##
**Sheaves in geometry and logic: a first introduction to topos theory.**
*(English)*
Zbl 0822.18001

Universitext. New York etc.: Springer-Verlag. xii, 627 p. (1992).

The summer of 1995 marked two anniversaries. The first is that of the 1945 paper of Eilenberg and MacLane which formally introduced the terms “category”, “functor”, and “natural transformation” to the world of mathematics [cf. S. Eilenberg and S. MacLane, Trans. Am. Math. Soc. 58, 231–294 (1945; Zbl 0061.09203)].

From the perspective of fifty years later, it seems quite safe to say that the 1945 paper, along with the ideas and seemingly endless supply of examples which flowed from it, changed forever the working vocabulary of mathematics. For some, the change was, and still is, viewed much as classically trained physicists viewed the “gruppenpest”: “Ignore it and perhaps it will go away”. For many, at least those who still see it as merely giving a formal meaning to Emmy Noether’s 1930’s then lonely and revolutionary, “Don’t just study the Betti numbers, but rather the groups and the boundary homomorphisms which give rise to them!”, it has only somewhat trivially changed the way that they have long studied and communicated “modern” mathematics. But for a smaller and younger group, that 1945 paper provoked a change not just in terminology or even methodology, but in their entire philosophical approach to mathematics.

To see how profound that change could be after only twenty-five years, one needs only to look at the second of these 1995 anniversaries, that of the 1970 International Congress of Mathematicians in Nice where Lawvere introduced his joint work with Tierney and the concept of an “elementary topos” [cf. F. W. Lawvere, Actes Congr. Int. Math. 1970, 1, 329–334 (1971; Zbl 0261.18010)]. Still in active development, the ideas embodied in the non set-theoretic, categorical language of “elementary topos theory” not only incorporate the already extraordinary set-based categorical view of geometry (as abstract sheaf theory) pioneered by the Grothendieck school, but even lead to a totally new way of looking at the logical foundations of mathematics itself.

Up to now the only truly comprehensive text treating both of these two quite different (“geometric” and “logical”) aspects of the theory was P. T. Johnstone’s “Topos theory”. London etc.: Academic Press (1977; Zbl 0368.18001). Unfortunately, at least for the beginner, it assumes from the outset a considerable and sophisticated knowledge of category theory and is written in a concise, telegraphic style which makes it superb as a reference but probably overwhelming to all but the most dedicated.

The text under review here, a collaborative effort of a world famous mathematician and expositor of mathematics coupled with one of the most active and gifted contributors to the field, is a dedicated attempt to fill this expository void with an up-to-date work, still introducing all major aspects of the subject, but designed to appeal to both the neophyte graduate student and the interested but non-specialist working mathematician. This is not at all an easy task and the full measure of their success will best be judged by their intended audience. But the material, in very large part, is there at the appropriate introductory level, and the care is evident. Especially at the beginning, the authors are acutely conscious of the fact that most readers are not going to have any in-depth knowledge of category theory and are likely to quickly loose interest if they are immediately faced with learning a huge amount of unmotivated preliminary material before they can even start. They have attempted to obviate this by either recalling the relevant definitions in the context as it arises and reproving a theorem which could have been cited as a corollary of a more general categorical one or, in case that this is really infeasible, have provided a direct page reference to the most widely available general text, S. MacLane’s “Categories for the working mathematician” [(1971; Zbl 0232.18001), reprint of the 3rd corr. print. 1975 (1995; Zbl 0818.18001)]. Occasionally, and quite beautifully, after a detailed in-context(!) proof of a specific theorem has been given, the authors demonstrate how the just proved theorem can be seen as a corollary of a really useful but purely categorical result. Students taught with this type of superior pedagogy are unlikely to forget either fact. This is also noteworthy in the authors lucid explanations of such things as the often confusing contrast of “internal” and “external” language and “local and global elements” often used in category theory and crucially important in the theory of topoi.

The overall style of the book is probably best described as “peripatetic”. Concepts and examples are introduced, partially developed in one context, and then returned to and amplified at a later point in perhaps another context. Although this is pedagogically quite sound and keeps the reader from being overwhelmed, in contrast to Johnstone, it has the effect of occasionally making the book quite maddening to use purely as a reference.

But then, in spite of its length (over 600 pages), it is not intended to be encyclopedic: the homological and homotopical algebra of topoi is deliberately (although “reluctantly”) omitted; the role of topos theory in theoretical computer science is not even mentioned, except as declared “not really relevant, at least for our purposes”, and some of its aspects most interesting to logicians, for instance those concerning recursive realizability and “the effective topos” only appear in passing mention. Inexplicably, to this reviewer at least, the counterpart of the classical fiber bundle (étale space) picture of topological sheaf theory, the theory of Grothendieck fibrations, never appears even in the exercises. Instead, for these and other omitted topics, an “Epilog” in the form of an annotated guide to further reading is provided.

What is provided in the text, however, is broad based and may roughly be described as five chapters devoted to “geometry” and five chapters devoted to “logic” although readers may disagree as to “which is which”, since this if often more a matter of the reader’s reaction to perceived differences in conventions of style than to actual differences in content.

In the first “geometric” part, it follows a roughly historical path: sheaves as fiber bundles (étale spaces), then sheaves for Grothendieck topologies on sites, and then elementary topoi and sheaves for Lawvere- Tierney topologies. The associated sheaf functor is constructed in the description at hand and its properties are developed. At each stage one or more examples are given to illustrate some of the many possible applications of the subject. The depth of these varies considerably: “sheaves and manifolds” will probably only serve to whet the reader’s appetite, whereas the Zariski site of finitely presented \(k\)-algebras is presented with sufficient elementary algebraic geometry background to make it really interesting to the beginner; continuous group actions on (discrete) topological spaces is a recurring and favorite example.

Chapter VI, “Topoi and Logic”, marks the rough transition to the applications of topos theory to logic and model theory and introduces the very real contributions of Lawvere and Tierney to the latter in showing how elementary topoi can be used as models for set theory and indeed replace sets as a foundation. That “Cohen forcing” is really topology in this new and broad sense is a remarkable result which deserves a much wider audience. Professional logicians may dislike the informality with which “the formal intuitionistic logic of elementary topoi” is treated in this text, but for those of us who have always wondered what Brouwer could have meant by “all functions are continuous”, the authors give a delightful (and quite concrete) topos-theoretic model of set theory in which this is true.

Other contributions to model theory which primarily explicate the view of Grothendieck topoi and geometric morphisms as providing “classifying topoi” for models of “geometric theories” occupy much of the remainder of the book. The very general theory, which would be book-length in its own right, is not covered in any real detail except for a brief exposition in the last chapter. Instead, the authors have chosen to take several concrete examples of easily understood and/or well known topoi and, in detail, show how the idea works. Thus the presheaf topos on the category of finitely presented rings as the classifying topos for ring objects serves as a typical “algebraic” example, while the Zariski topos for local rings and the much less obvious examples of simplicial sets for linear orders and \(G\)-sets for \(G\)-torsors (principal homogeneous spaces) serve to illustrate the depth of the idea.

An introduction to the theory of “locales” and “localic topoi”, where much of the current research in the field is concentrated, is given the chapter next to the end and represents, along with “geometric theories”, the major part of the subject not covered in Johnstone’s 1977 book. Amusingly, “Giraud’s theorem” which gave the first “recognition criteria” for Grothendieck topoi as abstract categories is given at the beginning of Johnstone’s book and as an appendix in MacLane-Moerdijk! But then this book was written not for the experts, but for almost everybody else.

From the perspective of fifty years later, it seems quite safe to say that the 1945 paper, along with the ideas and seemingly endless supply of examples which flowed from it, changed forever the working vocabulary of mathematics. For some, the change was, and still is, viewed much as classically trained physicists viewed the “gruppenpest”: “Ignore it and perhaps it will go away”. For many, at least those who still see it as merely giving a formal meaning to Emmy Noether’s 1930’s then lonely and revolutionary, “Don’t just study the Betti numbers, but rather the groups and the boundary homomorphisms which give rise to them!”, it has only somewhat trivially changed the way that they have long studied and communicated “modern” mathematics. But for a smaller and younger group, that 1945 paper provoked a change not just in terminology or even methodology, but in their entire philosophical approach to mathematics.

To see how profound that change could be after only twenty-five years, one needs only to look at the second of these 1995 anniversaries, that of the 1970 International Congress of Mathematicians in Nice where Lawvere introduced his joint work with Tierney and the concept of an “elementary topos” [cf. F. W. Lawvere, Actes Congr. Int. Math. 1970, 1, 329–334 (1971; Zbl 0261.18010)]. Still in active development, the ideas embodied in the non set-theoretic, categorical language of “elementary topos theory” not only incorporate the already extraordinary set-based categorical view of geometry (as abstract sheaf theory) pioneered by the Grothendieck school, but even lead to a totally new way of looking at the logical foundations of mathematics itself.

Up to now the only truly comprehensive text treating both of these two quite different (“geometric” and “logical”) aspects of the theory was P. T. Johnstone’s “Topos theory”. London etc.: Academic Press (1977; Zbl 0368.18001). Unfortunately, at least for the beginner, it assumes from the outset a considerable and sophisticated knowledge of category theory and is written in a concise, telegraphic style which makes it superb as a reference but probably overwhelming to all but the most dedicated.

The text under review here, a collaborative effort of a world famous mathematician and expositor of mathematics coupled with one of the most active and gifted contributors to the field, is a dedicated attempt to fill this expository void with an up-to-date work, still introducing all major aspects of the subject, but designed to appeal to both the neophyte graduate student and the interested but non-specialist working mathematician. This is not at all an easy task and the full measure of their success will best be judged by their intended audience. But the material, in very large part, is there at the appropriate introductory level, and the care is evident. Especially at the beginning, the authors are acutely conscious of the fact that most readers are not going to have any in-depth knowledge of category theory and are likely to quickly loose interest if they are immediately faced with learning a huge amount of unmotivated preliminary material before they can even start. They have attempted to obviate this by either recalling the relevant definitions in the context as it arises and reproving a theorem which could have been cited as a corollary of a more general categorical one or, in case that this is really infeasible, have provided a direct page reference to the most widely available general text, S. MacLane’s “Categories for the working mathematician” [(1971; Zbl 0232.18001), reprint of the 3rd corr. print. 1975 (1995; Zbl 0818.18001)]. Occasionally, and quite beautifully, after a detailed in-context(!) proof of a specific theorem has been given, the authors demonstrate how the just proved theorem can be seen as a corollary of a really useful but purely categorical result. Students taught with this type of superior pedagogy are unlikely to forget either fact. This is also noteworthy in the authors lucid explanations of such things as the often confusing contrast of “internal” and “external” language and “local and global elements” often used in category theory and crucially important in the theory of topoi.

The overall style of the book is probably best described as “peripatetic”. Concepts and examples are introduced, partially developed in one context, and then returned to and amplified at a later point in perhaps another context. Although this is pedagogically quite sound and keeps the reader from being overwhelmed, in contrast to Johnstone, it has the effect of occasionally making the book quite maddening to use purely as a reference.

But then, in spite of its length (over 600 pages), it is not intended to be encyclopedic: the homological and homotopical algebra of topoi is deliberately (although “reluctantly”) omitted; the role of topos theory in theoretical computer science is not even mentioned, except as declared “not really relevant, at least for our purposes”, and some of its aspects most interesting to logicians, for instance those concerning recursive realizability and “the effective topos” only appear in passing mention. Inexplicably, to this reviewer at least, the counterpart of the classical fiber bundle (étale space) picture of topological sheaf theory, the theory of Grothendieck fibrations, never appears even in the exercises. Instead, for these and other omitted topics, an “Epilog” in the form of an annotated guide to further reading is provided.

What is provided in the text, however, is broad based and may roughly be described as five chapters devoted to “geometry” and five chapters devoted to “logic” although readers may disagree as to “which is which”, since this if often more a matter of the reader’s reaction to perceived differences in conventions of style than to actual differences in content.

In the first “geometric” part, it follows a roughly historical path: sheaves as fiber bundles (étale spaces), then sheaves for Grothendieck topologies on sites, and then elementary topoi and sheaves for Lawvere- Tierney topologies. The associated sheaf functor is constructed in the description at hand and its properties are developed. At each stage one or more examples are given to illustrate some of the many possible applications of the subject. The depth of these varies considerably: “sheaves and manifolds” will probably only serve to whet the reader’s appetite, whereas the Zariski site of finitely presented \(k\)-algebras is presented with sufficient elementary algebraic geometry background to make it really interesting to the beginner; continuous group actions on (discrete) topological spaces is a recurring and favorite example.

Chapter VI, “Topoi and Logic”, marks the rough transition to the applications of topos theory to logic and model theory and introduces the very real contributions of Lawvere and Tierney to the latter in showing how elementary topoi can be used as models for set theory and indeed replace sets as a foundation. That “Cohen forcing” is really topology in this new and broad sense is a remarkable result which deserves a much wider audience. Professional logicians may dislike the informality with which “the formal intuitionistic logic of elementary topoi” is treated in this text, but for those of us who have always wondered what Brouwer could have meant by “all functions are continuous”, the authors give a delightful (and quite concrete) topos-theoretic model of set theory in which this is true.

Other contributions to model theory which primarily explicate the view of Grothendieck topoi and geometric morphisms as providing “classifying topoi” for models of “geometric theories” occupy much of the remainder of the book. The very general theory, which would be book-length in its own right, is not covered in any real detail except for a brief exposition in the last chapter. Instead, the authors have chosen to take several concrete examples of easily understood and/or well known topoi and, in detail, show how the idea works. Thus the presheaf topos on the category of finitely presented rings as the classifying topos for ring objects serves as a typical “algebraic” example, while the Zariski topos for local rings and the much less obvious examples of simplicial sets for linear orders and \(G\)-sets for \(G\)-torsors (principal homogeneous spaces) serve to illustrate the depth of the idea.

An introduction to the theory of “locales” and “localic topoi”, where much of the current research in the field is concentrated, is given the chapter next to the end and represents, along with “geometric theories”, the major part of the subject not covered in Johnstone’s 1977 book. Amusingly, “Giraud’s theorem” which gave the first “recognition criteria” for Grothendieck topoi as abstract categories is given at the beginning of Johnstone’s book and as an appendix in MacLane-Moerdijk! But then this book was written not for the experts, but for almost everybody else.

Reviewer: J.Duskin (Buffalo)

### MSC:

18-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

18B25 | Topoi |

03G30 | Categorical logic, topoi |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03E99 | Set theory |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

54B40 | Presheaves and sheaves in general topology |