A MODEL CATEGORY STRUCTURE ON THE CATEGORY OF SIMPLICIAL CATEGORIES JULIA E. BERGNER Abstract.In this paper we put a cofibrantly generated model category str* *uc- ture on the category of small simplicial categories. The weak equivalenc* *es are a simplicial analogue of the notion of equivalence of categories. 1.Introduction Simplicial categories, which in this paper we will take to mean categories en* *riched over simplicial sets, arise in the study of homotopy theories. Given any model category M, the simplicial localization of M as given in [3] is a simplicial ca* *tegory which possesses the homotopy-theoretic information contained in M. Finding a model category structure on the category of simplicial categories is then the f* *irst step in studying the homotopy theory of homotopy theories. In an early version of the preprint [1], Dwyer, Hirschhorn, and Kan present a cofibrantly generated model category structure on the category of simplicial ca* *te- gories, but as To"en and Vezzosi point out in their paper [9], this model categ* *ory structure is incorrect, in that some of the proposed generating acyclic cofibra* *tions are not actually weak equivalences. Here we complete the work of [1] by describ* *ing a different set of generating acyclic cofibrations which are in fact weak equiv* *alences and which, along with the generating cofibrations given in [1], enable us to pr* *ove that the desired model category structure exists. Note that the term "simplicial category" is potentially confusing. As we have already stated, by a simplicial category we mean a category enriched over simpl* *icial sets. If a and b are objects in a simplicial category C, then we denote by Hom * *C(a, b) the function complex, or simplicial set of maps a ! b in C. This notion is more restrictive than that of a simplicial object in the category of categories. Usi* *ng our definition, a simplicial category is essentially a simplicial object in the cat* *egory of categories which satisfies the additional condition that all the simplicial ope* *rators induce the identity map on the objects of the categories involved [2, 2.1]. We will assume that our simplicial categories are small, namely, that they ha* *ve a set of objects. A functor between two simplicial categories f : C ! D consist* *s of a map of sets f : Ob(C) ! Ob(D) on the objects of the two simplicial categories, and function complex maps f : Hom C(a, b) ! Hom D(fa, fb) which are compatible with composition. Let SCdenote the category whose objects are the small simplic* *ial categories and whose morphisms are the functors between them. This category SC is the underlying category of our model category structure. In a similar way, we can consider categories enriched over the category of to* *polog- ical spaces. Making slight modifications to the ideas from this paper, it is po* *ssible ____________ Date: June 24, 2004. 1 2 J.E. BERGNER to put an analogous model category structure on the category of small topologic* *al categories. Recall that a model category structure on a category C is a choice of three d* *is- tinguished classes of morphisms, namely, fibrations, cofibrations, and weak equ* *iv- alences. We use the term acyclic (co)fibration to denote a map which is both a (co)fibration and a weak equivalence. This structure is required to satisfy * *five axioms [4, 3.3]: o MC1: C is complete and cocomplete. In other words, C has all small limits and colimits. o MC2: If f and g are maps in C and their composite gf is defined, and if two of the three maps f, g, and gf are weak equivalences, then so is the third. o MC3: If a map f is a retract of g and g is a fibration, cofibration, or * *weak equivalence, then so is f. o MC4: Given a solid arrow commutative diagram of the form A _____//X>>~ |i|~ ~ |p| |fflffl~fflffl| B _____//Y a dotted arrow lift exists if either (i)i is a cofibration and p is an acyclic fibration, or (ii)i is an acyclic cofibration and p is a fibration. o MC5: Any map f can be factored in two ways: (i)f = pi where i is a cofibration and p is an acyclic fibration, and (ii)f = qj where j is an acyclic cofibration and q is a fibration. In axiom MC4, we say that i has the left lifting property with respect to p, * *or equivalently that p has the right lifting property with respect to i. Before defining these three classes of morphisms in SC, we need some notation. Suppose that C and D are two simplicial categories. Let ß0C denote the category* * of components of C, namely, the category in which the objects are the same as thos* *e of C and the morphisms are the path components of the simplicial sets of morphisms in SC. Explicitly, if a and b are objects of C, then Hom ß0C(a, b) = ß0Hom C(a, b). If f : C ! D is a map of simplicial categories, then ß0f : ß0C ! ß0D denotes the induced map on the categories of components of C and D. If C is a simplicial category, say that a morphism e 2 Hom C(a, b)0 is a homo* *topy equivalence if there is a map e0 2 Hom C(b, a)0 such that the composite e0e 2 Hom C(a, a)0 is in the same path component as the identity map on a and the composite ee02 Hom C(b, b)0 is in the same path component as the identity map on b. Alternatively stated, e is a homotopy equivalence if e becomes an isomorphism in ß0C. Now, given these definitions, our three classes of morphisms are defined as f* *ol- lows. (1) The weak equivalences are the maps f : C ! D satisfying the following two conditions: SIMPLICIAL CATEGORIES 3 o (W1) For any objects a1 and a2 in C, the map Hom C (a1, a2) ! Hom D(fa1, fa2) is a weak equivalence of simplicial sets. o (W2) The induced functor ß0f : ß0C ! ß0D is an equivalence of categories. (2) The fibrations are the maps f : C ! D satisfying the following two condi- tions: o (F1) For any objects a1 and a2 in C, the map Hom C (a1, a2) ! Hom D(fa1, fa2) is a fibration of simplicial sets. o (F2) For any object a1 in C, b in D, and homotopy equivalence e : fa1 ! b in D, there is an object a2 in C and homotopy equivalence d : a1 ! a2 in C such that fd = e. (3) The cofibrations are the maps which have the left lifting property with respect to the maps which are both fibrations and weak equivalences. The weak equivalences are often called DK-equivalences, as they were first de- scribed by Dwyer and Kan in [2]. They are a generalization of the notion of equ* *iv- alence of categories to the simplicial setting. We now state our main theorem. Theorem 1.1. There is a model category structure on the category of all small simplicial categories with the above weak equivalences, fibrations, and cofibra* *tions. We will actually prove the stronger statement that the above model category structure is cofibrantly generated. Recall that a cofibrantly generated model c* *ate- gory is one for which there are two specified sets of morphisms, one of generat* *ing cofibrations and one of generating acyclic cofibrations, such that a map is a f* *ibra- tion if and only if it has the right lifting property with respect to the gener* *ating acyclic cofibrations, and a map is an acyclic fibration if and only if it has t* *he right lifting property with respect to the generating cofibrations. For more details * *about cofibrantly generated model category structures, see [6, Ch. 11]. To prove the * *the- orem, we will use the following proposition, which is stated in more general fo* *rm by Hirschhorn [6, 11.3.1]. Proposition 1.2. Let M be a category with specified classes of weak equivalences and fibrations. Define a map to be a cofibration if it has the left lifting pro* *perty with respect to the acyclic fibrations, and suppose that, with these three classes o* *f mor- phisms, M satisfies model category axioms MC1, MC2, and MC3. Suppose further that there exist sets C and A of maps in M satisfying the following properties: (1) Both C and A permit the small object argument [6, 10.5.15]. (2) A map is a fibration if and only if it has the right lifting property wi* *th respect to the maps in A. (3) A map is an acyclic fibration if and only if it has the right lifting pr* *operty with the maps in C. (4) A map is an acyclic cofibration if and only if it has the left lifting p* *roperty with respect to the fibrations. Then there is a cofibrantly generated model category structure on M in which C * *is a set of generating cofibrations and A is a set of generating acyclic cofibrati* *ons. 4 J.E. BERGNER Let SSets denote the category of simplicial sets. Recall in SSets we have for* * any n 0 the n-simplex [n], its boundary `[n], and, for any 0 k n, V [n, k], which is ` [n] with the kth face removed. Given a simplicial set X, we denote by |X| its geometric realization. The standard model category structure on SSets is cofibrantly generated; the generating cofibrations are the maps `[n] ! [n] for n 0, and the generating acyclic cofibrations are the maps V [n, k] ! [n] for n 1 and 0 k n. More details on simplicial sets and the model category structure on them can be found in [5]. There is a functor (1) U : SSets ! SC which takes a simplicial set X to the category with objects x and y and with Hom (x, y) = X but no other nonidentity morphisms. We will say that a simplicial set K is weakly contractible if all the homotopy groups of |K| are trivial. We will refer to the model category structure on the category of simplicial c* *at- egories with a fixed set O of objects, denoted SCO , such that all the morphisms induce the identity map on the objects, as defined by Dwyer and Kan in [3]. The weak equivalences are the maps satisfying condition W1 and the fibrations are t* *he maps satisfying condition F1. We can then define our generating cofibrations and acyclic cofibrations as fo* *llows. The generating cofibrations are the maps o (C1) U `[n] ! U [n] for n 0, and o (C2) OE ! {x}, where OE is the simplicial category with no objects and {x} denotes the simplicial category with one object x and no nonidentity morphisms. The generating acyclic cofibrations are o (A1) the maps UV [n, k] ! U [n] for n 1, and o (A2) inclusion maps {x} ! H, where {x} is as in C2 and {H} is a set of representatives for the isomorphism classes of simplicial categories wit* *h two objects x and y, weakly contractible function complexes, and only counta* *bly many simplices in each function complex. Furthermore, we require that the inclusion map {x} q {y} ! H be a cofibration in SC{x,y}. The idea behind the set A2 of generating acyclic cofibrations is the fact that two simplicial categories can have a weak equivalence between them which is not* * a bijection on objects, much as two categories can be equivalent even if they do * *not have the same objects. We only require that our weak equivalences are surjective on equivalence classes of objects. Thus, we must consider acyclic cofibrations * *for which the object sets are not isomorphic. In section 2, we show that these proposed generating acyclic cofibrations sat* *isfy the necessary conditions to be a generating set. In section 3, we complete the * *proof of Theorem 1. In section 4, we prove a technical lemma that we needed in section 2. Acknowledgments. The author would like to thank Bill Dwyer for many helpful conversations about this paper. SIMPLICIAL CATEGORIES 5 2. The Generating Acyclic Cofibrations In this section, we will show that a map in SC satisfies properties F1 and F2* * if and only if it has the right lifting property with respect to the maps in A1 an* *d A2. We begin by stating some facts about the Dwyer-Kan model category structure SCO on the category of simplicial categories with a fixed set O of objects. The weak equivalences are the maps which satisfy property W1, and the fibrations are the maps which satisfy property F1. The cofibrations are the maps which have the left lifting property with respect to the acyclic fibrations. However, we would* * like a more explicit description of the cofibrations in this category, for which we * *need some definitions. If C is a simplicial category, then let Ck denote the (discr* *ete) category whose morphisms are the k-simplices of the morphisms of C. Definition 2.1. [3, 7.4] A map f : C ! D in SCO is free if (1) f is a monomorphism, (2) if * denotes the free product, then in each simplicial dimension k, the category Dk admits a unique free factorization Dk = f(Ck) * Fk, where Fk is a free category, and (3) for each k 0, all degeneracies of generators of Fk are generators of F* *k+1. Definition 2.2. [3, 7.5] A map f : C ! D of simplicial categories is a strong r* *etract of a map f0 : C ! D0if there exists a commutative diagram C0 ffi00 ffiffi00f0|| ffiffi0fflffl| fffiffiffif000D0 ffiffi>AAA000A>""" ffiffi""" AAA00 ffiffi""id A_,,_ D _____________//D Then, we have [3, 7.6] that the cofibrations of SCO are precisely the strong retracts of free maps. In particular, a cofibrant simplicial category is a retr* *act of a free category. Given these facts, we now continue with our discussion of the generating acyc* *lic cofibrations. Recall (1) the map U : SSets ! SC. We first consider the set A1 of maps UV [n, k] ! U [n] for n 1 and 0 k n. Using the model category structure on simplicial sets, we can see that a map of simplicial categories has the righ* *t lifting property with respect to the maps in A1 if and only if it satisfies the propert* *y F1. We then consider the maps in A2 which we would also like to be generating acyclic cofibrations and show that maps with the right lifting property with re* *spect to the maps in A1 and A2 are precisely the maps which satisfy conditions F1 and F2. The proof of this statement will take up the remainder of this section, and* * we will treat each implication separately. Proposition 2.3. Suppose that a map f : C ! D of simplicial categories has the right lifting property with respect to the maps in A1 and A2. Then f satisf* *ies condition F2. Before proving this proposition, we state a lemma. 6 J.E. BERGNER Lemma 2.4. Let F be a (discrete) simplicial category with object set {x, y} and one nonidentity morphism g : x ! y. Let E0be a simplicial category also with ob* *ject set {x, y}. Let i : F ! E0 send g to a homotopy equivalence in Hom E0(x, y). Th* *is map i can be factored as a composite F ! H ! E0in such a way that the composite map {x} ! F ! H is isomorphic to a map in A2. We will prove this lemma in section 4. We now prove Proposition 2.3 assuming Lemma 2.4. Proof of Proposition 2.3.Given objects a1 in C and b in D, we need to show that* * a homotopy equivalence e : fa1 ! b in D lifts to a homotopy equivalence d : a1 ! * *a2 for some a2 in C such that fa2 = b and fd = e. So, we begin by considering the objects a = fa1 and b in D. We first consider the case where a 6= b. Define E0 to be the full simplicial subcategory of D with objects a = fa1 and b, and let F be a simplicial category with objects a and b and a single nonidentity morphism g : a ! b. Let i : F ! E0 send g to a homotopy equivalence e : a ! b. By Lemma 2.4, we can factor this map as F ! H ! E0in such a way that the composite {a} ! F ! H is isomorphic to a map in A2. It follows that the composite {a1} ! {a} ! H is also isomorphic to a map in A2. Then consider the composite H ! E0 ! D where the map E0 ! D is the inclusion map. These maps fit into a diagram {a1}_____//C==- | -- | | - |f |fflffl- fflffl| H ______//D The lift exists because we assume that the map f : C ! D has the right lifting property with respect to all maps in A2. Now, composing the map F ! H with the lift sends the map g in F to a map d in C such that fd = e. The map d is a homotopy equivalence since all the morphisms of H are homotopy equivalences and therefore map to homotopy equivalences in C. Now suppose that a = b. Define E0to be the simplicial category with two objec* *ts a and a0such that each function complex of E0is the simplicial set Hom D(a, a) * *and compositions are defined as they are in D. We then define the map E0! D which sends both objects of E0 to a in D and is the identity map on all the function complexes. Given this simplicial category E0, the argument proceeds as above. We now prove the converse. Proposition 2.5. Suppose f : C ! D is a map of simplicial categories which satisfies properties F1 and F2. Then f has the right lifting property with resp* *ect to the maps in A2. Again, we state a lemma before proceeding with the proof of this proposition. Lemma 2.6. Suppose that A ! B is a cofibration, C ! D is a fibration, and B0 ! B is a weak equivalence in a model category M. Then in the following SIMPLICIAL CATEGORIES 7 commutative diagram A __=__//_A____//"C` | | | | | | fflffl|~fflffl|fflfflfflffl| B0 ____//_B____//D a lift B ! C exists if and only if a lift B0! C exists. Proof.If the lift B ! C exists, it follows that the lift B0! C exists via compo* *sition with the map B0! B. To prove the converse, we first note that the map B0 ! B can be factored as the composite B0,! B00i B of a cofibration and a fibration, where each is a weak equivalence because the * *map B0! B is. Therefore by model category axiom MC4, there is a lift in the diagram A _____//"B`0__//B0066n | n n n | | n n ~| fflffl|nn= fflfflfflffl| B _____________//_B It now suffices to show that there is a lift in the diagram B0"____________//_`C77nn ~|| n n n || fflffl|nn fflfflfflffl| B00_____//B____//_D However, this fact again follows from axiom MC4. We are now able to prove the proposition. Proof of Proposition 2.5.We need to show that there exists a lift in any diagram of the form {x} ____//_C>>_ | _ | | _ f| |fflffl_ fflffl| H _____//_D where {x} ! H is a map in A2. Since the map f satisfies property F2, then given an object a1 in C and a homotopy equivalence e : a = fa1 ! b, there exists an object a2 in C and a homotopy equivalence d : a1 ! a2 such that fd = e. Let g : x ! y be a homotopy equivalence in H. Let F denote the subcategory of H consisting of the objects x and y and g its only nonidentity morphism. Consid* *er the composite map {x} ! F ! H and the resulting diagram {x} _____//C|>>_ | _ | | _ | fflffl|_|| F | | | | | fflffl| fflffl| H ______//D 8 J.E. BERGNER Because the map F ! D factors through H, which consists of homotopy equiva- lences, the image of g in D is a homotopy equivalence. Thus, the existence of t* *he lift in the above diagram follows from the fact that the map f satisfies F2. Now, we need to show that the rest of H lifts to C. We begin by assuming that a 6= b and therefore a1 6= a2. Consider the full simplicial subcategory of C w* *ith objects a1 and a2, and denote by C0the isomorphic simplicial category with obje* *cts x and y. Define D0analogously where we take objects x and y rather than a and b. Now, we can work in the category SC{x,y}of simplicial categories with fixed obj* *ect set {x, y}. Note that the map C0! D0is still a fibration in SC, and in fact it * *is a fibration in SC{x,y}. Now define E to be the pullback in the diagram E _____//_C0 | | | | fflffl|fflffl| H _____//D0 Then the map E ! H is also a fibration in SC{x,y}[4, 3.14(iii)]. By Lemma 2.4, we can factor the map F ! E as the composite F ! H0! E for some simplicial category H0such that the composite {x} ! F ! H0is isomorphic to a map in A2. Then, note that the composite map H0 ! E ! H is a weak equivalence in SC{x,y}since all the function complexes of H and H0 are weakly contractible. Now, we have a diagram F __=___//F____//_E>>~ | | ~ | | | ~ | fflffl|~fflffl|fflffl|~ H0 ____//_H____//D in which the dotted arrow lift exists by Lemma 2.6. If a = b, then D0(and possibly C0) as defined above will have only one object* * x. If this is the case, then define the simplicial category D00with two objects x * *and y such that each function complex is the simplicial set Hom D0(x, x) (as in the p* *roof of Proposition 2.3). We can then factor the map H ! D0through the object D00, where the map D00! D sends both objects of D00to a in D and is the identity map on each function complex. If C0 also has one object, then we obtain a simplici* *al category C00in the same way. Then, we can repeat the argument above in the left-hand square of the diagram F _____//_C00__//_C0>> | __ | | | _ | | fflffl|_fflffl| fflffl| H ____//_D00___//D0 to obtain a lift H ! C00, and hence a lift H ! C0via composition. 3. The Model Category Structure In order to show that our proposed model category structure exists, we need to show that our definitions are compatible with one another. In particular, we ne* *ed to prove that the maps with the left lifting property with respect to the fibratio* *ns are exactly the acyclic cofibrations, and that the maps with the right lifting prop* *erty SIMPLICIAL CATEGORIES 9 with respect to the generating cofibrations are exactly the maps which are fibr* *ations and weak equivalences. Before proving these statements, however, we prove that first three model category axioms hold in SC. Proposition 3.1. With the given weak equivalences, fibrations, and cofibrations, SC satisfies model category axioms MC1, MC2, and MC3. Proof.It can be shown that the category of all simplicial categories has all co* *prod- ucts and all coequalizers, and therefore all colimits, and all products and equ* *alizers, and therefore all limits. Thus, SC satisfies MC1. MC2 and MC3 follow as usual, for example, as in [4, 8.10]. We first consider the sets C1 and C2. Suppose we have a map f : C ! D which is a fibration and a weak equivalence. Using simplicial set arguments, we can see * *that a map satisfies conditions F1 and W1 if and only if it has the right lifting pr* *operty with respect to the maps U `[n] ! U [n] for n 0, where U is the map (1) from simplicial sets to simplicial categories defined in the first section. However, the maps U `[n] ! U [n] only generate those cofibrations between simplicial categories with the same number of objects, a condition that we do not require on our cofibrations of simplicial categories. Therefore, we includ* *e as a generating cofibration the map OE ! {x} from the simplicial category with no objects to the single-object simplicial category with no nonidentity morphisms.* * In other words, we are including the addition of an object as a cofibration. Proposition 3.2. A map in SC is a fibration and a weak equivalence if and only if it has the right lifting property with respect to the maps in C1 and C2. Proof.First suppose that f : C ! D is both a fibration and a weak equivalence. * *By conditions F1 and W1, the map Hom C(a, b) ! Hom D(fa, fb) is an acyclic fibrati* *on of simplicial sets for any choice of objects a and b in C. In other words, ther* *e is a lift in any diagram of the form `[n]______//HomC(a, b) q88 | qq | | qq | fflffl|q fflffl| [n]_____//HomD(fa, fb) However, having this lift is equivalent to having a lift in the diagram U `[n]_____//C==z | z z || | z | fflffl|z fflffl| U [n] _____//D where the objects x and y of U `[n] map to a and b in C, and analogously for U * *[n] and D. Hence, f has the right lifting property with respect to the maps in C1. It remains only to show that f has the right lifting property with respect to* * the map OE ! {x}. However, this property is equivalent to f being onto on objects. Being onto on homotopy equivalence classes of objects follows from condition W2. Then suppose that e : a ! b is an isomorphism in D and there is an object a1 in* * C such that fa1 = a. Since e is a homotopy equivalence, by F2 there is a homotopy 10 J.E. BERGNER equivalence in C with domain a1 and which maps to e under f. In particular, the* *re is an object a2 in C mapping to b. Conversely, suppose that f has the right lifting property with respect to the maps in C1 and C2. Again, using the model category structure on simplicial sets, we have that the map Hom C (a, b) ! Hom D(fa, fb) is both a fibration and a weak equivalence, satisfying both F1 and W1. It follo* *ws that Hom ß0C(a, b) ! Hom ß0D(fa, fb) is an isomorphism. As above, having the right lifting property with respect to the map OE ! {x} is equivalent to being onto on objects. These two facts show then that ß0C ! ß0D is an equivalence of categories, proving condition W2. It remains to show that f satisfies property F2. By Proposition 2.3 and the f* *act that satisfying F1 is equivalent to having the right lifting property with resp* *ect to maps in A1, it suffices to show that f has the right lifting property with resp* *ect to the maps in A2. But, a map {x} ! H in A2 can be written as a (possibly infinite) composition of a pushout along OE ! {x} followed by pushouts along maps of the form U `[n] ! U [n], and f has the right lifting property with respect to all s* *uch maps since these are just the maps in C1 and C2. Proposition 3.3. A map in SC is an acyclic cofibration if and only if it has the left lifting property with respect to the fibrations. The proof will require the use of the following lemma: Lemma 3.4. Let A ! B be a map in A1 or A2 and i : A ! C a map in SC. Then in the pushout diagram A __i__//C | | | | fflffl|fflffl| B _____//D the map C ! D is a weak equivalence. Proof.First suppose that the map A ! B is in A2. Let O be the set of objects of C and define O0 to be the set O\{x}. (For simplicity of notation, we assume that ix = x.) Assume as before that x and y are the objects of H. We denote also by O0 the (simplicial) category with object set O0 and no nonidentity morphisms. Consider the diagram X = O q {y}_____//C q {y} = C0 | | | | fflffl| fflffl| H0= O0q H _________//_D and notice that D is also the pushout of this diagram. Since X (regarded as a s* *et) is the object set of any of these categories, note that the left hand vertical * *arrow is a cofibration in SCX. We factor the map X ! C0 as the composite of a cofibration and an acyclic fibration in SCX " ~ XØ____//C00_////_C0. SIMPLICIAL CATEGORIES 11 Since SCX is proper [3, 7.3], it follows from [6, 13.5.4] that the pushouts o* *f each row in the diagram H0oo____?X`____//_C0OOOOOO =|| |=| ~|| | |`Ø " | H0oo____?X ____//_C00 ~|| |=| =|| fflffl| fflffl|Øfflffl|" ß0H0oo___ X ____//_C00 are weakly equivalent to one another. In particular, the pushout of the bottom * *row is weakly equivalent to D. It remains to show that there is a weak equivalence * *of pushouts of the rows of the diagram " ß0H0 oo___XØ____//_C00 | | | | | | fflffl| fflffl|fflffl| ß0H0 oo___X ____//_C0. However, a calculation shows that the pushout of this bottom row is weakly equi* *v- alent in SC to the pushout of the diagram ß0Hoo___{x} ____//_C and therefore that the pushout of the top row is weakly equivalent to the pusho* *ut of the bottom row. It follows that the map C ! D is a weak equivalence in SC. For the maps in A1, we have pushout diagrams UV [n, k]_j__//C. | | | | fflffl| fflffl| U [n] _____//_D As before, define O to be the object set of C. Let O00= O\{x, y}. (Again, for notational simplicity we will assume that jx = x and jy = y.) Now we consider the diagram O00q UV [n, k]___//C | | | | fflffl| fflffl| O00q U [n]______//D in SCO. However, since the left vertical map is a weak equivalence and assuming that the top map is a cofibration (factoring if necessary as above), we can aga* *in use the fact that SCO is proper to show that C ! D is a weak equivalence in SCO and thus also in SC. Proof of Proposition 3.3.First suppose that a map C ! D is an acyclic cofibrati* *on. By the small object argument ([4, Sec. 7] or [6, Ch. 11]), we have a factorizat* *ion of the map C ! D as the composite C ! C0 ! D where C0 is obtained from C by a directed colimit of iterated pushouts along the maps in A1 and A2. Thus, by Lemma 3.4 above and the fact that a directed colimit of such maps is a weak equivalence, this map C ! C0is a weak equivalence. Furthermore, the map C0! D 12 J.E. BERGNER has the right lifting property with respect to the maps in A1 and A2. Thus, by Proposition 2.3, it is a fibration. It is also a weak equivalence since the map* *s C ! D and C ! C0 are, by axiom MC2. In particular, by the definition of cofibration, * *it has the right lifting property with respect to the cofibrations. Therefore, the* *re is a lift in the diagram C __~__//C0>> | ~~ | | ~ | fflffl|fflffl|=~ D _____//D Hence the map C ! D is a retract of the map C ! C0 and therefore also has the left lifting property with respect to fibrations. Conversely, suppose that the map C ! D has the left lifting property with respect to fibrations. In particular, it has the left lifting property with re* *spect to the acyclic fibrations, so it is a cofibration by definition. We again obta* *in a factorization of this map as the composite C ! C0! D where C0is obtained from C by iterated pushouts of the maps in A1 and A2. Once again, the map C0! D has the right lifting property with respect to the maps in A1 and A2 and thus i* *s a fibration by Proposition 2.3. Therefore there is a lift in the diagram C __~__//C0>> | ~~ | | ~ | fflffl|fflffl|=~ D _____//D Again using Lemma 3.4, the map C ! D is a weak equivalence because it is a retract of the map C ! C0. We have now proved everything we need for the existence of the model category structure on SC. Proof of 1.1.It remains to show that the four conditions of Proposition 1.2 are satisfied. It can be shown that both OE and {x} are small, and using the smalln* *ess of V [n, k] and [n] in SSets [7, 3.1.1], it can be shown that each U `[n] is s* *mall relative to the set C1 and each UV [n, k] is small relative to the set A1 [6, 1* *0.5.12]. Therefore, condition 1 holds. Condition 2 follows from Propositions 2.3 and 2.* *5. Condition 3 is proved in Proposition 3.2, and condition 4 is proved in Proposit* *ion 3.3. 4. Proof of 2.4 Recall that we have a (simplicial) category F with objects x and y and a sing* *le nonidentity morphism g : x ! y, and a simplicial category E0 also with objects x and y such that there is a map i : F ! E0which sends g to a homotopy equivalence x ! y in E0. We first replace E0 by its subcategory of weak equivalences which we denote by E. In order to make our constructions homotopy invariant, we take functorial cofibrant replacements eF! F and eE! E in the model category SC{x,y} as given in [3, 2.5], and in this construction eFis actually isomorphic to F. Now, take the localization F-1F (respectively eE-1eE) obtained by formally in- verting all the morphisms in each simplicial degree of F (respectively eE). Th* *ese localizations are the groupoid completions of F and eE, respectively. (In takin* *g a SIMPLICIAL CATEGORIES 13 functorial cofibrant replacement and then the groupoid completion, we have taken the simplicial localizations of F and E with respect to all the morphisms in ea* *ch as defined in [3].) We now have a diagram F oo=__F _____//F-1F | | | | | | fflffl|fflffl| fflffl| E oo___eE_____//eE-1eE. To assure that our next step is homotopy invariant, we factor the map F-1F ! eE-1eEas the composite p F-1F __i__//Z___//eE-1eE where i is an acyclic cofibration and p is a fibration in SC{x,y}. However, we * *will continue to write F-1F rather than Z to avoid more notation than necessary. We take the pullback of the bottom right hand corner of the above diagram and denote it G: G_____//F-1F | | | | fflffl| fflffl| eE____//eE-1eE Lemma 4.1. The composite map {x} ! F ! G is a weak equivalence in SC. Proof.Since the simplicial categories eE, eE-1eE, and F-1F all consist of homot* *opy equivalences, so must G. Therefore, all the morphisms of ß0G are isomorphisms. It then suffices to show that G has weakly contractible function complexes. B* *e- cause all of the morphisms of E, and hence also of eE, are homotopy equivalence* *s, the map eE! eE-1eEis a weak equivalence in SC{x,y}[3, 9.5]. Note that F-1F is the simplicial category in SC{x,y}with exactly one morphism between any two objects. In particular, F-1F has weakly contractible function complexes. Now, because all the categories have as objects x and y and all the maps invo* *lved are the identity on these objects, we can consider the above pullback diagram in SC{x,y}. Since this model category structure is proper [3, 7.3], every pullbac* *k of a weak equivalence along a fibration is a weak equivalence. The map eE! eE-1eE is a weak equivalence and the map F-1F ! eE-1eEis a fibration, so it follows th* *at the map G ! F-1F is a weak equivalence in SC{x,y}, and therefore G has weakly contractible function complexes. Thus, the map {x} ! G satisfies the conditions to be a weak equivalence in SC. However, not all the maps {x} ! G are isomorphic to maps in A2 because the simplicial categories G could have uncountable simplices in their function comp* *lexes. Furthermore, there is no reason to assume that the inclusion map {x} q {y} ! G * *is a cofibration in SC{x,y}. To complete the proof, we need to show that any acycl* *ic cofibration {x} ! G as above factors as a composite {x} ! H ! G where the inclusion map {x} ! H is in A2. Let H0 be the simplicial category F. Let i : H0 ! G be the inclusion map. We will construct a simplicial category H from H0 satisfying the necessary propert* *ies specified in A2. We first state the following lemma: 14 J.E. BERGNER Lemma 4.2. Let f : A ! B be a map of simplicial sets where B is weakly con- tractible, and let u : Sn ! |A| be a map of CW-complexes for some n 0. Then f can be factored as a composite A ! A0! B where A0is obtained from A by at- taching a finite number of nondegenerate simplices and the composite map of spa* *ces Sn ! |A| ! |A0| is null homotopic. Proof.We first assume that the map f is a cofibration; if not, we factor it as * *the composite A __i_//_B0p_//_B where in the model category structure on simplicial sets i is a cofibration and* * p is an acyclic fibration. Thus, we can assume that f is an inclusion map, replacing* * B by B0 if needed. Now consider the composite map of spaces Sn ! |A| ! |B|, which is necessarily null homotopic since B is weakly contractible. The composite map then factors through CSn, the cone on Sn, and we have a diagram Sn _____//_|A|___//|B|66mmm | mmmmm | mmmm fflffl|mmm CSn Now, since CSn is compact, its image will intersect only a finite number of c* *ells of |B| nontrivially. Then define A0to be a simplicial set such that |A0| contai* *ns |A| as well as all the cells in this image. Now, consider the categories H0 and G as described above and the inclusion map i : H0 ! G. Each of these categories has four function complexes to conside* *r. For the category H we call them Hj, and for G we call them Gj for 1 j 4. (The numbering is arbitrary but must match up between the two categories. So if H1 = Hom H(x, y), then we must have G1 = Hom G(x, y).) Identify n 0 such that all maps Sm ! |Hj| are null homotopic for all 0 m < n and all 1 j 4, but there is a map Sn ! |Hj| which is not null homotopic f* *or some j. We then apply Lemma 4.2 to the map Hj ! Gj and the map Sn ! |Hj|. Replace the function complex Hj with the simplicial set A0obtained from Lemma 4.2. This process may result in more maps Sm ! |A0| which are not null homotopic than for the original |Hj|, but only for m > n. Also, it will not have more than countably many more such maps than |Hj| did. Now that we have added simplices to our function complex, we include all necessary compositions of these morphis* *ms with the original morphisms of H0 to obtain a new simplicial category which we denote H1. There will be at most countably many new simplices added from these compositions. Repeat the above process with another map from Sn to a function complex of H1, again, where n is minimal, to obtain another category H2. Contin* *ue, perhaps countably many times, to obtain a category H such that for any n and any function complex H0of H, any map Sn ! |H0| is nullhomotopic. To show that it is possible to obtain such an H in this way, we need only show that there are at m* *ost countably many homotopy classes of maps from spheres to each function complex that need to be killed off. However, this fact follows from the following lemma: Lemma 4.3. Let A be a simplicial set with countably many simplices. Then for all n 0 there are at most countably many distinct homotopy classes of maps Sn ! |A|. SIMPLICIAL CATEGORIES 15 Proof.It suffices to show that there are at most countably many homotopy classes of maps from Sn into any finite CW complex X. For a simply connected CW complex X, an argument using Serre mod C theory [8] shows that all the homotopy groups of X are countable if and only if the homology groups of X are countable, which they are when X is finite. The case of a general CW complex X follows from this one using a universal cover argument. By construction, this simplicial category H is free, and therefore the map {x} q {y} ! H is a cofibration in SC{x,y}. Thus, we have obtained a factorization {x} ! H ! G. We are now able to complete the proof of 2.4. Proof of 2.4.Using the simplicial category H from above and the map H ! G, we obtain a composite map {x} ! F ! H ! G ! eE! E ! E0. In particular, we have a factorization F ! H ! E0. As we have shown above, the composite {x} ! F ! H is isomorphic to a map in A2. References [1]W.G. Dwyer, P.S. Hirschhorn, and D.M. Kan. Model Categories and More General* * Abstract Homotopy Theory, preprint, available at http://www-math.mit.edu/~psh. [2]W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra, Topology* * 19 (1980), 427-440. [3]W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, Journal of * *Pure and Applied Algebra 17 (1980) 267-284. [4]W.G. Dwyer and J. Spalinski, Homotopy theories and model categories, in Hand* *book of Alge- braic Topology, Elsevier, 1995. [5]P.G. Goerss and J.F. Jardine. Simplicial Homotopy Theory, Progress in Math, * *vol. 174, Birkhauser, 1999. [6]Philip S. Hirschhorn, Model Categories and Their Localizations, Mathematical* * Surveys and Monographs 99, AMS, 2003. [7]Mark Hovey, Model Categories, Mathematical Surveys and Monographs 63, AMS, 1* *999. [8]Serre, Jean-Pierre. Groupes d'homotopie et classes de groupes ab'eliens, Ann* *als of Mathemat- ics, 58 (1953), 258-294. [9]Bertrand To"en and Gabriele Vezzosi. Homotopical algebraic geometry I: topos* * theory, preprint available at http://arxiv.org/PS_cache/math/pdf/0207/0207028.pdf. Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 E-mail address: jbergner@nd.edu