SYMMETRIC SPECTRA
MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Contents
Introduction 2
1. Symmetric spectra 5
1.1. Simplicial sets 5
1.2. Symmetric spectra 6
1.3. Simplicial structure on Sp 8
1.4. Symmetric -spectra 10
2. The smash product of symmetric spectra 11
2.1. Symmetric sequences 11
2.2. Symmetric spectra 14
2.3. The ordinary category of spectra 18
3. Stable homotopy theory of symmetric spectra 19
3.1. Stable equivalence 19
3.2. Model categories 26
3.3. Level structure 31
3.4. Stable model category 33
4. Comparison with the Bousfield-Friedlander category 39
4.1. Quillen equivalences 39
4.2. The stable Bousfield-Friedlander category 40
4.3. The Quillen equivalence 42
4.4. Description of V 45
5. Additional properties of symmetric spectra 46
5.1. Level model structure 46
5.2. Stable cofibrations 51
5.3. Pushout smash product 52
5.4. Proper model categories 56
5.5. The monoid axiom 57
5.6. Semistable spectra 61
6. Topological spectra 63
6.1. Compactly generated spaces 63
6.2. Topological spectra 66
6.3. Stable model structure 67
6.4. Properties of topological symmetric spectra 73
References 76
____________
Date: March 4, 1998.
1991 Mathematics Subject Classification. 55P42, 55U10, 55U35.
The first two authors were partially supported by NSF Postdoctoral Fellowshi*
*ps.
The third author was partially supported by an NSF Grant.
1
2 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Introduction
Stable homotopy theory studies spectra as the linear approximation to spaces.
Here, "stable" refers to the consideration of spaces after inverting the suspen*
*sion
functor. This approach is a general one: one can often create a simpler category
by inverting an operation such as suspension. In this paper we study a particul*
*arly
simple model for inverting such operations which preserves product structures. *
*The
combinatorial nature of this model means that it is easily transported, and hen*
*ce
may be useful in extending the methods of stable homotopy theory to other setti*
*ngs.
The idea of a spectrum is a relatively simple one: Freudenthal's suspension
theorem implies that the sequence of homotopy classes of maps
[X; Y ] -![X; Y ] -!: :-:![nX; nY ] -!: : :
is eventually constant for finite-dimensional pointed CW-complexes X and Y , wh*
*ere
X = S1 ^ X is the reduced suspension of X. This suggests forming a stable
category where the suspension functor is an isomorphism. The standard way to
do this is to define a spectrum to be a sequence of pointed topological spaces
(or simplicial sets) Xn together with structure maps S1 ^ Xn -! Xn+1. This
was first done by Lima [Lim59 ] and later generalized by Whitehead [Whi62 ]. The
suspension functor is not an isomorphism in the category of spectra, but becomes
an isomorphism when we invert the stable homotopy equivalences. The resulting
homotopy category of spectra is often called the stable homotopy category and
has been extensively studied, beginning with the work of Boardman [Vog70] and
Adams [Ada74 ] and continuing to this day. Notice that this definition of a spe*
*ctrum
can be applied to any situation where one has an operation on a category that o*
*ne
would like to invert; however, this simplest construction does not preserve the*
* smash
product structure coming from spaces.
One of the stable homotopy category's basic features is that it is symmetric
monoidal. There is a smash product, built from the smash product of pointed
topological spaces and analogous to the tensor product of modules, that is asso-
ciative, commutative, and unital, up to coherent natural isomorphism. However,
the category of spectra defined above is not symmetric monoidal. This has been a
sticking point for almost forty years now. Indeed, it was long thought that the*
*re
could be no symmetric monoidal category of spectra; see [Lew91 ], where it is s*
*hown
that a symmetric monoidal category of spectra can not have all the properties o*
*ne
might like.
Any good symmetric monoidal category of spectra allows one to perform alge-
braic constructions on spectra that are impossible without such a category. Thi*
*s is
extremely important, for example, in the algebraic K-theory of spectra. In part*
*icu-
lar, given a good symmetric monoidal category of spectra, it is possible to con*
*struct
a homotopy category of monoids (ring spectra) and of modules over a given monoi*
*d.
In this paper, we describe a symmetric monoidal category of spectra, called
the category of symmetric spectra. The ordinary category of spectra as described
above is the category of modules over the sphere spectrum. The sphere spectrum
is a monoid in the category of sequences of spaces, but it is not a commutative
monoid, because the twist map on S1 ^ S1 is not the identity. This explains why
the ordinary category of spectra is not symmetric monoidal, just as in algebra *
*where
the usual internal tensor product of modules is defined only over a commutative
ring. To make the sphere spectrum a commutative monoid, we need to keep track
SYMMETRIC SPECTRA 3
of the twist map, and, more generally, of permutations of coordinates. We there*
*fore
define a symmetric spectrum to be a sequence of pointed simplicial sets Xn toge*
*ther
with a pointed action of the permutation group n on Xn and equivariant structure
maps S1 ^ Xn -! Xn+1. We must also require that the iterated structure maps
Sp ^ Xn -! Xn+p be px n-equivariant. This idea is due to the third author; the
first and second authors joined the project later.
At approximately the same time as the third author discovered symmetric spec-
tra, the team of Elmendorf, Kriz, Mandell, and May [EKMM97 ] also constructed
a symmetric monoidal category of spectra, called S-modules. Some generalizations
of symmetric spectra appear in [MMSS1 ]. These many new symmetric monoidal
categories of spectra, including S-modules and symmetric spectra, are shown to *
*be
equivalent in an appropriate sense in [MMSS1 ] and [MMSS2 ]. Another symmetric
monoidal category of spectra sitting between the approaches of [EKMM97 ] and *
*of
this paper is developed in [DS ]. We also point out that symmetric spectra are *
*part
of a more general theory of localization of model categories [Hir97]; we have n*
*ot
adopted this approach, but both [Hir97] and [DHK ] have influenced us consider*
*ably.
Symmetric spectra have already proved useful. In [GH97 ], symmetric spec-
tra are used to extend the definition of topological cyclic homology from rings
to schemes. Similarly, in [Shi97], B"okstedt's approach to topological Hochsch*
*ild
homology [B"ok85] is extended to symmetric ring spectra, without connectivity c*
*on-
ditions. And in [SS], it is shown that any linear model category is Quillen equ*
*ivalent
to a model category of modules over a symmetric ring spectrum.
As mentioned above, since the construction of symmetric spectra is combinator*
*ial
in nature it may be applied in many different situations. Given any well-behaved
symmetric monoidal model category, such as chain complexes, simplicial sets, or
topological spaces, and an endofunctor on it that respects the monoidal struc-
ture, one can define symmetric spectra. This more general approach is explored
in [Hov98b ]. In particular, symmetric spectra may be the logical way to constr*
*uct
a model structure for Voevodsky's stable homotopy of schemes [Voe97].
In this paper, we can only begin the study of symmetric spectra. The most sig-
nificant loose end is the construction of a model category of commutative symme*
*tric
ring spectra; such a model category has been constructed by the third author in
work in progress. It would also be useful to have a stable fibrant replacement *
*func-
tor, as the usual construction QX does not work in general. A good approximation
to such a functor is constructed in [Shi97].
At present the theory of S-modules of [EKMM97 ] is considerably more develo*
*ped
than the theory of symmetric spectra. Their construction appears to be signific*
*antly
different from symmetric spectra; however, there is work in progress [MMSS2 ] *
*show-
ing that the two approaches define equivalent stable homotopy categories and eq*
*uiv-
alent homotopy categories of monoids and modules, as would be expected. Each
approach has its own advantages. The category of symmetric spectra is technical*
*ly
much simpler than the S-modules of [EKMM97 ]; this paper is almost entirely s*
*elf-
contained, depending only on some standard results about simplicial sets and to*
*po-
logical spaces. As discussed above, symmetric spectra can be built in many diff*
*erent
circumstances, whereas S-modules appear to be tied to the category of topologic*
*al
spaces. There are also technical differences reflecting the result of [Lew91 ]*
* that
there are limitations on any symmetric monoidal category of spectra. For exampl*
*e,
the sphere spectrum S is cofibrant in the category of symmetric spectra, but is
not in the category of S-modules. On the other hand, every S-module is fibrant,*
* a
4 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
considerable technical advantage. Also, the S-modules of [EKMM97 ] are very w*
*ell
suited to the varying universes that arise in equivariant stable homotopy theor*
*y,
whereas we do not yet know how to realize universes in symmetric spectra. For a
first step in this direction see [SS].
Organization. The paper is organized as follows. We choose to work in the
category of simplicial sets until Section 6, where we discuss topological symme*
*tric
spectra. This is a significant technical simplification; while it is possible t*
*o develop
symmetric spectra in topological spaces from scratch, technical issues arise th*
*at
are not present when working with simplicial sets. In the first section, we def*
*ine
symmetric spectra, give some examples, and establish some basic properties. In
Section 2 we describe the closed symmetric monoidal structure on the category of
symmetric spectra, and explain why such a structure can not exist in the ordina*
*ry
category of spectra. In Section 3 we study the stable homotopy theory of symmet*
*ric
spectra. This section is where the main subtlety of the theory of symmetric spe*
*ctra
arises: we cannot define stable equivalence by using stable homotopy isomorphis*
*ms.
Instead, we define a map to be a stable equivalence if it is a cohomology isomo*
*rphism
for all cohomology theories. The main result of this section is that symmetric
spectra, together with stable equivalences and suitably defined classes of stab*
*le
fibrations and stable cofibrations, form a model category. As expected, the fib*
*rant
objects are the -spectra; i.e., symmetric1spectra X such that each Xn is a Kan
complex and the adjoint Xn -! XSn+1of the structure map is a weak equivalence.
In Section 4, we prove that the stable homotopy theories of symmetric spectra a*
*nd
ordinary spectra are equivalent. More precisely, we construct a Quillen equival*
*ence
of model categories between symmetric spectra and the model category of ordinary
spectra described in [BF78 ].
In Section 5 we discuss some of the properties of symmetric spectra. In par-
ticular, in Section 5.1, we tie up a loose end from Section 3 by establishing t*
*wo
different model categories of symmetric spectra where the weak equivalences are
the level equivalences. We characterize the stable cofibrations of symmetric sp*
*ectra
in Section 5.2. In Section 5.3, we show that the smash product of symmetric spe*
*ctra
interacts with the model structure in the expected way. This section is crucial*
* for
the applications of symmetric spectra, and, in particular, is necessary to be s*
*ure
that the smash product of symmetric spectra does define a symmetric monoidal
structure on the stable homotopy category. We establish that symmetric spectra
are a proper model category in Section 5.4, and use this to verify the monoid a*
*x-
iom in Section 5.5. The monoid axiom is required to construct model categories *
*of
monoids and of modules over a given monoid, see [SS97]. In Section 5.6, we defi*
*ne
semistable spectra, which are helpful for understanding the difference between *
*sta-
ble equivalences and stable homotopy equivalences. Finally, we conclude the pap*
*er
by considering topological symmetric spectra in Section 6.
Acknowledgments. The authors would like to thank Dan Christensen, Bill Dwyer,
Phil Hirschhorn, Dan Kan, Haynes Miller, John Palmieri, Charles Rezk, and Stefan
Schwede for many helpful conversations about symmetric spectra. We would also
like to thank Gaunce Lewis and Peter May for pointing out that topological spac*
*es
are more complicated than we had originally thought.
Notation. We now establish some notation we will use throughout the paper.
Many of the categories in this paper have an enriched Hom as well as a set valu*
*ed
Hom. To distinguish them: in a category C, the set of maps from X to Y is
denoted C(X; Y ); in a simplicial category C, the simplicial set of maps from X*
* to Y
SYMMETRIC SPECTRA 5
is denoted Map C(X; Y ) or Map (X; Y ); in a category C with an internal Hom, t*
*he
object in C of maps from X to Y is denoted Hom C(X; Y ) or Hom (X; Y ). In case*
* C
is the category of modules over a commutative monoid S, we also use Hom S(X; Y )
for the internal Hom.
1.Symmetric spectra
In this section we construct the category of symmetric spectra over simplicial
sets. We discuss the related category of topological symmetric spectra in Secti*
*on 6.
We begin this section by recalling the basic facts about simplicial sets in Sec*
*tion 1.1,
then we define symmetric spectra in Section 1.2. We describe the simplicial str*
*uc-
ture on the category of symmetric spectra in Section 1.3. The homotopy category
of symmetric -spectra is described in Section 1.4.
1.1. Simplicial sets. With the exception of Section 6 dealing with topological
spectra, this paper is written using simplicial sets. We recall the basics. Con*
*sult
[May67 ] or [Cur71] for more details.
The category has the ordered sets [n] = {0; 1; : :;:n} for n 0 as its objec*
*ts
and the order preserving functions [n] ! [m] as its maps. The category of simpl*
*icial
sets, denoted S, is the category of functors from opto the category of sets. Th*
*e set
of n-simplices of the simplicial set X, denoted Xn, is the value of the functor*
* X at
[n]. The standard n-simplex [n] is the contravariant functor (-; [n]). Varying n
gives a covariant functor [-]: ! S. By the Yoneda lemma, S([n]; X) = Xn
and the contravariant functor S([-]; X) is naturally isomorphic to X.
Let G be a discrete group. The category of G-simplicial sets is the category *
*SG
of functors from G to S, where G is regarded as a category with one object. A
G-simplicial set is therefore a simplicial set X with a left simplicial G-actio*
*n, i.e.,
a homomorphism G ! S(X; X).
A basepoint of a simplicial set X is a distinguished 0-simplex * 2 X0. The
category of pointed simplicial sets and basepoint preserving maps is denoted S*.
The simplicial set [0] = (-; [0]) has a single simplex in each degree and is the
terminal object in S. A basepoint of X is the same as a map [0] ! X. The
disjoint union X+ = X q [0] adds a disjoint basepoint to the simplicial set X.
For example, the 0-sphere is S0 = [0]+ . A basepoint of a G-simplicial set X is*
* a
G-invariant 0-simplex of X. The category of pointed G-simplicial sets is denoted
SG*.
The smash product X ^ Y of the pointed simplicial sets X and Y is the quotient
(X x Y )=(X _ Y ) that collapses the simplicial subset X _ Y = X x * [ * x Y
to a point. For pointed G-simplicial sets X and Y , let X ^G Y be the quotient
of X ^ Y by the diagonal action of G. For pointed simplicial sets X, Y , and Z,
there are natural isomorphisms (X ^ Y ) ^ Z ~= X ^ (Y ^ Z), X ^ Y ~=Y ^ X
and X ^ S0 ~=X. In the language of monoidal categories, the smash product is a
symmetric monoidal product on the category of pointed simplicial sets. We recall
the definition of symmetric monoidal product, but for more details see [ML71 , *
*VII]
or [Bor94, 6.1].
Definition 1.1.1.A symmetric monoidal product on a category C is: a bifunctor
: C x C ! C; a unit U 2 C; and coherent natural isomorphisms (X Y )
Z ~= X (Y Z) (the associativity isomorphism), X Y ~=Y X (the twist
isomorphism), and U X ~=X (the unit isomorphism). The product is closed if
6 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
the functor X (-) has a right adjoint Hom (X; -) for every X 2 C. A (closed)
symmetric monoidal category is a category C with a (closed) symmetric monoidal
product.
Coherence of the natural isomorphisms means that all reasonable diagrams built
from the natural isomorphisms also commute [ML71 ]. When the product is closed,
the pairing Hom (X; Y ): Copx C ! C is an internal Hom. For example, the smash
product on the category S* of pointed simplicial sets is closed. For X; Y 2 S*,*
* the
pointed simplicial set of maps from X to Y is MapS*(X; Y ) = S*(X^[-]+ ; Y ). F*
*or
pointed G-simplicial sets X and Y , the simplicial subset of G-equivariant poin*
*ted
maps is Map G(X; Y ) = SG*(X ^ [-]+ ; Y ).
1.2. Symmetric spectra. Let S1 be the simplicial circle [1]=@[1], obtained by
identifying the two vertices of [1].
Definition 1.2.1.A spectrum is
1. a sequence X0; X1; : :;:Xn; : :o:f pointed simplicial sets; and
2. a pointed map oe :S1 ^ Xn ! Xn+1 for each n 0.
The maps oe are the structure maps of the spectrum. A map of spectra f :X ! Y
is a sequence of pointed maps fn :Xn ! Yn such that the diagram
S1 ^ Xn --oe--!Xn+1
? ?
S1^fn?y fn+1?y
S1 ^ Yn --oe--!Yn+1
is commutative for each n 0. Let SpN denote the category of spectra.
Replacing the sequence of pointed simplicial sets by a sequence of pointed to*
*po-
logical spaces in 1.2.1 gives the original definition of a spectrum (due to Whi*
*tehead
and Lima). The categories of simplicial spectra and of topological spectra are
discussed in the work of Bousfield and Friedlander [BF78 ].
A symmetric spectrum is a spectrum to which symmetric group actions have
been_added. Let p be the group of permutations of the set _p= {1; 2; : :;:p}, w*
*ith
0= ;. As usual,_embed_pxq as the subgroup of p+q with p acting_on the first
p elements of p + qand q acting on the last q elements of p + q. Let Sp = (S1)^p
be the p-fold smash power of the simplicial circle with the left permutation ac*
*tion
of p.
Definition 1.2.2.A symmetric spectrum is
1. a sequence X0; X1; : :;:Xn; : :o:f pointed simplicial sets;
2. a pointed map oe :S1 ^ Xn ! Xn+1 for each n 0; and
3. a basepoint preserving left action of n on Xn such that the composition
oep = oe O (S1 ^ oe) O . .O.(Sp-1 ^ oe): Sp ^ Xn ! Xn+p;
i^oe
of the maps Si^ S1 ^ Xn+p-i-1 S---!Si^ Xn+p-i is p x n-equivariant
for p 1 and n 0.
SYMMETRIC SPECTRA 7
A map of symmetric spectra f :X ! Y is a sequence of pointed maps fn :Xn ! Yn
such that fn is n-equivariant and the diagram
S1 ^ Xn --oe--!Xn+1
? ?
S1^fn?y fn+1?y
S1 ^ Yn --oe--!Yn+1
is commutative for each n 0. Let Sp denote the category of symmetric spectra.
Remark 1.2.3. In part three of Definition 1.2.2, one need only assume that the
maps oe :S1 ^ Xn ! Xn+1 and oe2: S2 ^ Xn ! Xn+2 are equivariant; since the
symmetric groups p are generated by transpositions (i; i + 1), if oe and oe2 are
equivariant then all the maps oep are equivariant.
Example 1.2.4. The symmetric suspension spectrum 1 K of the pointed sim-
plicial set K is the sequence of pointed simplicial sets Sn ^ K with the natural
isomorphisms oe :S1 ^ Sn ^ K ! Sn+1 ^ K as the structure maps and the diag-
onal action of n on Sn ^ K coming from the left permutation action on Sn and
the trivial action on K. The composition oep is the natural isomorphism which is
pxn-equivariant. The symmetric sphere spectrum S is the symmetric suspension
spectrum of the 0-sphere; S is the sequence of spheres S0; S1; S2; : :w:ith the*
* nat-
ural isomorphisms S1 ^ Sn ! Sn+1 as the structure maps and the left permutation
action of n on Sn.
Example 1.2.5. The Eilenberg-Mac Lane spectrum HZ is the sequence of sim-
plicial abelian groups Z Sn, where (Z Sn)k is the free abelian group on the
non-basepoint k-simplices of Sn. We identify the basepoint with 0. The symmet-
ric group n acts by permuting the generators, and one can easily verify that the
evident structure maps are equivariant. One could replace Z by any ring.
Remark 1.2.6. Bordism is most easily defined as a topological symmetric spec-
trum, see Example 6.2.3. As explained in [GH97 , Section 6], many other examples
of symmetric spectra arise as the K-theory of a category with cofibrations and *
*weak
equivalences as defined by Waldhausen [Wal85, p.330].
A symmetric spectrum with values in a simplicial category C is obtained by
replacing the sequence of pointed simplicial sets by a sequence of pointed obje*
*cts in
C. In particular, a topological symmetric spectrum is a symmetric spectrum with
values in the simplicial category of topological spaces; see Section 6.
By ignoring group actions, a symmetric spectrum is a spectrum and a map of
symmetric spectra is a map of spectra. When no confusion can arise, the adjecti*
*ve
"symmetric" may be dropped.
Definition 1.2.7.Let X be a symmetric spectrum. The underlying spectrum UX
is the sequence of pointed simplicial sets (UX)n = Xn with the same structure
maps oe :S1 ^ (UX)n ! (UX)n+1 as X but ignoring the symmetric group actions.
This gives a faithful functor U :Sp ! SpN.
Since the action of n on Sn is non-trivial for n 2, it is usually impossible*
* to
obtain a symmetric spectrum from a spectrum by letting n act trivially on Xn.
However, many of the usual functors to the category of spectra lift to the cate*
*gory
of symmetric spectra. For example, the suspension spectrum of a pointed simplic*
*ial
set K is the underlying spectrum of the symmetric suspension spectrum of K.
8 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Many examples of symmetric spectra and of functors on the category of sym-
metric spectra are constructed by prolongation of simplicial functors.
Definition 1.2.8.A pointed simplicial functor or S*-functor is a pointed functor
R: S* ! S* and a natural transformation h: RX ^ K ! R(X ^ K) of bifunctors
such that the composition RX ^ S0 ! R(X ^ S0) ! R(X) is the unit isomorphism
and the diagram of natural transformations
(RX ^ K) ^ L_h^L_//R(X ^ K) ^ L
| |
| h|
fflffl| h fflffl|
RX ^ (K ^ L)_____//R(X ^ K ^ L)
is commutative. A pointed simplicial natural transformation, or S*-natural tran*
*s-
formation, from the S*-functors R to the S*-functor R0is a natural transformati*
*on
o :R ! R0such that oh = h0(o ^ K).
Definition 1.2.9.The prolongation of a S*-functor R: S* ! S* is the functor
R: Sp ! Sp defined as follows. For X a symmetric spectrum, RX is the
sequence of pointed simplicial sets RXn with the composition oe :S1 ^ R(Xn) !
R(S1 ^ Xn) -Roe-!R(Xn+1) as the structure maps and the action of n on R(Xn)
obtained by applying the functor R to the action of n on Xn. Since R is a S*-
functor, each map oep is equivariant and so RX is a symmetric spectrum. For f
a map of symmetric spectra, Rf is the sequence of pointed maps Rfn. Since R
is an S*-functor, Rf is a map of spectra. Similarly, we can prolong an S*-natur*
*al
transformation to a natural transformation of functors on Sp .
Proposition 1.2.10.The category of symmetric spectra is bicomplete (every small
diagram has a limit and a colimit).
Proof.For any small category I, the limit and colimit functors SI*! S* are poin*
*ted
simplicial functors; for K 2 S* and D 2 SetIthere is a natural isomorphism
K ^ colimD ~=colim(K ^ D)
and a natural map
K ^ limD ! lim(K ^ D):
A slight generalization of prolongation gives the limit and the colimit of_a di*
*agram
of symmetric spectra. |__|
In particular, the underlying sequence of the limit is (limD)n = limDn and the
underlying sequence of the colimit is (colimD)n = colimDn.
1.3. Simplicial structure on Sp . For a pointed simplicial set K and a sym-
metric spectrum X, prolongation of the S*-functor (-) ^ K :S* ! S* defines the
smash product X ^ K and prolongation of the S*-functor (-)K :S* ! S* defines
the power spectrum XK . For symmetric spectra X and Y , the pointed simplicial
set of maps from X to Y is Map Sp (X; Y ) = Sp (X ^ [-]+ ; Y ).
In the language of enriched category theory, the following proposition says t*
*hat
the smash product X ^ K is a closed action of S* on Sp . We leave the straight-
forward proof to the reader.
SYMMETRIC SPECTRA 9
Proposition 1.3.1.Let X be a symmetric spectrum. Let K and L be pointed
simplicial sets.
1. There are coherent natural isomorphisms X ^ (K ^ L) ~=(X ^ K) ^ L and
X ^ S0 ~=X.
2. (-) ^ K :Sp ! Sp is the left adjoint of the functor (-)K :Sp ! Sp .
3. X ^ (-): S* ! Sp is the left adjoint of the functor Map Sp (X; -): Sp !
S*.
The evaluation map X ^ MapSp (X; Y ) ! Y is the adjoint of the identity map
on Map Sp (X; Y ). The composition pairing
Map Sp (X; Y ) ^ MapSp (Y; Z) ! Map Sp (X; Z)
is the adjoint of the composition
X ^ MapSp (X; Y ) ^ MapSp (Y; Z) ! Y ^ MapSp (Y; Z) ! Z
of two evaluation maps. In the language of enriched category theory, a category
with a closed action of S* is the same as a tensored and cotensored S*-category.
The following proposition, whose proof we also leave to the reader, expresses t*
*his
fact.
Proposition 1.3.2.Let X, Y , and Z be symmetric spectra and let K be a pointed
simplicial set.
1. The composition pairing Map Sp (X; Y ) ^ MapSp (Y; Z) ! Map Sp (X; Z) is
associative.
2. The adjoint S0 ! Map Sp (X; X) of the isomorphism X ^ S0 ! X is a left
and a right unit of the composition pairing.
3. There are natural isomorphisms
Map Sp (X ^ K; Y ) ~=Map Sp (X; Y K) ~=Map Sp (X; Y )K :
Proposition 1.3.1 says that certain functors are adjoints, whereas Propositio*
*n 1.3.2
says more; they are simplicial adjoints.
The category of symmetric spectra satisfies Quillen's axiom SM7 for simplicial
model categories.
Definition 1.3.3.Let f :U ! V and g :X ! Y be maps of pointed simplicial
sets. The pushout smash product f g is the natural map on the pushout
f g :V ^ X qU^X U ^ Y ! V ^ Y:
induced by the commutative square
U ^ X -f^X---!V ^ X
? ?
U^g?y ?yV ^g
U ^ Y ----! V ^ Y:
f^Y
Let f be a map of symmetric spectra and g be a map of pointed simplicial sets.
The pushout smash product f g is defined by prolongation, (f g)n = fn g.
Recall that a map of simplicial sets is a weak equivalence if its geometric r*
*eal-
ization is a homotopy equivalence of CW-complexes. One of the basic properties *
*of
simplicial sets, proved in [Qui67, II.3], is:
10 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Proposition 1.3.4.Let f and g be monomorphisms of pointed simplicial sets.
Then f g is a monomorphism, which is a weak equivalence if either f or g is a
weak equivalence.
Prolongation gives a corollary for symmetric spectra. A map f of symmetric
spectra is a monomorphism if fn is a monomorphism of simplicial sets for each
n 0.
Definition 1.3.5.A map f of symmetric spectra is a level equivalence if fn is a
weak equivalence of simplicial sets for each n 0.
Corollary 1.3.6.Let f be a monomorphism of symmetric spectra and let g be a
monomorphism of pointed simplicial sets. Then f g is a monomorphism, which
is a level equivalence if either f is a level equivalence or g is a weak equiva*
*lence.
By definition, a 0-simplex of Map Sp (X; Y ) is a map X ^ [0]+ ! Y , but
X ^[0]+ ~=X and so a 0-simplex of Map Sp (X; Y ) is a map X ! Y . A 1-simplex
of Map Sp (X; Y ) is a simplicial homotopy H :X ^ [1]+ ! Y from H O (X ^ i0)
to H O (X ^ i1) where i0 and i1 are the two inclusions [0] ! [1]. Simpli-
cial homotopy generates an equivalence relation on Sp (X; Y ) and the quotient *
*is
ss0Map Sp (X; Y ). A map f :X ! Y is a simplicial homotopy equivalence if it has
a simplicial homotopy inverse, i.e., a map g :Y ! X such that gf is simplicially
homotopic to the identity map on X and fg is simplicially homotopic to the iden-
tity map on Y . If f is a simplicial homotopy equivalence of symmetric spectra,
then each of the maps fn is a simplicial homotopy equivalence, and so each of t*
*he
maps fn is a weak equivalence. Every simplicial homotopy equivalence is therefo*
*re
a level equivalence. The converse is false; a map can be a level equivalence a*
*nd
NOT a simplicial homotopy equivalence.
1.4. Symmetric -spectra. The stable homotopy category can be defined using
-spectra and level equivalences.
Definition 1.4.1.A Kan complex (see Example 3.2.6) is a simplicial set that
satisfies the Kan extension condition. An -spectrum is a spectrum X such that
for each n 0 the simplicial set Xn is a Kan complex and the adjoint Xn !
Map S*(S1; Xn+1) of the structure map S1 ^ Xn ! Xn+1 is a weak equivalence of
simplicial sets.
Let SpN SpN be the full subcategory of -spectra. The homotopy cat-
egory Ho (SpN) is obtained from SpN by formally inverting the level equiva-
lences. By the results in [BF78 ], the category Ho(SpN) is naturally equivalent*
* to
Boardman's stable homotopy category (or any other). Likewise, let Sp Sp
be the full subcategory of symmetric -spectra (i.e., symmetric spectra X for
which UX is an -spectrum). The homotopy category Ho(Sp ) is obtained from
Sp by formally inverting the level equivalences. Since the forgetful functor
U :Sp ! SpN preserves -spectra and level equivalences, it induces a functor
Ho(U): Ho (Sp ) ! Ho (SpN). As a corollary of Theorem 4.3.2, the functor
Ho(U) is a natural equivalence of categories. Thus the category Ho(Sp ) is nat-
urally equivalent to Boardman's stable homotopy category. To describe an inverse
of Ho(U), let 1 : SpN ! S* be the functor that takes a spectrum to the 0-space
of its associated -spectrum. For any spectrum E 2 SpN, the symmetric spectrum
V E = 1 (E^S) is the value of the prolongation of the S*-functor 1 (E^-) at the
SYMMETRIC SPECTRA 11
symmetric sphere spectrum S; the underlying sequence is V En = 1 (E ^Sn). The
functor V preserves -spectra, preserves level equivalences, and induces a funct*
*or
Ho(V ): Ho (SpN) ! Ho(Sp ) which is a natural inverse of Ho(U).
The category of symmetric -spectra has major defects. It is not closed under
limits and colimits, or even under pushouts and pullbacks. The smash product,
defined in Section 2, of symmetric -spectra is a symmetric spectrum but not an
-spectrum, except in trivial cases. For these reasons it is better to work with
the category of all symmetric spectra. But then the notion of level equivalence
is no longer adequate; the stable homotopy category is a retract of the homotopy
category obtained from Sp by formally inverting the level equivalences but many
symmetric spectra are not level equivalent to an -spectrum. One must enlarge
the class of equivalences. The stable equivalences of symmetric spectra are def*
*ined
in Section 3.1. By Theorem 4.3.2, the homotopy category obtained from Sp by
inverting the stable equivalences is naturally equivalent to the stable homotopy
category.
2.The smash product of symmetric spectra
In this section we construct the closed symmetric monoidal product on the cat*
*e-
gory of symmetric spectra. A symmetric spectrum can be viewed as a module over
the symmetric sphere spectrum S, and the symmetric sphere spectrum (unlike the
ordinary sphere spectrum) is a commutative monoid in an appropriate category.
The smash product of symmetric spectra is the tensor product over S.
The closed symmetric monoidal category of symmetric sequences is constructed
in Section 2.1. A reformulation of the definition of a symmetric spectrum is gi*
*ven
in Section 2.2 where we recall the definition of monoids and modules in a symme*
*tric
monoidal category. In Section 2.3 we see that there is no closed symmetric mono*
*idal
smash product on the category of (non-symmetric) spectra.
2.1. Symmetric sequences. Every symmetric spectrum has an underlying se-
quence X0; X1; : :;:Xn; : :o:f pointed simplicial sets with a basepoint preserv*
*ing
left action of n on Xn; these are called symmetric sequences. In this section we
define the closed symmetric monoidal category of symmetric sequences of pointed
simplicial sets.
` __
Definition_2.1.1.The category = n0 n has the finite sets n = {1; 2; : :;:n}
for n 0 (0 = ;) as its objects and the automorphisms of the sets __nas its map*
*s.
Let C be a category. A symmetric sequence of objects in C is a functor ! C, and
the category of symmetric sequences of objects in C is the functor category C .
A symmetric sequence X 2 S* is a sequence X0; X1; : :;:Xn; : :o:f pointed
simplicial sets with a basepoint preserving left actionQof n on Xn. The category
C is a product category. In particular, S*(X; Y ) = pSp*(Xp; Yp).
Proposition 2.1.2.The category S* of symmetric sequences in S* is bicomplete.
Proof.The category S* is bicomplete, so the functor category S* is bicomplete. *
* |___|
Definition 2.1.3.The tensor product XY of the symmetric sequences X; Y 2 S*
is the symmetric sequence
_
(X Y )n = (n)+ ^pxq (Xp ^ Yq):
p+q=n
12 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
The tensor product f g :X Y -! X0 Y 0of the maps f :X -! X0 and
g :Y -! Y 0in S* is given by (f g)(ff; x; y) = (ff; fpx; gqy) for ff 2 p+q, x *
*2 Xp
and y 2 Yq.
The tensor product of symmetric sequences has the universal property for "bi-
linear maps":
Proposition 2.1.4.Let X; Y; Z 2 S* be symmetric sequences. There is a natural
isomorphism
Y x
S*(X Y; Z) ~= Sp*q (Xp ^ Yq; Zp+q)
p;q
The twist isomorphism o :XY ! Y X for X; Y 2 S* is the natural map given
by o(ff; x; y) = (ffaeq;p; y; x) for ff 2 p+q, x 2 Xp, and y 2 Yq, where aeq;p2*
* p+q
is the (q; p)-shuffle given by aeq;p(i) = i + p for 1 i q and aeq;p(i) = i - *
*q for
q < i p + q. The map defined without the shuffle permutation is not a map of
symmetric sequences.
Remark 2.1.5. There is another way of describing the tensor product and the
twist isomorphism. The category is a skeleton of the category of finite sets a*
*nd
isomorphisms. Hence every symmetric sequence has an extension, which is unique
up to isomorphism, to a functor on the category of all finite sets and isomorph*
*isms.
The tensor product of two such functors X and Y is the functor defined on a fin*
*ite
set C as
_
(X Y )(C) = X(A) ^ Y (B):
A[B=C;A\B=;
For an isomorphism f :C ! D the map (X Y )(f) is the coproduct of the
isomorphisms X(A) ^ Y (B) ! X(fA) ^ Y (fB). The twist isomorphism is the map
that sends the summand X(A)^Y (B) of (X Y )(C) to the summand Y (B)^X(A)
of (Y X)(C) by switching the factors.
Lemma 2.1.6. The tensor product is a symmetric monoidal product on the
category of symmetric sequences S*.
_
Proof.The unit of the tensor product is the symmetric sequence (0; -)+ =
(S0; *; *; : :):. The unit isomorphism is obvious. The associativity isomorph*
*ism
is induced by the associativity isomorphism in S* and the natural isomorphism
_
((X Y ) Z)n ~= (n)+ ^pxqxr (Xp ^ Yq ^ Zr):
p+q+r=n
The twist isomorphism is described in Remark 2.1.5. The coherence of the natural
isomorphisms follows from coherence of the natural isomorphisms for the_smash_
product in S*. |__|
We now introduce several functors on the category of symmetric sequences.
Definition 2.1.7.The evaluation functor Evn :S* ! S* is given by Evn X = Xn
and Evn f = fn. The free functor Gn :S* ! S* is the left adjoint of the evaluat*
*ion
functor Evn. The smash product X ^ K of X 2 S* and K 2 S* is the symmetric
sequence (X ^ K)n = Xn ^ K with the diagonal action of n that is trivial on
K. The pointed simplicial set Map S*(X; Y ) of maps from X to Y is the pointed
simplicial set S*(X ^ [-]+ ; Y ).
SYMMETRIC SPECTRA 13
For each n 0, the free symmetric sequence is [n] = (__n; -) and the free
functor is Gn = [n]+ ^-: S* ! S*. So, for a pointed simplicial set K, (GnK)n =
(n)+ ^ K and (GnK)k = * for k 6= n. In particular, GnS0 = [n]+ , G0K =
(K; *; *; : :):and G0S0 is the unit of the tensor product .
We leave the proof of the following basic proposition to the reader.
Proposition 2.1.8.There are natural isomorphisms:
1. GpK GqL ~=Gp+q(K ^ L) for K; L 2 S*.
2. X G0K ~=X ^ K for K 2 S* and X 2 S*.
3. MapS*(GnK; X) ~=MapQS*(K; Xn) for K 2 S* and X 2 S*.
4. MapS*(X Y; Z) ~= p;qMappxq (Xp ^ Yq; Zp+q) for X; Y; Z 2 Sp .
A map f of symmetric sequences is a level equivalence if each of the maps fn *
*is a
weak equivalence. Since S* is a product category, a map f of symmetric sequences
is a monomorphism if and only if each of the maps fn is a monomorphism.
Proposition 2.1.9.Let X be a symmetric sequence, f be a map of symmetric
sequences and g be a map of pointed simplicial sets.
1. X (-) preserves colimits.
2. If f is a monomorphism then X f is a monomorphism.
3. If f is a level equivalence then X f is a level equivalence.
4. If g is a monomorphism then Gng is a monomorphism for n 0.
5. If g is a weak equivalence then Gng is a level equivalence for n 0.
Proof.Parts (1), (2) and (3) follow from the definition of and the correspondi*
*ng
properties for the smash product of pointed simplicial sets. For Parts_(4)_and *
*(5)
use the isomorphism GnK = [n]+ ^ K. |__|
By part three of Proposition 2.1.8, Map ([n]+ ; X) ~=Xn. As n varies, [-]+ is
a functor op ! S*, and for X 2 S*, the symmetric sequence Map S*([-]+ ; X) is
naturally isomorphic to X.
Definition 2.1.10.Let X and Y be symmetric sequences. The symmetric se-
quence of maps from X to Y is
Hom (X; Y ) = Map S*(X [-]+ ; Y ):
Theorem 2.1.11. The tensor product is a closed symmetric monoidal product on
the category of symmetric sequences.
Proof.The tensor product is a symmetric monoidal product by Lemma 2.1.6. The
product is closed if there is a natural isomorphism
S*(X Y; Z) ~=S* (X; Hom (Y; Z)):
for symmetric sequences X; Y and Z.
By Proposition 2.1.4, a map of symmetric sequences f :X Y ! Z is a col-
lection of p x q-equivariant maps fp;q:Xp ^ Yq ! Zp+q. This is adjoint to
a collection of p-equivariant maps gp;q:Xp ! Map q (Yq; Zp+q). So there is a
natural isomorphism
Y Y
S*(X Y; Z) ~= Sp*(Xp; Map q (Yq; Zp+q))
p q
14 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Q
By Proposition 2.1.8, the functor sending _pto qMap q (Yq; Zp+q) is the funct*
*or
sending _pto Map (Y [p]+ ; Z) which by definition is Hom (Y; Z). Combining the*
*__
isomorphisms gives the natural isomorphism that finishes the proof. |*
*__|
2.2. Symmetric spectra. In this section we recall the language of "monoids"
and "modules" in a symmetric monoidal category and apply it to the category
of symmetric sequences. In this language, the symmetric sequence of spheres
S = (S0; S1; : :;:Sn; : :):is a commutative monoid in the category of symmetric
sequences and a symmetric spectrum is a (left) S-module.
Consider the symmetric sphere spectrum S. By Proposition 2.1.4, the natural
p x q-equivariant maps mp;q:Sp ^ Sq ! Sp+q give a pairing m: S S ! S.
The adjoint G0S0 ! S of the identity map S0 ! Ev0S = S0 is a two sided unit of
the pairing. The diagram of natural isomorphisms
Sp ^ Sq ^ Sr----! Sp ^ Sq+r
?? ?
y ?y
Sp+q ^ Sr ----! Sp+q+r
commutes, showing that m is an associative pairing of symmetric sequences.
A symmetric spectrum X has an underlying symmetric sequence of pointed
simplicial sets {Xn} and a collection of pointed pxq-equivariant maps oep: Sp^
Xq ! Xp+q for p 1 and q 0. Let oe0: S0 ^ Xn ! Xn be the unit isomorphism.
Then the diagram
p^oeq p
Sp ^ Sq ^ XrS___//S ^ Xq+r
~=|| |oep|
fflffl|oep+q fflffl|
Sp+q ^ Xr_______//Xp+q+r
commutes for p; q; r 0. By Proposition 2.1.4, the equivariant maps oep give a
pairing oe :S X ! X such that the composition G0(S0) X ! S X ! X is
the unit isomorphism and the diagram
S S X _Soe_//S X
mX || |oe|
fflffl|oe fflffl|
S X _________//X
commutes.
In the language of monoidal categories, S is a monoid in the category of sym-
metric sequences and a symmetric spectrum is a left S-module. Moreover, S is a
commutative monoid, i.e., the diagram
S SE______o_____//S S;
EEE yyyy
m EEEE yymy
E""__yy
S
commutes, where o is the twist isomorphism. To see this, one can use either the
definition of the twist isomorphism or the description given in Remark 2.1.5. T*
*hen,
as is the case for commutative monoids in the category of sets and for commutat*
*ive
monoids in the category of abelian groups (i.e., commutative rings), there is a
SYMMETRIC SPECTRA 15
tensor product S, having S as the unit. This gives a symmetric monoidal product
on the category of S-modules. The smash product X ^ Y of X; Y 2 Sp is the
symmetric spectrum X S Y .
We review the necessary background on monoidal categories. Monoids and mod-
ules can be defined in any symmetric monoidal category, see [ML71 ]. Let be a
symmetric monoidal product on a category C with unit e 2 C. A monoid in C is
an object R 2 C, a multiplication : R R ! R, and a unit map j :e ! R such
that the diagram
R R R -mR---!R R
? ?
Rm ?y ?ym
R R ----!m R
commutes (i.e., m is associative) and such that the compositions e R -j1-!R
R -! R and R e j1--!R R -! R are the unit isomorphisms of the product .
The monoid R is commutative if = O o where o is the twist isomorphism of .
A left R-module is an object M with an associative multiplication ff: R M ! M
that respects the unit; right R-modules are left Rop-modules, where Rop is R wi*
*th
the multiplication O o. A map of left R-modules M and N is a map f :M ! N
in C that commutes with the left R-actions. The category of left R-modules is
denoted R-Mod. If C is complete, the module category R-Mod is complete and the
forgetful functor R-Mod ! C preserves limits. If C is cocomplete and the functor
R (-) preserves coequalizers (in fact it suffices that R (-) preserve reflexi*
*ve
coequalizers) then the module category R-Mod is cocomplete and the forgetful
functor R-Mod ! C preserves colimits.
The symmetric sequence of spheres S = (S0; S1; S2; : :):is a commutative mono*
*id
in the category of symmetric sequences.
Proposition 2.2.1.The category of symmetric spectra is naturally equivalent to
the category of left S-modules.
Proof.A pairing m: S X ! X is the same as a collection of px q-equivariant
maps mp;q:Sp ^ Xq ! Xp+q. If X is a left S-module, there is a spectrum for
which X is the underlying symmetric sequence and the structure maps are the maps
oe = m1;n:S1^Xn ! Xn+1. The compositions oep are the pxq-equivariant maps
mp;q. Conversely, for X a symmetric spectrum, the map of symmetric sequences
m: S X ! X corresponding to the collection of pxq-equivariant maps mp;q=
oep: Sp ^ Xq ! Xp+q, where oe0 is the natural isomorphism S0 ^ Xn ! Xn, makes
X a left S-module. These are inverse constructions and give a natural equivalen*
*ce_
of categories. |__|
The smash product on the category of symmetric spectra is a special case of t*
*he
following lemma.
Lemma 2.2.2. Let C be a symmetric monoidal category that is cocomplete and let
R be a commutative monoid in C such that the functor R (-): C ! C preserves
coequalizers. Then there is a symmetric monoidal product R on the category of
R-modules with R as the unit.
16 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Proof.Let Q be a monoid in C, M be a right Q-module, and N be a left Q-module.
The tensor product M Q N is the colimit in C of the diagram
_m1__
M Q N _____////M N:
1m
If M is an (R; Q)-bimodule (i.e., the right action of Q commutes with the left *
*action
of R) then this is a diagram of left R-modules. Since the functor R (-) preser*
*ves
coequalizers, the colimit M Q N is a left R-module. If M is a left module over *
*the
commutative monoid R, the composition M R o-!RM -ff!M is a right action of
R; since R is commutative, the two actions commute and M is an (R; R)-bimodule.
Hence, the tensor product M R N is a left R-module. The unit of the product
R is R as a left R-module. The associativity, unit, and twist isomorphisms of
the product on C induce corresponding isomorphisms for the product R on the __
category of left R-modules. Thus R is a symmetric monoidal product. |__|
Definition 2.2.3.The smash product X ^ Y of symmetric spectra X and Y is the
symmetric spectrum X S Y .
Apply Lemma 2.2.2 to the commutative monoid S in the bicomplete category of
symmetric sequences S* to obtain the following corollary.
Corollary 2.2.4.The smash product X ^ Y is a symmetric monoidal product on
the category of symmetric spectra.
Next, some important functors on the category of symmetric spectra.
Definition 2.2.5.The functor S (-): S* ! Sp gives the free S-module S X
generated by the symmetric sequence X. For each n 0, the evaluation functor
Evn: Sp ! S* is given by Ev nX = Xn and Ev nf = fn. The free functor
Fn :S* ! Sp is the left adjoint of the evaluation functor Ev n. The functor
Rn :S* ! Sp is the right adjoint of the evaluation functor Evn :Sp ! S*.
The functor S (-) is left adjoint to the forgetful functor Sp ! S*. The free
functor Fn is the composition S Gn of the left adjoints Gn :S* ! S* (Defini-
tion 2.1.7) and S (-): S* ! Sp . Thus, for X 2 Sp and K 2 S*, the left
S-module X ^ FnK is naturally isomorphic to the left S-module X GnK. In par-
ticular, X ^ F0K is naturally isomorphic to the symmetric spectrum X ^ K defined
by prolongation in Section 1.3. Furthermore F0K = S ^ K is the symmetric sus-
pension spectrum 1 K of K, and F0S0 is the symmetric sphere spectrum S. For_
a pointed simplicial set K, RnK is the symmetric sequence Hom S*(S; K(-;n)+),
which is a left S-module since S is a right S-module.
We leave the proof of the following proposition to the reader.
Proposition 2.2.6.There are natural isomorphisms:
1. Fm (K) ^ Fn(L) ~=Fm+n (K ^ L) for K; L 2 S*.
2. MapSp (S X; Y ) ~=Map S*(X; Y ) for X 2 S* and Y 2 Sp .
3. MapSp (FnK; X) ~=Map S*(K; EvnX) for K 2 S* and X 2 Sp .
Proposition 2.2.7.Let f be a map of pointed simplicial sets.
1. Fn :S* ! Sp preserves colimits.
2. If f is a monomorphism then Fnf is a monomorphism.
3. If f is a weak equivalence then Fnf is a level equivalence.
SYMMETRIC SPECTRA 17
Proof.Use the isomorphism Fnf = S Gnf and Proposition 2.1.9. |___|
The internal Hom on the category of symmetric spectra is a special case of the
following lemma.
Lemma 2.2.8. Let C be a closed symmetric monoidal category that is bicomplete
and let R be a commutative monoid in C. Then there is a function R-module
Hom R(M; N), natural for M; N 2 C, such that the functor (-) R M is left adjoint
to the functor Hom R(M; -)
Proof.Let R be a monoid in C and let M and N be left R-modules. Then
Hom R(M; N) is the limit in C of the diagram
__m*_
Hom C(M; N)__m*_////HomC(R M; N)
where m* is pullback along the multiplication m: R M ! M and m* is the
composition
Hom C(M; N) R----!HomC(R M; R N) m*--!HomC(R M; N)
If Q is another monoid and N is an (R; Q)-bimodule then this is a diagram of ri*
*ght
Q-modules and the limit Hom R(M; N) is a right Q-module. If R is a commutative
monoid and N is a left R-module then the left action is also a right action and
N is an (R; R)-bimodule. So Hom R(M; N) is a right R-module and hence a left
R-module. It follows from the properties of the internal Hom in C and the_defin*
*ition_
of R that (-) R M is left adjoint to Hom R(M; -). |__|
Definition 2.2.9.Let X and Y be symmetric spectra. The function spectrum
Hom S(X; Y ) is the limit of the diagram in Sp
_m*__
Hom (X; Y )m*__////Hom(S X; Y:)
Combining Lemmas 2.2.2 and 2.2.8:
Theorem 2.2.10. The smash product is a closed symmetric monoidal product on
the category of symmetric spectra. In particular, there is a natural adjunction
isomorphism
Sp (X ^ Y; Z) ~=Sp (X; Hom S(Y; Z)):
Proof.The smash product ^ is a symmetric monoidal product by Corollary 2.2.4._
The adjunction isomorphism follows from Lemma 2.2.8 |__|
The adjunction is also a simplicial adjunction and an internal adjunction.
Corollary 2.2.11.There are natural isomorphisms
Map Sp(X ^ Y; Z) ~=Map Sp (X; Hom S(Y; Z)):
and
Hom S(X ^ Y; Z) ~=Hom S(X; Hom S(Y; Z))
Remark 2.2.12. We use Proposition 2.2.6 to give another description of the fun*
*c-
tion spectrum Hom S(X; Y ). For a symmetric spectrum X, the pointed simplicial
set of maps Map Sp (FnS0; X) is naturally isomorphic to Map S*(S0; EvnX) = Xn.
The symmetric spectrum FnS0 is the S-module S[n]+ and as n varies, S[-]+
18 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
is a functor op ! Sp . The symmetric sequence Map Sp (S [-]+ ; X) is
the underlying symmetric sequence of X. In particular, the natural isomorphism
Xn = Map Sp (FnS0; X) is n-equivariant. Applying this to Hom S(X; Y ) and using
Corollary 2.2.11, we find that the underlying symmetric sequence of Hom S(X; Y )
is the symmetric sequence Map Sp (X ^ (S [-]+ ); Y ).
We must also describe the structure maps of X from this point of view. Re-
call that Map Sp (FnS0; X) = Xn, Map Sp (FnS1; X) = Map S*(S1; Xn). Let
: F1S1 ! F0S0 be the adjoint of the identity map S1 ! Ev1 F0S0 = S1. The
induced map Map Sp (; X): X0 ! Map S*(S1; X1) is adjoint to the structure map
S1 ^ X0 ! X1. The map
^ FnS0: F1S1 ^ FnS0 = Fn+1S1 ! F0S0 ^ FnS0 = FnS0
is 1 x n-equivariant; the induced map
Map Sp ( ^ FnS0; X): Xn ! Map S*(S1; Xn+1)
is 1 x n-equivariant and is adjoint to the structure map oe :S1 ^ Xn ! Xn+1.
In order to apply this to Hom S(X; Y ), use Proposition 2.2.6 and Corollary 2.2*
*.11
to find a natural isomorphism
Map Sp (X ^ Fn+1S1; Y ) ~=Map S*(S1; MapSp (X ^ Fn+1S0; Y ):
Using this natural isomorphism, we find that the structure maps of Hom S(X; Y )
are the adjoints of the maps
Map Sp (X ^ FnS0; Y ) ! Map Sp (X ^ Fn+1S1; Y )
induced by ^ FnS0.
For example, Hom S(FkS0; X) is the k-shifted spectrum; its underlying symmet-
ric sequence is the sequence of pointed simplicial sets
Xk; X1+k; : :;:Xn+k : : :
with n acting on Xn+k by restricting_the_action of n+k to the copy of n that
permutes the first n elements of n + k. The structure maps of the k-shifted spe*
*c-
trum are the structure maps oe :S1 ^ Xn+k ! Xn+k+1 of X. More generally,
Hom S(FkK; X) is the k-shifted spectrum of XK .
2.3. The ordinary category of spectra. An approach similar to the last two
sections can be used to describe (non-symmetric) spectra as modules over the sp*
*here
spectrum in a symmetric monoidal category. But in this case the sphere spectrum
is not a commutative monoid, which is why there is no closed symmetric monoidal
smash product of spectra.
We begin as in Section 2.1 by considering a category of sequences.
Definition 2.3.1.The category N is the category with the non-negative integers
as its objects and with the identity maps of the objects as its only maps. The
category of sequences SN*is the category of functors from N to S*. An object of*
* SN*
is a sequence X0; X1; : :;:Xn; : :o:f pointed simplicial sets and a map f :X -!*
* Y
is a sequence of pointed simplicial maps fn :Xn ! Yn.
Definition 2.3.2.The graded smash product of sequences X and Y is the sequence
X Y given in degree n by
_
(X Y )n = Xp ^ Yq
p+q=n
SYMMETRIC SPECTRA 19
Lemma 2.3.3. The category of sequences is a bicomplete category and the graded
smash product is a symmetric monoidal product on SN*.
Proof.Limits and colimits are defined at each object, so SN*has arbitrary limit*
*s and
colimits since S* does. The graded tensor product is a symmetric monoidal produ*
*ct
on SN*. The unit is the sequence S0; *; *; . ...The coherence isomorphisms for *
*the
graded smash product follow from the corresponding coherence isomorphisms in __
S*. The twist isomorphism takes (x; y) to (y; x). |_*
*_|
Proposition 2.3.4.The sequence S whose nth level is Sn is a monoid in the
category of sequences. The category of left S-modules is isomorphic to the ordi*
*nary
category of spectra, SpN.
Proof.By definition, a pairing of sequences X Y ! Z is the same as a collection
of maps Xp^ Yq ! Zp+q for p; q 0. The associative pairing Sp ^ Sq ! Sp+q gives
an associative pairing of sequences S S ! S with an obvious unit map. Thus S is
a monoid. Since Sp = S1 ^ : :^:S1, a left action of the monoid S on the sequenc*
*e_
X is the same as a collection of maps S1 ^ Xq ! Xq+1 for q 0. |__|
Suppose there were a closed symmetric monoidal structure on SpN with S as
the unit; since SpN is the category of left S-modules it would follow that S is*
* a
commutative monoid in the category of sequences. But the twist map on S1^ S1 is
not the identity map and so S is not a commutative monoid in SN*. Therefore the*
*re
is no closed symmetric monoidal smash product of spectra. Note that the category
of right S-modules is isomorphic to the category of left S-modules, even though
S is not commutative. On the other hand, S is a graded commutative monoid up
to homotopy, in fact up to E1 -homotopy. This observation underlies Boardman's
construction of handicrafted smash products [Ada74 ].
3.Stable homotopy theory of symmetric spectra
To use symmetric spectra for the study of stable homotopy theory, one should
have a stable model category of symmetric spectra such that the category obtain*
*ed
by inverting the stable equivalences is naturally equivalent to Boardman's stab*
*le
homotopy category of spectra (or to any other known to be equivalent to Board-
man's). In this section we define the stable model category of symmetric spectr*
*a.
In Section 4 we show that it is Quillen equivalent to the stable model category*
* of
spectra discussed in [BF78 ].
In Section 3.1 we define the class of stable equivalences of symmetric spectra
and discuss its non-trivial relationship to the class of stable equivalences of*
* (non-
symmetric) spectra. In Section 3.2 we recall the axioms and basic theory of mod*
*el
categories. In Section 3.3 we discuss the level structure in Sp , and in Sectio*
*n 3.4
we define the stable model structure on the category of symmetric spectra which*
* has
the stable equivalences as the class of weak equivalences. The rest of the sect*
*ion is
devoted to checking that the stable model structure satisfies the axioms of a m*
*odel
category.
3.1. Stable equivalence. One's first inclination is to define stable equivalence
using the forgetful functor U :Sp ! SpN; one would like a map f of symmetric
spectra to be a stable equivalence if the underlying map Uf of spectra is a sta*
*ble
equivalence, i.e., if Uf induces an isomorphism of stable homotopy groups. The
reader is warned: THIS WILL NOT WORK. Instead, stable equivalence is
20 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
defined using cohomology; a map f of symmetric spectra is a stable equivalence *
*if
the induced map E*f of cohomology groups is an isomorphism for every generalized
cohomology theory E. The two alternatives, using stable homotopy groups or using
cohomology groups, give equivalent definitions on the category of (non-symmetri*
*c)
spectra but not on the category of symmetric spectra.
It would be nice if the 0th cohomology group of the symmetric spectrum X with
coefficients in the symmetric -spectrum E could be defined as ss0Map Sp (X; E),
the set of simplicial homotopy classes of maps from X to E. But, even though the
contravariant functor E0 = ss0Map Sp (-; E) takes simplicial homotopy equiva-
lences to isomorphisms, E0 may not take level equivalences to isomorphisms. This
is a common occurrence in simplicial categories, but is a problem as every level
equivalence should induce an isomorphism of cohomology groups; a level equiva-
lence certainly induces an isomorphism of stable homotopy groups. We introduce
injective spectra as a class of spectra E for which the functor E0 behaves corr*
*ectly.
Definition 3.1.1.An injective spectrum is a symmetric spectrum E that has the
extension property with respect to every monomorphism f of symmetric spectra
that is a level equivalence. That is, for every diagram in Sp
X --g--! E
?
f?y
Y
where f is a monomorphism and a level equivalence there is a map h: Y ! E such
that g = hf.
Some examples of injective spectra follow. Recall that Rn :S* ! Sp is the
right adjoint of the evaluation functor Evn :Sp ! S*. Also recall that a Kan
complex has the extension property with respect to every map of pointed simplic*
*ial
sets that is a monomorphism and a weak equivalence.
Lemma 3.1.2. If the pointed simplicial set K is a Kan complex then RnK is an
injective spectrum. If X is a symmetric sequence and E is an injective spectrum
then Hom S(S X; E) is an injective spectrum.
Proof.Since Evn is left adjoint to Rn, the spectrum RnK has the extension prop-
erty with respect to the monomorphism and level equivalence f if and only if K *
*has
the extension property with respect to the monomorphism and weak equivalence
Evn f. Since K is a Kan complex, it does have the extension property with respe*
*ct
to Evn f. Hence RnK is injective.
Since (SX)^(-) is the left adjoint of Hom S(SX; -), the spectrum Hom S(S
X; E) has the extension property with respect to the monomorphism and level
equivalence f if and only E has the extension property with respect to the map
(S X) ^ f. There is a natural isomorphism of maps of symmetric sequences
(S X)^f ~=X f. Since f is a monomorphism and a level equivalence, X f is a
monomorphism and a level equivalence by Proposition 2.1.9. Thus (SX)^f is also
a monomorphism and level equivalence of symmetric spectra. So Hom S(S X;_E)_
is injective. |__|
In fact, injective spectra are the fibrant objects of a model structure on Sp
for which every object is cofibrant (Section 5.1). In particular, we will see *
*in
SYMMETRIC SPECTRA 21
Corollary 5.1.3, there are enough injectives; every symmetric spectrum embeds in
an injective spectrum by a map that is a level equivalence.
Definition 3.1.3.A map f :X ! Y of symmetric spectra is a stable equivalence
if E0f :E0Y ! E0X is an isomorphism for every injective -spectrum E.
There are two other ways to define stable equivalence.
Proposition 3.1.4.Let f :X ! Y be a map of symmetric spectra. The following
conditions are equivalent:
o E0f is an isomorphism for every injective -spectrum E;
o MapSp (f; E) is a weak equivalence for every injective -spectrum E;
o Hom S(f; E) is a level equivalence for every injective -spectrum E.
Proof.Let K be a pointed simplicial set and E be a symmetric -spectrum. The
adjoints of the structure maps of E are weak equivalences of Kan complexes. From
Remark 2.2.12, for k; n 0 Ev kHom S(FnK; E) = EKn+k. The adjoints of the
structure maps of Hom S(FnK; E) are the weak equivalences of Kan complexes
EKn+k! ES1^Kn+k+1induced by the adjoints of the structure maps of E. Therefore,
Hom S(FnK; E) is an -spectrum.
Now let E be an injective -spectrum. Using the natural isomorphism FnK ~=
SGnK and Lemma 3.1.2, Hom S(FnK; E) is an injective spectrum.nBy the preced-
ing paragraph, Hom S(FnK; E) is an -spectrum. Hence ES = Hom S(F0(Sn); E)
and the k-shifted spectrum Hom S(FkS0; E) are injective -spectra. Given a stable
equivalence f :X -! Y , we want to show that Map Sp (f; E) is a weak equivalenc*
*e.
Since
n
ssn MapSp (f; E) = ss0Map Sp (f; ES )
and the simplicial sets Map Sp (Y; E) and Map Sp (X; E) are Kan complexes by
Lemma 3.1.5, Map Sp (f; E) is a weak equivalence on the basepoint components.
We must extend this to all components. To do so, note that Map Sp (f; E)S1 is a
weak equivalence for any injective -spectrum E, since the loop space only depen*
*ds
on the basepoint component. Consider the commutative diagram
Map Sp (Y; E)----! Map Sp (Y; Hom S(F1S0; E))S1
? ?
MapSp(f;E)?y ?y
MapSp (X; E) ----! MapSp (X; Hom S(F1S0; E))S1
where the horizontal maps are induced by the map E -! Hom S(F1S0; E)S1 adjoint
to the structure map of E. Since E is an injective -spectrum, this map is a
level equivalence of injective spectra. By Lemma 3.1.6, it is a simplicial homo*
*topy
equivalence. Hence the horizontal maps in the diagram above are weak equivalenc*
*es.
Since the right-hand vertical map is a weak equivalence, so is the left-hand ve*
*rtical
map Map Sp (f; E). Thus the first two conditions in the proposition are equival*
*ent.
Since Evk Hom S(f; E) = Map Sp (f; Hom S(FkS0; E)), the second two conditions_
are equivalent. |__|
Lemma 3.1.5. Suppose X is a symmetric spectrum and E is an injective spectrum.
Then the pointed simplicial set Map Sp (X; E) is a Kan complex. In particular, *
*each
pointed simplicial set En is a Kan complex.
22 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Proof.Suppose that f :K -! L is a monomorphism and weak equivalence of sim-
plicial sets. We must show that Map Sp (X; E) has the extension property with
respect to f. By adjointness, this is equivalent to showing that E has the exte*
*nsion
property with respect to X ^f. But X ^f is a monomorphism and level equivalence
by Corollary 1.3.6 applied to the monomorphism * -!X and f, so E does have_the_
required extension property. |__|
The basic properties of injective spectra which are needed in the rest of this
section are stated in the following lemma.
Lemma 3.1.6. Let f :X ! Y be a map of symmetric spectra.
1. If E 2 Sp is an injective spectrum and f is a level equivalence then E0f *
*is
an isomorphism of sets.
2. If f :X ! Y is a map of injective spectra, f is a level equivalence if and*
* only
if f is a simplicial homotopy equivalence.
The proof uses the following construction.
Construction 3.1.7 (Mapping cylinder construction).The mapping cylinder con-
struction for maps of symmetric spectra is the prolongation of the reduced mapp*
*ing
cylinder construction for maps of pointed simplicial sets. Let i0 and i1 be the*
* two
inclusions [0] ! [1]. The cylinder spectrum Mf of a map f :X ! Y 2 Sp is
the corner of the pushout square
X ^ [0]+ = X --f--! Y
? ?
X^i0?y ?ys
X ^ [1]+ ----!gMf
Let i = g O (X ^ i1): X ! Mf. Let r :Mf ! Y be the map on the pushout
induced by the identity map on Y and the composition X ^ [1]+ ! X ! Y .
Then f = ri, i is a monomorphism, rs = idY , and there is a simplicial homotopy
from sr to the identity map of Mf.
Proof of Lemma 3.1.6.For part one of the lemma, let f :X ! Y be a level equiv-
alence and let Mf be the mapping cylinder of f. As above f = ri, i: X ! Mf
is a monomorphism, and r :Mf ! Y is a simplicial homotopy equivalence. Then
E0r is an isomorphism and, if E0i is an isomorphism, the composition E0f is an
isomorphism. The map i is a monomorphism which, by the 2-out-of-3 property, is
a level equivalence. By the extension property of E with respect to i, the map *
*E0i
is surjective . The inclusion of the boundary j :@[1] ! [1] is a monomorphism.
By Corollary 1.3.6, the map
i j :X ^ [1]+ qX^@[1]+ Mf ^ @[1]+ ! Mf ^ [1]+
is a monomorphism and a level equivalence. The extension property of E with
respect to i j implies that if g; h: Mf ! E are maps such that gi and hi
are simplicially homotopic, then g and h are simplicially homotopic. So E0i is a
monomorphism and hence E0i is an isomorphism.
For the second part of the lemma, if f is a simplicial homotopy equivalence,
each fn is a simplicial homotopy equivalence of simplicial sets and so each fn *
*is a
weak equivalence. Conversely, suppose f :X ! Y is a level equivalence of inject*
*ive
spectra. By part one, X0f :X0Y ! X0X is an isomorphism. The inverse image
SYMMETRIC SPECTRA 23
of the equivalence class of the identity map X ! X is an equivalence class of m*
*aps
Y ! X. Since Y is injective, Y 0f :Y 0Y ! Y 0X is an isomorphism. Hence each *
* __
map in the equivalence class is a simplicial homotopy inverse of f. *
*|__|
Restricting part one of Lemma 3.1.6 to injective -spectra gives:
Corollary 3.1.8.Every level equivalence of symmetric spectra is a stable equiva-
lence.
Next recall the definition of stable homotopy equivalence in the category of *
*(non-
symmetric) spectra SpN.
Definition 3.1.9.For each integer k the kth homotopy group of the spectrum (or
symmetric spectrum) X is the colimit
sskX = colimnssk+nXn:
of the directed system given by the compositions
ssk+nXn E-!ssk+n+1(S1 ^ Xn) ssk+n+1oe-----!ssk+n+1Xn+1
of the suspension homomorphism and the map induced by oe.
A map of spectra f 2 SpN is a stable homotopy equivalence if ss*f is an isomo*
*r-
phism. For example, every level equivalence in SpN is a stable homotopy equival*
*ence
as it induces an isomorphism of homotopy groups. We do not define stable equiv-
alence of symmetric spectra in this way; as the following example shows, a stab*
*le
equivalence of symmetric spectra need not induce an isomorphism of homotopy
groups.
Example 3.1.10. The map : F1S1 ! F0S0 (see 2.2.12) is the adjoint of the
identity map S1 ! Ev 1S = S1. The nth space of F1S1 is (n)+ ^n-1 Sn,
which is a wedge of n copies of Sn. One can calculate that ss0F1S1 is an in-
finite direct sum of copies of the integers Z, whereas ss0F0S0 is Z. So ss* is
not an isomorphism of homotopy groups and thus U is not a stable homotopy
equivalence of (non-symmetric) spectra. But is a stable equivalence of symmet-
ric spectra. For a symmetric -spectrum E, Map Sp (F1S1; E) = Map S*(S1; E1),
Map Sp (F0S0; E) = E0, and the induced map Map Sp (; E) = E0 ! ES11is ad-
joint to the structure map S1^E0 ! E1. So Map Sp(; E) is a weak equivalence for
every -spectrum E, including the injective ones, and so is a stable equivalenc*
*e.
By the same argument, the maps ^ FnS0 are stable equivalences.
The forgetful functor U :Sp ! SpN does not preserve stable equivalences. On
the other hand, the functor U does reflect stable equivalences.
Theorem 3.1.11. Let f be a map of symmetric spectra such that ss*f is an iso-
morphism of homotopy groups. Then f is a stable equivalence.
Proof.To ease notation, let RX = Hom S(F1S1; X), so that RnX = Hom S(FnSn; X)
for n 0 and Evk RnX = Map S*(Sn; Xn+k). In particular R0X = X and there
is a natural map *: X ! RX induced by the map : F1S1 -! F0S0 discussed
in Example 3.1.10 and Remark 2.2.12. The maps Rn(*): RnX ! Rn+1X give a
directed system. Let
R1 X = colimn0 RnX = colimn0 Hom S(FnSn; X)
and let rX :X ! R1 X be the natural map from X to the colimit.
24 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Let E be an injective -spectrum. Since E is an -spectrum, the map
Rn(*): RnE ! Rn+1E
is a level equivalence for each n and the map rE :E ! R1 E is a level equivalen*
*ce.
Since E is injective, the proof of part one of Lemma 3.1.6 applied to rE shows
that there is a map g :R1 E ! E such that the composition grE is simplicially
homotopic to the identity map on E, though the other composite rE g may not be
simplicially homotopic to the identity map on R1 E. There is a natural transfor-
mation E0X ! E0(R1 X) sending the map f :X ! E to the map gR1 f; there is
a natural transformation E0(R1 X) ! E0X induced by composition with the map
rX :X ! R1 X. Since grE is simplicially homotopic to the identity map on E, the
composition of the natural transformations is the identity natural transformati*
*on
of the functor E0. In the diagram
E0Y _____//E0R1 Y_____//E0Y
E0f || E0R1 f|| E0f ||
fflffl| fflffl| fflffl|
E0X _____//E0R1 X_____//E0X
the composition of the horizontal maps is the identity, showing that E0f is a r*
*etract
of E0R1 f.
Let X be a symmetric spectrum that is a level Kan complex, i.e., each Xn
is a Kan complex. Since the functor ssk commutes with filtered colimits, the ho-
motopy group ssk EvnR1 X of the pointed simplicial set Evn R1 X is naturally
isomorphic to the homotopy group ssk-nX of the symmetric spectrum X. A warn-
ing: even though the groups ssk EvnR1 X and ssk+1Ev n+1R1 X are abstractly
isomorphic, the structure map of the symmetric spectrum R1 X need not induce
an isomorphism between them. In particular, despite its similarity to the stand*
*ard
construction of the -spectrum associated to a (non-symmetric) spectrum, R1 X
need not be an -spectrum and rX :X ! R1 X need not induce an isomorphism
of homotopy groups.
Now, let f be a map of symmetric spectra such that ss*f is an isomorphism.
Assume as well that X and Y are level Kan complexes. Then R1 f is a level
equivalence. By Proposition 3.1.8, E0R1 f is an isomorphism for every injective
-spectrum E. Thus E0f, which is a retract of E0R1 f, is an isomorphism for
every injective -spectrum E, and so f is a stable equivalence.
To finish the proof, let f :X ! Y be a map of arbitrary symmetric spectra
such that ss*f is an isomorphism. For every simplicial set X there is a natural*
* weak
equivalence X ! KX where KX is a Kan complex. There are several such functors:
K can be Kan's Ex1 functor; K can be the total singular complex of the geometric
realization; or K can be constructed using a simplicial small object argument. *
*In
each case, K is a S*-functor and X ! KX is a S*-natural transformation. By
prolongation, for every symmetric spectrum X there is a natural level equivalen*
*ce
X ! KX where KX is a level Kan complex. In the commutative diagram
X ----! KX
? ?
f?y ?yKf
Y ----! KY
SYMMETRIC SPECTRA 25
the horizontal maps are stable equivalences by 3.1.8; the map Kf is a stable eq*
*uiv-_
alence by the preceding paragraph; hence f is a stable equivalence. *
*|__|
As a corollary some of the standard results about spectra translate into resu*
*lts
about symmetric spectra.
Definition 3.1.12.Let (X; A) be a pair of symmetric spectra where A is a sub-
spectrum of X. The kth relative homotopy group of the pair (X; A) is the colimit
ssk(X; A) = colimnssk+n(Xn; An; *):
of the relative homotopy groups of the pointed pairs of simplicial sets (Xn; An*
*; *).
Lemma 3.1.13 (Stable excision).Let (X; A) be a pair of symmetric spectra with
A a subspectrum of X. The map of homotopy groups ssk(X; A) ! ssk(X=A) is an
isomorphism.
Proof.Consider the diagram:
ssk+q(Xq; Aq)___//ssk+q+p(Sp ^ Xq; Sp ^_Aq)//_ssk+q+p(Xp+q; Ap+q):
| | |
| | |
fflffl| fflffl| fflffl|
ssk+q(Xq=Aq)______//ssk+q+p(Sp ^ Xq=Aq)____//_ssk+q+p(Xp+q=Ap+q)
Let (K; L) be a pair of pointed simplicial sets. By the homotopy excision theor*
*em
the map ssn(Sp ^ K; Sp ^ L) ! ssn(Sp ^ K=L) is an isomorphism when n < 2p. So
the middle vertical arrow in the diagram is an isomorphism when p > k + q and __
hence the map of colimits ssk(X; A) ! ssk(X=A) is an isomorphism. |_*
*_|
Theorem 3.1.14. 1.Let f :X ! B be a map of symmetric spectra such that
fn :Xn ! Bn is a Kan fibration for each n 0 and let F be the fiber over
the basepoint. Then the map X=F ! B is a stable equivalence.
2. A map f 2 Sp of symmetric spectra is a stable equivalence if and only if *
*its
suspension f ^ S1 is a stable equivalence.
3. For symmetric1spectra X and Y such that Y is a level Kan complex, a map
X ! Y S is a stable equivalence if and only its adjoint X ^ S1 ! Y is a
stable equivalence.
Proof.By stable excision, the map X=F ! B induces an isomorphism of homotopy
groups and hence is a stable equivalence by Theorem 3.1.11.
For1part two, let E be an injective -spectrum. By Lemma 3.1.2, the spectra
ES , Hom S(F1S0; E) and Hom S(F1S1; E) are injective -spectra. If f is a stable
equivalence of symmetric spectra then the map Map Sp (f; ES1) = Map Sp (f ^
S1; E) is a weak equivalence of simplicial sets and so f ^ S1 is a stable equiv*
*alence.
Conversely if f ^ S1 is a stable equivalence then the map
MapSp (f ^ S1; Hom S(F1S0; E)) = Map Sp (f; Hom S(F1S1; E))
is a weak equivalence. The map E ! Hom S(F1S1; E) is a level equivalence of
injective spectra and thus a simplicial homotopy equivalence, by Lemma 3.1.6. S*
*o,
for every symmetric spectrum X, the induced map
Map Sp (X; E) ! Map Sp (X; Hom S(F1S1; E))
is a simplicial homotopy equivalence. Therefore Map Sp (f; E) is a weak equiva-
lence, and so f is a stable equivalence.
26 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
For part three, let f :X ! Y S1be a map and fa :X ^ S1 ! Y be the adjoint
of f. The diagram
f^S1 1
X ^ S1F____________//Y S ^ S1
FFF vvvv
aFFFF vvevv
f ##Fzzvv
Y
is commutative where the map ev is the evaluation map. By part one applied to
the prolongation of the path fibration, the map evis a stable equivalence; by p*
*art
two, f is a stable equivalence if and only if f ^S1 is a stable equivalence. Th*
*erefore_
f is a stable equivalence if and only if fa is a a stable equivalence. *
* |__|
Once we have the stable model category of symmetric spectra, part three of th*
*is
theorem tells us that it really is stable; i.e., that the suspension functor - *
*^ S1 is
an equivalence of model categories.
3.2. Model categories. In this section we recall the definition and the basic p*
*rop-
erties of model categories; see [DS95 ], [Hov97 ], or [DHK ] for a more detail*
*ed intro-
duction.
Definition 3.2.1.An ordered pair (f; g) of maps in the category C has the lifti*
*ng
property if every commutative square
A _____//X
f || |g|
fflffl|fflffl|
B _____//Y
in C extends to a commutative diagram
A _____//X>>":
""
f ||"" |g|
fflffl|fflffl|""
B _____//Y
We also say that f has the left lifting property with respect to g and that g h*
*as
the right lifting property with respect to f. More generally, if I and J are cl*
*asses
of maps in C, the pair (I; J) has the lifting property if every pair (f; g) wit*
*h f 2 I
and g 2 J has the lifting property. We also say that I has the left lifting pro*
*perty
with respect to J and that J has the right lifting property with respect to I.
It would be more accurate to say that the pair (f; g) has the lifting-extensi*
*on
property but we prefer the shorter term.
Definition 3.2.2.Let f and g be maps in a category C. The map f is a retract of
g if it is a retract in the category of arrows, i.e., if there is a commutative*
* diagram
A ----! B ----! A
? ? ?
f?y g?y f?y
X ----! Y ----! X:
such that the horizontal compositions are the identity maps. A class of maps is
closed under retracts if whenever f is a retract of g and g is in the class the*
*n f is
in the class.
SYMMETRIC SPECTRA 27
Definition 3.2.3.A model category is a category M with three distinguished
classes of maps; the class of weak equivalences, the class of cofibrations, and*
* the
class of fibrations; that satisfy the model category axioms below. We call a map
that is both a cofibration and a weak equivalence a trivial cofibration, and we*
* call
a map that is both a fibration and a weak equivalence a trivial fibration.
M1 Limit axiom The category M is bicomplete (closed under arbitrary small
limits and colimits).
M2 Two out of three axiom. Let f and g be maps in M such that gf is defined. *
*If
two of f, g and gf are weak equivalences then the third is a weak equivale*
*nce.
M3 Retract axiom. The class of weak equivalences, the class of cofibrations, *
*and
the class of fibrations are closed under retracts.
M4 Lifting axiom. A cofibration has the left lifting property with respect to
every trivial fibration. A fibration has the right lifting property with r*
*espect
to every trivial cofibration.
M5 Factorization axiom. Every map f 2 M has a factorization f = pi where i is
a cofibration and p is a trivial fibration and a factorization f = qj wher*
*e j is
a trivial cofibration and q is a fibration.
Three classes of maps that satisfy axioms M2, M3, M4 and M5 are a model
structure on the category. One should keep in mind that a category can have more
than one model structure; there can even be distinct model structures with the
same class of weak equivalences.
A bicomplete category has an initial object ; and a terminal object *. In a m*
*odel
category, an object X is cofibrant if the unique map ; ! X is a cofibration and*
* an
object X is fibrant if the unique map X ! * is a fibration. A model category is
pointed if the unique map ; ! * is an isomorphism.
Proposition 3.2.4 (The Retract Argument).Let C be a category and let f = pi
be a factorization in C.
1. If p has the right lifting property with respect to f then f is a retract *
*of i.
2. If i has the left lifting property with respect to f then f is a retract o*
*f p.
Proof.We only prove the first part, as the second is similar. Since p has the r*
*ight
lifting property with respect to f, we have a lift g :Y -! Z in the diagram
X --i--! Z
? ?
f?y p?y
Y _______Y
This gives a diagram
X _______X _______X
? ? ?
f?y i?y f?y
Y --g--! Z ---p-! Y
where the horizontal compositions are identity maps, showing that f is_a_retrac*
*t of
i. |__|
The following proposition is a converse to the lifting axiom.
Proposition 3.2.5 (Closure property).In a model category:
28 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
1. The cofibrations are the maps having the left lifting property with respec*
*t to
every trivial fibration.
2. The trivial cofibrations are the the maps having the left lifting property*
* with
respect to every fibration.
3. The fibrations are the maps having the right lifting property with respect*
* to
every trivial cofibration.
4. The trivial fibrations are the maps having the right lifting property with*
* respect
to every cofibration.
Proof.Use the factorization axiom and the retract argument. |___|
In particular, any two of the three classes of maps in a model category deter*
*mine
the third. For example, a weak equivalence is a map that factors as a trivial
cofibration composed with a trivial fibration.
Example 3.2.6. We recall the standard model structure on the category of sim-
plicial sets [Qui67, II.3]. A weak equivalence is a map whose geometric realiza*
*tion
is a homotopy equivalence of CW-complexes. The cofibrations are the monomor-
phisms and every simplicial set is cofibrant. Recall, the standard n-simplex is
[n] = (-; __n). The boundary of [n] is the subfunctor @[n] [n] of non-
surjective maps. For 0 i n, the ith horn of [n] is the subfunctor i[n] @[n]
of maps for which i is not in the image. Geometrically, i[n] is obtained from t*
*he
boundary of [n] by removing the ith face. The fibrations are the Kan fibrations,
the maps that have the right lifting property with respect to the maps i[n] ! [*
*n]
for n > 0 and 0 i n; the fibrant simplicial sets are the Kan complexes, the
simplicial sets that satisfy the Kan extension condition. A map is a trivial fi*
*bration
(a fibration and a weak equivalence) if and only if it has the right lifting pr*
*operty
with respect to the maps @[n] ! [n]. It follows that the pointed weak equiva-
lences, the pointed monomorphisms, and the pointed (Kan) fibrations are a model
structure on the category of pointed simplicial sets.
When constructing a model category, the factorization axiom can be the hardest
to verify. After some preliminary definitions, Lemma 3.2.11 constructs functori*
*al
factorizations in the category of symmetric spectra.
Definition 3.2.7.Let I be a class of maps in a category C.
1. A map is I-injective if it has the right lifting property with respect to *
*every
map in I. The class of I-injective maps is denoted I-inj.
2. A map is I-projective if it has the left lifting property with respect to *
*every
map in I. The class of I-projective maps is denoted I-proj.
3. A map is an I-cofibration if it has the left lifting property with respect*
* to
every I-injective map. The class of I-cofibrations is the class (I-inj)-pr*
*ojand
is denoted I-cof.
4. A map is an I-fibration if it has the right lifting property with respect *
*to
every I-projective map. The class of I-fibrations is the class (I-proj)-in*
*jand
is denoted I-fib.
Injective and projective are dual notions; an I-injective map in C is an I-
projective map in Cop; an I-fibration in C is an I-cofibration in Cop. The cla*
*ss
I-injand the class I-projare analogous to the orthogonal complement of a set ve*
*c-
tors. That analogy helps explain the following proposition, whose proof we leave
to the reader.
SYMMETRIC SPECTRA 29
Proposition 3.2.8.Let I and J be classes of maps in a category C.
1. If I J then J-inj I-injand J-proj I-proj.
2. Repeating the operations: I I-cof, I I-fib, I-proj= (I-proj)-cof=
(I-fib)-proj, and I-inj= (I-inj)-fib= (I-cof)-inj.
3. The following conditions are equivalent:
o The pair (I; J) has the lifting property.
o J I-inj.
o I J-proj.
o The pair (I-cof; J) has the lifting property.
o The pair (I; J-fib) has the lifting property.
4. The classes I-injand I-projare subcategories of C and are closed under re-
tracts.
5. The class I-injis closed under base change. That is, if
A ----! B
? ?
f?y g?y
X ----! Y
is a pullback square and g is an I-injective map then f is an I-injective *
*map.
6. The class I-projis closed under cobase change. That is, if
A ----! B
? ?
f?y g?y
X ----! Y
is a pushout square and f is an I-projective map then g is an I-projective
map.
Corollary 3.2.9.Let I be a class of maps in a category C. The class I-cofis a
subcategory of C that is closed under retracts and cobase change. The class I-f*
*ibis
a subcategory of C that is closed under retracts and base change.
Another useful elementary lemma about the lifting property is the following.
Lemma 3.2.10. Let L: C ! D be a functor that is left adjoint to the functor
R: D ! C. If I is a class of maps in C and J is a class of maps in D, the pair
(I; RJ) has the lifting property if and only if the pair (LI; J) has the liftin*
*g property.
The next lemma is used repeatedly to construct factorizations.
Lemma 3.2.11 (Factorization Lemma).Let I be a set of maps in the category
Sp . There is a functorial factorization of every map of symmetric spectra as an
I-cofibration followed by an I-injective map.
The factorization lemma is proved using the transfinite small object argument.
We begin by showing that every symmetric spectrum is suitably small.
Recall that an ordinal is, by recursive definition, the well-ordered set of a*
*ll smaller
ordinals. In particular, we can regard an ordinal as a category. A cardinal is *
*an
ordinal of larger cardinality than all smaller ordinals.
Definition 3.2.12.Let fl be an infinite cardinal. An ordinal ff is fl-filtered *
*if every
set A consisting of ordinals less than ff such that supA = ff has cardinality g*
*reater
than fl.
30 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Every fl-filtered ordinal is a limit ordinal. In fact, since fl is infinite,*
* every fl-
filtered ordinal is a limit ordinal ff for which there is no countable set A of*
* ordinals
less than ff such that supA = ff. The smallest fl-filtered ordinal is the first*
* ordinal
of cardinality greater than fl. For example, !1 is the smallest @0-filtered ord*
*inal. If
fl < __fland ff is __fl-filtered, then ff is fl-filtered.
Define the cardinality of a spectrum X to be the cardinality of its underlyin*
*g set
qn qk (EvnX)k. Then the cardinality of X is always infinite, which is convenient
for the following lemma.
Proposition 3.2.13.Let X be a simplicial spectrum of cardinality fl. Let ff be
a fl-filtered ordinal and let D :ff ! Sp be an ff-indexed diagram of symmetric
spectra. Then the natural map
colimffSp (X; D) ! Sp (X; colimffD)
is an isomorphism.
Proof.Every symmetric spectrum has a presentation as a coequalizer
S S X _____////_S__X__//X
of free symmetric spectra in the category Sp . The symmetric spectra X and
S X have the same cardinality. So the proposition follows once it is proved
forQfree symmetric spectra. There is a natural isomorphism Sp (S X; Y ) =
p p
pS* (Xp; Yp). The functors S* (Xn; -) have the property claimed for Sp (X; -).
This fact is the heart of the proposition. To begin the proof of it, suppose *
*we
have a map f :Xn ! colimffD, where D is an ff-indexed diagram of p-simplicial
sets. For each simplex x of Xn, we can choose an ordinal fix < ff and a simplex
yx 2 Dfixsuch that f(x) is the image of yx. Because ff is fl-filtered, we can t*
*hen
find one ordinal fi < ff and a map g :X -! Dfifactoring f. The map g may not
be simplicial or equivariant, but, again using the fact that ff is fl-filtered,*
* we can
go out far enough in the colimit so that g will be both simplicial and equivari*
*ant.
We leave the details to the reader.
Since fl is infinite, for every countable set A of ordinals that are strictly*
* less than
ff, the ordinal supA is strictly less than ff. Therefore, the countable produc*
*t of
functors S*(X; -) has the property claimed for Sp (X; -) and the proposition_is
proved. |__|
Proof of Lemma 3.2.11.We begin by constructing a functorial factorization
X -Ig!Eg -Pg-!Y
of g :X ! Y such that Ig is an I-cofibration and g = P g O Ig. For a map
f :bf ! cf in I, let Df be the set of commutative squares
bf ----! X
? ?
f?y ?yg
cf ----! Y:
Let
B = qf2I qDf bf; C = qf2I qDf cf; and F = qf2I qDf f :B ! C:
SYMMETRIC SPECTRA 31
By the definition of Df, there is a commutative square
B ----! X
? ?
F?y ?yg
C ----! Y:
Let Eg be the pushout X qB C, Ig be the map X ! Eg = X qB C, and P g :Eg !
Y be the natural map on the pushout. By construction, the map Ig :X ! X qB C
is an I-cofibration and g = P gOIg. However, the map P g need not be an I-injec*
*tive
map.
Use transfinite induction to define functorial factorizations of g
ffg Pffg
X -I-!Effg --! Y
for every ordinal ff. The induction starts at 0 with E0g = X, I0g = idX , and P*
* 0g =
g. For a successor ordinal ff + 1, Eff+1g = E(P ffg), Iff+1g = I(P ffg) O Iffg,*
* and
P ff+1g = P (P ffg). For fi a limit ordinal, Efig = colimff 0
and 0 k r. Let F I = [n0 Fn(I ).
2. Let I@ denote the set of maps @[r]+ -! [r]+ for r 0. Let F I@ =
[n0 Fn(I@).
32 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Proposition 3.3.3.The level fibrations are the F I -injective maps. The level
trivial fibrations are the F I@-injective maps.
Proof.A map g is a level (trivial) fibration if and only if Evn g = gn is a (tr*
*ivial)
Kan fibration for each n 0. But Evn g is a (trivial) Kan fibration if and only*
* if it
has the right lifting property with respect to the class (I@) I . Then by adjun*
*ction,
g is a level (trivial) fibration if and only if g has the right lifting propert*
*y_with
respect to the class (F I@) F I . |__|
The level structure is not a model structure; it satisfies the two-out-of-thr*
*ee
axiom, the retract axiom, and the factorization axiom but not the lifting axiom.
A model structure is determined by any two of its three classes and so the level
structure is over determined. In Section 5.1 we prove there are two "level" mod*
*el
structures with the level equivalences as the weak equivalences: one that is ge*
*ner-
ated by the level equivalences and the level cofibrations and one that is gener*
*ated
by the level equivalences and the level fibrations. In any case, the level homo*
*topy
category obtained by inverting the level equivalences is not the stable homotopy
category of spectra.
The pushout smash product (Definition 1.3.3) has an adjoint construction.
Definition 3.3.4.Let f :U ! V and g :X ! Y be maps of pointed simplicial
sets. The map
Map (f; g): Map (V; X) ! Map (U; X) xMap(U;Y )Map(V; Y ):
is the map to the fiber product induced by the commutative square
*
Map (V; X)- f---!Map (U; X)
? ?
g*?y ?yg*
Map (V; Y )----! Map (U; Y ):
f*
Let f be a map of pointed simplicial sets and g be a map of symmetric spectra.
Then Hom (f; g) is the map of symmetric spectra that is defined by prolongatio*
*n,
Evn Hom (f; g) = Map (f; gn).
Proposition 3.3.5. 1. If f 2 S* is a monomorphism and g 2 S* is a Kan
fibration then Map (f; g) is a Kan fibration. If, in addition, either f o*
*r g is
a weak equivalence, then Map (f; g) is a weak equivalence.
2. If f 2 S* is a monomorphism and g 2 Sp is a level fibration then Hom (f;*
* g)
is a level fibration. If, in addition, either f is a weak equivalence or g*
* is a
level equivalence, then Hom (f; g) is a level equivalence.
Proof.Part one is a standard property of simplicial sets, proved in [Qui67, II.*
*3]._
Part two follows from part one by prolongation. |__|
Definition 3.3.6.Let f :U ! V and g :X ! Y be maps in a category C. Then
C (f; g) is the natural map of sets to the fiber product
C(V; X) ! C(U; X) xC(U;Y )C(V; Y )
SYMMETRIC SPECTRA 33
coming from the commutative square
*
C(V; X) --f--!C(U; X)
? ?
g*?y ?yg*
C(V; Y )----! C(U; Y )
f*
A pair (f; g) has the lifting property if and only if C (f; g) is surjective.
Definition 3.3.7.Let f :U ! V and g :X ! Y be maps of symmetric spectra.
Then Map (f; g) is the natural map to the fiber product
Sp (f ^ [-]+ ; g): Map (V; X) ! Map (U; X) xMap(U;Y )Map(V; Y )
Proposition 3.3.8.Let f and h be maps of symmetric spectra and g be a map of
pointed simplicial sets. There are natural isomorphisms
Sp (f g; h) ~=(S*) (g; Map (f; h)) ~=Sp (f; Hom (g; h))
In fact this proposition holds in any simplicial model category.
Proof.Let f :U ! V and h: X ! Y be maps in Sp and g :K ! L be a map
in S*. Using adjunction and the defining property of pushouts and of pullbacks,
each of the three maps in the proposition is naturally isomorphic to the map fr*
*om
Sp (V ^ L; X) to the limit of the diagram
Sp (U ^ L; X) Sp (V ^ K; X) Sp (V ^ L; Y )
QQ mm QQQ mm
| QQQQQmmmm QQQQmmmm |
| mmmmQQQQQ mmmQQQQQm |
fflffl|vvmmm Q(( vvmmm Q(( fflffl|
Sp (U ^ K; X) Sp (U ^ L; Y ) Sp (V ^ K; Y )
QQ mm
QQQQ | mmmm
QQQQ | mmmm
QQQ(( fflffl|vvmmm
Sp (U ^ K; Y )
|___|
Corollary 3.3.9.Let f and h be maps of symmetric spectra and g be a map of
pointed simplicial sets. The following are equivalent:
o (f g; h) has the lifting property.
o (g; Map (f; h)) has the lifting property.
o (f; Hom (g; h)) has the lifting property.
3.4. Stable model category. In this section we define the stable cofibrations
and the stable fibrations of symmetric spectra. The main result is that the cla*
*ss of
stable equivalences, the class of stable cofibrations, and the class of stable *
*fibrations
are a model structure on Sp .
Recall that f is a level trivial fibration if fn is a trivial Kan fibration f*
*or each
n 0.
Definition 3.4.1.A map of symmetric spectra is a stable cofibration if it has
the left lifting property with respect to every level trivial fibration. A map*
* of
symmetric spectra is a stable trivial cofibration if it is a stable cofibration*
* and a
stable equivalence. A symmetric spectrum X is stably cofibrant if * ! X is a st*
*able
cofibration.
34 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
The basic properties of the class of stable cofibrations are next.
Proposition 3.4.2. 1. The class of stable cofibrations is the class F I@-cof.
2. The class of stable cofibrations is a subcategory that is closed under ret*
*racts
and closed under cobase change.
3. If f is a cofibration of pointed simplicial sets and n 0 then Fnf is a st*
*able
cofibration. In particular, FnK is stably cofibrant for K 2 S*.
4. If f 2 Sp is a stable cofibration and g 2 S* is a cofibration then the pu*
*shout
smash product f g is a stable cofibration.
5. If f 2 Sp is a stable cofibration and h 2 Sp is a level fibration then
Map (f; h) is a Kan fibration.
Proof.The stable cofibrations are the maps having the left lifting property wit*
*h re-
spect to the level trivial fibrations, which by Proposition 3.3.3 are the F I@-*
*injective
maps. So the stable cofibrations are the F I@-cofibrations.
Every class I-cofhas the properties stated in part two.
Suppose g 2 Sp is a level trivial fibration, and f 2 S* is a cofibration. Th*
*en
f has the left lifting property with respect to the trivial Kan fibration Evn g*
*. By
adjunction, Fnf has the left lifting property with respect to g. Hence Fnf is a*
* stable
cofibration. In particular, for every pointed simplicial set K, the map * ! FnK*
* is
a stable cofibration, and so FnK is stably cofibrant.
Now suppose f 2 Sp is a stable cofibration and g 2 S* is a cofibration. Then,
given a level trivial fibration h 2 Sp , the map Hom (g; h) is a level trivial*
* fibration
by Proposition 3.3.5. Therefore the pair (f; Hom (g; h)) has the lifting prope*
*rty.
Then by Corollary 3.3.9, the pair (f g; h) has the lifting property, and so f *
* g
is a stable cofibration.
Finally, suppose f 2 Sp is a stable cofibration and h 2 Sp is a level fibra*
*tion.
Given a trivial cofibration g 2 S*, Hom (g; h) is a level trivial fibration by*
* Propo-
sition 3.3.5. Therefore, the pair (f; Hom (g; h)) has the lifting property. *
*Then,
by Corollary 3.3.9, the pair (g; Map (f; h)) has the lifting property. Theref*
*ore_
Map (f; h) is a Kan fibration. |__|
The next definition is natural in view of the closure properties in a model c*
*ate-
gory, see Proposition 3.2.5.
Definition 3.4.3.A map of symmetric spectra is a stable fibration if it has the
right lifting property with respect to every map that is a stable trivial cofib*
*ration.
A map of symmetric spectra is a stable trivial fibration if it is a stable fibr*
*ation and
a stable equivalence. A spectrum X is stably fibrant if the map X ! * is a stab*
*le
fibration.
Theorem 3.4.4. The category of symmetric spectra with the class of stable equiv-
alences, the class of stable cofibrations, and the class of stable fibrations i*
*s a model
category.
Proof.The category Sp is bicomplete by Proposition 1.2.10. The two out of three
axiom and the retract axiom are immediate consequences of the definitions. By
definition, (i; p) has the lifting property when i is a stable trivial cofibrat*
*ion and
p is a stable fibration. The lifting axiom for i a stable cofibration and p a s*
*table
trivial fibration is verified in Corollary 3.4.7. The two parts of the factori*
*zation_
axiom are verified in Corollary 3.4.6 and Lemma 3.4.8. |__|
SYMMETRIC SPECTRA 35
Lemma 3.4.5. A map is a stable trivial fibration if and only if it is a level *
*trivial
fibration.
Proof.Suppose g is a level trivial fibration. By definition, every stable cofib*
*ration
has the left lifting property with respect to g and in particular every stable *
*trivial
cofibration has the left lifting property with respect to g. So g is a stable f*
*ibration
which is a level equivalence and hence a stable equivalence. So g is a stable t*
*rivial
fibration.
Conversely, suppose g is a stable trivial fibration. Recall that at this poin*
*t we do
not know that g has the right lifting property with respect to stable cofibrati*
*ons.
By Lemma 3.2.11, g can be factored as g = pi with i an F I@-cofibration and p an
F I@-injective map. Since p is a level equivalence, it is a stable equivalence.*
* By the
two out of three property, i is a stable equivalence. Therefore, i is a stable *
*trivial
cofibration and has the left lifting property with respect to g. By the Retrac*
*t __
Argument 3.2.4, g is a retract of p and so g is a level trivial fibration. *
* |__|
Corollary 3.4.6.Every map f of symmetric spectra has a factorization f = pi as
a stable cofibration i followed by a stable trivial fibration p.
Proof.By the Factorization Lemma 3.2.11, every map f in Sp can be factored as
f = pi with i an F I@-cofibration and p an F I@-injective map. Then i is a stab*
*le
cofibration and p is level trivial fibration, which, by Lemma 3.4.5, means that*
*_p is
a stable trivial fibration. |__|
Corollary 3.4.7.A stable cofibration has the left lifting property with respect*
* to
every stable trivial fibration.
Proof.By Lemma 3.4.5 every stable trivial fibration is a level trivial fibratio*
*n. By
definition, stable cofibrations have the left lifting property with respect_to_*
*every
level trivial fibration. |__|
The following lemma will finish the proof of Theorem 3.4.4.
Lemma 3.4.8. Every map f of symmetric spectra has a factorization f = pi as a
stable trivial cofibration i followed by a stable fibration p.
To prove the lemma we need a set of maps J such that a J-cofibration is a
stable trivial cofibration and a J-injective map is a stable fibration. Using *
*the
Factorization Lemma with the set J will prove Lemma 3.4.8. The set J is defined
in 3.4.9 and Corollary 3.4.16 verifies its properties. This takes up the rest o*
*f the
section.
The maps ^ FnS0 used in the definition below appeared in the description
of the function spectrum in Remark 2.2.12. They are stable equivalences (see
Example 3.1.10) but are not stable cofibrations or even level cofibrations. We
modify them to get the set J.
Definition 3.4.9.Let : F1S1 ! F0S0 be the adjoint of the identity map S1 !
Ev1F0S0 = S1 and let n be the map ^ FnS0: Fn+1S1 ! FnS0, so that 0 = .
The mapping cylinder construction 3.1.7 gives a factorization n = rncn where
rn :Cn ! FnS0 is a simplicial homotopy equivalence and cn :Fn+1S1 ! Cn is
a level cofibration. For n 0, let Kn = cn I@, i.e., Kn is the set of maps cn *
* j
for j 2 I@. Let K = [nKn and finally, let J = F I [ K.
36 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Lemma 3.4.10. For each n 0, the map cn :Fn+1S1 ! Cn is a stable trivial
cofibration.
Proof.The map n is a stable equivalence (Example 3.1.10) and the simplicial
homotopy equivalence rn is a stable equivalence. Using the factorization n = rn*
*cn
and the two out of three property of stable equivalences, cn is a stable equiva*
*lence.
Next we show that cn is a stable cofibration. The mapping cylinder Cn can
also be defined as the corner in the pushout square
Fn+1S1 _ Fn+1S1- i0_i1---!Fn+1S1 ^ [1]+
? ?
n_1?y ?y
Fn+1S1 --kn--! FnS0 _ Fn+1S1 - gn---! Cn;
where kn is the inclusion on the second factor, i0 and i1 come from the two inc*
*lusions
[0] ! [1], and gn is the natural map to the pushout. Using the properties of
stable cofibrations in Proposition 3.4.2, we find that the map * ! FnS0 is a st*
*able
cofibration and, by cobase change, that kn is a stable cofibration. Let j be t*
*he
cofibration @[1]+ ! [1]+ . Then (* ! Fn+1S1)j = i0_i1 is a stable cofibration
and, by cobase change, gn is a stable cofibration. Thus the composition cn =_gn*
*kn
is a stable cofibration. |__|
Next we characterize the J-injective maps.
Definition 3.4.11.A commutative square of simplicial sets
X - ---! Z
? ?
p?y q?y
Y - ---! W
f
where p and q are fibrations is a homotopy pullback square if the following equ*
*ivalent
conditions hold:
o The induced map X -! Y xW Z is a weak equivalence.
o For every 0-simplex v 2 Y0, the map of fibers p-1v ! q-1fv is a weak
equivalence.
Lemma 3.4.12. A map of symmetric spectra p: E ! B is J-injective if and only
if p is a level fibration and the diagram
a 1
En --oe--!ESn+1
? ?
(*) pn?y ?ypn+1
a 1
Bn --oe--!BSn+1;
is a homotopy pullback square for each n 0, where the horizontal maps are the
adjoints of the structure maps.
Proof.Since J = F I [ K and K = [nKn, a map is J-injective if and only if it
is F I -injective and Kn-injective for each n 0. By Proposition 3.3.3, the F I*
* -
injective maps are the level fibrations. By definition, p 2 Sp is a Kn-inject*
*ive
map if and only if p has the right lifting property with respect to the class c*
*n I@.
Then, by Corollary 3.3.9, p is Kn-injective if and only if Map (cn; p) has the*
* right
SYMMETRIC SPECTRA 37
lifting property with respect to the class I@. Hence, p is Kn-injective if and *
*only if
Map (cn; p) is a trivial Kan fibration. If p is a level fibration, Map (cn; p*
*) is a Kan
fibration by Proposition 3.4.2. So, a level fibration p is Kn-injective if and *
*only if
Map (cn; p) is a weak equivalence. Taken together, p is J-injective if and onl*
*y if p
is a level fibration and Map (cn; p) is a weak equivalence for each n 0.
For each n 0, the map rn :Cn ! FnS0 has a simplicial homotopy inverse
sn :FnS0 ! Cn for which rnsn is the identity map on FnS0 (see 3.1.7). Then
Map (cn; p) is simplicially homotopic to Map (snn; p). Since FnS0 is a simpli-
cial deformation retract of Cn, n is a simplicial deformation retract of snn
and Map (n; p) is a simplicial deformation retract of Map (snn; p). Therefore,
Map (cn; p) is a weak equivalence if and only if Map (n; p) is a weak equival*
*ence.
The map
Map (n; p): Map (FnS0; E) ! Map (FnS0; B) xMap(Fn+1S1;B)Map(Fn+1S1; E)
is naturally isomorphic to the map
1
En ! Bn xBS1n+1ESn+1:
induced by the diagram (*). If p is a level fibration then by definition the di*
*agram
(*) is a homotopy pullback square if and only if the map Map (n; p) is a weak
equivalence. __
Combining the conclusions of the three paragraphs completes the proof. |_*
*_|
Corollary 3.4.13.The map F ! * is J-injective if and only if F is an -
spectrum.
We also get the following corollary, which is not needed in the sequel. Its p*
*roof
uses properness (see Section 5.4).
Corollary 3.4.14.A level fibration between two -spectra is J-injective.
Lemma 3.4.15. Let p: X ! Y be a map of symmetric spectra. If p is J-injective
and p is a stable equivalence then p is a level equivalence.
Proof.Suppose p: X ! Y is a J-injective stable equivalence. In particular, p is*
* a
level fibration. Let F be the fiber over the basepoint. Since the class J-injis*
* closed
under base change, the map F ! * is J-injective and F is an -spectrum. The
map p factors as X ! X=F ! Y . The map X=F ! Y is a stable equivalence by
Theorem 3.1.14. Since p: X ! Y is a stable equivalence, q :X ! X=F is a stable
equivalence.
A Barratt-Puppe type sequence for symmetric spectra is constructed by prolon-
gation to give the diagram
X ! X qF (F ^ [1]+ ) ! F ^ S1 ! X ^ S1 ! (X qF (F ^ [1]+ )) ^ S1:
Let E be an injective -spectrum. Since the map X qF (F ^ [1]+ ) ! X=F is a
level equivalence, after applying E0(-) to this sequence we can rewrite the ter*
*ms
involving the homotopy cofiber as E0(X=F ). This gives an exact sequence
0q E0(q^S1)
E0X -E-- E0(X=F ) E0(F ^ S1) E0(X ^ S1) ------ E0(X=F ^ S1):
Since q :X ! X=F is a stable equivalence, E0q is an isomorphism by definition, *
*and
E0(q^S1) is an isomorphism by part two of Theorem 3.1.14. Hence, E0(F ^S1) = *
for every injective -spectrum E, and so, by part two of Theorem 3.1.14, E0F = *
for every injective -spectrum E. By Corollary 5.1.3, there is a level equivalen*
*ce
38 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
F ! E where E is an injective spectrum; since F is an -spectrum, E is an inject*
*ive
-spectrum. By Lemma 3.1.6, E0E = E0F = *. So E is simplicially homotopic to
* and F is level equivalent to *.
This does not finish the argument as the base of the fibration Xn ! Yn need n*
*ot
be connected. Since p is J-injective
a 1
Xn --oe--!XSn+1
? ?
pn?y ?ypS1n+1
a 1
Yn --oe--!YnS+1;
is a homotopy1pullback square for each n 0. The proof is completed by showing
that pSn+1is a trivial Kan fibration for every n 0 which implies that pn is a
trivial Kan fibration for every n 0. For a pointed simplicial set K, let cK
denote the connected component of the basepoint. If E ! B is a pointed Kan
fibration, then cE ! cB is a Kan fibration; if the fiber over the basepoint * 2*
* B is
contractible then cE ! cB is a trivial Kan fibration.1In1particular, cXn ! cYn *
*is a
trivial Kan fibration and therefore, (cXn)S ! (cYn)S is a trivial fibration. *
*Since
KS1 = (cK)S1 for any pointed simplicial set K, pS1n:XS1n! YnS1is a trivial_Kan
fibration for every n 0. |__|
The next corollary finishes the proof of Lemma 3.4.8.
Corollary 3.4.16.The J-cofibrations are the stable trivial cofibrations and the
J-injective maps are the stable fibrations.
Proof.Every level trivial fibration is J-injective since it satisfies the condi*
*tion in
Lemma 3.4.12. Thus, a J-cofibration has the left lifting property with respect *
*to
every level trivial fibration, and hence a J-cofibration is a stable cofibratio*
*n. Let E
be an -spectrum. The maps p: E ! * and q = Hom (j; E): E[1] ! ExE, where
j :@[1]+ ! [1]+ is the inclusion, are J-injective by Lemma 3.4.12. Let E be an
injective -spectrum and f be a J-cofibration. Since f has the left lifting prop*
*erty
with respect to p: E ! *, E0f is surjective. Since f has the left lifting prope*
*rty
with respect to q :E[1] ! E x E, E0f is injective. So E0f is an isomorphism and
every J-cofibration is a stable trivial cofibration.
Conversely, let f be a stable trivial cofibration. By the Factorization Lemma
3.2.11, f factors as f = pi where i is a J-cofibration and p is a J-injective m*
*ap.
We have just seen that i is a stable equivalence. So, the J-injective map p is*
* a
stable equivalence and by Lemma 3.4.15, p is a level equivalence. Therefore the
stable cofibration f has the left lifting property with respect to the map p. B*
*y the
Retract Argument 3.2.4, f is a J-cofibration.
Let F be the class of stable fibrations. Since J-cofis the class of stable tr*
*ivial
cofibrations, one has by the definition of stable fibrations that F = (J-cof)-i*
*nj. But
(J-cof)-inj= J-injby Proposition 3.2.8 (2). In other words, the stable fibratio*
*ns_
are the J-injective maps. |__|
In particular, Lemma 3.4.12 characterizes the stable fibrations. The stably *
*fi-
brant objects are the -spectra by Corollary 3.4.13. Corollary 3.4.16 finishes t*
*he
proof of Lemma 3.4.8 and the verification of the axioms for the stable model ca*
*te-
gory of symmetric spectra.
SYMMETRIC SPECTRA 39
4. Comparison with the Bousfield-Friedlander category
The goal of this section is to show that the stable homotopy theory of symmet*
*ric
spectra and the stable homotopy theory of spectra are equivalent. We begin in
Section 4.1 by recalling the general theory of Quillen equivalences of model ca*
*te-
gories. In Section 4.2 we provide a brief recap of the stable homotopy theory *
*of
(non-symmetric) spectra. In Section 4.3 we show that the forgetful functor U fr*
*om
symmetric spectra to spectra is part of a Quillen equivalence. The left adjoint*
* V
of U plays very little role in this proof, beyond its existence, so we postpone*
* its
construction to Section 4.4.
4.1. Quillen equivalences. In this section, we briefly recall Quillen functors *
*and
Quillen equivalences between model categories.
Definition 4.1.1.Let C and D be model categories. Let L: C ! D and R: D ! C
be functors such that L is left adjoint to R. The adjoint pair of functors L an*
*d R
is a Quillen adjoint pair if L preserves cofibrations and R preserves fibration*
*s. We
refer to the functors in such a pair as left and right Quillen functors. A Qui*
*llen
adjoint pair is a Quillen equivalence if for every cofibrant object X 2 C and e*
*very
fibrant object Y 2 D, a map LX -! Y is a weak equivalence if and only if its
adjoint X -! RY is a weak equivalence.
The definition of a Quillen adjoint pair can be reformulated.
Lemma 4.1.2. Let L and R be a pair of functors between model categories such
that L is left adjoint to R.
1. L preserves cofibrations if and only if R preserves trivial fibrations.
2. L preserves trivial cofibrations if and only if R preserves fibrations.
This lemma is an immediate corollary of Lemma 3.2.10; see also [DS95 , 9.8]. A
useful lemma associated to these questions is Ken Brown's lemma.
Lemma 4.1.3 (Ken Brown's Lemma). Let F be a functor between model cate-
gories.
1. If F takes trivial cofibrations between cofibrant objects to weak equivale*
*nces,
then F preserves all weak equivalences between cofibrant objects.
2. If F takes trivial fibrations between fibrant objects to weak equivalences*
*, then
F preserves all weak equivalences between fibrant objects.
For the proof of this lemma see [DS95 , 9.9].
In particular, a left Quillen functor L preserves weak equivalences between c*
*ofi-
brant objects, and a right Quillen functor R preserves weak equivalences between
fibrant objects.
The following proposition is the reason Quillen equivalences are important.
Proposition 4.1.4.A Quillen adjoint pair of functors between model categories
induces an adjoint pair of functors on the homotopy categories which is an adjo*
*int
equivalence if and only if the adjoint pair of functors is a Quillen equivalenc*
*e.
For the proof of this proposition, see [DS95 , Theorem 9.7].
We now describe a useful sufficient condition for a Quillen adjoint pair to b*
*e a
Quillen equivalence.
40 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Definition 4.1.5.Suppose F :C ! D is a functor between model categories. For
any full subcategory C0of C, we say that F detects and preserves weak equivalen*
*ces
of C0if a map f in C0is a weak equivalence if and only if F f is.
In practice, very few functors detect and preserve weak equivalences on the w*
*hole
category. However, many functors detect and preserve weak equivalences between
cofibrant objects or fibrant objects, so the next lemma is often useful. Before*
* stating
it, we need a definition.
Definition 4.1.6.Suppose C is a model category. A fibrant replacement functor
on C is a functor K :C ! C whose image lies in the full subcategory of fibrant
objects, together with a natural weak equivalence i: X ! KX.
There is a dual notion of a cofibrant replacement functor, but we do not use
it. Fibrant replacement functors are usually obtained by using a version of the
Factorization Lemma 3.2.11 appropriate for C to functorially factor the map X -*
*! 1
into a trivial cofibration followed by a fibration. We have already used fibra*
*nt
replacement functors in S* in the proof of Theorem 3.1.11.
Lemma 4.1.7. Suppose L: C ! D is a left Quillen functor with right adjoint R,
and suppose K is a fibrant replacement functor on D. Suppose R detects and pre-
serves weak equivalences between fibrant objects and the composition X ! RLX -R*
*i!
RKLX is a weak equivalence for all cofibrant objects X of C. Then the pair (L; *
*R)
is a Quillen equivalence.
There is also a dual statement, but this is the criterion we use.
Proof.Suppose f :LX -! Y is a map, where X is cofibrant and Y is fibrant.
Consider the commutative diagram below.
X ----! RLX --Rf--! RY
? ?
RiLX?y ?yRiY
RKLX ----! > RKY
RKf
The top composite is the adjoint g :X -! RY of f. The map iY is a weak
equivalence between fibrant objects, so RiY is a weak equivalence. The composite
X -! RLX -RiLX--!RKLX is a weak equivalence by hypothesis. Thus g is a weak
equivalence if and only if RKf is a weak equivalence. But R detects and preserv*
*es
weak equivalences between fibrant objects, so RKf is a weak equivalence if and
only if Kf is a weak equivalence. Since i is a natural weak equivalence, Kf is *
*a_
weak equivalence if and only if f is a weak equivalence. |*
*__|
4.2. The stable Bousfield-Friedlander category. In this section we describe
the stable homotopy theory of (non-symmetric) spectra. The basic results are
proved in [BF78 ], but we use the approach of Section 3. In particular, we can
define injective spectra and -spectra just as we did for symmetric spectra.
Definition 4.2.1.Suppose f :X -! Y is a map of spectra.
1. The map f is a stable equivalence if E0f is an isomorphism for every injec*
*tive
-spectrum E.
2. The map f is a stable cofibration if f has the left lifting property with *
*respect
to every level trivial fibration.
SYMMETRIC SPECTRA 41
3. The map f is a stable fibration if f has the right lifting property with r*
*espect
to every map which is both a stable cofibration and a stable equivalence.
We can then attempt to carry out the program of Section 3 with these definiti*
*ons.
The category SpN of non-symmetric spectra is neither symmetric monoidal nor
closed, so we must be careful in some places. However, if X is a sequence and
Y is a spectrum, we can form the sequences X Y and Hom N(X; Y ), and these
are S-modules in a canonical way. With this minor change, all of the results of
Section 3 go through without difficulty. There is one important change: in the
proof of Theorem 3.1.11, we considered the spectrum R1 X, which looks like an -
spectrum but is not in general. The analogous spectrum in SpN is an -spectrum,
so we obtain the following result.
Theorem 4.2.2. A map in SpN is a stable equivalence if and only if it is a stab*
*le
homotopy equivalence.
We also obtain the usual characterizations of the stable trivial fibrations a*
*nd the
stable fibrations.
Proposition 4.2.3.A map in SpN is a stable trivial fibration if and only if it *
*is a
level trivial fibration.
Proposition 4.2.4.A map f :E -! B in SpN is a stable fibration if and only if
it is a level fibration and, for all n 0, the diagram
En ----! ES1n+1
?? ?
y ?y
Bn ----! BS1n+1
is a homotopy pullback square.
Corollary 4.2.5.A spectrum X in SpN is fibrant in the stable model category if
and only if X is an -spectrum.
Finally, we recover the theorem of Bousfield and Friedlander.
Theorem 4.2.6 ( [BF78 ]).The stable equivalences, stable cofibrations, and stab*
*le
fibrations define a model structure on SpN.
Note that the model structure we have just described is the same as the model
structure in [BF78 ], since the stable equivalences and stable cofibrations are*
* the
same.
In particular, we have the following characterization of stable cofibrations.
Proposition 4.2.7.A map f :X -! Y in SpN is a stable cofibration if and only if
f0: X0 -!Y0 is a monomorphism and the induced map XnqXn-1^S1(Yn-1^S1) -!
Yn is a monomorphism for all n > 0. In particular, X is cofibrant if and only if
the structure maps are monomorphisms.
This proposition is proved in [BF78 ]; the proof is very similar to the proof*
* of the
analogous fact for symmetric spectra, proved in Proposition 5.2.2.
42 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
4.3. The Quillen equivalence. The goal of this section is to show that the for-
getful functor U from symmetric spectra to spectra is part of a Quillen equival*
*ence.
Obviously this requires that U have a left adjoint V :SpN -! Sp . We will assume
the existence of V in this section, and construct V in Section 4.4.
Proposition 4.3.1.The functors U :Sp -! SpN and its left adjoint V are a
Quillen adjoint pair.
Proof.Proposition 4.2.4 implies that U preserves stable fibrations. The stable
trivial fibrations in Sp and in SpN are the level trivial fibrations, so U pre*
*serves_
stable trivial fibrations as well. |__|
Theorem 4.3.2. The functor U :Sp ! SpN and its left adjoint V form a Quillen
equivalence of the stable model categories.
We prove this theorem by using Lemma 4.1.7. In particular, we need to under-
stand stable equivalences between stably fibrant objects.
Lemma 4.3.3. Suppose f :X -! Y is a stable equivalence between stably fibrant
objects in either Sp or SpN. Then f is a level equivalence.
Proof.Factor f as a stably trivial cofibration, i, followed by a stable fibrati*
*on, p.
Since f is a stable equivalence, p is also. Hence, p is a level trivial fibrat*
*ion by
Lemma 3.4.5. Also, i is a stably trivial cofibration between stably fibrant obj*
*ects,
hence it is a strong deformation retract, see [Qui67, II p. 2.5]. To see this, *
*note
that i has the left lifting property with respect to X ! *, so the lift constru*
*cts
a homotopy inverse to i. Because the simplicial structure is given on levels, a
strong deformation retract here is a level equivalence. So both i and p are_le*
*vel
equivalences, hence so is f. |__|
Corollary 4.3.4.U :Sp -! SpN detects and preserves stable equivalences be-
tween stably fibrant objects.
Let L denote a fibrant replacement functor in Sp , obtained by factoring X -!*
* *
into a stable trivial cofibration followed by a stable fibration. By Lemma 4.1.*
*7 and
Corollary 4.3.4, to prove Theorem 4.3.2 it suffices to show that X ! ULV X is a
stable equivalence for all cofibrant (non-symmetric) spectra X. We prove this in
several steps.
Definition 4.3.5.Given a simplicial set X, define eFn(X) to be the (non-symmetr*
*ic)
spectrum whose mth level is Sm-n ^ X for m n and the basepoint otherwise,
with the obvious structure maps. This defines a functor eFn:S* -!SpN left adjoi*
*nt
to the evaluation functor Evn.
Note that eF0X = 1 X. Also, since U O Evn = Evn, the left adjoints satisfy
V O eFn= Fn.
Lemma 4.3.6. The map X -! ULV X is a stable equivalence when X = 1 Y =
eF0Y for any Y 2 S*.
Proof.Consider the functor on simplicial sets QZ = colimnKnZ, where K is
a simplicial fibrant replacement functor. Because Q is simplicial we can prolon*
*g it
to a functor on Sp . The map F0Y -! QF0Y induces an isomorphism on stable
homotopy. Also QF0Y is an -spectrum since QZ ! QZ is a weak equivalence
for any Z 2 S*. Hence QF0Y is level equivalent to LF0Y , so F0Y ! LF0Y induces
SYMMETRIC SPECTRA 43
an isomorphism in stable homotopy. Since eF0Y ! UF0Y is a level equivalence_and
UF0Y ! ULF0Y is a stable homotopy equivalence, the lemma follows. |__|
Because both Sp and SpN are stable model categories, the following lemma is
expected.
Lemma 4.3.7. Suppose X is a cofibrant spectrum in SpN. Then the map X -!
ULV X is a stable equivalence if and only if X ^ S1 -! ULV (X ^ S1) is a stable
equivalence.
Proof.For notational convenience, we write X for X ^S1 and X for XS1 in this
proof, for X a (possibly symmetric) spectrum. Consider the stable trivial cofib*
*ra-
tion V X -! LV X in Sp . By Theorem 3.1.14 part three, V X ! LV X
is also a stable equivalence. By the lifting property of the stable trivial co*
*fi-
bration V X ! LV X and the 2-out-of-3 property, there is a stable equivalence
LV X ! LV X. This map is a stable equivalence between stably fibrant objects,
so by Corollary 4.3.4, f :ULV X ! ULV X is a stable equivalence.
So g :X ! ULV X is a stable equivalence if and only if fg :X ! ULV X
is a stable equivalence. Since and U commute, gf is a stable equivalence if and
only if X ! ULV X is a stable equivalence by part three of Theorem 3.1.14
for (non-symmetric) spectra. But, since U commutes with , the left adjoint V
commutes with , so we have a natural isomorphism ULV X ! ULV X. This __
completes the proof. |__|
Lemma 4.3.8. Let f :X ! Y be a stable equivalence between cofibrant spectra in
SpN. Then X ! ULV X is a stable equivalence if and only if Y ! ULV Y is a
stable equivalence.
Proof.Consider the following commuting square.
X ----! Y
?? ?
y ?y
ULV X ----! ULV Y
Since V is a left Quillen functor by Proposition 4.3.1, it preserves trivial co*
*fibra-
tions. Hence, by Ken Brown's Lemma 4.1.3, V preserves stable equivalences be-
tween cofibrant objects. Hence V X ! V Y is a stable equivalence. L takes stable
equivalences to level equivalences, by Lemma 4.3.3. So ULV X ! ULV Y is a level
equivalence since U preserves level equivalences. Hence the top and bottom maps
are stable equivalences, so the right map is a stable equivalence if and_only_i*
*f the
left map is. |__|
Using the preceding three lemmas we can extend Lemma 4.3.6 to any cofibrant
strictly bounded below spectrum.
Definition 4.3.9.Define a spectrum X 2 SpN to be strictly bounded below if
there is an n such that for all m n the structure map S1 ^ Xm -! Xm+1 is an
isomorphism.
Lemma 4.3.10. Suppose X 2 SpN is cofibrant and strictly bounded below. Then
the map X -! ULV X is a stable equivalence.
44 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Proof.Suppose X is strictly bounded below at n. Then we have a map eFnXn -g!X
which is the identity on all levels n. In particular, g is a stable homotopy
equivalence. Applying Lemma 4.3.8, this shows that to prove the lemma it is
enough to show that eFnXn ! ULFnXn is a stable equivalence. But there is an
evident map eFnXn ^ Sn ~=eFn(Xn ^ Sn) ! 1 Xn which is the identity map above
level n - 1, and so is a stable equivalence. Lemmas 4.3.6, 4.3.7, and 4.3.8_com*
*plete
the proof. |__|
We now extend this lemma to all cofibrant objects, completing the proof of
Theorem 4.3.2. First, we need to recall a basic fact about simplicial sets. R*
*e-
call that the homotopy group ssnX of a pointed simplicial set X is defined to be
ss0Map S*(Sn; KX), where K is a fibrant replacement functor. This ensures that
weak equivalences are homotopy isomorphisms. If X is already a Kan complex, X
is simplicially homotopy equivalent to KX, and so ssnX ~=ss0Map S*(Sn; X). Since
the simplicial sets @[n]+ and [n]+ are finite, the colimit of a sequence of Kan
complexes is again a Kan complex. Since the simplicial sets Sn and Sn ^ [1]+
are finite, homotopy commutes with filtered colimits of Kan complexes, and in
particular with transfinite compositions of maps of Kan complexes.
In fact, homotopy commutes with transfinite compositions of arbitrary monomor-
phisms of simplicial sets. To see this, apply the geometric realization to get*
* a
sequence of cofibrations of CW complexes. Since homotopy commutes with such
transfinite compositions, the result follows.
Lemma 4.3.11. Suppose X is a cofibrant object of SpN. Then the map X -!
ULV X is a stable equivalence.
Proof.Let Xi denote the truncation of X at i. That is, we have Xin= Xn for
n i and Xin= Xi^ Sn-i for n i. Then the Xi are strictly bounded below and
cofibrant, and there are monomorphisms Xi -! Xi+1 with colimiXi = X. Thus
each map Xi -!ULV Xi is a stable equivalence.
We claim that the induced map X -! colimiULV Xi is a stable equivalence. To
see this, note that
ssnX = colimissn+iXi= colimissn+icolimjXji
Since homotopy commutes with transfinite compositions of monomorphisms, we
find that ssnX ~=colimjssnXj. Similarly, since homotopy of Kan complexes com-
mutes with arbitrary filtered colimits, we find ssn colimiULV Xi ~=colimissnULV*
* Xi.
It follows that X -! colimiULV Xi is a stable homotopy equivalence, as required.
We now examine the relationship between ULV X and colimiULV Xi. Since
V is a left adjoint, V X ~= colimiV Xi. Each map V Xi -! LV Xi is a stable
trivial cofibration; we claim that the induced map colimiV Xi -! colimiLV Xi
is a stable equivalence. To see this, we define a new sequence Y iand maps of
sequences V Xi -!Y iand Y i-! LV Xi inductively. Define Y 0= LV X0. Having
defined Y i, define Y i+1by factoring the map Y iqV XiV Xi+1 -! LV Xi+1 into
a stable trivial cofibration Y iqV XiV Xi+1 -!Y i+1followed by a stable fibrati*
*on
Y i+1-!LV Xi+1. Then the induced map colimiV Xi -!colimiY iis a stable trivial
cofibration, by a lifting argument. On the other hand, each map Y i-! LV Xi is
a stable equivalence, by the two out of three axiom. Since LV Xi and hence Y i
are stably fibrant, the maps Y i-! LV Xi are level equivalences, by Lemma 4.3.3.
Since homotopy of level Kan complexes commutes with filtered colimits, we find
SYMMETRIC SPECTRA 45
that colimY i-! colimiLV Xi is a level equivalence, and therefore that V X ~=
colimiV Xi -!colimiLV Xi is a stable equivalence.
We now claim that colimiLV Xi is an -spectrum, and thus is stably fibrant. In-
deed, colimiLV Xi is a level1Kan complex by the1comments preceding this lemma.
Similarly, (colimLV Xi)Sn+1= colimi((LV Xi)Sn+1). Since homotopy of Kan com-
plexes commutes with filtered colimits, it follows that colimiLV Xiis an -spect*
*rum.
Hence the stable equivalence V X -! colimiLV Xi extends to a stable equiva-
lence LV X -! colimiLV Xi. By Lemma 4.3.3, this map is actually a level equiv-
alence. Since U preserves level equivalences and colimits, the map ULV X -!
colimiULV Xi is also a level equivalence. We have seen above that the map
X -! colimiULV Xi is a stable equivalence, so X -! ULV X must also be a_stable_
equivalence. |__|
Remark 4.3.12. It follows from the results of Section 5.3 that the smash produ*
*ct
on Sp induces a smash product on Ho Sp . The handicrafted smash products
of [Ada74 ] induce a smash product on Ho SpN. We now consider to what extent the
equivalence RU :Ho Sp -! HoSpN induced by U preserves these smash products.
Since U is a simplicial functor, there is a natural isomorphism RU(X ^ Y ) ~=
(RU)(X) ^ (RU)(Y ) for all (arbitrary desuspensions of) suspension spectra X. On
the other hand, in either HoSp or HoSpN, X ^Y is determined by the collection *
*of
F ^Y for all finite spectra F mapping to X. To be precise, X^Y is the minimal w*
*eak
colimit [HPS97 ] of the F ^Y . As an equivalence of categories, RU preserves mi*
*nimal
weak colimits, so there is an isomorphism RU(X ^ Y ) ~= (RU)(X) ^ (RU)(Y ),
However, we do not know if this is natural, as the minimal weak colimit is only*
* a
weak colimit. This isomorphism is natural up to phantom maps, however.
4.4. Description of V . This short section is devoted to the construction of the
left adjoint V :SpN -! Sp to the forgetful functor U :Sp -! SpN.
Recall that, in any cocomplete symmetric monoidal category C, the free monoid
or tensor algebra generated by an object X is T (X) = e_X _X2 _. ._.Xn _. .,.
where e is the unit and _ is the coproduct. The multiplication on T (X) is the
concatenation Xn Xm ! X(n+m) . Similarly, the free commutative monoid
on an object X is Sym (X) = e _ X _ (X2 =2) _ . ._.(Xn =n) _ . ...
Recall that the evaluation functor Evn :S* -! S* has a left adjoint Gn, where
GnX is (n)+ ^ X at level n and the basepoint everywhere else. Similarly, the
evaluation functor Evn :SN*-! S* has a left adjoint eGn, where eGnis X at level*
* n
and the basepoint everywhere else.
Lemma 4.4.1. In the category SN*of sequences, the sphere spectrum S is the ten*
*sor
algebra on the sequence eG1S1 = (*; S1; *; : :;:*; : :):. In the category S* o*
*f sym-
metric sequences, the sphere symmetric spectrum S is the free commutative monoid
on the symmetric sequence G1S1 = (*; S1; *; : :;:*; : :):.
Proof.The first statement follows directly from the definitions. In the categor*
*y of
symmetric sequences, (G1S1)n = GnSn, so T (G1S1) is (n)+ ^ Sn in degree n.
Therefore Sym (G1S1) is Sn in degree n. Since we already know S is a commu-
tative monoid, the map G1S1 -! S induces a map Sym (G1S1) -! S which is_an
isomorphism. |__|
This lemma explains why left S-modules and right S-modules are equivalent in
the category of sequences, since this is true for any tensor algebra. This lemm*
*a also
46 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
explains why Remark 1.2.3 holds, since an analogous statement holds for any free
commutative monoid.
W Now, the forgetful functor U :S* -! SN*has a left adjoint G, defined by GX =
GnXn, so that the nth level of GX is just (n)+ ^Xn. The functor G is monoidal;
that is, there is a natural isomorphism G(X)G(Y ) -!G(X Y ) compatible with
the associativity and unit isomorphisms. However, G is definitely not a symmetr*
*ic
monoidal functor; this natural isomorphism is not compatible with the commuta-
tivity isomorphisms. This explains how S can be commutative in S* yet US = S
is not commutative in SN*.
Since G is a monoidal functor, G preserves monoids and modules, and so de-
fines a functor G: SpN -! T (G1S1)-mod, left adjoint to the forgetful functor
T (G1S1)-mod -! SpN. On the other hand, the map of monoids T (G1S1) -p!
Sym (G1S1) = S defines the usual adjoint pair of induction and restriction. In-
duction takes a (left) T (G1S1)-module X to S T(G1S1)X, where the tensor prod-
uct uses the right action of T (G1S1) on S determined by p. It follows that the
left adjoint V :SpN -! Sp of the forgetful functor U :Sp -! SpN is V (X) =
S T(G1S1)GX.
5. Additional properties of symmetric spectra
In this section we discuss some properties of the category of symmetric spect*
*ra.
In Section 5.1, we consider the level model structures on Sp . In particular, *
*we
show that every symmetric spectrum embeds in an injective spectrum by a level
equivalence, completing the proof that the stable structures define a model str*
*ucture
on Sp . In Section 5.2 we characterize the stable cofibrations. In Sections 5*
*.3
and 5.5, we study the relationship between the stable model structure on Sp and
the smash product. This is necessary for constructing model categories of monoi*
*ds,
algebras, and modules, as is done in [SS97]. In Section 5.4, we show that the s*
*table
model structure on Sp is proper. Finally, in Section 5.6 we define semistable
spectra and investigate their relationship to stable homotopy equivalences.
5.1. Level model structure. In this section we construct the two level model
structures on the category of symmetric spectra.
Definition 5.1.1.A projective cofibration of symmetric spectra is a map that has
the left lifting property with respect to every level trivial fibration. The pr*
*ojective
cofibrations are the stable cofibrations from Section 3.4. The projective level*
* struc-
ture on Sp is the class of level equivalences, the class of projective cofibra*
*tions,
and the class of level fibrations. An injective fibration of symmetric spectra *
*is a map
that has the right lifting property with respect to every level trivial cofibra*
*tion (the
adjective "injective" refers to the lifting properties of the map and not to it*
*s being
a monomorphism). The injective level structure is the class of level equivalenc*
*es,
the class of level cofibrations, and the class of injective fibrations.
Theorem 5.1.2. The projective level structure and the injective level structure*
* are
model structures on the category of symmetric spectra.
Proof.The category of symmetric spectra is bicomplete. The class of level equiv*
*a-
lences has the two-out-of-three property. The retract axiom holds by constructi*
*on
in both the projective and injective level structures.
We now prove the lifting and factorization axioms, beginning with the project*
*ive
level structure. We use the sets of maps F I@ and F I defined in Definition 3.*
*3.2.
SYMMETRIC SPECTRA 47
The lifting axiom for a projective cofibration and a level trivial fibration ho*
*lds
by definition. The other lifting and factorization axioms follow by identifying*
* the
respective classes in terms of F I@ and F I . By part 4 of Lemma 5.1.4, an F I -
cofibration is a projective cofibration which is a level equivalence and an F I*
* -
injective map is a level fibration. Since (J-cof; J-inj) has the lifting prope*
*rty for
any class J, the lifting axiom for a map that is both a level equivalence and a
projective cofibration and a map that is a level fibration follows by setting J*
* = F I .
Moreover, every map can be factored as the composition of an F I -cofibration a*
*nd
an F I -injective map. Similarly, every map can be factored as the composition
of an F I@-cofibration and an F I@-injective map, by Lemma 3.2.11. By part 3 of
Lemma 5.1.4, an F I@-cofibration is a projective cofibration and an F I@-inject*
*ive
map is a level trivial fibration.
Now consider lifting and factorization for the injective level model structur*
*e.
Here we use a set C containing a map from each isomorphism class of monomor-
phisms i: X ! Y with Y a countable symmetric spectrum, and a set tC containing
a map from each isomorphism class of level trivial cofibrations i: X ! Y with Y
a countable symmetric spectrum. The lifting axiom for a level trivial cofibrati*
*on i
and an injective fibration p holds by definition. By part 5 of Lemma 5.1.4, a C-
cofibration is a level cofibration and a C-injective map is an injective fibrat*
*ion that
is a level equivalence. Since (J-cof; J-inj) has the lifting property for any c*
*lass J,
the lifting axiom for a level cofibration and a map that is both an injective f*
*ibration
and a level equivalence follows with J = C. Also, every map can be factored as *
*the
composition of a C-cofibration followed by a C-injective map, by Lemma 3.2.11.
Similarly, every map can be factored as the composition of a tC-cofibration and
a tC-injective map. By part 6 of Lemma 5.1.4, a tC-cofibration is a level trivi*
*al_
cofibration and a tC-injective map is an injective fibration. *
*|__|
Corollary 5.1.3.Every symmetric spectrum embeds in an injective spectrum by a
map that is a level equivalence.
Proof.For a symmetric spectrum X, the map X ! * is the composition of a level
trivial cofibration X ! E and an injective fibration E ! *. The fibrant object_E
is an injective spectrum. |__|
Some parts of the next lemma have already been proved. They are repeated for
easy reference. Recall that Rn :S* ! Sp is the right adjoint of the evaluation
functor Evn :Sp ! S*.
Lemma 5.1.4.
1. Let K S* be the class of Kan fibrations and let RK = [nRnK. Then a
map is RK-projective if and only if it is a level trivial cofibration.
2. Let tK S* be the class of trivial Kan fibrations and let R(tK) = [nRn(tK).
Then a map is R(tK)-projective if and only if it is a level cofibration.
3. Let F I@ be the set defined in 3.3.2. Then a map is F I@-injective if and *
*only
if it is a level trivial fibration. A map is an F I@-cofibration if and on*
*ly if it
is a projective cofibration.
4. Let F I be the set defined in 3.3.2. Then a map is F I -injective if and *
*only
if it is a level fibration. A map is an F I -cofibration if and only if i*
*t is a
projective cofibration and a level equivalence.
5. Let C be a set containing a map from each isomorphism class of monomor-
phisms i: X ! Y with Y a countable symmetric spectrum. Then a map is
48 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
C-injective if and only if it is an injective fibration and a level equiva*
*lence. A
map is a C-cofibration if and only if it is a level cofibration.
6. Let tC be a set containing a map from each isomorphism class of level triv*
*ial
cofibrations i: X ! Y with Y a countable symmetric spectrum. Then a
map is tC-injective if and only if it is a injective fibration. A map is a*
* tC-
cofibration if and only if it is a level trivial cofibration.
Proof.Parts 1 and 2: By adjunction, a map g has the left lifting property with
respect to the class RK (R(tK)) if and only if for each n 0 the map Evn g has
the left lifting property with respect to K (tK). But Evn g has the left lifti*
*ng
property with respect to K (tK) if and only if Evn g is a trivial cofibration (*
*arbi-
trary cofibration), i.e., if and only if g is a level trivial cofibration (arbi*
*trary level
cofibration).
Part 3 is proved in Propositions 3.3.3 and 3.4.2.
Part 4: The first claim is proved in Proposition 3.3.3. Every F I -cofibrati*
*on
has the left lifting property with respect to level fibrations, so is in partic*
*ular a
projective cofibration by Part 3. Every map in F I is a level trivial cofibra*
*tion
by Proposition 2.2.7, so is RK-projective by Part 1. So every F I -cofibration *
*is
also RK-projective and hence is a level trivial cofibration by Part 1 again. So*
* in
particular it is a level equivalence.
Conversely, suppose f is a projective cofibration and a level equivalence. We*
* can
factor f as the composition of an F I -cofibration i and an F I -injective map *
*p, by
Lemma 3.2.11. By the two-out-of-three property, p is a level equivalence. There*
*fore
the projective cofibration f has the left lifting property with respect to the *
*level
trivial fibration p. By the Retract Argument 3.2.4, f is a retract of i, and so*
* is an
F I -cofibration.
For part 5, first note that, by part 2, every C-cofibration is a level cofibr*
*ation.
Conversely, suppose f :X -! Y is a level cofibration. Then f is a C-cofibration*
* if,
for every C-injective map g and commutative square
X ----! E
? ?
f?y ?yg
Y ----! Z
there is a lift h: Y ! E making the diagram commute. Let P be the partially
ordered set of partial lifts: an object of P is a pair (U; hU ) such that X U *
* Y
and the diagram
X ____//_E>>"
""
iU||hU""" g||
fflffl|"fflffl|"
U ____//_Z;
is commutative. We define (U; hU ) (V; hV ) if U V and hV extends hU . Every
chain in P has an upper bound and so Zorn's lemma gives a maximum (M; hM ).
Suppose M is strictly contained in Y . Then, by taking the subspectrum generated
by a simplex not in M, we find a countable subspectrum D (by Lemma 5.1.6 below)
such that the level cofibration D\M ! D is not an isomorphism. By construction,
the map D \ M ! D is isomorphic to a map in C. By cobase change, M ! D [ M
is a C-cofibration. Thus hM extends to a partial lift on D [ M, contradicting *
*the
maximality of (M; hM ). Therefore M = Y , and so f is a C-cofibration.
SYMMETRIC SPECTRA 49
We now identify C-inj. Since (C-cof)-inj= C-inj, every C-injective map has the
right lifting property with respect to every monomorphism. In particular, every
C-injective map is an injective fibration. Let f :E ! B be a map having the rig*
*ht
lifting property with respect to every monomorphism. Let s: B ! E be a lift in
the diagram
* ----! E
?? ?
y ?yf
B _______B
Then fs is the identity map on B. To study the composite sf, let j be the monom*
*or-
phism @[1] ! [1]. The diagram
E _ E -sf_1---!E
? ?
E^j+?y ?yf
E ^ [1]+ ----! B
is commutative since fsf = f and has a lift since E ^ j+ is a monomorphism. The
lift is a simplicial homotopy from sf to the identity on E. Therefore f is a si*
*mplicial
homotopy equivalence and in particular f is a level equivalence. Conversely sup*
*pose
f is both an injective fibration and a level equivalence. We can factor f as t*
*he
composition of a C-cofibration i and a C-injective map p. By the two-out-of-thr*
*ee
property, i is a level equivalence. The level trivial cofibration i has the lef*
*t lifting
property with respect to the injective fibration f. By the Retract Argument 3.2*
*.4,
f is a retract of p and so is a C-injective map.
The proof of part 6 is similar, though slightly more complex. By part 1, every
tC-cofibration is a level trivial cofibration. Conversely, suppose f :X ! Y is*
* a
level trivial cofibration. Then f is a tC-cofibration if, for every tC-injectiv*
*e map g
and commutative square
X ----! E
? ?
f?y ?yg
Y ----! Z
there is a lift h: Y ! E making the diagram commute. We again let P be the
partially ordered set of partial lifts: an object of P is a pair (U; hU ) such*
* that
X U Y , and the diagram
X ____//_E>>"
""
iU||hU""" g||
fflffl|"fflffl|"
U ____//_Z;
is commutative, but we also require that the inclusion iU :X ! U is a weak
equivalence. We define (U; hU ) (V; hV ) as before. Every chain in P has an up*
*per
bound (using the fact that a transfinite composition of level trivial cofibrati*
*ons is
a level trivial cofibration) and so Zorn's lemma gives a maximum (M; hM ). The
inclusion X ! M is a level trivial cofibration, so, by the two-out-of-three pro*
*perty,
the inclusion M ! Y is a weak equivalence. If M is strictly contained in Y ,
Lemma 5.1.7, proved below, applied to the countable subspectrum of Y generated
50 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
by a simplex not in M, gives a countable subspectrum D of Y such that the
monomorphism D \ M ! D is a weak equivalence but is not an isomorphism.
By construction, D \ M ! D is isomorphic to a map in tC. By cobase change,
M ! D[M is a tC-cofibration. So hM extends to a partial lift on D[M. This is a
contradiction since (M; hM ) is maximal. Thus M = Y , and so f is a tC-cofibrat*
*ion. __
Since (tC-cof)-inj= tC-inj, the tC-injective maps are the injective fibration*
*s. |__|
Corollary 5.1.5.Every injective fibration is a level fibration and every projec*
*tive
cofibration is a level cofibration.
Proof.By Proposition 2.2.7, every map in F I@ is a level cofibration. Therefor*
*e,
by part 2 of Lemma 5.1.4, every projective cofibration is a level cofibration. *
* By
Proposition 2.2.7, every map in F I is a level trivial cofibration. Therefore *
*every_
injective fibration is a level fibration. |*
*__|
The following lemmas are used in the proof of Lemma 5.1.4.
Lemma 5.1.6. Let X be a spectrum, and suppose x is a simplex of Xn for some
n 0. Then the smallest subspectrum of X containing x is countable.
Proof.First note that if L is a countable collection of simplices in a simplici*
*al set
K, then the smallest subsimplicial set of K containing L is also countable. Ind*
*eed,
we need only include all degeneracies of all faces of simplices in L, of which *
*there are
only countably many. Similarly, if L is a countable collection of simplices in *
*a n-
simplicial set K, then the smallest sub-n-simplicial set containing L is counta*
*ble.
Indeed, we only need to include the orbits of all degeneracies of all faces of *
*simplices
in L.
Now, let Yn denote the sub-n-simplicial set of Xn generated by x. We have
just seen that Yn is countable. We then inductively define Yn+k to be the small*
*est
sub-n+k-simplicial set of Xn+k containing the image of S1 ^ Yn+k-1. Then each __
Yn+k is countable, and the Yn+k define a subspectrum of X containing x. |_*
*_|
It follows in similar fashion that the smallest subspectrum of a spectrum X
containing any countable collection of simplices of X is countable.
We need a similar lemma for inclusions which are level equivalences. To prove
such a lemma, we need to recall from the comments before Lemma 4.3.11 that
homotopy of simplicial sets commutes with transfinite compositions of monomor-
phisms. The same methods imply that relative homotopy commutes with transfinite
compositions of monomorphisms.
Lemma 5.1.7. Let f :X ! Y be a level trivial cofibration of symmetric spectra.
For every countable subspectrum C of Y there is a countable subspectrum D of Y
such that C D and D \ X ! D is a level trivial cofibration.
Proof.Let K L be a pair of pointed simplicial sets and v be a 0-simplex of K.
For n 1, let ssn(L; K; v) denote the relative homotopy set of the pair with the
null element as the basepoint (ignore the group structure when n 2). To ease
notation let ss0(L; K; v) be the pointed set ss0L=ss0K. The inclusion K ! L is a
weak equivalence if and only if ssn(L; K; v) = * for every v 2 K0 and n 0.
Now, construct a countable spectrum F C such that the map ss*(Cn; Cn\Xn; v) !
ss*(F Cn; F Cn \ Xn; v) factors through the basepoint * for every 0-simplex v of
Cn \ Xn and integer n 0. Since ss*(Y; X; v) = * and ss* commutes with filtered
colimits, for each homotopy class ff 2 ss*(Cn; Cn \ Xn; v) there is a finite si*
*mplicial
SYMMETRIC SPECTRA 51
subset Kff Yn such that ss*(Cn; Cn \ Xn; v) ! ss*(Kff[ Cn; (Kff[ Cn) \ Xn; v)
sends ff to the basepoint. Since Cn is countable, the set ss*(Cn; Cn \ Xn; v) *
*is
countable. Let Bn be the union of Cn with all the finite simplicial sets Kff. T*
*he
Bn are countable simplicial sets and generate a countable subspectrum F C of Y
having the desired property.
Repeat the construction to get a sequence of countable subspectra of Y :
C ! F C ! F 2C ! . .!.F nC ! : : :
Let D = colimnF nC. The spectrum D is countable. Since relative homotopy
commutes with transfinite compositions of monomorphisms, the set ss*(Dn; Dn \
Xn; v) has only one element. Therefore the inclusion Dn \ Xn ! Dn is a weak __
equivalence, and so D \ X ! D is a level equivalence. |__|
5.2. Stable cofibrations. The object of this section is to give a characterizat*
*ion
of stable cofibrations in Sp . To this end, we introduce the latching space.
__ __
Definition_5.2.1.Define_S to be the symmetric spectrum such that Sn = Sn for
n > 0 and S0 = *. The structure maps are the evident ones. Given_a symmetric
spectrum X, define the nth latching space, LnX, to be Evn(X ^ S).
__
There is a map of symmetric spectra i: S ! S which is the identity on positive
levels. This induces a natural transformation LnX ! Xn of pointed n simplicial
sets.
The following proposition uses a model structure on the category of pointed n
simplicial sets. A map f :X ! Y of pointed n simplicial sets is a n-fibration
if it is a Kan fibration of the underlying simplicial sets. Similarly, f is a *
*weak
equivalence if it is a weak equivalence of the underlying simplicial sets. The *
*map
f is a n-cofibration if it is a monomorphism such that n acts freely on the
simplices of Y not in the image of f. It is well-known, and easy to check, th*
*at
the n-cofibrations, the n-fibrations, and the weak equivalences define a model
structure on the category of pointed n-simplicial sets.
Proposition 5.2.2.A map f :X ! Y in Sp is a stable cofibration if and only if
for all n 0 the induced map Evn(f i): Xn qLnX LnY ! Yn is a n-cofibration.
Proof.Suppose first that Evn(fi) is a n-cofibration for all n. Suppose g :E ! B
is a level trivial fibration. We show that f has the left lifting property with*
* respect
to g by constructing a lift using induction on n. A partial lift defines a comm*
*utative
square
Xn qLnX LnY ----! En
?? ?
y ?y
Yn ----! Bn
Since the left vertical map is a n-cofibration and the right vertical map is a
trivial n-fibration, there is a lift in this diagram and so we can extend our p*
*artial
lift. Hence f has the left lifting property with respect to g, and so f is a s*
*table
cofibration.
To prove the converse, note that Evn is a left adjoint as a functor to pointe*
*d n
simplicial sets. Since the class of stable cofibrations is the class F I@-cof, *
*it suffices
to check that Evn(f i) is a n-cofibration for f 2 F I@. More generally, suppose
g :A ! B is a monomorphism of pointed simplicial sets. Since Fm g = S Gm g,
52 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
we have Fm g i = Gm g i, where the second is taken in S*. One can easily
check that Evn(Gm g i) is an isomorphism when n 6= m and is the map (n)+ ^ g __
when n = m. In both cases, Evn(Gm g i) is a n-cofibration, as required. |_*
*_|
5.3. Pushout smash product. In this section we consider the pushout smash
product in an arbitrary symmetric monoidal category and apply our results to Sp*
* .
We show that the projective level structure and the stable model structure on Sp
are both compatible with the symmetric monoidal structure. A monoid E in Sp
is called a symmetric ring spectrum, and is similar to an A1 -ring spectrum. Th*
*us,
there should be a stable model structure on the category of E-modules. Similarl*
*y,
there should be a model structure on the category of symmetric ring spectra and*
* the
category of commutative symmetric ring spectra. These issues are dealt with more
fully in [SS97] and in work in progress of the third author. Their work depends
heavily on the results in this section and in Section 5.5. The results of this *
*section
alone suffice to construct a stable model structure on the category of modules *
*over
a symmetric ring spectrum which is stably cofibrant. This section also contains
brief descriptions of two other stable model structures on Sp .
Definition 5.3.1.Let f :U ! V and g :X ! Y be maps in a symmetric
monoidal category C. The pushout smash product
f g :V ^ X qU^X U ^ Y ! V ^ Y:
is the natural map on the pushout defined by the commutative square
U ^ X -f^X---!V ^ X
? ?
U^g?y ?yV ^g
U ^ Y ----! V ^ Y:
f^Y
If C is a closed symmetric monoidal category,
Hom (f; g): Hom (V; X) ! Hom (U; X) xHom(U;Y )Hom(V; Y ):
is the natural map to the fiber product defined by the commutative square
*
Hom (V; X)- f---!Hom (U; X)
? ?
g*?y ?yg*
Hom (V; Y )----! Hom (U; Y ):
f*
Definition 5.3.2.A model structure on a symmetric monoidal category is called
monoidal if the pushout smash product f g of two cofibrations f and g is a
cofibration which is trivial if either f or g is.
In our situation, this is the correct condition to require so that the model *
*struc-
ture is compatible with the symmetric monoidal structure. Since the unit, S, is*
* cofi-
brant in symmetric spectra this condition also ensures that the symmetric monoi*
*dal
structure induces a symmetric monoidal structure on the homotopy category. For
a more general discussion of monoidal model structures, see [Hov97 ].
Recall, from Definition 3.3.6, the map of sets C (f; g).
SYMMETRIC SPECTRA 53
Proposition 5.3.3.Let f; g and h be maps in a closed symmetric monoidal cate-
gory C. There is a natural isomorphism
C (f g; h) ~=C (f; Hom (g; h))
Proof.Use the argument in the proof of Proposition 3.3.8. |__*
*_|
Proposition 5.3.4.Let I and J be classes of maps in a closed symmetric monoidal
category C. Then
I-cof J-cof (I J)-cof
Proof.Let K = (I J)-inj. By hypothesis, (I J; K) has the lifting property.
By adjunction, (I; Hom (J; K)) has the lifting property. By Proposition 3.2.8,
(I-cof; Hom (J; K)) has the lifting property. Then (J; Hom (I-cof; K)) has t*
*he
lifting property, by using adjunction twice. Thus (J-cof; Hom (I-cof; K)) has *
*the
lifting property, by Proposition 3.2.8. By adjunction, (I-cof J-cof; K) has th*
*e_
lifting property. So I-cof J-cof (I J)-cofand the proposition is proved. |__|
Corollary 5.3.5.For classes I, J and K in Sp , if I J K-cofthen I-cof
J-cof K-cof.
We now examine to what extent the pushout smash product preserves stable
cofibrations and stable equivalences. To do so, we introduce a new class of map*
*s in
Sp .
Definition 5.3.6.Let M be the class of monomorphisms in the category of sym-
metric sequences S*. A map of symmetric spectra is an S-cofibration if it is an
S M-cofibration. A symmetric spectrum X is S-cofibrant if the map * ! X is
an S-cofibration. A map is an S-fibration if it has the right lifting property *
*with
respect to every map which is both an S-cofibration and a stable equivalence.
S
Note that every stable cofibration is an S-cofibration, since F I@ = S nGnI*
*@.
On the other hand, by Proposition 2.1.9, every element of S M is a monomor-
phism, and so every S-cofibration is a level cofibration. There is a model str*
*uc-
ture on Sp , called the S model structure, where the cofibrations are the S-
cofibrations and the weak equivalences are the stable equivalences. The fibrati*
*ons,
called S-fibrations are those maps with the right lifting property with respect*
* to
S-cofibrations which are also stable equivalences. Every S-fibration is a stabl*
*e fi-
bration. This model structure will be used in a forthcoming paper by the third
author to put a model structure on certain commutative S-algebras.
We mention as well that there is a third model structure on Sp where the weak
equivalences are the stable equivalences, called the injective (stable) model s*
*truc-
ture. The injective cofibrations are the level cofibrations and the injective *
*stable
fibrations are all maps which are both injective fibrations and stable fibratio*
*ns. In
particular, the fibrant objects are the injective -spectra. The interested rea*
*der
can prove this is a model structure using the methods of Section 3.4, replacing*
* the
set I with the union of I and the countable level cofibrations.
Theorem 5.3.7. Let f and g be maps of symmetric spectra.
1. If f and g are stable cofibrations then f g is a stable cofibration.
2. If f and g are S-cofibrations then f g is an S-cofibration.
3. If f is an S-cofibration and g is a level cofibration, then f g is a level
cofibration.
54 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
4. If f is an S-cofibration, g is a level cofibration, and either f or g is a*
* level
equivalence, then f g is a level equivalence.
5. If f is an S-cofibration, g is a level cofibration, and either f or g is a*
* stable
equivalence, then f g is a stable equivalence.
Proof.Parts 1 through 4 of the Proposition are proved using Corollary 5.3.5.
Part 1: Let I = J = K = F I@. Then K-cofis the class of stable cofibrations.
We have a natural isomorphism
Fpf Fqg = Fp+q(f g)
for f; g 2 S*. By Proposition 1.3.4, f g is a monomorphism when f and g are.
Part 3 of Proposition 3.4.2 the shows that I J K-cof. Now use the corollary.
Part 2: Let I = J = K = S M (recall that M is the class of monomorphisms
in S*). By definition, K-cofis the class of S-cofibrations. For f and g in S*, *
*we
have a natural isomorphism
S f S g = S (f g)
where the first is taken in Sp and the second is taken in S*. In degree n,
_
(f g)n = (p+q)+ ^pxq (fp gq)
p+q=n
For f; g 2 M, each map fp gq is a monomorphism, so it follows that f g is a
monomorphism. Thus I J K-cof. Now use the corollary.
Part 3: Let I = S M and let J = K be the class of level cofibrations. By
Part 2 of Lemma 5.1.4, K-cof= K. For f 2 S* and g 2 Sp , we have a natural
isomorphism of maps of symmetric sequences
(S f) g = f g
where the first is taken in Sp and the second is taken in S*. We have seen
in the proof of part 2 that f g is a monomorphism of symmetric sequences if f
and g are monomorphisms. Hence I J K-cof. Now use the corollary.
Part 4: First assume g is a level trivial cofibration. Let I = S M and let J *
*= K
be the class of level trivial cofibrations. By Part 1 of Lemma 5.1.4, K = K-co*
*f.
Proposition 1.3.4 and the method used in the proof of part 2 imply that, if f a*
*nd
g are monomorphisms of symmetric sequences and g is a level equivalence, then
f g is a level equivalence. Recall that, for h 2 S* and g 2 Sp , we have a nat*
*ural
isomorphism of maps of symmetric sequences
(S h) g = h g
where the first is taken in Sp and the second is in S*. Hence I J K-cof.
Now use the corollary to prove part 4 when g is a level equivalence.
It follows that, for any injective spectrum E and an arbitrary S-cofibration *
*h,
the map Hom S(h; E) is an injective fibration. Indeed, if g is a level cofibrat*
*ion and
a level equivalence, Sp (g; Hom S(h; E)) ~=Sp (g h; E), and we have just seen
that g h is a level cofibration and a level equivalence, so (g h; E) has the *
*lifting
property.
Now suppose f is an S-cofibration and a level equivalence. Then the map
Hom Sp (f; E) is an injective fibration and a level equivalence. Indeed, we ha*
*ve
EvkHom Sp (f; E) = Map Sp (f ^ (S [k]+ ); E), by Remark 2.2.12. Since S
[k]+ is S-cofibrant, and f is both a level equivalence and a level cofibration,*
* we
have just proved that f ^ (S [k]+ ) = f (* ! S [k]+ ) is a level equivalence
SYMMETRIC SPECTRA 55
and a level cofibration. This shows that ss0Ev kHom Sp (f; E) is an isomorphism;
smashing with F0Sn and using a similar argument shows that ssn EvkHom Sp (f; E)
is an isomorphism.
Thus every level cofibration g has the left lifting property with respect to *
*the map
Hom Sp (f; E). By adjunction, f g and f (g j) where j :@[1]+ ! [1]+ is
the inclusion, have the extension property with respect to every injective spec*
*trum
E. It follows that E0(f g) is an isomorphism for every injective spectrum E and
hence that f g is a level equivalence.
Part 5: Because we are working in a stable situation, a level cofibration i: *
*X !
Y is a stable equivalence if and only if its cofiber Ci= Y=X is stably trivial.*
* The
map f g is a level cofibration by part (3). By commuting colimits, the cofiber*
* of
f g is the smash product Cf ^ Cg of the cofiber Cf of f, which is S-cofibrant,
and the cofiber Cg of g. Let E be an injective -spectrum. We will show that
Hom S(Cf ^ Cg; E) is a level trivial spectrum, and thus that Cf ^ Cg is stably
trivial.
First suppose that f is a stable equivalence. Then Hom S(Cf; E) is a level tr*
*ivial
spectrum which is also injective, by part 4 and the fact that Cf is S-cofibrant.
Therefore Hom S(Cf ^ Cg; E) ~=Hom S(Cg; Hom S(Cf; E)) is a level trivial spec-
trum, so Cf ^ Cg is stably trivial and thus f g is a stable equivalence.
Now suppose that g is a stable equivalence, so that Cg is stably trivial. By
adjunction Hom S(Cf ^ Cg; E) = Hom S(Cg; Hom S(Cf; E)). We claim that D =
Hom S(Cf; E) is an injective -spectrum. Indeed, we have already seen that D is
injective. From Remark 2.2.12, we have
Evn D ~=Map Sp (Cf ^ FnS0; E) ~=Map Sp (Cf; Hom S(FnS0; E))
Similarly, we have
1 1
(Ev n+1D)S ~=MapSp (Cf; Hom S(Fn+1S ; E)):
Since E is an -spectrum, (FnS0 ^ )*: Hom S(FnS0; E) -! Hom S(Fn+1S1; E)
is a level equivalence. Since E is injective, both the source and target are i*
*n-
jective, and so this map is1a simplicial homotopy equivalence by Lemma 3.1.6.
Hence Evn D -! (Ev n+1D)S is still a level equivalence, so D = Hom S(Cf; E)
is an injective -spectrum. Since Cg is stably trivial, Hom S (Cf ^ Cg; E) ~=
Hom S(Cg; Hom S(Cf; E)) is a level trivial spectrum, so Cf ^ Cg is stably trivi*
*al_
and thus f g is a stable equivalence. |__|
Corollary 5.3.8.The projective model structure and the stable model structure on
Sp are monoidal.
It also follows that the S model structure on Sp is monoidal, once it is pro*
*ven
to be a model structure. Neither the injective level structure nor the injectiv*
*e stable
structure is monoidal.
Adjunction then gives the following corollary.
Corollary 5.3.9.Let f and g be maps of symmetric spectra.
1. If f is a stable cofibration and g is a stable fibration, then Hom (f; g)*
* is
a stable fibration, which is a level equivalence if either f or g is a sta*
*ble
equivalence.
2. If f is a stable cofibration and g is a level fibration, then Hom (f; g) *
*is a level
fibration, which is a level equivalence if either f or g is a level equiva*
*lence.
56 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
3. If f is an S-cofibration and g is an S-fibration, then Hom (f; g) is an S-
fibration, which is a level equivalence if either f or g is a stable equiv*
*alence.
4. If f is an S-cofibration and g is an injective fibration, then Hom (f; g)*
* is
an injective fibration, which is a level equivalence if either f or g is a*
* level
equivalence.
Corollary 5.3.10.If X is an S-cofibrant symmetric spectrum, the functor X ^ -
preserves level equivalences and it preserves stable equivalences.
Proof.Part four of Theorem 5.3.7 implies that X ^ - preserves level trivial cof*
*i-
brations. Lemma 4.1.3 then implies that it preserves level equivalences, since *
*every
symmetric spectrum is level cofibrant. An arbitrary stable equivalence can be f*
*ac-
tored as a stable trivial cofibration followed by a level equivalence. Part fi*
*ve of
Theorem 5.3.7 implies that X ^ - takes stable trivial cofibrations to stable_eq*
*uiv-
alences. |__|
5.4. Proper model categories. In this section we recall the definition of a pro*
*per
model category and show that the stable model category of symmetric spectra is
proper.
Definition 5.4.1. 1.A model category is left proper if for every pushout square
A --g--!B
? ?
f?y ?yh
X ----! Y
with g a cofibration and f a weak equivalence, h is a weak equivalence.
2. A model category is right proper if for every pullback square
A ----! B
? ?
h?y ?yf
X ----!gY
with g a fibration and f a weak equivalence, h is a weak equivalence.
3. A model category is proper if it is both left proper and right proper.
The category of simplicial sets is a proper model category [BF78 ] (see [Hir9*
*7] for
more details). Hence the category of pointed simplicial sets and both level mod*
*el
structures on Sp are proper.
Theorem 5.4.2. The stable model category of symmetric spectra is proper.
Proof.Since every stable cofibration is a level cofibration, the stable model c*
*ategory
of symmetric spectra is left proper by part one of Lemma 5.4.3. Since every sta*
*ble
fibration is a level fibration, the stable model category of symmetric spectra_*
*is right
proper by part two of Lemma 5.4.3. |__|
Lemma 5.4.3.
1. Let
A --g--! B
? ?
f?y ?yh
X ----! Y
SYMMETRIC SPECTRA 57
be a pushout square with g a level cofibration and f a stable equivalence.*
* Then
h is a stable equivalence.
2. Let
A --k--! B
? ?
f?y ?yh
X ----!g Y
be a pullback square with g a level fibration and h a stable equivalence. *
*Then
f is a stable equivalence.
Proof.Part one: Let E be an injective -spectrum. Apply the functor MapSp (-; E)
to the pushout square. The resulting commutative square
Map (A; E) Map(g;E)------Map(B; E)
x x
Map(f;E)?? ??Map(h;E)
Map (X; E) ---- Map (Y; E)
is a pullback square of pointed simplicial sets with Map (f; E) a weak equivale*
*nce, by
Proposition 3.1.4. We claim that Map (g; E) is a Kan fibration. Indeed, let k :*
*E !
* denote the obvious map, and let c denote a trivial cofibration of pointed sim*
*plicial
sets. Then Map (g; E) = Map (g; k). We must show that (c; Map (g; k)) has the
lifting property. By Corollary 3.3.9, this is equivalent to showing that (c g*
*; k)
has the lifting property. But, by Proposition 1.3.4, c g is a level equivalenc*
*e and
a level cofibration. Since E is injective, it follows that (c g; k) has the l*
*ifting
property, and so Map (g; E) is a Kan fibration. By properness for simplicial se*
*ts,
Map (h; E) is a weak equivalence. It follows that h is a stable equivalence.
Part two: Let F be the fiber over the basepoint of the map g :X ! Y . Since
k is a pullback of g, F is isomorphic to the fiber over the basepoint of the map
k :A ! B. The maps X=F ! Y , A=F ! B and B ! Y are stable equivalences;
so A=F ! X=F is a stable equivalence. Consider the Barratt-Puppe sequence
(considered in the proof of Lemma 3.4.15)
F ! A ! A=F ! F ^ S1 ! A ^ S1 ! A=F ^ S1 ! F ^ S2
and the analogous sequence for the pair (F; X). Given an injective -spectrum E,
apply the functor E0(-), and note that E0(Z ^ S1) ~=ss1Map (Z; E) is naturally
a group. The five-lemma then implies that f ^ S1: A ^ S1 -! X ^ S1 is a stable *
* __
equivalence. Part two of Theorem 3.1.14 shows that f is a stable equivalence. *
* |__|
5.5. The monoid axiom. In [SS97], techniques are developed to form model cat-
egory structures for categories of monoids, algebras, and modules over a monoid*
*al
model category. One more axiom is required which is referred to as the monoid
axiom. In this section we verify the monoid axiom for symmetric spectra. The re-
sults of [SS97] then immediately give a model structure on symmetric ring spect*
*ra.
See also [Shi97] for more about symmetric ring spectra. After proving the monoid
axiom, we discuss the homotopy invariance of the resulting model categories of
modules and algebras.
58 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Let K denote the class in Sp consisting of all maps f ^ X, where f is a stab*
*le
trivial cofibration and X is some symmetric spectrum. The following theorem is
the monoid axiom for symmetric spectra.
Theorem 5.5.1. Transfinite compositions of pushouts of maps of K are stable
equivalences. That is, suppose ff is an ordinal and A: ff ! Sp is a functor wh*
*ich
preserves colimits and such that each map Ai-! Ai+1 is a pushout of a map of K.
Then the map A0 -!colimi 1 such that Xn -! Xn+1 induces an isomorphism sskXn -! ssk+1Xn+1
for all k ffn for sufficiently large n. Then X is semistable.
Proof.By Lemma 5.6.3, ssk(R1 KX)n ! ssk+1(R1 KX)n+1 is a monomorphism
between two groups which are isomorphic. In the first case these groups are fin*
*ite,
so this map must be an isomorphism. Hence R1 KX is an -spectrum, so X is
semistable.
For the second part we also show that R1 KX is an -spectrum. Since for fixed k
the maps ssk+iXn+i ! ssk+1+iXn+1+iare isomorphisms for large i, ssk(R1 KX)n_!
ssk+1(R1 KX)n+1 is an isomorphism for each k and n. |__|
SYMMETRIC SPECTRA 63
The next proposition shows that stable equivalences between semistable spectra
are particularly easy to understand.
Proposition 5.6.5.Let f :X -! Y be a map between two semistable symmet-
ric spectra. Then f is a stable equivalence if and only if it is a stable homo*
*topy
equivalence.
Proof.Every stable homotopy equivalence is a stable equivalence, by Theorem 3.1*
*.11.
Conversely, if f is a stable equivalence, so is Lf. Since stable equivalences b*
*etween
stably fibrant objects are level equivalences by Lemma 4.3.3, Lf is in particul*
*ar a
stable homotopy equivalence. Since X and Y are semistable, both maps X ! LX
and Y ! LY are stable homotopy equivalences. Hence f is a stable homotopy_
equivalence. |__|
6.Topological spectra
In this final section, we describe symmetric spectra based on topological spa*
*ces.
The advantage of symmetric spectra over ordinary spectra is that symmetric spec-
tra form a symmetric monoidal category, whereas spectra do not. Thus, we are
interested in a setting where symmetric spectra of topological spaces also form
a symmetric monoidal category. This requires that we begin with a symmetric
monoidal category of pointed topological spaces. The category Top* of all point*
*ed
topological spaces is not symmetric monoidal under the smash product; the assoc*
*ia-
tivity isomorphism is not always continuous. So we begin in Section 6.1 by desc*
*rib-
ing compactly generated spaces, a convenient category of topological spaces whi*
*ch
is closed symmetric monoidal. Then in Section 6.2 we define topological spectra,
and in Section 6.3 we discuss the stable model structure on topological symmetr*
*ic
spectra. In Section 6.4 we show that the stable model structure is monoidal and
proper. We also consider monoids and modules over them, though here our results
are much less strong than in the simplicial case. Indeed, in this regard, we do*
* not
know more than we do in a general monoidal model category, where we can appeal
to the results of [Hov98a ].
6.1. Compactly generated spaces. In this section, we describe a closed sym-
metric monoidal category of topological spaces. There are many different choices
for such a category [Wyl73 ], and any one of them suffices to construct topolog*
*ical
symmetric spectra. However, in order to get model categories of modules over a
monoid, one needs especially good properties of the underlying category of topo-
logical spaces. For this reason, we work in the category of compactly generated
spaces, for which the basic reference is the appendix of [Lew78 ]. This is the *
*same
category used in [LMS86 ] and [EKMM97 ]. We also discuss the model structure *
*on
the category of compactly generated spaces.
Definition 6.1.1.Suppose X is a topological space. A subset A of X is called
compactly open if, for every continuous map f :K -! X where K is compact
Hausdorff, f-1 (A) is open. We denote by kX the set X with the topology consist*
*ing
of the compactly open sets, and we say that X is a k-space if kX = X. We denote
the full subcategory of Top* consisting of pointed k-spaces by K*. The space X
is weak Hausdorff if for every continuous map f :K -! X where K is compact
Hausdorff, the image f(K) is closed in X. We denote the full subcategory of K*
consisting of weak Hausdorff spaces by T*, and refer to its objects as compactly
generated pointed spaces.
64 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Note that every locally compact Hausdorff space is compactly generated. The
functor k :Top* -! K* is a right adjoint to the inclusion functor. It follows t*
*hat
K* is bicomplete. Colimits are taken in Top*, so in particular a colimit of k-s*
*paces
is again a k-space. One must apply k to construct limits. Thus, the product in
K* is the topological space k(X x Y ). If X is locally compact Hausdorff, then
k(X x Y ) = X x Y .
One can easily see that limits of weak Hausdorff spaces are again weak Hausdo*
*rff.
Thus, T* is complete and the inclusion functor T* -!K* preserves all limits. Fr*
*eyd's
adjoint functor theorem then guarantees the existence of a left adjoint w :K* -*
*!T*
to the inclusion functor. The space wX is the maximal weak Hausdorff quotient of
X, but it is very hard to describe wX in general. The functor w creates colimits
in T*, so that T* is cocomplete. However, very often the colimit of a diagram i*
*n T*
is the same as the colimit of the diagram in K*, so that we never need to apply*
* w
in practice. This is the case for directed systems of injective maps [Lew78 , P*
*ropo-
sition A.9.3], and for pushouts of closed inclusions [Lew78 , Proposition A.7.5*
*]. In
particular, cofibrations are both injective and closed inclusions [Lew78 , Prop*
*osi-
tion A.8.2], so these remarks will apply to transfinite compositions and pushou*
*ts of
cofibrations.
Both the category K of unpointed k-spaces and the category T of unpointed com-
pactly generated spaces are closed symmetric monoidal under the k-space version
of the product [Lew78 , Theorem A.5.5]. The right adjoint of X x - is the funct*
*or
kC(X; -). Here C(X; Y ) is the set of continuous maps from X to Y . A subbasis
for the topology on C(X; Y ) is given by the sets S(f; U), where f :K -! X is a
continuous map, K is compact Hausdorff, and U is open in Y . A map g :X -! Y is
in S(f; U) if and only if (gOf)(K) U. If X is locally compact Hausdorff, then *
*this
topology on C(X; Y ) is the same as the compact-open topology. However, there
is no guarantee that C(X; Y ) is a k-space, even if X is locally compact Hausdo*
*rff,
and so we must apply k. If X and Y are both second countable Hausdorff, and X is
locally compact, then kC(X; Y ) = C(X; Y ) [Lew78 , p. 161]. Fortunately, C(X; *
*Y )
is weak Hausdorff as long as Y is [Lew78 , Lemma A.5.2], so kC(X; Y ) is already
weak Hausdorff and there is no need to apply w to get the closed structure in T.
The product induces a smash product X ^ Y = k(X x Y )=(X _ Y ) on K*.
Then K* is a closed symmetric monoidal category; the right adjoint of X ^ - is
kC*(X; -), k applied to the subspace of kC(X; Y ) consisting of pointed continu*
*ous
maps. The subcategory T* of K* is closed under the smash product [Lew78 , Lemma
A.6.2], so T* is also a closed symmetric monoidal category. In fact, the funct*
*or
w :K* -! T* is symmetric monoidal. This fact does not appear in [Lew78 ]; its
proof uses adjointness and the fact that kC*(X; Y ) is already weak Hausdorff i*
*f Y
is so.
We now discuss the geometric realization. One source for the geometric realiz*
*a-
tion is [GZ67 , III]. Note that the geometric realization of a finite simplicia*
*l set is
compact Hausdorff, and so is a k-space. Since the geometric realization is a l*
*eft
adjoint, it commutes with colimits, and so the geometric realization of any sim*
*pli-
cial set is a k-space. Also, the geometric realization of any simplicial set is*
* a cell
complex, so is Hausdorff. We can therefore consider the geometric realization a*
*s a
functor to T*. We denote the geometric realization by | - |: S* -!T*, and its r*
*ight
adjoint, the singular functor, by Sing:T* -!S*.
SYMMETRIC SPECTRA 65
The geometric realization has a number of properties not expected in a general
left adjoint. For example, it preserves finite limits [GZ67 , III.3]. Gabriel a*
*nd Zis-
man prove this using a different category of topological spaces, but their proo*
*f relies
on first working with certain finite simplicial sets and their geometric realiz*
*ations,
and then using the closed symmetric monoidal structure to extend to general sim-
plicial sets. Thus their argument works in any closed symmetric monoidal catego*
*ry
of topological spaces that contains finite CW complexes. It follows easily from
this that the geometric realization preserves the smash product, and so defines*
* a
symmetric monoidal functor | - |: S* -!T*.
The categories T*, K* and Top* are all model categories, and both w :T* -!K*
and the inclusion K* -!Top *are Quillen equivalences. The easiest way to descri*
*be
the model structure is to define f to be a weak equivalence (fibration) if and *
*only
Singf is a weak equivalence (fibration) of simplicial sets. Then f is a cofibra*
*tion
if and only if it has the left lifting property with respect to all trivial fib*
*rations.
Of course, the weak equivalences are the maps which induce isomorphisms on ho-
motopy groups at all possible basepoints. All inclusions of relative CW-complex*
*es
are cofibrations, and any cofibration is a retract of a transfinite composition*
* of in-
clusions of relative CW-complexes. In fact, the cofibrations are the class |I@|*
*-cof
and the trivial cofibrations are the class |I |-cof. Every topological space is*
* fibrant,
and the geometric realization is a Quillen equivalence. Both the geometric real*
*iza-
tion and the singular functor preserve and reflect weak equivalences, and the m*
*aps
| SingX| -!X and K -! Sing|K| are weak equivalences.
In proving the claims of the last paragraph, one needs the small object argum*
*ent
in each of the categories involved. The most obvious smallness statement is the
following which states that all spaces are small relative to inclusions.
Lemma 6.1.2. Suppose K is a topological space with cardinality fl, and suppose
X :ff -! Top* is a colimit-preserving functor, where ff is a fl-filtered ordina*
*l, and
suppose that each map Xfi-! Xfi+1is an inclusion. Then the map
colimffTop*(K; X) -!Top *(K; colimffX)
is an isomorphism.
Proof.One can verify using transfinite induction that each map Xfi-! colimffX is
an inclusion. Given a map f :K -! colimffX, there is a fi < ff such that f fact*
*ors
as a map of sets through Xfi, since ff is fl-filtered. But then f is automatic*
*ally_
continuous as a map to Xfisince the map Xfi-! colimffX is an inclusion. |_*
*_|
For Lemma 6.1.3 we need the class of inclusions to be preserved under various
constructions. Inclusions are not well behaved in general. However, in T*, clos*
*ed
inclusions are well behaved: they are closed under coproducts [Lew78 , Propo-
sition A.7.1(c)], pushouts [Lew78 , Proposition A.7.5], and transfinite composi-
tions [Lew78 , Proposition A.9.3]. Furthermore, the smash product preserves clo*
*sed
inclusions [Lew78 , Proposition A.7.3]. Given this, the proof of the Factoriza*
*tion
Lemma 3.2.11 applies to prove the following analogue.
Lemma 6.1.3. Let I be a set of closed inclusions in the category T*. There is
a functorial factorization of every map in T* as an I-cofibration followed by an
I-injective map.
66 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
The model structure on T* is monoidal and proper. The fact that it is monoidal
follows immediately from the corresponding fact for simplicial sets, since the *
*geo-
metric realization is monoidal and the generating cofibrations and trivial cofi*
*bra-
tions are in the image of the geometric realization. Properness for T* is a li*
*ttle
harder: right properness is formal, since all objects are fibrant, but left pro*
*perness
requires some care with local coefficients [Hir97, Proposition 11.1.11]. The mo*
*noid
axiom holds in both T* and K*. We get a model category of monoids in T*, but not
in K*. This is the reason we choose to use T* here and is explained in [Hov98a *
*].
6.2. Topological spectra. In this section we define the category of topological
symmetric spectra. The basic definitions are completely analogous to the defini*
*tions
for simplicial symmetric spectra.
The basic definitions for topological spectra and topological symmetric spect*
*ra
are straightforward. Let S1 2 T* be the unit circle in R2 with basepoint (1; 0).
Definition 6.2.1.A topological spectrum is
1. a sequence X0; X1; : :;:Xn; : :i:n T*; and
2. a pointed continuous map oe :S1 ^ Xn ! Xn+1 for each n 0.
The maps oe are the structure maps of the spectrum. A map of topological spectra
f :X ! Y is a sequence of pointed continuous maps fn :Xn ! Yn such that the
diagram
S1 ^ Xn --oe--!Xn+1
? ?
S1^fn?y fn+1?y
S1 ^ Yn --oe--!Yn+1
is commutative for each n 0. Let SpNT*denote the category of topological spect*
*ra.
Let Sp = (S1)^p with the (continuous) left permutation action of p.
Definition 6.2.2.A topological symmetric spectrum is
1. a sequence X0; X1; : :;:Xn; : :i:n T*;
2. a pointed continuous map oe :S1 ^ Xn ! Xn+1 for each n 0; and
3. a basepoint preserving continuous left action of n on Xn such that the
composition
oep = oe O (S1 ^ oe) O . .O.(Sp-1 ^ oe): Sp ^ Xn ! Xn+p;
i^oe
of the maps Si^ S1 ^ Xn+p-i-1 S---!Si^ Xn+p-i is p x n-equivariant
for p 1 and n 0.
A map of topological symmetric spectra f :X ! Y is a sequence of pointed contin-
uous maps fn :Xn ! Yn such that fn is n-equivariant and the diagram
S1 ^ Xn --oe--!Xn+1
? ?
S1^fn?y fn+1?y
S1 ^ Yn --oe--!Yn+1
is commutative for each n 0. Let SpT* denote the category of symmetric spectra.
SYMMETRIC SPECTRA 67
Example 6.2.3. The real bordism spectrum MO is the sequence of spaces MOn =
Sn ^On (EOn)+ . Here we require that EG be a functorial construction of a free
contractible G-space, for example Segal's construction, [S68]. Since there are *
*com-
muting left and right actions of On on Sn, the symmetric group acts on the left*
*, via
inclusion into On, on Sn and on MOn. One can check that the usual structure maps
S1^ MOn -! MOn+1 make MO into a topological symmetric spectrum. Similarly,
the other bordism spectra such as MU are naturally topological symmetric spectr*
*a.
One can check that these examples are even commutative symmetric ring spectra.
With these definitions, most of the basic results of Sections 1 and 2 go thro*
*ugh
without difficulty, replacing simplicial sets by topological spaces. That is, t*
*he cate-
gory of topological symmetric spectra is the category of S-modules, where S is *
*the
topological symmetric sequence (S0; S1; S2; : :):, and Sn = (S1)^n with the per-
mutation action of n. The category of topological symmetric spectra is a closed
symmetric monoidal category.
For topological symmetric spectra X and Y , we can form the mapping space
CSpT*(X; Y ) whose points are maps of symmetricQspectra from X to Y , topolo-
gized as k applied to the subspace of k kC*(Xn; Yn). We get the usual adjunct*
*ion
properties. Similarly, we can form function objects Hom (X; Y ) and Hom S(X; Y*
* ).
Indeed, given topological symmetric sequences X and Y , the nth space of the sy*
*m-
metric sequence Hom (X; Y ) is CT*(X GnS0; Y ). A point is a map of symmetric
sequences that raises degree by n; the leftover n action defines a symmetric se-
quence. Similarly, given topological symmetric spectra X and Y , the nth space
of the symmetric spectrum Hom S(X; Y ) is CSpT*(X ^ FnS0; Y ). A point is a
map of symmetric spectra that raises degree by n, and the topology is k applied
to the subspace topology inherited from CT*(X GnS0; Y ). The structure map
S1 ^ CSpT*(X ^ FnS0; Y ) -! CSpT*(X ^ Fn+1S0; Y ) takes (t; f) to oet(f), where
t 2 S1 and f is a map of degree n, and oet(f)(x) = f(oeX (t; x)) = oeY (t; f(x)*
*).
6.3. Stable model structure. In this section we discuss the stable homotopy
theory of topological symmetric spectra. Rather than proceeding analogously to
Section 3, we use the geometric realization and singular functor to lift the st*
*able
model structure on simplicial symmetric spectra to topological symmetric spectr*
*a.
Recall that both the geometric realization and the singular functor reflect a*
*nd
preserve weak equivalences, and that maps A -!Sing|A| and the counit | SingX| -!
X of the adjunction are both natural weak equivalences.
The following proposition is then immediate.
Proposition 6.3.1.The geometric realization induces a functor | - |: Sp -!
SpT* such that |X|n = |Xn|. The geometric realization is symmetric monoidal.
The singular functor induces a functor Sing:SpT* -! Sp , right adjoint to | - |,
such that (SingX)n = SingXn. Both the geometric realization and the singular
complex detect and preserve level equivalences, and the unit and counit maps of*
* the
adjunction are level equivalences.
We then make the following definition.
Definition 6.3.2.Suppose f :X -! Y is a map of topological symmetric spectra.
1. Define f to be a stable equivalence if Singf is a stable equivalence in Sp*
* .
2. Define f to be a stable fibration if Singf is a stable fibration in Sp .
68 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
3. Define f to be a stable cofibration if f has the left lifting property wit*
*h respect
to all stable trivial fibrations (i.e., maps which are both stable equival*
*ences
and stable fibrations).
We will prove that these structures form a model structure on SpT*, called the
stable model structure. Here are some of their basic properties.
Lemma 6.3.3. Suppose f :X -! Y is a map of topological symmetric spectra.
1. If f is a level equivalence, then f is a stable equivalence.
2. The map f is a stable trivial fibration if and only if f is a level trivia*
*l fibration.
3. The map f is a stable fibration if and only if f is a level fibration and *
*the
diagram
Xn ----! XS1n+1
?? ?
y ?y
Yn ----! YnS1+1
is a homotopy pullback square. In particular, a topological1symmetric spec*
*trum
X is stably fibrant if and only if the map Xn -! XSn+1is a weak equivalence
for all n 0. Such X are called -spectra.
4. The map f is a stable equivalence between -spectra if and only if f is a l*
*evel
equivalence between -spectra.
5. The map f is a stable fibration between -spectra if and only if f is a lev*
*el
fibration between -spectra.
6. The map f is a stable cofibration if and only if f 2 |F I@|-cof.
7. If f is a stable cofibration, then f is a level cofibration.
Proof.Part 1 follows from the corresponding statement for simplicial symmetric
spectra. Part 2 follows in the same way. For part 3, note that | - | commutes w*
*ith
smashing with S1, so, by adjointness, Sing(XS1n+1) is isomorphic to (SingXn+1)S*
*1.
Since Singcommutes with pullbacks and preserves and reflects weak equivalences,
the square in question is a homotopy pullback square if and only if Singf is a *
*stable
fibration. Parts 4 and 5 follow immediately from the corresponding statements f*
*or
simplicial symmetric spectra. For part 6, note that g is a stable trivial fibra*
*tion if
and only if Singg is in F I@-inj, which, by adjointness, holds if and only if g*
* is in
|F I@|-inj. The result follows. For part 7, note that level cofibrations are th*
*e class of
R(tK)-projective maps, just as in partS2 of Lemma 5.1.4, where tK is the class *
*of
trivial fibrations in T*, and R(tK) = n Rn(tK), and Rn :T* -!SpT* is the right
adjoint to the evaluation functor Evn. Thus, it suffices to show that each map *
*in__
|F I@| is a level cofibration, but this is clear. *
* |__|
To characterize the stable cofibrations, we can introduce the latching space *
*LnX
and the natural transformation i: LnX -! Xn just as we did in Section 5.2. In t*
*he
category of n-spaces, a n-cofibration is an equivariant map f :X -! Y which
has the left lifting property with respect to all equivariant maps that are bot*
*h weak
equivalences and fibrations. In particular, if g :X -! Y is an equivariant rela*
*tive
CW complex on which n acts cellularly and also acts freely on the cells not in *
*X,
then f is a n-cofibration.
SYMMETRIC SPECTRA 69
Lemma 6.3.4. A map f :X -! Y of topological symmetric spectra is a stable
cofibration if and only if for all n 0 the induced map Evn(f i): XnqLnX LnY -!
Yn is a n-cofibration.
The proof of this lemma is exactly the same as the proof of Proposition 5.2.2.
In order to prove that these classes define a model structure on SpT*, the fi*
*rst
thing we need is a version of the Factorization Lemma 3.2.11. The following
proposition is the analogue of Proposition 3.2.13, and its proof is similar, gi*
*ven
Lemma 6.1.2. Define the cardinality of a topological symmetric spectrum X to be
the cardinality of the disjoint union of its spaces Xn.
Proposition 6.3.5.Let X be a topological symmetric spectrum of cardinality fl.
Let ff be a fl-filtered ordinal and let D :ff -! SpT* be colimit-preserving ff-*
*indexed
diagram of level inclusions. Then the natural map
colimffSpT*(X; D) -!SpT*(X; colimffD)
is an isomorphism.
This proposition immediately leads to a factorization lemma for topological s*
*ym-
metric spectra, analogous to the Factorization Lemmas 3.2.11 and 6.1.3 for simp*
*li-
cial symmetric spectra and for compactly generated spaces.
Lemma 6.3.6. Let I be a set of level closed inclusions in SpT*. There is a fun*
*cto-
rial factorization of every map of topological symmetric spectra as an I-cofibr*
*ation
followed by an I-injective.
We can now establish the projective and stable model structures on SpT*. We
begin with the projective model structure.
Theorem 6.3.7. The stable cofibrations, level fibrations, and level equivalence*
*s de-
fine a model structure on SpT*, called the projective level structure. The geom*
*etric
realization | - |: Sp -! SpT* defines a Quillen equivalence between the projec*
*tive
level structure on Sp and the projective level structure on SpT*.
Proof.Certainly SpT* is bicomplete, and the retract and two out of three axioms*
* are
immediate consequences of the definitions. The lifting axiom for stable cofibra*
*tions
and level trivial fibrations follows from the definitions as well. Adjointness *
*implies
that the level trivial fibrations are the class |F I@|-inj. Part two of Lemma *
*6.3.3
then implies that the stable cofibrations are the class |F I@|-cof. Lemma 6.3.6*
* pro-
duces the required factorization into a stable cofibration followed by a level *
*trivial
fibration.
Adjointness also implies that the level fibrations are the class |F I |-inj. *
* Each
map of |F I | is a stable cofibration and a level equivalence, and in particula*
*r a level
trivial cofibration. Just as in part 1 of Lemma 5.1.4, the level trivial cofibr*
*ations
are the RK-projective maps. It follows that every map of |F I |-cofis a level t*
*rivial
cofibration. We knew already that each map of |F I |-cofis a stable cofibratio*
*n.
Thus, Lemma 6.3.6 gives us a factorization of any map into a stable cofibration*
* and
level equivalence followed by a level fibration.
We are left with the lifting axiom for stable cofibrations and level equivale*
*nces,
and level fibrations. Let f be a stable cofibration and level equivalence. Wr*
*ite
f = pg, where g 2 |F I |-cofand p is a level fibration. Then p is a level equiv*
*alence,
so f has the left lifting property with respect to p. It follows from Propositi*
*on 3.2.4
70 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
that f is a retract of g, so f has the left lifting property with respect to le*
*vel
fibrations.
The singular functor is obviously a right Quillen functor, and reflects all l*
*evel
equivalences. Now let K be a fibrant replacement functor on SpT*, obtained, for
example, from the Factorization Lemma 6.3.6 applied to |F I |. Then in order to
show that the pair (|-|; Sing) is a Quillen equivalence, it suffices by Lemma 4*
*.1.7 to
show that the map X -! SingK|X| is a level equivalence for all simplicial symme*
*tric
spectra X. However, the map X -! Sing|X| is a level equivalence. Since the map
|X| -! K|X| is a level equivalence, so is the map Sing|X| -! SingK|X|. By the
two out of three axiom, the composite X -! SingK|X| is a stable equivalence,_as
required. |__|
We do not know if the injective level structure on SpT* is a model structure.
Theorem 6.3.8. The stable model structure on SpT* is a model structure. We
have the following properties:
1. A map f 2 Sp is a stable equivalence if and only if |f| is a stable equiv*
*alence
in SpT*;
2. A map g 2 SpT* is a stable equivalence if and only if Singg is a stable
equivalence in Sp ; and
3. The geometric realization and singular functor define a Quillen equivalence
between the stable model structure on simplicial symmetric spectra and the
stable model structure on topological symmetric spectra.
Proof.The category SpT* is bicomplete, and the two out of three and retract
axioms are immediate consequences of the definitions. The stable trivial fibrat*
*ions
and the level trivial fibrations coincide, by Lemma 6.3.3. Thus the lifting and
factorization axioms for stable cofibrations follow from the corresponding axio*
*ms
in the projective level structure.
The stable fibrations form the class |J|-inj, where J is the set of generating
stable trivial cofibrations used in Section 3.4. In Proposition 6.3.11, we show*
* that
every map of |J|-cofis a stable cofibration and a stable equivalence. Hence the
Factorization Lemma 6.3.6 gives us the required (functorial) factorization of an
arbitrary map into a stable trivial cofibration followed by a stable fibration.
Only one lifting property remains. Suppose i is a stable trivial cofibration.*
* We
can factor i as i = qi0, where q is a stable fibration and i0 2 |J|-cof. By the
preceding paragraph, i0 is a stable equivalence. By the two out of three axiom,*
* q
is a stable equivalence. Thus i has the left lifting property with respect to q*
*, and
so the Retract Argument 3.2.4 implies that i is a retract of i0. Thus i 2 |J|-c*
*of,
and so i has the left lifting property with respect to every stable fibration p*
*. This
completes the proof that SpT* is a model category.
Now, by definition, Sing:SpT* -!Sp is a right Quillen functor, and f 2 SpT*
is a stable equivalence if and only if Singf is a stable equivalence in Sp . N*
*ow
let K be a fibrant replacement functor on SpT*, obtained, for example, from the
Factorization Lemma 6.3.6 applied to |J|. Then in order to show that the pair
(| - |; Sing) is a Quillen equivalence, it suffices by Lemma 4.1.7 to show that*
* the
map X -! SingK|X| is a stable equivalence for all simplicial symmetric spectra *
*X.
However, the map X -! Sing|X| is a level equivalence. Since the map |X| -!K|X|
is a stable equivalence, so is the map Sing|X| -! SingK|X|. By the two out of
three axiom, the composite X -! SingK|X| is a stable equivalence, as required.
SYMMETRIC SPECTRA 71
Finally, we show that | - | detects and preserves stable equivalences. Given a
map f :X -! Y , consider the commutative square below.
X --f--! Y
?? ?
y ?y
Sing|X|- ---! Sing|Y |
Sing|f|
The vertical maps are level equivalences. Now Sing|f| is a stable equivalence i*
*f and
only if |f| is a stable equivalence. Hence f is a stable equivalence if and_onl*
*y if |f|
is a stable equivalence. |__|
We still owe the reader a proof that the maps of |J|-cofare stable cofibratio*
*ns
and stable equivalences. We begin with the following lemma.
Lemma 6.3.9. Suppose f :A -!B is an element of J, and suppose the following
square is a pushout square in SpT*.
|A| --|f|--!|B|
? ?
r?y ?ys
C ----!g D
Then g is a stable equivalence and a stable cofibration.
Proof.It is clear that |f|, and hence g, is a stable cofibration. Note that |f|*
* is itself
a stable equivalence, since Sing|f| is level equivalent to the stable equivalen*
*ce f,
so is a stable equivalence.
We first suppose that r is a level cofibration. Consider the pushout square *
*in
Sp below.
Sing|A|-Sing|f|---!Sing|B|
? ?
Singr?y ?ys0
SingC ----! E
g0
Since Singpreserves level cofibrations (since it preserves injections), each ma*
*p in
this square is a level cofibration. Furthermore, Sing|f| is a stable equivalenc*
*e. By
the properness result of Theorem 5.4.2, we find that g0 is a stable equivalence.
There is a map h: E -! SingD such that hg0 = Singg, so it will suffice to prove
that h is a level equivalence.
To see this, we apply the geometric realization again to obtain a pushout squ*
*are
| Sing|A|||-Sing|f||-----!| Sing|B||
? ?
| Singr|?y ?y|s0|
| SingC| ----! |E|
|g0|
Again, each of the maps in this diagram is a level cofibration. There is an obv*
*ious
map from this pushout square to our original pushout square induced by the natu*
*ral
level equivalence | SingX| -!X. This map is a level equivalence everywhere exce*
*pt
72 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
possibly the bottom right. It follows from a general model category argument th*
*at
the induced map |E| -!D is a level equivalence. To see this, we can work one le*
*vel
at a time. Let D denote the category with three objects i; j; k and two non-ide*
*ntity
maps i -! j and i -! k. Then the pushout defines a functor TD*-! T* left adjoint
to the diagonal functor. There is a model structure on TD* such that a map f
from Xj- Xi -!Xk to Yj- Yi -!Yk is a weak equivalence (fibration) if and
only if each map fi, fj, and fk is a weak equivalence (fibration). Cofibrations*
* are
defined by lifting, and one can check that f is a cofibration if and only if fi*
*and the
induced maps XjqXi Yi-! Yj and Xk qXi Yi-! Yk are cofibrations. The diagonal
is easily seen to be a right Quillen functor, so the pushout is a left Quillen *
*functor,
and hence preserves weak equivalences between cofibrant objects. Hence the map
|E|n -! Dn is a weak equivalence as desired.
Since the level equivalence |E| -! D is the composite |E| Singh----!Sing|D| -*
*! D,
it follows that Singh is a level equivalence. Thus h is a level equivalence as *
*required,
and the proof is complete in the case where r is a level cofibration.
In the general case, we can factor r = r00O r0, where r0 is a stable cofibrat*
*ion
and r00is a stable trivial fibration, and hence a level equivalence. Then we ge*
*t a
commutative diagram
|A| --|f|--!|B|
? ?
r0?y ?ys0
0
C0 --g--! D0
? ?
r00?y ?ys00
C ----!g D
where each square is a pushout square. Since r0 is a stable cofibration, and in
particular a level cofibration, g0 is a stable equivalence (and a level cofibra*
*tion).
The map r00is a level equivalence. Since the model category T* is proper, it fo*
*llows
that s00is a level equivalence. The two out of three property then guarantees_t*
*hat_
g is a stable equivalence as required. |__|
Now we need to understand stable equivalences and colimits.
Lemma 6.3.10. Suppose ff is an ordinal, and X :ff -! SpT* is a functor such
that each map Xi -! Xi+1 is a stable cofibration and a stable equivalence. Then
the induced map X0 -!colimiXi is a stable cofibration and a stable equivalence.
Proof.It is easy to see that X0 -! colimiXi is a stable cofibration. Consider
the functor SingX :ff -! Sp . Then each map SingXi -! SingXi+1 is a level
cofibration and a stable equivalence. Let Y = colimiSingXi. Then the map
SingX0 -! Y is a stable equivalence, by Lemma 5.5.6. Thus it suffices to show
that the map Y -f!SingcolimiXi is a level equivalence.
Note that |Y | = colimi| SingXi|. There is a natural map |Y | -! colimiXi
induced by the level equivalences | SingXi| -!Xi. Since homotopy groups of topo-
logical spaces commute with transfinite compositions of cofibrations, and trans*
*finite
compositions of cofibrations of weak Hausdorff spaces are still weak Hausdorff,*
* it
follows that |Y | -! colimiXi is a level equivalence. Since this map factors as
SYMMETRIC SPECTRA 73
|Y | |f|-!| SingcolimiXi| -! colimiXi, we see that |f| is a level equivalence._*
*Hence
f is a level equivalence as required. |__|
Proposition 6.3.11.Every map in |J|-cofis a stable cofibration and a stable
equivalence.
Proof.Lemma 6.3.9 and Lemma 6.3.10 show that every colimit of pushouts of maps
of |J| is a stable cofibration and a stable equivalence. Now consider an arbitr*
*ary
map f in |J|-cof. The proof of the Factorization Lemma 6.1.3 shows that we can
factor f = qi, where i is a colimit of pushouts of maps of |J| and q 2 |J|-inj.*
* Since
f has the left lifting property with respect to q, the Retract Argument 3.2.4 s*
*hows
that f is a retract of i. Since i is a stable cofibration and a stable equivale*
*nce,_so
is f. |__|
We note that it is also possible to develop the homotopy theory of topological
symmetric spectra in a parallel way to our development for simplicial symmetric
spectra. However, there are some difficulties with this approach. We do not know
if the injective level structure is a model structure. Therefore, in this devel*
*opment,
a map f would be defined to be a stable equivalence if and only if Map (cf; X) *
*is a
weak equivalence for every -spectrum X, where cf is a cofibrant approximation
to f in the projective level structure.
6.4. Properties of topological symmetric spectra. In this section we show
that the stable model structure on topological symmetric spectra is monoidal and
proper. We also describe what we know about monoids and modules. Our results
here are much less complete than in the simplicial case, however.
We begin with an analogue of Theorem 5.3.7. We do not discuss S-cofibrations
in topological symmetric spectra.
Theorem 6.4.1. Let f and g be maps of topological symmetric spectra.
1. If f and g are stable cofibrations, so is f g.
2. If f is a stable cofibration and g is a level cofibration, then f g is a *
*level
cofibration.
3. If f is a stable cofibration and g is a level cofibration, and either f or*
* g is a
level equivalence, then f g is a level equivalence.
4. If f and g are stable cofibrations, and either f or g is a stable equivale*
*nce,
then f g is a stable equivalence.
Proof.The class of stable cofibrations is |F I@|-cof. Thus, by Corollary 5.3.5*
*, for
part 1 it suffices to show that |F I@| |F I@| consists of stable cofibrations.*
* But
|f| |g| = |f g|, so the result follows from the corresponding result for simp*
*licial
symmetric spectra.
For part 2, we use Corollary 5.3.5 with I = |F I@| and J = K the class of lev*
*el
cofibrations. Then J-cof= J, since J = R(tK)-inj, as we have already seen in
the proof of Lemma 6.3.3. Any map in I is of the form S f, where f is a level
cofibration of topological symmetric sequences. Then, just as in the proof of p*
*art 2
of Theorem 5.3.7, we have (S f) g = f g, where the second is taken in the
category of topological symmetric sequences. But in T* , we have
_
(f g)n = (p+q)+ ^pxq (fp gq):
p+q=n
74 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
Each map fp gq is a cofibration, since T* is a monoidal model category, f g is
a level cofibration as required.
For part 3, if g is a level equivalence, then we can use the same method as in
part 2 to conclude that f g is a level equivalence for f 2 |F I@|. Since the c*
*lass of
level trivial cofibrations is RK-inj(by the analogue of part 1 of Lemma 5.1.4),*
* this
completes the proof in case g is a level equivalence.
To do the case when f is a level equivalence, we use Corollary 5.3.5 with I =
|F I |, J the class of level cofibrations, and K the class of level trivial cof*
*ibrations.
Then I-cofis the class of stable cofibrations and level equivalences, as these *
*are the
cofibrations in the projective level structure. Then the proof that IJ K = K-c*
*of
proceeds just as in part 2.
Finally, for part 4 we use Corollary 5.3.5 with I = |F I@|, and J = K = |J|,
where the second J is the set of generating stable trivial cofibrations in Sp .*
* Then,
as in part 1, we use the fact that the geometric realization commutes with the *
*box_
product and the corresponding fact for simplicial symmetric spectra. |*
*__|
Corollary 6.4.2.The stable model structure and the projective level structure on
topological symmetric spectra are monoidal.
Note that Theorem 6.4.1 is not as strong as Theorem 5.3.7, since Theorem 5.3.7
implies that f g is a stable equivalence if f is a stable cofibration, g is a *
*level
cofibration, and either f or g is a stable equivalence. We do not know if this *
*is true
for topological symmetric spectra.
We now show that the stable model structure is proper.
Lemma 6.4.3. 1. Let
A --g--! B
? ?
f?y ?yh
X ----! Y
be a pushout square of topological symmetric spectra with g a level cofibr*
*ation
and f a stable equivalence. Then h is a stable equivalence.
2. Let
A --k--! B
? ?
f?y ?yh
X ----!g Y
be a pullback square of topological symmetric spectra with g a level fibra*
*tion
and h a stable equivalence. Then f is a stable equivalence.
Proof.Part 2 follows immediately from the fact that Singpreserves pullbacks, le*
*vel
fibrations, and stable equivalences, the corresponding fact for simplicial symm*
*etric
spectra 5.4.3, and the fact that Sing detects stable equivalences. Part 1 would
also follow in the same way if Singpreserved level cofibrations and pushouts. T*
*he
functor Sing does preserve level cofibrations, since it preserves monomorphisms.
But Singonly preserves pushouts up to level equivalence. More precisely, suppose
C is the pushout of Singg and Singf in the diagram in part 1. Then there is a m*
*ap
C -j!SingY , and we claim this map is a level equivalence. To prove this, it su*
*ffices
SYMMETRIC SPECTRA 75
to prove that |j| is a level equivalence. Since the geometric realization does *
*preserve
pushouts, it suffices to verify that the map |C| -!Y induced by the adjunction *
*level
equivalences | SingD| -!D for D = A; X and B is a level equivalence. We can work
one level at a time. We are then reduced to showing that, if we have a diagram
Z ---- X --f--! Y
?? ? ?
y ?y ?y
Z0 ---- X0 ----! Y 0
f0
of topological spaces, where f and f0are cofibrations and the vertical maps are*
* weak
equivalences, the induced map W -! W 0on the pushouts is a weak equivalence.
This statement is actually true in any left proper model category, and in parti*
*cular_
holds in Top*. A proof of it appears as [Hir97, Proposition 11.3.1]. *
* |__|
Theorem 6.4.4. The stable model structure on topological symmetric spectra is
proper.
Proof.This now follows immediately from Lemma 6.4.3. |___|
We now discuss monoids and modules in topological symmetric spectra. Here we
run into a serious problem: we do not know whether the monoid axiom is true or
not. The general question of how to cope with monoids and modules in a monoidal
model category when the monoid axiom may not hold is considered in [Hov98a ],
where special attention is given to the example of topological symmetric spectr*
*a.
In particular, the following theorem is proved in [Hov98a ].
Theorem 6.4.5. Suppose R is a monoid in SpT*.
1. There is a monoid R0which is stably cofibrant in SpT* and a level equivale*
*nce
and homomorphism f :R0-! R.
2. If R is stably cofibrant, there is a model structure on the category of R-*
*modules
where a map is a weak equivalence or fibration if and only if it is a stab*
*le
equivalence or stable fibration in SpT*.
3. If R and R0are stably cofibrant in SpT* and f :R -!R0is a homomorphism
of monoids and a stable equivalence, then f induces a Quillen equivalence
between the corresponding module categories.
Note that, given the good properties of closed inclusions in T*, part 2 of Th*
*eo-
rem 6.4.5 is an immediate corollary of the fact that topological symmetric spec*
*tra
form a monoidal model category. Also, the proof of part 3 of Theorem 6.4.5 is q*
*uite
similar to the proof of Lemma 5.5.7.
We interpret this theorem as asserting that, in order to study modules over R,
one must first replace R by a stably cofibrant monoid R0.
Since we do not know whether the monoid axiom is true in topological symmetric
spectra, we cannot expect a model category of topological symmetric ring spectr*
*a.
We do have the following result, also taken from [Hov98a ].
Theorem 6.4.6. Let M(SpT*) denote the category of monoids in SpT* and ho-
momorphisms. Let Ho M(SpT*) denote the quotient category of M(SpT*) obtained
by inverting all homomorphisms which are stable equivalences in SpT*. Then the
category Ho M(SpT*) exists and is equivalent to the homotopy category of monoids
of simplicial symmetric spectra.
76 MARK HOVEY, BROOKE SHIPLEY, AND JEFF SMITH
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Department of Mathematics, Wesleyan University, Middletown, CT
E-mail address: hovey@member.ams.org
Department of Mathematics, University of Chicago, Chicago, IL
E-mail address: bshipley@math.uchicago.edu
Department of Mathematics, Purdue University, West Lafayette, IN
E-mail address: jhs@math.purdue.edu