Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Universitext) by Saunders MacLane (PDF)

Description

Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds. Sheaves also appear in logic as carriers for models of set theory. This text presents topos theory as it has developed from the study of sheaves. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.

User’s Reviews

Editorial Reviews: Review From the reviews:”A beautifully written book, a long and well motivated book packed with well chosen clearly explained examples. … authors have a rare gift for conveying an insider’s view of the subject from the start. This book is written in the best Mac Lane style, very clear and very well organized. … it gives very explicit descriptions of many advanced topics–you can learn a great deal from this book that, before it was published, you could only learn by knowing researchers in the field.” (Wordtrade, 2008)

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Topos theory now has applications in fields such as music theory, quantum gravity, artificial intelligence, and computer science. It has been viewed by some as being excessively abstract and difficult to learn, and this is certainly true if one attempts to learn it from the research literature. The use of this book to learn topos theory certainly puts this view to rest, as the authors have given the readers an introduction to topos theory that is crystal clear and nicely motivated from an historical point of view. Indeed the prologue to the book gives the reader a deep appreciation of the origins of the subject, and could even serve as an introduction to a class on algebraic geometry.An understanding of sheaf theory and category theory will definitely help when attempting to learn topos theory, but the book could be read without such a background. Readers who want to read the chapters on logic and geometric morphisms will need a background in mathematical logic and set theory in order to appreciate them. Topos theory has recently been used in research in quantum gravity. A reader interested in understanding how topos theory is used in this research should concentrate on the chapter on properties of elementary topoi, the one on basic categories of topoi, and the chapter on localic topoi.The authors introduce topos theory as a tool for unifying topology with algebraic geometry and as one for unifying logic and set theory. The latter application is interesting, especially for readers (such as this reviewer), who approach the book from the standpoint of the former. Indeed, the authors discuss a fascinating use of topos theory by Paul Cohen in his proof of the independence of the Continuum Hypothesis in Zermelo-Fraenkel set theory.The prologue for this book is excellent, and should be read for the many insights and motivations for the subject of topos theory. The elementary category theory needed is then outlined in the next section. A “topos” is essentially a category that allows the construction of pullbacks, products, and so on, with the philosophy being that objects are to be viewed not only as things but as also having maps (functors) between them. In the section on categories of functors, this viewpoint becomes very transparent due to the many examples of categories that are also topoi are discussed. These examples are presented first so as to motivate the general definition of topos later on. Some of these categories are very familiar, such as the category of sets, the category of all representations of a fixed group, presheaves, and sheaves. Of particular interest in this section is the discussion of the propositional calculus, and its representation as a Boolean algebra. Replacing the propositional calculus with the (Heyting) intuitionistic propositional calculus results in a different representation by a Heyting algebra. From the standpoint of ordinary topology, the Heyting algebra is significant in that the algebra of open sets is not Boolean, i.e. the complement (or “negation”) of an open set is closed and not open in general Instead it follows the rules of a Heyting algebra. This type of logic appears again when considering the subobjects in the sheaf category, which have a “negation” which belong to a Heyting algebra. Thus topos theory is one that follows more than not the Brouwer intuitionistic philosophy of mathematics. Recently, research in quantum gravity has indicated the need for this approach, and so readers interested in this research will find the needed background in this part of the book.After a straightforward overview of how sheaf theory fits into the topos-theoretic framework, the authors also discuss the role of the Grothendieck topology in sheaf theory. This involves thinking of an open neighborhood of a point in a space as more than just a monomorphism of that neighborhood into the space (all the open neighborhoods thus furnishing a “covering” of the space). This need was motivated by certain constructions in algebraic geometry and Galois theory, as the authors explain in fair detail. A covering of a space by open sets is replaced by a new covering by maps that are not monomorphisms. Starting with a category that allows pullbacks, an indexed family of maps to an object of this category is considered. If for each object in this category one uses a rule to select a certain set of such families, called the coverings of the object under this rule, then ordinary sheaf theory can be used on these coverings. If one desires to drop the requirement that the category have pullbacks, this can be done by introducing a category that comes with such “covering families.” This is the origin of the Grothendieck topologies, wherein the indexed families are replaced by the sieves that they generate. A Grothendieck topology on a category is thus a function that assigns to each object in the category a collection of sieves on the object (this function must have certain properties which are discussed by the authors). Several examples of categories with the Grothendieck topologies are discussed, one of these being a complete Heyting algebra. Another example discussed comes from algebraic topology, via its use of the Zariski topology for algebraic varieties. The discussion of this example is brilliant, and in fact could be viewed as a standalone discussion of algebraic geometry.When considering the notion of the Grothendieck topology, the authors define the notion of a `site’, which is essentially a (small) category along with a Grothendieck topology on the category. They then show how to define sheaves on a site, which then form a category. A `Grothendieck topos’ is then a category which is equivalent to the category of sheaves on some site. The authors then show, interestingly, that a complete Heyting algebra can be realized as a subobject lattice in a Grothendieck topos.

⭐Several folks have already commented on how this volume provides an excellent formal introduction to category theory and topoi. I have nothing to add to those excellent reviews and encourage others to read them. What I wish to do here is comment on a somewhat tangential subject of my own interest, and that is the philosophy of logic.This book makes clear the very significant connections between logic, what I would call “general spatial reasoning,” and category theory. For anyone interested in the underlying core and structure of formal reasoning beyond the questionable dogmas of Russell-Frege proof-theoretic approaches, this book is an absolute must have. The proof-theoretic methods that have swamped the thinking about logic in most philosophical circles has seriously undermined our understanding of the relevant issues by blinding scholars to the genuine wealth of ideas that exists within mathematics. I would argue that this book, in conjunction with such works as Corry’s “Rise of Modern Algebra” (see inserted link) is a fundamental step away from the shackles of the Russell-Frege vision of formal logic that dominates so much thought in the philosophy of logic. (Corry’s work places algebraic thinking within an historical context that the mere formal study of the subject tramples right over. Such historical context is an essential element in the philosophical — as opposed to purely formal — study of such topics.)The materials in _Sheaves_ are presented in an accessible way for the non-mathematician, *provided* that person still has a reasonably solid background in some such topic(s) as formal logic, model theory, abstract algebra, etc. The focus of the text on those relational structures known as “sheaves” provides an especially illuminating approach to the connections between algebraic logic, category theory, and such “purely” logical topics as proofs and models.Also, let me add that I am writing this review of the *Kindle* edition. Obviously the wood-pulp version is wonderful, but the eBook version is well formatted, with none of the crazy symbol and fornatting issues that dogged earlier attempts to migrate mathematical texts into an electronic format.So I recommend this book to anyone with even a passing interest in philosophical logic. The time has come to move beyond Russell-Frege, and this text is an excellent instrument for taking such a step!(Here is the link to Leo Corry’s magnificent study:)

⭐Clear, to the point, readable. Not too hard, not too easy. Assumes some general familiarity with categories, but not too much. Everything is developed at a hearty clip: plenty of curt examples; avoids tedium by getting to the point. Ideal if you’re mature and interested in mathematics in general, and have the time to set your own pace. Might be a bit challenging for inexperienced grad students on a forced march — or not. Doable.

⭐This is a really good book to learn sheaves, it has the “tree generations” of sheaves; with topological spaces, with Grothendieck topologies, and with Lawvere-Tirney topologies. Also a great book in topoi theory, from the basics up to the Cohen topos.

⭐I’ve just had occasion to return to this book after several years, and am astonished anew by how good it is (just as I was the last time I came back to it after several years). Normally one expects a tradeoff between clarity of exposition and depth of content, but in this case the authors manage to provide superlative levels of both. This is a great book.

⭐This book is written in the best Mac Lane style, very clear and very well organized. It also benefits from Moerdijk’s extensive work organizing the theory of Grothendieck toposes by elementary means. The reader should have basic graduate knowledge of algebra and topology. The book is long because it gives very explicit descriptions of many advanced topics–you can learn a great deal from this book that, before it was published, you could only learn by knowing researchers in the field.

⭐I thought that this book definitely deserved its own review on the French version of Amazon. I would not exactly recommend it as a first introduction to topos theory though, unless you already have solid intuitions in Category theory. Roughly speaking, the first half of the book focuses on functors and sheaves, while the second half deals with logic. The book is profound, the difficulty rises progressively, and it is superbly written. One closes the book with the feeling of having understood something deep and beautiful. 5 stars.独立命題（特に CH、AC）に興味があり、この本を手に取りました。まだ理解していない部分も多い（Chap. VII 以降は読んでもいない）のですが、途中まででもあまりに面白い本だったのでレビューを書こうと思いました。Chap. VI において CH や AC が成り立たないモデルを作るのですが、そこに至るまでの道のりはかなりボリュームがあります。しかし、そこで展開される主張や議論を説明しようと思うと、以前の章に書かれていた多くの内容が自ずと必要になってくるのが分かります。そうして分厚いように思えたこの本が、重要な内容がコンパクトにまとまった稀有な良書であると実感するようになりました。また、内容について確かに（程よく）難しいと感じる一方で、副題である「A First Introduction to Topos Theory」が嘘ではないことに気付きました。Categories for The Working Mathematician の和訳が「圏論の基礎」になったのが妥当だと感じられるのと同程度には「基礎的」な内容がしっかりと記述されており、topoi に関するより発展的な文献を読む際の力強い助けになってくれるだろうと感じました。オススメです！P.S. この本の内容がコンピュータと繋がるといった声があるみたいですが、自分にはどこをどう読めばそんな感想になるのか分かりませんでした。具体的には SGL 内のどの概念を指しているんでしょうか。感想文に記載されている「計算可能」という言葉から連想ゲームをしている気がしますが、そもそも計算可能（computable）という言葉はこの本には出てこないんですよね……。自分としてはトポス理論の基礎を身に付けたい人にお勧めします。

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