DERIVED CATEGORIES AND PROJECTIVE CLASSES
J. DANIEL CHRISTENSEN
Abstract.An important example of a model category is the category of unb*
*ounded chain com-
plexes of R-modules, which has as its homotopy category the derived cate*
*gory of the ring R. This
example shows that traditional homological algebra is encompassed by Qui*
*llen's homotopical alge-
bra. The goal of this paper is to show that more general forms of homolo*
*gical algebra also fit into
Quillen's framework. Specifically, any set of objects in a complete and *
*cocomplete abelian category
A generates a projective class on A, which is exactly the information ne*
*eded to do homological al-
gebra in A. The main result is that if the generating objects are "small*
*" in an appropriate sense, then
the category of chain complexes of objects of A has a model category str*
*ucture which reflects the
homological algebra of the projective class. The motivation for this wor*
*k is the construction of the
üp re derived categoryö f a ring R. Finally, we explain how the categor*
*y of simplicial objects in
a possibly non-abelian category can be equipped with a model category st*
*ructure reflecting a given
projective class.
Contents
Introduction 1
1. Projective classes 3
2. Cofibrantly generated model categories *
* 4
3. Derived categories 5
4. The pure and categorical derived categories *
* 11
5. Simplicial objects and the bounded below derived category *
* 12
References 13
Introduction
An important example of a model category is the category Ch of unbounded cha*
*in complexes
of R-modules, which has as its homotopy category the derived category D of the *
*ring R. The
formation of a projective resolution is an example of cofibrant replacement, an*
*d traditional derived
functors are examples of derived functors in the model category sense. This exa*
*mple shows that
traditional homological algebra is encompassed by Quillen's homotopical algebra*
*, and indeed this
unification was one of the main points of Quillen's work [14].
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 "pr*
*ojective objectä nd
"exact sequence." However, it is sometimes useful to impose different definitio*
*ns of these terms.
___________
Date: November 17, 1998.
1991 Mathematics Subject Classification. Primary 18E30; Secondary 18G35, 55U3*
*5, 18G25.
Key words and phrases. Derived category, chain complex, relative homological *
*algebra, projective class, pure homo-
logical algebra.
1
2 J. DANIEL CHRISTENSEN
If this is done in a way that satisfies some natural axioms, what is obtained i*
*s a "projective class,"
which is exactly the information needed to do homological algebra in A. Our mai*
*n result is that
given any projective class satisfying a set-theoretic hypothesis, the category *
*of chain complexes of
objects of A has a model category structure which reflects the homological alge*
*bra of the projective
class. The motivation for this work is the construction of the üp re derived ca*
*tegoryö f a ring R.
Pure homological algebra has applications to phantom maps in the stable homotop*
*y category and
in the (usual) derived category of a ring, and these connections will be descri*
*bed.
When A has enough projectives, the projective objects and exact sequences fo*
*rm a projective
class. Therefore the results of this paper apply to traditional homological alg*
*ebra as well. Even
in this special case, it is not a trivial fact that the category of unbounded c*
*hain complexes can be
given a model category structure, and indeed Quillen restricted himself to the *
*bounded below case.
I know of three other written proofs that the category of unbounded chain compl*
*exes is a model
category [7, 8, 10], which do the case of R-modules, but this was probably know*
*n to others as well.
An important corollary of the fact that a derived category D is the homotopy*
* category of a model
category is that D(X, Y ) is a set (as opposed to a proper class) for any two c*
*hain complexes X and
Y . This is not the case in general, and much work on derived categories ignore*
*s this possibility. It
is not just a pedantic point; if one uses the morphisms in the derived category*
* to index constructions
in other categories or to define cohomology groups, one needs to know that the *
*indexing class is
actually a set. Recently, the unbounded case has been handled under various ass*
*umptions on A.
(See Weibel [16] Remark 10.4.5, which credits Gabber, and Exercise 10.4.5, whic*
*h credits Lewis,
May and Steinberger [13]. See also Kriz and May [12, Part III].) The assumption*
*s that appear in the
present paper are different from those that have appeared before and the proof *
*is somewhat easier
(because of our use of the theory of cofibrantly generated model categories), s*
*o this paper may be
of some interest even in this special case.
Another consequence of the fact that Ch is a model category is the existence*
* of resolutions
coming from cofibrant and fibrant approximations, and the related derived funct*
*ors. Some of these
are discussed in [1] and [15]. We do not discuss these topics here, but just me*
*ntion that the model
category point of view can provide a framework for some of this material.
We also briefly discuss the category of non-negatively graded chain complexe*
*s. In this case
we describe a model category structure that works for an arbitrary projective c*
*lass, without any
set-theoretic hypotheses. More generally, we show that under appropriate hypoth*
*eses a projective
class on a possibly non-abelian category A determines a model category structur*
*e on the category
of simplicial objects in A. As an example, we deduce that the category of equiv*
*ariant simplicial
sets has various model category structures.
We now briefly outline the paper. In Section 1 we give the axioms for a proj*
*ective class and
mention a few examples. In Section 3 we describe the model category structure o*
*n the category of
chain complexes which takes into account a given projective class. Then we stat*
*e the set-theoretic
assumption needed and prove our main result, using the recognition lemma for co*
*fibrantly gener-
ated categories, which is recalled in Section 2. In Section 4 we give two examp*
*les, the traditional
derived category of R-modules and the pure derived category, and we describe wh*
*y the pure derived
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 possibl*
*y non-abelian category.
DERIVED CATEGORIES AND PROJECTIVE CLASSES *
* 3
I thank Haynes Miller for asking the question which led to this paper and Ma*
*rk Hovey, Haynes
Miller and John Palmieri for fruitful and enjoyable discussions.
1.Projective classes
In this section we explain the notion of a projective class, which is the in*
*formation necessary
in order to do homological algebra. Intuitively, a projective class is a choic*
*e of which sort of
"elements" we wish to think about.
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 ment*
*ion P , we call
such a map a generalized element of A. A map A ! B in any category is determine*
*d 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 a pointed category, i.e., a category in which initial and terminal *
*objects exist and
agree. In a pointed category, any initial (equivalently, terminal) object is ca*
*lled a zero object. Let
P be an object of A. A sequence
A -! B -! C
is said to be P -exact if the composite A ! C is the zero map (the unique map w*
*hich factors
through a zero object) and
A(P, A) -! A(P, B) -! A(P, C)
is an exact sequence of pointed sets (the base points being the zero maps). Th*
*e latter can be
rephrased as the condition that A ! B ! C induces an exact sequence of P -eleme*
*nts. A P-exact
sequence is one which is P -exact for all P in P. A map A ! B is P -epic (resp.*
* P-epic) if it
induces a surjection of P -elements (resp. P-elements).
A projective class on A is a collection P of objects of A and a collection E*
* of sequences
A ! B ! C in A such that
(i)E is precisely the collection of all P-exact sequences;
(ii)P is precisely the collection of all objects P such that each sequence in *
*E is P -exact;
(iii)any map A ! B can be extended to a sequence P ! A ! B in E with P in P.
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 class. *
*An object of P is called
a P-projective. Condition (iii) says that there are "enough P-projectives."
Example 1.1. Let A be the category of pointed sets, let P be the collection of *
*all pointed sets and
let E be the collection of all exact sequences of pointed sets. Then E is preci*
*sely the collection of
P-exact sequences, and P is a projective class.
Example 1.2. For an associative ring R, let A be the category of left R-modules*
*, let P be the
collection of all summands of free R-modules and let E be the collection of all*
* exact sequences of
R-modules. Then E is precisely the collection of P-exact sequences, and P is a *
*projective class.
4 J. DANIEL CHRISTENSEN
The above two examples are äc tegorical" projective classes in the sense tha*
*t the P-epimor-
phisms are just the epimorphisms and the P-projectives are the categorical proj*
*ectives, 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. 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 consist of all P*
*-exact sequences.
Then P is a projective class. A short exact sequence is P-exact iff it remains *
*exact after tensoring
with every right module.
Example 1.4. Let S be the homotopy category of spectra and let P consist of all*
* retracts of wedges
of finite spectra. Then P determines a projective class, and a map is a P-epimo*
*rphism iff its cofibre
is a phantom map.
Examples 1.2 and 1.3 will be discussed further in Section 4. Example 1.4 is *
*studied in [2], along
with similar examples.
We note that all of the examples mentioned above are determined by a set in *
*the sense of the
following lemma, whose proof is easy.
Lemma 1.5. Suppose F is any set of objects in a pointed category with coproduct*
*s. Let E be the
collection of F-exact sequences and let P be the collection of all objects P su*
*ch that every sequence
in E is P -exact. Then P is the collection of retracts of coproducts of objects*
* of F and (P, E) is a
projective class.
A projective class is precisely the information needed to form projective re*
*solutions and define
derived functors. For further information, we refer the reader to [6] for the g*
*eneral theory and to [2]
in the special case of a triangulated category.
2. Cofibrantly generated model categories
In this section we briefly recall the basics of cofibrantly generated model *
*categories. This ma-
terial will be used in the next section to prove our main result. For more deta*
*ils, see the books by
Dwyer, Hirschhorn and Kan [5] and Hirschhorn [9]. We assume knowledge of the ba*
*sics of model
categories, for which [4] is an excellent reference.
We will always assume our model categories to be complete and cocomplete.
In the following, a cardinal number is thought of as the first ordinal with *
*that cardinality.
Definition 2.1. Given an ordinal ~, a ~-sequence in a category M is a diagram
X0 -! X1 -! . .-.! X~ -! X~+1 -! . . .
indexed by the ordinals less than ~, such that for each limit ordinal fl less t*
*han ~ the natural map
colim~