ORTHOGONAL SPECTRA AND S-MODULES M. A. MANDELL AND J. P. MAY Contents 1. Right exact functors on categories of diagram spaces 2 2. The proofs of the comparison theorems 5 3. The construction of the functor N* 12 4. The functor M and its comparison with N 16 References 20 There are several symmetric monoidal categories of "spectra" that are model categories with homotopy categories equivalent to the stable homotopy category. The most highly developed, and the first to be made rigorous, is the category M* * = MS of S-modules of [1]. Its objects are quite complicated, but the complication encodes computationally important information. A second such category is the category S of symmetric spectra, which is due to Jeff Smith and whose rigorous development is given in [3]. That paper is written simplicially, but a logical* *ly independent topological treatment has since been given [6]. Symmetric spectra are far simpler objects than S-modules, but the simplicity comes at a price: the weak equivalences are not just the maps that induce isomorphisms on homotopy groups, and this makes the homotopy theory of symmetric spaces quite subtle. A third such category is the category I S of orthogonal spectra, which was first defined by the second author [7] and was fully developed in the papers [5] and [6]. Orthogonal spectra are just as simple to define as symmetric spectra, yet * *their weak equivalences are just the maps that induce isomorphisms on homotopy groups. Thus, philosophically, orthogonal spectra are intermediate between S-modules and symmetric spectra, enjoying some of the best features of both. It is proven in [6] that the categories of symmetric spectra and of orthogonal spectra are Quillen equivalent. It is proven by Schwede in [10] that the homoto* *py categories of symmetric spectra and S-modules are equivalent. With a slight cha* *nge of the model structure on symmetric spectra, his equivalence is also a Quillen equivalence. However, this does not give a satisfactory Quillen equivalence bet* *ween the categories of orthogonal spectra and S-modules since the resulting functor * *from orthogonal spectra to S-modules is the composite of a right adjoint (to symmetr* *ic spectra) and a left adjoint. We shall give a Quillen equivalence between the categories of orthogonal spec* *tra and S-modules such that the Quillen equivalence of [10] is the composite of the Quillen equivalence between symmetric spectra and orthogonal spectra of [6] and our new Quillen equivalence. Thus orthogonal spectra are mathematically as well ____________ Date: November 14, 1998. 1 2 M. A. MANDELL AND J. P. MAY as philosophically intermediate between symmetric spectra and S-modules. Our proofs will give a concrete Thom space level understanding of the difference be* *tween orthogonal spectra and S-modules. To separate formalities from substance, we begin in Section 1 by establishing* * a formal framework for constructing symmetric monoidal left adjoint functors whose domain is a category of diagram spaces. In fact, this elementary category theory sheds new light on the basic constructions that are studied in all work on diag* *ram spectra. Deferring the construction that gives substance to the theory to Secti* *on 3, we prove the following comparison theorems in Section 2. Theorem 0.1. There is a strong symmetric monoidal functor N : I S - ! M and a lax symmetric monoidal functor N# : M -! I S such that (N; N# ) is a Quillen equivalence between I S and M . The induced equivalence of homotopy categories preserves smash products. Theorem 0.2. The pair (N; N# ) induces a Quillen equivalence between the cate- gories of orthogonal ring spectra and S-algebras. Theorem 0.3. For an orthogonal ring spectrum R, the pair (N; N# ) induces a Quillen equivalence between the categories of R-modules and of NR-modules. Theorem 0.4. The pair (N; N# ) induces a Quillen equivalence between the cate- gories of commutative orthogonal ring spectra and commutative S-algebras. This last result is the crucial comparison theorem since most of the deepest applications of structured ring and module spectra concern E1 -ring spectra or, equivalently by [1], commutative S-algebras. The analogues of the previous four theorems with orthogonal spectra and S- modules replaced by symmetric spectra and orthogonal spectra were proven in [6]. This has the following immediate consequence, which reproves the results of [10* *]. Theorem 0.5. The analogues of the previous four theorems with orthogonal spect* *ra replaced by symmetric spectra are also true. The functor N that occurs in the results above has all of the formal and homo* *topi- cal properties that one might desire. However, a quite different and considerab* *ly more intuitive functor M from orthogonal spectra to S-modules is implicit in the earliest work in this direction [8]. The functor M gives the most natural way * *to construct Thom spectra as commutative S-algebras, and its equivariant version w* *as used in an essential way in the proof of the localization and completion theore* *m for complex cobordism given in [2]. We define M and compare it with N in Section 4. Since orthogonal spectra, like S-modules, are intrinsically coordinate-free, * *they are well adapted to equivariant generalization. The study of that generalizatio* *n is work in progress. It is a pleasure to thank our collaborators Brooke Shipley and Stefan Schwede. Like Schwede's paper [10], which gives a blueprint for Section 2 here, this pap* *er is an outgrowth of our joint work in [5, 6]. 1.Right exact functors on categories of diagram spaces To clarify our arguments, we first give the formal structure of our construct* *ion of the adjoint pair (N; N# ) in a suitably general framework. We consider categ* *ories DT of D-shaped diagrams of based spaces for some domain category D, and we ORTHOGONAL SPECTRA AND S-MODULES 3 show that, to construct left adjoint functors from DT to suitable categories C * *, we need only construct contravariant functors D -! C . The proof is an exercise in* * the use of representable functors and must be standard category theory, but we do n* *ot know a convenient reference. Similar constructions are used in algebraic geomet* *ry. Let T be the category of based spaces and D be any based topological category with a small skeleton skD. A D-space is a continuous based functor D -! T . Let DT be the category of D-spaces. As observed in [5, x2], the evident levelwi* *se constructions define limits, colimits, smash products with spaces, and function D-spaces that give DT a structure of complete and cocomplete, tensored and cotensored, topological category. We call such a category topologically bicompl* *ete. We fix a topologically bicomplete category C for the rest of this section. We w* *rite C ^A for the tensor of an object C of C and a based space A. Functors and natur* *al transformations are assumed to be continuous. Definition 1.1.A functor between topologically cocomplete categories is right e* *x- act if it commutes with colimits and tensors. For example, any functor that is a left adjoint is right exact. For a contravariant functor E : DT - ! C and a D-space X, we can construct the coend Z d (1.2) E D X = E(d) ^ X(d): Explicitly, E D X is the coequalizer of the diagram W "^id//_W d;eE(e) ^ D(d; e) ^ X(d)__//_dE(d) ^ X(d); id^" where the wedges run over pairs of objects and objects of skD and the parallel arrows are wedges of smash products of identity and evaluation maps of E and X. For an object d 2 D, we have a left adjoint Fd : T - ! DT to the functor given by evaluation at d. If d* is defined by d*(e) = D(d; e), then FdA = d* ^ A. In particular, FdS0 = d*. Definition 1.3.Let D = DD : D -! DT be the evident contravariant functor that sends d to d*. The following observation is [5, 2.6]. Lemma 1.4. The evaluation maps D(d; e) ^ X(d) -! X(e) of D-spaces X induce a natural isomorphism of D-spaces D D X -! X. Together with elementary categorical observations, this has the following imm* *e- diate implication. It shows that (covariant) right exact functors F : DT -! C determine and are determined by contravariant functors E : D -! C . Theorem 1.5. If F : DT - ! C is a right exact functor, then (FOD)D X ~=FX. Conversely, if E : D -! C is a contravariant functor, then the functor F : D -!* * C specified by FX = E D X is right exact and F O D ~=E. Notation 1.6. Write F $ F* for the correspondence between right exact functors F : DT - ! C and contravariant functors F* : D -! C . Thus, given F, F* = FOD, and, given F*, F = F*D (-). In particular, on representable D-spaces, Fd* ~=F*d. 4 M. A. MANDELL AND J. P. MAY Corollary 1.7.Natural transformations between right exact functors DT - ! C determine and are determined by natural transformations between the correspondi* *ng contravariant functors D -! C . Proposition 1.8.Any right exact functor F : DT - ! C has the right adjoint F# specified by (F# C)(d) = C (F*d; C) for C 2 C and d 2 D. The evaluation maps D(d; e) ^ C (F*d; C) -! C (F*e; C) of the functor F# are the adjoints of the composites F*e ^ D(d; e) ^ C (F*d; C)"^id-!F*d ^ C (F*d; C)-i!C; where " is an evaluation map of the functor F* and i is an evaluation map of the category C . Proof.We must show that (1.9) C (FX; C) ~=DT (X; F# C): The description of FX as a coend implies a description of C (FX; C) as an end constructed out of the spaces C (F*d^X(d); C). Under the adjunction isomorphisms C (F*d ^ X(d); C) ~=T (X(d); C (F*d; C)); this end transforms to the end that specifies DT (X; F# C). |_* *__| As an illustration of the definitions, we show how the prolongation and forge* *tful functors studied in [5] and [6] fit into the present framework. Example 1.10. A (covariant) functor : D -! D0 induces the forgetful functor U : D0T - ! DT that sends Y to Y O . It also induces the contravariant functor DD0O : D -! D0T . Let PX = (DD0O)D X. Then P is the prolongation functor left adjoint to U. Now let D be symmetric monoidal with product and unit uD . By [5, x3], DT is symmetric monoidal with unit u*D. We denote the smash product of DT by ^D . Actually, the construction of the smash product is another simple direct application of the present framework. Example 1.11. We have the external smash product Z : DT xDT - ! (DxD)T specified by (X Z Y )(d; e) = X(d) ^ Y (e) [5, 3.1]. We also have the contravar* *iant functor DD O : D x D -! DT . The internal smash product is given by (1.12) X ^D Y = (DD O ) DxD (X Z Y ): It is an exercise to rederive the universal property (1.13) DT (X ^D Y; Z) ~=(D x D)T (X Z Y ); Z O ) that characterizes ^D from this definition. Proposition 1.14.Let F* : D -! C be a strong symmetric monoidal contravari- ant functor. Then F : DT -! C is a strong symmetric monoidal functor and F# : C -! DT is a lax symmetric monoidal functor. ORTHOGONAL SPECTRA AND S-MODULES 5 Proof.We are given an isomorphism : F*uD -! uC and a natural isomorphism OE : F*d ^C F*e -! F*(d e): Since F*uD ~=Fu*D, we may view as an isomorphism Fu*D-! uC. By (1.9) and (1.13), we have C (F(X ^D Y ); C) ~=(DT x DT )(X Z Y; F# C O ): Commuting coends past smash products and using isomorphisms (F*d ^ X(d)) ^ (F*e ^ Y (e)) ~=F*(d e) ^ X(d) ^ Y (e) induced by OE, we obtain the first of the following two isomorphisms. We obtain the second by using the tensor adjunction of C and applying the defining univer* *sal property of coends. Z (d;e) C (FX ^C FY; C) ~= C ( F# (d e) ^ X(d) ^ Y (e); C) ~= (DT x DT )(X Z Y; F# C O ): There results a natural isomorphism FX ^C FY ~= F(X ^D Y ), and coherence is easily checked. For F# , the adjoint u*D-! F# uC of gives the unit map. Taking the smash products of maps in C and applying isomorphisms OE, we obtain maps C (F*(d); C) ^ C (F*(e); C0) -! C (F*(d e); C ^C C0) that together define a map F# C Z F# C0- ! F# (C ^C C0) O : Using (1.13), there results a natural map F# C ^D F# C0- ! F# (C ^C C0); and coherence is again easily checked. |___| 2. The proofs of the comparison theorems We refer to [5, 6] for details of the category I S of orthogonal spectra and * *to [1] for details of the category M = MS of S-modules. Much of our work depends only on basic formal properties. Both of these categories are closed symmetric monoi* *dal and topologically bicomplete. They are Quillen model categories, and their model structures are compatible with their smash products. Actually, in [6], the cate* *gory of orthogonal spectra is given two model structures. In one of them, the sphere spectrum is cofibrant, in the other, the "positive stable model structure", it * *is not. In [6], use of the positive stable model structure was essential to obtain an i* *nduced model structure on the category of commutative orthogonal ring spectra. It is a* *lso essential here, since the sphere S-module S is not cofibrant. We begin by giving a quick summary of definitions from [5], recalling how ort* *hog- onal spectra fit into the framework of the previous section. Let I be the symme* *tric monoidal category of finite dimensional real inner product spaces and linear is* *o- metric isomorphisms. We call an I -space an orthogonal space. The category I T of orthogonal spaces is closed symmetric monoidal under its smash products X ^ Y and function objects F (X; Y ). 6 M. A. MANDELL AND J. P. MAY The sphere orthogonal space SI has V th space the one-point compactification SV of V ; SI is a commutative monoid in I T . An orthogonal spectrum, or I - spectrum, is a (right) SI -module. The category I S of orthogonal spectra is cl* *osed symmetric monoidal. We denote its smash products and function spectra by X^I Y and FI (X; Y ) (although this is not consistent with the previous section). There is a symmetric monoidal category IS with the same objects as I such that the category of IS-spaces is isomorphic to the category of I -spectra; IS contains I as a subcategory. The construction of IS is given in [5, 6.1]. Its s* *pace of morphisms IS(V; W ) is (V *^ SI )(W ), where V *(W ) = I (V; W )+ . In Secti* *on 3, we shall give a concrete alternative description of IS in terms of Thom spac* *es, and we shall construct a coherent family of cofibrant (-V )-sphere S-modules N*V that give us a contravariant "negative spheres" functor N* to which we can apply the constructions of the previous section. Note that the unit of I is 0, the un* *it of I S is SI , and, as required for consistency, IS(0; W ) = SW . Theorem 2.1. There is a strong symmetric monoidal contravariant functor N* : IS -! M , such that N*(V ) is a cofibrant S-module for V 6= 0. Moreover, the evaluation map " : N*(V ) ^ SV = N*(V ) ^ IS(0; V ) -! N*(0) ~=S of the functor is a weak equivalence for all V 6= 0. Here N*(0) ~=S since N* is strong symmetric monoidal. Propositions 1.8 and 1.14 give the following immediate consequence. Theorem 2.2. Define functors N : I S -! M and N# : M -! I S by letting N(X) = N* IS X and (N# M)(V ) = M (N*(V ); M). Then (N; N# ) is an adjoint pair of functors such that N is strong symmetric monoidal and N# is lax symmetr* *ic monoidal. This gives the basic formal properties of N and N# , and we turn to their hom* *o- topical properties. For this, we need the precise understanding of homotopy gro* *ups given in the next few paragraphs. We understand prespectra in the coordinate-free sense of [4, 1]. Thus a presp* *ec- trum X consists of based spaces X(V ) and based maps oe : W-V X(V ) -! X(W ), where V ranges through the finite dimensional sub inner product spaces of a cou* *nt- ably infinite dimensional real inner product space U, which we may take to be U = R1 . The homotopy groups ss*(X) are the homotopy groups of the underlying coordinatized prespectrum with nth space Xn = X(Rn). A prespectrum X is an -spectrum if its adjoint structure maps "oe: X(V ) -! W-V X(W ) are weak equivalences; it is a positive -spectrum if these maps are weak equivalences for V 6= 0; it is a spectrum if all of these maps are homeomo* *r- phisms. Let P and S denote the categories of prespectra and spectra. There is an underlying prespectrum functor I S -! P and an underlying spectrum functor M -! S . The homotopy groups of an orthogonal spectrum or S-module are the homotopy groups of the underlying prespectrum or spectrum. A map of orthogonal spectra or S-modules is a (stable) weak equivalence if it induces an isomorphis* *m of homotopy groups. It is convenient to recast the homotopy groups of S-modules in terms of ho- motopy classes of maps of S-modules. There is a canonical cofibrant n-sphere ORTHOGONAL SPECTRA AND S-MODULES 7 S-module SnSdefined in [1, II.1.7]. By [1, 1.8], we have identifications ssn(M) = [SnS; M]; where the brackets refer to maps in the homotopy category of M . For n 0, SnS= SS ^ Sn, where SS is the canonical cofibrant approximation to S. By [4, II.3.6] and [1, I.6.1], we have compatible systems of isomorphisms SmS^S SnS~=Sm+nS~= SmS^ Sn which, up to homotopy, are well-defined, associative, commutative, and unital. * *In particular, for n > 0, S-nSis isomorphic to the nth smash power (S-1S)(n). We w* *ill see in the next section that N*(R) = S-1S. Therefore, since N* is strong symmet* *ric monoidal, N*(Rn) ~=(N*(R))(n)= (S-1S)(n)~=S-nS and ss-n (M) = [N*(Rn); M] for n > 0. To put this in a form suitable for analyzing colimits, we further recast it i* *n terms of the cofibrant (m - n)-spheres N*(Rn) ^ Sm , where n > 0 and m 0. Under the isomorphism S-1S^ S1 ~=SS, the evaluation map " : N*(R) ^ S1 -! N*(0) ~=S agrees with the canonical cofibrant approximation SS -! S; see Lemma 3.20 below. This gives a weak equivalence (2.3) n;m : N*(Rn+1) ^ Sm+1 ~=N*(Rn) ^ N*(R) ^ S1 ^ Sm - ! N*(Rn) ^ Sm ; where we use the last coordinates of Rn+1 and Rm+1 to specify the first iso- morphism. Inductively, these weak equivalences give canonical weak equivalences N*(Rn) ^ Sm ~=Sm-nS and thus give identifications ssm-n (M) ~=[N*(Rn) ^ Sm ; M]: Finally, we relate the S-modules N*(Rn) ^ Sm to representable IS-spaces and the structure maps of orthogonal spectra. Write Fn = FRn : T - ! I S , so that FnS0 = (Rn)*. By [6, 5.4 and 5.5], there is a map n : Fn+1S1 -! FnS0 whose induced map *n: Xn ~=I S (FnS0; X) -! I S (Fn+1S1; X) ~=Xn+1 is the adjoint structure map of the underlying coordinatized prespectrum of X, and n is a weak equivalence. In terms of represented IS-spaces, n is given by composition in IS: (Rn+1)* ^ S1 (Rn+1)* ^ IS(Rn; Rn+1) -! (Rn)*: Here (Rn+1)* ~=(Rn)* ^ (R)* by [5, I.3.8], and n agrees with id^1 under the isomorphism. Smashing with Sm , we obtain canonical maps (2.4) n;m : (Rn+1)* ^ Sm+1 ~=(Rn)* ^ (R)* ^ S1 ^ Sm - ! (Rn)* ^ Sm : Our functorial semantics show that Nn;m = n;m. Definition 2.5.For orthogonal spectra X, define a natural homomorphism : ss*(X) -! ss*(NX) as follows. An element OE 2 ssm-n (X) = colimssm (Xn) can be represented by a m* *ap f : FnSm - ! X for n sufficiently large. Let (OE) be the homotopy class of Nf : N*(Rn) ^ Sm ~=NFnSm - ! NX: 8 M. A. MANDELL AND J. P. MAY The discussion above ensures that Nf only depends on OE. For S-modules M, define a natural homomorphism # : ss*(M) -! ss*(N# M) as follows. For a map g : N*(Rn) ^ Sm - ! M, the adjoint of g is a map Sm - ! M (N*(Rn); M) = (N# M)(Rn): Let # (g) be the element of ssn-m (N# M) represented by this map. Lemma 2.6. The map on homotopy groups induced by the unit j : Id-! N# N of the (N; N# )-adjunction coincides with the composite # O . Proof.A check of definitions shows that # O sends the fundamental class of ssn-m (FnSm ) to the element of ssn-m (N# NFnSm ) represented by the unit map__ j : FnSm - ! N# NFnSm . The conclusion follows by naturality. |__| Lemma 2.7. For S-modules M, N# M is a positive -spectrum and the map # : ss*(M) -! ss*(N# M) is an isomorphism. Therefore a map f of S-modules is a weak equivalence if and only if N# f is a weak equivalence of orthogonal spectr* *a. Proof.For n > 0 and m 0, the discussion above and evident adjunctions give an isomorphism ssm-n (M) ~=[N*(Rn) ^ Sm ; M] ~=ssm (N# M)(Rn) that is formalized by the definition of # . This gives the first statement_and * *the second statement follows. |__| Lemma 2.8. The functor N# preserves q-fibrations. Proof.Let f : M -! N be a q-fibration of S-modules. We must show that N# f is a positive q-fibration of orthogonal spectra. According to [6, 6.5 and x9],* * this means that, for n > 0, the map f* : M (N*(Rn); M) -! M (N*(Rn); N) is a Serre fibration and the diagram M (N*(Rn); M) _"oe//_M (N*(Rn+1); M) f*|| |f*| fflffl| fflffl| M (N*(Rn); N)__"oe//_M (N*(Rn+1); N) is a homotopy pullback. The second condition is clear since N# M is a positive - spectrum. For the first condition, we recall from [1, VII.4.6] that f is a q-fi* *bration if and only if it satisfies the RLP (right lifting property) with respect to al* *l maps i0 : F 1nCSq -! F 1nCSq ^ I+ : Here 1n : T - ! S is left adjoint to evaluation at n (= Rn) and F : S -! M is a "free functor" whose definition will be recalled in the next section. We s* *hall see there that, for n > 0, N*(Rn) is isomorphic to F 1nS0. The functors F and 1ncommute with smash products with spaces. Now an easy adjunction argument shows that f* : M (N*(Rn); M) -! M (N*(Rn); N) satisfies the RLP with respect_ to the maps CSq -! CSq ^ I+ and is thus a Serre fibration. |__| Since N# preserves weak equivalences and q-fibrations, (N; N# ) is a Quillen * *ad- joint pair. By standard model theoretic arguments [6, 2.2, 2.3], the following * *result implies that it is a Quillen equivalence, as claimed in Theorem 0.1. ORTHOGONAL SPECTRA AND S-MODULES 9 Proposition 2.9.The unit j : X -! N# NX of the adjunction is a weak equiva- lence for all cofibrant orthogonal spectra X. In fact, to prove Theorem 0.1, we only need this for orthogonal spectra that * *are cofibrant in the positive stable model structure, but we shall prove it for ort* *hogonal spectra that are cofibrant in the stable model structure of [6, x6]. We refer t* *o positive cofibrant and cofibrant orthogonal spectra to distinguish between these classes. Proof.We first prove that j : FnS0 ! N# NFnS0 is a ss*-isomorphism for n 0. The (m + n)th space of FnS0 is O(m + n)+ ^O(m)Sm and, since NFnS0 = N*(Rn), the (m + n)th space of N# NFnS0 is M (N*(Rm+n ); N*(Rn)). It suffices to show that, for m 0, the map (2.10) j : O(m + n)+ ^O(m) Sm - ! M (N*(Rm+n ); N*(Rn)) is a (2m - 1)-equivalence. Classically, the inclusion of Sm in the left hand si* *de is a (2m - 1)-equivalence. For m + n > 0, the right hand side is homotopy equivalent* * to QSm . Indeed, the functor F cited in the proof of Lemma 2.8 carries CW spectra * *to weakly equivalent S-modules, and F has a right adjoint V that carries S-modules to weakly equivalent spectra; see [1, III.1.3]. Since N*(Rn) ~=F 1nS0, we see t* *hat M (N*(Rm+n ); N*(Rn)) ' (1nS0)m+n = QSm : Therefore there is some map Sm -! M (N*(Rm+n ); N*(Rn)) that is a (2m - 1)- equivalence. Thus, to see that the map (2.10) is a (2m - 1)-equivalence, we only need to see that the composite map (2.11) Sm = {1}+ ^ Sm ! O(m + n)+ ^O(m) Sm ! M (N*(Rm+n ); N*(Rn)) specifies a generator of ssm M (N*(Rm+n ); N*(Rn)). This holds since the map (2* *.11) is adjoint to the weak equivalence N*(Rm+n )^Sm - ! N*(Rn) obtained inductively from (2.3). By Lemmas 2.6 and 2.7, j : X -! N# NX is a weak equivalence if and only if : ss*(X) -! ss*(NX) is an isomorphism. By standard results on the homotopy groups of spectra and their analogues for the homotopy groups of orthogonal spectra pr* *oven in [6, x4], is an isomorphism for X if and only if is an isomorphism for X, a* *nd the class of orthogonal spectra for which is an isomorphism is closed under we* *dges, pushouts along h-cofibrations, sequential colimits, and retracts. Therefore is* * an isomorphism for all cofibrant orthogonal spectra X since it is an isomorphism_f* *or X = FnS0, n 0. |__| The proof of Theorem 0.2.The category of orthogonal ring spectra has two model structures. The respective weak equivalences and q-fibrations are created in t* *he category of orthogonal spectra with its stable model structure or its positive * *stable model structure. The category of S-algebras is a model category with weak equiv- alences and q-fibrations created in the category of S-modules. The pair (N; N#* * ) restricts to a Quillen adjoint pair relating the category of orthogonal ring sp* *ectra with its positive stable model structure to the category of S-algebras. To pro* *ve that (N; N# ) is a Quillen equivalence, we must show that j : R -! N# NR is a weak equivalence when R is a positive cofibrant orthogonal ring spectrum. More generally, if R is a cofibrant orthogonal ring spectrum, then the underlying or* *thog- onal spectrum of R is cofibrant (although not positive cofibrant) by [6, 8.6]._* *The conclusion follows from Proposition 2.9. |__| 10 M. A. MANDELL AND J. P. MAY The proof of Theorem 0.3.The category of R-modules is a model category with weak equivalences and q-fibrations created in the category of orthogonal spectra with its positive stable model structure. Moreover, by [6, 8.6(iv)], we may ass* *ume without loss of generality that R is a positive cofibrant orthogonal ring spect* *rum. The category of NR-modules is a model category with weak equivalences and q- fibrations created in the category of S-modules. The pair (N; N# ) restricts to a Quillen adjoint pair relating these two categories, and we must show that j : Y -! N# NY is a weak equivalence when Y is a positive cofibrant R-module. More generally, the underlying orthogonal spectrum of a cofibrant module over a cofi* *brant orthogonal ring spectrum is cofibrant (although not positive cofibrant) by[6,_8* *.6]. The conclusion follows from Proposition 2.9. |__| The proof of Theorem 0.4.The category of commutative orthogonal ring spectra has a model structure with weak equivalences and q-fibrations created in the ca* *t- egory of orthogonal spectra with its positive stable model structure [6, 10.1].* * The category of commutative S-algebras has a model structure with weak equivalences and q-fibrations created in the category of S-modules [1, VII.4.8]. The pair (N* *; N# ) restricts to a Quillen adjoint pair relating these categories, and we must prov* *e that j : R -! N# NR is a weak equivalence when R is a cofibrant commutative or- thogonal ring spectrum. Since the underlying orthogonal spectrum of R is not cofibrant, we must work harder here. We use the notations and results of [6, x1* *0], where the structure of cofibrant commutative orthogonal ring spectra is analyzed and the precisely analogous proof comparing commutative symmetric ring spectra and commutative orthogonal ring spectrum is given. We may assume that R is an AF +I-cell complex, where A is the free commutative orthogonal ring spectrum functor, and we claim first that j is a weak equivalen* *ce when R = AX for a positive cofibrant orthogonal spectrum X. It suffices to prove that j : X(i)=i -! N# N(X(i)=i) is a weak equivalence for i 1. On the right, N(X(i)=i) ~=(NX)(i)=i, and NX is a cofibrant S-module. Consider the commutative diagram j Ei+ ^i X(i) ____//_N# N(Ei+ ^i X(i)) ~= N# (Ei+ ^i (NX)(i)) q|| |N#Nq| |N#q| fflffl| fflffl| fflffl| X(i)=i _____j____//_N# N(X(i)=i) ~= N# ((NX)(i)=i): The q are the evident quotient maps, and the left and right arrows q are weak equivalences by [6, 10.9] and [1, III.5.1]. The top map j is a weak equivalence* * by induction up the cellular filtration of Ei, the successive subquotients of whic* *h are wedges of copies of i+ ^ Sn, using the fact that X(i)is positive cofibrant. By passage to colimits, as in the last step of the proof of Theorem 0.1, the result for general R will follow from the result for an AF +I-cell complex that* * is constructed in finitely many stages. We have proven the result when R requires only a single stage, and we assume the result when R is constructed in n stages. Thus suppose that R is constructed in n + 1 stages. Then R is a pushout (in the category of commutative orthogonal ring spectra) of the form Rn ^AX AY , where Rn is constructed in n-stages and X -! Y is a wedge of maps in F +I. By [6, (10.7)], R ~=B(Rn; AX; AT ), where T is a suitable wedge of orthogonal spectra FnS0. Since the simplicial bar construction is proper and since N commutes with ORTHOGONAL SPECTRA AND S-MODULES 11 geometric realization, we see by tracing through the cofibration sequences used* * in the proof of the invariance of bar constructions in [1, X.4] that it suffices t* *o show that j is a weak equivalence on the commutative orthogonal ring spectrum Rn ^S (AX)(q)^ AT ~=Rn ^S A(X _ . ._.X _ T ) of q-simplices for each q. By the definition of AF +I-cell complexes, we see th* *at this smash product (= pushout) can be constructed in n-stages, hence the conclusion_ follows from the induction hypothesis. |__| Remark 2.12.Consider the diagram _P___ __N__// I ooU_//I_Soo___ M ; N# where S is the category of symmetric spectra and U and P are the forgetful and prolongation functors of [5, 6] (see Example 1.10). Clearly (U O N# )(M)(n) ~=M ((S-1S)(n); M) where S(0)S= S. This is the right adjoint M -! S used by Schwede [10], and N O P is its left adjoint. Thus the adjunction studied in [10] is the composite* * of the adjunctions (P; U) and (N; N# ). There is a slight caveat. Schwede works simp* *li- cially, using the total singular complex to convert the topological model categ* *ory of S-modules to a simplicial model category and then comparing it to the simplicial model category of symmetric spectra of simplicial sets [3]. This has the effect* * of introducing unnecessary and distracting passages back and forth between spaces and simplicial sets at every step. Since, by [6], the model theory of diagram s* *pec- tra of topological spaces is now as well developed as the model theory of diagr* *am spectra of simplicial sets, as was not the case when [10] was written, there is* * no longer any point to this: it is much cleaner to work topologically, relying on * *the Quillen equivalence between symmetric spectra of simplicial sets and symmetric spectra of spaces of [3, x6] and, with multiplicative elaborations, [6, x13] to* * pass to the simplicial world when appropriate. The following parenthetical remarks complete our set of comparisons. Remarks 2.13.(i) The category of N -spectra, or coordinatized prespectra, was denoted P and given a positive stable model structure in [6, x9], and it was sh* *own there that the evident forgetful and prolongation functors give a Quillen equiv* *alence between that category and the category of symmetric spectra. As observed in [6, 6.12], our category P of coordinate-free prespectra is the category of V -spect* *ra, where V is the discrete subcategory of I consisting of its objects and identity morphisms. Exactly as in [6, x9], P has a positive stable model structure, and * *the evident forgetful and prolongation functors give a Quillen equivalence between * *it and the category of orthogonal spectra. (ii) The category of spectra was given a stable model structure in [1, VIIxx4, * *5]. Easy direct comparisons of definitions show that the adjoint pair (L; `) of [4,* * I.2.2] connecting P and S is a Quillen equivalence and the adjoint pair (F; V ) of [1, III.1.3] connecting S and M is a Quillen equivalence. 12 M. A. MANDELL AND J. P. MAY 3. The construction of the functor N* We prove Theorem 2.1 here. Implicitly, we shall give two constructions of the functor N*. The theory of S-modules is based on a functor called the twisted ha* *lf- smash product, denoted n, the definitive construction of which is due to Cole [* *1, App]. The theory of orthogonal spectra is the theory of diagram spaces with dom* *ain category IS. Both n and IS are defined in terms of Thom spaces associated to spaces of linear isometries. We first define N* in terms of twisted half-sm* *ash products. We then outline the definition of twisted half-smash products in term* *s of Thom spaces and redescribe N* in those terms. That will make the connection with the category IS transparent, since the morphism spaces of IS are Thom spaces closely related to those used to define the relevant twisted half-smash product* *s. Here we allow the universe U on which we index our coordinate free prespectra and spectra to vary. We write PU and S U for the categories of prespectra and spectra indexed on U. We have a forgetful functor ` : S U -! PU with a left adjoint spectrification functor L : PU - ! S U. We can describe L explicitly in the cases of interest to us. A prespectrum X is an inclusion prespectrum if its adjoint structure maps "oe: XV -! W-V XW are inclusions. Lemma 3.1. Let X be an inclusion prespectrum. Then LX(V ) = colimWV W-V X(W ): The V th map of the unit j : X -! `LX of the (L; `)-adjunction is the map from the initial term X(V ) into the colimit, and j is a weak equivalence of prespec* *tra. We have a suspension spectrum functor U that is left adjoint to the zeroth sp* *ace functor U . Let SU = U (S0). The functors U and U are usually denoted 1 and 1 , but we wish to emphasize the choice of universe rather than its infinite dimensionality. We write 1 and 1 when U = R1 , and we then write SU = S. More generally, for any finite dimensional sub inner product space V of U, we have a shift desuspension functor UV: T -! S U, denoted 1V when U = R1 , that is left adjoint to evaluation at V . Explicitly, UV(A) is the spectrificat* *ion of the evident prespectrum with W th space W-V A, where W-V A = * if V is not contained in W . For inner product spaces U and U0, let I (U; U0) be the space of linear isome* *tries U - ! U0, not necessarily isomorphisms. It is contractible when U0 is infinite dimensional [8, 1.3]. We have a twisted half-smash functor 0 I (U; U0) n (-) : S U- ! S U; whose definition we shall recall shortly. It is a "change of universe functor" * *that converts spectra indexed on U to spectra indexed on U0 in a well-structured way. Now fix U = R1 and consider the universes V U for V 2 I . Identify V with V R V U. In the language of [1], we define (3.2) N*(V ) = S ^L (I (V U; U) n VVU (S0)): To make sense of this, recall that we have the linear isometries operad L wit* *h nth space L (n) = I (Un ; U). The operad structure maps are given by compositions and direct sums of linear isometries, and they specialize to give a monoid stru* *cture on L (1), a left action of L (1) on L (2), and a right action of L (1) x L (1) * *on L (2). An L-spectrum is a spectrum with an action by L (1); see Definition 4.4 * *and ORTHOGONAL SPECTRA AND S-MODULES 13 Remark 4.5. By [1, Ix5], we have an "operadic smash product" (3.3) E ^L E0= L (2) nL (1)xL (1)E Z E0 between L-spectra E and E0, where EZE0is the external smash product indexed on U2 [1, Ix2]. The sphere S is an L-spectrum, and the action of L (1) by composit* *ion on I (V U; U) induces an action of L (1) on I (V U; U) n VVU (S0). An L-spectrum E has a unit map : S ^L E -! E that is always a weak equivalence and sometimes an isomorphism [1, I.x8 and IIx1]. An S-module is an L-spectrum E such that is an isomorphism. In particular, is an isomorphism when E = S, when E = S^L E0for any L-spectrum E0, and when E is the operadic smash product of two S-modules [1, I.8.2, II.1.2]. The smash product ^S is the restriction to S-modules of ^L . The functor J specified by (3.4) J(E) = S ^L E carries L-spectra to naturally weakly equivalent S-modules, and we have (3.5) N*(V ) = J(I (V U; U) n VVU (S0)): This makes sense of (3.2). It even makes sense when V = {0}. Here we interpret spectra indexed on the universe {0}as based spaces. The space I ({0}; U) is a p* *oint, namely the inclusion iU : {0}- ! U. The functor iU*= iU n (-) : T - ! S U is left adjoint to the zeroth space functor, hence iU*= U . Thus (3.5) specializes* * to give N*(0) = JS and, as we have noted, : JS -! S is an isomorphism. The evident homeomorphisms 0-V W0-W (V 0-V )(W0-W) V A ^ B ~= (A ^ B) for V V 0in V U and W W 0in W U, induce an isomorphism (3.6) VVU (A) Z WUW (B) ~=(VVW)UW (A ^ B) upon spectrification, where Z : S V U x S WU -! S (V W)U is the external smash product. Together with the formal properties [1, A.6.2 and A.6.3] of twisted half-smash products, the canonical homeomorphism L (2) xL (1)xL (1)I (V U; U) x I (W U; U) ~=I ((V W ) U; U) given by Hopkins' lemma [1, I.5.4], and the associative and unital properties o* *f ^L of [1, Ixx5,8], the isomorphisms (3.6) induce isomorphisms (3.7) OE : N*(V ) ^S N*(W ) -! N*(V W ): To see the motivation for the definition (3.5), identify Rn U with Un . For a spectrum E 2 S , L (1) n E is denoted LE and is the free L-spectrum generated by E. The functor F alluded to in Lemma 2.8 is specified by F E = JLE, and F E is weakly equivalent to E under mild hypotheses [1, I.4.6 and I.8.5]. In t* *he model structure on S-modules, the cofibrant S-modules are the retracts of F I- cell modules, where F I is the collection of maps F 1nSq -! F 1nCSq, n 0 and q 0. In particular, the canonical cofibrant sphere modules are defined by SS = F S and S-nS= F 1nS0 for n > 0. Thus, as used in the previous section, N*(R) = S-1Sand, for n 1, N*(Rn) ~=(S-1S)(n)~=S-nS= F 1nS0 14 M. A. MANDELL AND J. P. MAY (where the middle isomorphism is only canonical up to homotopy). If dimV = n, n > 0, then N*(V ) is isomorphic to N*(Rn) and is therefore cofibrant. Intuitiv* *ely, (3.5) gives a coordinate-free generalization of the definition of cofibrant neg* *ative sphere S-modules. We must still prove the contravariant functoriality in V of N*(V ), check the naturality of OE, and prove that the evaluation maps " : N*(V ) ^ SV -! N*(0) a* *re weak equivalences. While this can be done directly in terms of the definitions * *on hand, it is more illuminating to review the definition of the half-smash product and relate it directly to the morphism spaces of the category IS. We introduce a category of Thom spaces for this purpose. Definition 3.8.Let U and U0 be finite or countably infinite dimensional real in* *ner product spaces. Let0V and V 0be finite dimensional sub inner product spaces of U and U0. Let IVU;U;Vb0e the space of linear isometries f : U -! U0 such that f(V ) V00. For V W , let W - V denote the orthogonal complement0of V in W . Let EU;UV;Vb0e the subbundle of the product bundle IVU;U;Vx0V 0whose points are 0 U;* *U0 the pairs (f; x) such that x 2 V 0- f(V ). Let TVU;U;Vb0e the Thom space of EV;* *V 0; it is obtained by applying fiberwise0one-point compactification and identifying al* *l of the points at 1. The spaces TVU;U;Va0re the morphism spaces of a based topologi* *cal Thom category whose objects are the pairs (U; V ). Composition 0;U00 U;U0 U;U00 (3.9) O : TVU0;V^00TV;V 0-! TV;V 00 is defined0by (g; y)O(f; x)0= (gOf; g(x)+y). Points (idU; 0) give identity morp* *hisms. If IVU;U;Vi0s empty, TVU;U;Vi0s a point. For any U and any pair (U0; V 0), 0 0 V 0 (3.10) T0U;U;V=0I (U; U )+ ^ S : The category is symmetric monoidal with respect to direct sums of inner product spaces. On morphism spaces, the map 0 U2;U0 U1U2;U0U0 (3.11) : TVU1;U11;V10^ TV2;V220-! TV1V2;V1102V20 sends ((f1; x1); (f2; x2)) to (f1 f2; x1 + x2). Note that we have a trivializ* *ation isomorphism of bundles 0 U;U0 0 EU;UV;Vx0V ~=IV;V 0x V and thus an "untwisting isomorphism" 0 V U;U0 V 0 (3.12) TVU;U;V^0S ~=IV;V 0+^ S : The theory of orthogonal spectra is based on the full sub-category of whose objects are the pairs (V; V ). Here, if V V 0, then it is easily verified that 0 0 V 0-V TVV;V;V~0=O(V )+ ^O(V 0-VS) : Comparing with the definitions in [5, 6.1, 8.4], we obtain the following result. Proposition 3.13.The full subcategory of whose objects are the pairs (V; V ) is isomorphic as a based symmetric monoidal category to the category IS such that an orthogonal spectrum is a continuous based functor IS -! T . ORTHOGONAL SPECTRA AND S-MODULES 15 We regard this isomorphism of categories as an identification. In contrast, the twisted half-smash product is defined in terms of the full s* *ub category of whose objects are the pairs (U; V ) in which U is infinite dimensi* *onal. The following definition and lemma are taken from [1, A.4.1-A.4.3]. 0 Definition 3.14.Fix (U; V ) and U0. Define a prespectrum TVU;U;-indexed on U0 0 by letting its V 0th space be TVU;U;Va0nd letting its structure map for V 0 W 0* *be induced by passage to Thom spaces from the evident bundle map 0 0 0 U;U0 U;U0 EU;UV;V 0(W - V ) ~=EV;W0|I U;U0-! EV;W0: V;V 0 0 U;U0 For V W , define a map o : W-V TWU;U;--! TV;- of prespectra indexed on U0 by letting its V 0th map be induced by passage to Thom spaces from the evident bundle map 0 U;U0 U;U0 EU;UW;V 0(W - V ) ~=EV;V 0|I U;U0-! EV;V 0: W;V 0 0 U;U0 U;U0 Observe that TVU;U;-is an inclusion prespectrum and define MV;- = LTV;- . (That is, write M consistently for Thom spectra associated to Thom prespectra T .) Lemma 3.15. The spectrified map 0 W-V U;U0 U;U0 U;U0 Lo : W-V MU;UW;-~=L( TW;- ) -! LTV;- = MV;- is an isomorphism of spectra indexed on U0. The following is a special case of the definition of the twisted half smash p* *roduct given in [1, A.5.1]. Definition 3.16.Let E be a spectrum indexed on U. Define 0 I (U; U0) n E = colimVMU;UV;-^ EV where the colimit (in S U0) is taken over the maps 0 W-V U;U0 U;U0 W-V U;U0 MU;UV;-^ EV ~= MW;- ^ EV ~=MW;- ^ EV -! MW;- ^ EW induced by the structure maps of E. The following result of Cole [1, A.3.9] is pivotal. Proposition 3.17.For based spaces A, there is a natural isomorphism 0 I (U; U0) n UVA ~=MU;UV;-^ A of spectra indexed on U0. The proof is simply the observation that, in this case, the defining colimit * *stabi- lizes at the V th stage. Returning to the fixed choice of U = R1 and taking A =* * S0, this gives the alternative description (3.18) N*(V ) ~=JMVVU;U;-: We regard this isomorphism as an identification and use it to show the required functoriality of the N*(V ). 16 M. A. MANDELL AND J. P. MAY Definition 3.19.Tensoring linear isometries V -! W with idU, we obtain a map TVV;W;W-! TVV;U;WUW . Define the evaluation maps N*(W ) ^ IS(W; V ) -! N*(V ) of the contravariant functor N* to be the maps JMWU;UW;-^ TVV;W;Wid^-! JMWU;UW;-^ TVV;U;WUW ~= JL(TWWU;U;-^ TVV;U;WUW ) JL(O)-!JL(T V U;U V U;U V;- ) = JMV;- induced by composition in the category . The naturality of OE is now checked by rewriting its definition in terms of T* *hom complexes, using specializations of (3.11). Finally, we have the following lemm* *a. Lemma 3.20. The evaluation map " : N*(V ) ^ SV - ! N*(0) ~=S of the functor N* is a weak equivalence. When V = R, " factors as the composite of the canonic* *al isomorphism N*(R)^S1 ~=SS and the canonical cofibrant approximation SS -! S. Proof.Using the untwisting isomorphisms 0 TVV;U;UV^0SV ~=IVV;U;UV^0SV and applying L, we obtain an isomorphism of L-spectra MVVU;U;-^ SV ~=I (V U; U)+ ^ S: Applying J and using JS ~=S, we find by (3.18) that (3.21) N*(V ) ^ SV ~=J((I (V U; U)+ ^ S) ~=I (V U; U)+ ^ S: Under this isomorphism, the evaluation map corresponds to the homotopy equiva- lence induced by the evident homotopy equivalence I (V U; U)+ -! S0. When V = R, LS ~=L (1)+ ^ S and the isomorphism just given is the cited canonical_ isomorphism N*(R) ^ S1 ~=SS. |__| 4. The functor M and its comparison with N We begin with the composite pair of functors (4.1) I S _P___//P__L_//_S : Here P is the underlying prespectrum functor (denoted U in [5, 6]) that we have already used implicitly in defining the homotopy groups and weak equivalences of orthogonal spectra and L is the spectrification functor of [4, I.2.2]. The func* *tor M is the composite of three functors: (4.2) I S _P__//_P[L]_L_//_S [L]J__//M : The categories P[L] and S [L] are the categories of L-prespectra and L-spectra.* * We have already indicated what L-spectra are, and we shall define L-prespectra sho* *rtly. The functors P and L in (4.2) are restrictions of those of (4.1), and the funct* *or J is specified in (3.4). Thus, to construct the functor M, it remains to define P* *[L] and to show that the functors P and L induce functors from orthogonal spectra to L-prespectra and from L-prespectra to L-spectra. The arguments are already implicit in [8]. ORTHOGONAL SPECTRA AND S-MODULES 17 Definition 4.3.For a prespectrum X and a linear isometry f : U -! U, define a prespectrum f*X by (f*X)(V ) = X(fV ), with structure maps f oe X(fV ) ^ SW-V id^S//_X(fV ) ^ Sf(W-V_)__//X(fW ): Observe that f*X is a spectrum if X is a spectrum. Definition 4.4.An L-prespectrum is a prespectrum X together with maps (f) : X -! f*X of prespectra for all linear isometries f : U -! U such that (id) = id, (f0) O (f) = (f0O f), and the function : TVU;U;W^ X(V ) -! X(W ) specified by ((f; w); x)) = oe((f)(x); w) is a continuous map. An L-spectrum is an L-prespectrum that is a spectrum. Let P[L] and S [L] denote the categories of L-prespectra and L-spectra. Remark 4.5.We have a monad L in S specified by L(E) = L (1)nE. In [1, I.4.2], an L-spectrum was defined to be an algebra over this monad. By [9, XXII.5.3], t* *hat notion of an L-spectrum coincides with the notion that we have just defined. Wh* *ile we have nothing to add to this equivalence of definitions, we emphasize that it* * is central to the mathematics: it converts structures that are defined one isometr* *y at a time into structures that are defined globally in terms of spaces of isometri* *es. Lemma 4.6. The functor L : P -! S induces a functor P[L] -! S [L]. Proof.For a linear isometry f : U - ! U, the functor f* : P - ! P and its restriction f* : S -! S have left adjoints f*. The functor f* on spectra is def* *ined in terms of the functor f* on prespectra by f* = Lf*` [4, IIx1]. Let X be an L- prespectrum. The map (f) has an adjoint map f*X -! X; applying L, we obtain a map f*LX -! LX, and its adjoint gives an induced map (f) : LX -! f*LX. The properties (id) = id and (f0 O f) = (f0) O (f) are inherited from their prespectrum level analogues. Since the functor L is continuous and commutes with smash products with spaces, the continuity and equivariance condition on in__ Definition 4.4 are also inherited by LX. |__| Lemma 4.7. The functor P : I S -! P takes values in P[L]. Proof.We obtain (f) : X -! f*X by applying the functoriality of X and the naturality of oe to the restrictions of linear isometries f : U -! U to linear * *isometric isomorphisms f : V -! f(V ) for indexing spaces V . It is clear by functorialit* *y that (id) = idand (f0Of) = (f0)O(f). The continuity and equivariance condition on * * __ in Definition 4.4 follow from the continuity, naturality and equivariance of o* *e. |__| Remark 4.8.For general L-prespectra, the map (f) : X(V ) -! X(fV ) may de- pend on the linear isometry f : U -! U, not just on its restriction V -! f(V ).* * For those L-prespectra that come from orthogonal spectra, this map does depend sole* *ly on the restriction of f. For this reason, there is no obvious functor P[L] -! I* * S . The following lemmas give the basic formal properties of the functor M. Lemma 4.9. The functor M is right exact. 18 M. A. MANDELL AND J. P. MAY Proof.The functors P , L, and J are each right exact. This is obvious for P from the spacewise specification of colimits and smash products with based spaces, a* *nd__ it holds for L and J since these functors are left adjoints. * * |__| Lemma 4.10. There is a canonical isomorphism : M(SI ) -! S. Proof.Clearly P (SI ) is the usual sphere prespectrum and thus S = LP (SI )._As_ we have already used, JS ~=S by [1, I.8.2]. |__| Lemma 4.11. The functor M is lax symmetric monoidal. Proof.We have MSI ~=S, and we must construct a natural map OE : M(X) ^S M(X0) -! M(X ^I X0) for orthogonal spectra X and X0. The functor J commutes with smash products, in the sense that (JE) ^S (JE0) ~=J(E ^L E0) for L-spectra E and E0. Thus it suffices to construct a map of L-spectra OE : LP (X) ^L LP (X0) -! LP (X ^I X0); and OE is obtained by passage to coequalizers from a map : L (2) n LP (X) Z LP (X0) -! LP (X ^I X0): To construct , it suffices to construct maps (f) : LP (X)(V ) ^ LP (X0)(V 0) -! LP (X ^I X0)(f(V V 0)) for linear isometries f 2 L (2) such that the (f) satisfy analogs of the condit* *ions in Definition 4.4 [9, XXII.5.3]. The functoriality of X and X0 gives maps X(V ) ^ X0(V 0) -! X(f(V )) ^ X0(f(V 0)): The universal property (1.13) that relates the external and internal smash prod* *uct of orthogonal spectra gives a map X Z X0- ! (X ^I X0) O of (IS x IS)-spaces, and this gives maps X(f(V )) ^ X0(f(V 0)) -! (X ^I X0)(f(V V 0)): We obtain the required maps (f) from the composites X(V ) ^ X0(V 0) -! (X ^I X0)(f(V V 0)) by passing to prespectra and then to spectra, as in the proof of Lemma 4.6. The coherence properties of the maps OE obtained from these maps are shown by form* *al_ verifications from the properties of the various smash products. * *|__| Turning to homotopical properties, we have the following observation. Lemma 4.12. If X is a positive inclusion orthogonal spectrum, then there are natural isomorphisms ss*(X) ~=ss*(M(X)): Proof.We have a natural weak equivalence : M(X) = JLP (X) -! LP (X) for any X. When P X is a positive inclusion prespectrum, the natural map of_ prespectra P X -! LP (X) is a ss*-isomorphism by Lemma 3.1. |__| Now the following theorem compares M and N. ORTHOGONAL SPECTRA AND S-MODULES 19 Theorem 4.13. There is a symmetric monoidal natural transformation ff : N -! M such that ff : NX -! MX is a weak equivalence if X is cofibrant. Proof.Recall the definition M* = M O DIS : IS -! M (see Definition 1.3 and Notation 1.6). By Corollary 1.7, to construct ff, it suffices to construct a na* *tural transformation ff* : N* -! M*. Thus consider the orthogonal spectra V *specified by V *(W ) = IS(V; W ). By definition, M*V = MV *= JLP V *. By Proposition 3.13, for W U, P V *(W ) ~=TVV;W;W: For V W Z, the structural map agrees under this isomorphism with : TVV;W;W^ SZ-W ~=TVV;W;W^ T00;Z-W;Z-W-! TVV;Z;Z: We obtain a map of Thom spaces TVV;U;UW-! TVV;W;Wby restricting isometries f : V U -! U such that f(V ) W to V . These maps define a map of prespectra TVV;U;U--! P V *. Applying JL and using (3.18), there results a map of S-modules ff* : N*(V ) = JLTVV;U;U--! JLP V *= M*(V ): It is an exercise to verify from Proposition 3.13 and the definitions that thes* *e maps specify a natural transformation that is compatible with smash products. Using Theorem 1.5, define ff = ff* IS id: NX = N* IS X -! M* IS X ~=MX: Then ff is a symmetric monoidal natural transformation, and it remains to prove that ff : NX -! MX is a weak equivalence if X is cofibrant. It suffices to assu* *me that X is an F I-cell complex (see [6, xx1,6]). Since M and N are right exact,* * it follows by the usual induction up the cellular filtration of X, using commutati* *ons with suspension, wedges, pushouts, and colimits, that it suffices to prove that* * ff is a weak equivalence when X = V *. In this case, ff reduces to ff*. Again by suspension, it suffices to prove that V ff* : V N*(V ) -! V M*(V ) is a weak equivalence. We have an untwisting isomorphism (3.21) for the source * *of V ff* and an analogous isomorphism M(V *) ^ SV ~=I (V; U)+ ^ S for its target. Under these isomorphisms, V ff* is the smash product with S of * *the map I (V U; U) -! I (V; U) induced by restriction of linear isometries, and th* *is__ map is a homotopy equivalence since its source and target are contractible. * * |__| Remark 4.14.By Proposition 1.8, the functor M has right adjoint M# . However, M does not appear to preserve cofibrant objects and does not appear to be part * *of a Quillen equivalence. 20 M. A. MANDELL AND J. P. MAY References [1]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by* * M. Cole). Rings, modules, and algebras in stable homotopy theory. Surveys and Monographs in M* *athematics Vol. 47. 1997. American Mathematical Society. [2]J. P. C. Greenlees and J. P. May. Localization and completion theorems for * *MU-module spectra. Annals of Math. 146(1997), 509-544. [3]M. Hovey, B. Shipley, and J. Smith. Symmetric spectra. Preprint. 1998. [4]L. G. Lewis, Jr., J. P. May, and M. Steinberger (with contributions by J. E* *. McClure). Equivariant stable homotopy theory. Springer Lecture Notes in Mathematics Vo* *l. 1213. 1986. [5]M. A. Mandell, J. P. May, S. Schwede, and B. Shipley. Diagram spaces, diagr* *am spectra, and FSP's. Preprint. 1998. [6]M. A. Mandell, J. P. May, S. Schwede, and B. Shipley. Model categories of d* *iagram spectra. Preprint. 1998. [7]J. P. May. Pairings of categories and spectra. J. Pure and Applied Algebra.* * 19(1980), 299-346. [8]J. P. May (with contributions by F. 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