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The Topos of Music is the upgraded and vastly deepened English extension of the seminal German Geometrie der Töne. It reflects the dramatic progress of mathematical music theory and its operationalization by information technology since the publication of Geometrie der Töne in 1990. The conceptual basis has been vastly generalized to topos-theoretic foundations, including a corresponding thoroughly geometric musical logic. The theoretical models and results now include topologies for rhythm, melody, and harmony, as well as a classification theory of musical objects that comprises the topos-theoretic concept framework. Classification also implies techniques of algebraic moduli theory. The classical models of modulation and counterpoint have been extended to exotic scales and counterpoint interval dichotomies.

The probably most exciting new field of research deals with musical performance and its implementation on advanced object-oriented software environments. This subject not only uses extensively the existing mathematical music theory, it also opens the language to differential equations and tools of differential geometry, such as Lie derivatives. Mathematical performance theory is the key to inverse performance theory, an advanced new research field which deals with the calculation of varieties of parameters which give rise to a determined performance. This field uses techniques of algebraic geometry and statistics, approaches which have already produced significant results in the understanding of highest-ranked human performances.

The book's formal language and models are currently being used by leading researchers in Europe and Northern America and have become a foundation of music software design. This is also testified by the book's nineteen collaborators and the included CD-ROM containing software and music examples.

Customer Reviews (2)

A massive and massively impressive work
Category theory meets music meets cognitive theory. There's a huge conceit here, that the most powerful tools of mathematics can bring out further understanding of music. And I was unsure of it to start. But there's a payout here. The math illuminates musical structures in an elegant and deep manner. Along the way there's nearly as much to say about brain function and cognition as music. (Though they're clearly closely related.)

I'd say that the book's audience is necessarily limited, but that for those who can safely approach the book, it's nothing less than a remarkable achievement. I liken it to when, as a physics graduate student, I first encountered Arnold's Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics). Mathematical elegance applied to the world.

The shotgun approach to deep musical structure
A pompous and confused application of category theory to music.

The math is advanced, and so is the obscurity of the encyclopedic and voluminous prose. This is a pity, since Mazzola makes a number of fascinating and deep observations. Fighting through the dense, barely edited, text to find these gems, however, is an activity best left to the most masochistic of PhD candidates in music theory, especially given the heft and price of the book. ... Read more

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This book is an introduction to the theory of toposes, as first developed by Grothendieck and later developed by Lawvere and Tierney. Beginning with several illustrative examples, the book 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. This is the first text to address all of these various aspects of topos theory at the graduate student level. ... Read more

Customer Reviews (2)

Excellent
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.

Clear explicit descriptions
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 havebasic graduate knowledge of algebra and topology. The book is long becauseit gives very explicit descriptions of many advanced topics--you can learna great deal from this book that, before it was published, you could onlylearn by knowing researchers in the field. ... Read more

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Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Now available in this two volume set, it contains all the important information both volumes provides. Considered to be a complete benefit for all researchers and academics in theoretical computer science, logicians and philosophers who study the foundations of mathematics, and those working in differential geometry and continuum physics. ... Read more

Customer Reviews (1)

An easy call
Very simply: if you want to know a very great deal about topos theory, buy this book. I mean, seriously, if you plan to make real work on topos theory a part of your life, then grit your teeth and come up with the money. If you do not want to know a very great deal about it, do not buy this book. You can use it at the library as a reference.

If you merely want a professional understanding of what topos theory is, then read Johnstone's earlier TOPOS THEORY. That far shorter book gives a better overview. My Amazon review of it discusses others on the subject. Most are more accessible than Johnstone's books and go more into particular aspects of the theory.

This book is a reference on all the methods, and the latest results, in topos theory. If you want the definition of "split opfibration", it is here, along with some 80 pages of background, examples, and motivation. Johnstone does an heroic job of unifying the terminology and organizing the theorems.

More than that, Johnstone has written down an expert, encyclopedic view of the subject today. It is rare for a top mathematical researcher to give so deep an account of their field. It is rare for anyone to even work out such an explicit, coherent, extensive account of the whole. Not everyone will agree with his view. Some would like to see much less of such logical topics as "allegories", others would like to see the logic more formalized from the start. But Johnstone builds a case for his choices: partly implicit in his success at explaining things this way, and partly by explicit reasons.

If you want to know that much about the subject then you want to immerse yourself in this book. ... Read more

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Still the standard text
You understand topos theory when you understand this book. I cannot say the same of any other.

Mac Lane and Moerdijk SHEAVES IN GEOMETRY AND LOGIC is a beautifully written book, a long and well motivated book packed with well chosen clearly explained examples. Both those authors have a rare gift for conveying an insider's view of the subject from the start. My own book ELEMENTARY CATEGORIES, ELEMENTARY TOPOSES goes quickly and simply to the key ideas (if I say so myself). Barr and Wells TOPOSES, TRIPLES, AND THEORIES uses the most efficient tools to get at the central theorems.

But no other book goes as concisely and comprehensively to all the aspects of toposes as this one. Category theory, algebra, logic, arithmetic, geometry, and cohomology all come in, in a well chosen perspective. This book is hard to read in places, especially at the start. It cannot serve alone as an introduction unless you are really gifted. But it remains the best single text on the subject.

Johnstone has a three-volume set on the current state of topos theory due to appear later this year. It may well become the standard reference. But it will not replace this book.

A classic book on topos theory
This book, printed in 1978, is still an essential, I would say,unavoidable, part of a toposopher education. You have to read it rightafter S. MacLane's "Categories for the Working Mathematician". Ihave read the book three times. ... Read more

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Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas.Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and to thereby demonstrate the overall unity of the subject. The material is organized in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text. ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as l-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary topological space, and Grothendieck's theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety, this is completely accurateit is possible to recover a sober space from its associated site. However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies. Conversely, there are Grothendieck topologies which do not come from topological spaces. André Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they defined. His conjectures postulated that there should be a cohomology theory of algebraic varieties which gave numbe... More: http://booksllc.net/?id=12910 ... Read more

Customer Reviews (1)

Great book; out of print.
This is a very readable introduction to the subject. Too bad it's out of print. ... Read more

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Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientsts, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continum physics.This compendium contains material that was previously available only in specialist journals.This is likely to become the standard reference work for all those interested in the subject. ... Read more

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The road stretched in a pale, straight streak, narrowing to a mere thread at the limit of vision--the only living thing in the wild darkness. All was very still. It had been raining; the wet heather and the pines gave forth scent, and little gusty shivers shook the dripping birch trees. In the pools of sky, between broken clouds, a few stars shone, and half of a thin moon was seen from time to time, like the fragment of a silver horn held up there in an invisible hand, waiting to be blown. ... Read more

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"The road stretched in a pale, straight streak, narrowing to a mere thread at the limit of vision--the only living thing in the wild darkness. All was very still. It had been raining; the wet heather and the pines gave forth scent, and little gusty shivers shook the dripping birch trees. In the pools of sky, between broken clouds, a few stars shone, and half of a thin moon was seen from time to time, like the fragment of a silver horn held up there in an invisible hand, waiting to be blown." ... Read more

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High Quality Content by WIKIPEDIA articles! In category theory, a regular category is a category with finite limits and coequalizers of kernel pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images, without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic. ... Read more