; TeX output 2006.09.01:174794 9HDDtGGcmr17CommqutativeDiagramsinT[-E;gX(vqersion4)$ Gy!N cmbx12PaulTaylor2"߆T cmtt12www.cs.man.ac.uk/!", cmsy10pt&w_XQ cmr1215June1997Vi $t : cmbx9Abstract 9@#o cmr9TuAEX+andL5Aacmr6A TuAEXha9ve+bAecomestandardasaw9ay+ofwritingpapersin @Computer}=ScienceandCategoryTheory:.~Ev9eninsourceformtheyare@easier" tocompAoseandreadthanattemptstowritemathematicsin@ASCIAI.qInCategoryTheory\comm9utativeqdiagrams"areessen9tialfora@clear?visualunderstandingofthepapAer,Obutthegraphicscapabilitiesof@TuAEXxaresolimitedthatitisv9erydiculttodrawthemnicely:,Uifat@all. This+man9ualdescribAesanewbutreverse-compatibleversionofa@pac9k|ragetodrawsuchdiagrams,expressedinalanguageinwhichmany@usersha9vealreadyfounditveryeasytoexpressthemselves.!U}'-Nff cmbx121@ wIntros3ductionc'K`y cmr10InFpapGersinmathematicsandcomputersciencewhichemployCategoryTheory*, 'thereEismuchEbGene tinclarityif\commutativediagrams"areusedasmuch'asupGossibletoillustratede nitions,!equationsanduniversalproperties.kHereis'atypicalsuchdiagram:ritisoneoftheMacLane-Kellyequations.4d>y b> cmmi10A8 !", cmsy10(BQ(CDG))^1fe Ŝrassl*O line10-y(A8Bq)(CDG)_1fe Ŝrassl*Ϭ-.y((A8Bq)C)D؈=>yA8((BQC)DG)?sidJU8assldX?d9;*fe^1feŨrassl*Ϭ-.y(A8(BQC))DWassliz8idTX(6T9S**fe-2'This4manualdescribGesversion4oftheauthor'spackqagefordrawingdiagrams'linethisin(plain)TU>'ExXorL5ffٓRcmr7A͉TU>'EX. V*ersion3isalreadyverywidelyusedin'ther0CategoryTheoryandTheoreticalComputerSciencecommunities. sMost'ofItheunderlyingcoGdehasbeenrewritten,withagreatimprovementItothe'appGearanceEofthediagrams,butitremainscompatiblewiththepreviously'developGedandverypGopularstraightforwardlanguage.'2@ wDesignCriteriac'DrawingasuchadiagramusingtheL5ffA͉TU>'ExX2'ExX'doGcumentQ.haveloGokedasprofessionalaswhatchildrenbringhomefromtheir' rstdayatschoGol.rMyviewis,ifyouwant3': cmti10wysiwyg,usepGenandpaper! 𗍑6Besidestheawfulresults,]mouse-drivenmethoGdstakelongerandareless'pGortable.}If!youwanttowriteajointpapGerwithacolleagueontheotherside'ofatheworld,itisagreatdealsimplertosendasingleascii lebyelectronic'mailthantopackqagetwentyofthem(includingone lepGerdiagramaswell'asthemaintext)encoGdedinsomeweirdcommercialbinaryformat.6Onthebasisoftheseremarks,&thedesigncriteriaofthispackqageareas'follows:31.@TheEentirediagrammustformpartofthesourceofthedoGcumentitself.@Inx otherwords,theremustbGenopreprocessing(cf.eqnin4- cmcsc10Unix)or@inclusionof les(suchasPostScriptpictures).-32.@SimpleodiagramsmustbGeabletobedrawn\onthe y"andnotneedto@bGe[ydrawnonpaper rst..Obviously*,rWcomplexdiagramswillalreadyhave@bGeenworkedoutonpapGeranyway*.33.@ThelayoutofthesourcecoGdemustresembletheintendeddiagramasfar@assyntacticallypGossible.34.@TheremustbGenomeasuringoflabelstocalculateco-ordinatesorlengths@ofarrows.35.@TherewWmustbGeavqarietyofarrowstyles,,withfacilitiesforde ningnew@ones. >Diagonal+arrows(which,#throughlackofappropriateprimitives,@TU>'ExXpmakesverydiculttodraw)shouldbGeprovidedatvqariousslopGes,@albGeitwithlimitedchoiceandfeatures.36.@The2'ExXandwithL5ffA͉TU>'EX,@andnotrelyonnon-standardfonts^1 |sorlanguagefeatures.37.@F*uture{versionswhichimprovetheappGearanceofthediagramsmust,as@faraspGossible,hbecompatiblewithpastpapGerswrittenusingthepackqage@|butyouhackatyourownrisk!'Inradditiontherearestheticcriteria,ΜsomeofwhichmaybGeamatterof'opinion:6@ArrowsXshouldstretchtomeettheob8jectswhichareintendedtobGetheir@endpGoints.-6@ArrowsQ shouldbGealigned(bothhorizontallyandvertically)withthe@centresoftheob8jects.6@LabGels%onarrowsshouldnota ectthespacingofthediagramexceptto@avoidoverlapping.6@Stretchingofarrowsshouldnota ectthecenteringoftheirlabGels.'9Zff J= "5-:1L|{Ycmr8DiagonalɄlinescan5#fcmti8only|b'ExXasfollows:cݍv'\inputdiagrams '\def\Assl{{\rmassl}}\def\Id{{\rmid}}D㊍qɍ'\beginfdiagramg"qǍ(A*(B*(C*D))r&\rTo^\Assl&(A*B)*(C*D)\j&\rTo^\Assl4\E&((A*B)*C)*D\\|\dTo^{\Id*\Assl}r&&=\j&4\E&\uTo_{\Assl*\Id}\\(A*((B*C)*D)r&&?\rTo^\Assl\j&4\E&(A*(B*C))*D\\'\endfdiagramg 6InEMplainTU>'ExXyoudothesamething,Vwriting\diagramand\enddiagram 'whereverwehave\begin{diagram}and\end{diagram}.6InL5ffA͉TU>'ExX2"/Syoucanput^@\usepackage{diagrams}'in*>placeofthe\inputcommand,4butyouhavetorenameoraliasthe leto'diagrams.sty. 6Thebasicruleistodividethediagramintocells,/justlikethecellsofa9Z#[A8(BQ(CDG))v1fe6Ubުassl*Ϭ-y(A8Bq)(CDG) &1fe6Ubުassl*Ϭ-I((A8Bq)C)D '[󍍟v./v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.؈='[󍍟v./v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.& id0>8assl LЍJ?J~ϟLΉfeqVasslp8idٳ.n6mfe$vA8((BQC)DG)t1fe:| assl*Ϭ-H4b(A8(BQC))Dq蟻ȍp`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.ȍp`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.ꟻȍp`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.Fȍp`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.;ɍ'matrix,ݗandthentypGethecontentsofthecellswithcolumnsdelimitedby\&"'andQrowsby\\\".ThebGottomarrowextendsthroughtheemptycellseither'sideofit. NoticethatalthoughthematriximpGosesakindofco-ordinate'system,=the*widthsofthecolumnsandtheheightsoftherowsarevqariable'andchosenautomatically|byTU>'ExX,ݝinthesamewayasitdoGesformatrices,'tables,etc.6AtR rstyouwillprobablyneedtodrawthediagramcarefullyonpapGer'and.divideitintocellsinthiswaybGeforetypingitin.vThemaindicultyis'workingouthowmany&stoinsert;?forthisitisusefultoobservethatinthe'simplestc}'ases!^2^6@ob8jectsandverticalsgoinoGdd-numberedcolumns,andY6@horizontalsanddiagonalsgoineven-numbGeredcolumns.'Then,7Lofcourse,youneedanevennumbGerof&sbetweencolumnsofthe'sameparityandanoGddnumbGerbetweendi erentones.+ Afteralittlepractice'you'll}learnotherrulesofthumb,butevenifyoumakeamistake,theDVI'previewer@Fwillmakeitclearhowtocorrectit.EErrormessagesabGoutclashing'orunterminate}'darrowsindicatethatsomethingiswrongwithoutpreviewing. 6Eachh+cellshouldcontaineither {anob8ject(anordinarymathematical'expression,zset/ inmathsmoGde)orBamorphism(anarrowsuchas\rTo^f).'HorizontalandverticalarrowscannotbGemixedinonecell;moreoveronlyone'horizontal$arrowpGercellisallowed,nbutseethesection6onparallelmaps'bGelow.'aff J= "5-:2LܻTheb'ExXsense:ifyouwanttouse'theminsectionheadingsyoumust\protectthem.#č'4@ wLabs3els'EacharrowcarriesuptothreelabGels,rwhosepositionisspeci edanalogously'tosupGerscripts^3 |sby cI^abGove,_bGelow,rightand~middle.'F*orMreversecompatibility*,qabGove=leftandbGelow=rightforverticalarrows.6V*ery'oldversionsofthepackqageusedpGositionalarguments;Jthesearealsostill'suppGorted,6butmustb}'eenclosedinbraces,6e.g.\rTo{f}{g}butnot\rTofg.6Explicitly*,thelabGelsareplacedasfollows:6@forhorizon9talarrows,\rTo^f_gand\rTo{f}{g}giveꍑ f$ v! gw;6@forv9erticalarrows,\dTog,\dTo^f_gand\dTo{f}{g}givefu cmex10?y;g[ٲ;6@forpQositiv9egradientdiagonalarrows,[\ldTo^f_g,\ldTogand @\ldTo{f}{g}allgive^fK.xg(similarly\ruTo).6@butfornegativ9egradientarrows, fabGove=rightandbGelow=left, fso@\rdTo^f_g,\rdTofand\rdTo{f}{g}givegn'& ^f@(similarly\luTo);6@Usingtilde,thelabGelmayinsteadbreakthearrow: \rTo~fgives@A1fe 8oIffe*Ϭ-UXIϵBr.If2thearrowhadamiddle(e.g.+in\rCrossedInto,[page8),@theTlabGelwouldreplaceit. mThisissometimesusefultopreservethe@symmetryofadiagramwiththreeverticals.'X-ff J= "5-:3LܻHoÎwevertheyarenotqrecognisedsynÎtacticallyinthesamewayJ,andsoforinstance K cmsy8nnolimitsD willnotwÎork. Thesub-andsupY*oucanuse\!and\;commandstocompGensateforthis,@remembGeringthatshouldyousubsequentlychangetheshortfallamount@thenyoumustalsoadjustthecompGensation.@An alternativewayofdoingthisistode neanewarrowcommand:@section7explainshow.@Don'tr makeahabitofputting\relaxafterarrowcommands: itcan@lead#"togapsinshortdouble-shaftedarrows,kforreasonswhicharetoGo@complicatedtoexplainhere.6@F*orverticalanddiagonalarrowsadditionaltextwilllietothesideof(or@bGetween)ithearrows.ZnInthecaseofverticalsthisisatthesameheight@asthemiddleandlabGelsofthearrows;seesection6.6ByadefaultthelabGelsonthearrowsaresetin\textstyle(thesamesizeas'intext)whentheyarepartofamatrix,andin\scriptstyle(likesubscripts)'inAJMڟ1fe(ߒIύHn 5gX p1Qصfi6/odisplay*Ϭ-zõB'Similarly*,}ob8jectswithinadiagramaresetin\displaystyleunlessthe'objectstylejCoptionisused. [<(Intext,e.g.$A\rToB$,thesettingofthe'ob8jectsisoutofthecontrolofthediagramspackqage.ްUnlessyouhavedone'somespGecialprogrammingofyourown,theywillbGesetin\textstyle.)`ڠ94 '69v4.rLabGels9u '5@ wTheCubs3e'ThecubGeo ersagoodexampleofhowhorizontalandverticalarrowsstretch. 'NoticeyUhowthecellshavebGeensub-dividedfurthertogivethe\3D"y2e ect,Dand'thatthepGositionsofthelabelsonthemapsbetraythecellsinwhichthey'weredeclared.Becauseofthesub-division,theparityruleforthecolumnin'which0thehorizontalarrowsgohasbrokendown.6Y*oushouldnowformulate'for>yourselfanewruleofthumbforthecaseswhenyouneedone,two,three'orfour&sbGetweentwocommands.|qɍ'\beginfdiagramgRqǍ0|A'O&<&\rTo^{f'}&k&?B'6C&f<'&<\\O&\rdTo_a<&&k&\vLine^{h'}6C&\rdTo_bf<'&<\\\dTo^{g'}O&<&A& \rTo^fk&_\HonV6C&f<'&B<\\O&<& \dTo^g&k&\dTo6C&f<'&<\\0|C'O&?\hLine<& \VonH&\rTo^{k'}k&?D'6C&f<'&\dTo_h<\\O&\rdTo_c<&&k&6C&\rdTo_df<'&<\\O&<&C&k&\rTo^k6C&f<'&D<\\'\endfdiagramg6The^Pbrokenlines(h^0 ,andkP^0в)aresimplytwoarrows. 7bThecommands'\HonVU(\horizontalonvertical")and\VonHU(\verticalonhorizontal")allow'arrowsinonedirectiontopassthrough,Hwhilstobstructingarrowsintheother'directionGhasifanob8jectwerepresent.Theyeachhaveanoptionalargument:'\HonV[=]and\VonH[\|]shouldbGeusedtoallowextraspacefordoublelines.'An alternativeistodeclarethe\front"lineintheintersectioncell,whereit'will(6havethee ectofobstructingtheotherarrow,2CbutthenitslabGel(ifany)'wouldclashwiththeotherarrow.6There.isalsoacommand\HmeetVwhichcausessinglehorizontaland'verticallinestomeet,formingacorner.u}ESA01feb^cf^0*Ϭ-CBq0lq@r@ls@t @lu@#;aluRl}@ ~@l@쀟 @"l@!͵b"lRvA,w1fedߘcf*Ϭ-zh^09ILΉ1Lfe2afB<}C0wJg[ٟ^0/vX?9>pg|fe1fe.wRŸ1fe*kP^0*Ϭ-DG0pX?9Jp.HDfelq@r@ls@t @lu@cluRl}@ ~@l@쀟 @"l@8d"lR~C\gO*X?9C*i*feZܟ1feduk*Ϭ-2%8D9lh1*6X$?69O*i*fekЍ'Y*ouJshouldsoGonbeabletoreadandcomposethesourceofthesediagrams'asʤeasilyastheprintedversion,Malthoughonewouldnotnormallygotothe'troubleofaligningthe&sinthesourcecoGde!^4'X-ff J= "5-:4LܻTJodosoneatlyinthismanÎualrequiredspፍvAT,w1fe@ffe*Ϭ-XBфhdF"dGh q dFhhdF៕ ,w1fe@ ogfe*Ϭ-a`B<~Ch*Ǭ?c*-*feoaNk6`-*-*fe1ʍ6Horizon9talWarrowsmaybGe\piledontopofeachother. #tE ectively*,a'one-columndiagramiscreated,]whichmayhaveeither>ahorizontalarrowor'anoob8jectineachrow. nThisworksbGothindiagramsandintext. nThe'spacing<(\baselineskip)bGetween"Z1feD*Ϭ-E\newarrowfLinegt-----"Z1feHE\newarrowfEmbedgt>--->-#Դ1feCUN*-E\newarrowfOntogt----f>>g"Z1fe@*Ϭ-*-E\newarrowfEEmbeddgtf>>g---f>>g- -'԰1fe;UV*-*-E\newarrowfDotstogt....>".v.ፍv.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v.v./{-E\newarrowfDashtogtfgfdashgfgfdashg>"81fe81fe81fe81fe81fe81fe81fe81fe81fe81fe81fe81feH-E\newarrowfCorrespondsgt<--->"*&T1fe@N*-E\newarrowfCrossedIntogtfboldhookg-+->!|?ncmbsy5&?X1fe1Ǎ*Iϲ+vfem*Ϭ-E\newarrowfImpliesgt====f=>g"ō=㏍============4)E\newarrowfMapstogt|--->"ZB!fb"弟1feDDF*Ϭ-E\newarrowfIntogtC--->!|&?X1fe@ꪍ*Ϭ-E\newarrowfOpenintogtftriangleg--->"T.&\1fe@*Ϭ-E\newarrowfCongruentgt33333"sE\newarrowfTeXtogt----f->g"Z1fe?*I!E\newarrowfBackwardsgt<----"*&T1feD*E\newarrowfMultigt----o"Z1feC*IϸE\newarrowfCrosstogt----X"Z1feEhcIϸE\newarrowfPartialgt----fharpoong"Z1fe?*Iϵ*E\newarrowfTeXontogt----f->>g"Z1fe?*I. msam10'Note vFthatthe{->>}, {harpoon},{->} vFand{=>}headsuseTU>'ExX's'\rightharpoondown(+),l\rightarrow(!)and\Rightarrow())symbGols,'consisting ofheadswithshafts.Also,MStheboldbooktailusesL5ffA͉TU>'ExX's\boldmath:'please'seethenoteabGoutheads=littleveeonpage11.ThehookandCtails'arethesame.6Eachdeclarationbi&\newarrow{Name[}{tail}{ ller}{middle}{ ller}{he}'ad}'de neseightarrowcommands>qǍ<˸nrNameoright2[orʎeast <˸nlNameoleft9#orʎwest<˸ndNameodown0orʎsouth<˸nuNameoup<\orʎnorth<˸nruNameoright&uporʎnortheast<˸nrdNameoright&down orʎsoutheast<˸nluNameoleft&uporʎnorthwest<˸nldNameoleft&downorʎsouthwest֎9r,luJ+uqr,ru&)I&)@/\ӟ*@8}V@@'@N)W\ӟV`}*h'h) +l(֟-'1feL6L~!!feN)1fe *-t+r&) &)/\ӟV8}*@'N)@W\ӟ*@`}V@h'@h)Rr,ldJ+d㎍L̬?L~㎉feqr,rd'fromthe venamedcompGonents.6SomegbutnotallofthenamesgivenintheexamplesabGovearede ned 'inTthesourceofthepackqageasdistributed. HHoweverthisisonlyreally'intendedforreversecompatibility*,lbGecausejustasitisbettertode nea'macro\isomorphicforthesymbGolT͍+3=(ifthatiswhatyouuseittomean)'thantowrite\conginyourdoGcuments,soitisadvisabletode nethearrow'commandnamesinthe lewhereyoukeepyourprivqatecollectionofsymbGols. X94 'PaulT*aylor'sdiagramspackqageԄ399u 6F*orPexample, $ifinyoursub8jecttherearespGecialkindsoffunctions 'knownstandardlyasimmersions,dinclusionsandinternalisationsandwrittenas&X|+1fe a*Ϭ-UXIϲ,1- ߡfe˨*- \and cY. fe !*Ϭ-,thenyoushouldde ne\rImmerse,\rIncludeand'\rInternaliseD&insteadofusingthenames\rInto,U0\rEmbedand\rOpeninto.'Themottoiswhatyouse}'eiswhatyoumeanc! 7.ThisavoidsremembGeringor'gettingconfusedabGouttheso-calledstandardmacronames,andalsoenables'youtochangeyourmindabGoutthenotationifitisnotstandardbutsub8ject'toexpGerimentation.6Theh compGonentsasgiveninthe\newarrowcommandaremnemonics,qwhich'arethemselvesde nedusingthecommands33\newarrowhead,\newarrowtail,\newarrowfillerand\newarrowmiddle.'MostofthecharactersinthestandardTU>'ExXComputerMoGdernfontswhichare 'appropriateMforthesecompGonentshavealreadybGeenusedinthepackqageand'are5illustratedabGove.If5youwishtode neothercompGonents,,Ctheexamples'attheendofthesourceshowhowtodothis. {Ifyouaredoingthiswith'publiclyavqailablefonts,,{>>},<,{<<},x,+,O]and~ocanbeusedbothasheads'( fth2pGosition)andtails( rst). Thedeclaration<----de nes\rBackwards'tobGealeft-wardpointingarrowand\lBackwardstopGointright; thisis'notrecommended,Basyou'recertaintogetconfused. W*eshallseeinthe'nextsectionhowtochangethestyleofthefourarrowheadcompGonentsina'systematicQNway*,t?therebyavoidingtheuglymixtureofveeandL5ffA͉TU>'ExXarrowheads'obtainedusingearlierversionsofthispackqage.6InnDsomeoftheexamplesabGovenD(withthedotandrule llers)the ller'wasC(repGeatedasthemiddleandoneorbothofthetips,Swhereaswithothers'(dash)"weusedemptymiddleandtipsinstead. [Thiswasjusttomakethe'de nitionseasiertotypGe:\newarrowtreatsmiddlesandtipswhichagreewith'the+ llersasemptyones.ItispGossibletousetwodi erent llers,6hbutthisis'notrecommended.6Thecommands\HorizontalMapand\VerticalMaparestillsuppGorted,but'should^bGereplacedby\newarrow. 8TheseoughtonlytohavebGeenusedin'macro* les|notinthetextofdoGcuments|tode nefourarrowcommands'together(right,*left,upanddown).5Thetwentyargumentsde ningthe ve'compGonents9;ofeachofthesefourarrowscan(ifthey'renotalreadyprovided)'bGere-arrangedasthefourargumentsofeachof ve\newarrowhead,Aetc.,'declarationsVwiththesamee ect. Themnemonicsofthe vecompGonentsare'thenDgivento\newarrow. =Ifyouhaveanydicultydoingthisconversion,'pleasecontactme.#č'8@ wOptions'Thew\diagramor\begin{diagram}commandandthearrowcommandsmay'bGefollowedbyalistofoptionsinsquarebrackets,forexample\l\begin{diagram}[size=2em,textflow]\rTo[abut] S94 '10-v8.rOptions9u 'whichapplytothediagramorarrowtowhichtheyareattached.wOptionsmay 'bGe}&givenforanentiredoGcument,porwithinTU>'ExX'snormalscopingrules,bya'stand-alonecommandsuchas퍑fL\diagramstyle[centredisplay,PostScript=dvips]'Y*oumayuseanynumbGerof\diagramstylecommands. InL5ffA͉TU>'ExX2"2Wthese 'optionscanbGegivenwhenloading:QL\usepackage[centredisplay,PostScript=dvips]{diagrams}'thoughyoushouldavoidusingmacrosintheoptionvqalues(seepage28). 6Note:if)thetextofthe rstcellinadiagrambGeginswithasquarebracket,'theprogramwillattempttoreaditasanoption,Ajustassquarebrackets'within’L5ffA͉TU>'ExXarrayscancausetrouble.d'Thisusuallyleadstoacatastrophic'error, whichcanbGeavoidedusing\diagramstyle[nooptions].%Thisproblem'doGesn'tariseinL5ffA͉TU>'ExXifyouuse\begin{diagram}followedbya(spaceor)'newlinebGeforethetextofthediagram.6Someoptionstakeavqalue,usinganequalssign;thoseenclosedinsquare'bracketsbGelowareoptional,Etheothersmandatory*. *AMostoftheoptionsare'appropriatefordiagramsratherthanarrows.鍍"abut|PLeave-NnogapbGetween-Nmapsandob8jects;susefulfor(\Hasse")diagrams@displayingorderstructures(page17)."alignlabels|uThe:labGelsonhorizontalarrowsremaincentredinthecells@whereMtheyarede ned,despitedi erentsizeendpGoints;acf. 4midshaft.@This^isthedefault,anditisrecommendedfordiagramsofaregular@character.1#IfthereisinsucientspacetopGositionthelabelsinthisway*,@theyXarecentred(automaticallyandsilently)instead.yWhicheveroption@isused,themiddleofthearrow,ifpresent,isalwayspGositionedwiththe@labGels."amstex|[?EnablerecognitionofAU>'M S-TU>'ExX'scommutativediagramcommands;@seesection14."balance|`Thekwholediagramiscentredhorizontallywithoutregardtothe@left-andrightmostlabGels,[9sothatitsweightliesinthemiddleofthe@page.QThismisnotthedefault,bGecauseifyouwerenotawareofitthe@gapswouldcomeasasurprise,butitisrecommended."bottom|[?When(thediagramisplacedalongsidesomesimpletext,Sthebaseline@ofthattextisalignedwiththatofthebGottomrowofthediagram."centre@ ?Iϲand@fe|Q>FQ.^OtherSsimilaradjustmentsmay@alsobGemadetocharacterpositioning,andtotheslopesofdiagonals@in?(PostScriptmoGdetoavoid\stepping." Mostocelaserprintersare@300dpi,fbut@ifyouintendtouseaprinterofadi erentresolutionforyour@ nal`output,@xsetdpi=1270orwhatever|otherwisetheshaftswillbGe@toGowtlowinstead.Thesameappliesifyouintendtomagnifyorreduce;for@exampleqdpi=212shouldbGeusedfortwo-upqorA4-on-A5 nalprinting.@IniplainTU>'ExXyoumayset\magnificationatthebGeginningofthe le,@butTkdonotGCchangedpiasthiscompGensatesautomatically*.Thenopixel@optiondisablesthefudgefactor."dpm=@resolutionr|AsdpibutindotspGermetre;e.g.r1270dpi=50000dpm."eqno=@labelr|qPlacelab}'el(inmathsmoGde)besidethediagram,Casanequation@numbGer.rSeealsoLaTeXeqno."fixed|UF*orcediagonalarrowstobGesetonthe rstpassandnotstretched."flushleft[=@widthr]|The=oldversionofleftflush. SThealignmentison@theedgeofthegrid,'irrespGectiveofwhatverticalmapsarepresent(this@makesthediagramstickoutontheleftwithnogoGodvisualreason)."grid=@namer|lUsethegridname.rSeepage20."gridx=@namer|qUsethegridwiththehorizontalandverticalcompGonents@interchanged."h=@distancer|qSameasheight."hcentre@orrhcenter|{Sameasnobalance."heads=@namer|qSetthe>and{>>}arrowheadsandtailstothosede nedby@\newarrowhead{name[}and\newarrowtail{name}.XThefollowingstyles@arecurrentlyavqailable:ǍpYLaTeX.-ܟ1fe-UN*---fe%UV*-*-0(default;usesline10)Yvee.>1fe+UD|I>I>I>fe#UL|I>xI>Ylittlevee.>۟1fe,Ԅ+v >П >̟ >+yfe$Ԍ+v >+r >Yboldlittlevee.;*cmmib7>X1fe+f >= >9 >Ecfe#f >b >0(uses\boldmath)YtriangleT׵.1fe)*I.XI.TI.fe"*I.*I.Yo.1fe)*I*I*Ife"*I*IYO. 1fe$* ̟ ȟ #;feʍ* * YX8hc1fe,8ЍcII{I-fe$8؍cIcIY+.+1fe%*I+G#I+GI+"8fe*I+*I+Ycurlyvee.1fe+UD|IIIfe#UL|IxI0(usesAMSsymbGols)Yblacktriangle.I1fe&jtII͟IIɟII!㓄fertIIpII0(usesAMSsymbGols)Ylittleblack./u7msam7Ik1fe*t I1Ο I1ʟ Ife"t Ip I0(usesAMSsymbGols) 94 '12-v8.rOptions9u @TheZverticalcurlyveeheadscomefromtheAMSZsymbGols,Tandthe @horizontals,Kfrom 3cmsyintheComputerMoGdernfonts,KseemappGearto@bGe~slightlymis-alignedwiththeveeheadsfromcmmi.Ifyouusethem@butPforgetto\usepackage{amssymb},dyou'llgetan\undefinedcontrol@sequence"?errorinthemiddleofalotofgarbage(deeplynested):hit@returnseveraltimesandcarryon.@TheCboldlittleveeheadsrelyonL5ffA͉TU>'ExX's\boldmathcommand,and@defaulttolittleveeinplainTU>'ExX.\Theymaynotworkcorrectlyifyou@usexLeslieLampGort'soldL5ffA͉TU>'ExX2.09fontselection. 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(Notyet@implementedR|infactthePostScriptoptioncurrentlyalwayssdoGesthis@todiagonals,butnottoverticals."htriangleheight=@distancer|{Set6height=distanc}'eandthenwidthinsuch@ajwaythattheminimal35-gridwillmakeanequilateral.triangle,and@a5]7-gridmakesaregularhexagon(page19).tiIfthedistanc}'eXisnot@spGeci ed,theexistingheightisusedandthewidthadjustedaccordingly*."htrianglewidth=@distancer|;Setwidth=distanc}'e(andthenheighttomake@these gures."inline|[?UsethisoptiononindividualdiagramswhicharebGeingdisplayed@alongsideloneanother,iforexamplein$$...$$orL5ffA͉TU>'ExX'scenteror@displaymathNenvironments,bwhentheglobaldisplayoptionisflushleft@orcentredisplay."l>=@distancer|w;F*orces+arrows(particularlyhorizontalsandrotateddiagonals)@toyhaveatleastthespGeci edlength(default2em),toavoidgettingbirds'@feetinsteadofarrows:A1feۄ|I>FIB!.Sometimesthismakesthearrowover-print@an&ob8jectorappGeardisplaced;inthiscaseyou'llseean\over-full\hbox"@or\increasecellwidth"errormessage."labelstyle=@commandr|InsertsthiscommandineverylabGeltext;t\scriptstyle@isthecommonestchoiceapartfromthedefault."landscape|jIfUlPostScriptisenabled,rotatetheentirediagramby90^@anticloGckwise.OptionsD~whichrefertotheextremerowsandcolumnsare@moGdi edaccordingly*.kUsefulforbigdiagramswithlongob8jectsorlabels."large|USameassize=5em."LaTeXeqno|jUsenL5ffA͉TU>'ExX'srunningequationnumbGeringnforaneqno(q.v.).@Y*oucanputaL5ffA͉TU>'ExX\labelinsidethediagram."lefteqno|ePlace)MtheequationnumbGer,3as)Mgivenbyeqno,3ontheleftofthe@diagram. f94 'PaulT*aylor'sdiagramspackqage5139u "leftflush[=@widthr]|DisplayFhthediagramontheleftofthepageinstead @of]thecentre,;tcf. fleqn.sty.This]isconsideredgoGod]bookdesignby@certainpublishers.oIfwidth+isgiven,"theleftmostverticalofthediagram@is,(ifpGossible,alignedatthatdistancefromtheleftedgeofthepage.@Theideaisthatthearrows,ratherthanthetext,de nethevisual@extentܝofthediagram.Ifthe rstcolumncontainsmultipleverticals,the@leftmostofthemisusedforthealignment."Iftherearenoverticalsin@theleftmostcolumn,thealignmentisontheleftextremityofthetext."leftshortfall=@distancer|The>UgapbGetween>Uthearrow(towhichitis@appliedindividually)andtheob8jectonitsleftisasspGeci ed."lowershortfall=@distancer|;SimilarlybGelow."loose|UTherowsandcolumnsofthediagramhaveatle}'astޝtheheightand@widthddspGeci ed,}~butmaystretch,}~inthesamewayasthoseofanarray@or\table.ͅThisisthedefault,sbGecausetheresultsaremoreoflessright@inEOmostcircumstances,Vbutthiscanresultingapsinthediagram,soit@ish recommendedthattightbGeusedinthe nalversionofadocument,@withappropriatemanualadjustmentofthesizeofthegrid."middle|[?ThediagramiscentredbGothhorizontallyusingbalanceand@verticallyusingvmiddle,q.v."midhshaft@orrmidshaft|;LabGels#arecentredintheshaftsofhorizontal@arrows;$cf. f-alignlabels.ThishappGensautomaticallyforin-textor@one-linediagrams,orifthearrowistoGoshorttoalignthelabels."moreoptions|uAllowDanotherlistofoptionsinsquarebrackets.xThisisintended@forAmacroslike\def\funnydiagram{\diagram[options!,moreoptions]}to@allow\funnydiagram[options!]. "KThearrowcommandsalwaysallowany@numbGeroflistsofoptions."nobalance|jHorizontal>centringofdiagramsisdonewithrespGecttothe@extremities."nohcheck|eDisablescertainerrorchecking."nohug|UDo3notrotatethelabGelsonverticalanddiagonalmaps,-butprint@them*hhorizontally*. (Notyetimplemented|infactthePostScript@optioncurrentlyalways!rotateslabGelsondiagonals,butnotonverticals."nooptions|jUsedhin\diagramstyle,CAthisdisablestheparsingofoptions.@ThisCisforreversecompatibilityinthecasewhereseveraldiagramsbGegin@withasquarebracket."noorigin|eDisablesoriginandbalance."nopixel|`Disable thedpifudgefactor,:q.v.,by settingaveryhighresolution."noPostScript|zDisables]theuseofembGeddedPostScriptandfGthePS]and@noPSAoptions.TheCvqalue,ifany*,isignored,soyouneedonlyeditin/out@thepre xnowhilstleavingthechoiceofPostScripttranslatorintact."noPS|PDisablePostScriptonindividualmapsanddiagrams."notextflow|p?DisablesYtextflow. Thisisneededonsomeindividualdiagrams@(where]theyappGearasnounsinasentence,tforinstance)whenyouuse@\diagramstyle[textflow]."noTPIC|[?DisabletheuseofTPIC\specials.94 '14-v8.rOptions9u "objectstyle=@commandr|;Inserts ASthiscommandineveryob8jecttext; @\scriptstyleisthecommonestchoiceapartfromthedefault."origin|[?Makes thewidth,_heightanddepthofthewholediagramzero,@loGcating;Yitatthebaselineofthebottomrow,/inthecentreofthe@leftmosts column.uThisallowsittobGepositionedbytheuser;usefulfor@mixingdiagramswithL5ffA͉TU>'ExXpicturesandothergraphics."p=@distanceorrpilespacing=@distance|81Set=thedistancebGetween=parallel@verticals<(inthesamecell);ZthatbGetweentherowsofa\pileishalfof@thistoallowthingstobGeputbetweenparallelhorizontals."pixelsize=@distancer|AnticipateXthisoutputresolution;:pixelsize=.02true@mmisthesameasdpi=1270ordpm=50000."portrait|eDisablethelandscapeoption,q.v."PostScript=@authorr|Enable 7btheuseofembGeddedAdobePostScript@\specialccommandsinaformsuppGortedbyauthor'stranslator. EThe@followingarecurrentlyrecognisedbytheauthors'surnames,Ctheirprograms@or'ExX."PS|F?SuppGoseyouwanttousePostScriptforsomebutnotallmapsand@diagrams+pinthe nalversion. ThenusePS+#foreachofthembutput@\diagramstyle{noPostScript}inthepreambleduringdrafting."repositionpullbacks|tThepullbacksymbGolshavea xedpGositionrelative@tothecell,irrespGectivelyofwhereyouputthe\SEpbketc.command."righteqno|jPuttheequationnumbGer(eqno),ifany*,ontheright."rightshortfall=@distancer|;Analogoustoleftshortfall."ruleaxis|eSetzNtheheightoftheshaftofthehorizontalarrows;7unormally@\mathaxis."s=@distancer|qSameassize."scriptlabels|zLabGelsonmapsaresetin\scriptstyle;usefulifthey're@ratherlong."shortfall=@distancer|SetMthegapbGetweenMarrowsandtheob8jectstowhich@theypGoint."silent|[?Suppressfwarningsandsecond-passerrormessages. ,Onlyusethis@option_(ifeitheryoudon'tcarewhatthediagramloGokslikeoryou've@typGesetitandlookedattheresultalready*.rNotrecommended.94 'PaulT*aylor'sdiagramspackqage5159u "size=@distancer|SetbGothheight=distanc}'e[andwidth=distance.NIJMostgeometricalproblemswithdiagramscanbQesolv9edby P9enlargingthecells.rPleasetrythisandthetightoptionbGeforecontactingme."small|USameassize=2em."textflow|eThejtextwhichfollowsthediagramintheTU>'ExXsourceisbrought@uprto lloutthelineprecedingit;Musefultoavoidrthe\suchthatthe@diagramњ|commutes"clichGe. ThishasbGeenusedforseveraldiagramsin@thismanual."thick[=@breadthr]|;As L5ffA͉TU>'ExX\thicklines,hoptionallysettingthewidthof@orthogonalruledlines. $TheL5ffA͉TU>'ExXcommanditselfhasnoe ectwithin@diagrams.rThedefaultrulebreadthistwicethatforthethinoption."thin[=@breadthr]|Thisqissimilarlyanalogousto\thinlines.Thedefaultrule@breadthjisthesameasthatusedbyTU>'ExX,namely\fontdimen8\textfont3."tight|UF*orceallofthecellsinthegridtohaveexactlythesizeyouspGecify.@This+isrecommendedforthe nalversionofadoGcument,5butisnotthe@default\mbGecauseitmaycauseover-printing,}$whichrequirestheintervention@oftheusertocure(bysettingheightandwidth);cf.rloose."top|KAnalogoustobottom,exceptthatthetoprowisused."TPIC|PUseTPIC\specialcommandsinsteadofL5ffA͉TU>'ExXlinesegmentsto@drawdiagonallines. ThesearesuppGortedbyTextures,T*omasRokicki's@dvipsandPaulV*o8jta'sxdvi."uppershortfall=@distancer|;Analogoustoleftshortfall,exceptabGove."vcentre@orrvcenter|{V*ertical pGositioningisthesameaswithTU>'ExX's@\vcenteracommandorL5ffA͉TU>'ExX's[c]option.AJHoweveravmiddleiswhatis@usuallywanted."vmiddle|`IfadiagramwithanoGddnumberofrowsofconstantheight(the@usualcase)isplacedalongsidesomesimpletext,thebaselineofthetext@willdbGealignedwiththatofthemiddlerow,}irrespectivelydoftheheight@and depthofthelabGelsonthetopandbottomrows. =mIngeneral,Mthe@baselineaofthediagramishalf-wayabGetweenthoseofthetopandbGottom@rows."vtriangleheight=@distancer|{Set6height=distanc}'eandthenwidthinsuch@a )waythattheminimal5j*3-gridwillmakeanequilateral4triangle@anda985-gridmakesaregularhexagon(page19)."vtrianglewidth=@distancer|;Setwidth=distanc}'e(andthenheighttomake@these gures."w=@distanceorrwidth=@distance|ѸCSet1gthedistancebGetween1gthecentreofone@columnandthenexttotheright.M94 '16-v8.rOptions9u '9@ wApplicationtoadjunctions:1'Here0isasideapplicationofcommutative0diagramstodisplaying\adjoint 'correspGondences." 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ZThecorner'symbGolCisamacro(\SEpbk)placedinthesamecellastheob8ject,Tbutwhich'printswayoutsideit.pnqɍ'\beginfdiagramg:qǍU&r&I<;&s<"&< \\&\rdTo~{(x,y)}\rdTo(4,2)^x\rdTo(2,4)_yr&I<;&s<"&< \\&r&X\times_Z\SEpbkYI<;&\rTo_ps<"&X< \\&r&_\dTo^qI<;&s<"&\dTo_f< \\&r&,YI<;&\rTo^gs<"&Z< \\'\endfdiagramg*MZvUlt@lv@(x;y[ٲ)lv @lx@lxRltHluHlvHlwHlxHlyHlzHl{ Hl| Hl}Hl~Hxl~jLr@A 5A&iAy@ A2ڟ At A)9A_3fmAB<١AܟFLAӌvO A&)vyӌzO UʋkX¸Z ՟ [\B  ;YM1fe@ Z8p*Ϭ-㜵X<hYfq*X?9G*+fep1fe(ˍ8g*Ϭ-*Zlf*X"?9M*-*feO͍6F*or[reversecompatibility*, 2somecommandswith\compass"namesare'provided,Msuchqas\NW,\WNW,\NNNW,togetherwithdottedandothervqariants.'Howevertheauxiliary leextra-diagonals.texisobsoleteandmustnotb}'e'use}'d.6Themainuseofdiagonalsotherthanthesimplestonesisforillustrating'orderqstructures(\Hassediagrams"). pPosetsaretrivialcategories, butin'practicethey'rethemostdicultdiagramstodraw!@\begin{diagram}[height=2.5em,width=1.5em,abut]@o&&&&o&&&&o&&&&o\\@&\luLine\luLine(10,2)&&\ruLine&&\relax\luLine(6,2)&&@\ruLine(6,2)&&\luLine&&\ruLine\ruLine(10,2)&\\@&&o&&&&&&&&o&@\end{diagram}'Jff J= "5-:8LܻThe.koptiondiagonalbasecÎhangesthis;Eitisintendedforemulationofotherdiagrams pacÎkages.94 '18 10.rDiagonallines9u 'oo o4 oltT9@T T؟TTlpTfdlrTPȟP8Plt*PʟGP cPlvP̟rP"PlxVPΟP$ :Plz *PПGP&cPl|Plv9Bڟ lrfd̟ bw b"bA͟blxb#bΟ byb $bʟ"u" "A˟"lv"ї!"̟"w "" "l|T9HT TT TlxTfdl~9J"& +0lzfdltʟc Glv *̟ :"lxVΟ$rlzПc&Gl|*ҟ8#(,l~To o,Ǎ6The}obsoletecommand\DiagonalMapisalsostillsuppGorted,1butshouldbe 'replacedSbythenewcommands.lIfyouhaveusedthiswithfontsotherthan'line10ɲ(forexampleMichaelSpivqak'sfontslams1tolams5),a{pleasecontact'metoassistwithconversion. ܪ6Bydefault,{thediagonalarrowsaresetonthe rstpassusingL5ffA͉TU>'ExXline'segments.a_Bycontrast,thehorizontalsandverticalsuseruledlineswherever'pGossibleandareadjustedonasecondpassofthediagram(withinthe'\enddiagramcommand)tomeettheirendpGoints.6Whenthediagonalsaresetduringthe rstpass,ereverythinghappGens'withinthecellwheretheyaredeclared,wherethealgorithmhasnoaccessto'informationl1abGoutthesize(andexactwhereabouts)oftheob8jectsatwhichthe'arrows]aremeanttopGoint.5Thereforethelengthsofthediagonalsarechosen'somewhatarbitrarily*,thoughtheabutoptionwillmakethemtoucha\cdot.6T*o:getbGetterresultswehavetousemoresophisticatedmethoGds,Iandgo'outsidewhatisstrictly\standard"TU>'ExX."'11H&AlternativeTL32EdXnologyō'InordertogetthebGestresults,gdyoushouldbeawareofthewayinwhich'diagonalsareconstructed.[TU>'ExXisultimatelyonlycapableofpGositioning'charactersfromvqariousfontsanddrawingblackrectangleswithhorizontaland'vertical4sides:=thereisnoprimitivefordiagonallinesorforrotation. "This'meansthatwehaveOT6@eithertojuxtapGosecharacterswithalimitedchoiceofangles, asL5ffA͉TU>'ExX's@\lineFand\vectorcommandsdo,withtheresultthat(bGecauseofpixel@roundingerrors)theymaynotlineupcorrectly*,U6@ortousetheloGopholeintheDVIelanguageprovidedbythe\special@commandftotellsomepGost-processingfsoftware(suchasdvips)todothe@workforus,therebysurrenderingpGortability*,6@orhotoover-printhoalargenumbGerhoofdots,makingtheDVIhU leverylarge@since.atleast12bytesareneededforeachdot. ThisiswhatPП'IDCfTU>'ExX@doGes.OT'InYthispackqageoptionsareprovidedtoemploythe rsttwoofthesemethoGds.6@Byydefault,͗charactersfromL5ffA͉TU>'ExX's^9 line10fontareused. Arrowheads @fromB thisfontareusedirrespGectivelyofthosechosenbytheheadsoption.@Thenearest^10 ;uavqailableL5ffA͉TU>'ExXslopGeischosen,withthee ectthatarrows@may8sometimesfallshortorovershoGot:heightandwidthshouldbeset@withthisinmind.'ff J= "5-:9LܻTheݎcow-:10LܻTheapproÎximationusescontinuedfractions,|aliasM_Euclid'salgorithm,withacorrection.Thereisalsoco'ExX's`+\specialcommand.'ExXPostScripttranslator@bGecausethissortsthecharactersoneachpagebyfontandcannotdeal@with\specialcommands.6@IfN$theTPIC^12 optionisenabled,aasimplersetof\specialcommandsis@usedOtodrawdiagonallines,butrotationisnotavqailable. dvipsand@xdviunderstandthese,fbutotherpGost-processorsmaygiveerrormessages@andprintnothing.'NoadditionalmacropackqagesorspGecialknowledgeareneededtomakeuseof'thesemethoGds:rallofthenecessarycodeisbuiltintothediagramspackqage.6F*utureversionsmayexploittheseextensionstoprovideotherfeatures,Ȝsuch'asAcurvedlines;boindeedthisislikelytobGethemaindirectionoffurther'development.rOthermethoGdsmayalsobeprovided.rHowever6@ifR"noextensionoptionisselectedastandardDVIR le(usingtheline10@font)9willbGealwaysgenerated,givinganapproximationtothefeatures@requested.6@developmentofnewfeatureswillusePostScriptmoGde,zwithoutwhicha@\secondclassservice"willbGeprovided.#č'12H&Regular gures'The$di erencebGetween$theseoptionscanbeseenbytryingtodrawaregular'hexagon.pnqɍ'\beginfdiagramg[vtrianglewidth=1em,abut,tight]:qǍo|&&{}&<&\rLinew-:12LܻTPIC8isagraphicsprepro'ExXmethoGddoesnot|thoughitissurprisinglyaccurate!2{r[ލq1fe(Ruzu_c@dP-fd-TudTcF_TPTfdu-TzudTc@_TPTfdu_cFdP-fdq1fe([ލ}1fe(R u_cLdP-fd.-T3udT8cR_T=PTfdL5ffA͉TU>'ExX -TudTcL_TPTfd.3u_8cRd=P-fd}1fe(4鍑'Thevtrianglewidthoptionanditsvqariantssetthewidthandheightof'thefcells,butthearrayretainsthesymmetryofarectangle.uT*esselationsof'trianglesandhexagonsmaybGedrawninthisway*.6Drawing(awlxR슟@l@쌟@l @쎟@ڡڵb쎟R,xAy1fedߖdf*Ϭ-h^0ŹcLΉ"LfefBBQC0⍍K"g[ٟ^05V?VBp^fe`A1fewP1fe0.kP^0*Ϭ->0DG0p9?Źdp4H\feYlt@`u@hlv@ow @wlx@gcwlxR슟@l@쌟@l @쎟@ٸd쎟RC{^gL*?E*i*feޟ1fedu k*Ϭ-8Dh+*$?O*i*fedq΍'WithoutthePostScriptoptionthediagonalarrowswillnotbGesetcorrectly*.#č'13H&EmulationofTL32EdXercise18.46'ThisandthenextsectiondescribGehowtoconvertdiagramsinyourexisting 'doGcumentswhichweredrawnusingothercommutativediagrammacropackqages'tousethisone,<whereverpGossiblechangingonlythepreambleormacro le'andnotthetextitself.6ExerciseX18.46ofTheTKß'E-Xb}'ook l3providedXsomeideasforcommutative'diagrams,$fromwhichthepresentpackqagewasoriginallydevelopGed. The'followinginstructionsarebasedonwhatisgiventhere, butifyouhaveadded'otherarrowmacrosyouwillhavetoworkouthowtore-de nethemby'followingtheexamplesgiven.6IfyouhaveusedtheTKß'E-Xb}'ook macrosastheystand|with\matrix'enclosing1thediagram|youwill rsthavetodistinguishbGetweenthediagrams'andarealmatrices.(IfyougivearraysofnumbGerstothecommutativediagrams'packqage,’they6willcomeoutratherwidelyspaced.)ThisproGcessisunnecessary'ifOyoualreadyhaveamacrofordiagramsratherthanmatrices:justre-de ne'thatinasimilarway*.6In@_yourmacro leyouprobablyhavesomethinglikewhatfollowsthe'\iffalsencommandbGelow,butwithadditionalmacroswritteninasimilarway'(thisisthereasonforsettingitalloutingorydetailinthemanualrather'thanprovidinganextrainput leasinthecaseofsomeofthe\packqages"'describGed|inthenextsection).IfyoumakethefollowingadditionsyouwillbGe'able"toreverttotheoriginal(intheunlikelyeventthatyouarenotsatis ed)'bychangingthisto\iftrue.6Thisisnotparticularlydicult: D when,asaTU>'ExXnovice,Iwrotemy'Ph.D.lthesisinAugust1986,҇IhadsixtydiagramsdrawnusingtheTKß'E-Xb}'ook'macros,gtogetherAdwithseveralothersthathadtobGedrawnbyhand.Recovering'thatAfromanarchiveAtapGe,gIrecentlyfoundthatallbut veofthesixtycould'bGeAconvertedwithoutanychangeatalltothetext,jwhilstthehand-drawn'onescannowbGedrawnwiththeup-to-datepackqage.@CߤN cmtt9C\iffalse J%yourmacros,copiedoradaptedfromTeXercise18.46(page325)J\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}}^94 '22 13.rEmulationofTU>'ExXercise18.469u FC\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} J%J\let\cdmatrix\matrix%reverttousing\matrixfordiagrams@\elseJ%Replacementfortheaboveusing...J\inputdiagrams%PaulTaylor'sdiagrams.@%J\def\mapright#1{\global\matrixwascdtrue\rTo^{#1}\relax}J\def\mapleft#1{\global\matrixwascdtrue\lTo^{#1}\relax}J\def\mapup#1{\global\matrixwascdtrue\uTo>{#1}\relax}J\def\mapdown#1{\global\matrixwascdtrue\dTo>{#1}\relax}@%J%Maybeyouhavesomevariantslikethis:J\def\mapupbefore#1{\global\matrixwascdtrue\uTo<{#1}\relax}@%J%Thefollowingexamplesmayalsobeuseful:J\let\into\rIntoJ\def\horizadjoint#1#2{\pile{\lTo^{#1}\\\bot\\\rTo_{#2}}}J\def\vertadjoint#1#2{\dTo<{#1}\dashv\uTo>{#2}}@%J%Hereisareplacementfor\matrixwhichgivesacommutativeJ%diagram,includingsomeextramacrosforuseinsidethem.J\def\cdmatrix{\bgroupT\edef\matrixlineno{\the\inputlineno}\global\matrixwascdfalseT%S@\let\matrix\pile`%inner\matrixisprobablyparallelhorizontalsT%T%re-definethediagonalarrowsOm\let\nwarrow\luTo\let\nearrow\ruTo\let\swarrow\ldTo\let\searrow\rdToT%T\diagram[]%beginthediagram(withoutoptions)SN\getthematrix%ɨdeletethislineifyouuseamstexratherthanplainT}J\def\getthematrix#1{#1\endcdmatrix}J\newif\ifmatrixwascdJ\def\endcdmatrix{\enddiagram\egroup}@%J%ThefollowingtellsyouwhetheritwasactuallyadiagramorJ%amatrix.Removethiswhenyou'vechangedthose\matrixJ%commandsinyourtextwhichshouldbediagramsto\cdmatrix.J\def\endcdmatrix{\enddiagram\egroupT\expandafter\message{^^JThe\string\matrix\spaceatlines^\matrixlineno--\the\inputlineno\spacewasreallya^\ifmatrixwascddiagram\elsematrix\fi.^^J}}J\let\matrix\cdmatrix\let\endmatrix\endcdmatrix@%@\fi6ThecoGdeasshownaboveassumesyouhaveTU>'ExXversion3(toprovide '\inputlineno);mit4printsa\messagetotellyouwhethereach\matrixhas'usedanyarrowcommands.Withacleverchoiceofthismessageyoucaneven'getyoureditortomakethenecessarychangestothetextforyouinbatch'moGde!rDeletetheextracodeafteryouhavedonethis.6Beware5thatAU>'M S-TU>'ExXuses\matrix#1\endmatrixinsteadof\matrix#1.'In:thiscase, delete\getthematrixasindicated. Then(whereappropriate)'change\endmatrixaswellas\matrix. Thesameappliesifyouhaveused'thearrayenviromentinL5ffA͉TU>'ExX.oB94 'PaulT*aylor'sdiagramspackqage5239u '14H&Emulationofothermacropackages'ThereareseveralotherTU>'ExXmacropackqagesincirculationfordrawing 'commutativediagrams,ofvqaryingdegreesofsophistication.qThissection'describGesXKhowtoadaptthepreambleofanexistingdoGcumentwhichwas'written^gtousesuchmacrossothatitprintsdiagramslikethoseinthis'manual%|instead.WhenyoupublishdoGcumentspreparedinthisfashion,.please'remembGertoacknowledgetheauthorsofbGothpackqages, makingitclearwhich'youۚusedtotypGetheoriginalsourceandwhichproGducedthe nishedproduct.'American$MathematicalSoQciet9y(AU>'M S-TU>'ExX,PlandAU>'MS-L5ffA͉TU>'ExX'samscd.sty):'seeMichaelSpivqakbGelow.'Mic9haeldBarr'scatmacmacroswerebasedonaprincipleofoverlapping'squares. Whilst2thisispGerhapscloserconceptuallytothecategoricalideas'whicharebGeingexpressed,itisnotpossibletoemulatethelanguageusingthe'matrixsyntax.rThesimpleshapGemacroscan,however,bGereplacedby@\def\square[#1`#2`#3`#4;#5`#6`#7`#8]{%O\diagram[]^{#1}&\rTo^{#5}&{#2}\\^\dTo<{#6}&&\dTo>{#7}\\^{#3}&\rTo^{#8}&{#4}O\enddiagram}'and#similarly\atriangle,a,\btriangle,\dtriangle,\ptriangle,'\qtriangle, \Atriangle,\Ctriangle,\Driangle,\Vriangle,\Arianglepair,'\Vrianglepairand\recurse,whichareeasyexercises.'KarlBerry'seplain:rseeStevenSmithbGelow.'F rancisHcBorceux'sdiagrampackqageusesL5ffA͉TU>'ExX'spictureenvironmentinstead 'of-TU>'ExXarrays,WbutsinceitisdesignedinanarrayfashionitcanbGeinterpreted.'Insteadofthat le,use@\inputdiagrams@\inputBorceux-to-Taylor'ThisPisavqailablefromthesameplaceas(my)diagrams.texitself.Currently'the=curved, freeandmultiplearrowsarenotimplemented, andallsize'parametersareignored.'EitanGurari'sdratex:rIhaven'tloGokedintothispackqageyet.'DonaldKn9uth'sExercise18.46:rseesection13ofthismanual.'F rankMittelbac9h'sAU>'M S-L5ffA͉TU>'ExXamscd.sty:rseeMichaelSpivqakbGelow.'JohnReynolds:rIhaven'tloGokedintothispackqageyet.'Kris1Rose'sXY-pic&usesadi erentconventionforwheretodeclarearrows, 'namelyJinthecellswiththeirsources.ThisconventionJwillbGesupportedinthe'next#release.ThesyntaxofRose'sarrowcommandsisalsomorecomplicated.'IdonotintendtosuppGorthisfontsorturningcommands.{Z94 '24'14.rEmulationofothermacropackqages9u 'RainerSc9h@opf'sAU>'M S-L5ffA͉TU>'ExXamscd.sty:rseeMichaelSpivqakbGelow.j'Stev9enSmith:rIhaven'tloGokedintothispackqageyet.'Mic9haelSpiv\rak'sL1AlXqM .S-TU>'ExX:similarcommentsapplyhereastoKrisRose's 'packqage.'Mic9haelySpiv\rak'sAU>'M S-TU>'ExX(asdescribGedinTheJoyofTKß'E-X)hadsomevery'primitivecommutativediagrams,enclosedin\CD...\endCD. 6Y*ouOWcanobtainadramaticimprovementOWinthesediagramswithoutchanging'theXDtextofyourdoGcumentbyusingmypackqagewiththeamstexoption.Given'a(complete,correct)AU>'M S-TU>'ExXdoGcument,changetheheadertoreadjh\inputamstexh\inputdiagramsh\diagramstyle[amstex] 'If1fAU>'M S-TU>'ExXorAU>'MS-L5ffA͉TU>'ExX(amscd.sty)hadbGeenloaded,}thiswillreplace'thede nitions.6Y*oucanalsoincludeAU>'M S-TU>'ExXdiagramsextractedfromoldpapGersinnew'plainmETU>'ExXorL5ffA͉TU>'EXdoGcuments.dAInthiscase,donotloadAU>'M S-TU>'EX(unless'youwantitforsomeotherreason)butinsteadusejh\inputdiagramsh\def\CD{\diagram[amstex]} 'toBcon nethechangestothemeaningofthe@charactertothediagrams'themselves.rThein-linehorizontalarrowswillthennotbGeinterpreted.j'Timoth9yv\ranZandt'spstricks:rIhaven'tloGokedintothispackqageyet."+%'15H&Ffrequentlyaskedquestionsk0'W ouldn'titbQebettertodra9witwithamouse?rNo.'Ho9wdoIgetit?rSeesection18.'Isitcompatiblewith...?'ExX,whilstgivingassistanceto'colleagueswhousedL5ffA͉TU>'ExX. InDecembGer1991thelocalTU>'ExXsystemwas'convertedztouseF*rankMittelbachandRainerSchopf'sfontselection(NFSS1)'forL5ffA͉TU>'ExX.rNowweuseL5ffA͉TU>'ExX2"*S,whichincludesNFSS2. 6Consequently/thereisagreatdealofcollectiveexpGerienceinusingthe'diagrams packqageinalloftheseenvironments. CkIdonothaveexpGerienceof'AU>'M S-TU>'ExX,mAU>'MS-L5ffA͉TU>'ExX,meplain}HorcommercialTU>'EXpackqages,mbutdonotknow'ofBanyreasonwhyitshouldnotworkwiththem:\pleasetellmeifyou nd'anydiculties.6Itis,3regrettably*,notuncommonforpublisherstocopypartsofL5ffA͉TU>'ExXand'article.sty2intotheirownjournalandconferencestyles,?andnotkeepthese'upZYtodatewithrespGecttobug- xesandotherchanges. +}ManyZYofthebug'repGorts~whichIreceiveinconnectionwiththediagramspackqageareinfact'traceableatoerrorsofthiskindinpublishers'styles.uMypGolicyisthatIwillno'longer xbugsinsoftwarefromcommercialorganisationsforfree,butsuggest'someconsultancyarrangement.%BIf,however,itispGossibletomakethepackqage'morerobustandcircumventsuchbugsinothersoftwarethenIwilldothis.94 'PaulT*aylor'sdiagramspackqage5259u 6RichardKStallman'stexinfoisdesignedfordoGcumentingothersoftware,^in 'whichNTU>'ExX'sspGecialcharactersoftenhaveimpGortantmeanings.F*orthisreason'many*.ofthe\catcodeshavebGeenchanged,4andinparticular@isusedwhere'TU>'ExXE+uses\.TheusualTU>'EXmeaningsarerestoredwithin@tex..\Etex.Itis'pGossibletoloadthispackqagewithoutthis,butyoumustdoF@@catcode`@\=0\catcode`\%=14\inputdiagrams\catcode`\%=12\catcode`\\=13'Y*ouomustalsodo@catcode`@&=4bGeforeusing@diagram.l=Braces{}staythe 'same. A6UnfortunatelythepackqagedoGesnotseemtoworkwithT*eX-XeT,the'extension6toTU>'ExXfortypGesettingArabicandHebrew./Ibelievethisproblemis'insupGerable.F'DoQes9itusean9yspecialfon9ts?SNo.One9ofthedesigncriteriaofthepackqage'isjthatallofthecompGonentscomefromthestandardComputerModernfonts'thatHcomewithTU>'ExX,exceptthatthediagonalarrowheadcharacterscomefrom'L5ffA͉TU>'ExX'sline10font.rAsoneusersaid,F@\IVagreewithyouinbGeingagainsttheuseofadditionalfonts.$It@takesHsometimeandexpGeriencetoport,msay*,L1AlXqM .S-TU>'ExXHfontsto[my@`pGersonal' computer].IwouldmuchpreferembGeddedPostScript@commands.CustomvDVIvdriverssuchasxdvi,wontheotherhand,@arenotwidelyavqailable,atleastnotfor[mycomputer]."'ThereareoptionalwarrowheadsfromtheAMSBsymbGolsfonts,Fwhichwere'usedNwhenthiscopyofthemanualwasprinted.[TheymaybGeobtainedby'anonymousFTPfrome-math.ams.com. A6It}issometimesclaimedthatitisadvqantageoustohavespGeciallydesigned'fonts5inordertoensurethatthecompGonentsmatchupcorrectly*..Howeverthe'reason whytheyfrequentlydonotispixel-rounding,;evenwhentheDVI-driver'do}'esGthiscorrectlyaccordingtotherulesspeci edbyDonaldKnuth.^14ŲThedpi'optionhasbGeenincludedtocorrectforthis.6Ifyouhaveotherarrowheadfontsavqailable,suchasthoseprovidedbyKris'RoseforXY-pic%{andMichaelSpivqakforL1q% cmsy6AqM IS-TU>'ExX,youmayusethemifyou'writeyourown\newarrowheadcommands.F'Corruptionb9yelectronicmail.pڲIfyoucannotuseFTPƚandtherouteto'youubyelectronicmailpassesthroughnon-ASCIGI5machines(suchasthrough'BITNET),%the lemaysu ercorruption.(ThisdoGesn'tmatterverymuchwith'themanual,butthemacropackqageitselfhasalistofcharactersatthetop,'and$kuseofmostofthosewhichexpGeriencehasshowntobGevulnerablehas'bGeencon nedtothe rstsection,Dwhereverpossible. ySearchfortheword'ASCIGIifindoubt.'T<8E3sXscapacit9yexceeded.uTU>'ExXwasdesignedintheearly1980s,whenRAM'wasmeasuredinkilobytes,!anddoGesnothavedynamicmemoryalloGcation.'Althoughtex.websaysloudly\Don'tT*ouch!",thecompile-timeparameters'listed|onpage300ofTheTKß'E-Xb}'ookare'ExXincreasesthemsubstantially|inaccordance'ff J=w-:14LܻIfiyÎoudon'tbintheoutputofgftype-icmmi10.300gf.ThecÎharacterincmsyisonepixellowerthanthis.ݠ94 '26p15.rF*requentlyaskedquestions9u 'withtheRAMavqailablein1990shardwareandthatneededby1990ssoftware. 'TherenisnothingUnix-spGeci caboutthesechanges:theycanjustaswellbGe'madeinanyothercompilationenvironment. KCurrentlyitseemsOzTU>'ExXhas'notmadethechanges|pleasecomplainabGoutthistoitsauthor,bAndrew'T*revorrow,notme.6TheYincreaseinsizeoverYversion3islargelyduetothediagnostics,?0i.e.'helpingyoutousethepackqage!6Ifyouget\noroGomforanother\dimen"itprobablymeansyou'reusing'PП'IDCfTU>'ExX,whichuses110outoftheavqailable256;Iuse20.6TheproGcessofstretchingthearrowsinacompleteddiagramisquiteslow,'butiYatleastinthispackqageTU>'ExXspGendsthetimedoingsomethingusefulrather'thanparsing(L5ffA͉TU>'ExXpicture)\putcommandsinsidemacros.Theemulation'ofF*rancisBorceux'smacros,forinstance,is50%fasterthantheoriginal.hKris'Rose'sXY-pic$wtakesabGoutsixtimesaslongtodrawthesamediagram.'DefaultTarro9wheads. MoEvenwhenyoude neyourownarrowswithspGecial'arrowheads,youestillgetgetL5ffA͉TU>'ExXheadsondiagonals.#ThisisbGecauseL5ffATU>'ExX'headsareavqailableinavarietyofdirections,butyourspGecialonesareonly'de nedvEforup,down,leftvEandright.@T*ogetspGecialheadsondiagonals,itis'necessarytorotatethem,andtodothatyouneedthePostScriptoption.'Displa9y#options.ܲIf,,aswasadvisedinearlierversionsofthismanual,,you'enclose23thediagramin$$...$$ora,thenewoptionsinsection8for'pGositioningitcannotwork. /Exceptinthosecaseswhereyouwanttwoor'moreldiagramssidebyside,νorasmalldiagramin-lineinthetext,itisbGetter'toEremovesuchenclosings;kthenyoucanexpGerimentwiththeoptionsusing'\diagramstyle.L5ffA͉TU>'ExX'scenterenvironmentis,forthispurpGose,not{adisplay'butҳaparagraphoftext,`withsomestrangeresultsifyouusethetextflow'option;inYthiscaseyoushouldusetheinlineoptionontheindividual'diagramstomakethemappGearsidebyside.'Large0gapsbQet9ween0diagonalsandtheirendpoin9ts.ThishappGensif'you!trytousethedefaultLaTeX!linesegmentstodrawverysteeporvery'shallowdiagonals.rT*ryusingthePostScriptoption.6Bydefault, the(rowsand)columnsofthematrixformingthediagram'canؤstretchtoaccommoGdatelongpiecesoftextasob8jectsandarrowlabGels.'Thisisappropriateforrectangulardiagramsandrequiresnouserintervention.'If,however,youhavediagonals,thisstretchingcausesthemtofallshortofthe'ob8jectstowhichtheyaremeanttopGoint,becausecurrentlytheyare(unlike'thehorizontalsandverticals)rigid.x75KA>1{T{Tqq{Tq{Tq{T{Tq{Tq{T{T˖1fe/@<1A;0A J>*Ϭ-`A8N@@ʖA@B@ٖC @D@TfDRABah*?+pp`F.Dp`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F.p`F./ʍ'Anexampleprovidedbyauserisshown.rThesolutionistouseoptionslike@\begin{diagram}[tight,width=4em,height=3em]'SeeIHsection8fordetails.JSometimesthereisn'troGomonthepagetoincrease 'thewidthsuciently*.Tryusingthescriptlabelsoption,eor,failingthat,put'thewholediagraminthescopGeofaL5ffA͉TU>'ExX\smalloreven\tinydeclaration.[94 'PaulT*aylor'sdiagramspackqage5279u 6Thisureal-lifeexampleillustratesanothercommonproblem:thesymbGols< 'and >arer}'elationshnotbracketsi |andit'snotunusualtoseeline-breaks'thePlwrongsideofthem,eveninpublished,suppGosedlyproof-read,books.'Ifsyoudon'toftenusethestrictarithmeticalrelationsand nditabGoreto'typGe\langle..\rangle,putthefollowinginyourmacro le:@\mathcode`\<="4268%<=\langle@\mathcode`\>="5269%>=\rangle@\mathchardef\gt="313E %arithmetic@\mathchardef\lt="313C %strictorder'AsOanotherpieceofgeneraladvice,#manypGeopleuse\mboxwhenitis'completelyuunnecessary*.kAmongstotherthings,!itinhibitsthereductionofthe'contents}whenusedasasub-orsupGerscript.InL5ffA͉TU>'ExX2"*C',tryusingthe\text'and\ensuremathcommands.'\Badly\dra9wndiagonals"errormessage.Thiswarningisgivenif(a)you'use$diagonalswhicharesetonthe rstpass(i.e.L5ffA͉TU>'ExX,TPIC#orfixed)and'(b)Fsomeofthecolumnsaresigni cantlywiderthanwasspGeci edbythewidth'orsizeoption.ItindicatesthattheproblemabGove,[withbiggapsattheends'ofdiagonals,mayhaveoGccurred.Checkthis,andifnecessarysetthetight'optionDandspGecifythenecessary(increased)widthyourself.vAlternatively*,Uuse'the6PostScriptoptiontogetthediagonalssetonthesecondpass;zthenthey'willmeettheirendpGoints.'LabQelsonarro9wsover-printobjectsorarrowheadsbQecomedetached.'Y*ou're٦tryingtosqueezetoGomuch٦intothecolumn: Eincreasewidth(asittells'yout`todo).yIftheob8jectatoneendismuchlongerthanthatattheother,'themidshaftoptionmaybGeappropriate.'Mixedormissingarro9wheads.Theideaofprovidingtheheadsoptionis'thatyoushouldspGecifyatthebeginningofyourdocumentwhichstyleyou'want. SThedefaultisheads=LaTeX!sincethisprovidesconsistencybGetween'orthogonalanddiagonalarrowheadswhenrotationisnotavqailable.'Unde nedsym9bQols. ˑT*ousethecurlyandblackarrowheadsyouneed'theeAMSsymbGolsfonts;Ęifoneofthese(\curlyvee,N~\blacktriangleleft)'iskunde ned,ƑhitreturnseveraltimestocompletetherunofTU>'ExXonyour'doGcument,|then[gobackandinsert\usepackage{amssymb}orchangetheheads'declaration.6If{you ndasymbGolwithameaninglessnamelike\CD@gF8or\cD@hAis'unde ned,it_qmeansthatyouhaveusedinternalmacrosfromapreviousversion'ofthediagramspackqage. 2Pleaseremovethem: thenamesaredelibGerately'meaninglesstopGersuadeyounottousethem.'Cany9ouputdiagramswithindiagrams? ϝY*es,Rbutit'snotoftenthat'such_thingsareneeded.SRemembGerthat\pileisusedforparallelhorizontals.'Sometimesyoumaywantanarrayofdiagrams.6DuringV[developmentIfoundproblemswhenthe rstcelloftheinner'diagrambwasempty*,andversion3.22resultedinTU>'ExX'selusive\interwoven'alignmentd'ExX2"*S:@\usepackage[flushleft=\mainindent]{diagrams}'where F\mainindentisa\dimen;inthiscase\the\mainindentwillwork, 'bGecause\theexpandsthe\dimentoitsvqalue(TKß'E-Xb}'ook,ZEpage214). [}V*alues'for(optionstoindividualdiagramsandarrows,rorparsedby\diagramstyle'insteadof\usepackage,maysafelycontainmacroswhereappropriate.'Horizon9talbarrowsoverprintobjects. -Y*oumustnotuse\hfill,z\hss,'\hspace,\hidewidthorothersimilarcommandstotrytoalterthee ective'sizeoftheob8ject.'What$ifitstilldoQesn'tw9ork?)Ifyouhaveaproblemwhichisnotanswered'bythismanual,pleasecompileashort lecontainingyourproblemdiagram'and~banymacros(suchas\Asslinsection3)itcontains. Runitthrough'texrorlatextocheckrthatnode nitionsaremissing,andincludeanoteof'the"dateandversionnumbGerwhichyouareusing.eThensenditbyele}'ctronic'mail:rpleasedonotusethetelephoneorpGostalmail.6I\amkeentoknowabGoutanyadverseinteractionswithothersoftware,'anything~whichisnotwellexplainedinthismanual,oranycasesofmis-typing'in:which\scrollmodedoGesnotgettotheendofthedocumentbecauseofa'diagramserror.#č'16H&Conditionsofuse'Y*ou.mayfreelycopyandpassonthispackqageandincludeitincollectionsof'freesoftware,butmaynotalteritorchargeafeeforit.6Please{ ensurethatyouareregisteredwithmeasauser,sothatyoucan'bGe)informedofnewversions.Any)electronicmailmessagecontainingthewords'\commutative" or\diagram"automaticallyregistersyou,asdoGesquotingyour'electronicmailaddresswhenfetchingitbyFTPfromftp.dcs.qmw.ac.uk.6IfyouconsiderthispackqagegoGodenoughtouse,[thenitisgoGodenoughto'acknowledge.After.wall,:itisacademicprotoGcoltocreditpriororsimultaneous'discoveryoftechniquesrelatedtoyourown,vevenifyouwereunawareof'themordidnotrelyonthemwhenyoumadeyourowndiscovery*. Such'acknowledgement=isaconditionofuseofthispackqage.,)However=thiscondition'isZwaivedifuseamountstonomorethan vediagrams,{eachofwhichiseither'assquareoratriangle.w Thisacknowledgementsmust,jofcourse,bGeremovedif'thedoGcumentisre-typGesetbymethoGdswhichdonotusethispackqage.6No$~pGermissionwasinthepastgiventousethispackqageforcommercial'purpGoses.FThis{includesadocumentwhosecopyrightissededbytheauthor(for'vqaluableconsiderationornot)toanotherpGersonorbodywhichsubsequently'intendstocollectroyaltiesforitsreproGduction.$Thisappliestocertain'journals-,andconferenceproGceedings.Permissionisnowgrantedforitsusefor'thevproGductionofacademicresearchandtextbooks,Ԋjournalsandconference'proGceedings,sub8jecttotheconditionsthat6@acknowledgementbGegivenasabove,6@anup-to-dateversionofthepackqagebGeusedforthe nalproduction,-94 'PaulT*aylor'sdiagramspackqage5299u 6@andonecopyofthebGookbesenttomeonpublicationinlieuofroyalty*, @attheabGoveaddress.'Use7Rbycommercialorganisationsisconsidered(forthispurpGose)tobeacademic'ifetheresultsareintendedforpublicationinanacademicforum,concernpure'researchanddonotrelatetoanyparticularcommercialproGduct.6ThesoftwaremaynotbGeusedforanymilitarypurpGoseunderany'circumstances.6Nouwarrantyisgivenwiththissoftware. }!Itissupplied\asis",and'neitherHthesourcenorthismanualnoranythingelseshallbGetakenasa'representation\thatitwillpGerformanyparticularfunction,sissuitableforany'particularZpurpGoseorisofmerchantableZquality.Inexecutingthesoftware,{the'userگimplicitlyacceptstheabGoveگconditionsandindemni estheauthor,&Queen'Mary1andW*est eldCollegeandanypGersonthroughwhomthesoftwarewas'obtained,02against&liabilityfordirectorconsequentialdamagesarisingfromthe'useofthissoftware.6Wheneveryouusecomputersyoumustekeepatleasttwoback-upcopies'of/allofyour les,{withoneofthemwellawayfromthemachine,{incase'ofѐ reorma8jorfailure.!BeforesendinganydoGcumentsforpublicationorto'anPXexpGendiveprintermakeathoroughvisualcheckusingapreviewersuchas'xdviandalowresolutionprinter.#č'17H&Reversecompatibility'Compatibility,withpastandfutureversions(thenumbGersofsomeofwhich'havebGeenassignedp}'ostfacto)isasfollows. KNotethedi erencebetween'\should"and\must".'V ersion0:`N0Seesection13.'V ersion1:`N0Horizontalarrowsmadetostretchtoedgeofcell;1987{9.LV\rTo{f}{g}1works,}ebut\rTofgdoGesn't:~itmustbGechangedto V\rTo^f_g.LVthe1D\mkern-20mu\rTo{f}{g}\mkern-20muidiomformanualstretchingVofhorizontalarrowstomeetob8jectsmustbGeremoved.LV\VerticalMapHeight;and\VerticalMapDepthcommandsareobsoleteVandareignored:rtheyshouldbGeremoved.LVThree-/orfour-argumentusesof\HorizontalMap,\VerticalMapandV\DiagonalMapjmustbGechangedto ve-argumentusesor, Epreferably*,Vto\newarrow.LVNested[V\commdiagcommandsforparallelmapsmustbGechangedtoV\pile.LVParallel%mapsconstructedbyputtingthemintherowsorcolumnsVbGeforeandaftermustbemovedtothecorrectcelland(inthecaseVofhorizontals)putina\pile.'V ersion2:`N0Horizontaljarrowsmadetostretchtomeetob8jectsand\supGerscript"@labGelsW~introduced;areusedforleftandright,wrespGectively*.;Unquali ed_labelsVare`pGositionedastheyoriginallywere:̺the 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LVCurvedarrows(PostScriptonly).YLVSidewaysmovementofarrows.LVAnMalternativewayofspGecifyingthepositionsofarrows,l`namely Vatthetailwithsignedrelativeco-ordinatesforthehead,6toallowVemulationofKrisRose'sXY-pic$wandMichaelSpivqak'sL1AlXqM .S-TU>'ExX.YLVShiftingob8jects,byspGecifyingtheleftorrightwidth.LVOptiontorotatelabGelsondiagonalsorleavethemhorizontal.LVApplicationtoelectroniccircuitdiagrams. 6SuggestionsYforimprovementsYandfurtherapplications(e.g. J|proGofnets, 'Petrinetsandcircuitdiagrams)arewelcome."w'18H&Availability'Thevdiagrampackqageiscurrentlyavqailableasasingle112kb^15 printableTU>'ExX'source+ lewhichiscompatiblewithbGothL5ffA͉TU>'ExXandplainTU>'EX.LItloadsthe'L5ffA͉TU>'ExXMline10fontas\tenln(itsL5ffATU>'ExXname)andalsode nessome(exotic)'arrowcommandsintermstheAMSsymbGols,butifthesearetobeusedthen'thepackqageamssymbmustbGeloadedseparately*. ؍6It~iseasiesttogetthispackqagebyanonymousFTP.Thisstandsfor\ le'transfer protoGcol";itisaway inwhichyoucanlogoninteractivelytomy'computerandfetch(someof)my les. 3Averylargeamountofsoftwareis'nowހfreelyavqailablebythismethoGd, soitiswellworthputtingpressureon'yoursystemadministratortogetyourmachinesconnected.6TheauthoritativeversionofthepackqagemaybGefoundattheFTPaddress`7@ftp://ftp.dcs.qmw.ac.uk/pub/tex/contrib/pt/diagrams'ItmayalsobGeobtainedfromtheComprehensiveTU>'ExXArchiveNetwork(CT*AN) 'atanyofthesitesydFftp.adfa.oz.auAustralia*/pub/tex/ctanFftp.cs.rmit.edu.auAustralia*/tex-archiveFftp.muni.czCzechRepublic*/pub/tex/CT*ANFftp.tex.ac.ukEngland*/tex-archiveFftp.loria.frF*rance*/pub/unix/tex/ctanFftp.uni-bielefeld.deGermany*/pub/texFftp.uni-stuttgart.deGermany*/tex-archive(/pub/tex)Fftp.dante.deGermany*/tex-archiveFftp.center.osakqa-u.ac.jpJapan*/CT*ANFwuarchive.wustl.eduMissouri,USA*/packqages/T*eXFftp.cs.ruu.nlNetherlands*/pub/tex-archiveFftp.duke.eduNorthCarolina,USA*/tex-archiveFsunsite.unc.eduNorthCarolina,USA*/pub/packqages/T*eXFftpserver.nus.sgSingapGore*/pub/zi/T*eXFnic.switch.chSwitzerland*/mirror/texFdongpGo.math.ncu.edu.twT*aiwan*/tex-archiveFftp.uu.netVirginia,USA*/pub/text-proGcessing/T*eXFftp.shsu.eduT*exas,USA*/tex-archive'wheretheroGotofthearchiveisthegivendirectoryandmypackqageis`7@macros/generic/diagrams/Taylor/diagrams.tex'ff 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