Theory and Applications of Categories comma Vol period 20 comma 2008 comma pp period .. 1 endash 4 period \noindentEDITORIALTheory NOTICE and : Applications of Categories , Vol . 20 , 2008 , pp . \quad 1 −− 4 . MAX .. KELLY \ centerline {EDITORIAL NOTICE : } 5 ..Theory JUNE .. and 1930 Applications endash 26 of.. JANUARY Categories , 2007Vol . 20 , 2008 , pp . 1 – 4 . Gregory .. Maxwell .. Kelly was .. solelyEDITORIAL responsible .. for .. NOTICE introducing .. : category theory .. into \ centerlineAustralia at{MAX a t ime\quad whenKELLY the subj} ect was inMAX it s infancy KELLY period .... The Eilenberg hyphen Kelly monograph Closed Categories of 1 966 set5 the JUNE stage for two 1930 more – generations 26 JANUARY of Australian category 2007 \ centerlinetheorists period{Gregory5 \ ..quad ThisJUNE .. Maxwell research\quad .. stream Kelly1930 reached−− was26 ..\ solely maturityquad responsibleJANUARY with .. Max 2007 quoteright for} introducing s .. book .. Basic category .. Concepts .. of Enrichedtheory Category into Theory open parenthesis CUP 1 982 semicolon see TAC Reprints comma No period .. 1 0 closing parenthesis comma and\ hspace nowAustralia finds∗{\ f appli i l l at} hyphenGregory a t ime when\quad theMaxwell subj ect\quad was inKelly it s infancy was \quad .solely responsibleThe Eilenberg\quad - Kellyf o r \quad introducing \quad category theory \quad i n t o cationmonograph in many areas of mathematics comma theoretical physics comma computer architecture comma software \noindentdesignClosed commaAustralia Categories and information at of a 1 t management 966 ime set when the period the stage subj for ect two was more in generations it s infancy of Australian . \ h f i l l categoryThe Eilenberg − Kelly monograph Maxtheorists received all . his This schooling research at Bondi Beach stream where reached he was a student maturity of the with Marist Max ’ s book Basic \noindentBrothersConcepts throughoutClosed of Categories his Enriched primary andCategory of secondary 1 966 Theory set education the ( CUP stage period 1 982 .... for Max ; two see topped TAC more the Reprints generations NSW High , No . of Australian 1 0 ) , category t hSchool e oand r i s Leaving t nows . finds\quad Certificate appliThis Examination - \quad r e overalls e a r c h period\quad .... Hestream went on reached to win in\quad 1 95 1 thematurity University with \quad Max ’ s \quad book \quad Basic \quad Concepts \quad o f EnrichedMedalcation for MathematicsCategory in many areas Theory at the of University mathematics ( CUP 1 of 982 Sydney , ; theoretical see and TACto gain Reprintsphysics the .. James , computer , No King . of\quad architectureIr hyphen1 0 ) , , software and now finds appli − rawangdesign Travelling , and Scholarshipinformation tomanagement study at Cambridge . period .. There he obtained a BA with First \noindentClass HonourscationMax and receivedtwo in Wrightmany all areas quoteright his schooling of s mathematics Prizes at in Bondi .... 1 953 , Beach theoretical comma where .... a he Rayleigh physics was a Prize student , in computer .... of 1 the 955 commaMarist architecture .... and PhD , in software 1 957Brothers semicolon throughout the doctorate his was primary in algebraic and topology secondary under education the supervision . of Max Shaun topped Wylie the period NSW .... Max High \noindenthlineSchooldesign Leaving , Certificate and information Examination management overall . . He went on to win in 1 95 1 the University PublishedMedal on for 2008 Mathematics hyphen 0 1 hyphen at the 9 period University of Sydney and to gain the James King of Ir - \ hspacecirclecopyrt-crawang∗{\ f i Travelling l Ross l }Max Street received Scholarship comma 2008 all toperiod his study schooling Permission at Cambridge to at copy Bondi for . private There Beach use he where granted obtained he period was a BA a with student First of the Marist 1 Class Honours and two Wright ’ s Prizes in 1 953 , a Rayleigh Prize in 1 955 , and PhD in \noindent1 957Brothers ; the doctorate throughout was in algebraic his primary topology and under secondary the supervision education of . Shaun\ h f i l Wylie l Max . topped Max the NSW High \noindent School Leaving Certificate Examination overall . \ h f i l l He went on to win in 1 95 1 the University

\noindent Medal for Mathematics at the University of Sydney and to gain the \quad James King of Ir − rawang Travelling Scholarship to studyPublished at on Cambridge 2008 - 0 1 - 9. .\quad There he obtained a BA with First circlecopyrt − c Ross Street , 2008 . Permission to copy for private use granted . 1 \noindent Class Honours and two Wright ’ s Prizes in \ h f i l l 1 953 , \ h f i l l a Rayleigh Prize in \ h f i l l 1 955 , \ h f i l l and PhD in

\noindent 1 957 ; the doctorate was in algebraic topology under the supervision of Shaun Wylie . \ h f i l l Max

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ centerline { Published on 2008 − 0 1 − 9 . }

\noindent $ circlecopyrt −c $ Ross Street , 2008 . Permission to copy for private use granted . 1 2 .. MAX KELLY \noindentreturned to2 the\quad UniversityMAX KELLY of Sydney in early .. 1 957 as a Lecturer in Pure Mathematics and was promoted to Senior Lecturer in 1 96 1 and to Reader in 1 965 period .. In November 1 960 \noindent returned to the University of Sydney in early \quad 1 957 as a Lecturer in Pure Mathematics Max2 marriedMAX Imogen KELLY Datson period andMy wasreturned first contact promoted to with the to DrUniversity GSenior period ofM Lecturer Sydneyperiod Kelly in in early as 1 a 96 name 1 1 and 957was in asto preparing a Reader Lecturer for in inHonours 1 Pure 965 MathMathematics . \quad hyphenIn November and 1 960 Maxematics marriedwas .. promoted at the Imogen .. Leaving to Datson Senior Certificate Lecturer . Examination in 1 96 period 1 and .. to The Reader practice in was 1 965 to .. . attempt In November.. all past 1 960 papersMax period married .. I st Imogen il l have Datson the blue type. hyphen s et papers for .. 1 959 and 1 960 which declare Professor MyT first periodMy contact G firstperiod contact Room with as Dr with Chief G Dr . Examiner M G . . Kelly M and . Kelly Kelly as a as name one a name of was two was Assessorsin preparing in preparing period for for Honours Honours Math Math− - ematicsMaxematics Kelly\quad quoteright atat the the s love Leaving\quad of categoryLeaving Certificate theory Certificate also Examination consolidated Examination in. the The early practice 1 .960\quad s while wasThe tohe was practice attempt was all to \quad attempt \quad a l l past papersgivingpast lectures . papers\quad on homology .I st I stil theoryil l l have have period the the .. blue blue In an type attempt type - s− et tos papers understand et papers for the 1for s 959 ingular\quad and cohomology 11 960 959 which and declare1 960 which declare Professor Tr . ing GProfessor of . a Room product asT of . Chief G spaces . Room commaExaminer as .. Chief Max and Examinerfound Kelly that as he and could one Kelly notof astwo.. even one Assessors formulate of two Assessors the . questions . he wantedMax to Kelly ask without ’ s love some of basic category concepts theory from category also consolidated theory period in .. the Although early he1 960 heard s while he was Maxof Kelly functorsgiving ’ andlectures s naturallove on of transformations homology category theory theory at .. . Cambridge also In an consolidated attemptfrom the book to understand by in Eilenberg the early the and s ingular 1 960 cohomology s while he was givingSteenrodr ing lectures on of Foundations a product on homology of of Algebraicspaces theory, Topology Max . found comma\quad that heIn met he an Mac could attempt Lane not quoteright to even understand formulate s concept the of the categorical s questions ingular cohomology rproduct inghe of wanted first a productduring to ask lectures of without spaces of Michael some , Atiyah\ basicquad comma conceptsMax found while from Max that category was he a visitor could theory at not MIT .\ inquad lateAlthougheven formulate he heard the questions he1 962 wantedof period functors to .. Max ask and had without natural himself some sotransformations on developed basic concepts some at last ingCambridgefrom ideas category .. in from the field theory the semicolon book . \ byquad .. one EilenbergAlthough concept .. and he he heard ofcalled functorsSteenrod .. quotedblleft and on Foundations natural complex categoriestransformations of Algebraic quotedblright Topology at period\quad, he ..Cambridge While met Mac visit Lane ing from Tulane ’ s the concept .. book University of by categorical Eilenberg .. open parenthesis and New .. Orleans closingSteenrodproduct parenthesis on first Foundations .. in during .. 1 963 lectures hyphen of Algebraic of Michael Topology Atiyah , while , he Max met wasMac a Lane visitor ’ ats concept MIT in late of categorical1 962 product64 comma. Max first .. he had .. during met himself .. Eilenberg lectures so on .. developed at of .. Las Michael .. some Cruces Atiyah last .. giving ing , ideas .. while a .. series inMax the .. was of field lectures a ; visitor .. one on ..concept atdifferential MIT in he .. gradedlate 1categories 962called . \quad which “ complexMax Eilenberg hadcategories and himself Moore had so” . recentlyon While developed invented visit ing semicolonsome Tulane last these ing University were ideas the same\quad things ( Newin the Orleans field ; \quad one concept \quad he c aas l l complex e) d \ inquad categories 1‘‘ 963 complex -period 64 , .. categories Eilenberg he met insisted ’’ Eilenberg that . \quad Max remainWhile at in visit Las the US ing Cruces.. for Tulane another giving year\quad periodU a n i v series e r s i t y \quad ( New \quad Orleans ) \quad in \quad 1 963 − 64Indeed , of\quad lectures .. Eilenberghe \quadon commamet differential .. on\quad the ..Eilenberg spot graded comma .. categories\quad rang Alexat whichHeller\quad .. 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Aftertheist cigar he ed period asked s ix statements ..us The a next few lecture equivalent questions he finished to , the theit axiom becameproof using of clear choice a lemma we , explained I had have done notseen them no elsewhere topology , and proceeded period , so he to changed onMax theprove must spot havethe to equivalence forgotten t each the us precise. general He form completed of topology the lemma five . s of\ incequad the it wasimplicationsSoon not he the cameone in we that to were discuss one lecture product , totally\quad spaces and actedwithout surprised notes that and with we knew only a nothing few squats about staring the out axiom the windowof choice taking . \quad strongerImmediately puffs at the Dr Kelly l istcigar ed. s ix The statements next lecture equivalent he finished to the the proof axiom using of choice a lemma , I explained have not seen them elsewhere , and proceeded . to proveMax the must equivalence have forgotten . \quad the preciseHe completed form of the five lemma of the s ince implications it was not the in one that we one were lecture , totally without notes and with only a few squats staring out the window taking stronger puffs at the c i g a r . \quad The next lecture he finished the proof using a lemma I have not seen elsewhere . Max must have forgotten the precise form of the lemma s ince it was not the one we were EDITORIAL NOTICE .. 3 \ hspaceasked to∗{\ provef i l l in}EDITORIAL the final exam NOTICE period ..\quad Brian Day3 and I became Kelly quoteright s postgraduate students in 1 966 comma the year before he moved to the University of New South Wales as Professor of \noindent asked to prove in the final exam . \quad Brian Day and I became Kelly ’ s postgraduate students Pure Mathematics period .. Brian moved too comma .. while I stayed at .. Sydney periodEDITORIAL .. Our relationship NOTICE with3 inour 1asked supervisor 966 ,to the prove was year very in theformal before final in those exam he moved days . period Brian to the Day University and I became of New Kelly South ’ s postgraduate Wales as Professor students of PureOf coursein Mathematics 1 966 comma , the by year mid . \ beforequad .. 1 971Brian he comma moved moved when to theI too was University at , Macquarie\quad ofwhile andNew Max I South stayed had Wales returned at as\quad to Professor UNSWSydney of Pure . \quad Our relationship with ourfromMathematics supervisor .. Chicago comma was . very.. Brian the .. formal moved formality tooin .. thosehad , gone while days period I stayed . .. Max at .. had Sydney .. arranged .Our .. a .. relationship sabbatical .. at with .. 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\noindent $ This { f t p } : i t { / }ˆ{ em } { / f t p }ˆ{ may } . $ t ac $ be { . mta }ˆ{ a c c e s s e d } ./ { ca } pub ˆ{ @ } http { tac { / } / html }ˆ{ : / / www } { / volumes }ˆ{ . t ac . } { / }ˆ{ mta } { 20 / }ˆ{ . } { not }ˆ{ ca / t ac / } { i c e / 20 }ˆ{ or } { − not }ˆ{ by } { i }ˆ{ anonymous } { c e } . $ pdf \ h f i l l f t p \ h f i l l at THEORY AND APPLICATIONS OF CATEGORIES open parenthesis ISSN 1 20 1 hyphen 56 1 X closing parenthesis will disseminate articles\noindent that THEORY AND APPLICATIONS OF CATEGORIES ( ISSN 1 20 1 − 56 1 X ) will disseminate articles that significantlysignificantly advance advance the study the of study categorical of algebra categorical or methods algebra comma or that methods make significant , or that new make contribu significant hyphen new contribu − tions to mathematical science using categorical methods . \quad The scope of the j ournal includes : \quad a l l areas o f tionsTHEORY to mathematical AND APPLICATIONS science using categorical OF CATEGORIES methods period ( ISSN .. 1The 20 scope 1 - 56 of 1 theX ) j will ournal disseminate includes articles : .. all areas that of pure category theory , including higher dimensional categories ; applications of category theory to algebra , puresignificantly category theory advance comma the includingstudy of categorical higher dimensional algebra or categories methods semicolon , or that applications make significant of category new contribu theory to - algebra comma geometry and topology and other areas of mathematics ; \quad applications of category theory to computer geometrytions to and mathematical topology and science other areasusing of categorical mathematics methods semicolon . The .. applications scope of the of j category ournal includes theory to: computer all areas science , physics and other mathematical sciences ; contributions to scientific knowledge that make use of scienceof pure comma category physics theory and ,other including mathematical higher dimensional sciences semicolon categories contributions ; applications to of scientific category knowledge theory to thatalgebra make , use of categorical methods . categoricalgeometry methods and topology period and other areas of mathematics ; applications of category theory to computer science Articles appearing in the journal have been carefully and critically refereed under the responsibility of Articles, physics appearing and other in the mathematical journal have sciences been carefully ; contributions and critically to scientific refereed knowledge under the that responsibility make use of of categorical members of the Editorial Board . \quad Only papers j udged to be both significant and excellent are accepted membersmethods of the . Articles Editorial appearing Board period in the ..journal Only havepapers been j udged carefully to be and both critically significant refereed and under excellent the are responsibility accepted for publication . for publicationof members periodof the Editorial Board . Only papers j udged to be both significant and excellent are accepted for Full text of the j ournal is freely available in \quad . dvi , \quad P o s t s c r i p t \quad and PDF from the j ournal ’ s server at Fullpublication text of the . j Fullournal text is of freely the javailable ournal is in freely .. period available dvi comma in . dvi .. Postscript , Postscript .. and PDFand PDF from from the j the ournal j ournal quoteright s server at http : / / www . tac . mta . ca / tac / and by ftp . It is archived electronically and in printed paper format . http’ s: serverslash slash at http www : period / / www tac period . tac mta . period mta . ca ca slash / tac tac slash/ and and by by ftp ftp . 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Robert comma Rosebrugh McGill University , Mount : barr Allison at math University period mcgill :period rrosebrugh ca $ @ $ mta . ca TransmittingRichard Blute editors , Universit period e ´ d ’ Ottawa : rblute @ uottawa . ca Lawrence Breen , Universite ´ de Paris \noindent $ T−E−X $ nical editor . Michael Barr , McGill University : barr $ @ $ math . mcgill . ca Richard1 3 : Blutebreen comma@ math Universit . univ acute-e - pari d quoteright s 1 3 . Ottawa fr Ronald : rblute Brown at uottawa , University period of ca North Wales : ronnie Lawrence. profbrown Breen comma ( Universitat ) bt acute-e internet de Paris . 1 c 3 om : breenAurelio at math Carboni period , Universit univ hyphena ` dell pari Insubria s 1 3 period : aurelio fr \noindent Transmitting editors . Ronald. carboni Brown comma@ uninsubria University . of North it Valeria Wales de : ronnie Paiva period , Xerox profbrown Palo Alto .. Research open parenthesis Center : atpaiva closing@p parenthesisa−r c .. bt internet period c om . xerox . com Ezra Getzler , Northwestern University : getzler ( at ) northwestern ( dot ) edu \noindent Richard Blute , Universit $ \acute{e} $ d ’ Ottawa : rblute $@$ uottawa . ca AurelioMartin Carboni Hyland comma , University Universit of Cambridge a-grave dell : InsubriaM . Hyland : aurelio@dpm period−m s . carboni c m − ata uninsubriaac . uk periodP . T it. Johnstone Lawrence Breen , Universit $ \acute{e} $ de Paris 1 3 : breen. $@$ math . univ − p a r i s 1 3 . f r Valeria, University de Paiva of comma Cambridge Xerox : Paloptj Alto@ dpmms Research . Center c a − :m paivaac . at p uk to theAnders power Kock of a-r , University c period xerox of Aarhus period : com Ronald Brown , University of North Wales : ronnie. . profbrown \quad ( at ) \quad bt internet . c om Ezrako Getzler ck @ commaimf . Northwestern au . dk Stephen University Lack : getzler, University open of parenthesis Western Sydney at closing : parenthesiss . lack northwestern@ uws . edu open parenthesis dot closing Aurelio Carboni , Universit $ \grave{a} $ dell Insubria : aurelio . carboni $ @ $ uninsubria . it parenthesis. au eduF . William Lawvere , State University of New York at Buffalo : wlawvere @ ac su . buffalo Valeria de Paiva , Xerox Palo Alto Research Center : paiva $ @ p ˆ{ a−r }$ c . xerox . com Martin. edu HylandJean comma - Louis University Loday , of Universit Cambridgee ´ de : M Strasbourg period Hyland : loday at dp@ tomath the power . u of - m-m strasbg s period . fr c m-aIeke sub period ac period uk Ezra Getzler , Northwestern University : getzler ( at ) northwestern ( dot ) edu P periodMoerdijk T period , University Johnstone of Utrecht comma : Universitymoerdij k of Cambridge@ math . : uu ptj . at nldpmmsSusan period Niefield c a-m , Union sub period College ac :periodnief uk Martin Hyland , University of Cambridge : M . Hyland $ @ dp ˆ{ m−m }$ s . c $ m−a { . }$ ac . uk Andersi el Kock s @ commaunion University . edu ofRobert Aarhus Par : koe ´ ckDalhousie at imf period University au period : pare dk @ mathstat . dal . ca Jiri P . T . Johnstone , University of Cambridge, : ptj $ @ $ dpmms . c $ a−m { . }$ ac . uk StephenRosicky Lack , Masaryk comma University University of : ro Western s i cky Sydney@ math : .. s . period muni lack . at cz uwsBrooke period Shipley edu period , University au of Illinois Anders Kock , University of Aarhus : ko ck $ @ $ imf . au . dk F periodat Chicago William : bshipley Lawvere comma@ math State . uiUniversity c . edu of NewJames York Stasheff at Buffalo , University : wlawvere of North at ac su Carolina period : buffaloj ds period@ edu Stephen Lack , University of Western Sydney : \quad s . lack $@$ uws . edu . au Jeanmath hyphen . uncLouis . Loday edu commaRoss Street Universit , Macquarie acute-e University de Strasbourg : street : loday at@ math math . period mq . u hyphen edu . strasbg au Walter period fr F . William Lawvere , State University of New York at Buffalo : wlawvere $ @ $ ac su . buffalo . edu IekeTholen Moerdijk , York comma University University : tholen of Utrecht@ mathstat : moerdij . k at yorku math . period ca Myles uu period Tierney nl , Rutgers University : t i Jean − Louis Loday , Universit $ \acute{e} $ de Strasbourg : loday $@$ math . u − s t r a s b g . f r Susanerney Niefield@ math comma . Union rutgers College . edu : niefRobert i el s at F union . C . Waltersperiod edu , University of Insubria : robert . walters Ieke Moerdijk , University of Utrecht : moerdij k $ @ $ math . uu . nl Robert@ uninsubria Par acute-e . sub it commaR . J Dalhousie . Wood , UniversityDalhousie University : pare at mathstat : rj wood period@ mathstat dal period . ca dal . ca SusanJiri Rosicky Niefield comma , Masaryk Union CollegeUniversity : ro nief s i cky i at el math s $period @ $ muni union period . cz edu RobertBrooke ParShipley $ comma\acute University{e} { of, Illinois}$ Dalhousie at Chicago : University bshipley at math : pare period ui $ c@ period $ mathstat edu . dal . ca JiriJames Rosicky Stasheff comma , Masaryk University University of North Carolina : ro s : ij ds cky at math $ @ period $ math unc period . muni edu . cz BrookeRoss Street Shipley comma , Macquarie University University of Illinois : .. street at at math Chicago period :mq bshipley period edu period $ @ $ au math . ui c . edu JamesWalter Stasheff Tholen comma , University York University of : North tholen at Carolina mathstat period : j ds yorku $ period@ $ camath . unc . edu RossMyles Street Tierney ,comma Macquarie Rutgers University University : t i : erney\quad at mathstreet period $@$ rutgers period math edu . mq . edu . au WalterRobert FTholen period C , period York Walters University comma : University tholen of $ Insubria @ $ mathstat: robert period . yorku walters .at ca uninsubria period it MylesR period Tierney J period , Wood Rutgers comma University Dalhousie University : t i erney: rj wood $ at @ mathstat $ math period . rutgers dal period . ca edu Robert F . C . Walters , University of Insubria : robert . walters $ @ $ uninsubria . it R . J . Wood , Dalhousie University : rj wood $ @ $ mathstat . dal . ca