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Reviewer: Robert D. Tennent

This book attempts to reconcile two approaches to the foundations of mathematics: mathematical logic and category theory. The contents can be briefly summarized as follows. There is an introductory Part 0 on category theory, following which the authors discuss lambda calculi and their correspondence to Cartesian-closed categories in Part 1, intuitionistic higher-order theories and their correspondence to toposes in Part 2, and representability of numerical functions in various categories in Part 3. In Parts 1 and 2 there are introductory sections giving a perspective on the history of the correspondence to be discussed, as well as final sections of comments on specific sources. The pace of the exposition varies considerably. Part 0 is quite condensed and is essentially a summary of the terminology and notation used in the book. Part 1 is the best-written section; almost certainly, this material has been used as lecture notes. In Parts 2 and 3 the pace speeds up considerably, especially when the notion of a topos makes its first appearance. Many readers will want to refer to [1] for a more leisurely exposition. It should be noted by the prospective reader that this book appears in a series of studies in advanced mathematics and is not particularly oriented to the background or interests of computer scientists. What then is the significance of this work to computer science__?__ Well, on the one hand, mathematical logic is playing an increasingly important role in many branches of computer science, such as databases, artificial intelligence, programming languages, software engineering, complexity theory, and so on. On the other hand, category theory is a general approach to mathematical structure that has succeeded in formulating several concepts, such as natural transformations, universality, adjoints, and limits, that are important in almost every branch of mathematics. Thus, it follows that a work that uses category theory to analyze logic might also be of interest to computer scientists. This is certainly the case for the logic discussed in this book. The relevance of lambda calculi to programming-language theory is well established. For example, the procedure-definition mechanisms of LISP and ALGOL-60 are based on the untyped and typed variants, respectively, of these formal systems. One significance of the connection with Cartesian-closed categories is that when a new Cartesian-closed category is discovered, this immediately provides a possible interpretation for higher-order programming languages with procedural application and abstraction based on the lambda calculus. Examples may already be found in the computer science literature [2,3]. As for intuitionistic theories, they too are beginning to be important in computer science. An example is the “specification logic” of Reynolds [4,5], a formal system for reasoning about ALGOL-like languages with procedure mechanisms based on the typed lambda calculus. The fact that it is an intuitionistic theory was not initially realized by its designer, but the problem of finding an appropriate semantic interpretation was finally solved by exploiting the fact (discussed in Section 9 of Part 2 of this book) that a certain kind of functor category is a topos [6]. The key point about toposes is that any theorem about sets that can be proved constructively is immediately transferable to arbitrary toposes. Recent results on other systems of great interest to computer scientists, such as Martin-Lo¨f-style type theories [7,8,9] and higher-order (or polymorphic) lambda calculi [10,11,12], are too new to have been included in this book but are clear indications of the continuing importance that categorical logic will have in computer science. This book is essential reading for anyone interested in the subject.