An Introduction to Category Theory PDF
Preview An Introduction to Category Theory
“9781107010871book” — 2011/8/9 — 16:58 — page i — #1 AnIntroductiontoCategoryTheory Categorytheoryprovidesageneralconceptualframeworkthathasprovedfruitfulin subjectsasdiverseasgeometry,topology,theoreticalcomputerscienceand foundationalmathematics.Hereisafriendly,easy-to-readtextbookthatexplainsthe fundamentalsatalevelsuitablefornewcomerstothesubject. Beginningpostgraduatemathematicianswillfindthisbookanexcellent introductiontoallofthebasicsofcategorytheory.Itgivesthebasicdefinitions;goes throughthevariousassociatedgadgetry,suchasfunctors,naturaltransformations, limitsandcolimits;andthenexplainsadjunctions.Thematerialisslowlydeveloped usingmanyexamplesandillustrationstoilluminatetheconceptsexplained.Over200 exercises,withsolutionsavailableonline,helpthereadertoaccessthesubjectand makethebookidealforself-study.Itcanalsobeusedasarecommendedtextfora taughtintroductorycourse. “9781107010871book” — 2011/8/9 — 16:58 — page ii — #2 “9781107010871book” — 2011/8/9 — 16:58 — page iii — #3 An Introduction to Category Theory HAROLDSIMMONS UniversityofManchester “9781107010871book” — 2011/8/9 — 16:58 — page iv — #4 CAMBRIDGE UNIVERSITY PRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,Sa˜oPaulo,Delhi,Tokyo,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9781107010871 (cid:13)c H.Simmons2011 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Originalpublished2011 PrintedintheUnitedKingdomattheUniversityPress,Cambridge AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloguinginPublicationdata Simmons,Harold Anintroductiontocategorytheory/HaroldSimmons. p. cm. ISBN978-1-107-01087-1(hardback) 1. Categories(Mathematics) I. Title. QA169.S56 2011 5120.62–dc23 2011021721 ISBN978-1-107-01087-1Hardback ISBN978-0-521-28304-5Paperback Additionalresourcesforthispublicationatwww.cambridge.org/simmons CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. “9781107010871book” — 2011/8/9 — 16:58 — page v — #5 Contents Preface pagevii 1 Categories 1 1.1 Categoriesdefined 1 1.2 Categoriesofstructuredsets 8 1.3 Anarrowneednotbeafunction 16 1.4 Morecomplicatedcategories 27 1.5 Twosimplecategoriesandabonus 31 2 Basicgadgetry 34 2.1 Diagramchasing 34 2.2 Monicsandepics 37 2.3 Simplelimitsandcolimits 43 2.4 Initialandfinalobjects 45 2.5 Productsandcoproducts 47 2.6 Equalizersandcoequalizers 56 2.7 Pullbacksandpushouts 64 2.8 Usingtheoppositecategory 71 3 Functorsandnaturaltransformations 72 3.1 Functorsdefined 73 3.2 Somesimplefunctors 76 3.3 Somelesssimplefunctors 79 3.4 Naturaltransformationsdefined 90 3.5 Examplesofnaturaltransformations 93 4 Limitsandcolimitsingeneral 108 4.1 Templateanddiagram–afirstpass 109 4.2 Functorcategories 114 4.3 Problemandsolution 118 “9781107010871book” — 2011/8/9 — 16:58 — page vi — #6 vi Contents 4.4 Universalsolution 120 4.5 Ageometriclimitandcolimit 124 4.6 Howtocalculatecertainlimits 130 4.7 ConfluentcolimitsinSet 143 5 Adjunctions 148 5.1 Adjunctionsdefined 148 5.2 Adjunctionsillustrated 153 5.3 Adjunctionsuncoupled 164 5.4 Theunitandthecounit 170 5.5 Freeandcofreeconstructions 174 5.6 Contravariantadjunctions 186 6 Posetsandmonoidsets 190 6.1 Posetsandcompleteposets 190 6.2 Twocategoriesofcompleteposets 191 6.3 Sectionsofaposet 193 6.4 Thetwocompletions 195 6.5 Threeendo-functorsonPos 197 6.6 Longstringsofadjunctions 199 6.7 TwoadjunctionsforR-sets 202 6.8 Theupperleftadjoint 205 6.9 Theupperadjunction 209 6.10 Thelowerrightadjoint 212 6.11 Theloweradjunction 218 6.12 Somefinalprojects 221 Bibliography 223 Index 224 “9781107010871book” — 2011/8/9 — 16:58 — page vii — #7 Preface AsitsaysonthefrontcoverthisbookisanintroductiontoCategoryTheory.It givesthebasicdefinitions;goesthroughthevariousassociatedgadgetrysuch asfunctors,naturaltransformations,limitsandcolimits;andthenexplainsad- junctions.Thismaterialcouldbedevelopedin50pagesorso,buthereittakes some220pages.Thatisbecausetherearemanyexamplesillustratingthevar- iousnotions,someratherstraightforward,andotherswithmorecontent.More importantly,therearealsoover200exercises.Andperhapsevenmoreimpor- tantly,solutionstotheseexercisesareavailableonline. Thebookisaimedprimarilyatthebeginninggraduatestudent,butthatdoes not mean that other students or professional mathematicians will not find it useful. I have designed the book so that it can be used by a single student or smallgroupofstudentstolearnthesubjectontheirown.Thebookwillmake a suitable text for a reading group. The book does not assume the reader has abroadknowledgeofmathematics.Mostoftheillustrationsuserathersimple ideas,buteverynowandthenamoreadvancedtopicismentioned.Thebook canalsobeusedasarecommendedtextforataughtintroductorycourse. Everymathematicianshouldatleastknowoftheexistenceofcategorythe- ory, and many will need to use categorical notions every now and then. For thosegroupsthisisthebookyoushouldhave.Othermathematicianswilluse categorytheoryeveryday.Thatgrouphastolearnthesubjectsometime,and thisisthebooktostartthatprocess.Ofcourse,themoreadvancedtopicsare notdealtwithhere. Thebookhasbeendevelopedoverquiteafewyears.Severalshortcourses of about 10 hours have been taught (not always by me) using some of the material. In 2007, 2008, and 2009 I gave a course over the web to about a dozenuniversities.ThiswaspartofMAGIC,the Mathematics Access Grid Instruction and Collaboration “9781107010871book” — 2011/8/9 — 16:58 — page viii — #8 viii Preface cooperativeofquiteafewUniversityDepartmentsofMathematicsinEngland andWales.Thatwasaninterestingexperienceandhelpedmetosplitthemate- rialintosmallchunkseachoftherightlengthtofitintoonehour.(Thecourse isstillbeingtaughtbutsomeoneelsehastakenoverthewand.)Ofcourse,the orderinwhichmaterialistaughtneednotbethesameasthewrittenaccount. As someone once said, Mathematics is not a spectator sport. To learn and understand Mathematics you have to get stuck in and get your hands dirty. You have to do the calculations, manipulations, and proofs yourself, not just read the stuff and pretend you understand it. Thus I have included over 200 exercisestohelpwiththisprocess.Ihavealsowrittenamoreorlesscomplete setofsolutionstotheseexercises.Butthesearenotavailableinthebook,forit istooeasysimplytolookupasolution.Whenyoucan’tseehowtodoityou havetosweatabittofindasolution.Someoneelseoncesaidthathorsessweat, gentlemenperspire,andladiesglow.However,Ican’tremembermeetingmany horseswhocoulddomathematicsallthatwell.Inotherwords,althougheffort is important to learn mathematics you also need something else. You need helpeverynowandthen.Thatiswhythereareexercisesandsolutions.These solutionsareavailableat www.cambridge.org/simmons The book is divided into six Chapters, each chapter is divided into several Sections,andafewofthesearedividedintoBlocks(Subsections).Eachchap- tercontainsalistofItems,thatisDefinitions,Lemmas,Theorems,Examples, andsoon.Thesearenumberedbysection.ThusitemX.Y.Z isinChapterX, SectionY,andistheZthiteminthatsection.Whereasectionisdividedinto blockstheitemsarestillnumberedbytheparentsection. Each section contains a selection of Exercises. These are numbered sepa- ratelythroughoutthesection.ThusExerciseX.Y.Z isinChapterX,Section Y, and is the Zth exercise of that section. Again, where a section is divided intoblockstheexercisesarestillnumberedbytheparentsection. OccasionallyyouwillseeawordortwoINTHISFONT.Thisisamentionof aNOTIONthatisdealtwithinmoredetaillater.Youshouldremembertocome backtothisplacewhenyouunderstandthenotion. There are several other books available on this subject. Some of these are introductorytextsandsomearemoreadvanced.Ihavelistedsomeofthemin thebibliography.Noneoftheseareneededwhenreadingthisbook,butsome will certainly help broaden and advance your understanding of the subject. I haverefrainedfrompassingcommentonthesebooks,forIknowthatdifferent people have different tastes. However, you should look around for different accounts.Someofthesewillhelp.
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