of categories. This is definitely not the same as the usual definition of triangulated
categories, but it is closer than one might think at first glance. We also argue
that it is a better definition. Every triangulated category that arises in nature is
the homotopy category of a model category, so will be triangulated in our stronger
sense. We also consider generators in the homotopy category of a pointed model
category. These generators are extremely important in the theory of stable homo-
topy categories developed in [HPS97]. Our results are not completely satisfying,
but they do go a long way towards answering our original question: when is the
homotopy category of a model category a stable homotopy category?
Finally, we close the book with a brief chapter containing some unsolved or
partially solved problems the author would like to know more about.
Note that bold-faced page numbers in the index are used to indicate pages
containing the definition of the entry. Ordinary page numbers indicate a textual
I would like to acknowledge the help of several people in the course of writing
this book. I went from knowing very little about model categories to writing this
book in the course of about two years. This would not have been possible without
the patient help of Phil Hirschhorn, Dan Kan, Charles Rezk, Brooke Shipley, and
Jeff Smith, experts in model categories all. I wish to thank John Palmieri for count-
less conversations about the material in this book. Thanks are also due Gaunce
Lewis for help with compactly generated topological spaces, and Mark Johnson for
comments on early drafts of this book. And I wish to thank my family, Karen,
Grace, and Patrick, for the emotional support so necessary in the frustrating en-
terprise of writing a book.