Algebras and modules in monoidal model categories Stefan Schwede and Brooke E. Shipley1 8 9 Abstract: We construct model category structures for monoids 9 andmodulesinsymmetricmonoidalmodelcategories,withappli- 1 cations to symmetric spectra and Γ-spaces. n a 1991 AMS Math. Subj. Class.: primary 55U35, secondary 18D10 J 9 1 1 Summary ] T This paper gives a general approach for obtaining model category structures for algebras or A modulesoversomeothermodelcategory. Technically,whatwemeanbyan‘algebra’isamonoid . h in a symmetric monoidal category. Of course, the symmetric monoidal and model category at structures have to be compatible, which leads to the definition of a monoidal model category, m 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 pushouts of monoids [ (see Lemma 5.2) is then used to reduce the problem to standard model category arguments 1 based on Quillen’s “small object argument”. Our main result is stated in Theorem 3.1. v 2 8 Thisapproachwasdevelopedinparticulartoapplytothecategoryofsymmetricspectradefined 0 in [HSS] and to Γ-spaces in [Sch2]. In both of these categorieswe thus obtain model categories 1 for the associative monoids, the R-modules for any monoid R, and the R-algebras for any 0 commutative monoid R. A significant shortcut is possible if the underlying monoidal model 8 category has the special property that all objects are fibrant, see Remark 3.5. This is not true 9 / for our main examples, symmetric spectra and Γ-spaces. It does hold though, in the monoidal h model categories of simplicial abelian groups, chain complexes, or S-modules (in the sense of t a [EKMM]). m : We assume that the reader is familiar with the language of homotopical algebra (cf. [Q], [DS]) v and with the basic ideas concerning monoidal and symmetric monoidal categories (cf. [MacL, i X VII], [Bor, 6]) and triples (also called monads, cf. [MacL, VI.1], [Bor, 4]). r a Acknowledgments. We would first like to thank Charles Rezk for conversationswhich led us to the filtration that appears in Lemma 5.2. We also benefited from several conversations about thisprojectwithBillDwyer,MarkHoveyandManosLydakis. WewouldalsoliketothankBill 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. 1ResearchpartiallysupportedbyanNSFPostdoctoral Fellowship 1 2 Monoidal model categories Amonoidalmodelcategoryisessentiallya modelcategorywithacompatible closedsymmetric monoidalproduct. Thecompatibilityisexpressedbythepushoutproductaxiombelow. Inthis paper we always require a closed symmetric monoidal product although for expository 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 requirements analogous to the pushout product axiom or the simplicial axiom of [Q, II.2]. For example, closed simplicial model categories [Q, II.2] are such compatibly enriched categories over the monoidal model category of simplicial sets. Wealsointroducethemonoidaxiomwhichisthecrucialingredientforliftingthemodelcategory structure to monoids and modules. Examples of monoidal model categories satisfying the monoid axiom are given in Section 4. Definition 2.1 AmodelcategoryC isamonoidal model categoryifitisendowedwithaclosed 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 the map A∧L∪A∧K B∧K −−−→ B∧L isalsoacofibration. Ifinadditiononeofthe formermapsisaweakequivalence,soisthelatter 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-cof the class of reg maps obtained from the maps of I by cobase change and composition (possibly transfinite, see Appendix A.) These maps are referred to as the regular I-cofibrations. Definition 2.2 A monoidal model category C satisfies the monoid axiom if every map in ({acyc. cofibrations}∧C)-cofreg is a weak equivalence. Note that if C has the specialpropertythat everyobjectis cofibrant,then the monoidaxiomis a consequence of the pushout product axiom. However, this special situation rarely 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 generatingacycliccofibrations,andsimilarlyforacyclicfibrations. Thisisincontrasttogeneral model categorieswhere the lifting property has to be checkedagainstthe whole class of acyclic cofibrations. In cofibrantly generated model categories, the pushout product axiom 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 closed 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)-cofreg is 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) the 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 respect 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]× [A,X]. [A,Y] Hence, a map is in G(i) if and only if it has the left lifting property with respect 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 assumption; because of the closure properties it thus contains all cofibrations, see Lemma A.1. Reversing the roles of i and an arbitrary cofibration j : K −→ L we thus know that G(j) contains all generating cofibrations. Again by the closure properties, G(j) contains all cofibrations, which proves the pushoutproductaxiomfortwocofibrations. Theproofofthepushoutproductbeinganacyclic 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, every acyclic cofibration is a retract of a transfinite composition of cobase changes along the generating acyclic cofibrations. Since transfinite compositions of transfinite compositions are transfinite compositions,everymapin({acyc. cofibrations}∧C)-cofreg isthusaretractofamapin(J∧C)- cof . reg 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. Examples of model categories for which this theorem applies are given in section 4. We end this section with two theoremswhichcomparethehomotopycategoriesofmodulesoralgebrasoverweaklyequivalent monoids. We consider asymmetric monoidalcategorywith product ∧ andunit I. Amonoid is anobject R together with a “multiplication” map R∧R −→ R and a “unit” I −→ R which satisfy certainassociativityand unit conditions (see [MacL, VII.3]). R is a commutative monoid if the multiplicationmapisunchangedwhencomposedwiththetwist,orthesymmetryisomorphism, of R∧R. If R is a monoid, a left R-module (“object with left R-action” in [MacL, VII.4]) is an 3 objectN togetherwith anactionmapR∧N −→N satisfying associativityandunitconditions (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 product of two R-modulesisanotherR-moduleandsmashingoverRmakesR-modintoasymmetricmonoidal category with unit R. If C has equalizers, there is also an internal Hom object of R-modules, [M,N]R. Itistheequalizeroftwomaps[M,N] −−−−−−−−→→ [R∧M,N]. Thefirstmapisinducedby 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 category of R- modules. It is a formal property of symmetric monoidal categories (cf. [EKMM, VII 1.3]) that specifying an R-algebra structure on an object A is the same as giving A a monoid structure togetherwithamonoidmapf:R−→Awhichiscentralinthesensethatthefollowingdiagram commutes. swit-ch id∧-f R∧ A A ∧R A ∧ A f∧id mult. ? ? - A ∧ A A mult. Now we can state our main theorem. It essentially saysthat monoids,modules and algebrasin a cofibrantlygenerated,monoidalmodel categoryC againforma modelcategoryif the monoid axiomholds. (SeeAppendix Aforthe definitionofacofibrantlygeneratedmodelcategory.) To simplifytheexposition,weassumethatallobjectsinC aresmall(refertoAppendix 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. Inthe categoriesofmonoids,leftR-modules(whenRisafixedmonoid),andR-algebras(when R is a commutative monoid) a morphism is defined to be a fibration or weak equivalence 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 further 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 cofibrantly generated model category. 2. Let R be a commutative monoid in C. Then the category of R-modules is a cofibrantly 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 cofibrantly generated model category. If the unit I of the smash product is cofibrant in C, then every cofibration of R-algebras whose source is cofibrant in C is also a cofibration 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 Thefullstrengthofthemonoidaxiomisnotnecessarytoobtainamodelcategory 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 cofibrant in the underlying category C. For then the pushout product axiom implies that smashing with R preserves acyclic cofibrations. The following theorems concern comparisons of homotopy categories of modules and algebras. ThehomotopytheoryofR-modulesandR-algebrasshouldonlydependontheweakequivalence typeofthemonoidR. ToshowthisforR-moduleswemustrequirethatthefunctor−∧RN take any weak equivalence of right R-modules to a weak equivalence in C whenever N is 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 equivalences ∼ of right R-modules toweak equivalences in C. IfR−→S is a weak equivalence of monoids, then thetotalderived functorsofrestrictionandextensionofscalars induceequivalences ofhomotopy 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 acyclic 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 equivalence. [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 total derived functors of restriction and extension of scalars induce equivalences of homotopy categories Ho(R-alg) ∼= Ho(S-alg) . Proof: The proof is similar to the one of the previous theorem. Again the right adjoint restriction functor does not change underlying objects, so it preserves fibrations 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 property that all objects are fibrant. This greatly simplifies the situation. If there is also a simplicial or topological model category structure and if a simplicial (resp. topological) triple 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 againthat in our mainexamples, symmetric spectra andΓ-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-algebratriples. The categoryof commutativemonoids often has a model categorystruc- ture in the fibrant case (e.g. commutative simplicial rings or commutative S-algebras[EKMM, Cor. VII 4.8]). In contrast, for Γ-spaces and symmetric spectra, the category of commutative monoids can not form a model category with fibrations and weak equivalences defined 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 monoids. Evaluating this fibrant representative on 1+∈ Γop, or at level 0 respectively, would give a commutative simplicial monoid weakly equivalent to QS0. This would imply that the space QS0 is weakly equivalenttoaproductofEilenberg-MacLanespaces,whichisnotthecase. Thehomotopycat- egory of commutative monoids in symmetric spectra is still closely related to E -ring spectra, ∞ though. 4 Examples Simplicial sets. The category of simplicial sets has a well-known model category structure established by D. Quillen[Q,II 3,Thm.3]. The cofibrationsarethe degreewiseinjective maps,the fibrationsare theKanfibrationsandtheweakequivalencesarethemapswhichbecomehomotopyequivalences 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 category. The cartesianproductofsimplicial sets is symmetric monoidalwith unit the discrete 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 follows from the pushout product axiom. A monoid in the category of simplicial sets under cartesianproduct is just a simplicialmonoid,i.e., a simplicialobject ofordinaryunital andassociativemonoids. 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]. Γ-spaces and symmetric spectra These two examples are new. In fact, the justification for writing this paper is 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 Γ-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. The homotopy categoryof symmetric spectra is equivalent to the usual stable homotopy category of algebraic topology. In the case of Γ-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‘bravenewrings’,i.e.,ringsuptohomotopywithhighercoherenceconditions. Another approachto this idea consists of the S-algebras of [EKMM]. 6 Γ-spaces. Γ-spaceswere introducedby G. Segal[Se] who showedthat they give rise to a homo- topycategoryequivalenttotheusualhomotopycategoryofconnectivespectra. A.K.Bousfield andE.M.Friedlander[BF]consideredabiggercategoryofΓ-spacesinwhichtheonesintroduced by Segal appeared as the special Γ-spaces. Their category admits a simplicial model category structure with a notion of stable weak equivalence giving rise again to the homotopy theory of connectivespectra. ThenM.Lydakis[Ly]showedthatΓ-spacesadmitinternalfunctionobjects and a symmetric monoidal smash product with nice homotopical properties. Smallness and cofibrant generation for Γ-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, SpΣ, is described in [HSS]. There it is also shown that this category is a cofibrantly generated, monoidal model category, and that the associatedhomotopy categoryis equivalentto the usualhomotopycategoryofspectra. For symmetric spectra over the category of simplicial sets every object is small with respect to the whole category. The monoid axiom and the fact that smashing with a cofibrant left R-module preservesweakequivalencesbetweenrightR-modulesareverifiedin[HSS].Themonoidsinthis 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 fibrant. With thisspecialpropertyitiseasiertoliftmodelcategorystructuressincethe(oftenhardtoverify) 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 sometimes form model categories in these cases. The pushout product and monoid axioms also hold in these examples,butsincethefibrancypropertydeprivesthemoftheirimportance,wewillnotbother to prove them. Simplicial abelian groups. The model category structure for simplicial abelian groups was establishedbyQuillen[Q,II.6]. Theweakequivalencesandfibrationsaredefinedonunderlying simplicial sets. The cofibrations are the retracts of the free maps (see [Q, II p. 4.11, Rem. 4]). Thismodelcategoryis cofibrantlygeneratedandallobjects aresmall. The (degreewise)tensor productprovidesasymmetricmonoidalproductforsimplicialabeliangroups. Theunitforthis product is the integers, considered as a constant simplicial abelian group. A monoid 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 the standard 1-simplex into the underlying simplicial set of R is naturally a simplicial ring. The model categorystructure 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 commutative 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 maps which are surjective in positive degrees, and cofibrations are monomorphisms with degreewise projective cokernels. This model category is cofibrantly generated and every object is small. The cate- gory of unbounded chain complexes over k, although less well known, also forms a cofibrantly generatedmodelcategorywithweakequivalencesthehomologyisomorphismandfibrationsthe epimorphisms, see [HPS], remark after Thm. 9.3.1. The cofibrations here are still degreewise split injections, but their description is a bit more complicated than for bounded chain com- plexes. The followingremarksrefer to this categoryofZ-gradedchaincomplexesofk-modules. 7 The graded tensor product of chain complexes is symmetric monoidal and has adjoint internal hom-complexes. A monoidinthis symmetric monoidalcategoryis a differentialgradedalgebra (DGA). Everycomplex is fibrant and associative DGAs have path objects. To construct them, we need the following 2-term complex denoted I. In degree 0, I consists of a free k-module on two generators [0] and [1]. In degree 1, I is a free k-module on a single generator ι. The differential is given by dι = [1] − [0]. This complex becomes a coassociative and counital coalgebra when given the comultiplication ∆:I −→I⊗ I k defined by ∆([0])=[0]⊗[0], ∆([1])=[1]⊗[1], ∆(ι)=[0]⊗ι+ι⊗[1]. The counit map I −→k sends both [0] and [1] to 1∈k. The two inclusions k −→I given by the generators in degree 0 andthecounitaremapsofcoalgebras. NotethatthecomultiplicationofI isnotcocommutative (this is reminiscent of the failure of the Alexander-Whitney map to be commutative). For any coassociative, counital differential graded coalgebra C, and any DGA A, the internal Hom-chain complex Hom (C,A) becomes a DGA with multiplication Ch ∗ f ·g =µ ◦(f ⊗g)◦∆ A C where µ is the multiplication of A and ∆ is the comultiplication of C. In particular, A C Hom (I,A) is a DGA, and it comes with DGA maps from A and to A × A which make Ch itintoapathobject. Inthis waywerecoverthe modelcategorystructureforassociativeDGAs 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 fixed commutative DGA, and for modules over a fixed DGA A. We thus also get model categories in those 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, M , is described in [EKMM, VII 4.6]. This S model category structure is cofibrantly generated (see [EKMM, VII 5.6 and 5.8]). To ease notation, let F = S ∧ LΣ∞(−), the functor from topological spaces to M that is used to q L q S define the model categorystructureonS-modules. In ourterminology,the generating(acyclic) cofibrations are obtained by applying F to the generators for topological spaces, Sn −→CSn q (CSn −→CSn∧I ), whereCX isthe coneonX. The associativemonoidsarethe 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. This is where the “CofibrationHypothesis”comesin,see[EKMM,VII5.2]. TheunderlyingcategoryofS-modules isatopologicalmodelcategory,see[EKMM,VII4.4]andthetriplesinquestionarecontinuous. 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 commutativemonoidsdoesnotapplyherebecausetheunitofthesmashproductisnotcofibrant. 5 Proofs Proof of Theorem 3.1 (1). The category of R-modules is also the category of algebras over thetripleT whereT (M)=R∧M. ThetriplestructureforT comesfromthemultiplication R R R R∧R −→ R. This theorem is a direct application of Lemma A.3 since by the monoid axiom, the J -cofibrations are weak equivalences. T 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 productaxiomand the monoidaxiomfor the generating(acyclic) cofibrations. Every generating (acyclic) cofibration is induced from C by smashing 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 induced 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 acyclic cofibrations in the category of R-modules (called J above) consists of maps of J smashed with R. We thus T have the equality JT∧R(R-mod) = J∧ C. Since the forgetful functor R-mod −→ C preserves colimits (it has a right adjoint [R,−]), (JT∧(R-mod))-cofreg is 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 pushout in the monoidcategory. Thisfiltrationisalsoneededtoprovethestatementaboutcofibrantmonoids. The crucialstep only depends onthe weak equivalences and cofibrationsin 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 5.1 An applicable category is a symmetric monoidal categoryC equipped with two classesofmorphismscalledcofibrationsandweakequivalences,satisfyingthefollowingaxioms. • C has pushouts and filtered colimits. The monoidal product preserves colimits in each of its variables. • Any isomorphism is a weak equivalence and a cofibration. Weak equivalences are closed undercomposition. Cofibrationsandacycliccofibrationsareclosedundertransfinitecom- position and cobase change. • The pushout product and monoid axiom are satisfied. Of course, any monoidal model category which satisfies the monoid axiom is applicable. 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 andAppendixAstillmakessenseinanapplicablecategory. Inthefollowinglemma,letI (resp. J)bethe classofthosemaps betweenmonoidsinC whichareobtainedfromcofibrations(resp. acyclic cofibrations) in C by application of the free monoid functor, see (∗) below. 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, then 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, monoidal model category satisfying the monoid axiom. Also if I is cofibrant in C, then R, the unit for ∧R, is 9 cofibrantin R-mod. So wecanassumethatthe commutativemonoidRis actuallyequalto the unit I of the smash product, thus simplifying terminology from “R-algebras”to “monoids”. TouseLemmaA.3hereweneedtorecognizemonoidsin C asthe algebrasoverthefreemonoid triple T. For an object K of C, define T (K) to be T (K) = I ∐ K ∐ (K∧K) ∐ ... ∐ K∧n ∐ ... (∗) One can think of T(K) as the ‘tensor algebra’. Using that ∧ distributes over the coproduct, T (K) has a monoid structure given by concatenation. The functor T is left adjoint 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. Hence, 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 cofibrant monoid is a monoidM suchthattheunitmapI−→M isacofibrationofmonoids. SinceIiscofibrant,this implies thatthe unit mapis anunderlyingcofibration. Hence,M is cofibrantinthe underlying category C. Proof of lemma 5.2 The main ingredient is a filtration of a certain kind of pushout in the monoidcategory. Considera mapK −→L inC, a monoidX and amonoidmap T (K)−→X. We want to describe the pushout in the monoid category of the diagram - T (K) T (L) ? X The pushout P will be obtained as the colimit, in the underlying category C, of a sequence X =P −→P −→···−→P −→··· . 0 1 n IfonethinksofP asconsistingofformalproductsofelementsfromX andfromL,withrelations comingfromthe elements ofK andthe multiplicationin X,thenP consistsofthoseproducts n where the total number of factors from L is less than or equal to n. For ordinary monoids, this is infact a valid description,andwe will now translatethis idea into the element-free form which applies to general symmetric monoidal categories. As indicated above we set P = X and describe P inductively as a pushout in C. We first 0 n 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