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Ufdfeb*fdfei-KfdfenşkFfdfesfdfewgfdfez8ńfdfe|gRfdfe}fdfe~73fdfev~32fdfev~zfdfevθfdfevfdfev[nfdfev?6fdfev#LfdfevuQfdfevRfdfevIfdfevc6fdfevˍ:fdfev;fdfev<fdfevĆfdfew2fdfew!zfdfew7WfdfewMg5$fdfewefdfew~>fdfewOӼfdfewן0fdfewАfdfewzfdfex_7fdfex.DfdfexP*Äfdfexsӟfdfex5'fdfexJfdfex Dfdfey _Rfdfey6VfdfeyaPfdfey@fdfeycn&fdfey ^fdfez%NԄfdfezKr@ufdfez~32fdfe~32fdfe}|efdfe{fdfex^xfdfesNfdfen '.fdfegUfdfe^ɟfdfeU/fdfeJMfdfe>#.fdfe0,Kfdfe"Trfdfe{fdfeބfdfe(2fdfeO낄fdfe.܄fdfeן1@fdfe'Rfdfe~/s&fdfecfdfeHg4fdfe+fdfe 낄fdfe8-fdfe͑!fdfe;fdfekTjfdfed l=fdfe>Cfdfe5 fdfeߟfdfeAfdfe[fdfenP$fdfe@۟>fdfebfdfe~fdfe~̟%fdfe~~732fdfeѭ2713nonem(gbrc/k=graphicbraidcircled)cir-mcledbraid7 &e4JB.!P:AREIGISYPn|\nameKnsymrb.sbSorxedqxyGparameters"descriptionsGsymrb.IWdim ff-9􍍑ngibrcNyeTy32fdfeTyU/fdfeTxHvfdfeTv'NfdfeTs,rfdfeToRNfdfeTj܄fdfeTe'fdfeT^;"fdfeTWvZ܄fdfeTOTzNfdfeTFamfdfeTfdfePj(TfdfeP˙=OfdfePQ҄fdfePyşf<fdfePOz^fdfeP$VfdfeOpfdfeO˰fdfeOǪfdfeOobfdfeO@욄fdfeOfdfeNffdfeN!fdfeNz 3JfdfeNy32fdfeNG7DgfdfeNLUfdfeM;gfdfeMyfdfeMfdfeMV*fdfeM(fdfeL҄fdfeLJfdfeLfdfeLzqfdfeLQ nfdfeL()fdfeLП=fdfeKS/fdfeK hfdfeK~]fdfeKjrfdfeKG4fdfeK$ffdfeKCfdfeJhfdfeJfdfeJfdfeJ6fdfeJg>OfdfeJJgfdfeJ.fdfeJfdfeI(fdfeI՟لfdfeI|҄fdfeI4fdfeIzfdfeI6fdfeIoQ΄fdfeI[m1fdfeIHfdfeI63fdfeI$ZfdfeI߄fdfeI,fdfeH^fdfeH@5JfdfeHSfdfeH͒q fdfeHCfdfeH|fdfeH̺fdfeHfdfeHy UfdfeH`*fdfeHKfdfeHVkFfdfeHfdfeHfdfeH}ńfdfeH{RfdfeHz_fdfeHy3fdfeTy32fdfeTyYzfdfeTwnθfdfeTt;fdfeTonfdfeTi?6fdfeTbLfdfeTZQfdfeTQ RfdfeTF*IfdfeT:c6fdfeT,:fdfeTfdfeT ߟfdfeSĆfdfeS2fdfeS,zfdfeS WfdfeS5$fdfeSfdfeSz >fdfeS_̟ӼfdfeSDD0fdfeS'fdfeS tzfdfeR_7fdfeRnDfdfeR*ÄfdfeRHfdfeR_'fdfeR: JfdfeRDfdfeQ꼟RfdfeQVfdfeQ8PfdfeQj-@fdfeQfdfeKbfdfeLfdfeLGO%fdfeLy32fdfejLfeʟ fe fefefeʎv&e|&32fdfe|U/fdfe|rvfdfe|QNfdfe|Vrfdfe||Nfdfe|̟܄fdfe|B'fdfe|ޟ;"fdfe|Z܄fdfe|~zNfdfe|mfdfe|uNfdfe|kڄfdfe|_*fdfe|S,2fdfe|E0fdfe|7NYfdfe|(kvfdfe|%\fdfe|fdfe{ ;fdfe{亟Hfdfe{эfdfe{nrfdfe{/fdfe{ПJxfdfe{|:efdfe{dZfdfe{Lclfdfe{3>6fdfe{?̙fdfezf҄fdfez⒟fdfezJfdfez/fdfezZGfdfezk_mfdfezK"v΄fdfez*Q fdfezfdfey!fdfey˜fdfey^ބfdfeyyJfdfeyS\>fdfey,(Tfdfeyß=OfdfexFQ҄fdfexf<fdfexz^fdfex]fdfex1fdfexڟfdfew@Ǫfdfew̟bfdfewyF욄fdfewIfdfewffdfev=!fdfevJ3Jfdfev&32fdfevaDgfdfevNvUfdfevegfdfeu.yfdfeuџfdfeuTfdfeuafdfeu4ޟ҄fdfeuJfdfetПfdfetfdfet6nfdfeta)fdfet9=fdfet2S/fdfes6hfdfes~]fdfes̟rfdfes^fdfes]ޟffdfes<%CfdfesFhfdferAfdferfdferޟ6fdferhOfdfer̟gfdferh fdferM?fdfer32(fdferلfdfer҄fdfeq'4fdfeqӤfdfeqڟ6fdfeqQ΄fdfeqԟm1fdfeqfdfeqo]fdfeq]֟ZfdfeqM)߄fdfeq=Vfdfeq.fdfeq j5Jfdfeq&Sfdfeqq fdfep,Cfdfepfdfep˟̺fdfepʟfdfepգ UfdfepΊ*fdfepKfdfep€kFfdfepžfdfepޞfdfep ńfdfepޞRfdfepfdfep3fdfe|&32fdfe|zfdfe|θfdfe|efdfe|nfdfe|'?6fdfe|"Lfdfe|ПQfdfe|6Rfdfe|TIfdfe|s*c6fdfe|e:fdfe|W fdfe|G fdfe|5Ćfdfe|#/2fdfe|Vzfdfe{5Wfdfe{ޟ5$fdfe{.fdfe{6>fdfe{Ӽfdfe{}n0fdfe{`fdfe{Bzfdfe{#?_7fdfe{Dfdfez੟*Äfdfezrfdfez'fdfezsJJfdfezL32fdfep&32fdfepɟefdfepfdfepxfdfepbNfdfep%'.fdfep*Ufdfep|fdfepfdfepfdfep".fdfeq,KfdfeqBTrfdfeqC{fdfeq0ބfdfeqC2fdfeqV낄fdfeql܄fdfeqn1@fdfeqRfdfeqs&fdfeqVfdfeqޟ4fdferfdfer#낄fdferC -fdferc!fdfer;fdferڟTjfdferfdfet'bfdfetO,fdfety%fdfet32fdfe32g1Gnone"(gibrcI=graphicinrverseIbraid"circled)inrversecircledbraid9􍍑nguNFʄfeNyrpLfe ʟ fe fefefe ʎvfev&r31g1Gfng"(gu=graphicunit)unit*)ngcuNFʟfeNyrpLfe ʟ fe fefefe ʎvfev&r31g1Gfng"(gcu=graphicco-unit)counit*)ngmpNFʟfeNFʄfeNyhK SpLfe ʟ fe fefefe ʎvfevfev&hs5S31g1GfSg"(gmp~=graphicmorphism)mor-"phism.]-spSeci cationS4givrenby"theparameter,HngbmpNFʟfeNFʄfeIfe ʟ fe1S fefefe ʎpLfe ʟ fe fefefe ʎvfevfeqLfe ʟ fe1S fefefe ʎ31g1GfSg"(gbmp5=graphicborxed"morphism)FYbSorxedmorphism"-spSeci cationSӹgivrenbythe"parameter:HnglmptbNFʟfeNFʄfeIy32fd Iy32fd IFʟ fepLfe ʟ fe fefefe ʎvfevfeq&32fd q&32fd q fe31g1Gnone"(glmptbb=graphicleft[partof8]"morphismy [with]topbottom[con-"nectors])WleftpartoflongbSorxed"morphism-spSeci cationScanbe"insertedwithngnot-hastopand"bSottomconnectorsT9􍍑nglmptNFʟfeIy32fd Iy32fd IFʟ fepLfe ʟ fe fefefe ʎvfeq&32fd q&32fd q fe31g1Gnone"(glmptߚ=graphicleft[partof8]"morphism[with]top[connector])"left partoflongbSorxedmorphism"-hasonlytopconnectornglmpbNFʄfeIy32fd Iy32fd IFʟ fepLfe ʟ fe fefefe ʎvfeq&32fd q&32fd q fe31g1Gnone"(glmpb{V=graphicleft[partof8]"morphismhE[with]bottom[connec-"tor]);leftpartoflongbSorxedmor-"phism8d-hasonlybSottomconnec-"torF9􍍑nglmpIy32fd Iy32fd IFʟ fepLfe ʟ fe fefefe ʎq&32fd q&32fd q fe31g1Gnone"(glmpɎ=graphicleft[partof8]"morphism)DxleftpartoflongbSorxed"morphism-hasnoconnectorsngcmptbNFʟfeNFʄfeHy32fd Hy32fd pLfe ʟ fe fefefe ʎvfevfep&32fd p&32fd 31g1Gnone"(gcmptb)==graphiccenrter[part"of8]H^morphism[with]topbottom"[connectors])@scenrteroflongbSoxed"morphism-spSeci cationScanbe"insertedwithngnot-hastopand"bSottomconnectorsT9􍍑ngcmptNFʟfeHy32fd Hy32fd pLfe ʟ fe fefefe ʎvfep&32fd p&32fd 31g1Gnone"(gcmptp=graphiccenrter[partof8]"morphism[with]top[connector])"cenrteroflongbSoxedmorphism-"hasonlytopconnector!7 &eZGRAPHICAL!CALCULUSPR9OGRAM\5Y]mnameM"symrb.{bSorxedxy.parameters Odescription{"symrb./dim ff9􍍑ngcmpbPτfeJ32fd J32fd x`fe ʟ fe fefefe ʎ~τfex32fd x32fd 1N1.none O(gcmpb!,=graphiccenrter[partof8] OmorphismhE[with]bottom[connec- Otor])EcenrteroflongbSoxedmor- Ophism8d-hasonlybSottomconnec- OtorF9􍍑ngcmpJ32fd J32fd x`fe ʟ fe fefefe ʎx32fd x32fd 1N1.none O(gcmpod=graphiccenrter[partof8] Omorphism)#cenrteroflongbSoxed Omorphism-hasnoconnectors,HngrmptbPϟfePτfeI32fd I32fd Uϟ fex`fe ʟ fe fefefe ʎ~ϟfe~τfew32fd w32fd ϟ fe1N1.none O(grmptbz=graphicrighrt[partof8] Omorphismy [with]topbottom[con- Onectors])righrtpartoflongbSoxed Omorphism-spSeci cationScanbe Oinsertedwithngnot-hastopand ObSottomconnectorsT9􍍑ngrmptPϟfeI32fd I32fd Uϟ fex`fe ʟ fe fefefe ʎ~ϟfew32fd w32fd ϟ fe1N1.none O(grmpt=graphicrighrt[partof8] Omorphism[with]top[connector]) Orighrt.partoflongbSoxedmor- Ophism-hasonlytopconnector:HngrmpbPτfeI32fd I32fd Uϟ fex`fe ʟ fe fefefe ʎ~τfew32fd w32fd ϟ fe1N1.none O(grmpbn=graphicrighrt[partof8] OmorphismhE[with]bottom[connec- Otor])righrtpartoflongbSoxedmor- Ophism8d-hasonlybSottomconnec- OtorngrmpI32fd I32fd Uϟ fex`fe ʟ fe fefefe ʎw32fd w32fd ϟ fe1N1.none O(grmpԦ=graphicrighrt[partof8] Omorphism) righrtpartoflong ObSorxedmorphism-hasnoconnec- Otors89􍍑ngwmruhDΟfe33VΟfe33 L33fePτfel`fe$ʟ fe$ fefefe$ʎrΟfe33Οfe33 z33fe~τfe3N1.fngflCgfrSg O(gwmruhB=graphicwide OmultiplicationSh[with]half[length Ocounrt:forconnectors])widemul- Otiplication$withtopconnectors OatUR1thenthebSoxextendsbSelowthe&rpresenrtline.=(ItwillneverextendabSove.)=TheconnectorconnectsfromthetoppSosition0+t(fromabottomconnectorofanothersymrbol)tothebottompositionb(leadingtoatopconnectorofanothersymrbSol)..TThetopandbottompositionstandbhXarede nedwithrespSecttotheborxoftheconnectorasdescribedaborve. p(TheyhXmaybSenegativre.)8TheoverhangingbSoxdoSesnotdisturbthelowerlines.bT7 &eZGRAPHICAL!CALCULUSPR9OGRAM\7Y Sinceδmanryofsuchconnectorsareneededincomplicateddiagramsandsincetheyare&graphicallyvrerycomplicatedwealsohaveaverysimpleconnectorngclfyn9gconsist-ing justofavrerticallineconnectingpSosition1toposition1.VThefollorwingcommandsgivrethesameresult:s2C,߆T cmtt12\gcl{y}andC\gcn{1}{y}{1}{1}.The rstcommandshouldbSepreferredbecausethedvi-pagebuiltbryitismuchlesscomplicated. Theoriginalgraphicalequationonthe rstpageiswrittenass2,$$\gbeg{4}{6},\got{2}{A}\got{1}{M}\got{1}{N}\gnl,\gcmu\gcl{1}\gcl{3}\gnl,\gcn{1}{1}{1}{3}\glm\gnl,\gvac{1}\gibr\gnl,\gcn{2}{1}{3}{3}\glm\gnl,\gob{3}{M}\gob{1}{N},\gend=,\gbeg{4}{7},\got{2}{A}\got{1}{M}\got{1}{N}\gnl,\gcmu\gbr\gnl,\gcl{1}\gbr\gcl{1}\gnl,\glm\glm\gnl,\gcn{2}{1}{3}{3}\gcn{2}{1}{3}{1}\gnl,\gvac{1}\gibr\gnl,\gvac{1}\gob{1}{M}\gob{1}{N},\gend$$s;7 ,߆T cmtt12+o cmr9 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8@ cmti12N cmbx12': cmti10XQ cmr12K`y cmr10< lcircle10O line10xe