xviii Preface homotopy limit, and a powerful calculus for working with them was devel- oped. The language of homotopy limits and colimits and the techniques for manipulating them made it possible to easily state and conceptually prove many results that had previously seemed quite diﬃcult and inscrutable. Once the basic theory has been laid down, the most interesting and useful theorems are those that break the categorical barrier between domain and target. The basic example of such a theorem is the Blakers-Massey theorem, which compares homotopy pushout squares to homotopy pullback squares. Other excellent examples of duality-breaking theorems are the Hilton-Milnor theorem on the loop space of a wedge and Ganea’s theorem (which is dual to the most important special case of the Blakers-Massey theorem). All of these results were first proved with a great deal of technical finesse but can now be established easily using homotopy pushouts and pullbacks. The Aim of This Book. The aim of this book is to develop classical ho- motopy theory and some important developments that flow from it using the more modern techniques of homotopy limits and colimits. Thus homotopy pushouts and homotopy pullbacks play a central role. The book has been written with the theory of model categories firmly in mind. As is probably already evident, we make consistent and unapologetic use of the language of categories, functors, limits and colimits. But we are genuinely interested in the homotopy theory of spaces so, with the exception of a brief account of the abstract theory of model categories, we work with spaces throughout and happily make use of results that are special for spaces. Indeed, the third part of the book is devoted to the development of four basic properties that set the category of spaces apart from generic model categories. I have generally used topological or homotopy-theoretical arguments rather than algebraic ones. This almost always leads to simpler statements and simpler arguments. Thus my book attempts to upset the balance (ob- served in many algebraic topology texts) between algebra and topology, in favor of topology. Algebra is just one of many tools by which we understand topology. This is not an anti-algebra crusade. Rather, I set out hoping to find homotopy-theoretical arguments wherever possible, with the expecta- tion that at certain points, the simplicity or clarity afforded by the standard algebraic approach would outweigh the philosophical cleanliness of avoiding it. But I ended up being surprised: at no point did I find that ‘extra’ algebra made any contribution to clarity or simplicity. Omissions. This is a very long book, and many topics that were in my earliest plans have had to be (regretfully) left out. I had planned three chapters on stable homotopy, extraordinary cohomology and nilpotence and

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