CHAPTER 1.1
Homotopy theories and model categories
W. G. Dwyer and J. Spalinski
University of Notre Dame, Notre Dame, Indiana 46556 USA
Contents
1.Introduction::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
* 3
2.Categories :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
* 4
2.11.Colimits :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 7
2.18.Limits :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 9
2.24.Some remarks on limits and colimits:::::::::::::::::::::::::::::::::::::*
* 10
3. Model categories :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
* 12
4. Homotopy Relations on Maps ::::::::::::::::::::::::::::::::::::::::::::::: *
*17
4.1. Cylinder objects and left homotopy ::::::::::::::::::::::::::::::::::::*
*: 18
4.12. Path objects and right homotopies::::::::::::::::::::::::::::::::::::::*
* 21
4.20. Relationship between left and right homotopy ::::::::::::::::::::::::::*
*:: 23
5. The homotopy category of a model category ::::::::::::::::::::::::::::::::::*
*: 24
6. Localization of Categories :::::::::::::::::::::::::::::::::::::::::::::::::*
*: 28
7. Chain complexes :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
*29
7.4. Proof of MC1-MC3 ::::::::::::::::::::::::::::::::::::::::::::::::: 31
7.5. Proof of MC4(i) ::::::::::::::::::::::::::::::::::::::::::::::::::::: 31
7.7. Proof of MC4(ii)::::::::::::::::::::::::::::::::::::::::::::::::::::: 32
7.12.The small object argument::::::::::::::::::::::::::::::::::::::::::::: *
*33
7.18.Proof of MC5 ::::::::::::::::::::::::::::::::::::::::::::::::::::::: 35
8. Topological spaces ::::::::::::::::::::::::::::::::::::::::::::::::::::::::*
* 36
9. Derived functors :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
* 40
10.Homotopy pushouts and homotopy pullbacks :::::::::::::::::::::::::::::::::: *
* 46
10.4.Homotopy pushouts::::::::::::::::::::::::::::::::::::::::::::::::::: 47
10.8.Homotopy pullbacks :::::::::::::::::::::::::::::::::::::::::::::::::: 49
10.13.Other homotopy limits and colimits ::::::::::::::::::::::::::::::::::::*
*: 50
11.Applications of model categories :::::::::::::::::::::::::::::::::::::::::::*
*:: 51
References:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: *
* 55
The first author was partially supported by a grant from the National Science F*
*ounda-
tion.
HANDBOOK OF ALGEBRAIC TOPOLOGY
Edited by I.M. James
cO1995 Elsevier Science B.V. All rights reserved
1
2 Dwyer and Spalinski Chapter 1
Section 1 Homotopy theories 3
1.Introduction
This paper is an introduction to the theory of "model categories", which was de*
*vel-
oped by Quillen in [22] and [23]. By definition a model category is just an ord*
*inary
category with three specified classes of morphisms, called fibrations, cofibrat*
*ions
and weak equivalences, which satisfy a few simple axioms that are deliberately *
*rem-
iniscent of properties of topological spaces. Surprisingly enough, these axioms*
* give
a reasonably general context in which it is possible to set up the basic machin*
*ery of
homotopy theory. The machinery can then be used immediately in a large number
of different settings, as long as the axioms are checked in each case. Although*
* many
of these settings are geometric (spaces (x8), fibrewise spaces (3.11), G-spaces*
* [11],
spectra [5], diagrams of spaces [10] : :):, some of them are not (chain complex*
*es
(x7), simplicial commutative rings [24], simplicial groups [23] : :):. Certainl*
*y each
setting has its own technical and computational peculiarities, but the advantage
of an abstract approach is that they can all be studied with the same tools and
described in the same language. What is the suspension of an augmented commuta-
tive algebra? One of incidental appeals of Quillen's theory (to a topologist!) *
*is that
it both makes a question like this respectable and gives it an interesting answ*
*er
(11.3).
We have tried to minimize the prerequisites needed for understanding this pa-
per; it should be enough to have some familiarity with CW-complexes, with chain
complexes, and with the basic terminology associated with categories. Almost al*
*l of
the material we present is due to Quillen [22], but we have replaced his treatm*
*ent
of suspension functors and loop functors by a general construction of homotopy
pushouts and homotopy pullbacks in a model category. What we do along these
lines can certainly be carried further. This paper is not in any sense a survey*
* of
everything that is known about model categories; in fact we cover only a fracti*
*on of
the material in [22]. The last section has a discussion of some ways in which m*
*odel
categories have been used in topology and algebra.
Organization of the paper. Section 2 contains background material, principally a
discussion of some categorical constructions (limits and colimits) which come up
almost immediately in any attempt to build new objects of some abstract category
out of old ones. Section 3 gives the definition of what it means for a category*
* C to be
a model category, establishes some terminology, and sketches a few examples. In*
* x4
we study the notion of "homotopy" in C and in x5 carry out the construction of *
*the
homotopy category Ho(C ). Section x6 gives Ho(C )a more conceptual significance
by showing that it is equivalent to the "localization" of C with respect to the*
* class
of weak equivalences. For our purposes the "homotopy theory" associated to C is
the homotopy category Ho(C )together with various related constructions (x10).
Sections 7 and 8 describe in detail two basic examples of model categories, na*
*mely
the category Top of topological spaces and the category Ch Rof nonnegative chain
complexes of modules over a ring R. The homotopy theory of Top is of course fa-
miliar, and it turns out that the homotopy theory of Ch Ris what is usually cal*
*led
homological algebra. Comparing these two examples helps explain why Quillen
called the study of model categories "homotopical algebra" and thought of it as*
* a
4 Dwyer and Spalinski Chapter 1
generalization of homological algebra. In x9 we give a criterion for a pair of *
*functors
between two model categories to induce equivalences between the associated homo-
topy categories; pinning down the meaning of "induce" here leads to the definit*
*ion of
derived functor. Section 10 constructs homotopy pushouts and homotopy pullbacks
in an arbitrary model category in terms of derived functors. Finally, x11 conta*
*ins
some concluding remarks, sketches some applications of homotopical algebra, and
mentions a way in which the theory has developed since Quillen.
We would like to thank GianMario Besana and Krzysztof Trautman for help in
preparing this manuscript. We are also grateful for the comments of J. McClure,
W. Richter and J. Smith, which led among other things to simplifications in the
statement of 9.7 and in the proof of 10.7.
2.Categories
In this section we review some basic ideas and constructions from category theo*
*ry;
for more details see [17]. The reader might want to skip this section on first *
*reading
and return to it as needed.
2.1. Categories. We will take for granted the notions of category, subcategory,
functor and natural transformation [17, I]. If C is a category and X and Y are
objects of C , we will assume that the morphisms f : X ! Y in C form a set
HomC (X; Y ), rather than a class, a collection, or something larger. These mor-
phisms are also called maps or arrows in C from X to Y . Some categories that
come up in this paper are:
(i)the category Setwhose objects are sets and whose morphisms are functions
from one set to another,
(ii)the category Top whose objects are topological spaces and whose mor-
phisms are continuous maps,
(iii)the category Mod R whose objects are left R-modules (where R is an asso-
ciative ring with unit) and whose morphisms are R-module homomorphisms.
2.2. Natural equivalences. Suppose that F; F 0: C ! D are two functors, and th*
*at
t is a natural transformation from F to F 0. The transformation t is called a n*
*atural
equivalence [17, p. 16] if the morphism tX : F (X) ! F 0(X) is an isomorphism i*
*n D
for each object X of C. The functor F is said to be an equivalence of categorie*
*s if
there exists a functor G : D ! C such that the composites F G and GF are related
to the appropriate identity functors by natural equivalences [17, p. 90].
2.3. Full and faithful. A functor F : C ! D is said to be full (resp. faithful*
*) if for
each pair (X; Y ) of objects of C the map
Hom C(X; Y ) ! Hom D(F (X); F (Y ))
Section 2 Homotopy theories 5
given by F is an epimorphism (resp. a monomorphism) [17, p. 15]. A full subcate*
*gory
C0of C is a subcategory with the property that the inclusion functor i : C0! C *
*is
full (the functor i is always faithful). A full subcategory of C is determined *
*by the
objects in C which it contains, and we will sometimes speak of the full subcate*
*gory
of C generated by a certain collection of objects.
2.4. Opposite category. If C is a category then the opposite category C opis t*
*he
category with the same objects as C but with one morphism fop : Y ! X for each
morphism f : X ! Y in C [17, p. 33]. The morphisms of Cop compose according
to the formula fopgop= (gf)op. A functor F : Cop! D is the same thing as what
is sometimes called a contravariant functor C ! D. For example, for any category
C the assignment (X; Y ) 7! Hom C(X; Y ) gives a functor
Hom C(-; -) : Copx C ! Set :
2.5. Smallness and functor categories. A category D is said to be small if the
collection Ob (D ) of objects of D forms a set, and finite if Ob (D ) is a fini*
*te set
and D has only a finite number of morphisms between any two objects. If C is a
category and D is a small category, then there is a functor category CD in whic*
*h the
objects are functors F : D ! C and the morphisms are natural transformations;
this is also called the category of diagrams in C with the shape of D . For exa*
*mple,
if D is the category {a ! b} with two objects and one nonidentity morphism, then
the objects of C Dare exactly the morphisms f : X(a) ! X(b) of C and a map
t : f ! g in CD is a commutative diagram
X(a) -ta! Y (a)
f # g# :
X(b) -tb! Y (b)
In this case CD is called the category of morphisms of C and is denoted Mor (C *
*).
2.6. Retracts. An object X of a category C is said to be a retract of an object
Y if there exist morphisms i : X ! Y and r : Y ! X such that ri = idX. For
example, in the category Mod R an object X is a retract of Y if and only if the*
*re
exists a module Z such that Y is isomorphic to X Z. If f and g are morphisms
of C, we will say that f is a retract of g if the object of Mor (C ) represente*
*d by
f is a retract of the object of Mor (C ) represented by g (see the proof of the*
* next
lemma for a picture of what this means).
6 Dwyer and Spalinski Chapter 1
Lemma 2.7. If g is an isomorphism in C and f is a retract of g, then f is also
an isomorphism.
Proof. Since f is a retract of g, there is a commutative diagram
X -i! Y -r! X
f # g# f#
0 r0
X0 -i! Y 0 -! X0
in which the composites ri and r0i0are the appropriate identity maps. Since g is
an isomorphism, there is a map h : Y 0! Y such that hg = idYand gh = idY.0It__
is easy to check that k = rhi0is the inverse of f. |_|
2.8. Adjoint functors. Let F : C ! D and G : D ! C be a pair of functors. An
adjunction from F to G is a collection of isomorphisms
ffX;Y : Hom D(F (X); Y ) ~=Hom C(X; G(Y )); X 2 Ob(C ); Y 2 Ob(D )
natural in X and Y , i.e., a collection which gives a natural equivalence (2.2)*
* between
the two indicated Hom -functors C opx D ! Set (see 2.4). If such an adjunction
exists we write
F : C () D : G
and say that F and G are adjoint functors or that (F; G) is an adjoint pair, F *
*being
the left adjoint of G and G the right adjoint of F . Any two left adjoints of G*
* (resp.
right adjoints of F ) are canonically naturally equivalent, so we speak of "the*
*" left
adjoint or right adjoint of a functor (if such a left or right adjoint exists) *
*[17, p. 81].
If f : F (X) ! Y (resp. g : X ! G(Y )), we denote its image under the bijection
ffX;Y by f] : X ! G(Y ) (resp. g[ : F (X) ! Y ).
2.9. Example. Let G : Mod R ! Set be the forgetful functor which assigns to
each R-module its underlying set. Then G has a left adjoint F : Set ! Mod R
which assigns to each set X the free R-module generated by the elements of X.
The functor G does not have a right adjoint.
2.10. Example. Let G : Top ! Set be the forgetful functor which assigns to
each topological space X its underlying set. Then G has a left adjoint, which is
the functor which assigns to each set Y the topological space given by Y with t*
*he
discrete topology. The functor G also has a right adjoint, which assigns to eac*
*h set
Y the topological space given by Y with the indiscrete topology (cf. [17, p. 85*
*]).
Section 2 Homotopy theories 7
2.11.Colimits
We introduce the notion of the colimit of a functor. Let C be a category and D
a small category. Typically, C is one of the categories in 2.1 and D is from the
following list.
2.12. Shapes of colimit diagrams.
(i)A category with a set I of objects and no nonidentity morphisms.
(ii)The category D = {a b ! c}, with three objects and the two indicated
nonidentity morphisms.
(iii)The category Z+ = {0 ! 1 ! 2 ! 3 ! : :}:with objects the nonnegative
integers and a single morphism i ! j for i 6 j.
There is a diagonal or "constant diagram" functor
: C ! CD ;
which carries an object X 2 C to the constant functor (X) : D ! C (by defini-
tion, this "constant functor" sends each object of D to X and each morphism of D
to idX). The functor assigns to each morphism f : X ! X0 of C the constant
natural transformation t(f) : X ! X0 determined by the formula t(f)d = f for
each object d of D.
2.13. Definition. Let D be a small category and F : D ! C a functor. A colimit
for F is an object C of C together with a natural transformation t : F ! (C)
such that for every object X of C and every natural transformation s : F ! (X),
there exists a unique map s0: C ! X in C such that (s0)t = s [17, p. 67].
Remark. The universal property of a colimit implies as usual that any two colim*
*its
for F are canonically isomorphic. If a colimit of F exists we will speak of "th*
*e"
colimit of F and denote it colim(F ). The colimit is sometimes called the direct
limit, and denoted l!imF , l!imDF or colimDF . Roughly speaking, (colim(F )) is
the constant diagram which is most efficient at receiving a map from F , in the*
* sense
that any map from F to a constant diagram extends uniquely over the universal
map F ! (colim(F )).
2.14. Remark. A category C is said to have all small (resp. finite) colimits *
*if
colim(F ) exists for any functor F from a small (resp. finite) category D to C.*
* The
categories Set, Top and Mod R have all small colimits. Suppose that D is a small
category and F : D ! Set is a functor. Let U be the disjoint union of the sets
which appear as values of F , i.e., let U be the set of pairs (d; x) where d 2 *
*Ob(D )
and x 2 F (d). Then colim(F ) is the quotient of U with respect to the equivale*
*nce
relation "~" generated by the formulas (d; x) ~ (d0; F (f)(x)), where f : d ! d0
is a morphism of D . If F : D ! Top is a functor, then colim(F ) is an analogous
8 Dwyer and Spalinski Chapter 1
quotient space of the space U which is the disjoint union of the spaces appeari*
*ng as
values of F . If F : D ! Mod R is a functor, then colim(F ) is an analogous quo*
*tient
module of the module U which is the direct sum of the modules appearing as valu*
*es
of F .
Remark. If colim(F ) exists for every object F of CD, an argument from the univ*
*ersal
property (2.13) shows that the various objects colim(F ) of C fit together into*
* a
functor colim(-) which is left adjoint to :
colim: CD () C : :
We will now give some examples of colimits [17, p. 64].
2.15. Coproducts. Let D be the category of 2.12(i), so that a functor X : D ! C
is just a collection {Xi}i2Iof`objects of C. The colimit of`X is called the cop*
*roduct
of the collection and written iXi or, if I = {0; 1}, X0 X1. If C is Set or T*
*op
the coproduct is disjoint union; if C is Mod R, coproduct is direct sum.`If I =*
* {0; 1},
then the definition`of colimit (2.13) gives natural maps in0: X0 ! X0 X1 and
in1: X1`! X0 X1; given maps fi : Xi ! Y (i = 0, 1) there is a unique map
f : X0 X1 ! Y such that f . ini= fi(i = 0, 1). The map f is ordinarily denoted
f0+ f1.
2.16. Pushouts. If D is the category of 2.12(ii), then a functor X : D ! C is *
*a di-
agram X(a) X(b) ! X(c) in C. In this case the colimit of X is called the push*
*out
P of the diagram X(a) X(b) ! X(c). It is the result of appropriately gluing
X(a) to X(c) along X(b). The definition of colimit gives a natural commutative
diagram
X(b) -i! X(c)
j # j0# :
0
X(a) -i! P
Any diagram isomorphic to a diagram of this form is called a pushout diagram; t*
*he
map i0is called the cobase change of i (along j) and the map j0is called the co*
*base
change of j (along i). Given maps fa : X(a) ! Y and fc : X(c) ! Y such that
faj = fci, there is a unique map f : P ! Y such that fj0= fc and fi0= fa.
2.17. Sequential colimits. If D is the category of 2.12(iii), a functor X : D *
*! C
is a diagram of the following form
X(0) ! X(1) ! . .!.X(n) ! . . .
Section 2 Homotopy theories 9
in C; this is called a sequential direct system in C. The colimit of this direc*
*t system
is called the sequential colimit of the objects X(n), and denoted colimnX(n). I*
*f C is
one of the categories Set, Top or Mod R and each one of the maps X(n) ! X(n+1)
is an inclusion, then colimnX(n) can be interpreted as an increasing union of t*
*he
X(n); if C = Top a subset of this union is open if and only if its intersection*
* with
each X(n) is open.
2.18.Limits
We next introduce the notion of the limit of a functor [17, p. 68]. This is str*
*ictly
dual to the notion of colimit, in the sense that a limit of a functor F : D ! C*
* is the
same as a colimit of the "opposite functor" F op: Dop ! Cop. From a logical poi*
*nt
of view this may be everything there is to say about limits, but it is worthwhi*
*le to
make the construction more explicit and work out some examples.
Let C be a category and D a small category. Typically, C is as before (2.1) and
D is one of the following.
2.19. Shapes of limit diagrams.
(i)A category with a set I of objects and no nonidentity morphisms.
(ii)The category D = {a ! b c}, with three objects and the two indicated
nonidentity morphisms.
Let : C ! CD be as before (2.11) the "constant diagram" functor.
2.20. Definition. Let D be a small category and F : D ! C a functor. A limit
for F is an object L of C together with a natural transformation t : (L) ! F
such that for every object X of C and every natural transformation s : (X) ! F ,
there exists a unique map s0: X ! L in C such that t(s0) = s.
Remark. The universal property of a limit implies as usual that any two limits *
*for
F are canonically isomorphic. If a limit of F exists we will speak of "the" lim*
*it of F
and denote it lim(F ). The limit is sometimes called the inverse limit, and den*
*oted
limF , limDF or limDF . Roughly speaking, (lim(F )) is the constant diagram
which is most efficient at originating a map to F , in the sense that any map f*
*rom
a constant diagram to F lifts uniquely over the universal map (colim(F )) ! F .
2.21. Remark. A category C is said to have all small (resp. finite) limits if *
*lim(F )
exists for any functor F from a small (resp. finite) category D to C. The categ*
*ories
Set, Top and Mod R have all small limits. Suppose that D is a small category and
F : D ! Setis a functor. Let P be the product of the sets which appear as values
of F , i.e., let U be the set of pairs (d; x) where d 2 Ob(D ) and x 2 F (d), q*
* : U !
Ob(D ) the map with q(d; x) = d, and P the set of all functions s : Ob(D ) ! U
10 Dwyer and Spalinski Chapter 1
such that qs is the identity map of Ob (D ). For s 2 P write s(d) = (d; s1(d)),
with s1(d) 2 F (d). Then lim(F ) is the subset of P consisting of functions s w*
*hich
satisfy the equation s1(d0) = F (f)(s1(d)) for each morphism f : d ! d0 of D . *
*If
F : D ! Top is a functor, then lim(F ) is the corresponding subspace of the spa*
*ce
P which is the product of the spaces appearing as values of F . If F : D ! Mod R
is a functor, then lim(F ) is the corresponding submodule of the module U which*
* is
the direct product of the modules appearing as values of F .
Remark. If lim(F ) exists for every object F of CD , an argument from the unive*
*rsal
property (2.20) shows that various objects lim(F ) of C fit together into a fun*
*ctor
lim(-) which is right adjoint to :
: C () CD : lim:
We will now give two examples of limits [17, p. 70].
2.22. Products. Let D be the category of 2.19(i), so that a functor X : D ! C *
*is
just a collection {Xi}i2IofQobjects of C. The limit of X is called the productQ*
*of the
collection and written iXi or, if I = {0; 1}, X0 x X1 (the notation "X0 X1"
is more logical but seems less common). If C is Setor Top the product is what is
usually called direct product or cartesian product. If I = {0; 1}Qthen the defi*
*nition
of limit (2.20) gives natural maps pr0: X0 x X1 ! X0 and pr1: X0 X1 ! X1;
given maps fi : Y ! Xi (i = 0, 1) there is a unique map f : Y ! X0 x X1 such
that pri. f = fi(i = 0, 1). The map f is ordinarily denoted (f0; f1).
2.23. Pullbacks. If D is the category of 2.19(ii), then a functor X : D ! C is
a diagram X(a) ! X(b) X(c) in C . In this case the limit of X is called the
pullback P of the diagram X(a) ! X(b) X(c). The definition of limit gives a
natural commutative diagram
0
P -i! X(c)
j0# j# :
X(a) -i! X(b)
Any diagram isomorphic to a diagram of this form is called a pullback diagram; *
*the
map i0 is called the base change of i (along j) and the map j0 is called the ba*
*se
change of j (along i). Given maps fa : Y ! X(a) and fc : Y ! X(c) such that
ifa = jfc, there is a unique map f : Y ! P such that i0f = fc and j0f = fa.
2.24.Some remarks on limits and colimits
An object ; of a category C is said to be an initial object if there is exactly*
* one
map from ; to any object X of C. Dually, an object * of C is said to be a termi*
*nal
Section 2 Homotopy theories 11
object if there is exactly one map X ! * for any object X of C. Clearly initial*
* and
terminal objects of C are unique up to canonical isomorphism. The proof of the
following statement just involves unravelling the definitions.
Proposition 2.25.Let C be a category, D the empty category (i.e. the category
with no objects), and F : D ! C the unique functor. Then colim(F ), if it exist*
*s, is
an initial object of C and lim(F ), if it exists, is a terminal object of C.
Suppose that D is a small category, that X : D ! C is a functor, and that
F : C ! C 0is a functor. If colim(X) and colim(F X) both exist, then it is
easy to see that there is a natural map colim(F X) ! F (colimX). Similarly, if
lim(F ) and lim(F X) both exist, then it is easy to see that there is a natural
map F (limX) ! lim(F X). The functor F is said to preserve colimits if when-
ever X : D ! C is a functor such that colim(X) exists, then colim(F X) exists
and the natural map colim(F X) ! F (colimX) is an isomorphism. The functor F
is said to preserve limits if the corresponding dual condition holds [17, p 112*
*]. The
following proposition is a formal consequence of the definition of an adjoint f*
*unctor
pair.
Proposition 2.26.[17, p. 114-115] Suppose that
F : C () C0: G
is an adjoint functor pair. Then F preserves colimits and G preserves limits.
Remark. Proposition 2.26 explains why the underlying set of a product (2.22) or
pullback (2.23) in the category Mod R or Top is the same as the product or pull*
*back
of the underlying sets: in each case the underlying set (or forgetful) functor *
*is a right
adjoint (2.9, 2.10) and so preserves limits, e.g. products and pullbacks. Conve*
*rsely,
2.26 pins down why the forgetful functor G of 2.9 cannot possibly be a left adj*
*oint
or equivalently cannot possibly have a right adjoint: G does not preserve colim*
*its,
since for instance it does not take coproducts of R-modules (i.e. direct sums) *
*to
coproducts of sets (i.e. disjoint unions).
We will use the following proposition in x10.
Lemma 2.27. [17, p. 112] Suppose that C has all small limits and colimits and
that D is a small category. Then the functor category CD also has small limits *
*and
colimits.
Remark. In the situation of 2.27 the colimits and limits in C Dcan be computed
"pointwise" in the following sense. Suppose that X : D0 ! CD is a functor. Then
for each object d of D there is an associated functor Xd : D 0! C given by the
formula Xd(d0) = (X(d0))(d). It is not hard to check that for each d 2 Ob(D ) t*
*here
are natural isomorphisms (colimX)(d) ~=colim(Xd) and (limX)(d) ~=lim(Xd).
12 Dwyer and Spalinski Chapter 1
3. Model categories
In this section we introduce the concept of a model category and give some exam-
ples. Since checking that a category has a model category structure is not usua*
*lly
trivial, we defer verifying the examples until later (x7 and x8).
3.1. Definition. Given a commutative square diagram of the following form
A -f! X
i# p# (3.2)
B -g! Y
a lift or lifting in the diagram is a map h : B ! X such that the resulting dia*
*gram
with five arrows commutes, i.e., such that hi = f and ph = g.
3.3. Definition. A model category is a category C with three distinguished cla*
*sses
of maps:
(i)weak equivalences (!~),
(ii)fibrations (!!), and
(iii)cofibrations (,!)
each of which is closed under composition and contains all identity maps. A map
which is both a fibration (resp. cofibration) and a weak equivalence is called *
*an
acyclic fibration (resp. acyclic cofibration). We require the following axioms.
MC1 Finite limits and colimits exist in C (2.14, 2.21).
MC2 If f and g are maps in C such that gf is defined and if two of the three m*
*aps
f, g, gf are weak equivalences, then so is the third.
MC3 If f is a retract of g (2.6) and g is a fibration, cofibration, or a weak *
*equiva-
lence, then so is f.
MC4 Given a commutative diagram of the form 3.2, a lift exists in the diagram
in either of the following two situations: (i) i is a cofibration and p is an*
* acyclic
fibration, or (ii) i is an acyclic cofibration and p is a fibration.
MC5 Any map f can be factored in two ways: (i) f = pi, where i is a cofibration
and p is an acyclic fibration, and (ii) f = pi, where i is an acyclic cofibra*
*tion
and p is a fibration.
Remark. The above axioms describe what in [22] is called a "closed" model categ*
*ory;
since no other kind of model category comes up in this paper, we have decided to
leave out the word "closed". In [22] Quillen uses the terms "trivial cofibratio*
*n"
and "trivial fibration" instead of "acyclic cofibration" and "acyclic fibration*
*". This
conflicts with the ordinary homotopy theoretic use of "trivial fibration" to me*
*an a
fibration in which the total space is equivalent to the product of the base and*
* fibre;
in geometric examples of model categories, the "acyclic fibrations" of 3.3 usua*
*lly
Section 3 Homotopy theories 13
turn out to be fibrations with a trivial fibre, so that the total space is equi*
*valent
to the base. We have followed Quillen's later practice in using the word "acycl*
*ic".
The axioms as stated are taken from [23].
3.4. Remark. By MC1 and 2.25, a model category C has both an initial object
; and a terminal object *. An object A 2 C is said to be cofibrant if ; ! A
is a cofibration and fibrant if A ! * is a fibration. Later on, when we define *
*the
homotopy category Ho(C ), we will see that Hom Ho(C)(A; B) is in general a quot*
*ient
of Hom C(A; B) only if A is cofibrant and B is fibrant; if A is not cofibrant o*
*r B is
not fibrant, then there are not in general a sufficient number of maps A ! B in*
* C
to represent every map in the homotopy category.
The factorizations of a map in a model category provided by MC5 are not
required to be functorial. In most examples (e.g., in cases in which the factor*
*izations
are constructed by the small object argument of 7.12) the factorizations can be
chosen to be functorial.
We now give some examples of model categories.
3.5. Example. (see x8) The category Top of topological spaces can be given the
structure of a model category by defining f : X ! Y to be
(i)a weak equivalence if f is a weak homotopy equivalence (8.1)
(ii)a cofibration if f is a retract (2.6) of a map X ! Y 0in which Y 0is obta*
*ined
from X by attaching cells (8.8), and
(iii)a fibration if f is a Serre fibration (8.2).
With respect to this model category structure, the homotopy category Ho(Top ) is
equivalent to the usual homotopy category of CW-complexes (cf. 8.4).
The above model category structure seems to us to be the one which comes
up most frequently in everyday algebraic topology. It puts an emphasis on CW-
structures; every object is fibrant, and the cofibrant objects are exactly the
spaces which are retracts of generalized CW-complexes (where a "generalized CW-
complex" is a space built up from cells, without the requirement that the cells
be attached in order by dimension.) In some topological situations, though, weak
homotopy equivalences are not the correct maps to focus on. It is natural to ask
whether there is another model category structure on Top with respect to which
the "weak equivalences" are the ordinary homotopy equivalences. There is a beau-
tiful paper of Strom [26] which gives a positive answer to this question. If B *
*is a
topological space, call a subspace inclusion i : A ! B a closed Hurewicz cofibr*
*ation
if A is a closed subspace of B and i has the homotopy extension property, i.e.,*
* for
every space Y a lift (3.1) exists in every commutative diagram
B x 0 [ A x [0; 1]-! Y
# # :
B x [0; 1] -! *
14 Dwyer and Spalinski Chapter 1
Call a map p : X ! Y a Hurewicz fibration if p has the homotopy lifting propert*
*y,
i.e., for every space A a lift exists in every commutative diagram
A x 0 -! X
# p # :
A x [0; 1]-! Y
3.6. Example. [26] The category Top of topological spaces can be given the str*
*uc-
ture of a model category by defining a map f : X ! Y to be
(i)a weak equivalence if f is a homotopy equivalence,
(ii)a cofibration if f is a closed Hurewicz cofibration, and
(iii)a fibration if f is a Hurewicz fibration.
With respect to this model category structure, the homotopy category Ho(Top ) is
equivalent to the usual homotopy category of topological spaces.
Remark. The model category structure of 3.6 is quite different from the one of
3.5. For example, let W be the "Warsaw circle"; this is the compact subspace of
the plane given by the union of the interval [-1; 1] on the y-axis, the graph of
y = sin(1=x) for 0 < x 6 1, and an arc joining (1; sin(1)) to (0; -1). Then the
map from W to a point is a weak equivalence with respect to the model category
structure of 3.5 but not a weak equivalence with respect to the model category
structure of 3.6.
It turns out that many purely algebraic categories also carry model category
structures. Let R be a ring and Ch R the category of nonnegatively graded chain
complexes over R.
3.7. Example. (see x7) The category Ch Rcan be given the structure of a model
category by defining a map f : M ! N to be
(i)a weak equivalence if f induces isomorphisms on homology groups,
(ii)a cofibration if for each k > 0 the map fk : Mk ! Nk is a monomorphism
with a projective R-module (x7.1) as its cokernel, and
(iii)a fibration if for each k > 1 the map fk : Mk ! Nk is an epimorphism.
The cofibrant objects in Ch R are the chain complexes M such that each Mk is a
projective R-module. The homotopy category Ho(Ch R) is equivalent to the cate-
gory whose objects are these cofibrant chain complexes and whose morphisms are
ordinary chain homotopy classes of maps (cf. proof of 7.3).
Given a model category, it is possible to construct many other model categories
associated to it. We will do quite a bit of this in x10. Here are two elementary
examples.
3.8. Example. Let C be a model category. Then the opposite category Cop (2.4)
can be given the structure of a model category by defining a map fop : Y ! X to
Section 3 Homotopy theories 15
be
(i)a weak equivalence if f : X ! Y is a weak equivalence in C,
(ii)a cofibration if f : X ! Y is a fibration in C,
(iii)a fibration if f : X ! Y is a cofibration in C.
3.9. Duality. Example 3.8 reflects the fact that the axioms for a model cate-
gory are self-dual. Let P be a statement about model categories and P *the dual
statement obtained by reversing the arrows in P and switching "cofibration" with
"fibration". If P is true for all model categories, then so is P *.
Remark. The duality construction in 3.9 corresponds via 3.5 or 3.6 to what is
usually called "Eckmann-Hilton" duality in ordinary homotopy theory. Since there
are interesting true statements P about the homotopy theory of topological spac*
*es
whose Eckmann-Hilton dual statements P *are not true, it must be that there are
interesting facts about ordinary homotopy theory which cannot be derived from
the model category axioms. Of course this is something to be expected; the axio*
*ms
are an attempt to codify what all homotopy theories might have in common, and
just about any particular model category has additional properties that go beyo*
*nd
what the axioms give.
If C is a category and A is an object of C , the under category [17, p. 46] (or
comma category) A#C is the category in which an object is a map f : A ! X in
C. A morphism in this category from f : A ! X to g : A ! Y is a map h : X ! Y
in C such that hf = g.
3.10. Remark. Let C be a model category and A an object of C. Then it is possi*
*ble
to give A#C the structure of a model category by defining h : (A ! X) ! (A ! Y )
in A#C to be
(i)a weak equivalence if h : X ! Y is a weak equivalence in C,
(ii)a cofibration if h : X ! Y is a cofibration in C, and
(iii)a fibration if h : X ! Y is a fibration in C.
Remark. Let Top have the model category structure of 3.6 and as usual let * be *
*the
terminal object of Top, i.e., the space with one point. Then *#Top is the categ*
*ory
of pointed spaces, and an object X of *#Top is cofibrant if and only if the bas*
*epoint
of X is closed and nondegenerate [25, p. 380]. Thus (3.7) from the point of view
of model categories, having a nondegenerate basepoint is for a space what being
projective is for a chain complex!
3.11. Remark. In the situation of 3.10, we leave it to the reader to define th*
*e over
category C#A and describe a model category structure on it. If C is the categor*
*y of
spaces (3.5 and 3.6), the model category structure on C#A is related to fibrewi*
*se
homotopy theory [15].
16 Dwyer and Spalinski Chapter 1
In the remainder of this section we make some preliminary observations about
model categories.
3.12. Lifting Properties. A map i : A ! B is said to have the left lifting pro*
*perty
(LLP) with respect to another map p : X ! Y and p is said to have the right lif*
*ting
property (RLP) with respect to i if a lift exists (3.1) in any diagram of the f*
*orm
3.2.
Proposition 3.13.Let C be a model category.
(i)The cofibrations in C are the maps which have the LLP with respect to
acyclic fibrations.
(ii)The acyclic cofibrations in C are the maps which have the LLP with respe*
*ct
to fibrations.
(iii)The fibrations in C are the maps which have the RLP with respect to acyc*
*lic
cofibrations.
(iv)The acyclic fibrations in C are the maps which have the RLP with respect
to cofibrations.
Proof. Axiom MC4 implies that an (acyclic) cofibration or an (acyclic) fibration
has the stated lifting property. In each case we need to prove the converse. Si*
*nce
the four proofs are very similar (and in fact statements (iii) and (iv) follow *
*from (i)
and (ii) by duality), we only give the first proof. Suppose that f : K ! L has *
*the
LLP with respect to all acyclic fibrations. Factor f as a composite K ,! L0!~!L
as in MC5(i), so i : K ! L0is a cofibration and p : L0! L is an acyclic fibrati*
*on.
By assumption there exists a lifting g : L ! L0in the following diagram:
K -i! L0
f # p# ~:
L -id! L
This implies that f is a retract of i:
K -id! K -id! K
f # i# f #
L -g! L0 - p! L:
By MC3 we conclude that f is a cofibration. __|_|
Remark. Proposition 3.13 implies that the axioms for a model category are overd*
*e-
termined in some sense. This has the following practical consequence. If we are
trying to set up a model category structure on some given category and have cho-
sen the fibrations and the weak equivalences, then the class of cofibrations is*
* pinned
Section 4 Homotopy theories 17
down by property 3.13(i). Dually, if we have chosen the cofibrations and weak e*
*quiv-
alences, the class of fibrations is pinned down by property 3.13(iii). Verifyin*
*g the
axioms then comes down in part to checking certain consistency conditions.
Proposition 3.14.Let C be a model category.
(i)The class of cofibrations in C is stable under cobase change (2.16).
(ii)The class of acyclic cofibrations is stable under cobase change.
(iii)The class of fibrations is stable under base change (2.23).
(iv)The class of acyclic fibrations is stable under base change.
Proof. The second two statements follow from the first two by duality (3.9), so*
* we
only prove the first and indicate the proof of the second. Assume that i : K ,!*
* L
is a cofibration, and pick a map f : K ! K0. Construct a pushout diagram (cf.
MC1):
K -f! K0
i# j#
L -g! L0:
We have to prove that j is a cofibration. By (i) of the previous proposition it
is enough to show that j has the LLP with respect to an acyclic fibration. Let
p : E ! B be an acyclic fibration and consider a lifting problem
K0 -a! E
j # p# : (3.15)
L0 -b! B
Enlarge this to the following diagram
K -f! K0 -a! E
i# p# :
L -g! L0 -b! B
Since i is a cofibration, there is a lifting h : L ! E in the above diagram. By*
* the
universal property of pushouts, the maps h : L ! E and a : K0 ! E induce the
desired lifting in 3.15. The proof of part (ii) is analogous, the only differen*
*ce_being
that we need to invoke 3.13(ii) instead of 3.13(i). |_|
4. Homotopy Relations on Maps
In this section C is some fixed model category, and A and X are objects of C. O*
*ur
goal is to exploit the model category axioms to construct some reasonable homot*
*opy
18 Dwyer and Spalinski Chapter 1
relations on the set Hom C(A; X) of maps from A to X. We first study a notion of
left homotopy, defined in terms of cylinder objects, and then a dual (3.9) noti*
*on of
right homotopy, defined in terms of path objects. It turns out (4.21) that the *
*two
notions coincide in what will turn out to be the most important case, namely if*
* A
is cofibrant and X is fibrant.
4.1.Cylinder objects and left homotopy
4.2. Definition. A cylinder object for A is an object A ^ I of C together with*
* a
diagram (MC1, 2.15):
a i ~
A A ! A ^ I ! A
`
which factors the folding map idA+ idA: A A ! A (2.15). A cylinder object
A ^ I is called `
(i)a good cylinder object, if A A ! A ^ I is a cofibration, and
(ii)a very good cylinder object, if in addition the map A^I ! A is a (necessa*
*rily
acyclic) fibration.
If A^I is a cylinder object for A, we will denote the two structure maps A ! A*
*^I
by i0 = i . in0and i1 = i . in1(cf. 2.15).
4.3. Remark. By MC5, at least one very good cylinder object for A exists. The
notation A ^ I is meant to suggest the product of A with an interval (Quillen e*
*ven
uses the notation "A x I" for a cylinder object). However, a cylinder object A *
*^ I is
not necessarily the product of A with anything in C; it is just an object of C *
*with
the above formal property. An object A of C might have many cylinder objects
associated to it, denoted, say, A ^ I, A ^ I0,: :,:etc. We do not assume that t*
*here is
some preferred natural cylinder object for A; in particular, we do not assume t*
*hat
a cylinder object can be chosen in a way that is functorial in A.
Lemma 4.4. If A is cofibrant and A ^ I is a good cylinder object for A, then the
maps i0; i1 : A ! A ^ I are acyclic cofibrations.
Proof. It is enough to check this for i0. Since the identity map idA: A ! A fac*
*tors`
as A i0!A^I ~!A, it follows from MC2 that i0 is a weak equivalence. Since A A
is defined by the following pushout diagram (2.16)
; -! A
cofibration# in0#
`
A in1-!A A
Section 4 Homotopy theories 19
it follows from 3.14 that the map in0is a cofibration. Since i0 is thus the com*
*posite
a
A in0!A A ! A ^ I;
of two cofibrations, it itself is a cofibration. *
*__|_|
Definition. Two maps f; g : A ! X in C are said to be left homotopic (written
f l~g)`if there exists a cylinder object A ^ I for A such that the sum map f + *
*g :
A A ! X (2.15) extends to a map H : A ^ I ! X, i.e. such that there exists a
map H : A ^ I ! X with H(i0 + i1) = f + g. Such a map H is said to be a left
homotopy from f to g (via the cylinder object A ^ I). The left homotopy is said*
* to
be good (resp. very good) if A ^ I is a good (resp. very good) cylinder object *
*for A.
Example. Let C be the category of topological spaces with the model category
structure described in 3.5. Then one choice of cylinder object for a space A is
the product A x [0; 1]. The notion of left homotopy with respect to this cylind*
*er
object coincides with the usual notion of homotopy. Observe that if A is not a
CW-complex, A x [0; 1] is not usually a good cylinder object for A.
4.5. Remark. If f ~lg via the homotopy H, then it follows from MC2 that the
map f is a weak equivalence if and only if g is. To see this, note that as in t*
*he
proof of 4.4 the maps i0 and i1 are weak equivalences, so that if f = Hi0 is a *
*weak
equivalence, so is H and hence so is g = Hi1.
Lemma 4.6. If f ~lg : A ! X, then there exists a good left homotopy from f to
g. If in addition X is fibrant, then there exists a very good left homotopy fro*
*m f to
g.
`
Proof. The first statement follows from applying MC5(i) to the map A A ! A^I,
where A^I is the cylinder object in some left homotopy from f to g. For the sec*
*ond,
choose a good left homotopy H : A ^ I ! X from f to g. By MC5(i) and MC2,
we may factor A ^ I ~!A as
A ^ I ~,!A ^ I0!~!A:
Applying MC4 to the following diagram
A ^ I -H! X
# #
A ^ I0 -! *
gives the desired very good homotopy H0: A ^ I0! X. __|_|
Lemma 4.7. If A is cofibrant, then l~is an equivalence relation on Hom C(A; X).
20 Dwyer and Spalinski Chapter 1
Proof. Since we can take A itself as a cylinder`object`for A, we can take f its*
*elf
as a left homotopy between f and f. Let s : A A ! A A be the map which
switches factors (technically, s = in1+ in0). The identity (g + f) = (f + g)s s*
*hows
that if f ~lg, then g ~lf. Suppose that f ~lg and g ~lh. Choose a good (4.6)
left homotopy H : A ^ I ! X from f to g (i.e. Hi0 = f; Hi1 = g) and a good left
homotopy H0: A ^ I0! X from g to h (i.e. H0i00= g; H0i01= h). Let A ^ I00be the
pushout of the following diagram:
0
A ^ I -i1~A -i0~!A ^ I0 :
Since the maps i1 : A ! A^I and i00: A ! A^I0are acyclic cofibrations, it follo*
*ws
from 3.14 and the universal property of pushouts (2.16) that A ^ I00is a cylind*
*er
object for A. Another application of 2.16 to the maps H and H0 gives the_desired
homotopy H00: A ^ I00! X from f to h. |_|
Let ssl(A; X) denote the set of equivalence classes of Hom C(A; X) under the
equivalence relation generated by left homotopy.
4.8. Remark. The word "generated" in the above definition of ssl(A; X) is im-
portant. We will sometimes consider ssl(A; X) even if A is not cofibrant; in th*
*is
case left homotopy, taken on its own, is not necessarily an equivalence relatio*
*n on
HomC (A; X).
Lemma 4.9. If A is cofibrant and p : Y ! X is an acyclic fibration, then compo-
sition with p induces a bijection:
p* : ssl(A; Y ) ! ssl(A; X); [f] 7! [pf]:
Proof. The map p* is well defined, since if f; g : A ! Y are two maps and H is
a left homotopy from f to g, then pH is a left homotopy from pf to pg. To show
that p* is onto, choose [f] 2 ssl(A; X). By MC4(i), a lift g : A ! Y exists in *
*the
following diagram:
; -! Y
# p# ~ :
A -f! X
Clearly p*[g] = [pg] = [f]. To prove that p* is one to one, let f; g : A ! Y and
suppose that pf l~pg : A ! X. Choose (4.6) a good left homotopy H : A ^ I ! X
Section 4 Homotopy theories 21
from pf to pg. By MC4(i), a lifting exists in the following diagram
` f+g
A A -! Y
# p# ~
A ^ I -H! X:
and gives the desired left homotopy from f to g. __|_|
Lemma 4.10. Suppose that X is fibrant, that f and g are left homotopic maps
A ! X, and that h : A0! A is a map. Then fh l~gh.
Proof. By 4.6, we can choose a very good left homotopy H : A ^ I ! X between f
and g. Next choose a good cylinder object for A0:
a j ~
A0 A0,! A0^ I ! A0:
By MC4, there is a lifting k : A0^ I ! A ^ I in the following diagram:
`
` h h ` i
A0 A0 -! A A -! A ^ I
j # ~# :
A0^ I ~-! A0 -h! A
It is easy to check that Hk is the desired homotopy. __|_|
Lemma 4.11. If X is fibrant, then the composition in C induces a map:
ssl(A0; A) x ssl(A; X) ! ssl(A0; X); ([h]; [f]) 7! [fh]:
Proof. Note that we are not assuming that A is cofibrant, so that two maps A ! X
which represent the same element of ssl(A; X) are not necessarily directly rela*
*ted
by a left homotopy (4.8). Nevertheless, it is enough to show that if h l~k : A0*
*! A
and f ~lg : A ! X then fh and gk represent the same element of ssl(A0; X). For
this it is enough to check both that fh l~gh : A0! X and that gh l~gk : A0! X.
The first homotopy follows from the previous lemma. The second is obtained_by
composing the homotopy between h and k with g. |_|
4.12.Path objects and right homotopies
By duality (3.9), what we have proved so far in this section immediately gives
corresponding results "on the other side".
22 Dwyer and Spalinski Chapter 1
Definition. A path object for X is an object XI of C together with a diagram (2*
*.22)
X ~!XI p!X x X
which factors the diagonal map (idX; idX) : X ! X x X. A path object XI is
called
(i)a good path object, if XI ! X x X is a fibration, and
(ii)a very good path object, if in addition the map X ! XI is a (necessarily
acyclic) cofibration.
4.13. Remark. By MC5, at least one very good path object exists for X. The
notation XI is meant to suggest a space of paths in X, i.e., a space of maps fr*
*om
an interval into X. However a path object XI is not in general a function objec*
*t of
any kind; it is just some object of C with the above formal property.0An object*
* X
of C might have many path objects associated to it, denoted XI, XI ,: :,:etc.
We denote the two maps XI ! X by p0 = pr0. p and p1 = pr1. p (cf. 2.22).
Lemma 4.14. If X is fibrant and XI is a good path object for X, then the maps
p0; p1 : XI ! X are acyclic fibrations.
Definition. Two maps f; g : A ! X are said to be right homotopic (written f r~g)
if there exists a path object XI for X such that the product map (f; g) : A ! X*
* xX
lifts to a map H : A ! XI. Such a map H is said to be a right homotopy from f to
g (via the path object XI). The right homotopy is said to be good (resp.very go*
*od)
if XI is a good (resp. very good) path object for X.
Example. Let the category of topological spaces have the structure described in*
* 3.5.
Then one choice of path object for a space X is the mapping space Map ([0; 1]; *
*X).
Lemma 4.15. If f r~g : A ! X then there exists a good right homotopy from f to
g. If in addition A is cofibrant, then there exists a very good right homotopy *
*from
f to g.
Lemma 4.16. If X is fibrant, then r~is an equivalence relation on Hom C(A; X).
Let ssr(A; X) denote the set of equivalence classes of Hom C(A; X) under the
equivalence relation generated by right homotopy.
Lemma 4.17. If X is fibrant and i : A ! B is an acyclic cofibration, then compo-
sition with i induces a bijection:
i* : ssr(B; X) ! ssr(A; X) :
Lemma 4.18. Suppose that A is cofibrant, that f and g are right homotopic maps
from A to X, and that h : X ! Y is a map. Then hf r~hg.
Lemma 4.19. If A is cofibrant then the composition in C induces a map
ssr(A; X) x ssr(X; Y ) ! ssr(A; Y ).
Section 4 Homotopy theories 23
4.20.Relationship between left and right homotopy
The following lemma implies that if A is cofibrant and X is fibrant, then the l*
*eft
and right homotopy relations on Hom C(A; X) agree.
Lemma 4.21. Let f; g : A ! X be maps.
(i)If A is cofibrant and f l~g, then f r~g.
(ii)If X is fibrant and f r~g, then f l~g.
4.22. Homotopic maps. If A is cofibrant and X is fibrant, we will denote the
identical right homotopy and left homotopy equivalence relations on Hom C(A; X)
by the symbol "~" and say that two maps related by this relation are homotopic.
The set of equivalence classes with respect to this relation is denoted ss(A; X*
*).
Proof of 4.21. Since the two statements are dual, we only have to prove the fir*
*st
one. By 4.6 there exists a good cylinder object
a i0+i1 j
A A -! A ^ I ! A
for A and a homotopy H : A ^ I ! X from f to g. By 4.4 the map i0 is an acyclic
cofibration. Choose a good path object (4.13)
X q!XI (p0;p1)-!X x X
for X. By MC4 it is possible to find a lift K : A ^ I ! XI in the diagram
A -qf! XI
i0# # (p0;p1) :
A ^ I (fj;H)-!X x X
The composite Ki1 : A ! XI is the desired right homotopy from f to g. __|*
*_|
4.23. Remark. Suppose that A is cofibrant, X is fibrant, A ^ I is some fixed g*
*ood
cylinder object for A and XI is some fixed good path object for X. Let f; g : A*
* ! X
be maps. The proof of 4.21 shows that f ~ g if and only if f r~g via the fixed *
*path
object XI. Dually, f ~ g if and only if f l~g via the fixed cylinder object A ^*
* I.
We will need the following observation later on.
Lemma 4.24. Suppose that f : A ! X is a map in C between objects A and X
which are both fibrant and cofibrant. Then f is a weak equivalence if and only *
*if f
has a homotopy inverse, i.e., if and only if there exists a map g : X ! A such *
*that
the composites gf and fg are homotopic to the respective identity maps.
24 Dwyer and Spalinski Chapter 1
Proof. Suppose first that f is a weak equivalence. By MC5 we can factor f as a
composite
q p
A ,~!C !! X (4.25)
in which by MC2 the map p is also a weak equivalence. Because q : A ! C is a
cofibration and A is fibrant, an application of MC4 shows that there exists a l*
*eft
inverse for q, i.e. a morphism r : C ! A such that rq = idA. By lemma 4.17, q
induces a bijection q* : ssr(C; C) ! ssr(A; C), [g] 7! [gq]. Since q*([qr]) = [*
*qrq] = [q],
we conclude that qr r~1C and hence that r is a two-sided right (equivalently le*
*ft)
homotopy inverse for q. A dual argument (3.9) gives a two-sided homotopy inverse
of p, say s. The composite rs is a two sided homotopy inverse of f = pq.
Suppose next that f has a homotopy inverse. By MC5 we can find a factorization
f = pq as in 4.25. Note that the object C is both fibrant and cofibrant. By MC2,
in order to prove that f is a weak equivalence it is enough to show that p is a*
* weak
equivalence. Let g : X ! A be a homotopy inverse for f, and H : X ^ I ! X a
homotopy between fg and idX. By MC4 we can find a lift H0: X ^ I ! C in the
diagram
X qg-! C
i0# p# :
X ^ I H-! X
Let s = H0i1, so that ps = idX. The map q is a weak equivalence, so by the resu*
*lt
just proved above q has a homotopy inverse, say r. Since pq = f, composing on t*
*he
right with r gives p ~ fr (4.11). Since in addition s ~ qg by the homotopy H0, *
*it
follows (4.11, 4.19) that
sp ~ qgp ~ qgfr ~ qr ~ idC:
By 4.5, then, sp is a weak equivalence. The commutative diagram
C idC-!C -idC! C
# p # sp # p
X -s! C - p! X
shows that p is a retract (2.6) of sp, and hence by MC3 that the map_p_is a weak
equivalence. |_|
5. The homotopy category of a model category
In this section we will use the machinery constructed in x4 to give a quick con*
*struc-
tion of the homotopy category Ho(C ) associated to a model category C.
Section 5 Homotopy theories 25
We begin by looking at the following six categories associated to C.
Cc - the full (2.3) subcategory of C generated by the cofibrant objects in C.
Cf - the full subcategory of C generated by the fibrant objects in C.
Ccf - the full subcategory of C generated by the objects of C which are both fi*
*brant
and cofibrant.
ssC c- the category consisting of the cofibrant objects in C and whose morphisms
are right homotopy classes of maps (see 4.19).
ssC f- the category consisting of fibrant objects in C and whose morphisms are *
*left
homotopy classes of maps (see 4.11).
ssC cf- the category consisting of objects in C which are both fibrant and cofi*
*brant,
and whose morphisms are homotopy classes (4.22) of maps.
These categories will be used as tools in defining Ho(C ) and constructing a
canonical functor C ! Ho(C ). For each object X in C we can apply MC5(i) to
the map ; ! X and obtain an acyclic fibration pX : QX ~!!X with QX cofibrant.
We can also apply MC5(ii) to the map X ! * and obtain an acyclic cofibration
iX : X ~,!RX with RX fibrant. If X is itself cofibrant, let QX = X; if X is fib*
*rant,
let RX = X.
Lemma 5.1. Given a map f : X ! Y in C there exists a map "f: QX ! QY
such that the following diagram commutes:
"f
QX -! QY
pX# ~ pY# ~:
X -f! Y
The map "fdepends up to left homotopy or up to right homotopy only on f, and
is a weak equivalence if and only if f is. If Y is fibrant, then "fdepends up t*
*o left
homotopy or up to right homotopy only on the left homotopy class of f.
Proof. We obtain "fby applying MC4 to the diagram:
; -! QY
# ~ # pY :
QX f.pX-!Y
The statement about the uniqueness of "fup to left homotopy follows from 4.9. F*
*or
the statement about right homotopy, observe that QX is cofibrant, and so by 4.2*
*1(i)
two maps with domain QX which are left homotopic are also right homotopic. The *
* __
weak equivalence condition follows from MC2, and the final assertion from 4.11.*
* |_|
26 Dwyer and Spalinski Chapter 1
5.2. Remark. The uniqueness statements in 5.1 imply that if f = idX then "fis
right homotopic to idQX. Similarly, if f : X ! Y and g : Y ! Z and h = gf, then
"his right homotopic to "g"f. Hence we can define a functor Q : C ! ssC csending
X ! QX and f : X ! Y to the right homotopy class [f"] 2 ssr(QX; QY ).
The dual (3.9) to 5.1 is the following statement.
Lemma 5.3. Given a map f : X ! Y in C there exists a map f: RX ! RY such
that the following diagram commutes:
X -f! Y
iX# ~ iY# ~
f
RX -! RY:
The map f depends up to right homotopy or up to left homotopy only on f, and is
a weak equivalence if and only if f is. If X is cofibrant, then f depends up to*
* right
homotopy or up to left homotopy only on the right homotopy class of f.
5.4. Remark. The uniqueness statements in 5.3 imply that if f = idX then f is
left homotopic to idRX. Moreover, if f : X ! Y and g : Y ! Z and h = gf, then
his left homotopic to gf, Hence we can define a functor R : C ! ssC f sending
X ! RX and f : X ! Y to the left homotopy class [f] 2 ssl(RX; RY ).
Lemma 5.5. The restriction of the functor Q : C ! ssC cto Cf induces a functor
Q0: ssC f! ssC cf. The restriction of the functor R : C ! ssC f to C cinduces a
functor R0: ssC c! ssC cf.
Proof. The two statements are dual to one another, and so we will prove only the
second. Suppose that X and Y are cofibrant objects of C and that f; g : X ! Y
are maps which represent the same map in ssC c; we must show that Rf = Rg. It
is enough to do this in the special case f r~g in which f and g are directly re*
*lated_
by a right homotopy; however in this case it is a consequence of 5.3. *
*|_|
5.6. Definition. The homotopy category Ho(C ) of a model category C is the cat-
egory with the same objects as C and with
Hom Ho(C)(X; Y ) = Hom ssCcf(R0QX; R0QY ) = ss(RQX; RQY ) :
5.7. Remark. There is a functor fl : C ! Ho(C ) which is the identity on objec*
*ts
and sends a map f : X ! Y to the map R0Q(f) : R0Q(X) ! R0Q(Y ). If each
of the objects X and Y is both fibrant and cofibrant, then by construction the
Section 5 Homotopy theories 27
map fl : Hom C(X; Y ) ! Hom Ho(C)(X; Y ) is surjective and induces a bijection
ss(X; Y ) ~=Hom Ho(C)(X; Y ).
It is natural to ask whether or not dualizing the definition of Ho(C ) by repl*
*acing
the composite functor R0Q by Q0R would give anything different. The answer is
that it would not; rather than prove this directly, though, we will give a symm*
*etrical
construction of the homotopy category in the next section. There are some basic
observations about Ho(C )that will come in handy later on.
Proposition 5.8.If f is a morphism of C, then fl(f) is an isomorphism in Ho(C )
if and only if f is a weak equivalence. The morphisms of Ho(C )are generated un*
*der
composition by the images under fl of morphisms of C and the inverses of images
under fl of weak equivalences in C.
Proof. If f : X ! Y is a weak equivalence in C, then R0Q(f) is represented by a
map f0 : RQ(X) ! RQ(Y ) which is also a weak equivalence (see 5.1 and 5.3); by
4.24, then, the map f0 has an inverse up to left or right homotopy and represen*
*ts
an isomorphism in ssC cf. This isomorphism is exactly fl(f). On the other hand,*
* if
fl(f) is an isomorphism then f0 has an inverse up to homotopy and is therefore a
weak equivalence by 4.24; it follows easily that f is a weak equivalence.
Observe by the above that for any object X of C the map fl(iQX )fl(pX )-1 in
Ho(C ) is an isomorphism from X to RQ(X). Moreover, for two objects X and Y
of C, the functor fl induces an epimorphism (5.7)
Hom C(RQ(X); RQ(Y )) ! Hom Ho(C)(RQ(X); RQ(Y )) :
Consequently, any map f : X ! Y in Ho(C ) can be represented as a composite
f = fl(pY )fl(iQY )-1fl(f0)fl(iQX )fl(pX )-1
for some map f0: RQ(X) ! RQ(Y ) in C. __|_|
Proposition 5.8 has the following simple but useful consequence.
Corollary 5.9.If F and G are two functors Ho(C )! D and t : F fl ! Gfl is a
natural transformation, then t also gives a natural transformation from F to G.
Proof. It is necessary to check that for each morphism h of Ho(C )an appropriate
diagram D(h) commutes. By assumption D(h) commutes if h = fl(f) or h = fl(g)-1
for some morphism f in C or weak equivalence g in C. It is easy to check that if
h = h1h2, the D(h) commutes if D(h1) commutes and D(h2) commutes. The lemma
then follows from the fact (5.8) that any map of Ho(C )is a composite of_maps of
the form fl(f) and fl(g)-1. |_|
Lemma 5.10. Let C be a model category and F : C ! D be a functor taking weak
equivalences in C into isomorphisms in D . If f ~lg : A ! X or f r~g : A ! X,
then F (f) = F (g) in D .
28 Dwyer and Spalinski Chapter 1
Proof. We give a proof assuming f l~g, the other case is dual. Choose (4.6) a g*
*ood
left homotopy H : A ^ I ! X from f to g, so that A ^ I is a good cylinder object
for A:
a i0+i1 w
A A ,! A ^ I ~-!A :
Since wi0 = wi1(= idA), we have F (w)F (i0) = F (w)F (i1). Since w is a weak
equivalence, the map F (w) is an isomorphism and it follows that F (i0) = F (i1*
*)._
Hence F (f) = F (H)F (i0) is the same as F (g) = F (H)F (i1). |_|
Proposition 5.11.Suppose that A is a cofibrant object of C and X is a fibrant
object of C . Then the map fl : Hom C(A; X) ! Hom Ho(C)(A; X) is surjective, and
induces a bijection ss(A; X) ~=Hom Ho(C)(A; X).
Proof. By 5.10 and 5.8 the functor fl identifies homotopic maps and so induces a
map ss(A; X) ! Hom Ho(C)(A; X). Consider the commutative diagram
ss(RA; QX) -! ss(A; X)
fl# fl#
Hom Ho(C)(RA; QX) -! Hom Ho(C)(A; X)
in which the horizontal arrows are induced by the pair (iA; pX ). By 5.8 the lo*
*wer
horizontal map is a bijection, while by 4.9 and 4.17 the upper horizontal map is
a bijection. As indicated in 5.7, the left-hand vertical map is also a bijectio*
*n._The
desired result follows immediately. |_|
6. Localization of Categories
In this section we will give a conceptual interpretation of the homotopy catego*
*ry of
a model category. Surprisingly, this interpretation depends only on the class o*
*f weak
equivalences. This suggests that in a model category the weak equivalences carr*
*y the
fundamental homotopy theoretic information, while the cofibrations, fibrations,*
* and
the axioms they satisfy function mostly as tools for making various constructio*
*ns
(e.g., the constructions later on in x10). This also suggests that in putting a*
* model
category structure on a category, it is most important to focus on picking the *
*class
of weak equivalences; choosing fibrations and cofibrations is a secondary issue.
6.1. Definition. Let C be a category, and W C a class of morphisms. A functor
F : C ! D is said to be a localization of C with respect to W if
(i)F (f) is an isomorphism for each f 2 W , and
(ii)whenever G : C ! D 0is a functor carrying elements of W into isomor-
phisms, there exists a unique functor G0: D ! D0 such that G0F = G.
Section 7 Homotopy theories 29
Condition 6.1(ii) guarantees that any two localizations of C with respect to W*
* are
canonically isomorphic. If such a localization exists, we denote it by C ! W -1*
*C.
Example. Let Ab be the category of abelian groups, and W the class of morphisms
f : A ! B such that ker(f) and coker(f) are torsion groups. Let D be the catego*
*ry
with the same objects, but with Hom D(A; B) = Hom Ab(Q A; Q B). Define
F : Ab ! D to be the functor which sends an object A to itself and a map f to
Q f. It is an interesting exercise to verify directly that F is the localizati*
*on of
Ab with respect to W [12, p. 15].
Theorem 6.2. Let C be a model category and W C the class of weak equiv-
alences. Then the functor fl : C ! Ho(C ) is a localization of C with respect to
W .
More informally, Theorem 6.2 says that if C is a model category and W C is
the class of weak equivalences, then W -1C exists and is isomorphic to Ho(C ).
Proof of 6.2. We have to verify the two conditions in 6.1 for fl. Condition 6.1*
*(i)
is proved in 5.8. For 6.1(ii), suppose given a functor G : C ! D carrying weak
equivalences to isomorphisms. We must construct a functor G0: Ho(C ) ! D such
that G0fl = G, and show that G0is unique. Since the objects of Ho(C ) are the s*
*ame
as the objects of C, the effect of G0on objects is obvious. Pick a map f : X ! Y
in Ho(C ), which is represented by a map f0: RQ(X) ! RQ(Y ), well defined up to
homotopy (4.22). Observe by 5.10 that G(f0) depends only on the homotopy class
of f0, and therefore only on f. Define G0(f) by the formula
G0(f) = G(pY )G(iQY )-1G(f0)G(iQX )G(pX )-1 :
It is easy to check that G0is a functor, that is, respects identity maps and co*
*mpo-
sitions. If f is the image of a map h : X ! Y of C, then (5.1 and 5.3) after pe*
*rhaps
altering f0 up to right homotopy we can find a commutative diagram
X pX- QX iQX-!RQ(X)
h # "h# f0# :
Y pY- QY iQY-!RQ(Y )
Applying G to this diagram shows that G0(f) = G(h) and thus that G0 extends
G, that is, G0fl = G. The uniqueness of G0 follows immediately from the_second
statement in 5.8. |_|
7. Chain complexes
Suppose that R is an associative ring with unit and let Mod R denote the cate-
gory of left R-modules. Recall that the category Ch R of (nonnegatively graded)
chain complexes of R-modules is the category in which an object M is a collecti*
*on
30 Dwyer and Spalinski Chapter 1
{Mk}k>0 of R-modules together with boundary maps @: Mk ! Mk-1 (k > 1) such
that @2= 0. A morphism f : M ! N is a collection of R-module homomorphisms
fk : Mk ! Nk such that fk-1@ = @fk. In this section we will construct a model c*
*at-
egory structure (7.2) on Ch Rand give some indication (7.3) of how the associat*
*ed
homotopy theory is related to homological algebra.
7.1. Preliminaries. For an object M of ChR , define the k-dimensional cycle mo*
*d-
ule Cyk(M) to be M0 if k = 0 and ker(@ : Mk ! Mk-1) if k > 0. Define the
k-dimensional boundary module Bdk(M) to be image(@ : Mk+1 ! Mk). There are
homology functors Hk : ChR ! Mod R (k > 0) given by HkM = Cyk(M)=Bd k(M)
(we think of these homology groups as playing the role for chain complexes that
homotopy groups do for a space). A chain complex M is acyclic if H kM = 0
(k > 0). Recall that an R-module P is said to be projective [6] if the followin*
*g three
equivalent conditions hold:
(i)P is a direct summand of a free R-module,
(ii)every epimorphism f : A ! P of R-modules has a right inverse, or
(iii)for every epimorphism A ! B of R-modules, the induced map
Hom ModR(P; A) ! Hom ModR(P; B)
is an epimorphism.
The first goal of this section is to prove the following result.
Theorem 7.2. Define a map f : M ! N in Ch R to be
(i)a weak equivalence if the map f induces isomorphisms HkM ! HkN (k >
0),
(ii)a cofibration if for each k > 0 the map fk : Mk ! Nk is a monomorphism
with a projective R-module as its cokernel, and
(iii)a fibration if for each k > 0 the map fk : Mk ! Nk is an epimorphism.
Then with these choices Ch R is a model category.
After proving this we will make the following calculation. If A is an R-module,
let K(A; n) (n > 0) denote the object M of Ch R with Mn = A and Mk = 0 for
k 6= n (these are the chain complex analogues of Eilenberg-Mac Lane spaces).
Proposition 7.3.For any two R-modules A and B and nonnegative integers m,
n there is a natural isomorphism
Hom Ho(ChR)(K(A; m); K(B; n)) ~=Extn-mR(A; B) :
Here ExtkRis the usual Ext functor from homological algebra [6]. We take it to
be zero if k < 0.
Section 7 Homotopy theories 31
7.4.Proof of MC1-MC3
We should first note that the classes of weak equivalences, fibrations and cofi*
*brations
clearly contain all identity maps and are closed under composition. It is easy *
*to see
that limits and colimits in Ch Rcan be computed degreewise, so that MC1 follows
from the fact that Mod R has all small limits and colimits. Axiom MC2 is clear.
Axiom MC3 follows from the fact that in Mod R a retract of an isomorphism,
monomorphism or epimorphism is another morphism of the same type (cf. 2.7).
It is also necessary to observe that a retract (i.e. direct summand) of a proje*
*ctive
R-module is projective.
7.5.Proof of MC4(i)
We need to show that a lift exists in every diagram of chain complexes:
A -g! X
i# ~# p (7.6)
B -h! Y;
in which i is a cofibration and p is an acyclic fibration. By the definition of*
* fibration,
pk is onto for k > 0. But since (p0)* : H0(X) ! H0(Y ) is an isomorphism, an
application of the five lemma [17, p. 198] shows that p0 is also onto. Hence th*
*ere is
a short exact sequence of chain complexes
0 ! K ! X ! Y ! 0
and it follows from the associated long exact homology sequence [6] [25, p. 181]
that K is acyclic.
We will construct the required map fk : Bk ! Xk by induction on k. It is easy
to construct a plausible map f0, since, by 7.1 and the definition of cofibratio*
*n,
the module B0 splits up to isomorphism as a direct sum A0 P0, where P0 is a
projective module; the map f0 is chosen to be g0 on the factor A0 and any lifti*
*ng
P0 ! X0 of the given map P0 ! Y0 on the factor P0. Assume that k > 0 and that
for j < k maps fj : Bj ! Xj with the following properties have been constructed:
(i)@fj = fj-1@ 1 6 j < k,
(ii)pjfj = hj 0 6 j < k,
(iii)fjij = gj 0 6 j < k.
Proceeding as for k = 0 we can write Bk ~=AkPk and construct a map "fk: Bk !
Xk with properties (ii) and (iii) above. Let E : Bk ! Xk-1 be the difference map
@"fk- fk-1@, so that the map E measures the failure of "fkto satisfy (i). Then
(a)@. E = 0 because fk-1 satisfies (i),
(b) pk-1. E = 0 because pkf"k= hk commutes with @, and
(c)E . ik = 0 because "fkik = gk commutes with @.
32 Dwyer and Spalinski Chapter 1
It follows that E induces a map
E0: Bk=ik(Ak) ~=Pk ! Cyk-1(K) :
However, the chain complex K is acyclic and so the boundary map Kk ! Cyk-1(K)
is an epimorphism. Since Pk is a projective, E0can be lifted to a map E00: Pk !*
* Kk,
which, after precomposition with the surjection Bk ! Pk and postcomposition with
the injection Kk ! Xk, gives a map E000: Bk ! Xk. It is straightforward to check
that setting fk = f"k- E000gives a map Bk ! Xk which satisfies all conditions__
(i)-(iii). This allows the induction to continue. |_|
7.7.Proof of MC4(ii)
This depends on a definition and a few lemmas. Suppose that A is an R-module.
For n > 1 define the object Dn(A) of Ch Rto be the chain complex with
ae
Dn(A)k = 0A kk6==n;nn;-n1- 1;
The boundary map Dn(A)n ! Dn(A)n-1 is the identity map of A. The letter "D"
in this notation stands for "disk".
Lemma 7.8. Let A be an R-module and M an object of Ch R. Then the map
Hom ChR(Dn(A); M) ! Hom ModR(A; Mn)
which sends f to fn is an isomorphism.
This is obvious by inspection. In fact, the functor Dn(-) is left adjoint to t*
*he
functor from Ch Rto Mod R which sends M to Mn.
7.9. Remark. Lemma 7.8 immediately implies that if A is a projective R-module
then Dn(A) is what might be called a "projective chain complex", in the sense
that if p : M ! N is an epimorphism of chain complexes (or even an epimorphism
in degrees > 1), then any map Dn(A) ! N lifts over p to a map Dn(A) ! M.
Similarly, any chain complex sum of the form iDni(Ai) is a "projective chain
complex" as long as each Aiis a projective R-module.
Lemma 7.10. Suppose that P is an acyclic object of Ch R such that each Pk is
a projective R-module. Then each module CykP (k > 0) is projective, and P is
isomorphic as a chain complex to the sum k>1Dk(Cy k-1P ).
Proof. For k > 1 let P (k)be the chain subcomplex of P which agrees with P above
degree k - 1, contains Bdk-1P in degree k - 1, and vanishes below degree k - 1.
Section 7 Homotopy theories 33
The acyclicity condition gives isomorphisms P (k)=P (k+1)~=Dk(Cy k-1P ). It is *
*clear
that Cy0(P ) = P0 is a projective R-module, and so by 7.9 there is an isomorphi*
*sm
P = P (1)~=P (2) D1(Cy 0P ). Since any direct factor of a projective R-module
is projective, it follows that P (2)is a chain complex which satisfies the cond*
*itions
of the lemma but vanishes in degree 0. Repeating the above argument in degree 1
gives an isomorphism P (2)~=P (3) D2(Cy 1P ). The proof is now completed_by_
continuing along these lines. |_|
7.11. Remark. Lemma 7.10 implies that if P is an acyclic object of Ch R with
the property that each Pk is a projective R-module, then P is a "projective cha*
*in
complex" in the sense of 7.9.
Now we are ready to handle MC4(ii). We need to show that a lift exists in every
diagram of the form 7.6 in which i is an acyclic cofibration and p is a fibrati*
*on. By
the definition of cofibration, the map i is a monomorphism of chain complexes a*
*nd
the cokernel P of i is a chain complex with the property that each Pk is a proj*
*ective
R-module. By the long exact homology sequence [6] associated to the short exact
sequence
0 ! A ! B ! P ! 0
of chain complexes, P is acyclic. It follows from 7.11 that P is a "projective *
*chain
complex" in the sense of 7.9, so that B is isomorphic to the direct sum A P , *
*and
the desired lift can be obtained by using the map g on the factor A and, as far*
* as_
the other factor is concerned, picking any lift P ! X of the given map P ! Y . *
* |_|
7.12.The small object argument
It is actually not hard to prove MC5 in the present case by making very element*
*ary
constructions. We have decided, however, to give a more complicated proof that
works in a variety of circumstances. This proof depends on an argument, called
the "small object argument", that is due to Quillen and is very well adapted to
producing factorizations with lifting properties. For the rest of this subsecti*
*on we
will assume that C is a category with all small colimits.
Given a functor B : Z + ! C (ii) and an object A of C , the natural maps
B(n) ! colimB induce maps Hom C(A; B(n)) ! Hom C(A; colimB) which are
compatible enough for various n to give a canonical map (2.17)
colimnHomC(A; B(n)) ! Hom C(A; colimnB(n)) : (7.13)
7.14. Definition. An object A of C is said to be sequentially small if for eve*
*ry
functor B : Z+ ! C the canonical map 7.13 is a bijection.
34 Dwyer and Spalinski Chapter 1
7.15. Remark. A set is sequentially small if and only if it is finite. An R-mo*
*dule is
sequentially small if it has a finite presentation, i.e., it is isomorphic to t*
*he cokernel
of a map between two finitely generated free R-modules. An object M of Ch R is
sequentially small if only a finite number of the modules Mk are non zero, and *
*each
Mk has a finite presentation.
Let F = {fi : Ai ! Bi}i2Ibe a set of maps in C. Suppose that p : X ! Y is
a map in C, and suppose that we desire to factor p as a composite X ! X0! Y
in such a way that the map X0! Y has the RLP (3.12) with respect to all of the
maps in F. Of course we could choose X0 = Y , but the secondary goal is to find
a factorization in which X0 is as close to X as reasonably possible. We proceed*
* as
follows. For each i 2 I consider the set S(i) which contains all pairs of maps *
*(g; h)
such that the following diagram commutes:
Ai -g! X
fi# p# : (7.16)
Bi -h! Y
We define the Gluing Construction G1(F; p) to be the object of C given by the
pushout diagram
` ` +i+(g;h)g
i2I`(g;h)2S(i)Ai - ! X
fi# i1# :
` ` +i+(g;h)h1
i2I (g;h)2S(i)Bi - ! G (F; p)
This is reminiscent of a "singular complex" construction; we are gluing a copy *
*of
Bi to X along Ai for every commutative diagram of the form 7.16. As indicated,
there is a natural map i1 : X ! G1(F; p). By the universal property of colimits*
*, the
commutative diagrams 7.16 induce a map p1 : G1(F; p) ! Y such that p1i1 = p.
Now repeat the process: for k > 1 define objects Gk(F; p) and maps pk : Gk(F; p*
*) !
Y inductively by setting Gk(F; p) = G1(F; pk-1) and pk = (pk-1)1. What results
is a commutative diagram
X -i1! G1(F; p) -i2! G2(F; p)-i3! . . .ik-!Gk(F; p) -! . . .
p# p1# p2# pk# :
Y -=! Y -=! Y -=! . . .=-! Y -=! . . .
Let G1 (F; p), the Infinite Gluing Construction, denote the colimit (2.17) of t*
*he
upper row in the above diagram; there are natural maps i1 : X ! G1 (F; p) and
p1 : G1 (F; p) ! Y such that p1 i1 = p.
Proposition 7.17.In the above situation, suppose that for each i 2 I the object
Ai of C is sequentially small. Then the map p1 : G1 (F; p) ! Y has the RLP
(3.12) with respect to each of the maps in the family F.
Section 7 Homotopy theories 35
Proof. Consider a commutative diagram which gives one of the lifting problems in
question:
Ai -g! G1 (F; p)
fi# p1# :
Bi -h! Y
Since Aiis sequentially small, there exists an integer k such that the map g is*
* the
composite of a map g0: Ai! Gk(F; p) with the natural map Gk(F; p) ! G1 (F; p).
Therefore the above commutative diagram can be enlarged to another one
0 ik+1
Ai -g! Gk(F; p) -! Gk+1(F; p) -! G1 (F; p)
fi# pk# pk+1# p1 #
Bi -h! Y -=! Y -=! Y
in which the composite all the way across the top row is g. However, the pair (*
*g0; h)
contributes itself as an index in the construction of Gk+1(F; p) from Gk(F; p);
what it indexes is in fact a gluing of Bi to Gk(F; p) along Ai. By construction,
then, there exists a map Bi ! Gk+1(F; p) which makes the appropriate diagram
commute. Composing with the map Gk+1(F; p) ! G1 (F; p) gives a lifting_in the
original square. |_|
7.18.Proof of MC5
For n > 1, let Dn (the "n-disk") denote the chain complex Dn(R) (7.7) and for
n > 0 let Sn (the "n-sphere") denote the chain complex K(R; n) (7.3). There is
an evident inclusion jn : Sn-1 ! Dn which is the identity on the copy of R in
degree (n - 1). Let D0 denote the chain complex K(R; 0), let S-1 denote the zero
chain complex, and let j0 : S-1 ! D0 be the unique map. Note that the chain
complexes Dn and Sn are sequentially small (7.15).
The following proposition is an elementary exercise in diagram chasing.
Proposition 7.19.A map f : X ! Y in Ch R is
(i)a fibration if and only if it has the RLP with respect to the maps 0 ! Dn
for all n > 1, and
(ii)an acyclic fibration if and only if it has the RLP with respect to the m*
*aps
jn : Sn-1 ! Dn for all n > 0.
To verify MC5(i), let f : X ! Y be the map to be factored, and let F be the
set of maps {jn}n>0. Consider the factorization of f provided by the small obje*
*ct
argument (7.12):
X i1-!G1 (F; f) p1-!Y :
36 Dwyer and Spalinski Chapter 1
It is immediate from 7.17 and 7.19 that p1 is an acyclic fibration, so what we *
*have
to check is that i1 is a cofibration. This is essentially obvious; in each deg*
*ree n,
Gk+1(F; f) is by construction the direct sum of Gk(F; f) with a (possibly large)
number of copies of R; passing to the colimit shows that G1 (F; f)n is similarl*
*y the
direct sum of Xn with copies of R.
The proof of MC5(ii) is very similar: let f : X ! Y be the map to be factored,
let F0be the set of maps {0 ! Dn}n>1 and consider the factorization of f provid*
*ed
by the small object argument:
X i1-!G1 (F0; f) p1-!Y :
Again it is immediate from 7.17 and 7.19 that p1 is a fibration. We leave it to*
* the_
reader to check that i1 in this case is an acyclic cofibration. *
* |_|
Proof of 7.3. We will only treat the case in which m = 0 and n > 0; the general
case is similar. Use MC5(i) to find a weak equivalence P ! K(A; 0), where P is
some cofibrant object of Ch R. There are bijections
Hom Ho(C)(K(A; 0); K(B; n)) ~=Hom Ho(C)(P; K(B; n)) ~=ss(P; K(B; n))
where the first comes from the fact (5.8) that the map P ! K(A; 0) becomes an
isomorphism in Ho(C ), and the second is from 5.11. Let X denote the good path
object for K(B; n) given by
( B B i = n
Xi= B i = n - 1
0 otherwise
with boundary map Xn ! Xn-1 sending (b0; b1) to b1- b0. The path object struc-
ture maps q : K(B; n) ! X and p0; p1 : X ! K(B; n) are determined in dimen-
sion n by the formulas q(b) = (b; b) and pi(b0; b1) = bi. According to 4.23, tw*
*o maps
f; g : P ! K(B; n) represent the same class in ss(P; K(B; n)) if and only if th*
*ey
are related by right homotopy with respect to X, that is, if and only if there *
*is a
map H : P ! X such that p0H = f and p1H = g.
In the language of homological algebra, P is a projective resolution of A. A m*
*ap
f : P ! K(B; n) amounts by inspection to a module map fn : Pn ! B such
that fn@ = 0. Two maps f; g : P ! K(B; n) are related by a right homotopy
with respect to X if and only if there exists a map h : Pn-1 ! B such that
h@ = fn - gn. A comparison with the standard definition of Ext*R(A; -) in terms*
* of
a projective resolution of A [6] now shows that ss(P; K(B; n)) is in natural_bi*
*jective
correspondence with ExtnR(A; B). |_|
8. Topological spaces
In this section we will construct the model category structure 3.5 on the categ*
*ory
Top of topological spaces.
Section 8 Homotopy theories 37
8.1. Definition. A map f : X ! Y of spaces is called a weak homotopy equivalen*
*ce
[25, p. 404] if for each basepoint x 2 X the map f* : ssn(X; x) ! ssn(Y; f(x)) *
*is a
bijection of pointed sets for n = 0 and an isomorphism of groups for n > 1.
8.2. Definition. A map of spaces p : X ! Y is said to be a Serre fibration [25,
p. 375] if, for each CW-complex A, the map p has the RLP (3.12) with respect to
the inclusion A x 0 ! A x [0; 1].
Proposition 8.3.Call a map of topological spaces
(i)a weak equivalence if it is a weak homotopy equivalence,
(ii)a fibration if it is a Serre fibration, and
(iii)a cofibration if it has the LLP with respect to acyclic fibrations (i.e.*
* with
respect to each map which is both a Serre fibration and a weak homotopy equiva-
lence).
Then with these choices Top is a model category.
After proving this we will make the following calculation.
Proposition 8.4.Suppose that A is a CW-complex and that X is an arbitrary
space. Then the set Hom Ho(Top)(A; X) is in natural bijective correspondence wi*
*th
the set of (conventional) homotopy classes of maps from A to X.
Remark. In the model category structure of 8.3, every space is weakly equivalent
to a CW-complex.
We will need two facts from elementary homotopy theory (cf. 7.19). Let Dn
(n > 1) denote the topological n-disk and Sn (n > 0) the topological n-sphere.
Let D0 be a single point and S-1 the empty space. There are standard (boundary)
inclusions jn : Sn-1 ! Dn (n > 0).
Lemma 8.5. [14, Theorem 3.1, p. 63] Let p : X ! Y be a map of spaces. Then
p is a Serre fibration if and only if p has the RLP with respect to the inclusi*
*ons
Dn ! Dn x [0; 1], n > 0.
Lemma 8.6. Let p : X ! Y be a map of spaces. Then the following conditions are
equivalent:
(i)p is both a Serre fibration and a weak homotopy equivalence,
(ii)p has the RLP with respect to every inclusion A ! B such that (B; A) is a
relative CW-pair, and
(iii)p has the RLP with respect to the maps jn : Sn-1 ! Dn for n > 0.
This is not hard to prove with the arguments from [25, p. 376]. We will also n*
*eed
a fact from elementary point-set topology.
Lemma 8.7. Suppose that
X0 ! X1 ! X2 ! . .!.Xn ! . . .
38 Dwyer and Spalinski Chapter 1
is a sequential direct system of spaces such that for each n > 0 the space Xn i*
*s a
subspace of Xn+1 and the pair (Xn+1; Xn) is a relative CW-complex [25, p. 401].
Let A be a finite CW-complex. Then the natural map (7.13)
colimnHomTop(A; Xn) ! Hom Top(A; colimnXn)
is a bijection (of sets).
8.8. Remark. In the situation of 8.7, we will refer to the natural map X0 !
colimnXn as a generalized relative CW inclusion and say that colimnXn is obtain*
*ed
from X0 by attaching cells. It follows easily from 8.6 that any such generalized
relative CW inclusion is a cofibration with respect to the model category struc*
*ture
described in 8.3. There is a partial converse to this.
Proposition 8.9.Every cofibration in Top is a retract of a generalized relative
CW inclusion.
8.10. Proof of MC1-MC3. It is easy to see directly that the classes of weak
equivalences, fibrations and cofibrations contain all identity maps and are clo*
*sed
under composition. Axiom MC1 follows from the fact that Top has all small limits
and colimits (2.14, 2.21). Axiom MC2 is obvious. For the case of weak equivalen*
*ces,
MC3 follows from functoriality and 2.6. The other two cases of MC3 are similar,
so we will deal only with cofibrations. Suppose that f is a retract of a cofibr*
*ation
f0. We need to show that a lift exists in every diagram
A -a! X
f # p# (8.11)
B -b! Y
in which p is an acyclic fibration. Consider the diagram
A -i! A0 -r! A -a! X
f # f0# p#
B -j! B0 -s! B -b! Y
in which maps i, j, r and s are retraction constituents. Since f0 is a cofibrat*
*ion,
there is a lifting h : B0 ! X in this diagram. It is now easy to see that hj_is*
* the
desired lifting in the diagram 8.11. |_|
The proofs of MC4(ii) and MC5(ii) depend upon a lemma.
Lemma 8.12. Every map p : X ! Y in Top can be factored as a composite p1 i1 ,
where i1 : X ! X0is a weak homotopy equivalence which has the LLP with respect
to all Serre fibrations, and p1 : X0! Y is a Serre fibration.
Section 8 Homotopy theories 39
Proof. Let F be the set of maps {Dn x 0 ! Dn x [0; 1]}n>0. Consider the Gluing
Construction G1(F; p) (see 7.12). It is clear that i1 : X ! G1(F; p) is a relat*
*ive
CW inclusion and a deformation retraction; in fact, G1(F; p) is obtained from X*
* by
taking (many) solid cylinders and attaching each one to X along one end. It fol*
*lows
from the definition of Serre fibration that the map i1 has the LLP with respect*
* to
all Serre fibrations. Similarly, for each k > 1 the map ik+1 : Gk(F; p) ! Gk+1(*
*F; p)
is a homotopy equivalence which has the LLP with respect to all Serre fibration*
*s.
Consider the factorization
X i1-!G1 (F; p) p1-!Y
provided by the Infinite Gluing Construction. It is immediate that i1 has the L*
*LP
with respect to all Serre fibrations: given a lifting problem, one can inductiv*
*ely find
compatible solutions on the spaces Gk(F; p) and then use the universal property*
* of
colimit to obtain a solution on G1 (F; p) = colimkGk(F; p). The proof of Propos*
*i-
tion 7.17 shows that p1 has the RLP with respect to the maps in F and so (8.5)
is a Serre fibration; it is only necessary to observe that although the spaces *
*Dn are
not in general sequentially small, they are (8.7) small with respect to the par*
*ticular
sequential colimit that comes up here. Finally, by 8.7 any map of a sphere into
G1 (F; p) or any homotopy involving such maps must actually lie in Gk(F; p) for
some k; it follows that i1 is a weak homotopy equivalence because (by the remar*
*ks_
above) each of the maps X ! Gk(F; p) is a weak homotopy equivalence. |_|
Proof of MC5. Axiom MC5(ii) is an immediate consequence of 8.12. The proof of
MC5(i) is similar to the proof of 8.12. Let p be the map to be factored, let F *
*be
the set
F = {jn : Sn-1 ! Dn}n>0
and consider the factorization p = p1 i1 of p provided by the Infinite Gluing
Construction G1 (F; p). By 8.6 each map ik+1 : Gk(F; p) ! Gk+1(F; p) has the
LLP with respect to Serre fibrations which are weak homotopy equivalences; by
induction and a colimit argument the map i1 has the same LLP and so by definiti*
*on
is a cofibration. By 8.7 and the proof of 7.17, the map p1 has the RLP with res*
*pect
to all maps in the set F, and so (8.6) is a Serre fibration and a weak equivale*
*nce.
Proof of MC4. Axiom MC4(i) is immediate from the definition of cofibration. For
MC4(ii) suppose that f : A ! B is an acyclic cofibration; we have to show that f
has the LLP with respect to fibrations. Use 8.12 to factor f as a composite pi,*
* where
p is a fibration and i is weak homotopy equivalence which has the LLP with resp*
*ect
to all fibrations. Since f = pi is by assumption a weak homotopy equivalence, i*
*t is
clear that p is also a weak homotopy equivalence. A lift g : B ! A0exists in the
40 Dwyer and Spalinski Chapter 1
following diagram
A -i! A0
f # p# ~ (8.13)
B -id! B
because f is a cofibration and p is an acyclic fibration. (Recall that by defin*
*ition
every cofibration has the LLP with respect to acyclic fibrations). This lift g *
*expresses
the map f as a retract (2.6) of the map i. The argument in 8.10 above can now be
used to show that the class of maps which have the LLP with respect to all Serre
fibrations is closed under retracts; it follows that f has the LLP with respect*
*_to all
Serre fibrations because i does. |_|
Proof of 8.4. Since A is cofibrant (8.6) and X is fibrant, the set Hom Ho(Top)(*
*A; X)
is naturally isomorphic to ss(A; X) (see 5.11). It is also easy to see from 8.6*
* that the
product A x [0; 1] is a good cylinder object for A. By 4.23, two maps f; g : A *
*! X
represent the same element of ss(A; X) if and only if they are left homotopic v*
*ia
the cylinder object A x [0; 1], in other words, if and only if they are_homotop*
*ic in
the conventional sense. |_|
Proof of 8.9. Let f : A ! B be a cofibration in Top. The argument in the proof *
*of
MC5(i) above shows that f can be factored as a composite pi, where i : A ! A0
is a generalized relative CW inclusion and p : A0! B is an acyclic fibration. S*
*ince
f is a cofibration, a lift g : B ! A0exists in the resulting diagram 8.13, and_*
*this
lift g expresses f as a retract of i. |_|
9. Derived functors
Let C be a model category and F : C ! D a functor. In this section we define the
left and right derived functors of F ; if they exist, these are functors
LF; RF : Ho(C ) ! D
which, up to natural transformation on one side or the other, are the best poss*
*ible
approximations to an "extension of F to Ho(C )", that is, to a factorization of*
* F
through fl : C ! Ho(C ). We give a criterion for the derived functors to exist,*
* and
study a condition under which a pair of adjoint functors (2.8) between two model
categories induces, via a derived functor construction, adjoint functors betwee*
*n the
associated homotopy categories. The homotopy pushout and homotopy pullback
functors of x10 will be constructed by taking derived functors of genuine pusho*
*ut
or pullback functors.
9.1. Definition. Suppose that C is a model category and that F : C ! D is a
functor. Consider pairs (G; s) consisting of a functor G : Ho(C )! D and natural
Section 9 Homotopy theories 41
transformation s : Gfl ! F . A left derived functor for F is a pair (LF; t) of *
*this type
which is universal from the left, in the sense that if (G; s) is any such pair,*
* then
there exists a unique natural transformation s0: G ! LF such that the composite
natural transformation
0Ofl t
Gfl s-!(LF )fl -! F (9.2)
is the natural transformation s.
Remark. A right derived functor for F is a pair (RF; t), where RF : Ho(C )! D
is a functor and t : F ! (RF )fl is a natural transformation with the analogous
property of being "universal from the right".
Remark. The universal property satisfied by a left derived functor implies as u*
*sual
that any two left derived functors of F are canonically naturally equivalent. S*
*ome-
times we will refer to LF as the left derived functor of F and leave the natural
transformation t understood. If F takes weak equivalences in C into isomorphisms
in D , then there is a functor F 0: Ho(C )! D with F 0= F fl (6.2), and it is
not hard to see that in this case F 0itself (with the identity natural transfor*
*ma-
tion t : F 0fl ! F ) is a left derived functor of F . The next proposition show*
*s that
sometimes LF exists even though a functor F 0as above does not.
Proposition 9.3.Let C be a model category and F : C ! D a functor with the
property that F (f) is an isomorphism whenever f is a weak equivalence between
cofibrant objects in C . Then the left derived functor (LF; t) of F exists, and*
* for
each cofibrant object X of C the map
tX : LF (X) ! F (X)
is an isomorphism.
The proof depends on a lemma, which for future purposes we state in slightly
greater generality than we actually need here.
Lemma 9.4. Let C be a model category and F : Cc ! D (x5) a functor such that
F (f) is an isomorphism whenever f is an acyclic cofibration between objects of*
* Cc.
Suppose that f; g : A ! B are maps in C csuch that f is right homotopic to g in
C. Then F (f) = F (g).
Proof. By 4.15 there exists a right homotopy H : A ! BI from f to g such that BI
is a very good path object for B. Since the path object structure map w : B ! BI
is then an acyclic cofibration and B by assumption is cofibrant, it follows tha*
*t BI
is cofibrant and hence that F (w) is defined and is an isomorphism. The rest of
the proof is identical to the dual of the proof of 5.10. First observe that the*
*re are
equalities F (p0)F (w) = F (p1)F (w) = F (idB) and then use the fact that F (w)*
* is an
isomorphism to cancel F (w) and obtain F (p0) = F (p1). The equality F (f) =_F_*
*(g)
then follows from applying F to the equalities f = p0H and g = p1H. |_|
42 Dwyer and Spalinski Chapter 1
Proof of 9.3. By Lemma 9.4, F identifies right homotopic maps between cofibrant
objects of C and so induces a functor F 0: ssC c! D . By assumption, if g is a
morphism of ssC cwhich is represented by a weak equivalence in C then F 0(g)
is an isomorphism. Recall from 5.2 that there is a functor Q : C ! ssC cwith
the property (5.1) that if f is a weak equivalence in C then g = Q(f) is a right
homotopy class which is represented by a weak equivalence in C . It follows that
the composite functor F 0Q carries weak equivalences in C into isomorphisms in
D. By the universal property (6.2) of Ho(C ), the composite F 0Q induces a func*
*tor
Ho(C )! D, which we denote LF . There is a natural transformation t : (LF )fl !*
* F
which assigns to each X in C the map F (pX ) : LF (X) = F (QX) ! F (X). If X
is cofibrant then QX = X and the map tX is the identity; in particular, tX is an
isomorphism.
We now have to show that the pair (LF; t) is universal from the left in the se*
*nse
of 9.1. Let G : Ho(C )! D be a functor and s : Gfl ! F a natural transformation.
Consider a hypothetical natural transformation s0: G ! LF , and construct (for
each object X of C ) the following commutative diagram which in the horizontal
direction involves the composite of s0O fl and t;
s0QX tQX=id
G(QX) -! LF (QX) -! F (QX)
G(fl(pX))# # LF(fl(pX))=id # F(pX):
0 tX=F(pX)
G(X) sX-! LF (X) -! F (X)
If s0 is to satisfy the condition of 9.1, then the composite across the top row*
* of
this diagram must be equal to sQX , which gives the equality s0X= sQX G(flpX )-1
and proves that there is at most one natural transformation s0which satisfies t*
*he
required condition. However, it is obvious that setting s0X= sQX G(flpX )-1 does
give a natural transformation Gfl ! (LF )fl, and therefore (5.9) it also_gives a
natural transformation G ! LF . |_|
9.5. Definition. Let F : C ! D be a functor between model categories. A total
left derived functor LF for F is a functor
LF : Ho(C )! Ho(D )
which is a left derived functor for the composite flD . F : C ! Ho(D ). Similar*
*ly, a
total right derived functor RF for F is a functor RF : Ho(C )! Ho(D )which is a
right derived functor for the composite flD . F .
Remark. As usual, total left or right derived functors are unique up to canonic*
*al
natural equivalence.
9.6. Example. Let R be an associative ring with unit, and ChR the chain complex
model category constructed in x7. Suppose that M is a right R-module, so that M-
Section 9 Homotopy theories 43
gives a functor F : Ch R ! Ch Z. Proposition 9.3 can be used to show that the
total derived functor LF exists (see 9.11). Let N be a left R-module and K(N; 0)
(cf. 7.3) the corresponding chain complex. The final statement in 9.3 implies t*
*hat
LF (K(N; 0)) is isomorphic in Ho(Ch Z) to F (P ), where P is any cofibrant chain
complex with a weak equivalence P ~!K(N; 0). Such a cofibrant chain complex P
is exactly a projective resolution of N in the sense of homological algebra, an*
*d so
we obtain natural isomorphisms
HiLF (K(N; 0)) ~=TorRi(M; N) i > 0
where TorRi(M; -) is the usual i'th left derived functor of M R -. This gives o*
*ne
connection between the notion of total derived functor in 9.5 and the standard
notion of derived functor from homological algebra.
Theorem 9.7. Let C and D be model categories, and
F : C () D : G
a pair of adjoint functors (2.8). Suppose that
(i)F preserves cofibrations and G preserves fibrations.
Then the total derived functors
LF : Ho(C )() Ho(D ): RG
exist and form an adjoint pair. If in addition we have
(ii)for each cofibrant object A of C and fibrant object X of D , a map f : A !
G(X) is a weak equivalence in C if and only if its adjoint f[ : F (A) ! X is a *
*weak
equivalence in D ,
then LF and R G are inverse equivalences of categories.
Remark. In this paper we will not use the last statement of 9.7, but this crite*
*rion
for showing that two model categories have equivalent homotopy categories is us*
*ed
heavily by Quillen in [23]. There are various other structures associated to a *
*model
category besides its homotopy category; these include fibration and cofibration
sequences [22], Toda brackets [22], various homotopy limits and colimits (x10),*
* and
various function complexes [9]. All such structures that we know of are preserv*
*ed
by adjoint functors that satisfy the two conditions above.
9.8. Remark. Condition 9.7(i) is equivalent to either of the following two con*
*di-
tions:
(i0)G preserves fibrations and acyclic fibrations.
(i00)F preserves cofibrations and acyclic cofibrations.
Assume, for instance, that F preserves acyclic cofibrations. Let f : A ! B be
an acyclic cofibration in C and g : X ! Y a fibration in D . Suppose given the
44 Dwyer and Spalinski Chapter 1
commutative diagram on the left together with its "adjoint" diagram (2.8) on the
right:
[
A -u! G(X) F (A) u-! X
f # G(g)# F(f)# g# :
B -v! G(Y ) F (B) v[-! Y
Since F preserves acyclic cofibrations, a lift w : F (B) ! X exists in the righ*
*t-hand
diagram. Its adjoint w] : B ! G(X) is then a lift in the left-hand diagram. It *
*follows
that G(g) has the RLP with respect to all acyclic cofibrations in C, and theref*
*ore by
3.13(iii) that G(g) is a fibration. This gives 9.7(i). Running the argument in *
*reverse
and using 3.13(ii) shows the converse: if G preserves fibrations then F preserv*
*es
acyclic cofibrations.
The proof of 9.7 depends on a lemma that is also useful in verifying the hypot*
*heses
of 9.3.
Lemma 9.9. (K. Brown) Let F : C ! D be a functor between model categories.
If F carries acyclic cofibrations between cofibrant objects to weak equivalence*
*s, then
F preserves all weak equivalences between cofibrant objects.
Proof. Let f : A ! B be a weak equivalence in C between cofibrant`objects.
By MC5(i) we can factor the`coproduct (2.15) map f + idB: A B ! B as a
composite pq, where q : A B ! C is a cofibration and p : C ! B is an acyclic
fibration. It follows from the fact that A and B are cofibrant (cf. 4.4) that t*
*he
composite maps q . in0: A ! C and q . in1: B ! C are cofibrations. Since pq . i*
*niis
a weak equivalence for i = 0, 1 and p is a weak equivalence, it is clear from M*
*C2
that q . iniis a weak equivalence, i = 0, 1. By assumption, then F (q . in0), F*
* (q . in1)
and F (pq . in1) = F (idB) are weak equivalences in D. It follows that the maps*
*_F (p)
and hence F (pq . in0) = F (f) are also weak equivalences. |*
*_|
Proof of 9.7. In view of 9.8, 9.9 and the dual (3.9) of 9.9, Proposition 9.3 an*
*d its
dual guarantee that the total derived functors LF and RG exist. Since F is a l*
*eft
adjoint it preserves colimits (2.26) and therefore (2.25) initial objects. Sinc*
*e G is
a right adjoint it preserves limits and therefore terminal objects. It then fol*
*lows as
in 9.8 that F carries cofibrant objects in C into cofibrant objects in D, and t*
*hat G
carries fibrant objects in D into fibrant objects in C.
Suppose that A is a cofibrant object in C and that X is a fibrant object in D .
We will show that the adjunction isomorphism Hom C(A; G(X)) ~=Hom D(F (A); X)
respects the homotopy equivalence relation (4.21) and gives a bijection
ss(A; G(X)) ~=ss(F (A); X) : (9.10)
If f; g : A ! G(X) represent the same class in ss(A; G(X)), then f is left homo*
*topic
to g via a left homotopy H : A ^ I ! G(X) in which the cylinder object A ^ I is
Section 9 Homotopy theories 45
good (4.6) and hence cofibrant (4.4). It then follows from 9.8(i00) that F (A ^*
* I) is
a cylinder object for F (A) and hence that H[ : F (A ^ I) ! X is a left homotopy
between f[ and g[. Thus f[ ~ g[. A dual argument with right homotopies shows
that if f[ ~ g[ then f ~ g and establishes the isomorphism 9.10.
Let Q be the construction of 5.2 for C and S the construction of 5.4 for D. (We
have temporarily changed the letter denoting this functor from "R" to "S" in or*
*der
to avoid confusion with the notation for right derived functors). In view of the
construction of LF and RG given by the proof of 9.3 and its dual, the isomorphi*
*sm
9.10 gives for every object A of C and object X of D a bijection
*
Hom Ho(C)(A; RG (X)) (flpA)-!HomHo(C)(QA; G(SX))
-1)
~=HomHo(D)(F (QA); SX) ((fliX)-!H*omHo(D)(L F (A); X) :
It is clear that this bijection gives a natural equivalence of functors from Co*
*px D
to Sets, and the argument of 5.9 shows that it also gives a natural equivalence*
* of
functors Ho(C )opx Ho(D )! Sets. This provides the adjunction between LF and
RG .
Suppose that condition (ii) is satisfied. Let A be an cofibrant object of C. T*
*he
map i]F(A): A ! G(SF (A)) is then a weak equivalence in C because its adjoint
iF(A): F (A) ! SF (A) is a weak equivalence in D. Let
fflA = id]LF(A): A ! RG (L F (A))
denote the map in Ho(C )which is adjoint to the identity map of LF (A) in Ho(D *
*). It
follows from the above constructions that fflA is an isomorphism. Since every o*
*bject
of Ho(C )is isomorphic to A for a cofibrant object A of C, we conclude that ffl*
*A is an
isomorphism for any object A of Ho(C )and thus that the composite (R G)(L F ) is
naturally equivalent to the identity functor of Ho(C ). A dual argument shows t*
*hat
the composite (L F )(R G) is naturally equivalent to the identity functor of Ho*
*(D_).
This proves that LF and RG are inverse equivalences of categories. |*
*_|
9.11. Example. Let F : Ch R ! Ch Z be the functor of 9.6. In order to use 9.3
to show that the total derived functor LF exists, it is necessary to show that F
carries weak equivalences between cofibrant objects to weak equivalences. By 9.9
it is enough to check this for acyclic cofibrations between cofibrant objects. *
*Let
i : A ! B be a acyclic cofibration between cofibrant objects in Ch R. The quoti*
*ent
B=A is then an acyclic chain complex which satisfies the hypotheses of 7.10, so*
* that
by 7.11 there is an isomorphism B ~=A (B=A) and (7.10) a further isomorphism
between B=A and a direct sum of chain complexes of the form Dk(P ). Since F
respects direct sums we conclude that F (B) is isomorphic to the direct sum of
F (A) with a number of chain complexes of the form F (Dk(P )). By inspection
F (Dk(P )) is acyclic, and so F (i) is a weak equivalence.
46 Dwyer and Spalinski Chapter 1
10. Homotopy pushouts and homotopy pullbacks
The constructions in this section are motivated by the fact that pushouts and
pullbacks are not usually well-behaved with respect to homotopy equivalences. F*
*or
example, in the category Top of topological spaces, let Dn (n > 1) denote the
n-disk, jn : Sn-1 ! Dn the inclusion of the boundary (n - 1)-sphere, and * the
one-point space. There is a commutative diagram
Dn -jn Sn-1 -jn! Dn
# id# # (10.1)
* - Sn-1 -! *
in which all three vertical arrows are homotopy equivalences. The pushout (2.16)
or colimit of the top row is homeomorphic to Sn, the pushout of the bottom row
is the space "*", and the map Sn ! * induced by the diagram is not a homotopy
equivalence.
Faced with diagram 10.1, a seasoned topologist would probably say that the
pushout of the top row has the "correct" homotopy type and invoke the philosophy
that to give a pushout homotopy significance the maps involved should be replac*
*ed
if necessary by cofibrations. In this section we work in an arbitrary model cat*
*egory
C and find a conceptual basis for this philosophy. The strategy is this. Let D *
*be
the category {a b ! c} of 2.12 and CD the category of functors D ! C (2.5).
An object of CD is pushout data
X(a) X(b) ! X(c)
in C and a morphism f : X ! Y is a commutative diagram
X(a) - X(b) -! X(c)
fa# fb# fc# : (10.2)
Y (a) - Y (b) -! Y (c)
The pushout or colimit construction gives a functor colim: CD ! C. We will con-
struct a model category structure on CD with respect to which a weak equivalence
is a map f whose three components (fa; fb; fc) are weak equivalences in C. As 1*
*0.1
illustrates, in this setting the functor colim(-) is not usually homotopy invar*
*iant
(i.e., does not usually carry weak equivalences in C Dto weak equivalences is C*
* )
and so colim(-) does not directly induce a functor Ho(C D) ! Ho(C ). However, it
turns out that colim(-) does have a total left derived functor (9.5)
Lcolim: Ho(C D) ! Ho(C )
which in a certain sense (9.1) is the best possible homotopy invariant approxim*
*ation
to colim(-). We will call Lcolimthe homotopy pushout functor; it is left adjoin*
*t to
Section 10 Homotopy theories 47
the functor
Ho() : Ho(C )! Ho(C D)
induced by the "constant diagram" (2.11) construction : C ! C D. By 9.3,
computing Lcolim(X) for a diagram X involves computing colim(X0), where X0 is
a cofibrant object of CD which is weakly equivalent to X. It turns out that fin*
*ding
such a cofibrant X0involves replacing X(b) by a cofibrant object and replacing *
*the
maps X(b) ! X(a) and X(b) ! X(c) by cofibrations, and so in the end what we
do is more or less recover, in this abstract setting, the standard philosophy. *
*In fact,
it becomes clear (see 9.6) that this philosophy is no different from the philos*
*ophy
in homological algebra that a cautious practitioner should usually replace a mo*
*dule
by a projective resolution before, for instance, tensoring it with something.
Working dually gives a construction of the homotopy pullback functor. At the
end of the section we make a few remarks about more general homotopy colimits
or limits in C.
10.3. Remark. In the above situation, there is a natural functor Ho (C D) !
Ho(C )D, but this functor is usually not an equivalence of categories (and much
of the subtlety of homotopy theory lies in this fact). Consequently, the homoto*
*py
pushout functor Lcolimdoes not provide "pushouts in the homotopy category",
that is, it is not a left adjoint to constant diagram functor
Ho(C): Ho(C )! Ho(C )D:
10.4.Homotopy pushouts
Let C be a model category, D be the category {a b ! c} above, and C Dthe
category of functors D ! C. Given a map f : X ! Y of CD as in 10.2, let @b(f)
denote X(b) and define objects @a(f) and @c(f) of C by the pushout diagrams
X(b) -! X(a) X(b) -! X(c)
fb# # fb# # : (10.5)
Y (b) -! @a(f) Y (b)-! @c(f)
The commutative diagram 10.2 induces maps ia(f) : @a(f) ! Y (a), ib(f) : @b(f) !
Y (b), and ic(f) : @c(f) ! Y (c).
Proposition 10.6.Call a morphism f : X ! Y in CD
(i)a weak equivalence, if the morphisms fa, fb and fc are weak equivalences *
*in
C,
(ii)a fibration if the morphisms fa, fb and fc are fibrations in C, and
48 Dwyer and Spalinski Chapter 1
(iii)a cofibration if the maps ia(f), ib(f) and ic(f) are cofibrations in C.
Then these choices provide CD with the structure of a model category.
Proof. Axiom MC1 follows from 2.27. Axiom MC2 and the parts of MC3 dealing
with weak equivalences and fibrations are direct consequences of the correspond*
*ing
axioms in C. It is not hard to check that if f is a retract of g, then the maps*
* ia(f),
ib(f) and ic(f) are respectively retracts of ia(g), ib(g) and ic(g), so that th*
*e part
of MC3 dealing with cofibrations is also a consequence of the corresponding axi*
*om
for C. For MC4(i), consider a commutative diagram
(A(a) A(b) ! A(c))- ! (X(a) X(b) ! X(c))
f # p#
(B(a) B(b) ! B(c))- ! (Y (a) Y (b) ! Y (c))
in which f is a cofibration and p is an acyclic fibration. This diagram consist*
*s of
three slices:
A(a) -! X(a) A(b) -! X(b) A(c) - ! X(c)
fa # pa# fb# pb# fc# pc# :
B(a) -! Y (a); B(b) -! Y (b); B(c) - ! Y (c)
Since f is a cofibration and p is an acyclic fibration, we can obtain the desir*
*ed
lifting in the middle slice by applying MC4(i) in C ; this lifting induces maps
u : @a(f) ! X(a) and v : @c(f) ! X(c). Liftings in the other two slices can now
be constructed by applying MC4(i) in C to the squares
@a(f) -u! X(a) @c(f) -v! X(c)
ia(f)# pa# ic(f)# pc#
B(a) -! Y (a) B(c) -! Y (c)
in which each left-hand arrow is a cofibration. The proof of the second part of
MC4(ii) is analogous; in this case the fact that the maps ic(f) and ia(f) are
acyclic cofibrations follows easily from the fact that the class of acyclic cof*
*ibrations
in C is closed under cobase change (3.14).
To prove MC5(ii), suppose that we have a morphism f : A ! B. Use MC5(ii) in
C to factor the map fb : A(b) ! B(b) as A(b) ~,!Y !!B(b). Let X be the pushout
of the diagram A(a) A(b) ! Y and Z the pushout of Y A(b) ! A(c). There
is a commutative diagram
A(a) - A(b) -! A(c)
~ # ~ # ~#
X - Y -! Z
# # #
B(a) - B(b) -! B(c)
Section 10 Homotopy theories 49
in which the lower outside vertical arrows are constructed using the universal *
*prop-
erty of pushouts. Now use MC5(ii) in C again to factor the lower outside vertic*
*al
arrows as X ~,!X0!!B(a) and Z ~,!Z0!!B(c). It is not hard to see that the object
X0 Y ! Z0 of CD provides the intermediate object for the desired factorization_
of f. The proof of MC5(i) is similar. |_|
Proposition 10.7.The adjoint functors
colim: CD () C :
satisfy condition (i) of Theorem 9.7. Hence the total derived functors Lcolimand
R exist and form an adjoint pair
Lcolim: Ho(C D) () Ho(C ) : R :
Proof. This is clear from 9.8, since the functor preserves both fibrations_and
acyclic fibrations. |_|
This completes the construction of the homotopy pushout functor L colim:
Ho(C D) ! Ho(C ). According to 9.3, Lcolim(X) is isomorphic to colim(X) if X is
a cofibrant object of CD ; in general Lcolim(X) is isomorphic to colim(X0) for *
*any
cofibrant object X0 of CD weakly equivalent to X.
10.8.Homotopy pullbacks
The following results on homotopy pullbacks are dual (3.9) to the above ones on
homotopy pushouts, so we state them without proof.
Let C be a model category, let D be the category {a ! b c}, and C D the
category of functors D ! C. Given a map f : X ! Y of CD
X(a) -! X(b) - X(c)
fa# fb# fc# ; (10.9)
Y (a) -! Y (b) - Y (c)
let ffib(f) denote X(b) and define objects ffia(f) and ffic(f) of C by the pul*
*lback
diagrams
ffia(f)-! X(b) ffic(f)-! X(b)
# fb# # fb# : (10.10)
Y (a) -! Y (b) Y (c)- ! Y (b)
The commutative diagram 10.9 induces maps pa(f) : X(a) ! ffia(f), pb(f) : X(b) !
ffib(f), and pc(f) : X(c) ! ffic(f).
50 Dwyer and Spalinski Chapter 1
Proposition 10.11.Call a morphism f : X ! Y in CD
(i)a weak equivalence, if the morphisms fa, fb and fc are weak equivalences *
*in
C,
(ii)a cofibration if the morphisms fa, fb and fc are cofibrations in C, and
(iii)a fibration if the maps pa(f), pb(f) and pc(f) are fibrations in C.
Then these choices provide CD with the structure of a model category.
Proposition 10.12.The adjoint functors
: CD () C : lim
satisfy condition (i) of Theorem 9.7. Hence the total derived functors Rlim and*
* L
exist and form an adjoint pair
L : Ho(C D) () Ho(C ) : Rlim :
This completes the construction of the homotopy pullback functor R lim :
Ho(C D) ! Ho(C ). According to 9.3, R lim(X) is isomorphic to lim(X) if X is
a fibrant object of CD ; in general Rlim(X) is isomorphic to lim(X0) for any fi*
*brant
object X0 of CD weakly equivalent to X.
10.13.Other homotopy limits and colimits
Say that a category D is very small if it satisfies the following conditions
(i)D has a finite number of objects,
(ii)D has a finite number of morphisms, and
(iii)there exists an integer N such that if
A0 f1!A1 ! . .!.An
is a string of composable morphisms of D with n > N, then some fiis an identity
morphism.
Propositions 10.6 and 10.11 can be generalized to give two distinct model categ*
*ory
structures on the category CD whenever D is very small. These structures share *
*the
same weak equivalences (and therefore have isomorphic homotopy categories) but
they differ in their fibrations and cofibrations. One of these structures is ad*
*apted
to constructing Lcolimand the other to constructing R lim. We leave this as an
interesting exercise for the reader. The generalization of 10.6(iii) is as foll*
*ows. For
each object d of D , let @d denote the full subcategory of D #d (3.11) generate*
*d by
all the objects except the identity map of d. There is a functor jd : @d ! D wh*
*ich
sends an object d0! d of @d to the object d0of D. If X is an object of CD , let*
* X|@d
denote the composite of X with jd and let @d(X) denote the object of C given by
colim(X|@d). There is a natural map @d(X) ! X(d). If f : X ! Y is a map of CD ,
Section 11 Homotopy theories 51
define @d(f) by the pushout diagram
@d(X) -! X(d)
# #
@d(Y ) -! @d(f)
and observe that there is a natural map id(f) : @d(f) ! Y (d). Then the general-
ization of 10.6(iii) is the condition that the map id(f) be a cofibration for e*
*very
object d of D.
Suppose that D is an arbitrary small category. It seems unlikely that C D has
a natural model category structure for a general model category C. However, CD
does have a model category structure if C is the category of simplicial sets (1*
*1.1)
[3, XI, x8]. The arguments of x8 can be used to construct a parallel model cate*
*gory
structure on TopD . In these special cases the homotopy limit and colimit funct*
*ors
have been studied by Bousfield and Kan [3]; they deal explicitly only with the *
*case
of simplicial sets, but the topological case is very similar.
11. Applications of model categories
In this section, which is less self-contained than the rest of the paper, we wi*
*ll give
a sampling of the ways in which model categories have been used in topology and
algebra. For an exposition of the theory of model categories from an alternate *
*point
of view see [16]; for a slightly different approach to axiomatic homotopy theor*
*y see,
for example, [1].
11.1. Simplicial Sets. Let be the category whose objects are the ordered sets
[n]= {0; 1; : :;:n} (n > 0) and whose morphisms are the order-preserving maps
between these sets. (Here "order-preserving" means that f(i) 6 f(j) whenever
i 6 j). The category sSetof simplicial sets is defined to be the category of fu*
*nctors
op ! Set; the morphisms, as usual (2.5), are natural transformations. Recall f*
*rom
2.4 that a functor op ! Setis the same as a contravariant functor ! Set. For
an equivalent but much more explicit description of what a simplicial set is se*
*e [18,
p. 1]. If X is a simplicial set it is customary to denote the set X([n]) by Xn *
*and
call it the set of n-simplices of X.
A simplicial set is a combinatorial object which is similar to an abstract sim*
*plicial
complex with singularities. In an abstract simplicial complex [21, p. 15] [25, *
*p. 108],
for instance, an n-simplex has (n + 1) distinct vertices and is determined by t*
*hese
vertices; in a simplicial set X, an n-simplex x 2 Xn does have n+1 "vertices" i*
*n X0
(obtained from x and the (n + 1) maps [n]! [0]in op ) but these vertices are not
necessarily distinct and they in no way determine x. Let n denote the standard
topological n-simplex, considered as the space of formal convex linear combinat*
*ions
of the points in the set [n]. If Y is a topological space, it is possible to co*
*nstruct an
associated simplicial set Sing(Y ) by letting the set of n-simplices Sing(Y )n *
*be the
52 Dwyer and Spalinski Chapter 1
set of all continuous maps n ! Y ; this is a set-theoretic precursor of the sin*
*gular
chain complex of Y . The functor Sing: Top ! sSethas a left adjoint, which sends
a simplicial set X to a space |X| called the geometric realization of X [18, Ch*
*. III];
this construction is a generalization of the geometric realization construction*
* for
simplicial complexes. Call a map f : X ! Y of simplicial sets
(i)a weak equivalence if |f| is a weak homotopy equivalence (8.1) of topolog*
*ical
spaces,
(ii)a cofibration if each map fn : Xn ! Yn (n > 0) is a monomorphism, and
(iii)a fibration if f has the RLP with respect to acyclic cofibrations (equiv*
*alently,
f is a Kan fibration [18, x7]).
Quillen [22] proves that with these definitions the category sSetis a model cat*
*egory.
He also shows that the adjoint functors
|?| : sSet() Top : Sing
satisfy both conditions of Theorem 9.7 and so induce an equivalence of categori*
*es
Ho(sSet) ! Ho(Top ) (this is of course with respect to the model category struc*
*ture
on Top from x8). This shows that the category of simplicial sets is a good cate*
*gory
of algebraic or combinatorial "models" for the study of ordinary homotopy theor*
*y.
11.2. Simplicial Objects. There is an obvious way to extend the notion of simp*
*li-
cial set: if C is a category, the category sC of simplicial objects in C is def*
*ined to
be the category of functors op ! C (with natural transformations as the mor-
phisms). The usual convention, if C is the category of groups, for instance, is*
* to
call an object of sC a "simplicial group". The category C is embedded in sC by
the "constant diagram" functor (2.11) and in dealing with simplicial objects it*
* is
common to identify C with its image under this embedding. Suppose that C has an
"underlying set" or forgetful functor U : C ! Set(cf. 2.9). Call a map f : X ! Y
in sC
(i)a weak equivalence if U(f) is a weak equivalence in sSet,
(ii)a fibration if U(f) is a fibration in sSet, and
(iii)a cofibration if f has the LLP with respect to acyclic fibrations.
In [22, Part II, x4] Quillen shows that in all common algebraic situations (e.g*
*.,
if C is the category of groups, abelian groups, associative algebras, Lie algeb*
*ras,
commutative algebras, : :):these choices give sC the structure of a model categ*
*ory;
he also characterizes the cofibrations [22, Part II, p. 4.11].
Consider now the example C = Mod R. It turns out that there is a normaliza-
tion functor N : sMod R! ChR [18, x22] which is an equivalence of categories a*
*nd
translates the model category structure on sMod Rabove into the model category
structure on Ch R from x7. Thus the homotopy theory of sMod Ris ordinary ho-
mological algebra over R. For a general category C there is no such normalizati*
*on
functor, and so it is natural to think of an object of sC as a substitute for a*
* chain
complex in C , and consider the homotopy theory of sC as homological algebra,
or better homotopical algebra, over C . This leads to the conclusion (11.1) that
Section 11 Homotopy theories 53
homotopical algebra over the category of sets is ordinary homotopy theory!
11.3. Simplicial commutative rings. Let C be the category of commutative rings.
In [24] Quillen uses the model category structure on sC which was described abo*
*ve
in order to construct a cohomology theory for commutative rings (now called And*
*re-
Quillen cohomology). This has been studied extensively by Miller [19] and Goerss
[13] because of the fact that if X is a space the Andre-Quillen cohomology of
H*(X; Fp) plays a role in various unstable Adams spectral sequences associated *
*to
X. In this way the homotopical algebra of the commutative ring H*(X; Fp) leads
back to information about the homotopy theory of X itself; this is parallel to *
*the
way in which, if Y is a spectrum, the homological algebra of H*(Y ; Fp) as a mo*
*dule
over the Steenrod algebra leads to information about the homotopy theory of Y .
We can now answer a question from the introduction. Suppose that k is a field.*
* Let
C be the category of commutative augmented k-algebras and let R be an object
of C . Recall that C can be identified with a subcategory of sC by the constant
diagram construction. Topological intuition suggests that the suspension R of R
should be the homotopy pushout (x10) of the diagram * R ! *, where * is
a terminal object in sC. Since this terminal object is k itself, R should be the
homotopy pushout in sC of k R ! k. It is not hard to compute this; up to
homotopy R is given by the bar construction [19, Section 5] [13, p. 51] and the
i'th homotopy group of the underlying simplicial set of R is TorRi(k; k).
11.4. Rational homotopy theory. A simplicial set X is said to be 2-reduced if *
*Xi
has only a single point for i < 2. Call a map f : X ! Y between 2-reduced simpl*
*icial
sets
(i)a weak equivalence if H*(|f|; Q) is an isomorphism,
(ii)a cofibration if each map fk : Xk ! Yk is a monomorphism, and
(iii)a fibration if f has the RLP with respect to acyclic cofibrations.
In [23], Quillen shows that these choices give a model category structure on the
category sSet2 of 2-reduced simplicial sets. A differential graded Lie algebra X
over Q is said to be 1-reduced if X0 = 0. Call a map f : X ! Y between 1-reduced
differential graded Lie algebras over Q
(i)a weak equivalence if H*(f) is an isomorphism,
(ii)a fibration if fk : Xk ! Yk is surjective for each k > 1, and
(iii)a cofibration if f has the LLP with respect to acyclic fibrations.
These choices give a model category structure on the category DGL 1 of 1-reduced
differential graded Lie algebras over Q . By repeated applications of Theorem 9*
*.7,
Quillen shows [23] that the homotopy categories Ho(sSet2) and Ho(DGL 1) are
equivalent. It is not hard to relate the category sSet2to the category Top1 of *
*1-
connected topological spaces (there is a slight difficulty in that Top1 is not *
*closed
under colimits or limits and so cannot be given a model category structure). Wh*
*at
results is a specific way in which objects of DGL 1 can be used to model the ra*
*tional
homotopy types of 1-connected spaces. For a dual approach based on differential
54 Dwyer and Spalinski Chapter 1
graded algebras see [4] and for an attempt to eliminate some denominators [7].
There is a large amount of literature in this area.
11.5. Homology localization. Let h* be a homology theory on the category of
spaces which is represented in the usual way by a spectrum. Call a map f : X ! Y
in sSet
(i)a weak h*-equivalence if h*(|f|) is an isomorphism,
(ii)an h*-cofibration if f is a cofibration with respect to the conventional *
*model
category structure (11.1) on sSet, and
(iii)an h*-fibration if f has the RLP with respect to each map which is both a
weak h*-equivalence and an h*-cofibration.
Bousfield shows [2, Appendix] that these choices give a model category structure
on sSet, called, say the h*-structure. The hardest part of the proof is verifyi*
*ng
MC5(ii). Bousfield does this by an interesting generalization of the small obje*
*ct
argument (7.12). He first shows that there is a single map i : A ! B which is b*
*oth
a weak h*-equivalence and a h*-cofibration, such that f is a h*-fibration if an*
*d only
if f has the RLP with respect to i. (Actually he finds a set {iff}`of such test*
* maps,
but there is nothing lost in replacing this set by the single map ffiff.) Now*
* the
domain A of i is potentially quite large, and so A is`not necessarily sequentia*
*lly
small. However, if j is the cardinality of the set nAn of simplices of A, the
functor Hom sSet(A; -) does commute with colimits indexed by transfinite ordina*
*ls
of cofinality greater than j. Bousfield then proves MC5(ii) by using the general
idea in the proof of 7.17 but applying the gluing construction G({i}; -) transf*
*initely;
this involves applying the gluing construction itself at each successor ordinal*
*, and
taking a colimit of what has come before at each limit ordinal.
Let Ho denote the conventional homotopy category of simplicial sets (11.1).
Say that a simplicial set X is h*-local if any weak h*-equivalence f : A ! B
induces a bijection Hom Ho(B; X) ! Hom Ho(A; X). It is not hard to show that a
simplicial set which is fibrant with respect to the h*-structure above is also *
*h*-local.
It follows that using MC5(ii) (for the h*-structure) to factor a map X ! * as a
composite X ~,!X0!!* gives an h*-localization construction on sSet, i.e, gives *
*for
any simplicial set X a weak h*-equivalence X ! X0from X to an h*-local simplici*
*al
set X0. Since the factorization can be done explicitly with a (not so) small ob*
*ject
argument, we obtain an h*-localization functor on sSet. It is easy to pass from
this to an analogous h*-localization functor on Top. These functors extract fro*
*m a
simplicial set or space exactly the fraction of its homotopy type which is visi*
*ble to
the homology theory h*.
11.6. Feedback. We conclude by describing a way to apply the theory of model
categories to itself (see [8] and [9]). The intuition behind this application i*
*s the
idea that almost any simple algebraic construction should have a (total) derived
functor (x9), even, for instance, the localization construction (x6) which send*
*s a pair
(C ; W ) to the localized category W -1C. In fact it is possible to construct a*
* total left
Section References Homotopy theories 55
derived functor of (C ; W ) 7! W -1C, although this involves using Proposition *
*9.3
in a "meta" model category in which the objects themselves are categories enric*
*hed
over simplicial sets [17, p. 181]! If C is a model category with weak equivalen*
*ces W ,
let L(C ; W ) denote the result of applying this derived functor to the pair (C*
* ; W ).
The object L(C ; W ) is a category enriched over simplicial sets (or, with the *
*help of
the geometric realization functor, a category enriched over topological spaces)*
* with
the same collection of objects as C. For any pair of objects X, Y 2 Ob(C ) ther*
*e is
a natural bijection
ss0Hom L(C;W)(X; Y ) ~=Hom Ho(C)(X; Y )
which exhibits the set Hom Ho(C)(X; Y ) as just the lowest order invariant of a*
*n entire
simplicial set or space of maps from X to Y which is created by the localization
process. The homotopy types of these "function spaces" Hom L(C;W)(X; Y ) can be
computed by looking at appropriate simplicial resolutions of objects of C [9, x*
*4];
these function spaces seem to capture most if not all of the higher order struc*
*ture
associated to C which was envisaged and partially investigated by Quillen [22,
part I, p. 0.4] [22, part I, x2, x3].
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