This page intentionally left blank CAMBRIDGE TRACTS IN MATHEMATICS GeneralEditors B. BOLLOBA´ S, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO 184 AlgebraicTheories Algebraic Theories A Categorical Introduction to General Algebra J. ADA´ MEK TechnischeUniversita¨tCaroloWilhelminazuBraunschweig, Germany J. ROSICKY´ MasarykovaUniverzitavBrneˇ,CzechRepublic E. M. VITALE Universite´CatholiquedeLouvain,Belgium WithaForewordbyF. W. LAWVERE cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore, Sa˜oPaulo,Delhi,Dubai,Tokyo,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521119221 (cid:1)C J.Ada´mek,J.Rosicky´,andE.M.Vitale2011 Foreword(cid:1)C F.W.Lawvere2011 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2011 PrintedintheUnitedKingdomattheUniversityPress,Cambridge AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloguinginPublicationdata Ada´mek,Jiˇr´ı,dr. Algebraictheories:acategoricalintroductiontogeneralalgebra/J.Ada´mek, J.Rosicky´,E.M.Vitale. p. cm.–(Cambridgetractsinmathematics;184) Includesbibliographicalreferencesandindex. ISBN978-0-521-11922-1(hardback) 1.Categories(Mathematics) 2.Algebraiclogic. I.Rosicky´,Jiˇr´ı. II.Vitale,E.M. III.Title. IV.Series. QA169.A31993 2010 512(cid:2).62–dc22 2010018289 ISBN978-0-521-11922-1Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. ToSusy,Radka,andAle Contents Foreword pageix F.W.Lawvere Preface xv PART I: ABSTRACT ALGEBRAIC CATEGORIES 0 Preliminaries 3 1 Algebraictheoriesandalgebraiccategories 10 2 Siftedandfilteredcolimits 21 3 Reflexivecoequalizers 30 4 Algebraiccategoriesasfreecompletions 38 5 Propertiesofalgebras 46 6 Acharacterizationofalgebraiccategories 54 7 Fromfilteredtosifted 65 8 Canonicaltheories 74 9 Algebraicfunctors 80 10 Birkhoff’svarietytheorem 89 PART II: CONCRETE ALGEBRAIC CATEGORIES 11 One-sortedalgebraiccategories 103 12 Algebrasforanendofunctor 117 vii viii Contents 13 Equationalcategoriesof(cid:1)-algebras 127 14 S-sortedalgebraiccategories 139 PART III: SPECIAL TOPICS 15 Moritaequivalence 153 16 Freeexactcategories 163 17 Exactcompletionandreflexive-coequalizercompletion 182 18 Finitarylocalizationsofalgebraiccategories 195 Postscript 204 AppendixA Monads 207 AppendixB Abeliancategories 227 AppendixC Moreaboutdualitiesforone-sortedalgebraic categories 232 References 241 Listofsymbols 245 Index 247