MODEL STRUCTURES ON THE CATEGORY OF EX-SPACES



              MICHELE INTERMONT AND MARK W. JOHNSON


       Abstract.This paper describes several model structures on the categories*
 * of
       ex-spaces and ex-G-spaces when G is a compact Lie group. Two of these are
       of particular interest in that they have expected applications to the st*
 *udy of
       transfer maps and to parametrized spectra. These two structures are shown
       to coincide on the collection of Hurewicz fibrations, and an indication *
 *is also
       given, mainly via examples, of how they differ. The last two sections of*
 * this
       paper are mostly expository; they set forth the model category techniques
       needed to prove the main theorems.



                           1.Introduction

  The purpose of this article is to develop foundational material on the homoto*
 *py
of ex-spaces for several different applications.  While James has pursued some *
 *of
this foundational material in [3] and a variety of other works, he has avoided *
 *the
language of Quillen model categories.  Since this language is perhaps the close*
 *st
thing available to a standard language for modern homotopy theory, the current
article is intended to address this point. In fact, more than one reasonable no*
 *tion
for homotopy theory of ex-spaces is proposed here, which reflects the existence*
 * of
different model structures on the category of ex-spaces over a fixed base. An e*
 *ffort
is made to discuss the relationship between these structures, especially in Sec*
 *tion
6.
  Almost all of the model structures presented here arise in a two step process.
First, one "lifts" the model structure from the category of pointed spaces seve*
 *ral
times, using a different adjoint pair each time. (Lifting is a process which ex*
 *ploits an
adjoint pair and a known model category in order to build a new model structure*
 *.)
Then, one produces an "intersection" of these structures, which encodes informa*
 *tion
common to all of the structures.  Conditions on when such an intersection again
yields a model structure are given in Proposition 8.7. These conditions are sat*
 *isfied
in all cases of interest here.
  Two structures are distinguished. The "coarse" structure is familiar in the s*
 *ense
that weak equivalences are defined as weak equivalences on the total space and
similarly, cofibrations and fibrations are also defined by focusing on the tota*
 *l space.
The coarse structure is thus well-suited to retaining global information.  On t*
 *he
other hand, the "fine" structure involves building cell complexes in a local se*
 *nse,
hence this structure is well-suited to considering local information.  The clas*
 *s of
weak equivalences in the fine structure is defined in terms of spaces of local *
 *sections.

____________
   Date: December 1, 2000.
   1991 Mathematics Subject Classification. Primary: 55R70, 55U35; Secondary: 5*
 *5P91, 55U40.
   Key words and phrases. ex-spaces, Quillen model category, equivariant homoto*
 *py theory,
fibration.
                                  1



2                   M. INTERMONT AND M. W. JOHNSON


  There is an analog of Whitehead's Theorem for the "cell complexes" in each
structure. Between such objects, the model theoretic weak equivalences are shown
to coincide with the (pointed) fibrewise homotopy equivalences familiar from wo*
 *rk
of James and others. Also, for Hurewicz fibrations the two notions of weak equi*
 *v-
alences are shown to coincide in Theorem 6.5.  As a consequence, the homotopy
category associated to the coarse structure is essentially a subcategory of tha*
 *t as-
sociated to the fine structure.
  To some extent, the category of compactly generated, weak Hausdorff spaces
has become a standard convenient category of spaces in the language of Steenrod
[20]. However, the most naive category of such spaces over a base does not have
an exponential law. Two possible solutions are discussed here: restricting to o*
 *pen
maps and dropping the weak Hausdorff condition on the total space. The first re*
 *lies
upon work of Lewis in [14], while the latter depends upon work of Vogt [21], Bo*
 *oth
[1] and Day [4]. The primary focus here is on developing homotopy theory in the
context of Lewis's category of open ex-spaces.
  In spirit, this paper is divided into three main parts.  The first part, Sect*
 *ions
2 to 6, details the coarse and fine structures, as well as giving a brief synop*
 *sis
of Lewis's work. Throughout these sections, the reader will find several exampl*
 *es
and a minimum of model category technicalities. Section 7 constitutes the second
part.  Here, the generalizations to the equivariant setting are described.  Fin*
 *ally,
an introduction to the necessary facts on (topological) model categories is giv*
 *en in
the last two sections. Care has been taken throughout the earlier sections to a*
 *void
highly technical results or to defer the proofs to this third part of the paper*
 *. It is
hoped that this will make the results accessible to a broad audience. Those fam*
 *iliar
with model categories may find the last two sections of interest for its discus*
 *sion
of the notion of intersecting model structures, or the general approach of Theo*
 *rem
9.8.
  The authors wish to thank L. Gaunce Lewis, Jr. for suggesting the usefulness
of this work and for freely sharing his insight throughout its development. Tha*
 *nks
are also due to Wojciech Dorabiala for helpful conversations regarding the poss*
 *ible
geometric applications of this material.



                          2.Open Ex-Spaces

  The category o of compactly generated, weak Hausdorff topological spaces is
intended as the basic category of spaces throughout this article, unless otherw*
 *ise
specified, in order to avoid pathological topologies and to have an exponential*
 * law
on mapping spaces. A pointed space then refers to such a space together with a
chosen basepoint. The most straightforward notion of spaces over a base space B
would then be a space X together with a chosen map X ! B.  Unfortunately,
this notion proves to allow pathological behavior which destroys the exponential
law as well as damaging the relationship between spaces over B and spaces over
A associated to a map B ! A. In [14], Lewis shows that one possible solution is
restricting attention to X ! B where the map is an open map. This allows one to
recover the exponential law and gives three change of base functors associated *
 *to
any map B ! A which form two adjoint pairs. Another possible solution is to drop
the weak Hausdorff condition on the total space, as discussed in [1]. This dire*
 *ction
is addressed more in Section 4.



                        HOMOTOPY OF EX-SPACES                       3


  In [14], Lewis also discusses an analog of pointed spaces over a base. This c*
 *onsists
of an open map p : X ! B together with a choice of section s : B ! X, i.e. such
that ps = id. The idea is to think of the section as providing a continuous cho*
 *ice
of basepoint to the fibres. This is the category of open ex-spaces, where the p*
 *refix
"ex" is intended to make one think of cross-sections. Lewis shows that the cate*
 *gory
of open ex-spaces is closed symmetric monoidal under the fibrewise smash produc*
 *t.

Definition 2.1.  1.The category oB of ex-spaces over B, is the category whose
     objects are spaces over B, p : X ! B together with a choice of section
     s : B ! X of p. Morphisms are defined as maps which make the diagram

                                  B @
                              s""""  @@s0@
                                       @@
                            """""       @  
                          X @_____j______//Y
                             @@@       """
                              p@@@@ """p0"""

                                  B

     commute.
  2. The category O*(B) of open ex-spaces over B, is the full subcategory consi*
 *st-
     ing of those objects B s!X !pB for which p is an open map.

  Suppose f : B ! A is a map of spaces. Then given any X 2 oB , one may form
the following pushout diagram in spaces

                                  f
                             B ______//_A

                            s||       |s0|
                             |fflffl  |fflffl
                             X _____//f!(X)

which produces f!(X) 2 oA . The required map p0: f!(X) ! A is produced by the

universal property of the pushout from the maps X !p B !f A and id : A ! A,
which implies s0: A ! f!(X) is a section of p0as desired. Observe that if f : B*
 * ! A
and p : X ! B are open maps, then so is p0, because X q A ! A is an open map
by assumption, with the natural map X q A ! f!(X) clearly surjective.
  Similarly, given Y 2 oA , one may produce f*(Y ) 2 oB by the following pullba*
 *ck
diagram of spaces.

                            f*(Y )____//Y

                             p||       |p0|
                              fflffl|  fflffl|
                              B __f___//_A


As with f!, this construction produces an open ex-space provided f is an open
map and Y  2 O*(A).  The naturality of pushouts and pullbacks implies these
constructions are functorial.  These functors are related in a particularly use*
 *ful
way.

Lemma 2.2.  Given a map f : B ! A, the functors (f!; f*) form an adjoint pair
f! : oB ! oA and f* : oA ! oB (with f! the left adjoint).  If f is an open map,
then this adjoint pair restricts to an adjoint pair with f! : O*(B) ! O*(A) and



4                   M. INTERMONT AND M. W. JOHNSON


f* : O*(A) ! O*(B). Furthermore, both of the adjoint pairs are topological adjo*
 *int
pairs.

Proof.The adjunction is clear from the following commutative diagram and the
universal properties of pullbacks and pushouts.

                                  f
                         B|_______________//_Ay|

                         |             yyyy |
                         |           yy     |
                         |           __y    |
                         |      f!(X)       |
                         |      y<<  EE     |
                         |    yyy     EEE   |
                         |  yy          EE  |
                         fflffl|yy        ""fflffl|E
                         X|EE             <Y|<y

                         |  EEEE       yyyy |
                         |     EE    yy     |
                         |       ""  y      |
                         |      f*(Y )      |
                         |      y           |
                         |    yyy           |
                         |  yy              |
                         fflffl|__yyf       fflffl|
                         B _______________//_A

The statement that the adjoint pairs are topological adjoints may be summarized
by saying that the morphism sets in this category come equipped with topologies
so that the natural bijection one expects in an adjoint pair is, in fact, a nat*
 *ural
homeomorphism. This will be discussed further in Section 9.          __
                                                                    |__|


Example 2.3. Suppose A is a point. Since any map to a point is open, this impli*
 *es
O*(A) = oA = o*. Thus, the adjoint pair (f!; f*) associated to the collapse map
f : B ! * relates O*(B) to o*. The functor f!takes B !sX !pB to X=s(B) with
the collapsed image of s as basepoint.  For any pointed space M 2 o*, f*(M) is
given by B ~={m0} x B ! M x B ss2!B, henceforth denoted M x B.

  Proposition 3.2 and Remark 3.3 of [14] imply that the functor f* also has a r*
 *ight
adjoint in some cases.

Lemma 2.4.  Given an open map f : B ! A, there exists a topological adjoint pair
(f* : oA ! oB ; f* : oB ! oA ) (with f* the left adjoint). Furthermore this adj*
 *oint
pair determines a topological pair (f* : O*(A) ! O*(B); f* : O*(B) ! O*(A)).

  Returning to the example A = * discussed above, the right adjoint f* : O*(B) !
o* associated to f : B ! * sends B !s X !p B to the space SecB(X) of global
sections. That is, SecB(X) is the subspace of o(B; X) consisting of those s0: B*
 * ! X
such that ps0= id, given the section s associated to X as an object of O*(B) fo*
 *r a
basepoint.



                  3.Coarse Homotopy of Ex-Spaces

  The most natural place to begin in defining a model structure on the category
oB of ex-spaces over B or the category o=B of spaces over B is to simply forget
the role of B. The category o=B is equipped with a model category structure (see



                        HOMOTOPY OF EX-SPACES                       5


[10]) where a morphism

                                  f
                          X @____________//Y
                             @@        ""
                              @@@    """
                                @@  """"
                                  B

is a weak equivalence precisely when f is a weak equivalence of total spaces, a*
 *nd
similarly for cofibrations and fibrations. The analogous structure on oB is cal*
 *led
the coarse structure.  This model structure is cofibrantly generated, which is
an important notion throughout this article. Briefly, cofibrantly generated mod*
 *el
categories are equipped with a notion of cell complex which allows one to produ*
 *ce,
by a systematic method, the factorizations required in a model category.  This
is important if one is interested in producing similar model structures on more
complex categories such as ex-spectra. Furthermore, the coarse structure is (le*
 *ft)
proper, which allows one to apply localization techniques (see [9]) to produce a
stable model structure on the category of ex-orthogonal-spectra (see [17]).

Proposition 3.1.The category oB supports a cofibrantly generated, proper, topo-
logical model category structure with the weak eqivalences (cofibrations or fib*
 *rations)
defined as maps in oB which are weak homotopy equivalences (cofibrations or Ser*
 *re
fibrations) of total spaces in o.

  The generating cells in this case are simply Sn q B ! Dn+1 q B with any choice
of map Dn+1 ! B and the identity on B (to make the inclusion of that component
a section). Thus, the cardinalities of the sets map(Dn; B) determine the number*
 * of
generating cells in the coarse structure. Similarly, the generating acyclic cel*
 *ls are
of the form In q B ! In+1 q B with any choice of map In+1 ! B and the identity
on B. Notice that the form of the generating cells implies that cofibrant objec*
 *ts in
the coarse structure are essentially cell complexes (or retracts thereof).

Example 3.2. Construct an ex-space X(n) over the classifying space BU(n) of
the nth unitary group as follows. The canonical bundle EU(n) ! BU(n) is clearly
an open map. One logical way to produce an associated element of O*(BU(n)) is
by applying the fibrewise one-point compactification functor discussed in [3] t*
 *o this
canonical bundle. Clearly, this yields a sphere bundle X(n) ! BU(n) with section
(consisting of the points at infinity). However, considering the map f : BU(n) *
 *! *,
the pointed space f!(X(n)) should be more familiar as the Thom space of the
canonical bundle.

  Thus, Thom spaces provide a rich source of familiar examples of the change
of base functors.  It should also be emphasized that fibrewise operations arise*
 * in
a variety of geometric settings, which should be accessible using the techniques
developed here.
  There are several valuable technical properties of the coarse structure, disc*
 *ussed
in the following few lemmas. Possibly the most useful property of the coarse st*
 *ruc-
ture is left properness, which is important in the theory of localizing model c*
 *ate-
gories. (See [9].) The first result below is a standard fact concerning proper *
 *model
structures (see [10]).
  In order to state the next result properly, some terminology must be introduc*
 *ed.
Roughly speaking, an adjoint pair between model categories which descends to an
adjoint pair on the associated homotopy categories is a Quillen pair. A Quillen



6                   M. INTERMONT AND M. W. JOHNSON


equivalence refers to a Quillen pair where the induced adjoint pair on the asso*
 *ci-
ated homotopy categories is an equivalence of categories. (See [6] or [10] for *
 *formal
definitions.)

Lemma 3.3.  Given a map f : B ! A, the adjoint pair (f!; f*) forms a topological
Quillen pair between oB and oA equipped with the coarse structures. Furthermore,
if f is a weak equivalence then the pair is a topological Quillen equivalence.

Proof.By Lemma 8.10, it suffices to notice that f!sends Sn q B to Sn q A (as a
pushout) and similarly for Dn+1 q B. Hence, f!sends the generating cofibrations
in the coarse structure on oB to generating cofibrations in the coarse structur*
 *e on
oA . To verify the Quillen equivalence condition, consider the diagram exhibite*
 *d in
the proof of Lemma 2.2. It suffices to recall that o is a proper model category*
 * and
B ! X is a cofibration in the diagram while Y ! A is a fibration. This implies *
 *that
X ! f!(X) and f*(Y ) ! Y are weak equivalences, since f is a weak equivalence_
by assumption.                                                       |__|

Proposition 3.4.Suppose X and Y are both cofibrant and fibrant objects in the
coarse structure on oB . Then f : X ! Y is a coarse weak equivalence if and only
if f is a pointed fibre homotopy equivalence.

Proof.The point here is that the standard notion of pointed fibre homotopy and
the model theoretic notion of homotopy coincide under these hypotheses.  First,
because the objects in question are both cofibrant and fibrant, weak equivalenc*
 *es
are precisely the homotopy equivalences in the model category sense. Recall tha*
 *t a
homotopy between two maps X ! Y in a model category is defined as an extension
of the two maps to a so-called cylinder object.  A cylinder object is a choice *
 *of
factorization of the fold map X q X ! X as a cofibration followed by a weak
equivalence. The claim follows upon establishing that X q X ! X ^B (I+ x B) !
X yields such a factorization.  This is the content of Lemma 9.5 in the_current_
context.                                                              |__|

  The following is a standard consequence of the existence of a long exact sequ*
 *ence
in homotopy for a fibration.

Lemma 3.5.  Suppose X; Y 2 oB are such that p : X ! B and q : Y ! B are
Hurewicz fibrations. Then f : X ! Y is a coarse weak equivalence if and only if*
 * f
induces a weak equivalence on each fibre.

                    4.Generalization to k-spaces

  As mentioned before, there are two ways to construct a convenient category of
ex-spaces. The first approach, using the category of open ex-spaces from [14], *
 *has
already been discussed. Following earlier work of [21], an alternative approach*
 * of
relaxing the weak Hausdorff condition on the total space (while still only cons*
 *idering
bases B 2 o) is provided by [1] and [4]. Let K be the category of (not necessar*
 *ily
weak Hausdorff) k-spaces and let B 2 o  K. Then by results of [1] and [4] the
category KB of elements of K over and under B is a closed symmetric monoidal
category under the fibrewise smash product.
  As one expects, the category KB carries the coarse structure, with one possib*
 *ly
surprising property. The cellular replacements one builds in this structure wil*
 *l nat-
urally have weak Hausdorff total spaces. This follows from the fact that all of*
 * the
"generating cells" have weak Hausdorff total spaces, while the process of build*
 *ing



                        HOMOTOPY OF EX-SPACES                       7


cell complexes preserves this property. Thus, by considering cellular replaceme*
 *nts
one may recover the assumption of a weak Hausdorff total space for any homo-
topy invariant construction.  (Compare with a similar statement in the proof of
Proposition 5.8.)

Proposition 4.1.The category KB supports a cofibrantly generated, proper, topo-
logical model category structure with the weak eqivalences (cofibrations or fib*
 *rations)
defined as maps in KB which are weak homotopy equivalences (cofibrations or Ser*
 *re
fibrations) of total spaces in K.

  It is important to remember that colimits in K are formed as colimits in the
category of all topological spaces.  In particular, one can verify directly tha*
 *t the
pushout in K of a disjoint union of closed embeddings is again a closed embeddi*
 *ng.
Hence, the small object arguments in these situations rely on the following lem*
 *ma,
since relative cell complexes are colimits of the specified type.

Lemma 4.2.  Let X! = colimiXi where the maps Xi ! Xi+1 are closed embed-
dings and where Xi+1-Xiis T1 (ie, points are closed). If C is a compact Hausdor*
 *ff
space, then K(C; X!) ~=colimiK(C; Xi):

  Notice here that X0 is not assumed to be T1.

                   5. Fine Homotopy of Ex-Spaces

  For certain applications, the coarse structure on oB may prove sufficient; ho*
 *wever,
there is no reason to expect the coarse structure to induce a similar structure*
 * on
O*(B). This is a serious issue because oB is not a convenient category under the
fibrewise smash product. Section 4 outlined one solution to this problem by rel*
 *axing
the weak Hausdorff condition on the total space.  The solution proposed in this
section is to build a different model structure, the fine structure, on oB whic*
 *h is more
closely related to the local nature of the base space. The fine structure desce*
 *nds to
O*(B) quite readily, thereby producing a convenient category of ex-spaces in wh*
 *ich
to study homotopy-theoretic questions.  As mentioned in the introduction, there
is a notion of cell complex which yields a Whitehead Theorem comparing weak
equivalences in the fine structure to (pointed) fibrewise homotopy equivalences.
  In order to build a model structure which would reflect the local nature of B,
Lewis suggested that the authors consider constructing a model structure with w*
 *eak
equivalences defined in terms of "local spheres" - objects (Sn x U) q B in oB f*
 *or
open sets U.  One defines the homotopy groups associated to n and U in terms
of these objects. This leads to the fine structure (see Definition 5.3), where *
 *cell
complexes are built using the inclusion of local spheres into similarly defined*
 * local
disks.
  From an aesthetic point of view, the fine structure arises from "lifting" the*
 * model
structure from o* over a collection of adjoint pairs (see Theorem 9.8). The obv*
 *ious
method of making a pointed space into an ex-space is to simply apply the functor
? x B.  However, (?)+ x B and the forgetful functor from oB  to o are not an
adjoint pair. In fact, the forgetful functor has no left adjoint, which implies*
 * that
the coarse structure is not simply lifted over an adjoint pair in the usual fas*
 *hion.
This does not present a problem; however, it suggests lifting using ? x B and i*
 *ts
right adjoint. As mentioned previously, the right adjoint in question is simply*
 * the
space of global sections. The following simple example suggests global sections*
 * are
far from adequate.



8                   M. INTERMONT AND M. W. JOHNSON


Example 5.1. Let B = S1 and let W = S1 q S1.  Define W ! S1 to be the
double covering map on the first factor and the identity map on the second, wit*
 *h the
inclusion of S1 as the second factor for a section. Then the space of global se*
 *ctions
of W is a singleton, by consideration of fundamental groups. The projection of W
is an open map so W 2 O*(B) as well.

  Upon reflection, it seems that global sections are also insufficient to consi*
 *der the
local nature of the base. However, taken as a collection, the spaces of local s*
 *ections
over various open sets carry far more information.

Remark 5.2.  There is a topological sheaf of sections functor whose target is l*
 *og-
ically given a model structure where fibrations and weak equivalences are defin*
 *ed
entrywise as in [12]. This suggests defining weak equivalences as those maps wh*
 *ich
induce weak equivalences in each space of local sections.

  There is a strong analogy between the category oB and the category of G-spaces
for G a compact Lie group, mainly via the Borel construction. Thus, one looks to
the equivariant case for suggestions of how to construct the fine structure. Re*
 *call,
there are two standard notions of homotopy of G-spaces for G a compact Lie grou*
 *p.
The simpler of the two is similar to the coarse structure defined in the last s*
 *ection,
where the three classes are essentially defined by forgetting the G-action.  The
second structure, which reflects far more of the equivariant theory, involves l*
 *ooking
at homotopy groups associated to H and n for any closed subgroup H. In that cas*
 *e,
ssHn(X) may be defined in terms of Sn+^ G=H+ , or equivalently, as the standard
homotopy groups of the H fixed-point space ssn(XH ).
  In order to imitate the equivariant setting, it is important to identify the *
 *analog
of the fixed-point space functors associated to any closed subgroup H  G. This
role is played by the space of local sections functor, SecU(?), associated to a*
 *n open
subspace U  B.

Theorem 5.3.  Let U denote a set of open subspaces of B. Then oB forms a cofi-
brantly generated, right proper, topological model category where a weak equiva*
 *lence
(fibration) is defined as a morphism f in oB such that SecU(f) is a weak homoto*
 *py
equivalence (Serre fibration) in o* for each U 2 U.

Proof.This follows from Theorem 9.8 using the adjoint pairs {(i!(? x U); SecU(?*
 *))}
for the inclusion i : U ! B of an element U of U. By Lemma 9.10 (with G=e) oB __
is a topological category to which Theorem 9.8 applies.                      |_*
 *_|

Definition 5.4.The fine structure is the model structure given by the theorem
when U is taken to be the collection of all open subspaces of B. More generally,
the model structure given by the theorem for an arbitrary collection U of open
subspaces is called the U-structure.

  As suggested above, the generating cells in the U-structure are of the form

                    (Sn x U) q B ! (Dn+1 x U) q B

where U 2 U.  The sections are given by the obvious inclusion of B, while the
projections are given by projection onto the U factor in each product, followed*
 * by
the inclusion of U into B. Lewis [14] has shown projections are open maps (even
in the compactly generated product, which must be used here). Thus, each of the
generating cells in oB actually lives in O*(B). Recall from [14] that there is *
 *a right



                        HOMOTOPY OF EX-SPACES                       9


adjoint O : oB ! O*(B) to the full inclusion  : O*(B) ! oB . Hence, by Lemma
8.8, the following is a consequence of the theorem.

Corollary 5.5.Let U denote a set of open subspaces of B. Then O*(B) forms a
cofibrantly generated, right proper, topological model category where a weak eq*
 *uiv-
alence (fibration) is defined as a morphism f in oB such that SecU(f) is a weak
homotopy equivalence (Serre fibration) in o* for each U 2 U.

Remark 5.6.  It remains unclear whether the U-structure is left proper on either
oB or O*(B).  It is hoped that this and several other technical questions will *
 *be
addressed in a later paper.

Example 5.7. Let B = I and let X denote I x I equipped with the inclusion of
the base I x {0} as section and the vertical projection map so that X 2 O*(B).
This X is contractible as well as fibrewise contractible, cofibrant and fibrant*
 * in
both the coarse and fine structures. However, various subobjects of X are far m*
 *ore
complicated.                            S
  Let Y denote the subobject I x {0; 1}  {1} x I of X, which consists of three
sides of the square. Give Y the section I ! I x{1} so Y 2 oB . Then fibres over*
 * any
point other than 1 are S0, while the fibre at 1 is I. Notice that Y is not in O*
 **(B)
since any open subinterval of the fibre at 1 is not sent to an open set. The to*
 *tal
space of Y is contractible, but Y is not fibrewise contractible. The right adjo*
 *int
O applied to Y is the subobject I x {0; 1}, which is cofibrant in any U-structu*
 *re
where B is contained in U. In fact, I x {0; 1} acts as a cofibrant replacement *
 *for Y
in any such structure, as shown in the proof of the next result.

  Due to the destructive nature of O, the following is a pleasant surprise.

Proposition 5.8.The adjoint pair (; O) forms a Quillen equivalence between
O*(B) and oB in any U-structure.

Proof.The inclusion  preserves cofibrations and acyclic cofibrations by definit*
 *ion,
hence the adjoint pair forms a Quillen pair. In order to see that it forms a Qu*
 *illen
equivalence, choose a cofibrant X 2 O*(B) and fibrant Y 2 oB . It is necessary *
 *to
show that X ! O(Y ) is a weak equivalence in O*(B) if and only if (X) ! Y is
a weak equivalence in oB . In particular, it suffices to show O(Y ) ! Y is a we*
 *ak
equivalence in oB , because of the 2 of 3 property for weak equivalences and the
natural factorization X ! O(Y ) ! Y .
  To see this, consider a cofibrant replacement in the U-structure on oB W ! Y .
It is built from the generating cells using the small object argument.  Since t*
 *he
generating cells live in O*(B), which is closed under all colimits, this implie*
 *s W 2
O*(B) as well. As a right Quillen functor, O preserves acyclic fibrations, so t*
 *hat
O(W ) ! O(Y ) is a weak equivalence in O*(B), (hence also in oB ).  The claim
follows from noticing that the top horizontal map in the following diagram is an
isomorphism since W is in O*(B).

                                  ~=
                            O(W )_____//W

                            ~ ||        ~||
                              fflffl|   fflffl|
                            O(Y )_____//_Y

                                                                    |___|



10                  M. INTERMONT AND M. W. JOHNSON


  In particular, the proposition implies that the homotopy theory of O*(B) is
equivalent to that of oB in any U-structure.
  As with the coarse structure, one has a variety of useful technical consequen*
 *ces.
The most obvious of these is that all objects in any U-structure are fibrant.  *
 *It
is also convenient to know that the change of base functors associated to a map
f : B ! A induces adjoint pairs of associated homotopy categories under certain
conditions.

Lemma 5.9.  Suppose f : B ! A is an open map in o and all categories are taken
in their fine structures.

  1. The adjoint (f!; f*) forms a topological Quillen pair between oA and oB .
  2. If f is an injection, then the adjoint pair (f!; f*) forms a topological Q*
 *uillen
     pair between oB and oA .
  3. The adjoint pair (f*; f*) forms a topological Quillen pair between O*(A) a*
 *nd
     O*(B).
  4. If f is an injection, then the adjoint pair (f!; f*) forms a topological Q*
 *uillen
     pair between O*(B) and O*(A).

Proof.In each case, Lemma 8.10 reduces the problem to verifying that the left
adjoint applied to a generating cell is a cofibration and similarly for generat*
 *ing
acyclic cells. Recall that generating cells for fine structures on O*(B) or oB *
 *are of
the form

                    (Sn x U) q B ! (Dn+1 x U) q B:

  1. First, the open condition is required to see that the adjoint f* exists by*
 * [14].
     Notice f*((Sn xU)qA) ~=(Sn xf-1 (U))qB by consideration of the following
     pullback diagram

                    f*(Sn x U)_____//Sn x U____//Sn

                        |             |         ||
                        |             |         |
                        fflffl|       fflffl|   fflffl|
                     f-1 (U)_________//U_______//_* :

     Since f is continuous, f-1 (U) is open.  Thus, f* actually takes generating
     cells to generating cells, which are cofibrations as desired.
  2. If f is an injection, then for each open i : U ! B, the composite fi : U !*
 * A
     is also the inclusion of an open set. Hence, f!i!= j! with j : f(U) ! A the
     inclusion of an open set. In particular, this implies f!sends generating c*
 *ells
     in oB to generating cells in oA as desired.

  The open condition on f ensures that the adjoint pairs descend to the categor*
 *ies
of open ex-spaces.  Thus, the last two statements proceed exactly as in the fir*
 *st
two.                                                                 __
                                                                    |__|

  The following example, however, shows that the change of base functors, even
for a homotopy equivalence f, need not induce a Quillen equivalence. (Compare
with Lemma 3.3.)

Example 5.10.  Let Z denote the subset of I x I consisting of
               [        [                [                [

      [0; 1_3)   (2_3;x1]1_2(1_4; 3_4) x {0; 1}(1_4; 1_3)   (2_3;x3_4)I:



                        HOMOTOPY OF EX-SPACES                       11


Equip Z with the vertical projection and a choice of section "below the hole".
Then Z lies in O*(B) and is cofibrant (as well as fibrant) in the fine structur*
 *e. This
follows from the way Z is built as a cell complex. Begin with zero-cells over [*
 *0; 1_3),
(2_3; 1], and two zero-cells over (1_4; 3_4). Then attach one-cells joining the*
 * pieces over
(1_4; 1_3) and (2_3; 3_4).
  The most important fact about Z is that it is not weakly equivalent in the fi*
 *ne
structure to any object of the form M x B with M 2 o*. To see this, notice that
over the interval (1_3; 2_3) the space of sections of Z is the space S0, corres*
 *ponding
to the low or high components. However, over the interval (3_4; 1] there is a u*
 *nique
section.
  On the other hand, the space of sections of any M x B over an open interval
U is homeomorphic to map(U; M). For a contractible U (e.g.(1_3; 2_3) or (3_4; 1*
 *]) this
mapping space is homotopy equivalent to M itself. In particular, if Z were weak*
 *ly
equivalent to some M x B in the fine structure, the connectivity of the spaces *
 *of
sections would have to be the same over (1_3; 2_3) and over (3_4; 1], which is *
 *clearly not
the case.
  This example implies that the homotopy category of the fine structure on O*(B)
(or oB ) is not equivalent to the usual homotopy category of o* via either the *
 *adjoint
pair (f*; f*) or (f!; f*) for f : I ! *.  However, f is a homotopy equivalence
between cofibrant-fibrant spaces, whose fibre(s) are CW complex(es). Hence, the
fine structure on O*(B) depends on deeper information than the homotopy type of
B. Precisely what information is involved is currently under investigation.

  Another important technical fact is the following analog of Whitehead's Theo-
rem.

Proposition 5.11.Suppose X and Y are cofibrant objects in any U-structure on
oB (or O*(B)).  Then f : X ! Y  is a U-weak equivalence if and only if f is a
pointed fibre homotopy equivalence.

  The proof is the same as that for Proposition 3.4 and relies upon the topolog*
 *ical
nature of the model category structure, in order to apply Lemma 9.5.
  The reader may have been expecting U to be an open cover of B for the proposi-
tion to hold. However, this is not necessary, primarily because any cofibrant o*
 *bject
in the U-structure on oB (or O*(B)) will be trivial away from the portion of B
covered by U.



                    6. Comparison of Structures

  The following result establishes a valuable local to global principle for the*
 * fine
structure. The theorem (in a different language) is originally due to Dold in [*
 *5].

Theorem 6.1 (Dold's Theorem).Suppose X; Y 2 O*(B) (or oB ) and U is a nu-
merable open cover of B. Then a map f : X ! Y is a weak equivalence in the fine
structure on O*(B) (or oB ) if and only if i*(f) is a weak equivalence in the f*
 *ine
structure on O*(U) (or oU ) for each i : U ! B with U 2 U.

Proof.In the special case of the open inclusion i : U ! B, Lemma 5.9 implies th*
 *at
the restriction functor i* is both a left and right Quillen functor.  In partic*
 *ular,
i* preserves weak equivalences and cofibrant objects.  Thus, forming a cofibrant



12                  M. INTERMONT AND M. W. JOHNSON


replacement

                                  g
                              W _____//Z
                               |      |
                               |      |
                               fflffl|fflffl|
                              X __f__//Y


one has that i*(g) is a cofibrant replacement of i*(f) as well. In other words,*
 * one
need only consider the case where both X and Y are cofibrant in O*(B) (or oB ).
  Since X and Y cofibrant implies that i*(X) and i*(Y ) are both cofibrant and
fibrant, this reduces to considering pointed fibre homotopy equivalences by app*
 *ly-
ing Proposition 5.11 to the fine structure on O*(B) (or oB ) and O*(U) (or oU ).
However, this case is well-known; for example as Theorem 14.1 in [3] or Theorem_
1.5 in [16].                                                           |__|

Remark 6.2.  Notice that the theorem does not say that f is a fine weak equiv-
alence if SecU(f) is an ordinary weak homotopy equivalence for each U 2 U, as
evidenced by the example below.  Rather, by definition of the fine structure on
O*(U) (or oU ), one must consider SecV(f) for any open subspace V  B satisfying
V  U for some U 2 U.

Example 6.3. Consider the numerable open cover of S1 consisting of S1 alone,
together with the constant function at 1. Then SecB(f) is not sufficient to det*
 *er-
mine whether f is a weak equivalence in the fine structure.  Consider the object
W 2 O*(S1) from Example 5.1 and let Y 2 O*(S1) be the same space where the
projection map is taken to be the four-fold covering map on the first circle an*
 *d the
identity on the second circle. Then there is a map f : Y ! W in O*(S1) which
is a two-fold cover on the first circle and the identity on the second ("basepo*
 *int")
circle. Notice that the map SecS1(f) is the identity map on the basepoint (see *
 *5.1),
hence is a weak equivalence in o*.  To see that f is not a fine weak equivalenc*
 *e,
take V  S1 to be a quarter circle. Then SecV(Y ) consists of four points, while
SecV(W ) consists of two points.  Alternatively, to see that f is not a fine we*
 *ak
equivalence, recall that fundamental group computations imply f is not a coarse
weak equivalence and apply Theorem 6.5 below.

  Given a collection of open subspaces U of B, let S(U) denote the associated
sieve, i.e.  the collection of open subspaces V  of B such that V   U for some
U 2 U.

Corollary 6.4.Suppose U is a numerable open cover of B. Then the fine structure
on O*(B) (or oB ) is topologically Quillen equivalent to the S(U)-structure.

Proof.Since S(U) is contained in the collection of all open subspaces of B, it
should be clear that the property of being a fibration in the S(U)-structure is*
 * less
restrictive than that of being a fine fibration. Notice that the theorem implie*
 *s that
the two structures have the same class of weak equivalences, so the result_foll*
 *ows
from Lemma 8.9.                                                      |__|

  Notice the corollary implies all S(U)-structures for numerable open covers U *
 *pro-
duce equivalent homotopy theories, by transitivity of Quillen equivalences. How-
ever, the classes of fibrations need not coincide for different numerable cover*
 *s, so
the structures themselves may differ.



                        HOMOTOPY OF EX-SPACES                       13


  The next goal is to derive some comparison between the coarse and fine struc-
tures. Unfortunately, there is (as yet) no direct way to compare the two struct*
 *ures.
One would like to compare each of them to their intersection, however it is not*
 * clear
that their intersection exists.
  The following result, due to Lewis, implies that the coarse and fine structur*
 *es
essentially agree for Hurewicz fibrations. Thus, any discussion of transfers or*
 * other
geometric operations familiar for Hurewicz fibrations may be studied with either
model structure.
  Recall that a CW complex is locally finite provided each open cell intersects*
 * only
finitely many closed cells.

Theorem 6.5.  Suppose B is a locally finite CW complex and X; Y 2 oB such that
p : X ! B and q : Y ! B are Hurewicz fibrations. Then f : X ! Y is a coarse
weak equivalence if and only if f is a fine weak equivalence.

Proof.By Lemma 3.5, it suffices to verify that f is a fine weak equivalence pre*
 *cisely
when f induces a weak homotopy equivalence on each fibre.  Since B is locally
contractible (Theorem 1.3.2 in [7]), one may construct an open cover W of B
consisting of contractible open subspaces W  B.  Then Corollary 2.8.15 in [19]
implies that both X|W  and Y |W  are pointed fibre homotopy equivalent to trivi*
 *al
fibrations Fb x W  ! W and Fb0x W  ! W respectively.  Thus, for any open
V  W , one has the following commutative diagram with vertical maps homotopy
equivalences.


                       SecV(X) _______//SecV(Y )

                         ' ||             |'|
                           fflffl|        fflffl|
                      map(V; Fb)_____//map(V; Fb0)


The paracompactness of B (Theorem 1.3.5 of [7]) yields a numerable open cover U
of B which refines W, in the sense that each U 2 U is contained in some W 2 W.
  Suppose f is a fine weak equivalence and choose an element U 2 U such that
b 2 U. By the local contractibility of B, there exists a contractible open V  U
(hence V  W for some W 2 W) with b 2 V . Then a contracting homotopy of V
produces a natural homotopy equivalence map(V; Fb) ! Fb. Thus, SecV(f) a weak
homotopy equivalence implies that fb : Fb ! Fb0is a weak homotopy equivalence
as well.
  Now suppose fb : Fb ! Fb0is a weak homotopy equivalence for each b 2 B
and V 2 S(U).  By Dold's Theorem (6.1), it suffices to show SecV(f) is a weak
homotopy equivalence.  However, V  2 S(U) implies V   W for some W 2 W
once again. Thus, SecV(f) is homotopy equivalent to map(V; Fb) ! map(V; Fb0) as
above. Hence, it suffices to verify that V has the homotopy type of a CW comple*
 *x.
  The combination of Theorem 11.4 and the remarks preceding Theorem 10.4 in
[11] imply that B is a (metric) ANR (Absolute Neighborhood Retract) provided B
is metrizable, which is equivalent to the local finiteness condition by Proposi*
 *tion
1.5.17 in [7]. Now Proposition A.6.4 of [7] then implies that V is also a (metr*
 *ic)_
ANR, hence has the homotopy type of a CW complex by Theorem 5.2.1 of [7].  |__|


Remark 6.6.  Notice, the proof above implies that any open subspace of a locally
finite CW complex has the homotopy type of a CW complex.



14                  M. INTERMONT AND M. W. JOHNSON


  Consider an object X 2 oB .  Construct a new object N(X) 2 oB whose total
space is given by the following pullback diagram.

                           N(X)  ____//_P B

                             |         |ev
                             |         |  0
                             fflffl|   fflffl|
                             X ___p___//_B

The composition of the induced map N(X) ! P B followed by the evaluation at the
other end of the interval ev1 : P B ! B gives a map N(X) ! B and the universal
property of a pullback gives a section map. Thus, N(X) 2 oB . It is well known *
 *that
the composite N(X) ! B is, in fact, an Hurewicz fibration and that X includes as
a strong deformation retract of N(X). Thus, X is weakly equivalent to N(X) in t*
 *he
coarse structure with p0: N(X) ! B an Hurewicz fibration. By the functoriality
of N(?), X ! Y is a coarse weak equivalence precisely when N(X) ! N(Y ) is a
coarse weak equivalence. Using Theorem 6.5, this implies X ! Y is a coarse weak
equivalence if and only if N(X) ! N(Y ) is a fine weak equivalence. Hence, N(?)
induces a functor from the homotopy category associated to the coarse structure
into the homotopy category associated to the fine structure whose target is wit*
 *hin
the class of Hurewicz fibrations.


                           7. Ex-G-Spaces

  As one might expect, the results of Sections 3, 4, and 5 generalize to equiva*
 *riant
settings.  This section gives a brief discussion of the model structures on ex-*
 *G-
spaces. Throughout, G is taken to be a compact Lie group, and B is taken to be a
G-space. Let Go denote the category where the objects are G-spaces and the maps
are G-maps.

Definition 7.1.An ex-G-space is a G-space X together with G-maps s : B ! X
and p : X ! B such that ps = id. The category GoB is the category whose objects
are ex-G-spaces and whose morphisms are G-maps which make the diagram in
Definition 2.1 commute in the category Go.

  Recall that the coarse structure is defined by forgetting about the base spac*
 *e.
Thus, a coarse structure on GoB relies on a model structure on the category of
G-spaces. Such a structure exists by an unpointed version of Theorem 9.8. A pro*
 *of
of the pointed version of this theorem can also be found in [15].

Theorem 7.2.  There is a cofibrantly generated model structure on Go where a
morphism f is a weak equivalence (fibration) precisely when the H fixed point m*
 *ap
fH  is a weak homotopy equivalence (Serre fibration) for each closed subgroup H
G.

  In fact, there is another model structure on Go where one simply ignores the
G-action in defining weak equivalences and fibrations, but this structure on Go*
 * is
not of interest here.
  As suggested by the theorem, a map f : X ! Y in the coarse structure on GoB
is a weak equivalence (fibration) if the H fixed point map fH  : XH  ! Y H is a
weak homotopy equivalence (Serre fibration) for each closed subgroup H  G.
  The existence of the coarse structure on GKB , the equivariant version of KB ,
depends upon being able to apply Lemma 4.2 in order to show that the generating



                        HOMOTOPY OF EX-SPACES                       15


cells are small with respect to relative cell complexes in this context (see De*
 *finition
8.1). The generating cells are all of the form (Sn xG=H)qB ! (Dn+1 xG=H)qB
which are closed embeddings. Hence, the map on total spaces of any relative cell
complex is also a closed embedding as in Section 4. Since each of the maps in t*
 *he
relevant string will be an injection as well as a G-map, a factorization in K w*
 *ill also
be a factorization in GKB .
  In [17], May constructs a stable version of the coarse structure on GKB and
verifies that it satisfies a version of Spanier-Whitehead duality. A similar di*
 *scussion
based on the fine structure is currently under investigation.
  Turning attention to a fine structure on GoB , recall that the fine structure*
 * on oB
relied upon lifting the structure from o* over the adjoint pairs (i!(? x U); Se*
 *cU(?)).
From an equivariant point of view, one expects to consider the closed subgroups*
 * of
G in addition to the open subspaces U of B. However, an open subspace U  B
does not acquire an H-action by restriction, unless the subspace is assumed to *
 *be
H-invariant, i.e. hb 2 U for each b 2 U, h 2 H.
  Given a closed subgroup H  G and an H-invariant open subspace U  B, let
Sec(H;U)(X) denote the space of H-equivariant sections of X; that is, the H-maps
s0 : U ! X such that ps0 = i : U ! B.  Notice that H acts by conjugation on
the space map(U; X) of all maps (and also on map(U; U)), hence on the ordinary
space of sections SecU(X).  As expected, the ordinary section maps which are
also H-maps are the H fixed-points under this conjugation action.  In fact, the
functor X 7! Sec(H;U)(X) from GoB to o* has a left adjoint, given by the formula
Y 7! i!((Y ^ G=H+ ) x U).

Definition 7.3.A pair (H; U) is called an admissible pair if H  G is a closed
subgroup and U is an H-invariant, open subspace of B.

  Lifting over each of the adjoint pairs associated to admissible pairs (H; U) *
 *and
intersecting these lifted structures yields the following.

Theorem 7.4.  There exists a cofibrantly generated, right proper, topological m*
 *odel
structure on GoB (or GO*(B)) where a map f is a weak equivalence (fibration) if
and only if Sec(H;U)(f) is a weak homotopy equivalence (Serre fibration) for ea*
 *ch
admissible pair (H; U).

Proof.Using the set {(i!((?^G=H+ )xU); Sec(H;U)(?))|(H; U) is an admissible pai*
 *r},
Theorem 9.8 establishes the result.  That Theorem 9.8 itself applies follows_fr*
 *om
Lemma 9.10.                                                         |__|

               8. Lifting Model Category Structures

  This section is devoted to a technique of building new model categories out of
old, commonly called "lifting". Much of the material in this section is standar*
 *d and
included only for the convenience of the non-specialist. However, the definitio*
 *n of
intersecting and Proposition 8.7 (due to the second author) represents a new tw*
 *ist
on the standard lifting lemma. The reader is directed to [6], [8], [10] or Quil*
 *len's
original work [18] for any details which are omitted in these last two sections.
  The underlying idea of lifting model structures is quite simple.  Suppose one
is given a model category C, another category D, and an adjoint pair (L; R) with
L : C ! D the left adjoint. One can attempt to define a model structure on D by
defining a morphism f in D to be a weak equivalence (fibration) precisely when
R(f) is a weak equivalence (fibration). It is then necessary to define a cofibr*
 *ation in



16                  M. INTERMONT AND M. W. JOHNSON


D as an appropriate type of relative cell complex (or a retract thereof), where*
 * the
generating cells are taken to be the morphisms of the form L(j) with j a genera*
 *ting
cell in C. Recall that in any model structure, the class of cofibrations must c*
 *oincide
with the class of morphisms having the LLP (left lifting property) with respect*
 * to
all acyclic fibrations, i.e. fibrations which are also weak equivalences. In so*
 *me sense,
Lemma 8.4 below implies that lifting succeeds precisely when these two classes *
 *of
morphisms coincide.
  Of course, one must then verify Quillen's five axioms. However, there are act*
 *ually
only two key facts that need to be verified in most cases. General arguments are
given for each of these in topological situations.  (See the proof of Theorem 9*
 *.8
below.)
  By far the most technically difficult portion of these arguments is devoted to
constructing the factorizations required of a model structure.  To this end, one
normally introduces a notion of cell complex which is formalized as a cofibrant*
 *ly
generated model structure. In order to apply Quillen's small object argument,
the "boundaries" of the basic (or "generating") cells must be small in a techni*
 *cal
sense defined below.

Definition 8.1.Suppose K is a set of morphisms in C.

  1. A relative K-cell complex is the colimit of a (possibly transfinite) seque*
 *nce
     X0 ! X1 ! X2 ! : :X:ff! : :w:here each Xfl! Xfl+1fits into a pushout
     diagram

                           `     `kff`
                             Cff_____//Dff

                             |          |
                             |          |
                             fflffl|    fflffl|
                            Xfl______//Xfl+1


     with each kff2 K.
  2. The sources of K are K-small if for each source Cffof a morphism in K,
     there exists a cardinal ffsuch that for any relative K-cell complex Xffiwi*
 *th
     ffi  ffany morphism Cff! Xffimust factor as Cff! Xoe! Xffifor some
     oe < ffi.

  This condition of K-smallness proves to be the key to verifying the existence*
 * of
lifted structures (see the last portion of the proof of Proposition 8.7).

Remark 8.2.  In the usual case of topological spaces, the set K consists of the
inclusions of Sn ! Dn+1. Hence, the Cffare compact Hausdorff spaces, while the
sequence of maps Xi! Xi+1is known to consist of closed embeddings (as a relative
cell complex in the classical sense). In this case, one may choose each ff= ! a*
 *nd
the sources of K being K-small reduces to the standard fact that any compact
Hausdorff space mapping into an infinite union of closed subspaces actually maps
into some finite stage.

  Two different types of factorizations are required in a model structure. Henc*
 *e,
one needs to consider a similar cell complex construction where the relative co*
 *m-
plexes are also strings of weak equivalences. This leads to the notion of gener*
 *ating
acyclic cells and relative acyclic cell complexes.  Of course, the set of gener*
 *ating
acyclic cells will also need to satisfy a smallness condition as above.



                        HOMOTOPY OF EX-SPACES                       17


Definition 8.3.Suppose C is a model category where (L; R), L : C ! D, form an
adjoint pair. Then an R-fibration is a morphism f in D with R(f) a fibration in
C.  Similarly, an R-weak equivalence is a morphism f in D with R(f) a weak
equivalence in C.

  If C is a cofibrantly generated model category, let I denote the class of gen*
 *erating
cells while J denotes the class of generating acyclic cells. Then define L(I) =*
 * {L(i) :
i 2 I} and L(J) = {L(j) : j 2 J}.
  The following result describes when lifting proceeds to define a model struct*
 *ure
on D. As one might expect from the discussion above, the requirements are quite
technical. However, there is a general approach to verifying these conditions w*
 *hich
is presented in the next section.

Lemma 8.4 (Lifting Lemma). Suppose the sources of L(I) are L(I)-small, and the
sources of L(J) are L(J)-small. Then D is a cofibrantly generated model category
with the R-fibrations and R-weak eqivalences if and only if every morphism f in*
 * D
which has the LLP with respect to all R-fibrations is an R-weak equivalence.

  This argument has become standard. See [6], [8], [10], or [18]. It is general*
 *ized
in the proof of Proposition 8.7 below, hence is omitted now.
  As the reader has noticed by now, one may want to deal with more than a
single adjoint pair in a similar argument. This can be technically difficult, b*
 *ut is
made more manageable in most cases by the notion of intersecting model category
structures introduced below.

Definition 8.5.  1.Given a set of model structures on D, their intersection (if
     it exists) has weak equivalences (fibrations) defined as morphisms which a*
 *re
     weak equivalences (fibrations) in each of the model structures. One says t*
 *hat
     the intersection exists provided this induces another model structure on D.
  2. Given a set of cofibrantly generated model structures on D, let K denote t*
 *he
     union of the various sets of generating cofibrations, and L denote the uni*
 *on of
     the sets of generating acyclic cofibrations. Then these structures are ref*
 *erred
     to as relatively small provided the sources of K are K-small and the sourc*
 *es
     of L are L-small.

Remark 8.6.  The definition of relatively small is the translation of the small*
 *ness
condition from the statement of Lemma 8.4 to the current context. It is the des*
 *crip-
tion of when Quillen's small object argument applies using the union of the set*
 *s of
generating cells and generating acyclic cells (see part 5 in the proof of Propo*
 *sition
8.7).

  The intersection of structures is so-named because the class of weak equivale*
 *nces
in the intersection structure, or intersection weak equivalences, is precisely *
 *the
intersection of the classes of weak equivalences associated to the structures b*
 *eing
intersected. Similarly, the class of intersection fibrations is aptly named.
  Recall that the class of cofibrations in any model structure may be identifie*
 *d as
those morphisms satisfying the LLP with respect to all acyclic fibrations.  Thu*
 *s,
the union of the various classes of cofibrations will be contained in the class*
 * of
intersection cofibrations. Hence, it is reasonable to use the union K of the cl*
 *asses
of generating cells (or the union L of the classes of generating acyclic cells)*
 * to build
factorizations for the intersection structure. This leads one to define cofibra*
 *tions as
relative K-cell complexes (or retracts thereof) by analogy with the lifting sit*
 *uation



18                  M. INTERMONT AND M. W. JOHNSON


discussed previously. Once again, in order to have a model structure, this defi*
 *nition
must coincide with the definition of cofibrations via the LLP with respect to a*
 *cyclic
fibrations.

Proposition 8.7.Given a set of cofibrantly generated model structures on D, the*
 *ir
intersection forms a cofibrantly generated model category structure provided th*
 *ey are
relatively small and any morphism satisfying the LLP with respect to all inters*
 *ection
fibrations is an intersection weak equivalence.

Proof.The axioms (see Definition 3.3 of [6]) may be verified directly:

  1. The existence of limits and colimits is independent of what structure we c*
 *hoose
     to impose.
  2. The 2 of 3 property for weak equivalences follows from that of each struct*
 *ure
     together with the intersection definition.
  3. The class of cofibrations is defined as retracts of relative K-cell comple*
 *xes,
     hence is closed under retracts. Again, the definition and the same property
     for fibrations and weak equivalences in each structure implies that they a*
 *re
     closed under retracts as well.
  4. Let

                              A _____//X

                              i||     |p|
                               fflffl|fflffl|
                              B _____//Y

     be a commutative square. If i is a retract of a relative K-cell complex and
     p is an intersection acyclic fibration, then one must verify the existence*
 * of
     a lift.  Consider first the case that i : A ! B is a generating cofibratio*
 *n.
     Then in at least one of the model category structures, i has the LLP with
     respect to acyclic fibrations. So i has the LLP with respect to all inters*
 *ection
     acyclic fibrations by definition of intersection fibration and intersectio*
 *n weak
     equivalence. In building an arbitrary cofibration from generating cofibrat*
 *ions,
     the only operations used are pushouts (cobase change), coproducts, transfi*
 *nite
     composition and retracts. Each of these preserves the LLP with respect to a
     class of morphisms, so the case of an arbitrary i follows.
       Dually, assume that i is both an intersection weak equivalence and a ret*
 *ract
     of a relative K-cell complex while p is an intersection fibration in the d*
 *iagram
     above. It is shown below in (5) (independently) that i can be factored as a
     relative L-cell complex A ! W followed by an intersection fibration W ! B.
     Notice that the building argument just discussed implies that any relative
     L-cell complex has the LLP with respect to all intersection fibrations, he*
 *nce
     it is an intersection weak equivalence by assumption.  Rewriting the above
     diagram as

                           A AA__________//_X>>
                           || AA'AA   """""||

                           |    AA  """    |
                          '||     W>>      ||
                           |   """"|       |
                           |  ""   |       |
                           fflffl|"fflffl|"fflffl|=
                           B _____//B____//Y



                      HOMOTOPY OF EX-SPACES                       19


   makes it clear that the 2 out of 3 property discussed above forces W ! B
   to be an intersection acyclic fibration.  Now, the argument just completed
   implies that there is a lift B ! W . Finally, since A ! W is a relative L-ce*
 *ll
   complex it has the LLP with respect to all intersection fibrations. Composing
   two such lifts yields the desired lift B ! X in the original diagram.
5. Suppose f : X ! Y is a morphism in D. Construct a relative K-cell complex
   X = X0 ! Xflas follows. For limit ordinals ffi < fl, define Xffi= colimoe<ff*
 *iXoe.
   For successor ordinals define Xffi+1to be the pushout in the diagram

                         `     `kff`
                           Cff_____//Dff

                           |          |
                           |          |
                           fflffl|    fflffl|
                          Xffi_____//Xffi+1

   where the coproducts are indexed over the set of commutative squares of the
   form

                           Cff__kff//_Dff
                            |        |
                            |        |
                            fflffl|  fflffl|
                           Xffi____//Y :

   Now choose an upper bound ae for the set of {ff} associated to the various
   Cff. This is possible since there is only a set to consider. By the universal
   properties of colimits and pushouts, there is then a factorization of f as
   X ! Xae! Y with X ! Xaea relative K-cell complex. It remains to verify
   that the morphism Xae! Y is, in fact, an intersection acyclic fibration. Sin*
 *ce
   each of the original model structures was assumed to be cofibrantly generate*
 *d,
   it suffices to verify that Xae! Y has the RLP (right lifting property) with
   respect to each morphism in K. Consider any lifting square

                            Cff____//_Xae

                           kff||     ||
                             fflffl| fflffl|
                            Dff_____//_Y

   and notice the top horizontal map must factor through some earlier stage Xoe
   since ae > ff. However, the square

                           Cff_____//Xoe

                            |kff|    ||
                            fflffl|  fflffl|
                           Dff_____//_Y

   then commutes by assumption, which implies that it is one of the squares
   used to build Xoe+1from Xoe. Thus, Dff! Xoe+1! Xaeprovides the desired
   lift.
     The other factorization of f is produced similarly, using L in place of K,
   with one additional difficulty. (Notice, this requires the second condition *
 *for
   relatively small structures.) One must verify that the relative L-cell compl*
 *ex
   produced is actually an intersection weak equivalence. Once again, the build-
   ing argument above implies that the relative L-cell complex has the LLP with



20                  M. INTERMONT AND M. W. JOHNSON


     respect to all intersection fibrations. However, any morphism with the LLP
     with respect to all intersection fibrations is assumed to be an intersecti*
 *on
     weak equivalence, which completes the claim.

                                                                    |___|

  The hypotheses of this proposition include a smallness condition and a lifting
condition. Section 9 verifies that in certain topological situations, including*
 * all of
the situations arising in this paper, the hypotheses are satisfied.
  Before considering topological situations, three lemmas regarding Quillen pai*
 *rs
and Quillen equivalences are recorded.

Lemma 8.8.  Suppose  : E ! D is the inclusion of a subcategory such that there
exists a right adjoint R for . If D supports a cofibrantly generated model stru*
 *cture
whose generating cells live in E, then E is also a cofibrantly generated model *
 *category
with the three classes of maps inherited by inclusion.

Proof.The key observation here is that the inclusion  : E ! D being a left adjo*
 *int
implies that E is closed under colimits taken in D. Hence, all factorizations b*
 *uilt
using the small object argument again live in E. Also, whatever type of limits *
 *and
colimits exist in D exist formally in E as well. The remaining axioms follow_di*
 *rectly_
from those for the existing structure on D.                                |__|

  Recall, this fact was used to deduce that the fine structure exists on O*(B) *
 *once
it exists on oB . In particular, the lemma implies that the adjoint pair (; R) *
 *forms
a Quillen pair since the inclusion preserves (and reflects) cofibrations and ac*
 *yclic
cofibrations.

Lemma 8.9.  Suppose C is equipped with two different model category structures.
If the two structures have the same weak equivalences and one class of fibratio*
 *ns
is contained in the other, then the identity functor becomes a Quillen equivale*
 *nce
between the two structures.

Lemma 8.10.  Suppose (L; R), L : C ! D, form an adjoint pair between cofibrantly
generated model categories.  Then (L; R) is a Quillen pair if and only if L sen*
 *ds
generating cells to cofibrations and generating acyclic cells to acyclic cofibr*
 *ations.


                  9.Topological Model Categories

  In this section the discussion of model categories continues, turning attenti*
 *on to
topological model categories, where one has straightforward means of verifying *
 *the
conditions in either Lemma 8.4 or Proposition 8.7. The first portion of this se*
 *ction
introduces the technical notion of a topological category and some standard fac*
 *ts
concerning a notion of homotopy in such categories.  This is followed by several
examples detailing situations which have appeared throughout the paper. Finally,
the main existence theorem for model category structures is proven.
  Informally, a topological model category is a model category which is enriche*
 *d,
tensored and cotensored over o* in a way which interacts well with the model
structure. This is simply Quillen's notion of a simplicial model category with *
 *o*
playing the role of simplicial sets.
  Both [2] and [13] have excellent general treatments of categories enriched, t*
 *en-
sored and cotensored over o*. In this context, however, only the basics are nec*
 *essary.
A category D is enriched over o* provided the morphism sets are equipped with



                        HOMOTOPY OF EX-SPACES                       21


topologies and basepoints which make them objects of o*, such that the composi-
tion law becomes a continuous map D(B; C) ^ D(A; B) ! D(A; C). In addition, D
is said to be tensored and cotensored over o* if for each X; Y 2 D and M 2 o*,
there exist natural constructions X  M (called the tensor) and hom(M; Y ) (call*
 *ed
the cotensor) together with natural homeomorphisms

            o*(M; D(X; Y )) ~=D(X  M; Y ) ~=D(X; hom(M; Y )):

  By a standard abuse of notation, a topological category is a category D which
is enriched, tensored and cotensored over o*. However, the reader should be car*
 *eful
to notice that a topological model category is more than a topological category
equipped with a model structure; there is a restriction on how the two structur*
 *es
interact. The following two special cases give the essential nature of the cond*
 *ition.
Suppose j : M ! N is a cofibration in o* and X 2 C is a fibrant object. Then the
precomposition morphism - O j : hom(N; X) ! hom(M; X) must be a fibration in
C. Dually, suppose M 2 o* is cofibrant and f : X ! Y is a fibration in C, then *
 *the
postcomposition morphism f O - : hom (M; X) ! hom (M; Y ) must be a fibration
in C.

Example 9.1.    1.The category oB is a topological category.  Let B !s X !p

     B; B !tY !q B be objects in oB and let M be a pointed space. The tensor is
     defined by XM = X^B (M xB). The cotensor is most readily defined in two
     steps: form hom_(M; Y ) as the pullback of s0O - : o(M; Y ) ! o(M; B) along
     cB : B ! o(M; B), cB (b)(m) = b, and then define the cotensor hom(M; Y )
     as the pullback



                      hom (M; Y )____//hom_(M; Y )

                           |              |
                           |              |ff
                           fflffl|s0      fflffl|
                          B ____________//_Y :

     Here ff : hom_(M; Y )! o(M; Y ) ! o(*; Y ) ~=Y is induced by the inclusion*
 * of
     the basepoint of M. To check that these are indeed the correct constructio*
 *ns,
     first notice that oB (X; Y ) is a pointed space with the basepoint given by
     the map s0p : X ! B ! Y .  Considering the case of the tensor, a map in
     oB (X  M; Y ) is equivalent to a map g : X xB (M x B) ! Y subject to the
                                      0
     condition that X _B (M x B) ! B s!Y is the restriction of the map g. The
     space X xB (M x B) can be identified (as an element of oB ) with X x M by
     taking into account the fact that each of the subsquares below is a pullba*
 *ck
     square:

                 X xB (M x B)  _____//(M x B)____//M

                        |               |          |
                        |               |          |
                        fflffl|         fflffl|    fflffl|
                       X ______________//B_______//* :

     Here X x M is given projection (x; m) 7! p(x) and section b 7! (s(b); *).
     This means that the original map X  M ! Y is equivalent to a map "g:
     X x M ! Y  in oB , satisfying an additional condition as above.  Via the
     standard adjunction, o(X x M; Y ) ~= o(M; o(X; Y )), "gis equivalent to a
     pointed map M ! oB (X; Y ).



22                  M. INTERMONT AND M. W. JOHNSON


       Likewise, in the case of the cotensor, an element of oB (X; hom(M; Y ))
     is equivalent to a map f in oB (X; hom_(M; Y )) such that fff = s0p.  That
     hom_(M; Y ) is in oB follows quickly after noting that o(M; Y ) is in oB w*
 *ith the
     projection map sending k 7! qk(*) and the section sending b to the constant
     map on t(b).  The map f is equivalent to a map h 2 oB (X; o(M; Y )) such
     that h followed by evaluation at the basepoint of M is the map s0p.  Now
     h is equivalent to a map h_in oB (X x M; Y ) such that h_(x)(*) = s0p(x) f*
 *or
     all x 2 X which is finally equivalent to a map M ! oB (X; Y ) sending the
     basepoint in M to h(x; *). This is exactly the condition that says the map
     M ! oB (X; Y ) is actually a pointed map.
  2. The category GoB is also a topological category. The tensor and cotensor a*
 *re
     defined exactly as in oB , and one must simply note that there are appropr*
 *iate
     G-actions on all of the spaces. The pointed space M is considered as a poi*
 *nted
     G-space with trivial action, all mapping spaces are considered with actions
     given by conjugation, and all products have diagonal actions.  Finally, all
     pullbacks and pushouts are taken in the category Go.  As in the previous
     case, the basepoints are tedious to track down; however, it is not difficu*
 *lt to
     track the G-actions through the previous argument to verify that the neces*
 *sary
     adjunctions hold.

  Given two topological categories C and D, a functor F : C ! D is a topological
functor provided that all maps C(X; Y ) ! D(F (X); F (Y )) are continuous.  An
adjoint pair (L; R) between topological categories consists of topological func*
 *tors if
and only if the right adjoint R preserves cotensors. (See [2, II.6.3].) Dually,*
 * L and
R are topological functors if and only if the left adjoint L preserves tensors.*
 * Such
a pair is called a topological adjoint pair.

Example 9.2.    1.For each i : U ! B, the pair (i!(? x U); SecU(?)) is a topo-
     logical pair. The left adjoint can be viewed as a composite: ? x U : o* ! *
 *oU
     followed by i!: oU ! oB . By rewriting the pieces of the pushout which de-
     fines (M x U)  N, the functor ? x U can be shown to preserve tensors. That
     pushout is

             (M x U) _U (N x U) _____//_(M x U) xU (N x U)

                      |                       |
                      |                       |
                      fflffl|                 fflffl|
                     U _____________//(M x U) ^U (N x U) :

     As in Example 1, (M x U) xU (N x U) is isomorphic to (M x U) x N, hence
     to (M x N) x U. Since ? x U preserves pushouts in o, the pushout defining
     (M xU)_U (N xU) is isomorphic to the pushout defining (M _N)xU. This
     implies that the pushout defining (M x U) ^U (N x U) can be written as

                    (M _ N) x U _____//(M x N) x U

                          |                |
                          |                |
                          fflffl|          fflffl|
                      {*} x U _______//_(M ^ N) x U

     which gives the desired equality

    (M  N) x U = (M ^ N) x U ~=(M x U) ^U (N x U) = (M x U)  N:



                        HOMOTOPY OF EX-SPACES                       23


       It remains to show that i!is a topological functor. To see this, conside*
 *r its
     right adjoint, i*. The diagram

               i*hom_(M; Y )___//_hom_(M; Y_)__//o(M; Y )

                    |                |             |
                    |                |             |
                    |fflffl          fflffl|       fflffl|
                    U _____________//_B________//o(M; B)

     is a pullback diagram, as is the diagram

               hom_(M; i*Y )___//_o(M; i*Y_)__//_o(M; Y )

                    |               |             |
                    |               |             |
                    fflffl|         fflffl|       fflffl|
                    U __________//_o(M; U)___//_o(M; B) :

     The composites U ! o(M; B) along the bottom of each diagram are actu-
     ally the same map, as are the rightmost vertical maps. Hence, the pullbacks
     i*(hom_(M; Y ) and hom_(M; i*Y ) can be identified.  The functor i* applied
     to the pullback diagram which defines the cotensor hom (M; Y ) is again a
     pullback diagram, and in light of what has just been shown, this pullback
     can be identified with hom (M; i*Y ). Thus, i* preserves cotensors, which *
 *suf-
     fices. (Notice that this same argument actually establishes that f* preser*
 *ves
     cotensors for any open map f : B ! A.)
  2. The adjoint pair (i!(? ^ G=H+ x U); Sec(H;U)(?)) is a topological adjoint *
 *pair.
     This is similar to the previous example with all operations performed in t*
 *he
     category of G-spaces.

  Notice that the universal properties of tensors and cotensors imply that X S0*
 * ~=
X and hom (*; Y ) ~= * for any X; Y  2 D.  Clearly, one may define a homotopy
between morphisms f; g : X ! Y  in D as a morphism H in the commutative
diagram
                                ~=
                       X q X _____//X  (S0 _ S0)

                          |              |
                          |              (f;g)|
                          fflffl|H       fflffl|
                      X  I+  __________//Y :

Lemma 9.3.  Let (L; R) be a topological adjoint pair.  Then the right adjoint R
preserves null-homotopies in the sense above.

Proof.Notice that R must preserve the zero object and that the identity Z  I+ !
Z  I+ corresponds to a natural morphism Z ! hom (I+ ; Z  I+ ).  Applying R
(which commutes with cotensors) yields a morphism R(Z) ! hom(I+ ; R(Z  I+ ))_
which then corresponds to a natural map R(Z)  I+ ! R(Z  I+ ).            |__|

Lemma 9.4.  Let D be a topological category and let Y ! Z be null-homotopic in
o*. Then X  Y ! X  Z is null-homotopic.

Lemma 9.5.  Suppose X is a cofibrant object in a topological model category. Th*
 *en
X  I+ forms a cylinder object for X.

  Given a topological model category C and a set of topological adjoint pairs
{(Lff; Rff)|ff 2 A}, Lff: C ! D; indexed by A, an RA -weak equivalence is a



24                  M. INTERMONT AND M. W. JOHNSON


morphism f in D with Rff(f) a weak equivalence in C for each ff 2 A. Similarly,
an RA -fibration is a morphism f in D such that Rff(f) is a fibration in C for *
 *each
ff 2 A. (Compare with Definition 8.3.)

Definition 9.6.A regular topological category D is a topological category which
satisfies the following two conditions:

  1. Each object of D is small with respect to monomorphisms, and
  2. The natural map j : X ~=X  S0 ! X  I induced by the inclusion of the
     endpoints of the interval is a monomorphism.

  Before proceeding with the model structure existence theorem, one more defini-
tion is required. Note, this definition coincides with the usual notion of a de*
 *forma-
tion retract when the category D is o*.

Definition 9.7.Let D be a topological category.

  1. A morphism i : X ! Y in D is called the inclusion of a retract provided
     there exists a lift r in the diagram

                              X __=__//X>>_
                                   __
                              i||r___ ||
                               fflffl|fflffl|__
                              Y _____//* :

  2. An inclusion of a retract i : X ! Y is called the inclusion of a deformati*
 *on
     retract provided there exists a lift h in the diagram

                            j
              X  __i_//_Y____//hom(I+ ; Y )
               |           llll66
              i||     llhllll     |p|
               fflffl|llllll      fflffl|~=
               Y ______(=;ir)__//_Y x Y_____//_hom(S0+; Y )


     where precomposition by the map I+ ! S0 which collapses the interval to
     the non-basepoint gives j : Y ~=hom (S0; Y ) ! hom(I+ ; Y ).

  The following theorem is the main result of this section and provides the maj*
 *ority
of the model structures considered above.

Theorem 9.8.  Suppose C is a cofibrantly generated, topological model category
where every object is fibrant and {(Lff; Rff)|ff 2 A}, Lff: C ! D; forms a set
of topological adjoint pairs. If D is regular, then there exists a cofibrantly *
 *gener-
ated, topological model structure (where all objects are fibrant) on D consisti*
 *ng of
the RA -weak equivalences, and the RA -fibrations.

Proof.This model structure is constructed in two stages. First, one lifts over *
 *each
individual adjoint pair using Lemma 8.4; then, one intersects all of the result*
 *ing
structures using Proposition 8.7. Both stages have smallness conditions and lif*
 *ting
conditions that need to be verified.  The arguments presented below deal with
the first stage, given a fixed ff.  However, the second stage follows by the sa*
 *me
arguments, because the same cylinder object is used for each choice of ff.
  The first step toward verifying the smallness conditions of Lemma 8.4 is to
establish that the unique morphism Z  I ! * is an Rff-acyclic fibration for any
Z 2 D. As a right adjoint, Rffpreserves final objects. Since all objects in C a*
 *re



                        HOMOTOPY OF EX-SPACES                       25


fibrant by assumption, the unique morphism Z  I ! * is an Rff-fibration. Choose
a null homotopy of I and recall that by Lemma 9.4 the functor Z? preserves null
homotopies while the functor Rffpreserves null homotopies by Lemma 9.3. Hence
Z  I ! * is an Rff-weak equivalence as well.
  Now suppose i is a generating cofibration in C. Then i has the LLP with respe*
 *ct
to the map Rff(Lff(X)I) ! * by the previous paragraph with Z = Lff(X). Hence,
Lff(i) has the LLP with respect to Lff(X)  I ! *, by adjunction. However, the
fact that D is a regular topological category together with the existence of a *
 *lift in
the diagram


                         Lff(X)_____//Lff(X)88I
                                    rrr
                        Lff(i)|| rrrr    |
                            fflffl|rrr   fflffl||
                         Lff(Y )________//*


implies that Lff(i) is a monomorphism as the first half of a factorization of a
monomorphism.  Similarly, any relative cell complex in D built from the various
Lff(i) may be viewed as a sequence of monomorphisms, since the LLP with respect
to a class of maps is preserved under building relative cell complexes. Now, th*
 *e fact
that D is assumed to be a regular topological category implies that all objects*
 * are
small with respect to relative cell complexes built from the Lff(i). One procee*
 *ds
similarly to verify that all objects in D are small with respect to relative ce*
 *ll com-
plexes built from the Lff(j), where j varies over the generating acyclic cofibr*
 *ations
of C.
  The following argument, due to Quillen [18], serves to verify the lifting con*
 *dition
of Lemma 8.4.
  Notice that the natural morphism hom(I+ ; Y ) ! hom(S0; Y ) is an Rff-fibrati*
 *on.
This follows from considering the commutative diagram


                   Rffhom(I+ ; Y_)___//_Rffhom(S0; Y )

                       ~=||               ~=||
                         fflffl|            fflffl|
                  hom (I+ ; Rff(Y_))_//hom(S0; Rff(Y ))


where the vertical maps are isomorphisms because Rffpreserves cotensors.  The
fact that C is a topological model category together with the facts that S0 ! I+
is a cofibration in o* and Rff(Y ) is fibrant by assumption, imply that the bot*
 *tom
horizontal map is a fibration.
  Next, notice that the morphism X ! * is an Rff-fibration because each Rff
preserves the final object and all objects in C are assumed to be fibrant.  Thu*
 *s,
if g has the LLP with respect to any Rff-fibration then g is the inclusion of a
deformation retract. Hence Rff(g) is the inclusion of a deformation retract in *
 *C.
  Finally, since all objects in C are fibrant, the dual of Lemma 9.5 implies th*
 *at
any inclusion of a deformation retract in C is a homotopy equivalence in the mo*
 *del
category sense, hence is a weak equivalence. Also, the induced structure on D i*
 *s_
topological by definition, since Rffcommutes with cotensors.                 |_*
 *_|


Remark 9.9.  There is also a variation of this result which involves a set of a*
 *djoint
pairs with Rff: D ! Cff.



26                  M. INTERMONT AND M. W. JOHNSON


  Of course, it remains to show the categories considered earlier are regular t*
 *opo-
logical categories.  Keep in mind that monomorphisms and injections coincide in
each of these categories.


Lemma 9.10.  For any compact Lie group G, the category GoB is a regular topo-
logical category.


Proof.Recall the definition of tensors in GoB . To see that the map X ! X  I
is an injection it suffices to consider each fibre individually, where the map *
 *is the
injection Xb ! Xb^ I in o*.
  Suppose X 2 GoB , then Lemma 1.1.1 in [12] verifies that there exists a cardi*
 *nal
 with the following property. Given a map in o, X ! Yffiwith Yffithe colimit of*
 * a
sequence of monomorphisms in o and ffi > , there exists a factorization X ! Yoe!
Yffithrough an earlier stage of the colimiting system.  If X ! Yffiis a morphism
in GoB , one still has a map in o, g : X ! Yoeas above.  However, considering
underlying sets, Yoe! Yffia monomorphism in GoB implies that g is a morphism in
GoB as well.                                                         __
                                                                    |__|



                             References

 [1]P.I. Booth. The exponential law of maps II. Math. Z., 121:311-319, 1971.
 [2]F. Borceux. Handbook of Categorical Algebra. Vol 2. Cambridge University Pr*
 *ess, Cambridge,
   1994. Categories and structures.
 [3]M. Crabb and I. James. Fibrewise Homotopy Theory. Springer-Verlag, Berlin H*
 *eidelberg New
   York, 1998. Springer monographs in mathematics.
 [4]B. Day. A reflection theorem for closed categories. J. Pure Appl. Alg., 2:1*
 *-11, 1972.
 [5]A. Dold. Partitions of unity in the theory of fibrations. Ann. Math., 78(2)*
 *:223-255, 1963.
 [6]W.G. Dwyer and J. Spalinski. Homotopy theories and model categories. In Han*
 *dbook of
   Algebraic Topology, pages 73-126. North-Holland, Amsterdam, 1995.
 [7]R. Fritsch and R.A. Piccini. Cellular Structures in Topology. Cambridge Uni*
 *versity Press,
   Cambridge, 1990.
 [8]P. Goerss and J.F. Jardine. Simplicial Homotopy Theory. Birkhaueser, Basel,*
 * 1999.
 [9]P. Hirschhorn. Localization of Model Categories. In preparation, 2000.
[10]M. Hovey. Model Categories. American Mathematical Society, Providence, RI, *
 *1999.
[11]D.M. Hyman. A category slightly larger than the metric and CW-categories. M*
 *ich. Math. J.,
   15:193-214, 1968.
[12]M.W. Johnson. Enriched sheaves as a framework for stable homotopy theory. P*
 *hD Thesis,
   University of Washington, 1999.
[13]G.M. Kelly. Basic Concepts of Enriched Category Theory. Cambridge Universit*
 *y Press, Cam-
   bridge, 1982.
[14]L.G. Lewis Jr. Open maps, colimits, and a convenient category of fibre spac*
 *es. Top. and its
   Apps., 19:75-89, 1985.
[15]M.A. Mandell and J.P. May. Equivariant orthogonal spectra and S-modules. Pr*
 *eprint, 2000.
[16]J.P. May. Classifying spaces and fibrations. Am Math. Soc. Memo., 1(155), 1*
 *975.
[17]J.P. May. Notes on ex-spaces and ex-spectra. Preprint, 2000.
[18]D.G. Quillen. Homotopical Algebra. Springer-Verlag, Berlin, 1967. Lecture N*
 *otes in Mathe-
   matics, No. 43.
[19]E.H. Spanier. Algebraic Topology. McGraw-Hill, New York, 1966.
[20]N.E. Steenrod. A convenient category of topological spaces. Mich. Math. J.,*
 * 14:133-152, 1967.
[21]R.M. Vogt. Convenient categories of topological spaces for homotopy theory.*
 * Arch. Math.,
   22:545-555, 1971.



                      HOMOTOPY OF EX-SPACES                       27


Department of Mathematics, Kalamazoo College, Kalamazoo, MI 49006
E-mail address: intermon@kzoo.edu

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
E-mail address: johnson.295@nd.edu