I am chagrined to find myself as one of two category people in the bibliography of the work reviewed below and -- worse -- Peter says that they ascribed schizophrenic objects "not altogether appropriately" to me. He's right. CMP 1 663 208 (99:06) 18-02 (03G25 06D05 08C15 18A40) Clark, David M.(1-SUNYP); Davey, Brian A.(5-LTRB) Natural dualities for the working algebraist. (English. English summary) Cambridge Studies in Advanced Mathematics, 57. Cambridge University Press, Cambridge, 1998. xii+356 pp. $64.95. ISBN 0-521-45415-8 I began reading this book (conventionally enough) with the first two paragraphs of the Preface: "In 1936 Marshall Stone published a truly novel theorem ... What Stone discovered was a representation for all Boolean algebras which gave algebraists a usable understanding of their structure, using topological spaces to construct the representations ... In fact, Stone proved much more than a representation theorem ... In modern language, he proved that the category of Boolean algebras is dually equivalent to the category of Boolean spaces ...". Finding in these words what I presumed to be a conscious echo of the words: "This book is about a particular theorem -- the Stone representation theorem for Boolean algebras -- and some of the mathematical consequences which have developed from it ... Stone's key idea was the introduction of topology ... this was a really bold idea ... Moreover, [Stone's representation formed] one of the earliest nontrivial examples of an equivalence of categories ..." with which I had begun the Introduction to my own book Stone spaces, published in the same C.U.P. series in 1982 [Cambridge Univ. Press, Cambridge, 1982; MR 85f:54002], I naturally turned next to the Bibliography to check that my book was listed there. After all, though my book dealt with other subjects besides duality, it did have a whole chapter devoted to dualities of the type studied by Clark and Davey, and so it ought to be one of their standard references. My name does not appear in the Bibliography. Indeed, its only occurrence in the entire book is in the list of previous volumes in the series which appears opposite the title page. I mention this at the outset of this review, in order to give the reader a fair chance to make allowance for what he may construe as "sour grapes" on my part. For a related reason, I have departed from Mathematical Reviews tradition by writing the review in the first person; I cannot pretend to have achieved the standards of objectivity and dispassionateness that would be implied by a third-person review. However, it is not only the works of Johnstone which have been overlooked by Clark and Davey. I looked next in the Bibliography for John Isbell's ground-breaking 1972 paper on general functorial semantics [Amer. J. Math. 94 (1972), 535--596; MR 53 #580], and his later work characterizing concrete dualities in terms of commuting subtheories of a ruled theory: neither is present, and Isbell is represented only by a much less important 1980 paper on median algebra. Again, the 1982 paper in which Harold Simmons coined the term "schizophrenic object" [Topology Appl. 13 (1982), no. 2, 201--223; MR 83f:18006], for a set with two commuting algebraic structures, is not there, although Clark and Davey freely use this term in their text. Having discovered this, I began studying the Bibliography more systematically, and soon realized its salient feature: all works written by category-theorists, or making serious use of categorical ideas, are excluded from it -- with the twin exceptions of Saunders Mac Lane's classic Categories for the working mathematician [Springer, New York, 1971; MR 50 #7275] (which, after all, the authors could hardly have left out, given their indebtedness to his title), and of Peter Freyd's 1966 paper on algebra-valued functors [Colloq. Math. 14 (1966), 89--106; MR 33 #4116] (to which I shall return below). This, then, is a book on a categorical subject -- duality -- by authors who are not category theorists, and presumably intended for such a readership. However, the editors of Mathematical Reviews have asked a category-theorist to review it, so the review is written from that point of view. The subject of the book is the construction of dual equivalences, or more generally contravariant adjunctions, between algebraic categories $\scr A$ and categories $\scr X$ whose objects are compact topological spaces equipped with compatible algebraic and/or relational structure. Here one encounters the first restrictive feature of the authors' outlook: for them "algebraic" means "finitary algebraic" (an attitude already debunked by Marshall Stone in 1947), and so the idea of a compact topology as a kind of algebraic structure (indeed, the universal example of a structure which commutes with all finitary structures) cannot be expressed. Nor can the concomitant "unity of opposites" idea that $\scr A$ and $\scr X$ are two categories of the same kind -- they are doomed to remain forever separate, like the lovers on Keats's Grecian urn. Clark and Davey follow the common tradition in universal algebra that the underlying set of an algebra is not allowed to be empty. Having thus amputated the initial objects from many of their categories $\scr A$, they are forced to amputate the terminal objects from the dual categories $\scr X$, leading to unnecessary complications in the descriptions of these categories as quasivarieties. Since they have not amputated the terminal object from $\scr A$, they are forced (in appropriate cases) to allow the empty space as a member of $\scr X$. Thus, for them, a topological algebra does not necessarily have an underlying discrete algebra! As already mentioned, Clark and Davey make heavy use of schizophrenic objects. However, as far as most of the book is concerned, schizophrenic objects are simply a convenient ad hoc way of constructing a contravariant adjunction; the fact that every contravariant adjunction between categories of the type they consider is induced by mapping into a schizophrenic object is not mentioned until page 162, where it is ascribed (not altogether appropriately) to the paper of Freyd mentioned earlier, and stated in such a convoluted way as to be almost unrecognizable. Again, all the schizophrenic objects the authors consider are finite (and topologically discrete); thus, although they refer to Pontryagin duality in their Preface (and Pontryagin appears, along with Birkhoff and Stone, as one of the three mathematicians to whose memory the work is dedicated), they are able to describe only the special case of Pontryagin duality for abelian groups of some fixed finite exponent. All the above criticisms are, in a sense, trivial ones; but their cumulative effect is nontrivial. Category theory is often criticized for consisting entirely of trivialities; but, as Freyd long ago observed, its real function is to demonstrate that the trivial parts of mathematics are trivial for trivial reasons, and that is a valuable service which it performs for the mathematical community. Reading Clark and Davey's book, anyone new to the field would find it difficult or impossible to distinguish between the trivialities and the results with genuine content, since the former are so often presented in ad hoc ways that obscure the underlying pattern. This is not a book that I would recommend a graduate student to read. One might argue that the book's shortcomings are not of major importance in relation to its declared purpose. After all, it contains a wealth of detailed information on particular techniques for establishing duality theorems, and "working algebraists" (read: working universal-algebraists) will undoubtedly find it immensely useful to have all these techniques collected together in one place. I admit the force of that argument; but I also find it seriously worrying, for it carries the implication that universal-algebraists have given up the attempt to engage in dialogue with the rest of the mathematical community. As the authors remark at the end of their Preface, there is much that remains to be done in studying and classifying concrete dualities; but this book is not likely to inspire anyone outside the closed circle of universal-algebraists to take up the task. Reviewed by Peter Johnstone Copyright American Mathematical Society 2000