L-STABLE FUNCTORS GEORG BIEDERMANN Abstract.We generalize and greatly simplify the approach of Lydakis and Dundas-R"ondigs-Ostvaer to construct an L-stable model structure for sma* *ll functors from a closed symmetric monoidal model category V to a V-model category M, where L is a small cofibrant object of V. For the special ca* *se V = M = S* pointed simplicial sets and L = S1 this is the classical case of linear functors and has been described as the first stage of the Good* *willie tower of a homotopy functor. We show, that our various model structures are compatible with a closed symmetric monoidal product on small functor* *s. We compare them with other L-stabilizations described by Hovey, Jardine and others. This gives a particularly easy construction of the classical* * and the motivic stable homotopy category with the correct smash product. We establish the monoid axiom under certain conditions. Contents 1. Introduction 1 2. Closed symmetric monoidal model categories 4 3. Localization 7 4. Small functors 8 5. Fixing notation and first assumptions 9 6. The projective model structure 10 7. The homotopy model structure 11 8. The L-stable model structure 13 9. Functoriality 18 10. L-stabilization 20 11. Symmetric monoidal structures 22 12. The monoid axiom 26 13. L-spectra 32 14. Examples 37 References 38 1. Introduction Stable homotopy theory is concerned with solving homotopy theoretical proble* *ms up to a finite number of suspensions. So we want to work in a category, where t* *he suspension functor is inverted, i.e. is an equivalence of categories. Modern ac* *counts ____________ Date: April 19, 2007. 1991 Mathematics Subject Classification. 55P42, 55U35, 18D20, 55P91. Key words and phrases. stable functors, linear functors, spectra, Goodwillie* * calculus, stable homotopy theory, small functors, symmetric monoidal categories. 1 2 GEORG BIEDERMANN of homotopy theory use model structures and there are several constructions of model categories giving the right stable homotopy category. An early one was [3* *]. However, we also want to get the right symmetric monoidal product - the smash product - on the model level. Only much later, see [12] and [17], were model categories found, that also support a symmetric monoidal structure inducing the known smash product on the homotopy level. All of them involve a notion of spectrum. Spectra are closely connected to linear functors, i.e. generalized homology t* *heo- ries in the sense of Eilenberg and Steenrod. Every spectrum represents a homolo* *gy theory, and every homology theory is represented by a spectrum by Brown's rep- resentability theorem. It is very useful to observe here, that spectra themsel* *ves in all the different variants are simplicially enriched functors on one or anot* *her simplicially enriched category with values in simplicial sets. Manos Lydakis in [20] put a model structure on the category of functors from pointed finite simplicial sets to pointed simplicial sets, such that the homoto* *py types correspond exactly to linear functors i.e. reduced generalized homology theori* *es. Moreover he showed, that his category supports a symmetric monoidal product with a simplicial symmetric monoidal Quillen equivalence to the models for spectra. * *This was motivated by the Goodwillie tower of a homotopy functor [13], whose first s* *tage is exactly described by Lydakis' construction. He also showed, that the homotopy category associated to this functor category is symmetrically monoidally equiva* *lent to the classical stable homotopy category. Until now we have only talked about stabilization with respect to suspension: that is smashing with S1. But stabilization with respect to another object has gained interest. In motivic homotopy theory one considers the category of simpl* *icial (pre-)sheaves over smooth finite dimensional schemes over a base scheme. This category is a merger of simplicial sets and smooth schemes. Then one stabilizes* * it with respect to P1, which is the smash product of the simplicial circle S1 and * *the Tate circle A1- {0}. The applications of this idea by Voevodsky and many others are now famous. References include [21] and [18]. So it is legitimate to ask fo* *r a general theory of stabilization. We will address the following setting: Given a symmetric monoidal model cat- egory V, let us consider the category of V-model categories, whose morphisms are given by V-Quillen adjunctions. Now take a V-model category M and a V-small cofibrant object L. Tensoring with L induces a left V-Quillen functor on M, who* *se right adjoint will commute with filtered homotopy colimits. Under certain techn* *ical conditions we can associate to a V-category M another V-category, on which L ac* *ts as a Quillen equivalence. This will be called an L-stabilization of M. The spec* *ial case M = V = S*, pointed simplicial sets, and L = S1, the simplicial circle, br* *ings us back to the initial situation of stable homotopy theory. From the spectrum point of view this has been investigated by Hovey in [16]. We develop here Lydakis' point of view. Under certain conditions they turn out * *to be equivalent. To carry this out we study functor categories from certain full subcategories* * U of V to the given V-model category M. We first consider in section 6 the projec* *tive model structure on the category of small functors from U to M. Small functors were introduced to homotopy theory in [6] and to Goodwillie calculus in [1]. Th* *ey are exactly the right tool to deal with set-theoretic problems in functor categ* *ories L-STABLE FUNCTORS 3 with non-small source category. We use work of Day and Lack [8], that supplies the completeness of our functor categories with non-small source. In section 7 we localize the projective model structure, so that the objectwi* *se fibrant homotopy functors become the fibrant objects. A homotopy functor is a functor, that preserves weak equivalences. In this localization step we use the assumption, that every object in U is cofibrant. If U is small, this condition * *can probably be relaxed. But since we can make good use of it in the later developm* *ent, we just keep it. It is a technical weakness, that has to be dealt with in futur* *e work. Finally in section 8 we localize further, so that the objectwise fibrant L-st* *able homotopy functors become the fibrant objects. What are L-stable functors? By a general fact small functors are V-functors, so they come equipped with a natural map Xe( __) L ! Xe( __ L). Let us call the left V-Quillen functor obtained by tensoring with L simply L and its right adjoint R. A homotopy functor Xe:V ! M is L-stable if the adjoint map Xe! R O XeO L = V(L, Xe( __ L)) is an objectwise weak equivalence. Observe, that by [13, Thm 1.8] S1-stable fun* *ctors are exactly those with a Mayer-Vietoris sequence. In the L-stable structure eve* *ry small functor is weakly equivalent to an L-stable one. For L = S1 this is Lydak* *is' approach to the stable homotopy category. Lydakis has shown, how to provide a tensor product on the functor category, that induces the right symmetric monoidal structure on homotopy level. This is actually a special case of a more general procedure devised by Day in [7]. We p* *rove compatibility results with our various model structures on the functor categori* *es in section 11. We obtain in 11.9, that the category VU is a closed symmetric monoi* *dal model category. The category MU is a VU -category. For small source category U we can study functoriality with respect to M. We prove, that there is a 2-functor from V-model categories with certain propertie* *s to itself with values in L-stable categories, i.e. those on which L acts as a Qui* *llen equivalence. To study functoriality of our L-stabilization with respect to the * *object L is less straightforward as in [16]. With an auxiliary L-L0-stable model struc* *ture we show in 9.5, that the Quillen equivalence type of MU with the L-stable model structure depends only on the weak homotopy type of L. Finally we compare our L-stabilizations with symmetric L-spectra and Bousfiel* *d- Friedlander L-spectra constructed by Hovey in [16]. Using a certain subcategory SphVL= U resembling the spheres as a source category, we can prove, that there * *is a zig-zag of V-Quillen equivalences from our L-stable model structure on MSphVL to Hovey's symmetric L-spectra on M. This is only suboptimal, since there is a canonical candidate for a direct Quillen equivalence. By comparing L-stable fun* *c- tors to Bousfield-Friedlander L-spectra we can get sufficient conditions, when * *all three models are Quillen equivalent. This is discussed in sections 13 and 14. This is not the first time a generalization of Lydakis' work was attempted. F* *or certain small U and M = V and general L Dundas, R"ondigs and Ostvaer in [10] described L-stabilizations. Following them we can also prove the monoid axiom in certain settings 12.18. This article generalizes further and greatly simplifies* * their work on L-stable functors. 4 GEORG BIEDERMANN Acknowledgments: This work is a side track of my joint project with Boris Chorny and Oliver R"ondigs to define Goodwillie calculus of homotopy functors for general model categories. I am especially grateful to both of them to point out to me the right generators for the acyclic cofibrations in the various model categories. I would also like to thank Gerald Gaudens and Bjorn Dundas for help* *ful conversations. Finally I would like to thank Steve Lack for keeping me updated * *on his joint work on [8]. This article was written partly at the University of Western Ontario and part* *ly during my guest stay at the Fields Institute, Toronto. 2. Closed symmetric monoidal model categories We will first describe compatibility conditions between a model structure an* *d a closed symmetric monoidal structure. The basic references for enriched category theory are [2] and [19]. A good reference on enriched model categories is [15]. Let V be a closed symmetric monoidal category. We will denote the monoidal functor by and the unit by S. A V-category M is a category, which is enriched, tensored and cotensored over V, where these structures satisfy some adjunction relations. The enrichment given by the V-object of maps in M will be denoted by M(A, B) for objects A, B in M. The cotensor will be denoted by XA for X in M and A in V. Of course, for M = V we have V(A, X) = XA . Definition 2.1. Assume, that V and M are both complete and cocomplete. Let i: A ! B be a map in V and j :C ! D be map in M. The pushout product is the map i j :(A D) t(A C) (B C) ! B D in M obtained from the universal property of the pushout. There is an adjoint construction p i: XB ! Y BxY AXA for a map p: X ! Y in M. Definition 2.2. Let V be a closed symmetric monoidal category equipped with a model structure. Then V is a closed symmetric monoidal model category, if the t* *wo structures are compatible in the following sense: (i)For two cofibrations i and j in V their pushout product i j is a cofibrati* *on, which is acyclic if either i or j is acyclic. (ii)For a cofibrant replacement QS ! S of the unit S the induced map QS X ! S X ~=X is a weak equivalence for every object X in V. The dual formulation of (i) is that for a fibration p the map p i is a fib* *ration, which is acyclic if either i or p is acyclic. Later on we will assume, that all* * objects in V are cofibrant. In particular the unit S will be cofibrant turning conditio* *n (ii) in the previous definition redundant. Similar remarks apply for the next defini* *tion. Definition 2.3. Let V be a closed symmetric monoidal model category. Let M V-category equipped with a model structure. M is a V-model category, if the two structures are compatible in the following sense: (i)For a cofibrations i in V and a cofibration j in M their pushout product i * * j is a cofibration, which is acyclic if either i or j is acyclic. (ii)For a cofibrant replacement QS ! S of the unit S the induced map QS X ! S X ~=X is a weak equivalence for every object X in M. L-STABLE FUNCTORS 5 For the notion of cofibrantly generated model category see references. Lemma 2.4. Let I and J be sets of generating (acyclic) cofibrations for V. Supp* *ose I I cofand I J acofand that the unit S is cofibrant. Then V is a closed symmetric monoidal model category. The proof is a straightforward exercise in adjunctions and lifting propertie* *s. The precise form of the next definition can be read in [2, Def. 6.2.3.] or [19, p. * *9]. Definition 2.5. Let M and N be two V-categories. A V-functor Xe:M ! N is a function Ob(Xe): Ob (M) ! Ob(N ) together with maps M(A, B) ! N (Xe(A), Xe(B)) for all objects A, B in M, such that certain diagrams commute. They assert, that Xe behaves well with respect to composition and the identity. For a V-functor Xe:V ! M and objects K and L in V we have canonical maps S ! V(K L, K L) ~=V(K, K L)L! V(Xe(K), Xe(K L))L (2.1) ~=V(X (K) L, X(K L)). e e The resulting map (2.2) Xe( __) L ! Xe( __ L) will be called assembly map. The fact, that Xeis a V-functor can also be expres* *sed in terms of these maps. We will need these maps to define L-stability of functo* *rs. Remark 2.6. There is a notion of V-natural transformation, we refer the reader * *to [19, p.9]. For the definition of V-Quillen adjunction we refer to [15, Def. 4.2* *.18.]. It is explained there, that V-model categories together with V-Quillen adjunction * *and V-natural transformations form a 2-category, which we will denote by V-mod. We will usually refer to a morphism in this category by its left adjoint. Later on* * 5.2 we will define a full subcategory V-mod__of V-mod, that will be of primary interes* *t. We want to generalize the fact from simplicial model categories, that weak e* *quiv- alences between fibrant and cofibrant objects can be detected with the use of a* * sim- plicial interval 1. We will first introduce left V-homotopy and then V-homotopy equivalence, which actually should be called left V-homotopy equivalence. Definition 2.7. For the cofibrant unit S of V let Cyl(S) denote an object obtai* *ned by factoring the fold map S t S ! S into a cofibration i: S t S ! Cyl(S) follow* *ed by a weak equivalence p: Cyl(S) ! S. We have two inclusions S ! S t S, which we can compose with the map i to obtain two cofibrations i0 and i1: S ! Cyl(S). These are section of the map p and hence weak equivalences. We have the followi* *ng diagram: _i1_____________________________________* *_________ ___p_____________________________________* *_________ S t S_____//CS____//_Sff______________________________* *_____________________________________________________ i0 We emphasize, that for the moment there is a whole class of choices for Cyl(S). Lemma 2.8. Let V be a closed symmetric monoidal model category with cofibrant unit S and let M be a V-model category. Let CS be a cylinder object over S and let X be a cofibrant object in M. Then CS X becomes a cylinder object for X. 6 GEORG BIEDERMANN Proof.Since X is cofibrant, tensoring with X preserves cofibrations and acyclic cofibrations. So, if we tensor the diagram with X, we obtain a cofibration X t * *X ~= (S t S) X ! CS X and a map CS X ! S X ~=X, that has the acyclic cofibrations i0 idX and i1 idX :X ~=S X ! CS X as sections. By 2-out-of-3 CS X ! X is a weak equivalence. Hence CS X is a cylinder object for X. Definition 2.9. Two maps f and g :X ! Y in M are left V-homotopic if there exists a map H :CS X ! Y , called a left V-homotopy, such that the following diagram commutes X HH | HHHfH | HH fflffl|HH$$H CS OOX _____//Y::v | vvvv | vvvg | vv X where CS is some cylinder object for S. For example if S is the category of simplicial sets, then for cofibrant X S- homotopy is just simplicial homotopy. Lemma 2.10. If X is cofibrant, then the V-homotopy relation on M(X, Y ) is an equivalence relation. Proof.We first prove the special case X = ;. The V-homotopy relation is reflexi* *ve, since S is a cylinder object for S. It is symmetric by switching the inclusions* *. It is transitive, because the following pushout diagram again forms a cylinder obj* *ect for S: S | | fflffl| S ______//_CS | | | | fflffl|p.fflffl| S ____//_CS0____//CS00 The general result now follows by tensoring with X. Lemma 2.11. If X is cofibrant, then the V-homotopy relation on M(X, Y ) coin- cides with the left homotopy relation. Proof.This follows directly from lemma 2.8. Remark 2.12. The previous lemma 2.11 implies the V-enriched Whitehead lemma: a map between fibrant and cofibrant objects is a weak equivalence if and only if it is V-homotopy equivalence. Observe also, that V-enriched functors preserve V- homotopy and in particular V-homotopy equivalence. For later use we describe here the V-mapping cylinder as an analog of the si* *m- plicial mapping cylinder. L-STABLE FUNCTORS 7 Definition 2.13. Let f :A ! B be a map in a V-model category M between cofibrant objects. Then the V-mapping cylinder of f is defined by the following pushout square: i0 A _____//A Cyl(S) f|| || fflffl|p. fflffl| B ______//_CylV(f) Here Cyl(S) is a fixed cylinder object of the cofibrant unit S of V and i0: A ! A CS is induced by the inclusion i0 from 2.7. There are maps i1(f): A ! CylV(f) induced by the inclusion i1 from 2.7 and p(f): Cyl(f) ! B. Here i1(f) is a cofibration and and p(f) is a weak equivalence. We have a commutative diagram: f A _________________//_GB;; GG www GGG 'www i1(f)G##GG wwwp(f) CylV(f) We will usually drop the reference to V and simply denote the cylinder by Cyl(f* *). 3. Localization This short section is a quick crash course in the localization technique we * *will be using. Definition 3.1. Given an endofunctor F :C ! C in a model category C equipped with a coaugmentation ffl: id! F we call a map X ! Y in C an F -equivalence, if it induces a weak equivalence F X ! F Y . A map X ! Y is called an F -fibration, if it has the right lifting property with respect to all projective cofibration* *s, which are also F -equivalences. The following theorem is proved in [3, A.7] and [4, 9.3]. Theorem 3.2 (Bousfield-Friedlander). Suppose ffl: id! F is a coaugmented end- ofunctor of a right proper model category C satisfying the following axioms: (A.4):The functor F preserves weak equivalences. (A.5):The maps fflF(A), F fflA :F (A) ' F F (A) are weak equivalences for * *any object A 2 C. (A.6):Consider a pullback diagram W _____//Y | | | p| fflffl|ffflffl| X _____//Z where p is an F -fibration and f is an F -equivalence. Then W ! Y is an F -equivalence. Then the classes of cofibrations, F -equivalences and F -fibrations form a righ* *t proper model structure, which is simplicial or left proper, if the original model stru* *cture on C is simplicial or left proper. It possesses functorial factorization, if C * *does. There is the following characterization of F -fibrations. 8 GEORG BIEDERMANN Lemma 3.3. A map p: X ! Y in C is an F -fibration if and only if it is a fibrat* *ion, such that the diagram X _____//F X p || |Fp| fflffl| fflffl| Y _____//F Y is a homotopy pullback square in the underlying model structure. 4.Small functors We want to study categories of functors from full subcategories of V to V-mo* *del categories. In order to have the most flexibility we choose to work with small functors, since we do not want to restrict ourselves only to small source categ* *ories. Definition 4.1. A V-functor Xe: N ! M between two V-categories N and M is small if it is the V-left Kan extension from its restriction to some small f* *ull subcategory T of N . We call T the defining subcategory of Xe. Let MN denote the category of small functors from N to M. Remark 4.2. (i) Small functors are V-functors by our definition. (ii)Let i: U1 ,! U2 be an inclusion of full subcategories of V, then the restri* *ction functor i*: MU2 ! MU1 has a left adjoint LKani: MU1 ! MU2 given by V-left Kan extension and the category MU1 becomes a retract (up to natural equivalence) of the category MU2. In this way we will view the category MU as full subcategory of MV. So we can always assume, that our functors are actually defined on V. Definition 4.3. For an object V in V its covariant V-representable functor V ! V will be denoted by RV ( __) = V(V, __). This functor is obviously small with defining subcategory {V }. Remark 4.4. (i) If the unit S of V is cofibrant, all representables (ii)It is standard [19, Prop. 4.83], that a functor Xe:V ! M is small if and on* *ly if it is an enriched colimit of representables, i.e. there exists a small full sub* *category K of V such that Z K2K Xe~= RK XeK. Using this and [8, Prop. 8.3], that if W : V ! V and Y :M ! M are small, then X O W and Y O X are small, too. f e e f e e (iii)For any full subcategory U of V the category MU is cocomplete by definitio* *n. We also want it to be complete. This holds obviously, when U is small. But the remarkable result [8, Theorem 8.5] of Day and Lack shows, that this also holds, when U is cocomplete. Limits and colimits are both computed objectwise. L-STABLE FUNCTORS 9 Definition 4.5. Let Xeand Yebe functors in MU and let V be an object of V. We define a tensor : MU x V ! MU by (Xe V )(K) = Xe(K) V. The enrichment of MU is given by the V-object of natural transformations of V- functors Z V-nat (Xe, Ye) = V(X (K), Y(K)), K2K e e where the end is taken over a small full subcategory K of U defining Xe. The cotensor Vop x MU ! MU is given by (XeV)(K) = Xe(K)V = V(V, Xe(K)), where the right hand side is the cotensor of the underlying V-category M. Consistency would demand denoting the V-enrichment of MU by MU(Xe, Ye), but we save this notation for the enrichment of MU over itself 11.2. It is easy to check, that there are canonical isomorphisms V-nat (Xe, YeK) ~=V-nat (Xe K, Ye) ~=V(K, V-nat (Xe, Ye)), so that MU is indeed a V-category. 5.Fixing notation and first assumptions We reserve the letter V for a locally presentable cofibrantly generated symm* *etric monoidal model category. We denote the monoidal product by and the unit by S, which has to be cofibrant. Let IV and JV be sets of generating cofibrations and acyclic cofibration. We additionally assume that V possesses functorial fib* *rant replacements, i.e. for each object K there exists a functorial weak equivalence K ! Kfibinto a fibrant object Kfib. We choose one such replacement functor ( __)fiband we assume, that it is small, see definition 4.1. Definition 5.1. An object L in the symmetric monoidal category V is called V- small, if the V-enriched representable RL commutes with sequential V-colimits, * *i.e. if the canonical map colimn2NRL(Kn) = colimn2NV(L, Kn) ! V(L, colimn2NKn) = RL(colimn2NKn) is an isomorphism for every sequential system (Kn)n2N . The letter L will always stand for a V-small cofibrant object L in V. If the occasion arises, we will use L0for another such object with the same properties. We reserve the letter U for a full subcategory of V, that is either small or* * co- complete. We will gradually add more assumptions on U, as we go along, see 7.1, 8.1, 11.1 and 12.9. We reserve the letter M for a right proper locally presentable cofibrantly g* *en- erated V-model category. Let IM and JM be sets of generating cofibrations and acyclic cofibration. Again we assume that M possesses functorial small fibrant replacements and we choose such a replacement functor ( __)fib. Definition 5.2. We denote by V-mod__the full subcategory of V-mod 2.6 of right proper locally presentable cofibrantly generated V-model categories possessing a functorial small fibrant replacement. 10 GEORG BIEDERMANN Observe, that L induces left Quillen endofunctors of M and V K 7! L(K) = K L given by tensoring with L and denoted by the same letter. The right adjoint of L will be denoted by R(K) = KL . It commutes with filtered colimits and homotopy colimits, since L is small. Remark 5.3. Let ff be a regular cardinal. We remind the reader, that in an ff- presentable category ff-limits commute with ff-filtered colimits [2, Cor. 5.2.8* *]. In particular, finite homotopy limits commute with filtered homotopy colimits. 6.The projective model structure Following [6] we can equip MU with the projective model structure: weak equi* *v- alences and fibrations are given objectwise. Fibrations are detected by the cl* *ass RU JM , acyclic fibrations are detected by the class RU IM , where U runs through all objects of U. Here IM denotes a generating set of cofibrations for* * M, and JM denotes a set of generating acyclic cofibrations. Both classes are loca* *lly small, i.e. they satisfy the co-solution set condition, and all the source and * *target objects are small in the sense, that mapping out of them commutes with filtered colimits. Therefore we can use the generalized small object argument from [5] in the same way as in [6] to prove the existence of the projective model structure. Theorem 6.1. The category MU equipped with the projective model structure is class-cofibrantly generated V-model category. It is right or left proper, if M * *is so. If U is a small full subcategory of V, then MU is cofibrantly generated and hen* *ce possesses functorial factorization. Proof.We have to check, that MU is a V-model category. Let i: A ! B be a cofibration in V and let p: Xe! Yebe a fibration in MU. Then the induced map XeB! YeBxYeAXeA is an objectwise fibration, since cotensors are defined objectwise. It is an ac* *yclic fibration if either i or p are acyclic, for the same reason. If M is left proper, then MU is left proper, because weak equivalences and cofibrations are in particular objectwise and pushouts are computed objectwise. The same applies to right properness. If U is a small full subcategory, the generating classes of cofibrations and* * acyclic cofibrations are in fact sets, and so factorization is functorial. In general we do not expect to have functorial factorization. The reason is,* * that we do not know, how to choose the co-solution sets for the generalized small ob* *ject argument in a functorial way. This leaves all further localizations of the proj* *ective model structure on MU to be without functorial factorization, unless the source category is small. But we think, this is a small price to pay in order to handl* *e more general source categories than just small ones. Remark 6.2. If the unit S of V is cofibrant, all representable functors RK are projectively cofibrant. L-STABLE FUNCTORS 11 7.The homotopy model structure We will now outline, how to localize the projective model structure on MU to obtain the homotopy model structure, where the fibrant objects are exactly the projectively fibrant homotopy functors. Axiom 7.1. To construct the homotopy model structure on MU we have to intro- duce further assumption on U: (1)All objects in U are cofibrant. (2)It is closed under the fibrant approximation functor ( __)fib:U ! U. The last condition means, that for every object K there exists a functorial weak equivalence into a fibrant object Kfib, which is still in U. In the model category V we have a Whitehead lemma 2.12: a map between fibrant and cofibrant objects is a weak equivalence if and only if it is a V-ho* *motopy equivalence. If we have a weak equivalence w between fibrant objects in U, it is already a V-homotopy equivalence, since all objects in U are assumed to be cofibrant. Now let Xe be in MU. The weak equivalence w is mapped to a V- homotopy equivalence Xe(w), because small functors are V-functors. Altogether, if we precompose an arbitrary small functor Xe with a small fibrant replacement functor ( __)fib, the resulting functor Xe( __)fibwill be small as explained in* * remark 4.4 and it will be a homotopy functor. To obtain the homotopy model structure we just observe, that the functor Xe7! Xe( __)fibsatisfies the axioms of 3.2, a* *s was proved in [1, Prop. 3.3]. In the homotopy model structure on MU a map Xe! Yeis (1)a cofibration if and only if it is a projective cofibration, (2)a weak equivalences if and only if the induced map Xe( __)fib! Ye( __)fib is an objectwise weak equivalence. (3)a fibration if and only if the square Xe_____//X( __)fib e | | | | fflffl| fflffl| Ye_____//Y( __)fib e is an objectwise homotopy pullback square. Compare 3.3. Fibrant objects are exactly the objectwise fibrant homotopy functors. Definition 7.2. The functor ( __)fibis an endofunctor of U, that we assume to be small. Then the following functor denoted by ( __)h Xe7! ( __)fibO XeO ( __)fib= Xeh is a fibrant replacement functor in the homotopy model structure on MU. Theorem 7.3. The category MU equipped with the homotopy model structure is right proper V-model category. It is left proper if M is so. Proof.We have to show, that for a homotopy fibration p: Xe! Yein MU and a cofibration i: K1 ! K2 in V the induced map p i: XeK2! YeK2xYK1 X K1 e e 12 GEORG BIEDERMANN is a homotopy fibration. But this follows from the characterization of homotopy fibrations in 3.3 and the fact, that there is a natural isomorphism (YeK2xYK1 X K1)( __)fib~=YK2( __)fibxYK1( __)fibXK1( __)fib. e e e e e The assertion about left properness follows from theorem 3.2. The following fact is merely a tautology, but it is a useful different point * *of view. Lemma 7.4. A map is weak equivalence in the homotopy model structure if and only if its restriction to the category of fibrant objects is an objectwise wea* *k equiv- alence. Question 7.5. The assumption, that all objects in U be cofibrant, is unfortunat* *e. Is there a better way to construct the homotopy model structure without this assumption, in particular if U is not small? This needs further investigation. To proceed we will now find useful generators for the acyclic cofibrations i* *n the homotopy model structure. Definition 7.6. Let f :A ! B be an arbitrary acyclic cofibration in U and con- sider the induced map f* :RB ! RA of representable functors. Now factor f* into a projective cofibration j :RB ! Cyl(f*), which in the terminology of 2.13* * is given by j = i1(f*), followed by the objectwise fibration p: Cyl(f*) ! RA . Let us denote the set of all maps of the form j, where f runs through the set of all acyclic cofibrations between objects in U, by J0. Then form the set J00of pusho* *ut products j i, where j 2 J0 and i 2 IM . Finally define the set of maps in MU JhoMU:= J00[ JprojMU. Note that all maps in J00are projective cofibrations by 6.1. Theorem 7.7. A map in MU has the right lifting property with respect to the cla* *ss JhoMUif and only if it is a fibration in the homotopy model structure. We have not checked the co-solution set condition required for the generaliz* *ed small object argument of [5]. So we do not claim, that the homotopy model struc- ture is class-cofibrantly generated. There is merely a certain class of acyclic* * cofi- brations, that detect fibrations by the lifting property. However if U is small* *, the homotopy model structure is cofibrantly generated, because sources and targets * *of the generating set are small. Proof.We first prove, that a fibration in the homotopy model structure on MU has the right lifting property with respect to JhoMU. To streamline the argumen* *t we use right properness and the fact, that applying ( __)fibpreserves objectwise f* *ibra- tions, and reduce to the case of an objectwise Xe! Yefibration between objectwi* *se fibrant homotopy functors. Obviously such maps have the right lifting property with respect to JprojMU. Given a diagram (Cyl(f*) C) t(RB C) (RB D)____//X e (7.1) j |i| || fflffl| fflffl| Cyl(f*) D______________//Ye L-STABLE FUNCTORS 13 where j 2 J0 and i 2 IM . This is adjoint to the diagram: C __________//_XCyl(f*) | e | (7.2) i|| | fflffl| fflffl|* D _____//YeCyl(fx)Ye(B)Xe(B) Here XeCyl(f*)is the cotensor of MU over VU from 11.2. Note that the right hand map is an objectwise fibration by the fact 11.5, that MU with the projective st* *ruc- ture is a VU -model category. Now one can easily show, that there is a commutat* *ive square XeCyl(f*)oo____'________Xe(A) | | | |' * fflffl| fflffl| YeCyl(fx)Ye(B)Xe(B)o'o_ Ye(A) xYe(B)Xe(B) So a lifting exists in (7.2), hence also in the adjoint square (7.1). Conversely let Xe! Yebe a map with the right lifting property with respect to JhoMU. Then it is obviously an objectwise fibration. The previous diagrams show, that there is a weak equivalence Xe(A) ' Ye(A) xYe(B)Xe(B). We factor the map K ! Kfibinto an acyclic cofibration K ! K0 followed by an acyclic fibration K0 ! Kfib. This last map is a weak equivalence between fibrant and cofibrant objects, hence it is mapped to a weak equivalence by Xeand Ye. We obtain: Xe(K) ' Ye(K) xYe(K0)Xe(K0) ' Ye(K) xYe(Kfib)Xe(Kfib). So Xe! Yeis a fibration in the homotopy model structure by 3.3. 8.The L-stable model structure Now we will localize this structure further to obtain the L-stabilized or L- stabilized version. We need further assumptions on U. Let us list them here. Axiom 8.1. In order to gain a little bit more flexibility we will not just cons* *ider small functors defined on V, but also small functors from a full subcategory U * *of V. This full subcategory U should satisfy the following assumptions: (1)It is either small or cocomplete. (2)All objects in U are cofibrant. (3)It is closed under the fibrant approximation functor ( __)fib. (4)It contains the unit S. (5)It is closed under tensoring with L (and L0). In order to compare different linearizations we need a slightly more general* * no- tion. Remark 8.2. For a small functor Xe:U ! M and every object K in V there is an assembly map (8.1) (L O Xe)(K) = Xe(K) L ! Xe(K L) = (XeO L)(K), 14 GEORG BIEDERMANN since small functors are V-functors, compare (2.2). If L0! L is a morphism in V we have a commutative diagram: Xe(K) L0_____//Xe(K L0) | | (8.2) | | fflffl| fflffl| Xe(K) L_____//_Xe(K L) The composed diagonal map is called the L-L0-assembly map. Definition 8.3. A functor Xe:U ! M is called L-L0-stable, if it is a homotopy functor and the adjoint map of (8.2) tXf,K:Xe(K) ! (R0O XeO L)(K) is a weak equivalence for all objects K in U. If L = L0, (8.2) reduces to (8.1)* *. In that case such a functor is simply called L-stable. This is the case, we are mo* *stly interested in. Remark 8.4. For pointed simplicial sets V = S* the S1-stable functors are simply the linear functors in the sense of Goodwillie [13]. Definition 8.5. Let * be the terminal object of V and by abuse of language also of M. A functor Xe:V ! M is called reduced, if Xe(*) ' *. Remark 8.6. Note, that if the category V is pointed, in the sense that the init* *ial object is isomorphic to the terminal one *, then all small functors in MU are reduced. But we point out, that in general V does not have to be pointed for the construction of the L-stable model structures. Recall, that ( __)h is a fibrant replacement functor in the homotopy model s* *truc- ture, see 7.2. Definition 8.7. We define the following functor from MU to itself: 0 0n h n PLLXe= hocolimnR O Xe O L . Here, of course, R0is the right adjoint to L0given by cotensoring with the obje* *ct L0. For L = L0we will write PL or, if no0confusion can arise, we will drop0the refe* *rence to L and L0at all. The functor PLL = P is coaugmented. Let p: id! PLL denote the coaugmentation. There is a commutative diagram PLL0_____//PL (8.3) | | | | fflffl| fflffl| PL0 ____//_PLL0 of coaugmented functors. Remark 8.8. (i) If Xeis a homotopy functor, then P Xeis a homotopy functor as well. This follows from the assumption, that all objects are cofibrant. The fun* *ctor L preserves weak equivalences between cofibrant objects. R0and Xepreserve weak equivalences, because of the interspersed fibrant replacements. (ii)The functor P commutes with finite homotopy limits and filtered homotopy colimits because of 5.3. L-STABLE FUNCTORS 15 (iii)From the assumption, that R0 commutes with filtered homotopy colimits, it follows directly, that we have a natural equivalence P O R0( __)fib' R0O P . (iv)Therefore, for each Xethe functor P Xeis L-stable. (v) From 4.4 it follows, that P Xeis again small, if Xewas. Lemma 8.9. The functor P satisfies properties (A.4), (A.5) and (A.6) from 3.2. Proof.The fact, that P preserves projective weak equivalences, follows from the fact, that R0 preserves weak equivalences between fibrant objects. This is the reason, why a fibrant replacement functor had to be interposed. This proves (A.* *4). The proof of (A.5) is taken from [13]. It is clear, that the map pP Xeis a w* *eak equivalence, because P Xeis L-L0-stable and therefore the map R0nXehLn ! R0n+1XehLn+1 is a weak equivalence. Next we consider: P Xe! P (R0O XehO L) '!R0O P XeO L This composition is a weak equivalence, because P Xeis L-L0-stable. Therefore t* *he first map of the previous composition is a weak equivalence. This easily implie* *s, that P pXe is a weak equivalence proving (A.5). Finally we have to prove (A.6). But this follows easily from the fact, that P commutes with homotopy pullbacks by 8.8(ii) and that M is right proper. Remark 8.10. It is worth pointing out, that the functor P satisfies the axioms (A.4), (A.5) and (A.6) even before localizing to the homotopy model structure. This means, that there exists an L-stable model structure for non-homotopy func- tors. We do not know, how to interpret this. Now we can apply theorem 3.2 once again. We obtain the L-L0-stable model structure on MU. Cofibrations are still the projective ones. Before we summarize our findings in the next theorem, let us define explicitly the model structure * *and we prove another characterization of L-L0-stable fibrations. Definition 8.11. A map Xe! Yeis an L-L0-stable equivalence if and only if P Xe! P Yeis a homotopy weak equivalence or, in fact, a projective weak equivalence, * *since P Xeand P Yeare homotopy functors. L-L0-stable fibrations are homotopy fibratio* *ns Xe ! Ye, such that the diagram Xe_____//P Xe | | | | fflffl| fflffl| Ye_____//P Ye is a homotopy pullback diagram in the homotopy model structure. From now on we will also assume that L = L0 for ease on notation and re- mark, that the proofs are literally the same for the more general L-L0-stable m* *odel structure. 16 GEORG BIEDERMANN Lemma 8.12. A map Xe! Yeis an L-stable fibration if and only if it is a homotopy fibration such that the following diagram Xe_____//RX hL e | | | | fflffl| fflffl| Ye_____//RY hL e is a homotopy pullback square in the homotopy model structure. Proof.According to 3.3 an L-stable fibration is a map as described in definition 8.11. We have a diagram RXehL=______//_RP=XeL ___ ww;;w ____|___//__w ' || Xe P Xe | | fflffl|| fflffl| || RYehL ___|__//_RP|YeL@@ | CC | | | ' fflffl| fflffl| Ye ________//_P Ye where the front and the back square are homotopy pullbacks in the homotopy model structure. Since we have weak equivalences on the right hand side due to fact, * *that P Xeand P Yeare L-stable, the second square in the following diagram is a homot* *opy pullback: Xe ____//_RX hL____//P X e e | | | | | | fflffl| fflffl| fflffl| Ye_____//RY hL____//P Y e e The combined square is a homotopy pullback, so is the left hand square. Theorem 8.13. The category MU equipped with the L-stable model structure is a right proper V-model category. It is left proper if M is so. Proof.The existence of the model structure follows from 3.2 once again using the fact, that the functor P satisfies the necessary axioms by 8.9. This also cover* *s the left properness assertion. It remains to show, that for an L-stable fibration X* *e! Ye and a cofibration i: K1 ! K2 in V the induced map XeK2! YeK2xYK1 X K1 e e is an L-stable fibration. But this follows directly from the characterization o* *f L- stable fibrations in 8.12 and the fact, that there is a natural isomorphism R(XeK)L ~=(RXeL)K . General model categorical considerations give the following lemma. Lemma 8.14. For each functor Xethe functor P Xeis L-stable. Moreover, if Xe! Ye is a map in MU to an L-stable functor Ye, then there exists a zig-zag of maps ' P Xej Ce! Ye L-STABLE FUNCTORS 17 under Xe, where the first arrow is an objectwise acyclic fibration. We leave the precise uniqueness statement of the zig-zag to the reader. Now we will describe a certain class of L-stably acyclic cofibrations, that d* *etect L-stable fibrations by the lifting property. As for the homotopy model structu* *re we do not prove the co-solution set condition for this class and we do not clai* *m, that the L-stable model structure is class-cofibrantly generated. Again it will* * be cofibrantly generated as soon as U is small. Let V(L, RK ) denote the cotensor from 4.5 of L with the representable functor RK = V(K, __) for K in U. Adjoint to the identity of V(L, RK ) is the map oK :RK L L ~=V(L, RK ) L ! RK . Note that V(oK , Xe) = tXf,K, where tXf,Kis defined in 8.3. So a functor Xein M* *U is L-stable exactly if the cotensor 11.2 with this map is a weak equivalence for e* *very K. Observe, that oK is a map between projectively cofibrant objects, so we can form the V-mapping cylinder 2.13: j(K) p(K) RK L L _____//Cyl(oK_)__//RK Here j(K) is a projective cofibration and p(K) is an objectwise equivalence. Let J000be the class of maps of the form j(K) i, with j(K) as above for all K in U and where i runs through the class IM . Note, that all maps in J000are projecti* *ve cofibrations. Definition 8.15. We define a new class of maps by JL-stableMU:= J000[ JhoMU. Theorem 8.16. A map in MU has the right lifting property with respect to the class JL-stableMUif and only if it is a fibration in the L-stable model structu* *re. The proof is similar to the proof of 7.7 and will be omitted. Now we will prove, that the L-stabilization really deserves its name: the ac* *tion of L on MU induces an equivalence of the associated homotopy categories. Definition 8.17. Let N be a V-model category. If the functor L: N ! N given by tensoring with L is a V-left Quillen equivalence, we call N L-stable. Theorem 8.18. The action of L on MU given by the tensor from 4.5 is the left adjoint of a V-Quillen equivalence with right adjoint given by cotensoring with* * L. The category MU equipped with the L-stable model structure is L-stable. Again we will denote tensoring with L just by LXe = Xe L and cotensoring with RXe = XeL. Proof.It is obvious, that (L, R) form a Quillen pair. It remains to show, that * *(L, R) is a Quillen equivalence from MU to itself. Let Xe! RYebe an L-stable equivalen* *ce, where Xeis projectively cofibrant and Yeis L-stably fibrant. Let Xe,! P 0Xe'!P * *Xe be a factorization of the coaugmentation pXe into a projective cofibration foll* *owed by a projective weak equivalence. Since Yeis L-stable, it follows easily, that * *RYeis 18 GEORG BIEDERMANN L-stable. Then there is a map P Xe! RYe, that is a projective weak equivalence * *by our assumption on Yeand that makes the following diagram commutative: Xe_____//_Y>> zzz | ""e" zzz | "'"" __zzz' fflffl|"" P 0Xe____//_P Xe Since P 0Xeis projectively cofibrant, the adjoint map LP 0Xe! Yeis a projective weak equivalence and hence LP 0Xeis linear. The map LXe ! LP 0Xeis an L-stable acyclic cofibration, since Xe ! P 0Xeis one and L preserves them. Therefore the induced map P LXe ! LP 0Xeis a projective weak equivalence. We finally find, th* *at LXe ! Yeis an L-stable equivalence. The converse direction is similar. Remark 8.19. If Xeis an arbitrary functor in MU, then the assembly map Xe( __) L ! Xe( __ L) is an L-stable equivalence.: its adjoint is an L-stable equivalence, since the * *maps in the homotopy colimit for PL factor over each other, now the assertion follows f* *rom the previous theorem. 9.Functoriality In this section we study functoriality of our L-stabilization or L-stabiliza* *tion. We first consider the construction with M as variable. Then we prove independence from the weak homotopy type of L. Finally we observe, how the functor categories behave with respect to the source U. The tensor is a functor M x V ! M, that restricts to a functor M x U ! M. This induces a functor ~: M ! MU given by (9.1) ~(M) = M idU= M RS. Here we view the identity functor of U as the representable functor associated * *to the unit S restricted to U. There is a right adjoint ae: MU ! M, which is evaluating at S: (9.2) ae(Xe) = Xe(S) ~=V-nat (RS, Xe) Remember, that this is well defined, since by axiom 8.1(4) U contains the unit * *S. Recall also, that the action of L on MU is given by the tensor in 4.5 and that * *we have written: LXe = L O Xe= Xe L Observe, that for an object K in V we have ~(LM)(K) ~=M L K, while (L O ~(M))(K) ~=M K L ~=(~(M) O L)(K). So these functors are naturally isomorphic, but this isomorphism involves the t* *wist map of our symmetric monoidal structure. We have also isomorphisms Rae(Xe) ~=Xe(S)L ~=ae(RXe). We summarize: L-STABLE FUNCTORS 19 Lemma 9.1. For any U satisfying 8.1 the pair (~, ae) constitutes a morphism ~: M AE MU :ae in the category V-mod__of V-model categories 5.2. To study functoriality of our L-stabilization let : M1 ! M2 be a morphism in the category V-mod__. So is a left V-Quillen functor with right adjoint . Th* *ere are induced 2-functors *: MU1! MU2and *: MU2! MU1, defined by postcomposing with and . Note, that *Xe is small as long as Xe is so, because as a left V-Quillen functor commutes with V-left Kan extension* *s. Postcomposing with does not necessarily land in small functors again, therefo* *re we have to assume, that U is small to get a satisfactory theory. Postcomposing * *with a V-functor preserves preserves all kinds of composition laws of 1- and 2-morph* *isms in the category V-mod__, which means, that the associations stabUL:M 7! MU and ( , ) 7! ( *, *) constitute a 2-functor. Theorem 9.2. Let us fix V and a small U satisfying 8.1. Then for each small and cofibrant L we can associate to M in V-mod the category MU equipped with the L-stable model structure. This constitutes a 2-functor from the category V-mod* *__ 5.2 to itself. This 2-functor preserves Quillen equivalences. Proof.2-functoriality is just clear. Obviously, * preserves objectwise weak equivalences and fibrations, so it i* *s a right Quillen functor for the projective model structures on both sides. Now in* * all the localizations acyclic fibrations do not change, and weak equivalences betwe* *en fibrant objects are objectwise weak equivalences. So * preserves them, and by lemma 9.3 it is a right Quillen functor for the homotopy and the L-stable model structure. * and * are obviously V-functors. Finally, if ( , ) form a Quillen equivalence, then ( *, *) is a Quillen eq* *uivalence for the projective model structures. This descends obviously to the homotopy mo* *del structures, since *(Xe( __)fib) ~= *(Xe)( __)fib. For the L-stable model struc* *tures we have to check by 9.3, that * preserves L-stable fibrations between L-stable functors. Those are just objectwise fibrations, which * indeed preserves. We have used the following lemma due to Dugger [9]. A proof is also found in [14, Prop. 8.6.4.] Lemma 9.3. For a pair (L, R) of adjoint functors the following are equivalent: (i)(L, R) is a Quillen pair. (ii)The left adjoint L preserves cofibrations between cofibrant objects and all* * acyclic cofibrations. (iii)The right adjoint R preserves fibrations between all fibrant objects and a* *ll acyclic fibrations. Remark 9.4. For a morphism : M ! N in V-mod and an object Xe in MU there are natural transformations *(Xe( __)) L ! *(Xe( __) L) ! *(Xe( __ L)), 20 GEORG BIEDERMANN that are L-stable equivalences in N U. The proof is straightforward, if one use* *s the twist map L1 L2 ! L2 L1 correctly. Next we look at L as a variable. The following theorem is the only reason we introduced the L-L0-stable model structure for different L and L0. Theorem 9.5. If the map L0 ! L is a weak equivalence, the induced maps of coaugmented linearizations given by the identity functor on MU corresponding to the square 8.3 are Quillen equivalences. In particular we obtain, that the Quil* *len equivalence class of MU with the L-stable model structure only depends on the w* *eak homotopy type of L. Proof.This is obvious. Now let i: U1 ,! U2 be a full inclusion of full subcategories of V. We obser* *ved in 2, that we have an adjoint pair (9.3) LKani:MU1 AE MU2 :i*, where i* is the restriction functor and LKan iis the V-left Kan extension along* * i. Lemma 9.6. The pair from (9.3) is a V-Quillen pair for the projective, the homo- topy and the L-stable model structures. Proof.We just have to observe, that the the fibrancy conditions in all the model structure are given objectwise, so i* preserves them, and hence it is a right Q* *uillen functor. Observe, that under certain circumstances LKan ipreserves weak equivalences. Compare 12.10 and 12.11. The question, when this adjoint functor pair is a Quil* *len equivalence, will be briefly addressed at the end of section 10. 10.L-stabilization We will now define our L-stabilization or L-stabilization of a V-model categ* *ory M from V-mod__ with respect to L, prove an idempotency result 10.3. This is connected with L-spectra and we will examine this more closely in section 13. Before we prove theorem 10.3, let us observe a few things. Assume that (L, R) is already a Quillen equivalence. Then for all cofibrant M and fibrant N in V LM = M L ! N is a weak equivalence if and only if M ! RN = NL is a weak equivalence. The maps LK ! L(Kfib) and LK ! (LK)fibare also acyclic cofibrations. It follows, that the functorial maps Kfib! RL(Kfib) ! R(L(Kfib)fib) are weak equivalences. We can rephrase this by saying that the projectively fib* *rant replacement ( __)fibof the identity functor of V is L-stable. In fact, the ide* *ntity itself is L-stable. Definition 10.1. Consider the set of objects {LnS | n 2 N} and let SphVLbe the full subcategory of M obtained from this set and its image under the functor ( * *__)fib. Here we obviouslyVset L0S = S. For a V-model category M its L-stabilization is the category MSphL equipped with the L-stable model structure. Remark 10.2. (i) The category SphVLis the smallest allowable choice for U ac- cording to the axiom 8.1. At the end of this section we will address briefly t* *he question, whether one can use larger categories. L-STABLE FUNCTORS 21 (ii)If Xeis an L-stable functor, then the behavior of Xeon SphVLis determined up to weak equivalence by its value on S, because the maps (10.1) LXe(LnS) ! (Xe(Ln+1S)fib)fib Xe(Ln+1S) are weak equivalences. Theorem 10.3. Let M be L-stable, see 8.17, i.e. (L, R) is a Quillen equivalence on M. Then we have: (i)Let Xe:U ! M be a linearly fibrant functor. If ~(M) ! Xeis an L-equivalence, then M ! Xe(S) is a weak equivalence. (ii)The pair (~, ae) from 9.1, where U = SphVL, is a V-Quillen equivalence. Proof.Let (L, R) be a Quillen equivalence. The functor ~(M) is a projectively cofibrant L-stable homotopy functor. Indeed, the map K M L ! K L M ! (K L)fib M ! ((K L)fib M)fib is a weak equivalence for every K in V. Remember that all objects in U are cofi* *brant. So the adjoint map Kfib! R(LK)fibis a weak equivalence. So for every cofibrant M, every L-stably fibrant Xe and every map ~(M) ! Xe there is a commuting diagram ~(M) _______//Xe | | | | fflffl| fflffl| P ~(M) _____//P Xe consisting entirely of objectwise weak equivalences if and only if the map ~(M)* * ! Xe is an L-stable equivalence. So this is equivalent to M ! Xe(S) being a weak equivalence. This proves (i) and the first part of (ii). The second part of (* *ii) is obvious by (10.1). We saw, that SphVLis the smallest choice for U. The question, whether we can choose larger source categories, for which the previous statements are still tr* *ue, is an interesting one. An indication, what we can hope for, is given by the assumptio* *ns 12.9, we have to make for the monoid axiom. The largest choice for U in this se* *tting is the full subcategory of V given by the cofibrant V-finitely presentable obje* *cts. Let us call this choice Umax. In the case of V = S* and L = S1 the maximal choi* *ce is the category of finite pointed simplicial sets Sfin*and it follows from [20]* *, that restriction and left Kan extension give a Quillen equivalence fin SphV1 SS** o S* S . This is just another formulation of the well-known fact, that a generalized hom* *ology theory is completely determined by its values on the spheres. It is not always * *true, that Umax and SphVLyield Quillen equivalent functor categories in general. For a counter example see [10, p. 465] with V = S* and L = S0 t S0. So we would like to pose the following question. Question 10.4. Given V and L. What is the largest full subcategory U of V, such that the pair V U * LKani: MSphLAE M :i 22 GEORG BIEDERMANN is a V-Quillen equivalence, where i: SphVL! U is a full inclusion and where both sides carry the L-stable model structure. The answer to this question for the motivic case with L = P1 is open and is likely to involve Voevodsky's slice filtration. 11. Symmetric monoidal structures We are now going to describe, how to enrich the category MU over VU . In particular, for the case M = V the category VU will become again a closed sym- metric monoidal category. This enrichment is a straightforward generalization of the smash product constructed by Lydakis in [20] for the case V = S* of pointed simplicial sets and U = Sfin*of finite pointed simplicial sets. Axiom 11.1. Additionally to 8.1 we now have to make the following assumption. (6)U is closed under the symmetric monoidal product . First we define a new category U U: Ob (U U)= Ob(U) x Ob(U) Mor (U U)((A, B), (C, D))= V(A, C) V(B, D) Given two functors Ue:U ! V and Xe:U ! M there is a functor Xe` Ue:U U ! M defined by (11.1) (Xe` Ue)(A, B) = Xe(A) Ue(B). Further the monoidal product : U x U ! U factors over U U to give a functor : U U ! U, (A, B) 7! A B. The tensor product of Ueand Xeis then defined as the left Kan extension of Xe` * *Ue along : X `U U U _f__e//_M;;___ _____ || _________ fflffl|Xf_Ue_____ U Observe first, that this Kan extension exists for representable functors Ue= RA* * M and Xe= RB for objects A, B in U and M in M. There it is given simply by (RA M) RB = RA B M. But now we can extend it to all small functors by using the isomorphism from remark 4.4. The resulting functors are clearly small again. Definition 11.2. For two small functors F, G: U ! M we define a new functor MU(F, G): U ! V in the following way: MU(F, G)(K) := V-nat (F, G(K __)) There is also a cotensor (VU )opx MU ! MU given by Z HG (K) := H(K __)G(K), K2K L-STABLE FUNCTORS 23 where on the right hand side the cotensor is the underlying V-cotensor from 4.5 and K is a defining subcategory for G. These functors endow MU with a tensor, cotensor and an enrichment over VU . Lemma 11.3. For small functors F, H :U ! M and G: U ! V we have: MU(F, HG ) ~=MU(F G, H) ~=MU(F, H)G This proof is considerably easier than [20, Prop. 5.2]. Proof.Let F and G be given by the following colimits: Z A Z B F ~= RA F A and F ~= RB GB We have the following isomorphisms: Z A MU(F, HG ) ~= MU(RA F A, K 7! HG (K __)) Z A ZB ~= H(A B)FA GB Z A ZB ~= MU(RA B F A GB, H) ~=MU(F G, H) The second isomorphism is similar. Theorem 11.4. Let V is a closed symmetric monoidal model category. Then the category VU can be given the structure of a closed symmetric monoidal category. If M is V-category, then MU can be given the structure of a VU -category. The functor ~: M ! MU from 9.1 is the left adjoint of a symmetric monoidal V-adjunction. Proof.It follows from lemma 11.3, that VU is closed symmetric monoidal and that MU is a VU -category. It is clear, that (~, ae) form a V-adjunction. For object* *s M and N in M we also have the following natural isomorphisms: ~(M N) = (M N) RS ~=(M N) RS S ~=(M RS) (N RS) = ~(M) ~(N) These isomorphism commute with the twist map, which proves, that ~ is a sym- metric monoidal functor. The unit of the monoidal structure on VU is given by the inclusion functor RS :U ,! V, which for U = V is, of course, the identity. Observe, that the unit* * of VU is always projectively cofibrant by 6.2, because the unit S is cofibrant. In order to prove compatibility of the projective model structure of MU with its VU -enrichment, we have to remember, that V and M were assumed to be cofi- brantly generated with generating sets IV, JV and IM , JM respectively. Here, * *as everywhere, the I's are used for the cofibrations and the J's for the acyclic c* *ofi- brations. It is shown in [6], that VU and MU are class-cofibrantly generated wi* *th generating sets IMU = {RA i | i 2 IM , A 2 Ob(U)} 24 GEORG BIEDERMANN and JMU = {RA j | j 2 JM , A 2 Ob(U)}, where M = V is a special case. Remember again, that here we are using the generalized small object argument [5] to construct (non-functorial) factorizati* *ons, which accepts locally small classes rather than sets as input. Theorem 11.5. The category MU with its projective model structure is a VU -model category, where VU has the projective model structure. Proof.Using lemma 2.4 we have to check that for arbitrary objects K1 and K2 in U and arbitrary maps i01in IV and i02in IM the pushout product of i1 = RK1 i01:RK1 A ! RK1 B and i2 = RK2 i02:RK2 C ! RK2 D given by i1 i2: (RK1 K2 A D) t(RK1 K2 A C) (RK1 K2 B C) ! RK1 K2 B D is a projective cofibration. So, let us map it into an arbitrary acyclic proje* *ctive fibration Xe! Ye. But in the diagram (A D) t(A C) (B C)____//_X(K1 K2) e | | | | fflffl| fflffl| B D _____________//Ye(K1 K2) there exists a lifting, because the left hand side is a cofibration and the rig* *ht hand side is an acyclic fibration. This diagram is adjoint to our original lifting p* *roblem and proves the first part. The remaining properties to be shown are similar. Theorem 11.6. Let V be right proper. The category MU with its homotopy model structure is a VU -model category, where VU has either the projective or the ho* *mo- topy model structure. Proof.Cofibrations do not change. It remains to show, that fib IVU fiband IMU acof acof, where fiband acofare the classes of fibrations and acyclic c* *o- fibrations in the homotopy model structure on MU and acofis the class of acyclic cofibrations in the homotopy model structure on VU . Let Xe! Yebe a fibration in the homotopy model structure and let i: E ! F be in IV. We have to show, that for every object A in U the map A F RA F RA E (11.2) XeR ! Ye xYRA E X e e is a fibration in the homotopy model structure. We know already by 11.5, that it is an objectwise fibration. Precomposing with the functor ( __)fibcommutes with limits of the functor category. So it remains to show, that the diagram XeRA F __________________________//XeRA F( __)fib | | | | A fflffl|A A fflffl| A YeR F xYRA E X R E _____//(Y R F ( __)fib) x(YRA E ( __)fib)(X R E ( __)f* *ib) e e e e e is an objectwise homotopy pullback diagram. The functor XeRA F :U ! M is given by K 7! Xe(K A)F . L-STABLE FUNCTORS 25 So in the following diagram for every K and A the first square has to be a homo* *topy pullback square. XeF(K A) ___________//_YeF(K A) xYeE(K A)XeE(K A) | | | 1 | fflffl| fflffl| (11.3) XeF(Kfib A)_________//YeF(Kfib A) xYeE(Kfib A)XeE(Kfib A) | | | 2 | fflffl| fflffl| XeF(Kfib A)fib____//YeF(Kfib A)fibxYeE(Kfib A)fibXeE(Kfib A)fib Since the map XeF! YeFxYeEXeEis a fibration in the homotopy model structure by 7.3, the second square is a homotopy pullback. Using the fact, that ( __)fibass* *igns a weak equivalence Z ! Zfibto every Z and that every object in U is cofibrant, * *we can construct a weak equivalence (K A)fib' (Kfib A)fibunder K A. All our functors preserve this weak equivalence, because both objects are fibrant. Hence the outer square in diagram (11.3) is a homotopy pullback square, since it is w* *eakly equivalent to the following homotopy pullback square: XeF(K A)________//_YeF(K A) xYeE(K A)XeE(K A) | | (11.4) | | fflffl| fflffl| XeF(K A)fib____//YeF(K A)fibxYeE(K A)fibXeE(K A)fib Then square 1 of (11.3) is a homotopy pullback by a standard argument on homo- topy pullbacks. This proves fib IVU fib. Now for IMU acof acof. Let j :Ee ! Febe an acyclic cofibration in the homotopy model structure on VU . By 7.4 this implies, that the restriction of j* * to the category of fibrant objects is an acyclic projective cofibration. It follow* *s from 11.5, that the restriction of the map i j to the category of fibrant objects * *is an acyclic projective cofibration. This proves, that i j is acyclic cofibration * *in the homotopy model structure. Definition 11.7. Let M be a model category and a V-category for a closed sym- metric monoidal model category V with cofibrant unit. Then M is said to be a semi-V-model category, if the following properties are satisfied: (i)cofM cofV cofM (ii)acofM cofV acofM (iii)The map i j is an acyclic cofibration for every cofibration i in M and e* *very acyclic cofibration j between fibrant objects in V. Here cof and acof denote the classes of cofibrations and acyclic cofibrations w* *ith the respective category as subscript. Theorem 11.8. Let V be right proper. The category MU with its L-stable model structure is a VU -model category, where VU has either the projective or the h* *o- motopy model structure. It is a semi-VU -model category, if we equip VU with the L-stable model structure. If M is left proper and all maps in IM have cofibra* *nt source, we have a full VU -model category. 26 GEORG BIEDERMANN Proof.We will show first fib IVU fib, where fibis the class of L-stable fibra* *tions. A map p: Xe! Yein MU is an L-stable fibration if and only if p is a homotopy fibration, such that the square Xe_____//RX hL =: T X e e | | (11.5) | | fflffl| fflffl| Ye_____//RY hL =: T Y e e is a homotopy pullback square in the homotopy structure. Let i: RA K1 ! RA K2 be in IVU. Then the map XeK2 ! YeK2xYK1 X K1 e e is an L-stable fibration by 7.3. Now cotensoring with RA commutes with limits, as well as with T , since the right adjoint R commutes with cotensors. We deduc* *e, that for the map A K RA K RA K p i: XeR 2 ! Ye 2 xYRA K1 X 1 e e the diagram corresponding to (11.5) is a homotopy pullback. Therefore p i is a homotopy fibration. Now we have to show, that i j for i a projective cofibration in MU and j an acyclic cofibration in the L-stable model structure on VU between L-stably fibr* *ant objects is an L-stable equivalence. But we know this already, since j is simply* * an acyclic projective cofibration. Finally assume, that M is left proper, and recall, that by 8.13 in this case * *the L-stable model structure on MU is left proper. By assumption we also have, that the source and target of the generating cofibrations IM of M can be chosen to * *be cofibrant. Let j :Ce! Debe an acyclic L-stable cofibration in VU . Since M is r* *ight proper, the functor RA is projectively cofibrant for all objects A in U. By wha* *t we have shown already, the map RA Ce! RA De then is an L-stably acyclic cofibration in VU . Tensoring with a cofibrant obje* *ct M or N produces again an L-stably acyclic cofibration by 8.13. The fact, that the pushout product now is an L-stably acyclic cofibration, follows easily from the* * left properness of the L-stable model structure. For a comment about the cofibrancy assumption in 11.8 see question 13.6. For the special case M = V we do not need left properness. Corollary 11.9. If V is right proper, the category VU with its L-stable model structure is a closed symmetric monoidal model category. Proof.This follows from 11.8 and the fact, that the monoidal structure on VU is symmetric. 12. The monoid axiom The material in this section is adapted from [10] with only minor changes. L-STABLE FUNCTORS 27 Definition 12.1. Let C be a model category. For a class R of maps in C let R-ce* *ll be the class of maps obtained by all transfinite compositions of cobase changes* * of maps in R. The precise definition can be found in [14, 10.5.8]. Now let C be a symmetric monoidal model category with monoidal product . If O is a class of objects in C we denote by R O all maps of the form r idX:A X ! B X, where r runs through R and X trough O. We remind the reader that acofCstands for the class of acyclic cofibrations in C. The following axiom was introduced by Schwede and Shipley in [23] in order to study the homotopy theory of monoids and algebras and modules over them in a monoidal model category. Axiom 12.2. The model category C satisfies the monoid axiom, if all maps in {acofC Ob(C)}-cell are weak equivalences. Remark 12.3. (i) If the model category C is cofibrantly generated with generati* *ng set JC of acyclic cofibrations, then the monoid axiom holds in C, if all maps i* *n JC Ob (C) are weak equivalences. This follows, since in this case all acyclic cofi* *brations are retracts of maps in JC-cell. (ii)The monoid axiom is trivially satisfied, if all objects in C are cofibrant. The monoid axiom is easily proved for the projective and the homotopy model structure. We get into trouble to prove it for the L-stable model structure, wh* *ere we have to introduce a lot of very technical conditions. Theorem 12.4. If V satisfies the monoid axiom, the projective model structure on VU satisfies the monoid axiom. Proof.We have to show, that the maps in the class {RU JV|U 2 Ob(U} Ob(VU )-cell are weak equivalences. We can check the monoid axiom for every Xe in VU at a time. The functors RU are small in the sense, that mapping out of them commutes with filtered colimits, so compositions of objectwise weak equivalences always * *stay weak equivalences. We can then check the axiom objectwise, that means, that the class above is equal to the following class: [ [ {RU JV|U 2 Ob(U} (X (K))-cell Xf2Ob(VU)K2Ob(U) e Now the monoid axiom in VU follows from the monoid axiom in V. For the homotopy model structure we have to assume a small U, because we will check the monoid axiom on the generating acyclic cofibrations and we do not know, whether the homotopy model structure for non-small U is class-cofibrantly generated. Theorem 12.5. If V is satisfies the monoid axiom and U is small, the homotopy model structure on VU satisfies the monoid axiom. Proof.If U is small, the homotopy model structure is cofibrantly generated with the class JhoVU= J00[ JprojVUgiven in 7.6 as generating acyclic cofibrations. S* *ince we 28 GEORG BIEDERMANN already know, that the projective model structure satisfies the monoid axiom, it suffices to check, that the class J00 Ob(VU )-cell consists of weak equivalences in the homotopy model structure. So let f :A ! B be an acyclic cofibration in U and factor the induced map f* :RB ! RA as in 7.6: j * p RB _____//Cyl(f_)___//RA Let i: C ! D be a map in IV. The map ~ ~ (RB D) t (Cyl(f*) C) ! Cyl(f*) D (RB C) is a projective cofibration and a weak equivalence, when we evaluate it on the * *fibrant objects of U. So it is an acyclic projective cofibration on the full subcategor* *y Ufib offfibrantiobjectsbof U. But then the monoid axiom for the projective structure* * on VU gives the result. In order to prove the monoid axiom for the L-stable model structure we need some auxiliary facts and definitions. Definition 12.6. Let M be a cocomplete category and let Hom M ( __, __) denote the set of morphisms in M. An object A in M is finitely presentable, if the fun* *ctor Hom M (A, __) commutes with all filtered colimits. If M is a V-cocomplete V-category, an object A in M is V-finitely presentabl* *e, if the V-Hom functor RA = M(A, __) commutes with all filtered V-colimits. Remark 12.7. If the unit S of V is finitely presentable, then every V-finitely presentable object is finitely presentable. Lemma 12.8 ([10] Lemma 3.5). If sources and targets of the generating set IM of cofibrations of M are finitely presentable, the class of weak equivalences a* *nd the class of acyclic fibrations are closed under filtered colimits. Proof.Let H be a small index category for our colimit and equip MH with the projective model structure. Factor an objectwise weak equivalence into an acycl* *ic projective cofibration g followed by an objectwise acyclic fibration p. colimis* * a left Quillen functor, hence colimg is again an acyclic cofibration. The lemma follow* *s, if we can prove, that colimp is an objectwise acyclic fibration for an objectwi* *se acyclic fibration p: Xe! Ye. Consider a diagram A _____//colimXe i|| p|| fflffl| fflffl| B _____//colimYe where i 2 I. By adjointness the existence of a lift is equivalent to the surjec* *tivity of the map OE: Hom M (B, colimXe) ! Hom M (A, colimXe) x Hom M (B, colimY) HomM(A,colimYe) e L-STABLE FUNCTORS 29 where Hom M ( __, __) denotes the Hom-set of the category M. By assumption A and B are finitely presentable, and filtered colimits commute with pullbacks in* * the category of sets. So we can squeeze the colimit out and consider the maps OEh: Hom M (B, Xe(h)) ! Hom M (A, colimXe(h)) x Hom M (B, colimY(h)) Hom M(A,colimYe(h)) e for each h 2 H. Now OEh is surjective, since i is a cofibration and p(h): Xe(h)* * ! Ye(h) is an acyclic fibration. But the colimit of surjective maps of sets is surjecti* *ve. Axiom 12.9. We need two more assumptions on our categories U and V: (7)Every object in V is a filtered colimit of objects in U. (8)Every object of U is V-finitely presentable. Let u: U ! V be the inclusion functor. Then for any V-model category M we denote by LKan u:MU ! MV the V-left Kan extension along u, compare 4.2. Lemma 12.10 ([10] Lemma 4.9). Suppose that sources and targets in IM are finitely presentable and that U satisfies axioms (7) and (8). Then LKan upreser* *ves objectwise weak equivalences. Proof.Let f :Xe ! Yebe an objectwise weak equivalence in MU and let V 2 V. Since Xeis small, we can write: Z K2U LKanuf(V ) = RK (V ) f(K): Xe(V ) ! Ye(V ) By axiom (7) it is a filtered colimit of objects in U, so there exists a functor C :H ! V, such that H is filtered and colimh2HC(h) ~=V . We have with (8): Z K2U LKanuf(V )= RK (V ) f(K) Z K2U ~= V(K, colimC(h)) f(K) h2H Z K2U ~=colim V(K, C(h)) f(K) h2H ~=colimf(C(h)) h2H So LKan uf(V ) is a colimit of weak equivalence and hence itself a weak equival* *ence by 12.8. Curiously the following easy consequence does not appear in [10]. Corollary 12.11. Suppose that sources and targets in IM are finitely presentab* *le and that U satisfies axioms (7) and (8). Then LKan u preserves L-stable equiva- lences. Proof.By 9.6 the functor LKan uis a left Quillen functor from MU to MV, where both categories carry the L-stable model structure. If f is an L-stable equival* *ence, we factor it into an L-stably acyclic cofibration i followed by an objectwise a* *cyclic fibration p. Then LKan ui is an L-stably acyclic cofibration, since LKan uis a * *left Quillen functor, and LKan up is an objectwise equivalence by 12.10. Definition 12.12 ([10] Def. 4.6). A monoidal model category V is strongly left proper if the cobase change of a weak equivalence along any map in cof Ob (V)-c* *ell is a weak equivalence. Here cofis the class of cofibrations in V. 30 GEORG BIEDERMANN Remark 12.13. A strongly left proper monoidal model category is left proper. If a monoidal model category has only cofibrant objects, it is strongly left prope* *r. Lemma 12.14. A monoidal model category V is strongly left proper if and only if the following gluing lemma holds: Consider a diagram D X oo___C X ____//_S | | | | | | fflffl| fflffl| fflffl| D Y oo___C Y _____//T where all the vertical maps are weak equivalences and the two left hand vertical maps are maps in cof Ob(V). Then the induced vertical map on the pushout is a weak equivalence. The proof is analogous to the proof, that left properness is equivalent to t* *he statement, that the pushout is a weak equivalence if the two left hand vertical maps are cofibrations. It is left to the reader. Remark 12.15. For a functor Xein VU and an object A in U there is a comparison isomorphism Xe RA ~=XeO RA constructed in the following way: Z K2K Z K2K Xe RA ~= RK A Xe(K) ~= (RK O RA ) Xe(K) ~=X O RA e Here K is a defining subcategory for Xe and we use, that the monoidal product commutes with V-colimits. Lemma 12.16 ([10] Theorem 4.11). Let V be right proper and cofibrantly generate* *d, such that the sources and targets of the maps in IV are finitely presentable an* *d that tensoring with them preserves weak equivalences in V. We assume further, that V is strongly left proper and satisfies the monoid axiom. Let axioms (7) and (* *8) from 12.9 be satisfied. Then tensoring with a projectively cofibrant functor pr* *eserves objectwise weak equivalences. Proof.Let Xe! Yebe an objectwise weak equivalence. Using the comparison map from 12.15 we have for every K in U and M in M the following diagram: Xe (RK M) ______//_Ye (RK M) ~=|| ~=|| fflffl| fflffl| Xe(V(K, __)) M____//_Ye(V(K, __)) M The map Xe(V(K, __)) ! Ye(V(K, __)) is an objectwise weak equivalence by 12.10. If M is a source or target of a map in IV, the lower horizontal map is a weak equivalence by assumption and so is the upper. Now let Eebe a projectively cofibrant functor. It is a consequence of the sm* *all object argument, that Eeis a retract of the class IprojMU-cell. Further we kno* *w by 12.8, that objectwise weak equivalences are closed under transfinite compositio* *ns. So we can reduce to the case, where Eeis a finite composition of cobase changes L-STABLE FUNCTORS 31 of IprojMU. Here we can proceed by induction, where the induction start is expl* *ained above. For the induction step we have to look at the following diagram: F Am oo___F Am ____//_ m(R Dm ) Xe m(R Cm ) Xe Een Xe | | | | | | F fflffl| F fflffl| fflffl| m (RAm Cm ) Yeoo__ m(RAm Cm ) Ye_____//Een Ye The right vertical map is an objectwise weak equivalence by the induction assum* *p- tion. So are the other two vertical maps by the previous paragraph. The two left hand horizontal maps are not necessarily objectwise cofibrations, but they are coproducts of maps, that are objectwise in the class cof Ob (V). N* *ow, pushouts in the functor category are computed objectwise. So by the gluing lemma 12.14, which holds due to the strong left properness of V, the induced vertical map on the pushout Een+1 Xe! Een+1 Ye, which is computed objectwise, is an objectwise weak equivalence. Lemma 12.17. Let V and U be as in 12.16. Tensoring with a projectively cofibrant object preserves L-stable equivalences. Proof.We factor the L-stable equivalence into an L-stably acyclic cofibration f* *ol- lowed by an L-stably acyclic fibration, which is just an objectwise acyclic fib* *ration. Tensoring with the first map remains an L-stable equivalence because of the com- patibility result in 11.9. Tensoring with the second map remains an objectwise equivalence by 12.16. Theorem 12.18. Let V and U be as in 12.16. Then the L-stable model structure on VU satisfies the monoid axiom. Proof.We first prove, that tensoring a map in JL-stableMU= J000[ JhoMUdescribed in 8.15 with an arbitrary functor Xein VU remains an L-stable equivalence, and * *it suffices to do this for J000. Let ff: Ae! Xebe a projectively cofibrant replace* *ment for Xe. Let j :Q ! R be a map in J000. We have a diagram: e e Q A_____//Q X e | e e | e | | fflffl| |fflffl Re Ae____//Re Ye Since the source Q and the target R of j are projectively cofibrant, the horizo* *ntal maps are objectwiese weak equivaleences by 12.16. The left hand vertical map is an L-stable equivalence by 12.17, since Aeis projectively cofibrant and the cla* *im follows. Next we have to show, that the cobase change of a map in JL-stableVU Ob(VU ) is an L-stable equivalence. Let Xebe an arbitrary functor in VU and let Q ! R be a map in JL-stable e e VU . Let further f :Qe Xe! Yebe some map. We factor f into a projective cofibration i: Q X ! E followed by an objectwise acyclic fibrati* *on E ! Y . Then the pushout oef fe ied along i remains an L-stable equivalence e e Xf 32 GEORG BIEDERMANN because of left properness. Q X _____//_Eobj-'//_Y e e L-'e e L-' || || || fflffl|p. fflffl|p.fflffl| Re Xe_____//Pe1___//_Pe2 Now the map Ee! Pe1is objectwise in the class cof Ob (V)-cell. Since V is stron* *gly left proper, the cobase change Pe1! Pe2is an objectwise weak equivalence. This proves, that Ye! Pe2is an L-stable equivalence. 13. L-spectra In this section we would like to compare L-stable functors with L-spectra. T* *he first objective is to compare them with symmetric L-spectra constructed in [16], because they also have a monoidal product. Other references for symmetric spect* *ra are [17], [18] and [22]. We were not able to get a direct Quillen equivalence, * *but rather a zig-zag. To get better results in certain cases we also compare our L-* *stable functors to Bousfield-Friedlander-L-spectra, also constructed in [16]. Definition 13.1. Let SphL be the V-category with the natural numbers n 0 as objects and as morphisms ae m-n SphL(n, m) := L m-n* m ,,formelsne where k acts on Lk by permuting the tensor factors. Of course, we set L0 = S. A symmetric spectrum in M is a V-functor X :SphL ! M. We denote the category of symmetric L-spectra in M by Sp (M, L). There is an inclusion j :SphL ! SphVLof categories enriched over V, but this inclusion is not full. Hence the counit LKan jj* ! idof the induced V-Quillen p* *air SphV * (13.1) LKanj:Sp (M, L) = MSphLAE M L:j is not an isomorphism. Definition 13.2. The evaluation functor Evn :Sp (M, L) ! M sending X to Xn has a left adjoint Fn :M ! Sp (M, L). For example, F0 is given by M 7! (M, M L, M L 2, ...) = F0M. Lemma 13.3. For M = V the functor LKan j in (13.1) is strictly symmetric monoidal, the right adjoint j* is lax symmetric monoidal. Proof.We observe first, that every symmetric L-spectrum E over V can be written as a colimit E ~=colimi2IFkiVi, where Vi is an object of V. We can compute: k LKanjFkV ~=RL V L-STABLE FUNCTORS 33 The monoidal product on symmetric L-spectra is defined by extending the isomor- phism FkV F`W ~=Fk+`(V W ), while the monoidal product on small functors is defined by extending the isomor- phism (RA V ) (RB W ) ~=RA B (V W ). The claim easily follows. Definition 13.4. The L-stable model structure on Sp (M, L) is obtained by lo- calizing the level model structure with respect to the set of maps Fn+1(C L) ! Fn(C), where C runs through the sources and targets of the generating set of cofibrati* *ons IM of M. For details see [16, 8.7]. Here we use left properness of M to ensure, that all our C's are cofibrant. Theorem 13.5 (Hovey). Let C be a left proper cellular closed symmetric monoidal model category, let D be a left proper cellular C-model category and let L be a cofibrant C-small object. Suppose also, that the source and target of all maps * *in IC and ID are cofibrant. Then on Sp (D, L) there exists an L-stable model category, Sp (C, L) with its L-stable model structure becomes a closed symmetric monoidal model category and Sp (D, L) is a Sp (C, L)-model category. The previous theorem is proved in [16, 8.8] and [16, 8.11]. There is no assu* *mption on the cyclic permutation on L L L. The cofibrancy assumptions seem awkward. Question 13.6. In a left proper model category a map is a fibration (resp. acyc* *lic fibration) if and only if it satisfies the right lifting property with respect * *to all acyclic cofibrations (resp. cofibrations) between cofibrant objects. So can one always obtain from a given set of generators for the (acyclic) cofibrations a generati* *ng set, whose sources and targets are cofibrant? Definition 13.7. There are mutually inverse isomorphism of categories __I1_// Sp (M, L)UooI__ Sp (MU, L), 2 which merely change the priority of variables. Lemma 13.8. Suppose that V and M satisfy in addition to our usual assumptions from section 5 the conditions of Hovey's theorem 13.5. Suppose also, that U is small. We give Sp (M, L)U the L-stable model structure over the L-stable model structure of Sp (M, L) and Sp (MU, L) the L-stable model structure over the L- stable model structure of MU. Then the isomorphisms I1 and I2 in (13.1) are isomorphisms of model structures. This result permits us to identify the categories Sp (M, L)U and Sp (MU, L) with their L-stable model structure. Proof.Using the fact, that for a functor Xe:U ! V, an object M in M and any n 0 there is a natural isomorphism Fn(Xe M) ~=Xe FnM, 34 GEORG BIEDERMANN it is not difficult to check, that the generating sets of cofibrations and acyc* *lic cofi- brations of the two model structure are mapped to each other. It is proved in [16, section 8], that (F0, Ev0) is a V-Quillen adjunction, if* * the source category is already an L-stable model structure. It follows, that (13.2) F0: MU AE Sp (MU, L) :Ev 0 is a V-Quillen equivalence, provided we take the L-stable model structures ever* *y- where. Now we restrict to SphVLas source category. We remind the reader, that in this case the L-stable model structure on MSphVLis cofibrantly generated. Theorem 13.9. Let V be a pointed, right proper, strongly left proper, cellular,* * lo- cally presentable, closed symmetric monoidal model category. Let M be a proper, cellular, locally presentable V-model category. Let L be cofibrant and V-small.* * Sup- pose, that source and target of all maps in IV and IM are cofibrant. Then ther* *e is a zig-zag in V-mod__of V-Quillen equivalences ____I1____//_ V Sp (M, L)SphVLoo__I______Sp (MSphL, L) OO 2 OO ~||ae|| Ev0||F0|| |fflffl| fflffl|| Sp (M, L) MSphVL which is functorial in M. For V = M these Quillen equivalences are symmetric monoidal. Proof.We observed already, that the pair (F0, Ev0) is a V-Quillen equivalences * *and that the pair (I1, I2) is an isomorphism of model structures. But also (~, ae)* * is a Quillen equivalence by 10.3. It is tempting to complete the above diagram in the following way: ____I1____//_ V Sp (M, L)SphVLoo__I______Sp (MSphL, L) OO 2 OO (13.3) ~||ae|| Ev0||F0|| |fflffl| LKanj fflffl|| Sp (M, L) ________________//MSphVLoo_ j* Here the lower horizontal maps are the ones from (13.1). Unfortunately this dia* *gram is not commutative. It would be possible to prove, that the lower horizontal ma* *ps are a V-Quillen equivalence, if we had an answer to the following question. Question 13.10. Is the unit map E ! j* LKanjE in MSphL a stable equivalence of L-spectra, if E is cofibrant in Sp (M, L)? We have no doubt, that in all reasonable cases the answer to this question is affirmative. The problem is, that L-stable equivalences and L-stable fibrations* * of symmetric L-spectra are rather difficult to characterize. We point out however,* * that in the special cases or our examples in the next section 14 we get better resul* *ts. The reason is, that we can use ordinary Bousfield-Friedlander-L-spectra as step* *ping stone for the comparison, see remark 13.16. L-STABLE FUNCTORS 35 Definition 13.11. Let SphL be the V-category with objects the natural numbers n 2 N and morphisms ae m-n Sph L(n, m) = L * ,,formfornm < n. It comes with obvious functors i: SphL ! SphL and k :SphL ! SphVL. The category Sp(M, L) of V-functors from SphL to M is the category of Bousfield- Friedlander spectra. For details see [3] and [16]. First we recall theorems 4.12, 4.14 and 6.5 from [16]. Lemma 13.12 (Hovey). (i) Let M be a pointed proper locally presentable almost finitely generated model category. Let L in V be cofibrant and V-small. Then the L-stable model structure makes Sp(M, L) into a proper V-model category. (ii)The map X ! P X is an L-stable equivalence for all X in Sp(M, L). (iii)A map X ! Y in Sp(M, L) is an L-stable equivalence if and only if P X ! P X is a level equivalence. (iv)A map X ! Y in Sp(M, L) is an L-stable fibration if and only if it is a lev* *el fibration, such that the diagram X _____//P X | | | | fflffl| fflffl| Y _____//P Y is a homotopy pullback square in the level projective model structure. (v) The L-stable model structure gives Sp(M, L) the structure of a V-model cate- gory. Lemma 13.13. The functor k*: MSphVL! Sp(M, L) preserves and detects L- stable equivalences. Proof.Follows directly from 13.12(iii). We chose fibto be a small fibrant replacement functor in V or M and we abbre- viated by ( __)h the assignment Xe7! fibO XeO fib. Lemma 13.14 ([10] Cor. 7.4.). We take for granted our standard assumptions from section 5. Suppose that M is left proper and that the cyclic permutation on L L L is V-left homotopic to the identity. Then the unit map E ! k*(LKankE) in Sp(M, L) is a stable equivalence of Bousfield-Friedlander L-spectra for every cofibrant E. Proof.First one proves the claim for the special case E = SphL (n, __). In the diagram SphL(n, __)_1__//YYYYYk*(LKan kSphL_(n,2__))//_k*(LKan kSphL (n, __))h YYYYYYY | YYYYYYYYY |3 4YYYYYYYYYYYYY fflffl| YYYY,,YP (RLn) 36 GEORG BIEDERMANN we compute k*(LKan kSphL (n, __))h ~=(RLn)h and conclude, that map 3 is an L- stable equivalence. Map 2 is an objectwise equivalence. But also the composition SphL(n, __) Ln ! SphL(0, __) ! (SphL (0, __))h ! P (SphL (0, __)) is an L-stable equivalence. Since tensoring with L is a Quillen equivalence in * *the L-stable model structure, the adjoint map n SphL(n, __) ! V(Ln, P (SphL (0, __)) ~=P (RL ) is an L-stable equivalence. This is map 4 of the diagram. Here we are using the cyclic permutation assumption, because [16, 10.3] shows, that __ L is a Quillen equivalence only under this condition. By 2-out-of-3 this proves the special ca* *se, map 1 is an L-stable equivalence. For general cofibrant E we know, that E is a retract of the class [ IL-stableSp(M,L)-cell= ( Sph L(n, __) IM )-cell, n 0 where IL-stableSp(M,L)is the class of generating cofibrations for the L-stable * *model struc- ture on Sp(M, L) by [16, 1.8]. Here, for i 2 IM , we have: __ SphL(n, __) i = Fni, __ where F n:M ! Sp(M, L) is the V-left adjoint to the n-th evaluation functor Evn: Sp(M, L) ! M. Since L-stable equivalences of L-spectra are closed under sequential colimits, we can prove the claim by induction along pushouts. We use, that j* and LKan jcommute with colimits. We obtain diagrams F oo F // m SphL(nm , __) Dm____ m SphL(nm , __) Cm_________E | | | | | | F nfflffl| F nfflffl| fflffl| * m RL Dm oo___________ mRL Cm ________//_j (LKan jE) where the maps Cm ! Dm are in IM . All the vertical maps are L-stable equiva- lences. Since M is left proper, we can assume, that all Cm 's and Dm 's are cof* *ibrant. Hence the two left hand horizontal maps are cofibrations. Again by left propern* *ess we can conclude, that the pushout is an L-stable equivalence. Theorem 13.15. Under the assumptions from lemma 13.14 the pair of functors V k*: MSphL! Sp(M, L) :LKank is a V-Quillen equivalence. Proof.This follows directly from 13.13 and 13.14. Remark 13.16. We have a commutative diagram of V-Quillen pairs: ___________LKanj___________//_ V Sp (M, L) ~=MSphLTToo_________________________MSphL77n iiTTTTTTTTTT LKanknnnnnnnnn (13.4) TTTTTTTTTT nnnnnnnn LKani TTTT))TTTT wwnnnnnn Sp(M, L) ~=MSphL L-STABLE FUNCTORS 37 This shows, that under the assumptions from lemma 13.14 L-stable functors are V-Quillen equivalent to symmetric L-spectra if and only if Bousfield-Friedlande* *r- L-spectra are V-Quillen equivalent to symmetric L-spectra. See section 14 for applications. Of course, this is cheating and we believe that the pair (LKan j,* * j*) should be a Quillen equivalence under much more general conditions. In particul* *ar, the cyclic permutation condition should not play a role. 14.Examples Let us recall again the assumptions, under which the L-stable model structure on MU exists. o The category V has to be a locally presentable cofibrantly generated sym- metric monoidal model category equipped with a functorial small fibrant replacements. o The object L in V has to be V-small and cofibrant. If the unit S of V is small, every small object in V is V-small. o The category M should be a right proper locally presentable cofibrantly generated V-model category with a chosen functorial small fibrant replac* *e- ments. o For the L-stable model structure to exist on MU the category U has to satisfy the axioms 8.1. o In order to have a closed symmetric monoidal structure on VU we need 11.1. o Right properness of V and properness of M imply all the compatibility results. o Finally for the monoid axiom in VU we need, that V is strongly left prop* *er, satisfies the monoid axiom and sources and targets of IV are V-finitely presentable. Additionally U should satisfy 12.9. If V has only cofibra* *nt objects, it is strongly left proper and satisfies the monoid axiom. For our L-stabilization we considered the category SphVLas source category U. It satisfies axioms 8.1, 11.1 and, if L is V-finitely presentable, (8). Axiom * *(7) is obviously highly sensitive to the choice of L. Example 14.1. Let V = S* pointed simplicial sets and L = S1. In this case we know by [17, Theorem 4.2.5], that the diagram (13.4) consists of Quillen equiv- SphS*1 alences. So the homotopy category associated to the category S* S with the S1-stable model structure is the classical stable homotopy category with the co* *r- rect monoidal product. This was first proved in [20]. The monoid axiom holds for S1-stable functors, and was first proved in [10, lemma 6.30]. For symmetric spe* *ctra this was proved in [17, Theorem 5.4.1]. Example 14.2. Let S be a noetherian scheme of finite dimension. Let V be the category of A1-local motivic spaces over a scheme S. There are several ways to construct this model category, all of them Quillen equivalent to each other, an* *d all serve well for our purpose. See [21], [18] or [11]. Let L be the smash product * *P1 of the simplicial circle S1 with the Tate circle A1 - {0}. This object is cofibran* *t and and V-finitely presentable. It satisfies the cyclic permutation condition [18, * *3.13]. By [18, Theorem 4.31] we know, that (13.4) is a diagram of V-Quillen equivalenc* *es. Hence P1-stable functors supply a model for the stable motivic homotopy category together with the correct smash product. This was first observed in [10] and [1* *1]. 38 GEORG BIEDERMANN More generally, Jardine has constructed L-stable spectra and symmetric spectra over motivic spaces for L = S1 ^ G, where G is a compact object, see [18, 2.2]. They are shown to be Quillen equivalent in [18, Theorem 4.31]. Compact object in Jardine's sense are V-finitely presentable, and all objects are cofibrant in* * his setting. So our theory applies and again it follows, that our L-stable functors* * are Quillen equivalent to them. Example 14.3. In [10, section 9] it is shown, that G-equivariant stable homotopy theory can be described by this setup. References [1]G. Biedermann, B. Chorny, and O. R"ondigs. 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[23]S. Schwede and B. E. Shipley. Algebras and modules in monoidal model catego* *ries. Proc. London Math. Soc., 3(2):491-511, 2000. L-STABLE FUNCTORS 39 Department of Mathematics, Middlesex College, The University of Western On- tario, London, Ontario N6A 5B7, Canada E-mail address: gbiederm@uwo.ca