Conceptual Mathematics, 2nd Edition 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 general readers, but also to practicing mathematical scientists. This book provides a skeleton key, making 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. LAWVERE MASTER:Layout 1 23/5/09 08:46 Page ii LAWVERE MASTER:Layout 1 23/5/09 08:46 Page iii Conceptual Mathematics, 2nd Edition A first introduction to categories F. WILLIAM LAWVERE SUNYatBuffalo STEPHEN H. SCHANUEL SUNYatBuffalo LAWVERE MASTER:Layout 1 23/5/09 08:46 Page iv CAMBRIDGE UNIVERSITY PRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo,Delhi CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK www.cambridge.org Informationonthistitle:www.cambridge.org/9780521894852 Thisedition©CambridgeUniversityPress2009 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished1997 Secondedition2009 PrintedintheUnitedKingdom AcatalogrecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloginginPublicationData Lawvere,F.W. Conceptualmathematics:afirstintroductiontocategories/F.William Lawvere,StephenH.Schanuel.–2nded. p. cm. Includesindex. ISBN978-0-521-71916-2(pbk.)–ISBN978-0-521-89485-2(hardback) 1.Categories(Mathematics) I.Schanuel,S.H.(StephenHoel),1933– II.Title. QA169.L355 2008 9 512.62–dc22 2007043671 ISBN 978-0-521-89485-2hardback ISBN 978-0-521-71916-2paperback CambridgeUniversityPresshasnoresponsibilityfor thepersistenceoraccuracyofURLsforexternalor third-partyInternetWebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. LAWVERE MASTER:Layout 1 23/5/09 08:46 Page v to Fatima LAWVERE MASTER:Layout 1 23/5/09 08:46 Page vi LAWVERE MASTER:Layout 1 23/5/09 08:46 Page vii LAWVERE MASTER:Layout 1 23/5/09 08:46 Page viii viii Contents LAWVERE MASTER:Layout 1 23/5/09 08:46 Page ix Contents ix