HOMOTOPY THEORIES AND MODEL CATEGORIES
W. G. Dwyer and J. Spalinski
University of Notre Dame
Introduction
This paper is an introduction to the theory of "closed model categories", whi*
*ch was
developed by Quillen in [22] and [23]. By definition a closed model category i*
*s just an
ordinary category with three specified classes of morphisms, called fibrations,*
* cofibrations
and weak equivalences,which satisfy a few simple axioms that are deliberately r*
*eminiscent
of properties of topological spaces. Surprisingly enough, these axioms give a r*
*easonably
general context in which it is possible to set up the basic machinery of homoto*
*py theory.
The machinery can then be used immediately in a large number of different setti*
*ngs,
as long as the axioms are checked in each case. Although many of these setting*
*s are
geometric (spaces (x7), fiberwise spaces (2.12), G-spaces [11], spectra [5], di*
*agrams of
spaces [10] : :):,some of them are not (chain complexes (x6), simplicial commut*
*ative
rings [24], simplicial groups [23]: :):. Certainly each setting has its own te*
*chnical and
computational peculiarities,but the advantage of an abstract approach is that t*
*hey can all
be studied with the same tools and described in the same language. What is the *
*suspension
of an augmented commutativealgebra? One of incidental appeals of Quillen's theo*
*ry (to a
topologist!) is that it both makesa question like this respectable and gives it*
* an interesting
answer (10.3).
We have tried to minimize the prerequisites needed for understanding this pap*
*er; it
should be enough to have some familiarity with CW-complexes, with chain complex*
*es,
and with the basic terminology associated with categories. Almost all of the ma*
*terial we
present is due to Quillen [22],but we have replaced his treatment of suspension*
* functors
and loop functors by a general construction of homotopy pushouts and homotopy p*
*ullbacks
in a closed model category.What we do along these lines can certainly be carrie*
*d further.
This paper is not in any sense a survey of everything that is known about close*
*d model
categories; in fact we cover only a fraction of the material in [22]. The last *
*section has a
discussion of some ways in which model categories have been used in topology an*
*d algebra.
Organization!of!the paper. Section 1 contains background material,principally a*
* discussion
of!some categorical constructions (limits and colimits) which come up almost im*
*mediately
in!any!attempt to build new objects of some abstract category out of old ones. *
*Section
2!gives!the definition of what it means for a category C to be a closed model c*
*ategory,
!
The first author was supported in part by the National Science Foundation.
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*EX
2 W. DWYER AND J. SPALINSKI
establishes some terminology, and sketches a few examples. In x3 we study the n*
*otion of
"homotopy" in C and in x4 carry out the construction of the homotopy category H*
*o(C).
Section x5 gives Ho (C)a more conceptual significance by showing that it is equ*
*ivalent to
the "localization" of C with respect to the classof weak equivalences. For our *
*purposes
the "homotopy theory" associated to C isthe homotopy category Ho (C)together wi*
*th
various related constructions (x9).
Sections 6 and 7 describe in detail two basic examples of closed model catego*
*ries, namely
the category Top of topological spaces and the category ChR of nonnegative chai*
*n com-
plexes of modules over a ring R. The homotopy theory of Top is of course famil*
*iar, and
it turns out that the homotopy theory of Ch R is what is usually called homolog*
*ical alge-
bra. Comparing these two examples helps explain why Quillen called the study of*
* closed
model categories "homotopical algebra" and thought of it as a generalization of*
* homolog-
ical algebra. In x8 we give a criterion for a pair of functors between two clos*
*ed model
categories to induce equivalences between the associated homotopy categories; p*
*inning
down the meaning of "induce" here leads to the definition of derived functor. S*
*ection 9
constructs homotopy pushouts and homotopy pullbacks in an arbitrary closed mode*
*l cate-
gory in terms of derived functors.Finally, x10 contains some concluding remarks*
*, sketches
some applications of homotopical algebra, and mentions a way in which the theor*
*y has
developed since Quillen.