Math 549. Model categories
Textbook: Model categories
by Mark Hovey, Mathematical
Surveys and Monographs 63, AMS, Providence, RI, 1999 (x + 209 pages).
Some of the references cited in the book are available online.
Mac Lane's book Categories
for the Working Mathematician has some helpful background.
Here is its Amazon
[DS95] ( Homotopy theories and model categories by Dwyer
and Spalinski, 56 pages) can be found on Bill
Dwyer's home page as #75 in his publication list.
[DHK] (Model Categories and more General Abstract Homotopy
Theory by Dwyer, Hirschhorn and Kan, March 1997 draft, 101
pages, and January 2001 draft, 83 pages.) and [Hir97] (Localization of
Model Categories by Hirschhorn, April 2000 draft, 429 pages!) are available
online at Phil Hirschhorn's home
page. The two drafts of [DHK] are disjoint and are apparently subsets
of a much larger work in "what [they] like to think of as progress."
The chapters are numbered I-III, XI-XIV, and XXI-XXVII. The authors
are three of the most pleasant mathematicans you will ever meet.
(The Milnor Conjecture, by Vladimir Voevodsky) and [Wei97]
(Obituary Robert Thomason (1952-1995), by Charles Weibel) are in the K-theory
Topology Archive has
(Rings, modules, and algebras in stable homotopy theory by A. D. Elmendorf,
I. Kriz, M. A. Mandell, and J. P. May ),
(Monoidal model categories by Mark Hovey),
(Spectra and symmetric spectra in general model categories by Mark Hovey),
(Axiomatic Stable Homotopy Theory by Mark Hovey, John Palmieri and
(Symmetric spectra by Mark Hovey, Brooke Shipley, and Jeff Smith),
(Operads, algebras, modules, and motives by Igor Kriz and J.P. May ),
(Stable homotopy over the Steenrod algebra by John H. Palmieri ),
(Algebras and modules in monoidal model categories by Stefan Schwede and
Brooke E. Shipley),
and some information about [GJ97]
(Simplicial Homotopy Theory by P.G. Goerss and J.F. Jardine).
May's book Simplicial
Objects in Algebraic Topology is an early source on simplicial
Here is Hovey's description of the book:
Model categories are a tool for inverting certain
maps in a category in a controllable manner. As such, they are useful in
diverse areas of mathematics. The list of such areas is continually growing.
This book is a comprehensive study of the relationship
between a model category and its homotopy category. The author develops
the theory of model categories, giving a careful development of the main
examples. One highlight of the theory is a proof that the homotopy category
of any model category is naturally a closed module over the homotopy category
of simplicial sets.
Little is required of the reader beyond some category
theory and set theory, making the book accessible to graduate students.
The book begins with the basic theory of model categories and proceeds
to a careful exposition of the main examples, using the theory of cofibrantly
generated model categories. It then develops the general theory more fully,
showing in particular that the homotopy category of any model category
is a module over the homotopy category of simplicial sets, in an appropriate
sense. This leads to a simplification and generalization of the loop and
suspension functors in the homotopy category of a pointed model category.
The book concludes with a discussion of the stable case, where the homotopy
category is triangulated in a strong sense and has a set of small weak
I will describe what I see as the highlights of the
book, clarifying definitions and examples whenever possible.
This page was last revised on October 23, 2001.