This book provides a thorough and well-written guide to abstract homotopy theory. It could well serve as a graduate text in this topic, or could be studied independently by someone with a background in basic algebra, topology, and category theory.
        The book deals with a variety of approaches to abstract homotopy theory and explains very well the relationships among them. Most results are given clear proofs, while some proofs are relegated to well-outlined exercises. For only a few of the deepest results will the reader need to rely on an outside reference.
        The seven chapter titles suggest the scope of the book. They are as follows: 1. Abstract homotopy theory; 2. Homotopical algebra; 3. Case studies; 4. Groupoid enrichment and track homotopy; 5. Homotopy coherence; 6. Abstract simple homotopy theories; 7. Injective simple homotopy theories.
        One of the pervading themes is that of the cylinder, which is a functor ( ) ×I: C® C together with natural transformations e₀,e₁:Id_C ® ( ) and s:( ) ×I®Id_C such that se₀ = se₁ = Id_C. Homotopy theory is defined in terms of this concept. Cylinders on many different categories such as the category of groupoids or the category of chain complexes over a fixed abelian category, are described.
        Another theme is Kan conditions, which have to do with types of diagrams that can be filled in. The authors carefully explain which Kan conditions must be satisfied in order that various results can be deduced. One important example of this type is an abstract version of Dold's theorem. It says that if a cylinder satisfies Kan conditions DNE (2,1,1) and E(3,1,1) and preserves weak fibrations, then a map under an object A which is a homotopy equivalence is a homotopy equivalence under A.
        The notion of track homotopy is treated carefully. A track homotopy commutative square is a square of morphisms in C together with an equivalence class of homotopies between the two composites. These can be composed (similar to homotopy sum) horizontally or vertically, leading to a very delicate sort of calculus.
        In simple homotopy theory, one defines as simple the homotopy equivalences that can be obtained from elementary expansions. An obstruction theory for whether a homotopy equivalence is simple was developed by Whitehead.
        This idea is abstracted to category theory, beginning with C® C(S^-1), where S is a class of morphisms of C containing the isomorphisms and closed under composition. One goal is to determine A(X) and E(X), when X is an object of C. Here A(X) is the set of equivalence classes of morphisms in C (S^-1) with domain X, where the equivalence relation is defined so that following by a morphism in S does not change the equivalence class, while E(X) consists of the isomorphisms in A(X). Characterisations of E(X) are given in a number of different contexts.
        This review merely scratches the surface of the many variants of abstract homotopy theory which are discussed in this excellent book.

Donald M Davis 1-LEHI; Bethlehem, PA