Customer Reviews (10)

very solid introduction
i used this text as my first algebra book. if you just began math and want to see a solid introduction to modern math this is it; self-contained and introduces many important ideas that are commonly used in modern math but typically not taught in undergraduate math courses.

Rock solid
I had the first edition of this book as a student about 30 years ago and found that although the text was for the most part readable, the notion of universal element notion from category theory was introduced in a baffling way very early on and this was the stumbling block.
I've been rereading mathematics books out of pleasure of late and wanted to read this one, so I looked up the Amazon reviews. It was written that you should get the third edition. I ordered it and was not disappointed. The stumbling block has been recognized by the authors and pushed to much further in the book. A new chapter on Galois Theory has been added.
The contents of this book is more basic in general than say Lang's book and the reader is not left to fill in the missing pieces and it is generally more readable. What's more the exercises are well chosen to consolidate the learning.
The book is not encyclopedic, but to my mind constitutes a very solid modern grounding for any mathematical student.

Amazing Book
This is a fabulous book.I have had basically one year of effectively 'algebra for non-mathematicians' at a graduate level, and have a background in Physics and Engineering.I find this book to be the perfect next step in terms of understanding things a bit more deeply than Dummit&Foote, in particular because they bring in some wider issues and connect them to the basic material of group theory, ring theory, modules vector spaces etc.In other words it covers the same or similar material as Dummit & Foote, but adds in some wider examples (more lattice theory, operator theory) as well as a higher level of sophistication because it brings in a category theory perspective for example.It seems that this book is really oriented toward seeing the big mathematical picture implicit in lots of physics.So in that sense, for me, it's a fantastic mathematical physics book, without explicitly being so.

Graduate-Level Algebra emphasizing categorical ideas and applications outside algebra
Garrett Birkhoff and S. MacLane's _A Survey of Modern Algebra_ introduced U.S. undergraduates to the (axiomatic) algebra of Emmy Noether and Emil Artin, with elementary topics useful in applications in science and engineering. Birkhoff-MacLane has a place for algebraic number theory, but puts it in its place---Chapter 14! Birkhoff-MacLane features Birkhoff's interests in congruence relations (c.f., universal algebra), partially ordered sets (c.f., lattice theory), and linear algebra and geometry.

MacLane-Birkhoff's Algebra strives to teach algebra using the spirit and the ideas of category theory. Thus module theory is central to the text. However, this text is in theory accessible to undergraduate students, because the level of abstraction increases gradually, the examples are elementary, proofs are given in detail, and most problems can be solved easily (in the beginning chapters). These features make MacLane-Birkhoff a complement to Lang's Algebra, which uses category theory.

(Also, MacLane-Birkhoff does use ideas from lattice theory and universal algebra more than other texts and has a particularly detailed development of linear and multilinear algebra.)

For a more comprehensive graduate textbook, I would recommend Grillet's "Algebra" (which should replace Lang's book except in isolated populations of algebraic number theory).

Superb, if read with the right outlook
Birkhoff and MacLane collaborated for much of their careers, and their "A Survey of Modern Algebra", first published in 1941, was an easy-to-read, easy-to-teach-from, easy-to-learn-from early fruit of their collaboration. This jointly written book "Algebra", first published in 1967 and vastly improved in the 3rd Edition, can be far more difficult to tackle unless one goes at it with understanding of how to approach it. It mostly reflects MacLane's approach, rather than Birkhoff's, and MacLane was not only brilliant, but unusual among pure mathematicians, perhaps even idiosyncratic; he finally died at an advanced age a few months ago, and his passion for his field is reflected in the fact that he continued to advise graduate students well into his 90s, just as he had advised me (and criticized my thinking incessantly) as a graduate student more than 50 years before.

MacLane was far less interested in any particular topic in mathematics, although he was a master of many, than he was in how one should think about mathematics to understand it, do it on one's own, extend it, and most important of all, recognize when one had fully though through a problem and solved it, as contrasted to having merely produced a plausible discussion of it.

I know of no book on pure mathematics more worth reading than this one, but in contrast to some other reviewers who are probably clearer thinkers than I, I have to tackle it with great patience and care. The secret of grasping it without getting bogged down is to keep constantly in mind that MacLane filled in details without being much interested in them except as necessary completion of exposition. So, when you read it, do not concentrate on details; concentrate on overall structure of thought and exposition and then, later, come back to absorb details. That was how MacLane worked, and that was how he tried to teach his students to work. The key question always in his mind was: what formulation of axioms and structure is fruitful for attacking the topic at hand, and how can we use that formulation to create an inexorable train of thought leading to important results? This book, "Algebra" is very much a reflection of that way of thinking.

So, when you first read this book, skip freely over much of the development of particular topics. Instead, spend a great deal of time thinking about definitions, and about the precise way in which key theorems are stated. Spend time and effort exploring the question of why seemingly trivial variations of these would be less fruitful, or could even lead one into error. Skip from one part of the book to another, without getting bogged down in any one part. Ask yourself also why certain topics and certain cases are excluded. E.g. right at the beginning of the discussion of quadratic forms is a simple definition which begins: "If V is a finite dimensional vector space over a field F of characteristic not 2, ..." Pause right there and ponder over why fields of characteristic 2 are excluded from this definition; just skim the next ten pages without studying them. If you think hard enough to see why fields of characteristic 2 must be excluded from the discussion, the entire ensuing discussion of quadratic forms becomes crystal clear.

Once you have mastered the style in which the material is presented, you can quite easily come back and follow the details. And if you do that, I hope you will find this ook as rewarding as I have. ... Read more

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This text examines the field of homology. ... 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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Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and expoitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories. ... Read more

Customer Reviews (8)

Simply Great
Have you ever tried reading Descartes' "Geometry"? It's not a good place to learn about coordinate geometry. I tried. This was almost 10 years ago, but I still remember it pretty well. Ok, so maybe the experience was even a bit traumatic.

Usually when someone works out a theory, it takes a fresh perspective (or two, or ... you get it) to really digest it, and come up with a reasonable way of teaching it to newcomers. It's less evident nowadays, with improved communications technology and such, but people aren't exactly turning to Grothendieck's expositions as their intro to his geometry either. Mac Lane is an exception.

This book seems completely inapproachable. The title is scary. The topic is scary. Open to a random page and try to judge its accessibility: scary. Well, here's the real story: you need to know algebra through modules, and it'd be nice if this algebra background introduced "universals" like abelianization or free modules in a way that involved the diagrams and the unique mappings you get from the given ones. If this stuff makes any sense, you can read this book. It's not that scary. If you're up to the challenge, you might even enjoy it. This is actually my favorite book.

Here's the approach that I feel worked well for me:

- gloss over the set-theoretic foundations at first. Make sure you know the proper class/set and large/small category distinctions, but don't dwell on them much.

- focus on the examples that are familiar, but read through the others too. Mac Lane uses tons of examples to suit a variety of backgrounds, and his presentation is so clear that the theory can often explain the examples.

- trust the author. It may seem like product or comma categories deserve fuller treatment with more motivation. No. Let Mac Lane's 'minimalism' infect your thinking: it's no more complicated than what's on those pages. Make sure you *know* what's there, and you will come to *understand* the material as it is fleshed out through exercises or later writing.

The last point has been the most important for me. This book has been a great lesson in clear thinking, which is of extreme importance in mathematics. Why? It's complicated enough!

Poorly written standard text.
This book has everything you need, but it is written in an abstruse style in my opinion.

A Classic
Well, let us think about this a little bit...You want to learn Category theory, whether for some course or just for the fun of it, and now where do you turn in order to learn the necessary concepts.If you are a mathematician and have some experience, then you turn to the masters, the originators of the given subject and read their work.Sure, being the founder of a given subject does not imply that you are a good expositor and hence are capable of revealing the necessary concepts for the beginner-allow me to inform that Mac Lane is indeed as good as an expositor as he was a mathematician.For any doubters, I point you to the only other text you should read on Category theory, namely, "Category Theory" by Horst Herrlich and compare this text with Mac Lane's.Aside from that, and with respect to the text, for most beginners or interested readers I would suggest the following outline: Read 1.1-6; 2.1-3 & 8 possibly 2.4; all of 3; as for 4 skip section 3; 5.1-5; all of 8.Then, dependent upon your desires and or focus as well as your mathematical ability, it should become obvious which of the remaining topics should be read.Finally, the only other source I would recommend for learning Category theory can be found on-line using the keyword 'Awodey'.Anyways, Enjoy and good luck.

You may not need this unless you major in category theory.
I entirely agree with the reviewer Lucas Wilman.
As a book by the creator of category theory, it has extensively incorpoated relevant items.
However I don't think this is a *must read" unless you major in the subject: you will seldom need more than what is covered in a typical homological algebra course.
My inmpression is this book should be entitled "Categories for the starting/working category theorists".

Classic and worth it
It is difficult to make understand what "is" category theory. Is it a foundational discipline? Is it a discipline studying homomorphisms between algebras? Is it nonsense? Well, in my opinion this book does not help in gaining this kind of understanding. But all the stuff I read which have been written with that purpose in mind did not have any success - perhaps because I am not a mathematician, or perhaps because some concepts in category theory are really too abstract for anyone to give "an intuition" of them (you still can with functors and natural transformations, but try with adjointness...). This said, I found the book wonderful: Every concept is presented neatly. I use it as a reference each time I want a clear and rigorous definition of a concept. Sometimes this rigour helped me in gaining the famous intuition behind the concept. ... Read more

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This classic, written by two young instructors who became giants in their field, has shaped the understanding of modern algebra for generations of mathematicians and remains a valuable reference and text for self study and college courses. ... Read more

Customer Reviews (4)

too concise for its own good
Any college level math text should have AT LEAST one of the following, but hopefully both:

1) plenty of examples illustrating the use of the taught proofs, theorems, algorithms, procedures, etc.

2) Answers to AT LEAST some of the exercises in the back of the text.

Without one of these, how would you ever know you were grasping the material?

Unfortunately, this book comes up pretty short on #1 and completely empty on #2. there are many sections that are lacking examples, and there are NO solutions to the exercises.

the book reads more like an ordered collection of theorems and corollaries. Not so great if you wanna teach yourself. Probably more ideal for decorating a professor's bookshelf.

A classic algebra text!Wonderful book...
This is a classic book on Algebra.There is much that I like about it.It is exactly what the name suggests--a survey course.It briefly introduces all sorts of topics, including rings, fields, groups, galois theory, vector spaces, lattices, boolean algebras, and much more.It is written at a fairly elementary level and it generally doesn't go into a great amount of depth in each subject.Interestingly, many (more modern) algebra texts omit a number of rather basic topics in this book.Also, many modern books separate "linear algebra" from "abstract algebra", whereas this book takes a more integrated approach.

I find it exceptionally clear and easy to read.Many of the subjects are made particularly easy; there is a strong concrete flavour to the text.The authors provide good motivation for the material.

I think this book would make excellent reading material for someone who is planning to study algebra.I did not pick it up until early in graduate school, and I wish I had had access to it earlier, when I was first studying ring and field theory.It is a fantastic reference for intermediate students, since it covers just about all the basics of algebra, and does so in a very understandable way.I think this book would make a fine textbook for an undergraduate course as well.

This is how algebra texts ought to be written
I have just started reading this book, and already I am
enthralled by the beauty and elegance of the authors'
exposition. Assuming nothing more than an acquaintance with
school algebra and a little geometry, they develop
the basic properties of central algebraic structures, including
rings, groups and fields. These are treated by reference to
familiar examples, such as the ring of integers and the
rational, real and complex fields. Everything that one learned
in school algebra is to be found here, though, as is to be
expected, each topic is treated at a rigorous, mathematically
sophisticated level. In the first two chapters, the properties
of the integers and rational numbers are gradually examined,
ultimately down to the definition of addition and multiplication
on the basis of Peano postulates. The authors then consider
polynomials, the real and complex numbers, vector spaces, linear
algebra and other topics.
The writing style is clear, concise and elegant, with each new
concept being carefully defined as it is introduced. The proofs
achieve a satisfying balance between detail and brevity. Indeed,
reading the proofs and completing the exercises would do much, I
am sure, to enhance a reader's mathematical facility.

If you are interested in acquiring a deeper understanding of
algebra, this book should serve as an excellent introduction.

A smorgousborg of symmetries of the square
Modern algebra is an extraordinary topic and Birkhoff and MacLane do a superb job of exploring it. However, as is often the case with mathematical texts, the material can be somewhat dry. ... Read more

Product Description
This book has been designed to serve the needs of teachers of secondary mathematics. The editors & contributing authors..hope that it will also prove useful to scholars of varied interests. It has been written specifically to provide reference * background material for both the content & spirit of modern mathematics. ... Read more