x PREFACE
This book is also an exposition of model categories from the ground up. In
particular, this book should be accessible to graduate students. There are very few
prerequisites to reading it, beyond a basic familiarity with categories and functors,
and some familiarity with at least one of the central examples of chain complexes,
simplicial sets, or topological spaces. We also require some familiarity with the
basics of set theory, especially ordinal and cardinal numbers. Later in the book we
do require more of the reader; in Chapter 7 we use the theory of homotopy limits
of diagrams of simplicial sets, developed in [BK72]. However, the reader who gets
that far will be well equipped to understand [BK72] in any case.
This book is the author's attempt to understand the theory of model categories
well enough to answer one question. That question is: when is the homotopy
category of a model category a stable homotopy category in the sense of [HPS97]?
We do not in the end answer this question in quite as much generality as one would
like, though we come fairly close to doing so in Chapter 7. As I tried to answer this
question, it became clear that the theory necessary to do so was not in place. After
a long period of resistance, I decided it was important to develop the necessary
theory, and that the logical and most useful place to do so was in a book that
would assume almost nothing of the reader. A book is the logical place because
the theory I develop requires a foundation that is simply not in the literature. I
think this foundation is beautiful and important, and therefore deserves to be made
accessible to the general mathematician.
We now provide an overview of the book. See also the introductions to the in-
dividual chapters. The first chapter of this book is devoted to the basic definitions
and results about model categories. In highfalutin language, the main goal of this
chapter is to define the 2-category of model categories and show that the homotopy
category is part of a pseudo-2-functor from model categories to categories. This
is a fancy way, fully explained in Section 1.4, to say that not only can one take
the homotopy category of a model category, one can also take the total derived
adjunction of a Quillen adjunction, and the total derived natural transformation of
a natural transformation between Quillen adjunctions. Doing so preserves compo-
sitions for the most part, but not exactly. This is the reason for the word "pseudo".
In order to reach this goal, we have to adopt a different definition of model category
from that of [DHK], but the difference is minor. The definition of [DHK], on the
other hand, is considerably different from the original definition of [Qui67], and
even from its refinement in [Qui69].
After the theoretical material of the first chapter, the reader is entitled to some
examples. We consider the important examples of chain complexes over a ring,
topological spaces, and chain complexes of comodules over a commutative Hopf
algebra in the second chapter, while the third is devoted to the central example of
simplicial sets. Proving that a particular category has a model structure is always
difficult. There is, however, a standard method, introduced by Quillen [Qui67]
but formalized in [DHK]. This method is an elaboration of the small object argu-
ment and is known as the theory of cofibrantly generated model categories. After
examining this theory in Section 2.1, we consider the category of modules over a
Probenius ring, where projective and injective modules coincide. This is perhaps
the simplest nontrivial example of a model category, as every object is both cofi-
brant and fibrant. Nevertheless, the material in this section seems not to have
appeared before, except in Georgian [Pir86]. Then we consider chain complexes of
modules over an arbitrary ring. Our treatment differs somewhat from the standard
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