- Paperback: 376 pages
- Publisher: Cambridge University Press; 1st edition (November 28, 1997)
- Language: English
- ISBN-10: 0521478170
- ISBN-13: 978-0521478175
- Product Dimensions: 6.8 x 1.1 x 9.7 inches
- Shipping Weight: 1.4 pounds
- Average Customer Review: 17 customer reviews
- Amazon Best Sellers Rank: #794,708 in Books (See Top 100 in Books)
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Conceptual Mathematics: A First Introduction to Categories 1st Edition
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"Conceptual Mathematics provides an excellent introductory account to categories for those who are starting from scratch. It treats material which will appear simple and familiar to many philosophers, but in an unfamiliar way." Studies in History and Philosophy of Modern Physics
The idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best known names in categorical logic, this is the first book to apply categories to the most elementary mathematics.
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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.
I have several category theory books. The only book that I've found that really hits the mark is "Mathematical Physics" by Robert Geroch. Sophisticated yet accessible and genuinely fun to read. I'd rate Geroch at the first year graduate level for mathematics or physics majors. Might be too heavy for computer science because of the topic list.
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.
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