In the last fifty years, the use of the notion of 'category' has led to a remarkable unification and simplification of mathematics. Written by two of the best-known participants in this development, Conceptual mathe- matics is the first book to apply categories to the most elementary mathe- matics. It thus serves two purposes: to provide a skeleton key to mathematics for the general reader or beginning student; and to furnish an introduction to categories for computer scientists, logicians, physicists, linguists, etc. who want to gain some familiarity with the categorical method. Everyone who wants to follow the applications of mathematics to twenty-first century science should know the ideas and techniques explained in this book. .
Conceptual Mathematics A first introduction to categories F. WILLIAM LAWVERE State University of New York at Buffalo STEPHEN H. SCHANUEL State University of New York at Buffalo CAMBRIDGE UNIVERSITY PRESS .
Please read this We all begin gathering mathematical ideas in early childhood, when we discover that our two hands match, and later when we learn that other children also have grand- mothers, so that this is an abstract relationship that a child might bear to an older person, and then that 'uncle' and 'cousin' are of this type also; when we tire of losing at tic-tac-toe and work it all out, never to lose again; when we first try to decide why things look bigger as they get nearer, or whether there is an end to counting. As the reader goes through it, this book may add some treasures to the collection, but that is not its goal. Rather we hope to show how to put the vast storehouse in order, and to find the appropriate tool when it is needed, so that the new ideas and methods collected and developed as one goes through life can find their appropriate places as well. There are in these pages general concepts that cut across the artificial boundaries dividing arithmetic, logic, algebra, geometry, calculus, etc. There will be little discussion about how to do specialized calculations, but much about the ana- lysis that goes into deciding what calculations need to be done, and in what order. Anyone who has struggled with a genuine problem without having been taught an explicit method knows that this is the hardest part. This book could not have been written fifty years ago; the precise language of concepts it uses was just being developed. True, the ideas we'll study have been employed for thousands of years, but they first appeared only as dimly perceived analogies between subjects. Since 1945, when the notion of 'category' was first pre- cisely formulated, these analogies have been sharpened and have become explicit ways in which one subject is transformed into another. It has been the good fortune of the authors to live in these interesting times, and to see how the fundamental insight of categories has led to clearer understanding, thereby better organizing, and sometimes directing, the growth of mathematical knowledge and its applications. Preliminary versions of this book have been used by high school and university classes, graduate seminars, and individual professionals in several countries. The response has reinforced our conviction that people of widely varying backgrounds can master these important ideas. .
Note to the reader The Articles cover the essentials; the Sessions, tables, and exercises are to aid in gaining mastery. The first time we taught this course, the Articles were the written material given to the students, while Emilio Faro's rough notes of the actual class discussions grew into the Sessions, which therefore often review material previously covered. Our students found it helpful to go over the same ground from different viewpoints, with many examples, and readers who have difficulty with some of the exercises in the Articles may wish to try again after studying the ensuing Sessions. Also, Session 10 is intended to give the reader a taste of more sophisticated applica- tions; mastery of it is not essential for the rest of the book. We have tried in the Sessions to keep some of the classroom atmosphere; this accounts for the frequent use of the first person singular, since the authors took turns presenting the material in class. Finally, the expert will note that we stop short of the explicit definition of adjoint- ness, but there are many examples of adjunctions, so that for a more advanced class the concept could be explicitly introduced. .