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Conceptual Mathematics: A First Introduction to Categories [Kindle Edition]

F. William Lawvere , Stephen H. Schanuel
4.2 out of 5 stars  See all reviews (10 customer reviews)

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Book Description

In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics introduces this tool for the learning, development, and use of mathematics, to beginning students and also to practising mathematical scientists. This book provides a skeleton key that makes explicit some concepts and procedures that are common to all branches of pure and applied mathematics. The treatment does not presuppose knowledge of specific fields, but rather develops, from basic definitions, such elementary categories as discrete dynamical systems and directed graphs; the fundamental ideas are then illuminated by examples in these categories. This second edition provides links with more advanced topics of possible study. In the new appendices and annotated bibliography the reader will find concise introductions to adjoint functors and geometrical structures, as well as sketches of relevant historical developments.

Editorial Reviews


"This outstanding book on category theory is in a class by itself. It should be consulted at various stages of one's mastery of this fundamental body of knowledge."
George Hacken,

Book Description

Conceptual Mathematics introduces the concept of category to beginning students, general readers, and practicing mathematical scientists based on a leisurely introduction to the important categories of directed graphs and discrete dynamical systems. The expanded second edition approaches more advanced topics via historical sketches and a concise introduction to adjoint functors.

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Customer Reviews

Most Helpful Customer Reviews
36 of 36 people found the following review helpful
5.0 out of 5 stars A bird's eye view of the mathematical landscape August 18, 2010
By Michael
Over the last two years I have revisited different sections of this book and gain new profound insights with every read. With some dedication and time, this book will surely enrich your life! What this book offers is the truth. The concepts presented in this book are the underlying unifying ideas which make up mathematics itself in an even more general and profound sense than Set Theory (in fact, one of the authors has rigorously shown that set theory is a very special case of what is presented in this book). We can encounter categories not only at the microscopic level (where we define the fundamental ideas that allow us to construct mathematical concepts from the ground up), but at the macroscopic level as well (where complex constructions in distant fields become analogous to the microscopic building blocks). With these ideas we can show that multiplication and addition are actually more appropriately opposites of one another than addition and subtraction or multiplication and division. This book is the key to beginning a journey to discovering the true nature of mathematics. To continue (or supplement) your journey, also pick up a copy of Sets for Mathematics By F. William Lawvere and Robert Rosebrugh. With time and practice (attempt the exercises from both books!!!) you will be greatly rewarded. As a student of Mathematics, this has paid off in ways I never thought possible and continues to provide insight to nearly everything I learn in school and on my own.

A startling demonstration presented in this book is that Cantor's Diagonal Argument in generalized form not only proves that there are infinite different levels of infinity, but also Godel's Incompleteness Theorem!
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29 of 31 people found the following review helpful
Not long ago, I spoke with a professor at strong HBCU department. Her Ph.D. was nearly twenty years ago, but I shocked her with the following statement, "Most of our beginning graduate students [even those in Applied Mathematics] are entering with the basic knowledge and language of Category Theory. These days one might find Chemists, Computer Scientists, Engineers, Linguists and Physicists expressing concepts and asking questions in the language of Category Theory because it slices across the artificial boundaries dividing algebra, arithmetic, calculus, geometry, logic, topology. If you have students you wish to introduce to the subject, I suggest a delightfully elementary book called Conceptual Mathematics by F. William Lawvere and Stephen H. Schanuel" [Cambridge University Press 1997].
From the introduction: "Our goal in this book is to explore the consequences of a new and fundamental insight about the nature of mathematics which has led to better methods for understanding and usual mathematical concepts. While the insight and methods are simple ... they will require some effort to master, but you will be rewarded with a clarity of understanding that will be helpful in unraveling the mathematical aspect of any subject matter."
Who are the authors? Lawvere is one of the greatest visionaries of mathematics in the last half of the twentieth century. He characteristically digs down beneath the foundations of a concept in order to simplify its understanding. Though Schanuel has published research in diverse areas of Algebra, Topology, and Number Theory, he is known as a great teacher. The book is an edited transcript of a course taught by Lawvere and Schanuel to American undergraduate math students. The book was actually chosen as one of the items in the Library of Science Book Club.
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14 of 15 people found the following review helpful
5.0 out of 5 stars My favorite maths book! November 25, 2010
It has flaws, but is still one of the greatest maths book I've read. Aimed at high-school level and up, but towards the end it gets a bit complicated, so I doubt if a high school kid can fully understand it without consulting other books. But, most of the book is really easy to read, and the authors' effort to write such a book is admirable.

Lawvere is one of the developers of topos theory, where he found an axiomatization of the category of sets.

The last 2 sections are an introduction to topoi and logic. One key fact seems to be missing which caused me some perplexing: In the category of subobjects, 2 subobjects A and B has A > B if A includes B. Thus, the relation ">" creates a partial order amongst the subobjects. If A > B and B < A, then A = B, thus inducing an equivalence class, denoted by [A]. This is the reason why the subobject classifier has internal structure (different "shades" of truth values).

Also, the relation of topology to logic is analogous to the relation of classical propositional logic to the Boolean algebra of sets, with the sets replaced by open sets in topological space.

I've only read the 1st edition. The 2nd edition's first part is the same as the 1st edition, with additional advanced topics at the very end.
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9 of 9 people found the following review helpful
Such an excellent work as one is given to saying asto all productions of Lawvere's. This book, on the face of it, seems easy, even elementary. But there is, as Lawvere has said, an awful lot here. A book is elegant if it achieves to say a great deal with ease and a sense of depth of coverage.

The path to Cateogories and Toposes is via two book: Cat for working mathematicians and Sheaves in Geometry and Logic both by Mac Lane. But these are anything but easy or elementary.

There is a problem with mathematical texts of a pedagogic kind, one that this book avoids: their writers often confuse teaching with forma exposition. They don't "talk" to one but go off at their own formal tangents.
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Most Recent Customer Reviews
5.0 out of 5 stars Finally a less steep introduction to category theory
I've been trying to expose myself to category theory through the literature. The literature of course demands a sophistication I have yet to attain. Read more
Published 5 months ago by Amazon Customer
1.0 out of 5 stars I'm an electrical engineer but I couldn't understand this book
As and electrical engineer, I have a good background in mathematics - but this book was indecipherable to me. Read more
Published 23 months ago by
4.0 out of 5 stars Reflective Study of Mathematics
Dear All,

For what I think of as a student review of Conceptual Mathematics textbook, please see

conceptualmathematics.wordpress. Read more
Published on March 22, 2012 by Posina
3.0 out of 5 stars Belabored
I spent probably about 40 hours on this book. Its hard to judge since I read it before going to sleep, off and on, over the course of a year. The layout of the book is interesting. Read more
Published on March 1, 2012 by tcook
5.0 out of 5 stars Intuitive and insightful
This book is a simple introduction to category theory. It is strictly about categories, examples are easy to understand and relevant to beginners. Read more
Published on February 22, 2011 by Dmitri
4.0 out of 5 stars Very approachable
Unlike many category theory books that end up drowning you in a notational blizzard, this book starts off at a more basic level and carefully walks you upwards. Read more
Published on February 9, 2011 by Amazon Customer
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