Algebras and modules in monoidal model categories Stefan Schwede and Brooke E. Shipley1 Abstract: We construct model category structures for monoids and modules in symmetric monoidal model categories, with appli- cations to symmetric spectra and -spaces. 1991 AMS Math. Subj. Class.: primary 55U35, secondary 18D10 1 Summary This paper gives a general approach for obtaining model category structures for* * algebras or modules over some other model category. Technically, what we mean by an `algebr* *a' is a monoid in a symmetric monoidal category. Of course, the symmetric monoidal and model * *category structures have to be compatible, which leads to the definition of a monoidal m* *odel category, see Definition 2.1. To obtain a model category structure of algebras we have to* * introduce one further axiom, the monoid axiom (Definition 2.2). A filtration on certain pusho* *uts of monoids (see Lemma 5.2) is then used to reduce the problem to standard model category a* *rguments based on Quillen's "small object argument". Our main result is stated in Theore* *m 3.1. This approach was developed in particular to apply to the category of symmetric* * spectra defined in [HSS] and to -spaces in [Sch2]. In both of these categories we thus obtain m* *odel categories for the associative monoids, the R-modules for any monoid R, and the R-algebras* * for any commutative monoid R. A significant shortcut is possible if the underlying mon* *oidal model category has the special property that all objects are fibrant, see Remark 3.5.* * This is not true for our main examples, symmetric spectra and -spaces. It does hold though, in t* *he monoidal model categories of simplicial abelian groups, chain complexes, or S-modules (i* *n the sense of [EKMM]). We assume that the reader is familiar with the language of homotopical algebra * *(cf. [Q], [DS]) and with the basic ideas concerning monoidal and symmetric monoidal categories * *(cf. [MacL, VII], [Bor, 6]) and triples (also called monads, cf. [MacL, VI.1], [Bor, 4]). Acknowledgments. We would first like to thank Charles Rezk for conversations wh* *ich led us to the filtration that appears in Lemma 5.2. We also benefited from several conver* *sations about this project with Bill Dwyer, Mark Hovey and Manos Lydakis. We would also like * *to thank Bill Dwyer, Phil Hirschhorn, and Dan Kan for sharing the draft of [DHK] with us. In * *Appendix A we recall the notion of a cofibrantly generated model category from their book. _______________________________1 Research partially supported by an NSF Postdoctoral Fellowship 1 2 Monoidal model categories A monoidal model category is essentially a model category with a compatible clo* *sed symmetric monoidal product. The compatibility is expressed by the pushout product axiom b* *elow. In this paper we always require a closed symmetric monoidal product although for exposi* *tory ease we refer to these categories as just `monoidal' model categories. One could also * *consider model categories enriched over a monoidal model category with certain compatibility r* *equirements analogous to the pushout product axiom or the simplicial axiom of [Q, II.2]. F* *or example, closed simplicial model categories [Q, II.2] are such compatibly enriched categ* *ories over the monoidal model category of simplicial sets. We also introduce the monoid axiom which is the crucial ingredient for lifting * *the model category structure to monoids and modules. Examples of monoidal model categories satisf* *ying the monoid axiom are given in Section 4. Definition 2.1A model category C is a monoidal model category if it is endowed * *with a closed symmetric monoidal structure and satisfies the following pushout product axiom.* * We will denote the symmetric monoidal product by ^, the unit by I and the internal Hom * *object by [-; -]. Pushout product axiom. Let A ---! B and K ---! L be cofibrations in C. Then th* *e map A^ L [A^ K B ^K ---! B ^L is also a cofibration. If in addition one of the former maps is a weak equivale* *nce, so is the latter map. If C is a category with a monoidal product ^ and I is a class of maps in C, we * *denote by I ^C the class of maps of the form A^ Z - ! B ^Z for A -! B a map in I and Z an object of C. We also denote by I-cofregthe clas* *s of maps obtained from the maps of I by cobase change and composition (possibly tra* *nsfinite, see Appendix A.) These maps are referred to as the regular I-cofibrations. Definition 2.2A monoidal model category C satisfies the monoid axiom if every m* *ap in ({acyc. cofibrations}^ C)-cofreg is a weak equivalence. Note that if C has the special property that every object is cofibrant, then th* *e monoid axiom is a consequence of the pushout product axiom. However, this special situation rar* *ely occurs in practice. In Appendix A we recall cofibrantly generated model categories. In these model* * categories fibrations can be detected by checking the right lifting property against a set* * of maps, called generating acyclic cofibrations, and similarly for acyclic fibrations. This is * *in contrast to general model categories where the lifting property has to be checked against the whole* * class of acyclic cofibrations. In cofibrantly generated model categories, the pushout product a* *xiom and the monoid axiom only have to be checked for the generating (acyclic) cofibrations: 2 Lemma 2.3 Let C be a cofibrantly generated model category endowed with a close* *d symmetric monoidal structure. 1.If the pushout product axiom holds for the generating cofibrations and the * *generating acyclic cofibrations, then it holds in general. 2.Let J be a set of generating acyclic cofibrations. If every map in (J ^C)-c* *ofregis a weak equivalence, then the monoid axiom holds. Proof: For the first statement consider a map i : A -! B in C. Denote by G(i) t* *he class of maps j :K -! L such that the pushout product A^ L [A^ K B ^K - ! B ^L is a cofibration. This pushout product has the left lifting property with resp* *ect to a map f :X -! Y if and only if j has the left lifting property with respect to the map p : [B; X] -! [B; Y ] x[A;Y[]A; X]: Hence, a map is in G(i) if and only if it has the left lifting property with re* *spect to the map p for all f :X -! Y which are acyclic fibrations in C. G(i) is thus closed under cobase change, transfinite composition and retracts. * *If i : A -! B is a generating cofibration, G(i) contains all generating cofibrations by assum* *ption; because of the closure properties it thus contains all cofibrations, see Lemma A.1. Rev* *ersing the roles of i and an arbitrary cofibration j : K -! L we thus know that G(j) contains al* *l generating cofibrations. Again by the closure properties, G(j) contains all cofibrations, * *which proves the pushout product axiom for two cofibrations. The proof of the pushout product be* *ing an acyclic cofibration when one of the constituents is, follows in the same manner. For the second statement note that by the small object argument, Lemma A.1, eve* *ry acyclic cofibration is a retract of a transfinite composition of cobase changes along t* *he generating acyclic cofibrations. Since transfinite compositions of transfinite composition* *s are transfinite compositions, every map in ({acyc. cofibrations}^ C)-cofregis thus a retract of* * a map in (J ^C)- cofreg. __* *|_| 3 Model categories of algebras and modules In this section we state the main theorem, Theorem 3.1, which constructs model * *categories for algebras and modules. The proof of this theorem is delayed to section 5. Exampl* *es of model categories for which this theorem applies are given in section 4. We end this s* *ection with two theorems which compare the homotopy categories of modules or algebras over weak* *ly equivalent monoids. We consider a symmetric monoidal category with product ^ and unit I. A monoid i* *s an object R together with a "multiplication" map R ^R -! R and a "unit" I -! R which sati* *sfy certain associativity and unit conditions (see [MacL, VII.3]). R is a commutati* *ve monoid if the multiplication map is unchanged when composed with the twist, or the symmetry i* *somorphism, of R ^R. If R is a monoid, a left R-module ("object with left R-action" in [Mac* *L, VII.4]) is an 3 object N together with an action map R ^N -! N satisfying associativity and uni* *t conditions (see again [MacL, VII.4]). Right R-modules are defined similarly. Assume that C has coequalizers. Then there is a smash product over R, denoted M* * ^RN, of a right R-module M and a left R-module N. It is defined as the coequalizer, in * *C, of the two maps M ^R ^N ----!----!M ^N induced by the actions of R on M and N respectively* *. If R is a commutative monoid, then the category of left R-modules is isomorphic to the * *category of right R-modules, and we simply speak of R-modules. In this case, the smash prod* *uct of two R-modules is another R-module and smashing over R makes R-mod into a symmetric * *monoidal category with unit R. If C has equalizers, there is also an internal Hom object* * of R-modules, [M; N]R. It is the equalizer of two maps [M; N] ----!----![R ^M; N]. The first * *map is induced by the action of R on M, the second map is the composition of R ^- : [M; N] -! [R * *^M; R ^N] followed by the map induced by the action of R on N. For a commutative monoid R, an R-algebra is defined to be a monoid in the categ* *ory of R- modules. It is a formal property of symmetric monoidal categories (cf. [EKMM, V* *II 1.3]) that specifying an R-algebra structure on an object A is the same as giving A a mono* *id structure together with a monoid map f :R -! A which is central in the sense that the fol* *lowing diagram commutes. switch id^f R ^A _____Aw^R _____Aw^A | | f ^i|d |mult. |u |u A ^A __________________Awmult. Now we can state our main theorem. It essentially says that monoids, modules an* *d algebras in a cofibrantly generated, monoidal model category C again form a model category * *if the monoid axiom holds. (See Appendix A for the definition of a cofibrantly generated mode* *l category.) To simplify the exposition, we assume that all objects in C are small (refer to Ap* *pendix A) relative to the whole category. This last assumption can be weakened as indicated in A.5* *. The proofs will be delayed until the last section. In the categories of monoids, left R-modules (when R is a fixed monoid), and R-* *algebras (when R is a commutative monoid) a morphism is defined to be a fibration or weak equi* *valence if it is a fibration or weak equivalence in the underlying category C. A morphism is * *a cofibration if it has the left lifting property with respect to all acyclic fibrations. Theorem 3.1 Let C be a cofibrantly generated, monoidal model category. Assume f* *urther that every object in C is small relative to the whole category and that C satisfies * *the monoid axiom. 1.Let R be a monoid in C. Then the category of left R-modules is a cofibrantl* *y generated model category. 2.Let R be a commutative monoid in C. Then the category of R-modules is a cof* *ibrantly generated, monoidal model category satisfying the monoid axiom. 3.Let R be a commutative monoid in C. Then the category of R-algebras is a co* *fibrantly generated model category. If the unit I of the smash product is cofibrant i* *n C, then every cofibration of R-algebras whose source is cofibrant in C is also a cofibrat* *ion of R-modules. In particular, any cofibrant R-algebra is cofibrant as an R-module. If in part (3) of the theorem we take R to be the unit of the smash product, we* * see that in particular the category of monoids in C forms a model category. 4 Remark 3.2 The full strength of the monoid axiom is not necessary to obtain a m* *odel category of R-modules for a particular monoid R. In fact, to get hypothesis (1) of Lemma* * A.3 for R- modules, one need only know that every map in ({acyc. cofibrations}^ R) -cofreg is a weak equivalence. This holds, independent of the monoid axiom, if R is cof* *ibrant in the underlying category C. For then the pushout product axiom implies that smashin* *g with R preserves acyclic cofibrations. The following theorems concern comparisons of homotopy categories of modules an* *d algebras. The homotopy theory of R-modules and R-algebras should only depend on the weak * *equivalence type of the monoid R. To show this for R-modules we must require that the funct* *or -^ RN take any weak equivalence of right R-modules to a weak equivalence in C whenever N i* *s a cofibrant left R-module. In all of our examples this added property of the smash product * *holds. For the comparison of R-algebras, we also require that the unit of the smash product is* * cofibrant. Theorem 3.3 Assume that for any cofibrant left R-module N, -^ RN takes weak equ* *ivalences of right R-modules to weak equivalences in C. If R -~!S is a weak equivalence o* *f monoids, then the total derived functors of restriction and extension of scalars induce equiv* *alences of homotopy categories Ho(R-mod) ~= Ho (S-mod) : Proof: This is an application of Quillen's adjoint functor theorem ([Q, I.4 Thm* *. 3] or [DS, Thm. 9.7]). The weak equivalences and fibrations are defined in the underlying* * symmetric monoidal category, hence the restriction functor preserves fibrations and acycl* *ic fibrations. By assumption, for N a cofibrant left R-module N ~=R ^RN -! S ^RN is a weak equivalence. Thus if Y is a fibrant left S-module, an R-module map N * *-! Y is a weak equivalence if and only if the adjoint S-module map S ^RN -! Y is a weak e* *quivalence. [DS, Thm. 9.7] then gives the desired result. * * __|_| Theorem 3.4 Suppose that the unit I of the smash product is cofibrant in C and * *that for any cofibrant left R-module N, -^ RN takes weak equivalences of right R-modules to * *weak equiva- lences in C. Then for a weak equivalence of commutative monoids R -~! S, the to* *tal derived functors of restriction and extension of scalars induce equivalences of homotop* *y categories Ho(R-alg) ~= Ho (S-alg) : Proof: The proof is similar to the one of the previous theorem. Again the rig* *ht adjoint restriction functor does not change underlying objects, so it preserves fibrati* *ons and acyclic fibrations. Since cofibrant R-algebras are also cofibrant as R-modules (Thm. 3.* *1 (3)), for any cofibrant R-algebra the adjunction morphism is again a weak equivalence. So [DS* *, Thm. 9.7] applies one more time. _* *_|_| Remark 3.5 Some important examples of monoidal model categories have the proper* *ty that all objects are fibrant. This greatly simplifies the situation. If there is a* *lso a simplicial or topological model category structure and if a simplicial (resp. topological) tr* *iple T acts, then the category of T -algebras is again a simplicial (topological) category, so it* * has path objects. Hence hypothesis (2) of Lemma A.3 applies. One example of this situation is the* * category of S-modules in [EKMM]. Lemma A.3 (2) should be compared to [EKMM, Thm. VII 4.7]. 5 Remark 3.6 We point out again that in our main examples, symmetric spectra and * *-spaces, not all objects are fibrant, which is why we need a more complicated approach. * *In the fibrant case, one gets model category structures for algebras over all reasonable (e.g.* * continuous or simplicial) triples, whereas our monoid axiom approach only applies to the free* * R-module and free R-algebra triples. The category of commutative monoids often has a model c* *ategory struc- ture in the fibrant case (e.g. commutative simplicial rings or commutative S-al* *gebras [EKMM, Cor. VII 4.8]). In contrast, for -spaces and symmetric spectra, the category of* * commutative monoids can not form a model category with fibrations and weak equivalences def* *ined in the underlying category. For if such a model category structure existed, one could * *choose a fibrant replacement of the unit S0 inside the respective category of commutative monoid* *s. Evaluating this fibrant representative on 1+ 2 op, or at level 0 respectively, would give * *a commutative simplicial monoid weakly equivalent to QS0. This would imply that the space QS0* * is weakly equivalent to a product of Eilenberg-MacLane spaces, which is not the case. The* * homotopy cat- egory of commutative monoids in symmetric spectra is still closely related to E* *1 -ring spectra, though. 4 Examples Simplicial sets. The category of simplicial sets has a well-known model category structure estab* *lished by D. Quillen [Q, II 3, Thm. 3]. The cofibrations are the degreewise injective maps, * *the fibrations are the Kan fibrations and the weak equivalences are the maps which become homotopy* * equivalences after geometric realization. This model category is cofibrantly generated. The * *standard choice for the generating (acyclic) cofibrations are the inclusions of the boundaries * *(resp. horns) into the standard simplices. Here every object is small with respect to the whole ca* *tegory. The cartesian product of simplicial sets is symmetric monoidal with unit the di* *screte one-point simplicial set. The pushout product axiom is well-known in this case, (see [GZ,* * IV Prop. 2.2], [Q, II 3, Thm. 3]). Since every simplicial set is cofibrant, the monoid axiom f* *ollows from the pushout product axiom. A monoid in the category of simplicial sets under cartes* *ian product is just a simplicial monoid, i.e., a simplicial object of ordinary unital and asso* *ciative monoids. So the main theorem, Theorem 3.1 (3), recovers Quillen's model category structure * *for simplicial monoids [Q, II 4, Thm. 4, and Rem. 1, p. 4.2]. -spaces and symmetric spectra These two examples are new. In fact, the justification for writing this paper i* *s to give a unified treatment of why monoids and modules in these categories form model categories.* * Here we only give an overview; for the details the reader may consult [Se], [BF], [Ly] * *and [Sch2] in the case of -spaces, and [HSS] in the case of symmetric spectra. The particular* * interest in these categories comes from the fact that they model stable homotopy theory. Th* *e homotopy category of symmetric spectra is equivalent to the usual stable homotopy catego* *ry of algebraic topology. In the case of -spaces, one obtains the stable homotopy category of * *connective (i.e., (-1)-connected) spectra. Monoids in either of these categories are thus * *possible ways of defining `brave new rings', i.e., rings up to homotopy with higher coherence co* *nditions. Another approach to this idea consists of the S-algebras of [EKMM]. 6 -spaces. -spaces were introduced by G. Segal [Se] who showed that they give ris* *e to a homo- topy category equivalent to the usual homotopy category of connective spectra. * *A. K. Bousfield and E. M. Friedlander [BF] considered a bigger category of -spaces in which the* * ones introduced by Segal appeared as the special -spaces. Their category admits a simplicial mo* *del category structure with a notion of stable weak equivalence giving rise again to the hom* *otopy theory of connective spectra. Then M. Lydakis [Ly] showed that -spaces admit internal fun* *ction objects and a symmetric monoidal smash product with nice homotopical properties. Small* *ness and cofibrant generation for -spaces is verified in [Sch2], as well as the pushout * *product and the monoid axiom. The monoids in this setting are called Gamma-rings. Symmetric spectra. The category of symmetric spectra, Sp , is described in [HSS* *]. There it is also shown that this category is a cofibrantly generated, monoidal model cat* *egory, and that the associated homotopy category is equivalent to the usual homotopy category o* *f spectra. For symmetric spectra over the category of simplicial sets every object is small wi* *th respect to the whole category. The monoid axiom and the fact that smashing with a cofibrant le* *ft R-module preserves weak equivalences between right R-modules are verified in [HSS]. The * *monoids in this setting are called symmetric ring spectra. Fibrant examples: simplicial abelian groups, chain complexes and S-modules These are the examples of monoidal model categories in which every object is fi* *brant. With this special property it is easier to lift model category structures since the * *(often hard to verify) condition (1) of the lifting lemma A.3 is a formal consequence of fibrancy and * *the existence of path objects, see the proof of A.3. For example, the commutative monoids som* *etimes form model categories in these cases. The pushout product and monoid axioms also hol* *d in these examples, but since the fibrancy property deprives them of their importance, we* * will not bother to prove them. Simplicial abelian groups. The model category structure for simplicial abelian* * groups was established by Quillen [Q, II.6]. The weak equivalences and fibrations are defi* *ned on underlying simplicial sets. The cofibrations are the retracts of the free maps (see [Q, II* * p. 4.11, Rem. 4]). This model category is cofibrantly generated and all objects are small. The (de* *greewise) tensor product provides a symmetric monoidal product for simplicial abelian groups. Th* *e unit for this product is the integers, considered as a constant simplicial abelian group. A m* *onoid then is nothing but a simplicial ring. These have path objects given by the simplicial * *structure. This means that for a simplicial ring R the simplicial set Hom([1]; R) of maps of th* *e standard 1-simplex into the underlying simplicial set of R is naturally a simplicial rin* *g. The model category structure for simplicial rings and simplicial modules was established * *by Quillen in [Q, II.4, Thm. 4] and [Q, II.6]. Chain complexes. The category of non-negatively graded chain complexes over a c* *ommutative ring k forms a model category, see [Q, II p. 4.11, Remark 5], [DS, Section 7]. * *The weak equiv- alences are the maps inducing homology isomorphisms, the fibrations are the map* *s which are surjective in positive degrees, and cofibrations are monomorphisms with degreew* *ise projective cokernels. This model category is cofibrantly generated and every object is sma* *ll. The cate- gory of unbounded chain complexes over k, although less well known, also forms * *a cofibrantly generated model category with weak equivalences the homology isomorphism and fi* *brations the epimorphisms, see [HPS], remark after Thm. 9.3.1. The cofibrations here are sti* *ll degreewise split injections, but their description is a bit more complicated than for boun* *ded chain com- plexes. The following remarks refer to this category of Z-graded chain complexe* *s of k-modules. 7 The graded tensor product of chain complexes is symmetric monoidal and has adjo* *int internal hom-complexes. A monoid in this symmetric monoidal category is a differential g* *raded algebra (DGA). Every complex is fibrant and associative DGAs have path objects. To cons* *truct them, we need the following 2-term complex denoted I. In degree 0, I consists of a fr* *ee k-module on two generators [0] and [1]. In degree 1, I is a free k-module on a single ge* *nerator . The differential is given by d = [1] - [0]. This complex becomes a coassociative a* *nd counital coalgebra when given the comultiplication : I -! I k I defined by ([0]) = [0] [0]; ([1]) = [1] [1]; () = [0] + [1]. The counit m* *ap I -! k sends both [0] and [1] to 1 2 k. The two inclusions k -! I given by the generat* *ors in degree 0 and the counit are maps of coalgebras. Note that the comultiplication of I is n* *ot cocommutative (this is reminiscent of the failure of the Alexander-Whitney map to be commutat* *ive). For any coassociative, counital differential graded coalgebra C, and any DGA A,* * the internal Hom-chain complex HomCh(C; A)* becomes a DGA with multiplication f . g = A O (f g) O C where A is the multiplication of A and C is the comultiplication of C. In part* *icular, HomCh(I; A) is a DGA, and it comes with DGA maps from A and to A x A which make it into a path object. In this way we recover the model category structure for * *associative DGAs over a commutative ring, first discovered by J. F. Jardine [J]. Our approach is* * a bit more gen- eral, since we can define similar path objects for associative DGAs over a fixe* *d commutative DGA, and for modules over a fixed DGA A. We thus also get model categories in t* *hose cases. However, since the basic differential graded coalgebra I is not cocommutative, * *this does not provide path objects for commutative DGAs. S-modules. The model category of S-modules, MS, is described in [EKMM, VII 4.6* *]. This model category structure is cofibrantly generated (see [EKMM, VII 5.6 and 5.8])* *. To ease notation, let Fq = S ^L L1q(-), the functor from topological spaces to MS that * *is used to define the model category structure on S-modules. In our terminology, the gener* *ating (acyclic) cofibrations are obtained by applying Fq to the generators for topological spac* *es, Sn -! CSn (CSn -! CSn ^ I+), where CX is the cone on X. The associative monoids are the S* *-algebras. The difficult part for showing that model category structures can be lifted to * *the categories of modules and algebras in this case is verifying the smallness hypothesis. Thi* *s is where the "Cofibration Hypothesis" comes in, see [EKMM, VII 5.2]. The underlying category* * of S-modules is a topological model category, see [EKMM, VII 4.4] and the triples in questio* *n are continuous. Hence, Remark 3.5 applies to give path objects, recovering [EKMM, VII 4.7], in * *particular the model category structures for R-algebras and R-modules. Our module comparison * *theorem 3.3 recovers [EKMM, III 4.2]. Our method of comparing algebra categories over * *equivalent commutative monoids does not apply here because the unit of the smash product i* *s not cofibrant. 5 Proofs Proof of Theorem 3.1 (1). The category of R-modules is also the category of alg* *ebras over the triple TR where TR(M) = R^M. The triple structure for TR comes from the mul* *tiplication R ^ R -! R. This theorem is a direct application of Lemma A.3 since by the mono* *id axiom, the JT-cofibrations are weak equivalences. * * __|_| 8 Proof of Theorem 3.1 (2). The model category part is Theorem 3.1 (1). By Lemma * *2.3, it suffices to check the pushout product axiom and the monoid axiom for the genera* *ting (acyclic) cofibrations. Every generating (acyclic) cofibration is induced from C by smash* *ing with R, i.e. it is of the form R ^A -! R ^B for A -! B a(n) (acyclic) cofibration in C. In the pushout product of two such * *maps, one R smash factor cancels due to using ^R , so that the pushout product is again ind* *uced from a pushout product of (acyclic) cofibrations in C, where the pushout product axiom* * holds. This gives the pushout product axiom for ^R . If J is a set of generating acyclic cofibrations in C, the set of generating ac* *yclic cofibrations in the category of R-modules (called JT above) consists of maps of J smashed with * *R. We thus have the equality JT ^R(R-mod) = J ^C. Since the forgetful functor R-mod -! C p* *reserves colimits (it has a right adjoint [R; -]), (JT ^(R-mod))-cofregis a subset of (J* * ^C)-cofreg. The monoid axiom for C thus implies the monoid axiom for R-mod. * * __|_| Proof of Theorem 3.1 (3). This proof is much longer than the previous ones; it * *occupies the rest of the paper. The main ingredient here is a filtration of a certain p* *ushout in the monoid category. This filtration is also needed to prove the statement about co* *fibrant monoids. The crucial step only depends on the weak equivalences and cofibrations in the * *model category structure. Hence we formulate it in a more general context. The hope is that it* * can also be useful in a situation where one only has something weaker than a model category* *, without a notion of fibrations. The following definition captures exactly what is needed. Definition 5.1An applicable category is a symmetric monoidal category C equippe* *d with two classes of morphisms called cofibrations and weak equivalences, satisfying the * *following axioms. oC has pushouts and filtered colimits. The monoidal product preserves colimi* *ts in each of its variables. oAny isomorphism is a weak equivalence and a cofibration. Weak equivalences * *are closed under composition. Cofibrations and acyclic cofibrations are closed under t* *ransfinite com- position and cobase change. oThe pushout product and monoid axiom are satisfied. Of course, any monoidal model category which satisfies the monoid axiom is appl* *icable. We are essentially forgetting all references to fibrations since they play no role* * in the following filtration argument. Note that the notion of regular cofibrations as defined in* * Definition 2.2 and Appendix A still makes sense in an applicable category. In the following le* *mma, let I (resp. J) be the class of those maps between monoids in C which are obtained from cofi* *brations (resp. acyclic cofibrations) in C by application of the free monoid functor, see (*) b* *elow. Lemma 5.2 If C is an applicable category, any regular J-cofibration is a weak * *equivalence in the underlying category C. If the unit I of the smash product is cofibrant, th* *en any regular I-cofibration whose source is cofibrant in C is a cofibration in the underlying* * category C. Proof of Theorem 3.1 (3), assuming lemma 5.2. By the already established part * *(2) of Theorem 3.1, the category of R-modules is itself a cofibrantly generated, mo* *noidal model category satisfying the monoid axiom. Also if I is cofibrant in C, then R, the * *unit for ^R , is 9 cofibrant in R-mod. So we can assume that the commutative monoid R is actually * *equal to the unit I of the smash product, thus simplifying terminology from "R-algebras" to * *"monoids". To use Lemma A.3 here we need to recognize monoids in C as the algebras over th* *e free monoid triple T . For an object K of C, define T (K) to be T (K) = I q K q (K ^K) q : :q:K ^nq : : : (*) One can think of T (K) as the `tensor algebra'. Using that ^ distributes over t* *he coproduct, T (K) has a monoid structure given by concatenation. The functor T is left adj* *oint to the forgetful functor from monoids to C. Hence T is also a triple on the category * *C and the T - algebras are precisely the monoids. Because the monoidal product is closed symmetric, ^ commutes with colimits. He* *nce, the underlying functor of T commutes with filtered colimits, as required for Lemma * *A.3. The condition on the regular cofibrations is taken care of by Lemma 5.2. Let f : M * *-! N be a cofibration of monoids with M cofibrant in C. Every cofibration of monoids is a* * retract of a regular I-cofibration with I as in Lemma 5.2. Hence f is a retract of a regular* * I-cofibration with source cofibrant in C, hence is a cofibration in C. In particular, a cofib* *rant monoid is a monoid M such that the unit map I -! M is a cofibration of monoids. Since I is * *cofibrant, this implies that the unit map is an underlying cofibration. Hence, M is cofibrant i* *n the underlying category C. __|* *_| Proof of lemma 5.2 The main ingredient is a filtration of a certain kind of pus* *hout in the monoid category. Consider a map K -! L in C, a monoid X and a monoid map T (K) * *-! X. We want to describe the pushout in the monoid category of the diagram T (K)_____wT (L) | | | | | |u X The pushout P will be obtained as the colimit, in the underlying category C, of* * a sequence X = P0 -! P1 -! . .-.! Pn -! . .:. If one thinks of P as consisting of formal products of elements from X and from* * L, with relations coming from the elements of K and the multiplication in X, then Pn consists of * *those products where the total number of factors from L is less than or equal to n. For ordin* *ary monoids, this is in fact a valid description, and we will now translate this idea into t* *he element-free form which applies to general symmetric monoidal categories. As indicated above we set P0 = X and describe Pn inductively as a pushout in C.* * We first describe an n-dimensional cube in C; by definition, such a cube is a functor W : P({1; 2; : :;:n}) -! C from the poset category of subsets of {1; 2; : :;:n} and inclusions to C. If S * * {1; 2; : :;:n} is a subset, the vertex of the cube at S is defined to be W (S) = X ^C1 ^X ^C2 ^: :^:Cn ^X 10 with ae Ci = KL ifii62fSi 2 S: All maps in the cube W are induced from the map K -! L and the identity on the * *X factors. So at each vertex a total of n+ 1 smash factors of X alternate with n smash fac* *tors of either K or L. The initial vertex corresponding to the empty subset has all Ci's equal* * to K and the terminal vertex corresponding to the whole set has all Ci's equal to L. For exa* *mple for n = 2, the cube is a square and looks like X ^K ^X ^K ^X _____wX ^K ^X ^L ^X | | | | |u |u X ^L ^X ^K ^X _____wX ^L ^X ^L ^X: Denote by Qn the colimit of the punctured cube, i.e., the cube with the termina* *l vertex removed. Define Pn via the pushout in C Qn _____(Xw^L)^ n^ X | | | | | | |u |u Pn-1 _________Pn:w This is not a complete definition until we say what the left vertical map is. * *We define the map from Qn to Pn-1 by describing how it maps a vertex W (S) for S a proper sub* *set of {1; 2; : : :;:n}. Each of the smash factors of W (S) which is equal to K is fir* *st mapped into X. Then adjacent smash factors of X are multiplied. This gives a map W (S) -! X ^L ^X ^: :^:L ^X ; where the right hand side has |S |+ 1 smash factors of X and |S | smash factors* * of L. So the right hand side maps further to P|S|, hence to Pn-1 since S is a proper subset. We have to check that these maps on the vertices of the punctured cube W are co* *mpatible so that they assemble to a map from the colimit, Qn. So let S be again a prope* *r subset of {1; 2; : :;:n} and take i 62 S. We have to verify commutativity of the diagram W (S) _________(Xw^L)^ |S|^X _________P|S|w | | | | | | |u |u W (S [{i})_____(Xw^L)^ (|S|+1)^X_______P|S|+1:w By definition, W (S) and W (S [{i}) differ at exactly one smash factor in the 2* *i-th position which is equal to K for the former and equal to L for the latter. The upper lef* *t map factors as W (S)_________w(X ^L)^ a^ X ^K ^(X ^L)^ b^ X _____(Xw^L)^ |S|^X 11 where a (resp. b) is the number of elements in S which are smaller (resp. large* *r) than i; in particular a + b = |S |. The right map in this factorization pushes K into X an* *d multiplies the three adjacent smash factors of X. Hence the diagram in question is the com* *posite of two commutative squares W (S)_____w(X ^L)^ a^ X ^K ^(X ^L)^ b^ X ______P|S|w | | | | | | | | | |u |u |u W (S [{i})_________(Xw^L)^ (|S|+1)^X___________wP|S|+1: The right square commutes by the definition of P|S|+1. We have now completed the inductive definition of Pn. We set P = colimPn, the c* *olimit being taken in C. P comes equipped with C-morphisms X = P0 -! P and L ~=I ^L ^I -! X ^L ^X -! P1 -! P which make the diagram K _____wL | | | | | | |u |u X _____wP commute. There are several things to check: (i) P is naturally a monoid so that (ii) X -! P is a map of monoids and (iii)P has the universal property of the pushout in the category of monoids. Define the unit of P as the composite of X -! P with the unit of X. The multipl* *ication of P is defined from compatible maps Pn ^Pm -! Pn+m by passage to the colimit. The* *se maps are defined by induction on n + m as follows. Note that Pn ^Pm is the pushout i* *n C in the following diagram. Qn ^((X ^L)m ^ X) [(Qn ^Qm)((X ^L)n ^X) ^Qm _____w((X ^L)n ^X) ^((X ^L)m ^ X) | | | | | | | | | | | | | | |u |u (Pn-1 ^Pm ) [(Pn-1^ Pm-1)(Pn ^Pm-1_)__________________Pnw^ Pm The lower left corner already has a map to Pn+m by induction, the upper right c* *orner is mapped there by multiplying the two adjacent factors of X followed by the map (X ^L)n+* *m ^ X -! Pn+m from the definition of Pn+m . We omit the tedious verification that this i* *n fact gives a well defined multiplication map and that the associativity and unital diagrams commu* *te. Hence, P is a monoid. Multiplication in P was arranged so that X -! P is a monoid map. 12 For (iii) , suppose we are given another monoid M, a monoidal map X -! M, and a* * C-map L -! M such that the outer square in K _____wL4 | | | | 4 | | 4 |u |u 4 X fl___wP " 4 flfl " 4 flflfl"446] flffl M commutes. We have to show that there is a unique monoidal map P - ! M making t* *he entire square commute. These conditions in fact force the behavior of the comp* *osite map W (S) -! Pn -! P -! M. Since P is obtained by various colimit constructions fro* *m these basic building blocks, uniqueness follows. We again omit the tedious verificati* *on that the maps W (S) -! M are compatible and assemble to a monoidal map P -! M. Now that we have established that P is the pushout of the original diagram of m* *onoids, we continue with the homotopical analysis of the constructed filtration, i.e. we w* *ill verify that the regular J-cofibrations are weak equivalences. Assume now that K -! L is an acyc* *lic cofibration in C. The cube W used in the inductive definition of Pn has n + 1 smash factors* * of X at every vertex which map by the identity everywhere. Using the symmetry isomorphism for* * ^, these can all be shuffled to one side and we get that the map Qn -! (X ^L)^ n^ X is i* *somorphic to Qn ^ X ^(n+1)-! L^ n^ X ^(n+1): Here Qn is the colimit of a punctured cube analogous to W , but with all the sm* *ash factors of X in the vertices deleted. By iterated application of the pushout product ax* *iom, the map Qn -~!L^ nis an acyclic cofibration. So by the monoid axiom, the map Pn-1 -~!Pn* * is a weak equivalence. The map X = P0 ~-!P is an instance of a transfinite composite (ind* *exed by the first infinite ordinal) of the kind of maps considered in the monoid axiom, so * *it is also a weak equivalence. With the use of the filtration we just established that any pushout, in the cat* *egory of monoids, of a map in J is a countable composite of maps of the kind considered in the mo* *noid axiom. A transfinite composite of transfinite composites is again a transfinite compos* *ite. Because the forgetful functor from monoids to C preserves filtered colimits, this shows* * that regular J-cofibrations are weak equivalences. It remains to prove the statement about regular I-cofibrations under the assump* *tion that the unit I is cofibrant. We note that if in the above pushout diagram K -! L is a c* *ofibration and the monoid X is cofibrant in the underlying category, then Qn ^X ^(n+1)-! L^ n^X ^(n+1) is a cofibration in the underlying category (by several applications of the pus* *hout product axiom). Thus also the maps Pn-1 -! Pn and finally X = P0 -! P are cofibrations * *in the underlying category. Since the forgetful functor commutes with filtered colimi* *ts, transfinite composites of such pushouts in the monoid category are still cofibrations in th* *e underlying category C. __|* *_| 13 A Cofibrantly generated model categories We need to transfer model category structures to categories of algebras over tr* *iples. In [Q, p. II 3.4], Quillen formulates his small object argument, which is now the standar* *d device for such purposes. After Quillen, several authors have axiomatized and generalized the * *small object argument (see e.g. [Bl, Def. 4.4], [Cr, Def. 3.2] or [Sch1, Def. 1.3.1]). In ou* *r context we will need a transfinite version of the small object argument. An axiomatization suitable * *for our purposes is the `cofibrantly generated model category' of [DHK], which we now recall. If a model category is cofibrantly generated, its model category structure is c* *ompletely de- termined by a set of cofibrations and a set of acyclic cofibrations. The transf* *inite version of Quillen's small object argument allows functorial factorization of maps as cofi* *brations followed by acyclic fibrations and as acyclic cofibrations followed by fibrations. Most * *of the model cate- gories in the literature are cofibrantly generated, e.g. topological spaces and* * simplicial sets, as are all the examples that appear in this paper. Ordinals and cardinals. An ordinal fl is an ordered isomorphism class of well o* *rdered sets; it can be identified with the well ordered set of all preceding ordinals. For an o* *rdinal fl, the same symbol will denote the associated poset category. The latter has an initial obj* *ect ;, the empty ordinal. An ordinal is a cardinal if its cardinality is larger than that of an* *y preceding ordinal. A cardinal is called regular if for every set of sets {Xj}j2J indexed by a set* * J of cardinality less than Ssuch that the cardinality of each Xj is less than that of , then the* * cardinality of the union J Xj is also less than that of . The successor cardinal (the smalles* *t cardinal of larger cardinality) of every cardinal is regular. Transfinite composition. Let C be a cocomplete category and fl a well ordered s* *et which we identify with its poset category. A functor V : fl -! C is called a fl-sequence* * if for every limit ordinal fi < fl the natural map colimV |fi-! V (fi) is an isomorphism. The map* * V (;) -! colimflV is called the transfinite composition of the maps of V . A subcatego* *ry C1 C is said to be closed under transfinite composition if for every ordinal fl and eve* *ry fl-sequence V : fl -! C with the map V (ff) -! V (ff + 1) in C1 for every ordinal ff < fl, * *the induced map V (;) -! colimflV is also in C1. Examples of such subcategories are the cofibr* *ations or the acyclic cofibrations in a closed model category. Relatively small objects. Consider a cocomplete category C and a subcategory C1* * C closed under transfinite composition. If is a regular cardinal, an object C 2 C is c* *alled -small relative to C1 if for every regular cardinal and every functor V : -! C1 wh* *ich is a -sequence in C, the map colimHom C(C; V ) -! Hom C(C; colimV ) is an isomorphism. An object C 2 C is called small relative to C1 if there exi* *sts a regular cardinal such that C is -small relative to C1. I-injectives, I-cofibrations and regular I-cofibrations. Given a cocomplete ca* *tegory C and a class I of maps, we denote oby I-inj the class of maps which have the right lifting property with respe* *ct to the maps in I. Maps in I-inj are referred to as I-injectives. oby I-cof the class of maps which have the left lifting property with respec* *t to the maps in I-inj. Maps in I-cof are referred to as I-cofibrations. 14 oby I-cofreg I-cof the class of the (possibly transfinite) compositions of p* *ushouts of maps in I. Maps in I-cofregare referred to as regular I-cofibrations. Quillen's small object argument [Q, p. II 3.4] has the following transfinite an* *alogue. Note that here I has to be a set, not just a class of maps. Lemma A.1 [DHK] Let C be a cocomplete category and I a set of maps in C whose * *domains are small relative to I-cofreg. Then othere is a functorial factorization of any map f in C as f = qi with q 2 I-* *inj and i 2 I- cofregand thus oevery I-cofibration is a retract of a regular I-cofibration. Definition A.2[DHK] A model category C is called cofibrantly generated if it is* * complete and cocomplete and there exists a set of cofibrations I and a set of acyclic cofibr* *ations J such that othe fibrations are precisely the J-injectives; othe acyclic fibrations are precisely the I-injectives; othe domain of each map in I (resp. in J) is small relative to I-cofreg(resp* *. J-cofreg). Moreover, here the (acyclic) cofibrations are the I (J)-cofibrations. For a specific choice of I and J as in the definition of a cofibrantly generate* *d model category, the maps in I (resp. J) will be referred to as generating cofibrations (resp. g* *enerating acyclic cofibrations). In cofibrantly generated model categories, a map may be functori* *ally factored as an acyclic cofibration followed by a fibration and as a cofibration followed* * by an acyclic fibration. Let C be a cofibrantly generated model category and T a triple on C. We want to* * form a model category on the category of algebras over the triple T , denoted T -alg. Call a* * map of T -algebras a weak equivalence (resp. fibration) if the underlying map in C is a weak equiv* *alence (resp. fibration). Call a map of T -algebras a cofibration if it has the left lifting * *property with respect to all acyclic fibrations. The forgetful functor T -alg-! C has a left adjoint,* * the free functor F T. The following lemma gives two different situations in which one can lift a mode* *l category on C to one on T -alg. We make no great claim to originality for this lemma. Other l* *ifting theorems for model category structures can be found in [Bl, Thm. 4.14], [CG, Thm. 2.5], * *[Cr, Thm. 3.3], [DHK, II 8.2], [EKMM, VII Thm. 4.7, 4.9]. Let X be a T -algebra. We define a path object for X to be a T -algebra XI tog* *ether with T -algebra maps X _______XIw~_____XwxwX factoring the diagonal map, such that the first map is a weak equivalence and t* *he second map is a fibration in the underlying category C. Lemma A.3 Assume that the underlying functor of T commutes with filtered direc* *t limits. Let I (J) be a set of generating cofibrations (resp. acyclic cofibrations) for the * *cofibrantly generated model category C. Let IT (resp. JT) be the image of these sets under the free T* * -algebra functor. Assume that the domains of IT (JT) are small relative to IT-cofreg(JT-cofreg). * *Suppose 15 1.every regular JT-cofibration is a weak equivalence, or 2.every object of C is fibrant and every T -algebra has a path object. Then the category of T -algebras is a cofibrantly generated model category with* * IT (JT) the generating set of (acyclic) cofibrations. Proof: We refer the reader to [DS, 3.3] for the numbering of the model category* * axioms. All those kinds of limits that exist in C also exist in T -alg, and limits are crea* *ted in the underlying category C [Bor, Prop. 4.3.1]. Colimits are more subtle, but since the underlyi* *ng functor of T commutes with filtered colimits, they exist by [Bor, Prop. 4.3.6]. Model catego* *ry axioms MC2 (saturation) and MC3 (closure properties under retracts) are clear. One half of* * MC4 (lifting properties) holds by definition of cofibrations of T -algebras. The proof of the remaining axioms uses the transfinite small object argument, w* *hich exists here because of Lemma A.1, and the hypothesis about the smallness of the domains. We begin with the factorization axiom, MC5. Every map in IT and JT is a cofibr* *ation of T -algebras by adjointness. Hence any IT-cofibration or JT-cofibration is a co* *fibration of T - algebras. By adjointness and the fact that I is a generating set of cofibration* *s for C, a map is IT-injective precisely when the map is an acyclic fibration of underlying ob* *jects, i.e., an acyclic fibration of T -algebras. Hence the small object argument applied to th* *e set IT gives a (functorial) factorization of any map in T -alg as a cofibration followed by an* * acyclic fibration. The other half of the factorization axiom, MC5, needs hypothesis (1) or (2). A* *pplying the small object argument to the set of maps JT gives a functorial factorization of* * a map in T - alg as a regular JT-cofibration followed by a JT-injective. Since J is a genera* *ting set for the acyclic cofibrations in C, the JT-injectives are precisely the fibrations among* * the T -algebra maps, once more by adjointness. In case (1) we assume that every regular JT-cof* *ibration is a weak equivalence on underlying objects in C. We noted above that every JT-cofib* *ration is a cofibration in T -alg. So we see that the factorization above is an acyclic cof* *ibration followed by a fibration. In case (2) we can adapt the argument of [Q, II p.4.9] as follows. Let i : X -* *! Y be any JT-cofibration. We claim that it is a weak equivalence in the underlying catego* *ry. Since X is fibrant and fibrations are JT-injectives, we obtain a retraction r to i by lift* *ing in the square X _____wXid | | i | oeo | | oer | |uoe |uu Y _____w*: Y possesses a path object and i has the LLP with respect to fibrations. So a li* *fting exists in the square X _____wYi_____wY I | B BBC | | B B | | B | |uB B |uu Y ____________Ywx(Y:id;iOr) This shows that in the homotopy category of C, iOr is equal to the identity map* * of Y . Since maps in C are weak equivalences if and only if they become isomorphisms in the homot* *opy category 16 of C, this proves that i is a weak equivalence, and it finishes the proof of mo* *del category axiom MC5 under hypothesis (2). It remains to prove the other half of MC4, i.e., that any acyclic cofibration * *A v_____Bw~has the LLP with respect to fibrations. In other words, we need to show that the ac* *yclic cofibrations are contained in the JT-cofibrations. The small object argument provides a fact* *orization A v_____Ww~____wBw with A -! W a JT-cofibration and W -! B a fibration. In addition, W -! B is a w* *eak equivalence since A -! B is. Since A -! B is a cofibration, a lifting in A _____Ww v | | oeo|~ | oe | |uoe |uu B _____wBid exists. Thus A -! B is a retract of a JT-cofibration, hence a JT-cofibration. * * __|_| Remark A.4 Hypothesis (2) can be weakened to the existence of a fibrant replac* *ement functor in the category of T -algebras which interacts well with respect to the path ob* *ject, see [Sch2, Lemma A.3]. Quillen's argument in [Q, II p.4.9] in fact uses Kan's Ex1 functo* *r as such a fibrant replacement functor. Remark A.5 To simplify the exposition, we assume that every object of C is sma* *ll relative to the whole category C when we apply lemma A.3 in the rest of this paper. This* * holds for -spaces and symmetric spectra based on simplicial sets. If the underlying funct* *or of the triple T on C commutes with filtered direct limits, then so does the forgetful functor* * from T -algebras to C. Hence by adjointness, every free T -algebra is small relative to the who* *le category of T -algebras, so the smallness conditions of lemma A.3 hold. Of course, if one i* *s interested in a category where not all objects are small with respect to all of C one must veri* *fy those smallness conditions directly. 17 References [Bl] D. Blanc: New model categories from old, J. Pure Appl. Algebra 109 (1* *996), 37-60 [Bor] F. Borceux: Handbook of Categorical Algebra 2: Categories and Structu* *res, Encyclopedia of Mathematics and its Applications 51, Cambridge University Press (1* *994) [BF] A. K. 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Segal: Categories and cohomology theories, Topology 13 (1974), 293* *-312 Fakult"at f"ur Mathematik Department of Mathematics Universit"at Bielefeld University of Chicago 33615 Bielefeld, Germany Chicago, IL 60637, USA schwede@mathematik.uni-bielefeld.de bshipley@math.uchicago.edu 18