Conceptual Mathematics: A First Introduction to Categories [Paperback]

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.

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 ...