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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).

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 link.

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.

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