Cofibrations in Homotopy Theory
Andrei RadulescuBanu
86 Cedar St, Lexington, MA 02421 USA
Email address: andrei@alum.mit.edu
2000 Mathematics Subject Classification. Primary 18G55, 55U35, 18G10, 18G30,
55U10
Key words and phrases. cofibration category, model category, homotopy colimit
Abstract.We define AndersonBrownCisisnski (ABC) cofibration categories,
and construct homotopy colimits of diagrams of objects in ABC cofibracti*
*on
categories. Homotopy colimits for Quillen model categories are obtained *
*as a
particular case. We attach to each ABC cofibration category a right deri*
*vator.
A dual theory is developed for homotopy limits in ABC fibration categori*
*es
and for left derivators. These constructions provide a natural framework*
* for
'doing homotopy theory' in ABC (co)fibration categories.
Contents
Preface vii
Chapter 1. Cofibration categories 1
1.1. The axioms 2
1.2. Sums of objects in a cofibration category 4
1.3. Factorization lemmas 5
1.4. Extension lemmas 7
1.5. Cylinder and path objects 11
1.6. Elementary consequences of CF5 and CF6 12
1.7. Properness 16
Chapter 2. Relation with other axiomatic systems 21
2.1. Brown's categories of cofibrant objects 21
2.2. Quillen model categories 23
2.3. Baues cofibration categories 24
2.4. Waldhausen categories 25
Chapter 3. The homotopy category of a cofibration category 27
3.1. Universes and smallness 28
3.2. The homotopy category 28
3.3. Homotopic maps and ssM 29
3.4. Homotopy calculus of fractions 31
3.5. Fibrant and cofibrant approximations 38
3.6. Products of cofibration categories 46
3.7. Saturation 47
Chapter 4. Kan extensions. Total derived functors. 53
4.1. The language of 2categories 53
4.2. Adjoint functors 55
4.3. Kan extensions 56
4.4. Total derived functors 58
4.5. Left and right approximation functors. 59
4.6. Total derived functors in cofibration categories 68
Chapter 5. Review of category theory 71
5.1. Basic definitions and notations 71
5.2. Limits and colimits 72
5.3. Simplicial sets 73
5.4. The nerve of a category 74
5.5. Cofinal functors 74
5.6. The Grothendieck construction 76
v
vi CONTENTS
Chapter 6. Homotopy colimits in a cofibration category 79
6.1. Direct and inverse categories 79
6.2. Reedy and pointwise cofibration structures for direct diagrams 80
6.3. Colimits in direct categories (the absolute case) 82
6.4. Colimits in direct categories (the relative case) 93
6.5. Colimits in arbitrary categories 95
6.6. Homotopy colimits 102
6.7. The conservation property 110
6.8. Realizing diagrams 111
Chapter 7. Derivators 121
7.1. Prederivators 121
7.2. Derivators 122
7.3. Derivability of cofibration categories 124
7.4. Odds and ends 125
Bibliography 127
Index 129
Preface
Model categories, introduced by Daniel Quillen [Qui67 ], are a natural frame
work for doing homotopy theory in an axiomatic way. A Quillen model category
(M, W, Cof, Fib) consists of a category M, and three distinguished classes of m*
*aps:
the weak equivalences W, the cofibrations Cof and the fibrations Fib, subject t*
*o a
list of axioms (Def. 2.2.2).
Let us fix some notation. If D is a small category we denote MD the category*
* of
Ddiagrams in M. For a small functor u : D1 ! D2 we denote u* : MD2 ! MD1 the
functor defined by (u*X)d1= Xud1for objects d1fflD1. For a category with a class
of weak equivalences (M, W) we denote hoM = M[W1] its homotopy category,
and we will always consider the weak equivalences on MD to be pointwise, i.e. f*
* is
a weak equivalence in MD if fd is a weak equivalence in M for all objects dfflD.
A Quillen model category M admits homotopy pushouts and homotopy pull
backs. These are the total left (resp. right) derived functors of the pushout (*
*resp.
pullback) functor in M. If M is pointed one can construct the homotopy cofiber
and fiber of a map in hoM, and the suspension and loop space of an object in ho*
*M.
One can then form the cofibration sequence and the fibration sequence of a map *
*in
hoM.
Furthermore, a Quillen model category M admits all small homotopy colimits
and limits (also called the homotopy left and right Kan extensions). For a small
functor u : D1 ! D2, the homotopy colimit L colimu: ho(MD1) ! ho(MD2) is
the total left derived functor of the colimit functor colimu, and is left adjoi*
*nt to
hou* : ho(MD2) ! ho(MD1). Dually, the homotopy limit R limuis the total right
derived functor of limu, and is right adjoint to hou*.
For the general construction of homotopy colimits and limits, we recommend
the work of Dwyer, Kan, Hirschhorn and Smith [DKHS04 ], Hirschhorn [Hir00 ],
Chach'olski and Scherer [CS02 ] and Cisinski [Cis03 ]. Each of these references
presents a different perspective on homotopy (co)limits.
Once the construction of homotopy colimits and limits is understood, a very
interesting question we can ask is: What is really the role of cofibrations and
fibrations in a Quillen model category? The homotopy (co)limits are the total
derived functors of the (co)limit, so the definition of homotopy (co)limits dep*
*ends
just on the weak equivalences. The construction of homotopy colimits on the oth*
*er
hand requires the presence of cofibrations and weak equivalences  but does not
really involve fibrations. Dually, the construction of homotopy limits requires*
* the
presence of fibrations and weak equivalences, but not cofibrations.
It is therefore natural to ask if one can simplify Quillen's set of axioms, *
*and
separate a minimal set of axioms required by cofibrations and weak equivalences*
* in
order to still be able to construct homotopy colimits.
vii
viii PREFACE
In a very influential paper, Ken Brown [Bro74 ] formalized some of these obs*
*e
vations by defining categories of fibrant objects and working out in detail the*
*ir prop
erties. Reversing arrows, one defines categories of cofibrant objects, and Brow*
*n's
work carries over by duality to categories of cofibrant objects.
A category of cofibrant objects (M, W, Cof) consists of a category M, the cl*
*ass
of weak equivalences W and the class of cofibrations Cof, subject to a list of *
*axioms
(the duals of the axioms of [Bro74 ]). The axioms require in particular all obj*
*ects
to be cofibrant.
For a pointed category of cofibrant objects, Brown was able to construct ho
motopy cofibers of maps, suspensions of objects and the cofibration sequence of*
* a
map. These constructions exist in dual form for categories of fibrant objects.
Building on Brown's work, Don Anderson [And78 ] extended Brown's axioms
for a category of cofibrant objects by dropping the requirement that all objects
be cofibrant. Anderson called the categories defined by his new axioms left ho
motopical; our text changes terminology and calls them AndersonBrownCisinski
cofibration categories (or just cofibration categories for simplicity). The cof*
*ibration
category axioms we use are slightly more general than Anderson's.
Anderson's main observation was that the cofibration category axioms on M
suffice for the construction of a left adjoint of hou*, for any small functor u*
*. It is
implicit in his work that the left adjoint of hou* is a left derived of colimu.
Unfortunately for the history of this subject, Anderson's paper [And78 ] co*
*n
tains statements but omits proofs, and has a title ("Fibrations and Geometric
Realizations") that does not reflect the generality of his work. Also, Anderson*
* quit
mathematics shortly after his paper was published, the proofs of [And78 ] got *
*lost
and as a result his whole theory lay dormant for twenty five years.
We can be grateful to DenisCharles Cisinski [Cis02 ], [Cis03 ] for bringing*
* back
to light Brown and Anderson's ideas. Cisinski simplifies Anderson's arguments, *
*and
provides for cofibration categories a complete construction of homotopy colimits
along functors u : D1 ! D2 with D1 finite and direct.
Cisinski has also worked out the construction of homotopy colimits along ar
bitrary small functors u, as well as the end result regarding the derivability *
*of
cofibration categories (our Chap. 7). While this part of his work remains unpub
lished, he was kind enough to share with me its outline. I would like to thank
him for suggesting the correct formulation of axioms CF5CF6, and for patiently
explaining to me the finer points of excision.
The goal of these notes is then to work out a selfcontained account of homo
topy colimits from the axioms of an AndersonBrownCisinski cofibration categor*
*y,
and show that they satisfy the axioms of a right derivator. There are a number *
*of
properties of homotopy colimits that are a formal consequence of the right deri*
*vator
axioms, but they are outside of the scope of our text. We will instead try to i*
*nves
tigate the relation with the betterknown Quillen model categories, and compare
with other axiomatizations that have been proposed for cofibration categories.
While some of the proofs we propose may be new, the credit for this theory
should go entirely to Brown, Anderson and Cisinski. It was our choice in this t*
*ext to
make use of approximation functors and abstract Quillen equivalences, and for t*
*hat
we were influenced by the work of Dwyer, Kan, Hirschhorn and Smith [DKHS04 ].
PREFACE ix
Our treatment of direct and inverse categories bears the influence of Daniel Ka*
*n's
theory of Reedy categories outlined in [DKH97 ], [Hov99 ] and [Hir00 ].
I would like to thank Haynes Miller and Daniel Kan, my mentors in abstract
homotopy, for their gracious support and encouragement. I am grateful to Denis
Charles Cisinski, Philip Hirschhorn and Haynes Miller for the conversations we *
*had
on the subject of this text.
Outline. We start our exposition with the axioms of an AndersonBrown
Cisinski (ABC) cofibration category, and we proceed in Chap. 1 with their eleme*
*n
tary properties. We will make a distinction between ABC precofibration categori*
*es
(satisfying axioms CF1CF4) and ABC cofibration categories (satisfying axioms
CF1CF6).
This allows us going to Chap. 2 to compare ABC cofibration categories with
other axiomatic systems that have been proposed for categories with cofibration*
*s.
Most notably, we will show that any Quillen model category is an ABC cofibration
(and fibration) category.
In Chap. 3, we recall the theory of cylinders and homotopic maps in a pre
cofibration category. This allows us to abstract out the properties of cofibra*
*nt
approximation functors between precofibration categories, and later in Chap. 4 *
*we
abstract the cofibrant approximation functors to yet another level  these are *
*the
left approximation functors.
In Chap. 4, we recall the definition of Kan extensions and total derived fun*
*c
tors. The purpose of left approximation functors will be to construct total der*
*ived
functors, and to prove a Quillentype adjunction property between the total der*
*ived
functors of an adjoint pair of functors.
In Chap. 5, we recall over and under categories, and elementary properties of
limits and colimits.
In Chap. 6, we reach our main objective. We define the homotopy colimit
L colimu: ho(MD1) ! ho(MD2) as a left derived functor, and show that it exists
and it is left adjoint to hou* : ho(MD2) ! ho(MD1). We show that if D is a small
category and (M, W, Cof) is a cofibration category, then the diagram category MD
is again a cofibration category, with pointwise weak equivalences and pointwise
cofibrations.
Finally, in Chap. 7 we recall the notion of a right derivator, and show that*
* the
homotopy colimits in a cofibration category satisfy the axioms of a right deriv*
*ator.
The purpose of the last chapter is simply to assert the results of Chap. 6 with*
*in
the axiomatic language of derivators.
CHAPTER 1
Cofibration categories
This chapter defines AndersonBrownCisinski (or ABC) cofibration, fibration
and model categories. For simplicity, we refer to AndersonBrownCisinski cofib*
*ra
tion (fibration, model) categories as just cofibration (fibration, model) categ*
*ories,
when no confusion with Quillen model categories or Baues cofibration categories*
* is
possible.
What is an ABC cofibration category? It is a category M with two distinguish*
*ed
classes of maps, the weak equivalences and the cofibrations, satisfying a set o*
*f six
axioms which are denoted CF1CF6. An ABC fibration category is a category M
with weak equivalences and fibrations, satisfying the dual axioms F1F6. Am ABC
model category is a category M with weak equivalences, cofibrations and fibrati*
*ons
that is at the same time a cofibration and a fibration category.
Any Quillen model category is an ABC model category (Prop. 2.2.3). Dia
grams indexed by a small category in an ABC cofibration category form again a
cofibration category (Thm. 6.5.5), a property not enjoyed in general by Quillen
model categories.
The ultimate goal using the cofibration category axioms CF1CF6 is to con
struct (in Chap. 6) homotopy colimits in M indexed by small categories D, and
more generally to construct 'relative' homotopy colimits along small functors u*
* :
D1 ! D2.
The category M will not be assumed in general to be cocomplete. Under the
simplifying assumption that M is cocomplete, however, the homotopy colimit along
a functor u : D1 ! D2 is the left Kan extension along colimu: MD1 ! MD2. It is
a known fact that a colimit indexed by a small category D can be constructed in
terms of pushouts of small sums of objects in M. Furthermore, a 'relative' coli*
*mit
colimucan be described in terms of absolute colimits indexed by the over catego*
*ries
(u # d2) for d2fflD2 (see Lemma 5.2.1).
It is perhaps not surprising then that the cofibration category axioms speci*
*fy
an approximation property of maps by cofibrations (axiom CF4), as well as the
behaviour of cofibrations under pushouts (axiom CF3) and under small sums (axiom
CF5).
A large part of the theory can be developed actually from a subset of the ax*
*ioms,
namely the axioms CF1CF4. The theory of homotopic maps, of homotopy calculus
of fractions and of cofibrant approxiomation functors of Chap. 3 only requires
this smaller set of axioms. A category with weak equivalences and cofibrations
satisfying the axioms CF1CF4 will be called a precofibration category. From the
precofibration category axioms, it turns out that one can construct all the hom*
*otopy
colimits indexed by finite, direct categories [Cis02 ].
One of the ideas worth repeating is that while we need both the cofibrations
and the weak equivalences to construct homotopy colimits, the homotopy colimits
1
2 1. COFIBRATION CATEGORIES
are characterized in the end just by the weak equivalences. So when working wit*
*h a
cofibration category with a fixed class of weak equivalences, it would be desir*
*able to
try to work with a class of cofibrations as large as possible. This is where th*
*e concept
of left proper maps becomes useful (see Section 1.7). In a left proper precofib*
*ration
category (M, W, Cof), any cofibration A ! B with A cofibrant is a left proper
map, and the class of left proper maps denoted PrCof yields again a precofibrat*
*ion
category structure (M, W, PrCof ). But if M is a left proper CF1CF6 cofibration
category, then (M, W, PrCof ) may not satisfy the cofibration category axioms
CF5 and CF6. Dual results hold for right proper prefibration categories.
1.1.The axioms
Definition 1.1.1 (AndersonBrownCisinski cofibration categories).
An ABC cofibration category (M, W, Cof) consists of a category M, and two
distinguished classes of maps of M  the weak equivalences (or trivial maps) W *
*and
the cofibrations Cof, subject to the axioms CF1CF6. The initial object 0 of M
exists by axiom CF1, and an object A is called cofibrant if the map 0 ! A is a
cofibration.
The axioms are:
CF1: All isomorphisms of M are weak equivalences, and all isomorphisms
with the domain a cofibrant object are trivial cofibrations. M has an i*
*nitial
object 0, which is cofibrant. Cofibrations are stable under composition.
CF2: (Two out of three axiom) Suppose f and g are maps such that gf is
defined. If two of f, g, gf are a weak equivalence, then so is the thir*
*d.
CF3: (Pushout axiom) Given a solid diagram in M, with i a cofibration and
A, C cofibrant
Af_____//flfflCfflfflO
i OjO
fflfflfflffl
B ` ` `//D
then
(1) the pushout exists in M and j is a cofibration, and
(2) if additionally i is a trivial cofibration, then so is j.
CF4: (Factorization axiom) Any map f : A ! B in M with A cofibrant
factors as f = rf0, with f0 a cofibration and r a weak equivalence
CF5: If fi: Ai! Bi for ifflI is a set of cofibrations with Ai cofibrant, *
*then
(1) tAi, tBi exist and are cofibrant, and tfi is a cofibration.
(2) if additionally all fi are trivial cofibrations, then so is tfi.
CF6: For any countable direct sequence of cofibrations with A0 cofibrant
A0//a0_//A1//a1//_A2//a2//_...
(1) the colimit object colimAn exists and the transfinite composition
A0 ! colimAn is a cofibration.
(2) if additionally all aiare trivial cofibrations, then so is A0 ! col*
*imAn.
If (M, W, Cof) only satisfies the axioms CF1CF4, it is called a precofibrat*
*ion
category.
Pushouts are defined by an universal property, and are only defined up to
an unique isomorphism. Since all isomorphisms with cofibrant domain are trivial
1.1. THE AXIOMS 3
cofibrations, it does not matter which isomorphic representative of the pushout*
* we
choose in CF3. 0
In the axiom0CF4, the map f is sometimes0called a cofibrant replacement0of
f. If r : A ! A is a weak equivalence with A cofibrant, the object A is calle*
*d a
cofibrant replacement of A.
We would like to stress that the class of weak equivalences is not necessar
ily assumed to be saturated, although a cofibration category is still a cofibra*
*tion
category with respect to the saturation of weak equivalences (Thm. 3.7.2).
If A ! B is a cofibration with A cofibrant then B is cofibrant. 0But there
may exist cofibrations A ! B with A not cofibrant. If we0denote Cof the class
of cofibrations A ! B with A cofibrant, then (M, W, Cof ) is again a cofibration
category.
We will sometimes refer to a cofibration category as just M. We will also de*
*note
Mcofthe full subcategory of cofibrant objects of M.
The category0Mcof is a cofibration category, and in fact so is any0full0sub
category0M of M that includes Mcof, with the induced structure (M , W \ M ,
Cof \ M ). 0
If M is a precofibration category, then (M, W, Cof ), Mcofand any M0as above
are precofibration categories. We will sometimes refer to precofibration catego*
*ries
as CF1CF4 cofibration categories.
Definition 1.1.2 (AndersonBrownCisinski fibration categories).
An ABC fibration category (M, W, Fib) consists of a category M, and two distin
guished classes of maps  the weak equivalences W and the fibrations Fib, subje*
*ct
to the axioms F1F6. The terminal object 1of M exists by axiom F1, and an object
A of Fibis called fibrant if the map A ! 1 is a fibration.
The axioms are:
F1: All isomorphisms of M are weak equivalences, and all isomorphisms
with fibrant codomain are trivial fibrations. M has a final object 1, w*
*hich
is fibrant. Fibrations are stable under composition.
F2: (Two out of three axiom) Suppose f and g are maps such that gf is
defined. If two of f, g, gf are a weak equivalence, then so is the thir*
*d.
F3: (Pullback axiom) Given a solid diagram in M, with p a fibration and A,
C fibrant,
DO` ` `//B
q O p
fflfflfflfflOfflfflfflffl
C _____//A
then
(1) the pullback exists in M and q is a fibration, and
(2) if additionally p is a trivial prefibration, then so is q.
F4: (Factorization axiom) Any map f : A ! B in M with B fibrant factors
as f = f0s, with s a weak equivalence and f0 a fibration.
F5: If fi: Ai! Bi for ifflI is a set of fibrations with Bi fibrant, then
(1) xAi, xBi exist and are fibrant, and xfi is a fibration
(2) if additionally all fi are trivial fibrations, then so is xfi.
F6: For any countable inverse sequence of fibrations with A0 fibrant
4 1. COFIBRATION CATEGORIES
...a2_////_A2a1////_A1a0////_A0
(1) the limit object limAiexists and the transfinite composition limAn !
A0 is a fibration
(2) if additionally all ai are trivial fibrations, then so is limAn ! A*
*0.
If (M, W, Fib) only satisfies the axioms F1F4, it is called a prefibration *
*cate
gory.
The axioms are dual in the sense that (M, W, Cof) is a cofibration category *
*if
and only if (Mop, Wop, Cofop)0is a fibration category. *
* 0
In the axiom F4, the map f0 is called a fibrant replacement0of f. If r : A !*
* A
is a weak equivalence with A fibrant, the object A is called a fibrant replac*
*ement
of A. 0
If0we denote Fib the class of fibrations A ! B with B fibrant then (M, W,
Fib) again is a fibration category.
We will denote Mfibto be the full subcategory of fibrant objects0of a fibrat*
*ion
category M. The category Mfibas well as any full subcategory M of M that
includes Mfibsatisfy again the axioms of a fibration category.0
If M is a prefibration category, then so are (M, W, Fib), Mfiband any M0 as
above.
Definition 1.1.3 (AndersonBrownCisinski model categories).
An ABC model category (M, W, Cof, Fib) consists of a category M and three
distinguished classes of maps W, Cof, Fibwith the property that (M, W, Cof) is
an ABC cofibration category and that (M, W, Fib) is an ABC fibration category.
In all sections of this chapter except Section 1.6, we will do our work assu*
*m
ing only the precofibration category axioms CF1CF4 (and dually the prefibration
category axioms F1F4). In Section 1.6, we will assume that the full set of axi*
*oms
is verified.
1.2. Sums of objects in a cofibration category
In general, the objects of a precofibration category are not closed under fi*
*nite
sums. But finite sums of cofibrant objects exist and are cofibrant. Dually, i*
*n a
prefibration category finite products of fibrant objects exist and are fibrant.*
* In fact
we can prove the slightly more general statement:
Lemma 1.2.1.
(1) Suppose that M is a precofibration category. If fi: Ai! Bifor i = 0, ..*
*., n
are cofibrations with Ai cofibrant, then tAi, tBi exist and are cofibra*
*nt,
and tfi is a cofibration which is trivial if all fi are trivial.
(2) Suppose that M is a prefibration category. If fi: Ai! Bi for i = 0, ...*
*, n
are fibrations with Bi fibrant, then xAi, xBi exist and are fibrant, and
xfi is a fibration which is trivial if all fi are trivial.
Proof. We will prove (1), and observe that statement (2) is dual to (1). Us*
*ing
induction on n, we can reduce the problem to two maps f0 : A0 ! B0 and f1; A1 !
B1. If we prove the statement for f0, 1A1 and 1B0, f1 then the statement follows
for f0, f1. So it suffices to show that if f : A ! B is a (trivial) cofibration*
* and A,
C are cofibrant, then A t C, B t C exist and are cofibrant and f t 1C is a (tri*
*vial)
cofibration. From axiom CF3 (1) applied to
1.3. FACTORIZATION LEMMAS 5
0//_____//fflfflCfflffl
 
 
fflffl fflffl
A //___//A t C
we see that AtC exists and is cofibrant, and A ! AtC is a cofibration. Similarl*
*y,
B t C exists and is cofibrant, and from CF3 applied to
Af//___//flfflAftfClffl
f ft1C
fflffl fflffl
B //__//_B t C
we see that f t 1C is a cofibration which is trivial if f is trivial.
1.3.Factorization lemmas
The Brown Factorization Lemma is an improvement of the factorization axiom
CF4 for maps between cofibrant objects.
Lemma 1.3.1 (Brown factorization, [Bro74 ]).
(1) Let M be a precofibration category, and f : A ! B be a map between
cofibrant objects. Then f factors as f = rf0, where f0 is a cofibration*
* and
r is a left inverse to a trivial cofibration.
(2) Let M be a prefibration category, and f : A ! B be a map between fibrant
objects. Then f factors as f = f0s, where f0 is a fibration and s is a *
*right
inverse to a trivial fibration.
Proof. The statements are dual, so it suffices to prove (1). We need to con
struct f0, r and s with f = rf0 and rs = 1B .
If we apply the factorization axiom to f + 1B , we get a diagram
f+1B
A t B#_____________//#FB>>
FFF ~ """
FFF ""r
f0+s FF##0""
B
Since f0 + s is a cofibration and A, B are cofibrant, the maps f0 and s are cof*
*i
brations. The map r is a weak equivalence, and from the commutativity of the
diagram we have rs = 1B , therefore s is also a weak equivalence.
Remark 1.3.2. We have in fact proved a stronger statement. We have shown
that any map f : A ! B between cofibrant objects0in a precofibration category
factors as f = rf0, with rs = 1B where f0, f + s are cofibrations and s is a tr*
*ivial
cofibration.
Dually, any map f : A ! B between fibrant objects0in a prefibration category
factors as f = f0s, with rs = 1A where f0 and (f , r) are fibrations and r is a*
* trivial
fibration.
Next lemma is a relative version of the factorization axiom CF4 (resp. F4).
Lemma 1.3.3 (Relative factorization of maps).
(1) Let M be a precofibration category, and let
6 1. COFIBRATION CATEGORIES
f1
A1 _____//B1
a b
fflfflf2fflffl
A2 _____//B2
be a commutative diagram with A1, A2 cofibrant. Suppose that f1 = r1f01
is a factorization of f1 as a cofibration followed by a weak equivalenc*
*e.
Then there exists a commutative diagram
f01 0 r1
A1 //___//A1_~__//B1
 
a a0 b
fflfflf0fflffl2fflffl0r2
A2 //___//A2_~__//B2
where r2f02is a factorization of f20as a0cofibration followed by a weak
equivalence and such that A2 tA1 A1 ! A2 is a cofibration.
(2) The dual of (1) holds for prefibration categories.
Proof. To prove (1), in the commutative diagram
f01 0 r1
A1 //______//A1_______~_______//B1
 
a  b
fflffl fflffl0s 0r2 fflffl
A2 //___//A2 tA1 A1//_//_A2~__//_B2
the pushout A2 tA1 A01exists by CF3, and we construct the cofibration s and0the
weak equivalence r2 using the factorization axiom CF4 applied to A2 tA1 A1 !
B2.
The Brown Factorization Lemma has the following relative version:
Lemma 1.3.4 (Relative Brown factorization).
(1) Suppose that M is a precofibration category, and that
f1
A1 _____//B1
a b
fflfflf2fflffl
A2 _____//B2
is a commutative diagram with cofibrant objects.0Suppose0that f1 = r1f0*
*1,
r1s1 = 1 is a Brown factorization of f1, with f1, f1 + s1 cofibrations *
*and0
s1 a trivial cofibration.0Then0there exists a Brown factorization f2 = *
*r2f2,
r2s2 =01 with f2, f2 + s2 cofibrations and s2 a trivial cofibration and*
* a
map b such that in the diagram
1.4. EXTENSION LEMMAS 7
f01 0_r1_//
A1 //___//B1o~o_B1oo
 s1 
a b0 b
fflfflf0fflffl2fflffl0r2//_
A2 //___//B2o~o_B2oo
s2
we have that b0f01= f02a, br1 = r2b0, b0s1 = s2b, and that
A2 tA1 B01! B02and B2 tB1 B01! B02
are a cofibration (resp. a trivial cofibration).
(2) The dual of (1) holds for prefibration categories.
Proof. To prove (1), denote A3 = A2 tA1 B01and B3 = B2 tB1 B01.
f01 0_____r1_____//
A1 //___________//B1@oo__~______oB1o
""" @@ s1 
a """ @@@@ b
fflfflf0 """" f3 __@__r__//fflffl
A2 //___//A3____________//B3o~o_oB2o
s
We apply Lemma 1.3.3 to the commutative diagram
f01+s1 0 r1
A1 t B1//___//B1_~__//B1
f0atsb sb
fflffl f3+1B3 fflffl
A3 t B3_____________//B3
and we construct a commutative diagram
f1+1B1
A1 t B1#____________//_#GB1>>
 GGG "" 
 GG ~""" 
 f0+s GGG ""r1" 
 1 1 ##0 
f0atsb B1 sb
  
  
 b0 
fflffl  fflffl
A3 t B3 ___________//B3
##GG  ">>
GGG  ~ """
0 GG  "r3"
f3+s3 G##fflfflG0""
B2
We now set f02= f03f0, r2 = rr3 and s2 = s3s.
1.4. Extension lemmas
The Gluing Lemma describes the behavior of cofibrations and weak equivalences
under pushouts, and is one of the basic building blocks we will employ in the
construction of homotopy colimits.
Lemma 1.4.1 (Gluing Lemma).
8 1. COFIBRATION CATEGORIES
(1) Let M be a precofibration category. In the diagram
f12
A1//___________//BA2B
 BBBB  BBBB
 f BBB  BBB
 13 __  __
u1 A3 //___________//A4
   
   
 u3 u2 
fflffl  fflffl 
B1 //__________//B2 u4
BB  g12 BB 
BBB  BBB 
g13BB  BB 
B_fflffl_ B__fflffl
B3 //___________//B4
suppose that A1, A3, B1, B3 are cofibrant, that f12, g12 are cofibratio*
*ns,
and that the top and bottom faces are pushouts.
(a) If u1, u3 are cofibrations and the natural map
B1 tA1 A2 ! B2 is a cofibration, then u2, u4 and the natural map
B3 tA3 A4 ! B4 are cofibrations.
(b) If u1, u2, u3 are weak equivalences, then u4 is a weak equivalence.
(2) Let M be a prefibration category. In the diagram
B4____________////_BB3B
 BBBB  BBg31BB
 BBB  BBB
 __ g21  __
u4 B2 _____________////B1
   
   
 u2 u3 
fflffl  fflffl 
A4 ___________////_A3 u1
BB  BB 
BBB  BBf31B
BB  BB 
B_fflffl_ B__fflffl
A2 _____f21_____////A1
suppose that A1, A3, B1, B3 are fibrant, that f21, g21 are fibrations, *
*and
that the top and bottom faces are pullbacks.
(a) If u1, u3 are fibrations and the natural map
B2 ! B1 xA1 A2 is a fibration, then u2, u4 and the natural map
B4 ! B3 xA3 A4 are fibrations.
(b) If u1, u2, u3 are weak equivalences, then u4 is a weak equivalence.
Proof.0 The statements are dual, and we will prove only (1). Let B02= B1tA1
A2 and B4 = B3tA3 A4 be the pushout of the front and back faces of the diagram
of (1). These pushouts exist because of the pushout axiom CF3.
1.4. EXTENSION LEMMAS 9
f12
A1//___________//BA2B
 BBB  BBB
 BBB  BBB
 f13 B__  B__
 A3 //__________//A4
u1  0 
 u u2 
  3 fflffl u0
  0 4
  oB277 
  oooo ??? 
 0 ooo  00?? 
 g12oooo u2 ?? 
fflffl77ooo fflfflOOfflffl0
B1 //______g___//B2 nB477
@@@  12 @@@nnn
@@  0 nnnn@@ 00
g13@@  g34nnn @@ u4
@_fflffl_77nnnn@__fflffl
B3 //___________//B4
The maps g012and g034are0pushouts of0cofibrations, therefore0cofibrations.0Furt*
*her00
more,0we0observe that B4 = B3tB1 B2, and therefore u4 is the pushout of u2 along
B2 ! B4. 00
0 Let's0prove (a). If u1, u30and0u2 are cofibrations,0then0by the pushout axiom
u2 and u4 are cofibrations. u4 is a pushout of u2, therefore also a cofibration*
*. It
follows that u2, u4 are cofibrations.
Let's now prove (b), first under the assumption that
00
(1.1) u1, u3 and u2 are cofibrations
If they are,0since0u1, u2, u3 are weak equivalences we see that u1, u3 and thei*
*r00
pushouts u2, u4 must be trivial cofibrations. From the two out of three axiom,0*
*u20
is a weak equivalence, therefore a trivial cofibration,0and0so0its pushout u4 a*
*lso is
a trivial cofibration, which shows that u4 = u4u4 is a weak equivalence.
For general weak equivalences u1, u2, u3, we use the relative Brown factoriz*
*ation
lemma to construct the diagram
__r2_//
A2 //v2_//B02~ooB2oo_
OO OO w2 OO
f12 h12 g12
OO v1 O0Or1_//OO
A1 //___//B1o~o_B1oo
 w1 
f13 h13 g13
fflffl fflffl0rfflffl3//_
A3 //v3_//B3o~o_B3oo
w3
In this diagram:
(1) v1, w1, r1 are constructed as a Brown factorization of u1 as in Remark
1.3.2
(2) vi, wi, ri for i = 2, 3 are constructed as relative Brown factorization*
*s of
u2 resp. u3 over the Brown factorization v1, w1, r1.
The maps (w1, w2, w3) are trivial cofibrations and (u1, u2, u3) are weak equ*
*iv
alences, so (v1, v2, v3) are trivial cofibrations.
10 1. COFIBRATION CATEGORIES
Statement (b) is true for
 (v1, v2, v3) resp. (w1, w2, w3) because they satisfy property (1.1)
 therefore true for (r1, r2, r3) as a left inverse to (w1, w2, w3)
 therefore true for (u1, u2, u3) as the composition of (v1, v2, v3) and *
*(r1,
r2, r3).
As a corollary we have
Lemma 1.4.2 (Excision).
(1) Let M be a precofibration category. In the diagram below
f
Af__~__//flfflC
i
fflffl
B
suppose that A, C are cofibrant, i is a cofibration and f is a weak equ*
*iva
lence. Then the pushout of f along i is again a weak equivalence.
(2) Let M be a prefibration category. In the diagram below
B
p
f fflfflfflffl
C __~__//A
suppose that A, C are fibrant, p is a fibration and f is a weak equival*
*ence.
Then the pullback of f along p is again a weak equivalence.
Proof. Part (1) is a particular case of the Gluing Lemma (1) for f12= g12= *
*i,
g13= u3 = f, f13= u3 = 1A and u2 = 1B . Part (2) is dual.
It is now not hard to see that in the presence of the rest of the axioms, the
axiom CF3 (2) is equivalent to the Gluing Lemma (1) (b) and to excision. A dual
statement holds for the fibration axioms.
Lemma 1.4.3 (Equivalent formulation of CF3).
(1) If (M, W, Cof) satisfies the axioms CF1CF2, CF3 (1) and CF4, then
the following are equivalent:
(a) It satisfies CF3 (2)
(b) It satisfies the Gluing Lemma (1) (b)
(c) It satisfies the Excision Lemma 1.4.2 (1).
(2) If (M, W, Fib) satisfies the axioms F1F2, F3 (1) and F4, then the fol
lowing are equivalent:
(a) It satisfies F3 (2)
(b) It satisfies the Gluing Lemma (2) (b)
(c) It satisfies the Excision Lemma 1.4.2 (2).
Proof. It suffices to prove (1). We have proved (a) ) (b) as Lemma 1.4.1,
and we have seen (b) ) (c) in the proof of Lemma 1.4.2.
For (a) ( (c), suppose that A, C are cofibrant, that i : A ae B is0a trivial
cofibration and that f : A ! C is a map. Factor f as a cofibration f followed0*
*by0
a weak equivalence r, and using axiom CF3 (1) construct the pushouts g , s of f*
* , r
1.5. CYLINDER AND PATH OBJECTS 11
f0 r 0
Af//___//flfflC~//_fflfflCfflffl
i~ ~i0 j
fflfflfflfflgfflffl00s
B //___//B__~__//D
The maps g0, i0 and j are cofibrations by axiom CF3 (1). Using excision, i0 and*
* s
are weak equivalences, so by the 2 out of 3 axiom CF2 the map j is also a weak
equivalence.
1.5.Cylinder and path objects
We next define cylinder objects in a precofibration category, and show that
cylinder objects exist. Dually, we define and prove existence of path objects i*
*n a
prefibration category.
Definition 1.5.1 (Cylinder and path objects).
(1) Let M be a precofibration category, and A a cofibrant object of M. A
cylinder object for A consists of an object IA and a factorization of t*
*he
codiagonal r : A t A //i0+i1//_IAp~//_A, with i0+ i1 a cofibration and p
a weak equivalence.
(2) Let M be a prefibration category, and A a fibrant object of M. A path
object for A consists of an object AI and a factorization of the diago
(p0,p1)
nal : A _~i_//_AI___////_A x,Awith (p0, p1) a fibration and i a weak
equivalence.
Lemma 1.5.2 (Existence of cylinder and path objects).
(1) Let M be a precofibration category, and A a cofibrant object of M. Then
A admits a (nonfunctorial) cylinder object.
(2) Let M be a prefibration category, and A a fibrant object of M. Then A
admits a (nonfunctorial) path object.
Proof. To prove (1), observe that if A is cofibrant then the sum A t A exis*
*ts
and is cofibrant by Lemma 1.2.1, and we can then use the factorization axiom CF4
to construct a cylinder object A t A //i0+i1//_IAp~//_A. The statement (2) foll*
*ows
from duality.
Observe that for cylinder objects, the inclusion maps i0, i1 : A ! IA are
trivial cofibrations. For path objects, the projection maps p0, p1 : AI ! A a*
*re
trivial fibrations.
Lemma 1.5.3 (Relative cylinder and path objects).
(1) Let M be a precofibration category, and f : A ! B a map with A, B
cofibrant objects. Let IA be a cylinder of A. Then there exists a cylin*
*der
IB and a commutative diagram
A t A //___//IA_~__//A
ftf  If f
fflffl fflffl~fflffl
B t B //___//IB____//B
12 1. COFIBRATION CATEGORIES
with (B t B) tAtA IA//___//IBa cofibration.
(2) Let M be a prefibration category, and f : A ! B a map with A, B fibrant
objects. Let BI be a path object for of B. Then there exists a path obj*
*ect
AI and a commutative diagram
A __~__//AI____////A x A
f  fI fxf
fflfflfflffl~ fflffl
B _____//BI___////_B x B
with AI_____////BI xBxB (A x A)a fibration.
Proof. To prove (1), apply Lemma 1.3.3 to the diagram
A t A //___//IA_~__//A
ftf  f
fflffl r fflffl
B t B _____________//B
Statement (2) is dual to (1).
1.6.Elementary consequences of CF5 and CF6
In the previous sections we have proved a number of elementary lemmas that
are consequences of the precofibration category axioms CF1CF4. In this section,
we will do the same bringing in one by one the cofibration category axioms CF5
and CF6.
A word on the motivation behind the two additional axioms CF5CF6. In
the construction of homotopy colimits indexed by small diagrams in a cofibration
category, it turns out that the role of small direct categories (Def. 6.1.1) is*
* essential,
because an arbitrary small diagram can be approximated by a diagram indexed by
a small direct category (Section 6.5).
For a small direct category D, its degreewise filtration can be used to show*
* that
colimits indexed by D may be constructed using small sums, pushouts and countab*
*le
direct transfinite compositions (at least if the base category is cocomplete). *
*To put
things in perspective, the axiom CF3 is a property of pushouts, the axiom CF5 is
a property of small sums of maps and the axiom CF6 is a property of countable
direct transfinite compositions of maps.
Let us clarify for a moment what we mean in CF6 and F6 by transfinite direct
and inverse compositions of maps.
Definition 1.6.1. Let M be a category, and let k be an ordinal.
(1) A direct ksequence of maps (or a direct sequence of length k)
A0 _a01_//A1a12_//...__//_Ai___//... (i < k)
consists of a collection of objects Ai for i < k and maps ai1i2: Ai1! A*
*i2
for i1 < i2 < k, such that ai2i3ai1i2= ai1i3for all i1 < i2 < i3 < k.
The map A0 ! colimi>GG
fjjjjjjjj"""ffiffi
jjjjj r""" ffiffi
jjjj "0 ffiffi
AO______0____//YO~ff0iffiffi
f OO ffr1iffi
a0~ b0~ffiffi
 ffiffi
A0 //___f1___//_A1
Define0f0 as the pushout of f1. The map b0is a weak equivalence as the pushout *
*of
a , since M is left proper. The map r is a weak equivalence by the two out of t*
*hree
1.7. PROPERNESS 19
axiom, and the map f0 is left proper as the pushout of f1 which is a cofibration
with cofibrant domain (therefore left proper).
For (1) (c), let i : A ! B be a trivial left proper map and let f : A ! C be*
* a
map. In the diagram
f0 0 r
A _____//C__~__//C
i~ i0 j
fflfflfflffl0fflfflr0
B _____//D_____//D
using part (b) we have0factored f as rf0, with f0 left proper and r a weak0equi*
*va
lence. We define i and0j as the pushouts of i. We therefore0have that i and j a*
*re0
left proper. The map i is a weak equivalence since f is left proper. The map r*
* is
a weak equivalence since i is proper. From the two out of three axiom, the map j
is also a weak equivalence, therefore a trivial left proper map.
For (1) (d), the axioms CF1, CF2 and CF3 (1) are trivially verified. Part (c)
proves axiom CF3 (2), and part (b) proves axiom CF4.
It does not appear to be the case that (M, W, PrCof) necessarily satisfies C*
*F5
or CF6 if (M, W, Cof) does. The left proper maps appear to satisfy only the axi*
*oms
CF1CF4 with respect to the given class of weak equivalences W.
In the rest of the section, we will review briefly a number of elementary pr*
*op
erties of proper maps.
Proposition 1.7.5.
(1) Suppose that (M, W, Cof) is a left proper precofibration category. Then*
* a
weak equivalence is a trivial left proper map iff all its pushouts exis*
*t and
are weak equivalences.
(2) Suppose that (M, W, Fib) is a right proper prefibration category. Then a
weak equivalence is a trivial right proper map iff all its pullbacks ex*
*ist and
are weak equivalences.
Proof. We only prove (1).
Implication ) is a consequence of Thm. 1.7.4 (1) (c).
For (, suppose that i : A ! B is a weak equivalence whose pushouts remain
weak equivalences. We'd like to show that i is left proper. For any map f and w*
*eak
equivalence r, we construct the diagram with pushout squares
f r
A _____//C1_~__//_C2
i~ ~ i1 ~ i2
fflffl fflfflrfflffl0
B ____//_D1___//_D2
The maps i1, i2 and0are weak equivalences as pushouts of i. From the 2 out of 3
axiom, the map r is a weak equivalence, which shows that i is left proper.
Proposition 1.7.6.
(1) Suppose that (M, W, Cof) is a cocomplete, left proper precofibration ca*
*te
gory. If f, g are two composable maps such that gf is left proper and g*
* is
trivial left proper. Then f is left proper.
20 1. COFIBRATION CATEGORIES
(2) Suppose that (M, W, Fib) is a complete, right proper prefibration categ*
*ory.
If f, g are two composable maps such that gf is left proper and g is tr*
*ivial
left proper. Then f is left proper.
Proof. We only prove (1). In the diagram
A1 ____//_B1_r~//_C1
f  
fflffl fflfflrfflffl0
A2 ____//_B2___//_C2
g~ g0 g00
fflffl fflfflrfflffl00
A3 ____//_B3___//_C3
r is a weak equivalence0and all squares are pushouts.0To0prove that f is left p*
*roper,
we need to show0that r00is a weak equivalence. But r is a weak equivalence sin*
*ce
gf is proper. g and g are trivial0left proper as pushouts of g. It follows fr*
*om the
two out of three axiom that r is a weak equivalence.
Recall the definition of the retract of a map
Definition 1.7.7. A map f : A ! B in a category M is a retract of g : C ! D
if there exists a commutative diagram
__1A___________________________________________*
*________________________________________________________
_____________''____________________________________*
*_____________________________________________________________
A _____//C_____//A
f g f
fflffl fflfflfflffl
B _____//______77____________________________________*
*______________________________D//_B
_______________________________________________*
*_____________________________________________________________________________*
*_____________
1B
__
Note that the saturation W of the class of weak equivalences W is closed und*
*er
retracts.
Proposition 1.7.8. Suppose that (M, W) is a category with weak equivalences
and that W is closed under retracts.
(1) Assume that M is cocomplete. Then the class of left proper maps and that
of trivial left proper maps are both closed under retracts.
(2) Assume that M is complete. Then the class of right proper maps and that
of trivial right proper maps are both closed under retracts.
Proof. Follows directly from the definitions.
CHAPTER 2
Relation with other axiomatic systems
We would like to describe in this chapter how ABC cofibration categories cat*
*e
gories relate to Brown's categories of cofibrant objects, to Quillen model cate*
*gories
and to other axiomatizations that have been proposed for categories with cofibr*
*a
tions.
Aside from the goal of bringing together and comparing various axiomatizatio*
*ns
that have been proposed for (co)fibrations and weak equivalences, this allows u*
*s to
tap into a large class of examples of ABC model categories.
For example, simplicial sets sSets form a Quillen model category [Qui67 ],
with inclusions as cofibrations, with maps satisfying the Kan extension property
as fibrations and with maps whose geometric realization is a topological homoto*
*py
equivalence as weak equivalences. Any Quillen model category is an ABC model
category, and therefore sSets is an ABC model category. By Thm. 6.5.5, so is
any diagram category sSetsD for a small category D, and by Def. 6.3.8 and Thm.
6.5.6 so are the D2reduced D1diagrams sSets(D1,D2)for a small category pair
(D1, D2).
For considerations of space, we will not talk about categories with a natural
cylinder ([Kam72 ], [Shi89 ] and [KP96 ], which are a natural development of *
*the
ideas introduced in [Kan56 ]), although categories with a natural cylinder are*
* an
important part of the story of ABC cofibration categories.
2.1. Brown's categories of cofibrant objects
In his paper [Bro74 ], Brown defines categories of fibrant objects (and dual*
*ly
categories of cofibrant objectss). We list below Brown's axioms, stated in the
cofibration setting, slightly modified but equivalent to the actual axioms of [*
*Bro74 ].
Definition 2.1.1 (Categories of cofibrant objects). A category of cofibrant
objects (M, W, Cof) consists of a category M and two distinguished classes of
maps W, Cof  the weak equivalences and respectively cofibrations of M, subject
to the axioms below:
CFObj1: All isomophisms of M are trivial cofibrations. M has an initial
object 0, and all objects of M are cofibrant. Cofibrations are stable u*
*nder
composition.
CFObj2: (Two out of three axiom) If f, g are maps of M such that gf is
defined, and if two of f, g, gf are weak equivalences, then so is the t*
*hird.
CFObj3: (Pushout axiom) Given a solid diagram in M, with i a cofibration,
Af_____//flfflCfflffl
i j
fflfflfflffl
B ` ` `//D
21
22 2. RELATION WITH OTHER AXIOMATIC SYSTEMS
then the pushout exists in M and j is a cofibration. If additionally i *
*is a
trivial cofibration, then j is a trivial cofibration.
CFObj4: (Cylinder axiom) For any object A of M, the codiagonal r :
A t A ! A admits a factorization as a cofibration followed by a weak
equivalence.
The axioms for categories of fibrant objects are dual to those of Def. 2.1.1*
*, and
are denoted FObj1FObj4.
The precofibration categories (satisfying the minimal axioms CF1CF4 but not
the additional axioms CF5CF6) are essentialy a modification of Brown's categor*
*ies
of cofibrant objects  in the sense that we allow objects to be noncofibrant. *
*The
following lemma explains the precise relationship between precofibration catego*
*ries
and categories of cofibrant objects.
Proposition 2.1.2.
(1) Any category of cofibrant objects is a precofibration category. Convers*
*ely,
if M is a precofibration category, then Mcof is a category of cofibrant
objects.
(2) Any category of fibrant objects is a prefibration category. Conversely,*
* if
M is a prefibration category, then Mfibis a category of fibrant objects.
Proof. We only prove (1). Implication ( is an easy consequence of the ax
ioms. For the other direction ), the only axiom that needs to be proved is the
factorization axiom CF4.
To prove axiom CF4, we need to prove Brown's factorization lemma 1.3.1 in the
context of the axioms CFObj1CFObj4. We want to show that any map f : A ! B
factors as f = rf0, where f0 is a cofibration and rs = 1B for a trivial cofibra*
*tion s.
Choose a cylinder IA, and construct s as the pushout of the trivial cofibrat*
*ion
i0.
f
Af______//flfflBfflffl
i0~ ~ s
fflfflFfflffl
IA ____//_B0
Notice that i0 has p : IA ! A as a left inverse, and it follows that s has a
left inverse r. Let f0 be F i1, which satisfies f = rf0, and to complete the pr*
*oof it
remains to prove that f0 is a cofibration.
We notice that B0 is also the pushout of the diagram below
ft1
A tfA_____//flfflBftfAlffl
i0+i1 s+f0
fflfflF fflffl
IA _______//_B0
0
so f0 is A //___//B t A//s+f//_B0, therefore a cofibration.
2.2. QUILLEN MODEL CATEGORIES 23
2.2. Quillen model categories
Quillen's model categories involve both cofibrations and fibrations, and come
with builtin EckmanHilton duality between cofibrations and fibrations.
In preparation, recall the definition of the left (and right) lifting proper*
*ty of
maps.
Definition 2.2.1. For a solid commutative diagram in a category M
A _____//C>>"
i" " p
fflfflfflffl"
B _____//D
if a dotted arrow exists making the diagram commutative we say that i has the
LLP (left lifting property) with respect to p, and that p has the RLP (right li*
*fting
property) with respect to i.
Also, recall the definition Def. 1.7.7 of the retract of a map. We will use *
*the
(closed) Quillen model category axiom formulation from Hirschhorn's monograph
[Hir00 ].
Definition 2.2.2 (Quillen model categories). A Quillen model category (M,
W, Cof, Fib) consists of a category M and three distinguished classes of maps W,
Cof, Fib the weak equivalences, the cofibrations and respectively the fibratio*
*ns of
M, subject to the axioms below:
M1: M is complete and cocomplete.
M2: (Two out of three axiom) If f, g are maps of M such that gf is define*
*d,
and if two of f, g, gf are weak equivalences, then so is the third.
M3: (Retract axiom) Weak equivalences, cofibrations and fibrations are
closed under retracts.
M4: (Lifting axiom) Cofibrations have the LLP with respect to trivial fib*
*ra
tions, and trivial cofibrations have the LLP with respect to fibrations.
M5: (Factorization axiom) Any map of M admits a factorization as a cofi
bration followed by a trivial fibration, and a factorization as a trivi*
*al
cofibration followed by a fibration.
The retract axiom M3 states that given a commutative diagram
__1A___________________________________________*
*________________________________________________________
_____________''____________________________________*
*_____________________________________________________________
A _____//C_____//A
f g f
fflffl fflfflfflffl
B _____//______77____________________________________*
*______________________________D//_B
_______________________________________________*
*_____________________________________________________________________________*
*_____________
1B
if g is a weak equivalence (resp. cofibration, resp. fibration) then so is f.
The factorizations in M5 are not assumed to be functorial.
From the axioms M1M5 one can show that in a Quillen model category any
two of the following classes of maps of M  the cofibrations, the trivial cofib*
*rations,
the fibrations and the trivial fibrations  determine each other by the followi*
*ng
rules:
 A map is a cofibration , it has the LLP with respect to all trivial fibrat*
*ions
24 2. RELATION WITH OTHER AXIOMATIC SYSTEMS
 A map is a trivial cofibration , it has the LLP with respect to all fibrat*
*ions
 A map is a fibration , it has the RLP with respect to all trivial cofibrat*
*ions
 A map is a trivial fibration , it has the RLP with respect to all cofibrat*
*ions
Proposition 2.2.3. Any Quillen model category is an ABC model category.
Proof. A Quillen model category trivially satisfies the axioms CF2 and CF4.
The axioms CF1, CF3, CF5 and CF6 are satisfied since a map is a cofibration
(resp. trivial cofibration) iff it has the LLP with respect to all trivial fib*
*rations
(resp. fibrations). A dual argument shows that a Quillen model category satisfi*
*es
the axioms F1F6.
If M is a Quillen model category and D is a small category, then the category
of diagrams MD does not generally form a Quillen model category (except in im
portant particular cases, for example when M is cofibrantly generated or when D
is a Reedy category). But we will see further down (Thm. 6.5.5) that MD carries
an ABC model category structure, and in that sense one can always 'do homotopy
theory' on MD .
A Quillen model category M is called left proper if for any pushout diagram
j
Af_____//flfflCfflfflO
i OO
fflfflfflfflj0
B ` ` `//D
with i a cofibration and j a weak equivalence, the map j0 is a weak equivalence*
*. M
is called right proper if for any pullback diagram
q0
DO` ` `//B
O p
fflfflfflfflOfflfflfflfflq
C _____//A
with p a fibration and q a weak equivalence, the map q0 is a weak equivalence. M
is called proper if it is left and right proper.
From this definition and from Prop. 2.2.3 we immediately get
Proposition 2.2.4. Any proper Quillen model category is a proper ABC model
category.
2.3.Baues cofibration categories
We next turn our attention to the notion of (co)fibration category as defined
by Baues. We state below the axioms of a Baues cofibration category, in a sligh*
*tly
modified but equivalent form to [Bau88 ], Sec. 1.1.
Definition 2.3.1. A Baues cofibration category (M, W, Cof) consists of a
category M and two distingushed classes of maps W and Cof the weak equivalences
and the cofibrations of M  subject to the axioms below:
BCF1: All isomophisms of M are trivial cofibrations. Cofibrations are st*
*able
under composition.
BCF2: (Two out of three axiom) If f, g are maps of M such that gf is
defined, and if two of f, g, gf are weak equivalences, then so is the t*
*hird.
2.4. WALDHAUSEN CATEGORIES 25
BCF3: (Pushout and excision axiom) Given a solid diagram in M, with i a
cofibration,
f
Af_____//flfflCfflfflO
i OjO
fflfflfflfflg
B ` ` `//D
then the pushout exists in M and j is a cofibration. Moreover:
(a) If i is a trivial cofibration, then j is a trivial cofibration
(b) If f is a weak equivalence, then g is a weak equivalence.
BCF4: (Factorization axiom) Any map of M admits a factorization as a
cofibration followed by a weak equivalence.
BCF6: (Axiom on fibrant models) For each object A of M there is a triv
ial cofibration A ! B, with B satisfying the property that each trivial
cofibration C ! B admits a left inverse.
A Baues cofibration category may not have an initial object, but if it does *
*then
by BCF1 the initial object is cofibrant. We will not state the axioms for a Bau*
*es
fibration category  they are dual to the above axioms.
Proposition 2.3.2 (Relation with Baues cofibration categories).
(1) Any Baues cofibration category with an initial object is a left proper *
*pre
cofibration category.
(2) Any Baues fibration category with a terminal object is a right proper p*
*re
fibration category.
Proof. Easy consequence of the axioms and of Lemma 1.4.3.
2.4. Waldhausen categories
We list below the axioms we'll use for a Waldhausen cofibration category. Th*
*ese
axioms are equivalent to the axioms Cof1Cof3, Weq1 and Weq2 of [Wal85 ]. Ax
ioms for a Waldhausen fibration category will of couse be dual to the axioms be*
*low.
Definition 2.4.1. A Waldhausen cofibration category (M, W, Cof) consists
of a category M and two distinguished classes of maps W and Cof  the weak
equivalences and the cofibrations of M, subject to the axioms below:
WCF1: M is pointed. All isomophisms of M are trivial cofibrations. All
objects of M are cofibrant. Cofibrations are stable under composition.
WCF2: (Pushout axiom) Given a solid diagram in M, with i a cofibration,
Af_____//flfflCfflfflO
i OjO
fflfflfflffl
B ` ` `//D
then the pushout exists in M and j is a cofibration
WCF3: (Gluing axiom) In the diagram below
26 2. RELATION WITH OTHER AXIOMATIC SYSTEMS
f12
A1//___________//BA2B
 BBBB  BBBB
 f BBB  BBB
 13 __  __
u1 A3 //___________//A4
   
   
 u3 u2 
fflffl  fflffl 
B1 //__________//B2 u4
BB  g12 BB 
BBB  BBB 
g13BB  BB 
B_fflffl_ B__fflffl
B3 //___________//B4
if f12, g12 are cofibrations, u1, u2, u3 are weak equivalences and the *
*top
and bottom faces are pushouts, then u4 is a weak equivalence.
We have the following
Proposition 2.4.2 (Relation with Waldhausen cofibration categories).
(1) (a) If M is a pointed precofibration category, then Mcofis a Waldhausen
cofibration category.
(b) If M is a Waldhausen cofibration category satisfying the 2 out of 3
axiom CF2 and the cylinder axiom CFObj4, then it is a precofibration
category.
(2) (a) If M is a pointed prefibration category, then Mfibis a Waldhausen
fibration category.
(b) If M is a Waldhausen fibration category satisfying the 2 out of 3
axiom F2 and the path object axiom FObj4, then it is an prefibration
category.
Proof. Part (1) (a) follows from the Gluing Lemma 1.4.1, part (1) (b) from
Lemma 1.4.3 and Prop. 2.1.2, and part (2) is dual to the above.
CHAPTER 3
The homotopy category of a cofibration category
Our goal in this chapter is to describe the homotopy category of a cofibrati*
*on
category. All the definition and the results of this chapter actually only requ*
*ire the
smaller set of precofibration category axioms CF1CF4.
After a small discussion about categories and universes, we will recall the *
*defi
nition of the homotopy category hoM of a category M with a class of weak equiv
alences W. The homotopy category hoM is defined by a universal property, but
admits a description in terms of generators and relations, starting with the ob*
*jects
and maps of M and formally adding inverses of the maps in W.
For a precofibration category (M, W, Cof), recall that Mcofdenotes the full
subcategory of cofibrant objects of M. In Mcofwe define the homotopy relation
' on maps, and show that it is an equivalence relation. The localization of Mcof
modulo homotopy is denoted ssMcof. We show that the class of weak equivalences
between cofibrant objects admits a calculus of fractions in ssMcof. As a conseq*
*uence
we obtain a description of hoMcofwhereby any map in hoMcofcan be written (up
to homotopy!) as a 'left fraction' ft1, with t a weak equivalence. The theory *
*of
homotopic maps and calculus of fractions up to homotopy as described here is due
to Brown [Bro74 ].
We then show that the functor hoMcof! hoM is an equivalence of categories
(Anderson, [And78 ]). In fact,0we will develop an axiomatic description of cof*
*ibrant
approximation functors t : M ! M, which are modelled on the properties of the
inclusion Mcof! M. A cofibrant approximation functor induces an isomorphism
at the level of the homotopy category.
Cofibrant aproximation functors will resurface later in Section 6.5, when we*
* will
reduce the construction of homotopy colimits indexed by arbitrary small diagrams
to the construction of homotopy colimits indexed by small direct categories.
We will show that the homotopy relation extends to maps of M, by mandating
that f ' g if and only if f, g become equal in the homotopy category hoM. We pr*
*ove
that the new equivalence relation is backward compatible to homotopy relation on
maps of Mcof.
As an application of the theory developed in this chapter, we show that if Mk
is a small set of precofibration categories, then the functor ho(x Mk) ! xho Mk*
* is
an isomorphism of categories.
__ We will also prove for a cofibration category (M, W, Cof) that the_saturation
W of the weak equivalences yields again a cofibration category (M, W, Cof). This
is a result due to Cisinski, [Cis02 ].
27
28 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
3.1.Universes and smallness
If U U+ are two universes, a (U+ , U)category C has by definition a U+ 
small set of objects ObC and Usmall Homsets. A category C is Usmall if it is*
* a
(U, U)category. A functor f : C1 ! C2 is Usmall if C1, C2 are Usmall.
For a fixed universe pair U U+ , the U+ small sets are also referred to as
classes, while the Usmall sets are referred to simply as sets, or small sets. *
* We
will denote Cat(resp. CAT ) the category of Usmall categories and functors (re*
*sp.
(U+ , U) categories and functors).
3.2.The homotopy category
Suppose we have a (U+ , U)category M with a class of weak equivalences W.
The homotopy category hoM by definition is a U+ small category which comes
with a localization functor flM : M ! hoM, with the properties that
(1) flM sends weak equivalences to isomorphisms00
(2) for any other U+ small functor fl : M ! M that sends weak equivalence*
*s0
to isomorphisms, there0exists a unique U+ small functor ffi : hoM ! M
such that ffiflM = fl
The homotopy category is sometimes also denoted M[W1]. A theorem of
GabrielZisman [GZ67 ] states that the homotopy category exists, and is isomor*
*phic
to the category with
(1) objects ObM
(2) maps between X and Y the equivalence classes of zigzags
f1 f2 fn
X = A0 _____ A1 _____...____An = Y
where fi are maps in M going either forward or backward, and the maps
going backward are in W, where
(3) two zigzags are equivalent if they can be obtained one from another by*
* a
finite number of the following three operations and their inverses:
(a) skipping elements A 1A!A or A 1AA
(b) replacing A f!B g!C with A gf!C or A f B g C with A fgC
(c) skipping elements A f!B f A or A f B f!A
(4) composition of maps is induced by the concatenation of zigzags
__
The saturation W of the class of weak equivalences by definition_is the clas*
*s of
maps of M that become isomorphisms in hoM. The saturation W is closed under
composition, and includes the isomorphisms of M.
If M is a (U+ , U)category with a class of weak equivalences W, then hoM =
M[W1] is not necessarily a (U+ , U)small category. For example, if M is the
(U+ , U)category of sets Set, then W = Cof = Fib = Setgives M a structure of
both a cofibration and a fibration category, but Set[Set1] is only U+ small.
If a (U+ , U)category M admits a Quillen model category structure on the ot*
*her
hand, it is a known result that hoM is again a (U+ , U)category.
One benefit of the intrinsic description of the homotopy category in terms of
zigzags is that it shows that the homotopy category remains the same independe*
*nt
of the universe pair (U+ , U) that we start with.
3.3. HOMOTOPIC MAPS AND ssM 29
For the rest of the text, it is convenient to restrict ourselves to the case*
* when W
is closed under composition and includes the identity maps of M. In the language
of Def. 1.7.2, these are the categories with weak equivalences (M, W).
3.3. Homotopic maps and ssM
From this point on, we will restrict our focus to precofibration categories *
*and
(dually) to prefibration categories. Let M be a precofibration category, and l*
*et
f, g : A ' B be two maps with A, B cofibrant. A homotopy from f to g is a
commutative diagram
f+g
A tfA_____//flfflBfflffl
i0+i1 ~ b
fflfflH fflffl0
IA ______//B
with IA a cylinder of A and b a trivial cofibration. We thus have that Hi0 = bf
and Hi1 = bg. The map H is called the homtopy map between f and g, and we say
that the homotopy goes through the cylinder IA and through the trivial cofibrat*
*ion
b . We write f ' g to say that0f, g are homotopic.
A homotopy f ' g with B = B and b = 1B is called a strict homotopy. We
should be careful to point out that Brown uses the notation ' differently  to *
*denote
what we call strict homotopy.
We have defined the homotopy relation in terms of the weak equivalences and
cofibrations of M, but it turns out that the homotopy relation depends only on *
*the
weak equivalences. In fact, we will show that f ' g if and only if the maps f, g
become equal in hoM.
In view of this, we will make an abuse of language and use the same notation
' to denote homotopy in prefibration categories. Let M be a prefibration catego*
*ry,
and let f, g : B ' A be two maps with A, B fibrant. A homotopy from f to g is a
commutative diagram
___H__//_I
B0 A
b ~ (p0,p1)
fflfflfflfflfflfflfflffl(f, g)
B _____//A x A
with AI a path object of A and b a trivial fibration. If f, g are homotopic0we
therefore also write f ' g. A strict homotopy is a homotopy with B = B and
b = 1B .
Let A be a cofibrant object in the precofibration0category M. If IA is a cyl*
*inder
of0A, a refinement of IA consists of a cylinder I A and a trivial cofibration j*
* : IA !
I A such that the diagram below commutes
AtA##G0G 0
i0+i1xxxx GGi0+i1G
xx GGGG
xxx j ## 0
IA FF//____________//_I A
FF~F ~wwww
p FFF www0
F## wpw
A
30 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
If f ' g : A ! B through I0A with homotopy map H0, then H0j defines a
homotopy through IA. 0
Note0that0given any two cylinders0IA, I A, we can construct a0common0refine0
ment I A by factoring IA tAtA I A0!0A as a cofibration IA tAtA I A ! I A
followed by a weak equivalence I A ! A.
This allows us to prove the following lemma:
Lemma 3.3.1.
(1) Let M be a precofibration category, and f ' g : A ! B with A, B cofibra*
*nt.
Let IA be a cylinder of A. Then one can construct a homotopy f ' g
through IA.
(2) Let M be a prefibration category, and f ' g : B ! A with A, B fibrant.
Let AI be a path object for A. Then one can construct a homotopy f ' g
through AI.
Proof. We only prove (1).0 Assume that there exists a homotopy0f0 ' g
through0another cylinder I A. Construct a common refinement I A of IA and *
*00
I A. To prove (1), it suffices to construct a homotopy through the refinement I*
* A.
In the commutative diagram below
f+g
A tfAf_____//_lfflBfflffl
i00+i01 ~ b
0fflfflH0 fflffl0
I Af______//flfflBfflffl
j~ ~ b0
0fflffl0H0fflffl000
I A ______//B
H0,0B0 and0b0define the homotopy0f0' g, and j is the cylinder0refinement map fr*
*om
I A to I A.0We0construct0B00as the0pushout0of j and H , and we have the desired
homotopy H , B and b b through I A.
We can now prove
Theorem 3.3.2.
(1) If M is a precofibration category, then ' is an equivalence relation in
Mcof. Furthermore if f ' g : A ! B with A, B cofibrant then
(a) If h : B ! C with C cofibrant then hf ' hg
(b) If h : C ! A with C cofibrant then fh ' gh
(2) If M is a prefibration category, then ' is an equivalence relation in M*
*fib.
Furthermore if f ' g : A ! B with A, B fibrant then
(a) If h : B ! C with C fibrant then hf ' hg
(b) If h : C ! A with C fibrant then fh ' gh
Proof. We only prove (1). Clearly ' is symmetric and reflexive.
To see that ' is transitive, assume f ' g ' h : A ! B. From the previous
lemma, we may assume that both0homotopies go0through the same cylinder IA.
Denote these homotopies H1,0B1,0b1 and H2,0B2,0b2. Taking the pushout of b1 and
b2, we obtain homotopies H1, B , b and H2, B , b.
Notice that both diagrams below are pushouts
3.4. HOMOTOPY CALCULUS OF FRACTIONS 31
i0 At(i0+i1)
Af//__~___//_flfflIAfflffl A t AftfAl//__________//fflAftfIAlffl
i1~ ~ (i0+i1)tA i0tIA
fflffl~ fflffl fflffl IAti1 fflffl
IA //___//IA tA IA IA t A //__________//IA tA IA
The factorization r : A t A //i0ti1//_IA tApIA+p~//_Ais a cylinder: the map p +*
* p
is a weak equivalence because of the first diagram, and the map i0ti1 is a cofi*
*bration
because it is the second diagram precomposed with the cofibration (i0, i2) : At*
*A !
A t A t A. The commutative diagram below then defines a homotopy from f to h.
f+h
A tfAf______//_lfflBfflffl
i0ti1 ~b
fflfflH01+fflfflH02
IA tA IA _____//B0
To0prove (a), let IA, H, B0, b define a homotopy f ' g. In the diagram below
let C be the pushout of b, h.
f+g h
A tfA_____//flfflB//_fflfflCfflffl
i0ti1 ~ b ~ c
fflfflH fflffl0fflfflh00
IA ______//B_____//C
The outer rectangle defines a homotopy hf ' hg.
For (b), use Lemma 1.5.3 to construct relative cylinders IC, IA along h. From
Lemma 3.3.1, we can construct a homotopy f ' g through IA. Precomposing this
homotopy with IC ! IA yields a homotopy fh ' gh.
As a consequence of the previous theorem, for a precofibration category we c*
*an
factor Mcofmodulo ' and obtain a category ssMcof, with same objects as Mcof
and with HomssMcof(A, B) = HomMcof(A, B)='. For weak equivalences in ssMcof
we consider homotopy classes of maps that have one (and hence all) representati*
*ves
weak equivalece maps of M.
For a prefibration category, ssMfibdenotes the factorization of Mfibmodulo '.
Weak equivalences in ssMfibare by definition homotopy classes of maps that have
one (and hence all) representatives weak equivalence maps of M.
3.4. Homotopy calculus of fractions
We will show that for a precofibration category M, the category ssMcofadmits
a calculus of left fractions in the sense of GabrielZisman with respect to weak
equivalences. A nice way to phrase this is to say that Mcof admits a homotopy
calculus of left fractions. Dually, given a prefibration category M, its catego*
*ry of
fibrant objects Mfibadmits a homotopy calculus of right fractions.
We say that a category with weak equivalences (M, W) satisfies the 2 out of 3
axiom provided that for any composable morphisms f, g of M, if two of f, g, gf *
*are
in W then so is the third. Weak equivalences in a precofibration category M sat*
*isfy
the 2 out of 3 axiom, and so do weak equivalences in ssMcof.
32 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
For categories with weak equivalences (M, W) satisfying the 2 out of 3 axiom,
the GabrielZisman calculus of fractions takes a simplified form that we recall*
* below.
The general case of calculus of fractions  when weak equivalences do not neces*
*sarily
satisfy the 2 out of 3 axiom  is described in [GZ67 ] at pag. 12.
Theorem 3.4.1 (Simplified calculus of left fractions). Let (M, W) be a categ*
*ory
with weak equivalences satisfying the 2 out of 3 axiom. If any full diagram wit*
*h weak
equivalence map a
(3.1) A _____//_BO
a ~ ~ bOO
fflffl0fflffl0
A ` ` `//B
extends to a commutative diagram with b a weak equivalence, then
(1) Each map in HomhoM (A, B) can be written as a left fraction s1f
f 0 s
A _____//Boo~__B
with s a weak equivalence.
(2) Two fractions s1f,0t1g0are equal in hoM if and only if there exist we*
*ak
equivalences s , t as in the diagram below
(3.2) B000>>aaC
s0____ CCt0CC
___~_ ~ CCC
>B0>jjV~VVVVVgiii4B004i``A
f"""" iiiiiiiiVVVVVVVAAtAA
"""iiiiiiii VVVVVV~ AAA
iiii s VVVVV
A B
so that s0s = t0t and s0f = t0g.
If furthermore weak equivalences are left cancellable,0in the sense that for an*
*y pair
of maps f, g : A ' B and weak equivalence h : B ! B with hf = hg we have
f = g, then
(3) Two maps f, g : A ' B are equal in hoM if and only if f = g.
The dual result for right fractions is
Theorem 3.4.2 (Simplified calculus of right fractions). Let (M, W) be a cat
egory with weak equivalences satisfying the 2 out of 3 axiom. If any full diagr*
*am
with weak equivalence map a
(3.3) B0O` ` `//A0
b ~O ~a
fflfflOfflffl
B _____//_A
extends to a commutative diagram with b a weak equivalence, then
(1) Each map in HomhoM (A, B) can be written as a right fraction fs1
f
A oos~_A0 _____//B
with s a weak equivalence.
3.4. HOMOTOPY CALCULUS OF FRACTIONS 33
(2) Two fractions fs1,0gt10are equal in hoMfibif and only if there exist
weak equivalences s , t as in the diagram below
(3.4) A000B
s0____ BBt0BB
""_~___ ~ BB!!B
A0 UUUUUU itiiiiA00A
s""" iiiUUUUUUUiiii AAgA
""~ iiiii~i UUUUUU AA
""""ttiiiiii f UUUU**U__A
A B
so that ss0= tt0 and fs0= gt0.
If furthermore weak equivalences are right cancellable,0in the sense that for a*
*ny
pair of maps f, g : A ' B and weak equivalence h : A ! A with fh = gh we have
f = g, then
(3) Two maps f, g : A ' B are equal in hoM if and only if f = g.
The two statements being dual, we only need to supply a proof for the first
Thm. 3.4.1.
Proof. We construct a category C with objects ObM, and maps defined in
terms of fractions as explained below.
Fix two objects A and B. Consider the set of fractions s1f as in (1). Denote
~ the relation defined by (2) on the set of fractions s1f from A to B. The rel*
*ation
~ is clearly reflexive and symmetric.
To see that the relation is transitive, assume s11f1 ~ s21f2 ~ s31f3. We
get a commutative diagram
u1 000u2 00
B001___//_BOoo___B2O``A
AA t ">>"OO
t1 At2AAA 3""""t4
 A 0 "" 0
B01OOhhQB2==Q 6B36m
QQQQQmmmmmaaOOCCC
f1f2mmmmmms2Cs3CCCQQQQQ
mmf3mmm s1QQCQQ
A B
where the weak equivalences t1, t2 exist since s11f1 ~ s21f2, the weak equiv
alences t3, t4 exist since s21f2 ~ s31f3 and the weak equivalences u1, u2 exi*
*st
from the hypothesis (3.1) applied to t2, t3. The compositions u1t1, u2t4 satis*
*fy
u1t1f1 = u2t4f3 and u1t1s1 = u2t4s3 which shows that s11f1 ~ s31f3, and we
have proved that ~ is transitive.
We let HomC(A, B) denote the set of fractions from A to B modulo the equiva
lence relation ~. Given three objects A, B, C, we define composition HomC(A, B)x
HomC(B, C) ! HomC(A, C) as follows. Given fractions s1f, t1g
34 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
C00>>``A
g0"""" AAt0A
""" ~ AAA
" A 0
B0>>aaB =C=``@
f"""" BBsBB ___ @@t@
""" ~ BBB __g__ ~ @@@
_
A B C
we0use hypothesis0(3.1)0to construct an object C00, a map0g0and a0weak equivale*
*nce
t such that g s = t g, and then we define t1gOs1f as (t t)1(g f). The proof *
*that
the definition of composition does not depend on the choices involved is entire*
*ly
similar to the proof that ~ is transitive (we leave this verification to the re*
*ader).
Given an object A, the fraction (1A )11A is an identity element for the compos*
*ition.
Define a functor F : M ! C, by F (A) = A and by sending f : A ! B to
the fraction F (f) = (1B )1f. It is not hard to see that F is compatible with
composition, and that if s : A ! B is a weak equivalence in M then F (s) has
s11B as an inverse.
__ Since F sends weak equivalences to isomorphisms, it descends to a functor
F0: hoM ! C, and it is straightforward_to check that any other functor hoM !
C factors uniquely through F . It follows that the category C we constructed *
*is
equivalent to hoM, and the proof of (1) and (2) is complete.
If f, g0: A ' B are equal in hoM, then by (2) there exists a weak equivalence
h : B ! B such that hf = hg. If weak equivalences are left cancellable, then
f = g, and this proves (3).
The next result shows that given a precofibration category M, the category
ssMcofsatisfies the hypotheses of Thm. 3.4.1 up to homotopy.
Theorem 3.4.3.
(1) In a precofibration category
(a) Any full diagram with cofibrant objects and weak equivalence a
A _____//_BO
a ~ ~ bOO
fflffl0fflffl0
A ` ` `//B
extends to a (strictly)0homotopy commutative diagram with b a weak
equivalence and B cofibrant
(b) For any f, g : A ' B with A, B cofibrant 0 0
(i)If there is a weak equivalence a : A ! A with A cofibrant such
that fa ' ga, then f ' g 0 0
(ii)If there is a weak equivalence b : B ! B with B cofibrant su*
*ch
that bf ' bg, then f ' g
(2) In a prefibration category
(a) Any full diagram with fibrant objects and weak equivalence a
B0O` ` `//A0
bO~ ~a
fflfflO fflffl
B _____//_A
3.4. HOMOTOPY CALCULUS OF FRACTIONS 35
extends to a (strictly)0homotopy commutative diagram with b a weak
equivalence and B fibrant
(b) For any f, g : A ' B with A, B fibrant 0 0
(i)If there is a weak equivalence a : A ! A with A fibrant such
that fa ' ga, then f ' g 0 0
(ii)If there is a weak equivalence b : B ! B with B fibrant such
that bf ' bg, then f ' g
Proof. To prove (1) (a), denote f : A ! B and let IA be a cylinder of A.
The diagram
f
A __________//_B
a~ b
fflffl0 0 fflffl
A _____//A tA IA tA B
is strictly homotopy commutative. It remains to show that b is a weak equivalen*
*ce.
Denote F : IA ! IA tA B. The diagram
1tf
A tfA______//flfflAftfBlffl
i0+i1 ssIAtABO(i0t1B)
fflfflF fflffl
IA ______//IA tA B
is a pushout, so the right vertical map is a cofibration and therefore F i0 : A*
* !
IA tA B is a cofibration. 0
The map b factors as B ! IA tA B ! A tA IA tA B. The first factor
is a pushout of i1 : A ! IA, therefore a trivial cofibration. The second factor
is a pushout of the weak equivalence a by the cofibration F i0, therefore a weak
equivalence by excision. 0
To prove (1) (b) (i), pick (Lemma 1.5.3) relative cylinders0IA , IA over a. *
*By
Lemma 3.3.1, there0exists0a homotopy fa ' ga through IA and through a trivial
cofibration b . We get a commutative diagram
___ata___//_ __f+g_____//_
A0tfA0flffl~ A tfAflffl Bfflffl
i00+i01 i0+i1 ~ b00
fflffl0h1 0 fflffl h2 fflffl00
IA ___~__//_IA tA0tA0AftfA____//_lfflBfflffl
~j ~ b0
fflfflH fflffl0
IA ___________//_B
where the map j is a cofibration because IA0, IA are relative cylinders. But j *
*is
actualy a0trivial cofibration. To see that, notice that since a is a weak equiv*
*alence
and A, A are0cofibrant, a t a is also a weak equivalence and by excision so is*
* h1.
The map IA ! IA is a weak equivalence since a is, and by the 2 out of 3 Axiom
the map j is a weak equivalence.
36 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
We define B0 as the pushout0of0j0by h2.0 The map b0 is therefore a trivial
cofibration. We let b = b b , and IA, H, B , 1B0 defines a homotopy bf ' bg.
Let us now prove (1) (b) (ii). Pick a homotopy0bf ' bg going through the
cylinder IA and through the trivial cofibration b . In the diagram below
f+g b 0
A tfAf_____//_lfflB_~______//_fflfflBfflffl
i0+i1 b1 ~ b0
fflfflh1fflfflh2 h3 fflffl00
IA ______//B1//__//B2_~__//B
construct b1 as the pushout of i0+i1 and h2, h3 as the factorization of B1 ! B0*
*0as a
cofibration followed by a weak equivalence. Notice that h2b1 is a trivial cofib*
*ration,
and we have constructed a homotopy f ' g with homotopy map h2h1.
The proof of (2) is dual and is omitted.
The GabrielZisman left calculus of fractions applies therefore to the case *
*of
ssMcof, if M is a precofibration category.
Theorem 3.4.4.
(1) Let M be a precofibration category, and A, B be two cofibrant objects.
(a) Each map in HomhoMcof(A, B) can be written as a left fraction s1f
f 0 s
A _____//Boo~__B
with s a weak equivalence and B0 cofibrant.
(b) Two fractions s1f, t1g0are0equal in hoMcofif and only if there0ex*
*00
ist weak equivalences0s0,0t as0in the diagram (3.2) with B cofibr*
*ant
so that s s ' t t and s f ' t g.
(c) Two maps f, g : A ' B are equal in hoMcofif and only if they are
homotopic.
(2) Let M be a prefibration category, and A, B be two fibrant objects.
(a) Each map in HomhoMfib(A, B) can be written as a right fraction
fs1
f
A oos~_A0 _____//B
with s a weak equivalence and A0 fibrant.
(b) Two fractions fs1, gt1 are0equal0in hoMfibif and only if there000
exist weak0equivalences0s0, t0 as in the diagram (3.4) with A fibr*
*ant
so that ss ' tt and fs ' gt .
(c) Two maps f, g : A ' B are equal in hoMfibif and only if they are
homotopic.
Proof. This is a consequence of Thm. 3.4.1 and Thm. 3.4.3.
We can also prove a version Thm. 3.4.4 that describes hoMcof in terms of
fractions s1f with f a cofibration and s a trivial cofibration:
Theorem 3.4.5.
(1) Let M be a precofibration category, and A, B be two cofibrant objects.
(a) Each map in HomhoMcof(A, B) can be written as a left fraction s1f
3.4. HOMOTOPY CALCULUS OF FRACTIONS 37
f 0 s
A //___//Boo~__Boo
with f a cofibration and s a trivial cofibration
(b) Two such fractions s1f, t1g are equal0in0hoMcof if and only if
there0exist0trivial0cofibrations0s , t as in the diagram (3.2) suc*
*h that
s s ' t t and s f ' t g.
(2) Let M be a prefibration category, and A, B be two fibrant objects.
(a) Each map in HomhoMfib(A, B) can be written as a right fraction
fs1
f
A oos~oA0o____////_B
with f a fibration and s a trivial fibration
(b) Two fractions fs1, gt1 are0equal0in hoMfibif and only if there *
*0 0
exist trivial0fibrations0s , t as in the diagram (3.4) such that ss*
* ' tt
and fs ' gt .
Proof. We only prove (1). Denote ~ the equivalence relation defined by Thm.
3.4.4 (1) (b), and ~cofthe equivalence relation defined by the current Theorem's
(1) (b). It suffices to show that:
(1) Any fraction s1f with s a weak equivalence is ~ equivalent to a fracti*
*on
t1g, with g a cofibration and t a trivial cofibration
(2) If two fractions s1f, t1g with f, g cofibrations and s, t trivial cof*
*ibrations
are ~ equivalent, then they are ~cofequivalent.
To prove (1), construct the commutative diagram
B0<>aaCC
s0____ CCt0C
___~_ ~ CCC
>B0>jjV~VVVVVgiii4B004i``A
f"""" iiiiiiiiVVVVVVVAAtAA
"""iiiiiiii VVVVVV~ AAA
iiii s VVVVV
A B
construct s0, t0 as the pushouts0of0t, s. We therefore have s0s = t0t, and ther*
*efore
by Thm. 3.4.4 we get that s f and t g are equal in hoMcof. By Thm. 3.4.4 (1)
38 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
(c), we therefore get that s0f ' t0g, which means that the fractions s1f, t1g*
* are
~cofequivalent.
We have proved that in a precofibration category M, two maps between cofi
brant objects f, g : A ' B are homotopic if and only if they become equal in
hoMcof. But more is true  in fact they are homotopic if and only if they be
come equal in hoM. This will become apparent once we prove further down that
hoMcof! hoM is an equivalence of categories.
3.5. Fibrant and cofibrant approximations
We are interested in a class of precofibration category functors which induce
isomorphisms when passed to the homotopy category, and which serve as resolutio*
*ns
for computing left derived functors. These are the cofibrant approximation func*
*tors,
defined below. The cofibrant approximation functors should be thought of as an
axiomatization of the inclusion Mcof! M, where M is a precofibration category.
Definition 3.5.1. (Cofibrant approximation0functors) Let M be a precofi0
bration category. A functor t : M ! M is a cofibrant approximation if M is a
precofibration category with all objects cofibrant and
CFA1: t preserves the0initial object and cofibrations
CFA2: A map f of M is a weak equivalence if and only if tf is a weak
equivalence 0
CFA3: If A ! B, A ! C are cofibrations of M then the natural map
tB ttA tC ! t(B tA C) is an isomorphism 0 0
CFA4:0 Any map f : tA ! B factors as f = r O tf with f a cofibration of
M and r a weak equivalence of M.
A cofibrant approximation functor in particular sends any object to a cofibr*
*ant
object, and sends trivial cofibrations to trivial cofibrations. If M is a preco*
*fibration
category, then the inclusion Mcof! M is a cofibrant approximation.
The dual definition for prefibration categories is
Definition 3.5.2. (Fibrant0approximation functors) Let M be0a prefibration
category. A functor t : M ! M is a fibrant approximation if M is a prefibrati*
*on
category with all objects fibrant and
FA1: t preserves the0final object and fibrations
FA2: A map f of M is a weak equivalence if and only if tf is a weak
equivalence 0
FA3: If B ! A, C ! A are fibrations of M then the natural map t(B xA
C) ! tB xtA tC is an isomorphism 0 0 0
FA4: Any map f : A ! tB factors as f = tf O s with f a fibration of M
and s a weak equivalence of M.
To investigate properties of cofibrant and fibrant approximations we need the
notion of cofibrant (resp. fibrant) splitting. The notion is due to Cisinski, a*
*nd the
term is a translation of the french clivage.
Definition 3.5.3. Let t : M0! M be a cofibrant approximation of a precofi
bration category. A cofibrant splitting along t consists of the following data:
(1) For any object A of M, an object C(A) of M0 and a weak equivalence
pA : tC(A) ! A
3.5. FIBRANT AND COFIBRANT APPROXIMATIONS 39
(2) For any map f : A ! B, a commutative diagram
tC(A)__pA~//_fflfflA

ti(f) 
fflffl 
tD(f) f
OODDD 
tj(f)~ Doe(f)D
 DDD 
OO pB D"fflffl"
tC(B) _~___//B
with i(f) a cofibration and j(f) a trivial cofibration of M0(and theref*
*ore
oe(f) a weak equivalence in M).
The cofibrant splitting along t is normalized if for any object A of M we have
D(1A ) = C(A), i(1A ) = j(1A ) = 1C(A) and oe(1A ) = pA .
A cofibrant splitting along Mcof! M is simply called a cofibrant splitting of
M.
The dual definition for prefibration categories is
Definition 3.5.4. Let t : M0! M be a fibrant approximation of a prefibration
category. A fibrant splitting along t consists of the following data:
(1) For any object A of M, an object R(A) of M0 and a weak equivalence
iA : A ! tR(A)
(2) For any map f : A ! B, a commutative diagram
iA
A __~__//DtR(A)OO
 DDD OO
 DDD~ tq(f)
oe(f)D""D
f  tS(f)

 tp(f)
 
fflffliBfflfflfflffl
B __~__//tR(B)
with p(f) a fibration and q(f) a trivial fibration of M0(and therefore *
*oe(f)
a weak equivalence).
The fibrant splitting along t is normalized if for any object A of M we have S(*
*1A ) =
R(A), p(1A ) = q(1A ) = 1R(A) and oe(1A ) = iA .
A fibrant splitting along Mfib! M is just called a fibrant splitting of M.
Lemma 3.5.5.
(1) Any cofibrant approximation admits a normalized cofibrant splitting.
(2) Any fibrant approximation admits a normalized fibrant splitting.
Proof. We only prove (1). Let us construct0a normalized cofibrant splitting
along a cofibrant approximation t : M ! M.
For an object A of0M, we use the axiom CFA4 applied to t 0! A to construct
the object C(A) of M and the weak equivalence pA : tC(A) ! A.
For a map f : A ! B of M, if A = B and f = 1A then we just define
D(1A ) = C(A), i(1A ) = j(1A ) = 1C(A) and oe(1A ) = pA .
If f : A ! B is not an identity, we consider the map
40 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
fpA + pB : tC(A) t tC(B) ! B
We can use CFA3 and CFA4 to factor the map fpA + pB into
ti(f)+tj(f) oe(f)
tC(A) t tC(B)//_________//tD(f)_~__//B
where i(f) + j(f) is a cofibration in M0 and oe(f)0is a weak equivalence in M. *
*In
particular i(f) and j(f) are cofibrations of M . But oe(f) O tj(f) = pB and fro*
*m the
2 out of 3 axiom tj(f) and therefore j(f) are weak equivalences. The constructi*
*on
of the normalized splitting is complete.
Lemma 3.5.6.
(1) Any precofibration category M admits a normalized cofibrant splitting a*
*long
Mcof! M such that for A cofibrant we have C(A) = A and pA = 1A , and
for f : A ! B a cofibration between cofibrant objects we have D(f) = B,
i(f) = f and j(f) = oe(f) = 1B .
(2) Any prefibration category M admits a normalized fibrant splitting along
Mfib! M such that for A fibrant we have R(A) = A and iA = 1A and
for f : A ! B a fibration between fibrant objects we have S(f) = A,
p(f) = f and q(f) = oe(f) = 1A .
Proof. This proof is a slight variation on the proof of the previous lemma.
To prove (1), for cofibrant objects A of M just define C(A) = A and pA = 1A .
For noncofibrant objects A, use the factorization axtim CF4 to construct a weak
equivalence pA : C(A) ! A with C(A) cofibrant.
Let f : A ! B be a map in M. If A = B and f = 1A then we just define
D(1A ) = C(A), i(1A ) = j(1A ) = 1C(A) and oe(1A ) = pA .
If f is a nonidentity cofibration between cofibrant objects, then C(A) = A *
*and
C(B) = B. We define D(f) = B, i(f) = f and j(f) = oe(f) = 1B .
If f is neither an identity nor a cofibration between cofibrant objects, we *
*con
struct D(f), i(f), j(f), oe(f) factoring fpA +pB as a cofibration followed by a*
* weak
equivalence
i(f)+j(f) oe(f)
fpA + pB : C(A) t C(B)//_______//_D(f)_~_//_B
Since oe(f)j(f) = pB , by the 2 out of 3 axiom j(f) is a weak equivalence and t*
*he
construction of the normalized cofibrant splitting is complete. The proof of (2*
*) is
dual.
We will also need the lemmas below.
Lemma 3.5.7.
(1) Let t : M0! M be a cofibrant approximation0of a precofibration0category,
and let f, g : A ' B be maps of M . We then have f ' g in M if and
only if tf0' tg in Mcof.
(2) Let t : M ! M be a fibrant approximation0of a prefibration0category, and
let f, g : A ' B be maps of M . We then have f ' g in M if and only if
tf ' tg in Mfib.
3.5. FIBRANT AND COFIBRANT APPROXIMATIONS 41
Proof. The statement (2) is dual to (1), and we only prove (1). The maps
tf, tg are inside Mcof. Any cylinder A t A //i0+i1//_IAp~//_Ain M0yields a cyli*
*n
der tA t tA//ti0+ti1//_tIAtp~//_tAin Mcof. If we have a homotopy f ' g through
IA, B0, H and b then we get a homotopy tf ' tg through tIA, tB0, tH and tb.
Conversely, assume tf ' tg and let us prove that f ' g. We may assume
that f + g : A t A0! B is0a cofibration. Indeed, for general f, g we factor f +*
* g
as a cofibration f + g 0followed by0a weak equivalence r. The map tr is a0weak0
equivalence,0and0tr O tf ' tr O tg therefore from Thm. 3.4.3 we have tf ' tg*
* . If
we proved f ' g then by Thm. 3.3.2 it would follow that f ' g.
So assume that f + g : A t A ! B is a cofibration. Pick a cylinder IA,0and
construct using Lemma 3.3.1 a homotopy0tf ' tg through a map H : tIA ! B
and a weak equivalence b : tB ! B . We construct step by step the following0
commutative diagram, where the map b is the bottom composition tB ! B .
ti0+ti1
tA tftA//___//flffltIAQQfflffl
 QQQ
tf+tg th QQHQQQQQ
fflffltb1 fflffltb2 QQQQ((b30
tB //_____//tB1//__//tB2~___//B
Construct B1 = B tAtA IA with component maps b1 and h. From CFA3, tB1 is
the pushout of the left square0of the diagram, and using the pushout property we
construct the0map tB1 ! B . Finally we construct b3Otb2 as the CFA4 factorizati*
*on
of tB1 ! B .
The maps b1, b2, h are cofibrations. The map tb2O tb1 is a weak equivalence *
*by
the 2 out of 3 axiom CF2, and by CFA2 so is b2b1.
We get f ' g with homotopy map b2h through the trivial cofibration b2b1.
Recall that a functor u : M1 ! M2 is
(1) essentially surjective if any object of M2 is isomorphic to an object i*
*n the
image of u
(2) full if any map in HomM2 (uA, uB) is in the image of u, for all objects
A, B of M1
(3) faithful if u is injective on HomM1 (A, B) for all objects A, B of M1
The functor u is an equivalence of categories if and only if it is essential*
*ly
surjective, full and faithful.
Lemma 3.5.8.
(1) A cofibrant approximation t : M00! M of a precofibration category M
induces a faithful functor hoM0! hoMcof
(2) A fibrant approximation0t : M ! M of a prefibration category M induces
a faithful functor hoM ! hoMfib
Proof. The statements are dual, and we only prove (1). The image of t lies
inside Mcof, and0t sends weak equivalences to weak equivalences, therefore t in*
*duces
a functor hoM ! hoMcof. To show that the induced functor is0faithful, we have
to show cf. Thm. 3.4.4 that given two left fractions in hoM
42 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
f1 0s1 f2 00s2
A ____//_Bo~o__B and A _____//B o~o__B
if (ts1)1 O tf1 = (ts2)1 O tf2 in hoMcofthen s11f1 = s12f2 in hoM0.
Assume there exist weak equivalences __s1, __s2in Mcofsuch that
__
B>>``B
__s1""" BBB__s2
"""~"" ~BBBB
tB0==j~jVVVtf2i4tB004iiiVVaaD
tf1 iiiiiiVVVVVVV DDts2D
 iiiiiii VVVVVV~ DDD
iiiiii ts1 VVVVVD
tA tB
__s __ __ __
O1ts1_'_s2O_ts2 and s1O tf1 ' s2O tf2. We use the CFA3 and CFA4 axioms to
factor s1+ s2as
ts01+ts02 000 _s __
tB0t tB00//________//_tB____~______//B
0Since0__s1, __s2are weak0equivalences0and __s1= _sO ts01, __s2= _sO ts02it f*
*ollows that
ts1, ts2 and therefore s1, s2 are weak equivalences. 0 0
0On the other0hand, using Thm. 3.4.3 we get that ts10O ts10' ts2 O ts20and0
ts1Otf1 ' ts2Otf2. We apply Lemma 3.5.70and get that s1s1 ' s2s2 and s1f1 ' s2f*
*2.
In other words, s11f1 = s12f2 in hoM and the proof is complete.
Lemma 3.5.9.
(1) Let t : M0! M be a cofibrant approximation0of a precofibration category*
*.0
For any maps f, g : A ' B of M and weak equivalence b : tB ! B of M
with b O tf0= b O tg, we have that f ' g.
(2) Let t : M ! M be a fibrant0approximation of a prefibration0category. For
any maps f, g : A ' B of M and weak equivalence a : A ! tA of M
with tf O a = tg O a, we have that f ' g.
Proof. We only prove (1)  statement (2) is dual. The proof will resemble *
* 0
part of the proof of Lemma 3.5.7. Notice though that tA, tB are cofibrant, but B
is not necessarily cofibrant.
Just as in the proof of Lemma 3.5.7, we may assume that f + g : A t A ! B
is a cofibration. Pick a cylinder IA, and construct step by step the commutative
diagram below
ti0+ti1 tp
tA tftA//___//flffltIA~_______//fflfflA
tf+tg th bOtf=bOtg
fflffltb1 fflffltb2 b3 fflffl0
tB //_____//tB1//__//tB2~__//_B
In this diagram, the bottom horizontal composition is b : tB ! B0. We construct
B1 = B tAtA IA with component maps b1 and h. tB1 is the pushout of the
left square0of our diagram, and using the pushout property we construct the0map
tB1 ! B . We then construct b3 O tb2 as the CFA4 factorization of tB1 ! B .
3.5. FIBRANT AND COFIBRANT APPROXIMATIONS 43
The maps b1, b2, h are cofibrations. By the 2 out of 3 axiom CF2, the maps
tb2 O tb1 and therefore b2b1 are weak equivalences.
We get f ' g with homotopy map b2h through the trivial cofibration b2b1.
As a consequence,0we can prove for a cofibrant splitting along a cofibrant a*
*p
proximation t : M ! M that given a map f : A ! B in M and assuming0the data
C(A), C(B), pA , pB fixed, then the fraction j(f)1i(f) in hoM is independent *
*of
the choice of D(f), i(f), j(f) and oe(f). More precisely:
Lemma 3.5.10.
(1) Let t : M0! M be a cofibrant approximation of a precofibration category.
For any commutative diagram
tA0""!!DD
ti0 Dti00DDD
0 "" DD!!00
tD VVVVVoe0aaC tD=B=
CCC VVVV zzz BBoe00B
CCC VVVVVVzzz BB
0aaC==zz0VVVVVV0 B__B
tj tB0 tj VVV**VB
with i0, i00cofibrations and j0,0j00trivial0cofibrations0of0M0and0oe0,0*
*oe00weak0
equivalences0of M we have that j 1i = j 1i in hoM
(2) Let t : M ! M be a fibrant approximation of a prefibration category. For
any commutative diagram
tq0tA0==tq00aaDhhhhh4A4h>>
== aaDDhhhhhD """
hoe0hhhhhhhhhD"""D00D
hhhh D 00""oe
tD0 C tD
CC zzz
CCC zzz00
tp0 C!!!!C"tp"""zz
tB0
with p0, p00fibrations and q0, q00trivial0fibrations0of0M00and0oe0,0oe0*
*0weak0
equivalences of M we have that p q 1= p q 1 in hoM
Proof.0 Statement0(2)0is dual to (1), and we only prove (1).0 Construct0the0
sum D tB0 D with component maps the trivial cofibrations h and h . In the
commutative diagram below
stA0yy%%LL
ti0sssss LLLti00LL
ss LLL
0 yysssth0 0 00th0%%L000
tD //__//_t(DetB0eDS)SoootDo_KK99r
KKKK SSSSrrrr
KKKKK rrrr SSSSSoeSSS
tj0 ee0 99tj00rr SSS))
tB B
the bottom triangle is a pushout by0CFA3.0 The0map0oe0exists0by the universal
property of the pushout, since oe O tj = oe O tj , and is a weak equivalence *
*by the
44 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
2 out0of 3 axiom. Lemma 3.5.90(1)0applied0to0h0i0,0h00i00,0oe0implies that h0i0*
*' h00i00
in M , and we conclude that j 1i = j 1i in hoM .
We can now state the allimportant
Theorem 3.5.11 (Anderson).
(1) Given a precofibration category M, the inclusion iM : Mcof! M induces
an equivalence of categories hoiM : hoMcof! hoM
(2) Given a prefibration category M, the inclusion jM : Mfib! M induces
an equivalence of categories hojM : hoMfib! hoM
More generally we have
Theorem 3.5.12.
(1) A cofibrant approximation t : M0!0M of a precofibration category induces
an equivalence of categories hoM0! hoM.
(2) A fibrant approximation t : M0! M of a prefibration category induces an
equivalence of categories hoM ! hoM.
It should be noted that the last theorem is actually a particular case of an*
* even
more general result of Cisinski, for which we refer the reader to [Cis02 ], 3.1*
*2.
Proof of Thm. 3.5.12. Let us prove (1). By CFA2 the functor t sends weak 0
equivalences to weak equivalences and therefore descends to a functor hot : hoM*
* !
hoM.0Pick a normalized cofibrant splitting along t, and construct a functor s :*
* M !
hoM as follows:
(i)On objects, s(X) = C(X)
(ii)On maps, s(f) = j(f)1i(f)
Because the splitting is normalized, s sends identity maps to identity maps.*
* To
see that s preserves composition, let f : A ! B and g : B ! C. In the commutati*
*ve
diagram below
tC(A)x__pA~__//xA
ti(f)rrrr 
rrr 
xxrrr 
tD(f) UUUU f
0 nnnww eeLLUUUUUoe(f)UL 
tin(g)nnn LLLL UUUUU 
nnnn ~Ltj(f)LL UUUUU 
wwn ee pUU*fflffl*B
t(D(f) tC(B)D(g))ggP tC(B)yy __~__//B
PPPP~P ti(g)rrrrr 
0 PPPP rr 
tj (f) PPgg yyrrr 
tD(g)UUUU g
ffLLUUUUUoe(g)UL 
LL~L ~UUUUU 
tj(g)LLLL UUUUU 
ff pC UU*fflffl*
tC(C) __~____//C
the map j0(f) is a pushout of j(f), therefore a trivial cofibration, and i0(g) *
*is a
pushout of i(g) therefore a cofibration. By CFA3 the object t(D(f) tC(B)D(g))
is a pushout, and given that goe(f)tj(f) = gpB = oe(g)ti(g), from the universal
3.5. FIBRANT AND COFIBRANT APPROXIMATIONS 45
property of the pushout there exists a map h : t(D(f) tC(B) D(g)) ! C that
keeps the diagram commutative. The map h is a weak equivalence0by the02 out
of 3 axiom. From Lemma 3.5.10,0we get j(gf)1i(gf) = (j (f)j(g))1i (g)i(f) =
j(g)1i(g)j(f)1i(f) in hoM , therefore0s(gf) = s(g)s(f).
We conclude that s : M ! hoM is a functor. But s sends weak equivalences0
to isomorphisms, therefore descends to a functor hos : hoM ! hoM .
The maps pA define a natural isomorphism hot hos ! 1hoM . Therefore, the
functor hot is essentially surjective and full.
To construct a natural isomorphism hos0hot ! 1hoM0 it suffices to show that
hot is faithful. But hot factors as hoM ! hoMcof, which is faithful by Lemma
3.5.8, followed by hoiM : hoMcof! hoM. So it suffices to prove that hoiM is
faithful.
In case t = iM , we may apply the construction of the functor s for a normal*
*ized
cofibrant splitting satisfying the additional0properties of Lemma 3.5.6. In tha*
*t0case,
hos hoiM is the identity on the0objects of M and on the cofibration maps of M*
* .
By Thm. 3.4.5, any map in hoM can be written as a fraction0of cofibrations, and
therefore hos hoiM is the identity on any map of hoM . In particular, hoiM is
faithful, and the proof is complete.
Statement (2) follows from duality.
Remark 3.5.13. If t : M0! M is a cofibrant approximation of a precofibration
category M, suppose that M1 M is a subcofibration0category that includes
the image of t. Then the corestriction t1 : M ! M1 of t defines a cofibrant
approximation, and by Thm. 3.5.12 both hot, hot1 are equivalences of categories,
therefore hoM1 ! hoM is an equivalence of categories.
Remark 3.5.14. Suppose that t : M0!0M is a functor between precofibration
categories such that t restricted to Mcofis a cofibrant approximation. In view *
*of
Thm. 3.5.11 and Thm. 3.5.12,0it is0not hard to see that t induces a composite
equivalence of categories hoM hoMcof! hoM. The proper way to formulate
this is to say that the total left derived functor of t is an equivalence, whic*
*h we
prove as Thm. 4.6.3 in the next section.
Given a pair (M, W), we denote flM the localization functor flM : M ! hoM.
From Thm. 3.5.11 and Thm. 3.4.4 (c) we get:
Corollary 3.5.15.
(1) In a precofibration category M, two maps f, g : A ' B with A, B cofibra*
*nt
are homotopic if and only if flM f = flM g.
(2) In a prefibration category M, two maps f, g : A ' B with A, B fibrant a*
*re
homotopic if and only if flM f = flM g.
As a consequence, the homotopy relation in a precofibration category depends
only on the weak equivalences  and not on the choice of cofibrations. Dually, *
*in a
prefibration category the homotopy relation depends only on the weak equivalenc*
*es.
To see that, let (M, W) be a category with weak equivalences. Given two maps
f, g : A ' B we say that f ' g if flM f = flM g. This defines an equivalence
relation compatible with composition of maps. In case M is either a cofibration*
* or
a prefibration category, the new definiton of homotopic maps is compatible with
the old one.
46 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
We will denote ssM the factor category  with same objects as M, and with ma*
*ps
HomssM(A, B) = HomM (A, B)='. The new definition of ssM is again backward
compatible. With these notations we have:
Corollary 3.5.16.
(1) In a precofibration category M, the functors in the commutative diagram
pMcof qMcof
Mcof _____//ssMcof___//_hoMcof
iM ssiM hoiM
fflfflpM fflfflqM fflffl
M _______//_ssM_____//_hoM
have the property that qMcof, qM , ssiM are faithful and hoiM is an e*
*quiva
lence.
(2) In a prefibration category M, the functors in the commutative diagram
pMfib qMfib
Mfib ____//_ssMfib__//_hoMfib
jM  ssjM hojM
fflfflpM fflfflqM fflffl
M ______//_ssM______//hoM
have the property that qMfib, qM , ssjM are faithful and hojM is an e*
*quiv
alence.
3.6.Products of cofibration categories
Here is an application of homotopy calculus of fractions. If (Mk, Wk) for
kfflK is a set of categories with weak equivalences, one can form the product
(x kfflKMk, xkfflKWk) and its homotopy category denoted ho(x Mk). Denote pk :
xkfflKMk ! Mk the projection. The components (ho pk)kfflKdefine a functor
P : ho(x Mk) ! xho Mk
If each category Mk carries a (pre)cofibration category structure (Mk, Wk, Cofk*
*),
then (xMk, xWk, xCofk) defines the product (pre)cofibration category structure
on xMk. Dually, if each Mk carries a (pre)fibration category structure, then xMk
carries a product (pre)fibration category structure.
Suppose that Mk are precofibration categories, and that A = (Ak)kfflKis a
cofibrant object of xMk. Any factorization A t A ! IA ! A defines a cylinder in
xMk iff each component Ak t Ak ! (IA)k ! Ak is a cylinder in Mk.
If B = (Bk)kfflKis a second cofibrant object and f, g : A ' B is a pair of m*
*aps
in xMk, then any homotopy f ' g induces componentwise homotopies fk ' gk in
Mk. Conversely, any set of homotopies fk ' gk induces a homotopy f ' g.
Theorem 3.6.1. If Mk for kfflK are each precofibration categories, or are ea*
*ch
prefibration categories, then the functor
P : ho(x Mk) ! xho Mk
is an isomorphism of categories.
Proof. Assume that each Mk is a precofibration category (the proof for pre
fibration categories is dual). Our functor P is a bijection on objects, and we'*
*d like
to show that it is also fully faithful.
3.7. SATURATION 47
By Thm. 3.5.11 we have equivalences of categories hoMk ~=ho (Mk)cofand
ho(xMk) ~=ho(x(Mk)cof). It suffices therefore in our proof to assume that Mk =
(Mk)coffor all k.
To prove fullness of P , let Ak and Bk be objects of Mk. Denote A = (Ak)k and
B = (Bk)k. Any map OE : A ! B in xho Mk can be expressed on components, using
Thm. 3.4.4 (a) for each Mk, as a left fraction OEk = s1kfk, with weak equivale*
*nces
sk. The map OE therefore is the image via P of (sk)1(fk).
To prove faithfulness of P , suppose that OE, _ : A ! B are maps in ho(x Mk)
which have the same image via P . Using Thm. 3.4.4 (a) applied to xMk, we can
write OE as (sk)1(fk) and _ as (tk)1(gk), with sk, tk weak equivalences.0From*
*0Thm.
3.4.4 (b) applied to each Mk, we0can find0weak equivalences0sk,0tk such that we
have componentwise homotopies sksk0'0tktk and0skfk0' tkgk. The componentwise
homotopies induce homotopies s s ' t t and s f ' t g in xMk, so OE = _ and we
have shown that P is faithful.
3.7. Saturation
Another important consequence of homotopy calculus of fractions is the next
result, due to Cisinski [Cis02 ], which shows that the saturation of weak equiv*
*alences
in a cofibration category yields again a cofibration category. __
Given a category with weak equivalences (M, W), recall that W denotes the
saturation of W, i.e. the class of maps of M that become isomorphisms in hoM.
Lemma 3.7.1.
(1) Suppose that (M, W, Cof) is a precofibration category, and that f : A !*
* B
is a map with A, B cofibrant.
(a) f0has a left0inverse in0hoM if and only if there exists a cofibrati*
*on
f : B ! B such that f f is a weak equivalence. *
* 0
(b) f is an0isomorphism0in0hoM0if0and0only0if0there0exist0cofibrations *
*f :
B ! B , f : B ! B such that f f, f f are weak equivalences.
(2) Suppose that (M, W, Fib) is a prefibration category, and that f : A ! B
is a map with A, B fibrant.
(a) f0has a0right inverse in0hoM if and only if there exists a fibration
f : A ! A such that ff is a weak equivalence.
(b) f0is an0isomorphism0in0ho0M0if0and only0if0there0exist0fibrations
f : A ! A, f : A ! A such that ff , f f are weak equivalences.
Proof. We only prove (1). The implications (a) ((), (b) (() are clear.
To prove (a) ()), using Thm. 3.5.11 f has a left inverse in hoM if and only
if f has a left inverse in hoMcof.0From0Thm. 3.4.5, write the left inverse of f*
* in
hoMcofas a left0fraction s1f with f a cofibration0and s a trivial cofibratio*
*n.0
We get 1 = s1f f in hoMcof, therefore0s = f f in hoMcofwhich means s ' f f.
Since s is a weak equivalence, f f must be a weak equivalence. 0
Part (b) ()) is0a corollary of0(a)0applied first to the map f to construct f
then to the map f to construct f .
This allows us to prove the following result.
Theorem 3.7.2 (Cisinski). __
(1) If (M, W, Cof) is a (pre)cofibration category, then so is (M, W , Cof).
48 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
__
(2) If (M, W, Fib) is a (pre)fibration category, then so is (M, W_,_Fib).
(3) If (M, W, Cof, Fib) is an ABC model category, then so is (M, W , Cof,
Fib).
Proof. Part (2) is dual to_(1), and part (3) is a consequence of (1) and (2*
*).
Let's prove (1) (a) for (M, W , Fib) in the case when (M, W, Cof) is a precofib*
*ration
category.
(i) The axioms CF1, CF2, CF3 (1), CF4 are clearly satisfied.
(ii)_The axiom CF3 (2). Given a solid diagram in M, with A, C cofibrant and
i a W trivial cofibration,
Af_____//flfflCO
i OjO
fflfflfflffl
B ` ` `//D
then by the previous lemma there0exist0cofibrations0i0, i00such that i0i,0i00i0*
*0are0W
trivial0cofibrations.00Denote0j , j the pushouts of the cofibrations_i , i . *
*We get
that j j, j j are Wtrivial cofibrations, and thereore j is a W trivial cofib*
*ration.
Assume now that (M, W, Cof) satisfies CF5 and CF6.
(iii) The axioms CF5 (1) and CF6_(1). If (M, W, Cof) satisfies CF5 (1), resp.
CF6 (1), then clearly so does (M, W , Cof)..
(iv) The axiom CF5 (2). Suppose that the cofibration category_(M, W, Cof)
satisfies CF5. Assume that fi : Ai ! Bi for ifflI is a set of W trivial cofibr*
*ations
with Aicofibrant. The map tfiis a0cofibration0by0axiom0CF50(1).0By0the0previous*
*000
lemma, there exist cofibrations fi : Bi ! Bi, fi0: Bi0!0Bi0such that fifi, fi fi
are Wtrivial cofibrations._It follows that tfifi, tfi fi are Wtrivial cofibra*
*tions,
therefore tfi is a W trivial cofibration.
(v) The axiom CF6 (2). Suppose that the cofibration category (M, W, Cof)
satisfies CF6. We will use the equivalent formulation of CF6 (2) given by Lemma
1.6.6 (1) (iii). Consider a map of countable direct sequences of_cofibrations, *
*with
A0 cofibrant and all an, bn, fn cofibrations, such that fn are W trivial cofib*
*rations.
These are the full maps in the diagram below.
A0f//a0_//flfflA1//a1//_fflfflA2//a2//_fflffl...
f0 f1 f2
fflfflbfflffl0fflfflb1b2
B0f//___//flfflOB1////_fflfflOB2////_fflfflO...
f00OO f01OO f02OO
fflffl0bfflffl00fflffl0b010b02
B0 //```//B1//``//`B2//`//``...
We know from CF6 (1) that colimAn,_colimBn exist and are cofibrant. We want
to show that colimfn is a W weak equivalence. 0
0 By0induction, we will construct0the cofibrant objects Bn and the cofibrations
fn, bn with the property that fnfn are Wtrivial0cofibrations. Once the constru*
*ction
is complete, it will follow that colimfn0colimfn is a Wweak equivalence,0and0
applying the construction again to fn we get cofibrant objects Bn and cofibrati*
*ons
3.7. SATURATION 49
f00n, b00nsuch_that colimf00ncolimf0nis a Wweak equivalence. It will follow t*
*hat
colimfn is a W weak equivalence. 0
To complete the0proof,0it remains to construct the cofibrant objects Bn and
the cofibrations fn,0bn. We use induction on n. For n = 0, we use Lemma 3.7.1 to
construct B0 and f0. 0 0 0
Assume we constructed the object Bn and the cofibrations fn, bn1 such that
f0nfn is a Wweak equivalence.
We construct the following diagram
____0
fn+1 fn+1_____
Anf//an_//flfflAn+1//_fflfflBn+1////_Bn+1fflffl
_____
f0nfn~ fin~ f0n+1~
fflffl0flfflffln_ffin fflffl0
Bn //___//B0n+1//___~_______//_Bn+1
where:
(1) f0nfn is a Wtrivial cofibration 0
(2) fin is constructed as the pushout of fnfn, therefore is a Wtrivial cof*
*ibra
tion._fln is a pushout of an, therefore is a cofibration.
(3) f0n+1is_a cofibration constructed by Lemma 3.7.1 applied to fn+1, so th*
*at
f0n+1fn+1 is a Wtrivial cofibration __
(4) ffin is constructed_as the pushout of f0n+1fn+1, therefore a Wtrivial *
*cofi
____0
bration. fn+1 is a pushout of fin, therefore a Wtrivial cofibration.
________ 0
We define f0n+1= f0n+1f0n+1, which is a cofibration, and fn+1fn+1 is a Wtri*
*vial
cofibration. We also define b0n= ffinfln, which is a cofibration. The inductive*
* step is
now complete, and with it the proof of CF6 (2).
Let us adapt this discussion to the case of left proper cofibration categori*
*es,
and_show that if (M, W, Cof) is a left proper cofibration category then so is (*
*M,
W , Cof).
Lemma 3.7.3.
(1) Suppose that (M, W, Cof) is a left proper precofibration category, and
that f : A ! B be a map
(a) f has0a left inverse0in hoM0if and only if there exists a left prop*
*er
map f : B ! B such that f f is a weak equivalence.
(b) f is an0isomorphism0in0hoM0if0and0only0if there0exist0left0proper0
maps f : B ! B , f : B ! B such that f f, f f are weak
equivalences.
(2) Suppose that (M, W, Fib) is a right proper prefibration category, and t*
*hat
f : A ! B is a map.
(a) f has0a right0inverse in hoM0if and only if there exists a right pr*
*oper
map f : A ! A such that ff is a weak equivalence.
(b) f is an0isomorphism0in0hoM0if0and0only0if there0exist0right0proper0
maps f : A ! A, f : A ! A such that ff , f f are weak
equivalences.
50 3. THE HOMOTOPY CATEGORY OF A COFIBRATION CATEGORY
Proof. We only prove (1). The implications (a) ((), (b) (() are trivial.
Let us prove (a) ()). Construct the diagram
f0 0
B>_____//_B>OOOO
f """"" 
"" s2 0 
""f2  f2  0
AO_____//_B2___//_B2OOO
  OO
r1~ s1 
 f1  f01  0
A1 //__//_B1//_//_B1
as follows:
(1) A1 is a cofibrant replacement of A, f1 is a cofibrant replacement of fr*
*1.
(2) It follows that f1 has a left0inverse in0hoM. We use Lemma 3.7.1 to
construct a cofibration0f1 such0that f1f1 is a Wweak equivalence.0
(3) The maps f2, resp. f2 and f are pushouts of f1, resp. f1. These pushou*
*ts
can be constructed because M is left proper.
The maps r1 and s2s1 are Wweak equivalences, and all horizontal maps are left0
proper. It follows that all0vertical maps are Wweak equivalences. Since f1f1 i*
*s a
Wweak equivalence, so is f f 0
Part (b) is proved applying0(a) first0to0the map f to construct f and then a
second time to the map f to construct f .
Using the previous lemma, the statement below is immediate:
Proposition 3.7.4.
(1) In a left_proper_precofibration category (M, W, Cof), all the Wleft pr*
*oper
maps are W left proper
(2) In a right_proper prefibration category (M, W, Fib), all the Wright pr*
*oper
maps are W right proper
Proof. We only prove (1). Suppose that f : A ! B is a Wleft proper map.
In the diagram with full maps
r1 r2
A ____//_C1_r__//OC2```//OC3```//OC4O
f OO OO OO OO
fflffl fflfflr0fflfflr01fflfflr0fflffl2
B ` ``//D1` ` `//D2`` `//D3`` `//D4
__
suppose that r is a W weak equivalence. Use Lemma 3.7.3 to0construct0the0maps *
*r1,
r2 such that r1r and0r2r10are0Wweak0equivalences. Denote r , r1, r20the pushou*
*ts_
along f. The maps r1r and r2r1 are Wweak equivalences, therefore r is a W weak
equivalence.
We can now show the following result.
Theorem 3.7.5.
(1) If_(M, W, Cof) is a left proper (pre)cofibration category, then so is (*
*M,
W , Cof).
3.7. SATURATION 51
(2) If_(M, W, Fib) is a right proper (pre)fibration category, then so is (M,
W , Fib). __
(3) If (M, W, Cof, Fib) is a proper ABC model category, then so is (M, W ,
Cof, Fib).
Proof. Consequence of Thm. 3.7.2 and Prop. 3.7.4.
CHAPTER 4
Kan extensions. Total derived functors.
The purpose of this chapter is to introduce the apparatus of Kan extensions
and total derived functors, which will be used later to prove the existence and*
* basic
properties of homotopy colimits in cofibration categories.
We will use approximation functors (Def. 4.5.1) as a tool for stating and pr*
*oving
an existence theorem for total derived functors (Thm. 4.5.6), and a rather tech*
*nical
adjointness property of total derived functors (Thm. 4.5.10).
The approximation functors defined in this chapter should be thought of as
an axiomatization at the level of categories with weak equivalences (M, W) of t*
*he
(co)fibrant approximation functors defined in Section 3.5.
4.1. The language of 2categories
Recall that a 2category C is a category enriched over categories. This mean*
*s by
definition that for each two objects A, B of C the homset Hom(A, B) forms the *
*ob
jects of a category Hom (A, B). The the composition functor cABC : Hom (A, B) x
Hom (B, C) ! Hom (A, C) is required to be associative and to have 1A : ? !
Hom (A, A) as a left and right unit, where ? denotes the pointcategory.
The objects of a 2category C are called 0cells, the objects of Hom (A, B) *
*are
called 1cells and the morphisms of Hom (A, B) are called 2cells. A good refer*
*ence
on 2categories is the Borceux monography [Bor94 ].
This section describes the notation we use for compositions of 1cells and 2*
*cells
in a 2category. Each notation has a full form and a simple form. The simple fo*
*rm
of the notation is ambiguous, and is only used if it is clear from the context *
*which
functor or natural map operation we refer to.
We denote as usual 1cells f : A ! B with a single arrow. Between 1cells
f, g : A ! B, we denote 2cells as ff : f ) g, or just ff : f ! g if no confusi*
*on can
occur.
The composition of 1cells f, g
(4.1) A _f__//_B_g_//_C
is just the composition at the level of unenriched homsets Hom(, ) and is de
noted g O f : A ! C, or in simple form gf.
The composition of 2cells ff, fi
__f________//
ffff'
(4.2) A __g________//B
__h________//fiff'
is composition at the level of Hom (, ) and is denoted fi ff : f ) h, or in *
*simple
form fiff.
53
54 4. KAN EXTENSIONS. TOTAL DERIVED FUNCTORS.
The composition of a 2cell ff with a 1cell f
__g_______//_
(4.3) A _____f____//_B_h_______//_Cffff'
is just cABC (1f, ff), and we denote it ff.f : gf ) hf or in simple form fff. T*
*he
composition in the other direction
__f_______//_
(4.4) A __g_______//_B____h____//_Cffff'
is cABC (ff, 1h) and we denote it h.ff : hf ) hg, or in simple form hff.
The notations we have established up until now can be used to completely
describe compositions of 1 and 2cells. However it is convenient to introduce *
*the
? notation to denote thecomposition of adjacent of 2cells of planar diagrams.
We will denote the composition of 2cells ff : jf ) g and fi : h ) ij
(4.5) ?B?@
f """ @@@h
""  @@
""w  w @__
A @ " ffj"fi D>>
@@@  """
g @@  ""i
@OfflfflO""
C
as fi ? ff = (i.ff) (fi.f).
We use the same ? notation to denote the composition of 2cells ff : f ) jg
and fi : hj ) i
(4.6) ?B?@OO
f """ @@@h
""  @@
""w  w @__
A @@" ffj"fi"D>>
@@  """
g @@OO@"""i
C
as fi ? ff = (fi.g) (h.ff).
In particular, taking j to be an identity map in (4.5) or (4.6) we get the c*
*om
position of 2cells fi ? ff = cABC (ff, fi)
__f_______//___h_______//_
(4.7) A __g_______//_B_i_______//_Cffff'fiff'
Since cABC is a functor, (i.ff) (fi.f) = (fi.g) (h.ff) and the ? notation is co*
*nsistent
in (4.7) with (4.5) and (4.6).
In simple form, if no confusion is possible we denote fiff for fi ? ff.
There is a general theorem called the Pasting Theorem regarding compositions
of 2cells of planar diagrams, for which we refer the reader for example to [Po*
*w90 ]
.
Let us also recall that a (strict) 2functor f : C1 ! C2 is a function that *
*sends
1, 2 and 3cells of C1 to 1, 2 resp. 3cells of C2, with the property that it p*
*reserves
the various types of units and compositions of cells. If Cop denotes the opposi*
*te of
4.2. ADJOINT FUNCTORS 55
a 2category (with composition of 1 and 2cells having direction reversed), th*
*en a
contravariant 2functor0f : C1 ! C2 is just a (covariant) 2functor f : Cop1!0C*
*2.
A 2subcategory C of a 2category C consists of a subclass of objects ObC 0
ObC along with subcategories Hom C0(A, B) Hom C(A, B) for objects0A, BfflObC
that are stable under the composition rule cABC for0A, B, CfflObC and include*
* the
image of the unit 1A : ? ! Hom C(A, A) for AfflObC0. A 2subcategory is 2full *
*if
Hom C0(A, B) = Hom C(A, B) for any A, BfflObC .
The category of categories forms a 2category, with categories as 0cells, f*
*unc
tors as 1cells and natural maps as 2cells. We will use the notation introduce*
*d in
this section to denote compositions of functors and natural maps. In particula*
*r,
natural maps will be denoted with a double arrow ) (or simply with ! if no con
fusion is possible), and composition of natural maps viewed as adjacent 2cells*
* will
be denoted with ?.
4.2. Adjoint functors
Recall that an adjunction u1 a u2 between two functors u1 : A AE B : u2 is a
bijection of sets
i : HomB (u1A, B) ~=HomA (A, u2B)
natural in objects AfflA and BfflB. For example, if u1, u2 are equivalences of *
*cate
gories, then u1 a u2 (and u2 a u1).
Whenever we say that u1, u2 is an adjoint pair we refer to a particular bije*
*ction
i. The following proposition encodes the basic properties of adjoint functors t*
*hat
we will need.
Proposition 4.2.1. Suppose that u1 : A AE B : u2 is a pair of functors.
(1) An adjunction u1 a u2 is uniquely determined by natural maps OE : 1A )
u2u1 (the unit) and _ : u1u2 ) 1B (the counit of the adjunction) with t*
*he
property that both the following compositions are identities
u1 _u1OE+3__u1u2u1_u1+3__u1u2 _OEu2+3__u2u1u2u2_+3__u2
(2) If u1 a u2 is an adjunction with unit OE and counit _, then
(a) u1 (resp. u2) is fully faithful iff OE (resp. _) is a natural isomo*
*rphism.
(b) u1 and u2 are inverse equivalences of categories iff both OE and _ *
*are
natural isomorphisms.
Proof. See for example Mac Lane [Lan98 ].
Here is another way to state part (2) of the previous proposition. The proof*
* is
left to the reader.
Proposition 4.2.2. Suppose that u1 a u2 is an adjoint pair of functors as
above. Then the following statements are equivalent:
(1) (resp. (1r), resp. (1l)). For any objects AfflA, BfflB, a map u1A ! B is
an isomorphism iff (resp. if, resp. only if) its adjoint A ! u2B is an
isomorphism
(2) (resp. (2r), resp. (2l)). u1 and u2 are inverse equivalences of categor*
*ies
(resp. u2 is fully faithful, resp u1 is fully faithful).
56 4. KAN EXTENSIONS. TOTAL DERIVED FUNCTORS.
4.3.Kan extensions
For a category with weak equivalences (M, W), cf. Def. 1.7.2, we denote
flM : M ! hoM the localization functor.
A functor u : M1 ! M2 of categories with weak equivalences (M1,_W1), (M2,_W2)
descends to a functor hou : hoM1 ! hoM2 if and only if u(W1) W2 , where W2
denotes the saturation of W2 in M2.
In the general case however hou does not exist, and the best we can hope for
is the existence of a left or right Kan extension of flM2 u along flM1 .
Definition 4.3.1. Consider two functors u : A ! B and fl : A ! A0
(1) A left Kan extension of u along fl is a pair (Lflu, ffl) where Lflu : A*
*0! B
is a functor and ffl : Lflu O fl ) u is a natural map
A MMM
fl MMMuMM
fflfflMMMffl *
0 _________M&&M//_
A Lflu B
satisfying the0following universal property: if ( , ~) is another pair *
*of a
functor : A ! B and natural map ~ : O fl ) u
A MMMM
 MMuMM
fl ff___MMM____________________________________*
*________________________l *
fflffl0LflMM&&u&&_____________________________*
*_____________________________________________________________________________*
*______________________________
A __________88_________________________________*
*____________________________________________________Bffi *
__________________________________________*
*______________________________________________________________
then there exists a unique natural map ffi : ) Lflu with ffl ? ffi =0~
(2) A right Kan extension of u along fl is a pair (Rflu, ) where Rflu : A *
*! B
is a functor and : u ) Rflu O fl is a natural map
A MMM
fl MMMuMM
fflfflMMM +
0 _________M&&M//_
A Rflu B
satisfying the0following universal property: if ( , ~) is another pair *
*of a
functor : A ! B and a natural map ~ : u ) O fl
A MMMM
 MMuMM
fl ___MMM____________________________________*
*________________________+
fflffl0RflMM&&u&&_____________________________*
*_____________________________________________________________________________*
*______________________________
A __________88_________________________________*
*____________________________________________________Bffi +
__________________________________________*
*______________________________________________________________
then there exists a unique natural map ffi : Rflu ) with ffi ? = ~
The left Kan extension (Lflu, ffl) is also called in some references the lef*
*t derived
of u along fl and the right Kan extension (Rflu, ffl) the right derived of u al*
*ong fl.
Since it is defined by an universal property, if the left (or right) Kan extens*
*ion
exists then it is unique up to a unique isomorphism.
4.3. KAN EXTENSIONS 57
We next state a few simple properties of Kan extensions. First, we note that
the Kan extension of u along fl is independent on the choice of u and fl within*
* a
natural isomorphism class.
0 Proposition 4.3.2. Consider two0functor pairs0u, u0: A ' B and fl, fl0 : A '
A and natural isomorphisms u ~=u , fl ~=fl .
(1) The left Kan extension (Lflu, ffl) exists iff the left Kan extension (L*
*fl0u0, ffl0)
exists, and if they both exist they are naturally isomorphic. *
* 0 0
(2) The right Kan extension (Rflu, ) exists iff the right Kan extension (R*
*fl0u , )
exists, and if they both exist they are naturally isomorphic.
Proof. Immediate using the definitions.
Second, we prove an existence criterion for Kan extensions. If fl admits a l*
*eft
(resp. right) adjoint, then we show that u admits a left (resp. right) Kan exte*
*nsion
along fl.
Proposition 4.3.3. Consider two functors u : A ! B and fl : A ! A0.
(1) If fl admits0a left adjoint0fl0 with adjunction unit OE : 1A0 ) flfl0 a*
*nd counit
_ : fl fl ) 1A , then (ufl , u_)0is a left Kan extension of u along0fl
(2) If fl admits a right0adjoint fl with0adjunction unit OE : 1A ) fl fl *
*and
counit _ : flfl ) 1A0, then (ufl , uOE) is a right Kan extension of u *
*along
fl
Proof. We only prove (1).
OAOMMMM
0 MMMuM
flflu_____MMM______________________________________*
*_____________________*
fflffl0__0MM&&&&_________________________________*
*_____________________________________________________________________________*
*________________________
A _____ufl__88____________________________________*
*____________________________________________________Bffi *
_____________________________________________*
*___________________________________________________________
For any pair ( , ~) with : A0 ! B and ~ : 0fl ) u, we'd like to show that
there exists a unique natural map ffi : ) ufl with
(4.8) ~ = (u_) O (ffifl)
To show existence, we define ffi : ) flfl0 ) ufl0 as the composition
0
(4.9) ffi = (~fl ) O ( OE)
ffi defined by (4.9) satisfies (4.8), using the commutative diagram
0fl
fl__OEfl+3__ flfl0fl~fl+3__ufl0fl
EEEE
EEEEEEEfl_ u_
1 flEEOE&EEEff' 
~ ff'
fl________+3u
To show uniqueness, if a map ffi satisfies (4.8) then ffi satisfies (4.9) becau*
*se of the
commutative diagram
58 4. KAN EXTENSIONS. TOTAL DERIVED FUNCTORS.
____ffi+3__ufl0F
FFFF1ufl0FF
OE ufl0OEFFFFFFF
ff'0 ff'0 FOE&F0 0
flfl____+3_u0fl_flfl0+3__ufl
ffiflfl u_fl
We next show that Kan extensions commute with composition along the base
functor fl.
0Proposition04.3.4.0Consider0three functors u : A ! B, fl : A ! A0 and
fl : A ! A .
(1) If (Lflu, ffl) and (Lfl0Lflu, ffl0) exist, then (Lfl0Lflu, ffl ? ffl0)*
* is a left Kan
extension of u along fl0fl. 0 0
(2) If (Rflu, ) and (Rfl0Rflu, ) exist, then (Rfl0Rflu, ? ) is a left*
* Kan
extension of u along fl0fl.
Proof. Immediate using the universal property of the left (resp. right) Kan
extensions.
We state a corollary needed in the proof of Thm. 6.6.3.
Corollary 4.3.5. Consider two functors u :0A ! B0and0fl0: A0!0A0, and a
pair of inverse equivalences0of0categories0fl00:0A o A : fl , with natural is*
*omor
phisms OE : 1A0 ) fl fl and _ : 1A00) fl fl .
(1) The left Kan extension (Lflu, ffl) exists iff the left Kan extension (L*
*fl0flu, ffl0)
exists. If they both exist, then the latter is isomorphic to ((Lflu)fl0*
*0, ffl ? _).
(2) The right Kan extension (Rflu, ) exists iff the right Kan extension (R*
*fl0flu,
0) exists. If they both exist, then the latter is isomorphic to ((Rflu*
*)fl00,
OE ? ).
Proof. Consequence of Prop. 4.3.3 and Prop. 4.3.4.
4.4.Total derived functors
If we specialize the definition of the left and right Kan extensions to the *
*context
of categories with weak equivalences, we obtain the notion of total left and ri*
*ght
derived functors.
Definition 4.4.1. Suppose that (M1, W1), (M2, W2) are two categories with
weak equivalences, with localization functors denoted flM1 and respectively flM*
*1 ,
and suppose that u : M1 ! M2 is a functor.
(1) The total left derived functor of u, denoted (Lu, ffl), is the left Kan*
* extension
(LflM1(flM2 u), ffl) of flM2 u along flM1
M1 ___u___//M2
flM1 ff@Hl flM2
fflffl fflffl
ho M1 _Lu__//hoM2
4.5. LEFT AND RIGHT APPROXIMATION FUNCTORS. 59
(2) The total right derived functor of u, denoted (Ru, ), is the right Kan
extension (RflM1(flM2 u), ) of flM2 u along flM1
M1 ___u___//M2
flM1 j flM2
fflffl fflffl
ho M1 _Ru__//hoM2
The total left and derived functors (Lu, ffl), (Ru, ) are defined in terms *
*of the
localization functors flM1 , flM2 and therefore_will_not change if we replace i*
*n the
definition W1, W2 with their_saturations W1 , W2 .
Note that if u(W1) W2 then Lu = Ru = hou.
4.5. Left and right approximation functors.
Our next goal is to provide an existence result for total left (and right) d*
*erived
functors, which will be applied in the next section to the case of (co)fibration
categories.
Definition 4.5.1.0Consider0a functor t : M0! M between two categories with
weak equivalences (M , W ), (M, W) with the property that
(a) t sends weak equivalences0to weak equivalences and induces an equivalen*
*ce
of categories hot : hoM ! hoM.
The functor t is called a left approximation if
(1b) For0any object A of0M there exists a weak equivalence map tA0! A with
A an object of M
(1c) Any diagram of full maps with p a weak equivalence
tA0< deg(d2).
Let us recall the definition of the latching object of a diagram indexed by
a direct category, and dually the definition of the matching object of a diagram
indexed by an inverse category.
First we give the definition of the latching and matching categories:
Definition 6.1.2. Let D be a category, and d be an object of D.
(1) If D is direct, the latching category @(D # d) is the full subcategory *
*of the
over category (D # d) consisting of all objects except the identity of *
*d.
(2) If D is inverse, the matching category @(d # D) is the full subcategory*
* of
the under category (d # D) consisting of all objects except the identit*
*y of
d.
It is an easy consequence of the definitions0that0if D is direct, then @(D #*
* d)
is also a direct category, with deg(d ! d) = deg(d ). Dually, if the category D*
* is
inverse, then @(d # D) is an inverse category.
We next recall the definition of the latching (resp. matching) objects of a
diagram indexed by a direct (resp. inverse) category.
Definition 6.1.3. Let D be a category, and d be an object of D. Assume that
M is a category, and that X is a D diagram of M.
(1) If D is direct, the latching object of X at d is by definition
LXd = colim@(D#d)X
(2) If D is inverse, the matching object of X at d is by definition
MXd = lim@(d#D)X
The latching object LXd may not exist in general for all D diagrams of M;
but if the category M is cocomplete, then the latching object LXd always exists.
If D is direct or inverse and nfflZ+ then we will denote D>GG
fnnnnnn"""fflffl
nnnn r ""ffflfl
nnn "" fflffl
XO___0_//YO0~r1fflfflffl
f OO fflffl
a~ b~fflfflffl
 ff0l
X1 _f1_//_Y1
where X1 is a Reedy cofibrant replacement of X, r1f1 is a factorization0of fa0as
a Reedy cofibration f1 followed by a weak equivalence r1, and Y = X tX1 Y1.
The map f1 is in particular a pointwise cofibration. Its pushout f1 is therefor*
*e a
pointwise cofibration, and by excision the map b is a weak equivalence, so r is*
* a
weak equivalence0by the 2 out of 3 axiom.0We have thus constructed a factorizat*
*ion
f = rf as a pointwise cofibration f followed by a pointwise weak equivalence *
*r.
92 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
The proof of 6.2.4 part (1) is now complete, and part (2) is proved by duali*
*ty.
As a corollary of Thm. 6.2.4, we can construct the Reedy and the pointwise
cofibration category structures on restricted small direct diagrams in a cofibr*
*ation
category.
Definition 6.3.8. If (M, W) is a category with weak equivalences and (D1 , D*
*2)
is a category0pair, a D1 diagram X is called restricted with respect to D2 if f*
*or any
map d ! d of D2 the map Xd ! Xd0is a weak equivalence.
We will denote M(D1,D2)the full subcategory of D2restricted diagrams in MD1.
With this definition we have
Theorem 6.3.9.
(1) If (M, W, Cof) is a cofibration category and (D1, D2) is a small direct
category pair, then
(a) (M(D1,D2), WD1, CofD1reedy\M(D1,D2)) is a cofibration category  ca*
*lled
the D2restricted Reedy cofibration structure on M(D1,D2).
(b) (M(D1,D2), WD1, Cof(D1,D2)) is a cofibration category  called the *
*D2
restricted pointwise cofibration structure on M(D1,D2).
(2) If (M, W, Fib) is a fibration category and (D1, D2) is a small inverse
category pair, then
(a) (M(D1,D2), WD1, FibD1reedy\ M(D1,D2)) is a fibration category  cal*
*led
the D2restricted Reedy fibration structure.on M(D1,D2)
(b) (M(D1,D2), WD1, Fib(D1,D2)) is a fibration category  called the D2
restricted pointwise fibration structure on M(D1,D2).
Proof. We only prove part (1)  part (2) is dual.
(i) Axioms CF1 and CF2 are clearly verified for both the pointwise and the
Reedy restricted cofibration structures.
(ii) The pushout axiom CF3 (1). Given a pointwise cofibration i and a map f
with X, Y, Z pointwise cofibrant in M(D1,D2)
f
Xf_____//flfflZfflfflO
i OjO
fflfflfflffl
Y ` g``//T
then the pushout0j of i exists in MD1, and j is a pointwise cofibration. For any
map d ! d of D2 using the Gluing Lemma 1.4.1 applied to the diagram
6.4. COLIMITS IN DIRECT CATEGORIES (THE RELATIVE CASE) 93
fd
Xd_____________//_!!DZd!!C
 DDDD CCCC
 idDD  CCC
 D!!  !!
~  Yd_____________//Td
   
 ~ ~ 
fflffl  fflffl 
Xd0 ______ _____//Zd0 
!!CC  fd0 !!BB 
CCC  BBB 
i 0CC  BB 
d C!!fflffl B!!fflffl
Yd0______________//Td0
it follows that Td ! Td0is an equivalence, therefore T is a D2 restricted diagr*
*am.
Furthermore, j is a Reedy cofibration if i is one and X, Z are Reedy cofi
brant, by Thm. 6.2.4. This shows that the pushout axiom is satisfied for both t*
*he
pointwise and the Reedy restricted cofibration structures.
(iii) The axiom CF3 (2) is clearly verified for both the pointwise and the R*
*eedy
restricted cofibration structures.
(iv) The factorization axiom CF4. Let f : X ! Y be a map in M(D1,D2). If X *
* 0
is a pointwise0(resp.0Reedy) cofibrant diagram, by Thm. 6.2.4 f factors0as f = *
*rf
with f : X ! Y a pointwise (resp.0Reedy) cofibration and r : Y ! Y a pointwi*
*se
weak equivalence. In both cases Y is restricted, and CF4 is satisfied for both*
* the
pointwise and the Reedy restricted cofibration structures.
(v) Axiom CF5 for both the restricted pointwise and restricted Reedy cofibra
tion structures follows from the fact that if Xi, ifflI is a set of restricted *
*pointwise
(resp. Reedy) cofibrant diagrams, then tXiis a restricted pointwise (resp. Reed*
*y)
cofibrant diagram by Lemma 1.6.3.
(vi) Axiom CF6. Given a countable direct sequence of pointwise (resp. Reedy)
cofibrations with X0 pointwise (resp. Reedy) cofibrant
X0 //a01//_X1//a12//_X2//a23//_...
the colimit colimXn exists and is pointwise (resp. Reedy) cofibrant. If all Xn *
*are
restricted, from Lemma 1.6.5 colimXn is restricted. The axiom CF6 now follows
for both the restricted pointwise and restricted Reedy cofibration structures.
6.4. Colimits in direct categories (the relative case)
We have shown that given a cofibration category M and a small direct cate
gory D, then colimD carries Reedy cofibrations (resp. weak equivalences between
Reedy cofibrant diagrams) in MD to cofibrations (resp. weak equivalences between
cofibrant objects) in M. This result is extended below to the case of relative *
*col
imits from a small direct category to an arbitrary small category (Thm. 6.4.1) *
*and
between two small direct categories (Thm. 6.4.2).
Theorem 6.4.1.
(1) Let M be a cofibration category and u : D1 ! D2 be a functor between
small categories with D1 direct.
(a) If X is Reedy cofibrant in MD1, then colimuX exists and is pointwise
cofibrant in MD2
94 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
(b) If f : X ! Y is a Reedy cofibration in MD1, then colimuf is a
pointwise cofibration in MD2
(c) If f : X ! Y is a pointwise weak equivalence of Reedy cofibrant
objects in MD1, then colimuf is a pointwise weak equivalence in
MD2.
(2) Let M be a fibration category and u : D1 ! D2 be a functor between small
categories with D1 inverse.
(a) If X is Reedy fibrant in MD1, then limuX exists and is pointwise
fibrant in MD2
(b) If f : X ! Y is a Reedy fibration in MD1, then colimuf is a point
wise fibration in MD2
(c) If f : X ! Y is a pointwise weak equivalence of Reedy fibrant objec*
*ts
in MD1, then limuf is a pointwise weak equivalence in MD2.
Proof. We only prove statement (1)  statement (2) follows from duality.
Let us prove (a). To prove that colimuX exists, cf. Lemma 5.2.1 it suffices
to show for any object d2fflD2 that colim(u#d2)X exists. But the over category
(u # d2) is direct and the restriction of X to (u # d2) is Reedy cofibrant, the*
*refore
by Thm. 6.3.5 colim(u#d2)X exists and is cofibrant in M. It follows that colimuX
exists, and since colim(u#d2)X ~=(colimuX)d2we have that colimuX is pointwise
cofibrant in MD2.
We now prove (b). If f : X ! Y is a Reedy cofibration in MD1, then the
restriction of f to (u # d2) is Reedy cofibrant for any object d2fflD2, therefo*
*re by
Thm. 6.3.5 colim(u#d2)f is a cofibration in M. Since (colimuf)d2~= colim(u#d2)f
by the naturality of the isomorphism in Thm. 6.3.5, it follows that colimuf is
pointwise cofibrant in MD2.
To prove (c), assume that f : X ! Y is a pointwise weak equivalence be
tween Reedy cofibrant diagrams in MD1. The restrictions of X and Y to (u # d2)
are Reedy cofibrant for any object d2fflD2, and by Thm. 6.3.5 (1) (c) the map
colim(u#d2)X ! colim(u#d2)Y is a weak equivalence. In conclusion, the map
colimuX ! colimuY is a pointwise weak equivalence in MD2.
Theorem 6.4.2.
(1) Let M be a cofibration category and u : D1 ! D2 be a functor between
small direct categories.
(a) If X is Reedy cofibrant in MD1, then colimuX exists and is Reedy
cofibrant in MD2
(b) If f : X ! Y is a Reedy cofibration in MD1, then colimuf is a Reedy
cofibration in MD2
(c) If f : X ! Y is a pointwise weak equivalence of Reedy cofibrant
objects in MD1, then colimuf is a pointwise weak equivalence in
MD2.
(2) Let M be a fibration category and u : D1 ! D2 be a functor between small
inverse categories.
(a) If X is Reedy fibrant in MD1, then limuX exists and is Reedy fibrant
in MD2
(b) If f : X ! Y is a Reedy fibration in MD1, then limuf is a Reedy
fibration in MD2
6.5. COLIMITS IN ARBITRARY CATEGORIES 95
(c) If f : X ! Y is a pointwise weak equivalence of Reedy fibrant objec*
*ts
in MD1, then limuf is a pointwise weak equivalence in MD2.
Proof. We only prove statement (1)  statement (2) follows from duality.
Let us prove (a). We know from Thm. 6.4.1 (1) (a) that colimuX exists and
is pointwise cofibrant in MD2, and we would like to show that colimuX is Reedy
cofibrant. For that, fix an object d2 of D2, and let us try to identify the lat*
*ching
object colim@(D2#d2)colimuX. 0 0
The functor H : D2 ! Cat, Hd2 = (u # d2)Rrestricts to a functor @(D2 #
d2) ! Cat. The Grothendieck construction @(D2#d2)H has as objects 4tuples
(d01, d02, d2,0ud01! d02! d2) of objects d01fflD1, d02, d2fflD2 and maps ud01! *
*d02! d2
such that d2 ! d2 is a nonidentity map.
Denote @(u0# d2) the0full subcategory of the over category0(u # d2) consisti*
*ng
of triples (d1, d2, ud1 ! d2) withRa nonidentity map ud1 ! d2. We have an adjo*
*int
pair of functors F : @(u # d2) AE @(D2#d2)H : G, defined as follows:
F (d01, d2, ud01! d2) = (d01, ud01, d2, ud01! ud01! d2)
G(d01, d02, d2, ud01! d02! d2) = (d01, d2, ud01! d2)
id : 1@(u#d2)) GF
F G(d01,0d02,0d2,0ud01!0d02! d2) = (d01, ud01, d2, ud01!0ud01!0d2) )
(d1, d2, d2, ud1 ! d2 ! d2) given on 2nd component by ud1 ! d2.
The category @(u # d2) is direct, and the restriction of X to @(u # d2) is Reedy
cofibrant, therefore colim@(u#d2)X exists and is cofibrant.
G is a rightRadjoint functor, therefore right cofinal, and by Prop. 5.5.4 we
have that0colim @(D2#d2)HX exists0and is ~=colim@(u#d2)X. From Prop. 5.6.2,
colim(d2!d2)ffl@(D2#d2)colim(u#d2)X exists and is ~=colim@(u#d2)X. In conclusio*
*n,
the latching object colim@(D2#d2)colimuX exists and is ~=colim@(u#d2)X. In par
ticular the latching object is cofibrant.
The inclusion functor @(u # d2) ! (u # d2) is an open embedding, and from
Thm. 6.3.6 (1) (a) the latching map colim@(u#d2)X ! colim(u#d2)X is a cofibra
tion, therefore colimuX is Reedy cofibrant in MD2. The proof of statement (a) of
our theorem is complete.
Let us prove (b). If f : X ! Y is a Reedy cofibration, by Thm. 6.3.6 (1) (b)
the map
colim(u#d2)X tcolim@(u#d2)Xcolim@(u#d2)Y ! colim(u#d2)Y
is a cofibration, therefore colimuX ! colimuY is a Reedy cofibration in MD2.
Part (c) has been already proved as Thm. 6.4.1 (1) (c).
6.5. Colimits in arbitrary categories
Denote 0the subcategory of the cosimplicial indexing category , with same
objects as and with maps the orderpreserving0injective maps n1 ! n2.
If D is a category, we define D to be the category with objects the functo*
*rs
n ! D, and with maps (n1 ! D) ! (n2 ! D) the commutative diagrams
96 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
n1B_____f______//_n2
BB ___
BBB ___
B__B""__
D
where f is injective0and orderpreserving.
The category0 D is direct, and comes equipped with a terminal projection
functor pt : D ! D that sends n ! D to the image of the terminal object n of
the poset n. 0 0
The opposite category of D is denoted opD. It is an inverse0category,
and comes equipped with an initial projection functor pi : opD ! D that sends
n ! D to the image of the initial object 0 of the poset n.
Lemma 6.5.1. If u : D1 ! D2 is a functor and d2fflD2 is an object
(1) There is a natural isomorphism (upt# d2) ~= 0(u0# d2)
(2) There is a natural isomorphism (d2 # upi) ~= op(d2 # u)
Proof. Left to the reader.
Lemma 6.5.2. If D is a category and dfflD is an object
(1) (a) The category p1td has an initial object and hence has a contractib*
*le
nerve.
(b) The category (pt# d) has a contractible nerve.
(c) The inclusion p1td ! (pt# d) is homotopy right cofinal.
(2) (a) The category p1id has a terminal object and hence has a contractib*
*le
nerve.
(b) The category (d # pt) has a contractible nerve.
(c) The inclusion p1id ! (d # pi) is homotopy left cofinal.
Proof. We denote an object n ! D of 0D as (d0 ! ... ! dn), where di is
the image of iffln and di ! di+1 is the image of i ! i + 1. Each map i0: k ! n
determines a map (di0! ... ! dik) ! (d0 ! ... ! dn) in the category D.
We denote an object of (pt # d) as (d0 ! ... ! dn) ! d, where dn ! d is the
map pt(d0 ! ... ! dn) ! d. 0
The category p1td has the initial object (d)ffl D, which proves part (1) (*
*a).
For (1) (b), a contraction of the nerve of (pt# d) is defined by
________________________________________________*
*____________________________________________________________________________1*
*dOO
___________________________________________________*
*__________________________________________&&OOff'ffad
(pt# d)____________KS//_(pt# d)
___________________________________________________*
*__________________________________________________88OOOOfi
_______________________________________________*
*_________________________________________________________
cd
In this diagram, cd is the constant functor that takes as value the object (*
*d) 1d!
d. The functor ad sends an object (d0 ! ... ! dn) !f d to (d0 ! ... ! dn !f
_
d)_1d!d, and a map defined by i : k ! n to its extension i: k + 1! n + 1with
i(k + 1) = n + 1.
On an object (d0 ! ... ! dn) ! d, the natural map ff is given by the map
i : n ! n + 1, ik = k and the natural map fi is given by the map i : 0 ! n + 1,
i0 = n + 1.
6.5. COLIMITS IN ARBITRARY CATEGORIES 97
Let us prove (1) (c). Denote id : p1td ! (pt # d) the full inclusion funct*
*or.
Note that the images of ad and cd are inside id(p1td). Let x be an object of (*
*pt# d)
of the form (d0 ! ... ! dn) f!d.
If dn f!d is the identity map, then x is in the image of id therefore (x, x *
*1x!x) is
an initial object of (x # id). If dn f!d is not the identity map, then (adx, x *
*! adx)
defined by the map i : n ! n + 1, ik = k is an initial object in (x # id). In b*
*oth
cases, (x # id) is contractible therefore id is homotopy right cofinal.
The statements of part (2) follow from duality.
The category p1td0is direct for any object dfflD, since it is a subcategory*
* of the
direct category D. Dually, the category p1id is inverse for any object dfflD.
Lemma 6.5.3.
(1) Let M be a cofibration category and0D be a small category. Then for
any Reedy cofibrant diagram XfflM1D and any object dfflD, the restrict*
*ion
Xp1tdis Reedy cofibrant in Mpt d.
(2) Let M be a fibration category0and D be a small category. Then for any
Reedy fibrant diagram XfflM opDand1any object dfflD, the restriction
Xp1idis Reedy fibrant in Mpi d.
Proof. We only prove (1). For dfflD, fix an object d_= (d0 ! ... ! dn) ffl *
*0D
with dn = d. We need some notations. Assume that
i_= {i1, ..., iu}, j_= {j1, ..., jv}, k_= {k1, ..., kw}
is a partition of {0, ..., n} into three0(possibly empty) subsets with u+v+w = *
*n+1.
Denote n_i_,bj_the full subcategory of D. with objects dl0! ... ! dlxwith i_ *
* l_
and j_\ l_= ;, where l_= {l0, .., lx}. Although not apparent from the notation,*
* the
category n_i_,bj_depends on the choice of d_ffl 0D.
The category n_i_,bj_is direct, and has a terminal object denoted d_bj_. Den*
*ote @n_i_,bj_
the maximal full subcategory of n_i_,bj_without its terminal element.
Claim. The restriction of X to n_i_,bj_is Reedy cofibrant.
Taking in particular i_= {n} and j_= ;, the Claim implies that the Reedy
condition is satisfied for Xp1tdat d_. It remains to prove the Claim, and we *
*will
proceed by induction on n.
 The Claim can0be directly verified for n = 0. Assume that the Claim was
proved for n < n, and let's prove it for n 1.
 If j_6= ;, the claim follows from the inductive hypothesis for smaller *
*n. It
remains to prove the Claim for n_i_,b;_.
 We only need to prove the Reedy condition for X at the terminal object
d_b;_= d_of n_i_,b;_. The Reedy condition at any other object of n_i_,b*
*;_follows
from the inductive hypothesis for smaller n.
98 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
 If i_= {i1, ..., iu} is nonempty, in the diagram
(6.4) colim@n_{i1,...,iu}\{is},{dis}X//_fflfflcolim@n_{i1,...,iu},b;Xfflffl
 
 
fflffl fflffl
Xd_d{is}___________//colim@n_{i1,...,iu}\{is},b;X
the left vertical map is a cofibration with cofibrant domain, from the
inductive hypothesis for smaller n. If the top right colimit in the dia*
*gram
exists and is cofibrant, then the bottom right colimit exists, the diag*
*ram
is a pushout and therefore the right vertical map is a cofibration.
 The colimit colim@n_{0,...,n},b;X exists and is the initial object of M
 The map colim@n_;,b;X ! Xd_is a cofibration with cofibrant domain since
X is Reedy cofibrant in 0D.
 An iterated use of (6.4) shows that the colimit below exists and
colim@n_{i1,...,iu},b;X ! Xd_
is a cofibration with cofibrant domain. This completes the proof of the
Claim.
For a category D, the subcategories p1td 0D are disjoint0for dfflD, and *
*their
union0[dfflDp1td forms a0category that we will denote resD. We will also deno*
*te
ropesD the opposite of resD. 0
Given a cofibration0category0M we use the shorthand notation MreDsfor0the
category M( D, resD)of0 0resD restricted 0D diagrams in M. MreDsis a full
subcategory of M D , and (Thm. 6.3.9) it carries a restricted Reedy cofibration
structure as well as0a restricted pointwise cofibration0structure. Furthermore,*
* the
functor p*t: MD ! M D has its image inside MreDs.0 0 0
Dually, for a fibration0category M we denote MreopDsthe category M( opD, ro*
*pesD)
of restricted diagrams. MreopDscarries a restricted Reedy fibration structure0a*
*s well
as a restricted pointwise0fibration structure. The functor p*i: MD ! M opD has
its image inside MreopDs.
Proposition 6.5.4.
(1) Let M be a cofibration category and D be a small category.0 Then for
every restricted Reedy cofibrant diagrams X, X0fflMreDsand every diagram
Y fflMD
(a) A map X ! p*tY is a pointwise weak equivalence iff
its adjoint0colimptX ! Y is a pointwise weak equivalence.
(b) A map X ! X is a pointwise0weak equivalence iff
the map colimptX ! colimptX is a pointwise weak equivalence.
(2) Let M be a fibration category and D be a small0category. Then for every
restricted Reedy fibrant diagrams X, X0fflMreopDsand every diagram Y ff*
*lMD
(a) A map p*iY ! X is a pointwise weak equivalence iff
its adjoint Y ! limpiX is a pointwise weak equivalence.
6.5. COLIMITS IN ARBITRARY CATEGORIES 99
(b) A map X ! X0 is a pointwise0weak equivalence iff
the map limptX ! limptX is a pointwise weak equivalence.
Proof. We will prove (1), and (2) will follow from duality. Let d be an obj*
*ect
of D. The categories p1td and (pt # d) are direct, and since X is Reedy cofibr*
*ant
so are its restrictionsto1p1td(by Lemma 6.5.3) and to (pt# d).
The colimits colimpt dX, colim(pt#d)X thereforeexist.1 Since id : p1td !
(pt # d) is right cofinal, we have colimpt dX ~=colim(pt#d)X. We also conclude
that colimptX exists and colim(pt#d)X ~=(colimptX)d.
The diagram X is restricted and p1td has an initial object thatwe1will de
note e(d). We get a pointwise weak equivalence cXe(d)! Xp1tdin Mpt dfrom
the constant diagram to the restriction of X. But the diagram cXe(d)is Reedy
cofibrant since e(d) is initial in p1td. The map cXe(d)! Xp1tdis a pointwise
1d
weak equivalence between Reedy cofibrant diagrams in Mpt , therefore Xe(d)~=
1d p1d
colimpt cXe(d)! colim t X is a weak equivalence.
In summary, we have showed that the composition
1d (p #d) p
Xe(d)! colimpt X ~=colim t X ~=(colim tX)d
is a weak equivalence. We can now complete our proof.
To prove (a), the map colimptX ! Y is a pointwise weak equivalence iff the
map Xe(d)! Yd is a weak equivalence for all objects d of D. Since X is restrict*
*ed,
this last statement is true iff the map X ! p*tY0is a pointwise weak equivalenc*
*e.
To prove (b), the0map colimptX ! colimptX is a pointwise weak equivalence *
* 0
iff the map Xe(d)! Xe(d)is a weak equivalence for all objects d of D. Since X, X
are restricted, this last statement is true iff the map X ! X0 is a pointwise w*
*eak
equivalence.
As an application, we can now prove that the category of diagrams in a cofi
bration (resp. fibration) category admits a pointwise cofibration (resp. fibrat*
*ion)
structure. This result is due to Cisinski. We should note that the statement be*
*low
is not true if we replace "cofibration category" with "Quillen model category".
Theorem 6.5.5 (Pointwise (co)fibration structure).
(1) If (M, W, Cof) is a cofibration category and D is a small category then
(MD , WD , CofD ) is a cofibration category.
(2) If (M, W, Fib) is a fibration category and D is a small category then (*
*MD ,
WD , FibD) is a fibration category.
Proof. To prove (1), axioms CF1CF3 and CF5CF6 are easily verified. To
prove axiom CF4, we replay the argument in the proof of Thm. 6.2.4 (1) (b) part
(iv). Let f : X ! Y be a map of Ddiagrams with X pointwise0cofibrant. Let
a : X1 ! p*tX be a Reedy cofibrant replacement in MreDs. We factor X1 ! p*tY
as a Reedy cofibration f1 followed by a pointwise weak equivalence r1
f1 r1 *
X1 //__//_Y1_~_//_ptY
We then construct a commutative diagram
100 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
ggggggY33::uCC
gfgggggggggg uuu
ggggggg r uuu
gggggg uuu
XO//_______0________//gYO0~ 0
f OO r1
a0~ b0~
 
colimptX1 //colimptf1//_colimptY1
In this0diagram0a0resp. r01are the adjoints of a resp. r1, therefore by Prop. 6*
*.5.4 (1)
(a) a and r1 are weak equivalences. Since0f1 is a Reedy cofibration, colimptf1 *
*is a
pointwise0cofibration, and we construct f as the pushout of colimptf1.0It foll*
*ows
that f is a pointwise cofibration. By pointwise excision,0b and therefore r *
*are
pointwise weak equivalences. The factorization f = rf is the desired decomposi*
*tion
of f as a pointwise cofibration followed by a weak equivalence, and CF4 is prov*
*ed.
The proof of (2) is dual.
As an immediate corollary, we can show that for a small category pair (D1, D*
*2),
the category of reduced diagrams M(D1,D2)carries a pointwise cofibration struct*
*ure
if M is a cofibration category.
Theorem 6.5.6 (Reduced pointwise (co)fibration structure).
(1) If (M, W, Cof) is a cofibration category and (D1, D2) is a small catego*
*ry
pair, then (M(D1,D2), WD1, Cof(D1,D2)) is a cofibration category  call*
*ed
the D2restricted pointwise cofibration structure on M(D1,D2).
(2) If (M, W, Fib) is a fibration category and (D1, D2) is a small category
pair, then (M(D1,D2), WD1, Fib(D1,D2)) is a fibration category  called*
* the
D2 restricted pointwise fibration structure on M(D1,D2).
Proof. Entirely sililar to that of Thm. 6.3.9.
0 0
Denote MreDs,rcofthe full subcategory of MreDsof restricted Reedy cofibrant
diagrams for a cofibration category0M. The next proposition states that restric*
*ted
Reedy cofibrant diagrams in M D form a cofibrant0approximation of MD .
Dually for a fibration category M denote MreopDs,rfibthe full subcategory of
0op
MresD of restricted0Reedy fibrant diagrams. We show that restricted Reedy fi
brant diagrams in M opD form a fibrant approximation of MD .
Proposition 6.5.7. Let D be a small category.
0
(1) If M is a cofibration category then colimpt: MreDs,rcof! MD is a cofibr*
*ant
approximation for the pointwise cofibration0structure on MD .
(2) If M is a fibration category then limpi: MreopDs,rfib! MD is a fibrant
approximation for the pointwise fibration structure on MD .
Proof. We only prove (1), since the proof of (2) is dual.
The functor colimptsends Reedy cofibrations to pointwise cofibrations by Thm.
6.3.5 and preserves the initial object, which proves CFA1. CFA2 is a consequence
of Prop. 6.5.4 (1) (b). CFA3 is a consequence of Remark 5.6.4.
6.5. COLIMITS IN ARBITRARY CATEGORIES 101
For CFA4, let f : colimptX !0Y be a map in MD with X restricted and Reedy
cofibrant. Factor its adjoint f : X ! p*tY as a Reedy cofibration f1 followed *
*by a
pointwise weak equivalence r1
f1 0r1 *
X //___//Y__~__//ptY
We get the following CFA4 factorization of f
colimptf1 0r01
colimptX //_________//colimptY__~_//_Y
where r01is the adjoint of r1, therefore a weak equivalence.
Here is another application of Prop. 6.5.4:
Theorem 6.5.8. Let D be a small category.
(1) If M is a cofibration category then
0
ho p*t: hoMD ! hoMreDs
is an equivalence of categories.
(2) If M is a fibration category then
0op
hop*i: hoMD ! hoMresD
is an equivalence of categories.
Proof. Let us prove (1). We will apply the Abstract Partial Quillen Adjunc
tion Thm. 4.5.10 to
0 v1=colimpt
MreDs,rcof_________//MD
`" OO
t1 t2=1MD
0fflfflv2=p* 
oo______t___
MreDs MD
0 0
The functor t1 is the full inclusion of MreDs,rcofin MreDs. We have that v1, v2*
* is an
abstract Quillen partially equivalent pair with respect to t1, t2. Indeed:
(1) The functor pair v1, v2 is partially adjoint with respect to t1,0t2.
(2) t1 is a cofibrant approximation of the cofibration category MreDswith t*
*he
Reedy reduced structure. In particular t1 is a left approximation. The
functor t2 = 1MD is a right approximation.
(3) v1 preserves weak equivalences from Thm. 6.4.1, and so0does v2 = p*t.
(4) Prop. 6.5.4 (1) (a) states that for any objects XfflMreDs,rcof, Y fflMD*
* , a map
v1X ! t2Y is a weak equivalence iff its partial adjoint t1X ! v2Y is a
weak equivalence
In conclusion we have a pair of equivalences of categories
0 ________ho(v1)_s1_____//
ho MreDsoo___ho(v2)_s2~=ho(p*_ hoMD
t)
102 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
where s1 is a quasiinverse of hot1 and s2 is a quasiinverse of hot2, and ther*
*e
fore ho(v2) s2 is naturally isomorphic to ho(p*t). This proves that ho(p*t) is*
* an
equivalence of categories.
The proof of part (2) is dual.
6.6. Homotopy colimits
Suppose that (M, W) is a category with weak equivalences and suppose that
u : D1 ! D2 is a small functor. We denote flDi : MDi ! hoMDi, i = 1, 2 the
localization functors.
We define MD1colimuto be the full subcategory of MD1 of D1 diagrams X with
the property that colimuX exists in MD2. Denote icolimu: MD1colimu! MD1 the
inclusion. In general MD1colimumay be empty, but if M is cocomplete then MD1col*
*imu=
MD1. Let WD1colimube the class of pointwise weak equivalences of MD1colimu.
Dually, let MD1limube the full subcategory of MD1 of D1 diagrams X with the
property that limuX exists in MD2, let ilimu: MD1limu! MD1 denote the inclusion,
and let WD1limube the class of pointwise weak equivalences of MD1limu.
Definition 6.6.1.
(1) The homotopy colimit of u, if it exists, is the left Kan extension of f*
*lD2 colimu
along flD1icolimu
u
MD1colimu__colim___//_MD2
flD1icolimu fjjjjDLflu flD2
fflffl fflffl
hoMD1 __L_colimu//_hoMD2
and is denoted for simplicity (L colimu, fflu) instead of the more comp*
*lete
notation (LflD1icolimucolimu, fflu)
(2) The homotopy limit of u, if it exists, is the right Kan extension of fl*
*D2 limu
along flD1ilimu
u
MD1limu___lim_____//MD2
flD1ilimu jjjj fiu flD2
fflffl fflffl
hoMD1 ___R_limu_//_hoMD2
and is denoted for simplicity (R limu, u)
If (M, W, Cof) is a cofibration category, then we also define the class CofD*
*1colimu
of pointwise cofibrations f : X ! Y in MD1colimuwith the property that colimuf :
colimuX ! colimuY is well defined and pointwise cofibrant in MD2.
Dually, if (M, W, Fib) is a fibration category, then we define the class Fib*
*D1limu
of pointwise fibration maps f in MD1limuwith the property that limuf is well de*
*fined
and pointwise fibrant in MD2.
Lemma 6.6.2. Let u : D1 ! D2 be a small functor.
(1) If M is a cofibration category, then
(a) (MD1colimu,0WD1colimu, CofD1colimu) is a cofibration category
(b) colimpt: MreD1s,rcof! MD1colimuis a cofibrant approximation
6.6. HOMOTOPY COLIMITS 103
(c) hoMD1colimu! hoMD1 is an equivalence of categories.
(2) If M is a fibration category, then
(a) (MD1limu,0WD1limu, FibD1limu) is a cofibration category
opD D
(b) limpi: Mres,r1fib! Ml1imuis a fibrant approximation
(c) hoMD1limu! hoMD1 is an equivalence of categories.
Proof. We only prove0(1), since the proof of (2) is dual.
For an object XfflMreD1s,rcof, we have that the colimits colimptX, colimuptX
exist0and are pointwise cofibrant by Thm. 6.4.1 since X is Reedy cofibrant in
M D1. Since0colimuptX ~=colimucolimptX, we conclude that the functor
colimpt: MreD1s,rcof! MD1 has its image inside MD1colimu.
Let us prove (a). Axioms CF1CF2 and CF5CF6 are easily verified for MD1coli*
*mu.
The pushout axiom CF3 (1) follows from the fact that if
Xf_____//flfflZfflfflO
i OjO
fflfflfflffl
Y ` ` `//T
is a pushout in MD1 with X, Y, Z cofibrant in MD1colimuand i a cofibration in M*
*D1colimu,
then colimuX, colimuY, colimuZ are pointwise cofibrant and colimui is a point
wise cofibration in MD2, therefore by Remark 5.6.4 we have that colimuT exists
and is the pushout in MD2 of
colimuXf_____//flfflcolimuZfflfflO
colimui OcolimujO
fflffl fflffl
colimuY ` ` `//colimuT
The axiom CF3 (2) follows from a pointwise application of CF3 (2) in M.
Let us prove the factorization axiom CF4. We repeat the argument in the proof
of Thm. 6.5.5.
Let f : X ! Y be a map in MD1colimuwith0X cofibrant. Let a : X1 ! p*tX
be a Reedy cofibrant replacement in MreD1d. We factor X1 ! p*tY as a Reedy
cofibration f1 followed by a pointwise weak equivalence r1
f1 r1 *
X1 //__//_Y1_~_//_ptY
We then construct a commutative diagram
ggggggY33::uCC
gfgggggggggg uuu
ggggggg r uuu
gggggg uuu
XO//_______0________//gYO0~ 0
f OO r1
a0~ b0~
 
colimptX1 //colimptf1//_colimptY1
104 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
In this0diagram0a0resp. r01are the adjoints of a resp. r1, therefore by Prop. 6*
*.5.4 (1)
(a) a and r1 are weak equivalences. Since f10is a Reedy cofibration, colimptf1 *
*is a
cofibration in MD1colimu, and we construct f as the pushout of colimptf1. It f*
*ollows
that f0 is a cofibration of MD1colimu. By pointwise excision, b0 and therefore *
*r are
pointwise weak equivalences. The factorization f = rf0 is the desired decomposi*
*tion
of f as a pointwise cofibration followed by a weak equivalence in MD1colimu, an*
*d CF4
is proved. 0
Part (b) follows from Prop. 6.5.7 and the fact that colimpt: MreD1s,rcof! MD1
has its image inside MD1colimu. 0
To prove part (c), since both functors colimpt: MreD1s,rcof! MD1 and its
0
corestriction colimpt: MreD1s,rcof! MD1colimuare cofibrant approximations it fo*
*llows
0 0
(Thm. 4.6.3) that both induced functors hoMreD1s,rcof! hoMD1 and hoMreD1s,rcof!
hoMD1colimuare equivalences of categories. The functor hoMD1colimu! hoMD1 is
therefore an equivalence of categories.
We now state the main result of this section.
Theorem 6.6.3 (Existence of homotopy (co)limits).
(1) Let M be a cofibration category and u : D1 ! D2 be a small functor. Then
the homotopy colimit (L colimu, fflu) exists and
L colimu: hoMD1 AE hoMD2 : hou*
forms a naturally adjoint pair.
(2) Let M be a fibration category and u : D1 ! D2 be a small functor. Then
the homotopy limit (R limu, u) exists and
hou* : hoMD2 AE hoMD1 : R limu
forms a naturally adjoint pair.
Proof. Parts (1) and (2) are dual, and we will only prove part (1).
From Lemma 6.6.2 and Cor. 4.3.5, to prove the existence of the left Kan
extension (L colimu, fflu) it suffices to prove the existence of the total left*
* derived of
colimu: MD1colimu! MD2. But the latter is a0consequence of Thm. 4.6.2 applied
to the cofibrant approximation colimpt: MreD1s,rcof! MD1colimu.
To prove that L colimua hou* forms a naturally adjoint pair, we will apply
the Abstract Quillen Partial Adjunction Thm. 4.5.10 to
0 v1=colimupt
MreD1s,rcof________//_MD2OO

t1=colimpt t2=1MD2
fflfflv2=u* 
MD1 oo___________MD2
We have that v1, v2 is an abstract Quillen partially adjoint pair with respe*
*ct to
t1, t2:
(1) The functor pair v1, v2 is partially adjoint with respect to t1, t2.
(2) t1 is a cofibrant approximation of the cofibration category MD1 by Prop.
6.5.7. In particular t1 is a left approximation. The functor t2 = 1MD2 *
* is
a right approximation.
6.6. HOMOTOPY COLIMITS 105
(3) v1 preserves weak equivalences by Thm. 6.4.1, and so does v2 = p*t.
In conclusion we have a naturally adjoint pair
____ho(v1)_s1~=L_colimu//_
hoMD1 oo__ho(v___________*__hoMD2
2) s2~=ho(u )
where s1 is a quasiinverse of hot1 and s2 is a quasiinverse of hot2.
As a consequence of Thm. 6.6.3 we can verify that
Corollary 6.6.4. Suppose that u : D1 ! D2 and v : D2 ! D3 are two small
functors.
(1) If M is a cofibration category, then L colimvu~=L colimvL colimu.
(2) If M is a fibration category, then R limvu~=R limvR limu.
Proof. This is a consequence of the adjunction property of the homotopy
(co)limit and the fact that ho(vu)* ~=hou*hov*.
Consider a small diagram
(6.5) D1 __u__//D2
f OE=E g
fflffl fflffl
D3 __v__//D4
If M is a cofibration category, from the adjunction property of the homotopy co*
*limit
we get a natural map denoted
OEL colim: L colimuhof* ) hog*L colimv
and dually if M is a fibration category we get a natural map denoted
OER colim: hov*R limg) R limfhou*
Suppose that u : D1 ! D2 is a small functor. For any object d2fflD2 , the
standard over 2category diagram of u at d2 is defined as
p(u#d2)
(6.6) (u # d2)____//_e
iu,d2OEu?G,ded22
fflffl fflffl
D1 ___u__//_D2
In this diagram, the functor iu,d2is as defined in Section 5.1.2, and pD : D*
* ! e
denotes the terminal category projection. The functor ed2 embeds the terminal
category eas the object d2fflD2, and for an object (d1fflD1, f : ud1 ! d2) of (*
*u # d2)
the natural map OEu,d2(d1, ud1 ! d2) is f : ud1 ! d2. If M is a cofibration cat*
*egory,
we obtain a natural map
(6.7) Lcolim(u#d2)X ) (LcolimuX)d2
Dually, the standard under 2category diagram of u at d2 is
id2,u
(6.8) (d2 # u)____//D1
p(d2#u)OE?Gd2,uu
fflffl fflffl
e __ed2__//_D2
106 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
If M is a fibration category, we obtain a natural map
(6.9) (RlimuX)d2) Rlim(d2#u)X
The next theorem proves a base change formula for homotopy (co)limits. This
lemma is a homotopy colimit analogue to the well known base change formula for
ordinary colimits Lemma 5.2.1.
Theorem 6.6.5 (Base change property).
(1) If M is a cofibration category and u : D1 ! D2 is a small functor, then
the natural map (6.7) induces an isomorphism
Lcolim(u#d2)X ~=(LcolimuX)d2
for objects XfflMD1 and d2fflD2.
(2) If M is a fibration category and u : D1 ! D2 is a small functor, then t*
*he
natural map (6.9) induces an isomorphism
(RlimuX)d2~=Rlim(d2#u)X
for objects XfflMD1 and d2fflD2.
Proof. We only prove0(1).0 For a diagram XfflMD1, pick a0reduced Reedy
cofibrant replacement X ffl0resD1 of p*tX. We have colimuptX0 ~=LcolimuX. By
Lemma06.5.1, (upt# d2) ~= (u # d2). The restriction of X to the direct0catego*
*ry
(u # d2) is a Reedy cofibrant0replacement of the restriction of X to (u # d*
*2),
so colim(upt#d2)X0 ~=colim (u#d2)X0 ~=Lcolim(u#d2)X. Using Lemma 5.2.1, the
top map and therefore all maps in the commutative diagram
0
colim(upt#d2)X0____//_(colimuptX )d2
 
 
fflffl fflffl
Lcolim(u#d2)X______//(LcolimuX)d2
are isomorphisms, and the conclusion is proved.
Suppose that (M, W) is a category with weak equivalences, and that u : D1 !
D2 is a small functor. The next result describes a sufficient condition for (L *
*colimu,
fflu) to exist  without requiring M to carry a cofibration category structure.
In preparation, notice that the natural map (6.7) actually exists under the
weaker assumption that (M, W) is a category with weak equivalences, that Lcolimu
exists and is a left adjoint of hou*, and that Lcolim(u#d2)exists and is a left*
* adjoint
of hop*(u#d2). A dual statement holds for the map (6.9).
Theorem 6.6.6. Suppose that (M, W) is a pointed category with weak equiva
lences.
(1) If u : D1 ! D2 is a small closed embedding functor, then
(a) colimuand its left Kan extension (L colimu, fflu) exist, and L coli*
*mu
is a fully faithful left adjoint to hou*
(b) For any object d2 of D2, the functors colim(u#d2)and Lcolim(u#d2)
are well defined and the natural map (6.7) induces an isomorphism
Lcolim(u#d2)X ~=(LcolimuX)d2
for objects XfflMD1 and d2fflD2.
(2) If u : D1 ! D2 is a small open embedding functor, then
6.6. HOMOTOPY COLIMITS 107
(a) limuand its right Kan extension (R limu, u) exist, and R limuis a
fully faithful left adjoint to hou*
(b) For any object d2 of D2, the functors lim(d2#u)and Rlim(d2#u)are
well defined and the natural map (6.9) induces an isomorphism
(RlimuX)d2~=Rlim(d2#u)X
for objects XfflMD1 and d2fflD2.
Proof. We start with (1) (a). Denote u!: MD1 ! MD2 the 'extension by zero'
functor, that sends a diagram XfflMD1 to the diagram given by (u!X)d2= Xd2 for
d2ffluD1 and (u!X)d2= 0otherwise. The functor u!is a fully faithful left adjoin*
*t to
u*, and sends weak equivalences to weak equivalences. In particular, colimu~=u1
exists and is defined on the entire MD1, and L colimuas in Def. 6.6.1 exists an*
*d is
isomorphic to hou!. We apply the Abstract Quillen Adjunction Thm. 4.5.8 to
t1=id// u1=u!//_ t2=idoo
MD2 _____ MD2 uoo_*MD1_ _____MD1
2=u
observing that its hypotheses (1)(3) and (4l) apply. We deduce that L colimu~=
hou!is a fully faithful right adjoint to hou*.
To prove the isomorphism (1) (b), observe that (u # d2) has d2 as a terminal
object if d2ffluD1 and is empty otherwise, so colim(u#d2)X ~=Xd2if d2ffluD1 and*
* ~=0
otherwise. Both functors colim(u#d2)and colimupreserve weak equivalences, and
we have adjoint pairs Lcolim(u#d2)a hop*(u#d2)and Lcolimu a hou*. The natural
isomorphism colim(u#d2)X ~=(colimuX)d2 yields the desired isomorphism (6.7).
The proof of part (2) is dual.
The two lemmas below are part of the proof of Thm. 6.6.9 below. We keep
the notations used in Thm. 6.6.6 and its proof, and introduce a few new ones. M
denotes a pointed cofibration category, and u : D1 ! D2 a small closed embedding
functor. v : D2\D1 !0D2 denotes0the inclusion0functor  it is an open embedding.
We also denote V = v : (D2\D1) ! D2.
We denote u!= colimu: MD1 ! MD2 and v* = limu: MD2\D1 ! MD2  these
are the 'extension by zero' functors. We also denote v!= colimvand V!= colimV,
and we will0keep in mind that they are defined only on a full subcategory of MD*
*2\D1
resp. M (D2\D1).
We denote (MD2)0 the full subcategory of MD2 consisting of objects X with
the property that v!v*X exists and is pointwise cofibrant, and the map v!v*X ! *
*X0
is a pointwise cofibration. 0 is a cofibrant object, and we define the functor *
*u1 :
(MD2)0 ! MD2 as the pushout
v!v*X //____//XO
 O
 O
fflffl fflffl0
0 //``` `//u1X
Lemma 6.6.7.
(1) There exists a canonical partial adjunction
108 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
u01
(MD2)0`_____//"MD2OO
 1
  MD2
fflffl 
MD2 ou!u*oMD2_
(2) There exists a canonical partial adjunction
u*u01
(MD2)0`_____//"MD1OO
 1
  MD1
fflffl 
MD2 oou!__MD1
Proof. Denote fflX : v!v*X ae X the natural map defined for Xffl(MD2)0.
For any object ZfflMD2, the diagram
u!u*Z_______//Z
 
  Z
fflffl fflffl
0_______//v*v*Z
is both a pushout and a pullback. For Xffl(MD2)0, the maps X ! u!u*Z are in
a 11 correspondence with maps f : X ! Z such that Zf : X ! v*v*Z is null,
therefore in 11 correspondence with maps f : X ! Z such0that ffflX : v!v*X ! Z
is null, therefore in 11 correspondence0with maps u1X ! Z. This shows that we
have a natural bijection Hom(u1X, Z) ~=Hom(X, u!u*Z), which_proves (1).
__ For an object Y fflMD1_and a map u*u01X ! Y , denote YX fflMD2 the object wi*
*th
YXd= Yd for_dfflD1 and Y Xd=_(u01X)d_otherwise. We see that Hom(u*u01X, Y ) ~=
Hom(u01X, YX ) ~=Hom(X, u!u*Y X) ~=Hom(X, u!Y ), which completes the proof
of (2).
0
Lemma 6.6.8. For0any diagrams0Y, Y 0fflM D2 that are Reedy cofibrant, denote
X = colimptY and X = colimptY in MD2. Then
(1) The colimit v!v*X exists and is pointwise cofibrant, and v!v*X ae X is a
pointwise0cofibration 0
(2) If Y ! Y is a pointwise weak equivalence, then so is v!v*X ! v!v*X
Proof. The objects of 0D2 are all of the form
d_= (d0 ! ... ! di! d0i+1! ... ! d0n)
*
* 0
where d0, ..., difflD2\D1 and d0i+1, ..., d0nfflD1. As a consequence, given Y f*
*flM (D2\D1)
by Lemma 5.2.1 we have (V!Y )d_~=Yd0!...!di. The functor V! is thus defined on
0
the entire M (D2\D1) note that v!may not be defined on the entire MD2\D1.
The latching map of V!Y at d_is LYd_! Yd_if i = n, and Yd0!...!diid!Yd0!...!*
*di
0
if i < n. Based on this, we see that if Y fflM D2 is Reedy cofibrant then V *Y*
* is
Reedy cofibrant and V!V *Y ! Y is a Reedy cofibration.
6.6. HOMOTOPY COLIMITS 109
0
Pick Y fflMreD2s,rcofwith colimptY = X. Using Lemma 5.2.1 we see that v*X ~=
colimptV *Y .
0D \D V 0
Mrco2f 1 _____//MrcD2of
colimpt colimpt
fflfflv fflffl
MD2\D1 _______//MD2
The colimits colimptV *Y ~=v*X and colimptV!V *Y exist (the latter since V!V *Y
is Reedy cofibrant). By Lemma 5.2.2 we have that v!colimptV *Y ~=v!v*X exists
and is ~=colimptV!V *Y . Applying colimptto the Reedy cofibration V!V *Y ! Y
yields v!v*X ! X, which is a pointwise cofibration between pointwise cofibrant
objects by Thm.06.4.1. This proves part (1).
If Y ! Y is a pointwise0weak equivalence between Reedy cofibrant objects, *
* 0
then so is V!V *Y ! V!V *Y , so by Thm. 6.4.1 the map colimptV!V *Y ! colimptV!*
*V *Y
is a weak equivalence. This proves part (2).
Theorem 6.6.9.
(1) If M is a pointed cofibration category and u : D1 ! D2 is a small closed
embedding functor, then L colimuadmits a left adjoint.
(2) If M is a pointed fibration category and u : D1 ! D2 is a small open
embedding functor, then R limuadmits a right adjoint.
Proof. We only prove (1). We will apply the Abstract Partial Quillen Ad
junction Thm. 4.5.10 to
0 v1=u*u01colimpt
MreD2s,rcof_____________//_MD1OO

t1=colimpt t2=1MD1
fflffl v2=u! 
MD2 oo_________________MD1
We have that v1, v2 is an abstract Quillen partially adjoint pair with respect *
*to
t1, t2:
(1) From Lemma 6.6.8 (1) we have Im colimpt (MD2)0, so the functor v1
is correctly defined. Using Lemma 6.6.7 (2) we see that v1, v2 are part*
*ially
adjoint with respect to t1, t2.
(2) The functor t1 is a cofibrant approximation of MD2 by Prop. 6.5.7, ther*
*e
fore a left approximation. The functor t2 = 1MD1 is a right approximati*
*on,
and u2t2 preserves weak equivalences.
(3) By0Lemma 6.6.8 (2), the functor v!v*t1 preserves weak equivalences. But
u1t1Y is the pushout of v!v*t1Y ae t1Y0by v!v*t1Y ! 0, and an applica
tion of the Gluing Lemma shows that u1t1 and therefore v1 also preserve
weak equivalences.
(4) The functor v2 preserves weak equivalences.
We conclude that Rv2 ~=hou!~=L colimuadmits a left adjoint.
110 6. HOMOTOPY COLIMITS IN A COFIBRATION CATEGORY
6.7. The conservation property
Recall that a family of functors ui: A ! Bi, ifflI is conservative if for an*
*y map
ffflA with uif an isomorphism in Bi for all ifflI we have that f is an isomorph*
*ism
in A. A family of functors {ui} is conservative iff the functor (ui)i : A ! xiB*
*i is
conservative.
Theorem 6.7.1. Let D be a small category, and suppose that M is either
a cofibration or a fibration category. The projections pd : MD ! M on the
d componenent for all objects dfflD then induce a conservative family of functo*
*rs
ho(pd) : hoMD ! hoM.
Proof. Assume that M is a cofibration category (the proof for fibration cat
egories_is_dual). __
Let f : A ! B be a map in hoMD such that ho(pd)ffflho_M are isomorphisms
for all objects dfflD. We want to show that f is an isomorphism in hoMD .
Using the factorization axiom CF4 applied to the pointwise cofibration struc
ture (MD , WD , CofD), we may assume that A, B are pointwise cofibrant._ From
Thm. 3.4.5 applied to (MD , WD , CofD), we may further assume that f is the
image of a pointwise cofibration f of MD .
0Assume we proved our theorem for all direct small categories. The map p*tf in
M D satisfies the hypothesis0of our theorem and 0D is direct. It follows that*
* p*tf
is an isomorphism0in hoM D , so by Lemma 3.7.1 there exist pointwise cofibrati*
*ons0
f0, f00fflM D such0that f0p*f, f00f0 are pointwise weak0equivalences0in0M D .
But p*f is resD restricted, and therefore0so are f and f . By the same
Lemma 3.7.1 p*f is an isomorphism in ho MreDs, and by Thm. 6.5.8 f is an
isomorphism in hoMD
It remains to prove_our theorem in the case when D is direct._
As before, let f : A ! B be a map in hoMD such that ho(pd)ffflho_M are
isomorphisms for all objects dfflD, and we want to show that f is an isomorphis*
*m in
hoMD . Repeating the previous argument applied to the Reedy_cofibration struc
ture (MD , WD , CofDreedy), we may further assume that f is the image of a Reedy
cofibration f : A ! B. 0 0 0
We will construct a Reedy cofibration f : B ! B such that f ffflWD . Once0t*
*he
construction is complete, we0will0be0able0to0apply the0same0construction0to f *
*and
obtain a Reedy cofibration f : B ! B such that f f fflWD . As a consequence,
it will follow that f is an isomorphism in hoMD .
To summarize, given a Reedy cofibration f : A ! B with the property that0fd *
* 0
is an isomorphism0in hoM, it remains to construct0a Reedy cofibration f : B ! B
such that f ffflWD . We will construct f by induction0on degree.
For n = 0,0D0 is discrete0and the existence of f 0 follows from Lemma 3.7.1.
Assume now f 0, then there exists ffflHomhoMD (X,*
* Y )
such that f restricts to fn for all n 0.
Proof. Assume that M is a cofibration category and that D is a small direct
category. (The proof using the alternative hypothesis is dual).
We may assume that X, Y are Reedy cofibrant. We fix a sequence of cylinders
with respect to the Reedy cofibration structure InX and InY , and denote I1 X =
colim(X i0!IX i0!I2X...) and I1 Y = colim(Y !i0IY !i0I2Y...). Denote jn : X !
InX and kn : Y ! InY the trivial cofibrations given by iterated compositions of
i0. Using axiom CF6, the maps j1 = colimjn : X ! I1 X and k1 = colimkn :
Y ! I1 Y are trivial cofibrations.
By induction on n, we will construct a factorization in D n
an // oobn
InX n _____Zn __~__ InY n
with Zn Reedy cofibrant and bn a weak equivalence in D n , such that b1nan has
the homotopy type of fn, and a trivial Reedy cofibration cn1 in D