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Math 549. Model categories . Fall, 2001.

Textbook: Model categories by Mark Hovey, Mathematical Surveys and Monographs 63, AMS, Providence, RI, 1999 (x + 209 pages).

• The table of contents and list of errata are available online.
• Here are links to the book on Amazon and on Barnes & Noble.
• Here is a link to it at the AMS Bookstore. As an AMS member you can purchase it for $34 instead of the list price of $57.
• Here is the AMS review of the book.

• [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.

 
• [Voe97] (The Milnor Conjecture, by Vladimir Voevodsky) and [Wei97] (Obituary Robert Thomason (1952-1995), by Charles Weibel) are in the K-theory Preprint Archives.

 
• The Hopf Topology Archive has
• [EKMM97] (Rings, modules, and algebras in stable homotopy theory by A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May ),
• [Hov98a] (Monoidal model categories by Mark Hovey),
• [Hov98b] (Spectra and symmetric spectra in general model categories by Mark Hovey),
• [HPS97] (Axiomatic Stable Homotopy Theory by Mark Hovey, John Palmieri and Neil Strickland),
• [HSS98] (Symmetric spectra by Mark Hovey, Brooke Shipley, and Jeff Smith),
• [KM96] (Operads, algebras, modules, and motives by Igor Kriz and J.P. May ),
• [Pal97] (Stable homotopy over the Steenrod algebra by John H. Palmieri ),
• [SS97] (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 sets.


Course description:

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 generators.

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.