OBSTRUCTION THEORY IN MODEL CATEGORIES
J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
Abstract.Many examples of obstruction theory can be formulated as the
study of when a lift exists in a commutative square. Typically, one of *
*the
maps is a cofibration of some sort and the opposite map is a fibration, *
*and there
is a functorial obstruction class that determines whether a lift exists.*
* Work
ing in an arbitrary pointed proper model category, we classify the cofib*
*rations
that have such an obstruction theory with respect to all fibrations. Up *
*to weak
equivalence, retract, and cobase change, they are the cofibrations with *
*weakly
contractible target. Equivalently, they are the retracts of principal co*
*fibrations.
Without properness, the same classification holds for cofibrations with *
*cofibrant
source. Our results dualize to give a classification of fibrations that *
*have an
obstruction theory.
1. Introduction
The following extensionlifting problem is ubiquitous in modern homotopy theo*
*ry.
Consider a commutative square
A ____//_X
i p
fflffl fflffl
B ____//_Y
in which i is a cofibration (or less technically, some kind of monomorphism) wh*
*ile
p is a fibration (or some kind of epimorphism). When does a map B ! X exist
making both triangles commute?
Classical obstruction theory [14] [15] gives a detection principle for existe*
*nce (and
uniqueness) of lifts in the category of spaces in terms of homotopy theory.
In this paper, we show how some aspects of classical obstruction theory are e*
*n
tirely abstract, working the same for any model category. In addition to provid*
*ing
tools for lifting in nontopological contexts, this approach enlightens classic*
*al ob
struction theory by showing that much of it does not depend on specific propert*
*ies
of topological spaces.
We start with a pointed model category C. Examples include pointed topological
spaces, pointed simplicial sets, spectra, and chain complexes of modules over a*
* ring.
Fix a cofibration i. We say that i has an obstruction theory if there exists s*
*ome
object W in C such that for every commutative square as above, there is a well
defined weak homotopy class from W to the fibre of p (called the obstruction), *
*such
____________
Date: November 6, 2002.
1991 Mathematics Subject Classification. 55S35, 55U35, 18G55 (primary); 18G3*
*0, 55P42
(secondary).
Key words and phrases. obstruction theory, closed model category, simplicial*
* set, spectrum.
The first author was supported by an NSERC grant.
The third author was supported by an NSF Postdoctoral Research Fellowship.
1
2 J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
that a lift exists in the square if and only if the map from W to the fibre is *
*weakly null
homotopic. We also require that the obstruction class be functorial in the fibr*
*ation
p (see Definition 4.1).
Starting from this very general framework, we prove a theorem precisely class*
*ifying
the cofibrations that have obstruction theories. Assuming that C is both left a*
*nd right
proper, the class of cofibrations having obstruction theories is the smallest c*
*lass of
cofibrations containing all cofibrations with weakly contractible target and cl*
*osed
under retract, weak equivalence, and cobase change. In other words, they are the
retracts of principal cofibrations. Although W is not uniquely determined by t*
*he
definitions, when i is a principal cofibration, W can be taken to be a desuspen*
*sion
of the cofibre (see Remark 7.3). Moreover, the category of obstruction theorie*
*s is
contractible (see Subsection 4.2).
Our theorem explains why some kind of hypotheses are needed in classical obst*
*ruc
tion theory. It is not possible to give an obstruction theory for topological s*
*paces
without some kind of restriction to principal cofibrations or fibrations.
We obtain slightly weaker results when C is not proper. Without left properne*
*ss,
we must assume that the source of the cofibration is cofibrant. Without right p*
*roper
ness, we must assume that the targets of the fibrations are fibrant. Here the n*
*otion
of a fibrant obstruction theory becomes useful. The essential reason that fibra*
*ncy
and cofibrancy assumptions take the place of properness is that base changes al*
*ong
fibrations preserve weak equivalences between fibrant objects and cobase changes
along cofibrations preserve weak equivalences between cofibrant objects. The po*
*int
is that some result about base changes and cobase changes of weak equivalences *
*is
necessary; this result either comes by assumption from properness or from having
enough cofibrancy and fibrancy.
Obstruction theories in the sense of this paper have been used for simplicial*
* sets
with its usual model structure in [6] and [10]. The obstruction for lifting a s*
*tandard
generating cofibration with respect to a fibration is an element of a homotopy *
*group
of a fibre. See Section 8.1 for more details. This paper grew out of understand*
*ing
precisely how this special case works.
In a stable model category (see Section 8.3) we show that every cofibration h*
*as
an obstruction theory. In the special case of the model category of unbounded c*
*hain
complexes of objects in an abelian category, the results of the present paper p*
*rovide
a conceptual explanation for results such as [3, Lemma 2.3]. For spectra, obstr*
*uction
theory for the standard generating cofibrations is used in [12]. See Section 8.*
*3 for
more details. In addition, in the stable model category of prospectra, the res*
*ults of
this paper produce new tools for handling lifting problems, tools which were us*
*ed in
early versions of [4].
We work exclusively on the question of when cofibrations have obstruction the
ories, but everything dualizes to the study of fibrations that have coobstruct*
*ion
theories. A fibration p has a coobstruction theory if there exists some object*
* W in C
such that for every commutative square as above, there is a welldefined weak h*
*omo
topy class from the cofibre of i to W (called the obstruction), such that a lif*
*t exists
in the square if and only if the map from the cofibre to W is weakly nullhomot*
*opic.
We also require that the obstruction class be functorial in the cofibration i. *
* Co
obstruction theory is a key technique of [11]. Also, this notion of coobstruc*
*tion
theory is precisely what appears in [14, x 8.2] for the special case of a princ*
*ipal fibra
tion of topological spaces whose fibre is an EilenbergMac Lane space K(ß, n  *
*1).
In this case, W is the delooping K(ß, n) of the fibre, and the obstruction lies*
* in
OBSTRUCTION THEORY IN MODEL CATEGORIES 3
[B=A, K(ß, n)] = Hn(B, A; ß), where the cofibration is A ! B. We do not discuss
the iterative procedure that occurs when lifting against a tower of fibrations.
We are curious how many other examples of obstruction theory can be viewed as
special cases of our theory, but have not yet investigated this in detail.
A summary of the contents of the paper follows. We begin in Section 2 by intr*
*o
ducing the category of arrows ArC of a pointed model category C and describing *
*two
useful model structures on ArC. Then in Section 3 we establish some technical l*
*ifting
results. In Section 4 we give the definitions of obstruction theory, fibrant ob*
*struction
theory, and cofibrant fibrant obstruction theory and describe their elementary *
*prop
erties. Subsections 4.1 and 4.2 discuss a rigidification of the notion of obstr*
*uction
theory and the uniqueness of obstruction theories for a given cofibration.
In Sections 5 and 6 we exhibit several ways of producing cofibrations that ha*
*ve
obstruction theories. First, cofibrations with weakly contractible target have *
*obstruc
tion theories. Second, retracts and cobase changes preserve cofibrations that h*
*ave
obstruction theories. Third, and most difficult, weak equivalences preserve cof*
*ibra
tions that have obstruction theories. This allows us to prove the main classifi*
*cation
theorem in Section 7.
Section 8 contains some applications. We explain how to apply our notions of
obstruction theory to the unpointed category of simplicial sets or of topologic*
*al
spaces. We also show that every cofibration in a stable model category has an
obstruction theory.
We assume that the reader is familiar with model categories. The original ref*
*erence
is [13], but see [7], [8], or [5] for more modern treatments. We follow the not*
*ation
and conventions of [7] as closely as possible.
2. The Category of Morphisms
Throughout the entire paper, C denotes a pointed model category with functori*
*al
factorizations. Let ArC be the category of morphisms in C. It is a diagram cate*
*gory,
where the index category has two objects and a unique map between them. Thus
objects of ArC are morphisms in C, and morphisms in ArC are commuting squares
in C. When the meaning is clear, we write X for the identity object X ! X of Ar*
*C.
The category ArC supports two model structures: the injective structure and t*
*he
projective structure [2, p. 314]. In both structures, the weak equivalences are*
* level
wise weak equivalences. Therefore, the associated homotopy categories are ident*
*ical.
Definition 2.1. A weak equivalence (or more precisely a levelwise weak equiv
alence) (resp., injective cofibration, projective fibration) from a map f : X !
X0 to another map g : Y ! Y 0in the category ArC is a commutative square
X _____//_Y
f g
fflffl fflffl
X0 _____//Y 0
in which the horizontal maps are weak equivalences (resp., cofibrations, fibrat*
*ions).
The map f ! g is a projective cofibration if both X ! Y and X0qX Y ! Y 0
are cofibrations. The map f ! g is an injective fibration if both X0 ! Y 0and
X ! X0xY 0Y are fibrations.
4 J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
Both model structures are examples of Reedy model structures [7, Ch. 17]. One
way of obtaining the injective structure is by considering the opposite of the *
*projec
tive model structure on Ar(Cop).
Note that all projective cofibrations are injective cofibrations. Similarly, *
*all injec
tive fibrations are projective fibrations. A cofibration in C is an injective c*
*ofibrant
object if and only if it is a projective cofibrant object if and only if it has*
* cofibrant
source. Similarly, a fibration in C is projective fibrant if and only if it is *
*injective
fibrant if and only if it has fibrant target.
We restate here the following useful lemma about base changes and cobase chan*
*ges
of weak equivalences in model categories.
Lemma 2.2. Base changes along fibrations preserve weak equivalences between fi
brant objects. Dually, cobase changes along cofibrations preserve weak equivale*
*nces
between cofibrant objects.
Proof.See [7, Prop. 11.1.2].
The following lemma is used many times throughout the paper. It allows us to
unify arguments that either assume fibrancy or right properness.
Lemma 2.3. Let p ! p0 be a weak equivalence between fibrations p : X ! Y and
p0: X0! Y 0with fibres F and F 0respectively. If Y and Y 0are fibrant, or C is *
*right
proper, then the induced map F ! F 0is a weak equivalence.
Proof.Let P be the fibre product X0xY 0Y . Consider the diagram
X ____//_P____//X0
p  p0
fflffl fflffl fflffl
Y ____//_Y____//Y 0.
The map P ! X0is a weak equivalence: in the right proper case, this is by defin*
*ition,
and in the case where Y and Y 0are fibrant, this follows from Lemma 2.2. Theref*
*ore
the map X ! P is also a weak equivalence by the twooutofthree axiom. The fib*
*re
of P ! Y is isomorphic to F 0. Hence it suffices to assume that Y equals Y 0.
Consider the model category C # Y of objects over Y . Let R : C # Y ! C be the
functor taking a map A ! Y to *xY A. It is straightforward to check that R is r*
*ight
adjoint to the functor L : C ! C # Y taking an object A to the trivial map A ! *
*Y .
The functor L preserves cofibrations and acyclic cofibrations, so L and R for*
*m a
Quillen pair. This means that R preserves weak equivalences between fibrant obj*
*ects.
The fibrant objects of C # Y are the fibrations with target Y , so p and p0are *
*both
fibrant in this category. Therefore, Rp ! Rp0is a weak equivalence.
The right proper case also follows from [7, Prop. 11.2.9].
Frequently in abstract homotopy theory, one has a square
A ____//_X
i p
fflffl fflffl
B ____//_Y
commuting in C and wants to know whether a map B ! X exists making both
resulting triangles commute. We rewrite this problem in terms of the category A*
*rC.
OBSTRUCTION THEORY IN MODEL CATEGORIES 5
Lemma 2.4. Suppose given a square as above in the category C. A lift exists for
this square if and only if a lift exists in the square
* ____//_X
 
 
fflffl fflffl
i______//p
in ArC if and only if a lift exists in the square
i______//p
 
 
fflffl fflffl
B ____//_* .
Proof.The proof is a straightforward diagram chase.
We shall frequently switch between these three equivalent forms of the lifting
problem.
3. Lifting Results
We now study how lifts for a given cofibration carry over to another weakly e*
*quiv
alent cofibration.
Proposition 3.1. Let i : A ! B and i0: A0! B0 be cofibrations, and let i ! i0be
a weak equivalence in ArC. Also assume that A and A0are cofibrant or that C is *
*left
proper. Let p : X ! Y be any fibration. Then a lift exists in the square
(3.1) A0_____//X
fflffl 
i0 p
fflffl fflfflfflffl
B0 _____//Y
if and only if a lift exists in the square
(3.2) A __~_//_A0___//_X
fflffl 
i p
fflffl~ fflfflfflffl
B ____//_B0___//_Y.
Proof.The proof is very similar to the proof of [7, Prop. 11.1.16]. If a lift B*
*0! X
exists in the square 3.1, then the composition B ! B0! X is a lift for the squa*
*re 3.2.
The other implication is harder.
Suppose that a lift B ! X exists for the square 3.2. Let P be the pushout
A qB A0. Because of left properness or because of Lemma 2.2, the map B ! P is a
weak equivalence, so P ! B0 is a weak equivalence by the twooutofthree axiom.
Since j : A0! P is a cobase change of i, there is a lift P ! X in the square
A0 _____//X
fflffl 
j p
fflffl fflfflfflffl
P _____//Y.
Now consider the model category A0 # C # Y of objects under A0 and over Y .
Then P and B0 (equipped with the obvious structure maps from A0and to Y ) are
cofibrant objects of this category, while X is a fibrant object. Moreover, ther*
*e are
6 J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
morphisms P ! B0 and P ! X in A0# C # Y . The first map is a weak equivalence,
so there is also a map B0! X in A0# C # Y by [7, Cor. 7.6.5].
The point of this result is that finding lifts for i0 reduces to finding lift*
*s for i.
Given any lifting problem for i0(in other words, a square 3.1), we can consider*
* the
square 3.2 instead. However, finding lifts for i does not reduce to finding lif*
*ts for i0:
in general, not every square
A ____//_X
fflffl 
i p
fflffl fflfflfflffl
B ____//_Y
can be rewritten in the form 3.2, even if A is cofibrant.
For future reference, we record the dual result here.
Proposition 3.2. Let p : X ! Y and p0: X0 ! Y 0be fibrations, and let p ! p0be
a weak equivalence in ArC. Also assume that Y and Y 0are fibrant or that C is r*
*ight
proper. Let i : A ! B be any cofibration. Then a lift exists in the square
(3.3) A ____//_X
fflffl 
i p
fflffl fflfflfflffl
B ____//_Y
if and only if a lift exists in the square
(3.4) A _____//X__~__//X0
fflffl 
i p0
fflffl ~ fflfflfflffl
B ____//_Y____//Y 0.
Proof.The proof is dual to the proof of Proposition 3.1.
4.Obstruction Theory
Definition 4.1. A cofibration i has an obstruction theory if there exists an
object W such that for every fibration p with fibre F and every map i ! p in ArC
there exists a welldefined obstruction ff, which is a weak homotopy class from*
* W
to F (i.e., an element of [W, F ]), with the following two properties. First, a*
* lift exists
in the square
A ____//_X
fflffl 
i p
fflffl fflfflfflffl
B ____//_Y
if and only if ff is the trivial homotopy class. Second, ff is functorial in th*
*e following
sense. Given two fibrations p and p0 with fibres F and F 0respectively and a map
p ! p0, the obstruction ff0of the composition i ! p ! p0is the composition of t*
*he
obstruction ff of the map i ! p with the map F ! F 0.
Remark 4.2. The functoriality of obstructions can be reexpressed in terms of the
Grothendieck construction of the functor
C ! Sets: X 7! [W, X].
OBSTRUCTION THEORY IN MODEL CATEGORIES 7
Definition 4.3. A cofibration has a fibrant obstruction theory if the conditions
of Definition 4.1 (including functoriality) are satisfied for all maps i ! p fo*
*r which p is
a fibration with fibrant target. A cofibration has a cofibrant fibrant obstruct*
*ion
theory if the conditions of Definition 4.1 (including functoriality) are satisf*
*ied for
all injective cofibrations i ! p for which p is a fibration with fibrant target.
As in Remark 4.2, the functoriality of fibrant obstruction theories and cofib*
*rant
fibrant obstruction theories can be expressed in terms of Grothendieck construc*
*tions.
These three notions of obstruction theories have some obvious relationships. *
*If a
cofibration has an obstruction theory, then it necessarily has a fibrant obstru*
*ction
theory. Also, if it has a fibrant obstruction theory, then it has a cofibrant *
*fibrant
obstruction theory. Next we state some less obvious connections.
Proposition 4.4. If C is right proper, then a cofibration has a fibrant obstruc*
*tion
theory if and only if it has an obstruction theory.
Proof.The axioms for an obstruction theory are stronger than the axioms for a
fibrant obstruction theory, so one implication follows from the definitions.
Suppose that a cofibration i has a fibrant obstruction theory. Now consider a
map i ! p in which p is a fibration between not necessarily fibrant objects. Le*
*t ^p
be an injective fibrant replacement for p, so ^pis a fibration between fibrant *
*objects
and there is a weak equivalence p ! ^pin ArC. Let F and ^Fbe the fibres of p and
^prespectively, and let ^ffbe the obstruction for the composition i ! p ! ^p. *
*By
Lemma 2.3, the map F ! ^Fis a weak equivalence. Define the obstruction ff for
i ! p to be composition of ^ffwith the weak homotopy inverse of the map F ! ^F.
This definition of ff is functorial because injective fibrant replacements are *
*functorial.
By Proposition 3.2, a lift exists in the square i ! p if and only if a lift e*
*xists in
the square i ! p ! ^p; here we use that C is right proper. A lift exists in the*
* square
i ! p ! ^pif and only if ^ffis trivial. Since F ! ^Fis a weak equivalence, ^ffi*
*s trivial
if and only if ff is trivial.
The next proposition tells us that fibrant obstruction theories and cofibrant*
* fibrant
obstruction theories are actually equivalent.
Proposition 4.5. A cofibration has a fibrant obstruction theory if and only if *
*it has
a cofibrant fibrant obstruction theory.
Proof.The axioms for a fibrant obstruction theory are stronger than the axioms *
*for
a cofibrant fibrant obstruction theory, so one implication follows from the def*
*initions.
Suppose that a cofibration i has a cofibrant fibrant obstruction theory. For *
*every
map i ! p for which p is a fibration with fibrant target, take a functorial fac*
*torization
i ! p0! p
where the first map is an injective cofibration and the second map is an acyclic
injective fibration. Then p0 is a fibration with fibrant target, so there exis*
*ts an
obstruction ff0for i ! p0.
Let F 0and F be the fibres of p0 and p respectively. By Lemma 2.3, the map
F 0! F is a weak equivalence; here we use that p0and p have fibrant targets. De*
*fine
the obstruction ff for i ! p to be the composition of ff0with the map F 0! F . *
*This
definition is functorial since factorizations in the injective model structure *
*on ArC
are functorial.
8 J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
By Proposition 3.2, a lift exists for i ! p if and only if a lift exists for *
*i ! p0. By
assumption, a lift exists for i ! p0if and only if ff0is trivial. Since F 0! F *
*is a weak
equivalence, ff0is trivial if and only if ff is trivial.
The following proposition establishes a certain homotopy invariance property *
*of
obstructions. The result is not needed later, but we include it for completenes*
*s.
Proposition 4.6. Let i be a cofibration that has a fibrant obstruction theory, *
*and let
p be a fibration with fibrant target. Suppose given two maps f and g (i.e., com*
*muting
squares) from i to p that are right homotopic in the injective or projective mo*
*del
structure. Then the obstructions for f and for g are equal.
Proof.Let pI be a good path object for p in the injective model structure. This*
* is
also a good path object for p in the projective model structure. So in either *
*case
there exists a good right homotopy i ! pI between f and g [5, x 4.12]. This mea*
*ns
that there is a commutative diagram
@p@OO
f 


i_____//>>pI
>> 
g>>>
OEOE>fflffl
p.
Note that pI is a fibration with fibrant target because pI ! p x p is an inject*
*ive
fibration. There is an obstruction ff for the map i ! pI, which is a homotopy c*
*lass
into the fibre F Iof pI.
Let F be the fibre of p. By Lemma 2.3, the map F ! F Iis a weak equivalence
since p ! pI is a weak equivalence. Therefore, both maps F I! F are equal in the
homotopy category since they have the same right inverse.
Since the fibrant obstruction theory for i is functorial, the obstructions fo*
*r f and
g are the compositions of ff with the two maps F I! F . Therefore, the obstruct*
*ions
for f and g are equal.
4.1. Rigid Obstruction Theories. For any map f : X ! Y , we can choose a
functorial factorization of f into an acyclic cofibration X ! X0followed by a f*
*ibration
f0 : X0! Y . Write hofibfor the functor ArC ! C sending f to the fibre of f0. W*
*hen
f is a fibration, there is a natural weak equivalence fib(f) ! hofib(f), where *
*fib(f)
denotes the fibre of f (see the proof of Lemma 2.3).
Now consider a cofibration i that has an obstruction theory. Then we have an
obstruction ff in [W, hofib(i)] for the square i ! i0. Any other square i ! p, *
*where
p is a fibration, factors through the square i ! i0. Therefore, the class ff de*
*termines
by functoriality the entire obstruction theory for i.
This implies that any obstruction theory can be rigidified in the following w*
*ay.
Assume without loss of generality that W is cofibrant, and fix a representative*
* a :
W ! hofib(i) for the class ff. To each fibration p and square i ! p, assign the
map W ! hofib(p) which is the composite of a with the map hofib(i) ! hofib(p).
This is functorial at the model category level, and via the natural weak equiva*
*lences
fib(p) ! hofib(p) specializes to the obstruction theory we started with.
These ideas lead to the following definition. A rigid obstruction theory for a
cofibration i is a cofibrant object W and a map a : W ! hofib(i) such that for *
*each
OBSTRUCTION THEORY IN MODEL CATEGORIES 9
square i ! p with p a fibration, the square has a lift if and only if the compo*
*site
W ! hofib(i) ! hofib(p) is nullhomotopic. A map of strict obstruction theories
for i from a : W ! hofib(i) to a0 : W 0! hofib(i) is simply a map W ! W 0over
hofib(i).
4.2. Uniqueness. Our definition of obstruction theory does not uniquely determi*
*ne
W nor the assignment of obstruction classes to each square.
Given a cofibration i, we can form a category O(i) of obstruction theories fo*
*r i
in the following way. The objects are obstruction theories for i (i.e., pairs (*
*W, ff),
where W is an object of C and ff is a function that assigns an obstruction clas*
*s to
each square in a functorial way). A morphism (W, ff) ! (W 0, ff0) is a weak hom*
*otopy
class in C from W to W 0such that ff is given by composition of ff0 with this w*
*eak
homotopy class.
The category O(i) is contractible if it is nonempty. This is seen in the fo*
*llow
ing`way. First,`note that if (W, ff) and (W 0, ff0) are obstruction theories fo*
*r i, then
(W W 0, ff ff0) is also an obstruction theory`for i. Now, fix an obstructio*
*n the
ory a = (W, ff) and consider`the functor a  : O(i) ! O(i). There are natural
transformations idO(i)! a  ca, where ca : O(i) ! O(i) is the constant func*
*tor
sending everything to a. This shows that the identity map on the nerve of O(i) *
*is
nullhomotopic.
In this sense, obstruction theories for i are unique up to a contractible fam*
*ily
of choices. Similarly, one can define a category of rigid obstruction theories*
* (see
Subsection 4.1), and one again finds that this category is contractible if it i*
*s non
empty.
We will show in Remark 7.3 how to construct obstruction theories in practice.
5.Maps that Have Obstruction Theories
Having established the basic notions of obstruction theories, we study the ex*
*istence
of obstruction theories for certain kinds of cofibrations.
Proposition 5.1. Let i : A ! B be a cofibration such that A is cofibrant and B *
*is
weakly contractible and fibrant. Then i has a fibrant obstruction theory.
Corollary 6.4 will show that the assumption that B is fibrant is unnecessary.
Proof.Let p : X ! Y be a fibration with fibre F such that Y is fibrant. Suppose
given a square i ! p. Let p0: X0! B be the pullback fibration X xY B ! B. The
fibre of p0is also F . Lemma 2.3 applied to the square
X0 __=__//X0
p0 
fflfflfflfflfflfflfflffl
B __~___//*
implies that the map F ! X0 is a weak equivalence; here we use that B is fibran*
*t.
Define the obstruction for i ! p to be the homotopy class of the map A ! X0
composed with the weak homotopy inverse of F ! X0. This definition is functoria*
*l.
If a lift exists for i ! p, then the map A ! X0 factors through the contracti*
*ble
object B. It follows that the obstruction is nullhomotopic.
Now suppose that the obstruction is nullhomotopic. We need only show that the
square i ! p0 has a lift. Because A is cofibrant and X0 is fibrant, we have a *
*left
nullhomotopy of the map A ! X0. Thus, A ! X0 factors through a contractible
10 J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
object C. Here C is the cofibre of a cofibration A ! Cyl(A), where Cyl(A) is a
cylinder object for A. By factoring C ! X0 into an acyclic cofibration followed*
* by a
fibration, we may assume that C ! X0 is a fibration. This gives us a commutative
square
A ____//_C
fflffl 
i ~
fflfflfflfflfflffl
B _=__//_B
in which the right vertical arrow is an acyclic fibration because both C and B *
*are
weakly contractible. Hence a lift exists, and the composition B ! C ! X0 is the
desired lift.
Proposition 5.2. The class of cofibrations that have an obstruction theory (res*
*p.,
fibrant obstruction theory, cofibrant fibrant obstruction theory) is closed und*
*er cobase
change.
Proof.We prove the proposition for obstruction theories. The other cases are id*
*en
tical.
Let
A _____//A0
fflffl fflffl
i i0
fflffl fflffl
B ____//_B0
be a pushout square in which i (and hence i0also) is a cofibration. Suppose tha*
*t i
has an obstruction theory.
Given a map i0! p in which p : X ! Y is a fibration, define the obstruction ff
to be the obstruction for the composition i ! i0! p. This definition is functor*
*ial.
If B0 ! X is a lift for the square i0 ! p, then the composition B ! B0 ! X
is a lift for the square i ! i0 ! p. On the other hand, if B ! X is a lift for *
*the
square i ! i0! p, then the maps B ! X and A0! X induce a lift B0 ! X for
the square i0! p. Thus, a lift exists for the square i0! p if and only if a lif*
*t exists
for the square i ! i0! p. A lift exists for the square i ! i0! p if and only if*
* ff is
trivial.
Proposition 5.3. The class of cofibrations that have an obstruction theory (res*
*p.,
fibrant obstruction theory, cofibrant fibrant obstruction theory) is closed und*
*er re
tract.
Proof.We prove the proposition for obstruction theories. The other cases are id*
*en
tical.
Suppose that i : A ! B is a cofibration and that it has an obstruction theory*
*. Let
i0: A0! B0 be a retract of i. Then i0 is a cofibration since cofibrations are c*
*losed
under retract.
Given a map i0! p in which p : X ! Y is a fibration, define the obstruction ff
to be the obstruction for the composition i ! i0! p. This definition is functor*
*ial.
If B0! X is a lift for the square i0! p, then the composition B ! B0! X is a
lift for the square i ! i0! p. On the other hand, if B ! X is a lift for the sq*
*uare
i ! i0! p, then the composition B0! B ! X is a lift for the square i0! p. Thus,
a lift exists for the square i0! p if and only if a lift exists for the square *
*i ! i0! p.
A lift exists for the square i ! i0! p if and only if ff is trivial.
OBSTRUCTION THEORY IN MODEL CATEGORIES 11
6.Weakly Equivalent Cofibrations
The goal of this section is to show that a cofibration has an obstruction the*
*ory
if and only if a weakly equivalent cofibration has an obstruction theory (Propo*
*si
tion 6.3). Together with the results of the previous section, this establishes*
* the
characteristic properties of the class of cofibrations that have obstruction th*
*eories.
Note that we study fibrant obstruction theories in this section. When C is ri*
*ght
proper, Proposition 4.4 tells us that the same results hold for obstruction the*
*ories.
Proposition 6.1. Let i : A ! B and i0: A0! B0 be cofibrations, and let i ! i0be
a weak equivalence. Also assume that A and A0are cofibrant or that C is left pr*
*oper.
Then i has a fibrant obstruction theory if and only if i0 has a fibrant obstruc*
*tion
theory.
Proof.First suppose that i has a fibrant obstruction theory. Given a map i0 ! p
in which p : X ! Y is a fibration with fibrant target, define the obstruction f*
*f to
be the obstruction for the composition i ! i0! p. This definition is functorial*
*. By
Proposition 3.1, a lift exists in the square i0 ! p if and only if a lift exist*
*s in the
square i ! i0! p. A lift exists for the square i ! i0! p if and only if ff is t*
*rivial.
This finishes one implication.
Now suppose that i0has a fibrant obstruction theory. By Proposition 4.5, it s*
*uffices
to show that i has a cofibrant fibrant obstruction theory. Let i ! p be an inje*
*ctive
cofibration such that p : X ! Y is a fibration with fibrant target. Let p0 be *
*the
pushout pqii0: X qA A0! Y qB B0. The map p ! p0is a weak equivalence because
the map i ! p is an injective cofibration. Here we use that the injective struc*
*ture on
ArC is left proper or that i and i0are injective cofibrant so that Lemma 2.2 ap*
*plies
to the injective model structure on ArC.
The map p0is not in general a fibration, so let p00be an injective fibrant re*
*place
ment for p0. Then we have a commutative diagram
i_____//_p
~  ~
fflfflfflffl~
i0_____//p0___//p00.
Define the obstruction for i ! p to be the obstruction for i0! p00. This defini*
*tion is
functorial because the construction of p00is functorial.
The obstruction vanishes if and only if i0 ! p00has a lift. By Proposition 3*
*.1,
i0! p00has a lift if and only if i ! p00has a lift; here we use that C is left *
*proper
or that A and A0are cofibrant. By Proposition 3.2, i ! p00has a lift if and onl*
*y if
i ! p has a lift; here we use that p and p00have fibrant targets.
Recall that a projective cofibrant replacement ~f: ~X! ~Yof any map f : X ! Y
is a cofibration between cofibrant objects together with a commuting square
~X__~_////_X
fflffl
~f f
fflffl~fflffl
~Y____////_Y
in which the horizontal arrows are acyclic fibrations.
12 J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
Corollary 6.2. Let i be a cofibration, and suppose that i has cofibrant source *
*or that
C is left proper. Let ~ibe any projective cofibrant replacement for i. Then i*
* has a
fibrant obstruction theory if and only if ~idoes.
Proof.There is a weak equivalence ~i! i, so the result follows from Proposition*
* 6.1.
Proposition 6.3. Let i and i0be weakly equivalent cofibrations. Suppose that i *
*and
i0 have cofibrant sources or that C is left proper. Then i has a fibrant obstr*
*uction
theory if and only if i0has a fibrant obstruction theory.
The proof does not simply reduce to a repeated application of Proposition 6.1*
* to a
zigzag of weak equivalences because we need to know that the intermediate obje*
*cts
in the zigzag in ArC are cofibrations in order to apply that result.
Proof.Factor i0! * as
i0//~_//j__////_*
where the first map is an acyclic projective cofibration and the second is a pr*
*ojective
fibration. Note that j is a cofibration with cofibrant source. Then j is a proj*
*ective
fibrant replacement for i0, so there is an actual map ~i! j representing the we*
*ak
equivalence between i and i0, where ~iis a projective cofibrant replacement for*
* i.
By Corollary 6.2, i has a fibrant obstruction theory if and only if ~ihas a f*
*ibrant
obstruction theory. By Proposition 6.1, ~ihas a fibrant obstruction theory if *
*and
only if j has a fibrant obstruction theory. By Proposition 6.1 again, j has a f*
*ibrant
obstruction theory if and only if i0has a fibrant obstruction theory.
Corollary 6.4. Let i : A ! B be a cofibration such that B is weakly contractibl*
*e.
Suppose also that A is cofibrant or that C is left proper. Then i has a fibrant*
* obstruc
tion theory.
The difference between this result and Proposition 5.1 is that we don't assume
that B is fibrant here.
Proof.Let ~i: ~A! ~Bbe a projective cofibrant replacement for i. Let ~B! ^Bbe
an acyclic cofibration from ~Bto a fibrant object. Then the composite cofibrat*
*ion
A~! ^Bis weakly equivalent to i, and it has cofibrant source and fibrant and we*
*akly
contractible target. Proposition 6.1 and Proposition 5.1 finish the argument.
7. Classification of Cofibrations with Obstruction Theories
Theorem 7.1. Consider the smallest class of cofibrations with cofibrant source *
*that
contains all cofibrations with cofibrant source and weakly contractible target *
*and is
closed under retract, weak equivalence, and cobase change. This class coincides*
* with
the class of cofibrations with cofibrant source that have a fibrant obstruction*
* theory.
If C is right proper, then this class also coincides with the class of cofibrat*
*ions with
cofibrant source that have an obstruction theory.
Proof.The second claim follows from the first by Proposition 4.4. By Proposi
tions 5.2, 5.3, and 6.3 and Corollary 6.4, we need only show that if i is a cof*
*ibration
with cofibrant source that has a fibrant obstruction theory, then i is related *
*to a
cofibration with cofibrant source and weakly contractible target by a series of*
* weak
equivalences, retracts, and cobase changes.
OBSTRUCTION THEORY IN MODEL CATEGORIES 13
Let i : A ! B be a cofibration with a fibrant obstruction theory such that A *
*is
cofibrant. We will construct the following commutative diagram:
~F //__//_CF~
~ DDDDÆ%
fflfflfflfflfflffl
F //___//D_____//_G
 DDDD  
 OE&  
= ~ fflffl fflffl~ fflffl
A _____//_A//__//A0//___//C//___//C0
fflffl fflffl   
i ~i i0  p
fflffl fflffl fflfflfflfflfflfflfflfflfflffl
B //~//_Bf_=__//Bf_=__//Bf__=__//Bf.
Let B ! Bf be a fibrant replacement for B, and let ~i: A ! Bf be the composite,
which is also a cofibration. Let A ! A0! Bf be a factorization of ~iinto an acy*
*clic
cofibration followed by a fibration i0. Let F be the fibre of i0, and let ~F! F*
* be a
cofibrant replacement for F . Let ~F! CF~be a cofibration with weakly contracti*
*ble
target, and let D and C be the pushouts as indicated above. The map i0: A0! Bf
and the constant map D ! Bf agree on F and therefore induce a map C ! Bf.
Factor the map C ! Bf into an acyclic cofibration C ! C0 followed by a fibration
p : C0! Bf, and let G be the fibre of C0! Bf. The composite D ! Bf is constant,
so there is an induced map D ! G. This completes the construction of the above
diagram.
The map ~ihas a fibrant obstruction theory by Proposition 6.3 because i does.
Because obstructions are functorial, the obstruction for lifting the square ~i!*
* p is
the composite of the obstruction for lifting the square ~i! i0with the map F ! *
*G.
Since after inverting weak equivalences the map F ! G factors through the weakly
contractible object CF~, it is null in the homotopy category. Therefore the obs*
*truction
vanishes and a lift exists in the square ~i! p. The cofibration i is weakly equ*
*ivalent
to the cofibration ~i, which is a retract of the cofibration A ! C0 (because of*
* the
lift), which is weakly equivalent to A0 ! C, which is a pushout of the cofibrat*
*ion
F~! CF~, which has cofibrant source and weakly contractible target.
Theorem 7.2. Let C be left proper. Consider the smallest class of cofibrations *
*that
contains all cofibrations with weakly contractible target and is closed under r*
*etract,
weak equivalence, and cobase change. This class coincides with the class of cof*
*ibra
tions that have a fibrant obstruction theory. If C is also right proper, then t*
*his class
also coincides with the class of cofibrations that have an obstruction theory.
Proof.The first claim follows from Theorem 7.1, since Corollary 6.2 allows us to
assume that i has cofibrant source. Use Proposition 4.4 for the second claim.
Remark 7.3. The smallest class of cofibrations containing cofibrations with wea*
*kly
contractible target and closed under weak equivalences and cobase changes are t*
*he
principal cofibrations. That is, up to weak homotopy, they are the maps that
occur as a quotient map of some cofibre sequence. Therefore, the previous two
theorems tell us that the retracts of principal cofibrations are the cofibratio*
*ns that
have obstruction theories.
In the case of principal cofibrations, one can inspect the proofs of Proposit*
*ions 5.1,
5.2 and 6.1 and easily choose an object W . If i is principal, then there is a *
*cofibre
14 J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
sequence
A ____//_B___//C
in which the second map is weakly homotopic to i. Then A is one possible choice*
* of
the object W .
Corollary 7.4. Let B be a cofibrant object of C. If the cofibration * ! B has a
fibrant obstruction theory, then B is weakly equivalent to a retract of B, wh*
*ere
and refer to the suspension and loop functors on the homotopy category (see
Section 8.2).
Proof.This follows from the proof of Theorem 7.1. In the notation of that proof*
*, ~F
is B and C is B.
We shall see below that the converse is also true; retracts of suspensions al*
*ways
have obstruction theories.
Corollary 7.5. Let 2 n 1. In the category of pointed simplicial sets, the m*
*ap
* ! RP ndoes not have an obstruction theory.
Proof.The cohomology of RP nhas nontrivial cup products, but the cohomology of
a retract of a suspension has trivial cup products.
Similar considerations show that the map from a point to any space with non
trivial cup products (such as tori) does not have an obstruction theory.
It is possible to prove Corollary 7.5 directly, providing an independent veri*
*fication
of Theorem 7.1.
One consequence of this corollary is that cofibrations that have obstruction *
*theo
ries are not closed under composition. The map * ! RP 2is the composition of two
maps that both have obstruction theories.
8.Applications
8.1. Unpointed Spaces. Our first application concerns the category of simplicial
sets. When n 1, the generating cofibration @ n ! n has an obstruction theory
because the standard model structure on simplicial sets is right proper, because
n is weakly contractible, and because @ n is cofibrant. Strictly speaking, th*
*is
obstruction theory applies to the category of pointed simplicial sets. However,*
* one
can take any unpointed lifting problem and turn it into a pointed lifting probl*
*em by
choosing compatible basepoints (as long as n 1 so that @ n is nonempty).
We conclude that when n 1, obstructions to lifting squares of the form
@ n ____//_X
 
 p
fflffl fflffl
n _____//_Y
are elements of ßn1(F, *) for some *, where F is a fibre of the fibration p. W*
*hen
n = 0, there is no obstruction theory for the cofibration ; ! 0.
Theorem 8.1. Let n 1. A fibration p : X ! Y of unbased simplicial sets has the
right lifting property with respect to @ n ! n if and only if ßn1(F, *) is ze*
*ro for
every fibre F of p and every basepoint * of F .
OBSTRUCTION THEORY IN MODEL CATEGORIES 15
Proof.First suppose that every homotopy group vanishes. Then lifts exist because
the obstructions must be trivial.
Now suppose that p has the right lifting property. Consider a fibre F over an
element y of Y , and let * be any basepoint of this fibre. Since F is fibrant, *
*every
element of ßn1(F, *) is represented by an actual map f : @ n ! F . Take the sq*
*uare
@ n ____//_X
 
 p
fflffl fflffl
n _____//_Y
in which the bottom horizontal arrow is the constant map with value y and the t*
*op
horizontal arrow is the composition of f with the inclusion F ! X of the fibre.*
* A
lift exists in the square, and this shows that f is nullhomotopic.
8.2. Suspensions of Cofibrations. In any pointed model category, the suspen
sion A of any object A is the cofibre of a cofibration
iA : A ! CA,
where CA is a cone on A (i.e., a weakly contractible object together with a cof*
*ibra
tion from A). If C is left proper or A is cofibrant, a lifting argument combine*
*d with
the dual of the proof of Lemma 2.3 shows that if C0A is a fibrant cone on A, and
0A is the corresponding suspension, then there is a weak equivalence A ! 0A.
Thus the choice of cone object does not affect the weak homotopy type of A. If
f : A ! B is a map, and the cones are chosen so that f extends to a map CA ! CB
(e.g. if CB is fibrant, or the cones are chosen functorially), then we get an i*
*nduced
map f : A ! B. In fact, f is the suspension of f in the projective model
structure on ArC, and therefore the above argument shows that any two construc
tions of f yield weakly equivalent maps provided that f is a cofibration betwe*
*en
cofibrant objects or that C is left proper.
By choosing a cone functorially, one makes into a functor, and, by the dual
of Lemma 2.3, takes weak equivalences between cofibrant objects to weak equiv
alences. Thus the left derived functor of , defined by applying to a cofibr*
*ant
replacement, gives a welldefined functor on the homotopy category. If the mod*
*el
category is left proper, then the dual of Lemma 2.3 tells us that is homotopy
invariant on all objects.
Dually, the loop functor takes an object X to the fibre of a fibration
P X ! X,
where P X is a path object on X (i.e.,, a weakly contractible object with a fib*
*ration
to X). The other statements above dualize as well.
Theorem 8.2. Let C be a pointed model category, and let i : A ! B be any cofi
bration. Suppose that A is cofibrant or that C is left proper. Choose a model f*
*or i
such that it is a cofibration. Then i has a fibrant obstruction theory. If C i*
*s right
proper, then i has an obstruction theory.
Proof.The second claim follows from the first claim because of Proposition 4.4.*
* The
left proper case follows from the other case by Corollary 6.2. Hence we may sup*
*pose
that A is cofibrant.
By two applications of the dual to [13, Prop. I.3.3], B is the homotopy cofi*
*bre
of the map Ci ! A, where Ci is the mapping cone CA qA B. We may use
C(Ci) qCi A to compute B, where Ci ! C(Ci) is a cofibration with weakly
16 J. DANIEL CHRISTENSEN, WILLIAM G. DWYER, AND DANIEL C. ISAKSEN
contractible target. Thus i is weakly equivalent to a cobase change of the cof*
*ibration
Ci ! C(Ci). Now C(Ci) is weakly contractible and Ci is cofibrant since A is, so
Theorem 7.1 finishes the proof.
Remark 8.3. From the proofs of Propositions 5.1 and 5.2, it follows that the ob*
*struc
tions for lifting i are homotopy classes from the mapping cone Ci. This object*
* can
be thought of as the desuspension of the cofibre of i.
8.3. Stable Model Categories. Now we proceed to stable model categories.
A stable model category is a pointed model category in which the left derived
suspension (see Section 8.2) induces an automorphism of the homotopy category; *
*its
inverse is then the right derived loop functor.
Corollary 8.4. Let C be a stable model category. Every cofibration in C with co*
*fi
brant source has a fibrant obstruction theory. If C is left proper, then every *
*cofibration
in C has a fibrant obstruction theory. If C is left and right proper, then ever*
*y cofi
bration has an obstruction theory.
Proof.This follows from Theorem 8.2 and the fact that in a stable model categor*
*y,
every cofibration is weakly equivalent to the suspension of a cofibration.
Example 8.5. Let "spectra" refer to BousfieldFriedlander spectra [1] or symmet
ric spectra [9]. Both model categories are right proper. Consider the generat*
*ing
cofibration Fm @ n+! Fm n+of spectra. The meaning of Fm depends on which
category of spectra we are considering. The spectrum Fm @ n+is cofibrant, but t*
*his
fact is not essential because the stable model structures under consideration a*
*re left
and right proper. The desuspension of the cofibre of this generating cofibratio*
*n is
weakly equivalent to the sphere spectrum Snm1 . Therefore, obstructions for l*
*ifting
generating cofibrations of spectra are elements of stable homotopy groups.
Theorem 8.6. Let n 1 and m 0. A fibration p : X ! Y of spectra with
fibre Z has the right lifting property with respect to Fm @ n+! Fm n+if and on*
*ly if
ßnm1 Z is zero.
Proof.First suppose that every homotopy group vanishes. Then lifts exist because
the obstructions must be trivial.
Now suppose that p has the right lifting property. If @ n and n are based at
the 0th vertex, there is a pushout square
Fm @ n+_____//Fm @ n
 
 
fflffl fflffl
Fm n+______//Fm n,
so p has the right lifting property also with respect to the cofibration Fm @ n*
* !
Fm n.
Let ff be any element of ßnm1 Z. Since Fm @ n is a cofibrant model for the
sphere spectrum Snm1 , we can represent ff by an actual map f : Fm @ n ! Z.
Thus, we have a square
Fm @ n _____//X
 
 p
fflffl fflffl
Fm n______//Y
OBSTRUCTION THEORY IN MODEL CATEGORIES 17
in which the top horizontal map is the composition of f with the map Z ! X and
the bottom horizontal map is constant. This square has a lift l by assumption. *
*Since
the bottom map is constant, l factors through the fibre Z. Since the inclusion *
*Z ! X
is monic, this shows that f : Fm @ n ! Z factors through the weakly contractible
object Fm n. Hence f is weakly nullhomotopic, and ff = 0. This shows that
ßnm1 Z = 0.
Remark 8.7. The proof of Theorem 8.6 is more complicated than the proof of The
orem 8.1. The difference arises from the fact that the simplicial set n is we*
*akly
contractible, while the spectrum Fm n+is not.
Remark 8.8. Just as in Theorem 8.1, we must assume in Theorem 8.6 that n 1.
A fibration p : X ! Y with fiber Z has the right lifting property with respect *
*to
Fm @ 0+! Fm 0+if and only if the map Xm ! Ym of simplicial sets is surjective.
This does not guarantee that ßm1 Z is zero.
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[13]D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, *
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Department of Mathematics, University of Western Ontario, London, Ontario N6A
5B7, Canada
Email address: jdc@uwo.ca
Department of Mathematics, University of Notre Dame, South Bend, IN 46556
Email address: dwyer.1@nd.edu
Department of Mathematics, University of Notre Dame, South Bend, IN 46556
Email address: isaksen.1@nd.edu