QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL
ALGEBRA
J. DANIEL CHRISTENSEN AND MARK HOVEY
Abstract.An important example of a model category is the category of unb*
*ounded
chain complexes of R-modules, which has as its homotopy category the der*
*ived cat-
egory of the ring R. This example shows that traditional homological alg*
*ebra is
encompassed by Quillen's homotopical algebra. The goal of this paper is *
*to show
that more general forms of homological algebra also fit into Quillen's f*
*ramework.
Specifically, a projective class on a complete and cocomplete abelian ca*
*tegory A is
exactly the information needed to do homological algebra in A. The main *
*result is
that, under weak hypotheses, the category of chain complexes of objects *
*of A has
a model category structure that reflects the homological algebra of the *
*projective
class in the sense that it encodes the Extgroups and more general derive*
*d functors.
Examples include the üp re derived categoryö f a ring R, and derived ca*
*tegories
capturing relative situations, including the projective class for Hochsc*
*hild homology
and cohomology. We characterize the model structures that are cofibrantl*
*y gener-
ated, and show that this fails for many interesting examples. Finally, w*
*e explain how
the category of simplicial objects in a possibly non-abelian category ca*
*n be equipped
with a model category structure reflecting a given projective class, and*
* give examples
that include equivariant homotopy theory and bounded below derived categ*
*ories.
____________
Date: July 13, 2001.
1991 Mathematics Subject Classification. Primary 18E30; Secondary 18G35, 55U*
*35, 18G25, 55U15.
Key words and phrases. Derived category, chain complex, relative homological*
* algebra, projective
class, model category, non-cofibrant generation, pure homological algebra.
The first author was supported in part by NSF grant DMS 97-29992.
The second author was supported in part by NSF grant DMS 99-70978.
1
2 J. DANIEL CHRISTENSEN AND MARK HOVEY
Contents
Introduction 3
0. Notation and conventions 5
1. Projective classes 6
1.1. Definition and some examples 6
1.2. Homological algebra 8
1.3. Pullbacks 8
1.4. Strong projective classes 9
2. The relative model structure 10
2.1. Properties of the relative model structure *
* 16
2.2. Why projective classes? 22
3. Case A: Projective classes coming from adjoint pairs *
* 22
3.1. Examples 25
4. Case B: Projective classes with enough small projectives *
* 28
5. Cofibrant generation 30
5.1. Background 31
5.2. Projective classes with sets of enough small projectives *
* 32
5.3. The pure and categorical derived categories *
*35
5.4. Failure to be cofibrantly generated *
*36
6. Simplicial objects and the bounded below derived category *
* 39
6.1. Projective classes in pointed categories *
* 39
6.2. The model structure 40
References 41
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 3
Introduction
An important example of a model category is the category Ch(R) of unbounded *
*chain
complexes of R-modules, which has as its homotopy category the derived category*
* D(R)
of the associative ring R. The formation of a projective resolution is an exam*
*ple of
cofibrant replacement, traditional derived functors are examples of derived fun*
*ctors in
the model category sense, and Ext groups appear as groups of maps in the derived
category. This example shows that traditional homological algebra is encompasse*
*d by
Quillen's homotopical algebra, and indeed this unification was one of the main *
*points of
Quillen's influential work [Qui67].
The goal of this paper is to illustrate that more general forms of homologic*
*al algebra
also fit into Quillen's framework. In any abelian category A there is a natural*
* notion
of "projective objectä nd "epimorphism." However, it is sometimes useful to im*
*pose
different definitions of these terms. If this is done in a way that satisfies s*
*ome natural
axioms, what is obtained is a "projective class," which is exactly the informat*
*ion needed
to do homological algebra in A. Our main result shows that for a wide variety o*
*f projec-
tive classes (including all those that arise in examples) the category of unbou*
*nded chain
complexes of objects of A has a model category structure that reflects the homo*
*logical
algebra of the projective class in the same way that ordinary homological algeb*
*ra is
captured by the usual model structure on Ch(R).
When A has enough projectives, the projective objects and epimorphisms form a
projective class. Therefore the results of this paper apply to traditional hom*
*ological
algebra as well. Even in this special case, it is not a trivial fact that the *
*category of
unbounded chain complexes can be given a model category structure, and indeed Q*
*uillen
restricted himself to the bounded below case. We know of three other written pr*
*oofs that
the category of unbounded chain complexes is a model category [Gro, Hin97, Hov9*
*8],
which do the case of R-modules, but this was probably known to others as well.
An important corollary of the fact that a derived category D(A) is the homot*
*opy
category of a model category is that the group D(A)(X, Y ) of maps is a set (as*
* opposed
to a proper class) for any two chain complexes X and Y . This is not the case i*
*n general,
and much work on derived categories ignores this possibility. The importance o*
*f this
point is that if one uses the morphisms in the derived category to index constr*
*uctions
in other categories or to define cohomology groups, one needs to know that the *
*indexing
class is actually a set. Recently, D(A)(X, Y ) has been shown to be a set under*
* various
assumptions on A. (See Weibel [Wei94] Remark 10.4.5, which credits Gabber, and
Exercise 10.4.5, which credits Lewis, May and Steinberger [LMS86 ]. See also Kr*
*iz and
4 J. DANIEL CHRISTENSEN AND MARK HOVEY
May [KM95 , Part III].) The assumptions that appear in the present paper are di*
*fferent
from those that have appeared before and the proof is somewhat easier (because *
*of our
use of the theory of cofibrantly generated model categories), so this paper may*
* be of
some interest even in this special case.
Another consequence of the fact that Ch(R) is a model category is the existe*
*nce of
resolutions coming from cofibrant and fibrant approximations, and the related d*
*erived
functors. Some of these are discussed in [AFH97 ] and [Spa88]. We do not discus*
*s these
topics here, but just mention that these resolutions are immediate once you hav*
*e the
model structure, so our approach gives these results with very little work.
While our results include new examples of traditional homological algebra, o*
*ur focus is
on more general projective classes. For example, let A be an algebra over a com*
*mutative
ring k. We call a map of A-bimodules a relative epimorphism if it is split epi*
*c as a
map of k-modules, and we call an A-bimodule a relative projective if maps from *
*it lift
over relative epimorphisms. These definitions give a projective class, and Theo*
*rem 2.2
tells us that there is a model category, and therefore a derived category, that*
* captures
the homological algebra of this situation. For example, Hochschild cohomology g*
*roups
appear as Hom sets in this derived category (see Example 3.7).
We also discuss pure homological algebra and construct the üp re derived cat*
*egoryö f
a ring. Pure homological algebra has applications to phantom maps in the stable*
* homo-
topy category [CS98 ] and in the (usual) derived category of a ring [Chr98], co*
*nnections
to Kasparov KK-theory [Sch01], and is actively studied by algebraists and repre*
*sentation
theorists.
In the last section we describe a model category structure on the category o*
*f non-
negatively graded chain complexes that works for an arbitrary projective class *
*on an
abelian category, without any hypotheses. More generally, we show that under ap*
*pro-
priate hypotheses a projective class on a possibly non-abelian category A deter*
*mines a
model category structure on the category of simplicial objects in A. As an exa*
*mple,
we deduce that the category of equivariant simplicial sets has various model ca*
*tegory
structures.
We now briefly outline the paper. In Section 1 we give the axioms for a proj*
*ective class
and mention many examples that will be discussed further in Subsections 3.1 and*
* 5.3.
In Section 2 we describe the desired model structure coming from a projective c*
*lasses
and state our main theorem, which says that the model structure exists as long *
*as
cofibrant replacements exist. We also give two hypotheses that each imply the e*
*xistence
of cofibrant replacements. The first hypothesis handles situations coming from *
*adjoint
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 5
pairs, and is proved to be sufficient in Section 3, where we also give many exa*
*mples
involving relative situations. The second hypothesis deals with projective clas*
*ses that
have enough small projectives and is proved to be sufficient in Section 4. In S*
*ection 5
we prove that the model structure that one gets is cofibrantly generated if and*
* only if
there is a set of enough small projectives. We do this using the recognition le*
*mma for
cofibrantly generated categories, which is recalled in Subsection 5.1. This cas*
*e is proved
from scratch, independent of the main result in Section 2, since the proof is n*
*ot long. In
Subsection 5.3 we give two examples, the traditional derived category of R-modu*
*les and
the pure derived category. We describe how the two relate and why the pure deri*
*ved
category is interesting. In the final section we discuss the bounded below case*
*, which
works for any projective class, and describe a result for simplicial objects in*
* a possibly
non-abelian category.
We thank Haynes Miller for asking the question that led to this paper and Ha*
*ynes
Miller and John Palmieri for fruitful and enjoyable discussions.
0. Notation and conventions
We make a few blanket assumptions. With the exception of Section 6, A and B *
*will
denote abelian categories. We will assume that our abelian categories are bicom*
*plete;
this assumption is stronger than strictly necessary, but it simplifies the stat*
*ements of
our results. For any category C, we write C(A, B) for the set of maps from A to*
* B in C.
We write Ch(A) for the category of unbounded chain complexes of objects of A*
* and
degree zero chain maps. To fix notation, assume that the differentials lower de*
*gree. For
an object X of Ch(A), define ZnX := ker(d: Xn -!Xn-1) and BnX := im(d: Xn+1 -!
Xn), and write HnX for the quotient. A map inducing an isomorphism in Hn for al*
*l n is a
quasi-isomorphism. The suspension X of X has ( X)n = Xn-1 and d X = -dX .
The functor is defined on morphisms by ( f)n = fn-1. Given a map f :X -! Y of
chain complexes, the cofibre of f is the chain complex C with Cn = Yn Xn-1 and
with differential d(y, x) = (dy + fx, -dx). There are natural maps Y -! C -! X*
*, and
the sequence X -! Y -! C -! X is called a cofibre sequence.
Two maps f, g :X -! Y are chain homotopic if there is a collection of maps
sn: Xn -! Yn+1 such that f - g = ds + sd. We write [X, Y ] for the chain homoto*
*py
classes of maps from X to Y and K(A) for the category of chain complexes and ch*
*ain
homotopy classes of maps. Two complexes are chain homotopy equivalent if they a*
*re
isomorphic in K(A), and a complex is contractible if it is chain homotopy equiv*
*alent
to 0. K(A) is a triangulated category, with triangles the sequences chain homo*
*topy
6 J. DANIEL CHRISTENSEN AND MARK HOVEY
equivalent to the cofibre sequences above. The functors Hn(-), [X, -] and [-, Y*
* ] are
defined on K(A) and send triangles to long exact sequences.
For P in A, DkP denotes the (contractible) complex such that (DkP )n = P if *
*n = k
or n = k - 1 but (DkP )n = 0 for other values of n, and whose differential is t*
*he identity
in degree k. The functor Dk is left adjoint to the functor X 7! Xk, and right a*
*djoint
to the functor X 7! Xk-1. The path complex P X of a complex X is the contractib*
*le
complex such that (P X)n = Xn Xn+1, where d(x, y) = (dx, x - dy).
We assume knowledge of the basics of model categories, for which [DS95 ] is *
*an excel-
lent reference. We use the definition of model category that requires that the *
*category
be complete and cocomplete, and that the factorizations be functorial. Corresp*
*ond-
ingly, when we say that cofibrant replacements exist, we implicitly mean that t*
*hey are
functorial.
1. Projective classes
1.1. Definition and some examples. In this subsection we explain the notion of a
projective class, which is the information necessary in order to do homological*
* algebra.
Intuitively, a projective class is a choice of which sort of "elements" we wish*
* to think
about. In this section we focus on the case of an abelian category, but this de*
*finition
works for any pointed category with kernels.
The elements of a set X correspond bijectively to the maps from a singleton *
*to X,
and the elements of an abelian group A correspond bijectively to the maps from *
*Z to
A. Motivated by this, we call a map P -! A in any category a P -element of A. If
we don't wish to mention P , we call such a map a generalized element of A. A m*
*ap
A -!B in any category is determined by what it does on generalized elements. If*
* P is
a collection of objects, then a P-element means a P -element for some P in P.
Let A be an abelian category. A map B -! C is said to be P -epic if it induc*
*es a
surjection of P -elements, that is, if the induced map A(P, B) -!A(P, C) is a s*
*urjection
of abelian groups. The map B -!C is P-epic if it is P -epic for all P in P.
Definition 1.1.A projective class on A is a collection P of objects of A and a
collection E of maps B -!C in A such that
(i)E is precisely the collection of all P-epic maps;
(ii)P is precisely the collection of all objects P such that each map in E is *
*P -epic;
(iii)for each object B there is a map P -! B in E with P in P.
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 7
When a collection P is part of a projective class (P, E), the projective class *
*is unique, and
so we say that P determines a projective class or even that P is a projective c*
*lass. An
object of P is called a P-projective, or, if the context is clear, a relative p*
*rojective.
A sequence
A -!B -!C
is said to be P -exact if the composite A -!C is zero and
A(P, A) -!A(P, B) -!A(P, C)
is an exact sequence of abelian groups. The latter can be rephrased as the con*
*dition
that A -! B -! C induces an exact sequence of P -elements. A P-exact sequence is
one that is P -exact for all P in P.
Example 1.2. For an associative ring R, let A be the category of left R-module*
*s, let
P be the collection of all summands of free R-modules and let E be the collecti*
*on of all
surjections of R-modules. Then E is precisely the collection of P-epimorphisms,*
* and P
is a projective class. The P-exact sequences are the usual exact sequences.
Example 1.2 is a categorical projective class in the sense that the P-epimor*
*phisms
are just the epimorphisms and the P-projectives are the categorical projectives*
*, i.e.,
those objects P such that maps from P lift through epimorphisms.
Here are two examples of non-categorical projective classes.
Example 1.3. If A is any abelian category, P is the collection of all objects,*
* and E
is the collection of all split epimorphisms B -! C, then P is a projective clas*
*s. It is
called the trivial projective class. A sequence A -!B -! C is P-exact if and on*
*ly if
A -!ker(B -!C) is split epic.
Example 1.4. Let A be the category of left R-modules, as in Example 1.2. Let P
consist of all summands of sums of finitely presented modules and define E to c*
*onsist of
all P-epimorphisms. Then P is a projective class. A sequence is P-exact iff it *
*is exact
after tensoring with every right module.
Examples 1.2 and 1.4 will be discussed further in Subsection 5.3. Example 1*
*.3 is
important because many interesting examples are üp llbacksö f this projective *
*class
(see Subsection 1.3).
Let P be a projective class. If S is a subcollection of P (not necessarily a*
* set), and
if a map is S-epic iff it is P-epic, then we say that P is determined by S and *
*that S
8 J. DANIEL CHRISTENSEN AND MARK HOVEY
is a collection of enough projectives. Some projective classes, such as Example*
*s 1.2
and 1.4, are determined by a set, and the lemma below shows that any set of obj*
*ects
determines a projective class. The trivial projective class is sometimes not de*
*termined
by a set (see Subsection 5.4).
Lemma 1.5. Suppose F is any set of objects in an abelian category with coprodu*
*cts.
Let E be the collection of F-epimorphisms and let P be the collection of all ob*
*jects P
such that every map in E is P -epic. Then P is the collection of retracts of co*
*products of
objects of F and (P, E) is a projective class.
`
Proof.Given an object X, let P be the coproduct F indexed by all maps F -! X *
*and
all objects F in F. The natural map P -! X is clearly an F-epimorphism. Moreove*
*r, if
X is in P, then this map is split epic, and so X is a retract of a coproduct of*
* objects of
F. These two facts show that (P, E) is a projective class. *
* |___|
1.2. Homological algebra. A projective class is precisely the information neede*
*d to
form projective resolutions and define derived functors. All of the usual defin*
*itions and
theorems go through. A P-resolution of an object M is a P-exact sequence
. .-.!P2 -!P1 -!P0 -!M -! 0
such that each Pi is in P. If B is an abelian category and T :A -! B is an add*
*itive
functor, then the nth left derived functor of T with respect to P is defined by*
* LPnT (M) =
Hn(T (P*)) where P* is a P-resolution of M. One has the usual uniqueness of res*
*olutions
up to chain homotopy and so this is well-defined. From a P-exact sequence 0 -! *
*L -!
M -! N -! 0 one gets a long exact sequence involving the derived functors. T*
*he
abelian groups ExtnP(M, N) can be defined in the usual two ways, as equivalence*
* classes
of P-exact sequences 0 -! N -! L1 -! . .-.!Ln -! M -! 0, or as LPnT (M) where
T (-) = A(-, N).
For further details and useful results we refer the reader to the classic re*
*ference [EM65 ].
1.3. Pullbacks. A common setup in relative homological algebra is the following*
*. We
assume we have a functor U :A -!B of abelian categories, together with a left a*
*djoint
F :B -!A. Then U and F are additive, U is left exact and F is right exact.
If (P0, E0) is a projective class on B, we define P := {retracts of F P for *
*P in}P0and
E := {B -!C such that UB -!UC is in E0}. Then one can easily show that (P, E) i*
*s a
projective class on A and that a sequence is P-exact if and only if it is sent *
*to a P0-exact
sequence by U. (P, E) is called the pullback of (P0, E0) along the right adjoin*
*t U.
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 9
The most common case is when (P0, E0) is the trivial projective class (see E*
*xample 1.3).
Then for any M in A the counit F UM -! M is a P-epimorphism from a P-projective.
Example 1.6. Let R -!S be a map of associative rings. Write R-mod and S-mod for
the categories of left R- and S-modules. Consider the forgetful functor U :S-mo*
*d -!
R-mod and its left adjoint F that sends an R-module M to S R M. The pullback a*
*long
U of the trivial projective class on R-mod gives a projective class P. The P-pr*
*ojectives
are the S-modules P such that the natural map S R P -! P is split epic as a ma*
*p of
S-modules. The P-epimorphisms are the S-module maps that are split epic as maps*
* of
R-modules.
Example 1.7. As above, let R -! S be a map of rings. The forgetful functor
U :S-mod -! R-mod has a right adjoint G that sends an R-module M to the S-module
R-mod(S, M). We can pullback the trivial injective class along U to get an inj*
*ective
class on S-mod . (An injective class is just a projective class on the opposite*
* category.)
The relative injectives are the S-modules I such that the natural map I -! R-mo*
*d(S, I)
is split monic as a map of S-modules, and the relative monomorphisms are the S-*
*module
maps that are split monic as maps of R-modules.
We investigate these examples in detail in Section 3.
1.4. Strong projective classes. In Section 3 we will focus on projective classe*
*s that
are the pullback of a trivial projective class along a right adjoint. In this s*
*ubsection we
describe a special property that these projective classes have.
Definition 1.8.A projective class P is strong if for each P-projective P and ea*
*ch
P-epimorphism M -! N, the surjection A(P, M) -! A(P, N) of abelian groups is sp*
*lit
epic.
It is clear that a trivial projective class is strong, and that the pullback*
* of a strong
projective class is strong. The importance of strong projective classes comes f*
*rom the
following lemma.
Lemma 1.9. The following are equivalent for a projective class P on an abelian*
* category
A:
(i)P is strong.
(ii)For each complex C in Ch(A), if the complex A(P, C) in Ch(Ab ) has trivial*
* ho-
mology for each P in P, then it is contractible for each P in P.
(iii)For each map f in Ch(A), if the map A(P, f) in Ch(Ab ) is a quasi-isomorp*
*hism
for each P in P, then it is a chain homotopy equivalence for each P in P.
10 J. DANIEL CHRISTENSEN AND MARK HOVEY
Here A(P, C) denotes the chain complex with A(P, Ck) in degree k. We also us*
*e the
notation from Section 0.
Proof.(i) =) (ii): Assume P is a strong projective class and let C be a compl*
*ex in
Ch (A) such that A(P, C) has trivial homology for each P-projective P . Then fo*
*r each
k and each P we have a short exact sequence
0 -!A(P, ZkC) -!A(P, Ck) -!A(P, Zk-1C) -!0
of abelian groups. Because the projective class is strong, the sequence is spl*
*it. This
implies that the complex A(P, C) is isomorphic to kDk+1A(P, ZkC), and in parti*
*cular
that it is contractible.
(ii) =) (iii): Let f :X -! Y be a map in Ch(A) such that A(P, X) -! A(P, Y*
* ) is
a quasi-isomorphism for each P in P, and let C be the cofibre of f. Then by the*
* long
exact sequence, A(P, C) has trivial homology for each P in P. By (ii) this comp*
*lex is
contractible. This implies that A(P, X) -!A(P, Y ) is a chain homotopy equivale*
*nce.
(iii) =) (i): Let M -! N be a P-epimorphism with kernel L. Then the complex
L -!M -! N has trivial homology after applying A(P, -), for each P-projective P*
* . By
(iii), it is contractible after applying A(P, -), for each P . In particular, *
*A(P, M) -!
A(P, N) is split epic. |___|
2. The relative model structure
The object of this section is to construct a Quillen model structure on the *
*category
Ch (A) of chain complexes over A that reflects a given projective class P on A.
If X is a chain complex, we write A(P, X) for the chain complex that has the*
* abelian
group A(P, Xn) in degree n. This is the chain complex of P -elements of X.
Definition 2.1.A map f :X -! Y in Ch(A) is a P-equivalence if A(P, f) is a quas*
*i-
isomorphism in Ch(Z) for each P in P. The map f is a P-fibration if A(P, f) is*
* a
surjection for each P in P. The map f is a P-cofibration if f has the left lif*
*ting
property with respect to all maps that are both P-fibrations and P-equivalences*
* (the
P-trivial fibrations).
The motivation for this definition is that it implies that a complex
. .-.!P2 -!P1 -!P0 -!0 -!. . .
equipped with an augmentation P0 -! M to an object M is a cofibrant replacement
if and only if it is a P-resolution in the sense of Subsection 1.2. This impli*
*es that if
M and N are objects of A thought of as complexes concentrated in degree zero, t*
*hen
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 11
ExtnP(M, N) can be identified with maps from nM to N in the homotopy category *
*of
the model category Ch(A). This will be described in more detail in Subsection 2*
*.1.
The main goal of this section is then to prove the following theorem.
Theorem 2.2. Suppose P is a projective class on the abelian category A. Then t*
*he cat-
egory Ch(A), together with the P-equivalences, the P-fibrations, and the P-cofi*
*brations,
forms a Quillen model category if and only if cofibrant replacements exist. Wh*
*en the
model structure exists, it is proper. Cofibrant replacements exist in each of t*
*he following
cases:
A: P is the pullback of the trivial projective class along a right adjoint th*
*at preserves
countable sums.
B: There are enough ~-small P-projectives for some cardinal ~, and P-resoluti*
*ons can
be chosen functorially.
The words "enoughä nd "~-small" will be explained in Section 4.
We call this structure the relative model structure. We point out that we a*
*re
using the modern definition of model category [DHK97 , Hov98], so our factoriza*
*tions
will be functorial. Correspondingly, we require our cofibrant replacements to b*
*e functo-
rial as well. This theorem requires our blanket assumption that abelian categor*
*ies are
bicomplete.
In Subsection 2.1 we will describe further properties of these model structu*
*res, in-
cluding conditions under which they are monoidal. In Section 5 we show that if *
*there is
a set of enough small projectives, then the model structure is cofibrantly gene*
*rated. On
the other hand, we show in Subsection 5.4 that model categories coming from Cas*
*e A
are generally not cofibrantly generated.
In Subsection 2.2 we explain why it is no loss of generality to start with a*
* projective
class (P, E), rather than just an arbitrary class P of test objects.
Proof.Some of the properties necessary for a model category are evident from th*
*e def-
initions. It is clear that Ch(A) is bicomplete, since A is so. Also, P-equivale*
*nces have
the two out of three property, and P-equivalences, P-fibrations, and P-cofibrat*
*ions are
closed under retracts. Furthermore, P-cofibrations have the left lifting proper*
*ty with re-
spect to P-trivial fibrations, by definition. It remains to show that P-trivial*
* cofibrations
have the left lifting property with respect to P-fibrations, and that the two f*
*actoriza-
tion axioms hold. The remaining lifting property will be proved in Proposition *
*2.8, and
the two factorization axioms will be proved in Propositions 2.9 and 2.10, assum*
*ing that
12 J. DANIEL CHRISTENSEN AND MARK HOVEY
cofibrant replacements exist. Properness will be proved in Proposition 2.18, wh*
*ich also
defines the term.
That cofibrant replacements exist in cases A and B will be proved in Section*
*s 3 and
4, respectively. |___|
Note that our Theorem 2.2 will also apply when we have an injective class, t*
*hat is, a
projective class on Aop, by dualizing the definition of the model structure.
We begin the proof of Theorem 2.2 with a lemma that gives us a simple test o*
*f the
lifting property. We use the notation from Section 0.
Lemma 2.3. Suppose p: X -! Y is a P-fibration with kernel K, and i: A -! B is a
degreewise split inclusion whose cokernel C is a complex of relative projective*
*s. If every
map C -! K is chain homotopic to 0, then i has the left lifting property with *
*respect
to p.
Proof.We can write Bn ~= An Cn, where the differential is defined by d(a, c) =
(da+øc, dc) (we use the element notation for convenience, but it is not strictl*
*y necessary),
and ø :Cn -!An-1 can be any family of maps such that dø + ød = 0. Suppose we ha*
*ve
a commutative square as below.
A - -f--!X
? ?
i?y ?yp
B - ---!gY
In terms of the splitting Bn ~=An Cn, we have g(a, c) = pf(a)+ff(c), where the*
* family
ffn: Cn -! Yn satisfies dff = pfø + ffd. We are looking for a map h: B -! X mak*
*ing
the diagram above commute. In terms of the splitting, this means we are looking*
* for a
family of maps fin: Cn -!Xn such that pfi = ff and dfi = fø +fid. Since Cn is r*
*elatively
projective, and p is P-epic, there is a map fl :Cn -!Xn such that pfl = ff. The*
* difference
ffi = dfl - fø - fld may not be zero, but at least pffi = 0. Let j :K -! X deno*
*te the kernel
of p. Then there is a map F :Cn -! Kn-1 such that jF = ffi. Furthermore, one can
check that F d = -dF , so that F :C -! K is a chain map. By hypothesis, F is c*
*hain
homotopic to 0 by a map D :Cn -! Kn, so that Dd - dD = F . Define fi = fl + jD.
Then fi defines the desired lift, so i has the left lifting property with respe*
*ct to p. |___|
Now we study the P-cofibrations. A complex C is called P-cofibrant if the m*
*ap
0 -! C is a P-cofibration. A complex K is called weakly P-contractible if the m*
*ap
K -! 0 is a P-equivalence, or, equivalently, if all maps from a complex kP con*
*sisting
of a relative projective concentrated in one degree to K are chain homotopic to*
* 0.
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 13
Lemma 2.4. A complex C is P-cofibrant if and only if each Cn is relatively pro*
*jective
and every map from C to a weakly P-contractible complex K is chain homotopic to*
* 0.
Proof.Suppose first that C is P-cofibrant. If M -! N is a P-epimorphism, then t*
*he
map Dn+1M -! Dn+1N is a P-fibration. It is also a P-equivalence, since it is in*
* fact a
chain homotopy equivalence. Since C is P-cofibrant, the map
Ch (A)(C, Dn+1M) -!Ch (A)(C, Dn+1N)
is surjective. But this map is isomorphic to the map A(Cn, M) -! A(Cn, N), so C*
*n is
relatively projective.
If K is weakly P-contractible, then the natural map P K -! K is a P-trivial *
*fibration.
Since C is P-cofibrant, any map C -! K factors through P K, which means that it*
* is
chain homotopic to 0.
The converse follows immediately from Lemma 2.3, since the kernel of a P-tri*
*vial
fibration is weakly P-contractible. |*
*___|
Proposition 2.5.A map i: A -! B is a P-cofibration if and only if i is a degree*
*wise
split monomorphism with P-cofibrant cokernel.
Proof.Suppose first that i is a P-cofibration with cokernel C. Since P-cofibrat*
*ions are
closed under pushouts, it is clear that C is P-cofibrant. The map Dn+1An -! 0 *
*is a
P-fibration and a P-equivalence. Since i is a P-cofibration, the map A -!Dn+1An*
* that
is the identity in degree n extends to a map B -!Dn+1An. In degree n, this map *
*defines
a splitting of in.
Conversely, suppose that i is a degreewise split monomorphism and the cokern*
*el C of
i is P-cofibrant. We need to show that i has the left lifting property with res*
*pect to any
P-trivial fibration p: X -! Y . But this follows by combining Lemmas 2.3 and 2.*
*4. |___|
The next lemma provides a source of P-cofibrant objects, including the P-cel*
*lular
complexes.
Definition 2.6.Call a complex C purely P-cellular if it is a colimit of a colim*
*it-
preserving diagram
0 = C0 -!C1 -!C2 -!. . .
indexed by an ordinal fl, such that for each ff < fl the map Cff-! Cff+1is degr*
*eewise split
monic with cokernel a complex of relative projectives with zero differential. B*
*y öc limit-
preserving" we mean that for each limit ordinal ~ < fl, the map colimff<~Cff-! *
*C~ is
an isomorphism. We say C is P-cellular if it is a retract of a purely P-cellula*
*r complex.
14 J. DANIEL CHRISTENSEN AND MARK HOVEY
Lemma 2.7. (a)If D in Ch(A) is a complex of relative projectives with zero d*
*iffer-
ential, then D is P-cofibrant.
(b)If D in Ch(A) is a bounded below complex of relative projectives, then D i*
*s P-
cofibrant.
(c)If D in Ch(A) is P-cellular, then D is P-cofibrant.
Proof.(a) follows immediately from Lemma 2.4.
(b) Let D be a bounded below complex of relative projectives and write D n f*
*or the
truncation of D that agrees with D in degrees n and is 0 elsewhere. Then the *
*map
D n -! D n+1 is degreewise split monic and has a P-cofibrant cokernel (by (a)),*
* so is a
P-cofibration by Proposition 2.5. Since D n = 0 for n << 0 and 0 is P-cofibrant*
*, each
D n is P-cofibrant. Therefore, so is their colimit D.
(c) The proof is just a transfinite version of the proof of (b), combined wi*
*th the fact
that a retract of a cofibrant object is cofibrant. *
* |___|
We can now prove that the other lifting axiom holds.
Proposition 2.8.A map i: A -! B has the left lifting property with respect to P-
fibrations if and only if i is a P-trivial cofibration.
Proof.Suppose first that i has the left lifting property with respect to P-fibr*
*ations.
Then i is a P-cofibration, by definition, and the cokernel 0 -!C also has the l*
*eft lifting
property with respect to P-fibrations. In particular, since the map P C -! C i*
*s a P-
fibration, C is contractible. Hence i is a chain homotopy equivalence, and in p*
*articular
a P-trivial cofibration.
Conversely, suppose that i is a P-trivial cofibration with cokernel C. By Le*
*mma 2.3,
in order to show that i has the left lifting property with respect to P-fibrati*
*ons, it
suffices to show that every map from C to any complex K is chain homotopic to 0*
*. This
is equivalent to showing that C is contractible. Since i is degreewise split m*
*onic, for
each relative projective P there is a long exact sequence . .-.![ kP, A] -![ kP*
*, B] -!
[ kP, C] -! . ...Since i is a P-equivalence, [ kP, C] = 0 for each relative pro*
*jective P
and each k, and so P C -! C is a P-trivial fibration. Since C is P-cofibrant, t*
*he identity
map of C factors through P C, and so C is contractible. *
* |___|
Note that the proof shows that a P-trivial cofibration is in fact a chain ho*
*motopy
equivalence.
Now we proceed to prove the factorization axioms, under the assumption that *
*we
have cofibrant replacements.
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 15
Proposition 2.9.If every object A has a cofibrant replacement qA :QA -! A, then
every map in Ch(A) can be factored into a P-cofibration followed by a P-trivial*
* fibration.
Proof.Suppose f :A -!B is a map in Ch(A). Let C be the cofibre of f, so C = B *
* A
with d(b, a) = (db+fa, -da), and let E be the fibre of the composite g :QC -! C*
* -! A,
so E = A QC with d(a, q) = (da - gq, dq) (the desuspension of the cofibre). C*
*onsider
the diagram
A ---i-! E ----! QC --g--! A
flfl ? fl
fl qC?y flfl
A ----! B ----! C ----! A
f
whose rows are triangles in K(A). There is a natural fill-in map p: E -! B defi*
*ned by
p(a, q) = f(a) + ßB qC q, where ßB :C -! B is the projection. The map p makes t*
*he left-
hand square commute in Ch(A) and the middle square commute in K(A) (with the ch*
*ain
homotopy s(a, q) = (0, a)). The map i: A -!E is a P-cofibration since it is deg*
*reewise
split and its cokernel QC is P-cofibrant. Furthermore, since QC -! C is degree*
*wise
P-epic and ßB :C -! B is degreewise split epic, it follows that p is degreewise*
* P-epic.
Applying the functor [ kP, -] gives two long exact sequences, and from the five*
*-lemma
one sees that [ kP, p] is an isomorphism when P is a relative projective. Thus *
*f = pi is
the required factorization. |__*
*_|
Proposition 2.10. If every object A has a cofibrant replacement qA :QA -! A, th*
*en
every map in Ch(A) can be factored into a P-trivial cofibration followed by a P*
*-fibration.
Proof.It is well-known that we can factor any map f :A -!B in Ch(A) into a degr*
*eewise
split monomorphism that is also a chain homotopy equivalence, followed by a deg*
*reewise
split epimorphism. Since every degreewise split epimorphism is a P-fibration, w*
*e may as
well assume f is a degreewise split monomorphism and a chain homotopy equivalen*
*ce.
In this case, we apply Proposition 2.9 to factor f = pi, where p is a P-triv*
*ial fibration
and i is a P-cofibration. Since f is a chain homotopy equivalence, i must be a *
*P-trivial
cofibration, and so the proof is complete. *
*|___|
Note that the factorizations constructed in Proposition 2.9 and Proposition *
*2.10 are
both functorial in the map f, since we are implicitly assuming that cofibrant r*
*eplacement
is functorial.
The homotopy category of Ch(A), formed by inverting the P-equivalences, is c*
*alled
the derived category of A (with respect to P). It is denoted D(A), and a fundam*
*ental
result in model category theory asserts that D(A)(X, Y ) is a set for each X an*
*d Y . In
16 J. DANIEL CHRISTENSEN AND MARK HOVEY
Exercise 10.4.5 of [Wei94], Weibel outlines an argument that proves that D(A)(X*
*, Y )
is a set when there are enough (categorical) projectives, P is the categorical *
*projective
class, and A satisfies AB5. A connection between Weibel's hypotheses and Case B*
* is
that if A has enough projectives that are small with respect to all filtered di*
*agrams in
A, then AB5 holds. The smallness condition needed for our theorem is weaker tha*
*n this.
(See Section 4 for the precise hypothesis.)
2.1. Properties of the relative model structure. In this subsection, we investi*
*gate
some of the properties of the relative model structure. We begin by showing tha*
*t the
model category notions of homotopy, derived functor, suspension and cofibre seq*
*uence
agree with the usual notions. Then we study properness and monoidal structure. *
*We
discuss cofibrant generation in Section 5.
We assume throughout that P is a projective class on an abelian category A s*
*uch
that the relative model structure on Ch(A) exists.
We first show that the notion of homotopy determined from the model category
structure corresponds to the usual notion of chain homotopy.
Definition 2.11.([DS95 ] or [Qui67].) If M is an object in a model category C, *
*a good
` i p
cylinder object for M is an object M x I and a factorization M M -! M x I -! M
of the codiagonal map, with i a cofibration and p a weak equivalence. (Despite*
* the
notation, M x I is not in general a product of M with an object I.) A left homo*
*topy
between maps f, g :M -! N is a map H :M x I -! N such that the composite Hi is
` `
equal to f g :M M -! N, for some good cylinder object M x I.
The notion of good path object NI for N is dual to that of good cylinder obj*
*ect
and leads to the notion of right homotopy. The following standard result can be*
* found
in [DS95 , Section 4], for example.
Lemma 2.12. For M cofibrant and N fibrant, two maps f, g :M -! N are left hom*
*o-
topic if and only if they are right homotopic, and both of these relations are *
*equivalence
relations and respect composition. Moreover, if M x I is a fixed good cylinder *
*object for
M, then f and g are left homotopic if and only if they are left homotopic using*
* M x I;
similarly for a fixed good path object. *
*|___|
Because of the lemma, for M cofibrant and N fibrant we have a well-defined r*
*elation of
homotopy on maps M -! N. Quillen showed that the homotopy category of C, which
is by definition the category of fractions formed by inverting the weak equival*
*ences, is
equivalent to the category consisting of objects that are both fibrant and cofi*
*brant with
morphisms being homotopy classes of morphisms.
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 17
Now we return to the study of the model category Ch(A).
Lemma 2.13. Let M and N be objects of Ch(A) with M P-cofibrant. Two maps M -!
N are homotopic if and only if they are chain homotopic.
Proof.We construct a factorization M M -! M x I -! M of the codiagonal map
M M -! M in the following way. Let M x I be the chain complex that has Mn
Mn-1 Mn in degree n. We describe the differential by saying that it sends a gen*
*eralized
element (m, ~m, m0) in (M x I)n to (dm + ~m, -dm~, dm0- ~m). Let i: M M -! M *
*x I
be the map that sends (m, m0) to (m, 0, m0) and let p: M x I -! M be the map th*
*at
sends (m, ~m, m0) to m + m0. One can check easily that M x I is a chain complex*
* and
that i and p are chain maps whose composite is the codiagonal. The map i is deg*
*reewise
split monic with cokernel M, so it is a P-cofibration, since we have assumed t*
*hat M
is cofibrant. The map p is a chain homotopy equivalence with chain homotopy inv*
*erse
sending m to (m, 0, 0); this implies that it induces a chain homotopy equivalen*
*ce of
generalized elements and is thus a P-equivalence. Therefore M x I is a good cyl*
*inder
object for M.
It is easy to see that a chain homotopy between two maps M -! N is the same *
*as
a left homotopy using the good cylinder object M x I. By Lemma 2.12, two maps a*
*re
homotopic if and only if they are left homotopic using M x I. Thus the model ca*
*tegory
notion of homotopy is the same as the notion of chain homotopy when the source *
*is
P-cofibrant. |___|
There is a dual proof that proceeds by constructing a specific good path obj*
*ect NI
for N such that a right homotopy using NI is the same as a chain homotopy.
Corollary 2.14. Let A and B be objects of A considered as chain complexes conce*
*n-
trated in degree 0. Then D(A)(A, nB) ~=ExtnP(A, B).
See Subsection 1.2 for the definition of the Ext groups.
Proof.The group D(A)(A, nB) may be calculated by choosing a P-cofibrant replac*
*e-
ment A0 for A and computing the homotopy classes of maps from A0 to nB. (Re-
call that all objects are P-fibrant, so there is no need to take a fibrant repl*
*acement
for nB.) A P-resolution P of A serves as a P-cofibrant replacement for A, and*
* by
Lemma 2.13 the homotopy relation on Ch(A)(P, nB) is chain homotopy, so it foll*
*ows
that D(A)(A, nB) is isomorphic to ExtnP(A, B). |_*
*__|
More generally, a similar argument shows that the derived functors of a func*
*tor F
can be expressed as the cohomology of the derived functor of F in the model cat*
*egory
18 J. DANIEL CHRISTENSEN AND MARK HOVEY
sense. To make the story complete, we next show that the shift functor corres*
*ponds
to the notion of suspension that the category D(A) obtains as the homotopy cate*
*gory
of a pointed model category.
Definition 2.15.Let C be a pointed model category. If M is cofibrant, we define*
* the
`
suspension M of M to be the cofibre of the map M M -! M x I for some good
`
cylinder object M x I. (The cofibre of a map X -! Y is the pushout * X Y , whe*
*re *
is the zero object.) M is cofibrant and well-defined up to homotopy equivalenc*
*e.
The loop object N of a fibrant object N is defined dually. These operations*
* induce
adjoint functors on the homotopy category. A straightforward argument based on *
*the
cylinder object described above (and a dual path object) proves the following l*
*emma.
Lemma 2.16. In the model category Ch(A), the functor defined in Definition *
*2.15
can be taken to be the usual suspension, so that ( X)n = Xn-1 and d X = -dX .
Similarly, X can be taken to be the complex -1X. That is, ( X)n = Xn+1 and
d X = -dX . |___|
In particular, and are inverse functors. The second author [Hov98] has s*
*hown that
this implies that cofibre sequences and fibre sequences agree (up to the usual *
*sign) and
that and the cofibre sequences give rise to a triangulation of the homotopy c*
*ategory.
(See [Qui67, Section I.3] for the definition of cofibre and fibre sequences in *
*any pointed
model category.) Using the explicit cylinder object from the proof of Lemma 2.1*
*3, we
can be more explicit.
Corollary 2.17. The category D(A) is triangulated with the usual suspension. A *
*se-
quence L -!M -! N -! L is a triangle if and only if it is isomorphic in D(A) t*
*o the
usual cofibre sequence on the map L -!M (see Section 0). |*
*___|
Now we show that the model structures we construct are proper. A good refere*
*nce
for proper model categories is [Hir00, Chapter 11].
Proposition 2.18. Let P be any projective class on an abelian category A. Consi*
*der
the commutative square in Ch(A) below.
A - -f--!X
? ?
q?y ?yp
B - ---!gY
(a)If the square is a pullback square, p is a P-fibration, and g is a P-equiv*
*alence, then
f is a P-equivalence. That is, the relative model structure is right prope*
*r.
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 19
(b)If the square is a pushout square, q is a degreewise split monomorphism, a*
*nd f
is a P-equivalence, then g is a P-equivalence. In particular, the pushout *
*of a P-
equivalence along a P-cofibration is a P-equivalence. That is, the relativ*
*e model
structure is left proper.
A model category that is both left and right proper is said to be proper. No*
*te that
we don't actually need to know that our P-cofibrations, P-fibrations and P-equi*
*valences
give a model structure to ask whether the structure is proper.
Proof.Part (a) is an immediate consequence of [Hir00, Corollary 11.1.3], since *
*every
object is P-fibrant. For part (b), let C be the cokernel of q. Since pushouts a*
*re computed
degreewise, it follows that p is a degreewise split monomorphism with cokernel *
*C. Thus
we have a map of triangles
A - -f--!X
? ?
q?y ?yp
B - ---!gY
?? ?
y ?y
C - ---! C
id
in the homotopy category K(A). The top and bottom maps are P-equivalences, so t*
*he
middle map must be as well, by using the five-lemma and the long exact sequences
obtained by applying the functor [ kP, -]. |_*
*__|
We now consider monoidal structure. Monoidal model categories are studied
in [Hov98, Chapter 4]. We will assume that A is a closed monoidal category. T*
*hus
it is equipped with a functor : AxA -!A such that both A - and - A have right
adjoints for each A in A. In particular, what we need is that these functors pr*
*eserve
colimits. There is, of course, an induced closed monoidal structure on Ch(A), f*
*or which
we also use the notation .
Proposition 2.19. Let A be a closed monoidal abelian category with a projective*
* class
P such that cofibrant replacements exist and the unit is P-projective. Then the*
* relative
model structure on Ch(A) is monoidal if and only if the tensor product of two P*
*-cofibrant
complexes is always P-cofibrant.
Proof.There are two conditions that must hold for a model category to be monoid*
*al.
One of them is automatically satisfied when the unit is cofibrant. The unit of*
* the
monoidal structure on Ch(A) is 0S, where S is the unit of A, which has been as*
*sumed
to be P-projective. Thus the unit is P-cofibrant, by Lemma 2.7 (a),
20 J. DANIEL CHRISTENSEN AND MARK HOVEY
Therefore, the relative model structure is monoidal if and only if whenever *
*f :A -!B
and g :X -! Y are P-cofibrations, then the map
f g :(A Y ) qA X (B X) -!B Y
is a P-cofibration, and is a P-trivial cofibration if either f or g is a P-triv*
*ial cofibration.
It is easy to see that f g is a degreewise split monomorphism, with cokernel *
*C Z.
By taking f to be the map 0 -! C and g to be the map 0 -! Z, we see that if
the relative model structure is monoidal, then the tensor product of two P-cofi*
*brant
complexes is P-cofibrant. Conversely, if C Z is P-cofibrant whenever C and Z*
* are
P-cofibrant, the preceding paragraph implies f g is a P-cofibration whenever *
*f and
g are P-cofibrations. If either f or g is a P-trivial cofibration, then one of *
*C or Z is
P-trivially cofibrant, and hence contractible. It follows that C Z is contrac*
*tible, and
so f g is a P-trivial cofibration. *
*|___|
In particular, suppose the relative model structure is monoidal, and M, N ar*
*e P-
projective. Then S0M S0N ~=S0(M N) is P-cofibrant, and therefore M N is P-
projective. With this in mind, we say that a projective class P on a closed mon*
*oidal cat-
egory A is monoidal if the unit is P-projective and the tensor product of P-pro*
*jectives
is P-projective.
We do not know of any example of a monoidal projective class P where cofibra*
*nt
replacements exist, but the relative model structure is not monoidal. Certainly*
* this does
not happen in either Case A or Case B of Theorem 2.2. In Case A, we have the fo*
*llowing
corollary.
Corollary 2.20. Suppose F :B -! A is a monoidal functor between closed monoidal
abelian categories, with right adjoint U that preserves countable coproducts. T*
*hen the
relative model structure on Ch(A) is monoidal.
Proof.Proposition 3.3 says that QX QY is P-cofibrant for any X and Y , where Q
denotes the cofibrant replacement functor constructed in Section 3. If X and Y*
* are
already cofibrant, then lifting implies that X is a retract of QX and Y is a re*
*tract of
QY . Thus X Y is a retract of QX QY , so X Y is P-cofibrant. Proposition *
*2.19
completes the proof. |___|
In Case B, we will show in Corollary 4.4 that every P-cofibrant object is P-*
*cellular.
Thus the following corollary applies.
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 21
Corollary 2.21. Let A be a closed monoidal abelian category with a monoidal pro*
*jective
class P such that cofibrant replacements exist and every P-cofibrant object is *
*P-cellular
(Definition 2.6). Then the relative model structure on Ch(A) is monoidal.
Proof.Let A and B be P-cofibrant. By Proposition 2.19, it suffices to show that*
* A B is
P-cofibrant. By assumption, A is a retract of a transfinite colimit of a colimi*
*t-preserving
diagram
0 = A0 -!A1 -!. . .
such that for each ff, the map Aff-! Aff+1is degreewise split monic with cokern*
*el a
complex of relative projectives with zero differential. Since - B preserves *
*retracts,
degreewise split monomorphisms and colimits, it is enough to prove that iP B*
* is
P-cofibrant for each P-projective P and each i. Applying the same filtration ar*
*gument
to B, we find that it suffices to show that iP jQ = i+j(P Q) is P-cofibr*
*ant for
all P-projectives P and Q and integers i and j. But this follows immediately fr*
*om the
fact that P is monoidal and Lemma 2.7 (a). |__*
*_|
We can also prove a dual statement in Case A.
Proposition 2.22. Suppose U :A -! B is a monoidal functor of closed monoidal
abelian categories, with right adjoint F . Assume that U preserves countable pr*
*oducts.
Then the injective relative model structure on Ch(A) is monoidal.
By the "injective relative model structure", we mean the model structure obt*
*ained
by dualizing Theorem 2.2. The (trivial) cofibrations in this model structure ar*
*e called
the B-injective (trivial) cofibrations.
Proof.Suppose f :A -!B and g :X -! Y are B-injective cofibrations with cokernel*
*s C
and Z, respectively. This means that Uf and Ug are degreewise split monomorphis*
*ms.
Recall the definition of f g used in the proof of Proposition 2.19. Since U i*
*s monoidal
and preserves pushouts, U(f g) ~=Uf Ug. One can easily check that Uf Ug i*
*s a
degreewise split monomorphism, so f g is a B-injective cofibration. If f is a *
*B-injective
trivial cofibration, then the cokernel C of f has UC contractible. Let Z denot*
*e the
cokernel of g. Since the cokernel of Uf Ug is UC UZ, which is contractible,*
* f g
is a B-injective trivial cofibration. Since everything is cofibrant in the inje*
*ctive relative
model structure, the other condition in the definition of a monoidal model cate*
*gory is
automatically satisfied. |__*
*_|
Another important property of a model category is whether it is cofibrantly *
*generated.
This is the topic of Section 5.
22 J. DANIEL CHRISTENSEN AND MARK HOVEY
2.2. Why projective classes? The astute reader will notice that we haven't used*
* the
assumption that there is always a P-epimorphism from a P-projective. Indeed, al*
*l we
have used is that we have a collection P of objects such that, with the definit*
*ions at
the beginning of this section, cofibrant replacements exist. The following pro*
*position
explains why we start with a projective class.
Proposition 2.23. Let P be a collection of objects in an abelian category A suc*
*h that
cofibrant replacements exist in Ch(A). Then there is a unique projective class*
* (P0, E)
that gives rise to the same definitions of weak equivalence, fibration and cofi*
*bration.
Proof.Let E be the collection of P-epimorphisms and let P0 be the collection of*
* all
objects P such that every map in E is P -epic. Then P0 contains P, and a map is*
* P0-
epic if and only if it is P-epic. It follows that the P0-exact sequences are th*
*e same as
the P-exact sequences. A map f is a P-equivalence if and only if the cofibre o*
*f f is
P-exact. The same holds for the P0-equivalences, and since the two notions of e*
*xactness
also agree, the two notions of equivalence agree. Finally, since cofibrations a*
*re defined
in terms of the fibrations and weak equivalences, the two notions of cofibratio*
*n agree.
That the pair (P0, E) is a projective class follows immediately from the exi*
*stence of
cofibrant replacements.
Our work earlier in this section shows that the objects of P0are precisely t*
*hose objects
P that are cofibrant when viewed as complexes concentrated in degree 0. Thus t*
*he
projective class is the unique projective class giving rise to the same weak eq*
*uivalences,
fibrations and cofibrations. |_*
*__|
Thus by requiring our collection of objects to be part of a projective class*
*, we in effect
choose a canonical collection of objects determining each model structure we pr*
*oduce.
This means that the relevant question is: which projective classes give rise to*
* model
structures? Theorem 2.2 gives a necessary and sufficient condition, namely the *
*existence
of cofibrant replacements, and we know of no projective classes that do not sat*
*isfy this
condition.
An additional advantage of having a projective class is that it provides us *
*with the lan-
guage to state results such as: ExtnP(M, N) ~=HoCh (A)( nM, N) (see Corollary 2*
*.14).
3. Case A: Projective classes coming from adjoint pairs
In this section we prove Case A of Theorem 2.2.
Let U :A -! B be a functor of abelian categories, with a left adjoint F :B -*
*! A.
Then U and F are additive, U is left exact and F is right exact. Let P be the p*
*rojective
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 23
class on A that is the pullback of the trivial projective class on B (see Examp*
*le 1.3 and
Subsection 1.3). In this subsection we construct a cofibrant replacement functo*
*r for the
projective class P under the assumption that U preserves countable coproducts.
Because this projective class is strong, Lemma 1.9 tells us that the P-fibra*
*tions and
P-equivalences have alternate characterizations. A map f in Ch(A) is a P-fibrat*
*ion if
and only if A(P, f) is degreewise split epic for each P in P, and is a P-equiva*
*lence if
and only if A(P, f) is a chain homotopy equivalence for each P in P.
First we prove a lemma characterizing the fibrations and weak equivalences i*
*n this
model structure and giving us a way to generate cofibrant objects.
Lemma 3.1. (a)A map p: X -! Y is a P-fibration if and only if Up is a degree*
*wise
split epimorphism in Ch(B).
(b)A map p: X -! Y is a P-equivalence if and only if Up is a chain homotopy
equivalence in Ch(B).
(c)If i is a map in Ch(B) that is degreewise split monic, then F i is a P-cof*
*ibration.
(d)For any C in Ch(B), F C is P-cofibrant.
Proof.For part (a), note that the P-projectives are all retracts of F M for M 2*
* B.
Hence p is a P-fibration if and only if A(F M, p) = B(M, Up) is a surjection fo*
*r all
M 2 B. This is true if and only if Up is a degreewise split epimorphism. For pa*
*rt (b),
a similar argument shows that p is a P-equivalence if and only if B(M, Up) is a*
* quasi-
isomorphism for all M 2 B. We claim that this forces Up to be a chain homotopy
equivalence. Indeed, let C denote the cofiber of Up. Then B(M, C) is exact fo*
*r all
M 2 B. By taking M = ZnC, we find that C is exact and that Cn+1 -!ZnC is a split
epimorphism. It follows that C is contractible, as in the proof of Lemma 1.9. T*
*hus Up
is a chain homotopy equivalence.
For part (c), let C be the cokernel of i. Then F i is degreewise split moni*
*c with
cokernel F C. Suppose that p: X -! Y is a P-trivial fibration with kernel K. Th*
*en K is
P-trivially fibrant, so part (b) implies that UK is contractible. By Lemma 2.3,*
* to show
that F i is a P-cofibration, it suffices to show that every chain map F C -! K*
* is chain
homotopic to 0. By adjointness, it suffices to show that every chain map C -! *
*UK
is chain homotopic to 0. Since UK is contractible, this is clear. Part (d) foll*
*ows from
part (c), |___|
We now provide a construction, given a complex X, of a P-cofibrant complex QX
and a P-trivial fibration QX -! X. This comes out of the bar construction, whi*
*ch
we now recall from [ML95 , Section IX.6]. Given an object N of A, define the co*
*mplex
24 J. DANIEL CHRISTENSEN AND MARK HOVEY
BN by (BN)m = (F U)m+1 N when m -1 and 0 otherwise. We will define maps
s = sm :U(BN)m-1 -! U(BN)m and ffi = ffim :(BN)m -! (BN)m-1 . For m < 0 we
declare both to be zero. For m 0, sm :U(BN)m-1 = (UF )m UN -! U(BN)m =
UF (UF )m UN is defined to be adjoint to the identity map of F (UF )m UN. For m*
* 0,
we can then inductively define ffim :(BN)m = F U(F U)m N -! (BN)m-1 = (F U)m N
to be adjoint to the self-map 1 - sm-1 (Uffim-1 ) of U(F U)m N. Properties of *
*adjoint
functors then guarantee that (Uffim )sm + sm-1 (Uffim-1 ) = 1. Using this and t*
*he prop-
erties of adjoint functors, one deduces a sequence of implications ffim-1 ffim *
* = 0 =)
(Uffim )sm (Uffim ) = Uffim =) (Uffim )(Uffim+1 )sm+1 = 0 =) ffim ffim+1 = *
*0. Therefore
ffim-1 ffim = 0 for each m and so ffi makes BN into a chain complex.
The construction of BN, ffi, and s is obviously natural in N. Thus, given a *
*complex
X in Ch(A), we get a bicomplex (BX)m,n = (BXn)m , where
ffi :(BX)m,n -!(BX)m-1,n andd: (BX)m,n -!(BX)m,n-1
commute. Furthermore, s: U(BX)m,n -!U(BX)m+1,n and Ud also commute.
__ __ L
We can then define a total complex QX by (Q X)k = m+n=k(BX)m,n. The differ*
*en-
__
tial @ of QX takes the summand (BX)m,n into (BX)m-1,n (BX)m,n-1by (ffi, (-1)m d*
*).
__ __
Note that Q X is a filtered complex, with F iQX the subcomplex consisting of te*
*rms
__ __
with m i. In particular, F -1QX = -1X. We define QX to be QX= -1X, so that
(QX)k = (F U)Xk (F U)2Xk-1 . ...
The following proposition proves Case A of Theorem 2.2.
Proposition 3.2.There is a natural P-trivial fibration qX :QX -! X, and QX is
P-cofibrant.
Proof.We first show that QX is P-cofibrant. There is an increasing filtration
{F iQX}i 0on QX, where F 0QX = F UX and F iQX=F i-1QX = (F U)i+1 iX. Fur-
thermore, each inclusion F iQX -! F i+1QX is a degreewise split monomorphism. By
Lemma 3.1 (d), each quotient (F U)i+1 iX is P-cofibrant, so each map F i-1QX -!
F iQX is a P-cofibration. Hence F UX -! QX is a P-cofibration. Thus QX is P-
cofibrant.
The map qX is induced by ffi0: (F U)Xn -!Xn, adjoint to the identity. The ma*
*p qX
sends the other summands of (QX)n to 0. We leave to the reader the check that t*
*his is
a chain map. Since Uffi0 is a split epimorphism, UqX is a degreewise split epim*
*orphism,
so is a P-fibration by Lemma 3.1 (a). To show qX is a P-equivalence, it suffi*
*ces to
__ __
show that the fiber Q X is P-contractible, or, equivalently, that UQ X is contr*
*actible
(Lemma 3.1 (b)). The contracting homotopy is given by s. Indeed, on the summand
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 25
U(BX)m,n, the U(BX)m,n component of s(U@) + (U@)s is s(Uffi) + (Uffi)s = 1, and*
* the
U(BX)m+1,n-1component is s(-1)m (Ud) + (-1)m+1 (Ud)s = 0. (This is where we use
that U commutes with coproducts.) |___|
The following proposition implies that the relative model structure is monoi*
*dal in this
case, as explained in Corollary 2.20.
Proposition 3.3.Suppose F :B -!A is a monoidal functor of closed monoidal abeli*
*an
categories, with right adjoint U that preserves countable coproducts. Then, for*
* any X
and Y in Ch(A), QX QY is P-cofibrant.
Proof.Recall the filtration F iQX on QX used in the proof of Proposition 3.2. U*
*sing
this filtration, we find that QX QY is the colimit of F iQX QY , and each *
*map
F i-1QX QY -! F iQX QY is a degreewise split monomorphism with cokernel
(F U)i+1 iX QY . It therefore suffcies to show that this cokernel is P-cofibr*
*ant for all
i. A similar argument using the filtration on QY shows that it suffices to show*
* that
(F U)i+1( iX) (F U)j+1( jY ) ~=F (U(F U)i iX U(F U)j jY )
is P-cofibrant for all i, j 0. But this follow immediately from Lemma 3.1 (d)*
*. |___|
3.1. Examples.
Example 3.4. Let B be a bicomplete abelian category. Since the identity functo*
*r is ad-
joint to itself and preserves coproducts, we can apply Theorem 2.2 to the trivi*
*al projective
class P to conclude that Ch(B) is a model category. We call this the absolute m*
*odel
structure. The P-equivalences are the chain homotopy equivalences (Lemma 3.1), *
*the
P-fibrations are the degreewise split epimorphisms and the P-cofibrations are t*
*he de-
greewise split monomorphisms. Every object is both P-cofibrant and P-fibrant, a*
*nd the
homotopy category is the usual homotopy category K(B) in which chain homotopic *
*maps
have been identified. That this model structure exists was also shown in [Col]*
*. Note
that if B is closed monoidal, the absolute model structure is also monoidal, by*
* Corol-
lary 2.20. In particular, since every object is P-cofibrant, given a different*
*ial graded
algebra R 2 Ch(B), we get a model structure on the category of differential gra*
*ded R-
modules, where weak equivalences are chain homotopy equivalences (of the underl*
*ying
chain complexes) and fibrations are degreewise split epimorphisms.
Now let A and B be bicomplete abelian categories and let U :A -!B be a copro*
*duct
preserving functor with left adjoint F . By Theorem 2.2, Ch(A) has a relative *
*model
structure whose weak equivalences, cofibrations and fibrations are called B-equ*
*ivalences,
B-cofibrations and B-fibrations. This structure is a lifting of the absolute mo*
*del structure
26 J. DANIEL CHRISTENSEN AND MARK HOVEY
on Ch(B) in the sense that a map f in Ch(A) is a weak equivalence or fibration *
*if and
only if Uf is so in Ch(B). It is often the case that one wants to lift a model *
*structure
along a right adjoint. When the model structure is cofibrantly generated, neces*
*sary and
sufficient conditions for a lifting are known [DHK97 , 9.1], [Hir00, 13.4.2]. *
*Our main
theorem says that it is also possible to lift the absolute model structure on C*
*h(B), even
though it is not usually cofibrantly generated (see Subsection 5.4).
It follows from the above that F preserves cofibrations and trivial cofibrat*
*ions. The
adjoint functors F and U form a Quillen pair.
The category Ch(A) also has an absolute model structure. The identity functo*
*r sends
absolute fibrations and weak equivalences to B-fibrations and B-equivalences. T*
*hus the
identity functor is a right Quillen functor from the absolute model structure t*
*o the
relative model structure.
Example 3.5. Let R -!S be a map of rings. Write R-mod and S-mod for the catego*
*ries
of left R- and S-modules. Consider the forgetful functor U :S-mod -! R-mod and *
*its
left adjoint F that sends an R-module M to S R M. We saw in Example 1.6 that
this gives a projective class whose relative projectives are the S-modules P su*
*ch that
the natural map S R M -! M is split epic as a map of S-modules. The functor U
preserves coproducts, so Theorem 2.2 and Lemma 1.9 tell us that Ch(S) = Ch(S-mo*
*d )
has a model structure in which the weak equivalences are the maps that become c*
*hain
homotopy equivalences after forgetting the S-module structure. The fibrations a*
*re the
maps that in each degree are split epic as maps of R-modules. The cofibrations*
* are
defined by the left lifting property with respect to the trivial fibrations. Eq*
*uivalently,
by Proposition 2.5, they are the degreewise split monomorphisms whose cokernels*
* are
cofibrant. And by Lemma 2.4, a complex C is cofibrant if and only if each Cn i*
*s a
relative projective and every map from C to a complex K such that UK is contrac*
*tible
is chain homotopic to 0. Lemma 2.7 gives us a ready supply of cofibrant object*
*s. In
particular, a relative resolution of an S-module M is a cofibrant replacement, *
*so the
group HoCh (S)(M, iN) of maps in the homotopy category is isomorphic to the re*
*lative
Exti group [ML95 ].
When R and S are commutative, the functor F is monoidal, so this relative mo*
*del
structure is monoidal by Corollary 2.20. There is then a derived tensor product*
* X L Y
in HoCh (S), and if M and N are S-modules, Hi(M L N) is isomorphic to the rela*
*tive
Tori group [ML95 ].
Example 3.6. Again let R -! S be a map of rings and let U be the forgetful fun*
*ctor.
We saw in Example 1.7 that U has a right adjoint G that sends an R-module M to
QUILLEN MODEL STRUCTURES FOR RELATIVE HOMOLOGICAL ALGEBRA 27
R-mod(S, M) and so we get an injective class on S-mod by pulling back the trivi*
*al
injective class along U. U preserves products, so we can apply the duals of The*
*orem 2.2
and Lemma 1.9 to conclude that Ch (S) has a model structure with the same weak
equivalences as in the previous example. The cofibrations are the maps that in*
* each
degree are split monic as maps of R-modules, and the fibrations are the maps wi*
*th the
right lifting property with respect to with respect to the trivial cofibrations*
*. Fibrations
and fibrant objects can also be characterized by the duals of Lemmas 2.4 and 2.*
*7, and
the homotopy category again encodes the relative Ext groups. However, this rel*
*ative
model structure is not monoidal, even when R and S are commutative, since U is *
*not
monoidal.
Example 3.7. Suppose A is an algebra over a commutative ring k, and let C deno*
*te the
category of chain complexes of A-bimodules. Then there are forgetful functors f*
*rom A-
bimodules to left A-modules, right A-modules, and k-modules. Each of these pres*
*erves
coproducts, so we get three different relative model structures on C. When we *
*forget
to right or left A-modules, the cofibrant replacement functor Q applied to A gi*
*ves us
the usual un-normalized bar construction, with (QA)n = A n+2. The complex QA is
then also cofibrant in the relative model structure obtained by forgetting to k*
*-modules,
since it is a bounded below complex of relative projectives. The Hochschild coh*
*omology
HHn(A; N) of A with coefficients in a bimodule N (or a chain complex of bimodul*
*es
N) is then equal to HoC( nA, N), where we can use any of the three relative mod*
*el
structures above. We can extend this definition by replacing A with an arbitrar*
*y complex
of bimodules, but then the answer may depend on which relative model structure *
*we
use. It is most natural to use the relative model structure obtained from the f*
*orgetful
functor to k-modules, as then we can define Hochschild homology as well.
Indeed, the Hochschild homology groups are defined using the tensor product *
*M N
of bimodules, where we identify ma n with m an but also am n with m na.*
* We
emphasize that M N is only a k-module, not a bimodule. In particular, this te*
*nsor
product can't be associative, but it is commutative and also unital in a weak s*
*ense,
since M (A A) ~=M as a k-module. Furthermore, the functor - N has a right
adjoint that takes a k-module L to the bimodule Hom k(N, L), where, if g 2 Hom *
*(N, L),
(ga)(n) = g(an) and (ag)(n) = g(na). Both of these functors then extend to func*
*tors
on chain complexes. The proof of Proposition 2.19 applies to this case as well*
*, and
shows that if f and g are cofibrations in the relative model structure on C obt*
*ained by
forgetting to k-modules, then f g is a cofibration in the absolute model stru*
*cture on
chain complexes of k-modules. If either f or g is a trivial cofibration, so is *
*f g. This
28 J. DANIEL CHRISTENSEN AND MARK HOVEY
means the tensor product has a total left derived functor that is commutative a*
*nd unital
(in the above weak sense). We can therefore extend the usual definition of Hoch*
*schild
homology to complexes X and Y of bimodules, by defining HHn(X, Y ) = Hn(QX QY ).
In fact, it is possible to prove, using the technique of Proposition 2.19, that*
*, if X is
cofibrant, the functor X - takes weak equivalences in the relative model stru*
*cture to
chain homotopy equivalences. This implies that HHn(X, Y ) = Hn(QX Y ). This i*
*s a
direct generalization of the way Hochschild homology is defined in [ML95 , Sect*
*ion X.4].
4.Case B: Projective classes with enough small projectives
In this section we prove that cofibrant replacements exist in Case B. We beg*
*in by
introducing the terminology necessary for the precise statement of Case B.
We think of an ordinal as the set of all previous ordinals, and of a cardina*
*l as the first
ordinal with that cardinality.
Definition 4.1.Given a limit ordinal fl, the cofinality of fl, cofinfl, is the *
*smallest
cardinal ~ such that there exists subset T of fl of cardinality ~ with supT = f*
*l. The
cofinality of a successor ordinal is defined to be 1.
A colimit-preserving sequence from an ordinal fl to a category A is a diagram
X0 -!X1 -!. .-.!Xff-! Xff+1-! . . .
of objects of A indexed by the ordinals less than fl, such that for each limit *
*ordinal ~
less than fl the natural map colimff<~Xff-! X~ is an isomorphism.
For ~ a cardinal, an object P is said to be ~-small relative to a subcategor*
*y M
if for each ordinal fl with cofinfl > ~ and each colimit-preserving sequence X *
*:fl -! A
that factors through M, the natural map colimff