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Rating: 
Summary: A Good Introduction
Review: 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.)
Rating: 
Summary: Very uneven, but still useful
Review: 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.
Rating: 
Summary: Objects and maps are everywhere
Review: 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.
Rating: 
Summary: Heavy Hitter Strikes Out
Review: I sure hope Schanuel wrote this book and the publisher simply tacked on
Lawvere's name for marketing purposes. This text is a fantastic
example of why research mathematicians should not write for John Q.
Public. The random, pointless examples scattered throughout the book
remind me of the "word problems" that were so popular in high school
algebra texts written after the Chicago School hijacked the educational
textbook market. After teasing the reader with examples of real mathematics, e.g.
Pick's Formula, the authors stop short of actually proving a theorem
and scurry back to their shelter of objects and arrows where they can
safely field trivial questions by ersatz students with politically
correct names. Perhaps Category Theory is just not something that is accessible to the
general public? High school math teachers (I assume one intended
audience for the text) that can achieve even the slightest appreciation
of why Eilenberg and Mac Lane invented Category Theory are surely as
rare as rocking-horse poop. What I would really like to see from someone as eminent as Lawvere write a
first year graduate level book that covers elementary set theory and/or
logic using Category Theory. Translating Model Theory and Topoi(1.) to
this level would be a good start. College math professors are really
the only people in a position to understand and transmit this beautiful
theory to aspiring mathematicians. 1. Model Theory and Topoi, Lecture Notes in Mathematics 445,
Springer-Verlag 1975 Keith A. Lewis ...
Rating:
Summary: Heavy Hitter Strikes Out
Review: I sure hope Schanuel wrote this book and the publisher simply tacked on
Lawvere's name for marketing purposes. This text is a fantastic
example of why research mathematicians should not write for John Q.
Public. The random, pointless examples scattered throughout the book
remind me of the "word problems" that were so popular in high school
algebra texts written after the Chicago School hijacked the educational
textbook market. After teasing the reader with examples of real mathematics, e.g.
Pick's Formula, the authors stop short of actually proving a theorem
and scurry back to their shelter of objects and arrows where they can
safely field trivial questions by ersatz students with politically
correct names. Perhaps Category Theory is just not something that is accessible to the
general public? High school math teachers (I assume one intended
audience for the text) that can achieve even the slightest appreciation
of why Eilenberg and Mac Lane invented Category Theory are surely as
rare as rocking-horse poop. What I would really like to see from someone as eminent as Lawvere write a
first year graduate level book that covers elementary set theory and/or
logic using Category Theory. Translating Model Theory and Topoi(1.) to
this level would be a good start. College math professors are really
the only people in a position to understand and transmit this beautiful
theory to aspiring mathematicians. 1. Model Theory and Topoi, Lecture Notes in Mathematics 445,
Springer-Verlag 1975 Keith A. Lewis ...
Rating: 
Summary: Not for the Mathematically Mature
Review: 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.
Rating:
Summary: A retract in search of a section
Review: There is a wonderful course in category theory for high school students, just begging to be excavated from this multi-layered book.
Please don't be put off by the disjointed and uneasy combination of materials that cluster around certain themes. You know you will have a lot of work to do when the same definition (of monomorphism) is presented both on page 52 and also on page 336.
With all the elementary themes covered in many varying ways, it would be best to consider this book as having been structured as a retract for which your job will be to construct the appropriate section.
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