The first chapter presents some preliminary material on sets, functions, graphs, and graph homomorphisms. Chapters 2 and 3 introduce, respectively, the basic concepts of category and functor; many important examples and some simple constructions are described. The connection between functional programming languages and categories is explained. Chapter 4 deals with some fundamental ideas of category theory: diagrams, commutative diagrams, natural transformations, representable functors, universal objects and the Yoneda embedding. A weak version of a sketch, called a linear sketch (a graph with diagrams), is introduced. Products and sums in categories are discussed in Chapter 5; the important concept of a natural numbers object is defined here. An interpretation of formal languages and formal deduction systems as categories is provided. The subject of Cartesian closed categories is developed in Chapter 6; these structures have a special importance in computing science, due to their strong relationship with typed $\lambda$-calculus.
The notion of linear sketch is generalized in Chapter 7. Two major types of sketches are introduced and studied here: the FP (finite product) sketches and the FD (finite discrete) sketches. The first type is exemplified by the sketch for semigroups. The connection with the method of signatures and equations and the associated functional semantics are described. The second type is exemplified by the sketch for fields. The constructions of Chapter 5 are generalized in Chapter 8, where the general concepts of limit and colimit in a category are discussed. The theory of sketches is further developed in Chapters 9 and 10. The class of FL (finite limit) sketches is introduced as a generalization of the FP sketches. The consideration of homomorphisms of sketches leads to a discussion of the category of sketches; the constructions are applied to the description of parametrized data structures. Chapter 11 introduces fibrations and the Grothendieck construction; in particular, wreath products of categories are discussed. The basic theory of adjoint functors is developed in Chapter 12; as an application, locally Cartesian closed categories are described. Chapter 13 deals with the fixed points for endofunctors and with some related constructions; the concept of a triple is introduced. The final Chapter 14 contains an introduction to the theory of topoi; the authors emphasize here their point the topos semantics is a model for computation more appropriate than that of set theory.
Each section of the text contains a set of exercises that complement the exposition by providing further examples and results. Detailed solutions to all the exercises are collected in a large appendix following the main text.
The authors have succeeded in producing a book which is original in its contents and exposition, and accessible to a broad public of readers. People specializing in theoretical computing science will find here a solid category-theoretic foundation; while those interested mainly in category and topos theory could use it as an introduction to the authors' earlier, more specialized monograph [Toposes, triples and theories, Springer, New York, 1985; MR 86f:18001].
The text is suitable as a preparation for those readers who intend to proceed to more advanced books, such as that authored by A. Asperti and G. Longo [Categories, types, and structures, MIT Press, Cambridge, MA, 1991], or that by M. Barr and C. Wells [Category theory for computing science, Prentice Hall, New York, 1990; MR 92g:18001].
As is apparent from this description of the contents, McLarty treats standard topics in topos theory, also covered in other texts on the subject. Unlike most of these texts, however, the prerequisites for reading McLarty's book are kept to a minimum. In particular, his exposition should be accessible to mathematics undergraduates, and to interested philosophers, logicians and computer scientists with perhaps less background in mathematics. With this audience in mind, McLarty manages to cover a considerable range of topics in an elegant and clear fashion, thus giving these readers a good first impression of what the subject is about. Of course there is a price to pay: there are very few concrete mathematical examples of topoi in the book, the relation to topology and geometry hardly comes to the surface, and the internal logic is not put to actual use in independence proofs. In particular, the last chapters on synthetic differential geometry and on the effective topos (each about 10 pages long) do not aim to give more than a glimpse of the ideas involved.
However, these topics are treated extensively in other texts, and I am sure that McLarty's perspicuous treatment provides a good first introduction, making more advanced texts accessible.
In fact, the authors do more: they present category theory as a challenging mathematical subject, with (by comparison with other books on the subject) unprecedented rigour and brevity. Although no specific prerequisites are assumed, a considerable degree of mathematical maturity is needed to be able to follow the authors' pace. The pattern of their approach becomes clear right at the beginning of the book: derive representation theorems and thereby establish the connections to logic and geometry. To wit, from an expansion of the Cayley representation from groups to small categories they conclude (on p. 14): every universally quantified elementary sentence in the predicates of category theory true for the category of sets is true for all categories. For Cartesian (= finitely complete) and for regular categories, one has a similar result for Horn sentences in the respective defining sentences. Abelian categories are treated within the context of regular categories and are actually introduced by the desired logical characterization as bi-Cartesian categories which satisfy all Horn sentences in the bi-Cartesian predicates which hold for the category of abelian groups.
There are two characteristic features of the authors' presentation of topos-theoretic concepts. One is the aim to isolate properties which do not rely on the general notion of adjoint functor (just on adjoint maps between pre-ordered sets), and the other is the emphasis on the importance of relations. While the former aspect leads to the study of logos (a logos is a regular category in which the subobjects of an object form a lattice, and in which each inverse-image map has a right adjoint), the latter aspect gives a fairly easy notion of power object and the definition of topos as a Cartesian category in which every object has a power object. The second part of the book, much shorter than the first, is a natural extension of the relation-based approach used extensively in the first part. An allegory is a category in which one has, in addition to composition, a unary operation "reciprocation" $R\sp \circ$ and a partial binary operation "intersection" $R\cap S$, like in the category of sets with relations as morphisms (rather than with functions), for which a number of laws are required. Within the framework allegories, topos-theoretic results are derived, for instance: for every topos $\scr A$ there exists a Boolean topos $\scr B$ and a faithful representation $\scr A\to\scr B$. The book culminates in a discussion of the independence of the continuum hypothesis and of the axiom of choice from the axioms of set theory which differs substantially from Cohen's work.
\{Reviewer's remarks: In trying to find some negative points, the reviewer notes two strange features of the work: the absence of a list of references, and the occasional disregard for well-established terminology. The most extreme example of this kind of disregard is the use of the term "faithful functor" which, for the authors, includes reflection of isomorphisms, without any warning for the reader.\}
Das Buch behandelt im einzelnen zunachst (Kapitel 1:) die grundlegenden Begriffe Kategorie, Funktor, naturliche Transformation mit vielen Beispielen und geht dabei auf die fur manche Fragen wichtige Unterscheidung von kleinen und grossen Kategorien ein. Es folgen (Kapitel 2:) wesentliche Konstruktionen wie Produkte von Kategorien, Funktorkategorien, Kommakategorien, Quotientenkategorien und die Kategorien aller Kategoeien, (Kapitel 3:) universelle Pfeile und Yoneda's Basissatz, Produkte, Coprodukte und allgemeinere Limites, (Kapitel 4:) Adjungierte Funktoren und Cartesisch geschlossene Kategorien, (Kapitel 5:) Limites, ihre Beschreibung durch Produkte und Egalisatoren und ihr Verhalten unter adjungierten Funktoren, Freyd's Satze uber die Existenz adjungierter Funktoren, (Kapitel 6:) Monaden (Tripel), Algebren (Eilenberg-Moore) und freie Algebren (Kleisli), sowie der Triplierbarkeitssatz von Beck, (Kapitel 7:) monoidale Kategorien, Koharenz, Monoide in monoidalen Kategorien mit Anwendungen auf die homologische Algebra, geschlossene Kategorien, (Kapitel 8:) Abelsche Kategorien in der durch die Interessenverlagerung der letzten Jahre gebotenen Kurze, (Kapitel 9:) spezielle Satze uber Limites, Enden und Coenden (Tensorprodukte) und (Kapitel 10:) Kan-Erweiterungen mit dem letzten Abschnitt "All concepts are Kan extensions".
Der Inhalt umfasst damit alle gegenwartig interessanten Aspekte der Theorie mit Ausnahme der Topoi, die zur Zeit des Entstehens dieses Buches gerade von Lawvere und Tierney einem totalen Umbruch unterworfen wurden. Trotz seiner Intension als Einfuhrung leitet es den Leser auf allen behandelten Gebieten soweit, dass er in der Lage sein wird, die aktuelle Forschungsliteratur zu verstehen. Den einzelnen Kapiteln sind einfuhrende Zusammenfassungen vorausgestellt, die meisten Abschnitte enden mit Ubungsaufgaben und weitere Aufgaben findet der Leser im Text, wenn er die gegebenen Beispiele uberpruft. Empfohlen sei besonders auch die Lekture der "Notes" (informal remarks on the background and prospects of our subject) mit denen jedes Kapitel endet.
\{For the German translation, see \#7276 below.\}
In Chapter I, functor categories of the form ${\rm Set}\sp {\ssf C\sp {\rm op}}$ with $\ssf C$ a small category, usually called presheaf categories (the name is avoided as unmotivated at the beginning of the book), are discussed. It is established that they are topoi, and their topos structure is presented in detail. Several concrete examples and classes of presheaf categories (e.g., the category of $G$-sets, for a group $G)$ are described. There are also introductions to propositional logic through lattices and to quantifiers as adjoints.
The subject of Chapter II is categories of sheaves of sets over topological spaces. Sheaves are defined as presheaves with the collation property; the equivalence of sheaves and etale bundles is then established. The fact that sheaves form a topos, and the adjunction of the sheafification functor from presheaves and the inclusion into presheaves, are presented. There is a discussion of sheaves in relation to differentiable manifolds, and of sheaves of abelian groups.
Chapter III takes up Grothendieck topologies, sheaves on a site, and the category of the latter. There is an illuminating discussion of how an effort to complete an analogy between covering spaces and their associated groups on the one hand and Galois theory on the other leads to the idea of Grothendieck's "generalized spaces". Readers will find the detailed descriptions of the Zariski site and the site for the topos of continuous $G$-sets for a topological group $G$ very enlightening. The topos structure of the category of sheaves is displayed. Grothendieck's "double-plus" construction of the sheafification functor and its properties are given a carefully motivated exposition.
In Chapter IV, the central notion, that of an elementary topos, is reformulated, and the consequences of the definition, giving rise to a structure that reveals the topos as a universe of sets, are given a full exposition. The authors take pains to emphasize the elementary nature of the definition of topos (the fact that "topos" is a first-order concept), and the elementary, set-theory-free character of the development of the main properties of a topos. For example, after having shown the existence of finite colimits by R. Pare's method through a monad on the topos, a method that apparently steps "outside" the topos itself, they give details on how to reduce the proof to a purely elementary construction. The important issue of external versus internal statements is amply clarified.
Chapter V presents the basic constructions of topoi from given topoi. The fundamental one is the category of sheaves of a Lawvere-Tierney topology on a topos. With this, the idea of sheaves and sheafification appears for the third time, this time in the greatest generality. Group actions and category actions, generalizing the earlier $G$-sets and functor categories, are treated in the context of an arbitrary topos replacing sets, and a further generalization, the fact that coalgebras of a left exact comonad form a topos, is also given.
The chapter "Topoi and logic" (Chapter VI) discusses the independence of the continuum hypothesis and that of the axiom of choice, first proved by P. Cohen. The method for the first is closely related to Cohen's original method, but the independence of the axiom of choice is shown by a proof due to P. Freyd, which uses a genuine (non-posetal) Grothendieck topology. The detailed exposition of the Kripke-Joyal semantics, both in its form for a general topos, and in the form involving a site for a Grothendieck topology, is very welcome. As an application, Brouwer's continuity theorem ("every total real function is continuous") is shown to hold in the "gros topos" of sheaves over spaces with the open cover topology.
The last four chapters of the book deal with geometric morphisms. Chapter VII discusses, among other things, the connection between maps of sites and geometric morphisms between the corresponding sheaf topos, and the factorization of a geometric morphism into a surjection and an inclusion. Chapter VIII on "classifying topoi" is the most characteristic chapter of the book: it is a detailed study of several examples and classes of examples, including torsors, the Zariski topos as the classifying topos for local rings, and the topos of simplicial sets as that for linear orders (with endpoints). The chapter on "localic topoi" contains the classic embedding theorems of Barr and Deligne. There is a return to logic in the form of geometric logic in the last chapter.
Giraud's theorem, the abstract characterization of sheaf-toposes, is given in the appendix.
\{Reviewer's remarks: The book is a very welcome addition to the literature. Its approach to exposition is the direct opposite to that of the "classic" in the field: P. T. Johnstone's Topos theory [Academic Press, London, 1977; MR 57 #9791]. The latter is an extremely economical, concise book; as a consequence, it is not easy reading, although mastering it has its ample rewards. The present book, through its detailed discussions of points dealt with often in brief asides in Johnstone's book, through its many centrally important examples, and through its generally more friendly character will be a great help in the learning of the subject. The subjects treated in Johnstone's book which the present one does not cover (e.g., the Mitchell-Diaconescu generalization of Giraud's theorem) mostly belong to topos theory over a given but "arbitrary" base topos. I recommend the use of both books in a serious study of topos theory.
\{I feel that the present book would have been enhanced by a more carefully documented history. The results and proofs are rarely attributed, and historical remarks are essentially confined to the prologue and the epilogue. Also, in view of the fact that the book is a "first introduction", a somewhat more detailed indication of the actual state of the theory would have been useful.\}
The first volume (Basic category theory) starts with traditional topics: categories, functors, natural transformations, followed by chapters on limits and colimits, and on adjoint functors. But other more modern points of view intrude; for example, the tensor product of set-valued functors is defined in connexion with Kan extensions well in advance of the bicategory of profunctors. There is a technical chapter covering generators and projective objects, together with a little on factorization systems. The heart of the book is a thorough treatment of categories of fractions. As part of this development one gets orthogonality, factorization systems, and the connexion with reflective subcategories, all in a general form; the case of localisations (reflections which preserve finite limits) is given special attention, and the equivalences with formulations in terms of closure operations and in terms of bidense morphisms is spelt out in detail. Presheaf categories have not received much attention at this point, but this is remedied by a brief section which gives information on flat functors and on the Cauchy completion (the name deriving from an example in enriched category theory) or Karoubi envelope of a category. Two miscellaneous chapters follow. Chapter 7 is on 2-categories and the more general bicategories; the bicategory of profunctors (distributors) provides a good context in which to consider (amongst other topics) Cauchy completeness. The general theory of bicategories is not considered in detail, and there is still no accessible source where one can find the definitions of bicategory, morphism of bicategory, transformation of morphisms and modification together. Chapter 8 is on internal category theory, a topic not needed elsewhere in the volumes, though it is essential to topos theory over an arbitrary base, for which the best reference remains P. T. Johnstone's Topos theory [Academic Press, London, 1977; MR 57 #9791].
The second volume (Categories and structures) starts with two long chapters on abelian categories and on regular categories. The first chapter is a useful complement to P. Freyd's influential Abelian categories [Harper & Row, New York, 1964; MR 29 \#3517]; the treatment of localization is very elegant. The second is a less intense substitute for the section on regular categories in the book by Freyd and A. Scedrov [Categories, allegories, North-Holland, Amsterdam, 1990; MR 93c:18001]. The third chapter, on algebraic (that is, equational) theories, expounds a simple point of view on universal algebra due to Lawvere, and ends with a succinct discussion of standard Morita theory. The general categorical perspective on algebra follows in a thorough treatment of monads concluding with "A glance at descent theory". This deals with the case of ring homomorphisms only; more general settings (for example, those related to Tannaka duality) are not considered. Next on the agenda is the generalization from equational to essentially equational or finite limit theories. The categories of models for these are the finitely presentable categories, the theory of which can be generalised leading to the notion of accessible category. The treatment is succinct, but fuller accounts can be found in books by M. Makkai and R. Pare [Accessible categories: the foundations of categorical model theory, Contemp. Math., 104, Amer. Math. Soc., Providence, RI, 1989; MR 91a:03118] and J. Adamek and J. Rosicky [ Locally presentable and accessible categories, Cambridge Univ. Press, Cambridge, 1994; MR 95j:18001]. The remainder of the second volume is devoted to slight treatments of an assortment of special topics. First, enriched category theory, for which the basic reference remains G. M. Kelly's Basic concepts of enriched category theory [Cambridge Univ. Press, Cambridge, 1982; MR 84e:18001]. However, Borceux does give a brief account of change of base, which is not covered by Kelly. Secondly, topological categories. This chapter is mainly concerned to give the definition and main elementary properties of the Cartesian closed category of compactly generated spaces, a convenient category in particular as a basis for homotopy theory. Finally, fibrations. A categorical fibration is a precise abstraction of the notion of a category indexed over a category. There is no full treatment in the literature of this widely applied notion, so the sketch here is welcome.
The third volume (Categories of sheaves) starts with a succinct account of locale theory. While Johnstone's 1982 book [Stone spaces, Cambridge Univ. Press, Cambridge, 1982; MR 85f:54002] remains the best general reference for this area, Borceux does deal with open and etale maps, and with local compactness. All this is a prelude to a detailed treatment of categories of sheaves on a locale; this covers the traditional categories of sheaves on a topological space, as well as the Boolean-valued models of set theory. Next, sheaves on locales are generalised to sheaves on an arbitrary site (which include categories sets with a continuous group action), and the resulting Grothendieck toposes are studied using the general techniques related to localization from Volume 1. The treatment is thus an alternative to that in the book by S. Mac Lane and I. Moerdijk [Sheaves in geometry and logic, corrected reprint, Springer, New York, 1994; MR 96c:03119]. The elegant notion of classifying topos follows. Every Grothendieck topos is a classifying topos: the points of the topos (wherever they be taken) are the models for the theory classified. Borceux uses the sketch point of view, and makes the connection with the general material on accessible categories. At this point there is a change of flavour with the introduction of elementary toposes. The treatment is a little gentler than that of Johnstone's Topos theory [op. cit.], and after the basic material we are at once (and in great detail) given the internal logic of toposes. The detail will be welcomed by some, but may mislead others into thinking that the subject is dull. The volume ends with three disparate chapters, one on Boolean and De Morgan toposes, one on the axiom of infinity, and the final one on sheaves for a topology on an elementary topos.
The choice and emphasis of material in the three volumes is a personal and fairly conservative one. Expert readers may well quibble with aspects of the whole, but almost all will find chapters of value. But above all these volumes will be of enormous value to graduate students in pure or applied category theory.