SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL
CATEGORIES
MARK HOVEY
Abstract.We give two general constructions for the passage from unstable
to stable homotopy that apply to the known example of topological spaces*
*, but
also to new situations, such as the A1-homotopy theory of Morel-Voevodsk*
*y [19,
27]. One is based on the standard notion of spectra originated by Board-
man [28]. Its input is a well-behaved model category D and an endofunctor
T, generalizing the suspension. Its output is a model category SpN(D; T)
on which T is a Quillen equivalence. The second construction is based on
symmetric spectra [13], and applies to model categories C with a compati*
*ble
monoidal structure. In this case, the functor T must be given by tensori*
*ng
with a cofibrant object K. The output is again a model category Sp (C; K)
where tensoring with K is a Quillen equivalence, but now Sp (C; K) is ag*
*ain a
monoidal model category. We study general properties of these stabilizat*
*ions;
most importantly, we give a sufficient condition for these two stabiliza*
*tions to
be equivalent that applies both in the known case of topological spaces *
*and in
the case of A1-homotopy theory.
Introduction
The object of this paper is to give two very general constructions of the pas*
*sage
from unstable homotopy theory to stable homotopy theory. Since homotopy theory
in some form appears in many different areas of mathematics, this construction *
*is
useful beyond algebraic topology, where these methods originated. In particular,
the two constructions we give apply not only to the usual passage from unstable
homotopy theory of pointed topological spaces (or simplicial sets) to the stable
homotopy theory of spectra, but also to the passage from the unstable A1-homoto*
*py
theory of Morel-Voevodsky [19, 27] to the stable A1-homotopy theory. This examp*
*le
is obviously important, and the fact that it is an example of a widely applicab*
*le
theory of stabilization may come as a surprise to readers of [14], where specif*
*ic
properties of sheaves are used.
Suppose, then, that we are given a (Quillen) model category D and a functor
T :D -! D that we would like to invert, analogous to the suspension. We will
clearly need to require that T be compatible with the model structure; specific*
*ally,
we require T to be a left Quillen functor. We will also need some technical hyp*
*othe-
ses on the model category D, which are complicated to state and to check, but w*
*hich
are satisfied in almost all interesting examples, including A1-homotopy theory.*
* It is
well-known what one should do to form the category SpN(D; T ) of spectra, as fi*
*rst
written down for topological spaces in [2]. An object of SpN(D; T ) is a sequen*
*ce
____________
Date: September 24, 2000.
1991 Mathematics Subject Classification. 55U35, 18G55.
Key words and phrases. model category, homotopy category, stabilization, spe*
*ctra, symmetric
spectra.
1
2 MARK HOVEY
Xn of objects of D together with maps T Xn -! Xn+1, and a map f :X -! Y is
a sequence of maps fn :Xn -! Yn compatible with the structure maps. There is
an obvious model structure, called the projective model structure, where the we*
*ak
equivalences are the maps f :X -! Y such that fn is a weak equivalence for all *
*n. It
is not difficult to show that this is a model structure and that there is a lef*
*t Quillen
functor T :SpN(D; T ) -! SpN(D; T ) extending T on D. But, just as in the topo-
logical case, T will not be a Quillen equivalence. So we must localize the proj*
*ective
model structure on SpN(D; T ) to produce the stable model structure, with respe*
*ct
to which T will be a Quillen equivalence. A new feature of this paper is that w*
*e are
able to construct the stable model structure with minimal hypotheses on D, using
the localization results of Hirschhorn [11] (based on work of Dror Farjoun [7])*
*. We
must pay a price for this generality, of course. That price is that stable equi*
*valences
are not stable homotopy isomorphisms, but instead are cohomology isomorphisms
on all cohomology theories, just as for symmetric spectra [13]. If we put enough
hypotheses on D and T , then stable equivalences coincide with stable homotopy
isomorphisms. Using the Nisnevitch descent theorem, Jardine [14] has proved that
stable equivalences coincide with stable homotopy isomorphisms in the stable A1-
homotopy theory. His result does not follow from our general theorem, because t*
*he
hypotheses we need do not hold in the Morel-Voevodsky motivic model category.
However, Voevodsky (personal communication) has constructed a simpler model
category equivalent to the Morel-Voevodsky one that does satisfy our hypotheses.
As is well-known in algebraic topology, the category SpN(D; T ) is not suffic*
*ient
to understand the smash product. That is, if C is a symmetric monoidal model
category, and T is the functor X 7! X K for some cofibrant object K of C, it
almost never happens that SpN(C; T ) is symmetric monoidal. We therefore need
a different construction in this case. We define a category Sp (C; K) just as *
*in
symmetric spectra [13]. An object of Sp (C; K) is a sequence Xn of objects of
C with an action of the symmetric group n on Xn. In addition, we have n-
equivariant structure maps Xn K -! Xn+1, but we must further require that
the iterated structure maps Xn Kp -! Xn+p are n x p-equivariant, where
p acts on Kp by permuting the tensor factors. It is once again straightforward
to construct the projective model structure on Sp (C; K). The same localization
methods developed for SpN(D; T ) apply again here to give a stable model struct*
*ure
on which tensoring with K is a Quillen equivalence. Once again, stable equivale*
*nces
are cohomology isomorphisms on all possible cohomology theories, but this time *
*it
is very difficult to give a better description of stable equivalences even in t*
*he case
of simplicial symmetric spectra (but see [25] for the best such result I know).*
* We
point out that our construction gives a different construction of the stable mo*
*del
category of simplicial symmetric spectra from the one appearing in [13].
We now have competing stabilizations of C under the tensoring with K functor
when C is symmetric monoidal. Naturally, we need to prove they are the same in
an appropriate sense. This was done in the topological (actually, simplicial) c*
*ase
in [13] by constructing a functor SpN(C; T ) -! Sp (C; K), where K = S1 and T is
the tensor with S1 functor, and proving it is a Quillen equivalence. We are una*
*ble
to generalize this argument. Instead, following an idea of Hopkins, we construc*
*t a
zigzag of Quillen equivalences SpN(C; T ) -! E- Sp (C; K). However, we need to
require that the cyclic permutation map on K K K be homotopic to the identity
by an explicit homotopy for our construction to work. This hypothesis holds in *
*the
topological case with K = S1 and in the A1-local case with K equal to either the
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 3
simplicial circle or the algebraic circle A1- {0}. This section of the paper is*
* by far
the most delicate, and it is likely that we do not have the best possible resul*
*t.
We also investigate the properties of these two stabilization constructions. *
*There
are some obvious properties one would like a stabilization construction such as
SpN(D; T ) to have. First of all, it should be functorial in the pair (D; T ).*
* We
prove this for SpN(D; T ) and an appropriate analogue of it for symmetric spect*
*ra;
the most difficult point is defining what one should mean by a map from (D; T )*
* to
(D0; T 0). Furthermore, stabilization should be homotopy invariant. That is, if*
* the
map (D; T ) -! (D0; T 0) is a Quillen equivalence, the induced map of stabiliza*
*tions
should also be a Quillen equivalence. We also prove this for SpN(D; T ) and an
appropriate analogue of it for symmetric spectra; one corollary is that the Qui*
*llen
equivalence class of Sp (C; K) depends only on the homotopy type of K. Finally,
the stabilization map D -!SpN(D; T ) should be the initial map to a model categ*
*ory
E with an extension of T to a Quillen equivalence. However, this last statement
seems to be asking for too much, because the category of model categories is it*
*self
something like a model category. This statement is analogous to asking for an
initial map in a model category from X to a fibrant object, and such things do *
*not
usually exist. The best we can do is to say that if T is already a Quillen equi*
*valence,
then the map from D -! SpN(D; T ) is a Quillen equivalence. This gives a weak
form of uniqueness, and is the basis for the comparison between SpN(D; T ) and
symmetric spectra. See also see [22] and [23] for uniqueness results for the us*
*ual
stable homotopy category.
We point out that this paper leaves some obvious questions open. We do not
have a good characterization of stable equivalences or stable fibrations in eit*
*her
spectra or symmetric spectra, in general, and we are unable to prove that spect*
*ra or
symmetric spectra are right proper. We do have such characterizations for spect*
*ra
when the original model category D is sufficiently well-behaved, and the adjoin*
*t U
of T preserves sequential colimits. These hypotheses include the cases of ordin*
*ary
simplicial spectra and spectra in a new motivic model category of Voevodsky (but
not the original Morel-Voevodsky motivic model category). We also prove that
spectra are right proper in this situation. But we do not have a characterizati*
*on
of stable equivalences of symmetric spectra even with these strong assumptions.
Also, we have been unable to prove that symmetric spectra satisfy the monoid
axiom. Without the monoid axiom, we do not get model categories of monoids
or of modules over an arbitrary monoid, though we do get a model category of
modules over a cofibrant monoid. The question of whether commutative monoids
form a model category is even more subtle and is not addressed in this paper.
See [18] for commutative monoids in symmetric spectra of topological spaces.
There is a long history of work on stabilization, much of it not using model
categories. As far as this author knows, Boardman was the first to attempt to
construct a good point-set version of spectra; his work was never published (but
see [28]), but it was the standard for many years. Generalizations of Boardman's
construction were given by Heller in several papers, including [8] and [9]. Hel*
*ler has
continued work on these lines, most recently in [10]. The review of this paper *
*in
Mathematical Reviews by Tony Elmendorf (MR98g:55021) captures the response
of many algebraic topologists to Heller's approach. I believe the central idea*
* of
Heller's approach is that the homotopy theory associated to a model category D
is the collection of all possible homotopy categories of diagram categories ho *
*DI
and all functors between them. With this definition, one can then forget one had
4 MARK HOVEY
the model category in the first place, as Heller does. Unfortunately, the resul*
*ting
complexity of definition is overwhelming at present.
Of course, there has also been very successful work on stabilization by May a*
*nd
coauthors, the two major milestones being [16] and [6]. At first glance, May's
approach seems wedded to the topological situation, relying as it does on homeo-
morphisms Xn -! Xn+1. This is the reason we have not tried to use it in this
paper. However, there has been considerable recent work showing that this ap-
proach may be more flexible than one might have expected. I have mentioned [18]
above, but perhaps the most ambitious attempt to generalize S-modules has been
initiated by Mark Johnson [15].
Finally, we point out that Schwede [21] has shown that the methods of Bousfie*
*ld
and Friedlander [2] apply to certain more general model categories. His model
categories are always simplicial and proper, and he is always inverting the ord*
*inary
suspension functor. Nevertheless, the paper [21] is the first serious attempt t*
*o define
a general stabilization functor of which the author is aware.
This paper is organized as follows. We begin by defining the category SpN(D; *
*T )
and the associated projective model structure in Section 1. Then there is the b*
*rief
Section 2 recalling Hirschhorn's approach to localization of model categories. *
*We
construct the stable model structure modulo certain technical lemmas in Section*
* 3.
The technical lemmas we need assert that if a model category D is left proper
cellular, then so is the projective model structure on SpN(D; T ), and therefore
we can apply the localization technology of Hirschhorn. We prove these technical
lemmas, and the analogous lemmas for the projective model structure on symmetric
spectra, in an Appendix. In Section 4, we study the simplifications that arise
when the adjoint U of T preserves sequential colimits and D is sufficiently wel*
*l-
behaved. We characterize stable equivalences as the appropriate generalization *
*of
stable homotopy isomorphisms in this case, and we show the stable model structu*
*re
is right proper, giving a description of the stable fibrations as well. In Sect*
*ion 5,
we prove the functoriality, homotopy invariance, and homotopy idempotence of the
construction (D; T ) 7! SpN(D; T ). We investigate monoidal structure in Sectio*
*n 6,
showing that SpN(C; T ) is almost never a symmetric monoidal model category even
when C is so.
This demonstrates the need for a better construction, and Section 7 begins the
study of symmetric spectra. Since we have developed all the necessary techniques
in the first part, the proofs in this part are more concise. In Section 7 we di*
*scuss
the category of symmetric spectra. In Section 8 we construct the projective and
stable model structures on symmetric spectra, and in Section 9, we discuss some
properties of symmetric spectra. This includes functoriality, homotopy invarian*
*ce,
and homotopy idempotence of the stable model structure. We conclude the paper in
Section 10 by constructing the chain of Quillen equivalences between SpN(C; T )*
* and
Sp (C; K), under the cyclic permutation hypothesis mentioned above. Finally, as
stated previously, there is an Appendix verifying that the techniques of Hirsch*
*horn
can be applied to the projective model structures on SpN(D; T ) and symmetric
spectra.
Obviously, considerable familiarity with model categories will be necessary to
understand this paper. The original reference is [20], but a better introducto*
*ry
reference is [5]. More in depth references include [4], [11], and [12]. In part*
*icular,
we rely heavily on the localization technology in [11].
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 5
The author would like to thank Dan Dugger, Phil Hirschhorn, Mike Hopkins,
Dan Kan, Stefan Schwede, Brooke Shipley, Jeff Smith, Markus Spitzweck, and
Vladimir Voevodsky for helpful conversations about this paper. In particular, to
the author's knowledge, it is Jeff Smith's vision that one should be able to st*
*abilize
an arbitrary model category, a vision that could not be carried out without Phil
Hirschhorn's devotion to getting the localization theory of model categories ri*
*ght.
The idea of using almost finitely generated model categories in Section 4 is due
to Voevodsky, and the idea of using bispectra to compare symmetric spectra with
ordinary spectra (see Section 10) is due to Hopkins. The author also thanks the
referee for innumerable detailed and helpful suggestions.
1. Spectra
In this section and throughout the paper, D will be a model category and
T :D -! D will be a left Quillen endofunctor of D with right adjoint U. In this
section, we define the category SpN(D; T ) of spectra and construct the project*
*ive
model structure on SpN(D; T ).
The following definition is a straightforward generalization of the usual not*
*ion
of spectra [2].
Definition 1.1.Suppose T is a left Quillen endofunctor of a model category D.
Define SpN(D; T ), the category of spectra, as follows. A spectrum X is a seque*
*nce
X0; X1; : :;:Xn; : :o:f objects of D together with structure maps oe :T Xn -! X*
*n+1
for all n. A map of spectra from X to Y is a collection of maps fn :Xn -! Yn
commuting with the structure maps; this means that the diagram below
T Xn --oeX--!Xn+1
? ?
Tfn?y ?yfn+1
T Yn----!oeYn+1
Y
is commutative for all n.
Note that if D is either the model category of pointed simplicial sets or the*
* model
category of pointed topological spaces, and T is the suspension functor given by
smashing with the circle S1, then SpN(D; T ) is the Bousfield-Friedlander categ*
*ory
of spectra [2].
Definition 1.2.Given n 0, the evaluation functor Evn :SpN(D; T ) -! D takes
X to Xn. The evaluation functor has a left adjoint Fn :D -! SpN(D; T ) defined
by (FnA)m = T m-nA if m n and (FnA)m = 0 otherwise, where 0 is the initial
object of D. The structure maps are the obvious ones.
Note that F0 is an full and faithful embedding of the category D into SpN(D; *
*T ).
Lemma 1.3. The category of spectra is bicomplete.
Proof.Given a functor G from a small category I into SpN(D; T ), we define
(colimG)n = colimEvnOG and (limX)n = limEvn OG
Since T is a left adjoint, it preserves colimits. The structure maps of the col*
*imit
are then the composites
T (colimEvn OG) ~=colim(T O EvnOG) colim(oeOG)-------!colimEvn+1OG:
6 MARK HOVEY
Although T does not necessarily preserve limits, there is still a natural map
T (limH) -!limT H
for any functor H :I -!D. Then the structure maps of the limit are the composit*
*es
T (limEv nOG) -!lim(T O EvnOG) lim(oeOG)------!limEvn+1OG: |___|
Remark 1.4. The evaluation functor Evn :SpN(D; T ) -! D also has a right ad-
joint Mn :D -!SpN(D; T ). We define (MnA)i= Un-iA if i n, and (MnA)i= 1
if i > n, where 1 denotes the terminal object of D. The structure map T Un-iA -!
Un-i-1A is adjoint to the identity map of Un-iA when i < n. We leave it to the
reader to verify that Mn is right adjoint to Evn.
We wish to prolong the adjunction (T; U) to an adjunction of functors between
spectra. We will discuss prolonging more general adjunctions in Section 5.
Lemma 1.5. Suppose T is a left Quillen endofunctor of a model category D, with
right adjoint U. Define a functor T :SpN(D; T ) -! SpN(D; T ) by (T X)n = T Xn,
with structure map
T (T Xn) Toe--!T Xn+1;
where oe is the structure map of X. Define a functor U :SpN(D; T ) -! SpN(D; T )
by (UX)n = UXn, with structure map adjoint to
UXn Ueoe--!U(UXn+1)
where eoeis adjoint to the structure map of X. Then T is left adjoint to U.
Proof.We leave it to the reader to verify the functoriality of T and U. We show
they are adjoint. For convenience, let us denote the extensions of T and U to
functors of spectra by eTand eU. It suffices to construct unit maps X -! eUeTX *
*and
counit maps eTeUX -! X verifying the triangle identities, by [17, Theorem 4.1.2*
*(v)].
But we can take these unit and counit maps to be the maps which are the unit and
counit maps of the (T; U) adjunction in each degree. The reader should verify t*
*hat
these are maps of spectra. The triangle identities then follow immediately from*
*_the
triangle identities of the (T; U) adjunction. |_*
*_|
The following remark is critically important to the understanding of our appr*
*oach
to spectra.
Remark 1.6. The definition we have just given of the prolongation of T to an
endofunctor of SpN(D; T ) is the only possible definition under our very general
hypotheses. However, this definition does not generalize the definition of the
suspension when D is the category of pointed topological spaces and T A = A^S1.
Indeed, recall from [2] that the suspension of a spectrum X in this case is def*
*ined
by (X S1)n = Xn ^ S1, with structure map given by
Xn ^ S1 ^ S1 1^t--!Xn ^ S1 ^ S1 oe^1--!Xn+1 ^ S1;
where t is the twist isomorphism. On the other hand,_if_we apply our definition*
*_of
the prolongation of T above, we get a functor X 7! X S1 defined by (X S1)n =
Xn ^ S1 with structure map
Xn ^ S1 ^ S1 oe^1--!Xn+1 ^ S1:
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 7
This is a crucial and subtle difference whose ramifications we will study in Se*
*c-
tion 10.
We now show that SpN(D; T ) inherits a model structure from D, called the
projective model structure. The functor T :SpN(D; T ) -! SpN(D; T ) will be a l*
*eft
Quillen functor with respect to the projective model structure, but it will not*
* be
a Quillen equivalence. Our approach to the projective model structure owes much
to [2] and [13, Section 5.1]. At this point, we will slip into the standard mo*
*del
category terminology and notation, all of which can be found in [12], mostly in
Section 2.1.
Definition 1.7.A map f 2 SpN(D; T ) is a level equivalence if each map fn is a
weak equivalence in D. Similarly, f is a level fibration (resp. level cofibrati*
*on, level
trivial fibration, level trivial cofibration) if each map fn is a fibration (re*
*sp. cofibra-
tion, trivial fibration, trivial cofibration) in D. The map f is a projective c*
*ofibration
if f has the left lifting property with respect to every level trivial fibratio*
*n.
Note that level equivalences satisfy the two out of three property, and each *
*of the
classes defined above is closed under retracts. Thus we might be able to constr*
*uct a
model structure using these classes. To do so, we need the small object argumen*
*t,
and hence we assume that D is cofibrantly generated (see [12, Section 2.1] for a
discussion of cofibrantly generated model categories).
Definition 1.8.Suppose D is a cofibrantly generated model category with gen-
erating cofibrations I and generating trivial cofibrations J. Suppose T is a l*
*eft
Quillen endofunctor of D, and formSthe categorySof spectra SpN(D; T ). Define s*
*ets
of maps in SpN(D; T ) by IT = n FnI and JT = n FnJ.
The sets IT and JT will be the generating cofibrations and trivial cofibratio*
*ns
for a model structure on SpN(D; T ). There is a standard method for proving thi*
*s,
based on the small object argument [12, Theorem 2.1.14]. The first step is to s*
*how
that the domains of IT and JT are small, in the sense of [12, Definition 2.1.3].
Proposition 1.9.Suppose A is small relative to the cofibrations in D, and n 0.
Then FnA is small relative to the level cofibrations in SpN(D; T ). Similarly, *
*if A
is small relative to the trivial cofibrations in D, then FnA is small relative *
*to the
level trivial cofibrations in SpN(D; T ).
Proof.The main point is that Evn commutes with colimits. We leave the remainder_
of the proof to the reader. |__|
To apply this to the domains of IT, we need to know that the maps of IT-cof
are level cofibrations. See [12, Definition 2.1.7] for the definition of IT-co*
*f, and
similar notations such as IT-inj. Recall the right adjoint Mn of Evn constructe*
*d in
Remark 1.4.
Lemma 1.10. A map f in SpN(D; T ) is a level cofibration if and only if it has
the left lifting property with respect to Mnp for all n 0 and all trivial fibr*
*ations
p in D. Similarly, f is a level trivial cofibration if and only if it has the l*
*eft lifting
property with respect to Mnp for all n 0 and all fibrations p 2 D.
Proof.By adjunction, a map f has the left lifting property with respect to Mnp *
*if
and only if Evn f has the left lifting property with respect to p. Since a map *
*is a
cofibration (resp. trivial cofibration) in D if and only if it has the left lif*
*ting property_
with respect to all trivial fibrations (resp. fibrations), the lemma follows. *
* |__|
8 MARK HOVEY
Proposition 1.11.Every map in IT-cofis a level cofibration. Every map in JT-cof
is a level trivial cofibration.
Proof.Since T is a left Quillen functor, every map in IT is a level cofibration*
*. By
Lemma 1.10, this means that Mnp 2 IT-injfor all n 0 and all trivial fibrations
p. Since a map in IT-cofhas the left lifting property with respect to every map*
* in
IT-inj, in particular it has the left lifting property with respect to Mnp. Ano*
*ther
application of Lemma 1.10 completes the proof for IT-cof. The proof for_JT-cofi*
*s_
similar. |__|
Corollary 1.12.The domains of IT are small relative to IT-cof. The domains of
JT are small relative to JT-cof.
Proof.Since D is cofibrantly generated, the domains of I are small relative to *
*the
cofibrations in D, and the domains of J are small relative to the trivial cofib*
*rations
in D (see [12, Proposition 2.1.18]). Proposition 1.9 and Proposition 1.11_compl*
*ete
the proof. |__|
We remind the reader that a model structure is left proper if the pushout of
a weak equivalence through a cofibration is again a weak equivalence. Similarl*
*y,
a model structure is right proper if the pullback of a weak equivalence through*
* a
fibration is again a weak equivalence. A model structure is proper if it is bot*
*h left
and right proper. See [11, Chapter 11] for more information about properness.
Theorem 1.13. Suppose D is cofibrantly generated. Then the projective cofibra-
tions, the level fibrations, and the level equivalences define a cofibrantly ge*
*nerated
model structure on SpN(D; T ), with generating cofibrations IT and generating t*
*rivial
cofibrations JT. We call this the projective model structure. The projective mo*
*del
structure is left proper (resp. right proper, proper) if D is left proper (resp*
*. right
proper, proper).
Note that if D is either the model category of pointed simplicial sets or poi*
*nted
topological spaces, and T is the suspension functor, the projective model struc*
*ture
on SpN(D; T ) is the strict model structure on the Bousfield-Friedlander catego*
*ry of
spectra [2].
Proof.The retract and two out of three axioms are immediate, as is the lifting *
*ax-
iom for a projective cofibration and a level trivial fibration. By adjointness,*
* a map
is a level trivial fibration if and only if it is in IT-inj. Hence a map is a p*
*rojective
cofibration if and only if it is in IT-cof. The small object argument [12, The*
*o-
rem 2.1.14] applied to IT then produces a functorial factorization into a proje*
*ctive
cofibration followed by a level trivial fibration.
Adjointness implies that a map is a level fibration if and only if it is in J*
*T-inj. We
have already seen in Proposition 1.11 that the maps in JT-cofare level equivale*
*nces,
and they are projective cofibrations since they have the left lifting property *
*with
respect to all level fibrations, and in particular level trivial fibrations. He*
*nce the
small object argument applied to JT produces a functorial factorization into a
projective cofibration and level equivalence followed by a level fibration.
Conversely, we claim that any projective cofibration and level equivalence f *
*is
in JT-cof, and hence has the left lifting property with respect to level fibrat*
*ions.
To see this, write f = pi where i is in JT-cofand p is in JT-inj. Then p is a
level fibration. Since f and i are both level equivalences, so is p. Thus f has*
* the
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 9
left lifting property with respect to p, and so f is a retract of i by the retr*
*act
argument [12, Lemma 1.1.9]. In particular f 2 JT-cof.
Since colimits and limits in SpN(D; T ) are taken levelwise, and since every *
*projec-
tive cofibration is in particular a level cofibration, the statements about_pro*
*perness
are immediate. |__|
We also characterize the projective cofibrations. We denote the pushout of two
maps A -!B and A -!C by B qA C.
Proposition 1.14.A map i: A -! B of spectra is a projective cofibration if and
only if the induced maps i0: A0 -!B0 and jn :An qTAn-1 T Bn-1 -!Bn for n 1
are cofibrations in D. Similarly, i is a projective trivial cofibration if and *
*only if i0
and jn for n 1 are trivial cofibrations in D.
Proof.We only prove the cofibration case, leaving the similar trivial cofibrati*
*on
case to the reader. First suppose i: A -! B is a projective cofibration. We have
already seen in Proposition 1.11 that A0 -! B0 is a cofibration. We show that jn
is a cofibration by showing that jn has the left lifting property with respect *
*to any
trivial fibration p: X -! Y in D. So suppose we have the commutative diagram
below.
An qTAn-1 T Bn-1 ----! X
? ?
jn?y ?yp
Bn ----! Y
We must construct a lift in this diagram. By adjointness, it suffices to constr*
*uct a
lift in the induced diagram below,
A ----! MnX
? ?
i?y ?y
B ----! MnY xMn-1UY Mn-1UX
where Mn is the right adjoint of Evn. Using the description of Mn given in Re-
mark 1.4, one can check that the map MnX -! MnY xMn-1UY Mn-1UX is a level
trivial fibration, so a lift exists.
Conversely, suppose that i0 and jn are cofibrations in D for n > 0. We show
that i is a projective cofibration by showing that i has the left lifting prope*
*rty with
respect to any level trivial fibration p: X -! Y in SpN(D; T ). So suppose we h*
*ave
the commutative diagram below.
A --f--!X
? ?
i?y ?yp
B ----!gY
We construct a lift hn :Bn -! Xn, compatible with the structure maps, by induc-
tion on n. There is no difficulty defining h0, since i0 has the left lifting pr*
*operty
with respect to the trivial fibration p0. Suppose we have defined hj for j < n.*
* Then
10 MARK HOVEY
by lifting in the induced diagram below,
An qTAn-1 T Bn-1 (fn;oeOThn-1)---------!Xn
?? ?
y ?ypn
Bn ----!g Yn
n
we find the required map hn :Bn -! Xn. |___|
Finally, we point out that the prolongation of T is still a Quillen functor.
Proposition 1.15.Give SpN(D; T ) the projective model structure. Then the pro-
longation T :SpN(D; T ) -!SpN(D; T ) of T is a Quillen functor. Furthermore, the
functor Fn :D -!SpN(D; T ) is a Quillen functor.
Proof.The functor Evn obviously takes level fibrations to fibrations and level *
*trivial
fibrations to trivial fibrations. Hence Evn is a right Quillen functor, and so *
*its left
adjoint Fn is a left Quillen functor. Similarly, the prolongation of U to a fun*
*ctor
U :SpN(D; T ) -! SpN(D; T ) preserves level fibrations and level trivial fibrat*
*ions,_
so its left adjoint T is a Quillen functor. |_*
*_|
2.Bousfield localization
We will define the stable model structure on SpN(D; T ) in Section 3 as a Bou*
*sfield
localization of the projective model structure on SpN(D; T ). In this section w*
*e recall
the theory of Bousfield localization of model categories from [11].
To do so, we need some preliminary remarks related to function complexes. De-
tails can be found in [4], [11, Chapter 18], and [12, Chapter 5]. First of all,*
* given
an object A in a model category, we denote by QA a functorial cofibrant replace-
ment of A [12, p.5]. This means that QA is cofibrant and there is a natural tri*
*vial
fibration QA -! A. Similarly, RA denotes a functorial fibrant replacement of A,
so that RA is fibrant and there is a natural trivial cofibration A -! RA. By re-
peatedly using functorial factorization, we can construct, given an object A in*
* a
model category C, a functorial cosimplicial resolution of A. By mapping out of
this cosimplicial resolution we get a simplicial set Map `(A; X). Similarly, th*
*ere is
a functorial simplicial resolution of X, and by mapping into it we get a simpli-
cial set Map r(A; X). One should think of these as replacements for the simplic*
*ial
structure present in a simplicial model category. These function complexes will
not be homotopy invariant in general, so we define the homotopy function complex
as map (A; X) = Map r(QA; RX). Then map (A; X) is canonically isomorphic in
the homotopy category Ho SSet of simplicial sets to Map `(QA; RX), and defines
a functor Ho Copx HoC -! HoSSet . The homotopy function complex defines an
enrichment of Ho C over Ho SSet. In fact, Ho C is naturally tensored and coten-
sored over Ho SSet, as well as enriched over it. In particular, if is an arbi-
trary left Quillen functor between model categories with right adjoint , we have
map((L)X; Y ) ~=map (X; (R)Y ) in Ho SSet, where (L)X = QX is the total
left derived functor of and (R)Y = RY is the total right derived functor of .
Definition 2.1.Suppose we have a set S of maps in a model category C.
1. A S-local object of C is a fibrant object W such that, for every f :A -!B *
*in
S, the induced map map(B; W ) -!map (A; W ) is an isomorphism in HoSSet .
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 11
2. A S-local equivalence is a map g :A -! B in C such that the induced map
map(B; W ) -! map(A; W ) is an isomorphism in Ho SSet for all S-local ob-
jects W .
By [11, Theorem 3.3.8], S-local equivalences between S-local objects are in f*
*act
weak equivalences. In outline, one proves this by first reducing to the case wh*
*ere
f :A -! B is a cofibration and S-local equivalence between cofibrant S-local ob-
jects. Then, since f is a cofibration and A is fibrant, Map r(f; A): Map r(B; A*
*) -!
Map r(A; A) is a fibration of simplicial sets [12, Corollary 5.4.4]. Since f is*
* an S-
local equivalence and A is S-local, Map r(f; A) is also a weak equivalence, and*
* so
a trivial fibration of simplicial sets. In particular, Map r(f; A) is surjecti*
*ve. Any
preimage of the identity map is a homotopy inverse to f.
We will define cellular model categories, a special class of cofibrantly gene*
*rated
model categories, in the Appendix. The main theorem of [11] is that Bousfield
localizations of cellular model categories always exist. More precisely, Hirsch*
*horn
proves the following theorem.
Theorem 2.2. Suppose S is a set of maps in a left proper cellular model catego*
*ry C.
Then there is a left proper cellular model structure on C where the weak equiva*
*lences
are the S-local equivalences and the cofibrations remain unchanged. The S-local
objects are the fibrant objects in this model structure. We denote this new mod*
*el
category by LSC and refer to it as the Bousfield localization of C with respect*
* to
S. Left Quillen functors from LSC to D are in one to one correspondence with le*
*ft
Quillen functors : C -!D such that (Qf) is a weak equivalence for all f 2 S.
We will also need the following fact about localizations, which is implicit i*
*n [11,
Chapter 4].
Proposition 2.3.Suppose C and C0 are left proper cellular model categories, S is
a set of maps in C, and S0is a set of maps in C0. Suppose : C -!C0is a Quillen
equivalence with right adjoint , and suppose (Qf) is a S0-local equivalence for*
* all
f 2 S. Then induces a Quillen equivalence : LSC -! LS0C0 if and only if, for
every S-local X 2 C, there is a S0-local Y in C0such that X is weakly equivalen*
*t in
C to Y . This condition will hold if, for all fibrant Y in C0such that Y is S-l*
*ocal,
Y is S0-local.
Proof.Suppose first that does induce a Quillen equivalence on the localization*
*s,
and suppose that X is S-local. Then QX is also S-local, by [11, Lemma 3.3.1]. L*
*et
LS0 denote a fibrant replacement functor in LS0C0. Then, because is a Quillen
equivalence on the localizations, the map QX -! LS0QX is a weak equivalence
in LSC (see [12, Section 1.3.3]). But both QX and LS0QX are S-local, so QX -!
LS0QX is a weak equivalence in C. Hence X is weakly equivalent in C to Y ,
where Y is the S0-local object LS0QX.
The first step in proving the converse is to note that, since is a Quillen e*
*quiv-
alence before localizing, the map QX -! X is a weak equivalence for all fibrant
X. Since the functor Q does not change upon localization, QX -! X is a S0-local
equivalence for every S0-local object of C0. Thus is a Quillen equivalence af*
*ter
localization if and only if reflects local equivalences between cofibrant obje*
*cts,
by [12, Corollary 1.3.16].
Suppose, then, that f :A -! B is a map between cofibrant objects such that
f is a S0-local equivalence. We must show that map (f; X) is an isomorphism in
Ho SSet for all S-local X. Adjointness implies that map (f; Y ) is an isomorphi*
*sm
12 MARK HOVEY
for all S0-local Y , and our condition then guarantees that this is enough to c*
*onclude
that map (f; X) is an isomorphism for all S-local X. This completes the proof of
the converse.
We still need to prove the last statement of the proposition. So suppose X
is S-local. Then QX is also S-local, again by [11, Lemma 3.3.1], and, in C, we
have a weak equivalence QX -! RQX. Our assumption then guarantees that __
Y = RQX is S0-local, and X is indeed weakly equivalent to Y . |__|
The fibrations in LSC are not completely understood [11, Section 3.6]. The
S-local fibrations between S-local fibrant objects are just the usual fibration*
*s. In
case both C and LSC are right proper, there is a characterization of the S-local
fibrations in terms of homotopy pullbacks analogous to the characterization of *
*stable
fibrations of spectra in [2]. However, LSC need not be right proper even if C i*
*s, as
is shown by the example of -spaces in [2], where it is also shown that the expe*
*cted
characterization of S-local fibrations does not hold.
3. The stable model structure
Our plan now is to apply Bousfield localization to the projective model struc*
*ture
on SpN(D; T ) to obtain a model structure with respect to which T is a Quillen
equivalence. In order to do this, we will have to prove that the projective mod*
*el
structure makes SpN(D; T ) into a cellular model category when D is left proper
cellular. We will prove this technical result in the appendix. In this sectio*
*n, we
will assume that SpN(D; T ) is cellular, find a good set S of maps to form the *
*stable
model structure as the S-localization of the projective model structure, and pr*
*ove
that T is a Quillen equivalence with respect to the stable model structure.
Just as in symmetric spectra [13], we want the stable equivalences to be maps
which induce isomorphisms on all cohomology theories. Cohomology theories will
be represented by the appropriate analogue of -spectra.
Definition 3.1.A spectrum X is a U-spectrum if X is level fibrant and the adjoi*
*nt
Xn oee-!UXn+1 of the structure map of X is a weak equivalence for all n 0.
Of course, if D is the category of pointed simplicial sets or pointed topolog*
*ical
spaces, and T is the suspension functor, U-spectra are just -spectra. We will f*
*ind a
set S of maps of SpN(D; T ) such that the S-local objects are the U-spectra. To*
* do so,
note that if map (A; Xn) -! map(A; UXn+1) is an isomorphism in Ho SSet for all
cofibrant A in D, then Xn -! UXn+1 is a weak equivalence by [11, Theorem 18.8.7*
*].
Since D is cofibrantly generated, we should not need all cofibrant A, but only *
*those
A related to the generating cofibrations. This is true, but the proof is somewh*
*at
technical.
Proposition 3.2.Suppose C is a left proper cofibrantly generated model category
with generating cofibrations I, and f :X -! Y is a map in C. Then f is a weak
equivalence if and only if map (C; X) -!map (C; Y ) is an isomorphism in Ho SSet
for all domains and codomains C of maps of I.
This proof will depend on the fact that Map r(-; RZ) converts colimits in C
to limits of simplicial sets, cofibrations in C to fibrations of simplicial set*
*s, trivial
cofibrations in C to trivial fibrations of simplicial sets, and weak equivalenc*
*es be-
tween cofibrant objects in C to weak equivalences between fibrant simplicial se*
*ts.
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 13
These properties follow from [12, Corollary 5.4.4] and Ken Brown's lemma [12,
Lemma 1.1.12].
Proof.The only if half follows from [11, Theorem 18.8.7]. Conversely, suppose t*
*hat
map(C; f) is an isomorphism in Ho SSet for all domains and codomains of maps
of I. It suffices to show that map (A; f) is an isomorphism for all cofibrant o*
*bjects
A, by [11, Theorem 18.8.7]. But every cofibrant object is a retract of an I-ce*
*ll
complex (i.e. an object A such that the map 0 -! A is a transfinite composition
of pushouts of maps of I), so it suffices to prove that map (A; f) is an isomor*
*phism
for all cell complexes A. This is equivalent to showing that Map r(A; Rf) is a *
*weak
equivalence for all cell complexes A. Given a cell complex A, there is an ordin*
*al
and a -sequence
0 = A0 -!A1 -!: :-:!Afi-! : : :
with colimit A = A, where each map ifi:Afi-! Afi+1is a pushout of a map
of I. We will show by transfinite induction on fi that Map r(Afi; Rf) is a weak
equivalence for all fi . Taking fi = completes the proof.
The base case of the induction is trivial, since A0 = 0. For the successor
ordinal case, we suppose Map r(Afi; Rf) is a weak equivalence and prove that
Map r(Afi+1; Rf) is a weak equivalence. We have the pushout square below,
C ----! Afi
? ?
g?y ?yifi
D ----! Afi+1
where g is a map of I. We must first replace this pushout square by a weakly
equivalent pushout square in which all the objects are cofibrant, which we can *
*do
because C is left proper. Begin by factoring the composite QC -! C -! D into a
cofibration eg:QC -! eDfollowed by a trivial fibration eD-! D. In the terminolo*
*gy
of [11], egis a cofibrant approximation to g. By [11, Proposition 11.3.2], ther*
*e is a
cofibrant approximation eifi:fAfi-!]Afi+1to ifiwhich is a pushout of eg. That i*
*s,
we have constructed the pushout square below,
QC ----! fAfi
? ?
eg?y ?yeifi
eD ----! ]Afi+1
and a map from this pushout square to the original one that is a weak equivalen*
*ce
at each corner. By the properties of Map r(-; RZ) mentioned in the paragraph
preceding this proof, we have two pullback squares of fibrant simplicial sets as
below,
Map r(A]fi+1; RZ)----! Map r(De; RZ)
?? ?
y ?y
Map r(Affi; RZ)----! Map r(QC; RZ)
where Z = X and Z = Y , respectively. Here the vertical maps are fibrations.
There is a map from the square with Z = X to the square with Z = Y induced by
f. By hypothesis, this map is a weak equivalence on every corner except possibly
14 MARK HOVEY
the upper left. But then Dan Kan's cube lemma (see [12, Lemma 5.2.6], where the
dual of the version we need is proved, or [4]) implies that the map on the uppe*
*r left
corner Map r(A]fi+1; Rf) is also a weak equivalence. Since Map r(-; RZ) preserv*
*es
weak equivalences between cofibrant objects for any Z (see the paragraph preced*
*ing
this proof), it follows that Map r(Afi+1; Rf) is a weak equivalence.
We must still carry out the limit ordinal case of the induction. Suppose fi i*
*s a
limit ordinal and Map r(Afl; Rf) is a weak equivalence for all fl < fi. We must*
* show
that Map r(Afi; Rf) is a weak equivalence. For Z = X or Z = Y , the simplicial *
*sets
Map r(Afl; RZ) define a limit-preserving functor fiop -! SSet such that each map
Map r(Afl+1; RZ) -! Map r(Afl; RZ) is a fibration of fibrant simplicial sets, u*
*sing
the properties of Map r(-; RZ) mentioned in the paragraph preceding this proof.
There is a natural transformation from the functor with Z = X to the functor
with Z = Y , and by hypothesis this map is a weak equivalence at every stage. As
explained in Section 5.1 of [12], there is a model structure on functors fiop -*
*!SSet
where the weak equivalences and fibrations are taken levelwise. Both diagrams
Map r(Afl; RX) and Map r(Afl; RY ) are fibrant, since each simplicial set in th*
*em is
fibrant. The inverse limit is a right Quillen functor [12, Corollary 5.1.6], a*
*nd so
preserves weak equivalences between fibrant objects by Ken Brown's lemma [12,
Lemma 1.1.12]. Thus the inverse limit Map r(Afi; Rf) is a weak equivalence, as*
* __
required. This completes the transfinite induction and the proof. *
*|__|
Note that the left properness assumption in Proposition 3.2 is unnecessary wh*
*en
the domains of the generating cofibrations are themselves cofibrant, since ther*
*e is
then no need to apply cofibrant approximation.
In view of Proposition 3.2, we need to choose our set S so as to make
map (C; Xn) -!map (C; UXn+1)
an isomorphism in HoSSet for all S-local objects X and all domains and codomains
C of the generating cofibrations I. Adjointness implies that, if X is level fib*
*rant,
map(C; Xn) ~=map (FnQC; X) in Ho SSet, since FnQC = (LFn)C, where LFn is
the total left derived functor of Fn. Also, map (C; UXn+1) ~=map (Fn+1T QC; X).
In view of this, we make the following definition.
Definition 3.3.Suppose D is a left proper cellular model category with generati*
*ng
cofibrations I, and T is a left Quillen endofunctor of D. Define the set S of m*
*aps in
QC
SpN(D; T ) as {Fn+1T QC -in-!FnQC}, as C runs through the set of domains and
codomains of the maps of I and n runs through the non-negative integers. Here
the map iQCn is adjoint to the identity map of T QC, and so is an isomorphism in
degrees greater than n. Define the stable model structure on SpN(D; T ) to be t*
*he
localization of the projective model structure on SpN(D; T ) with respect to th*
*is
set S. We refer to the S-local weak equivalences as stable equivalences, and to*
* the
S-local fibrations as stable fibrations.
The referee points out that Definition 3.3 is an implementation of Adams' "ce*
*lls
now_maps later" philosophy [1, p.142]. Indeed, a map FnQC -e!X can be thought
of as a cell of the spectrum X, at least when C is a codomain of one of the gen*
*erating
cofibrations of I. Inverting the map Fn+1T QC -! FnQC is tantamount to allowing
a map from X to Y to be defined on the cell e only after applying T some number
of times.
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 15
Theorem 3.4. Suppose D is a left proper cellular model category and T is a left
Quillen endofunctor of D. Then the stably fibrant objects in SpN(D; T ) are the
U-spectra. Furthermore, for all cofibrant A 2 D and for all n 0, the map
A
Fn+1T A in-!FnA is a stable equivalence.
Proof.By definition, X is S-local if and only if X is level fibrant and
map (FnQC; X) -!map (Fn+1T QC; X)
is an isomorphism in Ho SSet for all n 0 and all domains and codomains C of
maps of I. By the comments preceding Definition 3.3, this is equivalent to requ*
*iring
that X be level fibrant and that the map map (C; Xn) -! map (C; UXn+1) be an
isomorphism for all n 0 and all domains and codomains C of maps of I. By
Proposition 3.2, this is equivalent to requiring that X be a U-spectrum.
Now, by definition, iAn is a stable equivalence if and only if map (iAn; X) i*
*s a
weak equivalence for all U-spectra X. But by adjointness, map (iAn; X) can be
identified with map (A; Xn) -! map (A; UXn+1). Since Xn -! UXn+1 is a weak
equivalence between fibrant objects, map (iAn; X) is an isomorphism in Ho SSet,_
by [12, Corollary 5.4.8]. |__|
We would now like to claim that the stable model structure on SpN(D; T ) that
we have just defined is a generalization of the stable model structure on spect*
*ra
of topological spaces or simplicial sets defined in [2]. This cannot be a triv*
*ial
observation, however, both because our approach is totally different and becaus*
*e of
Remark 1.6.
Corollary 3.5.If D is either the category of pointed simplicial sets or pointed
topological spaces, and T is the suspension functor given by smashing with S1, *
*then
the stable model structure on SpN(D; T ) coincides with the stable model struct*
*ure
on the category of Bousfield-Friedlander spectra [2].
Proof.We know already that the cofibrations are the same in the stable model
structure on SpN(D; T ) and the stable model structure of [2]. We will show that
the weak equivalences are the same. In any model category at all, a map f is a
weak equivalence if and only if map (f; X) is an isomorphism in Ho SSet for all
fibrant X, by [11, Theorem 18.8.7]. Construction of map(f; X) requires replacin*
*g f
by a cofibrant approximation f0 and building cosimplicial resolutions of the do*
*main
and codomain of f0. In the case at hand, we can do the cofibrant replacement and
build the cosimplicial resolutions in the projective model category of spectra,*
* since
the cofibrations do not change under localization. Thus map (f; X) is the same *
*in
both the stable model structure on SpN(D; T ) and in the stable model category *
*of
Bousfield and Friedlander. Since the stably fibrant objects are also the_same,_*
*the
corollary holds. |__|
We now begin the process of proving that the prolongation of T is a Quillen
equivalence with respect to the stable model structure on SpN(D; T ).
Lemma 3.6. Suppose D is a left proper cellular model category and T is a left *
*Quil-
len endofunctor of D. Then the prolongation of T to a functor T :SpN(D; T ) -!
SpN(D; T ) is a left Quillen functor with respect to the stable model structure.
Proof.In view of Hirschhorn's localization theorem 2.2, we must show that T (Qf)
is a stable equivalence for all f 2 S. Since the domains and codomains of the m*
*aps
16 MARK HOVEY
of S are already cofibrant, it is equivalent to show that T f is a stable equiv*
*alence
for all f 2 S. Since T Fn = FnT , we have T (iAn) = iTAn. In view of Theorem 3.*
*4,
this map is a stable equivalence whenever A, and hence T A, is cofibrant. Takin*
*g_
A = QC, where C is a domain or codomain of a map of I, completes the proof. |_*
*_|
We will now show that T is in fact a Quillen equivalence with respect to the
stable model structure. To do so, we introduce the shift functors.
Definition 3.7.Suppose D is a model category and T is a left Quillen endofunctor
of D. Define the shift functors s+ :SpN(D; T ) -!SpN(D; T ) and s- :SpN(D; T ) *
*-!
SpN(D; T ) by (s- X)n = Xn+1, (s+ X)n = Xn-1 for n > 0, and (s+ X)0 = 0, with
the evident structure maps. Note that s+ is left adjoint to s- .
Lemma 3.8. Suppose D is a left proper cellular model category and T is a left
Quillen endofunctor of D. Then:
(a) The shift functor s+ is a left Quillen functor with respect to the project*
*ive
model structure on SpN(D; T );
(b) The shift functor s+ commutes with T and s- commutes with U;
(c) We have s+ Fn = Fn+1 and Evn s- = Evn+1.
(d) The shift functor s+ is a left Quillen functor with respect to the stable *
*model
structure on SpN(D; T ).
Proof.For part (a), it is clear that s- preserves level equivalences and level *
*fi-
brations, so s- is a right Quillen functor with respect to the projective model
structure. Parts (b) and (c) we leave to the reader, except to note that adjoi*
*nt-
ness makes the two halves of part (b) equivalent, and similarly the two halves *
*of
part (c). For part (d), note that Theorem 2.2 implies that s+ defines a left Qu*
*illen
functor with respect to the stable model structure as long as s+ (iQCn) is a st*
*able
equivalence for all domains and codomains C of the generating cofibrations of D.
However, parts (b) and (c) imply that s+ (iQCn) = iQCn+1, which is certainly_a *
*stable
equivalence. |__|
We now prove that T is a Quillen equivalence with respect to the stable model
structure by comparing the T and U adjunction to the s+ and s- adjunction.
Theorem 3.9. Suppose D is a left proper cellular model category and T is a left
Quillen endofunctor of D. Then the functors T :SpN(D; T ) -! SpN(D; T ) and
s+ :SpN(D; T ) -! SpN(D; T ) are Quillen equivalences with respect to the stable
model structures. Furthermore, Rs- is naturally isomorphic to LT , and RU is
naturally isomorphic to Ls+ .
Proof.The maps Xn -! UXn+1 adjoint to the structure maps of a spectrum X
define a natural map of spectra X -! s- UX. This map is a stable equivalence
(in fact, a level equivalence) when X is a stably fibrant object of SpN(D; T ).*
* This
means that the total right derived functor R(s- U) is naturally isomorphic to t*
*he
identity functor on Ho SpN(D; T ) (where we use the stable model structure). On
the other hand, R(s- U) is naturally isomorphic to Rs- O RU and also to RU O
Rs- , since s- and U commute with each other. Thus the natural isomorphism
from the identity to R(s- U) gives rise to an natural isomorphism 1 -! Rs- O RU
and a natural isomorphism RU O Rs- -! 1. Therefore Rs- and RU are inverse
equivalences of categories, and so both s- and U are Quillen equivalences. Since
inverse equivalences of categories can always be made into adjoint equivalences,
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 17
Rs- and RU are in fact adjoint equivalences. Since LT and Rs- are both left
adjoint to RU, LT and Rs- are naturally isomorphic. Similarly, since Ls+ and __
RU are both left adjoint to Rs- , Ls+ and RU are naturally isomorphic. |_*
*_|
We note that Theorem 3.9, when applied to the Bousfield-Friedlander model
category of spectra_[2],_shows that the suspension functor without the twist (s*
*ee
Remark 1.6), X 7! X S1, is a Quillen equivalence. However, Theorem 3.9 does
not show that the suspension functor with the twist, X 7! X S1, is a Quillen
equivalence. Indeed, the maps Xn -! Xn+1 only define a map of spectra if we do
not put in the extra twist. We will discuss this issue further in Section 10. S*
*ee also
Remark 6.4.
4. The almost finitely generated case
The reader may well object at this point that we have defined the stable model
structure on SpN(D; T ) without ever defining stable homotopy groups. This is
because stable homotopy groups do not detect stable equivalences in general. The
usual simplicial and topological situation is very special. The goal of this se*
*ction is
to put some hypotheses on D and T so that the stable model structure on SpN(D; *
*T )
behaves similarly to the stable model structure on ordinary simplicial spectra.*
* In
particular, we show that, if D is almost finitely generated (defined below), se*
*quential
colimits in D preserve finite products, and U preserves sequential colimits, th*
*en the
usual 1 1 kind of construction gives a stable fibrant replacement functor. This
implies that a map f is a stable equivalence if and only if the analogue of 1 1*
* f
is a level equivalence. This allows us to characterize Ho SpN(D; T )(F0A; X) f*
*or
well-behaved A as the usual sort of colimit colimHo D(T nA; Xn). It also allows
us to prove that the stable model structure is right proper, under slightly more
hypotheses, so we get the expected characterization of stable fibrations.
Most of the results in this section do not depend on the existence of the sta*
*ble
model structure on SpN(D; T ), so we do not usually need to assume D is left pr*
*oper
cellular.
We now define almost finitely generated model categories, as suggested to the
author by Voevodsky.
Definition 4.1.An object A of a category C is called finitely presented if the
functor C(A; -) preserves direct limits of sequences X0 -!X1 -!: :-:!Xn -! : :.:
A cofibrantly generated model category C is said to be finitely generated if the
domains and codomains of the generating cofibrations and the generating trivial
cofibrations are finitely presented. A cofibrantly generated model category is *
*said
to be almost finitely generated if the domains and codomains of the generating
cofibrations are finitely presented, and if there is a set of trivial cofibrati*
*ons J0 with
finitely presented domains and codomains such that a map f whose codomain is
fibrant is a fibration if and only if f has the right lifting property with res*
*pect to
J0.
This definition differs slightly from other definitions. In particular, an o*
*bject
A is usually said to be finitely presented [26, Section V.3] if C(A; -) preserv*
*es all
directed (or, equivalently, filtered) colimits. We are trying to assume the min*
*imum
necessary. Finitely generated model categories were introduced in [12, Section *
*7.4],
but in that definition we assumed only that C(A; -) preserves (transfinitely lo*
*ng)
direct limits of sequences of cofibrations. The author would now prefer to call*
* such
18 MARK HOVEY
model categories compactly generated. Thus, the model category of simplicial se*
*ts is
finitely generated, but the model category on topological spaces is only compac*
*tly
generated. Since we will only be working with (almost) finitely generated model
categories in this section, our results will not apply to topological spaces. W*
*e will
indicate where our results fail for compactly generated model categories, and a
possible way to amend them in the compactly generated case.
The definition of an almost finitely generated model category was suggested by
Voevodsky. The problem with finitely generated, or, indeed, compactly generated,
model categories is that they are not preserved by localization. That is, if C *
*is a
finitely generated left proper cellular model category, and S is a set of maps,*
* then
the Bousfield localization [11] LSC will not be finitely generated, because we *
*lose
control of the generating trivial cofibrations in LSC. However, we will show in*
* the
following proposition that Bousfield localization sometimes does preserve almost
finitely generated model categories.
Recall from [12, Chapter 5] that it is possible to define X K for an object
X in a model category and a simplicial set K, even if the model category is not
simplicial.
Proposition 4.2.Let C be a left proper, cellular, almost finitely generated mod*
*el
category, and S be a set of cofibrations in C. Suppose that, for every domain *
*or
codomain X of S and every finite simplicial set K, X K is finitely presented.
Then the Bousfield localization LSC is almost finitely generated.
Proof.Since C is almost finitely generated, there is a set J0 of trivial cofibr*
*ations
so that a map p whose codomain is fibrant is a fibration if and only if p has t*
*he
right lifting property with respect to J0. Let (S) denote the set of maps
(A [n]) qAk[n] (B k[n]) -!B [n]
where A -!B is a map of S, n 0, [n] is the standard n-simplex, and k[n] for
0 k n is the horn obtained from [n] by removing the nondegenerate n-simplex
and the nondegenerate n - 1-simplex not containing vertex k. As explained in [1*
*1,
Proposition 4.2.4], a fibrant object X is S-local if and only if the map X -! 1*
* has
the right lifting property with respect to (S). Since an S-local fibration betw*
*een
S-local objects is just an ordinary fibration [11, Section 3.6], the set (S) [ *
*J0
will detect fibrations between fibrant objects in LSC, and therefore LSC is_alm*
*ost
finitely generated. |__|
Voevodsky has informed the author that he can make an unstable motivic model
category that is almost finitely generated. For the reader's benefit, we summar*
*ize
his construction. This summary will of necessity assume some familiarity with b*
*oth
the language of algebraic geometry and Voevodsky's central idea [27]. We begin
with the category E of simplicial presheaves (of sets) on the category of smooth
schemes over some base scheme k. There is a projective model structure on this
category, where a map of simplicial presheaves X -! Y is a weak equivalence
(resp. fibration) if and only if the map X(U) -! Y (U) is a weak equivalence
(resp. fibration) of simplicial sets for all U. The projective model structur*
*e is
finitely generated (using the fact that smooth schemes over k is an essentially
small category). There is an embedding of smooth schemes into E as representable
functors. We need to localize this model structure to take into account both the
Nisnevich topology and the fact that the functor X 7! X x A1 should be a Quillen
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 19
equivalence. To do so, we define a set S0 to consist of the maps X x A1 -!X for
every smooth scheme X and maps P -! X for every pullback square of smooth
schemes
B ----! Y
?? ?
y ?yp
A ----! X
j
where p is etale, j is an open embedding, and p-1(X - A) -! X - A is an iso-
morphism. Here P is the mapping cylinder (B q Y ) qBqB (A x [1]). We then
define S to consist of mapping cylinders on the maps of S0. The maps of S are t*
*hen
cofibrations whose domains and codomains are finitely presented (and remain so
after tensoring with any finite simplicial set), so the Bousfield localization *
*C = LSE
will be almost finitely generated.
There is then some work involving properties of the Nisnevich topology to show
that this model category is equivalent to the Morel-Voevodsky motivic model cat-
egory of [19], and to the model category used by Jardine [14]. Given this, if w*
*e let
T be the endofunctor of C which takes X to X x A1, then SpN(C; T ) is a model f*
*or
Voevodsky's stable motivic category.
The essential properties of almost finitely generated model categories that we
need are contained in the following lemma.
Lemma 4.3. Suppose C is an almost finitely generated model category.
1. If
X0 -!X1 -!: :-:!Xn -! : : :
is a sequence of fibrant objects, then colimXn is fibrant.
2. Suppose we have the commutative diagram below.
X0 ----! X1 ----! X2 ----! : : :----!Xn ----! : : :
? ? ? ?
p0?y p1?y p2?y pn?y
Y0 ----! Y1 ----! Y2 ----! : : :----! Yn ----! : : :
If each pn is a trivial fibration, so is colimpn. If each pn is a fibratio*
*n between
fibrant objects, so is colimpn.
Proof.Let J0denote a set of trivial cofibrations in C with finitely presented d*
*omains
and codomains that detect fibrations with fibrant codomain. For part (a), it su*
*ffices
to show that colimXn -! 1 has the right lifting property with respect to J0. But
this is clear, since any map from a domain of J0 to colimXn factors through som*
*e_
Xk, and Xk is fibrant. The second part is proved similarly. |_*
*_|
We now consider the stable model structure on SpN(D; T ) when D is an almost
finitely generated model category and T is a left Quillen endofunctor of D. In
analogy with ordinary Bousfield-Friedlander spectra, there is an obvious candid*
*ate
for a stable fibrant replacement of a spectrum X.
Definition 4.4.Suppose T is a left Quillen endofunctor of a model category D
with right adjoint U. Define : SpN(D; T ) -! SpN(D; T ) to be the functor s- U,
20 MARK HOVEY
where s- is the shift functor. Then we have a natural map X :X -! X, and we
define
2X n-1X nX
1 X = colim(X -X! X -X--!2X ---! : :-:----!nX ---! : :)::
Let jX :X -! 1 X denote the obvious natural transformation.
The following lemma, though elementary, is crucial.
Lemma 4.5. The maps X ; X :X -! 2X coincide.
Proof.The map X :Xn -! UXn+1 is the adjoint eoeof the structure map of X.
Hence X is just Uoeein each degree. Since the adjoint of the structure map_of UX
is just Uoee(see Lemma 1.5), X = X . |__|
We stress that Lemma 4.5 fails for symmetric spectra, and it is the major rea-
son we must work with finitely generated model categories rather than compactly
generated model categories. Indeed, in the compactly generated case, 1 is not a
good functor, since maps out of one of the domains of the generating cofibratio*
*ns
will not preserve the colimit that defines 1 X. The obvious thing to try is to
replace the functor by a functor W , obtained by factoring X -! X into a pro-
jective cofibration X -! W X followed by a level trivial fibration W X -! X. The
difficulty with this plan is that we do not see how to prove Lemma 4.5 for W . *
*An
alternative plan would be to use the mapping cylinder X -! W 0X on X -! X;
this might make Lemma 4.5 easier to prove, but the map X -! W 0X will not be a
cofibration. The map X -! W 0X may, however, be good enough for the required
smallness properties to hold. It is a closed inclusion if D is topological spac*
*es, for
example. The author knows of no good general theorem in the compactly generated
case.
This lemma leads immediately to the following proposition.
Proposition 4.6.Suppose T is a left Quillen endofunctor of a model category D,
and suppose that its right adjoint U preserves sequential colimits. Then the map
1 X :1 X -! (1 X) is an isomorphism. In particular, if X is level fibrant,
D is almost finitely generated, and U preserves sequential colimits, then 1 X i*
*s a
U-spectrum.
Proof.The map 1 X is the colimit of the vertical maps in the diagram below.
2X n-1X nX
X --X--! X -X---! 2X ----! : : :-----! nX ----! : : :
? ? ? ?
X ?y X ?y 2X ?y nX ?y
X ----! 2X ----! 3X ----! : : :----! n+1X -----! : : :
X 2X 3X nX n+1X
Since the vertical and horizontal maps coincide, the result follows. For the se*
*cond
statement, we note that if X is level fibrant, then each nX is level fibrant si*
*nce
is a right Quillen functor (with respect to the projective model structure). S*
*ince
sequential colimits in D preserve fibrant objects by Lemma 4.3, 1 X is level_
fibrant, and hence a U-spectrum. |__|
Proposition 4.7.Suppose T is a left Quillen endofunctor of a model category D
with right adjoint U. If D is almost finitely generated, and X is a U-spectrum,*
* then
the map jX :X -! 1 X is a level equivalence.
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 21
Proof.By assumption, the map X :X -! X is a level equivalence between level
fibrant objects. Since is a right Quillen functor, nX is a level equivalence as
well. Then the method of [12, Corollary 7.4.2] completes the proof. Recall th*
*at
this method is to use factorization to construct a sequence of projective trivi*
*al
cofibrations Yn -! Yn+1 with Y0 = X and a level trivial fibration of sequences
Yn -! nX. Then the map X -! colimYn will be a projective trivial cofibration.
Since sequential colimits in D preserve trivial fibrations by Lemma 4.3, the ma*
*p_
colimYn -! 1 X will still be a level trivial fibration. |*
*__|
Proposition 4.7 gives us a slightly better method of detecting stable equival*
*ences.
Corollary 4.8.Suppose T is a left Quillen endofunctor of a model category D with
right adjoint U. Suppose D is almost finitely generated and U preserves sequent*
*ial
colimits. Then a map f :A -! B is a stable equivalence in SpN(D; T ) if and only
if map (f; X) is an isomorphism in Ho SSet for all level fibrant spectra X such*
* that
X :X -! X is an isomorphism.
Proof.By definition, f is a stable equivalence if and only if map (f; Y ) is an*
* iso-
morphism for all U-spectra Y . But we have a level equivalence Y -! 1 Y by
Proposition 4.7, and so it suffices to know that map (f; 1 Y ) is an isomorphis*
*m__
for all U-spectra Y . But, by Proposition 4.6, 1 Y is an isomorphism. *
*|__|
This corollary, in turn, allows us to prove that 1 detects stable equivalenc*
*es.
The following theorem is similar to [13, Theorem 3.1.11].
Theorem 4.9. Suppose T is a left Quillen endofunctor of a model category D with
right adjoint U. Suppose that D is almost finitely generated and sequential col*
*imits
in D preserve finite products. Suppose also that U preserves sequential colimit*
*s. If
f :A -! B is a map in SpN(D; T ) such that 1 f is a level equivalence, then f is
a stable equivalence.
Before we can prove this theorem, however, we need to study how 1 interacts
with the enrichment map (X; Y ). This requires some model category theory based
on [12, Chapter 5].
Lemma 4.10. Suppose H :D -!E is a functor between model categories that pre-
serves fibrant objects, weak equivalences between fibrant objects, fibrations b*
*etween
fibrant objects, and finite products. Let C be a cofibrant object of D and let *
*X be a
fibrant object of D. Then there is a natural map map (C; X) -ae!map(HC; HX) in
Ho SSet.
Proof.This proof will assume familiarity with [12, Chapter 5]. In particular, *
*we
need the notions of a simplicial frame X* on a fibrant object X in a model cate*
*gory C
from [12, Section 5.2] and the associated functor SSet -!C that takes K to (X*)*
*K .
Here (X*)n = (X*)[n], and the recipe for building (X*)K from the (X*)[n]'s is
derived from the recipe for building K rom the [n]'s (see [12, Proposition 3.1.*
*5]).
Recall that a simplicial frame X* on X is a simplicial object in D with X0 ~=X *
*and
a factorization `oX -! X* -!roX into a weak equivalence followed by a fibration
in the category of simplicial objects in D. Here `oX is the constant simplicial
object on X, (roX)n+1 is the n + 1-fold product of X, the map `oX -! roX is the
diagonal map. The hypotheses on H guarantee that, if X* is a simplicial frame on
the fibrant object X, then H(X*) is a simplicial frame on HX. This is the key f*
*act
that this lemma relies on.
22 MARK HOVEY
Now, given a choice of functorial factorization on C, there is a canonical si*
*mplicial
frame XO associated to X, and the associated functor defines the cotensor actio*
*n of
Ho SSeton HoD by taking (K; X) to XK = (XO)K . For any other simplicial frame
X* there is a weak equivalence X* -!XO inducing an isomorphism (X*)K -! XK
in Ho C that is natural in K and only depends on the isomorphism (X*)0 -! X
(see [12, Lemma 5.5.2]). In particular, there is a weak equivalence of simplic*
*ial
frames H(XO) -! (HX)O inducing an isomorphism H(XK ) -! (HX)K in Ho E
that is natural in both X and K.
Finally, we have an adjointness isomorphism
Ho SSet(K; map(C; X)) ~=Ho D(C; XK ):
Thus, the identity map of map(C; X) gives us a map C -! Xmap(C;X)in HoD. Since
C is cofibrant and X is fibrant, this map is represented by a map C -! Xmap(C;X)
in D. By applying H, we get a map
HC -! H(Xmap(C;X)) ~=(HX)map(C;X):
Then, applying adjointness again, we get the desired natural map map_(C;_X) -!
map(HC; HX): |__|
With this lemma in hand, we can prove Theorem 4.9.
Proof of Theorem 4.9.We are given a map f such that 1 f is a level equiva-
lence. Since D is almost finitely generated and U preserves sequential colimit*
*s,
1 preserves level trivial fibrations. Therefore, 1 (Qf) is also a level equiv*
*a-
lence. Hence we may as well assume that f is a map of cofibrant spectra. Now,
suppose X is a U-spectrum such that the map X :X -! X is an isomorphism.
By Corollary 4.8, it suffices to show that map(f; X) is an isomorphism in Ho SS*
*et.
Since 1 f is a level equivalence, map(1 f; 1 X) is an isomorphism. It therefore
suffices to show that map (f; X) is a retract of map (1 f; 1 X). We will prove
this by showing that map(C; X) is naturally a retract of map(1 C; 1 X) for any
cofibrant spectrum C. Our hypotheses guarantee that 1 preserves level fibrant
objects, level fibrations between level fibrant objects, and all level trivial *
*fibrations,
since D is almost finitely generated. Furthermore, 1 preserves finite product*
*s,
since sequential colimits commute with finite products. Thus, Lemma 4.10 gives
us a natural map map (C; X) -! map(1 C; 1 X). There is also a natural map
ffC : map(1 C; 1 X) -!map (C; X) defined as the composite
1 X) map(C;j-1X)
map (1 C; 1 X) map(jC;---------!map(C; 1 X) -------! map(C; X);
where we have used the fact that iX is an isomorphism to conclude that jX is al*
*so
an isomorphism. Naturality means that, given a map g :C -! D, we have the
commutative diagram below.
1 g;1 X)
map (1 D; 1 X) -map(----------!map(1 C; 1 X)
? ?
ffD?y ?yffC
map(D; X) ------! map (C; X)
map(g;X)
We claim that the composite map (C; X) -!map (1 C; 1 X) -!map (C; X) is
the identity, so that map (C; X) is naturally a retract of map (1 C; 1 X). This
argument, which will depend heavily on the method of Lemma 4.10, will complete
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 23
the proof. Let fl :C -! Xmap(C;X)denote the adjoint of the identity of map(C; X*
*).
Then the composite map (C; X) -! map(1 C; 1 X) -! map(C; X) is adjoint to
the counter-clockwise composite in the following commutative diagram.
C --fl--! Xmap(C;X) _______ Xmap(C;X)
? ? ?
j?y ?yj ?yffi
~=
1 C ----! 1 (Xmap(C;X)) ----! (1 X)map(C;X) ---------! Xmap(C;X)
Qfl ~= (jmap(C;X)X)-1
The left square of this diagram is commutative because j is natural. There is
some choice of ffi that makes the right square commutative, but we claim that f*
*fi =
jmap(C;X)Xwill work. This completes the proof, and follows from [12, Lemma 5.5.*
*2],
because the natural isomorphism 1 (XK ) -! (1 X)K is induced by a map of *
*__
simplicial frames which is the identity in degree 0. |*
*__|
Corollary 4.11.Let T be a left Quillen endofunctor of a model category D with
right adjoint U. Suppose that D is almost finitely generated, that sequential c*
*olim-
its in D preserve finite products, and that U preserves sequential colimits. T*
*hen
jA :A -!1 A is a stable equivalence for all A 2 SpN(D; T ).
Proof.One can easily check that 1 jA is an isomorphism, using Proposition_4.6.
|__|
Finally, we get the desired characterization of stable equivalences.
Theorem 4.12. Let T be a left Quillen endofunctor of a model category D with
right adjoint U. Suppose that D is almost finitely generated, that sequential c*
*olimits
in D preserve finite products, and that U preserves sequential colimits. Let L0*
*denote
a fibrant replacement functor in the projective model structure on SpN(D; T ). *
*Then,
for all A 2 SpN(D; T ), the map A -! 1 L0A is a stable equivalence into a U-
spectrum. Also, a map f :A -! B is a stable equivalence if and only if 1 L0f is
a level equivalence.
Proof.The first statement follows immediately from Proposition 4.6 and Corol-
lary 4.11. By the first statement, if f is a stable equivalence, so is 1 L0f. S*
*ince
1 L0f is a map between U-spectra, it is a stable equivalence if and only if it_*
*is a
level equivalence. The converse follows from Theorem 4.9. |__|
Since we did not need the existence of the stable model structure to prove Th*
*eo-
rem 4.12, one can imagine attempting to construct it from the functor 1 L0. This
is, of course, the original approach of Bousfield-Friedlander [2], and this app*
*roach
has been generalized by Schwede [21]. Also, if there is some functor F , like h*
*omo-
topy groups, that detects level equivalences in D, then Theorem 4.12 implies th*
*at
F O 1 O L0detects stable equivalences in SpN(D; T ). The functor F O 1 O L0is a
generalization of the usual stable homotopy groups. One can see these generaliz*
*a-
tions in the following corollary as well.
Corollary 4.13.Let D be a left proper, cellular, almost finitely generated model
category where sequential colimits preserve finite products. Suppose T :D -! D
is a left Quillen functor whose right adjoint U commutes with sequential colimi*
*ts.
24 MARK HOVEY
Finally, suppose A is a finitely presented cofibrant object of D that has a fin*
*itely
presented cylinder object A x I. Then
Ho SpN(D; T )(FkA; Y ) = colimmHo D(A; Um Yk+m ):
for all level fibrant Y 2 SpN(D; T ).
Here we are using the stable model structure to form Ho SpN(D; T ), of course.
Proof.We have Ho SpN(D; T )(FkA; Y ) = SpN(D; T )(FkA; 1 Y )= ~, by Theo-
rem 4.12, where ~ denotes the left homotopy relation. We can use the cylinder
object Fk(A x I) as the source for our left homotopies. Then adjointness implies
that SpN(D; T )(FkA; 1 Y )=~= D(A; Evk1 Y )=~. Since A and AxI are finitely __
presented, we get the required result. |__|
By assuming slightly more about D, we can also characterize the stable fibrat*
*ions.
Corollary 4.14.Let D be a proper, cellular, almost finitely generated model cat*
*e-
gory such that sequential colimits preserve pullbacks. Suppose T :D -! D is a l*
*eft
Quillen functor whose right adjoint U commutes with sequential colimits. Then t*
*he
stable model structure on SpN(D; T ) is proper. In particular, a map p: X -! Y *
*is
a stable fibration if and only if p is a level fibration and the diagram
X ----! 1 L0X
? ?
f?y ?yLp
Y ----! 1 L0Y
is a homotopy pullback square in the projective model structure, where L0is a f*
*ibrant
replacement functor in the projective model structure.
Proof.We will actually show that, if p: X -! Y is a level fibration and f :B -!*
* Y
is a stable equivalence, the pullback B xY X -! X is a stable equivalence. This
means the the stable model structure on SpN(D; T ) is right proper, and then the
characterization of stable fibrations follows from [11, Proposition 3.6.8].
The first step is to use the right properness of the projective model structu*
*re
on SpN(D; T ) to reduce to the case where B and Y are level fibrant. Indeed, let
Y 0= L0Y , B0= L0B, and f0 = L0f. Then factor the composite X -! Y -! Y 0into
a projective trivial cofibration X -! X0 followed by a level fibration p0: X0 -*
*!Y 0.
Then we have the commutative diagram below,
B --f--! Y ---p- X
?? ? ?
y ?y ?y
B0 ----! Y 0---- X0
f0 p00
where the vertical maps are level equivalences. Proposition 11.2.4 and Corol-
lary 11.2.8 of [11], which depend on the projective model structure being right
proper, imply that the induced map B xY X -! B0xY 0X0 is a level equivalence.
Hence B xY X -! X is a stable equivalence if and only if B0xY 0X0 -! X0 is a
stable equivalence, and so we can assume B and X are level fibrant.
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 25
Now let S denote the pullback square below.
B xY X ----! X
?? ?
y ?yp
B ----! Y
f
nS
Then nS is a pullback square for all n, and there are maps nS ---! n+1S.
Since pullbacks commute with sequential colimits, 1 S is a pullback square. Fur-
thermore, 1 p is a level fibration, since sequential colimits in D preserve fib*
*rations
between level fibrant objects. Since f is a stable equivalence between level fi*
*brant
spectra, 1 f is a level equivalence by Theorem 4.12. So, since the projective m*
*odel
structure is right proper, the map 1 (B xY X -! X) is a level equivalence,_and_
thus B xY X -! X is a stable equivalence. |__|
5. Functoriality of the stable model structure
In this section, we consider the stable model structure on SpN(D; T ) as a fu*
*nctor
of the pair (D; T ). The most important result in this section is that SpN(D; T*
* ) is
Quillen equivalent to D if T is already a Quillen equivalence on D. This means
that the functor (D; T ) 7! (SpN(D; T ); T ) is idempotent up to Quillen equiva*
*lence.
This is as close we can get to our belief that, up to Quillen equivalence, SpN(*
*D; T )
should be the initial stabilization of D with respect to T .
We also show that SpN(D; T ) is functorial in the pair (D; T ), with a suitab*
*le def-
inition of maps of pairs. Under mild hypotheses, we show that SpN(D; T ) preser*
*ves
Quillen equivalences in the pair (D; T ). Applying this to the Bousfield-Friedl*
*ander
category of spectra of simplicial sets, we find that the choice of simplicial m*
*odel of
the circle has no effect on the Quillen equivalence class of the stable model c*
*ategory
of spectra.
Theorem 5.1. Suppose D is a left proper cellular model category, and suppose T
is a left Quillen endofunctor of D that is a Quillen equivalence. Then F0: D -!
SpN(D; T ) is a Quillen equivalence, where SpN(D; T ) has the stable model stru*
*cture.
Proof.By [12, Corollary 1.3.16], it suffices to check two conditions. We first *
*show
that Ev0: SpN(D; T ) -! D reflects weak equivalences between stably fibrant ob-
jects. We then show that the map A -! Ev 0LSF0A is a weak equivalence for
all cofibrant A 2 D, where LS denotes a stable fibrant replacement functor in
SpN(D; T ).
Suppose X and Y are U-spectra, and f :X -! Y is a map such that Ev0f = f0 is
a weak equivalence. We claim that f is a level equivalence, so that Ev0reflects*
* weak
equivalences between stably fibrant objects. Since f is a map of spectra, we ha*
*ve
oeY OT fn-1 = fnOoeX for n 1. Using adjointness, we find that UfnOeoeX= eoeYOf*
*n-1
for n 1. Since X and Y are U-spectra, we find that fn-1 is a weak equivalence
if and only if Ufn is a weak equivalence. On the other hand, since T is a Quill*
*en
equivalence on D, U reflects weak equivalences between fibrant objects of D, by*
* [12,
Corollary 1.3.16]. Therefore Ufn is a weak equivalence if and only if fn is a w*
*eak
equivalence. Altogether, fn-1 is a weak equivalence if and only if fn is a weak
equivalence. Since f0 is a weak equivalence by hypothesis, fn is a weak equival*
*ence
for all n, and so f is a level equivalence, as required.
26 MARK HOVEY
We now show that A -!Ev 0LSF0A is a weak equivalence for all cofibrant A 2 D.
Let R0 denote a fibrant replacement functor in the projective model structure on
SpN(D; T ). We claim that R0F0A is already a U-spectrum. Suppose for the moment
that this is true. Then we have the commutative diagram below.
F0A --i--! R0F0A
? ?
j?y ?y
LSF0A ----! 0
Since R0F0A is a U-spectrum, the right-hand vertical map is a stable fibration.*
* The
left-hand vertical map is a stable trivial cofibration, so there is a lift h: L*
*SF0A h-!
R0F0A with hi = j. Since i is a level equivalence and j is a stable equivalence*
*, h
is a stable equivalence. But both LSF0A and R0F0A are U-spectra, so h is a level
equivalence (see the discussion following Definition 2.1). This means that j is*
* also
a level equivalence. Hence the map
A = Ev0F0A -!Ev 0LSF0A
is a weak equivalence, as required.
It remains to prove that R0F0A is a U-spectrum when A is cofibrant. Let R
denote a fibrant replacement functor in D. Since T is a Quillen equivalence, and
T nA is cofibrant, the map
(F0A)n = T nA -!URT n+1A = UR(F0A)n+1
is a weak equivalence, by [12, Corollary 1.3.16]. In the commutative diagram be*
*low,
(F0A)n+1 ----! (R0F0A)n+1
?? ?
y ?y
R(F0A)n+1 ----! 0
the right-hand vertical map is a fibration, and the left-hand vertical map is a*
* trivial
cofibration. Thus there is a lift R(F0A)n+1 -!(R0F0A)n+1, which must be a weak
equivalence by the two-out-of-three property. Applying U, which preserves weak
equivalences between fibrant objects, we find that
(F0A)n -! UR(F0A)n+1 -!U(R0F0A)n+1
is a weak equivalence. This means that R0F0A is a U-spectrum, as required. |*
*___|
In particular, this theorem means that the passage (D; T ) 7! (SpN(D; T ); T )
is idempotent, up to Quillen equivalence. This suggests that we are doing some
kind of fibrant replacement of (D; T ) in an appropriate model category of model
categories, but the author knows no way of making this precise.
We now examine the functoriality of the stable model structure on SpN(D; T ).
We first consider what information we need to extend a functor on a category to*
* a
functor on the category of spectra.
Lemma 5.2. Suppose D and D0 are model categories equipped with left Quillen
endofunctors T :D -! D and T 0:D0 -! D0. Let U denote a right adjoint of
T and let U0 denote a right adjoint of T 0. Suppose : D0 -! D is a functor
and ae: U0 -!U is a natural transformation. Then there is an induced functor
SpN(; ae): SpN(D0; T 0) -! SpN(D; T ), sometimes denoted simply SpN() when the
choice of ae is clear, called the prolongation of .
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 27
Of course, the model structures are irrelevant to this lemma.
Proof.Given X 2 SpN(D0; T 0), we define (SpN(; ae)(X))n = Xn. The structure
map of SpN(; ae)(X) is adjoint to the composite
Xn eoe-!U0Xn+1 ae-!UXn+1;
where eoeis adjoint to the structure map of X. Given a map f :X -! Y , we_defin*
*e_
SpN(; ae)(f) by (SpN(; ae)(f))n = fn. |__|
Note that the natural transformation ae: U0 -! U is equivalent to a natural
transformation _ae:T -!T 0. Indeed, given ae, we define _aeto be the composite
T X -TjX---!T U0T 0X -TaeT0X---!T UT 0X -fflT0X---!T 0X:
where j denotes the unit and ffl the counit of the appropriate adjunctions. Co*
*n-
versely, given _ae, we can recover ae as the composite
_ae0 Uffl
U0X -jU0X---!UT U0X -U-U-X-!UT 0U0X ---! XUX:
We can describe the prolongation SpN(; ae) in terms of this associated natural
transformation _ae. Indeed, the structure map of SpN(; ae)(X) is the composite
_ae oe
T Xn -! T 0Xn -! Xn+1;
where oe is the structure map of X.
In practice, the functor : D0-! D that we wish to prolong will usually have a
left adjoint : D -!D0. We would like the prolongation SpN() to also have a left
adjoint. The following lemma deals with this issue.
Lemma 5.3. Suppose D and D0 are model categories equipped with left Quillen
endofunctors T :D -!D and T 0:D0-! D0. Let U denote a right adjoint of T and
let U0 denote a right adjoint of T 0. Suppose that : D0-! D is a functor with l*
*eft
adjoint , and ae: U0 -!U is a natural transformation. Then the prolongation
SpN(; ae): SpN(D0; T 0) -! SpN(D; T ) has a left adjoint e satisfying eFn ~=Fn.
If ae is a natural isomorphism, then e is a prolongation of .
The reason we do not denote e by SpN() is that e is not generally a prolon-
gation of .
Proof.Since Evn SpN(; ae) = Evn, if e exists we must have eFn ~=Fn. To
construct e, first note that the natural transformation ae has a dual, or conju*
*gate,
natural transformation o = Dae: T -! T 0, discussed in [12, p. 24] and in [17,
Section IV.7]. By iteration, we get induced natural transformations oq :T q-!
(T 0)q for all integers q 1. We now define (e X)n to be the coequalizer of two
maps
a a
ffn; fin : (T 0)pT qXr (T 0)pXq:
p+q+r=n p+q=n
On the summand (T 0)pT qXr, the top map ffn is (T 0)p applied to the iterated
structure map T qXr -! Xq+r of X. On the same summand, the bottom map fin
is (T 0)poq :(T 0)pT qXr -! (T 0)p+qXr. Note that T 0ffn is the retract of ffn*
*+1
consisting of those terms (T 0)pT qXr with p + q + r = n + 1 and p > 0. A simil*
*ar
statement is true for T 0fin. Since T 0preserves coequalizers, there is an indu*
*ced map
28 MARK HOVEY
T 0(e X)n -! (e X)n+1, which is the structure map of eX. We leave to the reader
the definition of ef for a map of spectra f :X -! Y .
The argument that e is left adjoint to SpN(; ae) is intricate, but straightfo*
*rward.
For example,`a map eX -! Y induces a map Xn -! Yn by restriction to the p = 0
summand in p+q=n(T 0)pXq. The adjoint of this is a map Xn -! Yn, which
induces a map of spectra X -! SpN(; ae)(Y`). To see that this is indeed a map of
spectra, consider the T Xn summand in p+q+r=n+1(T 0)pT qXr. Conversely, a
map X -f!SpN(; ae)Y`consists of maps fn :Xn -! Yn. The adjoints ffn:Xn -!
Yn define a map p+q=n(T 0)pXq gn-!Yn via the composite
0)pefq
(T 0)pXq (T-----!(T 0)pYq -!Yn;
where the second map is a composite of structure maps of Y . The fact that the
original map f is a map of spectra implies that these maps gn descend to define*
* a
map (e X)n -! Yn, which is automatically a map of spectra. We leave the rest of
the argument to the reader.
It remains to show that if ae is a natural isomorphism, then e is a prolongat*
*ion
of . If ae is a natural isomorphism, then o = Dae: T -! T 0 is also a natural
isomorphism. Let :T 0 -! T denote the inverse of o. Then, by Lemma 5.2,
there is a prolongation SpN(; ) of . We claim that SpN(; ) is left adjoint
to SpN(; ae). Indeed, there are natural candidates for the unit and counit of t*
*his
purported adjunction; namely, the maps which are levelwise the unit and counit
of the (; ) adjunction. We leave it to the reader to check that these maps are
maps of spectra and are natural. To ensure that they are the unit and counit of
an adjunction, we need to verify the triangle identities [17, Theorem 4.1.2(v)]*
*, but_
these follow immediately from the triangle identities of the (; ) adjunction. *
* |__|
In view of the preceding lemma, we make the following definition.
Definition 5.4.Suppose D and D0are left proper cellular model categories, T is
a left Quillen endofunctor of D, and T 0is a left Quillen endofunctor of D0. A *
*map
of pairs (; o): (D; T ) -! (D0; T 0) is a functor : D -! D0with a right adjoint*
* ,
and a natural transformation o :T -! T 0. We say that a map of pairs (; o) is
a Quillen map of pairs if is a left Quillen functor and oA is a weak equivalen*
*ce
for all cofibrant A 2 D.
Note that the natural transformation o in this definition is the natural tran*
*s-
formation Dae in Lemma 5.3. Our goal is for a Quillen map of pairs to induce a
corresponding map of pairs on the stable model category of spectra. For this to*
* be
true, we need some condition on o, and requiring oA to be a weak equivalence for
all cofibrant A seems to be the least we can assume.
Note that there is an obvious associative and unital composition of maps of p*
*airs.
Proposition 5.5.Suppose (; o): (D; T ) -! (D0; T 0) is a Quillen map of pairs.
Then there is an induced Quillen map of pairs
(e ; eo:(SpN(D; T ); T ) -!(SpN(D0; T 0); T 0);
where SpN(D; T ) and SpN(D0; T 0) are given the stable model structures, such t*
*hat
e O Fn ~= Fn. This induced map of pairs is compatible with composition and
identities.
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 29
Proof.We constructed e and its right adjoint SpN() = SpN(; ae) in Lemma 5.3.
Here denotes a right adjoint of , and ae is the dual natural transformation
Do :U0 -! U, where U is right adjoint to T and U0 is right adjoint to T 0.
Since SpN()(X)n = Xn, and is a right Quillen functor, SpN() preserves level
fibrations and level trivial fibrations. Hence e is a left Quillen functor with*
* respect
to the projective model structures. To show that e is a left Quillen functor wi*
*th
respect to the stable model structures, it suffices to show that e iAn is a sta*
*ble
equivalence for all cofibrant A, by Theorem 2.2. Since eFn ~=Fn by Lemma 5.3,
eFn+1T A ~=Fn+1T A and eFnA ~=FnA. Therefore, the map eiAndiffers from
the map iAn by the map Fn+1oA . Since iAn is a stable equivalence, and oA is a
weak equivalence by hypothesis, eiAnis a stable equivalence. Therefore, e is a *
*left
Quillen functor with respect to the stable model structures.
We define eoby defining its dual natural transformation
Deo= SpN(ae): SpN()U0 -!USpN();
Indeed, we just define SpN(ae) to be ae in each degree. We leave it to the rea*
*der
to verify that this defines a natural map of spectra SpN(ae). Since o is a weak
equivalence on all cofibrant objects of D, ae = Do is a weak equivalence on all
fibrant objects of D0. To see this, note that o induces a natural isomorphism i*
*n the
homotopy category. Adjointness implies that ae also induces a natural isomorphi*
*sm
in the homotopy category, and it follows that ae is a weak equivalence on all f*
*ibrant
objects of D. Thus SpN(ae) will be a level equivalence on all level fibrant obj*
*ects
of SpN(D0; T 0), so, by reversing the homotopy category argument just given, eo*
*is a
level equivalence on all cofibrant objects of SpN(D; T ). We leave it to the re*
*ader to
check compatibility of (e ; eo) with compositions and identities. *
* |___|
Proposition 5.5 and Theorem 5.1 give us a weak universal property of SpN(D; T*
* ).
Corollary 5.6.Suppose (; o): (D; T ) -!(D0; T 0) is a Quillen map of pairs such
that T 0is a Quillen equivalence. Give SpN(D; T ) its stable model structure. T*
*hen
there is a functor : Ho SpN(D; T ) -! HoD0 such that O LF0 = L: Ho D -!
Ho D0.
This corollary is trying to say that (SpN(D; T ); T ) is homotopy initial amo*
*ng
Quillen maps of pairs (D; T ) -!(D0; T 0) where T 0is a Quillen equivalence. Th*
*ough
the statement of the corollary is the best statement of this concept we have be*
*en
able to find, we suspect there is a better one.
Proof.By Proposition 5.5 there is a Quillen map of pairs
(e ; eo): (SpN(D; T ); T ) -!(SpN(D0; T 0); T 0)
induced by (; o). By Theorem 5.1, F0: D0-! SpN(D0; T 0) is a Quillen equivalenc*
*e.
Define to be the composite R Ev0OLe . |___|
Proposition 5.5 shows that the correspondence (D; T ) 7! (SpN(D; T ); T ) is *
*func-
torial. We would like to know that it is homotopy invariant. In particular, we
would like to know that if (; o) is a Quillen equivalence of pairs, then the in*
*duced
Quillen map of pairs (e ; eo) on spectra is a Quillen equivalence with respect *
*to the
stable model structures. Our proof of this seems to require some hypotheses.
30 MARK HOVEY
Theorem 5.7. Suppose (; o): (D; T ) -! (D0; T 0) is a Quillen map of pairs such
that is a Quillen equivalence. Suppose as well that either the domains of the
generating cofibrations for D can be taken to be cofibrant, or that oA is a weak
equivalence for all A. Then, in the induced Quillen map of pairs
(e ; eo): (SpN(D; T ); T ) -!(SpN(D0; T 0); T 0);
the Quillen functor e is a Quillen equivalence with respect to the stable model
structures.
Proof.We will first show that e is a Quillen equivalence with respect to the pr*
*o-
jective model structures. Use the same notation as in the proof of Proposition *
*5.5,
so that denotes the right adjoint of . Then reflects weak equivalences between
fibrant objects, by [12, Corollary 1.3.16]. Thus SpN() reflects level equivalen*
*ces
between level fibrant objects. By [12, Corollary 1.3.16], to show that e is a Q*
*uil-
len equivalence with respect to the projective model structures, it suffices to*
* show
that X -! SpN()R0eX is a level equivalence for all cofibrant X 2 SpN(D; T ),
where R0 is a fibrant replacement functor in the projective model structure on
SpN(D0; T 0). Let R denote a fibrant replacement functor in D0. By a lifting ar*
*gu-
ment, the weak equivalence (e X)n :(R0eX)n factors through the trivial cofibrat*
*ion
(e X)n -! R(e X)n. We therefore get a weak equivalence R(e X)n -! (R0eX)n,
and so a weak equivalence R(e X)n -! (SpN()R0eX)n. Therefore, it suffices
to show that Xn -! R(e X)n is a weak equivalence for all n and all cofibrant
X. Since Xn is cofibrant and is a Quillen equivalence, it suffices to show that
Xn -! (e X)n is a weak equivalence for all n and all cofibrant X.
Since every cofibrant X is a retract of a transfinite composition of pushouts*
* of
maps of IT, we can in fact assume that X is the colimit of a -sequence
0 = X0 -!X1 -!X2 -!: :-:!Xfi-! : :-:!X = X
for some ordinal , where each map Xfi-! Xfi+1is a pushout of a map of IT. We
will prove by transfinite induction on fi that Xfin-!(e Xfi)n is a weak equival*
*ence
for all n and all fi . The base case of the induction is trivial. The limit or*
*dinal
step of the induction follows from [11, Proposition 18.10.1], since each of the*
* maps
Xfin-!Xfi+1nand each of the maps (e Xfi)n -! (e Xfi+1)n is a cofibration of
cofibrant objects.
For the succesor ordinal step of the induction, suppose Xfi-! Xfi+1is a pusho*
*ut
of the map Fm C -Fm-f-!Fm D of IT. Then we have a pushout diagram
(Fm C)n ----! (Fm D)n
?? ?
y ?y
Xfin ----! Xfi+1n
and another pushout diagram
(e Fm C)n----! (e Fm D)n
?? ?
y ?y
(e Xfi)n----! (e Xfi+1)n
in D0. Note that (Fm C)n = T n-mC, where we interpret T n-mC to be the
initial object if n < m. Similarly, (e Fm C)n ~=(Fm C)n = (T 0)n-m C. Thus the
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 31
natural transformation o induces a map from the first of these pushout squares *
*to
the second. If C (and hence also D) is cofibrant, then this map of pushout squa*
*res
is a weak equivalence at both the upper left and upper right corners. Or, if oA*
* is
a weak equivalence for all A, then again this map is a weak equivalence at both
the upper left and upper right corners. It is also a weak equivalence at the lo*
*wer
left corner, by the induction hypothesis. Since both of the top horizontal maps*
* are
cofibrations in D0, Dan Kan's cube lemma [12, Lemma 5.2.6] implies that the map
is a weak equivalence on the lower right corner. This completes the induction.
We have now proved that e is a Quillen equivalence with respect to the projec*
*tive
model structures. We have already seen that e is a Quillen functor with respect*
* to
the stable model structures in Proposition 5.5. In view of Proposition 2.3, to *
*show
that e is a Quillen equivalence with respect to the stable model structures, it*
* suffices
to show that if Y is level fibrant in SpN(D0; T 0) and SpN()Y is a U-spectrum, *
*then
Y is a U0-spectrum. Since SpN()Y is a U-spectrum, the composite
Yn eoe-!U0Yn+1 Do--!UYn+1
is a weak equivalence for all n, where eoeis adjoint to the structure map of Y *
*and
Do is the natural transformation dual to o. Since oA is a weak equivalence for *
*all
cofibrant A, (Do)X is a weak equivalence for all fibrant X. This was explained *
*in
the last paragraph of the proof of Proposition 5.5. Therefore,
eoe:Yn -! U0Yn+1
is a weak equivalence. But reflects weak equivalences between fibrant objects,
by [12, Corollary 1.3.16]. Hence eoeis a weak equivalence for all n, and_so Y i*
*s a
U0-spectrum. |__|
As an example of Theorem 5.7, suppose we take a pointed simplicial set K weak*
*ly
equivalent to S1. Then there is a weak equivalence K -! RS1, where R is a fibra*
*nt
replacement functor on simplicial sets. This induces a natural transformation *
*of
left Quillen functors o : - ^K -! - ^ RS1. In Theorem 5.7, take D = D0equal to
the model category of pointed simplicial sets, take to be the identity, and ta*
*ke
o to be this natural transformation. Then we get a Quillen equivalence between
the stable model categories of spectra obtained by inverting K and inverting RS*
*1.
Therefore, the choice of simplicial circle does not matter, up to Quillen equiv*
*alence,
for Bousfield-Friedlander spectra.
6. Monoidal structure
In this section, we show that our stabilization construction preserves some m*
*o-
noidal structure. For example, if D is a simplicial model category, and T is a
simplicial functor, then the category SpN(D; T ) of spectra is again a simplici*
*al
model category, in both the projective and stable model structures, and the ext*
*en-
sion of T is again a simplicial functor. However, if C is a symmetric monoidal *
*model
category (see [12, Chapter 4], where these are called symmetric Quillen rings),*
* and
T :C -! C is a monoidal functor, the category SpN(C; T ) of spectra will almost
never be a monoidal category. This is the reason we need the symmetric spectra
introduced in the next section.
Throughout this section, C will be a symmetric monoidal model category, and
D will be a C-model category. This means that C is both a closed symmetric
monoidal category and a model category, and the model structure is compatible
32 MARK HOVEY
with the symmetric monoidal structure in a precise sense that we will recall be*
*low.
It also means that D is a right C-module with a compatible model structure. That
is, D is tensored, cotensored, and enriched over C. See [12, Chapter 4] for com*
*plete
definitions.
We remind the reader of the precise definition of the compatibility between t*
*he
monoidal structure and the model structure.
Definition 6.1.Suppose C is a monoidal category. Given maps f :A -! B and
g :C -! D in C, we define the pushout product f g of f and g to be the map
f g :(A D) qAC (B C) -!B D
induced by the commutative square below.
A C -f1---!B C
? ?
1g ?y ?y1g
A D ----! B D
f1
Note that f g also makes perfect sense in case f 2 D and g 2 C, since D is a
right C-module.
Definition 6.2.A symmetric monoidal model category is a symmetric monoidal
category C equipped with a model structure satisfying the following two conditi*
*ons.
1. If f and g are cofibrations in C, then f g is a cofibration. If, in addit*
*ion,
one of f or g is a trivial cofibration, so is f g.
2. Let QS -r! S be a cofibrant replacement for the unit S of the monoidal
structure, so that QS is cofibrant and r is a trivial fibration. Then, for*
* all
cofibrant X, the induced map
X QS -1r-!X S ~=X
is a weak equivalence.
Similarly, if D is a model category that is enriched, tensored, and cotensored *
*over
C, then D is a C-model category when the following two conditions are satisfied.
1. If f is a cofibration in D and g is a cofibration in C, then f g is a cofi*
*bration
in D. If, in addition, one of f or g is a trivial cofibration, so is f g.
2. For all cofibrant X in D, the induced map
X QS -1r-!X S ~=X
is a weak equivalence.
Note that a left Quillen functor between C-model categories is called a C-Qui*
*llen
functor if it preserves the action of C up to natural isomorphism. See [12, Def*
*ini-
tion 4.1.7 and Definition 4.2.18]. A C-Quillen functor that is a Quillen equiva*
*lence
is called a C-Quillen equivalence.
We then have the following theorem.
Theorem 6.3. Let C be a cofibrantly generated symmetric monoidal model cate-
gory, and suppose the domains of the generating cofibrations of C can be taken *
*to
be cofibrant. Suppose D is a left proper cellular C-model category, and that T *
*is a
left C-Quillen endofunctor of D. Then SpN(D; T ), with the stable model structu*
*re,
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 33
is again a C-model category, and the extension of T is a C-Quillen self-equival*
*ence
of SpN(D; T ). The functors Fn :D -!SpN(D; T ) are C-Quillen functors.
Proof.We define the action of C on SpN(D; T ) levelwise. That is, given X 2
SpN(D; T ) and K 2 C, we define (X K)n = Xn K. The structure map is given
by
T (Xn ^ K) ~=T Xn ^ K -oe^1-!Xn+1 ^ K;
where the first isomorphism comes from the fact that T preserves the C-action,
and oe is the structure map of X. One can easily verify that this makes SpN(D; *
*T )
tensored over C. Similarly, if we denote the cotensor of Z 2 D and K 2 C by ZK ,
we can define the cotensor XK of X 2 SpN(D; T ) and K 2 C by (XK )n = XKn.
The structure map T (XKn) -!XKn+1of XK is adjoint to the composite
T (XKn) ^ K ~=T (XKn^ K) T(ev)---!T Xn oe-!Xn+1
where ev: XKn^ K is the evaluation map, adjoint to the identity of XKn. This
makes SpN(D; T ) cotensored over C. Finally, if we denote the enrichment of Z a*
*nd
W in D by Map (Z; W ) 2 C, we define the enrichment Map (X; Y ) 2 C of X and Y
in SpN(D; T ) to be the equalizer of two maps
Y Y
ff; fi : Map (Xn; Yn) Map(Xn; UYn+1):
n n
Here ff is the product of the maps Map (Xn; Yn) Map-Xn;eoe-----!Map(Xn; UYn+1) *
*where
eoedenotes the adjoint of the structure map, and fi is the product of the maps
Map (Xn+1; Yn+1) Map(eoe;Yn+1)--------!Map(T Xn; Yn+1) ~=Map (Xn; UYn+1);
where the isomorphism exists since T preserves the action of C. This functor
Map (X; Y ) makes SpN(D; T ) enriched over C.
We must now check that these structures are compatible with the model struc-
ture. We begin with the projective model structure on SpN(D; T ). One can easily
check that if h is a map in D and g is a map in C, then Fnh g = Fn(h g). Thus,
if f = Fnh is one of the generating cofibrations of the projective model struct*
*ure
on SpN(D; T ), and g is a cofibration in C, then f g is a cofibration in SpN(D*
*; T ).
It follows that f g is a cofibration for f an arbitrary cofibration of SpN(D; *
*T )
(see [12, Corollary 4.2.5], [13, Corollary 5.3.5], or [24, Lemma 2.3]). A simil*
*ar argu-
ment shows that f g is a projective trivial cofibration in SpN(D; T ) if, in ad*
*dition,
either f is a projective trivial cofibration in SpN(D; T ) or g is a trivial co*
*fibration
in C. Finally, if QS -! S is a cofibrant approximation to the unit S in C, and *
*X is
cofibrant in SpN(D; T ), then each Xn is cofibrant in D, so the map X QS -! X is
a level equivalence as required. Thus SpN(D; T ) with its projective model stru*
*cture
is a C-model category.
Since the cofibrations in the projective and stable model structures on SpN(D*
*; T )
coincide, to show that SpN(D; T ) with its stable model structure is also a C-m*
*odel
category, we only need to show that, if f is a stable trivial cofibration in Sp*
*N(D; T )
and g is a cofibration in C, then f g is a stable equivalence. It suffices to *
*check
this for g :K -! L one of the generating cofibrations of C, again using [12, Co*
*rol-
lary 4.2.5]. In this case, by hypothesis, K and L are cofibrant in C. Thus the
functor - K is a Quillen functor with respect to the projective model structure
34 MARK HOVEY
on SpN(D; T ), and similarly for L. We will show that - K is a Quillen func-
tor with respect to the stable model structure as well. To see this, note that*
* if
iQCn: Fn+1T QC -! FnQC is an element of the set S, then iQCn K ~=iQCKn ,
since T preserves the C-action. In view of Theorem 3.4, the map iQCKn is a sta*
*ble
equivalence. Theorem 2.2 then implies that -K is a Quillen functor with respect
to the stable model structure on SpN(D; T ), and similarly for - L. Thus, if f*
* is
a stable trivial cofibration, so are f K and f L. It follows from the two out*
*_of_
three property that f g is a stable equivalence, as required. |*
*__|
Remark 6.4. Suppose that the functor T is actually given by T X = X K for
some cofibrant object K of C. We then have two different ways of tensoring with
K on SpN(D; T ). We have the functor X 7! X K that we have just constructed_
as part of the C-action on SpN(D; T ). We also have the functor X 7! X K which
is the extension of T to a Quillen equivalence_on the stable model structure on
SpN(D; T ). As explained in Remark 1.6, X K does not involve the twist map
t of the symmetric monoidal structure on C. However, the functor X K does
use the twist map as part of its structure map; indeed, in order to construct t*
*he
isomorphism T (X K) ~=T X K we need to permute the two different copies of
K. Therefore,_we_do not know that X 7! X K is a Quillen equivalence, even
though X 7! X K is. We will have to deal with this point more thoroughly in
Section 10, when we compare SpN(D; T ) with symmetric spectra.
Theorem 6.3 gives us a functorial stabilization. We first simplify the notati*
*on.
Suppose K is a cofibrant object of a symmetric monoidal model category C. Then
T = - K is a left Quillen functor on any C-model category D. In this case, we
denote SpN(D; T ) by SpN(D; T ).
Corollary 6.5.Suppose K is a cofibrant object of a cofibrantly generated symmet-
ric monoidal model category C where the domains of the generating cofibrations
can be taken to be cofibrant. Then the correspondence D 7! SpN(D; K), where
SpN(D; K) is given the stable model structure, defines an endofunctor of the ca*
*te-
gory of left proper cellular D-model categories.
Note that the "category" of left proper cellular C-model categories is not re*
*ally a
category, because the Hom -sets need not be sets. It is really a 2-category, an*
*d the
correspondence D 7! SpN(D; K) is actually a 2-functor. See [12] for a descripti*
*on
of this point of view on model categories.
Proof.Given a left proper cellular C-model category D, we have seen in Theorem *
*6.3
that SpN(D; K) is a C-model category. Given a left C-Quillen functor : D -! D0
between two left proper cellular C-model categories D and D0 with right adjoint
, there is a natural isomorphism o :(- K) -! (-) K. Taking ae = Do
in Lemma 5.3, we get an induced functor e: SpN(D; K) -!SpN(D0; K) with right
adjoint SpN(; ae). Since o is an isomorphism, e is a prolongation of , and so o*
*ne
can easily check that e preserves the action of C. Proposition 5.5 guarantees t*
*hat
e is a Quillen functor with respect to the stable model structures. *
*|___|
We now point out that, if C is a symmetric monoidal model category, and T
is a C-Quillen endofunctor of C, then the category SpN(C; T ) is almost never i*
*tself
monoidal, though, as we have seen, it has an action of C. To see this, first n*
*ote
that T is naturally isomorphic to - K for K = T S. Now consider the category
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 35
CN of sequences from C. An object of CN is a sequence Xn of objects of C, and a
map f :X -! Y is a sequence of maps fn :Xn -! Yn.` Then CN is a symmetric
monoidal category, where we define (X Y )n = p+q=nXp Yq. Furthermore, if
D is a C-model category, then DN is a right CN-module, using the same definitio*
*n of
the tensor product. The sequence F0S = (S; K; K K; : :;:Kn ; : :):is a monoid
in CN.
Lemma 6.6. Suppose C is a symmetric monoidal model category and T is a left
C-Quillen functor with K = T S. Then SpN(C; T ) is the category of right modules
in CN over the monoid F0S. Furthermore, if D is a C-model category, SpN(D; K)
is the category of right modules in DN over F0S.
We leave the proof of this lemma to the reader, as it is a matter of unwinding
definitions. The important corollary of this lemma is that the monoid F0S is al*
*most
never commutative, and therefore SpN(C; K) can not be a symmetric monoidal cat-
egory with unit F0S. Indeed, F0S is commutative if and only if the commutativity
isomorphism of C applied to K K is the identity. This happens only very rarely.
7.Symmetric spectra
We have just seen that the stabilization functor SpN(C; T ) is not good enoug*
*h in
case C is a symmetric monoidal model category and T is a C-Quillen functor, bec*
*ause
SpN(C; T ) is not usually itself a symmetric monoidal model category. In this s*
*ection,
we begin the construction of a better stabilization functor for this case. We *
*will
concentrate on the category theory in this section, leaving the model structure*
*s for
the next section. The terms used for the algebra of symmetric monoidal categori*
*es
and modules over them are defined in [12, Section 4.1].
Through most of this section, then, C will be a bicomplete closed symmetric
monoidal category with unit S, and K will be an object of C. The category D wil*
*l be
a bicomplete right C-module category; this means that D is enriched, tensored, *
*and
cotensored over C. Note that any C-functor T on C itself is of the form T (L) =*
* LK
for K = T S, so we will only consider such functors. Because of this, we will d*
*rop
the letter T from our notations and replace it with K.
This section is based on the symmetric spectra and sequences of [13]. The main
idea of symmetric spectra is that the associativity and commutativity isomorphi*
*sms
of C make Kn into a n-object of C, where n is the symmetric group on n letters.
We must keep track of this action if we expect to get a symmetric monoidal cate*
*gory
of K-spectra.
The following definition is [13, Definition 2.1.1].
`
Definition 7.1.Let = n0 n be the category whose objects are the sets
__n= {1; 2; : :;:n} for n 0, where _0= ;. The morphisms of are the isomorphis*
*ms
of __n. Given a category E, a symmetric sequence in E is a functor -! E. The
category of symmetric sequences is the functor category E .
A symmetric sequence X in a category E is a sequence X0; X1; : :;:Xn; : :o:f
objects of E with an action of n on Xn. It is sometimes more useful to consider*
* a
symmetric sequence as a functor from the category of finite sets and isomorphis*
*ms to
E. Since the category is a skeleton of the category of finite sets and isomorp*
*hisms,
there is no difficulty in such a change of viewpoint.
As a functor category, the category of symmetric sequences in E is bicomplete
if E is so; limits and colimits are taken levelwise. Furthermore, since our cat*
*egory
36 MARK HOVEY
C is closed symmetric monoidal, so is C , as explained in [13, Section 2.1]. Th*
*is
result is a special case of the much more general work of Day [3]. Recall that *
*the
monoidal structure is given by
a
(X Y )(C) = X(A) Y (B)
A[B=C;A\B=;
where we think of X, Y , and X Y as functors from finite sets to C. Equivalent*
*ly,
though less canonically, we have
a
(X Y )n = n xpxq (Xp Yq):
p+q=n
This notation may need some explanation. Given a set and an object A of a
cocomplete category E, x A is the coproduct of || copies of A. If is a group,
then x A has an obvious left -action; x A is the free -object on A. Note that
a -action on A is then equivalent to a map x A -! A satisfying the usual unit
and associativity conditions. Also, if admits a right action by a group 0, and
A is a left 0-object, then we can form x0 A as the colimit of the 0-action on
x A, where ff 2 0takes the copy of A corresponding to fi 2 to the copy of A
corresponding to fiff-1 by the action of ff.
The unit of the monoidal structure on C is the symmetric sequence (S; 0; 0; *
*: :):,
where 0 is the initial object of C. To define the closed structure on C , we f*
*irst
define Hom n (L; L0) for n-objects L and L0in C in the usual way, as an equaliz*
*er
of the two maps Hom (L; L0) -!Hom (n x L; L0) defined using the structure maps
of L and L0. The closed structure is then given by
Y
Hom (X; Y )k = Hom n (Xn; Yn+k):
n
Since our category D is enriched, tensored, and cotensored over C, D is enri*
*ched,
tensored, and cotensored over C , making D a right C -module category. Indeed,
the same definition as above works to define the tensor structure. The cotensor
structure is defined as follows. First we define Hom n (L; A) for n-objects L o*
*f C
and AQof D as an appropriate equalizer. Then, for X 2 D and Z 2 C , we define
XZk= nHom n (Zn; Xn+k). The enrichment Map (X; Y ) 2 C for X and Y in
D is defined similarly. In the same way, if D is an enriched monoidal category
over C (a C-algebra, in the terminology of [12, Sectin 4.1]), then D is an enr*
*iched
monoidal category over C .
Consider the free commutative monoid Sym(K) on the object (0; K; 0; : :;:0; :*
* :):
of C . One can easily check that Sym (K) is the symmetric sequence (S; K; K
K; : :;:Kn ; : :):where n acts on Kn by permutation, using the commutativity
and associativity isomorphisms.
Definition 7.2.Suppose C is a symmetric monoidal model category, D is a C-
model category, and K is an object of C. The category of symmetric spectra
Sp (D; K) is the category of modules in D over the commutative monoid Sym(K)
in C . That is, a symmetric spectrum X is a sequence of n-objects Xn 2 C and
n-equivariant maps Xn K -! Xn+1, such that the composite
Xn Kp -! Xn+1 Kp-1 -! : :-:!Xn+p
is n x p-equivariant for all n; p 0. A map of symmetric spectra is a collection
of n-equivariant maps Xn -! Yn compatible with the structure maps of X and Y .
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 37
Because Sym (K) is a commutative monoid, the category Sp (C; K) is a bi-
complete closed symmetric monoidal category, with Sym (K) itself as the unit (s*
*ee
Lemma 2.2.2 and Lemma 2.2.8 of [13]). We denote the monoidal structure by
X ^ Y = X Sym(K)Y , and the closed structure by Hom Sym(K)(X; Y ). Similarly,
Sp (D; K) is bicomplete, enriched, tensored, and cotensored over Sp (C; K) with
the tensor structure denoted X ^Y again, and, if D is a C-monoidal model catego*
*ry,
then Sp (D; K) will be a monoidal category enriched over Sp (C; K).
Of course, if we take C = SSet *and K = S1, we recover the definition of
symmetric spectra given in [13], except that we are using right Sym (K)-modules
instead of left Sym (K)-modules.
Definition 7.3.Given n 0, the evaluation functor Evn :Sp (D; K) -!D takes
X to Xn. The evaluation functor has a left adjoint Fn :D -! Sp (D; K), defined
by FnA = eFnA Sym(K), where eFnA is the symmetric sequence (0; : :;:0; n x
A; 0; : :):.
Note that F0A = (A; AK; : :;:AKn ; : :):, and in particular F0S = Sym(K).
In general, we have (FnA)m = m xm-n (AK(m-n) ) for m n. Also, if A 2 D
and L 2 C, there is a natural isomorphism FnA^Fm L ~=Fn+m (AL), just as in [13,
Proposition 2.2.6]. In particular, F0: C -! Sp (C; K) is a (symmetric) monoidal
functor, and so Sp (D; K) is naturally enriched, tensored, and cotensored over
C. In fact, this structure is very simple. Indeed, if X 2 Sp (D; K) and L 2 C,
X L = X Sym(K)F0L is just the symmetric sequence whose nth term is Xn L.
The structure map is the composite
Xn L K -1t-!Xn K L -!Xn+1 L:
Note the presence of the twist map t; this is required even when L = K to get
a symmetric spectrum, unlike the case of ordinary spectra. Similarly, XL =
Hom Sym(K)(F0L; X) is the symmetric sequence whose nth term is XLn, with the
twist map again appearing as part of the structure map.
Remark 7.4. The evaluation functor Evn has a right adjoint
Mn :D -!Sp (D; K);
just as in the spectrum case (see Remark 1.4). Indeed, we define
MnA = Hom (Sym (K); fMnA);
where fMnA is the symmetric sequence that is the terminal object in dimensions
other than n, and is the cofree n-object C(n; A) in dimension n. As an object
of C, C(n; A) is just the n!-fold product of A. Given ae 2 n and f 2 C(n; A),
the n-action is defined by (aef)(ae0) = f(ae0ae). Just as in Remark 1.4, MnA is*
* the
terminal object in dimensions greater than n.
8. Model structures on symmetric spectra
Throughout this section, C will denote a left proper cellular symmetric monoi*
*dal
model category, D will denote a left proper cellular C-model category, and K wi*
*ll
denote a cofibrant object of C. In this section, we discuss the projective and *
*stable
model structures on the category Sp (D; K) of symmetric spectra. The results in
this section are very similar to the corresponding results in Section 3, so we *
*will
leave most of the proofs to the reader.
38 MARK HOVEY
Definition 8.1.A map f 2 Sp (D; K) is a level equivalence if each map fn is a
weak equivalence in D. Similarly, f is a level fibration (resp. level cofibrati*
*on, level
trivial fibration, level trivial cofibration) if each map fn is a fibration (re*
*sp. cofibra-
tion, trivial fibration, trivial cofibration) in D. The map f is a projective c*
*ofibration
if f has the left lifting property with respect to every level trivial fibratio*
*n.
Then, just as in Definition 1.8, if we denote the generating cofibrationsSof D
by I andSthe generating trivial cofibrations by J, we define IK = nFnI and
JK = n FnJ.
We have analogues of 1.9-1.12 with almost the same proofs. The only real
difference is that it is less obvious that the maps of IK are level cofibration*
*s, and
that the maps of JK are level trivial cofibrations. If g :A -!B is a map in D, *
*and
m n, then (Fng)m is the map
m xm-n (g Km-n ) ~=g (m xm-n Km-n ):
As a map in D, this is the coproduct of m!=(m - n)! copies of g Km-n . Since K
is cofibrant, if g is a cofibration (trivial cofibration), then Fng is a level *
*cofibration
(level trivial cofibration). This uses the fact that D is a C-model category, a*
*nd also
that (Fng)m = 0 for m < n.
We then construct the projective model structure just as in the proof of Theo-
rem 1.13.
Theorem 8.2. The projective cofibrations, the level fibrations, and the level *
*equiv-
alences define a left proper cellular model structure on Sp (D; K).
The set IK is the set of generating cofibrations of the projective model stru*
*cture,
and JK is the set of generating trivial cofibrations. The cellularity of the pr*
*ojective
model structure is proved in the Appendix.
Note that Evn takes level (trivial) fibrations to (trivial) fibrations, so Ev*
*n is a
right Quillen functor and Fn is a left Quillen functor, with respect to the pro*
*jective
model structure.
Theorem 8.3. The category Sp (C; K), with the projective model structure, is a
symmetric monoidal model category. The category Sp (D; K), with its projective
model structure, is a Sp (C; K)-model category.
See the discussion following Definition 6.1 for the definition of a symmetric*
* mo-
noidal model category, or see [12, Chapter 4] for more detail.
Proof.We first show that the pushout product f g is a cofibration when f is
a cofibration in Sp (D; K) and g is a cofibration in Sp (C; K), and that f g
is a trivial cofibration when, in addition, one of f or g is a level equivalenc*
*e. As
explained in [12, Corollary 4.2.5], we may as well assume that f and g belong to
the sets of generating cofibrations or generating trivial cofibrations. In eith*
*er case,
we have f = Fm f0 and g = Fng0. But then f g = Fm+n (f0 g0). Since Fm+n is
a left Quillen functor, the result follows.
It remains to show that, if X is cofibrant in Sp (D; K) and Q(Sym (K)) is
a cofibrant replacement for the unit Sym (K) of Sp (C; K), then the map X
Q(Sym (K)) -!X Sym(K) ~=X is a level equivalence. Let QS denote a cofibrant
replacement for the unit S in C. Then we claim that F0QS is a cofibrant replace*
*ment
for F0S = Sym(K) in Sp (C; K). Indeed, F0QS is cofibrant, and Evn F0QS is just
QS Kn . Since K is cofibrant and C is a monoidal model category, the map
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 39
F0QS -! F0S is a level equivalence. Now, since X is cofibrant in Sp (D; K), each
Xn is cofibrant. Hence the map Xn QS -! Xn is a weak equivalence for all n, __
and so the map X ^ F0QS -! X is a level equivalence, as required. |_*
*_|
We point out that one can show that the projective model structure on Sp (C; *
*K)
satisfies the monoid axiom of [24], assuming that C itself does so. This means *
*there
is a projective model structure on the category of monoids in Sp (C; K) and on
the category of modules over any monoid. We do not include the proofs of these
statements since we have been unable to prove the analogous statements for the
stable model structure.
The projective cofibrations of symmetric spectra are more complicated than th*
*ey
are in the case of ordinary spectra. The idea is the same as that for ordinary *
*spectra.
Recall that an ordinary spectrum A is cofibrant if A0 is cofibrant and each map
An-1 K -! An is a cofibration (see Proposition 1.14). This then implies that
each map Ak K(n-k) -! An for k < n is a cofibration as well. However, in the
case of symmetric spectra, we need to guarantee that Ak K(n-k) -! An is a
cofibration in the category of k x n-k objects. We therefore introduce a more
complicated object, called the latching space LnA, which is an amalgam of all t*
*he
objects Ak K(n-k) , induced up from k x n-k-objects to n-objects.
_______
Definition 8.4.Define the symmetric spectrum Sym (K) in Sp (C; K) to be 0 in
degree 0 and Kn in degree n, for n > 0, with the obvious structure_maps. Define
the nth latching space LnA of A 2 Sp (D; K) by LnA = Evn(A ^ Sym (K)).
Note that LnA is precisely the colimit of the objects nxkxn-k (AkK(n-k) )
for k < n, where the colimit_is taken using the Sym (K)-module structure. The o*
*b-
vious map of spectra Sym (K)-! Sym (K) induces a n-equivariant natural trans-
formation i: LnA -! An. When restricted to n xkxn-k (Ak Kn-k ), this
map is just the structure map of A.
Note that the latching space is a n-object of D. There is a model structure
on n-objects of D where the fibrations and weak equivalences are the underlying
ones. This model structure is cofibrantly generated: if I is the set of genera*
*ting
cofibrations of D, then the set of generating cofibrations of Dn is the set n *
*x I.
Recall that, for an object N, n x N is the coproduct of n! copies of N, given t*
*he
obvious n-structure. The cofibrations in Dn are cofibrations f in D where n
acts freely away from the image of f.
Proposition 8.5.A map f :A -! B in Sp (D; K) is a projective cofibration if
and only if the induced map Evn(f i): An qLnA LnB -! Bn is a cofibration in
Dn for all n. Similarly, f is a projective trivial cofibration if and only if *
*Evn(f i)
is a trivial cofibration in Dn for all n.
Note the similarity of Proposition 8.5 to Proposition 1.14.
Proof.We only prove the cofibration case, as the trivial cofibration case is an*
*alo-
gous. Suppose first that each map Evn(f i) is a cofibration in Dn . We show
that f is a projective cofibration by showing f has the left lifting property w*
*ith
respect to every level trivial fibration p: X -! Y . Suppose we have a commutat*
*ive
40 MARK HOVEY
square as below.
A ----! X
? ?
f?y ?yp
B ----! Y
We construct a lift Bn -! Xn in this diagram by induction on n. When n = 0,
this is easy since f0 is a cofibration and p0 is a trivial fibration in D. Supp*
*ose we
have constructed compatible partial lifts Bk -! Xk for k < n. These partial lif*
*ts
assemble into a n-equivariant map LnB -! Xn. Combining this with the given
map An -! Xn, we get the commutative diagram of n-equivariant maps below.
An qLnA LnB ----! Xn
? ?
Evn(fi)?y ?ypn
Bn ----! Yn
Since Evn(f i) is a cofibration in Dn and pn is a trivial fibration, there is*
* a lift
Bn -! Xn in this diagram. This completes the induction step. The resulting map
B -! X is a map of spectra, since the structure maps of B are encoded into the
map LnB -! Bn, and gives us the desired lift.
To prove the converse, we need to show that if f is a cofibration of symmetric
spectra, then Evn(f i) is a n-cofibration. Since f is a retract of a transfini*
*te
composition of pushouts of maps of IK , and Evn(-i) preserves retracts, transfi*
*nite
compositions, and pushouts, we can assume f is a map in IK . Then we can write
f = Fm g for some integer m and some map g :C -! D in D. In this case, one can
check that the map LnFm C -! Fm C is an isomorphism except when n = m, in
which case it is the map from the initial object to m xC. It follows that_Evn(f*
*_i)
is an isomorphism when n 6= m, and Evm (f i) = m x g. |__|
We must now localize the projective model structure to obtain the stable model
structure.
Definition 8.6.A symmetric spectrum X 2 Sp (D; K) is an -spectrum if X is
level fibrant and the adjoint Xn -! XKn+1of the structure map of X is a weak
equivalence for all n.
Just as with Bousfield-Friedlander spectra, we would like the -spectra to be *
*the
fibrant objects in the stable model structure. We therefore invert analogous ma*
*ps.
Definition 8.7.Define the set of maps S in Sp (D; K) to be
QC
{Fn+1(QC K) in--!FnQC}
as C runs through the domains and codomains of the generating cofibrations of D,
and n 0. The map iQCn is adjoint to the map
QC K -! Evn+1FnQC = n+1 x (QC K)
corresponding to the identity of n+1. We then define the stable model structure
on Sp (D; K) to be the Bousfield localization with respect to S of the projecti*
*ve
model structure on Sp (D; K). The S-local weak equivalences are called the stab*
*le
equivalences, and the S-local fibrations are called the stable fibrations.
The following theorem is then analogous to Theorem 3.4, and has the same proo*
*f.
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 41
Theorem 8.8. The stably fibrant symmetric spectra are the -spectra. Further-
A
more, for all cofibrant A 2 D and for all n 0, the map Fn+1(A K) in-!FnA is
a stable equivalence.
Just as in Corollary 3.5, this theorem implies that, when D = SSet* or Top *,
Sp (C; K) is the same as the stable model category on (simplicial or topologica*
*l)
symmetric spectra discussed in [13].
The analog of Lemma 3.6 also holds, with the same proof, so that tensoring wi*
*th
K is a Quillen endofunctor of Sp (D; K). Of course, we want this functor to be
a Quillen equivalence. As in Definition 3.7, we prove this by introducing the s*
*hift
functors.
Definition 8.9.Define the right shift functor s- :Sp (D; K) -! Sp (D; K) by
s- X = Hom Sym(K)(F1S; X). Thus (s- X)n = Xn+1, where the n-action on Xn+1
is induced by the usual inclusion n -! n+1. The structure maps of s- X are the
same as the structure maps of X. Define the left shift functor s+ :Sp (D; K) -!
Sp (D; K) by s+ X = X Sym(K)F1S, so that s+ is left adjoint to s- . We have
(s+ X)n = n xn-1 Xn-1 for n > 0, and (s+ X)0 is the initial object of D. The
structure maps of s+ X are induced from the structure maps of X.
Note that adjointness gives natural isomorphisms
(s- X)K~=Hom Sym(K)(F0K; Hom Sym(K)(F1S; X))
~=Hom Sym(K)(F0K ^ F1S; X) ~=Hom Sym(K)(F1K; X):
A similar chain of isomorphisms shows that s- (XK ) is also naturally isomorphic
to Hom Sym(K)(F1K; X).
There is a map F1K -! F0S which is the identity in degree 1. By adjointness,
this map induces a map
X = Hom Sym(K)(F0S; X) -!Hom Sym(K)(F1K; X) = (s- X)K :
By applying Evn, we get a map Xn -! XKn+1. This map is adjoint to the structure
map of X, as is explained for simplicial symmetric spectra in [13, Remark 2.2.1*
*2].
Therefore, X is an -spectrum if and only if this map X -! (s- X)K is a level eq*
*uiv-
alence and X is level fibrant. Hence the same method used to prove Theorem 3.9
also proves the following theorem.
Theorem 8.10. The functors X 7! X K and s+ are Quillen equivalences with
respect to the stable model structure on Sp (D; K). Furthermore, Rs- is natural*
*ly
isomorphic to L(- K) and R((-)K ) is naturally isomorphic to Ls+ .
We have now shown that Sp (D; K) is a K-stabilization of D. However, for
symmetric spectra to be better than ordinary spectra, we must show that Sp (C; *
*K)
is a symmetric monoidal model category.
Theorem 8.11. Suppose that the domains of the generating cofibrations of both C
and D are cofibrant. Then the stable model structure makes Sp (C; K) into a sym-
metric monoidal model category, and the stable model structure makes Sp (D; K)
into a Sp (C; K)-model category.
Proof.We prove this theorem in the same way as Theorem 6.3. Since the cofibra-
tions in the stable model structure are the same as the cofibrations in the pro*
*jective
model structure, the only thing to check is that f g is a stable equivalence w*
*hen
42 MARK HOVEY
f and g are cofibrations and one of them is a stable equivalence. We may as
well assume that f :FnA -! FnB is a generating cofibration in Sp (D; K) and
g is a stable trivial cofibration in Sp (C; K), by [12, Corollary 4.2.5]; the a*
*rgu-
ment for f a stable trivial cofibration and g a generating cofibration in Sp (C*
*; K)
is the same. Then, by hypothesis, A and B are cofibrant in D. We claim that
FnA ^ (-): Sp (C; K) -! Sp (D; K) is a Quillen functor with respect to the sta-
ble model structures, and similarly for FnB ^ (-). Indeed, in view of Theorem 2*
*.2,
it suffices to show that FnA ^ iQCm is a stable equivalence for all m 0 and all
domains or codomains C of the generating cofibrations of C. But one can easily
check that FnA ^ iQCm = iAQCn+m. Then Theorem 8.8 implies that this map is a
stable equivalence, as required.
Thus, both functors FnA ^ (-) and FnB ^ (-) are Quillen functors with respect
to the stable model structures. Consider the commutative diagram below,
FnA ^ X -f^X---!FnB ^ X_______FnB ^ X
? ? ?
FnA^g?y ff?y ?yFnB^g
FnA ^ Y ----! P ----! FnB ^ Y
fg
where the left-hand square is a pushout square. Since g is a stable trivial cof*
*ibration,
so are FnA^g amd FnB ^g. Since the left-hand square is a pushout square ff is a*
*lso
a stable trivial cofibration. By the two out of three property of stable equiva*
*lences,_
f g is a stable equivalence, as required. |__|
The functor F0: C -! Sp (C; K) is a symmetric monoidal Quillen functor, so,
under the hypotheses of Theorem 8.11, Sp (D; K) is a C-model category as well.
In fact, we only need the domains of the generating cofibrations of C to be cof*
*i-
brant to conclude that Sp (D; K) is a C-model category, using the argument of
Theorem 8.11.
As we mentioned above, we do not know if the stable model structure satisfies
the monoid axiom. Given a particular monoid R, one could attempt to localize the
projective model structure on R-modules to obtain a stable model structure. How-
ever, for this to work one would need to know that the projective model structu*
*re
on R-modules is cellular, and the author does not see how to prove this. This p*
*lan
will certainly fail for the category of monoids, since the projective model str*
*ucture
on monoids will not be left proper in general.
We also point out that it may be possible to prove some of the results of Sec*
*tion 4
for symmetric spectra. Not all of those results can hold, since stable homotopy
isomorphisms do not coincide with stable equivalences even for symmetric spectra
of simplicial sets. Nevertheless, in that case, every stable homotopy isomorphi*
*sm
is a stable equivalence [13, Theorem 3.1.11]. Shipley [25] has constructed a fi*
*brant
replacement functor for simplicial symmetric spectra as well. We do not know if
these results hold for symmetric spectra over a general well-behaved almost fin*
*itely
generated model category.
9.Properties of symmetric spectra
In this section, we point out that the arguments of Section 5 also apply to
symmetric spectra. In particular, if (-) K is already a Quillen equivalence on
D, then F0: D -! Sp (D; K) is a Quillen equivalence. In this case, with some
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 43
additional mild hypotheses, the homotopy category of D is enriched, tensored, a*
*nd
cotensored over Ho Sp (C; K). We also show symmetric spectra are functorial in
an appropriate sense. In particular, we show that the Quillen equivalence class*
* of
Sp (D; K) is an invariant of the homotopy type of K.
Throughout this section, C will denote a left proper cellular symmetric monoi*
*dal
model category, D will denote a left proper cellular C-model category, and K wi*
*ll
denote a cofibrant object of C.
The proof of the following theorem is the same as the proof of Theorem 5.1.
Theorem 9.1. Suppose (-) K is a Quillen equivalence of D. Then F0: D -!
Sp (D; K) is a Quillen equivalence.
Corollary 9.2.Suppose that the domains of the generating cofibrations of both C
and D are cofibrant, and suppose that (-) K is a Quillen equivalence of D. Then
Ho D is enriched, tensored, and cotensored over Ho Sp (C; K).
Proof.Note that HoSp (D; K) is certainly enriched, tensored, and cotensored over
Ho Sp (C; K). Now use the equivalence of categories LF0: Ho D -!Ho Sp (D; K)__
to transport this structure back to Ho D. |__|
Recall that the homotopy category of any model category is naturally enriched,
tensored, and cotensored over Ho SSet [12, Chapter 5]. Corollary 9.2 is the fir*
*st
step to the assertion that the homotopy category of any stable (with respect to*
* the
suspension) model category is naturally enriched, tensored, and cotensored over*
* the
homotopy category of (simplicial) symmetric spectra. See [23] for further resul*
*ts
along these lines.
Symmetric spectra are functorial in a natural way. The following theorem is
analogous to Proposition 5.5. We use the notation Sp () for the map of symmetric
spectra induced by a functor because Sp () is always a prolongation of , unlike
the situation of Proposition 5.5.
Theorem 9.3. Suppose C is a left proper cellular symmetric monoidal model cate-
gory, and D and D0are left proper cellular C-model categories. Suppose also tha*
*t the
domains of the generating cofibrations of C, D, and D0can be taken to be cofibr*
*ant.
Then any C-Quillen functor : D -!D0extends naturally to a Sp (C; K)-Quillen
functor
Sp (): Sp (D; K) -!Sp (D0; K):
Furthermore, if is a Quillen equivalence, so is Sp ().
Proof.The functor induces a C -functor D -! (D0) , which takes the symmetric
sequence (Xn) to the symmetric sequence (Xn). It follows that induces a
Sp (C; K)-functor Sp (): Sp (D; K) -!Sp (C0; K), which takes the symmetric
spectrum (Xn) to the symmetric spectrum (Xn), with structure maps
Xn K ~=(Xn K) oe--!Xn+1;
where oe denotes the structure map of X. Let denote the right adjoint of . Sin*
*ce
preserves the tensor action of C, preserves the cotensor action of C. Then the
right adjoint of Sp () is Sp (), which takes the symmetric spectrum (Yn) to the
symmetric spectrum (Yn), with structure maps adjoint to the composite
Yn eoe-!YnK+1~=(Yn+1)K ;
44 MARK HOVEY
where eoedenotes the adjoint to the structure map of Y . Since Evn Sp () = Evn,
it follows that Sp ()Fn = Fn.
It is clear that Sp () preserves level fibrations and level equivalences, so *
*Sp ()
is a Quillen functor with respect to the projective model structures. In view *
*of
Theorem 2.2, to see that Sp () defines a Quillen functor with respect to the st*
*able
model structures, it suffices to show that Sp ()(iQCn) is a stable equivalence *
*for
all domains and codomains of C of the generating cofibrations of C. But one can
readily verify that Sp ()(iQCn) = iQCn , which is a stable equivalence as requi*
*red.
Thus Sp () is a Quillen functor with respect to the stable model structures.
If is a Quillen equivalence, one can check that Sp () is a Quillen equivalen*
*ce
with respect to the projective model structure. Indeed, in that case, reflects*
* weak
equivalences between fibrant objects ( [12, Corollary 1.3.16]), so Sp () reflec*
*ts level
equivalences between level fibrant objects. Let R0 denote a fibrant replacement
functor in the projective model structure and let R denote a fibrant replacement
functor in D0. Then, by using lifting as in the proof of Theorem 5.7, we find
that the map X -! Sp ()R0Sp ()X is a level equivalence if and only if Xn -!
R(Sp ()X)n = RXn is a weak equivalence for all n. Note that the additional
complexity of R0 coming from the symmetric group actions is irrelevant to this
reduction argument. The latter map is a weak equivalence since is a Quillen
equivalence.
To see that Sp () is still a Quillen equivalence with respect to the stable m*
*odel
structures, it suffices to show that Sp () reflects stably fibrant objects, in *
*view
of Proposition 2.3. Suppose X is level fibrant and Sp ()(X) is an -spectrum.
Then Xn -eoe!(XKn+1) is a weak equivalence for all n. Since reflects weak
equivalences between fibrant objects by [12, Corollary 1.3.16], this means_that*
* X
is an -spectrum, as required. |__|
Symmetric spectra are also functorial, in a limited sense, in the cofibrant o*
*bject
K.
Theorem 9.4. Suppose C is a left proper cellular symmetric monoidal model cat-
egory, and D is a left proper cellular C-model category. Suppose the domains of*
* the
generating cofibrations of C and D can be taken to be cofibrant. Finally, suppo*
*se
f :K -! K0 is a weak equivalence of cofibrant objects of C. Then f induces a
natural Quillen equivalence (-) Sym(K)Sym (K0): Sp (D; K) -!Sp (D; K0).
Proof.The map f induces a map of commutative monoids Sym (K) -! Sym(K0).
This induces the usual induction map
(-) Sym(K)Sym (K0): Sp (D; K) -!Sp (D; K0);
and its right adjoint, the restriction map that takes a Sym (K0)-module X to X *
*it-
self, thought of as a Sym(K)-module via the map Sym(K) -!Sym (K0). Restriction
obviously preserves level fibrations and level equivalences, so is a Quillen fu*
*nctor
with respect to the projective model structure. Also, restriction preserves the*
* eval-
uation functors Evn, so, by adjointness, FnA Sym(K)Sym (K0) = FnA. Here we
must interpret FnA as an object of Sp (D; K) on the left side of this equation *
*and
as an object of Sp (D; K0) on the right side. It follows that, if C is a domain*
* or
codomain of a generating cofibration of D, iQCnSym(K)Sym (K0) is the map
Fn+1(QC K) -!FnQC
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 45
in Sp (C; K0). The weak equivalence QC K -! QC K0 induces a level equiva-
lence Fn+1(QCK) -!Fn+1(QCK0). Since the map Fn+1(QCK0) -!FnQC is
a stable equivalence, so is iQCnSym(K)Sym (K0). Thus, by Theorem 2.2, induction
is a Quillen functor with respect to the stable model structures.
We now prove that induction is a Quillen equivalence between the projective
model structures. The proof of this is similar to the proof of Theorem 5.7. That
is, restriction certainly reflects level equivalences between level fibrant obj*
*ects. By
a lifting argument as in the proof of Theorem 5.7, it therefore suffices to sho*
*w that
the map X -! X Sym(K)Sym (K0) is a level equivalence for all cofibrant X. The
transfinite induction argument of Theorem 5.7 will prove this without difficult*
*y.
To show that induction is a Quillen equivalence between the stable model stru*
*c-
tures, we need only check that0restriction reflects stably fibrant objects. Thi*
*s follows_
from the fact that the map ZK -! ZK is a weak equivalence for all fibrant Z. *
* |__|
In particular, it does not matter, up to Quillen equivalence, what model of t*
*he
simplicial circle one takes in forming the symmetric spectra of [13].
10.Comparison of spectra and symmetric spectra
In this section, C will be a left proper cellular symmetric monoidal model ca*
*t-
egory, K will be a cofibrant object of C, and D will be a left proper cellular *
*C-
model category. Then we have two different stabilizations of D, namely the stab*
*le
model structures on SpN(D; K) and Sp (D; K), where SpN(D; K) is the category
of T -spectra SpN(D; T ) when T is the functor T X = X K. The object of this
section is to compare them. The quick summary of our result is that SpN(D; K)
and Sp (D; K) are related by a chain of Quillen equivalences whenever the cyclic
permutation self-map of K K K is homotopic to the identity. The precise
statement requires a few more hypotheses; see the statements of Theorem 10.1 and
Theorem 10.3.
This is not the ideal theorem; one might hope for a direct Quillen equivalence
rather than a zigzag of Quillen equivalences, and one might hope for weaker hy-
potheses, or even no hypotheses. However, some hypotheses are necessary, as
pointed out to the author by Jeff Smith. Indeed, the category HoSp (C; K) is sy*
*m-
metric monoidal, and therefore Ho Sp (C; K)(F0S; F0S), the set of self-maps of *
*the
unit object of Ho Sp (C; K), is a commutative monoid. If we have a chain of Qui*
*l-
len equivalences between the stable model structures on SpN(C; K) and Sp (C; K)
that preserves the functors F0, then Ho SpN(C; K)(F0S; F0S) would also have to *
*be
a commutative monoid. With sufficiently many hypotheses on C and K, for exam-
ple if C is the category of simplicial sets and K is a finite simplicial set, w*
*e have
seen in in Section 4 that this mapping set is the colimit colimHo C(Kn ; Kn ).
There are certainly examples where this monoid is not commutative; for example
K could be the mod p Moore space, and then homology calculations show this
colimit is not commutative. In fact, this monoid will be commutative if and only
if the cyclic permutation map of K K K becomes the identity in Ho C after
tensoring with sufficiently many copies of K. Hence we need some hypothesis on
the cyclic permutation map.
The central idea of this section is that commuting stabilization functors mus*
*t be
equivalent, an idea suggested to the author by Mike Hopkins in a different cont*
*ext.
The following theorem is a consequence of this idea. To make sense of it, recal*
*l_that
there are two different ways to tensor with K on SpN(D; K); the functor X 7! X *
* K
46 MARK HOVEY
that is a Quillen equivalence but does not involve the twist map, and the funct*
*or
X 7! X K that may not be a Quillen equivalence but does involve the twist map.
Theorem 10.1. Suppose C is a left proper cellular symmetric monoidal model cat-
egory, and that the domains of the generating cofibrations of C can be taken to
be cofibrant. Suppose D is a left proper cellular C-model category such that t*
*he
functor X 7! X K is a Quillen equivalence of the stable model structure on
SpN(D; K). Then there is a C-model category E together with C-Quillen equiv-
alences Sp (D; K) -! E- SpN(D; K), where Sp (D; K) and SpN(D; K) are
given the stable model structures. Furthermore, we have a natural isomorphism
[Ho Sp (D; K)](F0A; F0B) ~=[Ho SpN(D; K)](F0A; F0B) for A; B 2 D.
Proof.We take E = SpN(Sp (D; K); K) with its stable model structure. This
makes sense since Sp (D; K) is a C-model category, by the comments following
Theorem 8.11. By Theorem 5.1, F0: Sp (D; K) -! E is a C-Quillen equivalence.
On the other hand, since SpN(D; K) is a C-model category by Theorem 6.3, we
can also consider E0 = Sp (SpN(D; K); K) with its stable model structure. The
action of K on SpN(D; K) is then X 7! X K, as pointed out in Remark 6.4. By
hypothesis, this functor is already a Quillen equivalence, so Theorem 9.1 impli*
*es
that F0: SpN(D; K) -!E0is a C-Quillen equivalence.
We claim that E is isomorphic to E0as a model category. This is the precise s*
*ense
in which our two stabilization functors commute with each other, and obviously *
*will
complete the proof. An object of E is a set {Ym;n} of objects of D, where m; n *
* 0,
together with certain maps. There is an action of n on Ym;n, and there are n-
equivariant maps Ym;n K -! Ym+1;n and Ym;n K -ae!Ym;n+1. In addition, the
composite Ym;nKp -! Ym;n+p is nxp-equivariant, and there is a compatibility
between and ae, expressed in the commutativity of the following diagram.
Ym;n K K -1T---!Ym;n K K -ae1---!Ym;n+1 K
? ?
1 ?y ?y
Ym+1;n K _______ Ym+1;n K ----!ae Ym+1;n+1
An object of the category E0 is a set {Ym0;n} of objects of D for m; n 0 to-
gether with certain maps. In this case, we have an action of m on Ym0;nand
m -equivariant maps ae0: Ym0;n K -! Ym0+1;nand 0: Ym0;n K -! Ym0;n+1. The
composite Ym0;n Kp -! Ym0+p;nis m x p-equivariant, and there is a similar
compatibility relationship between ae0and 0. There is therefore an isomorphism *
*of
categories from E to E0that takes Y to Y 0, where Ym0;n= Yn;m, ae0= , and 0= ae.
There is a projective model structure on both E and E0, where a map f is a we*
*ak
equivalence (or fibration) if and only if fm;n is a weak equivalence (or fibrat*
*ion)
in D for all m; n. The isomorphism between E and E0 obviously preserves this
projective model structure. The stable model structure on both E and E0 is the
Bousfield localization of the projective model structure with respect to the ma*
*ps
Fm;n+1(QC K) -! Fm;nQC and Fm+1;n(QC K) -! Fm;nQC, where Fm;n is
left adjoint to the evaluation functor Evm;n, and C runs through the domains and
codomains of the generating cofibrations of D. The isomorphism between E and E0
preserves this set of maps, so must preserve the entire stable model structure.
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 47
The composites
D F0-!Sp (D; K) F0-!E
and
D F0-!SpN(D; K) F0-!E0~=E
are both naturally isomorphic to F0;0, since both are left adjoint to Ev0;0. Th*
*erefore
we have natural isomorphisms
[Ho Sp (D; K)](F0A; F0B)~=[Ho E](F0;0A; F0;0B)
~=[Ho E0](F0;0A; F0;0B)
~=[Ho SpN(D; K)](F0A; F0B)
which complete the proof. |___|
In particular, we have calculated [Ho SpN(D; K)](F0A; F0B) in Corollary 4.13;
when both the hypotheses of that corollary and the hypotheses of Theorem 10.1
hold, we get the expected result
[Ho Sp (D; K)](F0A; F0B) = colimHoD(A Kn ; B Kn )
for cofibrant A and B.
Theorem 10.1 indicates that we should try to prove that (-) K is a Quil-
len equivalence of SpN(D;_K). The only way the author can see to do this is by
comparing (-) K to (-) K, which we_know_is a Quillen equivalence. The ba-
sic idea is to compare X K K to X K K. Both of these spectra have the
same spaces, and their structure maps differ precisely by the cyclic permutation
self-map of K K K. So if we knew that this map were the identity, they
would be the same spectrum. The hope is then that, if we knew that the cyclic
permutation map were only homotopic to the identity, these two spectra would
still_be_equivalent in Ho SpN(D; K). One can, in fact, construct a map of spect*
*ra
X K K -! R0(X K K), where R0is a level fibrant replacement functor and X
is cofibrant, by inductively modifying the identity map. Unfortunately, the aut*
*hor
does not know how to do this modification in_a_natural way, so is unable to pro*
*ve
that the derived functors L(X K K) and L(X K K) are equivalent using this
method.
Instead, we will follow a suggestion of Dan Dugger. We will construct a new f*
*unc-
tor_F_on cofibrant objects X of SpN(D; K) and natural level equivalences_F_X -!
X K K and F X -! X K K. It will follow immediately that L(X K K) is
naturally equivalent to L(X K K), and therefore that X 7! X K is a Quillen
equivalence on SpN(D; K). Unfortunately, to construct F , we will need to make
some unpleasant assumptions that ought to be unnecessary.
Definition 10.2.Given a symmetric monoidal model category C whose unit S is
cofibrant, a unit interval in C is a cylinder object I for S such that there ex*
*ists a
map HI: I I -! I satisfying HIO (1 i0) = HIO (i0 1) = i0ss and HIO (1 i1) is
the identity. Here i0; i1: S -! I and ss :I -! S are the structure maps of I. G*
*iven
a cofibrant object K of C, we say that K is symmetric if there is a unit interv*
*al I
and a homotopy
H :K K K I -! K K K
from the cyclic permutation to the identity.
48 MARK HOVEY
Note that [0; 1] is a unit interval in the usual model structure on compactly
generated topological spaces, and [1] is a unit interval in the category of sim*
*plicial
sets. Indeed, the required map H1: [1] x [1] takes both of the nondegenerate
2-simplices 011 x 001 and 001 x 011 to 001. Similarly, the standard unit interv*
*al
chain complex of abelian groups is a unit interval in the projective model stru*
*cture
on chain complexes. Also, any symmetric monoidal left Quillen functor preserves
unit intervals. It follows, for example, that the unstable A1-model category of
Morel-Voevodksy has a unit interval.
Our goal, then, is to prove the following theorem.
Theorem 10.3. Let C be a symmetric monoidal model category with cofibrant unit
S, and let D be a left proper cellular C-model category. Suppose that K is a co*
*fibrant
object of C, and that either K is itself symmetric or the domains of the genera*
*ting
cofibrations of D are cofibrant and K is weakly equivalent to a symmetric objec*
*t of
C. Then the functor X 7! X K is a Quillen equivalence of SpN(D; K).
This theorem is not the best one ought to be able to do. For example, by
considering the analogous functors with more than three tensor factors of K, it
should be possible to show that the same theorem holds if there is a left homot*
*opy
between some even permutation of Kn and the identity. Also, it seems clear
that one should only need the cyclic permutation, or more generally some even
permutation, to be equal to the identity in Ho C. That is, we should not need
a specific left homotopy between an even permutation and the identity. But the
author does not know how to remove this hypothesis.
In any case, we have the following corollary.
Corollary 10.4.Let C be a left proper cellular symmetric monoidal model category
whose unit S is cofibrant, and whose generating cofibrations can be taken to ha*
*ve
cofibrant domains. Let D be a left proper C-model category. Suppose K is a cofi*
*brant
object of C, and either that K is itself symmetric, or else that the domains of*
* the
generating cofibrations of D are cofibrant and K is weakly equivalent to a symm*
*etric
object of C. Then there is a C-model category E and C-Quillen equivalences
Sp (D; K) -!E- SpN(D; K):
We will prove Theorem 10.3 in a series of lemmas.
Lemma 10.5. Let C be a symmetric monoidal model category whose unit S is
cofibrant. Suppose we have the (not necessarily commutative)square below
A --f--!X
? ?
r?y ?ys
B ----!gY
in a C-model category D, where A and B are cofibrant. Let H :A I -! Y be a
left homotopy from gr to sf, for some unit interval I. Then there is an object *
*B0
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 49
of D, a weak equivalence B0-q!B, a commutative square
A --f--! X
? ?
r0?y ?ys
B0 ----! Y
g0
such that qr0 = r, and a left homotopy H0: B0 I -! Y between gq and g0.
Furthermore, this construction is natural in an appropriate sense.
Naturality means that, if we have a map of such homotopy commutative squares
that preserves the homotopies, then we get a map of the resulting commutative
squares that preserves the maps q and H0. The precise statement is complicated,
and we leave it to the reader.
Proof.We let B0 be the mapping cylinder of r. That is, we take B0 to be the
pushout in the diagram below.
A --i0--!A I
? ?
r?y ?yh
B ----! B0
j
The map r0is then the composite hi1, and the map g0is the map that is g on B and
H on A I. It follows that g0r0= Hi1 = sf, as required. The map q :B0-! B is
defined to be the identity on B and the composite AI -ss!A r-!B on AI. Since
j is a trivial cofibration (as a pushout of i0), it follows that q is a weak eq*
*uivalence,
and it is clear that qr0= r. We must now construct the homotopy H0. First note
that B0 is cofibrant, since B is so and j is a trivial cofibration, and so B0 I*
* is a
cylinder object for B0. In fact, B0 I is the pushout of A I I and B I over
A I. Define H0 to be the constant homotopy B I -ss!B -g!Y on B I and the
homotopy
A I I -1HI--!A I -H!Y
on A I I. The fact that HI O (i0 1) = i0ss guarantees that H0 is well-define*
*d,
and the other conditions on HI guarantee that H0 is a left homotopy from gq to *
* __
g0. We leave the naturality of this construction to the reader. *
*|__|
We also need the following lemma about the behavior of unit intervals.
Lemma 10.6. Suppose C is a symmetric monoidal model category whose unit S
is cofibrant. Let I and I0 be unit intervals with structure maps i0; i1: S -! I*
* and
i00; i01:S -! I0, and define J by the pushout diagram below.
0
S --i0--!I0
? ?
i1?y ?yfi
I ----!ffJ
Then J is a unit interval.
50 MARK HOVEY
Proof.The reader is well-advised to draw a picture in the topological or simpli*
*cial
case, from which the proof should be clear. We think of J as the interval whose*
* left
half is I and whose right half is I0. The fact that J is a cylinder object for *
*S, with
structure maps j0 = ffi0 and j1 = fii01, is proved in [11, Lemma 7.3.11]. Becau*
*se
the tensor product preserves pushouts, we can think of J J as a square consist*
*ing
of a copy of I I in the lower left, a copy of I I0 in the upper left, a copy *
*of
I0 I is the lower right, and a copy of I0 I0 in the upper right. We define the
necessary map G: J J -! J by defining G on each subsquare. On the lower left,
we use the composite I I -H!I -ff!J, where H is the homotopy making I into a
unit interval. Similarly, on the upper right square, we use the composite fiH0.*
* On
the upper left square we use the constant homotopy ff(1 ssI0). On the lower ri*
*ght
square we use the composite
I0 I -ssI01---!I -i11--!I I -H!I -ff!J:
We leave it to the reader to check that this makes J into a unit interval. *
* |___|
The importance of these two lemmas for spectra is indicated in the following
consequence.
Lemma 10.7. Suppose C is a left proper cellular symmetric monoidal model cate-
gory with cofibrant unit S, unit interval I, and cofibrant object K. Let D be a*
* left
proper cellular C-model category. Suppose A; B 2 SpN(D; K), where A is cofibran*
*t,
and we have maps fn :An -! Bn for all n and homotopies Hn :AnKI -! Bn+1
from fn+1oeA to oeB (fn 1), where oe(-)is the structure map of the spectrum (-*
*).
Then there is a spectrum C, a level equivalence C -h!A, and a map of spectra
C -g! B such that gn is homotopic to fnhn. Furthermore, this construction is
natural in an appropriate sense.
Once again, the naturality involves the homotopies Hn as well as the maps fn.
We leave the precise statement to the reader.
Proof.We define Cn, hn, gn and a homotopy H0n: Cn In -! Bn from fnhn to gn,
where In is a unit interval, inductively on n, using Lemma 10.5. To get started,
we take C0 = A0, h0 to be the identity, g0 to be f0, and H00to be the constant
homotopy (with I0 = I). For the inductive step, we apply Lemma 10.5 to the
diagram
Cn K -gn1---!Bn K
? ?
oeA(hn1)?y ?yoeB
An+1 ----! Bn+1
fn+1
and the homotopy obtained as follows. We have a homotopy
01 oe
Cn K In 1T---!Cn In K -Hn---!Bn K --!BBn+1
from oeB (fn1)(hn1) to oeB (gn1). On the other hand, we also have the homotopy
Hn(hn 1) from fn+1oeA (hn 1) to oeB (fn 1)(hn 1). We can amalgamate these
to get a homotopy Gn :Cn K In+1 -!Bn+1 from fn+1oeA (hn 1) to oeB (gn 1),
and In+1 is still a unit interval by Lemma 10.6. Hence Lemma 10.5 gives us an
object Cn+1, a map oeC :Cn K -! Cn+1, and a map gn+1: Cn+1 -! Bn+1 such
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 51
that gn+1oeC = oeB (gn 1). It also gives us a map hn+1: Cn+1 -! An+1 such that
hn+1oeC = oeA (hn1) and a homotopy H0n+1:Cn+1In+1 -!Bn+1 from fn+1hn+1
to gn+1. This completes the induction step and the proof (we leave naturality_to
the reader). |__|
With this lemma in hand we can now give the proof of Theorem 10.3.
Proof of Theorem 10.3.We first reduce to the case where K is itself symmetric. *
*So
suppose the generating cofibrations of D have cofibrant domains, and suppose K0
is symmetric and weakly equivalent to K; this means there are weak equivalences
K -! RK -! RK0- K0, where R denotes a fibrant replacement functor. This
implies that the total left derived functors X 7! X L K and X 7! X L K0 are
naturally isomorphic on the homotopy category of any C-model category. In par-
ticular, it suffices to show that X 7! X K0is a Quillen equivalence on SpN(D; K*
*).
On the other hand, by Theorem 5.7, there are C-Quillen equivalences
SpN(D; K) -!SpN(D; RK) -!SpN(D; RK0)- SpN(D; K0):
It therefore suffices to show that X 7! XK0is a Quillen equivalence of SpN(D; K*
*0);
that is, we can assume that K itself is symmetric.
Let H denote the given homotopy from the cyclic permutation to the identity
of_K__K K. Let X be a cofibrant spectrum, let eoedenote the structure map of
X K K, and let oe denote the structure map of X K K. These two structure
maps differ by the cyclic_permutation,_and therefore we are in the situation of
Lemma 10.7, with A = X K K, B = X K K, fn equal to the identity map,
and Hn = (oeX 1 1)(1 H). It follows that we get a functor F defined on
__ __
cofibrant objects of SpN(D; K) and natural level equivalences F X -h!X K K
and F X -g!X K K, where the latter map is a level equivalence_since_gn is
homotopic to hn. Thus the total left derived functors of (-)__K K and (-)KK
are naturally isomorphic. Since we know already that (-) K K is a Quillen_
equivalence, so is (-) K K, and hence so is (-) K. |__|
Appendix A. Cellular model categories
In this section we define cellular model categories and show that the projec-
tive model structures on SpN(D; T ) and Sp (D; K) are cellular when the model
structure on D is cellular. This is necessary to ensure that the Bousfield loca*
*liza-
tions used in the paper do in fact exist. The definitions in this section are t*
*aken
from [11]. Throughout this section, then, T will be a left Quillen endofunctor *
*of
a model category D; when we refer to Sp (D; K), we will be thinking of D as a
C-model category, where C is some symmetric monoidal model category, and of K
as a cofibrant object of C.
A cellular model category is a special kind of cofibrantly generated model ca*
*te-
gory. Three additional hypotheses are needed.
Definition A.1.A model category E is cellular if there is a set of cofibrations
I and a set of trivial cofibrations J making E into a cofibrantly generated mod*
*el
category and also satisfying the following conditions.
1. The domains and codomains of I are compact relative to I.
2. The domains of J are small relative to the cofibrations.
3. Cofibrations are effective monomorphisms.
52 MARK HOVEY
The first hypothesis above requires considerable explanation, which we will p*
*ro-
vide below. We first point out that the second hypothesis will hold in the proj*
*ective
model structure on SpN(D; T ) or Sp (D; K) when it holds in D.
Lemma A.2. Suppose D is a cofibrantly generated model category with generating
cofibrations I and generating trivial cofibrations J, and T is a left Quillen e*
*ndo-
functor of D. Suppose the domains of J are small relative to the cofibrations i*
*n D.
Then the domains of the generating trivial cofibrations JT of the projective mo*
*del
structure on SpN(D; T ) are small relative to the cofibrations in SpN(D; T ). S*
*imi-
larly, if D is a C-model category, K is a cofibrant object of C, and T = (-) K,
then the domains of the generating trivial cofibrations JK of the projective m*
*odel
structure on Sp (D; K) are small relative to the cofibrations in Sp (D; K).
Proof.For SpN(D; T ), this follows immediately from the definition of JT, Propo*
*si-_
tion 1.9, and Proposition 1.11. The proof for Sp (D; K) is similar. *
*|__|
We now discuss the third hypothesis.
Definition A.3.Suppose E is a category. A map f :X -! Y in E is an effective
monomorphism if f is the equalizer of the two obvious maps Y Y qX Y .
Proposition A.4. Suppose D is a cofibrantly generated model category and T is
a left Quillen endofunctor of D. Suppose that cofibrations are effective monomo*
*r-
phisms in D. Then level cofibrations, and in particular projective cofibrations*
*, are
effective monomorphisms in SpN(D; T ). Similarly, when D is a C-model category
and T = (-)K for some cofibrant K 2 C, level cofibrations are effective monomor-
phisms in Sp (D; K).
Proof.This is immediate, since colimits and limits in SpN(D; T ) and Sp (D;_K)
are taken levelwise. |__|
We must now define compactness. This will involve some preliminary definition*
*s.
These definitions are based on the idea of CW-complexes, but are of necessity
somewhat technical in the general case.
Definition A.5.Suppose I is a set of maps in a cocomplete category E. A relative
I-cell complex is a map which can be written as the transfinite composition of
pushouts of coproducts of maps of I. That is, given a relative I-cell complex *
*f,
there is an ordinal and a colimit-preserving functor X : -! E and a collection
{(T fi; efi; hfi)fi<} satisfying the following properties.
1. f is isomorphic to the transfinite composition of X.
2. Each T fiis a set.
3. Each efiis a function efi:T fi-!I.
4. Given fi < and i 2 T fi, if efii:Ci-! Di is the image of i under efi, the*
*n hfii
is a map hfii:Ci-! Xfi.
5. Each Xfi+1is the pushout in the diagram
` `efii`
TfiCi?----! TfiDi?
` hfi? ?
iy y
Xfi ----! Xfi+1
SPECTRA AND SYMMETRIC SPECTRA IN GENERAL MODEL CATEGORIES 53
The ordinal together with the colimit-preserving functor X and`the collection
{(T fi; efi; hfi)fi<} is called a presentation of f. The set fiTfiis the set *
*of cells of
f, and given a cell e, its presentation ordinal is the ordinal fi such that e 2*
* T fi.
The presentation ordinal of f is .
We also need to define subcomplexes of relative I-cell complexes.
Definition A.6.Suppose E is a cocomplete category and I is a set of maps in E.
Given a presentation , X : -!C, and {(T fi; efi; hfi)fi<} of a map f as a relat*
*ive
I-cell complex, a subcomplex of f (or really of the presentation of f), is a co*
*llection
{(Tefi; eefi; ehfi)fi<} such that the following properties hold.
1. Every eT fiis a subset of T fi, and eefiis the restriction of efito eT.fi
2. There is a colimit-preserving functor Xe: -! E such that Xe0 = X0 and
a natural transformation Xe -! X such that, for every fi < and i 2 eT fi,
the map ehfii:Ci -!Xefiis a factorization of hfii:Ci -!Xfithrough the map
eXfi-!Xfi.
3. For all fi < , ^Xfi+1is the pushout in the diagram
` `effii`
eTfiCi----! eTfiDi
` ehfi?? ??
iy y
Xefi ----! Xefi+1
Given a subcomplex`of f, the size of that subcomplex is the cardinality of its
set of cells fi