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Reviewer: Van Van Dyke Parunak

One can erect a system of mathematics on various foundations. Euclid provided the first known formal system, based on geometric notions such as points in a space, straight lines, and distance. Classical probability theory is grounded in the idea of sets, set membership, and operations between sets such as union and intersection. Category theory offers yet another foundation, based on the idea of morphisms, that is, mappings among objects. Category theory is a relative newcomer, invented in the 1940s to help explain the common features among structures in different branches of mathematics. Since a mathematical theory is itself an abstraction, discussions of the relations among such theories are even more abstract, leading some of the originators of the formalism to describe it fondly as “general abstract nonsense.” Apart from tentative applications in physics (in representing Feynman diagrams) and computer science (to provide a formal definition of types), in both cases dealing with highly theoretical problems, the theory has not yet attracted a following beyond the circle of pure mathematicians. Spivak's book promises to change this state of affairs in two ways. First, it abounds in concrete examples and informal explanations (offering not only definitions, propositions, and theorems, but also examples, slogans(!), and numerous exercises, all worked out in detail). It often presents the same concept several times, first with concrete examples and then more abstractly. As a result, even readers with little exposure to abstract mathematics can follow it with a little application. Second, it motivates the theory by the running example of “ologs,” or “otology logs,” a knowledge representation formalism that maps readily to category theory on the one hand and to databases on the other. This example is broadly applicable in any scientific domain, addressing common problems such as analyzing the relation between theory and experimental data, schema alignment, and ontology mapping. Ordinary engineers must grapple with such problems every day, and category theory offers powerful tools to help them. This combination of pedagogical accessibility and a hook to realistic, common problems results in a volume that will appeal both to budding mathematicians who seek an easy introduction to the theory, and to anyone who must deal with structured information in the real world. The first four chapters introduce the reader to the history of category theory, and some of the basic concepts, using the category of sets for formal exposition and the olog as a running example. After introducing the reader to products, coproducts, and the associated (co)limits, it works through a series of mathematical structures, including monoids, groups, graphs, and orders, culminating in a category-theoretical exposition of the notion of a database. The title of chapter 4 captures the spirit of these chapters: “Categories and Functors, Without Admitting It.” With this tutorial foundation in place, chapters 5 and 6 go back over the same ground, but with more rigor. Now Spivak not only admits to categories, functors, and natural transformations, but defines them formally, with links back to the more intuitive explanations in the first four chapters. The culmination of chapter 5 is the proof that categories and schemata are equivalent, thus urging anybody who is designing a database schema to leverage category theory in the process. Chapter 6 formalizes other notions introduced in the first four chapters, such as limits and colimits, and chapter 7 introduces higher-level topics including adjoint functors, monads, and operads. By this time, the reader is not only comfortable enough to move on to the more formal literature in category theory, but also motivated by an understanding of how the theory can impact practical issues in managing structured information. The volume is intended to be read sequentially. The index is quite sparse, hinting that this is a textbook, not a reference work. Those who take the time to work through the exercises will come away with a clear intuition into the power of category theory and strong motivation to apply it to their own applications. More reviews about this item: Amazon Online Computing Reviews Service