There are as many ways of approaching the concept of a topos as there are for a blind man to describe an elephant after touching one part of its anatomy (the reviewer is indebted to A. Joyal for this comparison). So it is appropriate that, in the recent flood of new books on topos theory, different writers have chosen very different ways of presenting the subject. The present book represents what must surely be an extreme point in the spectrum of possible approaches: the one which makes the greatest possible use of formal languages. The material, throughout the book, is of a highly technical nature, which presents the reviewer with a problem: any attempt at detailed description of the book’s contents is going to prove unenlightening to the average reader of Zentralblatt, so the best he can do is to try to describe the authors’ motives in writing it - – but, since he is not himself one of the authors, he stands in danger of misrepresenting those motives. I apologize in advance to Messrs Chapman and Rowbottom if I have succumbed to this danger; but here, for what it’s worth, is my attempt at explaining why one might want to write a book of this type (and why someone else might want to read it).
It is by now well understood that one can describe a topos \({\mathcal E}\) “from the inside” by means of a formal language \(L({\mathcal E})\) (sometimes called the Mitchell-Bénabou language) and that one can employ essentially set-theoretic (albeit constructive) modes of reasoning to prove things about \({\mathcal E}\) within this language. Since many mathematicians feel more comfortable with st theory than with category theory, this has led to the idea that one might take the language rather than the topos as the primitive notion, and to the development by various people (Coste, Fourman, Boileau-Joyal, Zangwill, \(\ldots\)) of appropriate “local set theories”, expressible in such a language, whose models are essentially the same thing as toposes. There is no doubt that the use of such theories has been beneficial, in that it has enabled many people whose background is in formal logic to gain an entrée to topos theory.
All this is fine as long as one is interested only in what goes on inside a single topos (or, more generally, a pair of toposes linked by a logical functor). But this is not enough: the geometric origins of topos theory, and in particular the idea of a topos as a generalized topological space, demand the consideration of geometric morphisms (or generalized continuous maps) between toposes, and the functors encoding such morphisms are not normally logical. And these “geometric” aspects of topos theory insist on interacting with the “logical” side which views a topos as a category of sets: in particular, there is an important intuition which says that when one is studying toposes over a given topos \({\mathcal E}\) (that is, equipped with specified geometric morphisms into \({\mathcal E})\), one might as well consider \({\mathcal E}\) to be “the” topos of sets and the toposes over it to be “generalized Grothendieck toposes” defined by sites within \({\mathcal E}\).
The Mitchell-Bénabou language of \({\mathcal E}\) is quite inadequate to handle this intuition: it deals very well with “small” categories (that is, internal categories in \({\mathcal E})\), but not with “large” ones like toposes over \({\mathcal E}\). The category-theorist’s response to this problem is to develop a theory of indexed (or fibred) categories over \({\mathcal E}\), which does allow one to think of \({\mathcal E}\) as the category of sets; but this approach requires a fair amount of 2-categorical expertise, and is thus not to everyone’s taste. There is therefore a perceived need for a formal language which is adequate to describe indexed categories over \({\mathcal E}\) as well as internal categories within \({\mathcal E}\).
This is what Chapman and Rowbottom have set out to provide, taking as their starting-point the “local set theory” developed by Rowbottom’s former student Zangwill, and described in detail in the book by J. L. Bell which appeared in this series four years ago [Toposes and local set theories: an introduction (Oxford Logic Guides, 14) (1988; Zbl 0649.18004)]. The book under review can thus be viewed as “volume 2” of Bell’s work; in particular, any prospective reader of it would be well advised to study Bell’s book first. The extent to which Chapman and Rowbottom have succeeded in their aims may be judged by the fact that their book concludes (if one discounts the Appendix) with an account of the relative Giraud theorem (the theorem which precisely expresses the connection between elementary toposes over a base and internal sites in the base) which is conducted entirely within the formal language they have developed. It is surely fair to say that if their language can cope with this theorem then it can cope with anything which is likely to arise.
However, the next question must be whether this approach makes the relative Giraud theorem any easier to understand. Whilst the answer one gives to this question will inevitably depend on one’s own background, the reviewer suspects that most people will answer “no”: the amount of technical machinery that must be developed along the way (and virtually all the machinery in the book is used in the final chapter) is quite formidable. Moreover, it is perhaps unfortunate that the theorem which the authors have chosen as their goal does not seem to have any significant applications which can readily be described in their language: thus the book is liable to leave a reader who has not previously encountered the more advanced 2-categorical aspects of toposes and geometric morphisms wondering whether all the effort was worth while.
Nevertheless, the book does have its virtues. By providing a language adequate to handle indexed categories over a topos (and much more besides), the authors have done much to “demystify” indexed category theory and make it accessible to those for whom the sight of a 2- categorical diagram is liable to induce instant incomprehension. All such people (and there are many) should make the effort to master the technicalities in this book, if they wish to get closer to an understanding of what really goes on in the 2-category of toposes.