Y 1 X1 XZ 177 2 255 Zs Y2 YY Y U ZwhereparallelmorphismsarecombinedwithProposition7Inaunitarypretabularallegorythetwointerpretationsaboveofarelationalcompositeare ID: 471847
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Onlyonepartoftheproofisnon-trivial,butwepostponethewholethinguntilafterthenecessarylemmas.FreydandScedrovgiveaconstruction[FS90,B.3]ofthefreeallegoryonaregulartheory,whichallowsustointerpretanyformulaofregularlogicinaunitarypre-tabularallegory(relativetosomegiveninterpretationofthebasicsorts,termsandpredicates).SupposewehavepredicatesR(x;y)andS(y;z),interpretedasr:X#Yands:Y#Zrespectively.ThentheirrelationalcompositeisgivenbytheformulaSR(x;z)=9 :zR(x; )^xS( ;z)Thiscanbeinterpretedintwodi erentways:asthecompositesr:X#Z,ormore`literally'asXr// Y 1'' X1// XZ 177 2// 255 Zs// Y2// YY// Y// U// Zwhereparallelmorphismsarecombinedwith\.Proposition7.Inaunitarypre-tabularallegory,thetwointerpretationsaboveofarelationalcompositeareequal;thatis,sr=((1r1\2s2)\2)1Proof.Firstnotethatontheleft-handsidesr=sr\�=sr\21,andthatontheright((1r1\2s2)\2)1=(�YZ(r1\s2)\2)1where�YZ=:Y#U#Zisthetopelement.Inonedirectionwehavesr\21=(sr1\2)1modularlaw=(sr1\2\2)1(s(r1\s2)\2)1modularlaw(�YZ(r1\s2)\2)1Intheother,(�YZ(r1\s2)\2)1(�YZs(sr1\2)\2)1modularlaw(�YY(sr1\2)\2)1=(21(sr1\2)\2)1�=21=(2(1sr1\12)\2)1mapsdistribute=2(1sr1\12\22)1modularlaw=2(1sr1\(1\2)2)1mapsdistribute=2(1sr1\2)1seebelow=2(1sr1\2)1modularlaw=(sr1\2)11=2=1=sr\21modularlaw2 Proof.Byprop.7,theleft-handsideisAB && AB12// ABAB34// 1244 3433 AB// A// ABFactoringthedouble(co)projectionsthroughthecanonicalisomorphismABAB=AABB,weget(1(1\2)\24)13=(11(BB)\24)13lemma11=(11\13)12mod.,=1=(11\23)12lemma8=1312idem Lemma11.Let2;3:ABB!B,andwrite12=h1;2i,13=h1;2iasbefore.Then2\3=2(12\13)=2(1AB)whereinthemiddleandontheright2:AB!B.Proof.Fortheleft-handequalitywehave2(12\13)=2(11\22\23)lemma8=2(11\22)\3modularlaw=211\2\3modularlaw=�\2\3=2\3Fortheright,1AB=h1;h1;1i2i,whichis11\22\32bylemma8,theoppositeofwhichcomposedwith2istherstlineabove. Nowwemayproceedwithourpostponedproof.Proofofprop.6.Bothallegoriesandbicategoriesofrelationsarelocallypar-tiallyordered2-categoriesequippedwithanidentity-on-objectsinvolution.SupposeBisabicategoryofrelations.TheFrobeniuslawimpliesthemod-ularlaw[CW87,remark2.9(ii)].ThetensorunitI,theterminalobjectofMap(B),isaunit:thereisauniquemapX!IforanyX,and1IisthetopelementofB(I;I)byprop.2.Theproductprojectionstabulatethetopelements,soBispre-tabular.Conversely,letAbeaunitarypre-tabularallegory,andrefertoprop.2.Map(A)hasniteproducts,andlocalniteproductsaregivenbythedeni-tionofanallegoryandthepresenceoftheunit;theidentityontheunitisbydenitionthetopelementoftherelevanthomset.4 andthenthemodularlawagaintoturntheaboveintoXr// Y Zs// Y '' XX0ZZ0 :: 55 )) $$ YY0// U// ZZ0X0r0// Y0 77 Z0s0// Y0 ?? Butnowwemayusethesymmetryof\toswapthemorphismcontainingswiththatcontainingr0:theresultingmorphism(after\ingwith2andcomposingwiththecoprojectionoutofXX0)isexactlytheinterpretationafterprop.7of(1s1\2s01)(1r1\2r02)Thus isfunctorial. References[CW87]AurelioCarboniandR.F.C.Walters.CartesianbicategoriesI.JournalofPureandAppliedAlgebra,49:11{32,1987.[FS90]PeterFreydandAndreScedrov.Categories,Allegories.North-Holland,1990.6