A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY STEFAN SCHWEDE AND BROOKE SHIPLEY 1. Introduction Roughly speaking, the stable homotopy category is obtained from the homotopy * *category of topological spaces by inverting the suspension functor, yielding a `linear' * *approximation to the homotopy category of spaces. The isomorphism classes of objects in the stab* *le homotopy category represent the generalized cohomology theories, defined by the Eilenber* *g-Steenrod ax- ioms [ES ] without the dimension axiom (which distinguishes `ordinary' from `ge* *neralized' coho- mology theories). The first construction of the full stable homotopy category was given by Boar* *dman [Bo]. More recently many models for the stable homotopy category have been found, mos* *t of which have the additional structure of a closed model category in the sense of Quille* *n [Q ]. In [Mar ], H. R. Margolis introduced a short list of axioms and conjectured that they char* *acterize the stable homotopy category up to an equivalence of categories. In Theorem 3.2 we* * prove that these axioms indeed uniquely specify the stable homotopy category whenever ther* *e is some underlying Quillen model category. We also prove a more structured version, the Uniqueness Theorem below, which * *states that the model category of spectra itself is uniquely determined by certain equivale* *nt conditions, up to so called Quillen equivalence of model categories (a particular adjoint p* *air of functors which induces equivalences of homotopy categories, see Definition 2.4). This im* *plies that the higher order structure including cofibration and fibration sequences, homotopy * *(co-)limits, Toda brackets, and mapping spaces is uniquely determined by these conditions. This * *is of interest due to the recent plethora of new model categories of spectra [EKMM , HSS, MMS* *S , Ly]. The Uniqueness Theorem provides criteria on the homotopy category level for decidin* *g whether a model category captures the stable homotopy theory of spectra; the search for s* *uch intrinsic characterizations was another main motivation for this project. A model category is stable if the suspension functor is invertible up to homo* *topy. For stable model categories the homotopy category is naturally triangulated and comes with* * an action by the graded ring sss*of stable homotopy groups of spheres, see 2.3. The Uniquene* *ss Theorem shows that this sss*-triangulation determines the stable homotopy theory up to Quille* *n equivalence. Uniqueness Theorem. Let C be a stable model category. Then the following four c* *onditions are equivalent: (1)There is a chain of Quillen equivalences between C and the model category o* *f spectra. (2)There exists a sss*-linear equivalence between the homotopy category of C a* *nd the homotopy category of spectra. (3)The homotopy category of C has a small weak generator X for which [X; X]Ho(* *C)*is freely generated as a sss*-module by the identity map of X. (4)The model category C has a cofibrant-fibrant small weak generator X for whi* *ch the unit map S -! Hom (X; X) is a ss*-isomorphism of spectra. ____________ Date: April 26, 2000; 1991 AMS Math. Subj. Class.: 55U35, 55P42. Research supported by a BASF-Forschungsstipendium der Studienstiftung des deu* *tschen Volkes. Research partially supported by NSF grants. 1 2 STEFAN SCHWEDE AND BROOKE SHIPLEY The Uniqueness Theorem is proved in a slightly more general form as Theorem 5* *.3. The extra generality consists of a local version at subrings of the ring of rational numb* *ers. Our reference model for the category of spectra is that of Bousfield and Friedlander [BF , De* *f. 2.1]; this is probably the simplest model category of spectra and we review it in Section 4. * *There we also discuss the R-local model structure for spectra for a subring R of the ring of * *rational numbers, see Lemma 4.1. The notions of `smallness' and `weak generator' are recalled in * *3.1. The unit map is defined in 5.2. Our work here grows out of recent developments in axiomatic stable homotopy t* *heory. Mar- golis' axiomatic approach was generalized in [HPS ] to study categories which s* *hare the main formal properties of the stable homotopy category, namely triangulated symmetri* *c monoidal categories with a weak generator or a set of weak generators. Hovey [Ho , Ch. 7* *] then studied properties of model categories whose homotopy categories satisfied these axioms* *. Heller has given an axiomatization of the concept of a "homotopy theory" [He1], and then c* *haracterized the passage to spectra by a universal property in his context, see [He2, Sec. 8* *-10]. The reader may want to compare this with the universal property of the model category of s* *pectra which we prove in Theorem 5.1 below. In [SS] we classify stable model categories with a small weak generator as mo* *dules over a ring spectrum, see Remark 5.4. The equivalence of parts (1) and (4) of our Uniquenes* *s Theorem here can be seen as a special case of this classification. In general, though, there* * are no analogues of parts (2) and (3) of the Uniqueness Theorem. Note, that here, as in [SS], w* *e ignore the smash product in the stable homotopy category; several comparisons and classifi* *cation results respecting smash products can be found in [Sch2, MMSS ]. 2.Stable model categories Recall from [Q , I.2] or [Ho , 6.1.1] that the homotopy category of a pointed* * model category supports a suspension functor with a right adjoint loop functor . Definition 2.1.A stable model category is a pointed, complete and cocomplete ca* *tegory with a model category structure for which the functors and on the homotopy category * *are inverse equivalences. The homotopy category of a stable model category has a large amount of extra * *structure, some of which will play a role in this paper. First of all, it is naturally a triang* *ulated category (cf. [Ve]). A complete reference for this fact can be found in [Ho , 7.1.6]; we sket* *ch the constructions: by definition the suspension functor is a self-equivalence of the homotopy cate* *gory and it defines the shift functor. Since every object is a two-fold suspension, hence an abelia* *n co-group object, the homotopy category of a stable model category is additive. Furthermore, by [* *Ho , 7.1.11] the cofiber sequences and fiber sequences of [Q , I.3] coincide up to sign in the s* *table case, and they define the distinguished triangles. Since we required a stable model category t* *o have all limits and colimits, its homotopy category will have infinite sums and products. Apart from being triangulated, the homotopy category of a stable model catego* *ry has a natural action of the ring sss*of stable homotopy groups of spheres. Since this action * *is central to this paper, we formalize and discuss it in some detail. For definiteness we set sssn* *= colimk[Sn+k; Sk], where the colimit is formed along right suspension - ^ 1S1 : [Sn+k; Sk] -! [Sn+* *k+1; Sk+1]. The ring structure is given by composition of representatives. Definition 2.2.A sss*-triangulated category is a triangulated category T with b* *ilinear pairings sssn T (X; Y ) ----! T (X[n]; Y ) ; ff f 7-! ff . f for all X and Y in T and all n 0, where X[n] is the n-fold shift of X. Furth* *ermore the pairing must be associative, unital and composition in T must be bilinear in t* *he sense that A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY 3 (ff . g) O f[n] = ff_. (g O f) = g O (ff . f). A sss*-exact functor between sss* **-triangulated categories is a functor L : T -! T together with a natural isomorphism o : L(X)[1] ~=L(X[1]) * *such that o (L; o) forms an exact functor of triangulated categories, i.e., for every d* *istinguished triangle X -! Y -! Z -! X[1] in T the sequence L(X) -! L(Y ) -! L(Z) -! L(X)[1] is a __ * * o-1 distinguished triangle in T, where the third map is the composite L(Z) -! L* *(X[1]) -! L(X)[1]; o (L; o) is sss*-linear, i.e., for all X and Y in T and n 0 the following di* *agram commutes sssn T (X; Y ) ______________________________Tw(X[n];.Y ) | | Id L | |L | | __ |u __ __ |u sssn T(L(X); L(Y ))____Tw(L(X)[n];.L(Y ))u___T_(L(X[n]);_L(YO))o where o : L(X)[n] -! L(X[n]) is the n-fold iterate of instances of the isom* *orphism o. A sss*-linear equivalence between sss*-triangulated categories is a sss*-exact * *functor which is an equivalence of categories and whose inverse is also sss*-exact. Every triangulated category can be made into a sss*-triangulated category in * *a trivial way by setting ff . f = 0 whenever the dimension of ff is positive. Now we explain how* * the homotopy category of a stable model category is naturally a sss*-triangulated category. Construction 2.3.Using the technique of framings, Hovey [Ho , 5.7.3] constructs* * a pairing ^L : Ho(C) x Ho(S*) ----! Ho(C) which makes the homotopy category of a pointed model category C into a module (* *in the sense of [Ho , 4.1.6]) over the symmetric monoidal homotopy category of pointed simpl* *icial sets under smash product. In particular, the pairing is associative and unital up to coher* *ent natural iso- morphism, and smashing with the simplicial circle S1 is naturally isomorphic to* * suspension as defined by Quillen [Q , I.2]. If C is stable, we may take X[1] := X ^L S1 as th* *e shift functor of the triangulated structure. We define the action sssn [X; Y ]Ho(C) ------! [X[n]; Y ]Ho(C) as follows. Suppose ff : Sn+k -! Sk is a morphism in the homotopy category of * *pointed simplicial sets which represents an element of sssn= colimk[Sn+k; Sk] and f : X* * -! Y is a morphism in the homotopy category of C. Since C is stable, smashing with Sk is * *a bijection of morphism groups in the homotopy category. So we can define ff.f to be the uniqu* *e morphism in [X ^L Sn; Y ]Ho(C)such that (ff.f)^L 1Sk = f ^L ff in the group [X ^L Sn+k; Y ^* *L Sk]Ho(C). Here and in the following we identify the n-fold shift X[n] = (. .(.(X ^L S1) ^L S1)* * . .).^L S1 with X ^L Sn under the associativity isomorphism which is constructed in the proof o* *f [Ho , 5.5.3] (or rather its pointed analog [Ho , 5.7.3]); this way we regard ff . f as an el* *ement of the group [X[n]; Y ]Ho(C). Observe that sss*acts from the left even though simplicial set* *s act from the right on the homotopy category of C. By construction ff . f = (ff ^ 1S1) . f, so the morphism ff . f only depends * *on the class of ff in the stable homotopy group sssn. The sss*-action is unital; associativity can be* * seen as follows: if fi 2 [Sm+n+k ; Sn+k] represents another stable homotopy element, then we have (fi . f) ^L ff=(1Y ^L ff) O ((fi . f) ^L 1Sn+k) = (1Y ^L ff) O (f ^L* * fi) = f ^L (ff O fi) = ((ff O fi) . f) ^L 1Sk 4 STEFAN SCHWEDE AND BROOKE SHIPLEY in the group [X ^L Sm+n+k ; Y ^L Sk]Ho(C). According to the definition of ff . * *(fi . f) this means that ff . (fi . f) = (ff O fi) . f. Bilinearity of the action is proved in a si* *milar way. Definition 2.4.A pair of adjoint functors between model categories is a Quillen* * adjoint pair if the right adjoint preserves fibrations and trivial fibrations. An equivalent* * condition is that the left adjoint preserves cofibrations and trivial cofibrations. A Quillen ad* *joint pair induces an adjoint pair of functors between the homotopy categories [Q , I.4 Thm. 3], t* *he total derived functors. A Quillen functor pair is a Quillen equivalence if the total derived * *functors are adjoint equivalences of the homotopy categories. The definition of Quillen equivalences just given is not the most common one;* * however it is equivalent to the usual definition by [Ho , 1.3.13]. Suppose F : C -! D is * *the left adjoint of a Quillen adjoint pair between pointed model categories. Then the total left* * derived functor LF : Ho(C)-! Ho(D) of F comes with a natural isomorphism o : LF (X)^LS1 -! LF (* *X^LS1) with respect to which it preserves cofibration sequences, see [Q , I.4 Prop. 2]* * or [Ho , 6.4.1]. If C and D are stable, this makes LF into an exact functor with respect to o. It s* *hould not be surprising that (LF; o) is also sss*-linear in the sense of Definition 2.2, but* * showing this requires a careful review of the definitions which we carry out in Lemma 6.1. Remark 2.5. In Theorem 5.3 below we show that the sss*-triangulated homotopy ca* *tegory de- termines the Quillen equivalence type of the model category of spectra. This i* *s not true for general stable model categories. As an example we consider the n-th Morava K-th* *eory spec- trum K(n) for n > 0 and some fixed prime p. This spectrum admits the structure * *of an A1 -ring spectrum [Ro ], and so its module spectra form a stable model category. The co* *efficient ring K(n)* = Fp[vn; v-1n], with vn of degree 2pn - 2, is a graded field, and so the * *homotopy cate- gory of K(n)-modules is equivalent, via the homotopy group functor, to the cate* *gory of graded K(n)*-modules. Similarly the derived category of differential graded K(n)*-modu* *les is equiva- lent, via the homology functor, to the category of graded K(n)*-modules. This d* *erived category comes from a stable model category structure on differential graded K(n)*-modul* *es with weak equivalences the quasi-isomorphisms. The positive dimensional elements of sss*a* *ct trivially on the homotopy categories in both cases. So the homotopy categories of the model* * categories of K(n)-modules and differential graded K(n)*-modules are sss*-linearly equival* *ent. However, the two model categories are not Quillen equivalent; if they were Quillen equiv* *alent, then the homotopy types of the function spaces would agree [DK , Prop. 5.4]. But all fun* *ction spaces of DG-modules are products of Eilenberg-MacLane spaces, and this is not true for K* *(n)-modules. 3.Margolis' uniqueness conjecture H. R. Margolis in `Spectra and the Steenrod algebra' introduced a set of axio* *ms for a stable homotopy category [Mar , Ch. 2 x1]. The stable homotopy category of spectra sat* *isfies the axioms, and Margolis conjectures [Mar , Ch. 2, x1] that this is the only model, i.e., t* *hat any category which satisfies the axioms is equivalent to the stable homotopy category. As pa* *rt of the structure Margolis requires the subcategory of small objects of a stable homotopy categor* *y to be equivalent to the Spanier-Whitehead category of finite CW-complexes. So his uniqueness que* *stion really concerns possible `completions' of the category of finite spectra to a triangul* *ated category with infinite coproducts. Margolis shows [Mar , Ch. 5 Thm. 19] that modulo phantom * *maps each model of his axioms is equivalent to the standard model. Moreover, in [CS ], Ch* *ristensen and Strickland show that in any model the ideal of phantoms is equivalent to the ph* *antoms in the standard model. A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY 5 For objects A and X of a triangulated category T we denote by T (A; X)* the g* *raded abelian homomorphism group defined by T (A; X)m = T (A[m]; X) for m 2 Z, where A[m] is * *the m- fold shift of A. If T is a sss*-triangulated category, then the groups T (A; * *X)* form a graded sss*-module. Definition 3.1.An object G of a triangulated category T is called a weak genera* *tor if it detects isomorphisms, i.e., a map f : X -! Y is an isomorphism if and only if it induce* *s an isomorphism between the graded abelian homomorphism groups T (G; X)* and T (G; Y )*. An obj* *ect G of T is small if for any family of objects {Ai}i2Iwhose coproduct exists the canonic* *al map M a T (G; Ai) ----! T (G; Ai) i2I i2I is an isomorphism. A stable homotopy category in the sense of [Mar , Ch. 2 x1] is a triangulated* * category S endowed with a symmetric monoidal, bi-exact smash product ^ such that: o S has infinite coproducts, o the unit of the smash product is a small weak generator, and o there exists an exact and strong symmetric monoidal equivalence R : SW f -!* * S small between the Spanier-Whitehead category of finite CW-complexes ([SW ], [Mar * *, Ch. 1, x2]) and the full subcategory of small objects in S . The condition that R is strong monoidal means that there are coherently unita* *l, associative, and commutative isomorphisms between R(A ^ B) and R(A) ^ R(B) and between R(S0)* * and the unit of the smash product in S . Hence a stable homotopy category S become* *s a sss*- triangulated category as follows. The elements of sssnare precisely the maps fr* *om Sn to S0 in the Spanier-Whitehead category. So given ff 2 sssn= SW (Sn; S0) and f : X -! Y in S* * we can form f ^ R(ff) : X ^ R(Sn) -! Y ^ R(S0). Via the isomorphisms X ^ R(Sn) ~=X[n] ^ R(S* *0) ~=X[n] and Y ^ R(S0) ~=Y we obtain an element in S (X[n]; Y ) which we define to be ff* * . f. This sss*-action is unital, associative and bilinear because of the coherence condit* *ions on the functor R. As a consequence of our main theorem we can prove a special case of Margolis'* * conjecture, namely we can show that a category satisfying his axioms is equivalent to the h* *omotopy category of spectra if it has some underlying model category structure. Note that we do * *not ask for any kind of internal smash product on the model category which occurs in the follow* *ing theorem. Theorem 3.2. Suppose that S is a stable homotopy category in the sense of [Mar * *, Ch. 2 x1] which supports a sss*-linear equivalence with the homotopy category of some sta* *ble model category. Then S is equivalent to the stable homotopy category of spectra. Proof.Let C be a stable model category which admits a sss*-linear equivalence * *: S -! Ho(C). The image X 2 Ho(C)under of the unit object of the smash product is a small we* *ak generator for the homotopy category of C. Because the equivalence is sss*-linear, X sati* *sfies condition (3) of our main theorem, and so C is Quillen equivalent to the model category of sp* *ectra. Thus the homotopy category of C and the category S are sss*-linearly equivalent to the o* *rdinary_stable homotopy category of spectra. * * |__| 4.The R-local model structure for spectra In this section we review the stable model category structure for spectra def* *ined by Bousfield and Friedlander [BF , x2] and establish the R-local model structure (Lemma 4.1). A spectrum consists of a sequence {Xn}n0 of pointed simplicial sets together* * with maps oen : S1 ^ Xn -! Xn+1. A morphism f : X -! Y of spectra consists of maps of po* *inted 6 STEFAN SCHWEDE AND BROOKE SHIPLEY simplicial sets fn : Xn -! Yn for all n 0 such that fn+1 O oen = oen O (1S1^ f* *n). We denote the category of spectra by Sp. A spectrum X is an -spectrum if for all n the si* *mplicial set Xn is a Kan complex and the adjoint Xn -! Xn+1 of the structure map oen is a weak * *homotopy equivalence. The sphere spectrum S is defined by Sn = Sn = (S1)^n, with structu* *re maps the identity maps. The homotopy groups of a spectrum are defined by ss*X = colimissi+*|Xi| A morphism of spectra is a stable equivalence if it induces an isomorphism of h* *omotopy groups. A map X -! Y of spectra is a cofibration if the map X0 -! Y0 and the maps Xn [S1^Xn-1S1 ^ Yn-1 ----! Yn for n 1 are cofibrations (i.e., injections) of simplicial sets. A map of spect* *ra is a stable fibration if it has the right lifting property (see [Q , I p. 5.1], [DS , 3.12] or [Ho , * *1.1.2]) for the maps which are both cofibrations and stable equivalences. Bousfield and Friedlander show in [BF , Thm. 2.3] that the stable equivalence* *s, cofibrations and stable fibrations form a model category structure for spectra. A variation * *of their model category structure is the R-local model structure for R a subring of the ring o* *f rational numbers. The R-local model category structure is well known, but we were unable to find * *a reference in the literature. A map of spectra is an R-equivalence if it induces an isomorphi* *sm of homotopy groups after tensoring with R and is an R-fibration if it has the right lifting* * property with respect to all maps that are cofibrations and R-equivalences. Lemma 4.1. Let R be a subring of the ring of rational numbers. Then the cofibr* *ations, R- fibrations and R-equivalences make the category of spectra into a model categor* *y, referred to as the R-local model category structure. A spectrum is fibrant in the R-local mode* *l structure if and only if it is an -spectrum with R-local homotopy groups. We use `RLP' to abbreviate `right lifting property'. For one of the factoriza* *tion axioms we need the small object argument (see [Q , II 3.4 Remark] or [DS , 7.12]) relative to * *a set J = Jlv[Jst[JR of maps of spectra which we now define. We denote by [i], @[i] and k[i] respect* *ively the simplicial i-simplex, its boundary and its k-th horn (the union of all (i - 1)-* *dimensional faces except the k-th one). A subscript `+' denotes a disjoint basepoint. We denote* * by FnK the spectrum freely generated by a simplicial set K in dimension n, i.e., (FnK)j = * *Sj-n^ K (where Sm = * for m < 0). Hence FnK is a shift desuspension of the suspension spectrum* * of K. First, Jlvis the set of maps of the form Fn k[i]+ ----! Fn [i]+ for i; n 0 and 0 k i. Then Jstis the set of maps Zjn^ @[i]+ [Fn+jSj^ @[i]+Fn+jSj ^ [i]+ ----! Zjn^ [i]+ for i; j; n 0. Here, Zjn= (Fn+jSj ^ [1]+) [Fn+jSjx1FnS0 is the reduced mapping cylinder of the map : Fn+jSj -! FnS0 which is the iden* *tity in spectrum levels above n + j. Note that is a stable equivalence, but it is not* * a cofibration. Since both source and target of are cofibrant, the inclusion of the source int* *o the mapping cylinder Zjnis a cofibration. It is shown in [Sch1, Lemma A.3] that the stable* * fibrations of spectra are precisely the maps with the RLP with respect to the set Jlv[ Jst. For every natural number k we choose a finite pointed simplicial set Mk which* * has the weak homotopy type of the mod-k Moore space of dimension two. We let JR be the set o* *f maps Fnm Mk ----! Fnm C(Mk) A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY 7 for all m; n 0 and all natural numbers k which are invertible in R, where C(Mk* *) denotes the cone of the Moore space. Now we prove a sequence of claims: (a)A map X -! * has the RLP for the set J = Jlv[Jst[JR if and only if X is an * *-spectrum with R-local homotopy groups. (b)A map which is an R-equivalence and has the RLP for J is also an acyclic fi* *bration in the stable model structure. (c)Every map can be factored as a composite p O i where p has the RLP for J an* *d i is a cofibration and an R-equivalence and is built from maps in J by coproducts,* * pushouts and composition. (d)A map is an R-fibration if and only if it has the RLP for J. (a) The RLP for (Jlv[Jst) means that X is stably fibrant, i.e., an -spectrum.* * For -spectra the lifting property with respect to the map Fnm Mk -! Fnm C(Mk) means precisel* *y that every element in the mod-k homotopy group [Fnm Mk; X]Ho(Sp)~= ss0m map(Mk; Xn) ~= ssm+2-n (X; Z=k) is trivial. Since this holds for all m; n 0 and all k which are invertible in * *R, the map X -! * has the RLP for J if and only if X is an -spectrum with R-local homotopy groups. (b) Suppose f : X -! Y is an R-equivalence and has the RLP for J. Then f is i* *n particular a stable fibration and we denote its fiber by F . There exists a long exact seque* *nce connecting the homotopy groups of F , X and Y . Since f is an R-equivalence, the localized hom* *otopy groups R ss*F of the fiber are trivial. As the base change of the map f, the map F -!* * * also has the RLP for J. By (a), F is an -spectrum whose homotopy groups are R-local. Hen* *ce the homotopy groups of the fiber F are trivial, so the original map f is also a sta* *ble equivalence. (c) Every object occurring as the source of a map in J is a suspension spectr* *um of a finite simplicial set, hence sequentially small in the sense of [Q , II 3.4 Remark] or* * [DS , Def. 7.14]. Thus Quillen's small object argument (see [Q , II 3.4 Remark] or [DS , 7.12]) provid* *es a factorization of a given map as a composite p O i where i is built from maps in J by coproduc* *ts, pushouts and composition, and where p has the RLP for J. Since every map in J is a cofib* *ration, so is i. Cofibrations of spectra give rise to long exact sequences of homotopy groups* *, and homotopy groups of spectra commute with filtered colimits of cofibrations. So to see th* *at i is an R- equivalence it suffices to check that the maps in J are R-equivalences. The ma* *ps in Jlv are levelwise equivalences, the maps in Jstare stable equivalences, hence both are * *R-equivalences. Since the stable homotopy groups of the Moore space Mk are k-power torsion, the* * maps in JR are also R-equivalences. (d) We need to show that a map has the RLP for J if and only if it has the RL* *P for the (strictly bigger) class of maps j which are cofibrations and R-equivalences. Th* *is follows if any such j is a retract of a map built from maps in J by coproducts, pushouts and c* *omposition. We factor j = p O i as in (c). Since j and i are R-equivalences, so is p. Since p * *also has the RLP for J, it is a stable acyclic fibration by (b). So p has the RLP for the cofibr* *ation j, hence j is indeed a retract of i. Proof of Lemma 4.1.We verify the model category axioms as given in [DS , Def. 3* *.3]. The category of spectra has all limits and colimits (MC1), the R-equivalences satis* *fy the 2-out- of-3 property (MC2) and the classes of cofibrations, R-fibrations and R-equival* *ences are each closed under retracts (MC3). By definition the R-fibrations have the RLP for ma* *ps which are both cofibrations and R-equivalences. Furthermore a map which is an R-equivale* *nce and an R-fibration is an acyclic fibration in the stable model structure by claim (b) * *above, so it has the RLP for cofibrations. This proves the lifting properties (MC4). The stable * *model structure 8 STEFAN SCHWEDE AND BROOKE SHIPLEY provides factorizations of maps as cofibrations followed by stable acyclic fibr* *ations. Stable acyclic fibrations are in particular R-equivalences and R-fibrations, so this is also a* * factorization as a cofibration followed by an acyclic fibration in the R-local model structure. Th* *e claims_(c) and (d) provide the other factorization axiom (MC5). * * |__| Lemma 4.2. Let C be a stable model category, G : C -! Spa functor with a left a* *djoint and R a subring of the rational numbers. Then G and its adjoint form a Quillen adjo* *int pair with respect to the R-local model structure if and only if the following three condi* *tions hold: (i)G takes acyclic fibrations to level acyclic fibrations of spectra, (ii)G takes fibrant objects to -spectra with R-local homotopy groups and (iii)G takes fibrations between fibrant objects to level fibrations. Proof.The `only if' part holds since the R-local acyclic fibrations are level a* *cyclic fibrations, the R-fibrant objects are the -spectra with R-local homotopy groups (claim (a) * *above), and R- fibrations are in particular level fibrations. For the converse suppose that G * *satisfies conditions (i) to (iii). We use a criterion of Dugger [Du , A.2]: in order to show that G* * and its adjoint form a Quillen adjoint pair it suffices to show that G preserves acyclic fibrat* *ions and it preserves fibrations between fibrant objects. The R-local acyclic fibrations are precisel* *y the level acyclic fibrations, so G preserves acyclic fibrations by assumption (i). We claim that * *every level fibration f : X -! Y between -spectra with R-local homotopy groups is an R-fibration. Giv* *en this, G preserves fibrations between fibrant objects by assumptions (ii) and (iii). To prove the claim we choose a factorization f = p O i with i : X -! Z a cofi* *bration and R-equivalence and with p : Z -! Y an R-fibration. Since Y is R-fibrant, so is Z* *. Hence i is an R-equivalence between -spectra with R-local homotopy groups, thus a level equiv* *alence. Hence i is an acyclic cofibration in the strict model (or level) model structure for * *spectra of [BF , 2.2], so that the level fibration f has the RLP for i. Hence f is a retract of the R-* *fibration_p, and so it is itself an R-fibration. * * |__| 5.A universal property of the model category of spectra In this section we formulate a universal property which roughly says that the* * category of spectra is the `free stable model category on one object'. The following theor* *em associates to each cofibrant and fibrant object X of a stable model category C a Quillen a* *djoint functor pair such that the left adjoint takes the sphere spectrum to X. Moreover, this * *Quillen pair is essentially uniquely determined by the object X. Theorem 5.3 gives conditions u* *nder which the adjoint pair forms a Quillen equivalence. We prove Theorem 5.1 in the final sec* *tion 6. Theorem 5.1. (Universal property of spectra) Let C be a stable model category a* *nd X a cofibrant and fibrant object of C. (1)There exists a Quillen adjoint functor pair X ^ - : Sp -! C and Hom (X; -) * *: C -! Sp such that the left adjoint X ^ - takes the sphere spectrum, S, to X. (2)If R is a subring of the rational numbers and the endomorphism group [X; X]* *Ho(C)is an R-module, then any adjoint functor pair satisfying (1) is also a Quillen pa* *ir with respect to the R-local stable model structure for spectra. (3)If C is a simplicial model category, then the adjoint functors X ^ - and Ho* *m (X; -) of (1) can be chosen as a simplicial Quillen adjoint functor pair. (4)Any two Quillen functor pairs satisfying (1) are related by a chain of natu* *ral transforma- tions which are weak equivalences on cofibrant or fibrant objects respectiv* *ely. Now we define the unit map and deduce the R-local form of our main uniqueness* * theorem. A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY 9 Definition 5.2.Let X be a cofibrant and fibrant object of a stable model catego* *ry C. Choose a Quillen adjoint pair X ^ - : Sp -! C and Hom (X; -) : C -! Sp as in part (1) * *of Theorem 5.1. The unit map of X is the map of spectra S ----! Hom (X; X) which is adjoint to the isomorphism X ^ S ~=X. By the uniqueness part (4) of Th* *eorem 5.1, the spectrum Hom (X; X) is independent of the choice of Quillen pair up to stab* *le equivalence of spectra under S. Theorem 5.3. Let R be a subring of the ring of rational numbers and let C be a * *stable model category. Then the following four conditions are equivalent: (1)There is a chain of Quillen equivalences between C and the R-local stable m* *odel category of spectra. (2)There exists a sss*-linear equivalence between the homotopy category of C a* *nd the homotopy category of R-local spectra. (3)The homotopy category of C has a small weak generator X for which [X; X]Ho(* *C)*is freely generated as an R sss*-module by the identity map of X. (4)The model category C has a cofibrant-fibrant small weak generator X for whi* *ch the groups [X; X]Ho(C)*are R-modules and the unit map S -! Hom (X; X) induces an isomo* *rphism of homotopy groups after tensoring with R. Furthermore, if X is a cofibrant and fibrant object of C which satisfies condit* *ions (3) or (4), then the functors Hom (X; -) and X ^ - of Theorem 5.1 (1) form a Quillen equivalence* * between C and the R-local model category of spectra. Remark 5.4. In [SS] we associate to every object of a stable model category an * *endomorphism ring spectrum. The spectrum Hom (X; X) given by Theorem 5.1 (1) is stably equiv* *alent to the underlying spectrum of the endomorphism ring spectrum. Moreover, the unit map a* *s defined in 5.2 corresponds to the unit map of ring spectra. So condition (4) of the ab* *ove theorem means that the endomorphism ring spectrum of X is stably equivalent, as a ring * *spectrum, to the R-local sphere ring spectrum. This expresses the equivalence of conditions * *(1) and (4) as a corollary of the more general classification result of [SS] for stable model ca* *tegories with a small weak generator. The special case in this paper, however, has a more direct proo* *f. Proof of Theorem 5.3.Every Quillen equivalence between stable model categories * *induces an equivalence of triangulated homotopy categories. The derived functor of a left * *Quillen functor is also sss*-linear by Lemma 6.1; if it is an equivalence, then the inverse equ* *ivalence is also sss*- linear. So condition (1) implies (2). Now assume (2) and let X be a cofibrant a* *nd fibrant object of Ho(C) which in the homotopy category is isomorphic to the image of the local* *ized sphere spectrum under any sss*-linear equivalence. With this choice, condition (3) hol* *ds. Given condition (3), we may assume that X is cofibrant and fibrant and we cho* *ose a Quillen adjoint pair X ^ - and Hom (X; -) as in part (1) of Theorem 5.1. Since the grou* *p [X; X]Ho(C) is an R-module, the functors form a Quillen pair with respect to the R-local mo* *del structure for spectra by Theorem 5.1 (2). By Lemma 6.1 the map X ^L - : [S; S]Ho(SpR)*----![X; X]Ho(C)* induced by the left derived functor X ^L - and the identification X ^L S ~=X is* * sss*-linear (note that the groups on the left hand side are taken in the R-local homotopy c* *ategory, so that [S[n]; S]Ho(SpR)is isomorphic to R sssn). Source and target of this map are fr* *ee R sss*-modules, and the generator IdS is taken to the generator IdX . Hence the map X ^L - is a* *n isomorphism. For a fixed integer n, the derived adjunction and the identification X[n] ~=X ^* *L S[n] provide 10 STEFAN SCHWEDE AND BROOKE SHIPLEY an isomorphism between [X[n]; X]Ho(C)and [S[n]; RHom (X; X)]Ho(SpR)under which * *X ^L - corresponds to [S[n]; S]Ho(SpR)-! [S[n]; RHom (X; X)]Ho(SpR)given by compositio* *n with the unit map. For every spectrum A the group [S[n]; A]Ho(SpR)is naturally isomorphi* *c to R ssnA, so this shows that the unit map induces an isomorphism of homotopy groups after* * tensoring with R, and condition (4) holds. To conclude the proof we assume condition (4) and show that the Quillen funct* *or pair Hom(X; -) and X ^ - of Theorem 5.1 (1) is a Quillen equivalence. Since the grou* *p [X; X]Ho(C) is an R-module, the functors form a Quillen pair with respect to the R-local mo* *del struc- ture for spectra by Theorem 5.1 (2). So we show that the adjoint total derived* * functors RHom (X; -) : Ho(C) -! Ho (SpR) and X ^L - : Ho (SpR) -! Ho(C) are inverse equi* *va- lences of homotopy categories. Note that the right derived functor RHom (X; -) * *is taken with respect to the R-local model structure on spectra. For a fixed integer n, the derived adjunction and the identification X ^L S[n* *] ~=X[n] provide a natural isomorphism (*) ssn RHom (X; Y ) ~= [S[n]; RHom (X; Y )]Ho(SpR)~= [X[n]; Y ]Ho(C): So the functor RHom (X; -) reflects isomorphisms because X is a weak generator.* * Hence it suffices to show that for every spectrum A the unit of the adjunction of derive* *d functors A -! RHom (X; X ^L A) is an isomorphism in the stable homotopy category. Consi* *der the full subcategory T of the R-local stable homotopy category with objects those s* *pectra A for which A -! RHom (X; X ^L A) is an isomorphism. Condition (4) says that the uni* *t map S -! Hom (X; X) is an R-local equivalence, so T contains the (localized) spher* *e spectrum. Since the composite functor RHom (X; X ^L -) commutes with (de-)suspension and * *preserves distinguished triangles, T is a triangulated subcategory of the homotopy catego* *ry of spectra. As a left adjoint the functor`X ^L - preserves coproducts.`By formula (*) above* * and since X is small, the natural map IRHom (X; Ai) -! RHom (X; IAi) is a ss*-isomorphism o* *f spectra for any family of objects Aiin Ho(C). Hence the functor RHom (X; -) also preser* *ves coproducts. So T is a triangulated subcategory of the homotopy category of spectra which is* * also closed un- der coproducts and contains the localized sphere spectrum. Thus, T is the whole* * R-local_stable homotopy category, and this finishes the proof. * * |__| 6.Construction of homomorphism spectra In this last section we show that the derived functor of a left Quillen funct* *or is sss*-linear, and we prove Theorem 5.1. Lemma 6.1. Let F : C -! D be the left adjoint of a Quillen adjoint pair between* * stable model categories. Then the total left derived functor LF : Ho(C)-! Ho(D) is sss*-exac* *t with respect to the natural isomorphism o : LF (X) ^L S1 -! LF (X ^L S1) of [Ho , 5.6.2]. Proof.To simplify notation we abbreviate the derived functor LF to L and drop t* *he superscript Lover the smash product on the homotopy category level. By [Ho , 5.7.3], the le* *ft derived functor L is compatible with the actions of the homotopy category of pointed simplicial* * sets - Hovey summarizes this compatibility under the name of `Ho (S*)-module functor' [Ho , * *4.1.7]. The isomorphism o : L(X) ^ S1 -! L(X ^ S1) is the special case K = S1 of a natural * *isomorphism oX;K : L(X) ^ K ----! L(X ^ K) for a pointed simplicial set K which is constructed in the proof of [Ho , 5.6.2* *] (or rather its pointed analog in [Ho , 5.7.3]). It is important for us that the isomorphism o * *is associative (this is part of being a `Ho (S*)-module functor'), i.e., that the composite L(A) ^ K ^ M -oA;K^1M-------!L(A ^ K) ^ M --oA^K;M------!L(A ^ K ^ M) A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY 11 is equal to oA;K^M (as before we suppress the implicit use of associativity iso* *morphisms such as (A ^ K) ^ M ~=A ^ (K ^ M)). In particular the map oX;Sn: L(X) ^ Sn -! L(X ^ Sn)* * is equal to the n-fold iterate of instances of o-;S1. Now let f : X -! Y be a morphism in the homotopy category of C and let ff : S* *n+k -! Sk represent a stable homotopy element. We have to show that ff . L(f) = L(ff . f)* * O oX;Sn in the group [L(X) ^ Sn; L(Y )]Ho(D). By the definition of ff . L(f) this means proving (1) L(f) ^ ff = (L(ff . f) O oX;Sn) ^ 1Sk in the group [L(X) ^ Sn+k; L(Y ) ^ Sk]Ho(D). Since oY;Sk: L(Y ) ^ Sk -! L(Y ^ * *Sk) is an isomorphism we may equivalently show equation (1) after composition with oY;Sk.* * We note that (2) oY;SkO (L(f) ^ ff)= L(f ^ ff) O oX;Sn+k (3) = L((ff . f) ^ 1Sk) O oX^Sn;SkO (oX;Sn^ 1Sk) (4) = oY;SkO (L(ff . f) ^ 1Sk) O (oX;Sn^ 1Sk) = oY;SkO ((L(ff . f) O oX;Sn) ^ 1Sk) ; which is what we had to show. Equations (2) and (4) use the naturality of o. Eq* *uation (3) uses_ the defining property of the morphism ff . f and the associativity of o. * * |__| Now we prove Theorem 5.1. We start with Proof of Theorem 5.1 (2).By assumption the group [X; X]Ho(C)is a module over a * *subring R of the ring of rational numbers. Since Hom (X; -) is a right Quillen functor, * *it satisfies the conditions of Lemma 4.2 for Z. For fibrant Y , the n-th homotopy group of the * *-spectrum Hom(X; Y ) is isomorphic to the group [S[n]; RHom (X; Y )]Ho(Sp). By the deriv* *ed adjunction this group is isomorphic to the group [X ^L S[n]; Y ]Ho(C)~=[X[n]; Y ]Ho(C), wh* *ich is a module over the R-local endomorphism ring [X; X]Ho(C). Hence the homotopy groups of th* *e -spectrum Hom(X; Y ) are R-local. Thus Hom (X; -) satisfies the conditions of Lemma 4.2 f* *or R and it_is a right Quillen functor for the R-local model structure. * * |__| Now we construct the adjoint functor pair Hom (X; -) and X ^ - in the case of* * a simplicial stable model category. This proves part (3) of Theorem 5.1 and also serves as a* * warm-up for the general construction which is very similar in spirit, but involves more tec* *hnicalities. Construction 6.2.Let C be a simplicial stable model category and X a cofibrant * *and fibrant object of C. We choose cofibrant and fibrant models !nX of the desuspensions of* * X as follows. We set !0X = X and inductively choose acyclic fibrations 'n : !nX -! (!n-1X) w* *ith !nX cofibrant. We then define the functor Hom (X; -) : C -! Sp by setting Hom (X; Y )n = map C(!nX; Y ) where `map C' denotes the simplicial mapping space. The spectrum structure maps* * are adjoint to the map map C(!n-1X; Y )-mapC(f'n;Y-)-------!mapC(!nX ^ S1; Y ) ~= mapC(!nX; Y ) where f'nis the adjoint of 'n. The functor Hom (X; -) has a left adjoint X ^ - : Sp -! C defined as the coeq* *ualizer _ _ (*) !nX ^ S1 ^ An-1 ----!----!!nX ^ An ----! X ^ A : n n 12 STEFAN SCHWEDE AND BROOKE SHIPLEY The two maps in the coequalizer are induced by the structure maps of the spectr* *um A and the maps e'n: !nX ^ S1 -! !n-1X respectively. The various adjunctions provide bije* *ctions of morphism sets C(X ^ S; W ) ~= Sp(S; Hom(X; W )) ~= S*(S0; Hom(X; W )0) ~= C(X; W ) natural in the C-object W . Hence the map X ^ S -! X corresponding to the ident* *ity of X in the case W = X is an isomorphism; this shows that the left adjoint takes the sp* *here spectrum to X. Since !nX is cofibrant the functor mapC(!nX; -) takes fibrations (resp. acycl* *ic fibrations) in C to fibrations (resp. acyclic fibrations) of simplicial sets. So the functo* *r Hom (X; -) takes fibrations (resp. acyclic fibrations) in C to level fibrations (resp. level acy* *clic fibrations) of spectra. Since C is stable, f'nis a weak equivalence between cofibrant objects, so for f* *ibrant Y the spectrum Hom (X; Y ) is an -spectrum. Hence Hom (X; -) satisfies the conditions* * of Lemma 4.2 for R = Z, and so Hom (X; -) and X ^ - form a Quillen adjoint pair. Since t* *he functor Hom(X; -) is defined with the use of the simplicial mapping space of C, it come* *s with a natural, coherent isomorphism Hom (X; Y K) ~=Hom (X; Y )K for a simplicial set K. So Hom* * (X; -) and its adjoint X ^ - form a simplicial Quillen functor pair which proves part (3) * *of Theorem 5.1. It remains to construct homomorphism spectra as in part (1) of Theorem 5.1 fo* *r a general stable model category, and prove the uniqueness part (4) of Theorem 5.1. Reade* *rs who only work with simplicial model categories and have no need for the uniqueness state* *ment may safely ignore the rest of this paper. To compensate for the lack of simplicial mapping spaces, we work with cosimpl* *icial frames. The theory of `framings' of model categories goes back to Dwyer and Kan, who us* *ed the termi- nology (co-)simplicial resolutions [DK , 4.3]; we mainly refer to Chapter 5 of * *Hovey's book [Ho ] for the material about cosimplicial objects that we need. If K is a pointed sim* *plicial set and A a cosimplicial object of C, then we denote by A ^ K the coend [ML , IX.6] Z n2 A ^ K = An ^ Kn ; which is an object of C. Here An ^ Kn denotes the coproduct of copies of An ind* *exed by the set Kn, modulo the copy of An indexed by the basepoint of Kn. Note that A ^ [m]+ is* * naturally isomorphic to the object of m-cosimplices of A; the object A ^ @[m]+ is also ca* *lled the m-th latching object of A. A cosimplicial map A -! B is a Reedy cofibration if for a* *ll m 0 the map A ^ [m]+ [A^@[m]+ B ^ @[m]+ ----! B ^ [m]+ is a cofibration in C. Cosimplicial objects in any pointed model category admit* * the Reedy model structure in which the weak equivalences are the cosimplicial maps which are le* *velwise weak equivalences and the cofibrations are the Reedy cofibrations. The Reedy fibrati* *ons are defined by the right lifting property for Reedy acyclic cofibrations or equivalently with * *the use of matching objects; see [Ho , 5.2.5] for details on the Reedy model structure. If A is a c* *osimplicial object and Y is an object of C, then there is a simplicial set C(A; Y ) of C-morphisms def* *ined by C(A; Y )n = C(An; Y ). There is an adjunction bijection of pointed sets C(A ^ K; Y ) ~=S*(K* *; C(A; Y )). If A is a cosimplicial object, then the suspension of A is the cosimplicial object A* * defined by (A)m = A ^ (S1 ^ [m]+) : Note that A and A^S1 have different meanings: A^S1 is (naturally isomorphic to)* * the object of 0-cosimplices of A. There is a loop functor for cosimplicial objects which * *is right adjoint to ; we do not use the precise form of Y here. For a cosimplicial object A and * *an object Y A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY 13 of C there is an adjunction isomorphism C(A; Y ) ~= C(A; Y ) : A cosimplicial object in C is homotopically constant if each cosimplicial str* *ucture map is a weak equivalence in C. A cosimplicial frame (compare [Ho , 5.2.7]) is a Reedy * *cofibrant and homotopically constant cosimplicial object. The following lemma collects from [* *Ho , Ch. 5] those properties of cosimplicial frames which are relevant to our discussion. Lemma 6.3. Let C be a pointed model category. (a)The suspension functor for cosimplicial objects preserves Reedy cofibration* *s, Reedy acyclic cofibrations and level equivalences between Reedy cofibrant objects. (b)If A is a cosimplicial frame, then so is A. (c)If A is a cosimplicial frame, then the functor C(A; -) takes fibrations (re* *sp. acyclic fibra- tions) in C to fibrations (resp. acyclic fibrations) of simplicial sets. (d)If Y is a fibrant object of C, then the functor C(-; Y ) takes level equiva* *lences between Reedy cofibrant cosimplicial objects to weak equivalences of simplicial sets. Proof.(a) For a cosimplicial map f :A -! B the map in C (A) ^ [m]+ [(A)^@[m]+ (B) ^ @[m]+ ----! (B) ^ [m]+ is isomorphic to the pushout product f i [Ho , 4.2.1] of f with the inclusion * *i of S1 ^ @[m]+ into S1 ^ [m]+. So if f is a Reedy cofibration, then f i is a cofibration in C* * by [Ho , 5.7.1]; hence A -! B is a Reedy cofibration. In cosimplicial level m, the map f is give* *n by the map f ^ (S1 ^ [m]+). If f is a Reedy acyclic cofibration, then f ^ (S1 ^ [m]+) * *is an acyclic cofibration in C by [Ho , 5.7.1]; hence f is also a level equivalence. Suspensi* *on then preserves level equivalences between Reedy cofibrant objects by Ken Brown's lemma [Ho , 1* *.1.12]. (b) If A is a cosimplicial frame, then A is again Reedy cofibrant by part (a)* *. A simplicial face map di: [m - 1] -! [m] induces an acyclic cofibration d*i: (A)m-1 = A ^ (S1 ^ [m - 1]+) ----! A ^ (S1 ^ [m]+) = (A)m by [Ho , 5.7.2], so A is also homotopically constant. (c) This is the pointed variant of [Ho , 5.4.4 (1)]. (d) If A -! B is a Reedy acyclic cofibration, then for every cofibration of p* *ointed simplicial sets K -! L the map A ^ LA^K B ^ K -! B ^ L is an acyclic cofibration in C by [* *Ho , 5.7.1]. By adjointness the induced map C(B; Y ) -! C(A; Y ) is an acyclic fibration of * *simplicial sets. By Ken Brown's Lemma [Ho , 1.1.12], the functor C(-; Y ) thus takes level equiv* *alences between_ Reedy cofibrant objects to weak equivalences of simplicial sets. * * |__| The following lemma provides cosimplicial analogues of the desuspensions !nX * *of Construc- tion 6.2. Lemma 6.4. Let Y be a cosimplicial object in a stable model category C which is* * Reedy fibrant and homotopically constant. Then there exists a cosimplicial frame X and a leve* *l equivalence X -! Y whose adjoint X -! Y is a Reedy fibration which has the right lifting pr* *operty for the map * -! A for any cosimplicial frame A. Proof.Since C is stable there exists a cofibrant object X0 of C such that the s* *uspension of X0 in the homotopy category of C is isomorphic to the object Y 0of 0-cosimplices. * *By [DK , 4.5] or [Ho , 5.2.8] there exists a cosimplicial frame X with X0 = X0. Since X is Re* *edy cofibrant, the map d0 q d1 : X0 q X0 -! X1 is a cofibration between cofibrant objects in C* *; since X is also homotopically constant, these maps express X1 as a cylinder object [Q , I * *1.5 Def. 4] for X0. The 0-cosimplices of X are given by the quotient of the map d0 q d1, hence* * (X )0 is a 14 STEFAN SCHWEDE AND BROOKE SHIPLEY model for the suspension of X0 in the homotopy category of C. Since (X )0 is co* *fibrant and Y 0 is fibrant, the isomorphism between them in the homotopy category can be realiz* *ed by a weak equivalence j0 : (X )0 -~!Y 0in C. Since Y is Reedy fibrant and homotopically c* *onstant, the map Y -! cY 0is a Reedy acyclic fibration, where cY 0denotes the constant cosim* *plicial object. Since X is Reedy cofibrant, the composite map 0 X ----!c(X )0 --cj--!cY 0 can be lifted to a map j : X - ! Y . The lift j is a level equivalence since j0* * is an equivalence in C and both X (by 6.3 (b)) and Y are homotopically constant. The adjoint X -* *! Y of j might not be a Reedy fibration, but we can arrange for this by factoring it as * *a Reedy acyclic cofibration X -! X followed by a Reedy fibration : X -! Y , and replacing j b* *y the adjoint ^ : X -! Y of the map ; by Lemma 6.3 (a) the map X -! X is a level equivalenc* *e, hence so is ^. Now suppose A is a cosimplicial frame and g : A -! Y is a cosimplicial map wi* *th adjoint ^g: A -! Y . We want to construct a lifting, i.e., a map A -! X whose composit* *e with : X -! Y is g. We choose a cylinder object for A, i.e., a factorization A_A -* *! A x I -! A of the fold map as a Reedy cofibration followed by a level equivalence. The sus* *pension functor preserves Reedy cofibrations and level equivalences between Reedy cofibrant obj* *ects by Lemma 6.3 (a), so the suspended sequence A_A -! (A x I) -! A yields a cylinder object* * for A. In particular the 0-th level of (A x I) is a cylinder object for (A)0 = A ^ * *S1 in C. By [Ho , 6.1.1] the suspension map : [A0; X0] -! [A0 ^L S1; X0 ^L S1] in the* * homotopy category of C can be constructed as follows. Given a C-morphism f0 : A0 -! X0, * *one chooses an extension f : A -! X to a cosimplicial map between cosimplicial frames. The * *map f ^ S1 : A ^ S1 -! X ^ S1 then represents the class [f0] 2 [A0 ^L S1; X0 ^L S1]. Composi* *tion with the 0-th level ^ 0: X ^ S1 -! Y 0of the level equivalence ^ : X -! Y is a bijec* *tion from [A0^L S1; X0 ^L S1] to [A0^L S1; Y 0]. Since C is stable, the suspension map is* * bijective, which means that there exists a cosimplicial map f : A -! X such that the maps ^ 0O (* *f ^ S1) and ^g0 represent the same element in [A ^ S1; Y 0]. The map f need not be a lift of the original map g, but we can find a lift in* * the homotopy class of f as follows. Since A ^ S1 is cofibrant and Y 0is fibrant, there exis* *ts a homotopy H1 : ((AxI))0 -! Y 0from ^ 0O(f^S1) to ^g0. Evaluation at cosimplicial level ze* *ro is left adjoint to the constant functor, so the homotopy H1 is adjoint to a homotopy ^H1: (A x * *I) -! cY 0of cosimplicial objects. Since Y is Reedy fibrant and homotopically constant, the * *map Y -! cY 0 is a Reedy acyclic fibration. So there exists a lifting H2 : (A x I) -! Y in th* *e commutative square ^O(f)_^g A_A ______wY v| | | |~ |u |uu (A x I) _____cYw0^ H1 which is a homotopy from ^O (f) to ^g. Taking adjoints gives a map ^H2: A x I -* *! Y which is a homotopy from O f to g. Since X -! Y is a Reedy fibration and the front * *inclusion i0 : A -! A x I is a Reedy acyclic cofibration, we can choose a lifting H3 : A * *x I -! X in the A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY 15 commutative square f A _______wX v| | | | i0| | | | |u |uu A x I_____wY^H 2 The end of the homotopy H3, i.e., the composite map H3 O i1 : A -! X, is then a* * lift_of the original map g : A -! Y since ^H2O i1 = g. * * |__| Construction 6.5.Let C be a stable model category and X a cofibrant and fibrant* * object of C. We define Reedy fibrant cosimplicial frames !nX as follows. As in [Ho , 5.2.* *8] we can choose a cosimplicial frame !0X with (!0X)0 = X and a Reedy acyclic fibration '0 : !0X* * -! cX which is the identity in dimension zero. Then !0X is Reedy fibrant since X is * *fibrant in C. By Lemma 6.4 we can inductively choose cosimplicial frames !nX and level eq* *uivalences ^'n: (!nX) -! !n-1X whose adjoints 'n : !nX -! (!n-1X) are Reedy fibrations wi* *th the right lifting property for cosimplicial frames. By Lemma 6.3 (a), preserve* *s Reedy acyclic cofibrations, so preserves Reedy fibrations. Hence (!n-1X) and thus !nX is Re* *edy fibrant. We then define the functor Hom (X; -) : C -! Sp by setting Hom (X; Y )n = C(!nX; Y ) : The spectrum structure maps are adjoint to the map C(!n-1X; Y ) ---C(^'n;Y-)-----!C( (!nX); Y ) ~= C(!nX;:Y ) The left adjoint X ^ - : Sp -! C of Hom (X; -) is defined by the same coequaliz* *er diagram (*) as in Construction 6.2, except that an expression like !nX ^ An now refers to t* *he coend of a cosimplicial object with a simplicial set. Also the isomorphism between X ^S an* *d X is obtained by the same representability argument as in 6.2. Since !nX is a cosimplicial frame, the functor C(!nX; -) takes fibrations (re* *sp. acyclic fi- brations) in C to fibrations (resp. acyclic fibrations) of simplicial sets by L* *emma 6.3 (c). So the functor Hom (X; -) takes fibrations (resp. acyclic fibrations) in C to leve* *l fibrations (resp. level acyclic fibrations) of spectra. Since ^'nis a level equivalence between c* *osimplicial frames, Lemma 6.3 (d) shows that the map C('^n; Y ) is a weak equivalence for fibrant Y* * ; thus the spec- trum Hom (X; Y ) is an -spectrum for fibrant Y . So Hom (X; -) and its adjoint * *form a Quillen pair by Lemma 4.2 for R = Z. This proves part (1) of Theorem 5.1. Proof of Theorem 5.1 (4).Let H : Sp -! C be any left Quillen functor with an is* *omorphism H(S) ~=X, and let G : C -! Sp be a right adjoint. We construct natural transfo* *rmations : Hom (X; -) -! G and : H -! (X ^ -) where Hom (X; -) and X ^ - are the Quill* *en pair which were constructed in 6.5. Furthermore, will be a stable equivalence * *of spectra for fibrant objects of C and will be a weak equivalence in C for every cofibrant s* *pectrum. So any two Quillen pairs as in Theorem 5.1 (1) can be related in this way via the pair* * Hom (X; -) and X ^ - of Construction 6.5. We denote by Fn the cosimplicial spectrum given by (Fn)m = Fn[m]+ and we deno* *te by Ho the functor between cosimplicial objects obtained by applying the left Qu* *illen functor H levelwise. The functor Ho is then a left Quillen functor with respect to the Re* *edy model struc- tures on cosimplicial spectra and cosimplicial objects of C. We inductively cho* *ose compatible maps n : Ho(Fn) -! !nX of cosimplicial objects as follows. Since Fn is a cosi* *mplicial 16 STEFAN SCHWEDE AND BROOKE SHIPLEY frame, Ho(Fn) is a cosimplicial frame in C. The map '0 : !0X -! cX is a Reedy a* *cyclic fibration, so the composite map ~= Ho(F0) ----! cH(F0S0) ----! cX admits a lift 0 : Ho(F0) -! !0X which is a level equivalence between cosimplic* *ial frames. The map 'n : !nX -! (!n-1X) has the right lifting property for cosimplicial fr* *ames, so we can inductively choose a lift n : Ho(Fn) -! !nX of the composite map Ho(Fn) ----! Ho(Fn-1) -(-n-1)----! (!n-1X) : We show by induction that n is a level equivalence. The map n ^ S1 is a weak * *equivalence in C since the other three maps in the commutative square n^S1 H(FnS1) ~=Ho(Fn) ^ S1 _____w!nX ^ S1 | | | |^'0n | | |u |u H(Fn-1S0) ~=Ho(Fn-1)0 ______(!n-1X)0w 0 n-1 are. The map n ^ S1 is a model for the suspension of ( n)0. Since C is stable * *and ( n)0 is a map between cofibrant objects, ( n)0 is a weak equivalence in C. Since Ho(Fn) a* *nd !nX are homotopically constant, the map n : Ho(Fn) -! !nX is a level equivalence. The adjunction provides a natural isomorphism of simplicial sets G(Y )n ~=C(H* *o(Fn); Y ) for every n 0, and we get a natural transformation n : Hom (X; Y )n = C(!nX; Y ) ---C(-n;Y-)----!C(Ho(Fn); Y ) ~= G(Y )n : By the way the maps n were chosen, the maps n together constitute a map of spe* *ctra Y : Hom(X; Y ) -! G(Y ), natural in the C-object Y . For fibrant objects Y , Y is a* * level equivalence, hence a stable equivalence, of spectra by Lemma 6.3 (d) since n is a level equ* *ivalence between cosimplicial frames. Now let A be a spectrum. If we compose the adjoint H(Hom (X; X ^ A)) -! X ^ * *A of the map X^A : Hom (X; X ^ A) -! G(X ^ A) with H(A) -! H(Hom (X; X ^ A)) com- ing from the adjunction unit, we obtain a natural transformation A : H(A) -! 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Verdier, Des categories derivees des categories abeliennes, Asteris* *que 239 (1997). With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis, xii+253 p* *p. Fakult"at f"ur Mathematik, Universit"at Bielefeld, 33615 Bielefeld, Germany E-mail address: schwede@mathematik.uni-bielefeld.de Department of Mathematics, University of Chicago, Chicago, IL 60637, USA E-mail address: bshipley@math.uchicago.edu