Customer Reviews

Conceptual Mathematics: A First Introduction to Categories

5 star:		(7)
4 star:		(4)
3 star:		(2)
2 star:		(1)
1 star:		(1)

 

 

Lawvere and Schanuel have created a book at once accessible and stimulating at a great many levels. It discusses the concepts of Category Theory in a simulated "classroom" setting, addressing common questions of students at crucial points in the book. It also wanders in a care-free manner through an amazing number of topics. The book is interesting to non-mathematicians at a philosophical level, and to (beginning) mathematicians as an introduction to an exciting new area of mathematics. The authors have a great attitude, and offer great starting-points for investigation.

I read it as a first year pure math undergraduate, and though it was at times at too low a level (the 'tests,' for instance, are very easy reviews of basic ideas), it never became boring. For me, it read 'like a novel' (and a page-turner, at that). My only gripe is the lack of an annotated "further reading" section, which would have rounded out the book.

Many of the reviews evaluate the book from the perspective of graduate students in mathematics want to learn categories, and it's certainly the wrong choice for that purpose. If you think of this as a serious math textbook, then it fails in that goal: significant proofs are the exception rather than the rule; very few, and trivial, exercises; very lacking in depth.

This is a great book because it provides a motivation for investigating categories. It helped me when I was in the position of hearing from a lot of places that subjects I was interested in often used category theory. I tried to read a few "real" books about category theory, and didn't get very far because they did not make the connections I was looking for. I accumulated three or four such books, all with bookmarks at about page 50 to 75. This book taught me relatively little about the theory of categories or the body of knowledge about them, but it provided a wealth of connections between categories and other topics, which made me better able to finish a couple of the real books and figure out what I needed to know there.

My advice, if you're in anything like that situation, is to read this book. Just don't take it too seriously, and don't try to milk more out of it than is really there. Then go learn more about category theory from elsewhere.

Highly intuitive introduction to this abstract, but highly practical area of mathematics with one glaring fault. First the good news. I have never seen a more carefully explained introduction into an area of mathematics. Many examples and explanations of the principles behind and applications of concept analysis. However, the glaring fault is organization. Details are given without adequate tie in to how they relate to others. The text bounces from one area to the next so it is easy to lose sight of the whole picture. On balance its strengths far outweigh its weaknesses so I recommend it without reservation.

As a first introduction to Categories, this book is well written, clever, simple and very clear. However, I was disappointed with it. From the notoriety of the authors and the, yes, cool illustrations I assumed it would be a gem. However, it fell short. I've been toying with Category Theory for a few years, and every time I try to get into a book on Categories I get stumped at the notions of Functors and Natural Transformations. This book, however, dealt with neither at length, despite the fact that Category Theory originated around the notion of Natural Transformations in the first place. (As I understand it at least.) That said, there are many very cool passages in the book, including a functional analysis of a Chinese restaurant and an elegent exposition of Brouwer's Fixed Point Theorem.

Still, for my purposes, I prefer Robert Goldblatt's "Topoi: The Categorical Analysis of Logig" and Michael Barr's "Category Theory for Computing Science". As both are intended for non Category Theorists, both build their presentations of Category Theory from sratch. Sadly, I think both are out of print. Not for the faint of heart, I'm told Saunders Mac Lane's "Categories for the Working Mathematician" is the classic. (It's on my wish list.)

As a topic in itself, category theory should need not to wait until grad-level to be described just because that may be when category theory's power can really begin to be exploited, but unfortunately, most of the category theory books I have looked at presume that level of mathematics.

Similar to what other reviewers noted, I would also say that this book demonstrates the potential of creating a good high-school/undergrad level intro to category theory. But unfortunately, that potential is not quite realized here.

There are hokey intermittent "conversations with students", as a tool to describe ideas, that are more distraction than aid. Some of the examples given are rather condescending in their simplicity. Yet, at other times the authors seem to breeze through more difficult topics with little or no examples. And the organization seems erratic - there is no clear sense of a gameplan as to where they are leading the reader or how all the concepts fit together.

Functors are surprisingly almost glossed over, as if they were relatively unimportant. There are exercises throughout the book, but with no answers provided, they are not really very helpful.

Having said all that, with some focused effort on the reader's part, the ideas do come forth, and admittedly, the authors do cover a fairly broad spectrum of aspects of category theory. This is certainly a non-trivial topic to try and teach, and an introductory book cannot be faulted for not carrying every notion to the nth-degree of either breadth or depth.

Category Theory is one of those topics that (to me) appears 'ho-hum' until you see it actually applied to various topics. The authors have necessarily had to perform a balancing act between describing concepts while not getting caught up in excessively complex examples. I think this will leave many readers less than satisfied, but realistically, the book would have been twice as long had they really delved deeper into examples (or they would have had to be very terse in the actual descriptions of category theory, which is the choice most authors writing for a more mathematically-inclined audience seem to make - e.g., _Mathematical Physics_ by Geroch (good book!) or _Basic Category Theory for Computer Scientists_ by Pierce).

If you are mathematically astute, you probably will find this book tedious. But if you are not a grad+ math major, then this book may well be worth the effort as a way to begin to learn a very profound and powerful set of tools and concepts.

In the preface of this book, the author comments that this book has been used successfully in high schools, colleges, graduate schools, and by professors. After reading this book, I can believe it. This book is simply a gem.

Mind you, although this book is very easy to read, some of the concepts contained within it are very abstract and can be very difficult to fully comprehend. While a high school student will surely get something out of this book, it would be hard to understand everything in it without knowing a fair amount of mathematics.

I would recommend this book to any mathematician. It is an absolute must-read. The author makes the claim that working through this book will improve your ability to categorize (no pun intended!) your mathematical knowledge so as to better know how to approach problems. From my experience, this claim is true. This book somehow teaches some of the things about problem-solving that many people believe cannot be taught.

This book looks deceptively simple, especially relative to beasts such as MacLane's "Categories for the Working Mathematician". However, I find that I keep coming back to this book, sometimes after several months. In particular, I have found that reading this book has opened the door to understanding some of the advanced mathematics books that previously seemed inaccessible to me, such as Lang's "Algebra".

Excellent book for non-professional mathematicians, like me (I'm a software engineer), who wants to understand modern mathematics and apply its ideas in analysis of complex problems. Lots of pictures and diagrams (compared to terse wording in other mathematical books) really help to understand and master the subject. I think most of negative reviews come from professional mathematicians, but they don't need this book.

It seems clear to me that this book is aimed at people who don't have much of a background in advanced mathematics. I could not read more than sixty pages of the book because it proceeded too slowly. I must definitely point out that this is not a fault of the book as much as it is my fault for not reading a different book instead. I do think that the book does an admirable job at teaching category theory to those who don't have much background in advanced mathematics. It proceeds logically and explains concepts in a manner as to build up the readers intuition for the subject. Thus, I'm compelled to rate the book highly even though I could not gain much out of it. The main reason I write this review is so that others do not make the same mistake that I did. For those who have some mathametical training, I have heard of another highly acclaimed book called Category Theory for the Working Mathematician.

When I saw the one star review, and the general disagreement about the merits of this book I couldn't help but jump in. clearly this is not a work aimed at the professional mathematicians who will find it only tedious and repetitive. No, it has been designed for the many folks, pretty much like me I assume, who have some background in undergraduate mathematics and would love to learn more about the fascinating, and to them "new" field of category theory. It takes such folks by the hand and not only explains clearly the basic concepts, but, much more importantly explains WHY such seemly obvious and highly abstract concepts become exceptionally useful, and productive of insights, when applied to a huge number of ostensibly largely unrelated areas within mathematics and its many practical applications, and THAT, after all, is what Category Theory, much like Abstract Algebra and Algebraic Topology before it, is really all about. So sure, if you already know a great deal about the many general features thematized by Category Theory that are useful everywhere from Boolean Logic to AI, to Lord knows what all else, this is not the best book for you to start with -- though, provided one isn't an elitisit killjoy, it might still interest you to see how two very highly respected scholars have attempted to lay the field out in such a way that it can engage folks who are just beginning to move into such areas of mathematical abstraction. Anyway, I give it five stars, since my only reservation is the minor complaint that at times the examples -- the much maligned "word puzzles" -- aren't quite as stimulating or "right on the money" as they perhaps could be. It is also true that, provided one has sufficient mathematical background, it is actually easier to progress more deeply into the field more quikly using Steven Awodey's now already standard "Category Theory." It provides most of the same materials, and a good deal more, a bit more parsimoniously and directly, without requiring the reader to take nearly so many "detours," some of which seem rather obvious, although it is also a bit less thorough in explaining what the long term theoretical payoff will be. All things considered, however, mthematicis pedagogy today still cries out for MORE, not fewer, books like "Conceptual Mathematics," and fortunately a new breed of fine authors is now beginning to work the boundaries between rigorous mathematics and clear popular presentation, a need that was barely recognized only thirty years ago, except by a few intrepid souls including Gamow, Conway, Rucker, Barrow etc.. All in all, at least IMHO, these two authors, both first rate Category Theorists, are to be greatly commended, not condemned, for having written a work which works to make their field more readily comprehensible to others with lesser immediate preparation for it.

The theoretical physicist John Baez wrote, "[Conceptual Mathematics] may seem almost childish at first, but it gradually creeps up on you. Schanuel has told me that you must do the exercises--if you don't, at some point the book will suddenly switch from being too easy to being way too hard! If you stick with it, by the end you will have all the basic concepts from topos theory under your belt, almost subconsciously."

Conceptual Mathematics has only two prerequisites: Basic high-school algebra, and willingness to work through the material carefully. In return, this book offers a solid introduction to Cartesian closed categories and topoi. Major topics include sections and retractions, initial and terminal objects, products and coproducts, exponentiation, and subobject classifiers.

These topics are illustrated using a variety of basic categories, each of which the authors introduce from scratch. These categories include sets, dynamic systems, and graphs, plus many variations of these categories. The self-contained nature of these examples is the book's greatest strength--almost every other introduction to category theory assumes prior knowledge of either topology, logic, or theoretical computer science.

But why take the time to study Cartesian closed categories and topoi? An example may help.

In computer science, the best-known Cartesian closed category is the lambda calculus, which lies at the heart of functional programming languages like Haskell and Scheme. But Cartesian closed categories appear everywhere in mathematics, logic and theoretical physics. And these connections between subjects can be exploited: For example, there's a program named Djinn, which translates Haskell type signatures into statements in intuitionist logic (using the Curry-Howard-Lambek correspondence). From there, Djinn runs a theorem prover, and then translates the output back into Haskell functions satisfying the original type signatures. In other words, by exploiting the connection between type systems and logic, it becomes possible to use tools from one field to solve problems in another.

A word of caution, however: Conceptual Mathematics omits several central topics in category theory, including functors, natural transformations, and adjoints. In many cases, it lays extensive groundwork for these topics, but never gets around to covering the topics themselves. So if you want to go beyond a basic introduction to closed Cartesian categories and topoi, you're going to need another book.

Despite these limitations, however, Conceptual Mathematics is an enjoyable--and uniquely accessible--introduction to category theory.