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Special Topics in Mathematics for Computer Scientists: Sets, Categories, Topologies, and Measures

Ernst-Erich Doberkat
Publication Date: 
Number of Pages: 
[Reviewed by
Ittay Weiss
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A title such as Special Topics in Mathematics for Computer Scientists leaves much freedom for the author as to what to include in the book. The subtitle informs us that sets, categories, topologies, and measures will be encountered, framing the aims toward foundations and logic in set theory, algebraic aspects of category theory, and topological and measure theoretic issues in analysis. The author took upon himself to weave these topics into a coherent whole spanning over 600 pages, designed to please computer scientists. I am not a computer scientist but I do have some understanding of the subject that goes a bit deeper than simply knowing where the “on” switch is. Thus it is with great interest that I sat down to read the book.

As I am now sitting down to describe the book, starting with its contents, I must admit it is tempting to list what the book does not cover rather than what it does cover, simply since that would save me quite a bit of work. The book is absolutely packed with ideas and theorems, classical results, well-knowns as well as obscure pieces of mathematics, familiar and less familiar axioms, and generally many many facts, ranging from trivial to deep and pivotal.

I will first attempt to briefly describe the contents of the book. The aim is Giry’s monad, which dictates the need for category theory (to say what a monad is) and of measure theory (the habitat of Giry’s monad). But the author spares no ink as each chapter rolls on and on, discussing chains of inter-related ideas, at times discussing trivialities and other times taking a deep breath and diving deep.

The book is probably the most original I’ve seen in terms of its contents. So for instance in Chapter 1, The Axiom of Choice and Some of Its Equivalents, one finds the expected appearance of Zorn’s Lemma, as well as less obvious appearances such as the Stone representation theorem, not to mention whole sections on \(\sigma\)-algebras and Banach-Mazur games. Chapter 2 is a brief introduction to categories, functors, and natural transformations, going on to monads, adjunctions, monad algebras, coalgebras, and modal logic; this is probably the least unorthodox part of the book. Chapter 3, on topological spaces, quickly gets to convergence through filters and separation properties, revisits Banach-Mazur games, and goes on to the Stone-Weierstrass theorem. Chapter 4 covers measurable sets, real-valued functions, Borel sets, Polish spaces, Souslin’s separation theorem, Neumann’s selectors, measurable selections, the Daniell integral, the Riesz representation theorem, Fubini’s theorem, projective limits, the weak topology, the Hutchinson metric, and a spoonful of Hilbert space theory. Is your head spinning? So is mine.

The thread of ideas and presentation is clearly derived and dictated by the author’s renowned expertise in the area, producing a very unique point of view on an active area of research. I like very much that the author introduces what needs to be introduced whenever that is required, regardless of how easy or complicated that may be, simply jumping to it. The result of this approach is the exciting medley of topics which forms the book, an incarnation of what would happen if Giry’s monad had the power to attract to it the mathematical results and ideas needed to make its acquaintance assuming little to no preliminary knowledge, and would then proceed to taking a stroll in the park. What one would find in the park would resemble the contents of the book. As one reads the book, one feels Giry’s monad’s presence, as if it’s just been here a moment ago. Unfortunately, just like the scene left in the park, the book is often a bit messy, but not in a terribly harmful way. Regrettably, there are some stylistic issues related to language and editing which if the reader is willing to be patient and forgiving about, then she will learn a great of deal of mathematics presented in a way never encountered before. This is quite a unique book.

Ittay Weiss is a Teaching Fellow at the University of Portsmouth, UK.

See the table of contents in the publisher's webpage.