A MODEL STRUCTURE FOR INCLUSION PRESPECTRA IN

        WHICH THE FIBRANT OBJECTS ARE SPECTRA



                            MICHAEL COLE



                              Contents

  1.  Introduction                                                    1
  2.  Categories of prespectra and spectra                                1
  3.  Homotopy groups of prespectra                                    3
  4.  Model structures for spaces, prespectra, and spectra                   4
  5.  The spectral model structure for inclusion prespectra                  5
  6.  Spectral maps of prespectra                                       6
  7.  Spectral maps of inclusion prespectra                               8
  8.  Spectral fibrations and cofibrations                                11
  9.  The proofs of the model structure                                 13
  References                                                         14



                           1.Introduction

  Among the many possible ways to construct a category equivalent to Boardman's
stable category, the approach of Lewis and May [5] is very convenient for point*
 * set
analysis of spectra. Their category S of spectra has good formal properties bef*
 *ore
passage to the stable category and it arises from an easily understood category
of prespectra P.  In contrast to other categories of spectra, limit and function
spectrum constructions in S are simple and concrete, but colimit and smash prod*
 *uct
constructions are subtle due to the mysterious behavior of the spectrification *
 *functor
L: P ! S.
  There is an old idea, mentioned to me by Mark Hovey, that it might be pos-
sible to construe spectrification as fibrant replacement with respect to a suit*
 *able
Quillen closed model structure on the category P of prespectra. In this paper we
give a partial solution to this problem. The category I of inclusion prespectra*
 * is
intermediate between P and S. We define a Quillen closed model structure on I
for which the fibrant objects are the spectra. The homotopy category associated
to this model structure is equivalent to the stable category.


____________
   Date: Nov 8, 1999.
   1991 Mathematics Subject Classification. Primary 55P42, 55U35.
   Key words and phrases. model category, homotopy category, spectrum, inclusio*
 *n prespectrum,
spectrification.
                                  1




              2. Categories of prespectra and spectra

  Let T denote the category of based compactly generated weak Hausdorff spaces.
If V and W are real inner product spaces we write V  W or W  V to denote
that V  is a finite dimensional subspace of W .  We write V  < W or W > V  to
exclude the possibility that V = W . When V  W we will let W - V denote the
orthogonal complement of V in W .  Let U be a universe_i.e.  U is a countably
infinite dimensional real inner product space. A prespectrum E indexed on U is a
collection of based spaces {EV 2 T }V <U together with based maps

                 oeV;W :W-V EV ! EW;   V  W < U

that satisfy the evident transitivity condition. A map of prespectra f :D ! E is
a collection of based maps {fV :DV ! EV }V <U that strictly commute with the
structure maps for D and E.  Since we will not need to consider more than one
universe, we leave U out of our notation and write simply P for this category.
  A prespectrum E is said to be an inclusion prespectrum if the adjoint structu*
 *re
maps
                 eoeV;W:EV ! W-V EW;   V  W < U

are inclusions (k-ified if necessary since we are working with compactly genera*
 *ted
spaces). A prespectrum E is said to be a spectrum if each adjoint structure map
eoeV;Wis a homeomorphism. We write I and S respectively for the full subcategor*
 *ies
of P consisting of the inclusion prespectra and spectra.  Observe that we have
inclusions of categories S  I  P.
  The technical lynch-pin of the theory of spectra is the following.

Proposition 2.1.There exists a functor L0:I ! S that is left adjoint to the
inclusion S ! I.  There exists a functor L00:P ! I that is left adjoint to the
inclusion I ! P. Hence the composite functor L = L0O L00:P ! S is left adjoint
to the inclusion S ! P.

  The functor L0 was constructed by Peter May [7].  The functor L00is due to
Gaunce Lewis [4]. Generally speaking, the technical complexities that arise in *
 *the
point set analysis of spectra are due to the mysterious behavior of the Lewis f*
 *unctor
L00.

Proposition 2.2.The categories P, I, and S, are bicomplete and are enriched,
tensored, and cotensored over T . In detail, let A denote any of these categori*
 *es.
The hom sets A(D; E) have natural basepoints and natural topologies (as spaces *
 *in
T ). There exist bifunctors

                  ^: A x T -! A      (smashproduct)

           F (-; -): T opx A -! A      (function(pre)spectrum)

that satisfy natural adjunction isomorphisms

              A(D ^ X; E) ~=A(D; F (X; E)) ~=T (X; A(D; E)):

  For prespectra, the limits, colimits, smash products, and function spectra are
defined spacewise in the evident way. The prespectrum level constructions of li*
 *mits
and function prespectra preserve inclusion prespectra and spectra.  Colimits and
smash products with spaces are defined for I (respectively S) by applying the
functor L00(respectively L) to the constructions in P. See [5] for more details.
                                  2




  In our subsequent work we will need the explicit construction of the May func*
 *tor
L0. Let D be any prespectrum. We define the prespectrum L0D by

                    (L0D)V = colimV Z<U Z-V DZ

where the colimit is taken over the maps

                   Z1-VeoeZ1;Z2Z -VZ -Z       Z -V
          Z1-V DZ1 ---------!   1   2  1DZ2 ~= 2   DZ2

for V  Z1 < Z2 < U. Notice that if D is an inclusion prespectrum then the maps
in the colimit are inclusions. The adjoint structure maps eoeV;Wfor L0D are giv*
 *en
by the compositions

           (L0D)V = colimZZ-V DZ ~=colimZW-V Z-W DZ

               -! W-V colimZ Z-W DZ = W-V (L0D)W:

Observe that if D is an inclusion prespectrum then L0D is a spectrum since the
functor W-V  commutes with colimits of sequences of inclusions.
  Thus we have a functor L0:P ! P. The evident maps i0D:D ! L0D define a
natural transformation i0:idP ! L0. Restricted to inclusion prespectra, the May
construction defines a functor L0:I ! S that is left adjoint to the inclusion S*
 * ! I.
For an inclusion prespectrum D and a spectrum E the adjunction isomorphism is
described by
                                          0)*
                  S(L0D; E) = I(L0D; E) (iD---!I(D; E):


                 3. Homotopy groups of prespectra

  We recall the definition of the homotopy groups ssnD of a prespectrum D where
n is any integer (positive, zero, or negative).  Let [ ; ] denote based homotopy
classes of maps of based spaces.

Definition 3.1.For a prespectrum D and n  0,

                     ssnD = colimZ<U [SZ ^ Sn; DZ]

where the colimit is taken over the evident maps. For an indexing space V < U we
denote
                    ss-V D = colimV Z<U [SZ-V ; DZ]:

It turns out that, up to natural isomorphism, the functor ss-V depends only on *
 *the
dimension of V . Thus we may unambiguosly define for n  0

                     ss-n D = ss-V D;   dim V = n:

We have functors ssn :P ! Ab where Ab denotes abelian groups and n is any
integer.

  We call a map of prespectra f :D ! E a ss*-isomorphism if for each n 2 Z,
f*: ssnD ! ssnE is an isomorphism. The following facts are well known and, in a*
 *ny
case, are easily proved.

Proposition 3.2.Let E be a spectrum. If n  0 and V < U is any indexing space,
or if n < 0 but n + dim V  0, then there is a natural isomorphism

                          ssnE ~=ssn+dimV EV

where the right side denotes the usual space level homotopy group of the based *
 *space
EV .
                                  3




Corollary 3.3.A map f :D ! E of spectra is a ss*-isomorphism if and only if
it is a level weak equivalence (meaning that for each V < U, fV :DV ! EV is a
weak equivalence of spaces).

Proposition 3.4.If D is an inclusion prespectrum then the natural map iD :D !
LD is a ss*-isomorphism.

Proof.This follows from the construction of the May functor and the fact that t*
 *he __
space level functor ss* commutes with colimits of sequences of inclusions.     *
 *  |__|

Corollary 3.5.A map f :D ! E of inclusion prespectra is a ss*-isomorphism if
and only if Lf :LD ! LE is a level weak equivalence of spectra.

Proof.Apply the preceeding results to the diagram

                                 iD
                             D _____//LD

                            f ||      |Lf|
                              fflffl| fflffl|
                             E __iE_//LE:


                                                                    |___|


      4. Model structures for spaces, prespectra, and spectra

  Recall from [8] that a closed model structure on a category A consists of thr*
 *ee
classes of morphisms of A which we will denote by W, F, and C (weak equivalence*
 *s,
fibrations, and cofibrations respectively). We also write AF  for the class of *
 *acyclic
fibrations (W \F) and AC for the class of acyclic cofibrations (W \C). These cl*
 *asses
are required to satisfy axioms that are deliberately reminiscent of properties *
 *of space
level homotopy equivalences, fibrations, and cofibrations.  There is a "homotopy
category" Ho A associated to a closed model structure on A. Ho A is equivalent
to the localization A[W-1] of A at the weak equivalences. Excellent expositions*
 * of
the theory of model structures can be found in [2] and [3].
  The standard model structure for based spaces is the following.

Proposition 4.1.In the category T let WT denote the class of weak equivalences
in the usual sense (maps that induce isomorphism in homotopy groups). Let FT
denote the class of Serre fibrations.  Let CT denote the class of maps that have
the LLP (left lifting property) with respect to WT \ FT . These classes compris*
 *e a
Quillen closed model structure on T .

  All objects are fibrant with respect to this model structure. The cofibrant o*
 *bjects
are the retracts of cell complexes. Any of the various methods of CW or cellular
approximation of spaces defines a cofibrant replacement functor.
  The Lewis-May construction of the stable category closely parallels the above
approach to based spaces. Although Lewis and May did not use the language of
model categories in their work, their construction may be stated as follows.

Proposition 4.2.In the category S let WS denote the class of maps of spectra
that are level weak equivalences (equivalently ss*-isomorphisms). Let FS denote*
 * the
class of level Serre fibrations. Let CS denote the class of maps that have the *
 *LLP
with respect to WS \ FS. These classes form a Quillen closed model structure on
S. The associated homotopy category is equivalent to the stable category.
                                  4




  All spectra are fibrant with respect to this model structure. There is a theo*
 *ry of
cellular spectra and CW spectra that closely imitates the space level theory. T*
 *he
cofibrant objects are the retracts of cellular spectra.
  Finally, we consider two model structures on P. The simpler of the two is cal*
 *led
the level model structure.

Proposition 4.3.In the category P let W` denote the class of level weak equiv-
alences. Let F` denote the class of level Serre fibrations. Let C` denote the c*
 *lass
of maps that have the LLP with respect to W`\ F`. These classes form a Quillen
closed model structure on P.

  The level model structure again has the property that all objects are fibrant.
The level structure has the virtue of simplicity but, unfortunately, the associ*
 *ated
homotopy category is NOT equivalent to the stable category.  It does, however,
restrict to the "right" model structure for spectra. The level structure is a u*
 *seful
stepping stone for the proofs of the stable model structure on P which consists*
 * of
the following.

Proposition 4.4.In the category P let Wstdenote the class of ss*-isomorphisms.
Let Cstdenote the class of maps that have the LLP with respect to Wst\F`. Let F*
 *st
denote the class of maps that have the RLP with respect to Wst\ Cst. These clas*
 *ses
form a Quillen closed model structure on P. The associated homotopy category is
equivalent to the stable category.

  Recall that a prespectrum D is said to be an -spectrum _ more logically, D
should be called an -prespectrum _ if the adjoint structure maps eoeV;W:DV !
W-V DW are weak equivalences. It can be shown that the fibrant objects with
respect to the stable model structure are the -spectra. More generally, we have
the following fact.

Proposition 4.5.A map of prespectra f :D ! E is a stable fibration if and only
if it is a level fibration and for each V  W < U the square

                        DV  _____//W-V DW

                       fV ||          |W-V|fW
                          fflffl|     fflffl|
                        EV  _____//W-V EW

is a weak homotopy pullback.

  The stable model structure is a topological analogue of the Bousfield-Friedla*
 *nder
model structure which was originally conceived for prespectra made out of simpl*
 *icial
sets [1].  An elegant treatment of the level and stable model structures can be
achieved by slightly adapting the results of Mandell, May, Schwede, and Shipley
[6].  Their paper also gives far-reaching generalizations of these model struct*
 *ures
which apply to other categories of prespectra.



     5. The spectral model structure for inclusion prespectra

  We define now our proposed model structure for inclusion prespectra which we
shall call the spectral model structure. Most of the remainder of this paper wi*
 *ll be
devoted to the proof of the following.
                                  5




Proposition 5.1.For the category I of inclusion prespectra let Wspecdenote the
class of ss*-isomorphisms. Let Fspecdenote the class of maps that are level Ser*
 *re
fibrations and that have the property that whenever V  W < U the square


                        DV  _____//W-V DW

                       fV ||          |W-V|fW
                          fflffl|     fflffl|
                        EV  _____//W-V EW

is a pullback of spaces. Let Cspecdenote the class of maps that have the LLP wi*
 *th
respect to Wspec\ Fspec. These classes constitute a Quillen closed model struct*
 *ure
on I. The associated homotopy category is equivalent to the stable category.

  Observe that if D is any inclusion prespectrum (indeed, any prespectrum) then
the final map D ! * is always a level Serre fibration. Hence D ! * is a spectral
fibration if and only if the pullback condition is satisfied _ equivalently, if*
 * and only
if D is a spectrum. It follows that the fibrant objects with respect to the spe*
 *ctral
model structure are the spectra.
  Note the similarities between the spectral model structure for I and the stab*
 *le
model structure for P. If we weaken our definition of spectral fibration and re*
 *quire
only that the square is a weak homotopy pullback, we recover the class of stable
fibrations. It can be shown that the stable and level model structures for P re*
 *strict
to model structures on I. We have the following comparison of model structures
on I.

                           W`  Wst= Wspec

                           Fspec Fst F`

                         AF spec AF st= AF `

                            C` = Cst Cspec

                         AC ` ACst ACspec:



                   6.Spectral maps of prespectra

  We formalize the pullback condition in our definition of spectral fibration.

Definition 6.1.A map of prespectra f :D ! E is spectral if whenever V  W <
U the square

                        DV  _____//W-V DW

                       fV ||          |W-V|fW
                          fflffl|     fflffl|
                        EV  _____//W-V EW

is a pullback of spaces.

  Thus a spectral fibration is a spectral map of inclusion prespectra that is a*
 * level
fibration. Notice that if E is a spectrum, then f is spectral if and only if D *
 *is also
a spectrum. In particular, the final map D ! * is spectral if and only if D is a
spectrum.
  The following proposition will not be explicitly needed in the sequel, but we
state it anyway since it explains intuitively the relevance of spectral maps to*
 * model
                                  6




structures. If  denotes a commutative square


                              X __h__//D>>"

                             f || ""  |g|
                               fflffl|fflffl|"
                              Y __k__//E


in any category (without the dashed arrow as yet), let lift denote the set of l*
 *ifts
for  (morphisms Y ! D that complete the diagram). Thus

               lift = {`: Y ! D | ` O f = h and g O `}=:k

If  is a commutative square of prespectra and V < U we write V for the square
in T consisting of the V th spaces.

Proposition 6.2.If  is a commutative square


                              X __h__//D

                             f ||     |g|
                               fflffl|fflffl|
                              Y __k__//E

of prespectra and the map g is spectral, then there exist canonical transformat*
 *ions

                       OEV;W :liftW -! liftV

defined for V  W < U that satisfy the transitivity condition

                          OEV;W O OEW;Z = OEV;Z:

Thus {liftV }V <U is an inverse system of sets with respect to the maps OEV;W. *
 *If
 V denotes the evident map lift ! liftV then OEV;W O  W  =  V . The map

                     { V }: lift -! limV <UliftV

is an isomorphism.

  Intuitively this proposition states that if g is spectral, then the set of li*
 *fts for
depends only on the cofinal properties of the prespectra X and Y and the map f.
This is analogous to the fact that if E is a spectrum and D is a prespectrum th*
 *en

                     P(D; E) ~=limV <UT (DV; EV )

where the limit is taken over suitable maps. Thus spectral maps of prespectra a*
 *re
to lifts of squares of prespectra as spectra are to maps of prespectra. We omit*
 * the
proof of Proposition 6.2 since we will not need to explicitly use it.
  We will, however, need the facts that the class of spectral maps is closed un*
 *der
retracts and pullbacks.

Proposition 6.3.A retract of a spectral map is spectral.

Proof.Suppose that f :X ! Y is a retract of the spectral map g :D ! E. Consider
the diagrams

         XV  ____//_W-V XW              DV  ____//_W-V DW

        fV ||          W-V|fW|         gV||          |W-V|gW
           fflffl|     fflffl|           fflffl|     fflffl|
         Y V _____//W-V Y W             EV _____//W-V EW
                                  7




and note that the left square is a retract of the right square. It is easily pr*
 *oved_that
in any category the retract of a pullback square is a pullback square.         *
 * |__|


  The following lemma either is well known or should be well known. Its proof is
straightforward.

Lemma 6.4.  Given a commutative cubical diagram in any category

                      A _____//PPPCPPPP
                       |   PPPPP|  PPPPP
                       |      |PPPP    PPP
                       fflffl|fflffP''Pl| PP((P
                      B _____//PPPPDPPPPA0__//_C0
                            PPP    PPPPP|     |
                              PPPP    |PPPP   |
                                  PP''PfflPP((ffl|fflffl|
                                     B0 ____//_D0

if the squares A0B0C0D0, AA0CC0, and BB0DD0 are pullbacks, then the square
ABCD is also a pullback.

Proposition 6.5.If a square of prespectra

                              X _____//D

                             f ||     |g|
                               fflffl|fflffl|
                              Y _____//E

is a pullback and g is spectral, then f is spectral.

Proof.Apply the preceeding lemma to the diagram

          XV  ___________//WWWWWWYWVWWWWW
            |       WWWWWWWWW|     WWWWWWWWW
            |              |WWWWWW         WWWWWW
            fflffl|        fflffl|WWWWWW++W      WWWWWW++W
       W-V XW    _____//WWWWW-V YWWWWW   DV ___________//EV
                     WWWWWW         WWWWWWWWW|           |
                          WWWWWWW         | WWWWWW       |
                                WWWWW++   fflffl|WWW++   fflffl|
                                     W-V DW    _____//W-V EW

(using the fact that the functor W-V  preserves pullbacks).                  |_*
 *__|


  We list without proof some further properties of spectral maps.

Proposition 6.6.Let  be a small category, let D; E : ! P be functors, and
suppose that f :D ! E is a natural transformation such that for each  2  the
map f : D  ! E  is spectral. Then the induced map on limits

                        {f }: limD  -! limE

is spectral. If f :D ! E and g :E ! F are maps of prespectra and g is spectral,
then f is spectral if and only if g O f is spectral. If X 2 T is a space and f *
 *:D ! E
is a spectral map then the map

                        f*: F (X; D) -! F (X; E)

is spectral.

                                  8




              7. Spectral maps of inclusion prespectra

  We collect here some results that we will use in our proofs of the model stru*
 *cture.
We begin with a lemma on pullbacks of spaces.

Lemma 7.1.  Suppose that we are given sequences of spaces in T

                    A1 -! A2 -! . .-.! An -! : : :

                    B1 -! B2 -! . .-.! Bn -! : : :

                    C1 -! C2 -! . .-.! Cn -! : : :

                    D1 -! D2 -! . .-.! Dn -! : : :

and natural transformations f :A ! B, g :A ! C, h: B ! D, and k :C ! D
such that for each n the square

                                  gn
                             An _____//Cn

                            fn||       kn||
                              fflffl|  fflffl|
                             Bn __hn_//Dn


is a pullback in T . If the maps Bn ! Bn+1 and Cn ! Cn+1 are inclusions (k-ified
if necessary) then the maps An ! An+1 are also inclusions. If, moreover, the ma*
 *ps
Dn ! Dn+1 are monomorphic then the square

                                 {gn}
                        colimAn _____//colimCn

                        {fn}||           |{kn}|
                            fflffl|      fflffl|
                        colimBn _{hn}//_colimDn


is a pullback.

Proof.If B and C are sequences of inclusions, then in the diagram

                            {fn;gn}
                      An _____________//_Bn x Cn
                       |                   |
                       |                   |
                       fflffl|             fflffl|
                     An+1 _{f_______//_Bn+1 x Cn+1
                             n+1;gn+1}

the upper and right arrows are inclusions. Therefore the left arrow is an inclu*
 *sion
so we see that the sequence A also consists of inclusions.
  Now assume further that the maps Dn ! Dn+1 are monomorphic. Consider a
test diagram

                    X UUUUUI6
                      66I6IUUUUUUfiU
                        66 Ifl   UUUU
                         66  $$I     UUUU**
                         ff666colimAn{gn}//_colimCn
                            66 |             |
                             66|{fn}         |{kn}
                              6aeaefflffl|   fflffl|
                            colimBn _{hn}//_colimDn

                                  9




for which we must prove that the dashed arrow uniquely exists.  Since we are
working with compactly generated spaces, an elementary argument shows that it
suffices to prove the result for the case that X is compact.  We will need to u*
 *se
that fact that the colimit in T of a sequence of monomorphisms is the same as t*
 *he
colimit in the category of all topological spaces (for proof see [4], Appendix *
 *A).
Hence the natural maps into the colimits


                           Bm -! colimBn

                           Cm -! colimCn

                           Dm -! colimDn


are monomorphic.
  Now assume that X is compact.  There is a m large enough that ff factors
through a map eff:X ! Bm and fi factors through a map efi:X ! Cm . Since the
mth square is a pullback, there is a unique map efl:X ! Am that completes the
diagram

                        X BRRR1
                          11BRRRRRefiR1
                            1B1BefRRRRl
                            11  !! ___RR((R//_
                            eff11Am  gm  Cm
                              11 |        |
                               11|fm      |km
                                1fflffl|  fflffl|
                                Bm __hm_//Dm :


Then the composite

                        X -efl-!Am -! colimAn

is the solution fl to our original diagram and it is unique.                   *
 * |___|


  This lemma has the following consequence.


Proposition 7.2.The May functor L0:I ! S preserves finite limits.


Proof.We consider first pullbacks.  Let us have a pullback diagram of inclusion
prespectra.
                                  g
                              X _____//Z

                             f ||     |k|
                               fflffl|fflffl|
                              Y __h__//W


For V < U choose a cofinal sequence V1 < V2 < : :o:f indexing spaces in U such
that V  V1. Then we may apply Lemma 7.1 with


                           An = Vn-V XVn

                           Bn = Vn-V Y Vn

                           Cn = Vn-V ZVn

                           Dn = Vn-V W Vn
                                  10




Recalling the definition of L0we see that colimAn = (L0X)V , colimBn = (L0Y )V ,
colimCn = (L0Z)V , and colimDn = (L0W )V . Hence the square

                                 L0g
                            L0X _____//L0Z

                           L0f||       |L0k|
                              fflffl|  fflffl|
                            L0Y _L0h_//L0W


is a pullback. It follows now that L0preserves finite products since the isomor*
 *phism
L0(Y xZ) ~=L0Y xL0Z is just the above case for pullbacks when W = *. Therefore_
L0preserves all finite limits.                                             |__|

  A significantly stronger result, due to Gaunce Lewis, states that the spectri*
 *fica-
tion functor from injection prespectra to spectra preserves finite limits [5].
  Henceforward, for notational convenience we will write LD rather than L0D
even when D is understood to be an inclusion prespectrum. In our proofs of the
factorization axioms we will use the following special case of Proposition 7.2.

Corollary 7.3.Given a pullback square of inclusion prespectra of the form

                                  g
                             X  _____//_Z

                             f||       k||
                              fflffl|  fflffl|
                             D  __iD//_LD;


the map Lg :LX ! LZ is an isomorphism of spectra.

Proof.After spectrifying we see that Lg is a pullback of LiD , which is the_ide*
 *ntity
map on LD. A pullback of an isomorphism is an isomorphism.               |__|

Proposition 7.4.A map of inclusion spectra f :X ! Y is spectral if and only if
the square
                                 iX
                             X _____//LX

                            f ||      |Lf|
                              fflffl| fflffl|
                             Y __iY_//LY;

is a pullback.

Proof.Any map of spectra is spectral, so if the square is a pullback, it follow*
 *s by
Proposition 6.5 that f is spectral. Suppose now that f is spectral. We apply le*
 *mma
7.1 with

                           An = XV

                           Bn = Y V

                           Cn = Vn-V XVn

                           Dn = Vn-V Y Vn

where V1 < V2 < : :i:s a cofinal sequence of indexing spaces in U. Then colimAn*
 * =
XV , colimBn = Y V , colimCn = (LX)V , and colimDn = (LY )V . The conclusion_
follows.                                                              |__|
                                  11




               8.Spectral fibrations and cofibrations

  We begin with a well known fact about spaces.

Lemma 8.1.  Given a diagram in T

                 X1 _____//X2____//_:_:_://_Xn____//: : :

                 f1||      f2||            |fn|
                   fflffl| fflffl|         fflffl|
                  Y1_____//_Y2___//: :_:__//Yn____//: : :

if the maps Xn ! Xn+1 and Yn ! Yn+1 are inclusions and each fn is a (acyclic)
Serre fibration, then the induced map on colimits

                      {fn}: colimXn -! colimYn

is a (acyclic) Serre fibration.

Proof.Serre fibrations are the maps that have the RLP with respect to the level
zero inclusions (Dn)+ ! (Dn x I)+ and acyclic Serre fibrations are characterized
by the RLP with respect to the standard inclusions (Sn-1)+ ! (Dn)+ .  Since
(Sn-1)+ , (Dn)+ , and (Dn xI)+ are compact, the result follows easily from the *
 *fact_
that compact spaces are small with respect to sequences of inclusions in T .   *
 *  |__|

Proposition 8.2.If a map of inclusion prespectra f :D ! E is a level (acyclic)
fibration then Lf :LD ! LE is an (acyclic) fibration of spectra.

Proof.For V < U choose a cofinal sequence V1  V2  : :o:f indexing spaces in U
such that V  V1. Consider the diagram

                    V1-VDV1   _____//V2-VDV2 _____//: : :

                 V1-VfV1 ||             V2-VfV2||
                         fflffl|        fflffl|
                    V1-VEV1   _____//V2-VEV2 _____//: : :

Now we use the fact that the functors Vn-V preserve (acyclic) Serre fibrations._
The conclusion follows from Lemma 8.1.                                  |__|

  Combining Proposition 7.4 and Proposition 8.2 we have the following.

Corollary 8.3.A map of inclusion spectra f :D ! E is a spectral (acyclic) fibra-
tion if and only if f is the pullback of an (acyclic) fibration of spectra.

  Now we consider spectral cofibrations and spectral acyclic cofibrations.

Lemma 8.4.  If f :X ! Y  is a map of inclusion prespectra (or even arbitrary
prespectra) and g :D ! E is a map of spectra, then f has the LLP with respect to
g if and only if Lf has the LLP with respect to g.

Proof.By adjunction, the diagrams


                    X __h__//D>>"      LX _Lh__//D==-

                   f ||"`"  |g|       Lf||L-`-  |g|
                     fflffl|fflffl|"    fflffl|-fflffl|
                    Y __k__//E         LY _Lk__//E

are equivalent.                                                        |___|

                                  12




Corollary 8.5.A map of inclusion prespectra f :D ! E is a spectral (acyclic)
cofibration if and only if Lf :LD ! LE is a (acyclic) cofibration of spectra.


Proof.If f is in Cspecthen, by definition, f has the LLP with respect to AF spe*
 *c.
Certainly an acyclic fibration of spectra, regarded as a map of inclusion presp*
 *ectra,
is in AF spec. Thus f has the LLP with respect to acyclic fibrations of spectra*
 * and
therefore Lf does also. Hence Lf is a cofibration of spectra. Conversely, if Lf*
 * is
a cofibration of spectra, then Lf has the LLP with respect to acyclic fibration*
 *s of
spectra and therefore f does also.  Since spectral acyclic fibrations are pullb*
 *acks
of acyclic fibrations of spectra, f must have the LLP with respect to AF spec.
Therefore f is in Cspec. Moreover, we know from Corollary 3.5 that f is in Wspe*
 *cif_
and only if Lf is a weak equivalence of spectra.                            |__|



               9. The proofs of the model structure

  We are ready to prove Proposition 5.1. First note that our classes Wspec, Fsp*
 *ec,
and Cspeccontain all identity maps. Clearly Wspecsatisfies the 2 out of 3 prope*
 *rty.


Proposition 9.1.The classes Wspec, Fspec, and Cspecare closed under retracts.


Proof.This is obvious for Wspec.  Retracts of level fibrations are level fibrat*
 *ions
and, by Proposition 6.3, retracts of spectral maps are spectral.  Hence Fspecis
closed under retracts. The class Cspecis necessarily closed under retracts sinc*
 *e_it is
defined by the LLP with respect to the class AF spec.                       |__|


  We consider now the lifting properties. One we have as a matter of definition*
 * of
the class Cspec. The other is given by the following.


Proposition 9.2.The class AC spec= Wspec\ Cspechas the LLP with respect to
Fspec.


Proof.Let f 2 AC specand g 2 Fspec.  Since g is a pullback of the fibration of
spectra Lg it suffices to show that f has the LLP with respect to Lg.  But by
Lemma 8.4 this is equivalent to requiring that Lf has the LLP with respect to
Lg.  By Corollary 8.5 Lf is an acyclic cofibration of spectra and the conclusio*
 *n_
follows.                                                              |__|


  It remains to prove the factorization axioms.  The proofs of the two cases are
logically identical, so we give details for just one of the cases.


Proposition 9.3.Any map of inclusion prespectra f :D ! E admits a factoriza-
tion

                            D -i-!X -p-!E

where i is in AC specand p is in Fspec.


Proof.Consider a factorization of Lf in the category of spectra


                           LD -j-!Z -q-!LE
                                  13




where j is an acyclic cofibration of spectra and q is a fibration of spectra. W*
 *e have
a diagram
                          D0PPPA
                            0APPPPjOiDPP00
                             00AAi  PPP
                              00   ___PPP((//_
                             f 00X   ff  Z
                                0 |       |
                                00|p      |q
                                 0fflffl| fflffl|
                                 E __iE_//LE

where the square is a pullback and i: D ! X is the unique dashed arrow completi*
 *ng
the diagram. By Corollary 8.3 p is in Fspec. By Corollary 7.3 Lff: LX ! Z is an
isomorphism of spectra. It follows that

                    j = L(j O iD ) = L(ff O i) = Lff O Li:

Therefore Li = (Lff)-1 O j. Since j is an acyclic cofibration of spectra and (L*
 *ff)-1
is an isomorphism of spectra, Li is an acyclic cofibration of spectra. It follo*
 *ws_from
Corollary 8.5 that i is in AC spec.                                         |__|

  An identical argument establishes that:

Proposition 9.4.Any map of inclusion prespectra f :D ! E admits a factoriza-
tion
                            D -i-!X -p-!E

where i is in Cspecand p is in AF spec.

  This completes the demonstration that our spectral structure is a Quillen clo*
 *sed
model structure for I.  Moreover, it is easily argued that the associated homo-
topy category is equivalent to the stable category. Simply note that the inclus*
 *ion
prespectra that are both fibrant and cofibrant are the retracts of spectra that*
 * are
cellular in the sense of [5]. The model theoretic homotopy relation on the cate*
 *gory
of these objects agrees with the usual topological notion of homotopy defined in
terms of cylinders E ^ I+ .

                             References

[1]A.K. Bousfield and E.M. Friedlander. Homotopy Theory of -spaces, Spectra, an*
 *d Bisimplicial
  Sets. In Geometric Applications of Homotopy Theory (Proc. Conf. Evanston, Ill*
 *, 1977). M.G.
  Barratt and M.E. Mohawald, eds. Springer Lecture Notes in Math, Vol. 658, 197*
 *8.
[2]W.G. Dwyer and J. Saplanski. Homotopy theories and Model Categories. In Hand*
 *book of
  Algebraic Topology, ed. I.M. James. Elsevier, 1995.
[3]M. Hovey. Model Categories. Mathematical Surveys and Monographs, vol. 63. Am*
 *erican Math-
  ematical Society, Providence, RI, 1999.
[4]L.G. Lewis. The Stable Category and Generalized Thom Spectra. Thesis, Univer*
 *sity of Chicago,
  1978.
[5]L.G. Lewis, J.P. May, and M. Steinberger (with contributions by J.E. McClure*
 *). Equivariant
  Stable Homotopy Theory. Springer Lecture Notes in Mathematics, Vol. 1213, 198*
 *6.
[6]M.A. Mandell, J.P. May, S. Schwede, and B. Shipley. Model Categories of Diag*
 *ram Spectra.
  Preprint 1998.
[7]J.P. May. E1 Ring Spaces and E1 Ring Spectra. Springer Lecture Notes in Math*
 *ematics,
  Vol. 577, 1977.
[8]D.G. Quillen. Homotopical Algebra. Springer Lecture Notes in Mathematics, Vo*
 *l. 43, 1967.

  Department of Mathematics, Hofstra University, Hempstead, NY 11549
  E-mail address: matmzc hofstra.edu
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