I am grateful to the M.I.T. Mathematics Department for putting up with me
over all these years, and making me always feel welcome. Although there has been
a gradual change in the group of topologists there since I was a graduate student,
what has been constant is the warm, supportive attitude that has allowed me always
to think of it as home.
This project grew out of my reading an early draft of Emmanuel Dror Farjoun's
book [24]; I am grateful to him for his preprints and helpful communications.
Drafts of this book been have been circulating for a number of years, originally
under the title Localization, Cellularization, and Homotopy Colimits and more re-
cently as Localization of Model Categories, and a number of people have provided
helpful comments. Paul Goerss helped correct a problem with the proof of the
existence of the localization model category structure for cellular model categories.
Garth Warner gave several early drafts an amazingly careful review; he found many
obscurities and outright errors, and provided a level of analysis that usually takes
several years and a group of reviewers. Daniel Biss, Dan Christensen, Dan Dug-
ger, Bill Dwyer, Lars Hesselholt, Mike Hopkins, Mark Hovey, Dan Isaksen, Haynes
Miller, George Peschke, Charles Rezk, and Jeff Smith offered corrections and help-
ful suggestions. Of course, any mistakes or omissions that remain are entirely my
Above all, though, I owe a great deal of thanks to Dan Kan. He is responsible
for my first becoming interested in this subject, and I am indebted to him for
countless helpful conversations and suggestions. His view of mathematics has been
a lasting influence upon me, and it is a pleasure to thank him for his support and
encouragement over the years.
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