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