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*
*gSteenrod 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 cofibrantfibrant 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 BASFForschungsstipendium 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 Rlocal 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 selfequivalence of the homotopy cate*
*gory and it defines
the shift functor. Since every object is a twofold suspension, hence an abelia*
*n cogroup 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 nfold 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
__ *
* o1
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 nfold 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 nfold 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 nth Morava Kth*
*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; v1n], 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 quasiisomorphisms. 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
DGmodules are products of EilenbergMacLane 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 SpanierWhitehead category of finite CWcomplexes. 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, biexact 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 SpanierWhitehead category of finite CWcomplexes ([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
SpanierWhitehead 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 Rlocal 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 Rlocal 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^Xn1S1 ^ Yn1 ! 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 Rlocal model structure for R a subring of the ring o*
*f rational numbers.
The Rlocal model category structure is well known, but we were unable to find *
*a reference in
the literature. A map of spectra is an Requivalence if it induces an isomorphi*
*sm of homotopy
groups after tensoring with R and is an Rfibration if it has the right lifting*
* property with respect
to all maps that are cofibrations and Requivalences.
Lemma 4.1. Let R be a subring of the ring of rational numbers. Then the cofibr*
*ations, R
fibrations and Requivalences make the category of spectra into a model categor*
*y, referred to as
the Rlocal model category structure. A spectrum is fibrant in the Rlocal mode*
*l structure if and
only if it is an spectrum with Rlocal 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 isimplex, its boundary and its kth horn (the union of all (i  1)*
*dimensional faces
except the kth 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 = *
*Sjn^ 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 modk 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 Rlocal homotopy groups.
(b)A map which is an Requivalence 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 Requivalence and is built from maps in J by coproducts,*
* pushouts and
composition.
(d)A map is an Rfibration 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 modk homotopy group
[Fnm Mk; X]Ho(Sp)~= ss0m map(Mk; Xn) ~= ssm+2n (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 Rlocal homotopy groups.
(b) Suppose f : X ! Y is an Requivalence 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 Requivalence, 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 Rlocal. 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 Requivalences. The ma*
*ps in Jlv are
levelwise equivalences, the maps in Jstare stable equivalences, hence both are *
*Requivalences.
Since the stable homotopy groups of the Moore space Mk are kpower torsion, the*
* maps in JR
are also Requivalences.
(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 Requivalences. 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 Requivalences, 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 Requivalences satis*
*fy the 2out
of3 property (MC2) and the classes of cofibrations, Rfibrations and Requival*
*ences are each
closed under retracts (MC3). By definition the Rfibrations have the RLP for ma*
*ps which are
both cofibrations and Requivalences. Furthermore a map which is an Requivale*
*nce and an
Rfibration 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 Requivalences and Rfibrations, so this is also a*
* factorization as a
cofibration followed by an acyclic fibration in the Rlocal 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 Rlocal 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 Rlocal homotopy groups and
(iii)G takes fibrations between fibrant objects to level fibrations.
Proof.The `only if' part holds since the Rlocal acyclic fibrations are level a*
*cyclic fibrations,
the Rfibrant objects are the spectra with Rlocal 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 Rlocal 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 Rlocal homotopy groups is an Rfibration. 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
Requivalence and with p : Z ! Y an Rfibration. Since Y is Rfibrant, so is Z*
*. Hence i is an
Requivalence between spectra with Rlocal 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 Rfibration. *
* __
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
Rmodule, then any adjoint functor pair satisfying (1) is also a Quillen pa*
*ir with respect
to the Rlocal 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 Rlocal 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 Rlocal 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 Rlocal 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 cofibrantfibrant small weak generator X for whi*
*ch the groups
[X; X]Ho(C)*are Rmodules 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 Rlocal 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 Rlocal 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 Rmodule, the functors form a Quillen pair with respect to the Rlocal 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 Rlocal 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 Rmodule, the functors form a Quillen pair with respect to the Rlocal 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 Rlocal 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 Rlocal 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 Rlocal 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*
* Rlocal_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 nfold 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 nth 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 Rlocal endomorphism ring [X; X]Ho(C). Hence the homotopy groups of th*
*e spectrum
Hom(X; Y ) are Rlocal. Thus Hom (X; ) satisfies the conditions of Lemma 4.2 f*
*or R and it_is
a right Quillen functor for the Rlocal 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*
* warmup 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 ! (!n1X) 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(!n1X; 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 ^ An1 !!!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 ! !n1X 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 Cobject 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 mcosimplices of A; the object A ^ @[m]+ is also ca*
*lled the mth
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 Cmorphisms 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 0cosimplices 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)m1 = 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 0cosimplices. *
*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 0cosimplices 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 0th 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 Cmorphism 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 0th 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) ! !n1X whose adjoints 'n : !nX ! (!n1X) 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 (!n1X) 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(!n1X; 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 ! (!n1X) 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(Fn1) (n1)! (!n1X) :
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(Fn1S0) ~=Ho(Fn1)0 ______(!n1X)0w 0
n1
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 Cobject 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) ! X*
* ^ A
between the left Quillen functors. The transformation induces a natural trans*
*formation
L : LH  ! X ^L  between the total left derived functors. For any Y in Ho(C)*
* the
map (LA)* : [X ^L A; Y ]Ho(C)! [LH(A); Y ]Ho(C)is isomorphic to the bijection *
*(RY )* :
[A; RHom (X; Y )]Ho(D)! [A; RG(Y )]Ho(D). Hence LA is an isomorphism in the ho*
*motopy __
category of C and so the map A is a weak equivalence in C for every cofibrant s*
*pectrum A. __
References
[Bo] J. M. Boardman, Stable homotopy theory, Mimeographed notes, Johns Hopkins*
* University, 196970.
[BF] A. K. Bousfield and E. M. Friedlander, Homotopy theory of spaces, spectr*
*a, and bisimplicial sets,
Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1*
*977), II (M. G. Barratt and
M. E. Mahowald, eds.), Lecture Notes in Math. 658, Springer, Berlin, 1978*
*, pp. 80130.
[CS] J. D. Christensen and N. P. Strickland, Phantom Maps and Homology Theorie*
*s, Topology 37 (1998),
339364.
[Du] D. Dugger, Replacing model categories with simplicial ones, Trans. Amer. *
*Math. Soc., to appear.
[DK] W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Top*
*ology 19 (1980), 427440.
[DS] W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, Han*
*dbook of algebraic topology
(Amsterdam), NorthHolland, Amsterdam, 1995, pp. 73126.
[ES] S. Eilenberg and N. E. Steenrod, Axiomatic approach to homology theory, P*
*roc. Nat. Acad. of Sci.
U.S.A. 31 (1945), 117120.
A UNIQUENESS THEOREM FOR STABLE HOMOTOPY THEORY 17
[EKMM]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, a*
*nd algebras in stable
homotopy theory. With an appendix by M. Cole, Mathematical Surveys and Mo*
*nographs, 47, American
Mathematical Society, Providence, RI, 1997, xii+249 pp.
[He1] A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (1988), no. 383, v*
*i+78 pp.
[He2] A. Heller, Stable homotopy theories and stabilization, J. Pure Appl. Alge*
*bra 115 (1997), no. 2, 113130.
[Ho] M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63,*
* American Mathematical
Society, Providence, RI, 1999, xii+209 pp.
[HPS] M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy*
* theory, Mem. Amer. Math.
Soc. 128 (1997), no. 610.
[HSS] M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, J. Amer. Math. Soc*
*. 13 (2000), 149208.
[Ly] M. Lydakis, Simplicial functors and stable homotopy theory, Preprint, Uni*
*versit"at Bielefeld, 1998.
[ML] S. Mac Lane, Categories for the working mathematician, Graduate Texts in *
*Math. 5, Springer, New
YorkBerlin, 1971, ix+262 pp.
[MMSS]M. A. Mandell, J. P. May, S. Schwede and B. Shipley, Model categories of *
*diagram spectra, Proc. London
Math. Soc., to appear.
[Mar] H. R. Margolis, Spectra and the Steenrod algebra. Modules over the Steenr*
*od algebra and the stable homo
topy category, NorthHolland Mathematical Library 29, NorthHolland Publi*
*shing Co., AmsterdamNew
York, 1983, xix+489 pp.
[Q] D. G. Quillen, Homotopical algebra, Lecture Notes in Math. 43, SpringerV*
*erlag, 1967.
[Ro] A. Robinson, Obstruction theory and the strict associativity of Morava K*
*theories, Advances in homo
topy theory (Cortona, 1988), Cambridge Univ. Press 1989, 143152.
[Sch1]S. Schwede, Stable homotopy of algebraic theories, Topology, to appear, h*
*ttp://hopf.math.purdue.edu/
[Sch2]S. Schwede, Smodules and symmetric spectra, Preprint, 1998, http://hopf.*
*math.purdue.edu/
[SS] S. Schwede and B. Shipley, Classification of stable model categories, Pre*
*print, 1999,
http://www.mathematik.unibielefeld.de/"schwede
[SW] E. H. Spanier and J. H. C. Whitehead, A first approximation to homotopy t*
*heory, Proc. Nat. Acad. Sci.
U.S.A. 39 (1953), 655660.
[Ve] J.L. 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
Email address: schwede@mathematik.unibielefeld.de
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
Email address: bshipley@math.uchicago.edu