Preface xix another on Goodwillie calculus. But in the book that emerged it seemed thematically appropriate to draw the line at stable homotopy theory, so space and thematic consistency drove these chapters to the cutting room floor. Problems and Exercises. Many authors of textbooks assert that the only way to learn the subject is to do the exercises. I have taken this to heart, and so there are no outright proofs in the book. Instead, theorems are followed by multi-part problems that guide the readers to find the proofs for themselves. To the expert, these problems will read as terse proofs, perhaps suitable for exposition in a journal article. Reading this text, then, is a preparation for the experience of reading research articles. There are also a great many other problems incorporated into the main flow of the text, problems that develop interesting tangential results, explore applications, or carry out explicit calculations. In addition, there are numerous exercises. These are intended to help the student develop some habits of mind that are extremely useful when reading mathematics. After definitions, the reader is asked to find examples and nonexamples, to explore how the new concept fits in with previous ideas, etc. Other exercises ask the reader to compare theorems with previous results, to test whether hypotheses are needed, or can be weakened, and so on. Audience. This book was written with the idea that it would be used by students in their first year or two of graduate school. It is assumed that the reader is familiar with basic algebraic concepts such as groups and rings. It is also assumed that the student has had an introductory course in topology. It would be nice if that course included some mention of the fundamental group, but that is not necessary. Teaching from This Book. This book covers more topics, in greater depth, than can be covered in detail in a typical two-semester homotopy theory or algebraic topology sequence. That being said, a good goal for a two-semester course would be to cover the high points of Parts 1 – 4 in the first semester and Parts 5 – 6 in the second semester, followed by some or all of Part 7 if time permits. Here’s some more detail. The first semester would start with a brief (one day) introduction to the language of category theory before heading on to Part 2 to develop the basic theory of cofibrations, fibrations, and homotopy limits and colimits. Part 1 is an overview of the basics of category theory and shouldn’t be covered in its own right at all refer back to it as needed to bring in more advanced category-theoretical topics. Chapters 3 and 4, in which the category of

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