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Reviewer: William A Fahle

Category theory is a relatively new branch of mathematics involving the study of collections of functions among some objects. As a generalization of group theory, it is a useful topic of study for theoretical computer scientists. Results in category theory will have widespread application to all sorts of common objects. The seminal text on category theory is by Saunders Mac Lane [1]. Awodey's Category theory is a text for the rest of us. It assumes only a basic knowledge of mathematics, such as linear algebra and discrete mathematics. When Awodey introduces other ideas, he provides good basic definitions. Examples are drawn from many areas of mathematics, but they are primarily illustrative and optional for grasping the main ideas of the text. The first chapter defines categories as collections of objects and the arrows (or functions) between those collections. It gives numerous examples. Computer scientists familiar with abstract thinking will have little trouble tackling this section. Subsequent chapters (2 through 6) give theorems based on these definitions and expand the definitions to create abstract algebras of categories. The concept of duality takes up one full chapter, as do the concepts of limits and colimits. The author explores the connection between groups and category theory, and defines and examines exponentials, all leading up to the general theory of categories. A chapter also introduces the concept of naturality, which describes how functions commute and compose, and gives some properties independent of the actual underlying functions. Later chapters cover the categories of diagrams (chapter 8), including the useful Yoneda lemma and its applications. Monads and algebras are discussed in the final chapter (10). The crux of the book appears in the chapter (9) about adjoints. The author claims that adjoints capture an important mathematical aspect of functions and logic that is both fundamental and not captured elsewhere in mathematics. The full definition is given here, and it literally takes the rest of the book to provide all the lead-in definitions, so this section is difficult to summarize. The concepts in this book are at once advanced and fundamental. To the extent that they are fundamental, they are essential to a working computer scientist, and the treatment here is an excellent and methodical introduction. Online Computing Reviews Service