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Review by: Kenneth P Bogart. Monthly 78 1 , Professors Mac Lane and Birkhoff have once again set the pace and tone for instruction in abstract algebra for some time to come. I chose their book for my introductory course in abstract algebra for honours majors at Dartmouth.


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Since these students have a personal commitment to mathematics and frequently go on to graduate work, I felt strongly that their introduction to algebra should be couched in modern terminology and that their course should include relatively new algebraic concepts e. When I chose a text, I felt this one was the only one that satisfied these requirements.

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Although I strongly urge undergraduates who already know some algebra and first year graduate students to study this book carefully, I do not recommend it for an introductory course. Until we find a good way of teaching sophistication and sophisticated ideas in freshman calculus I am not sure we even should , students will have too high a hurdle to jump in starting this book "cold. Review by: Wilhelm Magnus. Mathematics of Computation 22 , In spite of the similarity of the titles and the coincidence of the names although not the sequence of names of the authors, this is not a new edition of the 'Survey of Modern Algebra' Macmillan Co.

The motivation for it is summarized in the first paragraph of the Preface: "Recent years have seen striking developments in the conceptual organization of mathematics. These developments use certain new concepts such as 'module,' 'category,' and 'morphism' which are algebraic in character and which indeed can be introduced naturally on the basis of elementary materials. The efficiency of these ideas suggests a fresh presentation of algebra.

But most of these chapters are used now also for the purpose of introducing and illustrating the concepts of "category," "functor," and "universal element" which, in the penultimate chapter on 'Categories and adjoint functors' become the main topic of the book. Review by: Ancel C Mewborn. Monthly 74 10 , It is natural to compare this book with the authors' A Survey of Modern Algebra since it is directed to the same audience. Most of the topics covered in the earlier book appear in some form in the present book; there is one notable exception: Galois Theory is omitted.

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Despite the large intersection of topics covered in Survey and in Algebra , the new book is not just an updating of the old. The treatment of many topics and the general tone differ greatly from that of Survey.

The notation and terminology are "categorical," as are many proofs. Universal properties and duality are introduced in the first chapter and play an essential role throughout most of the book. Many concepts are defined in terms of, or are related to, universal mapping properties; and these properties are used when the concepts are applied. Review by: Charles W Curtis. Mathematical Reviews MR 35 The concrete examples which underlie undergraduate algebra courses - integers, real and complex numbers, polynomials, vectors, matrices, determinants - are neither modern nor abstract.

A textbook in which these objects are presented becomes a book on "modern algebra'' by the kind of scaffolding it erects to exhibit these objects. An earlier generation of mathematicians learned, for example in the authors' A survey of modern algebra ; ; that these objects are examples of groups, rings, fields, and vector spaces, and saw that the standard constructions in algebra often reduce to finding the right equivalence relation.

It is refreshing to see in this book the standard material of undergraduate algebra presented systematically from a new point of view. This time the axiomatic systems and equivalence relations are there, but are governed in turn by the language of categories, functors, universal objects, and dualities.

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From the Preface of the 2nd edition. The treatment of many of the topics in the first edition has been simplified - and clarified - in this second edition. The material on universal constructions, formerly introduced at the end of the first chapter, has now been assembled in Chapter IV, at a point where there are at hand many more effective examples of these constructions. A great many points in the exposition have been clarified, for instance in a simpler construction of the integers, a more elementary description of polynomials, and a more direct treatment of dual spaces.

The chapter on special fields now includes power series fields and a treatment of the p-adic numbers. There is a wholly new chapter on Galois theory; in exchange, the chapter on affine geometry has been dropped. New exercises have been added and some old slips have been excised. Review of 3rd edition by: David Singmaster. The Mathematical Gazette 75 , Birkhoff and Mac Lane's A Survey of Modern Algebra appeared in and made the ideas of modern algebra accessible to several generations of graduate and undergraduate students.

The first edition of Algebra appeared in and was intended to incorporate the new ideas of algebra, such as module, tensor product, category and morphism.

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I well remember the surprise of opening the first edition and finding Chapter I was Sets, Functions, and Universal Elements, with commutative diagrams already on p 7 and the universal mapping property of the cartesian product on p The logical sequence of the rest of the text was fairly natural: integers, groups, rings, fields, modules, vector spaces and linear algebra, structure of groups, lattices, categories, multilinear algebra - the novelty of the work was the use of the categorial viewpoint from the beginning.

In the second edition of , the authors reorganised their approach, changing the first chapter to Sets, Functions, and Integers and deferring the introduction of categorical ideas until Chapter IV. This seemed pedagogically sounder, as then the reader will have a supply of examples on which to base the generalisations.

The chapters were somewhat reordered and the chapter on affine and projective spaces was replaced by one on Galois theory. In the present third edition, a number of misprints have been corrected, but the main text is otherwise identical to the second edition. Category theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by mathematicians working in a variety of other fields of mathematical research. This occurs at several levels.

On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows.

The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid - a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of generalized monoid. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure given by a tensor product lead inter alia to the study of more convenient categories of topological spaces.

Review by: Alex Heller. American Scientist 61 3 , This is a valuable book. Category theory has developed vigorously over a quarter of a century and has become an indispensable tool in many parts of mathematics. Now one of its cofounders presents us with a definitive synthesis designed to furnish to the working mathematician, i.

This is a far-from-trivial undertaking. The main ideas of category theory are interconnected in a remarkably complicated way.


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  2. SHEAVES IN GEOMETRY AND LOGIC - A FIRST INTRODUCTION TO TOPOS THEORY.
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Thus the cycle of notions - monoid, category, monad, algebra - might be explicated starting. The same is true of the cycle-limit, universal construction, representable functor, adjoint functor.


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The relations between these cycles are more complex still. Professor Mac Lane has provided us not only with a clue through this labyrinth, but with a synopsis of its plan. From the Introduction. This book is intended to describe the practical and conceptual origins of Mathematics and the character of its development - not in historical terms, but in intrinsic terms. Thus we ask: What is the function of Mathematics and what is its form? In order to deal effectively with this question, we must first observe what Mathematics is. Hence the book starts with a survey of the basic parts of Mathematics, so that the intended general questions can be answered against the background of a careful assembly of the relevant evidence.

In brief, a philosophy of Mathematics is not convincing unless it is founded on an examination of Mathematics itself. Wittgenstein and other philosophers have failed in this regard. Review by: Penelope Maddy. The Journal of Symbolic Logic 53 2 , The jacket blurb for Mac Lane's encyclopedic survey characterizes it as "a background for the philosophy of Mathematics. Instead, the reader is treated to a masterful guided tour of the subject, from the natural, rational, ordinal, and cardinal numbers, Euclidean and non-Euclidean geometries, real and complex numbers, functions, transformations, groups, and the calculus, to linear algebra, differential geometry, mechanics, complex analysis, topology, set theory, logic, and category theory.

These elegant and concise chapters clearly benefit from Mac Lane's own lifetime of productive engagement with his subjects. Appearances aside, however, this is not a book of popularized mathematics. The presentation is too dense for the uninitiated amateur; there is little concern for pedagogy, few concessions to the reader's fatigue or attention span, and little comic or other relief.

History and anecdote are conspicuously absent, with a few notable exceptions, including a delightful ode on the Riemann conjecture, set to the tune of 'Sweet Betsy from Pike'. Review by: Stephen L Bloom. This is a fascinating book.

Its aim is to "capture in words a description of the form and function of Mathematics as a background for the Philosophy of mathematics". In brief, the plan of the book is to give a rather detailed summary of a good part of current topics in Mathematics, and use this material to evaluate various philosophical claims. It is claimed that the reader need have only "some acquaintance with Mathematics", but the level of sophistication needed to appreciate the discussion varies a great deal.

Sheaves in Geometry and Logic : A First Introduction to Topos Theory

However, for a mathematician, most of the discussions will be delightful, and informative. Although motivated by philosophical concerns, the book is best read as a source for excellent descriptions of actual mathematical fields of practice, of the roots of the major problems that motivate many areas of research, and of revelations of many interconnections among the various branches of Mathematics. Review by: F Gareth Ashurst.