp e ct to inner ho r n inclusions Λ n i → ∆ n , 0 < i < n . It turns out that there is a unique model structure o n s S et where the c o fibrations are the monomorphis ms and the fibrant o b jects ar e the quasi-categ ories; this will be ca lled the Joyal mo del structur e and denoted s S e t J . T he w eak eq uiv alences in s S et J will here b e called Joyal e quivalenc es . The existence of the J oy al mo del structur e will not b e needed in this pap er, although it provides some useful context. The notions of quasi-categor ies and Joy al eq uiv alences, how ever, will b e used in several places. See Section 2.3 for additional background. There is a functor, constructed in [L], which sends an y simplicial set K to a corres p o nding simplicial ca tegory C ( K ) ∈ s C at . This is the left adjoint in a Quillen pair C : s S et J ⇄ s C at : N , where N is called the coheren t nerv e . The functor N can b e describ ed quite explicitly (see Section 2), but the functor C is in compar ison a little mysterious. In [L] each C ( K ) is defined as a cer tain colimit in the catego ry s C at , but colimits in s C at are notoriously difficult to understand. Our main go al in this pap er is to give a differ ent mo del for the functor C . Define a nec klace (which we picture as “ unfastened”) to be a simplicial s et of the form ∆ n 0 ∨ ∆ n 1 ∨ · · · ∨ ∆ n k where each n i ≥ 0 and where in ea ch wedge the final vertex of ∆ n i has be e n glued to the initia l vertex of ∆ n i +1 . The first and last vertex in any necklace T are denoted α T and ω T , resp ectively (or just α a nd ω if T is obvious fro m context). If S and T are t w o necklaces, then by S ∨ T we mean the necklace obtained in the evident wa y , by gluing the final v ertex ω S of S to the initial vertex α T of T . W rite N ec for the categor y who se ob jects are necklaces a nd where a mo rphism is a map of simplicia l sets which preser ves the initial and final vertices. Let S ∈ s S et and let a, b ∈ S 0 . If T is a necklace, we use the no ta tion T → S a,b to indicate a morphism of simplicial sets T → S whic h sends α T to a and ω T to b . Let ( N ec ↓ S ) a,b denote the evident categ ory whose o b jects are pair s [ T , T → S a,b ] where T is a necklace. Note that for a, b, c ∈ S , there is a functor ( N ec ↓ S ) b,c × ( N ec ↓ S ) a,b − → ( N ec ↓ S ) a,c which sends the pair [ T 2 , T 2 → S b,c ] × [ T 1 , T 1 → S a,b ] to [ T 1 ∨ T 2 , T 1 ∨ T 2 → S a,c ]. Let C nec ( S ) be the function which assigns to any a, b ∈ S 0 the simplicial set C nec ( S )( a, b ) = N ( N ec ↓ S ) a,b (the classical nerve of the 1- c ategory ( N e c ↓ S ) a,b ). RIGIDIFICA TION OF QUASI-CA TEGORIES 3 The above pa irings of ca tegories induce s pair ings o n the nerves, which makes C nec ( S ) into a simplicial ca tegory with ob ject set S 0 . Theorem 1 .2. Ther e is a natur al zig-zag of we ak e quivalenc es of simplicial c ate- gories b etwe en C nec ( S ) and C ( S ) , for al l simplicial set s S . In the a bove result, the weak equiv alences for simplicial categor ies are the so- called “DK-e quiv alences ” used b y Bergner in [B]. See Section 2 for this notion. In this pap er we also give an explicit des cription of the mapping s paces in the simplicial catego ry C ( S ). A r ough statement is given b elow, but see Section 4 for more details. Theorem 1 .3. L et S b e a s implicial set and let a , b ∈ S . Then the mapping sp ac e X = C ( S )( a, b ) is the simplicial set whose n - simplic es ar e t riples subje ct to a c ertain e quivalenc e re lations. The triples c onsist of a ne cklac e T , a map T → S a,b , and a flag − → T = { T 0 ⊆ · · · ⊆ T n } of vertic es in T . F or the e quivalenc e r elation, se e Cor ol lary 4.4. The fac e maps and de gener acy maps ar e obtaine d by r emovi ng or r ep e ating elements T i in the flag. The p airing C ( S )( b, c ) × C ( S )( a, b ) − → C ( S )( a, c ) sends the p air of n - simplic es ([ T → S ; − → T i ] , [ U → S, − → U i ]) to [ U ∨ T → S, − − − − − → U i ∪ T i ] . Theorem 1.2 turns o ut to b e very useful in the s tudy of the functor C . There a re many to o ls in classical homotopy theor y for understanding the ho mo topy types of nerves of 1-categor ie s, and via T he o rem 1 .2 these to ols can be applied to understand mapping spaces in C ( S ). W e demonstr ate this tec hnique in Section 6 by proving, in a new w ay , the fo llowing tw o prop erties o f C found in [L]. Theorem 1 .4. L et X and Y b e simplicial sets. (a) The natur al map C ( X × Y ) → C ( X ) × C ( Y ) is a we ak e quivalenc e of simplicial c ate gories; (b) If X → Y is a Joyal e quivalenc e then C ( X ) → C ( Y ) is a we ak e quivalenc e. 1.5. Notation and T erminolo g y . W e will sometimes use s S et K to refer to the usual mo de l structure on simplicial sets, which we’ll ter m the Kan mo del stru c- tur e . The fibrations ar e the Kan fibrations, the w eak equiv alences (calle d Ka n equiv alences fro m now on) are the maps which induce homotopy equiv alences on geometric rea liz ations, and the cofibrations are the monomorphis ms . W e will often be working with the category s S et ∗ , ∗ = ( ∂ ∆ 1 ↓ s S et ). Note that N ec is a full subcateg ory of s S et ∗ , ∗ . An ob ject of s S et ∗ , ∗ is a simplicia l set X with tw o distinguished points a and b . W e sometimes (but not alwa ys) write X a,b for X , to r emind us that things are taking place in s S et ∗ , ∗ instead of s S et . If C is a (simplicial) ca tegory containing ob jects X and Y , w e wr ite C ( X, Y ) for the (simplicial) set of mo rphisms from X to Y . 2. Ba ckg round on quasi-ca tegories In this se ction we give the background on q uasi-catego ries and simplicial cate- gories needed in the re s t of the pap er. 4 DANIEL DUGGER AND DA VID SPIV AK 2.1. Simplicial categories. A simplicial categor y is a catego ry enriched over sim- plicial sets; it can also b e thought of as a simplicia l ob ject of C at in which the categorie s in each level hav e the same o b ject set. W e use s C at to denote the ca te- gory of simplicia l categor ies. A cofibrantly-generated mo del structure o n s C at was developed in [B]. A map of s implicial ca tegories F : C → D is a weak equiv alence (sometimes called a DK-e quivalenc e ) if (1) F or all a, b ∈ ob C , the map C ( a, b ) → D ( F a, F b ) is a K a n equiv alence of simplicial sets; (2) The induced functor of ordinar y ca tegories π 0 F : π 0 C → π 0 D is surjective on isomorphism c lasses. Likewise, the map F is a fibration if (1) F or all a, b ∈ o b C , the map C ( a, b ) → D ( F a, F b ) is a Kan fibra tion of simplicial sets; (2) F or a ll a ∈ ob C and b ∈ o b D , if e : F a → b is a map in D whic h b ecomes an isomorphism in π 0 D , then there is an o b ject b ′ ∈ C and a map e ′ : a → b ′ such that F ( e ′ ) = e and e ′ bec omes an is omorphism in π 0 C . The cofibrations are the maps which hav e the left lifting prop erty with resp ect to the acy c lic fibrations. Remark 2. 2 . The seco nd part of the fibra tio n condition seems a little awkward at first. In this pap er we will actually hav e no need to think a b o ut fibrations of simplicial categ o ries, but have included the definition for completeness. Bergner writes down sets of generating co fibrations and acyclic cofibr ations in [B]. 2.3. Quasi-categories and Jo y al equiv alences. As mentioned in the intro duc- tion, there is a unique model structure o n s S et with the pr op erties that (i) The cofibr a tions are the monomorphisms; (ii) The fibrant ob jects are the quasi-categ ories. It is easy to s ee that ther e is at mo st one such structure. T o do this, let E 1 be the 0-coskeleton—see [AM], for instance—of the set { 0 , 1 } (note that the geo metric realization of E 1 is ess ent ially the standard mo del for S ∞ ). The map E 1 → ∗ has the rig ht lifting pro per ty with r e s pe c t to all mono morphisms, and so it will b e an a cyclic fibration in this s tructure. Therefore X × E 1 → X is also an ac yclic fibration for any X , and hence X × E 1 will be a cylinder ob ject for X . Since every ob ject is cofibrant, a ma p A → B will b e a weak equiv alence if and only if it induces bijections [ B , Z ] E 1 → [ A, Z ] E 1 for every quasi-categor y Z , wher e [ A, Z ] E 1 means the co equalizer of s S et ( A × E 1 , Z ) ⇒ s S et ( A, Z ). Therefore the weak equiv alences are determined b y prop erties (i)–(ii), and since the c o fibrations and weak equiv alences are determined so are the fibrations. Motiv ated by the a bove discuss io n, we define a map of simplicial sets A → B to b e a Jo y al equiv alence if it induces bijections [ B , Z ] E 1 → [ A, Z ] E 1 for e very quasi-categ ory Z . That there actua lly exists a mo del stucture satisfying (i) a nd (ii) is not s o clea r, but it was estalished by Joy a l (see [J1] or [J2], or [L] for ano ther pro of ). F or this reason, we will call it the Joy al mo del structure a nd denote it b y s S et J . The weak equiv alences are defined differently in b oth [J2] and [L], but of course turn out to be equiv alent to the definition we hav e ado pted here. RIGIDIFICA TION OF QUASI-CA TEGORIES 5 In the r est of the pap er w e will never use the Joy al mo del structure, only the notion of Joyal equiv alence. 2.4. Bac kground on C and N . Giv en a simplicial catego ry S , one can construct a simplicial s et called the c oher ent nerve of S [L, 1 .1.5]. W e will now describ e this construction. Recall the adjoint functors F : G rph ⇄ C at : U . Here C at is the categor y of 1- categorie s, and G rp h is the ca teg ory of graphs : a graph consists of an ob ject set and mor phism sets, but no comp os ition law. The functor U is a forg e tful functor , and F is a free functor . Given any category C we ma y then consider the co monad resolution ( F U ) • ( C ) given by [ n ] 7→ ( F U ) n +1 ( C ). This is a s implicial ca tegory . There is a functor of simplicia l categ ories ( F U ) • ( C ) → C (where the latter is considered a discrete simplicial ca tegory). This functor induces a weak equiv a lence on all mapping spaces, a fact whic h can b e seen b y applying U , at which point the comonad resolution pic ks up a con tracting homotopy . Note tha t this means that the simplicial mapping spaces in ( F U ) • ( C ) are all homotopy discrete. Recall that [ n ] denotes the categ ory 0 → 1 → · · · → n , wher e there is a unique map fro m i to j whenev er i ≤ j . W e let C (∆ n ) denote the simplicia l categor y ( F U ) • ([ n ]). The mapping spaces in this simplicial category can be ana lyzed c o m- pletely , and are as follows. F or each i and j , let P i,j denote the p os et o f a ll subsets of { i , i + 1 , . . . , j } containing i and j (o rdered b y inclusio n). Note that the nerve of P i,j is isomorphic to the cube (∆ 1 ) j − i − 1 if j > i , ∆ 0 if j = i , a nd the emptyset if j < i . The nerves of the P i,j ’s na tur ally form the mapping spac e s of a s implicial category with ob ject s et { 0 , 1 , . . . , n } , using the pairing s P j,k × P i,j → P i,k given by union of sets. Lemma 2. 5. Ther e is an isomorphism of simplicial c ate gorie s C (∆ n ) ∼ = N P . Remark 2.6. The pro of of the above lemma is a bit of a n aside fro m the ma in thrust of the pap er, so it is given in Appendix A. In fact we co uld have define d C (∆ n ) to b e N P , which is what Lurie do es in [L], and av oided the lemma en tirely; the constr uction ( F U ) • ([ n ]) will never aga in b e used in this paper . Nevertheless, the identification of N P with ( F U ) • ([ n ]) seems informative to us. F or any simplicia l ca tegory D , the coherent nerv e of D is the simplicial set N D g iven by [ n ] 7→ s C at ( C (∆ n ) , D ) . It was prov en b y Lurie [L] that N D is always a q uasi-catego ry; see also Lemma 6.5 below. The functor N has a left a djoint denoted C : s S et → s C at . Any simplicial s e t K may b e wr itten as a colimit o f simplices via the for m ula K ∼ = colim ∆ n → K ∆ n , and consequently one has C ( K ) ∼ = colim ∆ n → K C (∆ n ) (2.6) where the colimit takes place in s C at . This formula is a bit unwieldy , how ev er, in the se ns e that it does not give muc h concrete informa tio n ab out the mapping spaces in C ( K ). The p oint of the next three sections is to o bta in suc h concr ete information, via the use o f necklaces. 6 DANIEL DUGGER AND DA VID SPIV AK 3. Necklaces A necklace is a simplicia l set obtained b y stringing simplices together in suc- cession. In this section we establish some basic facts ab out them, as well as facts ab out the more general category of o r dered simplicial sets. When T is a necklace we a re a ble to give a complete description of the ma pping spa ces in C ( T ) as nerves of certain p ose ts , gener alizing what was said for C (∆ n ) in the last section. See Prop ositio n 3 .7. As briefly discuss ed in the in tro duction, a nec klace is defined to be a simplicial set of the form ∆ n 0 ∨ ∆ n 1 ∨ · · · ∨ ∆ n k where each n i ≥ 0 and where in ea ch wedge the final vertex of ∆ n i has be e n glued to the initial vertex of ∆ n i +1 . W e say that the necklace is in preferred form if either k = 0 or each n i ≥ 1. Let T = ∆ n 0 ∨ ∆ n 1 ∨ · · · ∨ ∆ n k be in pre fer red for m. Each ∆ n i is called a b ead of the nec klace. A jo in t of the necklace is either an initial or a final vertex in some bea d. Th us, every necklace has a t least one v ertex, one bea d, and one joint; ∆ 0 is not a b ead in any necklace ex cept in the necklace ∆ 0 itself. Given a necklace T , wr ite V T and J T for the sets of vertices and joints of T . Note that V T = T 0 and J T ⊆ V T . Both V T and J T are totally ordered, b y saying a ≤ b if ther e is a directed path in T fro m a to b . The initial and fina l v ertices of T are denoted α T and ω T (and we s ometimes drop the subscr ipt); note that α T , ω T ∈ J T . Every nec klace T comes with a particular ma p ∂ ∆ 1 → T which sends 0 to the initial vertex of the nec klace, and 1 to the final v ertex. If S a nd T are tw o nec klaces, then by S ∨ T we mean the necklace obtained in the ev ident w ay , by gluing the final vertex of S to the initia l vertex o f T . Le t N ec denote the full sub categ ory of s S et ∗ , ∗ = ( ∂ ∆ 1 ↓ s S e t ) whose ob jects are necklaces ∂ ∆ 1 → T . W e sometimes talk ab out N ec as thoug h it is a sub category o f s S et . A simplex is a nec klace with one be ad. A spi ne is a nec klace in whic h every bea d is a ∆ 1 . Every nec klace T has an asso ciated simplex and spine, whic h w e now define. Le t ∆[ T ] be the simplex whos e vertex set is the sa me as the (order ed) vertex set of T . Likewise, let Spi[ T ] b e the longest spine inside of T . Note tha t there are inclusions Spi[ T ] ֒ → T ֒ → ∆[ T ]. The as signment T → ∆[ T ] is a functor, but T → Spi [ T ] is no t (for instance, the unique map o f necklaces ∆ 1 → ∆ 2 do es not induce a map on spines). 3.1. Ordered simplicial sets. If T → T ′ is a map of necklaces, then the image of T is also a necklace. T o pr ove this, as well a s for several other rea sons scattered thoughout the pap er, it turns out to b e very conv enient to w ork in somewhat gr eater generality . If X is a simplicial set, define a relation on its 0-simplices b y saying that x  y if there exists a spine T and a map T → X sending α T 7→ x a nd ω T 7→ y . In other words, x  y if there is a directed path from x to y inside of X . Note that this relation is clea rly reflexive a nd transitive, but no t necessarily a n tisymmetric: that is, if x  y a nd y  x it need not b e true that x = y . Definition 3. 2. A simplicia l set X is or der e d if (i) The r elation  define d on X 0 is antisymmetric, and RIGIDIFICA TION OF QUASI-CA TEGORIES 7 (ii) A n simplex x ∈ X n is determine d by its se quenc e of vertic es x (0)  · · ·  x ( n ) ; i.e. no two distinct n -simplic es have identic al vertex se quenc es. Note the r ole of deg enerate simplices in condition (ii). F or exa mple, notice that ∆ 1 /∂ ∆ 1 is no t an order ed simplicial set. The following notion is also useful: Definition 3.3. L et A and X b e simplicial s et s. A map A → X is c al le d a simple inclusion if it has the right lifting pr op erty with r esp e ct to the c anonic al inclusions ∂ ∆ 1 ֒ → T for al l ne cklac es T . (Note that such a map r e al ly is an inclusion, b e c ause it has the lifting pr op erty for ∂ ∆ 1 → ∆ 0 ). The notion of simple inclusion says that if there is a “path” (in the sense of a necklace) in X that star ts a nd ends in A , then it m ust lie entirely in A . As an example, four out of the fiv e inclusions ∆ 1 ֒ → ∆ 1 × ∆ 1 are simple inclusions. Lemma 3.4. A simple inclusion A ֒ → X has the right lifting pr op ert y with r esp e ct to the maps ∂ ∆ k ֒ → ∆ k for al l k ≥ 1 . Pr o of. Suppose g iven a square ∂ ∆ k / /   A   ∆ k / / X . By restr icting the map ∂ ∆ k → A to ∂ ∆ 1 ֒ → ∂ ∆ k (given by the initia l and final vertices o f ∂ ∆ k ), we get a c orresp onding lifting sq ua re with ∂ ∆ 1 ֒ → ∆ k . Since A → X is a simple inclusion, this new square ha s a lift l : ∆ k → A . It is not immediately cle a r that l r estricted to ∂ ∆ k equals our o riginal ma p, but the tw o maps a re equal after compo sing with A → X ; since A → X is a monomorphism, the tw o ma ps ar e themselves eq ual.  Lemma 3.5. L et X and Y denote or der e d simplicial sets and let f : X → Y b e a map. (1) The c ate gory of or der e d simplicial sets is close d under taking finite limits. (2) Every ne ckla c e is an or der e d simplicia l set. (3) If X ′ ⊆ X is a simplicial subset, then X ′ is also or der e d. (4) The map f is c ompl etely determine d by the map f 0 : X 0 → Y 0 on vertic es. (5) If f 0 is inje ctive t hen so is f . (6) The image of an n - simplex x : ∆ n → X is of the form ∆ k ֒ → X for some k ≤ n . (7) If T is a ne ckla c e and y : T → X is a map, then its image is a ne cklac e. (8) Supp ose that X ← A → Y is a diagr am of or der e d simplicial sets, and b oth A → X and A → Y ar e simple inclusions. Then the pushout B = X ∐ A Y is an or der e d simplic ial set, and the inclusions X ֒ → B and Y ֒ → B ar e b oth simple. Pr o of. F or (1), the terminal ob ject is a point with its unique ordering. Giv en a diagram of the form X − → Z ← − Y , 8 DANIEL DUGGER AND DA VID SPIV AK let A = X × Z Y . It is clear that if ( x, y )  A ( x ′ , y ′ ) then b oth x  X x ′ and y  Y y ′ hold, a nd so an tisymmetry of  A follows from that of  X and  Y . Condition (ii) from Definition 3.2 is eas y to chec k. Parts (2)– (5) ar e easy , and left to the r eader. F or (6), the seq uence x (0) , . . . , x ( n ) ∈ X 0 may hav e duplicates; let d : ∆ k → ∆ n denote a ny face such that x ◦ d contains all vertices x ( j ) and has no duplicates. Note that x ◦ d is an injection b y (5). A certain degeneracy of x ◦ d has the same vertex sequence as x . Since X is order ed, x is this degener acy o f x ◦ d . Hence, x ◦ d : ∆ k ֒ → X is the imag e of x . Claim (7) follo ws from (6). F or cla im (8) we fir st show that the maps X ֒ → B and Y ֒ → B are simple inclusions. T o see this, suppo se that u, v ∈ X are vertices, T is a necklace, a nd f : T → B u,v is a map; we wan t to show that f factor s through X . Note that any simplex ∆ k → B either factors through X or throug h Y . Suppo se that f do es not factor through X . F rom the set of beads of T which do no t factor through X , take any maximal subset T ′ in which a ll the b eads are adjacent. Then we have a necklace T ′ ⊆ T s uch that f ( T ′ ) ⊆ Y . If there ex ists a bead in T prior to α T ′ , then it must map in to X since T ′ was maxima l; so f ( α T ′ ) lies in X ∩ Y = A . Likewise, if there is no bea d prior to α T ′ then f ( α T ′ ) = u and so aga in f ( α T ′ ) lies in X ∩ Y = A . Similar remarks apply to show that f ( ω T ′ ) lies in A . At this p oint the fact that A ֒ → Y is a simple inclusion implies that f ( T ′ ) ⊆ A ⊆ X , whic h is a contradiction. So in fact f factored thro ugh X . W e hav e shown that X ֒ → B (and dually Y ֒ → B ) is a s imple inclusion. Now w e show that B is order e d, s o supp ose u , v ∈ B are such that u  v and v  u . Ther e there are spines T and U and maps T → B u,v , U → B v, u . Consider the comp osite spine T ∨ U → B u,u . If u ∈ X , then since X ֒ → B is a simple inclusion it follows that the image of T ∨ U maps en tirely into X ; so u  X v and v  X u , which means u = v beca use X is or dered. The same argument works if u ∈ Y , so this verifies antisymmetry of  B . T o v erify condition (ii) o f Definition 3.2, suppo se p, q : ∆ k → B are k -simplices with the s ame seq uence of vertices; we wish to show p = q . W e know that p factor s through X or Y , and so do es q ; if b oth factor thr ough Y , then the fact that Y is ordered implies that p = q (similar ly for X ). So w e may assume p factors thr o ugh X a nd q factors through Y . By induction o n k , the restrictions p | ∂ ∆ k = q | ∂ ∆ k are equal, hence factor throug h A . By Lemma 3.4 applied to A ֒ → X , the map p factors through A . Therefore it also facto rs through Y , and now we a re done b ecause q also factors through Y and Y is or der ed.  3.6. Categorification of nec klaces. Le t T b e a nec klace. O ur next go al is to give a co mplete descr iption o f the simplicial categor y C ( T ). The ob ject set of this category is precisely T 0 . F or vertices a, b ∈ T 0 , let V T ( a, b ) deno te the set of vertices in T b etw een a and b , inclusive (with r esp ect to the relation  ). Let J T ( a, b ) denote the union o f { a, b } with the set of join ts b etw een a and b . There is a unique subnecklace of T with joint s J T ( a, b ) and vertices V T ( a, b ); let e B 0 , e B 1 , . . . e B k denote its b e ads. Ther e ar e canonical inclusions of each e B i to T . Hence, there is a natur al map C ( e B k )( j k , b ) × C ( e B k − 1 )( j k − 1 , j k ) × · · · × C ( e B 1 )( j 1 , j 2 ) × C ( e B 0 )( a, j 1 ) → C ( T )( a, b ) RIGIDIFICA TION OF QUASI-CA TEGORIES 9 obtained by fir st including the e B i ’s in to T and then using the comp os itio n in C ( T ) (where j i and j i +1 are the join ts of B i ). W e will see tha t this map is a n is omorphism. Note that ea ch of the sets C ( e B i )( − , − ) has a n easy descriptio n, as in Lemma 2.5; from this one may extrap ola te a corresp onding description fo r C ( T )( − , − ), to b e explained next. Let C T ( a, b ) denote the p oset whos e elements are subsets of V T ( a, b ) which con- tain J T ( a, b ), order ed by inclus io n. There is a pair ing of categor ies C T ( b, c ) × C T ( a, b ) → C T ( a, c ) given by union of subsets. Applying the nerve functor, we obtain a simplicial categ ory N C T with ob ject set T 0 . F or a, b ∈ T 0 , an n - s implex in N C T ( a, b ) can be s e en as a flag of sets − → T = T 0 ⊆ T 1 ⊆ · · · ⊆ T n , where J T ⊆ T 0 and T n ⊆ V T . Prop ositio n 3 .7. L et T b e a ne cklac e. Ther e is a natur al isomorphism of simplicial c ate gories b etwe en C ( T ) and N C T . Pr o of. W rite T = B 1 ∨ B 2 ∨ · · · ∨ B k , where the B i ’s are the b eads of T . Then C ( T ) = C ( B 1 ) ∐ C ( ∗ ) C ( B 2 ) ∐ C ( ∗ ) · · · ∐ C ( ∗ ) C ( B k ) since C preserves colimits. Note that C ( ∗ ) = C (∆ 0 ) = ∗ , the categ ory with o ne ob ject and a single morphism (the ident it y). Note that we have isomorphisms C ( B i ) ∼ = N C B i by Lemma 2.5. W e there fore get maps of catego ries C ( B i ) → N C B i → N C T , and it is rea dily c heck e d these extend to a ma p f : C ( T ) → N C T . T o see that this functor is an isomorphism, it suffices to show that it is fully faithful (as it is c le arly a bijection on o b jects). F or any a, b ∈ T 0 we will construct a n in v erse to the map f : C ( T )( a, b ) → N C T ( a, b ), when b > a (the ca s e b ≤ a be ing obvious). Let B r and B s be the bea ds co n taining a and b , resp ectively (if a (r esp. b ) is a join t, let B r (resp. B s ) b e the latter (resp. for mer ) of the t w o b eads which contain it). Let j r , j r +1 . . . , j s +1 denote the o rdered elemen ts of J T ( a, b ), indexed so that j i and j i +1 lie in the b ea d B i ; note that j r = a and j s +1 = b . An y simplex x ∈ N C T ( a, b ) n can b e uniquely written a s the comp osite of n - simplices x s ◦ · · · ◦ x r , where x i ∈ N C T ( j i , j i +1 ) n . Now j i and j i +1 are v ertices within the same bead B i of T , therefore x i may b e rega rded as an n -simplex in C ( B i )( j i , j i +1 ). W e then get ass o ciated n -simplices in C ( T )( j i , j i +1 ), and taking their comp os ite gives an n -s implex ˜ x ∈ C ( T )( a, b ). W e define a map g : N C T ( a, b ) → C ( T )( a, b ) by sending x to ˜ x . One r eadily chec ks that this is well-defined and compatible with the simplicial o p er ators, and it is a ls o clear that f ◦ g = id. T o see that f is an isomorphis m it suffices to now show that g is sur jective. But upo n ponder ing colimits of categor ies, it is clea r tha t every map in C ( T )( a, b ) can be written as a comp osite of maps from the C ( B i )’s. It follows at o nce that g is surjective.  Corollary 3. 8. L et T = B 0 ∨ B 1 ∨ · · · ∨ B k b e a ne cklac e. L et a, b ∈ T 0 b e su ch that a < b . L et j r , j r +1 , . . . , j s +1 b e the elements of J T ( a, b ) (in or der), and let B i denote the b e ad c ontaining j i and j i +1 , for r ≤ i ≤ s . Then the m ap C ( B s )( j s , j s +1 ) × · · · × C ( B r )( j r , j r +1 ) → C ( T )( a, b ) is an isomorph ism. Ther efo r e C ( T )( a, b ) ∼ = (∆ 1 ) N wher e N = | V T ( a, b ) − J T ( a, b ) | . In p articular, C ( T )( a, b ) is c ont r actible if a ≤ b and empty otherwise. 10 DANIEL DUGGER AND DA VID SPIV AK Pr o of. F ollows a t once from the pre v ious lemma.  Remark 3.9. Given a necklace T , there is a heur istic wa y to understand faces (b oth co dimension one and higher) in the cubes C ( T )( a, b ) in terms o f “paths” fro m a to b in T . T o cho o se a face in C ( T )( a, b ), o ne cho oses three subsets Y , N , M ⊂ V T ( a, b ) which cover the set V T ( a, b ) and are mutually disjoint. The set Y is the set of vertices which we req uire our path to go through – it must contain J T ( a, b ); the set N is the set of vertices which we require our path to not go throug h; and the set M is the s et of vertices for which we leav e the question op en. Such choices determine a unique face in C ( T )( a, b ). The dimension of this face is precisely the num ber of vertices in M . 4. The ca tegorifica tion functor By this p oint, we fully understand C (∆ n ) as a simplicial categ ory . Recall that C : s S et → s C at is defined for S ∈ s S et by the for m ula C ( S ) = colim ∆ n → S C (∆ n ) . The trouble with this for mula is that g iven a diag ram X : I → s C at of simplicial categorie s, it is generally quite difficult to understa nd the mapping spaces in the colimit. In our case, howev er, something sp ecial happ ens b ecause the simplicial categorie s C (∆ n ) ar e “directed” in a certain s ense. It tur ns out by making use of necklaces one can write down a precise desc ription o f the mapping spa ces for C ( S ); this is the goal of the present section. Fix a simplicial set S and elemen ts a, b ∈ S 0 . F o r a ny nec klace T and map T → S a,b , there is a n induced map C ( T )( α, ω ) → C ( S )( a, b ). Let ( N ec ↓ S ) a,b denote the category whose ob jects ar e pairs [ T , T → S a,b ] and whose morphisms are maps of nec klaces T → T ′ giving commutativ e triangles ov er S . Then we obtain a map colim T → S ∈ ( N ec ↓ S ) a,b h C ( T )( α, ω ) i − → C ( S )( a, b ) . (4.1) Let us write E S ( a, b ) for the domain of this map. Note tha t there are compo sition maps E S ( b, c ) × E S ( a, b ) − → E S ( a, c ) (4.2) induced in the following w ay . Given T → S a,b and U → S b,c where T and U ar e necklaces, o ne obtains T ∨ U → S a,c in the eviden t manner . The comp osite C ( U )( α U , ω U ) × C ( T )( α T , ω T ) / / C ( T ∨ U )( ω T , ω U ) × C ( T ∨ U )( α T , ω T ) µ   C ( T ∨ U )( α T , ω U ) induces the pairing of (4.2). One readily chec ks that E S is a simplicial catego r y with o b ject set S 0 , and (4.1) yields a map o f simplicial catego ries E S → C ( S ). Moreov er, the construction E S is clea rly functorial in S . Here is our first r esult: Prop ositio n 4. 3. F or every simplicial set S , the map E S → C ( S ) is an isomor- phism of simplicia l c ate gori es. RIGIDIFICA TION OF QUASI-CA TEGORIES 11 Pr o of. First note that if S is itself a necklace then the iden tit y map S → S is a terminal ob ject in ( N ec ↓ S ) a,b . It follows at o nce that E S ( a, b ) → C ( S )( a, b ) is an isomorphism for all a and b . Now let S b e a n arbitrary simplicial set, and c ho o se v ertices a , b ∈ S 0 . W e will show that E S ( a, b ) → C ( S )( a, b ) is a bijection. Consider the comm utativ e diagra m of simplicial sets  colim ∆ k → S E ∆ k  ( a, b ) t / / ∼ =   E S ( a, b )    colim ∆ k → S C (∆ k )  ( a, b ) C ( S )( a, b ) . The b ottom eq uality is the definition of C . The left-hand map is an iso morphism by our rema r ks in the first paragr aph. It follows that the top map t is injective. T o complete the proo f it therefo re suffices to show that t is s urjective. Cho ose an n -simplex x ∈ E S ( a, b ) n ; it is repr esented b y a necklace T , a ma p f : T → S a,b , and an element ˜ x ∈ C ( T )( α, ω ). W e hav e a commut ative dia gram  colim ∆ k → T C (∆ k )  ( α, ω ) / / C ( T )( α, ω ) (colim ∆ k → T E ∆ k ) ( α, ω ) / / O O f   E T ( α, ω ) O O E f   (colim ∆ k → S E ∆ k ) ( a, b ) t / / E S ( a, b ) . The n -simplex in E T ( α, ω ) repr e sented b y [ T , id T : T → T ; ˜ x ] is sent to x under E f . It suffices to show that the middle horizo nt al ma p is surjective, for then x will be in the image of t . But the top ma p is an isomor phism, and the vertical arrows in the top row are isomorphisms by the remar ks fr om the first pa r agra ph. Th us, we are done.  Corollary 4.4. F or any simplicial set S and element s a, b ∈ S 0 , the simplicial set C ( S )( a, b ) admits the fol lowing description. An n -simplex in C ( S )( a, b ) c onsists of an e quivalenc e class of triples [ T , T → S, − → T ] , wher e • T is a n e cklac e; • T → S is a map of simplicial sets which sends α T to a and ω T to b ; and • − → T is a fl ag of sets T 0 ⊆ T 1 ⊆ · · · ⊆ T n such that T 0 c ontains t he joints of T and T n is c ontaine d in the set of vertic es of T . The e quivale nc e r elatio n is gener ate d by c onsidering ( T → S ; − → T ) and ( U → S ; − → U ) to b e e quival ent if ther e exists a map of ne cklac es f : T → U over S with − → U = f ∗ ( − → T ) . The i th fac e (r esp. de gener acy) map omits (r esp. r ep e ats) the set T i in the flag. That is, if x = ( T → S ; T 0 ⊆ · · · ⊆ T n ) r epr esen t s an n -simplex of C ( S )( a, b ) and 0 ≤ i ≤ n , t hen s i ( x ) = ( T → S ; T 0 ⊆ · · · ⊆ T i ⊆ T i ⊆ · · · ⊆ T n ) and d i ( x ) = ( T → S ; T 0 ⊆ · · · ⊆ T i − 1 ⊆ T i +1 ⊆ · · · ⊆ T n ) . 12 DANIEL DUGGER AND DA VID SPIV AK Pr o of. This is a stra ightf orward interpretation of the c o limit app ear ing in the def- inition of E S from (4.1). Recall that ev ery colimit ca n b e wr itten as a co equalizer colim T → S ∈ ( N ec ↓ S ) a,b h C ( T )( α, ω ) i ∼ = co eq h a T 1 → T 2 → S C ( T 1 )( α, ω ) ⇒ a T → S C ( T )( α, ω ) i , and that ele ments of C ( T ) ar e identified with flag s of subsets of V T , cont aining J T , by Lemma 3.7.  Our next goal is to simplify the equiv a lence r elation appear ing in Cor ollary 4.4 somewhat. This analysis is somewhat cumbers ome, but culminates in the impor tant Prop ositio n 4 .10. Let us b egin by introducing some terminology . A flagge d nec klace is a pair [ T , − → T ] where T is a necklace and − → T is a flag of subsets of V T which all contain J T . The length o f the flag is the n um be r o f s ubset symbols, or one less than the nu m be r of subsets. A morphism of flag ged necklaces [ T , − → T ] → [ U, − → U ] exists o nly if the flags hav e the sa me leng th, in which case it is a map of necklaces f : T → U such that f ( T i ) = U i for all i . Finally , a flag − → T = ( T 0 ⊆ · · · ⊆ T n ) is called flanked if T 0 = J T and T n = V T . Note that if [ T , − → T ] and [ U, − → U ] are both flanked, then every morphism [ T , − → T ] → [ U, − → U ] is surjective (b ecause its image will b e a subnecklace of U having the same join ts and vertices as U , hence it must be all of U ). Lemma 4. 5. U nder the e quivalenc e r elation of Cor ol lary 4.4, e ach of the triples [ T , T → S, − → T ] is e quivale nt to one in which the flag is fl anke d. Mor e over, two flanke d triples ar e e quival ent (in the sense of Cor ol lary 4.4) if and only if they c an b e c onne cte d by a zig-zag of morphisms of flagge d ne cklac es in which every triple of the zig-zag is flanke d. Pr o of. Suppose g iven a flagged necklace [ T , T 0 ⊆ · · · ⊆ T n ]. There is a unique subnecklace T ′ ֒ → T whose set of jo in ts is T 0 and whose vertex set is T n . Then the pair ( T ′ , T 0 ⊆ · · · ⊆ T n ) is flanked. This assignment, which we call flankific ation , is actually functorial: a morphism of flagged necklaces f : [ T , − → T ] → [ U, − → U ] must map T ′ int o U ′ and therefore gives a mor phism [ T ′ , − → T ] → [ U ′ , − → U ]. Using the equiv a lence r elation of Corollar y 4.4, each triple [ T , T → S, − → T ] will b e equiv alent to the flanked triple [ T ′ , T ′ → T → S, − → T ] via the map T ′ → T . If the flanked triple [ U, U → S, − → U ] is equiv alent to the flanked triple [ V , V → S, − → V ] then there is a zig - zag of maps b etw een triples whic h star ts at the first and ends at the second, by Corollar y 4 .4. Applying the flankification functor gives a corre sp o nding zig-zag in which every ob ject is flanked.  Remark 4. 6. By the previous lemma, we ca n alter our mo del for C ( S )( a, b ) so that the n -simplices are equiv alence c la sses of triples [ T , T → S, − → T ] in whic h the flag is flanked, and the equiv alence relatio n is given by maps (which are nece s sarily surjections) of flanked triples. Under this mo del the deg e ne r acies a nd inner faces are giv en by the sa me description as b efore: r ep eating or omitting o ne o f the subsets in the flag. The outer faces d 0 and d n are now mor e complicated, how ever, be c ause omitting the first or last subset in the flag may produce one which is no longer flanked; one must fir st remov e the subset and then apply the flankifica tion functor from Lemma 4.5. This mo del for C ( S )( a, b ) was o riginally shown to us by Jacob Lurie; it will play only a very minor ro le in what follows. RIGIDIFICA TION OF QUASI-CA TEGORIES 13 Our next task will b e to analyze surjections of flag g ed triples . Let T b e a necklace and S a simplicia l set. Say tha t a map T → S is totally nondegenerate if the image of each b ead of T is a nondegenera te simplex of S . No te a totally nondegenera te map need not be an injection: for example, let S = ∆ 1 /∂ ∆ 1 and consider the nondegenerate 1-simplex ∆ 1 → S . Recall that in a simplicial s e t S , if z ∈ S is a degenera te simplex then there is a unique nondege ner ate simplex z ′ and a unique degeneracy o per ator s σ = s i 1 s i 2 · · · s i k such that z = s σ ( z ′ ); see [H, Lemma 15.8.4]. Using this, and the fact that degeneracy oper ators cor r esp ond to surjections o f simplices, one finds that for any map T → S there is a nec klace T , a map T → S which is totally nondegenera te, and a s ur jection of necklaces T → T making the evident triang le commute; mor eov er, these three things are unique up to isomorphism. Prop ositio n 4.7. L et S b e a simplicial set and let a, b ∈ S 0 . (a) Supp ose that T and U ar e ne cklac es, U u − → S and T t − → S ar e two maps, and that t is t otal ly nonde gener ate. Then t her e is at most one surje ction f : U ։ T such that u = t ◦ f . (b) Supp ose that one has a diagr am U g     f / / / / T   V / / S wher e T , U , and V ar e fl agge d ne cklac es, T → S is total ly n onde gener ate, and f and g ar e surje ctions. Then t her e ex ists a unique map of flagge d ne cklac es V → T making the diagr am c ommu t e. Pr o of. W e first make the obser v ation that if A → B is a sur jection of nec klaces and B 6 = ∗ then every be a d of B is surjected on by a unique b e a d of A . Also , each b ead of A is either colla psed onto a joint o f B or else mapp ed surjectively onto a bea d of B . F or (a), note that we may assume T 6 = ∗ (or els e the claim is trivial). Assume there are tw o distinct surjections f , f ′ : U → T such that tf = tf ′ = u . Let B b e the first bead of U on which f and f ′ disagree. Let j denote the initial v ertex of U , a nd let C b e the b ead of T whose initial vertex is f ( j ) = f ( j ′ ). If u ma ps B to a po in t in S then B cannot surject o nto the b ea d C (using that T → S is totally nondegenerate); so B must be colla psed to a p o int by b oth f and f ′ . Alternatively , if u do es not ma p B to a p oint then B must surject o nt o the b ead C via b oth f and f ′ ; this identifies the simplex B → U → S with a degenera cy of the nondegenera te simplex C → S . Then by uniqueness of degenera c ies we ha ve that f a nd f ′ m ust coincide on B , which is a contradiction. Next we turn to part (b). No te that the map V → T will necessa rily b e sur jective, so the uniqueness part is guaranteed by (a); we need only show exis tence . Observe that if B is a b ead in U whic h maps to a point in V then it maps to a p oint in T , b y the r easoning a b ov e. It now follows that there exists a necklace U ′ , obtained by collapsing ev ery bea d of U that maps to a point in V , and a 14 DANIEL DUGGER AND DA VID SPIV AK commutativ e diagr am U f / / / / A A A A A g     T   U ′ f ′ > > > > } } } } } g ′ ~ ~ ~ ~ } } } } } V / / S Replacing U , f , and g by U ′ , f ′ , a nd g ′ , and dropping the primes, we can now as sume that g induces a one-to- one c orresp ondence b etw e e n beads of U and bea ds of V . Let B 1 , . . . , B m denote the b ea ds o f U , a nd let C 1 , . . . , C m denote the bea ds of V . Assume that we hav e constructed the lift l : V → T on the b eads C 1 , . . . , C i − 1 . If the bea d B i is mapp ed by f to a p oint, then evidently we can define l to map C i to this sa me p oint and the diagram will commute. Other wise f maps B i surjectively onto a certain b ead D inside of T . W e hav e the diag r am B i f / / / / g     D t   C i v / / S where here f and g ar e sur jections betw een simplices and therefore represent de- generacy ope r ators s f and s g . W e hav e that s f ( t ) = s g ( v ). But the simplex t of S is no ndegenerate by assumption, therefor e by [H, Lemma 15.8 .4] we m ust have v = s h ( t ) for s ome degener a cy op erator s h such that s f = s g s h . The op erator s h corres p o nds to a surjection of simplices C i → D mak ing the above square com- m ute, and we define l on C i to co incide with this map. Contin uing by inductio n, this pro duces the desir ed lift l . It is eas y to see that l is a map o f flagged necklaces, as l ( V i ) = l ( g ( U i )) = f ( U i ) = T i .  Corollary 4. 8 . L et S b e a simplicial set and a, b ∈ S 0 . Under the e quivalenc e r elation fr om Cor ol lary 4.4, every triple [ T , T → S a,b , − → T ] is e quivalent to a unique triple [ U, U → S a,b , − → U ] which is b oth flanke d and t otal ly nonde gener ate. Pr o of. Let t = [ T , T → S a,b , − → T ]. Then t is clea rly equiv a lent to at lea st one flanked, totally nondegener ate triple b eca use we can r eplace t w ith [ T ′ , T ′ → S a,b , − → T ] (flanki- fication) and then with [ T ′ , T ′ → S a,b , − → T ′ ] (defined above Pro p o sition 4.7). Now suppo s e that [ U, U → S a,b , − → U ] and [ V , V → S a,b , − → V ] are b oth flank ed, totally nondegenerate, a nd equiv alent in C ( S )( a, b ) n . Then b y Lemma 4.5 there is a zig-za g of maps b etw een flanked necklaces (over S ) co nnecting U to V : W 1 | | | | z z z z z z z z     < < < < < < < W 2                < < < < < < < · · ·                ; ; ; ; ; ; ; W k            # # # # H H H H H H H H U = U 1 U 2 U 3 · · · U k U k +1 = V Using Pro po sition 4.7, we inductiv ely construct sur jections of flank ed necklaces U i → U over S . This pro duces a surjection V → U ov er S . Similar ly , we obtain a s urjection U → V ov er S . By P r op osition 4.7(a) these ma ps must b e inv erses of each other; that is, they are isomo rphisms.  RIGIDIFICA TION OF QUASI-CA TEGORIES 15 Remark 4.9. Again, as in Rema rk 4 .6 the a b ove corolla ry shows that we can describ e C ( S )( a, b ) as the simplicial set whose n -simplices are triples [ T , T → S, − → T ] which are bo th flanked and totally nondegenera te. The degenerac ies and inner faces are again easy to descr ib e—they are rep etition or omission of a set in the flag—but for the outer faces one m ust first omit a set and then mo dify the triple appropriately . The us e fulness o f this descriptio n is limited b eca use of these complica tio ns with the outer faces, but it do es make a brief app ear a nce in Cor ollary 4.13 b elow. The following result is the culmination of o ur work in this section, and will turn out to be a key step in the pro of of our ma in theorems. Fix a simplicial set S and vertices a, b ∈ S 0 , and let F n denote the c a tegory of flagged triples ov er S a,b that hav e length n . That is, the ob jects of F n are triples [ T , T → S a,b , T 0 ⊆ · · · ⊆ T n ] and mor phisms a re maps of necklaces f : T → T ′ ov er S such that f ( T i ) = ( T ′ ) i for all i . Prop ositio n 4 . 10. F or e ach n ≥ 0 , the nerve of F n is homotopy discr ete in s S e t K . Pr o of. Recall fr om Lemma 4.5 that there is a functor φ : F n → F n which sends any triple to its ‘flankification’. There is a natur al transformatio n from φ to the ident it y , and the image of φ is the sub catego ry F ′ n ⊆ F n of flanked triples . It will therefore suffice to prov e that (the nerve of ) F ′ n is homotopy dis crete. Recall fr om Coro llary 4.8 tha t every c o mpo nent of F ′ n contains a unique tr iple t which is both flanked and totally nondegenerate. Mor eov er, follo wing the pro of of that cor ollary one sees that every triple in the same co mpo nent as t admits a unique map to t —that is to say , t is a fina l ob ject for its c o mpo nent. There fore its comp onent is con tractible. This completes the pr o of.  4.11. The functor C applied to o rdered simplicial se ts . Note that even if a simplicial se t S is small—say , in the se ns e that it has finitely many nondeg enerate simplices—the spa ce C ( S )( a, b ) may b e quite large. This is due to the fact that there a re infinitely many necklaces mapping to S (if S is nonempt y). F or certain simplicial sets S , how ever, it is po ssible to restr ict to necklaces whic h lie inside of S ; this cuts down the po s sibilities. The following results and subsequent example demonstrate this. R ecall the definition o f o rdered simplicial sets fro m (3.2). Lemma 4. 12. L et D b e an or der e d simplic ial set and let a, b ∈ D 0 . Then every n -simplex in C ( D )( a, b ) is r epr esente d by a unique triple [ T , T → D , − → T ] in which T is a ne cklac e, − → T is a flanke d fl ag of length n , and the map T → D is inje ctive. Pr o of. By Corollar y 4 .8, every n -simplex in C ( D )( a, b ) is repres ent ed b y a unique triple [ T , T → D , − → T ] whic h is b oth flanked a nd totally non- de g enerate. It suffices to show that if D is order ed, then any totally non-degener ate map T → D is injectiv e. This follows from Lemma 3 .5(6).  Corollary 4.13. L et D b e an or der e d simplicial set, and a, b ∈ D 0 . L et M D ( a, b ) denote the simplicial set for which M D ( a, b ) n is the set of triples [ T , T f − → D a,b , − → T ] , wher e f is inje ctive and − → T is a flanke d flag of length n ; fac e and b oundary maps ar e as in Rema rk 4.6. Then ther e is a natu r al isomorphism C ( D )( a, b ) ∼ = − → M D ( a, b ) . 16 DANIEL DUGGER AND DA VID SPIV AK Pr o of. This follows immediately from Lemma 4.12.  Example 4.14. Consider the simplicial set S = ∆ 2 ∐ ∆ 1 ∆ 2 depicted as follows: 1 3 0 2 • • • • ✲ ❄ ❄ ✲    ✒ W e will describ e the ma pping space X = C ( S )(0 , 3) by giving its non- degenerate simplices and face maps. By Lemma 4.12, it suffices to consider flanked nec klaces that inject in to S . There are only fiv e such necklaces that ha ve endp oints 0 a nd 3. These are T = ∆ 1 ∨ ∆ 1 , which maps to S in tw o different wa ys f , g ; a nd U = ∆ 1 ∨ ∆ 1 ∨ ∆ 1 , V = ∆ 1 ∨ ∆ 2 , and W = ∆ 2 ∨ ∆ 1 , each of which maps uniquely into S 0 , 3 . The image of T 0 under f is { 0 , 1 , 3 } a nd under g is { 0 , 2 , 3 } . The images of U 0 , V 0 , and W 0 are all { 0 , 1 , 2 , 3 } . W e find tha t X 0 consists of thre e elements [ T ; { 0 , 1 , 3 } ], [ T ; { 0 , 2 , 3 } ] and [ U ; { 0 , 1 , 2 , 3 } ]. There are tw o nondeg e ne r ate 1-simplices , [ V ; { 0 , 1 , 3 } ⊂ { 0 , 1 , 2 , 3 } ] and [ W ; { 0 , 2 , 3 } ⊂ { 0 , 1 , 2 , 3 } ]. These connect the three 0- s implices in the ob vious wa y , resulting in tw o 1-simplices with a common final vertex. Ther e a r e no higher non-degenera te simplices. Thus C ( S )(0 , 3) lo oks like • • / / o o • . 5. Homotopical models for ca tegorifica tion In the last s ection we gav e a very explicit description of the mapping spaces C ( S )( a, b ), for ar bitrary simplicia l sets S and a, b ∈ S 0 . While this description was explicit, in some wa ys it is not v ery useful from a homotopical standp oint—in practice it is hard to use this des cription to identify the homotopy type of C ( S )( a, b ). In this section w e will discuss a functor C nec : s S et → s C at that ha s a simpler description than C a nd whic h is more ho motopical. W e prove that for any simplicial set S there is a natural zigza g o f weak equiv alences b etw een C ( S ) and C nec ( S ). V ar iants of this constr uction are a lso introduced, leading to a co lle c tion of functor s s S et → s C at all of which are weakly equiv alent to C . Let S ∈ s S et . A choice of a, b ∈ S 0 will b e re g arded as a map ∂ ∆ 1 → S . Let ( N e c ↓ S ) a,b be the ov ercategory for the inclusion functor N ec ֒ → ( ∂ ∆ 1 ↓ S ). Finally , define C nec ( S )( a, b ) = N ( N ec ↓ S ) a,b . This is a simplicial categ ory in a n evident wa y . Remark 5.1. B oth the functor C and the functor C nec hav e distinct adv antages and disadv antages. The main adv an tage to C is that it is left adjoint to the coherent nerve functor N (in fact it is a left Q uillen functor s S et J → s C at ); as such, it preserves colimits. How ever, as mentioned above, the functor C can be difficult to use in pr actice b ecause the mapping spa ces hav e an awkward description. It is at this p o int that our functor C nec bec omes useful, b ecause the mapping spaces are g iven as ner ves o f 1-ca tegories. Man y to ols are av a ilable for deter mining when a mor phism b et ween nerves is a Kan equiv alence. This will b e an imp ortant po int in [DS], where we show the C functor gives a Quillen equiv a le nc e b etw een s S et J and s C at . See also Section 6 b elow. RIGIDIFICA TION OF QUASI-CA TEGORIES 17 Our main theorem is that there is a simple zig z a g o f weak equiv ale nc e s b etw een C ( S ) and C nec ( S ); that is, there is a functor C hoc : s S et → s C at and natural w eak equiv alences C ← C hoc → C nec . W e b egin b y describing the functor C hoc . Fix a simplicial s e t S . Define C hoc ( S ) to have o b ject set S 0 , and for every a, b ∈ S 0 C hoc ( S )( a, b ) = ho colim T ∈ ( N ec ↓ S ) a,b C ( T )( α, ω ) . Note the similarities to Theo rem 5 .2, where it was s hown that C ( S )( a, b ) ha s a similar description in which the ho colim is r e pla ced by the colim. In our definition of C hoc ( S )( a, b ) we mea n to use a pa rticular model for the ho motopy co limit, namely the diag o nal of the bis implicia l set whose ( k , l )-simplices are pairs ( F : [ k ] → ( N ec ↓ S ) a,b ; x ∈ C ( F (0 ))( α, ω ) l ) , where F (0) deno tes the necklace obtained b y a pplying F to 0 ∈ [ k ] and then applying the for getful functor ( N ec ↓ S ) a,b → N ec . The compo s ition law for C hoc is defined just as for the E S construction fro m Section 4. W e pr o ceed to establish natural transformations C hoc → C nec and C hoc → C . Note tha t C nec ( S )( a, b ) is the homoto py colimit of the cons tant functor {∗ } : ( N ec ↓ S ) a,b → s S et which s ends everything to a p oint. The map C hoc ( S )( a, b ) → C nec ( S )( a, b ) is the map of homotopy colimits induced by the evident map of dia- grams. Since the spaces C ( T )( α, ω ) are all con tractible simplicial sets (see Co rol- lary 3.8), the induced map C hoc ( S )( a, b ) → C nec ( S )( a, b ) is a Kan equiv alence. W e th us obtain a natural weak equiv alence of simplicial categ ories C hoc ( S ) → C nec ( S ). F or any dia g ram in a mo del categor y there is a canonical natural transfor ma- tion from the homotopy co limit to the colimit of that diagr am. Hence there is a morphism C hoc ( S )( a, b ) → colim T ∈ ( N ec ↓ S ) a,b C ( T )( α, ω ) ∼ = C ( S )( a, b ) . (F or the isomorphism we are using Prop os itio n 4.3.) As this is natural in a, b ∈ S 0 and natural in S , we have a natural trans formation C hoc → C . Theorem 5.2. F or every simplicia l set S , the maps C ( S ) ← C hoc ( S ) → C nec ( S ) define d ab ove ar e we ak e quivalenc es of simplicial c ate gorie s. Pr o of. W e hav e alr eady establishe d that the natural tra nsformation C hoc → C nec is a n ob ject wise equiv a lence, so it suffices to show that for each simplicial set S and ob jects a, b ∈ S 0 the natura l map C hoc ( S )( a, b ) → C ( S )( a, b ) is als o a Ka n equiv alence. Recall that C hoc ( S )( a, b ) is the diagonal of a bisimplicial set whose l th ‘horizontal’ row is the nerve N F l of the category of flagg ed necklaces ma pping to S , where the flags hav e length l . Also reca ll from Corollary 4.4 that C ( S )( a, b ) is the simplicial set which in level l is π 0 ( N F l ). But Pr op osition 4.1 0 says that N F l → π 0 ( N F l ) is a Kan equiv alence, for every l . It follows that C hoc ( S )( a, b ) → C ( S )( a, b ) is also a Kan equiv ale nc e .  5.3. Other mo dels for categorification. One can imagine v aria tions of our bas ic construction in which one replaces ne cklaces with other co nv enien t simplicia l s ets— which we mig h t term “g adgets”, for la ck of a b etter word. W e will see in Sec tio n 6, for instance, that us ing pr o du ct s of necklaces leads to a nice theo rem ab out the categorific a tion of a pro duct. In [DS] several key ar guments will hinge on a clever 18 DANIEL DUGGER AND DA VID SPIV AK choice o f wha t gadg ets to use. In the materia l b elow we give some basic r equirements of the “ga dg ets” which will ensur e they give a model equiv alent to that of necklaces. Suppo se P is a sub catego ry of s S e t ∗ , ∗ = ( ∂ ∆ 1 ↓ s S e t ) cont aining the terminal ob ject. F or a ny simplicia l set S and vertices a, b ∈ S 0 , let ( P ↓ S ) a,b denote the ov ercategory whose o b jects are pairs [ P , P → S ], where P ∈ P and the map P → S sends α P 7→ a and ω P 7→ b . Define C P ( S )( a, b ) = N ( P ↓ S ) a,b . The ob ject C P is simply a n assignment whic h takes a simplicial set S with t w o distinguished vertices and pro duces a “ P -mapping spac e.” How ev er, if P is clos e d under the wedge op era tion (i.e. for any P 1 , P 2 ∈ P one has P 1 ∨ P 2 ∈ P ), then C P may b e g iven the structure of a functor s S et → s C at in the evident w ay . Definition 5. 4. We c al l a sub c ate gory G ⊆ s S et ∗ , ∗ a c ate gor y of gadgets if it satisfies the fol lowing pr op ert ies: (1) G c ontains the c ate gory N ec , (2) F or every obje ct X ∈ G and every ne cklac e T , al l maps T → X ar e c ontaine d in G , and (3) F or any X ∈ G , the simplicial set C ( X )( α, ω ) is c ontr actib le. The c ate gory G is said to b e close d under w e dges if it is also tru e that (4) F or any X, Y ∈ G , the we dge X ∨ Y also b elongs to G . The above definition can be gener alized somewha t by allowing N ec → G to b e an arbitrary functor ov er a natural transfor mation in s S et ; we do not need this generality in the pres ent pap er. Prop ositio n 5.5. L et G b e a c ate gory of gadgets. Then for any simplicial set S and any a, b ∈ S 0 , the natur al map C nec ( S )( a, b ) − → C G ( S )( a, b ) (induc e d by the inclusion N ec ֒ → G ) is a Kan e quiva lenc e. If G is close d u n der we dges then the map of simplicial c ate gories C nec ( S ) → C G ( S ) is a we ak e quivalenc e. Pr o of. Let j : ( N ec ↓ S ) a,b → ( G ↓ S ) a,b be the functor induced by the inclusion map N ec ֒ → G . The map in the statement of the prop osition is just the nerve of j . T o verify that it is a Kan equiv a lence, it is enoug h b y Quillen’s Theorem A [Q] to v erify that all the ov ercategor ies of j are contractible. So fix an ob ject [ X , X → S ] in ( G ↓ S ) a,b . The overcategory ( j ↓ [ X, X → S ]) is precisely the category ( N ec ↓ X ) α,ω , the nerve o f whic h is C nec ( X )( α, ω ). B y Theorem 5.2 and our assumptions o n G , this is contractible. The sec o nd statement o f the result is a direct conseque nc e of the first.  6. Prope r ties of ca tegorifica tion In this s ection we establis h tw o main prop erties of the c a tegorifica tion functor C . First, we pr ov e tha t there is a natural weak equiv alence C ( X × Y ) ≃ C ( X ) × C ( Y ). Second, we pr ov e that whenev er S → S ′ is a Joy a l equiv alence it follo ws that C ( S ) → C ( S ′ ) is a weak equiv alence in s C at . These prop erties are also pr ov en in [L], but the pr o ofs we g ive here are of a different natur e and make central use of the C nec functor. RIGIDIFICA TION OF QUASI-CA TEGORIES 19 If T 1 , . . . , T n are nec klaces then they a re, in particular, ordere d simplicial sets in the sense of Definition 3.2. So T 1 × · · · × T n is also order e d, b y L e mma 3.5. Let G b e the full sub categor y o f s S et ∗ , ∗ = ( ∂ ∆ 1 ↓ s S e t ) whose ob jects are pro ducts of necklaces with a map f : ∂ ∆ 1 → T 1 × · · · × T n that has f (0)  f (1). Prop ositio n 6.1. The c ate gory G is a c ate gory of gadgets in the sense of Defini- tion 5.4. F or the pro o f o f this one needs to verify that C ( T 1 × · · · × T n )( α, ω ) ≃ ∗ . This is not difficult, but is a bit of a distraction; we prove it later as Prop os ition A.4. Prop ositio n 6 .2. F or any simplicial sets X and Y , b oth C ( X × Y ) and C ( X ) × C ( Y ) ar e simplici al c ate gories with obje ct set X 0 × Y 0 . F or any a 0 , b 0 ∈ X and a 1 , b 1 ∈ Y , the natur al map C ( X × Y )( a 0 a 1 , b 0 b 1 ) → C ( X )( a 0 , b 0 ) × C ( Y )( a 1 , b 1 ) induc e d by C ( X × Y ) → C ( X ) and C ( X × Y ) → C ( Y ) is a Kan e quivalenc e. Con- se quent ly, t he map of simplicia l c ate gories C ( X × Y ) → C ( X ) × C ( Y ) is a we ak e quivalenc e in s C at . Pr o of. Let G denote the ab ov e categ ory of gadgets, in which the o b jects ar e pro ducts of nec klaces. By Theorem 5.2 a nd Prop osition 5.5 it suffices to prove the re sult fo r C G in place of C . Consider the functors ( G ↓ X × Y ) a 0 a 1 ,b 0 b 1 φ / / ( G ↓ X ) a 0 ,b 0 × ( G ↓ Y ) a 1 ,b 1 θ o o given by φ : [ G, G → X × Y ] 7→  [ G, G → X × Y → X ] , [ G, G → X × Y → Y ]  and θ :  [ G, G → X ] , [ H , H → Y ]  7→ [ G × H , G × H → X × Y ] . Note that we are using that the subca tegory G is closed under finite pro ducts. It is very easy to see that there is a natural transformatio n id → θ φ , obtained by using diag onal ma ps , a nd a natural transformation φθ → id, obtained b y us ing pro- jections. As a consequence, the maps θ and φ induce inv erse homotopy equiv alences on the ne r ves. This completes the pro o f.  Let E : S et → s S et denote the 0-coskeleton functor (see [AM ]). F or any simplicial set X and set S , we hav e s S et ( X, E S ) = S et ( X 0 , S ). In particula r, if n ∈ N we denote E n = E { 0 , 1 , . . . , n } . Lemma 6.3 . F or any n ≥ 0 , the simplicia l c ate gory C ( E n ) is c ont r actible in s C at — that is to say, al l the mapping sp ac es in C ( E n ) ar e c ontr actible. Pr o of. By Theor em 5.2 it is sufficien t to prov e that the ma pping space C nec ( E n )( i, j ) is contractible, for every i, j ∈ { 0 , 1 , . . . , n } . This mapping spa ce is the nerve of the ov ercategory ( N ec ↓ E n ) i,j . 20 DANIEL DUGGER AND DA VID SPIV AK Observe that if T is a necklace then an y map T → E n extends uniquely ov er ∆[ T ]. This is b ecaus e maps into E n are determined b y wha t they do on the 0 - skeleton, and T ֒ → ∆[ T ] is an iso morphism on 0-skeleta. Consider tw o functors f , g : ( N ec ↓ E n ) i,j → ( N ec ↓ E n ) i,j given by f : [ T , T x − → E n ] 7→ [∆[ T ] , ∆[ T ] ¯ x − → E n ] and g : [ T , T x − → E n ] 7→ [∆ 1 , ∆ 1 z − → E n ] . Here ¯ x is the unique extension of x to ∆[ T ], and z is the unique 1-simplex o f E n connecting i to j . Observe that g is a cons ta nt functor. It is easy to see that there ar e natural tra nsformations id → f ← g . The functor g facto rs through the termina l catego ry {∗} , s o after taking ner ves the identit y map is null homotopic. Hence ( N ec ↓ E n ) i,j is co nt ractible.  F or completeness (and b ecause it is s hort) we include the following lemma, es- tablished in [L, Pro o f o f 2.2 .5.1]: Lemma 6. 4. The functor C : s S et → s C at takes monomorphism to c ofibr ations. Pr o of. Ev ery cofibration in s S e t is obtaine d b y co mpo sitions and cobase changes from bo undary inclusions of simplices. It ther efore suffices to show that for each n ≥ 0 the map f : C ( ∂ ∆ n ) → C (∆ n ) is a cofibra tion in s C at . Let 0 ≤ i, j ≤ n . If i > 0 o r j < n then ( N ec ↓ ∆ n ) i,j ∼ = ( N ec ↓ ∂ ∆ n ) i,j , where b y f ( i, j ) : C ( ∂ ∆ n )( i, j ) → C (∆ n )( i, j ) is an iso morphism by P rop osition 4.3. F or the remaining cas e i = 0 , j = n , the map f (0 , n ) is the inclusion of the b o unda ry of a cub e b : ∂ ((∆ 1 ) n − 1 ) → (∆ 1 ) n − 1 . Let U : s S et → s C at denote the functor which sends a simplicial set S to the unique simplicial categor y U ( S ) w ith tw o ob jects x, y and mor phisms Hom( x, x ) = Hom( y , y ) = {∗} , Ho m( y , x ) = ∅ , and Hom( x, y ) = S . In view of the genera ting cofibrations for s C at (see [B]), it is e a sy to show that U preserves cofibrations. Hence U ( b ) is a cofibration. Notice that f is the pushout o f U ( b ) along the obvious map U [ ∂ ((∆ 1 ) n − 1 )] → C ( ∂ ∆ n ) s ending x 7→ 0 and y 7→ n . Thus, f is a cofibra tion.  Lemma 6. 5. If D is a simplicia l c ate gory t hen N D is a quasi-c ate gory. Pr o of. By adjo intness it suffices to show that e a ch C ( j n,k ) is an acyclic cofibr ation in s C at , where j n,k : Λ n k ֒ → ∆ n is an inner horn inclusion (0 < k < n ). It is a cofibration b y Lemma 6.4, so we m ust o nly verify that it is a weak equiv a lence. Just a s in the proo f of (6.4) a bove, C (Λ n k )( i, j ) → C (∆ n )( i, j ) is an isomorphism unless i = 0 and j = n . It only r emains to s how that C (Λ n k )(0 , n ) → C (∆ n )(0 , n ) is a Kan eq uiv alence.. An a nalysis as in Example 4.14 identifies C (Λ n k )(0 , n ) with the result o f removing one face from the b oundary of (∆ 1 ) n − 1 , which clear ly has the same homoto p y t ype as the cube (∆ 1 ) n − 1 .  Prop ositio n 6.6. If S → S ′ is a map of simplicial sets which is a Joyal e quival enc e then C ( S ) → C ( S ′ ) is a we ak e quivalenc e of simplicial c ate gories. RIGIDIFICA TION OF QUASI-CA TEGORIES 21 Pr o of. F or any simplicial set X , the map C ( X × E n ) → C ( X ) induced b y pro jection is a w eak eq uiv alence in s C at . This follows by com bining Prop osition 6 .2 with Lemma 6.3: C ( X × E n ) ∼ − → C ( X ) × C ( E n ) ∼ − → C ( X ) . Since X ∐ X ֒ → X × E 1 is a cofibratio n in s S et , C ( X ) ∐ C ( X ) = C ( X ∐ X ) → C ( X × E 1 ) is a cofibra tion in s C at , b y Lemma 6.4. It follows that C ( X × E 1 ) is a cylinder ob ject for C ( X ) in s C at . So if D is a fibr ant simplicial ca tegory we may compute homotopy classes of maps [ C ( X ) , D ] as the co equalizer co eq  s C at ( C ( X × E 1 ) , D ) ⇒ s C at ( C ( X ) , D )  . But using the adjunction, this is isomo rphic to co eq  s S et ( X × E 1 , N D ) ⇒ s S et ( X , N D )  . The ab ov e co equalizer is [ X , N D ] E 1 , and w e hav e identified [ C ( X ) , D ] ∼ = [ X , N D ] E 1 . (6.7) Now let S → S ′ be a Joyal equiv a le nce. T he n C ( S ) → C ( S ′ ) is a map b etw een cofibrant ob jects of s C at . T o prov e that it is a weak equiv alenc e in s C at it is sufficient to prov e that the induced map on homotopy classes [ C ( S ′ ) , D ] → [ C ( S ) , D ] is a bijection, for every fibran t ob ject D ∈ s C a t . Since N D is a quasi- categor y by Lemma 6.5 and S → S ′ is a Joy al equiv alence, w e hav e that [ S ′ , N D ] E 1 → [ S, N D ] E 1 is a bijection; the res ult then follo ws b y (6.7).  Remark 6.8. In fa ct it tur ns out that a map of simplicial sets S → S ′ is a Joy al equiv alence if and only if C ( S ) → C ( S ′ ) is a weak equiv alence of simplicial categorie s. This was prov en in [L], and will b e reprov en in [DS] using an extension of the metho ds from the present pap er. Appendix A. Leftover pr oofs In this section we give t w o pr o ofs whic h w ere pos tp oned in the b o dy o f the paper . A.1. Products of nec klaces. Our fir st goal is to prov e Prop os ition 6.1. Let T 1 , . . . , T n be necklaces, and co ns ider the pro duct X = T 1 × · · · × T n . The main thing we need to prove is that w hene ver a  X b in X the mapping space C ( X )( a, b ) ≃ ∗ is contractible. Definition A.2. An or der e d simplicia l set ( X ,  ) is c al le d str ongly or der e d if, for al l a  b in X , t he mapping sp ac e C ( X )( a, b ) is c ontr actible. Note that in a ny ordered simplicial set X with a , b ∈ X 0 , w e have a  b if and only if C ( X )( a, b ) 6 = ∅ . Thu s if X is strongly ordered then its structure as a simplicial category , up to w eak equiv alence, is completely determined b y the ordering on its vertices. W e also po int out that every necklace T ∈ N ec is strongly ordered by Corollar y 3.8. 22 DANIEL DUGGER AND DA VID SPIV AK Lemma A. 3 . Supp ose given a diagr am X f ← − A g − → Y wher e X , Y , and A ar e stro ngly or der e d simplicial sets and b oth f and g ar e simple inclusions. L et B = X ∐ A Y and assum e the fol lowing c onditions hold: (1) A has finitely many vertic es; (2) Give n any x ∈ X , the set A x  = { a ∈ A | x  B a } has an initial element (an element which is smal ler than every other element). (3) F or any y ∈ Y and a ∈ A , if y  Y a then y ∈ A . Then B is stro ngly or der e d. Pr o of. By Lemma 3.5(8), B is an order ed simplicial set and the maps X ֒ → B and Y ֒ → B are simple inclus io ns. W e must show that for u, v ∈ B 0 with u  v , the mapping space C ( B )( u, v ) is contractible. Supp ose that u and v ar e both in X ; then since X ֒ → B is simple, any necklace T → B u,v m ust factor throug h X . It follows that C ( B )( u, v ) = C ( X )( u, v ), whic h is contractible since X is strongly ordered. The case u, v ∈ Y is ana logous. W e cla im we cannot have u ∈ Y \ A and v ∈ X \ A . F or if this is so and if T → B is a spine connecting u to v , then there is a last v ertex j of T that maps into Y . The 1-simplex leaving that vertex then cannot be long entirely to Y , hence it b elongs entirely to X . So j is in b oth X and Y , and hence it is in A . Then we hav e u  j and j ∈ A , which by assumption (3) implies u ∈ A , a co nt radiction. The only re maining c a se to a nalyze is when u ∈ X and v ∈ Y \ A . Consider the po set A 0 of vertices of A , under the r elation  . Let P denote the collection of linearly o rdered subse ts S of A 0 having the pr op erty tha t u  a  v for a ll a ∈ S . That is, each element of P is a chain u  a 1  · · ·  a n  v where each a i ∈ A . W e regard P as a categ ory , where the ma ps ar e inclusions. Also let P 0 denote the sub c ategory of P consisting of a ll subsets except ∅ . Define a functor D : P op → C at by sending S ∈ P to { [ T , T ֒ → B u,v ] | S ⊆ J T } , the full sub categor y o f ( N ec ↓ B ) u,v spanned by ob jects T m − → B u,v for whic h m is an injection and S ⊆ J T . Let us adopt the notation M S ( u, v ) = co lim T ∈ D ( S ) C ( T )( α, ω ) . Note that ther e is a natural map M ∅ ( u, v ) − → colim T ∈ ( N ec ↓ S ) u,v C ( T )( α, ω ) ∼ = C ( B )( u, v ) . The first ma p is not a priori an isomorphism b ecause in the definition of D ( ∅ ) we require that the map T → B b e an injection. How ever, using Lemma 4.12 (or Corollar y 4 .13) it follows at once that the ma p actua lly is an isomorphism. W e claim that for each S in P 0 the “la tc hing” map L S : colim S ′ ⊃ S M S ′ ( u, v ) → M S ( u, v ) is an injection, wher e the colimit is o ver sets S ′ ∈ P whic h strictly co ntain S . T o see this, supp ose that one has a tr iple [ T , T ֒ → B u,v , t ∈ C ( T )( α, ω ) n ] giving an n -simplex of M S ′ ( u, v ) and a nother triple [ T ′ , T ′ ֒ → B u,v , t ′ ∈ C ( U )( α, ω ) n ] giving an n -simplex of M S ′′ ( u, v ). If these b ecome iden tical in M S ( u, v ) then it must be RIGIDIFICA TION OF QUASI-CA TEGORIES 23 that they hav e the sa me flankification ¯ T = ¯ U and t = t ′ . Note that every join t of T is a joint o f ¯ T , so the joints of ¯ T include b oth S ′ and S ′′ . Because the jo ints of any necklace ar e linearly ordere d, it follows that S ′ ∪ S ′′ is linearly or dered. Since T → ¯ T is an injection, we may consider the triple [ ¯ T , ¯ T ֒ → B u,v , t ] as an n - simplex in M S ′ ∪ S ′′ ( u, v ), which maps to the tw o orig inal triples in the colimit; this proves injectivit y . W e cla im that the latching map L ∅ : colim S ∈ P op 0 M S ( u, v ) → M ∅ ( u, v ) is an isomorphism. Injectivit y w as established above. F or surjectivity , o ne needs to prov e that if T is a ne cklace and T ֒ → B u,v is an inclusion, then T m ust cont ain at least one vertex of A as a joint. T o see this, recall that every simplex of B either lies entirely in X or entirely in Y . Since v / ∈ X , there is a last joint j 1 of T which maps into X . If C denotes the b ea d whos e initial v ertex is j 1 , then the image of C c an not lie ent irely in X ; so it lies entirely in Y , which means that j 1 belo ngs to bo th X and Y —hence it be lo ngs to A . F ro m here the argument pro ceeds as follo ws. W e will show: (i) The na tural map ho co lim S ∈ P op 0 M S ( u, v ) → co lim S ∈ P op 0 M S ( u, v ) is a Kan equiv alence; (ii) Each M S ( u, v ) is contractible, hence the ab ov e homoto p y colimit is Kan equiv- alent to the nerve of P op 0 ; (iii) The nerve of P 0 (and hence also P op 0 ) is co ntractible. This will prov e that M ∅ ( u, v ) = C ( B )( u, v ) is contractible, as desir ed. F or (i) we refer to [D, Section 13] and use the fact that P op 0 has the structure of a directed Reedy catego ry . Indeed, w e can assig n a degree function to P that sends a set S ⊆ A 0 to the nonnega tive integer | A 0 − S | ; all non-identit y morphisms in P op 0 strictly increase this degree. By [D, P rop osition 1 3.3] (but with T op replac e d by s S e t ) it is enoug h to show that all the latching maps L S are co fibr ations, and this has already b een es tablished ab ov e. F or claim (iii), wr ite θ fo r the initial vertex of A u  . Define a functor F : P 0 → P 0 by F ( S ) = S ∪ { θ } ; note tha t S ∪ { θ } will be linearly ordered, so this makes sense. Clearly there is a natural transfor mation from the iden tit y functor to F , and a lso from the constant { θ } functor to F . It r eadily follows that the iden tit y map on N P 0 is homotopic to a co nstant map, he nc e N P 0 is contractible. Finally , for (ii) fix some S ∈ P 0 and let u = a 0 ≺ a 1 ≺ . . . ≺ a n ≺ a n +1 = v denote the co mplete set of elements of S ∪ { u, v } . A nec klace T ֒ → B u,v whose joint s include the elements of S can b e split along the joints, and thus uniquely written as the wedge of necklaces T i ֒ → B a i ,a i +1 , one for ea ch 0 ≤ i ≤ n . Under this identification, one has C ( T )( α, ω ) ∼ = C ( T 0 )( α 0 , ω 0 ) × · · · × C ( T n )( α n , ω n ) . Thu s D ( p ) is isomorphic to the catego r y ( N ec ↓ m X ) u,a 1 × ( N ec ↓ m A ) a 1 ,a 2 × · · · × ( N ec ↓ m A ) a n − 1 ,a n × ( N ec ↓ m Y ) a n ,v , where ( N ec ↓ m X ) s,t denotes the categor y whose ob jects ar e [ T , T → X s,t ] where the map T → X is a monomo r phism. Now, it is a gene r al fact ab out colimits taken in the categ ory of (simplicial) sets, that if M i is a catego ry and F i : M i → s S et is a functor, for each i ∈ { 1 , . . . , n } , 24 DANIEL DUGGER AND DA VID SPIV AK then there is an iso morphism of s implicial sets colim M 1 ×···× M n ( F 1 × · · · × F n ) ∼ = − →  colim M 1 F 1  × · · · ×  colim M n F n  . (A.2.2) Applying this in our ca se, we find that M S ( u, v ) ∼ = C ( X )( u, a 1 ) × C ( A )( a 1 , a 2 ) × · · · × C ( A )( a n − 1 , a n ) × C ( Y )( a n , v ) . Note that this is alwa ys contractible, since X , A , a nd Y a re strongly ordered. This prov es (ii) and completes the argument.  Prop ositio n A.4. L et T 1 , . . . , T m b e ne cklac es. Then t heir pr o duct T 1 × · · · × T m is a str ongly or der e d simplicial set. Pr o of. W e b egin with the case P = ∆ n 1 × · · · × ∆ n m , where each necklace is a simplex, a nd s how that P is stro ngly order ed. It is ordered by Lemma 3.5, so choose vertices a, b ∈ P 0 with a  b . If T is a necklace, any map T → ∆ j extends uniquely to a ma p ∆[ T ] → ∆ j . It follows that any map T → P a,b extends uniquely to ∆[ T ] → P a,b . Consider the tw o functor s f , g : ( N ec ↓ P ) a,b → ( N ec ↓ P ) a,b where f sends [ T , T → P ] to [∆[ T ] , ∆[ T ] → P ] a nd g is the constant functor sending everything to [∆ 1 , x : ∆ 1 → P ] where x is the unique edge of P connecting a and b . Then clear ly there are natural transfor mations id → f and g → f , showing that the three maps id, f , and g induce homotopic maps on the nerves. So the ident it y induces the null map, hence C nec ( P )( a, b ) = N (( N ec ↓ P ) a,b ) is contractible. The result for P now follows by Theo rem 5.2. F or the general case, a s sume by induction that we know the result for all pro ducts of necklaces in w hich at most k − 1 of them ar e not equal to b eads. The case k = 1 was handled b y the pr e vious par agraph. Consider a pro duct Y = T 1 × · · · × T k × D where each T i is a nec klace and D is a pro duct of beads. W r ite T k = B 1 ∨ B 2 ∨· · ·∨ B r where each B i is a b ead, and let P j = ( T 1 × · · · × T k − 1 ) × ( B 1 ∨ · · · ∨ B j ) × D . W e k now by induction that P 1 is strong ly order ed, and w e will prov e by a second induction that the same is true for ea ch P j . So a ssume that P j is strong ly ordered for some 1 ≤ j < r . Let us denote A = ( T 1 × · · · × T k − 1 ) × ∆ 0 × D and Q = ( T 1 × · · · × T k − 1 ) × B j +1 × D . Then we hav e P j +1 = P j ∐ A Q , and w e know that P j , A , and Q are strongly ordered. Note that the maps A → P j +1 and A → Q are simple inclusions: they are the pro ducts of ∆ 0 → B j (resp. ∆ 0 → B j +1 ) with ident it y maps, and any inclusion ∆ 0 → ∆ m is clearly simple. It is e a sy to ch eck that hypothesis (1)– (3 ) of Lemma A.3 ar e satisfied, a nd so this finishes the pro of.  Pr o of of Pr op osition 6.1. This follows immediately fro m Prop os ition A.4.  RIGIDIFICA TION OF QUASI-CA TEGORIES 25 A.5. The category C (∆ n ) . Our final goal is to give the pro of of Lemma 2.5. Recall that this says there is an isomorphism C (∆ n )( i, j ) → N ( P i,j ) for n ∈ N and 0 ≤ i, j ≤ n , where P i,j is the p oset o f subsets of { i, i + 1 , . . . , j } containing i and j . Pr o of of L emma 2.5. The result is obvious when n = 0, so w e assume n > 0 . Let X = ( F U ) • ([ n ])( i, j ) and Y = P i,j . F or each ℓ ∈ N , we will provide an iso morphism X ℓ ∼ = Y ℓ , and these will be compatible with fa c e and degener acy maps. One under s tands X 0 = F U ([ n ])( i, j ) a s the set of free comp ositions o f sequences of morphisms in [ n ] which star t at i and end at j . By keeping track of the set of ob jects involv ed in this chain, we identify X 0 with the set of s ubs ets of { i , i +1 , . . . , j } which contain i and j . This gives an iso morphism X 0 → Y 0 . Similarly , for ℓ > 0 one has that X ℓ is the s et of free comp ositions of s equences of mor phisms in X ℓ − 1 . It is r eadily seen that X ℓ (even when ℓ = 0) is in one-to- one cor r esp ondence with the s e t of ways to “parenthesize” the sequence i, . . . , j in such a w ay that ev ery element is contained in ( ℓ + 1)-many pa rentheses (and no closed parenthesis directly follows an op en parenthesis). Given such a parenthesized sequence, one can rank the par ent heses by “ int eriority” (so that in terior parent heses hav e higher r ank). The fa c e and degenera cy maps on X a re given by dele ting or rep eating all the parentheses of a fixed rank. Under this description, a vertex in a n ℓ - s implex of X is given b y choo sing a rank and then ignoring all parentheses except those of that rank. Then b y taking only the last elemen ts be fo re a clo se-parenthesis, w e get a subset of { i + 1 , . . . , j } containing j ; b y unioning with { i } , we get a w ell-defined element of Y 0 . Given tw o ranks, the subs e t of { i + 1 , . . . , j } c o rresp onding to the higher rank will c o ntain the subset co rresp onding to the lower ra nk. One also sees immediately that an ℓ -simplex in X is determined b y its set of vertices, and so we can iden tify X ℓ with the set of seq ue nce s S 0 ⊆ S 1 ⊆ · · · ⊆ S ℓ ⊆ { i, i + 1 , . . . , j } containing i and j . This is precisely the set of ℓ -simplices of Y , so we have our isomorphism. It is clear ly compatible with face and deg eneracy maps.  References [AM] M. Ar tin and B. Mazur, ´ Etale homotopy , Lecture Notes in Math. 10 0 , Springer-V erlag, Berlin-New Y ork, 1969. [B] J. Bergner, A mo del c ate gory struct ur e on the c ate g ory of simplicial c atego ries , T r ans. Amer. Math. Soc. 3 59 (2007), no. 5, 2043–2058. [CP] J-M Cordier, T. Porter. Homotopy c oher ent ca te g ory the ory T rans. Amer. M ath. So c. 3 49 (1997), no. 1, 1-54. [D] D. Dugger, A primer on homotopy c olimits , preprint, http://mat h.uoregon.edu/ ∼ ddug ger . [DS] D. Dugger and D. Spiv ak, Mapping sp ac es in q uasic-ate g ories , preprint , 2009. [H] P . J. Hirschhorn, Mo del c atego ries and lo c alizations , M athematical Surve ys and Mono- graphs 99 , American Mathematical Society , Pro vidence, RI, 2003. [J1] A. Jo y al, Quasi-c atego ries and Kan c omplexes , J. Pure Appl. A l gebra 17 5 (2002), no. 1–3, 207–222. [J2] A. Jo y al, The the ory of quasi-c ategories , prepri n t. [L] J. Luri e, Higher top os the ory , Annals of Mathematics Studies 17 0 , Pri nceton University Press, Princeton, NJ, 2009. [Q] D. Quillen, Higher algebra ic K -the ory. I. , Algebraic K -theory , I: Higher K -theories (Proc. Conf., Batelle Memorial Inst., Seattle, W ash., 1972) , pp. 85–147. Lecture Notes i n M ath. 341 , Springer, B er lin, 1973. 26 DANIEL DUGGER AND DA VID SPIV AK Dep art ment of M a them a tics , University of Oregon, Eugen e, OR 9 7403 Dep art ment of M a them a tics , University of Oregon, Eugen e, OR 9 7403 E-mail addr ess : ddugg er@uoreg on.edu E-mail addr ess : dspiv ak@uoreg on.edu