# Conceptual Mathematics: A First Introduction to Categories

May 27, 2010

Authors: F. William Lawvere and Stephen H. Schanuel

Edition: Second

Publisher: Cambridge University Press

Pages: 408

Price: £70.00 and £29.99

ISBN 9780521894852 and 719162

Category theory was first introduced in 1945 by Samuel Eilenberg and Saunders Mac Lane when they were laying down the foundations of algebraic topology. It is not normally explicit in the undergraduate curriculum, although the use of arrows to represent functions between sets is a notation from category theory. A category consists of certain objects and functions between the objects: functions are basic, objects arise as domains and co-domains of functions.

The authors of this book argue that "making (categorical concepts) more explicit helps us to go beyond elementary algebra into more advanced mathematical sciences", and they set out to make "their simplicity ... accessible" to students.

In the categorical approach, properties of functions are described in terms of their universal properties rather than in terms of how they treat elements of sets. For example, a first introduction to sets and functions might define an injection f: A - B to be a function such that, if f(x) = f(y) where x and y are elements of A, then x = y.

The categorical approach is to characterise an injection as a monomorphism: a function that can have the "universal property" of cancellation on the left, ie, if fg = fh where g and h are functions X - A, then g = h. Proofs are based on these universal properties rather than arguments about elements.

This book begins with a leisurely introduction to sets and functions and then moves on to consider universal constructions in other categories, selecting categories whose objects are sets with some simple additional structure such as an endomorphism.

I found it an extremely stimulating text and it has encouraged me to consider whether arguments based on universal properties (which, in my experience of teaching topology, many students find difficult) could be introduced at an earlier stage.

However, I am not sure that this book's mixture of an informal approach with quite sophisticated ideas works. For example, early on we are told that a "point" in a set is a function to the set from a singleton set. What will a reader who has just been told that a set is just a collection of objects make of this?

The definition falls into place 200 pages later when we meet the definition of a point in an object in a general category as a function from a "terminal" object. In fact, the level of this book is uneven, with some simple topics dealt with in enormous detail, but then quite difficult ideas passed over quite quickly. A number of technical terms are used without explanation. Finally, it is not really a textbook on category theory, since there is no real mention of functors and natural transformations.

Who is it for? An interesting supplementary read for graduate students embarking on a study of category theory. The simple categories explored provide interesting settings for exploring key ideas.

Presentation An interesting approach, with its seven "articles" summarising the main ideas supplemented by 35 "sessions" exploring examples at more leisure.

Would you recommend it? Only to strong students or to lecturers wishing to reflect on how categorical ideas could be introduced at an undergraduate level.