E-Book Overview
Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course.
E-Book Content
An Introduction to Category Theory Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course. An Introduction to Category Theory HAROLD SIMMONS University of Manchester CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107010871 c H. Simmons 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Original published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Simmons, Harold An introduction to category theory / Harold Simmons. p. cm. ISBN 978-1-107-01087-1 (hardback) 1. Categories (Mathematics) I. Title. QA169.S56 2011 5120 .62–dc23 2011021721 ISBN 978-1-107-01087-1 Hardback ISBN 978-0-521-28304-5 Paperback Additional resources for this publication at www.cambridge.org/simmons Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page vii 1 Categories 1.1 Categories defined 1.2 Categories of structured sets 1.3 An arrow need not be a function 1.4 More complicated categories 1.5 Two simple categories and a bonus 1 1 8 16 27 31 2 Basic gadgetry 2.1 Diagram chasing 2.2 Monics and epics 2.3 Simple limits and colimits 2.4 Initial and final objects 2.5 Products and coproducts 2.6 Equalizers and coequalizers