A1local symmetric spectra
J.F. Jardine1
Mathematics Department
University of Western Ontario
London, Ontario N6A 5B7
Canada
Introduction
This paper is about importing the stable homotopy theory of symmetric spectra
[4] and more generally presheaves of symmetric spectra [8] into the MorelVoevo*
*dsky
stable category [9], [11], [12]. Loosely speaking, the latter is the result of *
*formally
inverting the functor X 7! T ^X on the category of pointed simplicial presheave*
*s on
the smooth Nisnevich site of a field within the MorelVoevodsky A1local homoto*
*py
theory, where T is defined to be the quotient of schemes A1=(A1  0). The Morel
Voevodsky stable homotopy theory is exotic in at least two ways: it lives with*
*in
a localized homotopy theory of simplicial presheaves, and the object T is not a
circle in any sense, but is rather weakly equivalent within the A1local theory*
* to
an honest suspension S1 ^ Gm of the scheme underlying the multiplicative group.
Smashing with T is thus a combination of topological and geometric suspensions.
The MorelVoevodsky stable category is fundamental for Voevodsky's proof of t*
*he
Milnor Conjecture [11]. It arises from a suitable notion of stable equivalence,*
* sub
sumed by a proper closed simplicial model structure on the category of presheav*
*es of
T spectra on a smooth Nisnevich site. A presheaf of T spectra X consists of p*
*ointed
simplicial presheaves Xn , n 0, together with bonding maps T ^ Xn ! Xn+1 .
A symmetric object in this category, or rather a presheaf of symmetric T spe*
*ctra,
is a presheaf of T spectra Y , equipped with symmetric group actions n x Y n !
Y n in all levels such that all composite bonding maps T ^p^ Xn ! Xp+n are
(pxn)equivariant. The main new results of this paper assert that the category *
*of
presheaves of symmetric T spectra carries a notion of stable equivalence withi*
*n the
A1local theory which is part of a proper closed simplicial model structure (Th*
*eorem
4.18), and such that the forgetful functor to presheaves of T spectra induces *
*an
equivalence of the stable homotopy category for presheaves of symmetric T spec*
*tra
with the MorelVoevodsky stable category (Theorem 5.14). This collection of res*
*ults
gives a category which models the MorelVoevodsky stable category, and also has
a symmetric monoidal smash product.
___________________________1
Partially supported by NSERC.
1
These results are simple enough to state, but a bit complicated to demonstrate
in that their proofs involve some fine detail from the construction of the More*
*l
Voevodsky stable category. It was initially expected, given the experience of *
*[8],
that the passage from presheaves of T spectra to presheaves of symmetric T sp*
*ectra
would be essentially axiomatic, following the lines of the original proof of [4*
*]. This
remains true in a gross sense, but many of the steps in the proofs of [4] and [*
*8] involve
standard results from stable homotopy theory which cannot be taken for granted
in the A1local context. In particular, the construction of the MorelVoevodsky
stable category is quite special: one proves it by verifying the BousfieldFrie*
*dlander
axioms A4  A6 [1], but the proofs of these axioms involve Nisnevich descent in*
* a
rather nontrivial way, and essentially force one to introduce variants of the *
*notion
of what I call a pseudoflasque simplical presheaf. The class of pseudoflasque
simplicial presheaves contains all fibrant objects, but is also closed under fi*
*ltered
colimit and the "T loop" functor; the key technical point is that these constr*
*uctions
also preserve pointwise weak equivalences.
To illustrate the difficulty, there is a natural level fibrant model : X ! JX
of a presheaf of T spectra X in the A1local theory: one inductively construc*
*ts
A1trivial cofibrations Xn ! JXn where JXn is fibrant, just as in the Bousfield
Friedlander paper [1]. The T loops functor X 7! T X is right adjoint to smashi*
*ng
with T , and is defined by internal hom. There is finally a presheaf of T spe*
*ctra
QT X which is defined levelwise by setting QT Xn to be the filtered colimit of *
*the
system
2oe*
Xn oe*!T Xn+1 Toe*!2TXn+2 T! . .;.
and there are natural maps
X ! JX j!QT JX:
The catch is that the BousfieldFriedlander script (particularly axiom A5) requ*
*ires
one to show that this composite map induces an A1equivalence in all levels upon
applying the composite functor QT J. This is equivalent to the assertion that Q*
*T :
QT QT JX ! QT JQT JX is an A1equivalence in all levels, and a naive approach
would be to try to show that T is a level A1equivalence. More properly (for
an inductive argument), one would want to show that the functor T preserves
A1equivalences of pointed simplicial presheaves which are globally fibrant for*
* the
Nisnevich topology in the traditional sense ... but nobody knows how to do this.
This is where Nisnevich descent comes in: it implies that the map : QT JX !
JQT JX is a pointwise weak equivalence in all levels. But now all objects in si*
*ght
are pseudoflasque and T is compact in a suitable sense, so nT is a pointwise w*
*eak
equivalence in all levels, and the difficulty is overcome. The problem about t*
*he
T loops functor T and the Nisnevich descent trick involved in its solution hav*
*e no
analogues in the general stable homotopy theory of presheaves of spectra.
2
There is a satisfying theory of pseudoflasque simplicial presheaves and comp*
*act
objects _ the latter are closed under finite smash product and homotopy cofibre,
and include all schemes and finite simplicial sets. The upshot is that the T th*
*at we
have been referring to belongs to a broader class of compact objects which incl*
*udes
finite simplicial sets, and for compact T the category of presheaves of T spec*
*tra on
a smooth Nisnevich site has a proper closed simplicial model structure associat*
*ed
to an adequate notion of stable equivalence. Explicitly, a map g : X ! Y of
presheaves of T spectra is a stable equivalence if it induces a level A1equiv*
*alence
g* : QT JX ! QT JY . These ideas are the subject of the first two sections of t*
*his
paper, and the main result of Section 2 is Theorem 2.11, which asserts the exis*
*tence
of the proper closed simplicial model structure.
Theorem 2.11 is proved without reference to stable homotopy groups. This is
achieved in part by using an auxilliary closed model structure for presheaves o*
*f T 
spectra, for which the cofibrations (respectively weak equivalences) are maps w*
*hich
are cofibrations (respectively A1equivalences) in each level. The fibrant obje*
*cts for
the theory are called injective objects, and one can show (Corollary 2.12) that*
* naive
homotopy classes of maps taking values in objects W which are both injective and
stably fibrant for the theory detects stable equivalences. This idea was lifted*
* from
[4], and appears again for presheaves of symmetric T spectra in Section 4. It *
*is an
important technical device if one does not assume that the compact object T has
a coH structure.
The coHstructure of the original object
T = A1=(A1  0) ' S1 ^ Gm
becomes important in the remaining sections. It is crucial for the development *
*of the
stable homotopy theory of presheaves of symmetric T spectra (eg. Proposition 4*
*.14,
proof of Theorem 4.18) to know that fibre sequences and cofibre sequences coinc*
*ide
up to stable equivalence _ this is the first major result of Section 3 (Lemma 3*
*.12,
Corollary 3.13). The section closes by proving the assertion (Lemma 3.14, Corol*
*lary
3.17) that the functors X 7! X ^ T and Y 7! T Y are inverse to each other on the
stable category. The method of proof involves long exact sequences in presheaves
of weighted stable homotopy groups. These groups were introduced in [11]; the
construction given here depends on knowing that a presheaf of T spectra X is a
piece of a type of asymmetric bispectrum object for which one suspends by the
simplicial circle S1 in one direction and by the scheme Gm in the other.
The last two sections of this paper contain the main results: the proper clos*
*ed
simplicial model structure for stable equivalence of presheaves of symmetric T 
spectra is Theorem 4.18, and the equivalence of stable categories is Theorem 5.*
*14.
With all of the material in the previous sections in place, and subject to bein*
*g careful
about the technical difficulty underlying the stability functor for the categor*
*y of
presheaves of T spectra that is discussed above, the derivation of the proper *
*closed
3
simplicial model structure for presheaves of symmetric T spectra follows the m*
*ethod
developed in [4] and [8]. The demonstration of the equivalence of stable catego*
*ries
is also by analogy with the methods of those papers, but one has to be a bit mo*
*re
careful again, so that it is necessary to discuss presheaves of T bispectra in*
* a limited
way.
There is an extra bit of geometry required for Sections 4 and 5 (so that stab*
*le
homotopy groups of presheaves of T bispectra behave well _ see the development
preceding Lemma 5.2), in that one has to know that the obvious action of the cy*
*clic
permutation of order 3 induces the identity map on the 3fold smash T ^3 in the
A1local homotopy category. This result has been announced by Voevodsky, for
example in [12], and appears here as Lemma 4.7; the proof involves an old and
simple idea from [6].
I have been concealing some notational complexity. In particular, the A1loc*
*al
theory for simplicial presheaves on a smooth Nisnevich site is constructed by f*
*or
mally inverting some (really, any) rational point f : * ! A1 of the affine line*
* A1
in the homotopy category of simplicial presheaves _ this is done along the line*
*s of
either [2] of [9]. It is traditional in homotopy theory to say that the locali*
*zation
theory arising from formally inverting a cofibration f is the flocal theory, a*
*nd one
speaks of fequivalences and ffibrations in the same way. This notational conv*
*en
tion pervades this paper: the A1local theory is called the flocal theory alm*
*ost
everywhere, with exceptions inserted merely for pedagogical emphasis.
Secondly, all references to "the" Nisnevich site are a bit bogus. There are *
*no
big sites in this paper _ all sites are assumed to be small. One achieves this*
* for
geometric sites by imposing a bound by a fixed infinite cardinal on all objects*
*, which
cardinals are large enough to catch particular groupings of schemes of interest,
according to methods developed in [5]. In particular, passage between one big
cardinal and another affects neither cohomology nor homotopy type. Typically,
one assumes that ff is an infinite cardinal which is an upper bound for the set*
* of
morphisms of a site, and then choose other cardinals and such that = 2 and
> 2ffto make the localization theories work. This is particularly important for
the construction of controlled stably fibrant models that are required to const*
*ruct
finjective and stably ffibrant presheaves of symmetric T spectra.
This work owes an enormous debt to the work of Fabien Morel, Jeff Smith and
Vladimir Voevodsky, and to conversations with all three; I would like to thank
them. In particular, many of the central results of the first three sections o*
*f this
paper were announced in some form in [11], while the Nisnevich descent technique
that is so important here was brought to my attention by Morel, and appears in
[9].
The conversations that I speak of took place at a particularly stimulating me*
*et
ing on the homotopy theory of algebraic varieties at the Mathematical Sciences
Research Institute in Berkeley in May, 1998. The idea for this project was esse*
*n
4
tially conceived there. The appendix of the paper was mostly written a few weeks
prior during a visit to Universite Paris VII. I thank both institutions for the*
*ir
hospitality and support.
1. Preliminaries.
1.1. A1local homotopy theory.
We shall assume throughout this paper that f is a rational point f : * ! A1 of
the affine line A1 in the category of smooth schemes (Smk)Nis of finite type o*
*ver
a field k, equipped with the Nisnevich topology. The empty scheme ; is a member
of this category, and it represents the empty presheaf ; on (Smk)Nis.
The localization theory arising from formally inverting the map f in the cate*
*gory
of simplicial presheaves on (Smk)Nis is usually called the A1local homotopy t*
*heory
for the field k. I shall refer to it as the flocal theory, to make the notatio*
*n easier
to deal with.
In the case of interest, one says that a simplicial presheaf X on the Nisnevi*
*ch
site is flocal if it is globally fibrant in the usual sense, and has the right*
* lifting
property with respect to all simplicial presheaf inclusions
(A1 x A) [A B (f;j)!A1 x B
arising from f : * ! A1 and all cofibrations j : A ! B. A simplicial presheaf m*
*ap
g : X ! Y is said to be an flocal equivalence if it induces a weak equivalenc*
*e of
simplicial sets
g* : hom (Y; Z) ! hom (X; Z)
in function complexes for every flocal object Z. A map p : U ! V is an ffibra*
*tion
if it has the right lifting property with respect to all maps which are simulta*
*ne
ously flocal equivalences and cofibrations. The homotopy theory arising from t*
*he
following theorem is the A1local homotopy theory of Morel and Voevodsky:
Theorem 1.1. The category SPre (Smk)Nis of simplicial presheaves of simplicial
presheaves on the smooth Nisnevich site of a field, together with the classes of
cofibrations, flocal equivalences and ffibrations, satisfies the axioms for a*
* proper,
closed simplicial model category.
The simplicial structure is the standard one for simplicial presheaves: the fun*
*ction
complex hom (X; Y ) for simplicial presheaves X and Y has nsimplices consist*
*ing
of all simplicial presheaf maps X x n ! Y . Most of Theorem 1.1 is derived in
[2], meaning that all except the properness assertion is proved there. Morel a*
*nd
Voevodsky demonstrate the Theorem 1.1 in [9] _ an alternative proof of properne*
*ss
appears in the Appendix of this paper. The proofs in [2] and the Appendix hold *
*for
arbitrary choices of rational point * ! A of any simplicial presheaf on any sma*
*ll
Grothendieck site C.
5
At that level of generality, suppose ff is an infinite cardinal which is an u*
*pper
bound for the cardinality of the set Mor (C) of morphisms of C. As before, pick*
* a
rational point f : * ! A, and suppose that A is ffbounded in the sense that all
sets of simplices of all sections A(U) have cardinality bounded above by ff.
Pick cardinals and such that
= 2 > > 2ff:
In [2] it is shown that there is a functor X 7! LX defined on simplicial preshe*
*aves
X together with a natural transformation jX : X ! LX which is an ffibrant
model for X, such that the following properties hold:
L1: L preserves weak equivalences.
L2: L preserves cofibrations.
L3: Let fi be any cardinal with fi ff. Let {Xj} be the filtered system of
subobjects of X which are fibounded. Then the map
lim!L(Xj) ! LX
j
is an isomorphism.
L4: Let fl be an ordinal number of cardinality strictly greater than 2ff. L*
*et
X : fl ! S be a diagram of cofibrations so that for all limit ordinals s*
* < fl
the induced map
lim!X(t) ! X(s)
t 2ffand set = 2 . The axioms sE1  sE7 of [2] and
their consequences apply to categories of presheaves of T spectra. We verify *
*the
bounded cofibration axiom sE7; the remaining axioms are easily verified, giving
statement (2).
Recall that the classes of cofibrations and fequivalences of simplicial pres*
*heaves
on (Smk)Nis together satisfiy the bounded cofibration condition for the cardin*
*al
in the sense that given a diagram
Xy
(2.2) i
u
A y________wjY
such that the cofibration i is an fequivalence and the subobject A of Y is 
bounded, there is a bounded suboject B of Y with A B, with B \ X ,! B an
fequivalence.
Suppose now that the objects and maps of diagram (2.2) are in the category of
presheaves of T spectra, where i is a level fequivalence and a level cofibrat*
*ion and
A is bounded. There is a simplicial presheaf B0 with A0 B0 Y 0 such that
B0 is bounded and the cofibration B0 \ X0 ,! B0 is an fequivalence. Write j0
for the inclusion B0 ,! Y 0and use the diagram
T ^ A0 ________wT ^ B0
  0
oe oe . (T ^ j )
u u
A1 ____________wjY 1
__1
to show that there is a bounded subobject A Y 1such that the map
A1 [T^A0 T ^ B0 ! Y 1
__1 __1
factors through A . There is a bounded subobject B1 Y 1with A B1 such
that the cofibration B1 \ X1 ,! B1 is an fequivalence. This is the beginning of
18
an inductive construction which produces a bounded subobject B of the presheaf
of spectra Y with A B such that the level cofibration B \ X ,! B is a level
equivalence.
Insofar as the factorization axiom CM5 in part (2) of Lemma 2.1 is covertly
proved by using a small object argument, there is a natural finjective model c*
*on
struction: there is a natural map of presheaves of T spectra iX : X ! IX, such*
* that
iX is a level cofibration and a level fequivalence, and IX is finjective. Mo*
*re gen
erally, any level fequivalence X ! Y with Y finjective is said to be an finj*
*ective
model for X.
There is a natural level ffibrant model jX : X ! JX, meaning that jX is
a cofibration and a level fequivalence and JX is level ffibrant. This can be
constructed directly from the small object arguments for the flocal theory, or*
* by
using the controlled ffibrant object construction X 7! LX of [2].
The smooth Nisnevich site (Smk)Nis is a geometric site consisting of schemes
and all their subschemes, and has a topology which is at least as fine as the Z*
*ariski
topology.
I say that a simplicial presheaf X on (Smk)Nis is fpseudoflasque if
(1) X is pseudoflasque, and
(2) every map X(U) ! (A1xU) induced by the projection of schemes A1xU !
U is a weak equivalence of simplicial sets.
Every ffibrant (or flocal) simplicial presheaf on (Smk)Nis is fpseudoflasq*
*ue,
and the class of fpseudoflasque simplicial presheaves is closed under filtere*
*d col
imits.
A pointed simplicial presheaf T on the smooth Nisnevich site is said to be f
compact if the following conditions hold:
C1: All inductive systems Y1 ! Y2 ! : :o:f pointed simplicial presheaves ind*
*uce
isomorphisms
Hom *(T; lim!Yi) ~=lim!Hom *(T; Yi):
i i
C2: If X is fpseudoflasque, then so is Hom *(T; X).
C3: The functor Hom *(T; ) takes pointwise weak equivalences of fpseudo
flasque simplicial presheaves to pointwise weak equivalences.
The following result generates examples of fcompact simplicial presheaves:
Lemma 2.3.
(1) All pointed schemes U in the underlying site (Smk)Nis are fcompact.
(2) All finite pointed simplicial sets K are fcompact.
19
(3) If A ,! B is an inclusion of schemes, then the quotient B=A is fcompact.
(4) If S and T are fcompact, then S _ T and S ^ T and are fcompact.
(5) If g : S ! T is a map of fcompact simplicial presheaves, then the point*
*ed
mapping cylinder Mg and the homotopy cofibre Cg are fcompact.
Proof: For (1), we know that there is an isomorphism
Hom (U; X)(V ) ~=X(U x V )
and so the functor X 7! Hom *(U; X) preserves filtered colimits of simplicial
presheaves. It follows as well that all maps
Hom (U; X)(V ) ! Hom (U; X)(V x A1)
induced by projection are weak equivalences of simplicial sets (see Corollary 1*
*.8).
There is a fibre sequence
(2.4) Hom *(U; X) ! Hom (U; X) ! Hom (*; X)
if X is pseudoflasque. It follows that Hom *(U; X) is fpseudoflasque if X i*
*s f
pseudoflasque and U is a scheme (see Corollary 1.9). The functor X 7! Hom (U;*
* X)
preserves pointwise weak equivalences of simplicial presheaves; use the fibre s*
*e
quence (2.4) to show that the functor X 7! Hom *(U; X) preserves pointwise weak
equivalences of fpseudoflasque simplicial presheaves.
Statement (2) is proved by first observing that there is a natural isomorphism
Hom *(K; X) ~=hom *(K; X):
The functor X 7! hom *(K; X) plainly preserves filtered colimits since K is a *
*finite
simplicial set. This immediateley gives C3, and C2 follows from Lemma 1.10, and
the functor X 7! hom *(K; X) preserves pointwise weak equivalences of pointed
presheaves of Kan complexes.
Statement (3) is a consequence of Lemma 1.7, and smash product part of state
ment (4) is an adjointness argument.
Suppose that X is fpseudoflasque. The diagram
S _ S _________wS _ T
 
 
u u
S ^ 1+_________w Mg
20
that defines the pointed mapping cylinder Mg induces a pullback diagram
Hom *(Mg; X) _________wHom *(S ^ 1+; X)
(2.5)  
u u
Hom *(S _ T; X) _________wHom *(S _ S; X)
and the map
Hom *(S ^ 1+; X) ! Hom *(S _ S; X)
is pseudoflasque, by the pointed version of Lemma 1.10. Hom *(Mg; X) is there*
*fore
pseudoflasque. The composite
Hom *(Mg; X) ! Hom *(S _ T; X) ! Hom (T; X)
is also pseudoflasque, and so the pointwise homotopy fibre Hom *(Cg; X) is ps*
*eudo
flasque. The objects other than Hom *(Mg; X) in the pointwise fibre square (2.*
*5)
take the projections U xA1 ! U to weak equivalences. Properness for simplicial *
*sets
therefore implies that the simplicial presheaves Hom *(Mg; X) and Hom *(Cg; X)
are fpseudoflasque. One shows similarly that the functors Hom *(Mg; ) and
Hom *(Cg; ) preserve pointwise weak equivalences of fpseudoflasque objects.
Both functors preserve filtered colimits, since they are built in finitely many*
* steps
from functors that do the same. We have proved statement (5).
Remark 2.6. One can show that statement (3) of Lemma 2.3 follows from state
ment (5), but the presented proof is easier. Statement (3) implies that the Mor*
*el
Voevodsky object T = A1=(A1  0) is fcompact.
Suppose henceforth that T is an fcompact pointed simplicial presheaf on the
smooth Nisnevich site (Smk)Nis.
The T loops functor T Y is defined for pointed simplicial presheaves Y in te*
*rms
of internal hom by
T Y = Hom *(T; Y ):
The T loops functor is right adjoint to smashing with T , and so the bonding m*
*aps
oe : T ^ Xn ! Xn+1 of a presheaf of T spectra X can equally well be specified*
* by
their adjoints oe* : Xn ! T Xn+1 . Just as ordinary stable homotopy theory, the*
*re
is a "fake T loop spectrum" T X with
(T X)n = T (Xn );
21
and having bonding maps adjoint to the maps
T (oe*) : T (Xn ) ! T (T (Xn+1 )):
The maps oe* determine a natural morphism of T spectra
oe* : X ! T X[1];
and the spectrum QT X is defined to be the inductive colimit of the system
2oe*[2]
X oe*!T X[1] Toe*[1]!2TX[2] T! . . .
Write jX : X ! QT X for the associated canonical map. We shall be particularly
interested in the composite map
X jX!JX jJX!QT JX;
which will be denoted by "jX.
A map g : X ! Y of presheaves of T spectra is said to be a stable fequivale*
*nce
if it induces a level fequivalence
QT J(g) : QT JX ! QT JY:
Observe that g is a stable fequivalence if and only if it induces a level equi*
*valence
IQT J(g) : IQT JX ! IQT JY:
More usefully, perhaps, it is a consequence of Corollary 1.4 that g is a stable*
* f
equivalence if and only if the induced map QT J(g) is a pointwise equivalence of
fpseudoflasque simplicial presheaves in all levels.
A stable ffibration is a map which has the right lifting property with respe*
*ct to
all maps which are cofibrations and stable fequivalences.
We shall prove the following statements:
A4 Every level fequivalence is a stable fequivalence
A5 The maps
"jQTJX; QT J("jX) : QT JX ! (QT J)2X
are stable fequivalences.
A6 Stable fequivalences are closed under pullback along stable ffibration*
*s.
Stable fequivalences are closed under pushout along cofibrations.
22
Lemma 2.7. The statements A4 and A5 hold for presheaves of T spectra on
(Smk)et.
Proof: If g : X ! Y is a level fequivalence between T spectra such that X and
Y are level ffibrant, then g is a pointwise weak equivalence of fpseudoflas*
*que
objects (g is even a homotopy equivalence) in all levels, and so all nTg and QT*
* g
are level pointwise equivalences by C2 and C3. This proves A4.
The map QT J(jX ) : QT JX ! QT J2X is a level fequivalence by A4. There is
a commutative diagram
QT J(jJX )
QT J2Xu___________w QT JQTuJX
 
QT (jJX ) QT (jQTJX )
 
QT JX _____________wQT QT JX
QT (jJX )
The vertical map QT (jJX ) is a level fequivalence because jJX is a pointwise*
* weak
equivalence of fpseudoflasque simplicial presheaves in each level, and QT pre*
*serves
such by C2 and C3. All maps QT (jZ ) are isomorphisms by C1 and a cofinality
argument. The map jQTJX is a pointwise weak equivalence of fpseudoflasque
simplicial presheaves in each level by Corollary 1.4, and so the map QT (jQTJX *
* )
has the same property by C2 and C3. It follows that QT J(jJX ) and QT J("jX) are
level fequivalence.
There is a commutative diagram
JQTuJXn ________woe*Hom*(T; JQTuJXn+1 )
j  
QTJX ' T (jQTJX )
 
QT JXn _________woe*Hom*(T; QT JXn+1 )
The map jQTJX is a level pointwise equivalence by Corollary 1.4, the lower map
oe* is an isomorphism by a cofinality argument and C1, and the map T (jQTJX ) *
*is
a pointwise weak equivalence of fpseudoflasque simplicial presheaves by C2 and
C3. It follows that all maps oe* : JQT JXn ! T JQT JXn+1 are pointwise weak
equivalences, and so the map
jJQTJX : JQT JX ! QT JQT JX
is a level fequivalence. In particular, the composite
jQTJX jJQTJX
QT JX ! JQT JX ! QT JQT JX
is a level fequivalence.
23
Lemma 2.8. The class of stable fequivalences is closed under pullback along le*
*vel
ffibrations.
Proof: Suppose given a pullback diagram
A xY X ________wg*X

 p
 
u u
A ___________wgY
in which g is a stable fequivalence and p is a level ffibration. We want to s*
*how
that g* is a stable fequivalence.
By properness of the flocal level structure (Theorem A.6) and A4, we can
assume that all objects are level ffibrant. Every level fequivalence C ! D of*
* level
ffibrant objects consists of pointwise weak equivalences Cn ! Dn of fpseudo
flasque simplicial presheaves, so QT takes each level fequivalence of level f*
*fibrant
objects to a map of T spectra which consists of pointwise weak equivalences in*
* all
levels. Thus, it suffices to assume that all objects are level ffibrant and sh*
*ow that
QT (g*) is a level fequivalence.
All induced maps
g* : QT An ! QT Y n
are pointwise weak equivalences. The maps
p* : QT Xn ! QT Y n
are filtered colimits of pointwise Kan fibrations, and are therefore pointwise *
*Kan
fibrations. Finally, QT preserves pullbacks and the ordinary simplicial set cat*
*egory
is proper, so the maps
QT (g*) : QT (A xY X)n ! QT Xn
are pointwise weak equivalences of simplicial presheaves.
Every stable ffibration is a level ffibration, because every level fequiva*
*lence
is a stable fequivalence. Lemma 2.8 therefore implies the first statement of A*
*6.
The statements A4 and A5 together imply a BousfieldFriedlander recognition
principle for stable ffibrations (Lemma A.9 of [1]):
24
Lemma 2.9. A map p : X ! Y is a stable ffibration if p is a level ffibration *
*and
the diagram
X ________w"jXQT JX

p  p
  *
u u
Y ________w"jQT JY
Y
is level homotopy cartesian.
In particular, a presheaf of T spectra X is stably ffibrant if X is level f*
*fibrant
and the maps oe* : Xn ! T Xn+1 are fequivalences (or pointwise weak equiva
lences). We shall need the converse assertion:
Lemma 2.10. Suppose that X is stably ffibrant. Then X is level ffibrant, and
all maps oe* : Xn ! T Xn+1 are pointwise weak equivalences.
Proof: The composite
iQTJX
X jX!JX jIX!QT JX ! IQT JX
is a stable fequivalence by Lemma 2.7, and the object IQT JX is stably ffibra*
*nt
since all maps
oe* : IQT JXn ! T IQT JXn+1
are pointwise weak equivalences by Corollary 1.4 (see also the argument for Lem*
*ma
2.8). Write X : X ! IQT JX for this composite.
Factorize X as
X ___________wXIQT JX
')
ff 'ss
'
Z
where ss is a level ffibration and a level fequivalence, and ff is a cofibrat*
*ion. Then
ss is a stable ffibration (since it has the right lifting property with respec*
*t to all
cofibrations). It follows that Z is stably ffibrant and all maps oe* : Zn ! T *
*Zn+1
are pointwise weak equivalences. Also, the map i : X ! Z is a cofibration and a
stable fequivalence. The object X is therefore a retract of Z, and so the maps
oe* : Xn ! T Xn+1 are pointwise weak equivalences.
25
Theorem 2.11. Suppose that T is an fcompact object on the smooth Nisnevich
site (Smk)Nis. Then the category of presheaves of T spectra on that site, tog*
*ether
with the classes of cofibrations, stable fequivalences and stable ffibrations*
*, sat
isfies the axioms for a proper closed simplicial model category.
Proof: We know from [1] and Lemma 2.7 that the category of presheaves of T 
spectra satisfies the closed model axioms CM1  CM4, and the cofibrationtrivial
fibration part of the factorization axiom CM5. We also know (Lemma A.8 of [1])
that a map p : X ! Y is a stable fequivalence and a stable ffibration if and *
*only
if it is a level fequivalence and a level ffibration.
It is a consequence of Corollary 2.9 and Lemma 2.10 that a level ffibration
between stably ffibrant objects must be a stable ffibration.
To prove the remaining part of CM5, suppose given a map g : X ! Y of
T spectra. Form the diagram
X _____________________________wX4IQT JX 4
 4  4
 4f46f*  4f46f
 
g  _______________________*
 Y xIQTJY Z  wZ
 g*
 
 p*  p
u u
Y _____________________________wYIQT JY
where p is a level ffibration and ff is a cofibration and a level fequivalenc*
*e. Then
Z is level ffibrant, and the maps ff : IQT JXn ! Zn are pointwise equivalences
of fpseudoflasque simplicial presheaves, so it follows from Lemma 2.10 that Z*
* is
stably ffibrant. Thus, p is a stable ffibration.
The map * is a stable fequivalence by Lemma 2.8, so that ff* is a stable
fequivalence. Factorize ff* as
0
X _____________wff4W
4
ff*446 ss
u
Y xIQTJY Z
where ff0is a cofibration and ss is a level ffibration and a level fequivalen*
*ce. Then
ff0 is also a stable fequivalence, and ss is a stable ffibration, so f = (p*s*
*s) . ff0 is
26
a factorization of f as a stable ffibration following a cofibration which is a*
* stable
fequivalence, giving CM5.
Part of the properness assertion was proved in Lemma 2.8. For the cofibration
statement, form a pushout diagram
A ___________wgC
 
j  
u u
B ________wg*B [A C
where j is a cofibration and g is a stable fequivalence. We must show that g*
is a stable equivalence. By properness of the level structure (Theorem A.6) and
by taking a suitable factorization in the level structure, we can assume that g*
* is a
cofibration. But then it's a standard fact about closed model categories that t*
*rivial
cofibrations are closed under pushout.
We must finally verify Quillen's axiom SM7. Suppose that i : K ! L is a cofi
bration of pointed simplicial sets and that ff : A ! B is a cofibration of pres*
*heaves
of T spectra. We must show that the cofibration
(A ^ L) [(A^K) (B ^ K) ! B ^ L
is a stable fequivalence if either j is a stable fequivalence or i is a weak *
*equivalence
of simplicial sets. The case where i is a weak equivalence is a consequence of *
*the
levelwise structure. The remaining case is verified by showing that the cofibra*
*tion
ff ^ L : A ^ L ! B ^ L is a stable fequivalence if ff is a stable fequivalenc*
*e.
From Corollary 2.9 and Lemma 2.10, one sees that if W is both stably ffibrant
and finjective, then so is the presheaf of T spectra hom *(L; W ). It will t*
*herefore
follow that ff ^ L is a stable fequivalence if we can show that a map g : X ! *
*Y is
a stable fequivalence if and only if it induces a bijection
*
[Y; W ] g!~=[X; W ]
in morphisms in the level homotopy category for all finjective stably ffibrant
objects W .
Level homotopy classes of maps [X; W ] coincide with morphisms in the stable
category if W is finjective and stably ffibrant. In effect, the morphisms in *
*the sta
ble category from X to W coincide with naive homotopy classes of maps ss(X0; W )
for some choice of trivial level ffibration p : X0 ! X, where X0 is cofibrant.*
* But
ss(X; W ) = [X; W ] = [X0; W ] = ss(X0; W )
27
in the level homotopy category since every object in the finjective structure *
*is
cofibrant and W is finjective.
It follows that any stable fequivalence g : X ! Y induces a bijection
~=
g* : [Y; W ] ! [X; W ]
of level homotopy classes for all finjective stably ffibrant objects W .
Conversely, suppose all such maps g* are bijections, and form the diagram
g**
[IQT JY; W ] ________w[IQT JX; W ]
* ~ ~ *
Y = = X
u u
[Y; W ] _____________wg*[X; W ]
Then the induced map g* : IQT JX ! IQT JY induces bijections g**for all f
injective stably ffibrant objects W . The presheaves of T spectra IQT JX and
IQT JY are finjective and stably ffibrant, and so the map g* must be a homoto*
*py
equivalence.
Here's a corollary of the proof of Theorem 2.11 that we shall use repeatedly:
Corollary 2.12. A map g : X ! Y is a stable fequivalence if and only if it
induces bijections ~
g* : [Y; W ] =![X; W ]
of level (equivalently, stable) homotopy classes for all stably ffibrant finj*
*ective
objects W .
Remark 2.13. Corollary 2.12 is not expressed in terms of function spaces, becau*
*se
there is nothing in the assumptions for Theorem 2.11 which would guarantee that
either T W or hom (X; W ) has an Hspace structure.
Theorem 2.11 has analogues outside the flocal setting. One can, in particula*
*r,
define a pointed simplicial presheaf S on the smooth Nisnevich site to be compa*
*ct
if the following hold:
(1) All inductive systems Y1 ! Y2 ! : :i:nduce isomorphisms
Hom *(S; lim!Yi) ~=lim!Hom *(S; Yi):
i i
(2) If X is pseudoflasque, then so is Hom *(S; X).
(3) The functor Hom *(S; ) preserves pointwise weak equivalences of pointed
pseudoflasque simplicial presheaves.
28
Then just as before, examples of compact simplicial presheaves include all poin*
*ted
finite simplicial sets and all schemes in the smooth Nisnevich site, and there *
*is
an analogue of Lemma 2.3. Level cofibrations and level fibrations of presheaves
of Sspectra determine proper closed simplicial model structures as in Lemma 2.1
(actually, for all pointed simplicial presheaves S on all Grothendieck sites), *
*and so
one is entitled to say that a map g : X ! Y of presheaves of Sspectra is a sta*
*ble
equivalence if it induces a level equivalence g* : QS JX ! QS JY . Cofibrations*
* of
presheaves of Sspectra are defined by the level fibration structure just as be*
*fore,
and stable fibrations are defined by a lifting property. We then have the follo*
*wing
result:
Theorem 2.14. Suppose that S is a compact pointed simplicial presheaf on the
smooth Nisnevich site (Smk)Nis. Then the classes of cofibrations, stable equi*
*v
alences and stable fibrations together determine a proper closed simplicial mod*
*el
category structure for the category of presheaves of Sspectra on this site.
The proof of this result proceeds by exact analogy with the proof of Theorem 2.*
*11
_ one simply removes all references to f. The case corresponding to S = S1 was
discussed in the Introduction.
There is a further generalization of Theorem 2.14 for any geometric site C co*
*n
sisting of schemes and their subschemes, and having a topology at least as fine*
* as
the Zariski topology, in the presence of a suitable analogue of Lemma 1.3.
Any map : S ! T of pointed simplicial presheaves on the site (Smk)Nis
induces a functor
* : PreSpt T(Smk)Nis ! PreSpt S(Smk)Nis;
by precomposing the bonding maps with . More precisely, for any presheaf of
T spectra X, *X is the presheaf of Sspectra with (*X)n = Xn , and having
bonding maps given by the composites
S ^ Xn ^1!T ^ Xn oe!Xn+1 :
There is homotopical content to this construction when S and T are fcompact
and is an fequivalence:
Proposition 2.15. Suppose that : S ! T is an fequivalence of fcompact
objects on the site (Smk)Nis. Then the functor * induces an equivalence of sta*
*ble
homotopy categories
* : Ho (Pre SptT (Smk)Nis) ! Ho (Pre SptS (Smk)Nis):
Proof: Write oe for the bonding maps of *X. The functor * clearly preserves
level fequivalences, level ffibrations and level cofibrations. If X is level *
*ffibrant,
29
there is a diagram
Xn ________woeAT Xn+1 _________wT2oeTXn+2 . . .
A
oe AAC * *
u u
S Xn+1 ________wS'oeS T Xn+2 . . .
' ' 
S oe '') S *
u
2SXn+2 . . .
All vertical maps are pointwise weak equivalences, so there are induced natural
pointwise weak equivalences * : QT Xn ! QS Xn for level ffibrant objects X. It
follows that g : X ! Y is a stable fequivalence of presheaves of T spectra if*
* and
only if *g : *X ! *Y is a stable fequivalence of presheaves of Sspectra. In
particular, * induces a functor
* : Ho (Pre SptT (Smk)Nis) ! Ho (Pre SptS (Smk)Nis):
on homotopy categories. It also follows, using Lemma 2.9, that * preserves stab*
*le
fibrations.
To go further, we must presume that is a cofibration as well as an fequival*
*ence.
This suffices, by Lemma 2.3.5.
Given this new assumption, one can further show that * preserves cofibrations.
In effect, given a cofibration i : A ! B of presheaves of T spectra, there is *
*a pushout
diagram
(S ^ Bn ) [(S^An) (T ^ Bn ) ________w(S ^ Bn ) [(S^An) An+1
 
(; i)* *
u u
T ^ Bn _________________w(T ^ Bn ) [(T^An) An+1
in which (; i)* is a cofibration. The canonical map (S ^ Bn ) [(S^An) An+1 ! Bn*
*+1
for *i is the composite
(S ^ Bn ) [(S^An) An+1 *! (T ^ Bn ) [(T^An) An+1 ! Bn+1 ;
30
and so *i is a cofibration of presheaves Sspectra if i is a cofibration of pre*
*sheaves
of T spectra. __
Every stably ffibrant presheaf of Sspectra_X is of the form_Xn= *X for some
stably fibrant presheaf of T spectra X . To see this, let X = Xn , and choo*
*se
bonding maps __oe: T ^ Xn ! Xn+1 making the following diagram commute:
S ^ Xn ________woeXn+1
 hhj_
^ 1  h oe
u h
T ^ Xn
One gets away with this because ^ 1 is an ftrivial cofibration. It follows t*
*hat
every stably ffibrant presheaf of Sspectra X is stably fequivalent to a pres*
*heaf
of T spectra *Y , where Y is a stably ffibrant and cofibrant presheaf of T s*
*pectra.
To finish off the proof, the idea is to show that : S ! T induces a weak
equivalence of Kan complexes
hom (A; X) *! hom (*A; *X)
for all cofibrant A and stably ffibrant X. Computing in ss0 then implies that
induces bijections
~= * *
* : [Y; X] ! [ Y; X]
for all stably ffibrant, cofibrant objects X and Y . The desired result then f*
*ollows
from basic category theory.
We show that * is a weak equivalence of Kan complexes by showing that, given
any solid arrow diagram
@n __________w hom (A; X)
 OP
 O 
 O O 
u O u
n ________w hom (*A; *X)
a dotted arrow exists such that
(1) the upper triangle commutes, and
(2) the lower triangles commute up to homotopy which is constant on @n.
31
This homotopy lifting property is implied by the following: given any solid arr*
*ow
commutative diagrams
*ff*
A ________wffX *A ________w X
 iij *  hhj
j  i g j  h f
ui uh
B *B
the dotted arrow g exists, making the diagram of presheaves of T spectra commu*
*te,
and there is a homotopy *g ' f which is constant at *ff on *A. This last
property is proved by an inductive homotopy extension argument which depends
on the assumption that is an ftrivial cofibration, and it is left to the read*
*er.
The commutativity of the diagram (1.2) for the controlled ffibrant model con
struction X 7! LX of [2] implies that this construction can be promoted to the
category of presheaves of T spectra. More explicitly, there is a natural level*
* fibrant
model jX : X ! LX defined for presheaves of T spectra such that the map jX is*
* a
level cofibration and a level fequivalence. The standard properties of the fun*
*ctor L
(see Section 1.1) pass to the spectrum level, and so the functor L is an exampl*
*e of a
functor F : PreSpt T(Smk)Nis ! PreSpt T(Smk)Nis which satisfies the following:
L1: F preserves level weak equivalences.
L2: F preserves level cofibrations.
L3: Let fi be any cardinal with fi ff. Let {Xj} be the filtered system of
subobjects of X which are fibounded. Then the map
lim!F (Xj) ! F X
j
is an isomorphism.
L4: Let fl be an ordinal number of cardinality strictly greater than 2ff. L*
*et
X : fl ! PreSpt T(Smk)Nis be a diagram of level cofibrations so that for
all limit ordinals s < fl the induced map
lim!X(t) ! X(s)
t > 2ff;
where ff is an upper bound on the cardinality of the set of morphisms of (the c*
*hosen
approximation for) the smooth Nisnevich site.
Remark 2.16. If the presheaf of T spectra X has extra structure, such as a sym
metric structure, then that structure is preserved by the functor X 7! LX: the
pairings
LXn ^ L OE!L(Xn ^ L)
satisfy properties (2) and (3) above, and are natural in L and Xn so that they
respect all symmetric group actions.
Say that a map g : X ! Y of presheaves of T spectra is an F equivalence if*
* it
induces a level weak equivalence F g : F X ! F Y .
Proposition 2.17. Suppose that the functor
F : PreSpt T(Smk)Nis ! PreSpt T(Smk)Nis
satisfies the conditions L1  L7 above. Then the class of cofibrations of presh*
*eaves
of T spectra which are F equivalences satisfies the bounded cofibration condi*
*tion
for the cardinal .
Proof: The class of maps of presheaves of T spectra which are level cofibratio*
*ns
and level weak equivalences satisfies the bounded cofibration condition for the*
* car
dinal . To see this, recall that the category of simplicial presheaves satisfi*
*es the
bounded cofibration condition with respect to the cardinal , since is an upper
bound for the cardinality of the set of morphisms of the underlying site [2, Le*
*mma
2.3]. Then use the argument for Lemma 2.1.2.
Suppose that i : X ,! Y is a cofibration in the category of presheaves of T 
spectra, and that j : A ,! Y is a subobject of Y . Then the restriction X \ A !*
* A
is a cofibration of presheaves of T spectra (so that the statement of the Prop*
*osition
makes sense). The claim for ordinary presheaves of spectra (ie. T = S1) was pro*
*ved
in Lemma 3.1 of [2]. There is nothing special about the simplicial circle S1 in*
* that
argument, so the same argument obtains here.
33
Alternatively, the key is to show that the map
j* : (T ^ An) [(T^(An\Xn)) (An+1 \ Xn+1 ) ! (T ^ Y n) [(T^Xn) Xn+1
is an inclusion in all presheaves of simplices for all n. But
(T ^ An) [(T^(An\Xn)) (An+1 \ Xn+1 ) = ((T  *) x (An  Xn )) t (An+1 \ Xn+1 );
at the simplex level, while
(T ^ Y n) [(T^Xn) Xn+1 = ((T  *) x (Y n Xn )) t Xn+1 ;
and the map between the two is obvious.
Let X ! Y be an F equivalence and a cofibration of presheaves of T spectra,
and let A Y be a bounded subobject. Inductively define a chain of bounded
subobjects A = A0 A1 A2 . . .Y over , and a chain of subobjects
F (A) = F (A0) X1 F (A1) X2 F (A2) . .F.Y;
also over , with the property that the cofibration
F X \ Xs ! Xs
is a level weak equivalence. Set B = lim!s 2 , just as in the proof of Lemm*
*a 6
of [8].
It follows from Theorem 4.2 and Proposition 4.4 that any morphism f : X ! Y
of presheaves of symmetric T spectra may be successively factored
X ________wi1hhXs'________wi2Xsi
h h h '
h h h 'p'1 p2
f hj ') u
Y
54
where
(1) i1 is a level cofibration which has the left lifting property with respe*
*ct to all
stable ffibrations, and induces a trivial fibration
i*1: hom (Xs; W ) ! hom (X; W )
for each stably ffibrant W , and p1 is a stable ffibration;
(2) i2 is a level cofibration and a level fequivalence, and p2 is an finje*
*ctive
fibration.
In particular, Up2 is a level ffibration, which is level fequivalent to the s*
*table
fibration Up1, so that p2 is a stable ffibration by Lemma 2.9 as well as an f
injective fibration of presheaves of symmetric spectra. By specializing to Y *
*= *,
we obtain a natural construction
X i1!Xs i2!Xsi
of an finjective stably ffibrant model Xsi for a given presheaf of symmetric *
*T 
spectra X.
Say that a map f : X ! Y of presheaves of symmetric T spectra is a stable
fequivalence if it induces a weak equivalence of Kan complexes
f* : hom (Y; W ) ! hom (X; W )
for each finjective stably ffibrant object W . The maps i1 and i2 above are b*
*oth
stable fequivalences. Following the script of [8] we can also show
Lemma 4.5. Suppose that the objects X and Y are stably ffibrant and finjectiv*
*e.
Then a map g : X ! Y is a stable fequivalence if and only if it is a level f
equivalence.
Proof: If g is a stable fequivalence, then a little fun with function complexes
shows that g is a homotopy equivalence, and hence a homotopy equivalence in all
levels. The converse is clear, but depends on the fact that the function compl*
*ex
hom (X; W ) is an Hspace if W is stably ffibrant and finjective.
Corollary 4.6. Suppose that X and Y are stably ffibrant objects. Then a map
g : X ! Y is a stable fequivalence if and only if it is a level fequivalence.
We're going to need the following:
Lemma 4.7 (Voevodsky). The cyclic permutation oe = (1; 2; 3) 2 3 induces the
identity morphism on T 3in the pointed flocal homotopy category.
55
Proof: First of all, there is an isomorphism of pointed presheaves
An=(An  0) ^ A1=(A1  0) ~=An+1 =(An+1  0);
since
((An  0) x A1) [ (An x (A1  0)) = An+1  0
inside An+1 . It follows that there is an isomorphism
T n~= An=(An  0):
There is a pointed algebraic group action
Gln x T n! T n
in the presheaf category which is induced by the standard action Gln x An ! An.
It follows that any rational point g 2 Gln(k) induces a morphism of presheaves
g : T n! T n:
In particular, the permutation matrix corresponding to the element oe = (1; 2; *
*3)
induces the map
oe : T 3! T 3
in the statement of the Lemma.
Generally, if the element g has determinant 1, then g is a product of element*
*ary
transformations, and so there is an algebraic path
!g : A1 ! Gln
such that !(1) = g and !(0) = e. The element oe 2 Gl3(k) has determinant 1, so
there is a composite (pointed) algebraic homotopy
A1 x T 31x!oe!Gl3 x T 3! T 3
from oe : T 3! T 3to the identity on T 3(see also Theorem 1.1 of [6]). The maps*
* oe
and the identity therefore coincide in the flocal homotopy category.
Suppose that Z is a presheaf of symmetric T spectra and that K is a pointed
simplicial presheaf. The presheaf of symmetric T spectra
ZK = Hom *(K; Z)
is defined in levels by
Hom *(K; Z)n = Hom *(K; Zn);
56
where hom *denotes pointed internal hom, as in Section 3. The structure map
T p^ Hom *(K; Zn) oe!Hom *(K; Zp+n )
is the unique pointed simplicial set map making the diagram
T p^ Hom *(K; Zn) ^ K ________woeH^oKm*(K; Zp+n ) ^ K
p  
T ^ ev ev
u u
T p^ Zn _______________________woeZp+n
commute, where ev is the evaluation map wherever it appears. This construction
is natural in K and Z, and there are natural isomorphisms
Hom *(K ^ L; Z) ~=Hom *(K; Hom *(L; Z))
for all pointed simplicial presheaves K, L, and presheaves of symmetric T spec*
*tra
Z.
There is a natural shift operator Z 7! Z[1] for presheaves of symmetric T sp*
*ectra
Z. In effect, Z[1] is the object defined in levels by Z[1]m = Z1+m , where oe *
*2 m
acts on Z[1]m as 1 oe 2 m+1 . In other words, 1 oe(1) = 1 and
1 oe(i) = 1 + oe(i  1)
for i > 1. The structure map oe* : T p^ Z[1]m ! Z[1]p+m is defined to be the
composite
T p^ Z1+m oe!Zp+1+m c(p;1)1!Z1+p+m ;
where c(p; 1) 2 p+1 is the cyclic permutation of order p + 1. One checks that o*
*e*
is p x m equivariant. Define the shifted spectrum Z[s] inductively by Z[s] =
Z[s  1][1], or directly.
The standard maps oe* : Zn ! Hom *(T; Z1+n ) which are adjoint to the com
posites
Zn ^ T o!T ^ Zn oe!Z1+n
together determine a natural map of of presheaves of symmetric T spectra
oe* : Z ! Hom *(T; Z[1]) = Hom *(T; Z)[1]:
Write T Z[1] = Hom *(T; Z)[1].
57
Suppose that Z is a presheaf of symmetric T spectra which is level ffibrant.
Flipping loop factors defines a natural isomorphism
o* : 2TZ[2] ! 2TZ[2];
and there is an isomorphism (1; 2) : Z[2] ! Z[2] which consists of maps (1; 2) :
Z2+n ! Z2+n induced by the transposition (1; 2) 2 n+2 . Write "oefor the bonding
maps of T Z[1]. Then there is a natural commutative diagram
T oe*[1] 2
T Z[1]a___________weT Z[2]
aeae
o"e aeaeo (1; 2)*o*
* u
2TZ[2]
which translates into a diagram of simplicial presheaves
T Zn+1 ae_________wT2oe*TZn+2
aeae
(4.8) "oeaeaeo (1; 2)*o*
* u
2TZn+2
for each n.
Lemma 4.9. Suppose that Z is a presheaf of symmetric T spectra which is level
ffibrant. Then the map oe* : Z ! T Z[1] induces an fstable equivalence UZ !
UT Z[1] of presheaves of T spectra.
Proof: It suffices to show that the diagram
Zn __________woe*T Zn+1 ________wT2oe*TZn+2 ________w. . .

oe*  2 
 T oe* T oe*
u u u
T Zn+1 ________w"oe2TZn+2 ________w3TZn+3 ________w. . .
* T "oe*
induces an isomorphism in colimits of presheaves of homotopy groups.
58
The induced map in the colimit is plainly a monomorphism. There is a commu
tative diagram
2Toe* 3 n+3
T Zn+1 ________wTfoe*l2TZn+2 ________wT Z
flflflffl * *
"oe* u(1;22)*ooe u(1; 2)*o
2TZn+2 ________wTfl3*TZn+3
flflflffl *
T "oe* uT ((1; 2)*o )
3TZn+3
The composite
T ((1; 2)*o*)(1; 2)*o* = T (o*)o*
is induced by precomposition with the map (1; 2; 3) : T 3! T 3. This map repres*
*ents
the identity map in the flocal homotopy category over each kscheme U by Lemma
4.7, and so the induced map in homotopy groups
[Si ^ T 3; Zn+3 U ] ! [Si ^ T 3; Zn+3 U ]
is the identity, for all kschemes U. The map T (o*)o* therefore induces the
identity in all presheaves of homotopy groups. It follows that
T "oe*. "oe*= 2Toe* . T oe* : ss*(T Zn+1 ) ! ss*(3TZn+3 ):
For a level ffibrant object Z as in the statement of the Lemma 4.9, define t*
*he
presheaf of symmetric T spectra QT Z to be the filtered colimit of the system
""oe*
(4.10) Z oe*!T Z[1] "oe*!2TZ[2] ! . . .
Lemma 4.11. Suppose that Z is a level ffibrant presheaf of symmetric T spectr*
*a.
Then there is a natural isomorphism
QT Zn ~=QT UZn :
Proof: To extend the notation for the bonding map "oeof T Z[1] given above,
write
oe~(n)*= o^e~(n1)*: nTZ[n] ! n+1TZ[n + 1];
59
so that "oe*= oe~(1)*and ""oe*= oe~(2)*in the sequence (4.10). Repeated instanc*
*es of
the diagram (4.8) give a commutative diagram
~(k)
kTZn+k a__________woe*ek+1TZn+k+1
A aeae ~(k1)
AA Taeaeooe* (1; 2)*o*
A u
A k+1TZn+k+1
A
k A (1; 2)*T o*
T oe* A u
A .
A ..
A 
AC (1; 2)*k1To*
u
k+1TZn+k+1
Write k+1 for the composite of the vertical maps in the diagram. The morphism
k+1 is a composite of instances of isomorphisms of the form iTo* or (1; 2)*, and
therefore commutes (up to interpretation of notation) with the morphism k+1Toe*.
Now suppose given natural isomorphisms fli : iTZn+i ! iTZn+i such that the
diagram
~(1) oe~(2) oe~(k1)
T Zn+1 ________woe*2TZn+2 ________w*._._.____w*kTZn+k
 fl fl
1  2 k
u u u
T Zn+1 ________w2TZn+2 ________w . . ._______wkTZn+k
T oe* 2Toe* k1Toe*
commutes, and all isomorphisms fli are compositions of instances of jTo* or (1;*
* 2)*.
In particular, presume that fl2 = o*(1; 2)* : 2TZn+2 ! 2TZn+2 . Then the isomor
phism flk commutes with kToe* : kTZn+k ! k+1TZn+k+1 , and so we are entitled
to set flk+1 to be the composite
k+1TZn+k+1 k+1!k+1TZn+k+1 flk!k+1TZn+k+1
60
and get a natural commutative diagram
~(k)
kTZn+k f________woe*lk+1TZn+k+1
flfl
 flffl k+1
 kToe* k+1 n+k+u1
flk T Z
 flk
u u
kTZn+k ________wk+1TZn+k+1
kToe*
The natural isomorphism flk+1 is a composite of instances of the isomorphisms
iTo* and (1; 2)*.
Formally, there is a map c : QT X ^ K ! QT (X ^ K) which fits into a natural
commutative diagram
QT Xu^ K ________wcQT (X ^ K);
 NNP
flX ^ K  N NflX^K
 N
X ^ K
for all presheaves of symmetric T spectra X and pointed simplicial sets K. It
follows that the functor QT prolongs to a simplicial functor
QT : hom (X; Y ) ! hom (QT X; QT Y ):
Proposition 4.12. Suppose that ff : X ! Y is a map of presheaves of symmetric
T spectra such that Uff : UX ! UY is a stable fequivalence of presheaves of
T spectra. Then ff is a stable fequivalence of presheaves of symmetric T spe*
*ctra.
Proof: We can assume that X and Y are level ffibrant. If W is a stably f
fibrant and finjective object, then the canonical map flW : W ! QT W is a lev*
*el
fequivalence, and hence induces a weak equivalence
fl*W : hom (QT W; W ) ! hom (W; W ):
It follows that there is a map gW : QT W ! W such that the composite gW flW is
simplicially homotopic to the identity 1W on W .
The composite
gW* fl*
hom (X; W ) QT!hom (QT X; QT W ) ! hom (QT X; W ) !Xhom (X; W )
61
is induced by composition with gW flW , and is therefore homotopic to the ident*
*ity
on hom (X; W ). The composition and the homotopy are natural in X. If ff :
X ! Y induces a stable fequivalence Uff : UX ! UY , then the induced map
QT ff : QT X ! QT Y is a level fequivalence by Lemma 4.9, and so the maps
QT ff* : hom (QT Y; W ) ! hom (QT X; W )
and hence
ff* : hom (Y; W ) ! hom (X; W )
are weak equivalences of pointed simplicial sets.
Remark 4.13. Notice that Lemma 4.9 is not involved in the proof of Proposition
4.12.
We are now ready to invoke the results of Section 3 to prove the following:
Proposition 4.14. Suppose that p : X ! Y is a map of presheaves of symmetric
T spectra which is both a stable ffibration and a stable fequivalence. Then *
*p is
a level fequivalence.
Proof: It suffices to show that the fibre F of p is level contractible. If so,*
* the
underlying map Up of presheaves of T spectra is a stable ffibration and a sta*
*ble f
equivalence by a long exact sequence argument in bigraded stable homotopy groups
(3.10), and is therefore a level fequivalence (see the proof of Theorem 2.11).
To see that F is level contractible, use Lemma 3.12 to replace the fibre sequ*
*ence
by the cofibre sequence
(4.15) F i!X ss!X=F:
More precisely, Lemma 3.12 guarantees that the map of presheaves of T spectra
underlying the canonical map p* : X=F ! Y is a stable fequivalence, and so p* *
*is
a stable fequivalence of presheaves of symmetric T spectra by Proposition 4.1*
*2.
The cofibre sequence (4.15) induces fibration sequences
* i*
(4.16) hom (X=F; W ) ss!hom (X; W ) ! hom (F; W )
for all finjective stably ffibrant finjective objects W , in which all maps *
*ss* are
weak equivalences. The canonical map W ! T W [1] is a level fequivalence of
finjective stably ffibrant presheaves of symmetric T spectra. The fibre sequ*
*ence
(4.16) can therefore be delooped, and so hom (F; W ) is an Hspace having triv*
*ial
homotopy groups _ it must therefore be contractible. This means that the map
F ! * is a stable fequivalence of stably ffibrant objects, so it is a level w*
*eak
equivalence by Corollary 4.6.
62
Corollary 4.17. A map p : X ! Y of presheaves of symmetric T spectra is
a stable ffibration and a stable fequivalence if and only if it is both a lev*
*el
ffibration and a level fequivalence.
Proof: One direction is Proposition 4.14; the other follows from the definition*
* of
stable fequivalence of presheaves of symmetric T spectra.
Say that a map i : A ! B of presheaves of symmetric T spectra is a stable f
cofibration if it has the left lifting property with respect to all morphisms p*
* : X ! Y
which are simultaneously stable ffibrations and stable fequivalences. In view*
* of
Corollary 4.17, the maps
T Gn(A+ ) ! T Gn(LU r+)
induced by the inclusions A LU r are stable fcofibrations for all r and objec*
*ts
U 2 C. Here, LU denotes the left adjoint to the Usections functor for simpli*
*cial
presheaves.
Theorem 4.18. The category PreSpt T(Smk)Nis of presheaves of symmetric T 
spectra on the smooth Nisnevich site for a field k, and the classes of stable f
equivalences, stable ffibrations and stable fcofibrations, together satisfy t*
*he ax
ioms for a proper closed simplicial model category.
Proof: On account of Proposition 4.4, every map g : X ! Y of presheaves of
symmetric spectra has a factorization
X ________wj4Z
4 
(4.19) g 446 p
u
Y
such that p is a stable ffibration, and j has the left lifting property with r*
*espect to
all stable ffibrations and induces trivial fibrations j* : hom (Z; W ) ! hom *
* (X; W )
for all stably fibrant objects W . In particular, j is a stable fequivalence *
*and a
stable fcofibration. The map j is a level cofibration, by Lemma 4.3.
A transfinite small object argument says that g : X ! Y has a factorization
X ________wi4U
4
g 446 q
u
Y
63
such that i has the left lifting property with respect to all maps which are si*
*mul
taneously level fibrations and level weak equivalences, and q has the right lif*
*ting
property with respect to all morphisms
T Gn(A+ ) ! T Gn(LU r+)
corresponding to cofibrations A ,! LU n of simplicial presheaves for all n and
objects U 2 C. In particular, q is a level trivial ffibration and hence a sta*
*ble
ffibration as well as a stable fequivalence by Corollary 4.17. The map i has
the left lifting property with respect to all maps which are stable ffibration*
*s and
stable fequivalences, also by Corollary 17, so that i is a stable fcofibratio*
*n. It is
a consequence of the small object argument that the map i is a level cofibratio*
*n.
The factorization axiom CM5 has therefore been demonstrated. The existence
of the factorization (4.19) implies that every map which is a stable fcofibrat*
*ion
and a stable fequivalence has the left lifting property with respect to all st*
*able
ffibrations and is a level cofibration, by a standard argument. We have proved
CM4, and the axioms CM1  CM3 are obvious.
If i : K ,! L is an inclusion of simplicial sets and p : X ! Y is a stable f*
*fibration
of presheaves of symmetric T spectra, then the induced map
(i*; p*) : hom *(L+ ; X) ! hom *(K+ ; X) xhom *(K+;Y )hom *(L+ ; Y )
is a stable ffibration, which is trivial if i is a weak equivalence or p is a *
*stable
equivalence. This is on account of the corresponding statement for presheaves
of spectra and Corollary 4.17, and implies the simplicial model axiom SM7 for
Pre SptT (Smk)Nis.
Stable fequivalences are closed under pushout along stable fcofibrations. *
*To
see this, apply the functors hom ( ; W ) corresponding to stably ffibrant fi*
*njective
objects W , and use the fact that all stable fcofibrations are level cofibrati*
*ons, along
with properness for simplicial sets.
To see that stable fequivalences are preserved by pullback along stable f
fibrations, consider a pullback square
X1 ________wg*X2
p  p
1  2
u u
Y1 ________wgY2
where p2 is a stable ffibration and g is a stable fequivalence. The fibration*
*s p1
64
and p2 have a common fibre F , and there is a commutative diagram
X1=F ________w"gX2=F
 
 
u u
Y1 ___________wgY2
in which the vertical maps induce stable fequivalences of presheaves of the un*
*der
lying T spectra, by Lemma 3.12. Apply the functor hom ( ; W ) as above to the
diagram of cofibre sequences
F ________wX1 ________wX1=F

=  g  
 * "g
u u u
F ________wX2 ________wX2=F
The resulting comparison diagram of fibre sequences shows that the induced map
*
hom (X2; W ) g*!hom (X1; W )
is a weak equivalence. To see this, deloop the fibre sequences using the canoni*
*cal
level fequivalence
W ! T W [1] ' Gm W [1]
of stably ffibrant finjective objects, so that g**is a map of Hspaces which *
*induces
an isomorphism of groups in ssi for i 0.
5. Equivalence of stable categories.
The purpose of this section is to show that the homotopy categories associated
to the stable closed model structures for presheaves of T spectra and presheav*
*es of
symmetric T spectra are equivalent on the smooth Nisnevich site.
The equivalence of homotopy categories is induced by the functors U (which
forgets the symmetry) and its left adjoint V . As in [4] and [8], the proof of*
* the
equivalence of homotopy categories boils down to showing that any stable ffibr*
*ant
model j : V X ! (V X)s associated to a cofibrant presheaf of T spectra X induc*
*es
a stable fequivalence given by the composite
X j!UV X Uj!U(V X)s:
65
The idea of proof is to use a layer filtration for X, and then show that the re*
*sult
for all of the layers implies the statement for X.
The identity functor X 7! X and the functor X 7! U(V X)s both preserve stable
fequivalence. Each of the layers is a shifted suspension object up to stable e*
*quiv
alence, so we inductively prove the claim for shifted suspensions, beginning wi*
*th
ordinary presheaves of suspension T spectra 1TK associated to pointed simplici*
*al
presheaves K.
The canonical map j : 1TK ! UV (1TK) is an isomorphism, so it suffices to
find a stably ffibrant model
V (1TK) ~=T K j!(T K)s
for the presheaf of symmetric T spectra T K such that the map j induces a stab*
*le
equivalence Uj : U(T K) ! U(T K)s of presheaves of T spectra.
The construction that we use involves presheaves of T bispectra. A presheaf *
*of
T bispectra X consists of pointed simplicial presheaves Xr;s, r; s 0, togethe*
*r with
bonding maps
oeh : T ^ Xr;s! Xr+1;s and oev : T ^ Xr;s! Xr;s
such that the diagram
T ^ Xr;s+1______________________________woehXr+1;s+1uu
 
T ^ oev 
 
T ^ T ^ Xr;s[ oev
[[]~= 
o ^ 1 
T ^ T ^ Xr;s__________wTT^^oeXr+1;s
h
where o : T ^T ! T ^T is the isomorphism which flips smash factors. A presheaf *
*of
T bispectra may alternatively be viewed as a "T spectrum object" in the categ*
*ory
of presheaves of T spectra, in the sense that the collections of objects Xr;* *
*form
presheaves of T spectra for all r 0, and the horizontal bonding maps oeh dete*
*rmine
morphisms oeh* : Xr;*^ T ! Xr+1;* given in levels by the composites
Xr;s^ T o!~=T ^ Xr;soeh!Xr+1;s:
There is a second way to interpret X as a T spectrum object, by starting with *
*the
presheaves of T spectra X*;sand taking bonding maps X*;s^ T ! X*;s+1 induced
by the maps oeh.
66
These ideas are completely analogous to the fundamental ideas underlying or
dinary bispectra [5]. Perhaps much of that machinery can be pushed through to
presheaves of T bispectra  the trick for the moment is to avoid doing so.
Maps g : X ! Y of presheaves of T bispectra consist of collections of morphi*
*sms
g : Xr;s ! Y r;swhich preserve all structure. A map g : X ! Y is said to
be a level fequivalence (respectively ffibration) if each of the component ma*
*ps
g : Xr;s! Y r;sis an fequivalence (respectively ffibration). It is an easy ex*
*ercise,
using the level model structure for presheaves of T spectra, to show that ther*
*e is a
level fequivalence i : X ! Y for every object X, such that Y is level ffibran*
*t.
Suppose that X is level ffibrant. The map oeh* : Xr;*^T ! Xr+1;*of presheaves
of T bispectra has an adjoint oeh* : Xr;*! T Xr+1;*, and so there are commutat*
*ive
diagrams
Xr;s____________woeh*T Xr+1;s

oe  
v*  (oev)*
u u
T Xr;s+1________w2TXr+1;s+1
T oeh*
One has to be careful here (compare with Section 3.2): the map (oev)* is the ad*
*joint
of the canonical choice of bonding map oev : T ^ T Xr+1;s ! T Xr+1;s+1 for
the presheaf of T spectra T Xr+1;s, and a bit of calculation shows that there *
*is a
commutative diagram
T Xr+1;sA________wT2oev*TXr+1;s+1
A
(oe A AC o*
v)* u
2TXr+1;s+1
where o* is induced by flipping the loop factors. It follows that composing two
instances of these diagrams give a picture
Xr;s____________woeh*T Xr+1;s
 T (T (oev*)oev*)
 u
T (oev*)oev* 3 r+1;s+2
 T X
 *
u uc
2TXr;s+2________w3TXr+1;s+2
2Toeh*
67
where c* = T (o*)o* is induced in loop factors by the cyclic permutation c =
(1; 2; 3) of order 3.
Lemma 4.7 implies that the map c* induces the identity in presheaves of homo
topy groups. We therefore have a commutative diagram of presheaves of groups
ssjXr;s________wssj2TXr+2;s _____w. . .
u u
(5.1) ssj2TXr;s+2____w ssj4TXr+2;s+2 ____w. . .
u u
.. .
. ..
in which the horizontal morphisms induced by maps 2nT(T (oeh)oeh) and the verti*
*cal
maps are induced by maps 2nT(T (oev)oev)
Write ssjQXr;sfor the filtered colimit of the diagram (5.1), and say that a m*
*ap
g : X ! Y of level ffibrant presheaves of T bispectra is a stable fequivale*
*nce if
it induces isomorphisms of presheaves of groups
ssjQXr;sg*!~=ssjQY r;s
for all j, r and s. One expands the definition of stable fequivalence to arbit*
*rary
presheaves of T bispectra by declaring a map to be a stable fequivalence if t*
*he
induced map on level fibrant models is a stable fequivalence.
It is plain, for example that the presheaves of groups ssjQXr;sare filtered c*
*olim
its of presheaves of stable homotopy groups corresponding to both horizontal and
vertical choices of presheaves of T spectra. This leads immediately to the fol*
*lowing
Lemma 5.2. Suppose that g : X ! Y is a map of presheaves of T bispectra such
that either all maps g : Xr;*! Y r;*, r 0, or all maps g : X*;s! Y *;s, s 0, *
*are
stable fequivalences of presheaves of T spectra. Then g is a stable fequival*
*ence
of presheaves of T bispectra.
A presheaf of T bispectra Y is said to be stably ffibrant if it is level f*
*fibrant
and all bonding maps oeh : Y r;s! T Y r+1;sand oev : Y r;s! T Y r;s+1are f
equivalences (hence pointwise equivalences).
Every presheaf of T spectra Z has an associated presheaf of T bispectra 1TZ
consisting of the objects
Z; Z ^ T; Z ^ T 2; : : :
in the obvious way. The technical device that begins the proof of the main resu*
*lt
of this section is the following:
68
Lemma 5.3. Let Z be a presheaf of T spectra and suppose that Y is a stably f
fibrant presheaf of T bispectra. Suppose that the morphism g : 1TZ ! Y is a
stable fequivalence. Then the map g : Z ! Y 0at level 0 is a stable fequivale*
*nce
of presheaves of T spectra, and Y 0is a stably ffibrant presheaf of T spectr*
*a.
Proof: We can suppose that there is a level ffibrant model j : 1TZ ! X for
1TZ such that the map g factors through j. Make the suspension index of 1 ZT
the horizontal index, so that
(1TZ)r;s= Zs ^ T r:
The map of presheaves of T spectra
Xr;*T(oeh*)oeh*!2TXr+2;*
is a stable fequivalence by Lemma 3.14, and so there is an isomorphism
ssjQT Xr;*~= lim!ssj2nTXr;s+2n~= ssjQXr;s:
There is a similar isomorphism
ssjQT Y r;*~=lim!ssj2nTY r;s+2n~=ssjQY r;s:
since Y is stably ffibrant. The morphisms
ssjQXr;s! ssjQY r;s
are isomorphisms of presheaves of groups by assumption, so in particular the map
ssjQT X0;s! ssjQT Y 0;s
is an isomorphism as well.
Lemma 5.4. Suppose that K is a pointed simplicial presheaf, and let i : T K !
(T K)s be a stable ffibrant model for the presheaf of symmetric T spectra T K.
Then i induces a stable fequivalence Ui : U(T K) ! U(T K)s of presheaves
of T spectra.
Corollary 5.5. Suppose that K is a pointed simplicial presheaf. Then the map
1TK j*!UV (1TK)s
is a stable fequivalence.
69
Proof or Lemma 5.4: It suffices, by formal nonsense, to find just one stable
ffibrant model for T K which satisfies the statement of the lemma.
There is a T spectrum object 1T(T K) in the category of presheaves of sym
metric T spectra, given by
1T(T K)n = (T K) ^ T n:
Suppose that the suspension degree is horizontal, and so the presheaf of T bis*
*pectra
underlying this object is specified in bidegrees by
U(1T(T K))r;s= T s^ K ^ T r:
The functor QT and the level ffibrant model functor L are both simplicial func*
*tors,
so the maps of presheaves of T spectra
T s^ K ^ T *! LQT L(T s^ K ^ T *)
determine a map
1T(T K) ! LQT L(1T(T K))
of T spectrum objects in the category of presheaves of symmetric T spectra wh*
*ose
underlying map of presheaves of T bispectra consists of stable ffibrant models
T s^ K ^ T *! LQT L(T s^ K ^ T *)
in each vertical degree. By Lemma 3.14, the vertical bonding map
LQT L(T s^ K ^ T *) ! T LQT L(T s+1^ K ^ T *)
is a stable fequivalence and hence a level fequivalence, so that the presheaf*
* of
T bispectra U(LQT L(1T(T K))) is stable ffibrant. In particular, the presheaf
of symmetric T spectra LQT L((T K)^T 0) is stable ffibrant, as is its underly*
*ing
presheaf of T spectra. Finally, Lemma 5.3 implies that the map of presheaves *
*of
T spectra
U((T K) ^ T 0) ! U(LQT L((T K) ^ T 0))
is a stable fequivalence.
Lemma 5.6. A map g : X ! Y of presheaves of symmetric T spectra is a stable
fequivalence if and only if the suspension g ^ T : X ^ T ! Y ^ T is a stable
fequivalence.
70
Proof: If g is a stable fequivalence, then g ^ T is a stable fequivalence, on
account of the isomorphisms
hom (X ^ T; W ) ~=hom (X; T W )
and the fact that the functor T preserves stably ffibrant finjective objects.
If g ^ T is a stable fequivalence, then the natural stable fequivalence W !
T W [1] induces a diagram
*
hom (Y; W ) __________________wghom(X; W )
 
'  '
u * u
hom (Y; T W [1]))____________wghom(X; T W [1])
~  ~
=  =
u u
hom (Y ^ T; W [1]) ___________w*hom(X ^ T; W [1])
(g ^ T )
If g ^ T is a stable fequivalence, then (g ^ T )* is a weak equivalence for al*
*l stably
ffibrant finjective W , so g* is a weak equivalence for all such W .
Corollary 5.7. The composite
j* : X j!T (X ^ T ) Tj!T (X ^ T )s
is a stable fequivalence of presheaves of symmetric T spectra, for any choice*
* of
stably ffibrant model j for X ^ T .
Proof: There is a diagram
X ^ T ________wj*a^eTT (X ^ T )s ^ T
aeae
j aeaeo ev
u
(X ^ T )s
and the evaluation map ev is a stable fequivalence of the underlying presheaves
of T spectra by Corollary 3.17. Now use the previous lemma.
71
Now for some elementary category theory. There are natural isomorphisms
~=
: T UX ! U(T X)
and ~
^ : UX ^ T =! U(X ^ T )
such that the diagram
T (UX ^ T )
jUX hhj 
h T ^
h u
(5.8) UXA T U(X ^ T )
A
UjX AAC 
u
UT (X ^ T )
commutes. These isomorphisms are actually identities, and the commutativity of
the diagram just represents the fact that T and smashing with T mean exactly the
same thing in both presheaves of T spectra and presheaves of symmetric T spec*
*tra.
The natural isomorphism induces a natural isomorphism
V (X ^ T ) *! V X ^ T
in the standard way. At the same time, there is a natural map
"^ : V (X ^ T ) ! V X ^ T
which is adjoint to the composite
X ^ T j^T!UV X ^ T ^!U(V X ^ T )
One calculates directly, using the diagram (5.8), to show that the maps "^ and
* coincide. The point of this construction, for us, is that the following diag*
*ram
commutes as a consequence of the definition of the map "^:
X ^ T4_________wjU^VTX ^ T
4 
4 ^
4 u
(5.9) j 4 U(V X ^ T )
4 
46 U1*
u
UV (X ^ T )
72
Lemma 5.10. The natural map j* : X ! U(V X)s is a stable fequivalence if and
only if the map j* : X ^ T ! U(V (X ^ T ))s is a stable fequivalence.
Proof: The map j* : X ! U(V X)s is a stable fequivalence if and only if the
composite
X ^ T j*^T!U(V X)s ^ T ^!~=U((V X)s ^ T ) Uj!U((V X)s ^ T )s
is a stable fequivalence. To see this, use the commutativity of the diagram (5*
*.8)
to see that the adjoint of the composite
U(V X)s ^ T ^!U((V X)s ^ T ) Uj!U((V X)s ^ T )s
is the map Uj* : U(V X)s ! UT ((V X)s^ T )s, and is a stable fequivalence since
the map j* : (V X)s ! T (V X)s ^ T )s is a stable fequivalence of stably ffib*
*rant
presheaves of symmetric T spectra by Corollary 5.7. You will also find yourse*
*lf
using Corollary 3.18.
Construct a diagram of stable fequivalences
V X ^uT _________wj(^VTX)s ^ T ________wj((V X)s ^ T )s

 hhj
*  h "j
 h
V (X ^ T ) ________wjV (X ^ T )s
in the category of presheaves of symmetric T spectra, and observe that the map
"jmust be a level fequivalence, so that U"jis a level fequivalence. Now use t*
*he
diagram (5.9) to show that there is a commutative diagram
X ^ T __________wj*U^(TV X)s ^ T ________w^U((V X)s ^ T ) ________wUjU((V X)*
*s ^ T )s
 fl flflffl
j  flflflfl
 flflfl fl
u flflfl U"j
UV (X ^ T ) ________wUjUV (X ^ T )s
It follows that the top composite (and hence the map j* : X ! U(V X)s) is a sta*
*ble
fequivalence if and only if the composite
X ^ T j!UV (X ^ T ) Uj!UV (X ^ T )s
73
is a stable fequivalence.
There are canonical stable fequivalences
1TK[n] ^ T n! 1TK
and
1TXn [n] ! FnX
where
FnX : X0; X1; : :;:Xn ; T ^ Xn ; T 2^ Xn ; : : :
is the nth stage of the layer filtration for a presheaf of T spectra X. The fo*
*llowing
is then a consequence of Corollary 5.5 and Lemma 5.10:
Corollary 5.11.
(1) Suppose that K is a pointed simplicial presheaf. Then the map
j* : 1TK[n] ! UV (1TK[n])s
is a stable fequivalence.
(2) Suppose that X is a presheaf of T spectra. Then the map
j* : FnX ! UV (FnX)s
is a stable fequivalence for all n 0.
Lemma 5.12. Suppose that
X0 ! X1 ! . . .
is an inductive system of presheaves of T spectra such that all maps
j* : Xn ! U(V Xn)s
are stable fequivalences. Then the map
j* : lim!Xn ! UV (lim!Xn)s
n n
is a stable fequivalence.
74
Proof: It is, first of all, enough to assume that all Xn are stably ffibrant. *
*Recall
that we can find natural stable ffibrant models j : Xn ! (Xn)s (actually a
stable ffibrant model for the inductive system) such that the induced map lim*
*!j :
lim!Xn ! lim!(Xn)s is a cofibration and a stable fequivalence. In the diagram
V (lim!Xn) __________wjV (lim!Xn)s
 
V (lim!j) V (lim!j)s
u u
V (lim!(Xn)s) ________wjV (lim!(Xn)s)s
the map V (lim!j) is a cofibration and a stable fequivalence, so that the ind*
*uced
map V (lim!j)s of stable ffibrant models is a level fequivalence. The funct*
*or U
preserves level fequivalences, so that in the diagram
lim!Xn __________wj*UV (lim!Xn)s
 
lim!j UV (lim!j)s
u u
lim!(Xn)s ________wj*UV (lim!(Xn)s)s
one sees that one instance of j* is a stable fequivalence if and only if the o*
*ther is.
Now suppose that all Xn are stably ffibrant. There is a diagram
limUj
lim!UV (Xn)'_____________________w!lim!U(V (Xn))s'
lim j NP '~ '~
! N ')= ')=
N U(lim j)
lim!Xn' U(lim!V (Xn)) ______________________w!U(lim!V (X*
*n)s)
' [ A
j ') [~= A U"j
[^ AD
UV (lim!Xn) ______________________wUjUV (lim!Xn)s
where the displayed isomorphisms are canonical and "jis chosen to make the fol
lowing diagram commute:
lim j
lim!V (Xn) ________w!lim!V (Xn)s
~  "
=  j
u u
V (lim!Xn) ________wjV (lim!Xn)s
75
Recall in particular that the map lim!j is a cofibration and a stable fequiva*
*lence,
so that the map "jis a stable fequivalence. This map "jmust also be a level
fequivalence since all objects V (Xn)s are stably ffibrant.
Finally all maps j* : Xn ! UV (Xn)s are stable fequivalences by assumption.
We can factorize the natural transformation j* so that there are commutative
diagrams
Xn h h
 h hihj

j*
 4 Zn
 4 4
u 47 p
UV (Xn)s
the maps i are ftrivial cofibrations such that lim!i is a ftrivial cofibrati*
*on, and the
maps p are stable ffibrations. In effect, choose i and p inductively by factor*
*izing
the map
Zn [Xn Xn+1 ! UV (Xn+1 )s
induced by the composite
Zn p!UV (Xn)s ! UV (Xn+1 )s
as an ftrivial cofibration jn : Zn [Xn Xn+1 ! Zn+1 followed by a stable ffibr*
*ation
p : Zn+1 ! UV (Xn+1 )s. Each p is a stable fequivalence of stable ffibrant ob*
*jects
and is therefore a pointwise equivalence in all levels, so that lim!p is a poi*
*ntwise
equivalence in all levels. It follows that the map lim!j* is a stable fequiv*
*alence,
and so the composite map
lim!Xn j!UV (lim!Xn) Uj!UV (lim!Xn)s
is a stable fequivalence.
Corollary 5.11 and Lemma 5.12 together imply the following:
Proposition 5.13. The natural map j* : X ! U(V X)s is a stable fequivalence
for all presheaves of T spectra X.
Theorem 5.14. The functors U and V induce an adjoint equivalence of stable
homotopy categories
U" : Ho (Pre SptT (Smk)Nis) o Ho (Pre SptT (Smk)Nis) : "V
76
Proof: The functor V preserves ftrivial cofibrations and stable fequivalences
between cofibrant objects. Define a functor
V" : Ho (Pre SptT (Smk)Nis) ! Ho (Pre SptT (Smk)Nis)
by setting "VX = V (Xc) where Xc ss!X is a choice of cofibrant model for X. The
map ss is chosen to be a level ffibration and a level fequivalence; V takes *
*level
fequivalences to stable fequivalences, so that V (ss) is a stable fequivalen*
*ce.
The functor U preserves level fequivalences and stable ffibrations. Define*
* a
functor
U" : Ho (Pre SptT (Smk)Nis) ! Ho (Pre SptT (Smk)Nis)
by setting "UY = U(Ys) where j : Y ! Ys is a choice of stable ffibrant model f*
*or
Y .
From the definitions, "U"VX = UV (Xc)s, and there is a natural map X ! "U"VX
in the homotopy category given by the maps
X ssXc j*!U(V (Xc))s
The map j* is a stable fequivalence by Proposition 5.13, so that the map X !
U""VX is a natural isomorphism in the homotopy category.
Similarly, "V"UY = V (U(Ys)c), and there are maps
V (U(Ys)c) V(ss)!V U(Ys) ffl!Ys j Y
I claim that the map ffl : V U(Ys) ! Ys is a stable fequivalence. To see this,*
* form
the diagram
V U(Ys) ________wj(V U(Ys))s
N
 N
ffl N N "
u NQ j
Ys
where the map "jexists, making the diagram commute, since j is an ftrivial cof*
*i
bration and Ys is stably ffibrant. Apply the functor U to see the diagram
U(Ys)a________wjeU(V U(Ys)) ________wUjU(V U(Ys))s
aeae N
1 aeaeo Uffl N N
u NQN U"j
U(Ys)
77
The composite Uj.j : U(Ys) ! U(V U(Ys))s is a stable fequivalence by Propositi*
*on
5.13, so that U"jis a stable fequivalence of presheaves of T spectra. But the*
*n "jis a
stable fequivalence of presheaves of symmetric T spectra, and so ffl : V U(Ys*
*) ! Ys
is a stable fequivalence.
We have seen that the natural maps X ! U""VX and "V"UY ! Y are isomor
phisms in the respective homotopy categories _this gives the desired equivalence
of homotopy categories. One can manually show that "V is left adjoint to "U, a*
*nd
that we've already found the canonical maps for the adjunction.
Appendix: Properness
Suppose that C is a small Grothendieck site, and let ff be a cardinal which i*
*s an
upper bound for the cardinality of the set Mor (C) of morphisms of C. Suppose t*
*hat
I is a simplicial presheaf on C having a rational point f : * ! I. This map f i*
*s a
cofibration, and we are entitled to a corresponding flocalization homotopy the*
*ory
for the category SP re(C), according to the results of [2].
By this, one means in part that there is a natural transformation jX : X ! LX,
where LX is a globally fibrant simplicial presheaf such that LX ! * has the rig*
*ht
lifting property with respect to all inclusions
(A.1) * x LU n [*xY I x Y I x LU n
arising from all subobjects Y LU n. Further, LX is constructed from X via
a transfinite small object argument which is based on the inclusions (A.1), and*
* is
subject to controls on cardinality in such a way that the properties L1  L7 of
Section 1 hold for choices of cardinals and such that = 2 and > 2ff.
One says that a simplicial presheaf Z is flocal if Z is globally fibrant, an*
*d the
map Z ! * has the right lifting property with respect to all inclusions (A.1). *
* It
follows that Z ! * has the right lifting property with respect to all inclusions
(* x B) [(*xA) (I x A) I x B
arising from cofibrations A ! B. It follows, in particular, that the map
f* : hom (I x Y; Z) ! hom (* x Y; Z)
is a weak equivalence for all simplicial presheaves Y if Z is flocal, and henc*
*e that
all induced maps
hom (I x LU n; Z) ! hom ((I x Y ) [(*xY )(* x LU n); Z)
are trivial fibrations of simplicial sets.
78
By construction, the simplicial presheaf LX is flocal, and the map jX : X !
LX induces a trivial fibration
j*X: hom (LX; Z) ! hom (X; Z)
for all flocal objects Z.
A simplicial presheaf map g : X ! Y is an fequivalence if the induced map
g* : hom (Y; Z) ! hom (X; Z)
is a weak equivalence of simplicial sets for all flocal objects Z. The origina*
*l map
f : * ! I is an fequivalence. More generally, we have seen that the maps
f x 1Y : * x Y ! I x Y
and the inclusions
(* x B) [(*xA) (I x A) I x B
are fequivalences. The canonical map jX : X ! LX is also an fequivalence.
A map p : X ! Y is an ffibration if it has the right lifting property with
respect to all cofibrations of simplicial presheaves which are fequivalences. *
*It is a
consequence of Theorem 4.6 of [2] that the category SPre (C) with the cofibrati*
*ons,
fequivalences and ffibrations, together satisfy the axioms for a closed simpl*
*icial
model category.
The goal of this section is to show that the flocal closed model structure on
SPre (C) is proper, for any such map f : * ! I.
Let D be a simplicial presheaf, and write f : D ! D x I for the composite
D ~=D x * 1Dxf!D x I:
Lemma A.2. Suppose given maps
D f!D x I g!X
and a global fibration ss : U ! X, and suppose that X is ffibrant. Then the
induced map
f* : U xX D ! U xX (D x I)
is an fequivalence.
79
Proof: To make the notation easier, given a map ff : V ! X, write Vff= U xX V
for the pullback of ff along ss : U ! X. In this notation, the statement of the
Lemma is the assertion that the induced map
f* : Dgf ! (D x I)g
is an fequivalence.
The object X is ffibrant and the projection map pr : D x I ! D is an f
equivalence, so there is a simplicial homotopy
0 1 d1
D x I ________wd(D x I) x u________D x I
 A A
pr  A
 h A A g
u uAD
D _______________wgfX:
Pulling back along the global fibration ss : U ! X gives a diagram
d0* 1 d1*
Dgf ____________w(D x )h(fx1) u__________Dgf
 
f*  f*
u u
(D x I)gf.pr________w(D x I x 1)h u________(D x I)g
d0* d1*
All of the maps labelled dffl*are local weak equivalences, since ss is a global*
* fibration
and the ordinary closed model structure for SPre (C) is proper. It therefore su*
*ffices
to show that the map f* : Dgf ! (D x I)gf.pris an fequivalence.
But the map gf . pr factors through the projection map pr, so that there is an
isomorphism ~
: (D x I)gf.pr=! Dgf x I
and a commutative diagram
Dgffl
fl
 flflf
f*  flflffl
u ~
(D x I)gf.pr________w=Dgf x I
80
The map f* is therefore an fequivalence.
An elementary ftrivial cofibration is a member of the saturation of the fami*
*ly
of cofibrations
(* x LU n) [(*xY )(I x Y ) I x LU n
and all maps
C ,! D
which are cofibrations and local weak equivalences, where D is ffbounded. An
finjective fibration is a map p : X ! Y which has the right lifting property *
*with
respect to all morphisms in the set of cofibrations of this form.
Lemma A.3.
(1) An finjective fibration p is a global fibration.
(2) The class of finjective fibrations is closed under composition.
(3) A simplicial presheaf Z is flocal if and only if the map Z ! * is an f
injective ffibration.
(4) Every simplicial presheaf map g : X ! Y has a factorization
Xh____________wgY
hhj ')
j ' 'q
W
where q is an finjective ffibration and j is an elementary fcofibrati*
*on and
an fequivalence.
(5) Every elementary fcofibration is an fequivalence.
Proof: Part (4) is the consequence of a standard transfinite small object argu
ment.
The family of maps having the left lifting property with respect to all finj*
*ective
ffibrations is a saturated class containing the generating elementary fcofibr*
*ations,
so that the elementary fcofibrations have the left lifting property with respe*
*ct to
all injective fibrations. It follows from the factorization statement (4) that*
* every
elementary fcofibration is a retract of an elementary fcofibration which is an
fequivalence. But then every elementary fcofibration is an fequivalence, giv*
*ing
(5).
Now we can list some consequences of Lemmas A.2 and A.3:
81
Lemma A.4. Suppose given maps
C j!D g!X
and a global fibration ss : U ! X, and suppose that X is ffibrant and j is an
elementary fcofibration. Then the induced map
j* : U xX C ! U xX D
is an fequivalence.
Proof: The class of cofibrations C ,! D ! X over X which pull back to f
equivalences U xX C ! U xX D is saturated by exactness of pullback, and con
tains all ordinary trivial cofibrations since the standard closed model structu*
*re on
SPre (C) is proper.
In any diagram
Y _______________wLU n[

  [
f f* [
u u [ f
I x Y ________wNN(I x Y ) [Y LU [n '
N N N N N [ '
N N N N N [] '
N N NNP '')
I x LU n
g

u
X
the maps f and f* pull back to fequivalences along ss by Lemma A.2, and so
pulls back to an fequivalence along ss. This means that all generators of the *
*family
of elemetary fcofibrations pull back to fequivalences along ss, so all elemen*
*tary
fcofibrations pull back to fequivalences along ss.
Corollary A.5. Suppose given a pullback diagram
A xX U ________wg*U

 ss
 
u u
A ___________wgX
where X is ffibrant, g is an fequivalence and ss is a global fibration. Then*
* the
induced map g* is an fequivalence.
82
Proof: Find a factorization
Ah____________wgX
hhj ')
j ' 'q
W
of g, where j is an elementary fcofibration and q is an finjective fibration.*
* Then
W is ffibrant by Lemma A.3, and the fact that the classes of ffibrant objects*
* and
finjective objects coincide. Thus q is an fequivalence of ffibrant objects, *
*and is
therefore an ordinary local weak equivalence, and hence pulls back to a local w*
*eak
equivalence along the global fibration ss. But then the elementary fcofibratio*
*n j
pulls back to an fequivalence by Lemma A.4.
Theorem A.6 (Properness). Suppose given a diagram
A xX U ________wg*U

 ss
 
u u
A ___________wgZ
such that ss is an ffibration and g is an fequivalence. Then the induced map *
*g*
is an fequivalence.
Proof: Form a diagram
U _________wiV
 p
ss 
u u
Z ________wjLZ
such that i is a cofibration and an fequivalence, LZ is ffibrant, p is an ff*
*ibration
and j is a cofibration and an fequivalence. Consider the pullback diagram
Z xLZ V _________wj*V

p  p
*  
u u
Z ___________wjLZ
83
The map j* : Z xLZ V ! V is an fequivalence by Corollary A.5. The induced
comparison
Ua_________weZ xLZ V
ae N
ssaeo Np*
NQ
Z
is an fequivalence of ffibrant objects in SPre (C) # X, hence a homotopy equi*
*v
alence, and so the map is a local weak equivalence. Properness for the standard
closed model structure on SPre (C) implies that the induced map
A xZ U *! A xLZ V
is a local weak equivalence. Thus, in the diagram
A xZ U ____________wg*U

 
*  
u u
A xLZ V ________w0Z xLZ V
g
the map g* is an fequivalence if and only if g0 is an fequivalence. But the m*
*aps
j*g0 and j* are fequivalences by Corollary A.5, so g0 is an fequivalence.
Theorem A.6 is not the full properness assertion for the flocal theory, but *
*it
is the heart of the matter. The second half of the properness axiom asserts th*
*at
the class of fequivalences is closed under pushout along cofibrations. This me*
*ans
that, given a pushout diagram
A ___________wgC
 
i 
u u
B ________wg*B [A C
with i a cofibration and g an fequivalence, the map g* should be an fequivale*
*nce.
This is easily proved: the functor hom ( ; W ) takes pushouts of simplicial pre*
*sheaves
to pullbacks of simplicial sets, and the map i* : hom (B; W ) ! hom (A; W ) i*
*s a
fibration and g* : hom (C; W ) ! hom (A; W ) is a weak equivalence if W is f*
*local.
Properness for ordinary simplicial sets implies that the induced map
g**: hom (B [A C; W ) ! hom (B; W )
is a weak equivalence of simplicial sets. This is true for all flocal objects*
* W , so
that g* is an fequivalence.
84
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