Proof-Theoretical CoherenceKostaDosenandZoranPetri cRevisedVersionSeptember2007Theversionpostedherediersfromtheversionprintedin2004byKingsCollegePub-lications(CollegePublications, London). Besidessomerelativelyslightadditionsandcorrections, includingasmall numberof additional references, amajorcorrectioncon-cerningcoherencefordicartesianandsesquicartesiancategories, postedalreadyintherevised versions of May 2006 and March 2007, may be found in 9.6. The present versiondiersfromtheversionofMarch2007byhavingasimplerproofofcoherenceforlatticecategories in 9.4, and a major correction concerning coherence for lattice categories withzero-identityarrowsin 12.5.PrefaceThisisabookincategorial(orcategorical)prooftheory,aeldofgeneralprooftheoryattheborderbetweenlogicandcategorytheory. Inthiseldthe language, more than the methods, of category theory is applied to proof-theoretical problems. Propositionsareconstruedasobjectsinacategory,proofsasarrowsbetweentheseobjects,andequationsbetweenarrows,i.e.commutingdiagramsofarrows,arefoundtohaveproof-theoreticalmean-ing. Theyprovideareasonablenotionof identityof proofs byequatingderivationsthatarereducedtoeachotherinacut-eliminationornormal-ization procedure, or they may be involved in nding a unique normal formforderivations.Toenter intocategorial proof theoryonecrosses what shouldbethewatershed between proof theory and the rest of logic. We are not interestedanymoreinprovabilityonlynamely, intheexistenceof proofswhichcorrespondstoaconsequencerelationbetweenpremisesandconclusions.Wehaveinsteadaconsequencegraph, wheretheremaybemorethanonedierent proof with the same premise and the same conclusion. We describetheseapparentlydierentproofs,codethembytermsforarrows,andndthat some descriptions standforthe same proof,i.e.the same arrow,whileothersdonot. Ourconsequencegraphisacategory, oftenof akindthatcategoristshavefoundimportantfortheirownreasons.On the other hand,in categorial proof theory proof-theoretical,syntac-tical, methods are applied to problems of category theory. These are mainlymethodsofnormalizinginthestyleofGentzenorofthelambdacalculus.(In this book, conuence techniques like those in the lambda calculus domi-nate in the rst part, while cut elimination dominates in the second, bigger,part.) Thissyntactical standpointissomethingthatmanycategoristsdonot favour. Insteadof dealingwithlanguage, theyprefer toworkas iftheydealtwiththethingsthemselves. Wendthatforsomeproblemsofcategorytheory,andinparticularforso-calledcoherenceproblems,whichmakethesubjectmatterof thisbook, payingattentiontolanguageisofgreathelp.Thetermcoherencecovers incategorytheorywhat fromalogicalpointof viewwouldbecalledproblemsof completeness, axiomatizabilityanddecidability. Dierentauthorsputstressondierentthings. Forourownpurposes wewill xaparticular notionof coherence, whichagreescompletely with Mac Lanes usage of the term in [99], the primordial paperoncoherence.Inthe1960s, atthesametimewhencoherencestartedbeinginvesti-gated in category theory, the connection between category theory and logiciii Prefacewasestablished, mainlythroughLawveresideas(see[94]). Therootsofcategorialprooftheorydatefromthesameyearstheycanbefoundinaseriesof papersbyLambek: [84], [85], [86] and[87]. LambekintroducedGentzensproof-theoretical methodsincategorytheory, whichMacLaneandKellyexploitedin[81] tosolveamajorcoherenceproblem(seealso[101]).There are not many books in categorial proof theory. The early attemptto present matters in [127] has shortcomings. Proofs are not systematicallycoded by terms for arrows;only the sources and targets of arrows are men-tionedmostof thetime, andtoomuchworkislefttothereader. Someclaimsareexcessivelydiculttoverify,andsomearenotcorrect(see[69],[12], Section3, and[14], Section1). Lambeks andScotts book[90] isonlypartlyaboutcategorial prooftheoryandcoherence, understoodasadecidabilityproblemforequalityof arrowsincartesianclosedcategories.(Justashortchapterof [128], Chapter8, touchesuponthistopic.) Theonlyremainingbookincategorialprooftheoryweknowabout,[38],isde-voted to showing that cut elimination characterizes fundamental notions ofcategorytheory,inparticularthenotionofadjunction. Somepartsofthatbook(Sections4.10and5.9)areaboutcoherence.Papers in categorial logic often touch upon this or that point of catego-rial prooftheory, butarenotveryoftenspecicallywithintheeld. Andeven when they are within this eld,some authors prefer to advertise theirworkassemantical. Itshouldbeclear,however,thatthisisnotseman-tics in the established model-theoretical sensethe sense in which the wordwas used in logic in the twentieth century. We nd this semantics of proofsmoreproof-theoreticalthanmodel-theoretical.We will try to cover with the references in our book not the whole liter-atureofcategorial prooftheory, butonlypapersrelevanttotheproblemstreated. To acknowledge more direct inuences,we would like,however,tomentionattheoutsetafewauthorswithwhomwehavebeenincontact,andwhoseideasaremoreorlessclosetoours.First, Jim Lambeks pioneering and more recent work has been for us, asformanyothers,asourceofinspiration. MaxKellyspapersoncoherence(see [77], [78], [54] and[79]) are less inuencedbylogic, thoughlogicalmattersareimplicitinthem. Sergei Solovievscontributionstocategorialprooftheory(see[118],[119]and[120])andDjordjeCubrics (see [28],[29]and [30]) are close to our general concerns, though they do not deal exactlywiththesubject matter of this book; thesameapplies tosomeworkofAlexSimpson(inparticular,[117]).WeextendRobertSeelysandRobinCockettscategorialpresentationofafragmentoflinearlogic,baseduponwhattheycalllinear,aliasweak,distribution(see[22]; otherpaperswillbecitedinthebodyofthebook),Preface iiiwhich we call dissociativity. This is an associativity principle involving twooperations, whichinthecontext of lattices delivers distribution. WhileCockettandSeelyareconcernedwithdissociativityasitoccursinlinearlogic,andenvisagealsoapplicationsinthestudyofintuitionisticlogic,wehavebeenorientedtowards thecategoricationof classical propositionallogic. ThesubtitleofourbookcouldbeGeneralprooftheoryofclassicalpropositional logic. We wouldhave put this subtitle were it not thatagreat part of thebookis about fragments of this proof theory, whicharefragmentsof theproof theoryof otherlogicstoo, andarealsoof anindependent interest for category theory. Besides that,we are not sure ourtreatmentofnegationinthelastchapterisasconclusiveaswhatprecedesit. (Wealsopreferashorterandhandiertitle.)Proofs intheconjunctive-disjunctive fragment oflogic,which is relatedto distributive lattices,may, but need not,be taken to be the same in clas-sicalandintuitionisticlogic,andtheyarebetternottakentobethesame.Classical prooftheoryshouldbebasedonplural (multiple-conclusion)se-quents, whileintuitionisticprooftheory, thoughitmaybepresentedwithsuchsequents, ismoreoften, andmorenaturally, presentedwithsingular(single-conclusion) sequents. By extending Cocketts and Seelys categorialtreatment of dissociativity, wepresent inthecentral part of thebookacategorication, i.e. ageneralizationincategorytheory, of thenotionofdistributivelattice, whichgivesaplausiblenotionof identityof proofsinclassical conjunctive-disjunctivelogic. ThisnotionisrelatedtoGentzenscut-eliminationprocedureinaplural-sequentsystem. Bybuildingfurtheronthat, attheendof thebookweprovideaplausiblecategoricationofthenotionof Booleanalgebra, whichgivesanontrivial notionof identityofproofsforclassicalpropositionallogic,alsorelatedtoGentzen.It is usually considered that it is hopeless to try to categorify the notionof Booleanalgebra, becauseall plausiblecandidatesbasedonthenotionofbicartesianclosedcategory(i.e.cartesianclosedcategorywithniteco-products)leduptonow toequatingallproofswiththesamepremisesandconclusions. In our Boolean categories, which are built on another base, thisis not the case. The place where in our presentation of the matter classicalandintuitionisticprooftheorypartwaysisinunderstandingdistribution.In intuitionistic proof theory distribution of conjunction over disjunction isanisomorphism, whiledistributionofdisjunctionoverconjunctionisnot.This is how matters stand in bicartesian closed categories. We take that inclassicalprooftheoryneitherofthesedistributionsisanisomorphism,andrestoresymmetry,typicalforBooleannotions.Wereachour notionof Booleancategoryverygradually. This grad-ual approachenables us toshortencalculations at latter stages. More-over, alongthewayweprovecoherenceforvariousmoregeneral notionsiv Prefaceof category, enteringintothenotionof Booleancategoryorrelatedtoit.Coherence is understood in our book as the existence of a faithful structure-preserving functor from a freely generated category, built out of syntacticalmaterial,intothecategorywhosearrowsarerelationsbetweenniteordi-nals. Thisisalimitednotionofcoherence, andourgoal istoexplorethelimitsof thisparticularnotionwithintherealmof classical propositionallogic. Weareawarethatothernotionsof coherenceexist, andthatevenour notioncanbegeneralizedbytakinganother categoryinsteadof thecategorywhosearrowsarerelationsbetweenniteordinals. Theseothernotionsandthesegeneralizationsare,however,outsidetheconnesofourbook,and we will mention them only occasionally (see,in particular, 12.5and 14.3)MacLanesprimordial coherenceresultsformonoidal andsymmetricmonoidal categoriesin[99] areperfectlycoveredbyournotionof coher-ence. Whentheimageof thefaithful functorisadiscretesubcategoryofthecategorywhosearrowsarerelationsbetweenniteordinals,coherenceamounts to showing that the syntactical category is a preordering relation,i.e. thatall diagramscommute. Thisisthecasesometimes, butnotal-ways,andnotinthemostinterestingcases. MacLanescoherenceresultsare scrutinized in our book, and new aspects of the matter are made mani-fest. We also generalize previous results of [72] (Section 1) on strictication,i.e. on producing equivalent categories where some isomorphisms are turnedintoidentityarrows. Ourstricticationisuseful,becauseitfacilitatestherecordingoflengthycalculations.Forcategorieswithdissociativity,whichcoverproofsinthemultiplica-tiveconjunctive-disjunctivefragmentoflinearlogic,andalsoproofsintheconjunctive-disjunctivefragmentof classical logic, weprovidenewcoher-enceresults, andweprovecoherencefor our Booleancategories. Thesecoherencetheorems,whicharethemainresultsofthebook,yieldasimpledecisionprocedurefortheproblemwhetheradiagramofcanonicalarrowscommutes,i.e.fortheproblemwhethertwoproofsareidentical.Themostoriginal contributionof ourbookmaybethatwetakeintoaccountunion, oraddition, of proofsinclassical logic. Thisoperationonproofswiththesamepremiseandsameconclusionisrelatedtothemixprincipleof linearlogic. ItplaysanimportantroleinourBooleancate-gories, and brings them close to linear algebra. Taking union of proofs intoaccountsavesGentzenscut-eliminationprocedureforclassical logicfromfallingintotriviality, asfarasidentityofproofsisconcerned. Thismodi-edcuteliminationisthecornerstoneoftheproofofourmaincoherencetheoremforclassicalpropositionallogic.Wetakeintoaccountalsothenotionofzeroproof, anotionrelatedtounion of proofsa kind of dual of it. With union of proofs hom-sets becomePreface vsemilatticeswithunit,butweenvisagealsothattheybejustcommutativemonoids,asinadditiveandabeliancategories. Zeroproofs,whicharelikealeapfromanypremisetoanyconclusion, aremappedintotheemptyrelationinestablishingcoherence. Althoughtheyenableustoproveany-thingasfarasprovabilityisconcerned,theyareconservativewithrespecttothepreviouslyestablishedidentityofproofsinlogic. Wewillshowthatenvisaging zero proofs is useful. It brings logic closer to linear algebra, andfacilitatescalculations. Wendalsothatthenotionofzeroproofmaybepresent in logic even when we do not allow passing from any premise to anyconclusion,butrestrictourselvestothetypesoftheacceptabledeductionsconnectingpremises andconclusions, i.e. sticktoprovabilityinclassicallogic. Negationmaybetiedtosuchrestrictedzeroproofs.Zeroproofs resemblewhat Hilbert calledideal mathematical objects,likeimaginarynumbers or points at innity. If our concernis not withprovability, but withproofsnamely, identityof proofszeroproofs areuseful andharmless. Wedont thinkwehaveexhaustedtheadvantagesoftakingthemintoaccountingeneralprooftheory. Webelieve,however,wehavefullledtoagreatextentthepromisesmadeintheprogrammaticsurvey[40] (summarizeduptoapointintherstchapterof thebook),whichprovidesfurtherdetailsaboutthecontextofourresearch.We suppose our principal public should be a public of logicians, such aswe are, but we would like no less to have categorists as readers. So we havestrivedtomakeourexpositionself-contained, bothonthelogical andonthe categorial side. This is why we go into details that logicians would takeforgranted,andintootherdetailsthatcategoristswouldtakeforgranted.Only for the introductory rst chapter, whose purpose is to give motivation,andforsomeasides, inparticularattheveryend, werelyonnotionsnotdenedinthebook,butinthestandardlogicalandcategorialliterature.Wesupposethattheresultsofthisbookshouldbeinterestingnotonlyfor logic and category theory, but also for theoretical computer science. Wedonot control verywell, however, thequicklygrowingliteratureinthiseld, andwewill refrainfromenteringintoit. Wedonotpretendtobeexpertsinthatarea. Someoftheinvestigationsofproofsofclassicallogicthatappearedsince1990inconnectionwithmodaltranslationsintolinearlogicorwiththelambda-mucalculus, inwhichthemotivation, thestyleandthejargonof computersciencedominate, seemtobeconcernedwithidentityofproofs, butitisnotcleartoushowexactlytheseconcernsarerelatedtoours. Weleaveforotherstojudge.This is more a research monograph than a textbook, but the text couldserve nevertheless as the base for agraduate course incategorial prooftheory. Weprovideafterthenalchapteralistofproblemsleftopen. Toassistthereader, wealsoprovideattheendof thebookalistof axiomsvi Prefaceanddenitions, andalist of categories treatedinthe book(whicharequitenumerous), togetherwithchartsforthesecategoriesindicatingthesubcategoryrelationsestablishedbyourcoherenceresults.WewouldliketothankinparticularAlexSimpsonandSergeiSolovievforencouraginganduseful commentsonthepreprintof thisbook, whichwasdistributedsinceMay2004. Wewouldliketothankalsoothercol-leagues who read this preprint and gave compliments on it,or helped us inanothermanner.DovGabbaywasextremelykindtotakecareof thepublishingof thebook. WeareverygratefultohimandtoJaneSpurrfortheireortsandeciency.The results of this book were announced previously in a plenary lectureattheLogicColloquiuminM unsterinAugust2002, andinatalkattheInternational CongressMASSEEinBorovetsinSeptember2003,withthesupportoftheAlexandervonHumboldtFoundation. WeareindebtedtoSlobodan Vujosevic and Milojica Jacimovic for the invitation to address theEleventhCongressof theMathematiciansof SerbiaandMontenegro, heldinPetrovacinSeptember2004,withatalkintroducingmatterstreatedinthebook. WehadtheoccasiontogivesuchintroductorytalksalsoattheLogic Seminar in Belgrade in the last two years and, thanks to MariangiolaDezani-Ciancaglini andtheTypesprojectof theEuropeanUnion, attheTypesconferenceinJouy-en-JosasinDecember2004.WewouldliketothankwarmlytheMathematicalInstituteoftheSer-bianAcademyof SciencesandArtsinBelgradeandtheFacultyof Phi-losophyoftheUniversityofBelgradeforprovidingconditionsinwhichwecouldwritethisbook. OurworkwasgenerouslysupportedbyaprojectoftheMinistryofScienceofSerbia(1630: RepresentationofProofs).Belgrade,December2004CONTENTSPreface iChapter1. Introduction 11.1. Coherence 11.2. Categorication 61.3. TheNormalizationConjectureingeneralprooftheory 101.4. TheGeneralityConjecture 151.5. Maximality 241.6. Unionofproofsandzeroproofs 261.7. Strictication 29Chapter2. SyntacticalCategories 332.1. Languages 342.2. Syntacticalsystems 362.3. Equationalsystems 392.4. Functorsandnaturaltransformations 422.5. Denableconnectives 442.6. Logicalsystems 472.7. Logicalcategories 512.8. C-functors 532.9. ThecategoryRel andcoherence 59Chapter3. Strictication 653.1. Stricticationingeneral 653.2. Directstrictication 783.3. Stricticationanddiversication 84Chapter4. AssociativeCategories 874.1. Thelogicalcategories K 884.2. Coherenceofsemiassociativecategories 894.3. Coherenceofassociativecategories 934.4. Associativenormalform 964.5. Stricticationofassociativecategories 984.6. Coherenceofmonoidalcategories 1014.7. Stricticationofmonoidalcategories 103viiviii ContentsChapter5. SymmetricAssociativeCategories 1075.1. Coherenceofsymmetricassociativecategories 1075.2. ThefaithfulnessofGH 1105.3. Coherenceofsymmetricmonoidalcategories 112Chapter6. BiassociativeCategories 1156.1. Coherenceofbiassociativeandbimonoidalcategories 1156.2. Formsequences 1176.3. Coherenceofsymmetricbiassociativecategories 1176.4. Coherenceofsymmetricbimonoidalcategories 1196.5. ThecategoryS121Chapter7. DissociativeCategories 1277.1. Coherenceofdissociativecategories 1287.2. Netcategories 1327.3. Coherenceofnetcategories 1337.4. Netnormalform 1427.5. Coherenceofsemidissociativebiassociativecategories 1437.6. Symmetricnetcategories 1457.7. CuteliminationinGDS 1487.8. InvertibilityinGDS 1567.9. Linearlydistributivecategories 163Chapter8. MixCategories 1678.1. Coherenceofmixandmix-dissociativecategories 1678.2. Coherenceofmix-biassociativecategories 1698.3. Coherenceofmix-netcategories 1738.4. Coherenceofmix-symmetricnetcategories 1768.5. Coherenceofmix-symmetricbiassociativecategories 182Chapter9. LatticeCategories 1859.1. Coherenceofsemilatticecategories 1859.2. Coherenceofcartesiancategories 1919.3. Maximalityofsemilatticeandcartesiancategories 1949.4. Coherenceoflatticecategories 1999.5. Maximalityoflatticecategories 2059.6. Coherencefordicartesianandsesquicartesiancategories 2079.7. Relativemaximalityofdicartesiancategories 213Contents ixChapter10. Mix-LatticeCategories 21910.1. Mix-latticecategoriesandanexample 21910.2. Restrictedcoherenceofmix-latticecategories 22310.3. Restrictedcoherenceofmix-dicartesiancategories 227Chapter11. DistributiveLatticeCategories 23111.1. DistributivelatticecategoriesandtheirGentzenization 23211.2. Cuteliminationin D 24611.3. Coherenceofdistributivelatticecategories 26311.4. Legitimaterelations 26811.5. Coherenceofdistributivedicartesiancategories 270Chapter12. Zero-LatticeCategories 27512.1. Zero-latticeandzero-dicartesiancategories 27612.2. Coherenceofzero-latticeandzero-dicartesiancategories 28212.3. Maximalityofzero-latticeandzero-dicartesiancategories 28512.4. Zero-latticeandsymmetricnetcategories 28612.5. Zero-identityarrows 287Chapter13. Zero-MixLatticeCategories 29513.1. Coherenceofzero-mixlatticecategories 29613.2. Zero-mixlatticeanddistributivelatticecategories 30113.3. Coherenceofzero-mixdicartesiancategories 30413.4. ThecategorySemilat306Chapter14. CategorieswithNegation 30914.1. DeMorgancoherence 31014.2. Booleancoherence 31614.3. Booleancategories 32214.4. Concludingremarks 328ProblemsLeftOpen 331ListofEquations 332ListofCategories 345Charts 354Bibliography 359Index 371Chapter1IntroductionInthisintroductorychapterweprovideinaninformalmannermotivationfor the main themes of the book, without giving an exhaustive summary ofits content (such summaries are provided at the beginning of every chapter).Agreatdealofthechapter(1.3-6)isbasedonthesurvey[40].While inthe bodyof the book, startingfromthe next chapter, ourexposition, except for some asides, will be self-contained, both from a logicaland from a categorial point of view, here we rely on some acquaintance withprooftheory(whichthereadermayhaveacquiredinclassictextslike[60],[111] and[82], Chapter15, orinthemorerecenttextbook[128]), andonsomenotionsof categorytheory(whichmaybefoundin[100] and[90]).Many, but not all, of the notions we needfor this introductionwill bedenedlaterinthebook.Tohavereadthepresentchapterisnotessentialforreadingtherestofthebook. AreaderimpatientformoreprecisioncanmovetoChapter2,wherethebookreallystarts,andreturntothisintroductionlateron.1.1. CoherenceIt seems that what categorists call coherence logicians would, roughly speak-ing, call completeness. This is the questionwhether we have assumedfor aparticular brandof categories all theequations betweenarrows weshouldhaveassumed. Completenessneednotbeunderstoodhereascom-pleteness withrespect tomodels. Wemayhaveasyntactical notionof12 CHAPTER1. INTRODUCTIONcompletenesssomethingakintothe Post completeness of the classicalpropositionalcalculusbutoftensomesortofmodel-theoreticalcomplete-ness is implicit in coherence questions. Matters are made more complicatedbythefactthatcategoristsdonotliketotalkaboutsyntax, anddonotperceivetheproblemasbeingoneofndingamatchbetweensyntaxandsemantics. Theydonottalkofformalsystems,axiomsandmodels.Moreover, questions that logicians wouldconsider tobequestions ofdecidability, whichisnotthesameascompleteness, areinvolvedinwhatcategorists call coherence. A coherence problem often involves the questionof decidingwhether twoterms designatethesamearrow, i.e. whether adiagramof arrowscommuteswewill call thisthecommutingproblemandsometimesitmayinvolvethequestionofdecidingwhetherthereisinacategoryanarrowofagiventype, i.e. withagivensourceandtargetwewillcallthisthetheoremhoodproblem(cf.[38],Sections0.2and4.6.1).Coherenceisunderstoodmostlyassolvingthecommutingproblemin[90](see p. 117, which mentions [84] and [85] as the origin of this understanding).The commuting problem seemsto beinvolvedalsointheunderstanding ofcoherenceof[79](Section10).Completenessanddecidability,thoughdistinct,are,ofcourse,notfor-eigntoeachother. Acompleteness proof withrespect toamanageablemodelmayprovide,moreorlessimmediately,toolstosolvedecisionprob-lems. Forexample, thecompletenessproof fortheclassical propositionalcalculuswithrespecttothetwo-elementBooleanalgebraprovidesimme-diatelyadecisionprocedurefortheoremhood.The simplest coherence questions are those where it is intended that allarrowsofthesametypeshouldbeequal,i.e.wherethecategoryenvisagedisapreorder. Theoldestcoherenceproblemisofthatkind. Thisproblemhastodowithmonoidal categories, andwassolvedbyMacLanein[99](where early related work by Stashe and D.B.A. Epstein is mentioned; see[122]forhistoricalnotes,andalso[123],AppendixB,co-authoredwithS.Shnider). Themonoidal categoryfreelygeneratedbyasetofobjectsisapreorder. So Mac Lane could claim that showing coherence is showing thatall diagramscommute. WeprovideinChapter4adetailedanalysisofMacLanescoherenceresultformonoidalcategories.In cases where coherence amounts to showing preorder, i.e. showing that1.1. Coherence 3fromagivensetofequations, assumedasaxioms, wecanderiveall equa-tions (providedtheequatedterms areof thesametype), fromalogicalpointofviewwehavetodowithaxiomatizability. Wewanttoshowthatadecidablesetofaxioms(andwewishthissettobeassimpleaspossible,preferablygivenbyanitenumberofaxiomschemata)deliversallthein-tendedequations. Ifpreorderisintended,thenallequationsareintended.Axiomatizability is in general connected with logical questions of complete-ness, and a standard logical notion of completeness is completeness of a setofaxioms. Wherealldiagramsshouldcommute, coherencedoesnotseemtobeaquestionofmodel-theoreticalcompleteness,buteveninsuchcasesitmaybeconceivedthatthemodelinvolvedisadiscretecategory(cf.theendof 2.9).Categorists are interestedinaxiomatizations that permit extensions.Theseextensionsareinanewlanguage, withnewaxioms, andsuchex-tensionsoftheaxiomsofmonoidalcategoriesneednotyieldpreordersanymore. Categoristsarealsointerested,whentheylookforaxiomatizations,innding the combinatorial building blocks ofthe matter. The axioms aresuch building blocks, as in knot theory the Reidemeister moves are the com-binatorial building blocks of knot and link equivalence (see [97], Chapter 1,oranyothertextbookinknottheory).InMacLanes secondcoherenceresult of [99], whichhas todowithsymmetricmonoidal categories, it is not intendedthat all equations be-tweenarrowsofthesametypeshouldhold. WhatMacLanedoescanbedescribedinlogical termsinthefollowingmanner. Ontheonehand, hehasanaxiomatization, and, ontheotherhand, hehasamodel categorywherearrowsarepermutations; thenheshowsthathisaxiomatizationiscompletewithrespecttothismodel. Itisnowonderthathiscoherenceproblemreducestothecompletenessproblemfortheusualaxiomatizationofsymmetricgroups.Algebraists do not speak of axiomatizations, but of presentations by gen-eratorsandrelations. All theaxiomatizationsinthisbookwill bepurelyequationalaxiomatizations,asinalgebraicvarieties. Such were the axiom-atizationsof[99]. Categoriesarealgebraswithpartial operations, andwearehereinterestedintheequationaltheoriesofthesealgebras.InMacLanescoherenceresultsformonoidalandsymmetricmonoidal4 CHAPTER1. INTRODUCTIONcategories one has to deal only with natural isomorphisms. Coherence ques-tionsintheareaof n-categoriesareusuallyrestrictedlikewisetonaturalisomorphisms(see[96]). However, inthecoherenceresultforsymmetricmonoidal closedcategoriesof[81] therearealreadynatural anddinaturaltransformationsthatarenotisomorphisms.Anatural transformationistiedtoarelationbetweentheargument-places of the functor in the source and the argument-places of the functor inthetarget. Thisrelationcorrespondstoarelationbetweenniteordinals,andincomposingnaturaltransformationswecomposetheserelations(see2.4and 2.9). Withdinaturaltransformationsthematterismorecompli-cated,andcompositionposesparticularproblems(see[109]). Inthisbookwe deal with natural transformations, and envisage only in some commentscoherence for situations where we do not have natural transformations. Ourgeneral notionofcoherencedoesnot, however, presupposenaturalityanddinaturality.Our notion of coherence result is one that covers Mac Lanes and Kellyscoherenceresults mentioneduptonow, but it is moregeneral. Wecallcoherence a result that tells us that there is a faithful functor G from a cat-egory Sfreelygeneratedinacertainclassofcategoriestoamanageablecategory M. Thiscallsforsomeexplanation.Itisdesirable,thoughperhapsnotabsolutelynecessary,thatthefunc-torGbestructure-preserving, whichmeansthatitpreservesstructureatleastuptoisomorphism(see 1.7below, and, inparticular, 2.8). Inallcoherenceresults wewill consider, thefunctor Gwill preservestructurestrictly, i.e. onthenose. Thecategories Sand Mwill beinthesameclassofcategories,andGwillbeobtainedbyextendinginauniquewayamapfromthegeneratorsof Sinto M.The category Mis manageable when equations of arrows, i.e. commutingdiagrams of arrows, are easier to consider in it than in S. The best is if thecommutingproblemisobviouslydecidablein M,whileitwasnotobviousthatitissuchin S.Withour approachtocoherenceweareorientedtowards solvingthecommutingproblem, andwearelessinterestedinthetheoremhoodprob-lem. In this book, we deal with the latter problem only occasionally, mostlywhen we need to solve it in order to deal with the commuting problem (see1.1. Coherence 54.2, 7.1, 7.3-5, 8.2-3and 11.4). This shouldbe stressedbecauseotherauthorsmaygiveamoreprominentplacetothetheoremhoodprob-lem. We nd that the spirit of the theoremhood problem is not particularlycategorial: thisproblemcanbesolvedbyconsideringonlycategoriesthatarepreorders. Andordinary,orperhapslessordinary,logicalmethodsforshowingdecidabilityoftheoremhoodareheremoreuseful thancategorialmethods. Forthecategoriesinthisbook,thedecidabilityofthetheorem-hoodproblemisshownbysyntactical orsemantical logical tools. Amongthelatterwealsohavesometimessimplytruthtables. Wehaveusedonpurpose the not very precise term manageable for the category Mto leaveroom for modications of our notion of coherence, which would be orientedtowards solving another problem than the commuting problem. Besides thetheoremhoodproblem,onemayperhapsalsoenvisagesomethingelse,butour ocial notion of coherence is oriented towards the commuting problem.In this book, the manageable category M will be the category Rel witharrows being relations between nite ordinals, whose connection with natu-raltransformationswehavementionedabove. ThecommutingprobleminRel is obviously decidable. We do, however, consider briey categories thatmay replace Relin particular, the category whose arrows are matrices (see12.5).Thefreelygeneratedcategory Swill bethemonoidal categoryfreelygeneratedbyasetof objects, orthesymmetricmonoidal categoryfreelygeneratedbyasetofobjects,ormanyothersofthatkind. Thegeneratingsetofobjectsmaybeconceivedasadiscretecategory. Inourunderstand-ing of coherence, replacing this discrete generating category by an arbitrarycategory would prevent us to solve coherencesimply because the commut-ingprobleminthearbitrarygeneratingcategorymaybeundecidable. Farfromhavingmoregeneral, stronger, results if thegeneratingcategoryisarbitrary,wemayendupbyhavingnoresultatall.The categories Sin this book are built ultimately out of syntactic mate-rial,aslogicalsystemsarebuilt. Categoristsarenotinclinedtoformulatetheircoherenceresultsinthewaywedoinparticular, theydonotdealoften with syntactically built categories (but cf. [131], which comes close tothat). If,however,moreinvolvedandmoreabstractformulationsofcoher-ence that may be found in the literature (for early references on this matter6 CHAPTER1. INTRODUCTIONsee[80])havepractical consequencesforsolvingthecommutingproblem,ourwayofformulatingcoherencehastheseconsequencesaswell.That there is a faithful structure-preserving functor G from the syntac-ticalcategory Stothemanageablecategory Mmeansthatforallarrowsfandgof Swiththesamesourceandthesametargetwehavef= gin S i Gf= Ggin M.The direction from left to right in this equivalence is contained in the func-torialityofG,whilethedirectionfromrighttoleftisfaithfulnessproper.If S is conceivedas a syntactical system, while Mis a model, thefaithfulnessequivalencewehavejuststatedislikeacompletenessresultinlogic. Theleft-to-rightdirection, i.e. functoriality, issoundness, whiletheright-to-leftdirection,i.e.faithfulness,iscompletenessproper.Inthisbookwewillsystematicallyseparatecoherenceresultsinvolvingspecial objects(suchasunitobjects, terminal objectsandinitial objects)fromthosenot involvingthem. Theseobjects tendtocausediculties,and the statements and proofs of the coherence results gain by having thesediculties kept apart. When coherence is obtained both in the absence andinthepresenceofspecialobjects,ourresultsbecomesharper.1.2. CategoricationBycategoricationonecanunderstand,verygenerally,presentingamath-ematicalnotioninacategorialsetting,whichusuallyinvolvesgeneralizingthenotionandmakingnerdistinctions. Inthisbook, however, wehavesomething more specic in mind. We say that we have a categorication ofthe notion of algebraic structure in which there is a preordering,i.e. reex-ive and transitive, relation R when we replace R with arrows in a category,andobtaintherebyamoregeneral categorial notioninsteadof theinitialalgebraicnotion. Iftheinitialalgebraicstructureisacompletelyfreealge-braofterms, likethealgebraofformulaeofapropositional language, theelementsofthealgebrajustbecomeobjectsinafreecategoryintheclassofcategoriesresultingfromthecategorication. Otherwise,somesplittingoftheobjectsisinvolvedincategorication.1.2. Categorication 7Categoricationisnotatechnical notionwewill relyonlater, andsowewill nottrytodeneitmoreprecisely. Whatwehaveinmindshouldbeclearfromthefollowingexamples.Bycategorifyingthealgebraof formulaeof conjunctivelogicwiththeconstanttrueproposition, wherethepreorderingrelationRisinducedbyimplication, we mayendupwiththe notionof cartesiancategory. Wemay end up with the same notion by categorifying the notion of semilatticewithunit, wheretherelationRisthepartial orderingof thesemilattice.Asemilatticewithunitisacartesiancategorythatisapartial order, i.e.inwhichwheneverwehavearrowsfromatobandviceversa, thenaandbarethesameobject. Inthesamesense,thenotionofmonoidalcategoryisacategoricationof thenotionof monoid, andthenotionof symmet-ricmonoidal categoryis acategoricationof thenotionof commutativemonoid,thepreorderingrelationRinthesetwocasesbeingequality.Thereareotherconceptionsof categoricationexceptthatone. Onemaycategorifyanalgebrabytakingitsobjectstobearrowsofacategory.Thenotionof categoryisacategoricationinthissenseof thenotionofmonoid, monoidsbeingcategorieswithasingleobject. Inthatdirection,oneobtains moreinvolvednotions of categoricationinthen-categorialsetting(see[2]and[27]).Themotivationforcategoricationmaybeinternaltocategorytheory,but it maycome fromother areas of mathematics, like algebraic topol-ogyandmathematicalphysicsinparticular,quantumeldtheory(manyreferencesaregivenin[2]). Ourmotivationcomesfromproof theory, aswewill explaininlattersectionsof thisintroduction. Wearereplacingaconsequencerelation,whichisapreorderingrelation,byacategory,wherearrowsstandforproofs. Incomparingourapproachtoothers, notethatthesloganReplaceequalitybyisomorphisms!,whichissometimesheardinconnectionwithcategorication,doesnotdescribeexactlywhatwearedoing. Our sloganReplacepreorder byarrows! implies, however, theother one,and so the same categorial notions,like,for example,the notionofmonoidalcategory,mayturnupunderbothslogans.Inthis bookonemaynd, inparticular, categorications, inour re-stricted sense, of the notions of distributive lattice and Boolean algebra. Al-ternatively, these may be taken as categorications of conjunctive-disjunctive8 CHAPTER1. INTRODUCTIONlogic, or of the classical propositional calculus. Previously, a categoricationof the notion of distributive lattice was obtained with so-called distributivecategories, i.e. bicartesian categories with distribution arrows from a(bc)to (a b) (a c) that are isomorphisms (see [95],pp. 222-223 and Session26,and[20]). Bicartesianclosedcategories,i.e.cartesianclosedcategorieswithnitecoproducts(see[90], SectionI.8), aredistributivecategoriesinthissense.Inourcategoricationofthenotionofdistributivelattice,distributionarrows of the type above neednot be isomorphisms. This rejectionofisomorphismisimposedbyourwishtohavecoherencewithrespecttothecategoryRel of the precedingsection, since the relationunderlyingthefollowingdiagram:( a b ) ( a c )a ( b c )( a b ) ( a c )

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``namelytherelationunderlyingthediagramontheleft-handsidebelow,isnottheidentityrelationunderlyingthediagramontheright-handside:( a b ) ( a c )( a b ) ( a c )

( a b ) ( a c )( a b ) ( a c )Our categoricationof the notionof distributive lattice is basedonarrowsfroma (b c)to(a b) c, whichCockettandSeelystudiedintheir categorial treatment of afragment of linear logic(see[22]; furtherreferencesaregivenin 7.1and 7.9). Atrst, theycalledtheprincipleunderlyingthesearrowsweakdistribution,andthenchangedthistolineardistribution in [25]. Since this is a principle that delivers distribution in thecontext of lattices, but is in fact an associativityinvolving two operations,wehavecoinedthenamedissociativityfor it, toprevent confusionwithwhat is usually called distribution. Cockett and Seely were concerned with1.2. Categorication 9establishingsomesortofcoherencefordissociativitywithrespecttoproofnets.Before appearing in proof nets and in categories, dissociativity was pre-guredinuniversalalgebraandlogic(see 7.1forreferences). Dissociativ-ityisrelatedtothemodularitylawof lattices(see 7.1), andwewill seein 11.3howinacontextthatisacategoricationofthenotionoflatticethis two-sorted associativity delivers distribution arrows of the usual types,froma (b c)to(a b) (a c)andfrom(a b) (a c)toa (b c)(thearrowsoftheconversetypesarethereanyway),ofwhichneitherneedtobeanisomorphism. Thearrowsfrom(a b) (a c)toa (b c)neednotbeisomorphismsinbicartesianclosedcategoriestoo.ThecategoricationofthenotionofBooleanalgebra isusuallydeemedtobeahopelesstask(see 14.3), becauseitisassumedthiscategorica-tionshouldbebasedonthenotionofbicartesianclosedcategory. Inthatnotion, aswesaidabove, wehavearrowscorrespondingtodistributionofconjunctionoverdisjunctionthatareisomorphisms. Natural assumptionsinthiscontextleadtotriviality, i.e.tocategoriesthatarepreorders. Ourcategorication of the notion of Boolean algebra is not trivial in this sense.Itincorporatesthenotionofbicartesiancategory(i.e.categorywithniteproductsandcoproducts),butdoesnotadmitcartesianclosure. Itsessen-tial ingredientisourcategoricationof distributivelattices, inwhichthearrows corresponding to distribution of conjunction over disjunction are notisomorphisms.We think it is a prejudice to assume that there must be an isomorphismbehinddistributionof conjunctionoverdisjunction. Itwouldlikewisebeaprejudicetoassumethatbehindtheidempotencylawa a=aortheabsorptionlawa (a b) = aoflatticeswemusthaveisomorphisms. Thecategorication of the notion of lattice in bicartesian categories is not underthespell of thelattertwoassumptions, buttheisomorphismcorrespond-ingtodistributionofconjunctionoverdisjunctionisusuallypresupposed.This is presumably because in the category Set of sets with functionsthecentral categorythereisdistributionof cartesianproduct over disjointunionisanisomorphism. Inthecategoricationofthenotionofdistribu-tive lattice withdistributive categories, where a (b c) is isomorphicto(a b) (a c), it is not requiredthat a (b c) beisomorphicto10 CHAPTER1. INTRODUCTION(a b) (a c),presumablybecausethelatterisomorphism neednotexistinSet. Weassumeneitheroftheseisomorphismsinourcategoricationofthenotionofdistributivelattice.1.3. TheNormalizationConjectureingeneral prooftheoryCategoricationisinterestingforusbecauseofitsconnectionwithgeneralprooftheory. ThequestionWhatisaproof?wasconsideredbyPrawitzin[112] (SectionI) tobetherst questionof general proof theory. Tokeepupwiththetradition, wespeakofproof, thoughwecouldaswellreplacethistermbythemoreprecisetermdeduction, sincewehaveinminddeductiveproofs fromassumptions (includingtheemptycollectionofassumptions). TogetherwiththequestionWhatisaproof?, Prawitzenvisagedthefollowingas oneof therst questions tobeconsideredingeneralprooftheory(see[112],p.237):In the same way as one asks when two formulas dene the sameset or two sentences express the same proposition, one asks whentwoderivationsrepresentthesameproof; inotherwords, oneasksforidentitycriteriaforproofsorforasynonymity(orequivalence)relationbetweenderivations.AnanswertothequestionofidentitycriteriaforproofsmightleadtoananswertothebasicquestionWhatisaproof?. Aproofwouldbetheequivalenceclassof aderivation. TherelatedquestionWhatisanalgo-rithm? couldbeansweredbyananalogousfactoringthroughanequiv-alencerelationonrepresentationsofalgorithms. (Moschovakisstressedin[107], Section 8, the fundamental interest of identity criteria for algorithms.)Prawitzdidnotonlyformulatethequestionofidentitycriteriaforproofsveryclearly,butalsoproposedaprecisemathematicalanswertoit.Prawitzconsideredderivations innatural deductionsystems andtheequivalence relation between derivations that is the reexive, transitive andsymmetricclosureof theimmediate-reducibilityrelationbetweenderiva-tions. Of course, onlyderivationswiththesamepremisesandthesameconclusionmaybeequivalent. Prawitzsimmediate-reducibilityrelationis1.3. TheNormalizationConjectureingeneralprooftheory 11the one involvedinreducingaderivationtonormal formamatter hestudiedpreviouslyin[111]. As it is well known, theideaof this reduc-tionstemsfromGentzensthesis[60]. Aderivationreducesimmediatelytoanotherderivation(see[112],SectionII.3.3)whenthelatterisobtainedfromtheformer either byremovingamaximumformula(i.e. aformulawithaconnectivethatistheconclusionofanintroductionofandthemajor premise of an elimination of ), or by performing one of the permuta-tivereductions tied to the eliminations ofdisjunction andofthe existentialquantier, whichenablesustoremovewhatPrawitzcallsmaximumseg-ments. There are some further reductions,which Prawitz called immediatesimplications; theyconsistinremovingeliminationsofdisjunctionwherenohypothesis is discharged, andthere are similar immediate simplica-tionsinvolvingtheexistential quantier, andredundantapplicationsoftheclassical absurdityrule. Prawitzalsoenvisagedreductions hecalledimmediateexpansions, whichleadtotheexpandednormal formwherealltheminimumformulaeareatomic(minimumformulaearethosethatareconclusionsofeliminationsandpremisesofintroductions).Prawitzformulatesin[112] (SectionII.3.5.6)thefollowingconjecture,for which he gives credit (in Section II.5.2) to Martin-Lof, and acknowledgesinuencebyideasofTait:Conjecture. Twoderivationsrepresentthesameproofifandonlyiftheyareequivalent.WecallthisconjecturetheNormalizationConjecture.Thisconjecture, togetherwithanotherconjecture, whichwill becon-sideredinthenextsection, wasexaminedinthesurvey[40]. Thepresentsectionandthenextthreesectionsgiveanupdated,somewhereshortenedand somewhere expanded,variant of that survey,to which we refer for fur-ther, especiallyhistorical andphilosophical, remarks. (Someotherbitsofthatsurveyarein 14.3, whereamistakenstatementisalsocorrectedattheendofthesection.)ThenormalizationunderlyingtheNormalizationConjectureneednotbeunderstoodalwaysintheprecisesenseenvisagedbyPrawitz. Forintu-itionisticlogicPrawitzsunderstandingofnormalization, whichisderivedfromGentzen, isperhapsoptimal. Thereare, however, otherlogics, and,12 CHAPTER1. INTRODUCTIONinparticular, thereisclassical logic, towhichnatural deductionisnotsocloselytied, andfor whichwemaystill haveanotionof normalization,perhapsrelatedtoPrawitzs, butdierent. Whatcomestomindimmedi-ately for classical logic is Gentzens plural, i.e. multiple-conclusion, sequentsystems(seebelow)andcuteliminationforthem.Presumably, thenotionof normalizationwecanenvisageintheNor-malizationConjecturecannotbebasedonanarbitrarynotionof normalform. Itisdesirablethatthisnormalformbeunique,atleastuptosomesupercialtransformations(likealphaconversioninthelambdacalculus).But uniqueness shouldnot be enough. This normal formandthe lan-guageforwhichitisformulatedmustbesignicant,whereitisdiculttosaywhatsignicantmeansexactly. Thenormal formandthelanguageforwhichitisformulatedshouldnotbejustatechnical device, buttheymustbedeeplytiedtothelogic, andexhibititsessential features. Inthecaseof Prawitzsnormal formforderivationsinintuitionisticnatural de-duction, besides philosophical reasons havingtodowiththemeaningoflogical connectives, there are important ties with independently introducedmathematicalstructures.TheNormalizationConjecturewasformulatedbyPrawitzatthetimewhen the Curry-Howard correspondence between derivations in natural de-ductionandtypedlambdatermsstartedbeingrecognizedmoreandmore(though the label Curry-Howard was not yet canonized). Prawitzs equiv-alence relationbetweenderivations corresponds tobeta-etaequalitybe-tween typed lambda terms, if immediate expansions are taken into account,andtobetaequalityotherwise.Besides derivations in natural deduction and typed lambda terms, whereaccording to the Curry-Howard correspondence the latter can be conceivedjustascodesfortheformer, thereareother, moreremote, formal repre-sentationsofproofs. TherearerstGentzenssequentsystems,whicharerelatedtonatural deduction, butareneverthelessdierent, andtherearealsorepresentations of proofs as arrows incategories. The sources andtargets of arrows are takentobe premises andconclusions respectively,andequalityof arrows withthesamesourceandtarget, i.e. commutingdiagramsofarrows,shouldnowcorrespondtoidentityofproofsviaacon-jectureanalogoustotheNormalizationConjecture.1.3. TheNormalizationConjectureingeneralprooftheory 13The fact provedbyLambek(see [87] and[90], Part I; see also[39],[37] and [43]) that the category of typed lambda calculuses with functionaltypesandniteproducttypes,basedonbeta-etaequality,isequivalenttothe category of cartesian closed categories, and that hence equality of typedlambdatermsamountstoequalitybetweenarrowsincartesianclosedcat-egories,lendsadditionalsupporttotheNormalizationConjecture. Equal-ityofarrowsinbicartesianclosedcategoriescorrespondstoequivalenceofderivationsinPrawitzssenseinfull intuitionisticpropositional logic(see[109], Section 3, for a detailed demonstration that the equations of bicarte-sian closed categories deliver cut elimination for intuitionistic propositionallogic). The notion of bicartesian closed category is a categorication in thesense of the preceding section of the notion of Heyting algebra. The partialorderofHeytingalgebrasisreplacedbyarrowsinthiscategorication.Incategorytheory, theNormalizationConjectureistiedtoLawverescharacterizationof theconnectivesof intuitionisticlogicbyadjointsitua-tions. Prawitzsequivalenceof derivations, initsbeta-etaversion, corre-spondstoequalityofarrowsinvariousadjunctionstiedtologicalconnec-tives (see [94], [38], Section 0.3.3, [41] and [39]). Adjunction is the unifyingconceptforthereductionsenvisagedbyPrawitz.The fact that equalitybetweenlambdaterms, as well as equalityofarrowsincartesianclosedcategories,wererstconceivedforreasonsinde-pendentofproofsisremarkable. Thistellsusthatweareinthepresenceof asolidmathematical structure, whichmaybeilluminatedfrommanysides.PrawitzformulatedtheNormalizationConjecturehavinginmindnat-ural deduction, andsomainlyintuitionisticlogic. For classical logicweenvisagesomethingelse. OurcategoricationofthenotionofBooleanal-gebra, as the categorication of the notion of Heyting algebra with bicarte-sianclosedcategories, covers anotionof identityof proofs suggestedbynormalizationviacuteliminationinaplural-sequentsystem(seeChapters11and14). Thisisinspiteofthefactthatforusdistributionofconjunc-tionoverdisjunctiondoesnotgiverisetoisomorphisms, asinbicartesianclosedcategories. Thisdisagreementovertheisomorphismofdistributionmaybeexplainedasfollows.14 CHAPTER1. INTRODUCTIONClassical and intuitionistic logic do not dier with respect to the conse-quencerelationbetweenformulaeintheconjunction-disjunctionfragmentof propositional logic. Inotherwords, theydonotdierwithrespecttoprovablesequentsof theformA BwhereAandBareformulaeof theconjunction-disjunctionfragment. But, thoughthese two logics do notdierwithrespecttoprovability, theymaydierwithrespecttoproofs.Thestandardsequentformulationof classical logic, theformulationthatimposesitself byitssymmetryandregularity, isbasedonplural sequents ,wheremaybeacollectionwithmorethanoneformula,whereasthe standard sequent formulation of intuitionistic logic is based on singular,i.e. single-conclusion,sequents ,where cannot have more than oneformula, whilecan. Therearepresentationsof intuitionisticlogicwithplural sequents (see[103] and[32], Section5C4, withdetailedhistoricalremarksonpp. 249-250; cf. also[31], wheretheideaisalreadypresent),buttheyarenotstandard, andtheydonotcorrespondtonatural deduc-tion,asthosewithsingularsequentsdo. Moreover,intheseplural-sequentformulations of intuitionistic logic, a restriction based on singularity is keptforintroductionof implicationontheright-handside, whichcorrespondstothedeductiontheorem. Thedeductiontheoremenablesthedeductivemetalogictobemirroredwiththehelpof implicationintheobject lan-guage,andwhenitcomestothismirroring,plural-sequentformulationsofintuitionistic logic avow that their deductive metalogic is based on singularsequents.The connectionof intuitionistic logic withnatural deduction, wheretherearepossiblyseveral premises, butnevermorethanoneconclusion,goesverydeep. Therearemanyreasonstoholdthatthemeaningofintu-itionisticconnectivesisexplainedintheframeworkof natural deduction,assuggestedbyGentzen(see[60], SectionII.5.13). Singularsequentsareasymmetric, i.e. they have a plurality of premises versus a single conclusion.Theasymmetriesofintuitionisticlogic, and, inparticular, theasymmetrybetween conjunction and disjunction, can be explained by the asymmetry ofsingularsequentsthatunderlythislogic. Onecansupposethattheasym-metryofbicartesianclosedcategories,whichconsistsinhavinga (b c)isomorphicto(a b) (a c) without havinga (b c) isomorphicto(a b) (a c),hasthesameroots.1.4. TheGeneralityConjecture 15Thedissociativityprincipleof thearrowthatgoesfroma (b c)to(a b) c(see 1.2and 7.1)deliversarrowsthatgofroma (b c)to(a b) (a c)andfrom(a b) (a c)toa (b c)(see 11.3;wehavearrowsof theconversetypeswithoutassumingdistribution), butneitherof thesearrowsneedtobeisomorphisms. Sosymmetry, whichistypicalforBooleannotions,isrestored. (Anotherpossibilitytorestoresymmetrywould be to take that a(bc) is isomorphic to (ab)(ac) and a(bc)isisomorphicto(a b) (a c),whichisnotthecaseinSet,butwewillnotexplorethatpossibilityinthisbook.)Thedissociativityprinciple,whichisanessentialingredientofourcat-egoricationof thenotionsof distributivelatticeandBooleanalgebra, isbuiltintotheplural-sequentformulationofclassicallogic. Itistiedtothecutruleofpluralsequents(see 11.1,andalso 7.7).PrawitzenvisagedtheNormalizationConjectureforclassicallogic,butinanaturaldeductionformulation,i.e.withsingularsequents. Thisisnotthesameasconsideringthisconjecturewithpluralsequents.1.4. TheGeneralityConjectureAtthesametimewhenPrawitzformulatedtheNormalizationConjecture,inaseriesof papers ([84], [85], [86] and[87]) Lambekwasengagedinaproject where arrows in various sorts of categories were construed as repre-senting proofs. The source of an arrow corresponds to the premise, and thetarget to the conclusion. (Proofs where there is a nite number of premisesdierent fromone are representedbyproofs withasingle premise withthe help of connectives like conjunction and the constant true proposition.)With this series of papers Lambek inaugurated the eld of categorial prooftheory.ThecategoriesLambekconsideredin[84] and[85] arerstthosethatcorrespondtohis substructural syntacticcalculus of categorial grammar(thesearemonoidalcategorieswherethefunctorsa . . .and. . . ahaverightadjoints). Next, heconsideredmonads, whichbesidesbeingfunda-mentalforcategorytheory,coverproofsinmodallogicsoftheS4kind. In[86] and[87], Lambekdealtwithcartesianclosedcategories, whichcoverproofsintheconjunction-implicationfragmentof intuitionisticlogic. He16 CHAPTER1. INTRODUCTIONalsoenvisagedbicartesianclosedcategories,whichcoverthewholeofintu-itionisticpropositionallogic.Lambeks insight is that equations betweenarrows incategories, i.e.commuting diagrams of arrows, guarantee cut elimination, i.e. compositionelimination, inanappropriatelanguagefornamingarrows. (In[38] itisestablished that for some basic notions of category theory, and in particularforthenotionofadjunction,theequationsassumedarenecessaryandsuf-cient for composition elimination.) Since cut elimination is closely relatedtoPrawitzs normalizationof derivations, theequivalencerelationenvis-agedbyLambekshouldberelatedtoPrawitzs. (AnearlypresentationoftheconnectionbetweenPrawitzandLambekisin[105].)Thenormalizationofcuteliminationdoesnotinvolveonlyeliminatingcuts, but also equations between cut-free terms for arrows, which may guar-anteetheiruniqueness. (Thisislikeaddingtheetaequationstothebetaequationsinthetypedlambdacalculusandnaturaldeduction.)Lambeks work is interesting not only because he worked with an equiva-lence relation between derivations amounting to Prawitzs, but also becauseheenvisagedanotherkindof equivalencerelation. Lambeksideaisbestconveyed by considering the following example. In [86] (p. 65) he says thattherstprojectionarrowk1p,p:p p pandthesecondprojectionarrowk2p,p: p p p, which correspond to two derivations of conjunction elimina-tion, have dierent generality, because they generalize tok1p,q: pq p andk2p,q: p q qrespectively,andthelattertwoarrowsdonothavethesametarget; on the other hand,k1p,q: p q p andk2q,p: q p p do not have thesamesource. Theideaofgeneralitymaybeexplainedroughlyasfollows.Weconsidergeneralizationsofderivationsthatdiversifyvariableswithoutchangingtherulesofinference. Twoderivationshavethesamegeneralitywheneverygeneralizationof oneof themleadstoageneralizationof theother, sothat inthetwogeneralizations wehavethesamepremiseandconclusion(see[84],p.257). Intheexampleabove,thisisnotthecase.Generalityinduces anequivalencerelationbetweenderivations. Twoderivations are equivalent if andonlyif theyhave the same generality.LambekdoesnotformulatesoclearlyasPrawitzaconjectureconcerningidentitycriteriaforproofs, buthesuggeststhattwoderivationsrepresent1.4. TheGeneralityConjecture 17thesameproofifandonlyiftheyareequivalentinthenewsense. WewillcallthisconjecturetheGeneralityConjecture.Lambeksownattemptsatmakingthenotionofgeneralityprecise(see[84], p. 316, wherethetermscopeisusedinsteadof generality, and[85], pp. 89, 100) need not detain us here. In [86] (p. 65) he nds that theseattemptswerefaulty.Thesimplestwaytounderstandgeneralityistousegraphswhosever-ticesareoccurrencesof propositional lettersinthepremiseandthecon-clusionofaderivation. Weconnectbyanedgeoccurrencesoflettersthatmustremainoccurrencesofthesameletteraftergeneralizing, anddonotconnectthosethatmaybecomeoccurrencesofdierentletters. Sofortherstandsecondprojectionabovewewouldhavethetwographs pk1p,pp pk2p,ppp p

Whenthepropositional letterpisreplacedbyanarbitraryformulaAwehaveanedgeforeachoccurrenceofpropositionalletterinA.The generality of a derivation is such a graph. According to the Gener-alityConjecture, therstandsecondprojectionderivationsfromp ptoprepresentdierentproofsbecausetheirgeneralitiesdier.One denes an associative composition of such graphs, and there is alsoan obvious identity graph with straight parallel edges, so that graphs makeacategory, whichwecall thegraphical category. If ontheotherhanditistakenforgrantedthatproofsalsomakeacategory, whichwewill callthesyntactical category, withcompositionof arrowsbeingcompositionofproofs, andidentityarrowsbeingidentityproofs(anidentityproof com-posed with any other proof, either on the side of the premise or on the sideof theconclusion, isequal tothisotherproof), thentheGeneralityCon-jecturemayberephrasedastheassertionthatthereisafaithful functorfromthesyntacticalcategorytothegraphicalcategory. SotheGeneralityConjectureisanalogoustoacoherencetheoremof categorytheory. Themanageablecategoryisagraphicalcategory.Thecoherenceresultof [81] provestheGeneralityConjectureforthe18 CHAPTER1. INTRODUCTIONmultiplicative conjunction-implication fragment of intuitionistic linear logic(moduloaconditionconcerningthemultiplicativeconstanttrueproposi-tion,i.e. the unit with respect to multiplicative conjunction),and,inspiredby Lambek, it does so via a cut-elimination proof. The syntactical categoryin this case is a free symmetric monoidal closed category, and the graphicalcategoryisofakindstudiedin[54]. Thegraphsofthisgraphicalcategoryare closely related to the tangles of knot theory. In tangles, as in braids, wedistinguish between two kinds of crossings, but here we need just one kind,inwhichit is not distinguishedwhichof thetwocrossededges is abovetheother. (Forcategoriesoftanglessee[134],[129]and[73],Chapter12.)TangleswiththissinglekindofcrossingarelikegraphsoneencountersinBraueralgebras(see[15]and[132]). Hereisanexampleofsuchatangle:

__` (p(q(r r))) (s s)((p q) p) pTangles without crossings at all serve in [38] (Section 4.10; see also [42])toobtainacoherenceresult for thegeneral notionof adjunction, whichaccording to Lawveres Thesis underlies all the connectives of intuitionisticlogic,aswementionedintheprecedingsection. Intermsofcombinatoriallow-dimensional topology, themathematical contentofthegeneral notionof adjunctioniscaughtbytheReidemeistermovesof planarambientiso-topy. Ananalogous coherenceresult for self-adjunctions, whereasingleendofunctor is adjoint to itself,is proved in [49]. Through this latter resultwereachthetheoryof Temperley-Liebalgebras, whichplayaprominentroleinknottheoryandlow-dimensional topology, duetoJones represen-tationof Artins braidgroups inthesealgebras (see[74], [97], [110] andreferencestherein).In[49] onendsalsocoherenceresultsfor self-adjunctionswherethegraphicalcategoryisthecategoryofmatrices,i.e.theskeletonofthecate-goryofnite-dimensionalvectorspacesoveraxedeldwithlineartrans-1.4. TheGeneralityConjecture 19formationsasarrows. Tangleswithoutcrossingsmaybefaithfullyrepre-sentedinmatricesbyarepresentationderivedfromtheorthogonal groupcase of Brauers representation of Brauer algebras (see also [132], Section 3,and[70], Section3). This representationis basedonthe fact that theKronecker product of matrices gives rise to a self-adjoint functor in the cat-egory of matrices, and this self-adjointness is related to the fact that in thiscategory,aswellasinthecategoryRel,whosearrowsarebinaryrelationsbetweenniteordinals,niteproductsandcoproductsareisomorphic.Graphs like graphs of the tangle type were tied to sequent derivations ofclassical logic in [18] and [19], but without referring to categories, coherenceorthequestionofidentitycriteriaforproofs.In[108] thereareseveral coherenceresults, whichextend[99], forthemultiplicative-conjunctionfragments of substructural logics. But less usconcentratenowoncoherenceresultsforclassicalandintuitionisticlogic.The Normalization Conjecture and the Generality Conjecture agree onlyfor limited fragments of these two logics. They agree for purely conjunctivelogic, with or without the constant true proposition (see [46] and 9.1-2below). Proofs inconjunctivelogic are the same for classical andintu-itionisticlogic. HeretheNormalizationConjectureistakeninitsbeta-etaversion. Byduality,thetwoconjecturesagreeforpurelydisjunctivelogic,withorwithouttheconstantabsurdproposition . Ifwehavebothcon-junction and disjunction, but do not yet have distribution, and have neithernor ,thenthetwoconjecturesstillagreeforbothlogics,providedthegraphicalcategoryisthecategoryRel whosearrowsarerelationsbetweenniteordinals(see[48] and 9.4). Andhereitseemswehavereachedthelimitsof agreementasfarasintuitionisticlogicisconcerned. Withmoresophisticated notions of graphs, matters may stand dierently, and the areaofagreementforthetwoconjecturesmayperhapsbewider,butitcanbeevennarrower,aswewillseebelow.It may be questioned whether the intuitive idea of generality is caught bythe category Rel in the case of conjunctive-disjunctive logic. The problem isthat ifwp: p pp is a component of the diagonal natural transformation,andk1q,p: q q p is a rst injection, then in categories with nite products20 CHAPTER1. INTRODUCTIONandcoproductswehave(1qwp)k1q,p=k1q,pp,wheretheleft-handsidecannotbefurthergeneralized,buttheright-handsidecanbegeneralizedtok1q,pr. Theintuitiveideaofgeneralityseemstorequire that inwp: p pp we should not have only a relation between thedomainandthecodomain, asontheleft-handsidebelow, butanequiv-alencerelationontheunionof thedomainandthecodomain, asontheright-handside:_

``

``pp ppp p(see [50], and also [51]). With such equivalence relations, we can still get co-herence for conjunctive logic, and for disjunctive logic, taken separately, butforconjunctive-disjunctivelogictheleft-to-rightdirection, i.e. thesound-ness part, of coherence would fail (see 14.3). So for conjunctive-disjunctivelogictheideaofgeneralitywithwhichwehavecoherenceisnotquitetheintuitiveideasuggestedbyLambek,butonlysomethingclosetoit,whichinvolves the categorial notionof natural transformation(cf. the endof14.3).Evenwhenwestaywithintheconnes of thecategoryRel, our un-derstandingof generalitydoes not matchexactlytheintuitivenotionofgeneralityforconjunctive-disjunctivelogic. Intuitively, therelationsRofRel corresponding to generality should satisfy difunctionality in the sense of[114]; namely, we should have RR1R R. But this requirement is notsatised for our images in Rel under G of proofs in conjunctive-disjunctivelogic,evenintheabsenceofdistribution(seetheendof 14.3). GeneralityiscaughtbyRel onlyforfragmentsoflogic. Altogether, generalityservesonlyasaloosemotivationfortakingRel asourgraphical category. RealgroundsforRel areinthenotionofnatural transformation, whichhastodowithpermutingrulesinderivations.The Normalization Conjecture and the Generality Conjecture agree nei-therfortheconjunction-disjunctionfragmentofintuitionisticlogicwith 1.4. TheGeneralityConjecture 21and (see[47]and 9.6),norfortheconjunction-implicationfragmentofthislogic. Wedonothavecoherenceforcartesianclosedcategoriesifthegraphsinthegraphical categoryaretakentobeof thetangletypeKellyand Mac Lane had for symmetric monoidal closed categories combined withthe graphs we have in Rel for cartesian categoriesboth the soundness partand the completeness part of coherence fail (for soundness see a counterex-amplein 14.3, with p preplacedbyp p, andforcompletenesssee[125]). The soundness part of coherence fails also for distributive bicartesiancategories, andafortioriforbicartesianclosedcategories. Theproblemisthat in these categories distribution of conjunction over disjunction is takentobeanisomorphism, andRel doesnotdeliverthat, aswehaveseenin1.2.Theproblemwiththesoundnesspartofcoherenceforcartesianclosedcategories maybe illustratedwithtypedlambdaterms inthe followingmanner. Bybetaconversionandalphaconversion, wehavethefollowingequation:xx, xyy= yy, zzforyandzoftypep,andxoftypepp(whichcorrespondstop p). Theclosed terms on the two sides of this equation are both of type pppp. Thetypeof thetermontheleft-handsidecannotbefurthergeneralized, butthe type of the term yy, zz, can be generalized to ppqq. The problemnoted here does not depend essentially on the presence of surjective pairing , and of product types; it arises also with purely functional types. Thisproblem depends essentially on the multiple binding of variables,which wehaveinxx, x; thatis, itdependsonthestructural ruleof contraction.Thisthrowssomedoubtontheright-to-leftdirectionoftheNormalizationConjecture, whichPrawitz foundrelativelyunproblematic. It might beconsidered strange that two derivations represent the same proof if, withoutchanginginferencerules,onecanbegeneralizedinamannerinwhichtheothercannotbegeneralized.TheareaofagreementbetweentheNormalizationConjectureandtheGenerality Conjecture may be wider for classical logic, provided normaliza-tion is understood in the sense of cut elimination for plural sequent systemsandgeneralityisunderstoodinthesenseof thecategoryRel. Itextends22 CHAPTER1. INTRODUCTIONrsttoconjunctive-disjunctivelogicwithoutdistribution(see[48]and 9.4below). Next, inconjunctive-disjunctivelogicwithdistribution, withorwithout and , the agreement also holds (see Chapter 11). And it coversalso the whole classical propositional calculus, with a particular way of un-derstanding normalization involving zero proofs (see 1.6 and 14.2-3). Wedonotpretendthisparticularwayof understandingnormalizationinthepresence of negation is the only possible one, but in the absence of negationwe feel pretty secure,and the match between the two conjectures is indeedverygood. Gentzenscuteliminationprocedureforplural-sequentsystemsneeds onlyto be modiedin a natural way by admitting union ofproofs,arulethatinthiscontextamountstothemixruleoflinearlogic(seeChap-ters8and10). AdmittingunionofproofssavesGentzenscut-eliminationprocedure from falling into preorder and triviality. Our cut-elimination pro-cedurediersalsofromGentzensinthewayhowittreatsthestructuralruleofcontraction, butinthisrespectitismoreinthespiritofGentzen.(Wewill pointdownatappropriateplacesin 11.1-2howourprocedureisrelatedtoGentzens.)Zero proofs (which were mentioned already in the preface) come up withnegation. Their appearance is imposed by our wish to have coherence withrespect to Rel. With other graphical categories they may disappear, but atthecostofmanyproblems(whichwediscussin 14.3). Inparticular, thematchbetweentheNormalizationandtheGeneralityConjectureswouldbe impaired (see 14.3). The price we have to pay with our categoricationof thenotionof Booleanalgebrais that not all connectives will betiedtoadjointfunctors, asrequiredbyLawvere. Conjunctionanddisjunctionaretiedtotheusual adjunctionswiththediagonal functor(theproductbifunctor is right-adjoint to the diagonal functor, and the coproduct bifunc-tor is left-adjoint to the diagonal functor), but distribution is an additionalmatter, not delivered by these adjunctions, and classical negation and impli-cation do notcome withthe usualadjunctions. (There are perhaps hiddenadjunctionsof somekindhere.) Anotherpricewehavetopaywithzeroproofs is that all theorems, i.e. all propositions proved without hypotheses,will havezeroproofs. Sothetheoremsof classical propositional logic, incontradistinction to their intuitionistic counterparts, do not serve to encodethedeductivemetatheoryofclassicalpropositionallogic. Thismetatheory1.4. TheGeneralityConjecture 23exists,nevertheless,anditscategoricationisnotgivenbycategoriesthatarepreorders.Whenwecomparethetwoconjecturesweshouldsaysomethingabouttheircomputational aspects. WiththeNormalizationConjecturewehaveto rely inintuitionistic logic onreductionto a unique normal forminthetypedlambdacalculusinordertocheckequivalenceof derivationsintheconjunction-implicationfragment of intuitionisticpropositional logic.Nothingmorepracticalthanthatisknown,andsuchsyntacticalmethodsmaybetiresome. Outsideoftheconjunction-implicationfragment, inthepresenceofdisjunctionandnegation,suchmethodsbecomeuncertain.MethodsforcheckingequivalenceofderivationsinaccordancewiththeGeneralityConjecture, i.e. methodssuggestedbycoherenceresults, oftenhaveaclearadvantage. Thisisliketheadvantagetruthtableshaveoversyntactical methodsof reductiontonormal forminordertochecktauto-logicality. However,thesemanticalmethodsdeliveredbycoherenceresultshavethisadvantageonlyifthegraphicalcategoryissimpleenough,asourcategoryRel is. Whenwe enter intocategories suggestedbyknot the-ory, thissimplicitymaybelost. Then, onthecontrary, syntaxmayhelpustodecideequalityinthegraphical category. TheNormalizationCon-jecturehas madeaforayintheoretical computer science, intheareaoftyped programming languages. It is not clear whether one could expect theGeneralityConjecturetoplayasimilarrole.Thereexiveandtransitiveclosureoftheimmediate-reducibilityrela-tioninvolvedinnormalizationmaybedeemedmoreimportant thantheequivalencerelationengenderedbyimmediatereducibility, whichwehaveconsidered up to now. This matter leads outside our topic, which is identityof proofs,but it is worth mentioning. We may categorify the identity re-lation between proofs, and consider not only other relations between proofs,butmapsbetweenproofs. Theproperframeworkfordoingthatseemstobetheframeworkof weak2-categories, wherewehave2-arrowsbetweenarrows; or we could even go to n-categories, where we have n+1-arrows be-tween n-arrows (one usually speaks of cells in this context). Composition of1-arrowsisassociativeonlyuptoa2-arrowisomorphism,andanalogouslyforotherequationsbetween1-arrows. Identityof 1-arrowsisreplacedby2-arrows satisfying certain coherence conditions. In the context of the Gen-24 CHAPTER1. INTRODUCTIONerality Conjecture, we may also nd it natural to consider 2-arrows insteadofidentity. Theorientationwouldherebegivenbypassingfromagraphwithvariousdetourstoagraphthatismorestraight,whichneednotbetakenanymoreasequaltotheoriginalgraph.Withallthiswewouldenterintoaverylivelyeldofcategorytheory,interacting with other disciplines, mainly topology (see [96] and papers citedtherein). The eld looks very promising for general proof theory, both fromPrawitzsandfromLambekspointofview,but,asfarasweknow,ithasnotyetyieldedtoprooftheorymuchmorethanpromises.1.5. MaximalityThefragmentsoflogicmentionedintheprecedingsectionwheretheNor-malization Conjecture and the Generality Conjecture agree for intuitionisticlogic all possess a property called maximality. Let us say a few words aboutthisimportantproperty.For the whole eld of general proof theory to make sense, and in partic-ularforconsideringthequestionof identitycriteriaforproofs, weshouldnothavethatanytwoderivationswiththesamepremiseandconclusionareequivalent. Otherwise,oureldwouldbetrivial.Now, categories with nite nonempty products, cartesian categories andcategories with nite nonempty products and coproducts have the followingproperty. Take, for example, cartesian categories, and take any equation inthe language of free cartesian categories that does not hold in free cartesiancategories. If a cartesian category satises this equation, then this categoryisapreorder. Wehaveanexactlyanalogouspropertywiththeothersortsofcategorieswementioned(see 9.3and 9.5). Thispropertyisakindofsyntacticalcompleteness, analogoustothePostcompletenessoftheusualaxiomatizations of theclassical propositional calculus. Anyextensionoftheequationspostulatedleadstocollapse.Translatedintologicallanguage,thismeansthatPrawitzsequivalencerelation for derivations in conjunctive logic, disjunctive logic and conjunctive-disjunctivelogicwithoutdistributionandwithout and , whichinallthesecasesagreeswithourequivalencerelationdenedviageneralityinthesenseofRel,ismaximal. Anystrengthening,anyaddition,wouldyield1.5. Maximality 25thatanytwoderivationswiththesamepremiseandthesameconclusionareequivalent.If the right-to-left direction of the Normalization Conjecture holds, withmaximalitywecanecientlyjustifytheleft-to-rightdirection,whichPra-witz foundproblematic in[112], andabout whichKreisel was thinkingin[83]. Inthe footnote onp. 165of that paper Kreisel mentions thatBarendregtsuggestedthisjusticationviamaximality. Supposetheright-to-left directionof theNormalizationConjectureholds, supposethat forsomepremiseandconclusionthereismorethanoneproof, andsupposetheequivalencerelationismaximal. Theniftwoderivationsrepresentthesameproof, theyareequivalent. Becauseif theywerenotequivalent, wewouldneverhavemorethanoneproof withagivenpremiseandagivenconclusion. Nothingcanbemissingfromourequivalencerelation,becausewhatever is missing, bymaximality, leads tocollapseonthesideof theequivalence relation,and,by the right-to-left direction of the conjecture,italsoleadstocollapseonthesideofidentityofproofs.Prawitzin[112]founditdiculttojustifytheleft-to-rightdirectionoftheNormalizationConjecture, andKreisel waslookingformathematicalmeans that would provide this justication. Maximality is one such means.Establishingtheleft-to-rightdirectionoftheNormalizationConjecturevia maximality is like proving the completeness of the classical propositionalcalculus with respect to any kind of nontrivial model via Post completeness(whichisprovedsyntacticallybyreductiontoconjunctivenormal form).Actually, the rst proof of this completeness with respect to tautologies wasgivenbyBernaysandHilbertexactlyinthismanner(see[135], Sections2.4and2.5;seealso[66],SectionI.13,and 9.3below).Maximality for the sort of categories mentioned above is proved with thehelpofcoherencein[46]and[48](whichisestablishedproof-theoretically,bynormalization, cut eliminationandsimilar methods; see Chapter 9).Coherenceis helpful inprovingmaximality, but maximalitycanalsobeprovedbyothermeans,asthisisdoneforcartesianclosedcategoriesviaatypedversionofBohmstheoremin[121],[117]and[45]. Thisjustiestheleft-to-rightdirectionof theNormalizationConjecturealsofortheimpli-cational andtheconjunction-implicationfragmentsof intuitionisticlogic.Themaximalityof bicartesianclosedcategories, whichwouldjustifythe26 CHAPTER1. INTRODUCTIONleft-to-rightdirectionoftheNormalizationConjectureforthewholeofin-tuitionisticpropositionallogicis,asfarasweknow,anopenproblem. (Auseformaximalitysimilartothatpropoundedhereandin[45]and[46]isenvisagedin[133].)In [38] (Section 4.11) it is proved that the general notion of adjunction isalso maximalin some sense. The maximality we encountered above,whichinvolvesconnectivestiedtoparticularadjunctions,cannotbederivedfromthe maximality of the general notion of adjunction, but these matters shouldnotbeforeigntoeachother.Sincewendmaximalityaninterestingproperty, wepayattentiontoitinthisbookwherewecouldestablishitwiththehelpofourcoherenceresults, and where it is not a trivial property. Besides the maximality resultsfrom Chapter 9, mentioned above, there are analogous results in 12.3, 12.5and 13.3. Wealsopayattentiontomaximalityincaseswhereitcannotbeestablished(see 10.3and 11.5). Insomecaseswhereitdoesnothold,westillhaverelativemaximalityresults(see 9.7, 11.5and 12.5).1.6. UnionofproofsandzeroproofsGentzensplural-sequentsystemforclassical logichasimplicitlyaruleofunion,oraddition,ofderivations,whichisderivedasfollows:contractionsf : A BRCf : A B, Cg : A BLCg : C, A Bcut(RCf, LCg): A, A B, Bf g : A BHereRCfandLCgareobtainedfromfandgrespectivelybythinningontherightandthinningontheleft,andcut(RCf, LCg)maybeconceivedasobtained by applying to fand ga limit case of Gentzens multiple-cut rulemix, where the collection of mix formulae is empty. A related principle wasconsideredunderthenamemixinlinearlogic(see 8.1).Inacut-eliminationprocedurelikeGentzens, f gisreducedeithertof ortog(see[60], SectionsIII.3.113.1-2). If wehavef g=f andf g=g, thenwegetimmediatelyf=g, thatiscollapseandtriviality.1.6. Unionofproofsandzeroproofs 27In[64] (AppendixB.1byY. Lafont; seealso[67], Section1)thisistakenassucientgroundtoconcludethatcuteliminationintheplural-sequentsystemforclassical logicmustleadtopreorderandcollapse. (In[64], theinevitabilityof thiscollapseiscomparedtotheargumentpresentedafterProposition 1 of 14.3, which shows that a plausible assumption about clas-sicalnegationaddedtobicartesianclosedcategoriesleadstopreorder,butthese are dierent matters.)To evade collapse we may try keeping only oneof the equations f g= fand f g= g, and reject the other; then we mustalsorejectthecommutativityof , butitseemssuchdecisionswouldbearbitrary. (Forsimilarreasons, evenwithoutassumingthecommutativityof , theassumptionsof [127], p. 232, C.12, leadtopreorder.) Thereis,however,awaytoevadecollapseherethatisnotarbitrary. Themodica-tionofGentzenscut-eliminationprocedureexpoundedinChapter11(seealso 12.5)andourcoherenceresults(moreprecisely,theeasy,soundness,i.e.functoriality,partsoftheseresults)testifytothat.The Generality Conjecture tells us that we should have neither f g= fnor f g= g. The union of two graphs may well produce a graph dieringfromeachofthegraphsenteringintotheunion. Italsotellsusthatunionof proofs shouldbeassociativeandcommutative. Theidempotencylawf f= fis imposed by Rel, but it stands apart, and with another graphicalcategory, we may do without it (see 12.5). Without idempotency, union ofproofs is rather addition of proofs. Our way out of the problematic situationGentzen found himself in is to take into account union or addition of proofs.(Besides[40], section7, thepaper[5], whichdealswithcuteliminationinanelogic,alsomakesasimilarsuggestion.)If wehaveunionof proofs, itisnatural toassumethatwealsohaveforeveryformulaAandeveryformulaBazeroproof 0A,B :A B,withanemptygraph,whichwithunionofproofsmakesatleastacommutativemonoid; withidempotency, it gives the unit of asemilattice. We mayenvisagehavingzeroproofs0A,B: A BonlyforthoseAandBwherethere is also a nonzero proof from A to B, as we do in our categorication ofthe notion of Boolean algebra, but the more sweeping assumption involvingeveryAandeveryBmakessensetoo.We should immediately face the complaint that with such zero proofs wehaveenteredintoinconsistency,sinceeverythingisprovable. Thatistrue,28 CHAPTER1. INTRODUCTIONbut not all proofs have been made identical, and we are here not interestedinwhatisprovable, butinwhatproofsareidentical. If ithappensandwiththe GeneralityConjecture it will happenindeedthat introducingzeroproofsisconservativewithrespecttoidentityof proofsthatdonotinvolve zero proofs, then it is legitimate to introduce zero proofs, provided itis useful for some purpose. This is like extending our mathematical theorieswithwhatHilbertcalledideal objects; likeextendingthepositiveintegerswithzero,orlikeextendingtherealswithimaginarynumbers.Theuseof unionof proofsisthatitsavestheagreementbetweentheNormalizationandGeneralityConjecturesinthepresenceofdistribution,as we saidin 1.4. The use of zeroproofs is that it does the same inthe presence of negation. The idempotencyof unionis essential intheabsenceof zeroproofs, but not intheirpresence. Without idempotencyourgraphicalcategoryinthecaseofconjunctive-disjunctivelogicturnsuptobeacategorywhosearrowsarematrices,ratherthanthecategoryRel.Compositionbecomesmatrixmultiplication,andunionismatrixaddition.Andinthepresenceofzeromatrices,weobtainauniquenormalformlikein linear algebra: every matrix is the sum of matrices with a single 1 entry.Anumberoflogicianshavesoughtalinkbetweenlogicandlinearalge-bra, and here is such a link. We have it not for an alternative logic, but forclassical logic. Wehaveit, however, notatthelevel ofprovability, butatthelevelofidentityofproofs.Theuniquenormal formsuggestedbylinearalgebraisnotunrelatedto cut elimination. Inthe graphical categoryof matrices the result ofcuteliminationisobtainedbymultiplyingmatrices, andtheequationsofthiscategoryyieldacut-eliminationprocedure. Theyyielditevenintheabsence of zeroproofs, providedwe have 1 +1 =1. Unlike Gentzenscut-eliminationproceduresforclassical logic, thenewprocedureadmitsacommutativeadditionorunionofproofswithoutcollapse. So, inclassicallogic, the Generality Conjecture is not foreign to cut elimination, and henceitisnotforeigntotheNormalizationConjecture,providedweunderstandtheequivalencerelationinvolvedinthisconjectureinamannerdierentfromPrawitzs.Thisneednotexhausttheadvantagesofhavingzeroproofs. Theymaybe usedalsotoanalyze disjunctionelimination. Without pursuingthis1.7. Strictication 29topicveryfar, letusnotethatpassingfromA BtoAinvolvesazeroproof from Bto A, and passing from ABto Binvolves a zero proof fromAtoB. IfnextweareabletoreachCbothfromAandfromB,wemayadd our two proofs from ABto C, and so to speak cancel the two zeroproofs.Logicianswere, andstill are, interestedmostlyinprovability, andnotinproofs. Thisissoeveninprooftheory. Whenweaddressthequestionof identityof proofs wehavecertainlyleft therealmof provability, andenteredintotherealmof proofs. Thisshouldbecomeclearinparticularwhenweintroducezeroproofs.1.7. StricticationStricticationisinversetocategorication. Whilecategoricationusually(butnotalways)involvessplittingobjects,stricticationinvolvesidentify-ingobjects. Factoringasetthroughanequivalencerelation,i.e.replacingtheobjectsofasetbyequivalenceclassesofobjectsofthisset,isasimpleexampleof strictication. Logicians areveryusedtoakindof stricti-cationthat maybecalledlindenbaumization, bywhichthealgebraofformulaeofconjunctivelogicisreplacedbyafreelygeneratedsemilattice,or the algebra of formulae of intuitionistic propositional logic is replaced byafreelygeneratedHeytingalgebra, orthealgebraofformulaeofclassicalpropositionallogicisreplacedbyafreelygeneratedBooleanalgebra. Theequivalence relation involved in these strictications is mutual implication.Inthisbookweare, however,interestedinstricticationofcategories.Precise notions of strictication,which we need for our work, will be intro-duced in Chapter 3. Let us say for the time being that the simpler of thesenotions is a kind of partial skeletization of a category. An equivalence rela-tion, induced by a subcategory that is a groupoid and a preorder, is used toreplace the objects of the category by equivalence classes of objects. In theother, moregeneral andmoreinvolvednotion, thepartial skeletizationisapplied to a category generated out of a given category. (We are aware thispreliminaryroughdescriptionof themattercannotbeveryinformative.)After strictication, objects are replaced by equivalence classes, which maycorrespondtosequences,ormultisets,orsets,orstructuresofthatkind.30 CHAPTER1. INTRODUCTIONTheideais toobtainastrictiedcategoryequivalent totheinitiallygivencategoryinwhichcomputationsareeasiertorecord, becausesomearrowsthatwerenotidentityarrows, like, typically, associativityisomor-phisms, arereplacedbyidentityarrows(seeChapters5-8and11). Thisequivalence ofcategories isnotmeanttobeanyequivalence,butanequiv-alenceviafunctorsthatpreserveaparticularcategorial structureatleastuptoisomorphism. Forthatwewill denepreciselywhatitmeansforafunctor to preserve a structure, such as interests us, up to isomorphism (see2.8).Wewereinspiredbypreviousattemptstodenethisnotionoffunctorformonoidal categories, andbytheensuingstricticationresultsofJoyalandStreet in[72] (Section1) andof Mac Lanein[102] (Sections XI.2-3). Wedonot, however, ndthesedenitions andresults sucient forour purposes, evenwhenonlythe monoidal structure is strictied. Weneedsomethingmore general. We envisage strictifyingstructures otherthanjustmonoidal,andwewillhaveoccasioninthisbooktostrictifyalsowithrespecttosymmetry(see 6.5, 7.6and 8.4). Anotherlimitationofpreviousstricticationresultsformonoidal categoriesisthattheydonottake into account that the monoidal structure may be just a part of a morecomplexambientstructure, andthatthefunctorsinvolvedinequivalenceshouldpreservethisambient,notstrictied,structureuptoisomorphism.Tohavejustthemonoidal structurepreservedisratheruselessfromourpointofview(see 3.1).Ourresultsonstricticationwill bemuchmoregeneral, buttheyarenot such that they could not be further generalized. In particular, in den-ingthecategorial structurepreservedbyourfunctorsuptoisomorphismwehavepresupposedthatthisstructureisdenedonlywithcovariantbi-naryendofunctors. Anatural generalizationistotakehereintoaccountalso endofunctors of arbitrary arity, covariant in some argument-places andcontravariantinothers. Wesupposethatourresultscanbeextendedtocoversuchsituationstoo. Fortheapplicationsweneeditwas, however,enoughtocover thesimpler situation, excludingcontravariance, andwedidjustthat. Wewereafraidofcomplicatingfurtheramatteralreadyfullof details, to prove results for which we have no immediate application. (AsMacLanesaysin[100], p. 103: ... goodgeneral theorydoesnotsearch1.7. Strictication 31forthemaximumgenerality,butfortherightgenerality.)Soournotionof logical systeminthenextchapterinvolvesonlycon-junction and disjunction as binary connectives, together with the constantsand . Implicationisexcluded, andnegationisleftfortheendof thebook. Tocovertheseotherconnectives, wewouldneedtoextendourno-tionoflogical structuretopermitcontravariance. Weassumethiscanbedone in a straightforward manner at the cost of complicating notation. Werefrain, however, fromdoingsointhisbook, whosecentral pieceisaboutconjunctive-disjunctivelogic,andwherenegationappearsonlyattheend.Anyway,asfarasstricticationgoes,thislimitednotionoflogicalsystemissucientforourpurposes.Classical implication, denedintheusual wayintermsof disjunctionandnegation, does not comeout as averyimportant connectiveinourproof-theoretical perspective. It is not much of an implication, if the role ofimplicationistohelpinmirroringthedeductivemetatheoryintheobjectlanguage. Intuitionisticimplicationplaysthatrolebetter.Our results on strictication are still somewhat more general than whatwe strictly need. In strictifying a binary connective like conjunction, purelyconjunctiveformulaemaybereplacedbyequivalenceclasses that corre-spondtosequences,ormultisets,orsets,oftheatomicformulaejoinedbyconjunction. Forourpurposes, wecouldhavestucktothersttwostric-tications, butwithourgeneral treatmentwecoveralsothethird. Withthat,westaywithinthelimitsofcovariance.Strictication,thoughaninterestingtopiconitsown,isnotabsolutelyessential forourmaintopiccoherence. Itisforusjustatool, wecouldhavedispensedwithinprinciple. Thatwould, however, beatthecostofmakingalreadyprettylongrecordsevenlonger. Sostricticationisforusaratherusefultool.Itisatoolmoreusefulforrecordingcomputationsthanfordiscoveringhow they shouldbe done. Blurring distinctions may sometimes hinder thisdiscovery.Itisremarkablethatthegeneralnotionofstricticationmaybefoundimplicit inGentzens sequent systems,as we willtry to explainin 11.1,inthecentralchapterofthebook.Chapter2SyntacticalCategoriesIn this chapter, which is of a preliminary character, we dene the notions ofsyntacticalcategoriesneededforourwork. Inparticular,weintroducethenotion of logical category (which should not be confused with the homony-mous notionof [104], Section3.4). Logical categories areobtainedfromlogical systemsinapropositional languagebyreplacingderivationswithequivalenceclassesofderivations. Theequivalenceproducingtheseclassesisofgeneralmathematicalinterest,butithasalsoproof-theoreticalmean-ing, sothat theequivalenceclasses maybeidentiedwithproofs. Thispresupposessomenotionsoflogicandcategorytheory,whichwillbedulydened.Manyof thesenotions arequitestandard, andwegoover themjusttoxterminology. Somethinglessstandardmaybefoundinthesectionon denable connectives,where some intricacies inherent in this notion aremade manifest. Anewmatter is alsodetaileddenitions of notions offunctorspreservingthestructureof alogical category. Weareinterestedinparticularinthoseof thesefunctorsthatpreservethestructureuptoisomorphism. These denitions prepare the ground for Chapter 3. We treatthese matters in generality greater than we strictly need after that chapter.Itisnotessentialtomasterallthedetailswegointoinordertofollowtheexpositionlateron.Afterthesesyntacticalmatters,weintroduceattheendofthechaptera category that will serve as the main model of our logical categories. Thismodel,whichisintherealmofasemanticsofproofs,andnotintherealm3334 CHAPTER2. SYNTACTICALCATEGORIESof theusual semantics of propositions, is thecategorywhosearrows arerelationsbetweenniteordinalsacategorytiedtothenotionofnaturaltransformation. This categorywill servefor our coherenceresults. Oursyntaxislinkedtothismodel byfunctorsthatpreservethestructureonthenose,i.e.uptoanisomorphismthatisidentity.2.1. LanguagesAlanguageis aset of words, eachwordbeinganite(possiblyempty)sequenceofsymbols. Asymbolisamathematicalobjectofanykind. Thelengthof awordisthenumberof occurrencesof symbolsinit, andthisisthemoststandardmeasureof thecomplexityof aword. Inparticularcases,however,wemayrelyonvariousothermeasuresofcomplexity,like,forexample,thenumberofoccurrencesofsomeparticularkindofsymbol.Weintroducerst several languages of thekindlogicians call propo-sitional languages. Suchlanguagesaregeneratedfromaset Pofsymbolscalled letters; logicians would call them propositional letters or propositionalvariables. Sometimeswerequirethat Pbeinnite(seetheendof 2.8),but Pcanalsobenite, andevenempty. Sincenothinginparticularisassumed about P, the symbols of Pcan be arbitrary mathematical objects,and the denitions of notions built on P(such as that of logical system andlogical category;see 2.6-7 below) do not depend on the particular Pthatwaschosen.Let be a symbol of the kind called in logic n-ary connective, for n 0.A 0-ary, i.e. nullary, connective is more commonly known as a propositionalconstant; 1-ary are unary connectives and 2-ary connectives are binary con-nectives. We assume, as usual, that Pis disjoint from the set of connectives.Then a language L such as we need is built up with inductive clauses of thefollowingkind:(P) P L,() ifA1, . . . , An L,thenA1. . . An L.It is assumed here that is an n-ary connective. If n = 0, then A1. . . Anistheemptysequence, and L. Wehaveananalogousconventionforall2.1. Languages 35sortsofsequencesthatwillappearinthiswork: ifn = 0,thenx1. . . xnorx1, . . . , xnistheemptysequence,and {x1, . . . , xn}istheemptyset .The elements of L are called formulae; logicians would say propositionalformulae. Weusep,q, . . . ,sometimeswithindices,asvariablesforletters,i.e. elementsof P, andA, B, . . . , sometimeswithindices, asvariablesforformulae. The elements of Pandnullaryconnectives are calledatomicformulae. Theletterlengthof aformulaisthenumberof occurrencesoflettersinit.Wereservefornullaryconnectivesand forbinaryconnectives. Theformulapqp, whichisinthePolish, prex, notation, ismorecommonlywritten((p q) p), andwewill favour this common, inx, notationforbinaryconnectives. Polishnotationishandyfordealingwithn-arycon-nectives where n 3, but in the greatest part of this work we will have justnullary and binary connectives. A unary connective appears in Chapter 14.(Notationfor unaryconnectives that wouldnot bePolish, likeHilbertsnegationA, is uncommoninpropositional logic; for nullaryconnectivesthereisnoalternative.) Weassumethatwehaveasauxiliarysymbolstherightparenthesis)andtheleftparenthesis(,whichareneitherlettersnorconnectives,withwhosehelpweformulatetheclauseifA, B L, then(A B) L.This clause replaces () for binaryconnectives. As usual, we take theoutermostparenthesesofformulaeforgranted,andomitthem.Consider a binary relation Ton a set of elements called nodes such thatwhen xTywe say x is the predecessor of y, or yis the successor of x. A pathfromanodextoanodeyisasequencex1. . . xn,withn 1,suchthatxis x1and yis xn, while for every i {1, . . . , n1} we have xiTxi+1. A rootisanodewithoutpredecessors, andaleafanodewithoutsuccessors. Wesay that a node is of n-ary branching, with n 0, when it has n successors.Soleavesareofnullarybranching.A nitetree is such a relation Twhere the set of nodes is nite, there isexactly one root and every node except the root has exactly one predecessor.Itisclearthatineverynitetreethereisexactlyonepathfromtheroottoeachnode.36 CHAPTER2. SYNTACTICALCATEGORIESTheheightofanodeinanitetreeisthenumberofnodesinthepathfrom the root to this node. A nite tree is planar when all nodes of the sameheightn 1arelinearlyorderedbyarelation1andfn : B B, wehavebytheinductionhypothesisthatfn1 . . .f1 :A Bisequal to(h)1gforg :A Candh :B C.If fnisab-term, then, forh : B C, bytheDirectednessLemmawehavehfn=h, andf=h1gfollows. Iffnisab-term, thenbytheDirectednessLemmawehaveh f1n= h,andf= h1gfollowsagain.For f: A Bweobtaininthesamemanner f=h1g, andsof= f. OnemightsupposethatSemiassociativeCoherencecanbeinferreddi-rectlyfromAssociativeCoherence. Thiswouldbesoif wecouldndanindependentproof thatAisisomorphictoasubcategoryofA, aproofthatwouldnotrelyonSemiassociativeCoherence. Infact, weuseSemi-associative CoherencetoconcludethatAisisomorphictoasubcategoryofA. ThatAisisomorphictoasubcategoryofAamountstoshowingthatforfandgarrowtermsof C(A)wehavef= ginAif= ginA. Thatf= ginAimpliesf= ginAisclearwithoutappealingtoco-herence, but for the converse implication we use Semiassociative Coherence(cf. 14.4).IntheproofofAssociativeCoherenceabove,werelyessentiallyonthenormal formof formulae, anduse bothb-terms andb-terms. Thisis whyfor theproof of SemiassociativeCoherencewecouldnot relyontheDirectednessLemma,butweneededthe