On the structure of simplicial categories associated to quasi-categories

ON THE STR UCTURE OF SIMPLICIAL CA TEGORIES ASSOCIA TED TO QUASI-CA TEGORIES EMIL Y RIEHL DEP AR TMENT OF MA THEM A TICS, UNIVERSITY OF CHICAGO 5734 S. UNIVERSITY A V E., CHICAGO, IL 60637 E-MAIL: ERIEHL@M ATH.UCHICAGO. EDU Abstract. The homotop y coheren t nerv e fr om simpli cial catego ri es to simpli- cial s ets and its left adjoint C are imp ortant to the study of ( ∞ , 1)-catego ri es because they pro vide a means for comparing t wo mo dels of their r espective ho- motop y theories, giving a Quillen equiv alence b etw een the mo del structures for quasi-categories and simpli cial categ ories. The functor C also gives a cofibran t replacemen t for ordinary categories, regarded as trivial simpli cial categories. How ever, the hom-spaces of the s implicial category C X arisi ng fr om a quasi- category X are not w ell understoo d. W e sho w that when X is a quasi-category , all Λ 2 1 horns in t he hom-spaces of i ts simpli cial category can be filled. W e prov e, unexpectedly , that for any simpli cial set X , the hom-spaces of C X are 3-cosk eletal. W e charact erize the quasi- categories whose simplicial categories are lo cally quasi, finding explicit examples of 3-dimensional horns that cannot be fill ed in all other cases. Finally , we sho w that when X is the nerve of an ordinary catego ry , C X is isomorphic to the simplicial category obtained from the standard free si mplicial resolution, showing that the tw o known cofibrant “simplicial thick enings” of ordinary categories co incide, and furthermore its hom-spaces are 2-coskelet al. 1. Introduction In rece nt years, many a dv ances have b een made in the study of ( ∞ , 1)- c ate gories , lo osely defined to be categ ories enriched in ∞ -group oids or spaces . Mo dels of ( ∞ , 1)-categor ies abound, but in this pap er we restrict our attention to t wo of the simples t: quasi-categ ories, whic h ar e simplicial sets that satisfy a particular horn-filling prop erty , and simplicially enriched categories (henceforth, simplicia l c ate gories ). The catego ries sSet and s C at each b ear a mo del structure such that the fibr a nt ob jects are the mo dels of ( ∞ , 1)- c ategories — in the la tter cas e, the simplicial catego r ies whose ho m-spaces are K a n complexes. F urthermor e, there is a Quillen equiv alence b etw een them, with the r ight adjoint the homotopy coherent nerve N : sCat → s Set of Jean-Mar c Co rdier [3] (cf. the surv ey article [2] o r [15, chapters 1 a nd 2]). This adjunction provides the primary means of tra nslating b etw een these tw o mo dels, which accounts for its imp or tance. Interesting examples of quasi-ca tegories are often presented as homotopy co herent nerves of fibra nt simplicial categor ies: the “quasi-ca tegory of spaces” is one exa mple. More exo tically , a ny lo cally presentable quasi-categ ory is eq uiv alent to the homotopy coher ent ner ve of a combinatorial simplicial mo del ca teg ory [1 5, prop os ition A.3.7.6 ]. Conv ersely , the left adjoint Date : Decem b er 1, 2010. 1 2 EMIL Y RIE HL C : sSet → sCat “rig idifies” a simplicial set X , in which one can define a natu- ral notion of simplicia l “hom-space” betw een t w o v ertices , in to a category whose simplicial enric hment is given strictly . The v ertices o f the hom-spaces of C X ar e “comp osable” paths of edges in X , mea ning that the tar g et of each edg e in the se- quence is the source of the next, not necessar ily that any “ c omp osite” edge exists. If X is a quasi-c a tegory , then these edges can b e comp osed, and the 1-simplices in the hom-spa ces corres po nd to hig her simplices in X , which “witness” the sense in which a particular edge is a comp osite of a path given by others. It is tempting to describ e these 1-simplices as homotopies b etw e en the v ario us paths, bu t this isn’t an exact analog y because the 1 -simplices are directed: the sour ce is alw ays a shorter sequence of edges than the targ et. Instead, we prefer to think of the 1- simplices as “factorisa tions,” which exhibit how one path of e dg es ca n b e deformed into a long er one. The higher simplices o f the hom-spaces of C X are “ higher homotopies” that exhibit coherence relatio ns b etw een the “factor isations” r elating the v arious paths. By a gener al catego r ical principle [12, pr op osition 3 .1.5], the homo to py coherent nerve and its left adjoin t C are determined by a cosimplicial ob ject in sCat , that is, a functor C ∆ − : ∆ → s Cat , where ∆ is the us ual catego ry of finite non-empty ordinals [ n ] = { 0 , . . . , n } . The homoto py coherent nerve of a simplicial categ o ry C is the simplicial s et with n -simplices the simplicial functors C ∆ n → C , a nd the functor C is the left Kan e xtension of this functor along the Y oneda embedding ∆ → sSet . This can be computed b y a a familiar co end fo rmula, which we describ e in the next s ection. It follows that C X is the s implicial category “ freely g enerated” by X , in the sense that n -simplices in X cor resp ond to simplicial functors C ∆ n → C X , which, we shall s e e b elow, should b e thought of as homotopy coherent diagr ams in C X . Thus, to understand the adjunction C ⊣ N , we must fir s t build in tuition for the C ∆ n . There a r e many w ays to describ e the simplicial ca teg ories C ∆ n , one of whic h employs the free simplicial resolution constr uction of [3], [9 ], and elsewher e. By rep eatedly applying the free category como nad on Cat induced by the free-forgetful adjunction F ⊣ U b etw een categories and reflexive, directed graphs to the poset category [ n ], one o btains a s implicial ob ject in Cat : F U [ n ] F ηU / / F U F U [ n ] F U ǫ o o ǫF U o o F U F ηU / / F ηU F U / / F U F U F U [ n ] F U ǫF U o o ǫF U F U o o F U F U ǫ o o · · · Each of these ca tegories has the sa me o b jects as [ n ]. The ar rows of F U [ n ] are sequences of composable no n-identit y morphisms in [ n ]. The a rrows of F U F U [ n ] are a gain such sequences but with every morphism app earing in exactly one set of parentheses. The arrows of F U F U F U [ n ] are s equences of comp osa ble non-identit y morphisms with e very morphism app earing in exactly tw o sets of parentheses, and so forth. The face ma ps ( F U ) k ǫ ( F U ) j remov e the parentheses that are contained in exa ctly k others; F U · · · F U ǫ co mpo ses the mor phisms inside the innermos t parentheses. The degenera c y maps F ( U F ) k η ( U F ) j U double up the pa rentheses that are contained in exactly k other s; F · · · U F η U inserts pare n theses aro und each individual mo rphism. A simplicial ob ject in Cat determines a simplicia l catego ry exactly when each of the cons tituen t functors acts as the identit y on ob jects, as is the case here . Hence, this c onstruction sp ecifies a simplicial categ ory , which is C ∆ n . The vertices of SIMPLICIAL CA TEGORIES ASSOCIA TED TO QUASI-CA TEGORIES 3 the hom-spaces a re the s equences of comp osable non-identit y morphisms, i.e., the arrows of F U [ n ]; the 1-simplices of the hom-spaces are the a r rows o f F U F U [ n ]; and s o for th. More geometrically , C ∆ n is the simplicia l category with ob jects 0 , . . . , n and hom-spaces C ∆ n ( i, j ) defined to b e (o rdinary) nerves of certain p osets P i,j . If j < i , then P i,j and hence C ∆ n ( i, j ) is empty . Otherwise P i,j is the po s et of subsets of the in terv a l { k | i ≤ k ≤ j } ⊂ [ n ] that contain bo th endp oints. F o r j = i and j = i + 1, this pos et is t he terminal category . F or j > i + 1, a quic k calculation shows that P i,j is isomor phic to the pro duct of the category [1] with its e lf j − i − 1 times. Hence C ∆ n ( i, j ) =      (∆ 1 ) j − i − 1 when j > i, ∆ 0 when j = i, ∅ when j < i. F or pro o f that these tw o descriptions coincide, see [6, § 2 ]. Here is some int uition for these definitions. The hom-space C ∆ n ( i, j ) par ametrises paths from i to j in the p oset [ n ]. The vertices count the n umber of dis tinct paths: if j = i + 2, then there ar e tw o options — one which passes thro ugh the ob ject i + 1 and o ne which avoids it — and, a ccordingly , the simplicial s et C ∆ n ( i, i + 2) = ∆ 1 has tw o vertices. The higher dimensional data is de s igned such that homotopy c oher ent diagra ms [ n ] → C , studied extensively by Cordier and T imo th y Porter (but see a lso [15, section 1.2.6]), corresp ond to simplicial functors C ∆ n → C . W e illustrate the cas e where n = 3 and C = T op sSet , the simplicially enriched ca tegory of compactly generated spaces, with simplicial enric hment giv en b y applying the total singular co mplex functor to each hom-spa c e (cf. [4, sectio n 1]). A homotopy c ommu t ative diagram in T op sSet picks out spaces X , Y , Z , and W and functions X f / / j 8 8 l > > Y g / / k 7 7 Z h / / W such that there e x ist homotopies j ≃ g f , k ≃ hg , l ≃ k f , l ≃ hj , and l ≃ hg f . Such a diagra m is homotopy c oher ent if one can chose the ab ove homotopies in suc h a wa y that the comp osite homotopies l ≃ hg f are homotopic in the sense illustrated below l / /   B B B B B B B B B B B B ≃ ≃ k f   hj / / hg f The data specifying these each of these homo to pies is precisely the image of the simplicial map ∆ 1 × ∆ 1 = C ∆ 3 (0 , 3) → T op sSet ( X, W ). A more mo dern treatment of these same ideas is g iven in the stra ightening construction of [15, chapters 2 and 3], which a sso ciates c ontra v a riant simplicial (or ma r ked simplicial) functors with domain C X to righ t fibra tions (Car tesian fi- brations) ov er X , which can b e thought of as co ntra v a r iant K an-complex-v alued (quasi-categ o ry-v alued) pseudofunctors . The functor C fig ures pro minent ly in this 4 EMIL Y RIE HL corres p o ndence. The work contained in this pa p e r was motiv ated by our attendant desire to be able to compute pa rticular examples of this constr uc tio n. In the next section, we unravel the definition of the simplicial categor y C X as- so ciated to a simplicial set X a nd describ e the lower dimensiona l s implices of the hom-spaces C X ( x, y ). The in tuition provided by this calculatio n is satisfyingly co n- firmed by recent work of Dugger and Spiv a k [6], which identifi es the n -s implices o f C X ( x, y ) with necklaces in X , accompanied by ce r tain vertex data. In Se c tion 3, we use this c hara cterization to prov e that a ll Λ 2 1 horns in C X ( x, y ) can be filled, when X is a quasi-categ ory . In Sectio n 4, we demonstra te that the necklace repre- sentation is even mo re useful in highe r dimensions, proving the surpr ising fact that for any simplicial s et X , C X ( x , y ) is 3-coskeletal, which says that sufficient ly high dimensional simplicial spheres in these hom-space s can b e filled uniquely . In light of these results, one might hop e tha t the simplicial ca tegory ass o ciated to a quasi-ca tegory is lo cally quasi; how ever, this is seldom the case. In Section 5, we show that if C X is lo ca lly qua si, then X is the (or dina ry) ner ve of a c a tegory . W e c o nsider the ca se when X is the nerv e of a category in Section 6; C X is then its cofibrant replacement. W e prov e tha t the hom-spaces of C X a re 2-coskeletal, but that for mos t catego ries, there a re Λ 3 1 and Λ 3 2 horns in certain hom-spaces that cannot be filled. Finally , we show that the simplicial categor y obtained b y applying the free simplicial resolution construction descr ib ed ab ov e to an y small category A is iso morphic to the simplicial category C N A , where N : Cat → sSet is the o rdinary nerve functor. In other words, t hese constructions coincide for all categorie s, not just for the p oset categ o ries [ n ]. 2. U nderst anding the hom-sp aces C X ( x, y ) Although we are most interested in unders ta nding the simplicial catego ry as- so ciated to a quas i-categor y , a ll o f the re s ults in this se c tio n apply for a gener ic simplicial set X . T he notation for s implicia l sets thro ughout this pap er is consis- ten t with [11]. By definition C X = Z [ n ] ∈ ∆ a X n C ∆ n = co eq   a f : [ m ] → [ n ] a X n C ∆ m / / / / a [ n ] a X n C ∆ n   . It suffices to r estrict the interior copro ducts to the non-degene r ate simplices of X and the left outer copro duct to the gener ating coface maps d i : [ n − 1] → [ n ]. W r ite ˜ X n for the non-degener a te n -simplices of X . Then C X = colim                ` ˜ X 1 C ∆ 0 d 1   d 0   d 0 / / d 1 / / ` X 0 C ∆ 0 ` ˜ X 2 C ∆ 1 / / / / / /       ` ˜ X 1 C ∆ 1 ` ˜ X 2 C ∆ 2 · · ·                1 1 Some non-degenerate n -si mplices may hav e degenerate n − 1-si mplices as faces, so we cannot tec hnically restrict the f ace maps to maps d i : ˜ X n → ˜ X n − 1 . Instead, one must at tach eac h SIMPLICIAL CA TEGORIES ASSOCIA TED TO QUASI-CA TEGORIES 5 The ob jects of C X ar e the vertices of X . The simplicial categor ies C ∆ 0 and C ∆ 1 are the fr ee simplicia l ca tegories on the p oset categ ories [0 ] and [1] r esp ectively , and the free simplicia l c a tegory functor is a left adjoint and so commutes with colimits. Hence, if X is 1 -skeletal so that ˜ X n = ∅ for all n > 1, then C X is the free simplicia l category on the g raph with vertex set X 0 and edge set ˜ X 1 . Concretely , this means that the hom-spa ces C X ( x, y ) are a ll dis crete simplicia l s e ts containing a vertex for each path o f edges from x to y in X . In general, for each 2-simplex of X with b ounda r y as shown z g   ? ? ? ? ? ? ? x f ? ?         j / / y there exists a 1-s implex from the vertex j to the vertex g f in C X ( x, y ). F urthermore, for each vertex in some hom-space r epresenting a s equence o f paths containing j , there is a 1-simplex co nnec ting it to the vertex representing the same seq ue nce , except with g f in place of j . How ever, the 2-skeleton of X do es not determine the 1 -skeleta of the hom-spaces. F or exa mple, for each 3 - simplex σ of X as depicted b elow z g   k   @ @ @ @ @ @ @ @ x f > > ~ ~ ~ ~ ~ ~ ~ ~ j @ @ @ @ @ @ @ @ l / / y w h ? ? ~ ~ ~ ~ ~ ~ ~ ~ there is an edge fro m l to hg f in C X ( x, y ). In g e neral, ther e is an edge b etw een the vertices represented by paths p 1 . . . p r and q 1 . . . q r of e dg es from x to y in X if and only if each edge p in the first pa th that do es not app ear in the second is replaced by a sequence o f n -edges that a pp ea r a s the spine of some n -simplex of X with p as its diagonal . Here , the spine of an n - simplex is the s e quence of edges b etw e e n the adjacent vertices, using the usua l or dering o f the vertices, and the diagonal is the edge [0 , n ] from the initial vertex to the final one. In this wa y , each edge of C X ( x, y ) corr esp onds to a n e cklac e ∆ n 1 ∨ · · · ∨ ∆ n r → X in X . B y ∆ n ∨ ∆ k we alwa ys mean that the final vertex of the n -simplex is identified with the initial v ertex of t he k -simplex. A necklace is comprised o f a seque nce of b e ads , the ∆ n i ab ov e, that ar e strung together along the joins , defined to b e the union of the initial and final vertices of each b ead. When sp e aking collo quia lly , we may ass o ciate a b ead of the necklace with its image, a simplex o f the appro priate dimension in X , and the faces of the b ead with the corr esp onding faces o f the simplex. By a theorem of Daniel Dugg e r and David Spiv ak, necklaces can b e used to characterize the higher dimensio nal simplices of the ho m-spaces C X ( x, y ) as well, provided we keep tra ck of a dditional vertex data. degenerate f ace to the unique low er-dimensional non-degenerate simplex i t r epresen ts, but this tec hnicality w i ll not affect our conceptual discussi on. 6 EMIL Y RIE HL Theorem 2.1 (Dugger , Spiv ak [6, corollary 4.8]) . L et X b e a simplici al set with vertic es x and y . An n -simplex in C X ( x, y ) is un iquely r epr esent e d by a t riple ( T , f , ~ T ) , wher e T is a ne cklac e; f : T → X is a map of simplic ial sets that sends e ach b e ad of T to a non-de gener ate simplex of X and t he endp oints of the ne cklac e to the vertic es x and y , r esp e ctively; and ~ T is a flag of sets J T = T 0 ⊂ T 1 ⊂ T 2 ⊂ · · · ⊂ T n − 1 ⊂ T n = V T of vertic es V T of T , wher e J T is t he set o f joins of T . Necklaces f : T → X with the proper t y descr ib e d ab ov e are called total ly non- de gener ate . Note that the map f need not b e injective. If x = y is a vertex with a non-degenera te edge e : x → y , the map e : ∆ 1 → X defines a totally non-degenerate necklace in X . Dugger and Spiv ak prefer to c hara cterize the simplices of C X ( x, y ) as equiv a- lence classes of triples ( T , f , ~ T ), which are not necessarily totally non-degener ate [6, cor ollary 4.4, lemma 4.5 ]. Howev er, it is always p o s sible to replace an arbitrar y triple ( T , f , ~ T ) b y its unique tota lly no n- degenerate quotient. Lemma 2.2 (Dugger, Spiv ak [6, prop osition 4 .7]) . L et X b e a simpl icial set and supp ose T is a ne cklac e; f : T → X is a map of simplicial sets; and ~ T is a flag of sets J T = T 0 ⊂ T 1 ⊂ T 2 ⊂ · · · ⊂ T n − 1 ⊂ T n = V T of vertic es V T of T , wher e J T is t he set of joins of T . Then ther e is a un ique quotient ( T , f , ~ T ) of this triple such that f factors thr ough f via a s urje ct ion T ։ T and ( T , f , ~ T ) is total ly non-de gener ate. Pr o of. By the Eilenberg- Zilb er lemma [10, prop os ition I I.3.1, pp. 26-27 ], any sim- plex σ ∈ X n can be wr itten uniquely as ǫσ ′ where σ ′ ∈ X m is non-degenerate, with m ≤ n , and ǫ : X m → X n is a simplicia l op erator cor resp onding to a surjec- tion [ n ] → [ m ] in ∆ . The nec klace T agrees with T at each bead whose imag e is non-degener ate. If σ is a degenerate n - simplex in the im ag e of f , th en to for m T we r eplace t he bead ∆ n of T corresp o nding to σ b y the b ead ∆ m , where m is determined by the Eilen b erg- Zilb er decompo sition of σ , descr ibed a bove. Define f : T → X restricted to this ∆ m to equa l σ ′ . The morphism ǫ defines a surjective map of simplicial sets ∆ n → ∆ m , which defines the quotient map T ։ T at this bea d in s uch a wa y that f factors thro ugh f alo ng this map. Let ~ T b e the fla g of sets o f vertices of T g iven by the dir e c t image o f ~ T under T ։ T . The resulting triple ( T , f , ~ T ) is totally non-degenerate and unique suc h that f fa c tors thro ugh f .  With the aid of Lemma 2.2, the face maps d i : C X ( x, y ) n → C X ( x, y ) n − 1 can also be described in the la nguage of flag s a nd necklaces. In what f ollows, d i ~ T denotes the flag of s ets J T = T 0 ⊂ · · · ⊂ c T i ⊂ · · · ⊂ T n = V T with T i remov ed from the sequence . SIMPLICIAL CA TEGORIES ASSOCIA TED TO QUASI-CA TEGORIES 7 Theorem 2. 3 (Dugger, Spiv a k [6, remar ks 4.6 and 4.9]) . The fac es of an n -simplex ( T , f , ~ T ) ar e u niquely r epr esente d by t he t riples d i ( T , f , ~ T ) =      ( T ∗ , f | T ∗ , d 0 ~ T ) i = 0 ( T , f , d i ~ T ) 0 < i < n ( T ′ , f | T ′ , d n ~ T ) i = n wher e T ′ = T | T n − 1 is the maximal su bne cklac e of T with vertic es T n − 1 and T ∗ is the maximal subne cklac e of T with joins T 1 . If the triples ( T ∗ , f | T ∗ , d 0 ~ T ) and ( T ′ , f | T ′ , d n ~ T ) ar e total ly non-de gener ate, then these ar e the zer oth and n -th fac es, r esp e ctively; otherwise, these fac es ar e given by the u nique quotient s of L emm a 2.2. W e call T ∗ the T 1 - splitting of T . Each b ead of T is replaced by a necklace with the same spine whos e be a ds are each faces of the or iginal bea d. The vertices of each new bea d will be a consecutive subset of vertices of the b ead o f T w ith initial and final vertices in T 1 . The s um of the dimensions of these new b eads will e q ual the dimension of the origina l b ead. Example 2. 4. W e use Theorem 2.3 to compute the faces of the 3 -simplex (∆ 6 , σ : ∆ 6 → X, { 0 , 6 } ⊂ { 0 , 3 , 4 , 6 } ⊂ { 0 , 1 , 3 , 4 , 6 } ⊂ [6]) . The nec klace ∆ 3 ∨ ∆ 1 ∨ ∆ 2 is the { 0 , 3 , 4 , 6 } - splitting of the necklace ∆ 6 , so the zeroth face is (∆ 3 ∨ ∆ 1 ∨ ∆ 2 , σ | ∼ : ∆ 3 ∨ ∆ 1 ∨ ∆ 2 → X, { 0 , 3 , 4 , 6 } ⊂ { 0 , 1 , 3 , 4 , 6 } ⊂ [6]) where the map σ | ∼ is given on the first b ead by restricting σ to the face containing the vertices 0, 1, 2, and 3 ; on the second b ead by restricting to the fa c e containing the vertices 3 a nd 4; and on the third be ad by restricting to the face containing the vertices 4 , 5 , and 6. The first and second faces are (∆ 6 , σ : ∆ 6 → X, { 0 , 6 } ⊂ { 0 , 1 , 3 , 4 , 6 } ⊂ [6]) and (∆ 6 , σ : ∆ 6 → X, { 0 , 6 } ⊂ { 0 , 3 , 4 , 6 } ⊂ [6]) . Note that the necklace parts o f these faces are the same. The third face is (∆ 4 , d 2 d 5 σ : ∆ 4 → X, { 0 , 4 } ⊂ { 0 , 2 , 3 , 4 } ⊂ [4]) where we hav e chosen to rename the vertices in the flag. The simplicia l map d 2 d 5 σ : ∆ 4 → X is the r e s triction of σ to the face containing all vertices except for 2 a nd 5 . W e note an obvious corollar y to T he o rem 2.3, which will b e frequently explo ited. Corollary 2.5 . The ne cklac es r epr esen ting the inner fac es of an n -simplex ( T , f , ~ T ) ar e al l e qu al t o T . Pr o of. If 0 < i < n , the tr iple ( T , f , d i ~ T ) is totally non- degenerate a nd represents the i -th face o f ( T , f , ~ T ).  When des cribing n -simplices of C X ( x, y ) as triples ( T , f , ~ T ), we frequently define the necklace as a s ubsimplicial set of X , in which ca se the map f is under sto o d a nd may b e omitted fro m o ur notatio n. In dimensions 0 a nd 1 , the flag ~ T is completely determined b y the nec klace T , s o w e ma y represent the simplices of C X ( x, y ) b y necklaces alone. 8 EMIL Y RIE HL 3. Fil ling Λ 2 1 horns in C X ( x, y ) Recall, a qu asi-c ate gory is a simplicial set X s uch that ev ery inner ho rn in X has a fi l ler , i.e., any diagram Λ n k / /   X ∆ n > > } } } } has the indica ted extensio n. H ere, Λ n k is the simplicial subset of the standard (represented) n -simplex ∆ n generated by all of the ( n − 1)-dimensio nal faces except for the k -th, where 0 < k < n . Horn filling conditions in quasi-ca tegories guarantee that simplices in each di- mension can be comp osed. In this section, w e will use the characteriza tion of simplices in the hom-spa ces C X ( x, y ) a s necklaces to prov e the following theore m, which says that the “facto risations” relating paths o f edg es in a q ua si-catego ry can be comp osed. Theorem 3.1. L et X b e a quasi-c ate gory and let x and y b e any two vertic es. Then every horn Λ 2 1 → C X ( x, y ) has a fil ler ∆ 2 → C X ( x, y ) . Our pro of will use a lemma due to Andr´ e Joyal describing maps constructed from joins of s implicia l sets, where the join, denoted ⋆ , is the res triction of the Day tensor pro duct [5 ] o n aug mented s implicial sets a rising fro m the mo no idal s tr ucture on ∆ + , the category of all finite ordinals a nd order pr e s erving maps. The reader who wis hes to verify the pro ofs of Coro llaries 3.4 and 3.5 b elow should s ee [1 3] for an e x plicit definition. Recall a monomor phis m o f simplicia l sets is mid ano dyne if it is in the saturated class generated by the inner ho rn inclusions Λ n k → ∆ n and left ano dyne if it is in the saturated class generated by the ho rn inclusions with 0 ≤ k < n . The left ano dyne maps are precis ely those ma ps whic h ha ve the left lifting prop erty with resp ect to the left fibr ations , which are the maps that hav e the right lifting prop erty with resp ect to the appro pr iate horn inclusions. Lemma 3.2 (Joy al [1 5, lemma 2.1.2.3 ]) . If u : X → Y and v : Z → W ar e monomorphisms of s implicial sets s u ch that v is left ano dyne, then the map u ˆ ⋆v : X ⋆ W ∪ X ⋆Z Y ⋆ Z → Y ⋆ W is m id ano dyne. W e will apply Lemma 3.2 in the case wher e v is the inclusion i 0 : ∆ 0 → ∆ n of the initial vertex of ∆ n . Lemma 3. 3 . The map i 0 : ∆ 0 → ∆ n is left ano dyne. Pr o of. It suffices to show that i 0 lifts against any left fibration. Given a le ft fibration p : X → Y and a lifting pr oblem ∆ 0 i 0   a / / X p   ∆ n b / / Y SIMPLICIAL CA TEGORIES ASSOCIA TED TO QUASI-CA TEGORIES 9 we may lift the edge s [0 , k ] of the n -s implex b : ∆ n → Y to X b eca use we can solve lifting problems of the form ∆ 0 = Λ 1 0 a / /   X p   ∆ 1 b · i [0 ,k ] / / ; ; w w w w w Y This allows us to lift all 2-dimensional faces o f b containing the vertex 0 by filling Λ 2 0 horns. In turn, this allows us to lift all 3-dimensional faces o f b containing the vertex 0 by filling Λ 3 0 horns, and inductively we c a n lift a ll ( n − 1)-dimens ional faces of b containing 0 b y filling Λ n − 1 0 horns. Thes e faces form a Λ n 0 horn in X whos e image under p is the cor resp onding horn of b in Y . W e lift ag ainst the inclusion Λ n 0 → ∆ n to o btain the des ired lift of our n -simplex b .  Corollary 3.4. The inclusion ∆ k ∨ ∆ n → ∆ k + n is m id ano dyne. Pr o of. Apply Lemma 3.2 with u : ∅ → ∆ k − 1 and v = i 0 : ∆ 0 → ∆ n .  Corollary 3.5. The inclusion ∆ k ∪ [0 ,k ]=[0 , 1] ∆ n → ∆ k + n − 1 is m id ano dyne. Pr o of. Apply Lemma 3.2 with u = i 0 : ∆ 0 → ∆ k − 1 and v = i 0 : ∆ 0 → ∆ n − 1 .  W e now have a ll o f the to ols necessar y to prov e Theor em 3 .1. Pr o of of The or em 3.1. By Theorem 2.1, a horn Λ 2 1 → C X ( x, y ) is sp ecified by neck- laces T a nd U suc h that d 0 T = d 1 U . W e use Theo rem 2.3 to co mpute these fa ces. Let U ′ be the subnecklace of U who s e vertices e q ual the joins J U of U ; we call U ′ the diagonal of U . T he necklace U ′ is co mpr ised o f the path of edges which form the diagona l edges of the be ads of U ; for ea ch b ea d ∆ k of U , the [0 , k ] edge of that bea d app ear s in U ′ . By Theor em 2.3, d 1 U = U ′ , the quotient which collapses each degenerate edge of U ′ to a vertex. Let T ∗ to b e the subnecklace of T whose joins include all of the v ertices o f T ; we call T ∗ the spine of T . The necklace T ∗ is compr ised of the path of edges [ i, i + 1] b etw een adjacently num b er ed vertices in each b ea d of T . B y Theorem 2 .3, d 0 T = T ∗ , the quotient which collaps e s each degener ate edge of T ∗ to a vertex. The equation T ∗ = U ′ says that the spine o f T equals the diago na l of U , mo d- ulo any degenerate edges. W e cons truct a simplicial subset A of X that contains the da ta of T and U ; b y co nstruction a ny necklace S repr esenting a 2-simplex in C X ( x, y ) with T and U as second a nd zeroth faces will c o ntain A . Start with the necklace U and insert degenera te edges b etw een the joins of U if needed unt il the diagonal of this new necklace is a n expansion of the spine of T ; we might call this new necklace “ s tretched”. It won’t necessarily eq ua l the spine of T b ecause it may hav e s ome extra deg e nerate edges, which are the diag o nals of bea ds o f U . T o for m A , we glue the b ea ds of T into this necklace alo ng their spines. This ca n be do ne directly whenever the s pine of the b ead σ of T do e s not encounter any degenera te edges ar ising from the diagonal o f U . If the stretched version of U con tains a n extra degenerate edge at the lo cation co rresp onding to the i -th vertex of s ome b ead σ of T , we replace σ b y the de g enerate s implex s i σ , which can b e glued in a s des c rib ed ab ov e. The res ulting simplicial set A is obta ined by gluing a “ thic kened” version o f T , w he r e some b e ads hav e b een expanded to degener ate b eads to accommo date any 10 EMIL Y RIE HL degenerate diagonals of U , to the s tretched necklace U , which has “ga ps” b etw een bea ds o ccurring wherever there ar e degenera te edges in the spine o f T . T = ∆ 2 · g & & N N N N N N N N · / / f 8 8 p p p p p p p p · U = ∆ 3 ∨ ∆ 3 ·   : : : : :   ·   : : : : :   · f / / A A        = = = = · g / / A A        = = = = · · @ @     · @ @     A = ∆ 2 ∪ Λ 2 1 (∆ 3 ∨ ∆ 3 ) · = = = = = = = = g = = =   = = = = & & N N N N N N N N   . . . . . . . . . ·   = = = = 8 8 p p p p p p p p ·   . . . . . . . . .       · H H          · & & N N N N N N N N · / / H H          8 8 p p p p p p p p f        @ @         · W e wish to fill in A to obtain a new necklace S in X o n the vertices of A , whose joins contain the joins of T and a ny extr a copies of these vertices that a ppea r from degenerate dia gonals of U . T o construct such a necklace S , we need to “fill” each simplicial subset b etw een our s o-designated joins t o a simplex of the appropriate dimension; this co nstructs the b eads one at a time. Inductively , there ar e o nly tw o t yp es of fillings necessar y : for one , we m ust expa nd tw o adjace n t b eads ∆ k and ∆ n to form a single b ead ∆ k + n . F or the other we m ust expand t w o o verlapping bea ds, where the diagonal of one is the first e dg e a long the spine of another. Such extensions ar e p ossible b ecause the maps of 3.4 and 3.5 and its du al are mid a no- dyne, and X is a quasi- categor y . This construction defines a totally no n-degenera te necklace S in X which contains A . Let ~ S b e the flag J S ⊂ S 1 ⊂ V S , where S 1 is the v ertex s et of T plus, for each degenerate edge of U , an extra co p y of the cor resp onding vertex. W e claim that the triple ( S, k , ~ S ) is a filler for the horn Λ 2 1 → C X ( x, y ). The necklace repres e nting the face d 2 ( S, ~ S ) = ( S ′ , d 2 ~ S ) is obtained b y r estricting S to S 1 , which contains the vertices of T tog ether with some extr a copies arising from deg e nerate edges of U . This amo un ts to restricting A to the same vertices, and so it is clear that the totally non-degenera te quotient S ′ is T . Hence, ( S ′ , d 2 ~ S ) = ( T , ~ T ). Similar ly , the necklace S ∗ of d 0 ( S, ~ S ) = ( S ∗ , d 0 ~ S ) contains the b ea ds of U but also some degener ate edges betw een b eads. This says ex actly that ( S ∗ , d 0 ~ S ) = ( U, ~ U ), as desir e d.  Recall from the introduction that the 1- simplices in the hom-spa ces o f C X can be thoug ht of as “facto r isations,” which ar en’t reversible. As a result, we would not exp ect that Λ 2 0 or Λ 2 2 horns in the ho m-spaces can b e filled in genera l, even if X is a K an complex: if these hor ns co uld b e filled, it w ould follow that every 1-simplex would b e inv ertible up to homoto py . It is easy to find ex amples of outer horns that cannot b e filled: for any non-deg enerate 2-simplex in C X ( x, y ), the repr esentativ es of the outer faces are prop er subnec klac e s of the necklace representing the inner face. Ex changing the fir st face for either the zeroth o r the s econd, one o bta ins a Λ 2 2 or Λ 2 0 horn that has no filler. Unexpe c tedly , many hig her horns in these ho m-spaces can b e fille d uniquely , without any hypotheses o n X , by an immediate co rollar y to the main theo rem of the next se ction. 4. For any X , all C X ( x, y ) are 3-coskelet al Theorems 2.1 and 2.3 can also b e used to prov e the following. SIMPLICIAL CA TEGORIES ASSOCIA TED TO QUASI-CA TEGORIES 11 Theorem 4.1. F or any simplicial set X and vertic es x and y , the hom-sp ac e C X ( x, y ) is 3-c oskeletal. Recall, a simplicial set X is n - c oskeletal if any sphere ∂ ∆ k → X , with k > n , can be filled uniquely to a k -simplex ∆ k → X , where ∂ ∆ k is the simplicial subset of ∆ k generated by all of its ( k − 1)-dimensional faces (cf. [8, sectio n 0.7]). Mor ally , a n n -coskeletal Kan c o mplex co rresp onds to an ( n − 1)- type from cla ssical ho motopy theory . Pr o of. W e m ust show that for n ≥ 4 any sphere ∂ ∆ n → C X ( x, y ) has a unique filler. Using Theorem 2 .1, an n -spher e in C X ( x, y ) is a collectio n ( T 0 , ~ T 0 ) , . . . , ( T n , ~ T n ) of tota lly no n-degenerate necklaces in X accompa nied by flag s ~ T i of vertices J i ⊂ T 1 i ⊂ · · · ⊂ T n − 2 i ⊂ V i satisfying the r elations d i ( T j , ~ T j ) = d j − 1 ( T i , ~ T i ) for all i < j. By Corollary 2.5 the necklaces T i are equal for 0 < i < n ; we call this common necklace S . F urthermor e, w hen n ≥ 4, the relations b etw een the inner faces define a fla g ~ S o f vertices J S ⊂ S 1 ⊂ · · · ⊂ S n − 1 ⊂ V S such tha t d i ~ S = ~ T i , for 0 < i < n . It is clear that ( S, ~ S ) is the only p oss ible filler for this sphere . It remains to show that d 0 ( S, ~ S ) = ( T 0 , ~ T 0 ) a nd d n ( S, ~ S ) = ( T n , ~ T n ). Using any inner face i , we compute that (1) d i − 1 ( T 0 , ~ T 0 ) = d 0 ( T i , ~ T i ) = d 0 d i ( S, ~ S ) = d i − 1 d 0 ( S, ~ S ) . W e may choose i > 1 so that (1 ) is an inner face r elation; it follows from Corolla r y 2.5 that T 0 is the necklace of d 0 ( S, ~ S ) and d i − 1 ~ T 0 = d i − 1 d 0 ~ S . By choosing a different inner fa ce i > 1, we learn that the omitted sets of the flag s ~ T 0 and d 0 ~ S also a gree, so we c onclude tha t ~ T 0 = d 0 ~ S , and hence that d 0 ( S, ~ S ) = ( T 0 , ~ T 0 ). Similarly , using any inner face j , w e compute that (2) d j ( T n , ~ T n ) = d n − 1 ( T j , ~ T j ) = d n − 1 d j ( S, ~ S ) = d j d n ( S, ~ S ) . W e may c ho os e j < n − 1 so that (2 ) is an inner fac e relation; it follows from Corollar y 2.5 that T n is the necklace o f d n ( S, ~ S ) and d j ~ T n = d j d n ~ S . W e choos e any other inner face j < n − 1 to conclude that the flag ~ T n = d n ~ S , and hence that d n ( S, ~ S ) = ( T n , ~ T n ).  It is well-known tha t n -coskeletal s implicia l se ts have unique filler s for all horns of dimensio n at least n + 2 (cf., e.g., [7, s ection 2 .3]). Corollary 4.2. F or n > 4 , every horn Λ n k → C X ( x, y ) has a unique fil ler. Pr o of. When n > 4, the map sk 3 Λ n k → sk 3 ∆ n induced by the inclusio n is an isomorphism. Hence, an y map sk 3 Λ n k → C X ( x, y ) ca n b e extended uniquely to a map sk 3 ∆ n → C X ( x, y ). By a djunction, any hor n Λ n k → cosk 3 C X ( x, y ) ∼ = C X ( x, y ) has a uniq ue filler.  12 EMIL Y RIE HL Example 4.3. Le t X b e a simplicial set with t w o v ertices x a nd y , t wo edges f and g f ro m x to y , and a 2 -simplex α and a 3-s implex σ as shown. x f   g   ? ? ? ? ? ? ? x f   > > > > > > > > g / / y x g / / f   ? ? ? ? ? ? ? ? s 0 x ? ?         y y s 0 y ? ?        y s 0 y ? ?        The necklaces T 0 = T 3 = ∆ 2 mapping onto α and T 1 = T 2 = ∆ 3 mapping o nt o σ , with flags ~ T 0 { 0 , 2 } ⊂ { 0 , 1 , 2 } ⊂ { 0 , 1 , 2 } ~ T 1 { 0 , 3 } ⊂ { 0 , 2 , 3 } ⊂ { 0 , 1 , 2 , 3 } ~ T 2 { 0 , 3 } ⊂ { 0 , 1 , 3 } ⊂ { 0 , 1 , 2 , 3 } ~ T 3 { 0 , 2 } ⊂ { 0 , 2 } ⊂ { 0 , 1 , 2 } define a 3-sphere in C X ( x, y ). This spher e cannot b e filled b ecause there is no flag ~ S with d 1 ~ S = ~ T 1 and d 2 ~ S = ~ T 2 . Hence, C X ( x, y ) is not 2-coskeletal, a fact that can also b e verified b y direct co mputation o f C X . This shows that the result of Theor e m 4.1 is the stro ngest p ossible. 5. C X is not l ocall y quasi, if the quasi-ca tegor y X is not a ca tegor y In light of the prior results, one might hop e that all inner horns in the hom-s paces of the simplicial categ ory asso ciated to a quas i- categor y can b e filled. How ever, we will s how in this section tha t for any q uasi-categ o ry X that is not the nerve of an o rdinary categor y , we can find a hom-spa ce in C X that is not a quasi-ca tegory , proving the following theorem. Theorem 5.1. If X is a quasi-c ate gory and its simplici al c ate gory C X is lo c al ly quasi, then X is isomorphic to the nerve of a c ate gory. More explicitly , we will show that if X is any quas i-categor y that fails to satisfy either o f the following conditions (1) X is 2-coskeletal (2) X ha s unique fillers for Λ 2 1 , Λ 3 1 , and Λ 3 2 horns then ther e is a Λ 3 1 horn in some hom-space of C X that canno t b e filled. By a lemma below, any q uasi-categ ory tha t s atisfies both of these conditions is isomorphic to the nerve of a categor y . Thus, if X is a quasi-categ o ry such that the simplicial ca tegory C X is lo ca lly quasi, then X m ust b e the nerve of a catego ry . In the following section, w e complete our characteriza tion of the quas i-categor ies who se simplicia l categorie s are lo cally quasi, by considering the cas e wher e X is isomorphic to the nerve o f a c ategory . 2 2 Note that the simplicial category C Λ 2 1 is lo cally quasi, its three non-empt y hom-spaces b ei ng trivial, but Λ 2 1 is not itself a quasi- category . W e don’t concern ourselves with the tec hnicalities of determining which non-quasi-categories hav e lo cally quasi s i mplicial categories as a fluke. SIMPLICIAL CA TEGORIES ASSOCIA TED TO QUASI-CA TEGORIES 13 It is well-known that a simplicial set is isomo rphic to the nerve of a category if a nd only if every inner horn has a unique filler. A third equiv alent condition is given in the following lemma, though w e o nly pro ve the implication needed here. F or a full pro o f see [14, pro po sition 1.13]. Lemma 5. 2 . L et X b e a 2-c oskeletal simplicial set such that for n = 2 or 3 every inner horn Λ n k → X has a unique fi l ler. Then X is isomorphic to the nerve o f a c ate gory. Pr o of. By the a rgument given for Corollar y 4.2, any horn Λ n k → X with n > 3 has a uniq ue filler.  The following lemmas give explicit Λ 3 1 horns that c a nnot b e filled. Lemma 5.3. L et X b e a quasi-c ate gory with distinct n -simplic es σ and τ with the same b oundary, for some n ≥ 3 . L et x and y b e the c ommon initial and t erminal vertic es, re sp e ct ively, of t hese simplic es. Then ther e is a horn Λ 3 1 → C X ( x, y ) with no fil ler. Pr o of. Degenera te simplices with given bo undaries a re unique, so at lea st one o f σ or τ is non- degenerate; a s sume it is τ . Let S and T equal ∆ n with maps S → X and T → Y that send the unique b ead to σ a nd τ , resp ectively . P ick some pro p er subset J of vertices of ∆ n that co ntains b oth the initial a nd final vertices and a t least one other. Let U b e the maximal s ubnecklace of ∆ n that has these vertices as jo ins; i.e., let U be t he J - splitting of ∆ n . There is a natura l map U → X that factors through b oth S and T b ecaus e the image of U in X sits inside the common bo undary o f σ and τ . W e claim that U , S , a nd T form, resp ectively , the zeroth, second, and third fa c es of a hor n Λ 3 1 → C X ( x, y ) when given fla gs ~ U J ⊂ [ n ] ⊂ [ n ] ~ S { 0 , n } ⊂ J ⊂ [ n ] ~ T { 0 , n } ⊂ J ⊂ [ n ] where we replace ( U, ~ U ) and ( S, ~ S ) by their tota lly no n-degenera te quotients if necessary ; we igno re this p ossibility in our no tation as it do es not change the argument in an y substantial wa y . T o pro ve this, w e m ust c heck that d 0 S = d 1 U , d 0 T = d 2 U , and d 2 S = d 2 T . The firs t tw o equations say that the J -splittings of S and T ar e U , which is true by definition. The final equation says that S a nd T ar e the same when r estricted to J , which is true b eca us e the image of this r estriction lies in the common bo undary of σ and τ . A filler o f this ho rn would necessarily hav e ( S, ~ S ) as its second fac e . By Cor ollary 2.5, any such 3 - simplex m ust be given b y the necklace S itself together with the flag { 0 , n } ⊂ J ⊂ S 2 ⊂ [ n ] . But the third face any such 3 - simplex can’t p oss ibly be r epresented by the necklace T . Hence, this ho rn ha s no filler.  Lemma 5 .4. L et X b e a quasi-c ate gory with distinct 3-simplic es σ and τ su ch that d 0 σ = d 0 τ and d 2 σ = d 2 τ . L et x and y b e the c ommon initial and terminal vertic es, r esp e ctively, of these simplic es. Then ther e is a horn Λ 3 1 → C X ( x, y ) with n o fil ler. 14 EMIL Y RIE HL Pr o of. The argument given in the ab ov e pro of with n = 3 and J = { 0 , 1 , 3 } a pplies in this case. The conditions tha t U , S , and T fo rm a Λ 3 1 horn a mount to the requirement that the simplices σ and τ in the ima ges o f S and T resp ectively share common zeroth and second faces.  Pr o of of The or em 5.1. If X is not 2-coskeletal, then either there exis t distinct n - simplices with the same b oundary , in w hich case w e may apply Lemma 5.3, or there exis ts an n -s phere in X that cannot b e filled for some n > 2. In the latter case, choose 0 < i < n and fill the Λ n i horn con tained in the n -sphere to o btain, as the i -th face of the filler, and ( n − 1)-simplex , necessarily distinct from the one app earing in the sphere, but with the same b oundary . Unless n was equal to 3, we hav e fo und distinct simplices o f dimensio n 3 or higher with the same b oundar y , and we ca n apply Lemma 5 .3 to s how tha t C X is not lo cally quasi. In the remaining case, w e have tw o distinct 2-simplices α and β with common bo undary . W e construct the 3-simplices σ and τ of Lemma 5.4 by filling t wo Λ 3 1 - horns with second face α , third face either α o r β , and zero th face the appr opriate degeneracy . Applying Lemma 5.4, we co nclude that C X is not lo cally qua si if X is not 2 -coskeletal. Alternatively , if X ha s so me Λ 3 2 or Λ 3 1 horn with tw o distinct fillers, it is clear that Lemma 5.4 or the dual a rgument can b e applied. If X has so me Λ 2 1 horn with distinct fillers α and β , w e rep eat the co nstruction just given, noting tha t it do es not matter if d 1 α 6 = d 1 β . Thus, if X do es not hav e unique fillers for low dimensional inner ho rns, then C X is not lo cally quas i.  6. S implicial ca tegories associa ted to ca tegories It r emains to consider the simplicial catego ries C N A asso cia ted to (o rdinary) nerves of categ o ries A ; C N A is often referr ed to as the “simplicial thick ening” of A . More sp ecifically , C N A is the cofibrant r eplacement of A , reg arded as a trivial s implicia l c ategory , in the usual mo del s tr ucture for simplicia l c a tegories [1]. In this s ection, we show that the hom-spaces of C X are 2-co skeletal when X is isomorphic to the nerve of a ca tegory . W e then provide examples of Λ 3 1 and Λ 3 2 horns in some ho m- spaces that cannot b e filled, whenever the ca tegory has a non- trivial factorization of an identit y mo rphism. Finally , we show that the simplicial category C N A is isomorphic to ano ther known cofibr ant replacement o f A in sCat : namely , the standa r d fre e simplicia l r esolution of Sec tion 1. Throughout this section, we ass ume tha t X is iso morphic to the ner ve of a category . W e b egin by noting an obvious lemma with a useful cor ollary . Lemma 6.1. A simplex in X is de gener ate if and only if its spine c ontains a de gener ate e dge. Corollary 6.2. If T → X is a total ly non-de gener ate ne cklac e and J T ⊂ K ⊂ V T is a c ol le ction of vert ic es of T c ontaining the joins, then the K -splitting of T is total ly non-de gener ate. Pr o of. The spine of T equals the spine of its K -splitting.  Remark 6.3. In particular, if ( T , ~ T ) is tota lly non-degenerate, its zer oth f ace is the T 1 -splitting of T with flag d 0 ~ T . In pr actice this that we can r ecov er m uch of the data of the fla g ~ T from its zeroth face, as well as from the inner faces. SIMPLICIAL CA TEGORIES ASSOCIA TED TO QUASI-CA TEGORIES 15 Theorem 6.4. F or any simplic ial set X that is t he nerve of a c ate gory and any obje cts x and y , the hom-sp ac e C X ( x, y ) is 2-c oskeletal. Pr o of. By Theorem 4.1, it remains to show that any sphere ∂ ∆ 3 → C X ( x, y ) has a unique filler . Recall, a 3-spher e in C X ( x, y ) is a colle c tion ( T 0 , ~ T 0 ) , . . . , ( T 3 , ~ T 3 ) of tota lly no n-degenerate necklaces in X accompa nied by flag s ~ T i of vertices J i ⊂ T 1 i ⊂ V i satisfying the r elations d i ( T j , ~ T j ) = d j − 1 ( T i , ~ T i ) for all i < j. Corollar y 2 .5 and the relation b etw een the inner faces implies that the necklaces T 1 and T 2 are e q ual; we call this commo n necklace S . By Remark 6 .3, the relatio ns betw een the zero th, first, and sec ond fac e s define a fla g ~ S o f vertices J 1 = J 2 ⊂ T 1 2 = J 0 ⊂ T 1 0 = T 1 1 ⊂ V 0 = V 1 = V 2 such that d i ~ S = ~ T i for i < 3. I t is clear t hat ( S, ~ S ) is the only possible filler for this spher e. The rela tio n (1) with i = 2 implies tha t T 0 is the necklace o f d 0 ( S, ~ S ); hence, d 0 ( S, ~ S ) = ( T 0 , ~ T 0 ). The r elation (2) with j = 1 implies that T 3 is the necklace of d 3 ( S, ~ S ). By Remark 6.3, w e may use j = 0 in (2) to co nclude tha t ~ T n = d 3 ~ S , and hence that d 3 ( S, ~ S ) = ( T 3 , ~ T 3 ).  Example 6.5. Suppo s e X is the nerve of a categor y with non- ident ity morphis ms s : x → y a nd r : y → x such tha t rs is the identit y at x . Let T b e the necklace ∆ 3 and let U b e the necklace ∆ 2 ∨ ∆ 1 whose images in X hav e spines sr s . Then there is a 2- simplex α in C X ( x, y ) with zeroth face U , first face T , and second face a degeneracy , but there is no 2 -simplex with the p ositions of U a nd T reversed. The simplex α can be glued to dege nerate s implices to fo r m horns , depicted b elow, that hav e no filler. Λ 3 1 s U   U   ? ? ? ? ? ? ? Λ 3 2 s s 0 s   T   ? ? ? ? ? ? ? s s 0 s ? ?        T   ? ? ? ? ? ? ? U / / srs s U / / s 0 s   ? ? ? ? ? ? ? s 0 s ? ?        srs srs s 0 ( srs ) ? ?        s U ? ?        Remark 6.6. When X is isomor phic to the ner ve of a c ategory such that identit y morphisms cannot b e factor ed, or equiv alently , so that there do not exis t r a nd s a s ab ov e, then any restr ic tio n of a totally non-degenera te necklace in X is totally non- degenerate. By ar guments similar to t hose giv en ab ove, all Λ 2 1 , Λ 3 1 , a nd Λ 3 2 horns in ho m-spaces of C X c a n b e filled uniquely , and it follows from Lemma 5 .2, that the C X ( x, y ) are themselves ner ves of categor ie s in this case. The p oset categ ories [ n ] do satisfy this condition, but most interesting exa mples do not. W e conclude with a theorem that is most likely k nown somewhere, given the ubiquit y of the simplicial resolution constr uction describ ed in the introduction, but which, with Theo rem 2.1, admits a particular ly simple pr o of. 16 EMIL Y RIE HL Theorem 6. 7. F or any c ate gory A , the simplicia l c ate gory C N A is isomorphic to the simplicia l c ate gory obtaine d as the st andar d fr e e simplicia l r esolution of A . Pr o of. The ob jects o f both simplicial categories are the ob jects of A . It remains to show that the hom-spac es coincide. A nec klace in the ner ve of a categor y is uniquely determined by its spine a nd the set of joins; i.e., a necklace is a sequence of comp osa ble non-identit y mo rphisms each cont ained in one set of par entheses, indicating which mor phisms a re g roup ed tog ether to form a b ead. An n -simplex in a hom-ob ject of the standa rd free simplicial res olution is a sequence of comp osable non-identit y morphisms, each con tained within ( n − 1) sets o f parentheses. The morphisms in the sequence descr ib e the spine of a necklace and the lo cations o f each level of parentheses de fines the co r resp onding set in the fla g of vertex data; by Theorem 2.1, this exactly sp ecifies an n -simplex in the co rresp onding hom-o b ject of C N A .  A cknow le dgments. The author would like to thank Dominic V er it y for enduring several co nv ersations on this topic and b eing a very generous host. She would also like to thank her advisor , Peter May , who suggested the title for this pap er, and the anonymous reviewer, who suggested several ways to improv e its exp osition. The author is gr ateful for supp ort from the National Science F ounda tio n, who se Graduate Resea rch F ellowship allow ed her to visit Macquarie Universit y , where this work to o k place. References [1] J. Bergner. A model cat egory structure on the category of simplicial categories. T r ans. Am er. Math. So c . 359 (2007), 2043-2058. [2] J. Bergner. A survey of ( ∞ , 1)-catego ries. T owar ds higher c ate gories. IMA V ol. Math. Appl. 152 (Springer 2010), pp. 69-83. [3] J-M. Cordier. Sur la notion de diagramme homot opiquement coh´ erent . Cahier T op. et Ge om. Diff. 1 XXI II (1982), 93-112. [4] J-M. Cordier and T. Por ter. V ogts theorem on categories of homotop y coherent diagrams. Math. Pr o c. Cambridge Philos. So c. 1 00 (1986), 6590. [5] B. Da y . On closed categories of functors. R ep orts of the Midwest Cate gory Seminar IV Lecture Notes in Math. vol. 137 (Springer 1970), pp. 1-38. [6] D. Dugger and D. Spiv ak. Rigidi fication of quasi-categories. Preprint (2009). [7] J. Duskin. Simpli cial m atrices and the nerves of weak n -categories I: Nerves of bicategories. CT2000 Confer enc e (Como). The ory Appl. Cate g. (10) 9 (2001/20 02), 198-308. [8] J. Duskin. S implicial metho ds and the inte rpr etati on of “triple” c ohomolo gy . Mem. Amer. Math. So c. vol. 3, issue 2, no. 16, (1975). [9] W. Dwyer and D. Kan. Simplicial lo calizations of categories. J. Pur e and Appl. Algebr a 17 (1980), 267-284. [10] P. Gabriel and M. Zisman. Calculus of fr actions and homotopy the ory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35 (Springer-V erlag, 1967). [11] P. Goerss and J. J ardine. Simplicial homotopy the ory. Progress in Mathematics 174 (Birkhuser V erlag, 1999). [12] M. Hovey. Mo del c ate gories. Mathematical surve ys and monographs. vol. 63 (American Mathematical So ciet y , 1999). [13] A. J oy al. Quasi-categories and Kan complexes. J. Pur e Appl. Algebr a 175 (2002), 207-222. [14] A. Joy a l. The theory of quasi-categories I. In pr ogress (2008). [15] J. Lurie. Higher T op os The ory. Annals of mathematics s tudies. no. 170 (Princeton Universit y Press, 2009).