SIMPLICIAL MONOIDS AND SEGAL CATEGORIES JULIA E. BERGNER Abstract.Much research has been done on structures equivalent to topolog- ical or simplicial groups. In this paper, we consider instead simplicial* * monoids. In particular, we show that the usual model category structure on the ca* *tegory of simplicial monoids is Quillen equivalent to an appropriate model cate* *gory structure on the category of simplicial spaces with a single point in de* *gree zero. In this second model structure, the fibrant objects are reduced Se* *gal categories. We then generalize the proof to relate simplicial categories* * with a fixed object set to Segal categories with the same fixed set in degree z* *ero. 1.Introduction There has been much work done showing the equivalences of topological (or simplicial) groups, group-like A1 -spaces, and loop spaces. References for this* * work include [22], [23], and the more recent [1] and [2]. In this paper, we show th* *at there are some analogous comparisons to be made when we work with simplicial monoids rather than simplicial groups. We show that, from the perspective of homotopy theory, simplicial monoids are essentially the same as reduced Segal categories (Definition 1.2). In order to describe the problem more fully, we begin with some notation and terminology. Let denote the cosimplicial category, or category whose objects are finite ordered sets [n] = {0, . .,.n} for n 0 and whose morphisms are ord* *er- preserving maps between them. Then op is the opposite of this category and is called the simplicial category. Recall that a simplicial set X is a functor op* * ! Sets. We will denote the category of simplicial sets by SSets. (In the course o* *f this paper we will sometimes refer to simplicial sets as spaces, due to their homoto* *py- theoretic similarity with topological spaces [15, 3.6.7].) Induced from the map* *s in op are the face maps di : Xn ! Xn-1 and degeneracy maps si : Xn ! Xn-1. A few of the simplicial sets we will use are the n-simplex [n] for each n 0,* * its boundary `[n], and the boundary with the kth face removed, V [n, k]. More detai* *ls about simplicial sets can be found in [12, I]. We denote by |X| the topological* * space given by geometric realization of the simplicial set X [12, I.2]. More generally, a simplicial object in a category C is a functor op ! C. In particular, a functor op! SSets is a simplicial space or bisimplicial set [12,* * IV]. Given a simplicial set X, we will also use X to denote the constant simplicial * *space with the simplicial set X in each degree. By Xt (which should be thought of as "X-transposed") we will denote the simplicial space such that (Xt)n is the cons* *tant ____________ Date: August 29, 2005. 2000 Mathematics Subject Classification. Primary 18G30; Secondary 18E35, 18C* *10, 55U40. Key words and phrases. simplicial monoids, Segal categories, simplicial cate* *gories, model categories. 1 2 J.E. BERGNER simplicial set Xn, or the simplicial set which has theosetpXn at each level. We* * will denote the category of all simplicial spaces by SSets . In this paper, we consider a very specific kind of simplicial space, namely, a reduced Segal category. We first consider the more basic definition of a Segal precategory. Definition 1.1. A Segal precategory is a simplicial space X such that X0 is a discrete simplicial set. If X0 consists of a single point, then we will call X * *a reduced Segal precategory. Now note that for any simplicial space X there is a Segal map 'k : Xk ! X1_xX0_._.x.X0X1_-z_______" k for each k 2. (The precise definition of this map is given in section 2.) Definition 1.2. [14, x2] A Segal category X is a Segal precategory such that the Segal map 'k is a weak equivalence of simplicial sets for each k 2. A reduced Segal category is a Segal category such that X0 consists of a single point. Now, let M be a simplicial monoid, by which we mean a simplicial object in the category Mon of monoids, or a functor op! Mon. We will use an alternate viewpoint in which we use algebraic theories to define simplicial monoids. We b* *egin with the definition of an algebraic theory. Some references for algebraic theo* *ries include chapter 3 of [6], the introduction to [1], and section 3 of [4]. Definition 1.3. An algebraic theory T is a small category with finite products * *and objects denoted Tn for n 0. For each n, Tn is equipped with an isomorphism Tn ~=(T1)n. Note in particular that T0 is the terminal object in T . Here we will consider one particular theory, the theory of monoids, which we denote TM . To describe this theory, we first consider the full subcategory of * *the category of monoids whose objects Tn are the isomorphism classes of free monoids on n generators. We then define the theory of monoids TM to be the opposite of this category. Thus Tn, which is canonically the coproduct of n copies of T1 in Mon, becomes the product of n copies of T1 in TM . It follows that there is a projection map pi : Tn ! T1 for each i 2 in addition to other monoid maps. In fact, there are such projection maps in any algebraic theory. We use them to ma* *ke the following definition. Definition 1.4. [1, 1.1] Given an algebraic theory T, a strict simplicial T-alg* *ebra (or simply T-algebra) A is a product-preserving functor A : T ! SSets. Here, "product-preserving" means that for each n 0 the canonical map A(Tn) ! A(T1)n, induced by the n projection maps Tn ! T1, is an isomorphism of simplicial sets. In particular, A(T0) is the one-point simplicial set [0]. In general, a T-algebra A defines a strict algebraic structure on the space A* *(T1) corresponding to the theory T [1, x1]. So, a TM -algebra A defines a monoid str* *ucture on the space A(T1). It turns out that the category of simplicial monoids is equ* *ivalent to the category of TM -algebras [18, II.4]. We can also consider the case where the products are not preserved strictly, * *but only up to homotopy. SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 3 Definition 1.5. [1, 1.2] Given an algebraic theory T, a homotopy T-algebra is a functor X : T ! SSets which preserves products up to homotopy. The functor X preserves products up to homotopy if, for each n 0 the canonical map X(Tn) ! X(T1)n induced by the projection maps pi : Tn ! T1 is a weak equivalence of simplicial sets. In particular, we assume that X(T0) is weakly equivalent to [0]. Now that we have established our definitions for simplicial monoids, we turn * *to reduced Segal categories. As with the simplicial monoids, we would like to think of our reduced Segal categories as objects in a category of diagrams of a parti* *cular shape. In this case our diagram will be the simplicial category op. Because op is not an algebraic theory, and in particular because [n] is not isomorphic to * *the product of n copies of [1], we cannot use the same terminology of algebras as we did for the theories. We first define n specified maps [1] ! [n] in . Let ffi : [1] ! [n] be the m* *ap such that ffi(0) = i and ffi(1) = i + 1, defined for each 0 i n - 1. Using * *these maps, we will require the following product property on the diagrams: Definition 1.6. A functor F : op ! SSets is special if F ([0]) = [0] and if t* *he canonical map F ([n])_'_//_F ([1])n induced by the map a (ffi: [1] ! [n]) 0 i n-1 is a weak equivalence of simplicial sets. Thus, a reduced Segal category is a special functor F : op! SSets. Note that unlike in the case of homotopy T-algebras, we actually require F ([0]) to be [* *0], so that we actually get a reduced Segal category and not just a simplicial space equivalent to one. The precise relationship that we would like to prove between the simplicial monoids (or TM -algebras) and Segal categories is a Quillen equivalence of model categories. (We will give a brief overview of model categories and Quillen equi* *va- lences in the next section.) In this paper, we will prove the existence of a model category structure AlgT* *M on the category of TM -algebras. There is another model category structure LSSp*,f* *on the category of reduced Segal precategories such that the fibrant objects are r* *educed Segal categories. (We cannot obtain a model category structure on the category of reduced Segal categories because this category is not closed under limits and colimits.) Our main result is the following theorem: Theorem 1.7. The model category structure AlgTM is Quillen equivalent to the model category structure LSSp*,f. We prove this theorem in section 4 after reviewing model categories in sectio* *n 2 and setting up our particular model category structures in section 3. In sectio* *n 5, we generalize this result to an analogous statement for simplicial categories w* *ith a fixed object set O and Segal categories with the same fixed set O in degree zer* *o. In [5], we prove the more general result that there is a Quillen equivalence betwe* *en a model category structure on the category of all small simplicial categories (wh* *ere 4 J.E. BERGNER the weak equivalences are a simplicial version of equivalences of categories [3* *, x1]) and a model category structure on the category of all Segal precategories (with analogous weak equivalences). The techniques of this paper should carry over to a similar result for simpli* *cial groups by modifying the category op. Let I[n] denote the category with n + 1 objects and exactly one isomorphism between any two objects [19, x6]. Analogous* *ly to the way we form from the categories [n], we define the category I whose objects are the I[n] for all n 0 and whose morphisms are order-preserving map* *s. (Here "order-preserving" should be taken to mean "just like the maps in ," sin* *ce the ordering of objects in I[n] is less clear than in [n].) Now, a special functor I op ! SSets is a reduced Segal groupoid. We can then consider the theory of groups TG and strict TG -algebras, which are essent* *ially simplicial groups. The arguments of this paper should give us the result that t* *he model category structure on TG -algebras is Quillen equivalent to an appropriate model structure on "reduced Segal pregroupoids." The explicit argument will be included in a later paper. The idea here, for both monoids and groups, is that when we consider diagrams* * of spaces given by a particular algebraic theory, from a homotopy-theoretic perspe* *ctive we can actually consider diagrams of spaces given by some simpler, pared-down diagram in the sense that op can be considered a simplification of TM . Similar work has been done by Segal [21] for groups and abelian monoids, and by Bousfie* *ld [7] for groups and n-fold loop spaces. We hope to find more such examples in the future. Acknowledgments. I am grateful to Bill Dwyer for many helpful conversations about this paper. 2.A Review of Model Categories Recall that a model category structure on a category C is a choice of three d* *is- tinguished classes of morphisms: fibrations, cofibrations, and weak equivalence* *s. A (co)fibration which is also a weak equivalence will be called an acyclic (co)fi* *bration. With this choice of three classes of morphisms, C is required to satisfy five a* *xioms MC1-MC5 [11, 3.3]. An object X in C is fibrant if the unique map X ! * from X to the terminal object is a fibration. Dually, X is cofibrant if the unique map OE ! X from the* * initial object to X is a cofibration. The factorization axiom (MC5) guarantees that each object X has a weakly equivalent fibrant replacement bXand a weakly equivalent cofibrant replacement eX. These replacements are not necessarily unique, but th* *ey can be chosen to be functorial in the cases we will use [15, 1.1.3]. The model category structures which we will discuss are all cofibrantly gener* *ated. A cofibrantly generated model category C is a model category for which there are two sets of morphisms, one of generating cofibrations and one of generating acy* *clic cofibrations, such that a map is a fibration if and only if it has the right li* *fting property with respect to the generating acyclic cofibrations, and a map is an a* *cyclic fibration if and only if it has the right lifting property with respect to the * *generating cofibrations [13, 11.1.2]. (Recall that if the dotted arrow lift exists in the * *solid arrow SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 5 diagram A _____//X>>" |i|" " |p| |fflffl"fflffl| B _____//Y then we say that i has the left lifting property with respect to p and that p h* *as the right lifting property with respect to i.) To state the conditions we will* * use to determine when we have a cofibrantly generated model category structure, we make the following definition. Definition 2.1. [13, 10.5.2] Let C be a category and I a set of maps in C. Then an I-injective is a map which has the right lifting property with respect to ev* *ery map in I. An I-cofibration is a map with the left lifting property with respect* * to every I-injective. Using this definition, we will use the following theorem to determine when we have a cofibrantly generated model category structure. Theorem 2.2. [13, 11.3.1] Let M be a category which has all finite limits and c* *ol- imits. Suppose that M has a class of weak equivalences which satisfies the two-* *out- of-three property (model category axiom MC2) and which is closed under retracts. Let I and J be sets of maps in M which satisfy the following conditions: (1) Both I and J permit the small object argument [13, 10.5.15]. (2) Every J-cofibration is an I-cofibration and a weak equivalence. (3) Every I-injective is a J-injective and a weak equivalence. (4) One of the following conditions holds: (i)A map that is an I-cofibration and a weak equivalence is a J-cofibra* *tion, or (ii)A map that is both a J-injective and a weak equivalence is an I- injective. Then there is a cofibrantly generated model category structure on M in which I * *is a set of generating cofibrations and J is a set of generating acyclic cofibrati* *ons. We will refer to the standard model category structure on the category SSets of simplicial sets. In this case, a weak equivalence is a map of simplicial se* *ts f : X ! Y such that the induced map |f| : |X| ! |Y | is a weak homotopy equivalence of topological spaces. The cofibrations are monomorphisms, and the fibrations are the maps with the right lifting property with respect to the acy* *clic cofibrations [12, I.11.3]. This model category structure is cofibrantly generat* *ed; 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. Each model structure in this paper will have the additional structure of a si* *m- plicial model category. A simplicial category is a category with a simplicial * *set of morphisms between any two objects. A simplicial model category, then, is a model category M which is also a simplicial category, satisfying some axioms [1* *3, 9.1.6]. (Notice that this terminology is somewhat confusing in that simplicial * *ob- jects in the category of categories are not necessarily simplicial categories, * *nor are simplicial objects in the category of model categories actually simplicial mode* *l cat- egories, at least not without imposing further conditions.) The important part * *of this structure is that it enables us to talk about the function complex, or sim* *plicial 6 J.E. BERGNER set Map (X, Y ) for any pair of objects X and Y in M. For example, the model category SSets is a simplicial model category. For simplicial sets X and Y , t* *he simplicial set Map (X, Y ) is given by Map (X, Y )n = Hom (X x [n], Y ). However, a function complex is not necessarily homotopy invariant, so we have t* *he following definition. Definition 2.3. [13, 17.3.1] If M is a simplicial model category, then the homo- topy function complex Map h(X, Y ) is given by Map (Xe, bY), where eXis a cofib* *rant replacement of X in M and bYis a fibrant replacement for Y . More generally, a homotopy function complex Map h(X, Y ) in a (not necessarily simplicial) model category M, or in a category with weak equivalences, is the s* *im- plicial set Map (X, Y ) given by the simplicial set of morphisms between X and Y in the hammock localization LH M of M [9]. Many of the model category structures that we will work with will be obtained by localizing a given model category structure with respect to a set of maps. To make sense of this notion, we make the following definitions. Definition 2.4. Suppose that S = {f : A ! B} is a set of maps in a model category M. An S-local object X is a fibrant object of M such that for any f : A ! B in S, the induced map on homotopy function complexes f* : Map h(B, W ) ! Map h(A, W ) is a weak equivalence of simplicial sets. A map g : X ! Y in M is then an S-loc* *al equivalence if for every local object W , the induced map on homotopy function complexes g* : Map h(Y, W ) ! Map h(X, W ) is a weak equivalence of simplicial sets. We now define the localization of a model category M. The following theorem holds for all model categories M which are left proper and cellular. We will n* *ot define these properties here but refer the reader to [13, 13.1.1, 12.1.1] for d* *etails. All the model categories we will work with can be shown to satisfy these proper* *ties. Theorem 2.5. [13, 4.1.1] Let M be a left proper cellular model category. There * *is a model category structure LSM on the underlying category of M such that: (1) The weak equivalences are the S-local equivalences. (2) The cofibrations are precisely the cofibrations of M. (3) The fibrations are the maps which have the right lifting property with r* *espect to the maps which are both cofibrations and S-local equivalences. (4) The fibrant objects are the S-local objects. (5) If M is a simplicial model category, then its simplicial structure induc* *es a simplicial structure on LSM. In particular, given an object X of M, we can talk about its functorial fibra* *nt replacement LX in LSM. The object LX is an S-local object which is fibrant in M, and we will call it the localization of X in LSM. We now state the definition of a Quillen pair of model category structures. R* *ecall that for categories C and D a pair of functors F : C____//Do:oR_ SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 7 is an adjoint pair if for each object X of C and object Y of D there is an isom* *orphism ' : Hom D (F X, Y ) ! Hom C(X, RY ) [16, IV.1]. The adjoint pair is sometimes written as the triple (F, R, '). Definition 2.6. [15, 1.3.1] If C and D are model categories, then an adjoint pa* *ir (F, R, ') between them is a Quillen pair if one of the following equivalent sta* *tements holds: (1) F preserves cofibrations and acyclic cofibrations. (2) R preserves fibrations and acyclic fibrations. We will use the following theorem to show that we have a Quillen pair of loca* *lized model category structures. Theorem 2.7. [13, 3.3.20] Let C and D be left proper, cellular model categories and let (F, R, ') be a Quillen pair between them. Let S be a set of maps in C a* *nd LSC the localization of C with respect to S. Then if LF S is the set in D obtai* *ned by applying the left derived functor of F to the set S [13, 8.5.11], then (F, R* *, ') is also a Quillen pair between the model categories LSC and LLFS D. We now have the following definition of Quillen equivalence, which is the sta* *n- dard notion of "equivalence" of model category structures. Definition 2.8. [15, 1.3.12] A Quillen pair is a Quillen equivalence if for all* * cofi- brant X in C and fibrant Y in D, a map f : F X ! Y is a weak equivalence in D if and only if the map 'f : X ! RY is a weak equivalence in C. We will use the following proposition to prove that our Quillen pairs are Qui* *llen equivalences. Recall that a functor F : C ! D reflects a property if, for any morphism f of C, whenever F f has the property, then so does f. Proposition 2.9. [15, 1.3.16] Suppose that (F, R, ') is a Quillen pair from C to D. Then the following statements are equivalent: (1) (F, R, ') is a Quillen equivalence. (2) F reflects weak equivalences between cofibrant objects, and for every fi* *brant Y in D the map F (RY )c ! Y is a weak equivalence. (3) R reflects weak equivalences between fibrant objects, and for every cofi* *brant X in C the map X ! R(F X)f is a weak equivalence. 3. Model Category Structures In this section, we set up the model category structures that we will need to prove Theorem 1.7. Because we will need to consider a model structure on the category of simplic* *ial monoids, we now describe a model structure on the category of T-algebras for any algebraic theory T. Proposition 3.1. [18, II.4], [20, 3.1] Let T be an algebraic theory and AlgT th* *e cate- gory of T-algebras. Then there is a cofibrantly generated model category struct* *ure on AlgT in which the weak equivalences and fibrations are levelwise weak equivalen* *ces of simplicial sets and the cofibrations are the maps with the left lifting prop* *erty with respect to the maps which are fibrations and weak equivalences. 8 J.E. BERGNER We would also like to have a model category structure for homotopy T-algebras. However, we will not have a model category structure on the category of homotopy T-algebras itself; the category of homotopy T-algebras does not have all coprod- ucts. We will have a model category structure on the category of all T-diagrams* * of simplicial sets in which homotopy T-algebras are the fibrant objects. To obtain* * this structure, we begin by considering the category of all functors T ! SSets, which we denote by SSetsT. Theorem 3.2. [12, IX 1.4] There is a model category structure on SSetsT in which the weak equivalences are the levelwise weak equivalences, the fibrations are t* *he levelwise fibrations, and the cofibrations are the maps which have the left lif* *ting property with respect to the maps which are both fibrations and weak equivalenc* *es. The desired model structure can be obtained by localizing the model structure on SSetsT with respect to a set of maps. We will summarize this localization he* *re; a complete description is given by Badzioch [1, x5]. In an algebraic theory T, consider the functor Hom T(Tk, -). We then have maps a pk : Hom T(T1, -) ! Hom T(Tk, -) k induced from the projection maps in T. We then localize the model category stru* *c- ture on SSetsT with respect to the set S = {pk|k 0}. We denote the resulting model category structure LSSetsT. Proposition 3.3. [1, 5.5]. The fibrant objects in LSS etsT are the homotopy T- algebras which are fibrant in SSetsT. We now have the following result by Badzioch. Theorem 3.4. [1, 6.4] Given an algebraic theory T, there is a Quillen equivalen* *ce of model categories between AlgT and LSSetsT. By applying this Quillen equivalence to the theory of monoids TM , we have reduced the proof of our main theorem to finding a Quillen equivalence between LSSetsTM and an appropriate model structure for reduced Segal categories. As with the homotopy T-algebras, we will need to consider the category of reduced Segal precategories and find a model structure in which the fibrant objects are Segal categories. We consider the general case of the category whose objects are Segal precateg* *ories with a fixed set O in degree zero and whose morphisms are object-preserving on that set. We will prove the existence of a model structure on this category in * *which the weak equivalences are levelwise weak equivalences of simplicial sets. We gi* *ve the proof here for any set O, since we will need it for the more general constr* *uction in the last section of the paper. Thus, let SSpO denote the category whose obje* *cts are simplicial spaces with a fixed set O in degree zero and whose morphisms are maps of simplicial spaces which are the identity map on O. This model category structure on SSpO will be analogous to the projective mod* *el category structure on simplicial spaces in which the fibrations and weak equiva* *lences are levelwise fibrations and weak equivalences of simplicial sets [12, IX.1.4].* * How- ever, this structure must be modified so that the objects defining the generati* *ng cofibrations and generating acyclic cofibrations will be in the category SSpO .* * We begin by understanding limits and colimits in SSpO . SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 9 Lemma 3.5. SSpO has finite limits. Proof.Suppose that for each object ff of an index category D, Xffis a simplicial space with the set O in degree zero. In the category of all simplicial spaces, * *we have the limit limDXff. However, if D is not connected, then this limit will not be * *in the category SSpO since limD(Xff)0 = Oss0D. However, if diag: O ! limD(Xff)0 is the diagonal map, we can define the limit in SSpO , denoted limODXff, as the pullba* *ck in the diagram limODXff______//limDXff | | | | fflffl|diag fflffl| O ________//limD(Xff)0 This new object now satisfies the universal property of limits when we require * *the maps involved to be the identity on degree zero and hence in our category SSpO . Lemma 3.6. SSpO has finite colimits. Proof.As with the limits, begin by considering the ordinary colimit in simplici* *al spaces. Again, let Xff, indexed by the objects ff of D, be objects in SSpO . No* *te that again we have a problem in degree zero if our`index category D has more th* *an one component, since in this case colimD(Xff)0 = ss0DO. Then if we consider t* *he fold map fold: colimD(Xff)0 ! O, we can define the colimit in SSpO , denoted colimODXffas the pushout in the dia* *gram colimD(Xff)0__fold__//O | | | | fflffl| fflffl| colimDXff______//colimODXff where the left-hand vertical map is the inclusion map. Similarly to the case f* *or limits, this new simplicial space satisfies the universal property for colimits. Proposition 3.7. There is a model category structure on SSpO , which we denote SSpO,f, in which the weak equivalences are levelwise weak equivalences of simpl* *icial sets, the fibrations are the levelwise fibrations of simplicial sets, and the c* *ofibra- tions are the maps with the left lifting property with respect to the maps whic* *h are fibrations and weak equivalences. The first step towards defining this model category structure is finding cand* *idates for the generating cofibrations and generating acyclic cofibrations.oRecallptha* *t for the projective model category structure on the category SSets of simplicial spaces, in which the weak equivalences and fibrations are levelwise, the genera* *ting acyclic cofibrations are of the form V [m, k] x [n]t! [m] x [n]t [12, IV.3.1]. (Recall that by [n]t we denote the simplicial space which is the constant simplicial set [n]k in degree k.) The first problem in using these maps for SSpO,f is that [n]t is not going to be in SSpO for all values of n. Instead, we need to define a separate n-simplex* * for 10 J.E. BERGNER any n-tuple x0, . .,.xn of objects in O, denoted [n]tx0,...,xn, so that the ob* *jects are preserved. Setting x_= (x1, . .,.xn), we write this simplex as [n]tx_. (The va* *lues of the xi can repeat in a particular x_.) Note that this object [n]tx_also nee* *ds to have all elements of O as 0-simplices, so we add any of these elements that have not already been included in the xi's, plus their degeneracies in higher degree* *s. Now we consider the modified generating acyclic cofibrations V [m, k] x [n]tx_! [m] x [n]tx_. However, these objects are still not in SSpO,f because the simplicial set in de* *gree zero for each is not discrete. This problem can be fixed by collapsing the copi* *es of [m] and V [m, k] in degree zero to their respective xi's. Note that the degene* *racies then get collapsed as well. More explicitly, we define the object (Rm,n,k)x_to * *be the pushout of the diagram V [m, k] x ( [n]tx_)0//_V [m, k] x [n]tx_ | | | | fflffl| fflffl| ( [n]tx_)0_________//_(Rm,n,k)x_. Similarly, we define the object (Qm,n)x_to be the pushout of the diagram [m] x ( [n]tx_)0__// [m] x [n]tx_ | | | | fflffl| fflffl| ( [n]tx_)0________//_(Qm,n)x_. Now we are able to define the set of maps Jf = {(Rm,n,k)x_! (Qm,n)x_} where m, n 1, 0 k m, and x_= (x0, . .,.xn) 2 On+1. This set Jf will be a * *set of generating acyclic cofibrations for SSpO,f. Similarly, we can define the set If = {(Pm,n)x_! (Qm,n)x_} for all m, n 0 and x_2 On+1, where (Pm,n)x_is the pushout of the diagram ` [m] x ( [n]tx_)0_//`[m] x [n]tx_ | | | | fflffl| fflffl| ( [n]tx_)0________//_(Pm,n)x_. We will show that these maps are a set of generating cofibrations for SSpO,f. Proof of Theorem 3.7.Lemmas 3.5 and 3.6 show that our category has finite lim- its and colimits. The two-out-of-three property and the retract axiom for weak equivalences follow as usual; see for example [11, 8.10]. It now suffices to ch* *eck the conditions of Theorem 2.2. To prove statement (1), notice that the maps in the sets If and Jf are modified versions of the generating cofibrations in the proj* *ec- tive model category structure on simplicial spaces, which permit the small obje* *ct argument [13, 10.5.15]. Hence the ones in If and Jf do also. SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 11 Notice that the If-injectives are precisely the levelwise acyclic fibrations,* * and that the Jf-injectives are precisely the levelwise fibrations. Thus, we have sa* *tisfied conditions (3) and (4)(ii). Now notice that the If-cofibrations are precisely the cofibrations, by our de* *f- inition of cofibration. Furthermore, the Jf-cofibrations are the maps with the left lifting property with respect to the fibrations. In particular, they are * *the acyclic cofibrations in the projective model category structure on simplicial s* *paces which are between Segal precategories and leave the degree zero space fixed. Si* *nce the cofibrations can be described analogously, condition (2) holds, completing * *the proof. We now want to localize this model category so that the fibrant objects are Segal categories with the set O in degree zero. If we can find an appropriate m* *ap to localize with respect to, then the desired model category structure will fol* *low from Theorem 2.5. We first consider the map ' used by Rezk [19, x4] to localize simplicial spac* *es to obtain more general Segal spaces, then modify it so that the objects are in SSp* *O,f. Rezk defines a map ffi : [1] ! [k] in such that 0 7! i and 1 7! i + 1 for e* *ach 0 i k - 1. Then for each k he defines the object k-1[ G(k)t= ffi [1]t i=0 and the inclusion map 'k : G(n)t ! [k]t. His localization is with respect to t* *he coproduct of inclusion maps a ' = (G(k)t! [k]t). k 0 However, in our case, the objects G(k)t and [k]t are not in the category SSp* *O . As before, we can replace [k]t with the objects [k]tx_, where x_= (x0, . .,.x* *k). Then, define k-1[ G(k)tx_= ffi [1]txi,xi+1. i=0 Now, we need to take coproducts not only over all values of k, but also over all k-tuples of vertices. We first define for each k 0 the map a 'k = (G(k)tx_! [k]tx_). x_2Ok+1 Then the map ' looks like a a ' = ('k : (G(k)tx_! [k]tx_)). k 0 x_2Ok+1 When the set O is not clear from the context, we will write 'O to specify that * *we are in SSpO . For any simplicial space X, there is a weak equivalence of simplicial sets a Map h( G(k)tx_, X) ! X1 xhX0. .x.hX0X1 x_2Ok+1 ________-z_______"k 12 J.E. BERGNER where the right-hand side is the homotopy limit of the diagram X1 _d0_//_X0od1oX1___d0_//._.d.0//_X0d1ooX1_ with k copies of X1. However, in this case, since X0 is discrete we can take t* *he limit X1_xX0_._.x.X0X1_-z_______" k on the right hand side. Then for any simplicial space X there is a map a a 'k = Map h('k, X) : Map h( [k]tx_, X) ! Map h( G(k)tx_, X). x_ x_ More simply written, this map is 'k : Xk ! X1_xX0_._.x.X0X1_-z_______" k and is often called a Segal map. Localizing with respect to this map ', then, will result in a model category structure in which the local objects are those objects X of SSpO,f for which ea* *ch Segal map Xk ! X1_xO_._.x.OX1_-z______"is a weak equivalence of simplicial sets. k Proposition 3.8. Localizing the model category structure on SSpO,f with respect to the map 'O results in a model category structure LSSpO,f on simplicial spaces with a fixed set O in degree zero in which the weak equivalences are the 'O -lo* *cal equivalences, the cofibrations are those of SSpO,f, and the fibrations are the * *maps with the right lifting property with respect to the cofibrations which are 'O -* *local equivalences. Furthermore, the fibrant objects are the Segal categories which * *are fibrant in SSpO,f. Proof.The proof follows from Theorem 2.5 and the above argument. For making some of our calculations, we will find it convenient to work in a model category structure SSpO,cin which the weak equivalences are again given by levelwise weak equivalences of simplicial sets, but in which the cofibrations, * *rather than the fibrations, are levelwise. Theorem 3.9. There is a model category structure SSpO,con the category of Segal precategories with a fixed set O in degree zero in which the weak equivalences * *and cofibrations are levelwise, and in which the fibrations are the maps with the r* *ight lifting property with respect to the acyclic cofibrations. To define sets Ic and Jc which will be our candidates for generating cofibra- tions and generating acyclic cofibrations, respectively, we first recall the ge* *nerating cofibrations and acyclic cofibrations in the Reedy (or injective) model category structure on simplicial spaces, in which the weak equivalences and cofibrations* * are levelwise. The generating cofibrations are the maps `[m] x [n]t[ [m] x `[n]t! [m] x [n]t for all m, n 0, and similarly the generating acyclic cofibrations are the maps V [m, k] x [n]t[ [m] x `[n]t! [m] x [n]t SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 13 for all n 0, m 1, and 0 k m [19, 2.4]. To modify these maps, we begin by considering the categoryoSeCatpof all Segal precategories and the inclusion functor SeCat ! SSets . This functor has a le* *ft adjoint which we call the reduction functor. Given a simplicial space X, we den* *ote its reduction by (X)r. Reducing X essentially amounts to collapsing the space X0 to its set of components and making the appropriate changes to degeneracies in higher degrees. So, we start by reducing the objects defining the Reedy generat* *ing cofibrations and generating acyclic cofibrations to obtain maps of the form ( `[m] x [n]t[ [m] x `[n]t)r ! ( [m] x [n])r and (V [m, k] x [n]t[ [m] x `[n]t)r ! ( [m] x [n]t)r Then, in order to have our maps in SSpO , we define a separate such map for each choice of vertices x_in degree zero and adding in the remaining points of O if necessary. As above, we use [n]tx_to denote the object [n]twith the (n+1)-tup* *le x_of vertices. We then define sets Ic = {( `[m] x [n]tx_[ [m] x `[n]tx_)r ! ( [m] x [n]tx_)r} for all m 0 and n 1, and Jc = {(V [m, k] x [n]tx_[ [m] x `[n]tx_)r ! ( [m] x [n]tx_)r} for all m 1, n 1, and 0 k m, where the notation (-)x_indicates the vert* *ices. Given these maps, we are now able to prove the existence of the model category structure SSpO,c. Proof of Theorem 3.9.The proofs that SSpO,c has finite limits and colimits and satisfies the two out of three and retract axioms, as well as condition (1), ar* *e the same as for SSpO,f. From the definitions of Ic and Jc, it follows that the Ic-injectives are the * *maps of Segal precategories with O in degree zero which are Reedy fibrations and that the Jc-injectives are the maps of Segal precategories with O in degree zero whi* *ch are Reedy acyclic fibrations. Furthermore, it follows from these facts that the* * Ic- cofibrations are precisely the cofibrations and that the Jc-cofibrations are pr* *ecisely the acyclic cofibrations. These observations imply that the conditions of Theor* *em 2.2 are satisfied. We can then localize SSpO,cwith respect to the map ' to obtain a model catego* *ry structure which we denote LSSpO,c. At first glance, one might wonder if the weak equivalences in LSS pO,f and LSSpO,c are actually the same, since each of these model structures is obtained via localization of a different model structure on Segal precategories. Howeve* *r, since the weak equivalences before localization are the same in each case, this* * lo- calization is independent of the underlying model structure [5, x7]. Therefore,* * the weak equivalences are actually the same in these two localized structures. (Not* *ice that here we need to use the more general notion of homotopy function complex in a category with specified weak equivalences, rather than in a simplicial mod* *el category, since we are working in two different model categories at once.) We then have the following result. 14 J.E. BERGNER Proposition 3.10. The adjoint pair given by the identity functor induces a Quil* *len equivalence of model categories LSSpO,f_____//LSSpO,c.oo_ Proof.Since the cofibrations in LSSpO,f are monomorphisms, the identity functor LSS pO,f! LSSpO,c preserves both cofibrations and acyclic cofibrations, so this adjoint pair is a* * Quillen pair. It remains to show that for any cofibrant X in LSS pO,f and fibrant Y in LSSpO,c, the map F X ! Y is a weak equivalence if and only of the map X ! RY is a weak equivalence. However, this fact follows since the weak equivalences a* *re the same in each category. Before proceeding to the main proof, we need one more model structure. Let * denote the set with one element. The objects of LSS p*,fhave a single point in degree zero, whereas the objects in LSSetsTM have an arbitrary simplicial s* *et in degree zero. To simplify matters, we define a model structure analogous to LSSetsTM but on the category of functors TM ! Sets which send T0 to [0]. Proposition 3.11. Consider the category SSetsTM* of functors TM ! SSets such that the image of T0 is [0]. There is a model category structure on SSetsTM* in which the weak equivalences and fibrations are defined levelwise and the cofibr* *ations are the maps with the left lifting property with respect to the acyclic fibrati* *ons. Proof.Limits and colimits exist in SSetsTM* by analogous arguments to the ones given in Lemmas 3.5 and 3.6. By taking sets of generating cofibrations and gene* *r- ating acyclic cofibrations for SSetsTM and modifying them in the same way as we did for SSp*,f, we can obtain generating sets for SSetsTM*. The proof follows j* *ust as the proof of Theorem 3.7. Now, to obtain a localized model category LSS etsTM*, we need to modify the maps a pk : Hom T(T1, -) ! Hom T(Tk, -) k that we used to obtain LSSetsTM from SSetsTM . Since Hom T(Tk, -)0 ~= [0] for all`k 0, the only change we need to make to these maps is to take the coprodu* *ct kHom T(T1, -) in the category SSetsTM* (as in Lemma 3.6). We then localize SSetsTM* with respect to the set of all such maps to obtain a model structure LSSetsTM*. Since a fibrant and cofibrant object X in LSSetsTM has X0 weakly equivalent to [0], it is not too surprising that we have the following result: Proposition 3.12. There is a Quillen equivalence of model categories LSS etsTM*____//LSSetsTMo.o_ Proof.There is an inclusion map I : LSSetsTM*! LSSetsTM whose right adjoint C collapses the space in degree zero to a point. Since the cofibrations of LSSe* *tsTM* are also cofibrations in LSS etsTM and the weak equivalences are defined in the same way in each model structure, I preserves cofibrations and acyclic cofibrat* *ions. Hence, this adjoint pair is a Quillen pair. SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 15 By the same argument, I also reflects weak equivalences between cofibrant ob- jects. To prove that we have a Quillen equivalence, it remains to show that for any fibrant object Y in LSSetsTM , the map I(CY )c ! Y is a weak equivalence. However, Y is a homotopy TM -algebra and hence Y0 is weakly equivalent to [0]. Therefore, up to homotopy Y is not changed by applying the functor C, and it follows that this maps is in fact a weak equivalence. 4. Proof of Theorem 1.7 For the rest of this section, we will set O = *, the one-element set. In the * *previous section, we used Theorem 3.4 to see that the model category structure on AlgTM , which is equivalent to the category of simplicial monoids, is Quillen equivalent to LSSetsTM . This model category is in turn Quillen equivalent to LSSetsTM* by Proposition 3.12. So, to prove Theorem 1.7 it suffices to show that there is a * *Quillen equivalence between the model categories LSSp*,fand LSSetsTM*. Let L1 be the functorial fibrant replacement functor (or localization) for LS* *Sp* given by taking a colimit of pushouts along the generating acyclic cofibrations* *, and let L2 be the analogous fibrant replacement functor for LSSetsTM*. We would like to know what the localization L1 does to an n-simplex [n]t*. N* *ote that we will use this notation in the sense of the previous section: we are spe* *cifying that we are working in the category of reduced Segal precategories, namely in t* *he case where O = *. To make the calculations about our localizations, we will use* * the model structure LSSp*,c, since in this case all objects are cofibrant and in pa* *rticular all monomorphisms are cofibrations. Lemma 4.1. Let L be a localization functor on a model category M. Given a small diagram of objects Xffof M, L(hocolimXff) ' LhocolimL(Xff). Proof.It suffices to show that for any local object Y , there is a weak equival* *ence of simplicial sets Map h(L(hocolimffLXff), Y ) ' Map h(LhocolimffXff, Y ). This fact follows from the following series of weak equivalences: Map h(LhocolimffLXff, Y')Map h(hocolimffLXff, Y ) ' holimffMaph(LXff, Y ) ' holimffMaph(Xff, Y ) ' Map h(hocolimffXff, Y ) ' Map h(LhocolimffXff, Y ). We now consider the simplicial space nerve(Tn)t, which is the nerve of the fr* *ee monoid on n generators, considered as a transposed simplicial space. Proposition 4.2. Let Fn denote the free monoid on n generators. Then in LSSp*,c, L1 [n]t*is weakly equivalent to nerve(Tn)t for each n 0. 16 J.E. BERGNER Proof.Note that when n = 0, [0]t*is isomorphic to nerve(T0)t, which is already a Segal category. Now consider the case where n = 1. We want to show that the map [1]t*! nerve(T1)t obtained by localizing with respect to the map '* is a weak equivale* *nce in LSSp*,c. In order to do this, we define a filtration of nerve(T1)t. Let the * *k-th stage of the filtration be X k(nerve(T1)t)j = {(xn1| . .|.xnj)| nj k}. Thus we have [1]t*= 1 2 . . . k . .n.erve(T1)t. Note that the fact that [1]t*= 1 can be observed from looking at the bar con- struction notation of each as a simplicial set (which we then view as a simplic* *ial space by transposing it). Each has one nondegenerate 1-simplex which we denote by x. Note that 1 has no nondegenerate 2-simplices. However, we want to define the "composite" of x with itself, a 1-simplex which we denote by x2, and add a nondegenerate 2-simplex [x, x] whose boundary consists of the 1-simplices x, x * *and x2. More formally, take the object G(2)t*as defined in the last section (which lives in 2) and add another 1-simplex x2 and the 2-simplex [x, x] in 2. We can describe this passing from 1 to 2 by the following pushout diagram: G(2)t*____// 1 | | | | fflffl| fflffl| [2]t*____// 2 Since we are working in SSp*,c, the left-hand vertical map is an acyclic cofibr* *ation, and therefore 1 ! 2 is an acyclic cofibration also [11, 3.14]. Similarly, to obtain 3 we will add an extra 1-simplex, denoted x3, in order * *to add a 3-simplex [x, x, x]. However, when taking the pushout, we do not want to start with G(3)*, since we have already added two of the new 1-simplices when we localized to obtain 2. So, we define ( [3]t*) 2 to be the piece of [3]t*that * *we already have in 2. Then our pushout looks like: ( [3]t*) 2____// 2 | | | | fflffl| fflffl| [3]t*______// 3 The map ( [3]t*) 2 ! [3]t*is a weak equivalence in LSSp*,cas follows. We have maps fi t G(3)t*ff_//_( [3]t*)_2//_ [3]*. Taking the function complex Map (-, X) for any local X for any of the three abo* *ve spaces yields X1 x X1 x X1 ' X3. The map ff is a weak equivalence since it is just a patching together of two localizations coming from the map G(2)t*! [2]t* **, which is a weak equivalence since it is one of the maps with respect to which w* *e are localizing. The composite map fiff is also a weak equivalence for the same reas* *on. Thus, fi is also a weak equivalence by the two-out-of-three property. Again, si* *nce SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 17 ( [3]t*) 2 ! [3]t*is an acyclic cofibration in LSS p*,c, the map 2 ! 3 is an acyclic cofibration also. For greater values of i, define ( [i + 1]t*) i to be the piece of [i + 1]t*t* *hat we already have from previous steps of the filtration. Note that it is always two * *copies of [i]t*attached along a copy of [i-1]t*, so the same argument as for i = 2 s* *hows that the map ( [i + 1]t*) i ! [i + 1]t*is a weak equivalence. Hence, for each * *i we obtain i+1via the pushout diagram ( [i + 1]t*)_i___//_ i | | | | fflffl| fflffl| [i + 1]t*____//_ i+1 Now that we have defined each stage of our filtration, using the bar construc* *tion notation shows how to map our new local object to nerve(T1)t. For example, [x, x, x2] 7! (x, x, x2) 2 T1 x T1 x T1. Using Lemma 4.1 we have that nerve(T1)t' L1(nerve(T1)t) ' L1(hocolim( i)) ' L1(hocolimL1( i)) ' L1(hocolimL1( 1)) ' L1L1( 1) ' L1( 1) ' L1( [1]t*). Now, for n = 2 (i.e. starting with [2]t*), we have three 1-simplices, which * *we will call x, y, and xy, and one nondegenerate 2-simplex [x, y]. Because we now have two variables, we need to define the filtration slightly differently as i* * = {[w1, . .,.wk]|l(w1. .w.k) i} where the wj's are words in x and y and l denot* *es the length of a given word. Note that by beginning with 1 we start with fewer simplices than those of the 2-simplex we are considering, but by passing to 2 * *we obtain the xy and [x, y] as well as additional nondegenerate simplices. In fact* *, we are actually starting the filtration with 1 = G(2)t*. The localizations procee* *d as in the case where n = 1, enabling us to map to nerve(T2)t. For n 3, the same argument works as for n = 2, with the filtrations being defined by the lengths of words in n letters. The resulting object is a reduced* * Segal category weakly equivalent to [n]t*. Hence, we have that for any n, L1 [n]t*is weakly equivalent to nerve(Tn)t. We now define a map J : op ! TM induced by the nerve construction on a monoid M. We will actually define the map Jop : ! TopM. (Note that TopMis just the full subcategory of the category of monoids whose objects are the isomorphi* *sm classes of free monoids.) For an object [n] of , define Jop([n]) = Tn where Tn denotes the free monoid on n generators, say x1, . .,.xn. In particular, Jop([0]) = *, the trivial mon* *oid. There are coface and codegeneracy maps induced from the nerve construction on M as follows. 18 J.E. BERGNER Taking the nerve of a simplicial monoid M and regarding it as a constant sim- plicial space results in a simplicial space which at level k looks like nerve(M)k = Mk = Hom Mon(Tk, M). By Yoneda's Lemma, the face operators di: nerve(-)k ! nerve(-)k-1 are induced by monoid maps Tk-1 ! Tk, and similarly for the degeneracy maps si: nerve(-)k ! nerve(-)k+1. Thus the simplicial diagram nerve(-) of representable functors Hom (Tk, -) gives rise to a cosimplicial diagram of representing objects Tk. In particular, the c* *oface maps are defined by: 8 >xkxk+1 i = k :xk+1 i > k and the codegeneracy maps are defined analogously. To obtain a simplicial diagr* *am of free monoids, we simply reverse the direction of the arrowsotopobtain a func* *tor J : op ! TM . This map induces a map J* : SSetsTM ! SSets which can be restricted to a map J* : SSetsTM*! SSp*. We now state two definitions in the following general context. Let p : C ! D, and let G : C ! SSets be a functor. Definition 4.3. If d is an object of D, then the over category or category of o* *bjects over d, denoted (p # d), is the category whose objects are pairs (c, f) where c* * is an object of C and f : p(c) ! d is a morphism in D. If c0 is another object of C, a morphism in the over category is given by a map p(c) ! p(c0) inducing a commutative triangle p(c)B | BBB | BBB | !!B | d | _==_ | __ | __ fflffl|__ p(c0) Definition 4.4. [13, 11.8.1] Let p, c, and G be defined as above, and let f : p* *(c) ! d be an object in (p # d). The left Kan extension over p is a functor p*G : D ! S* *Sets defined by (p*G)(d) = colim(p#d)((c, f) 7! G(c)). Note. Ordinarily, we would have to take a homotopy left Kan extension, where we replace the colimit in the definition with a homotopy colimit, to make sure that our calculations were homotopy invariant. However, since we are making our calculations in LSSp*,cwhere all objects are cofibrant, and since left Kan exte* *nsions agree with homotopy left Kan extensions on cofibrant objects [10, 3.7], the lef* *t Kan extension is sufficient. SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 19 Proposition 4.5. [13, 11.9.3] The functor SSetsC ! SSetsD given by sending G the left Kan extension p*G is left adjoint to the functor SSetsD ! SSetsC given* * by composition with p. In our specific case, we define J* : SSp* ! SSetsTM*to be the left Kan extens* *ion over J which is, by definition, the left adjoint to J*. Note that even if G is* * a reduced Segal category, J*(G) is not necessarily local in LSS etsTM*. To obtai* *n a TM -algebra, we must apply the localization functor L2. Define M[k] to be the TM -space with n copies of Tk at level n. Let G be the reduced Segal category nerve(Tk)t. Lemma 4.6. In LSSetsTM*, L2J*(G) is weakly equivalent to M[k]. Proof.It suffices to show that for any local object X in LSSetsTM*, Map hLSSetsTM*(L2J*G, X) ' X(Tk) since M[k] is precisely X(Tk). This fact can be shown in the following argument: MaphLSSetsTM*(L2J*G, X)' Map hLSSetsTM*(J*G, X) ' Map hLSSp*,c(G, J*X) ' Map hLSSp*,c(L1 [k]t*, J*X) ' Map hLSSp*,c( [k]t*, J*X) ' J*X[k] ' X(Tk). Note that this lemma, combined with Lemma 4.2, shows that L2J*( [k]t*) ' M[k]. (Actually, this fact is true even without localizing the left-hand side b* *ecause [k]t*is free.) Proposition 4.7. For any object X in SSp*,c, we have that L1X is weakly equiv- alent to J*L2J*X. ` Proof.First note that X ' hocolim op([n] ! i [ni]t) where the values of i depend on n. We begin by looking at L1X. Using Lemma 4.1, we have the following: a L1X ' L1hocolim op([n] 7! [ni]t*) a ' L1hocolim opL1([n] 7! [ni]t*) ' L1hocolim op([n] 7! nerve(TP ni)t) However, hocolim op([n] 7! nerve(TP ni)t) is already local, a fact which follows from the fact that the homotopy colimit can be taken at each level, yielding a * *Segal precategory in LSSp*,cwhich is still a Segal category. 20 J.E. BERGNER Working from the other side of the desired equation, we obtain, using the fact that left adjoints commute with colimits and homotopy colimits: a J*L2J*X ' J*L2J*hocolim op([n] 7! [ni]t*) a ' J*L2hocolim opJ*([n] 7! [ni]t*) a ' J*L2hocolim opL2J*([n] 7! [ni]t*) X ' J*L2hocolim op([n] 7! M[ ni]) In TM , this looks like the diagram in op, but now we have maps in the theory of monoids rather than simplicial maps. Thus, the restriction map J* will give * *the same objects but with simplicial maps, as wished. We begin by showing that we have a Quillen pair even before we apply the localization functors. Notice that we return to using our model structure SSp** *,f with levelwise fibrations. Proposition 4.8. The adjoint pair J* : SSp*,f___//SSetsTM*:oJ*o_ is a Quillen pair. Proof.In each case the fibrations and weak equivalences are defined levelwise. * *Since the right adjoint J* preserves the spaces at each level, it must preserve both * *fibra- tions and acyclic fibrations. We then need to show that this Quillen pair induces a Quillen equivalence be- tween the localized model category structures in order to prove our main theore* *m. While we will be working with LSS p*,frather than LSS p*,c, the calculations we made above will still hold because the weak equivalences are the same in each c* *ase (see the end of section 3). Proof of Theorem 1.7.We first need to show that we still have a Quillen pair ev* *en after localizing each category. This fact follows from Theorem 2.7 after we not* *ice that the left derived localizing set LJ*' [13, 8.5.11] is the same as the set o* *f maps S = {p0, p1, . .}.that we localize`with respect to in order to obtain LSSetsTM** *, as follows. Consider`the maps 'k : k [1]t*! [k]t*for each k. If we localize this map, we obtain knerve(T1)t! nerve(Tk)t, where Tk denotes the free monoid on k generators. Then we can apply the functor J* to obtain a map a J*( nerve(T1)t) ! J*(nerve(Tk)t) k ` which we can then localize to obtain, by`Lemma 4.6, the map kM[1] ! M[k]. However, this is precisely the map pk : kHom (T1, -) ! Hom (Tk, -). We now need to show that we have a Quillen equivalence. First, we need to know that the right adjoint J* reflects weak equivalences between fibrant objec* *ts. In each of the two localized model categories LSS p*,fand LSS etsTM*, an object is fibrant if and only if it is local and fibrant in the unlocalized model cate* *gory. Therefore, in each case a weak equivalence between fibrant objects is a levelwi* *se weak equivalence. Since J* does not change the spaces at each level, it must re* *flect weak equivalences between fibrant objects. SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 21 Finally, by Proposition 4.7, L1X ' J*L2J*X for any functor X : op! SSets, and in particular for any cofibrant X. Notice that our Quillen equivalences compose: LSS p*,fAE LSSetsTM*AE LSSetsTM AE AlgTM (where the left adjoint functors are the topmost maps). Therefore, we actually * *have a single Quillen equivalence LSSp*,fAE AlgTM . 5.A Generalization to Simplicial Categories and Segal Categories We would like to generalize our result to the category of simplicial categori* *es which have a set O of objects, where the morphisms are required to be the ident* *ity on this set, since a simplicial category is a generalization of a simplicial mo* *noid. Definition 5.1. A simplicial category is a category enriched over simplicial se* *ts, i.e. a category in which the morphisms between any two objects form a simplicial set. This use of the term "simplicial category" is potentially confusing. We do not mean that a simplicial category is simply a simplicial object in the category of categories. It is actually a simplicial object in the category of categories in* * which the categories at each level have the same objects. In particular, the face and degeneracy maps are the identity on the objects. Analogously, we will generalize to Segal categories with the same set O in de* *gree zero. As we have seen, any Segal category can be described as a functor from the simplicial category op to the category SSets of simplicial sets satisfying* * two conditions: discreteness of the zero space and a product condition up to homoto* *py (Definition 1.2). However, we will find it more convenient, when dealing with S* *egal categories with a fixed object set, to use a larger indexing category which spe* *cifies the objects. Let O be a set. We define the category opOas follows. The objects are given by [n]x0,...,xnwhere n 0 and (x0, . .x.n) 2 On+1. The [n] should be thought of as in the simplicial category op; however, recall that when we work with Segal categories we will require all morphisms to preserve the objects. Therefore, we* * need to have a separate [n] for each possible (n+1)-tuple of objects in O. The morph* *isms in opOare those of op but depend on the choice of (x0, . .,.xn). Specifically* *, the face maps are di: [n]x0,...,xn! [n - 1]x0,...,bxi,...,xn and the degeneracy maps are si: [n]x0,...,xn! [n + 1]x0,...,xi-1,xi,xi,xi+1,...,xn. We will sometimes refer to our Segal precategories with O in degree zero as op* *O- spaces. Note that if O is the one-object set, then opOis just op. Now we can use this notation to describe Segal categories. Definition 5.2. Given a category C, a functor F : opO! C is O-special if for e* *ach n 2 the map F ([n]x0,...,xn)'//_F ([1]x0,x1) xF[0]x1. .x.F[0]xn-1F ([1]xn-1,xn) is a weak equivalence of simplicial sets. 22 J.E. BERGNER Now, we can think of a Segal category with O in dimension zero as an O-special functor opO! SSets. Recall from section 3 that we have model category structur* *es LSSpO,fand LSSpO,con the category of Segal precategories with O in degree zero, in each of which the fibrant objects are Segal categories. We would like to think of the category of simplicial categories with object s* *et O as a diagram category as well. To do so, we need to define the notion of a multi-sorted algebraic theory. To see more details, see [4]. Definition 5.3. Given a set S, an S-sorted algebraic theory (or multi-sorted th* *eory) T is a small category with objects Tff_nwhere ff_n=< ff1, . .,.ffn > for ffi 2 * *S and n 0 varying, and such that each Tff_nis equipped with an isomorphism Yn Tff_n~= Tffi. i=1 For a particular ff_n, the entries ffi can repeat, but they are not ordered. T* *here exists a terminal object T0 (corresponding to the empty object of S). In particular, we can talk about the theory of O-categories, which we will de* *note by TOCat. To define this theory, first consider the category OCat whose objects* * are the categories with a fixed object set O and whose morphisms are the functors w* *hich are the identity map on the objects. The objects of TOCat are categories which * *are freely generated by directed graphs with vertices corresponding to the elements* * of the set O. This theory will be sorted by pairs of elements in O, corresponding * *to the morphisms with source the first element and target the second. In other wor* *ds, this theory is (O x O)-sorted [4, 3.5]. (In the one-object case, we have the or* *dinary theory of monoids.) We can then say that a simplicial category with object set O is a strict TOCat-algebra, where strict and homotopy T-algebras are defined f* *or multi-sorted theories T just as for ordinary algebraic theories. Again, we have a model structure AlgT on the category of all T-algebras and a model structure SSetsT on the category of all functors T ! SSets which can be localized as before to obtain a model category structure LSSetsT in which the l* *ocal objects are homotopy T-algebras [4]. For TOCat,`we can define a category SSetsT* *OCatO of functors TOCat ! SSets which send T0 to O [0]. Making modifications as in the case of LSSetsTM*, we can define a model structure LSSetsTOCatOwhich is Qui* *llen equivalent to LSSetsTOCat. Furthermore, Badzioch's rigidification theorem 3.4 for algebraic theories can* * be generalized to the case of multi-sorted theories, and therefore we have that th* *ere is a Quillen equivalence of model categories between AlgT and LSSetsT for any multi- sorted theory T [4, 5.1]. Applying this result to the multi-sorted theory TOCat and using the Quillen equivalence in the previous paragraph, we have reduced the problem to finding a Quillen equivalence between LSSpO,f and LSSetsTOCatO. We begin as before by defining a mapoofpdiagrams J : opO! TOCatwhich then induces a map J* : SSetsTOCatO! SSets O . First consider an arbitrary category C with object set O. Then its nerve will* * be a Segal category with the set O in dimension zero. Since it is a Segal category, * *we can view it as an O-special functor opO! SSets with the specification that [0]xi7!* * xi for any xi2 O. Now, given an integer n 0 and x_= (x0, . .,.xn) 2 On+1, define Tn,x_to be the free category with object set O and morphisms freely generated by the set SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 23 {xi-1 ! xi|1 i n}. However, note that this free category is, in light of t* *he definition of multi-sorted theory, just the object Tff_where ff_= (ff1, . .,.ff* *n) and ffi= (xi-1, xi) 2 O x O. (We will find both notations useful.) We then obtain a simplicial diagram as follows: a a Hom (T0, C) ( Hom (Tff1, C) W Hom (Tff_, C) . . . ff1 ff_=(ff1,ff2) Because we have a coproduct at each level, this diagram is not representable by a cosimplicial diagram. However, if we split up the coproducts and consider it * *as a diagram of objects Hom (Tff_, C) varying over ff_and n, we get a map J : opO! TOCat. Then we define the map op J* : SSetsTOCatO! SSets O to be the one induced by this map J above. Putting our model structures on these categories we get J* : LSSetsTOCatO! LSSpO,f. We will denote by L1 the localization of simplicial spaces with O in dimension zero, namely the functorial fibrant replacement functor in LSSpO,c. (As above, * *we will make our calculations in this category rather than in LSSpO,f.) We have an analogous localization in the category LSSetsTOCatO, which we will denote by L2. From there, we can apply the methods used in the previous section. We first consider what happens when we localize an n-simplex [n]x0,...,xn. If xi= xj fo* *r all 0 i, j n then we get the free monoid on n generators, with object xi. Howev* *er, if xi6= xj for some 0 i, j n, we get the nerve of the free category generat* *ed by the directed graph x0 ! x1 ! . .!.xn. For example, if xi 6= xj for all 0 i, j n, then the localization only consi* *sts of including in composites, say 1-simplices xi! xi+2. If n = 1 and x0 6= x1, then * *the localization does not change the Segal category since there are no "composable" 1-simplices. We will denote the free category by Tn,x_, as before, and its nerv* *e by Gn,x_= nerve(Tn,x_)t. Now, define J* be the left Kan extension of the map J*. However, note that again we must apply the localization functor L2 to assure that we get a homotopy TOCat-algebra in LSSetsTOCatO. As before, it suffices to show that for any opO* *-space X, we have that L1X = J*L2J*X. We will first need a lemma. Analogous to M[k] above, define C[n]x_to be the TOCat-diagram in of simplicial sets which has the same simplicial set at each level as Gn,x_, but with theory * *maps between them rather than the simplicial maps. Lemma 5.4. In LSSetsTOCatO, L2J*(Gn,x_) is weakly equivalent to C[n]x_. Proof.It suffices to show that for a local object X in LSSetsTOCatO, we have th* *at Hom LSSetsTOCatO(L2J*(Gn,x_), X) ' X(Tn,x_). 24 J.E. BERGNER This can be proved as follows: Hom LSSetsTOCatO(L2J*(Gn,x_),'X)HomLSSetsTOCatO(J*(Gn,x_), X) ' Hom LSSpO,c(Gn,x_, J*X) ' Hom LSSpO,c(L1 [n]x_, J*X) ' Hom LSSpO,c( [n]x_, J*X) ' J*(X)[n]x_ ' X(Tn,x_) Note that in TOCat, we would usually write Tff_instead of Tn,x_, where ff_= (ff* *1, . .,.ffn) and ffi= (xi-1, xi) for 1 i n. Now, using this lemma, the following proposition suffices to prove the theore* *m. Proposition 5.5. Given any opO-space X, L1X and J*L2J*X are weakly equiv- alent. Proof.Given any opO-space X, it can be written as the homotopy colimit X ' hocolim op([n] 7! qni,x_ [ni]x_) where the ni depend on n and x_= (x1, . .,.xni). We then have the following: L1X ' L1hocolim op([n] 7! qni,x_ [ni]x_) ' L1hocolim opL1([n] 7! qni,x_ [ni]x_) ' L1hocolim op([n] 7! qni,x_Gni,x_) Working from the other side, we have J*L2J*X ' J*L2J*hocolim op([n] 7! qni,x_ [ni]x_) ' J*L2hocolim opJ*([n] 7! qni,x_ [ni]x_) ' J*L2hocolim opL2J*([n] 7! qni,x_ [ni]x_) ' J*L2hocolim op([n] 7! qni,x_C[ni]x_) But these two are equal, by the above lemma. Now, using the same arguments as in the last section, we can prove the follow* *ing theorem: Theorem 5.6. The adjoint pair J* : LSSpO,f___//_LSSetsTOCatO:oJ*o_ is a Quillen equivalence. In particular, composing with the Quillen equivalences given by the generaliz* *a- tions of Theorem 3.4 and Proposition 3.12, there is a Quillen equivalence LSSpO,fAE AlgTOCat. SIMPLICIAL MONOIDS AND SEGAL CATEGORIES 25 References [1]Bernard Badzioch, Algebraic theories in homotopy theory, Ann. of Math. (2) 1* *55 (2002), no. 3, 895-913. [2]Bernard Badzioch, Kuerak Chung, and Alexander A. 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