A GENERALIZATION OF QUILLEN'S SMALL OBJECT
ARGUMENT
BORIS CHORNY
Abstract.We generalize the small object argument in order to allow for i*
*ts
application to proper classes of maps (as opposed to sets of maps in Qui*
*llen's
small object argument). The necessity of such a generalization arose wi*
*th
the appearance of several important examples of model categories which w*
*ere
proven to be non-cofibrantly generated [2, 7, 8, 17]. Our current appro*
*ach
allows for the construction of functorial factorizations and localizatio*
*ns in the
equivariant model category on diagrams of spaces [10] and in two differe*
*nt
model structures on the category of pro-spaces [11, 17].
The examples above suggest a natural extension of the framework of cof*
*i-
brantly generated model categories. We introduce the concept of a class-
cofibrantly generated model category, which is a model category generated
by classes of cofibrations and trivial cofibrations satisfying some reas*
*onable
assumptions.
1.Introduction
Quillen's definition of a model category has been slightly revised over the l*
*ast
decade. The changes applied to the first axiom MC1 requiring the existence of
all finite limits and colimits, and to the last axiom MC5 requiring the existen*
*ce of
factorizations. The modern approaches to the subject [14, 16] demand the existe*
*nce
of all small limits and colimits in MC1. This gives some technical advantages w*
*hile
treating transfinite constructions, such as localizations, in model categories.*
* The
modern version of the axiom MC5 requires the factorizations to be functorial.
Functoriality of the factorizations is a very important part of the structure*
* of
Quillen's model category. Most examples of model categories have functorial fac-
torizations, and many works on abstract homotopy theory assume that condition.
For example, there are two recent constructions of homotopy limits and colimits*
* in
abstract model categories equipped with functorial factorizations [5, 14].
The most widely known model category without functorial factorizations is the
category of pro-spaces or, more generally, of pro-objects (in the sense of Grot*
*hen-
dieck) in a proper model category C [11, 19] and its Bousfield localization mod*
*elling
the 'etale homotopy theory [4, 17]. The construction of functorial factorizatio*
*ns in
these model categories was one of our main goals during the work on this paper.
However, this task would not be accomplished without an observation that the we*
*ll-
known theorem of C.V. Meyer [21] implies immediately the existence of a functor*
*ial
replacement of a pro-map by a levelwise pro-map. Unfortunately this construction
depends on the choice of a functor which is inverse to the equivalence of categ*
*ories
____________
Date: January 28, 2004.
1991 Mathematics Subject Classification. Primary 55U35; Secondary 55P91, 18G*
*55.
Key words and phrases. model category, functorial factorization, pro-categor*
*y.
1
2 BORIS CHORNY
constructed by Meyer. It makes our construction less explicit and perhaps not
applicable for concrete computations. We provide a partial compensation for this
drawback by giving in Appendix A an explicit construction of functorial fibrant
replacement in the pro-categories with the strict model structure or, more prec*
*isely,
by proving that Isaksen's construction of (non-functorial) factorizations becom*
*es
functorial when applied just to fibrant replacements. The purpose of the append*
*ix
is also to discuss whether the construction of factorizations in [19] can be ma*
*de
functorial, provided that now we have the functorial levelwise replacements. We*
* do
not arrive at a definite conclusion to this question, but we show the specific *
*point
in the proof which distinguishes between the simpler case of fibrant replacemen*
*ts
and the case of general factorizations.
The main tool for the construction of functorial factorizations in model cate*
*gories
and localizations thereof is Quillen's small object argument [14, 16, 22]. Howe*
*ver,
in its original form, the argument is applicable neither to the category of dia*
*grams
with the equivariant model structure [10], nor to pro-categories, since it allo*
*ws for
the application in cofibrantly generated model categories only. We propose here
a generalization which may be used in a wider class of model categories. The
collections of generating cofibrations and generating trivial cofibrations may *
*now
form proper classes, satisfying the conditions of the following theorem:
Theorem 1.1 (The generalized small object argument). Suppose C is a category
containing all small colimits, and I is a class of maps in C satisfying the fol*
*lowing
conditions:
(1) There exists a cardinal ~, such that each element A 2 dom (I) is ~-small
relative to I-cof;
(2) There exists a functor S :Map C ! Map C equipped with an augmenta-
tion t: S ! IdMapC, such that S(f) 2 I-coffor every f 2 Map C and
any morphism of maps i ! f with i 2 I factors through the natural map
t(f): S(f) ! f.
Then there is a functorial factorization (fl, ffi) on C such that, for all morp*
*hisms f
in C, the map fl(f) is in I-cof and the map ffi(f) is in I-inj.
We say that a class I of maps in a category C permits the generalized small o*
*bject
argument if it satisfies conditions (1) and (2) of Theorem 1.1.
This theorem is the second attempt by the author to generalize the small obje*
*ct
argument. The previous version appeared in the study of the equivariant localiz*
*a-
tions of diagrams of spaces [6]. The specific properties of the equivariant mo*
*del
category of D-shaped diagrams of spaces and also the non-functorial factorizati*
*on
technique developed by E. Dror Farjoun in [10] suggested the rather complicated
technical notion of instrumentation. It is essentially a straightforward üf nct*
*orial-
izationö f Dror Farjoun's ideas. The classes of generating cofibrations and g*
*en-
erating trivial cofibrations of diagrams satisfy the conditions of instrumentat*
*ion,
but we were unable to apply the same version of the argument to the category of
pro-spaces. The conditions of Theorem 1.1 on the class I of maps generalize tho*
*se
of instrumentation, as we explain in Section 3.
Therefore, this paper shows that the two rather different homotopy theories of
pro-spaces and of diagrams of spaces fit into a certain joint framework. In ord*
*er to
describe the similarity between the two cases let us give the following
Definition 1.2. A model category C is called class-cofibrantly generated if
GENERALIZED SMALL OBJECT ARGUMENT 3
(1) there exists a class I of maps in C (called a class of generating cofibr*
*ations)
that permits the generalized small object argument and such that a map is
a trivial fibration if and only if it has the right lifting property wit*
*h respect
to every element of I, and
(2) there exists a class J of maps in C (called a class of generating trivial
cofibrations) that permits the generalized small object argument and such
that a map is a fibration if and only if it has the right lifting proper*
*ty with
respect to every element of J.
Obviously, the class-cofibrantly generated model categories are equipped with
functorial factorizations. The categorical dual to a class-cofibrantly generat*
*ed
model category is called class-fibrantly generated.
The purpose of this paper is to show that the equivariant model structure on
the diagrams of spaces is class-cofibrantly generated and the both known model
structures on the category of pro-spaces are class-fibrantly generated.
The applications of Quillen's small object argument are not limited to abstra*
*ct
homotopy theory. A similar argument is used, for example, in the theory of cat-
egories to construct reflections in a locally presentable category with respect*
* to a
small orthogonality class [3, 1.36]. Recently another generalization of the sm*
*all
object argument was considered by the category theorists J. Ad'amek , H. Herrli*
*ch,
J. Rosick'y and W. Tholen [1]. Their version of the argument applies to the "in-
jective subcategory problem" in locally ranked categories - a generalization of*
* the
notion of a locally presentable category which includes topological spaces. We *
*hope
that our generalization of the small object argument will be applicable to the *
*ö r-
thogonal subcategory problemä nd "injective subcategory problem" with respect
to some reasonable classes of morphisms.
The rest of the paper is organized as follows: Section 2 is devoted to the proof
of the generalized cosmall object argument. Next, we review some of our previous
results about the diagrams of spaces in Section 3 and show how they fit into the
newly established framework. After providing the necessary preliminaries on pro-
categories in Section 4 we give our main applications of the generalized (co)sm*
*all
object argument in Section 5. Appendix A is devoted to an alternative, explicit
construction of a functorial fibrant replacement in pro-C. This construction is*
* based
on the construction of factorizations given by D. Isaksen [19]. We also discuss*
* the
difficulty which arises while trying to check whether the construction of gener*
*al
factorizations in [19] is functorial.
Acknowledgements. I would like to thank Dan Isaksen for many fruitful conversa-
tions and suggestions for improving this paper. In particular I owe him the ide*
*a of
the proof of Theorem 5.3.
2.Proof of the generalized small object argument
Proof of Theorem 1.1.Given a cardinal ~ such that every domain of I is ~-small
relative to I-cof, we let ~ be a ~-filtered ordinal.
To any map f :X ! Y we will associate a functor Zf :~ ! C such that Zf0= X,
and a natural transformation æf: Zf ! Y factoring f, i.e., for each fi < ~ the
4 BORIS CHORNY
triangle
f
X ?``````````//`???"Y
?? """
?ØØ " æffi
Zffi
is commutative. Each map iffi:Zffi! Zffi+1will be a pushout of a map of the form
S(f), i.e., iffi2 I-cof, since I-cof is closed under pushouts.
We will define Zf and æf: Zf ! Y by transfinite induction, beginning with
Zf0= X and æf0= f. If we have defined Zfffand æffffor all ff < fi for some limit
ordinal fi, define Zffi= colimff> q | `
| >> q |fflO qq |
| >> k| q |
| >>> fflffl|qq |
{fi}|| >>> ZkDD g2M|
| >>> | ||
| >> |pk |
fflffl| >>> fflfflfflffl| fflfflfflffl|
{Yi}_____________>>__________________//_Yk_________//_BDD<<
?? >> x
?? >> x x
?? >> x
?? >ØØ x
?? x X x
?? i i x
?? | x
?? xiqi x
?? |||fflffl|||||||x||||
?? ||| ||| x
?? |xiZi||||| x x
??u||||uu|||x|
u??|x|p||||x|uuuu
uuuuu??i|i|x||||||
uuu |?ØØfflfflfflffl|x|||||||
S({fi}) |xiYi||||||||||||||||||||
In order to verify the factorization property, fix an arbitrary map {fi} ! g *
*with
g 2 M being a fibration between constant objects. It follows immediately from
the definition of the morphism set between two pro-objects that any map into a
constant pro-object factors through a map of the form 'ifor some i 2 I. Applying
this to the category pro-Map (C) ~=Map (pro-C), we find an index k 2 I such that
the fixed map {fi} ! g factors through Xk ! Yk.
In the diagram above the maps xiZi! Zk and xiYi! Yk are projections. The
dashed map xiYi! B is the composition of the projection with the map Yk ! B.
Finally, the dashed map Zk ! A is a lifting in the commutative square, which
exists in the model category C.
Now we are able to apply the generalized cosmall object argument to produce
for every map in pro-C its functorial factorization into a trivial cofibration *
*followed
by a fibration.
To obtain the second factorization we repeat the construction above for the c*
*lass
N of trivial fibrations between constant objects and factorize all the maps Xi!*
* Yi
into cofibrations followed by trivial fibrations.
Hence the cosmall object argument may be applied to provide every map f
with a functorial factorization into a cofibration followed by a trivial fibrat*
*ion in
pro-C.
GENERALIZED SMALL OBJECT ARGUMENT 11
Let S denote the category of simplicial sets with the standard model structur*
*e.
Our next goal is to construct functorial factorizations in the localized model *
*struc-
ture of [17]. From now on the words cofibration, fibration and weak equivalence
refer to Isaksen's model structure.
Since the procedure of (left Bousfield) localization preserves the class of c*
*ofibra-
tions and, hence, the class of trivial fibrations, the functorial factorization*
* into a
cofibration followed by a trivial fibration was constructed in the theorem abov*
*e.
We keep the class N of generating trivial fibrations the same as in the strict *
*model
structure. The class of generating fibrations L is defined to be the class of *
*all
co-n-fibrations (see [17, Def. 3.2]) between constant pro-objects for all n 2 N.
Proposition 5.2. The class of trivial cofibrations equals L-proj.
Proof.Every element of L-proj is a strong fibration [17, Def. 6.5]; therefore e*
*very
trivial cofibration has the left lifting property with respect to L by [17, Pro*
*p. 14.5].
Conversely, if a map i has the left lifting property with respect to L, then i *
*has the
left lifting property with respect to the class L0of all retracts of L-cocell c*
*omplexes.
By [19, Prop.5.2] L0contains all strong fibrations. But then [17, Prop. 6.6] im*
*plies
that L0contains all fibrations. Therefore, i must be a trivial cofibration.
Theorem 5.3. Isaksen's model structure on pro-S may be equipped with functorial
factorizations.
Proof.It suffices to construct a functorial factorization of every morphism of *
*pro-S
into a trivial cofibration followed by a fibration. We apply the same construct*
*ion
as in Theorem 5.1 to the class L, except for the factorizations of the levelwise
representation {fi}.
Apply first the Marde~si'c functor in order to guarantee that our pro-system *
*is
indexed by a cofinite strongly directed set. Since the Marde~si'c functor is na*
*turally
isomorphic to the identity, we abuse notation and keep calling the indexing cat*
*egory
I. We construct the factorizations of the maps fi by induction on the number n(*
*i)
of predecessors of i and factor fi into an n(i)-cofibration qi followed by a co*
*-n(i)-
fibration, which is possible by [17, Prop. 3.3]. This an induction on the set *
*of
natural numbers, since I is now cofinite.
For any element g :A ! B of L, there is a number n 2 N such that g is a
co-n-fibration. We may always enlarge k such that n(k) n and hence qk will be
an n-cofibration by [17, Lemma 3.6]. Finally, the lift in the commutative squar*
*e in
the diagram in the proof of Theorem 5.1 exists by [17, Def. 3.2].
Appendix A. An explicit construction of a functorial fibrant
replacement in pro-C
The purpose of this appendix is to give an explicit construction of functorial
fibrant replacements in the strict model category on pro-C. More precisely, the
purpose is to prove that the construction of fibrant replacements in [19] is fu*
*nctorial.
We do not know whether the construction of arbitrary factorizations is functori*
*al
in [19], but in view of Remark A.2 it seems unlikely.
Proposition A.1. If C is a proper model category with functorial factorizations,
then there exists a functorial fibrant replacement X,~!^Xin the category pro-C *
*with
the strict model structure.
12 BORIS CHORNY
Proof.Given a pro-object X, we apply first the Marde~si'c functor in order to r*
*eplace
it by an isomorphic pro-object M(X) indexed by a cofinite strongly directed set.
Since M( . ) is naturally isomorphic to the identity functor, we suppress the n*
*otation
and quietly assume that all pro-objects are indexed by cofinite strongly direct*
*ed
sets.
Our objective is to find for every pro-object X indexed by a cofinite strongly
directed set S, a functorial factorization of the map f :X ! * into a trivial c*
*ofi-
bration i: X,~!^Xfollowed by a fibration p: ^Xi * in the strict model structure*
* on
pro-C. We apply the factorization algorithm of [19], with a mild alteration, on*
* the
simplest level representation of f: (idS, {Xs ! *}).
We define X^ by induction. The only difference between the current construc-
tion and [11, x4.3] [19] is our vision of the bonding maps of X^. In the origi*
*nal
construction they were induced by the fibrations pk below.
For every s0 2 S such that there are no s 2 S with s s0, define ^Xs0by appl*
*ying
the functorial factorization of C on the map Xs0! *:
"~
Xs0 Øis__//^Xs0`p////_lim{s l > m. We define X^kand ik
by applying the functorial factorization of C on the map Xk ! lims