Categories for the Working Mathematician

TheUniversityofChicago’sSaundersMacLane,oneofthemostinfluentialAmericanmathematiciansofthe20thcenturyandarecipientoftheNationalMedalofScience,diedThursday,April14,inSanFranciscoafteralongillness.Hewas95.“HewasoneofthemostimportantfiguresintheUniversityofChicagoMathematicsDepartment,orindeedinAmericanmathematics,”DavidEisenbudwroteofMacLaneintheprefaceofthelatter’sautobiography,whichwillbepublishedinlateMaybyAKPetersLtd.Eisenbud,whoreceivedhisPh.D.inMathematicsfromtheUniversityofChicagounderMacLane’ssupervisionin1970,isthepresidentoftheAmericanMathematicalSocietyanddirectoroftheMathematicalSciencesResearchInstituteinBerkeley,Calif.PeterMay,ProfessorinMathematicsattheUniversityofChicago,describedthemultifacetedMacLaneas“extremelyenergetic,dynamic,clear-headed,opinionated,araconteur.”Inhisresearch,“hewasextraordinarilyperceptiveandoriginal,andhewasespeciallystrongasaphilosopherofmathematics,”Maysaid.“WithSammyEilenberghecreatedanewwayofthinkingaboutmathematics.Inalandmark1945paper,theyintroducedandnamedtheconceptsof‘categories,’‘functors’and‘naturaltransformations.’Thelanguagetheyintroducedtheretransformedmodernmathematics,”hesaid.“Infact,averygreatdealofmathematicssincethenwouldquiteliterallyhavebeenunthinkablewithoutthatlanguage.”F.WilliamLawvere,EmeritusprofessorofmathematicsattheStateUniversityofNewYorkatBuffalo,said,“Categorytheoryisstillexplodinginitsinfluenceafter60years,illuminatingandguidingthedevelopmentofpracticallyeveryoneofthemanyvariedfieldsofmathematics.”Categorytheorywasfirstdevelopedasalanguagetodescribetransformationsfromoneareaofmathematicsintoanother,althoughitlaterdevelopedintoafieldofstudyinitsownright.SaidMay,“MacLanewasoneofthepioneersofalgebraictopology,asubjectinwhichonetransforms,ordescribes,spatialstructures,whichonefirstseesintermsofshapes—spaceswithholes,likeadoughnut,say—intoalgebraicstructures,withadditionandmultiplication,inwhichonecandocalculations.”Thedevelopmentofcategorytheoryandalgebraictopologywasaccompaniedbythedevelopmentofanothersubject,homologicalalgebra,akindofalgebrathatplaysaprominentroleinalgebraictopologyandotherbranchesofmathematics.“MacLaneplayedamajorroleinestablishingthefoundationsoftheseandrelatedareasofmodernmathematics,andhewrotefamoustextsthataresomeofthemostreadableaccountsofthesefields,”Maysaid.Categorytheorywasatfirstperceivedbysomemathematiciansastooabstractforpracticalmathematics,Maysaid.Consequently,MacLanetitledhisintroductorybooktothefieldCategoriesfortheWorkingMathematician.PeterJohnstone,professorofthefoundationsofmathematicsattheUniversityofCambridge,alsolaudedMacLaneforhislegacyasamathematicalhistorian.“Inthemanypapershewroteonhistoricaltopicsinhislateryears,hehasleftauniquebodyofmaterialforfuturehistoriansof20th-centurymathematics,writtenbysomeonewhowasthereatthetimeandwhoknewwhatitwasliketobeworkingatthecuttingedgeofmathematicalresearch,”Johnstonesaid.MacLanealsosteerednationalscienceandmathematicspolicythroughhisworkonnumerousboards,andguideddozensofstudentstomathematicalcareers.Inthesupportofscientificresearch,MacLaneservedasvicepresidentoftheNationalAcademyofSciencesandtheAmericanPhilosophicalSociety,andaspresidentoftheMathematicalAssociationofAmericaandtheAmericanMathematicalSociety.AspresidentoftheMathematicalAssociationofAmericainthe1950s,hebeganthefirstactivityofthatorganizationtowardimprovementintheteachingofmodernmathematics.HewasamemberoftheNationalScienceBoardfrom1974to1980,providingsciencepolicyadvicetotheU.S.government.In1976,heledadelegationofmathematicianstothePeople’sRepublicofChinatoexaminetheconditionsaffectingthedevelopmentofmathematicsinthatnation.“Saunders’principalcharacteristicwashisimmenselystrongsenseofdutyandservice,”Johnstonesaid.“Hefeltitwashisdutytotheprofessiontowhichhewasprivilegedtobelong,eventhoughI’msurehefounditlesscongenialworkthanactuallydoingmathematics.”ForatimetherewerethreemembersoftheUniversityofChicagoMathematicsDepartmentwhohadreceivedtheirPh.D.sunderMacLane’ssupervision.ThefirstofthethreewasAlfredPutnam,whostudiedunderMacLaneatHarvardUniversityin1942.TheothertwowereJohnThompson,whoreceivedhisPh.D.in1959,andArunasLiulevicius,whodidlikewisein1960,bothfromtheUniversityofChicago.ThompsonreceivedtheFieldsMedal,sometimesreferredtoastheNobelPrizeofmathematics,in1970.AnotherprominentstudentofMacLane’s,IrvingKaplansky,alsoservedontheUniversityofChicagofaculty.KaplanskywasMacLane’sfirstPh.D.student,receivinghisdegreeatHarvardin1941.KaplanskyservedasChairmanofChicago’sDepartmentofMathematicsandasdirectorofBerkeley’sMathematicalSciencesResearchInstitute,andisamemberoftheNationalAcademyofSciences.MacLanesupervised39Ph.D.studentsatChicago,arecordsurpassedbyonlythreeotherprofessorsinthehistoryoftheUniversity’sMathematicsDepartment.HislastPh.D.student,SteveAwodey,completedhisdegreeinlogicin1997.Theninhis80s,MacLanewasstillenergetic,alwaysinsistingupontakingthestairstohisthird-floorofficeintheRyersonPhysicalLaboratory,Awodeyrecalled.MacLanewas“alinktoapastworldofmathematicsthat’sgonenow,”saidAwodey,anassociateprofessorinphilosophyatCarnegieMellonUniversity.MacLanestudiedlogicandmathematicsasaPh.D.studentunderseveralworld-renownedmathematiciansintheearly1930sinGöttingen,Germany,includingDavidHilbert,EmmyNoetherandHermannWeyl.AtthetimeGöttingenwastheworld’scenterofmathematicalresearch,andMacLaneoftentoldstoriesofthegreatfiguresheencounteredthere.MacLanewasborninNorwich,Conn.,onAug.4,1909.Heearnedhisbachelor’sdegreefromYaleCollegein1930,andhismaster’sfromtheUniversityofChicagoin1931.Hisfirstscientificpaperwaspublishedduringthisperiod,inphysics.Itwasco-authoredbyIrvingLangmuir,whoreceivedthe1932NobelPrizeinchemistry.ForhisPh.D.,whichMacLanereceivedin1934fromtheMathematischesInstitutofGöttingen,Germany,hestudiedundertwoprominentmathematicians,HermannWeylandPaulBernays.MacLaneworkedasamathematicsinstructoratHarvardandCornelluniversitiesandattheUniversityofChicagofrom1934to1938.HeservedagainontheHarvardfacultyfrom1938to1947,andatChicagobeginningin1947.In1944and1945,healsodirectedColumbiaUniversity’sAppliedMathematicsGroup,whichwasinvolvedinthewareffort.MacLanewasChairmanoftheMathematicsDepartmentatChicagofrom1952to1958,takingthereinsfromMarshallStoneatatimemanyconsiderthehighpointinthedepartment’shistory.HewasappointedtheMaxMasonDistinguishedServiceProfessorinMathematicsin1963,andbecameProfessorEmeritusin1982.Hewastheauthororco-authorofmorethan100researchpapersandsixbooks:ASurveyofModernAlgebra(1941);Homology(1963);Algebra(1967);CategoriesfortheWorkingMathematician(1971);Mathematics,FormandFunction(1985);andSheavesinGeometryandLogic:AFirstIntroductiontoToposTheory,withIekeMoerdijk,(1992).HisASurveyofModernAlgebra,writtenwithGarrettBirkhoff,wasforyearstheleadingtextbookinitsfieldandwasrepublishedbyAKPetersLtd.in1997.MacLanewaselectedtotheNationalAcademyofSciencesin1949.Hereceivedthenation’shighestawardforscientificachievement,theNationalMedalofScience,in1989.MacLanereceivedtwoGuggenheimFellowshipsandvisitedAustraliaasaFulbrightScholar.HealsoreceivedhonorarydegreesfromPurdueUniversity,YaleUniversityandtheUniversityofGlasgow,amongothers.OtherhonorsincludeboththeChauvenetPrizeandtheDistinguishedServiceawardoftheMathematicalAssociationofAmerica,theSteeleCareerPrizeoftheAmericanMathematicalSociety,andhonoraryfellowshipintheRoyalSocietyofEdinburgh.TheUniversityofChicagoAlumniAssociationpresentedMacLanetheNormanMacleanFacultyAwardin2003.TheMacleanAwardrecognizesemeritusprofessorsorseniorfacultymemberswhohavemadeoutstandingcontributionstoteachingandtotheexperienceofstudentlifeoncampus.MacLaneissurvivedbyhiswidow,Osa,ofSanFrancisco,andtwodaughters,GretchenMacLaneofNewYorkCityandCynthiaHayofLondon;onegrandson,WilliamHay,ofLondon;threestepchildren,WilliamSegalofWashington,D.C.;AndrewSegalofSantaFe,N.M.;andKarenSegalofSanFrancisco;,andfivestep-grandchildren.MacLane’sfirstwife,DorothyJonesMacLane,diedin1985.ServicesforthefamilywereheldinSanFranciscoApril19.TheMathematicalSciencesResearchInstituteinBerkeleywillholdamemorialat4:30p.m.Wednesday,May4.