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On a categorical generalization of the concept of fuzzy set [Elektronische Ressource] / Sergejs Solovjovs

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ONACATEGORICAL GENERALIZATION OF THECONCEPT OF FUZZY SETSergejs SolovjovsDissertationzur Erlangung des Grades eines Doktorsder Naturwissenschaften– Dr. rer. nat. –Vorgelegt im Fachbereich 3 (Mathematik & Informatik)der Universitat¨ Bremenim Juni 2007Datum des Promotionskolloquiums: 22. Juni 2007Erster Gutachter: Prof. Dr. Hans-Eberhard Porst (Universitat¨ Bremen)ˇZweiter Gutachter: Prof. Dr. Alexander Sostak (Universitat¨ Lettlands)ContentsAbstract vAcknowledgements viiIntroduction 11 Categories of lattice-valued sets as categories of arrows 51.1 The category X(A)ofA-valuedobjects.................... 51.1.1 Someexamples.................. 71.1.2 Basicsubcategories............................ 91.2 Topological properties of X(A)............. 10∗ ◦1.3 A relation between the functors (−) and (−) ................ 182 On a generalization of Goguen’s category Set(L) 212.1 The category X(A)anditstopologicalproperties............... 212.2 An example of a non-topological category X(A)........ 232.3 On concrete cartesian closedness of X(A)................... 262.4 X(A)isnotatopos ....................... 282.5 On representability of partial morphisms in X(A)............... 292.6 X(A)isaquasitopos....................... 322.6.1 The inner structure of X(A)................... 322.6.2 ArelationbetwenthestructuresgeneratedbyΩandΔ... 362.7 Some remarks on representability of partial morphisms in X(A)....... 373 Aspects of comma categories 41∗3.

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Published 01 January 2007
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ONA

CATEGORICALGENERALIZATIONOFTHE
CONCEPTOFFUZZYSET

SergejsSolovjovs

ionatDissert

zurErlangungdesGradeseinesDoktors
derNaturwissenschaften
–Dr.rer.nat.–

VorgelegtimFachbereich3(Mathematik&Informatik)
derUniversit¨atBremen
imJuni2007

Datum

esd

Promotionskolloquiums:

ErsterGutachter:

ZweiterGutachter:

22.

inuJ

2007

Prof.Dr.Hans-EberhardPorst

Prof.Dr.AlexanderˇSostak

(Universit¨at

(Universit¨at

)enemBr

Lettlands)

tsentCon

Abstract

tsledgemenwknoAc

ntioductroIn

1Categoriesoflattice-valuedsetsascategoriesofarrows
1.1ThecategoryX(A)ofA-valuedobjects....................
1.1.1Someexamples..............................
1.1.2Basicsubcategories............................
1.2TopologicalpropertiesofX(A).........................
1.3Arelationbetweenthefunctors(−)∗and(−)◦................
2OnageneralizationofGoguen’scategorySet(L)
2.1ThecategoryX(A)anditstopologicalproperties...............
2.2Anexampleofanon-topologicalcategoryX(A)................
2.3OnconcretecartesianclosednessofX(A)...................
2.4X(A)isnotatopos...............................
2.5OnrepresentabilityofpartialmorphismsinX(A)...............
2.6X(A)isaquasitopos...............................
2.6.1TheinnerstructureofX(A).......................
2.6.2ArelationbetweenthestructuresgeneratedbyΩandΔ.......
2.7SomeremarksonrepresentabilityofpartialmorphismsinX(A).......
3Aspectsofcommacategories
3.1DefinitionofthecommacategoryX∗(A)anditsalgebraicproperties....
3.2AfactorizationstructureforsourcesonX∗(A)................
3.3CoalgebraicpropertiesofX∗(A)........................
3.4Afactorizationstructure∗forsinksonX∗(A).................
3.5MonadicpropertiesofX(A)..........................
3.6SomeremarksonthemonadT.........................

iii

v

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557901812112326282922323367314417405456585

4Onfuzzificationofalgebraicandtopologicalstructures
4.1Fuzzificationmachineryforalgebraicstructures.......
4.2Fuzzificationmachineryfortopologicalstructures......

5Quantalemodules
5.1DefinitionofthecategoryQ-ModofQ-modules..
5.2Fromquantalemodulestotopologicalspaces....
5.3Q-Modisamonadicconstruct............
5.4OnsomespecialmorphismsinQ-Mod........
5.5Q-Modisnotanabeliancategory..........
5.6Quantalemodulesdonotformatopos........
5.7Tensorproductofquantalemodules.........
5.8Quantale-valuedpower-setfunctors..........
5.9FactorizationstructuresonQ-Mod..........
5.10Completionofpartiallyorderedsets.........

yBibliograph

iv

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Abstract

Thetheoryoffuzzystructureshasbeendevelopingratherfastrecently.Inparticular,
[22,23]presentdifferentapproachestothefoundationsoffuzzysets.Byanalogywiththe
aforesaidconceptwepresentthenotionofafuzzyobjectinacategory.
AfuzzysetintroducedbyZadehin[62]isasetequippedwithafunctiontotheunit
finterval[0,1],i.e.,justamapX✲[0,1].In[18]Goguenreplacedtheunitintervalby
fanarbitrary(butfixed)partiallyorderedsetLandconsideredmapsoftypeX✲L
calledL-fuzzysets.LateronsomeauthorsputastructureonX,e.g.,Rosenfeldin[45]
usedgroups.Someobviousgeneralizationsarise,namely:
•considerso-calledlattice-valuedsets,i.e.,allowchangeofbasisL;
•usedifferentlattice-theoreticalstructuresinsteadofaposet,e.g.,quantalesoreven
quantaloids(see,e.g.,[46,47]);
•usedifferentmathematicalstructuresinsteadofaset.
Bearingtheaforesaidideasinmindweproceedasfollows:givenaconcretecategory(A,U)
foverX,consideranA-valuedobjecttobeanU-structuredarrowX✲UA.IfAis
equippedwiththestructureof2-categoryandthefunctorUisadjointonegetsaconcrete
categoryX(A)ofA-valuedobjects.Ouraimistostudypropertiesofthiscategory.
InthefirstchapterweshowthenecessaryandsufficientconditionsforX(A)tobe
topological.ThenexttwochaptersaredevotedtothestudyoftwosubcategoriesofX(A).
OneofthemgeneralizestheGoguen’scategorySet(L)ofL-fuzzysetswithafixedbasisL
(see,e.g.,[18]),theotheristhecommacategory(idX↓U).Theformersubcategorygives
risetoafuzzificationprocedureofalgebraicandtopologicalstructuresconsideredinthe
subsequentchapter.Thelastchapterisdevotedtothecategoryofquantalemodules(see,

v

e.g.,[46,47])

constructions

asaparticularrealizationof

nda

stsulre

mrof

het

category

thecategoryA.Thechapter

ofmodules

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tsledgemenwknoAc

Iwouldliketothankallwhohelpedmeinpreparationofthethesis.Specialthanksaredue
tomysupervisorprof.A.ˇSostakswhowasthefirstmantoguidemethroughmyresearch.
I’malsogratefultotheteamoftheUniversityofBremen,Germany.Inthefirstplaceto
professorsH.-E.PorstandH.Herrlichaswellasallparticipantsoftheseminar”KatMAT”
formanyusefulsuggestionsandremarks.SpecialthanksaretoC.Schubert,C.Dzierzonand
K.Freundfortheirpatienceinansweringmyquestions.Aninterestingexampleconcerning
T0-spaceswassuggestedbyprof.R.-E.HoffmannwhenIwasparticipatingathisseminar
”OrderedSetsandLattices”.
Duringthe”SummerSchoolonGeneralAlgebraandOrderedSets2005”prof.J.Paseka
ofMasarykUniversityinBrno,ChechRepublicgavemesomegoodadvicesonquantale
modulesaswellassentmehishabilitationthesisonthistopic.
ThefinancialsideofmyresearchwassuppliedbytheEuropeanSocialFund(ESF).

Riga,Latvia
200727,yMa

vii

SergejsSolovjovs

ductiontroIn

Itisawell-acceptedobservationthattherealworldproblemsinvolvedifferentkindsof
vagueenvironments.Tofacethechallengeweneednewmathematicalconceptsor,quoting
Zadeh[61],
”...aradicallydifferentkindofmathematics,themathematicsoffuzzyorcloudyquan-
titieswhicharenotdescribedintermsofprobabilitydistributions.”

ThiswasZadeh’smainmotivationwhenhestartedfuzzysetsin[62]asfollows:
”Afuzzyset(class)Ain[agivenset]Xischaracterizedbyamembership(characteristic)
functionfA(x)whichassociateswitheachpointxinXarealnumberintheinterval
[0,1],withthevalueoffA(x)atxrepresentingthe”gradeofmembership”ofxinA.”

ThenextstepwasdonebyGoguenin[18]wherehewrote:
”Afuzzysetisasettogetherwithafunctiontoatransitivepartiallyorderedset
(hereaftercalledaposet);afuzzysetisthereforeasortofgeneralizedcharacteristic
function.WehabituallydenotetheposetbyLandcallthefuzzysetanL-fuzzysetor
.”set-Lan

ByanalogywiththecategorySetofsetsGoguenconsidersthecategorySet(L)ofL-sets
withafixedbasisL.
Anotherimportantnotionof[18]isthePrincipleofFuzzificationwhichsaysthat”a
fuzzy(orL-fuzzyorL-)somethingisanL-setofsomethings(i.e.,anL-fuzzysetontheset
ofsomethings)”.AsaresultRosenfeldin[45]fuzzifiedthenotionsofgroupoidandgroup,
butChang[11]andLowen[32]consideredfuzzytopologicalspaces.
StartingwithHutton[27],Rodabaugh[40],Eklund[15],andothers,thefollowingideas
eisar

•considerso-calledlattice-valuedsets,i.e.,allowthechangeofbasisLalongwiththe
changeofset;

1

2

•usedifferentlattice-theoreticalstructuresinsteadofposets,e.g.,quantalesoreven
quantaloids(see,e.g.,[46,47]);

•fuzzifydifferentmathematicalstructures,e.g.,modules,fields(incaseoftherealline
Rtheconceptoffuzzynumberappears[42])orcategories[59].

Thisleadstoanaturalgeneralizationofthenotionoffuzzyset,namely,tothenotionof
fuzzyobjectincategoryintroducedinthisthesisasfollows.Startwithaconcretecategory
(A,U)overX.
Definition.Afuzzy(A-fuzzyorA-valued)objectinthecategoryXisanU-structured
arrowXf✲UA.

Followingthestandardterminologyofthefuzzycommunity(see,e.g.,[25])wepreferthe
termA-valued(cf.lattice-valued)object.IfAisequippedwiththestructureof2-category
andthefunctorUisadjointonegetstheconcretecategoryX(A)ofA-valuedobjects
(Definition1.1.3).Thedefinitionallowsonetoconsiderdifferentrealizationsofthecategory
A,forexample,suchunexpectedcategoriesasthecategoryTopoftopologicalspacesor
thecategoryGrpofgroups(Examples1.1.9and1.1.12).Ouraimistostudypropertiesof
thecategoryX(A).
InthefirstchapterweshowthenecessaryandsufficientconditionsforX(A)tobe
topological(Proposition1.2.24).Inaword,theexistenceofcertainfunctorsisrequired.
SincethepropertyofbeingtopologicalprovidesthecategoryX(A)withmanyfeaturesof
itsbasecategory,i.e.,thecategoryX×A,onecanputtherequirementsasaxiomson
X(A).Thechapterendsbyrelationsbetweentheaforesaidfunctors.
ThenexttwochaptersaredevotedtothestudyoftwosubcategoriesofX(A).Thefirst
onegeneralizesGoguen’scategorySet(L)(andthereforeisdenotedbyX(A)),thesecond
oneisthecommacategory(idX↓U)(denotedbyX∗(A)).Weshowthenecessaryand
sufficientconditionsforX(A)tobeaquasitopos(Proposition2.6.1).Itfollowsthatas
suchthecategoryhasanadditionalrichinnerstructure(Definitions2.6.8–2.6.16).The
mainresultsforthecategoryX∗(A)includethenecessaryandsufficientconditionstobe
algebraic(coalgebraic)andmonadic(Propositions3.1.14,3.3.9and3.5.3).Wealsoconsider
factorizationstructuresonX∗(A)(Propositions3.2.6and3.4.5).

3

WecontinuebyconsideringageneralizationoftheaforesaidPrincipleofFuzzification.
Theissuehasarichhistorysincethenotionoffuzzysetinducedfuzzificationofdiffer-
entmathematicalstructures.WehavealreadymentionedL-topologicalspaces[11,24,32]
andL-groups[45].Bothapproachesarefixed-basedanduseimplicitlyGoguen’scategory
Set(L).Avariable-basisapproachoverthecategoryofsemi-quantales(whichisgood
enoughsincethecategoriesoffuzzifiedstructuresaretopologicalovertheirgroundcat-
egories)isconsideredin[39,41].Withthehelpofinternalcategorytheory(see,e.g.,
[33,12,13,14,8])onegetsafuzzificationmachineryofalgebraicstructuresinanysuffi-
cientlygoodcategory(existenceofsomelimitsisrequired),e.g.,intheaforesaidcategory
Set(L).Sometimes,however,thegroundcategoryissupposedtobeatopos(see,e.g.,[34])
thatisnotalwaystrueforthecategorySet(L)(see,e.g.,Proposition2.4.3).
Withtheaforesaidremarksinmindweintroduceafuzzificationmachineryoverthe
categoryX(A).Followingthehistoricalmoveweconsiderthefixed-basisapproachand
thereforeusethecategoryX(A)forafixedA-objectA.Thefuzzificationprocedureis
basedonthecategoriesofgeneralizedalgebraicandtopologicalstructuresAlg(T)and
Spa(T)(see,e.g.,[2]).AsaresultwegetthecategoriesBτ(A)andBP(A)ofrespectiveA-
valuedstructures(themeaningofindicesτandPisexplainedinthethesis).Theattentive
readerwillnoticethattheformercategorygeneralizestheapproachusedin[45],thelatter
category,however,fallsoutoftheaforesaidfuzzificationschemes.
InthelastchapterwepresentarealizationofthecategoryAthroughthecategory
Q-Modofquantalemodules(see,e.g.,[1,29,35,37,38,46,47])motivatedbyconstructions
andresultsfrommodulesoveraring.Thebriefhistoryofthecategoryisasfollows.
Thetermquantalewasintroducedin[35]inconnectionwithcertainaspectsofC∗-
algebras.Ontheotherhand,theconceptwasexpectedtorelatetothesemanticsofnon-
commutativelogics,forexample,thatofquantummechanics.Itdenotesanalgebraic
structure,sayQ,withtwoproperties:

•Qisacompletelattice;

•Qisequippedwithajoin-preservingassociativebinaryoperation.

Inparticular,eachframe(andthereforeeachcompleteBooleanalgebra)isaquantale.Other
examplesincludethepower-setofasemigroupaswellasthesetofallrelationsonaset.

4

Whilethestudyofsuchstructuresgoesbackuptothe1930’s,therehasrecentlybeenmuch
interestinquantalesinavarietyofcontexts.ThemosttopicalconnectioniswithGirard’s
linearlogic(see,e.g.,[17]).Inparticular,[1]enunciatesthefollowingslogan:Quantalesare
tolinearlogicasframesaretointuitionisticlogic.
Thefirstlatticeanalogyofringmodulewasintroducedin[29]byA.JoyalandM.Tierney
inconnectionwithanalysisofdescenttheory.Althoughtheyworkedwithcommutative
structures,mostoftheirresultsarealsovalidfornon-commutativecase.Theideaof
quantalemoduleappearedin[1]ofS.AbramskyandS.Vickersasthekeynotionfor
treatmentofprocesssemantics.
Itisthepurposeofthelastchaptertomakeafurthercontributiontothetheoryofquan-
talemodules.Itpresentsthecollectionofresultsweobtainedwhilestudyingthecategory
Q-Mod.Inparticular,weshowthatthecategoryQ-Modismonadic(Proposition5.3.6),
semiadditivebut(ingeneral)notabelian(Proposition5.5.2),andmonoidalw.r.t.tensor
product(Proposition5.7.15).SinceQ-ModgeneralizesthecategoryJCPosofcomplete
latticesandjoin-preservingmaps(JCPosand2-Modbeingisomorphic)andthereisa
naturalrelationbetweenJCPosandTop,weconsiderarelationbetweenQ-Modandthe
categoryQ-TopofQ-topologicalspaces[24](Proposition5.2.6).Thechapterendsbya
generalizationofthemethodofcompletionofposetsthroughlower-sets(Propositions5.10.2
.10.14).5dan

1Chapter

Categoriesoflattice-valuedsetsas
categoriesofarrows

Inthischapterweintroducethemainsubjectofourstudy,i.e.,thecategoryX(A)(see,
e.g.,[54]).ItisageneralizationofthecategorySet(JCPos)oflattice-valuedsubsetsofsets
(see,e.g.,[55])definedinourmaster’sthesisasfollows(noticethatJCPosisthecategory
ofcompletelatticesandjoin-preservingmaps[2]):
Objects:MapsXα✲AwhereXisasetandAisacompletelattice.
Morphisms:Set×JCPos-morphisms(X,α,A)(f,ϕ✲)(Y,β,B)suchthatϕ◦αβ◦f.
WereplaceSet(resp.JCPos)byanabstractcategoryX(resp.A)andtrytofind
suchpropertiesofbothcategorieswhichmakeX(A)toresembleSet(JCPos).Ourmain
achievementisProposition1.2.24whichshowsthenecessaryandsufficientconditionsfor
X(A)tobetopological.

1.1ThecategoryX(A)ofA-valuedobjects

Suppose(A,U)isaconcretecategoryoverXsuchthatthefollowingconditionsarefulfilled:
1Aisa2-category;

2Uisadjoint.
Foranintroductionintothetheoryof2-categoriessee,e.g.,[33,8].Forshortnesssakewe
ττwritef=⇒ginsteadof”thereexistsa2-cellf=⇒g”.Withthehelpof2wechoosean
adjointsituation(η,):FU:A✲X.

5

6

ConsidertheclassFofallstructuredarrowswithdomaininX,i.e.,ofalltriples
α(X,α,A)withX∈Ob(X),A∈Ob(A)andX✲UA∈Mor(X).Fromnowonweuse
thefollowingagreementswithoutfurtherreference:
α•theelementsofFaredenotedbyX✲UAorjustbyαaloneifthecontextisclear;
α•everyX-morphismoftheformX✲UAisconsideredasanelementofF,in
particular,thesetX(X,UA)isconsideredasasubfamilyofF.
Weintroducearelation””onF.
α✲Definition1.1.1.TakeanytwoFelementsX✲UA.By2wehavetwoA-morphisms
βατFX✲✲A,whereα=A◦Fα.Defineαβiffα=⇒β.
βBelowisonepropertyofthedefinedrelation.
αLemma1.1.2.SupposeXf✲Y✲✲UAUϕ✲UBareX-morphisms.Ifαβ,then
βα◦fβ◦fandUϕ◦αUϕ◦β.
τProof.α◦f=A◦F(α◦f)=A◦Fα◦Ff=α◦Ff=⇒βτ◦Ff=β◦fandUϕ◦α=
B◦F(Uϕ◦α)=(B◦FUϕ)◦Fα=(ϕ◦A)◦Fα=ϕ◦α=⇒ϕ◦β=Uϕ◦β.
hUϕNoticethatifwereplaceUA✲UBbyanarbitraryX-morphismUA✲UB,then
αβdoesnotimplyh◦αh◦β.
Nowthemaindefinition.
Definition1.1.3.ThecategoryX(A)isdefinedasfollows:
Objects:Theabove-mentionedclassFofallstructuredarrowswithdomaininX.
Morphisms:X×A-morphisms(X,α,A)(f,ϕ✲)(Y,β,B)suchthatUϕ◦αβ◦f.
ThefactthatX(A)-morphismsareclosedundercompositioncanbeeasilycheckedwith
thehelpofLemma1.1.2.
WeconsiderthecategoryX(A)asaconcretecategoryovertheproductcategoryX×A
inthefollowingway.
|−|Definition1.1.4.DefinetheforgetfulfunctorX(A)✲X×Aasfollows:
|(X,α,A)(f,ϕ✲)(Y,β,B)|=(X,A)(f,ϕ✲)(Y,B).

Remark1.1.5.X(A)canbeconsideredtobethecategoryofA-valuedobjects.

7

xampleseomeS1.1.1InthissubsectionwegivesomeexamplesofthecategoriesoftheformX(A).
Example1.1.6.AnycategoryXisa2-categoryinatrivialwaywithX(X,Y)considered
asdiscretecategories.SincetheidentityfunctorXidX✲Xisadjointwegetacategory
X(X)whichisisomorphictothecommacategory(idX↓idX).
InasimilarwayonecanconsiderthecategoryA(A)forany2-categoryA.
Example1.1.7.Theconstruct(JCPos,U)satisfiesbothproperties1and2sincefree
objectsarejustpower-setsP(X).Moreover,onecaneasilyseethatforeverytwomaps
α✲X✲UA,itfollowsthatαβintheusual(pointwise)senseiffαβinthesenseof
β1.1.1.tioninDefiExample1.1.8.ConsiderthecategorySetRelofsetsandrelations.Onecaneasilyshow
thatthecategoryisisomorphictotheKleislicategoryofthepower-setmonad.Thisfact
inducesthefollowingdefinitionoftheforgetfulfunctorSetRelU✲Set:
Xρ✲Y→P(X)fρ✲P(Y):S→{y∈Y|thereexistsx∈Ssuchthatxρy}.
ThenUisafaithfulfunctorwithaleftadjointSetF✲SetRelgivenasfollows:
Xf✲Y→Xρf✲Ywithxρfyifff(x)=y.
ηNoticethatFisanon-fullembedding.AddnaturaltransformationsidSet✲UFwith
ηXX✲UFX:x→{x}aswellasFU✲idSetRelwhereFUXX✲Xisde-
finedasfollows:SXxiffx∈S.Yougetanadjointsituation(η,):FU:
SetRel✲Set.SinceSetRelisa2-categoryyougetacategorySet(SetRel).Notice
αthatgivenX✲✲UYitfollowsthatαβiffα(x)⊆β(x)forallx∈X.
βExample1.1.9.Considertheconstruct(Top,U)oftopologicalspacesandcontinuous
maps.U-freeobjectsarejustdiscretetopologicalspaces.Moreover,thehomotopyclasses
ofhomotopiesas2-cellsmakeTopa2-category.Thus,onecanconsiderthecategory
).pTo(SetTheconstructTopgivesrisetoonemoreexample.
Example1.1.10.ConsiderthecategoryProstofpreorderedsets(i.e.,setssuppliedwitha
reflexiveandtransitiverelation)andisotonemaps.Accordingto[21]thereexistsafunctor
UTop✲Prostdefinedasfollows:
ff(X,τ)✲(Y,σ)→(X,)✲(Y,),wherexyiffx∈cl{y}.

8

UhasafullandfaithfulleftadjointProstF✲Topgivenasfollows:
ff(X,)✲(Y,)→(X,τ)✲(Y,σ),whereτhas{↑x|x∈X}asasubbase.
ηAddnaturaltransformationsidProstη✲UFwith(X,)(X,✲)UF(X,):x→xas
wellasFU✲idTopwithFU(X,τ)(X,τ✲)(X,τ):x→x.Yougetanadjointsituation
(η,):FU:Top✲Prost.ConsideringTopasa2-categoryasinExample1.1.9
onegetsthecategoryProst(Top).
Example1.1.11.Considertheconstruct(Cat,U)ofsmallcategoriesandfunctors.One
caneasilycheckthatthefunctorUisadjoint(see,e.g.,Example18.2in[2]).Moreover,
naturaltransformationsbetweenfunctorsmakeCata2-category.Thus,onecanconsider
thecategorySet(Cat).
Example1.1.12.Considertheconstruct(Grp,U)ofgroupsandgrouphomomorphisms.
SinceUisadjointitremainstobeshownthatGrpcanbeconsideredasa2-categorythat
fcanbedoneasfollows.GiventwogrouphomomorphismsA✲✲Bdefinea2-cellf=τ⇒g
gasanelementτ∈Bsuchthatforeveryelementa∈τA,f(a)·τ=υτ·g(a),where·denotesthe
compυτositionlawofthegroupA.Given2-cellsf=⇒gandh=⇒kdefinethecomposition
h◦f=⇒k◦gasanelementh(τ)·υ=υ·k(τ).
ThenextexampleisinasenseageneralizationofExample1.1.7.Itismotivatedby
[47]andrequirestwoadditionaldefinitions.
Definition1.1.13.AquantaloidisacategoryQsuchthat
•forA,B∈Ob(Q),thehom-setQ(A,B)isacompletelattice;
•compositionofmorphismsinQpreservesjoinsinbothvariables.
Inotherwords,aquantaloidisacategoryenrichedinthemonoidalcategoryJCPos(see,
e.g.,[30]aswellasSection5.7inChapter5ofthisthesis).
Definition1.1.14.LetQandSbequantaloids.Aquantaloidhomomorphismisafunctor
FQ✲Sinducingonhom-setsajoin-preservingmapQ(A,B)✲S(F(A),F(B)).
Example1.1.15.LetQtldsdenotethecategoryofsmallquantaloids(noticethatthe
assumptionofsmallnessisnotcrucialhereandisusedonlytoavoiddealingwithquasicat-
egoriesinsteadofcategories).ThereexiststheobviousforgetfulfunctorQtldsU✲Cat.
GivenacategoryX,definethefreequantaloidF(X)overXasfollows.Theobjectsof
F(X)arepreciselythoseofX.GivenF(X)-objectsX,YletF(X)(X,Y)=P(X(X,Y)).
TSGivenF(X)-morphismsX✲Y✲ZletT◦S={t◦s|t∈T,s∈S}.This
operationpreservesunionsineachvariableandthereforeyieldsaquantaloid.Define

9

CatF✲Qtlds:XG✲Y→F(X)Gˆ✲F(Y)withGˆ(XS✲Y)=G(X)G[S✲]G(Y).
AddnaturaltransformationsidCatη✲UFwithXηX✲UFX:Xf✲Y→X{f}✲Y
andFU✲idQtldswithFUQA✲Q:AS✲B→AS✲B.Onegetsanadjoint
situation(η,):FU:Qtlds✲Cat.
NaturaltransformationsbetweenfunctorsmakeQtldsa2-categoryandpartialorder
onhom-setsofquantaloidsmakesitevena3-category.Thus,onecanconsiderthecategory
Cat(Qtlds).
Example1.1.15willbestudiedmorecloselyinthenextchapter.Noticethatthe2-
categoriesconsideredinExamples1.1.7and1.1.8havethin2-cells.Allotherexamples
havenosuchproperty.

1.1.2Basicsubcategories
BelowarelistedseveralsubcategoriesofX(A)whichwillbestudiedinthenextchapters.
Example1.1.16.Thecategory(idX↓U)isanonfullsubcategoryofX(A).Recallthat
itisdefinedasfollows:
Objects:X(A)-objects.
Morphisms:X(A)-morphisms(X,α,A)(f,ϕ✲)(Y,β,B)suchthatUϕ◦α=β◦f.
Thecategory(idX↓U)willbedenotedbyX∗(A).
Example1.1.17.FixanA-objectAanddefineacategoryX(A)asfollows:
Objects:X(A)-objects(X,α,A).
Morphisms:X(A)-morphisms(X,α,A)(f,idA✲)(Y,β,A).
SincetheobjectAisfixedwewilldenoteobjectsandmorphismsofX(A)by(X,α)andf
respectively.
NoticethatincaseofthecategorySet(JCPos)thesubcategorySet(A)isthecategory
ofA-fuzzysubsetsofsetsdefinedin[18].ThusX(A)isageneralizationofSet(A).
Example1.1.18.FixanX-objectXanddefineacategoryX(A)asfollows:
Objects:X(A)-objects(X,α,A).
Morphisms:X(A)-morphisms(X,α,A)(idX,ϕ✲)(X,β,B).
SincetheobjectXisfixedwewilldenoteobjectsandmorphismsofX(A)by(α,A)andϕ
respectively.IfIisaninitialobjectofX,thenI(A)isisomorphictoA.

10

1.2TopologicalpropertiesofX(A)
InthissectionwearegoingtofindnecessaryandsufficientconditionsforX(A)tobe
topologicaloverX×A.
Startwiththesufficientones.ThestatementthatX(A)istopologicaloverX×Acan
canbeprovedintwodifferentwaysprovidingeitherinitialorfinallifts.Wewillconsider
initialonesfirst.
Introducethefollowingrequirements.
3ForeveryX×A-object(X,A),(X(X,UA),)isacompletelattice.
4ForeveryX-morphismYf✲XandeveryA-objectAitfollowsthatthemap
X(X,UA)−◦f✲X(Y,UA)ismeet-preserving.
5ThereexistsafunctorA(−)✲∗Xop:Aϕ✲B→B∗ϕ∗✲A∗withthefollowing
properties:
(i)A∗=UAandB∗=UB;
(ii)idUAϕ∗◦Uϕ;
(iii)Uϕ◦ϕ∗idUB;
(iv)foreveryA-morphismBϕ✲AandeveryX-objectXitfollowsthatthemap
∗X(X,UA)ϕ◦−✲X(X,UB)isorder-preserving.
Firstofalltwosimplepropertiesofthefunctor(−)∗.
Lemma1.2.1.5determinesA(−)✲∗Xopuniquely.
∗Proof.IfbothA(−)✲✲XophavetherequiredpropertiesandAϕ✲BisanA-morphism,
)−(thenϕ(ϕ∗◦Uϕ)◦ϕ=ϕ∗◦(Uϕ◦ϕ)ϕ∗.Similarϕ∗ϕandthereforeϕ∗=ϕ.
∗∗Lemma1.2.2.LetA(−)✲Xopbeafunctionsuchthat(Aϕ✲B)∗=B∗ϕ✲A∗.If
(−)∗satisfies(i)–(iv)of5,thenitisnecessarilyafunctor.
Proof.IfAidA✲AisanA-identity,thenidUA(idA)∗◦UidA=(idA)∗=UidA◦(idA)∗
idUAandthereforeidUA=(idA)∗.Thus(−)∗preservesidentities.
IfAϕ✲Bψ✲CareA-morphisms,then(ψ◦ϕ)∗ϕ∗◦Uϕ◦(ψ◦ϕ)∗ϕ∗◦
ψ∗◦Uψ◦Uϕ◦(ψ◦ϕ)∗=ϕ∗◦ψ∗◦U(ψ◦ϕ)◦(ψ◦ϕ)∗ϕ∗◦ψ∗.Ontheotherhand,
ϕ∗◦ψ∗(ψ◦ϕ)∗◦U(ψ◦ϕ)◦ϕ∗◦ψ∗=(ψ◦ϕ)∗◦Uψ◦Uϕ◦ϕ∗◦ψ∗(ψ◦ϕ)∗◦Uψ◦ψ∗(ψ◦ϕ)∗.
Thus,(ψ◦ϕ)∗=ϕ∗◦ψ∗andtherefore(−)∗preservescomposition.

11

Nowsomeexamples.
Example1.2.3.LookatthecategoryJCPosandyouwilleasilyfindtherequiredfunctor
(−)∗(see,e.g.,Chapter0.3in[16]):
JCPos✲Setop:A✲B→UB✲UA:b→ϕ−1[↓b],
(−)∗ϕϕ∗
i.e.,ϕ∗isjusttheupperadjoint(inthesenseofpartiallyorderedsets)ofϕ.Noticethatin
generalϕ∗doesnotpreservearbitraryjoinshoweveritdoespreservearbitrarymeetsand
thereforeisorder-preserving.
∗)−(Example1.2.4.IncaseofthecategorySetReldefinethefunctorSetRel✲Setopas
follows(noticethatpower-setsarecompletelattices):
ρfρX✲Y→P(Y)✲P(X):S→T:=(Uρ)−1[↓S].
ThelattersetTcanbewrittenas{x∈X|{y∈Y|xρy}⊆S}.
Thenextpropositionisanimmediateconsequenceofassumption5.
)ϕf,(Proposition1.2.5.AnX×A-morphism|(X,α,A)|✲|(Y,β,B)|isanX(A)-morphism
iffαϕ∗◦β◦f.
Proof.IfUϕ◦αβ◦f,thenαϕ∗◦Uϕ◦αϕ∗◦β◦f.Conversely,αϕ∗◦β◦f
impliesUϕ◦αUϕ◦ϕ∗◦β◦fβ◦f.
Wearegoingtoshowthat3–5guaranteetheexistenceofinitialliftsinX(A).
ThefirstpropositionisimmediatefromthedefinitionofX(A).
Proposition1.2.6.ThecategoryX(A)isamnestic.
Nowwecanprovethefollowing.
Proposition1.2.7.Theconcretecategory(X(A),|−|)istopological.
Proof.Inviewoftheprecedingpropositionitwillbeenoughtoshowtheexistenceof
(fi,ϕi)
α=[ϕi∗◦αi◦fi].Proposition1.2.5givesusaliftSˆofS.Weshowthattheliftisinitial.
initiallifts.LetS=((X,A)✲|(Xi,αi,Ai)|)i∈Ibea|−|-structuredsource.Define
I∈iTakeasourceT=((Y,β,B)(gi,ψi✲)(Xi,αi,Ai))i∈IinX(A)andanX×A-morphism
)ϕf,(ϕi∗◦Uψi◦βϕi∗◦αi◦gi=ϕi∗◦αi◦fi◦fimpliesUϕ◦β(ϕi∗◦αi◦fi◦f)=
|(Y,β,B)|✲|(X,α,A)|suchthat|T|=|S|ˆ◦(f,ϕ).SinceUϕ◦βϕi∗◦Uϕi◦Uϕ◦β=
I∈i(ϕi∗◦αi◦fi)◦f=α◦f,then(f,ϕ)isanX(A)-morphism.
I∈i

12

AsanimmediateconsequencewegetthatbothcategoriesSet(JCPos)andSet(SetRel)
ological.optearNowconsiderfinallifts.Forthemomentwewillforgetaboutassumptions3–5and
showthesecondwaytoprovethatX(A)istopologicaloverX×A.
Startwiththefollowingdefinition.
Definition1.2.8.WesaythatacategoryCisacompletequasilatticeprovidedthatthe
followingconditionsarefulfilled:
(i)IfAf✲BandBg✲AareC-morphisms,thenA=B.
(ii)Foreveryfamily(Ai)i∈IofC-objects,thereexistsa(necessarilyunique)C-objectA
tthahsucfi(a)foreveryi∈I,thereexistsaC-morphismAi✲A;
(b)ifBisanotherC-objectwithproperty(a),thenthereexistsaC-morphism
f✲.BAbyi∈IAi.
TheobjectAdefinedin(ii)willbecalledthequasijoinofthefamily(Ai)i∈Ianddenoted
Noticethateachcompletequasilatticehastheuniqueinitialobject,i.e.,thequasijoin
oftheemptyfamily.
Asimplercharacterizationofcompletequasilatticesisgivenbythenextlemma.
Lemma1.2.9.GivenacategoryCdefinearelation””ontheclassofitsobjectsas
follows:ABiffC(A,B)isnotempty.ThenCisacompletequasilatticeiff(Ob(C),)
isa(possiblylarge)completelatticewithquasijoinsbecomingjoins.
Considertwosimpleexamplesofcompletequasilattices.
Example1.2.10.Takeacompletepo-monoid(X,+,)(see[7])withthepropertythat
0=⊥,where0istheidentityof(X,+)and⊥isthelowerboundof(X,).Definea
categoryCasfollows:
Objects:ThesetX.
Morphisms:GivenC-objectsx,yletC(x,y)={a|a∈X,x+ay}.
Composition:GivenC-morphismsxa✲yandyb✲xdefineb◦a=a+b.
FortheconcreterealizationconsiderthesetN∗=N{∞}withtheextensionofthe
OnecaneasilyseethatCisacompletequasilattice.
usualoperationsinthefollowingway:x∞andx+∞=∞+x=∞forallx∈X.Then
N∗hastherequiredpropertiesandgivesrisetothecategoryC.

13

Xsuchthat=ρi.DefineagraphGasfollows:
Example1.2.11.Takeacompletelattice(X,).Let(ρi)i∈Ibeafamilyofrelationson
I∈iObjects:ThesetX.
Morphisms:GivenG-objectsx,yletG(x,y)={ρi|i∈I,xρiy}.
LetCbethe”pathcategoryofG”.OnecaneasilyseethatCisacompletequasilattice.
Fortheconcreterealizationconsiderthechain2.TakeI={1,2,3}anddefineρ1=
{(0,0)},ρ2={(0,1)},ρ3={(1,1)}.Then(ρi)i∈Ihastherequiredpropertiesandgives
risetothecategoryC.
Introducenewrequirements.
6ForeveryX×A-object(X,A),A(FX,A)isacompletequasilattice.
7ForeveryA-morphismAϕ✲BandeveryX-objectXitfollowsthatthemap
A(FX,A)ϕ◦−✲A(FX,B)isquasijoin-preserving.
◦◦8ThereexistsafunctorX(−)✲Aop:Xf✲Y→Y◦f✲X◦withthefollowing
properties:
(i)X◦=FXandY◦=FY;
(ii)Ff◦f◦=τ⇒idFY;
(iii)idFX=υ⇒f◦◦Ff;
(iv)foreveryX-morphismXf✲YandeveryA-objectAitfollowsthatthemap
A(FX,A)−◦f✲◦A(FY,A)isquasijoin-preserving.
SimilartoLemmas1.2.1and1.2.2onecanprovethefollowing.
Lemma1.2.12.(i)–(iii)of8determineX(−)✲◦Aopuniquely.
◦◦Lemma1.2.13.LetX(−)✲Aopbeafunctionsuchthat(Xf✲Y)◦=Y◦f✲X◦.If
(−)◦satisfies(i)–(iii)of8,thenitisnecessarilyafunctor.
Noticethatwedonotusecondition(iv)of8inLemmas1.2.12and1.2.13.
Nowsomeexamples.
Example1.2.14.IncaseofthecategoryJCPosdefinetherequiredfunctorasfollows:
Set(−)✲◦JCPosop:Xf✲Y→P(Y)f−✲1P(X):S→f−1[S].

14

Example1.2.15.IncaseofthecategorySetReldefinetherequiredfunctorasfollows:
1−ρ◦Set(−)✲SetRelop:Xf✲Y→Yf✲X.
Recallthatyρf−1xifff(x)=y.
Thenextpropositionisanimmediateconsequenceofassumption8.
)ϕf,(Proposition1.2.16.AnX×A-morphism|(X,α,A)|✲|(Y,β,B)|isanX(A)-morp-
τhismiffϕ◦α◦f◦=⇒β.
Proof.IfUϕ◦αβ◦f,thenϕ◦α=Uϕ◦α=τ⇒β◦f=β◦Ffandthereforeϕ◦α◦f◦=υ⇒
υτβ◦Ff◦f◦=⇒β.Conversely,ϕ◦α=⇒ϕ◦α◦f◦◦Ff=⇒β◦FfimpliesUϕ◦αβ◦f.

Nowwecanprovethefollowing.
Proposition1.2.17.InthecategoryX(A)every|−|-costructuredsinkhasauniquefinal
lift.Proof.InviewofProposition1.2.6itwillbeenoughtoshowtheexistenceoffinallifts.Let
S=(|(Xi,αi,Ai)|(fi,ϕ✲i)(X,A))i∈Ibea|−|-costructuredsink.Defineα=i∈I[ϕi◦αi◦fi◦].
Proposition1.2.16givesusaliftSˆofS.Weshowthattheliftisfinal.
TakeasinkT=((Xi,αi,Ai)(gi,ψi✲)(Y,β,B))i∈IinX(A)andanX×A-morphism
|(X,α,A)|(f,ϕ✲)|(Y,β,B)|with|T|=(f,ϕ)◦|S|ˆ.Sinceϕ◦ϕi◦αi◦fi◦◦f◦=ψi◦αi◦gi◦=τi⇒β
impliesϕ◦α◦f◦=ϕ◦i∈I(ϕi◦αi◦fi◦)◦f◦=i∈I(ϕ◦ϕi◦αi◦fi◦◦f◦)=υ⇒β,then(f,ϕ)is
anX(A)-morphism.
Weshowedthatrequirements6–8independentfromrequirements3–5imply
thatX(A)istopologicaloverX×A.
Noticethat3–5implythefollowing(seeProposition21.13in[2]).
)ϕf,(✲Propmonomorphiositionsm1iff.2.18.(f,ϕAn)isX(anAe)x-trmemorphial(rsmegu(lX,ar)α,mA)onomorphi(Y,smβ,iBn)iXs×anAexandtremαal=(rϕ∗eg◦uβl◦far).
Lemma1.2.19.AnX∗(A)-morphism(X,α,A)(f,ϕ✲)(Y,β,B)isaninitialX(A)-morp-
hismiffα=ϕ∗◦Uϕ◦α.
Proof.Thenecessity:α=ϕ∗◦β◦f=ϕ∗◦Uϕ◦α.Thesufficiency:α=ϕ∗◦Uϕ◦α=
ϕ∗◦β◦f.
ApplyingLemma1.2.19tothecategorySet(JCPos)wegetthatUϕ◦α=β◦ftogether
withinjectivityofϕimplyinitialityof(f,ϕ).
Similarfrom6–8onegets

15

Proposition1.2.20.AnX(A)-morphism(X,α,A)(f,ϕ✲)(Y,β,B)isanextremal(regular)
epimorphismiff(f,ϕ)isanextremal(regular)epimorphisminX×Aandϕ◦α◦f◦=β.
Lemma1.2.21.AnX∗(A)-morphism(X,α,A)(f,ϕ✲)(Y,β,B)isafinalX(A)-morphism
iffβ◦Ff◦f◦=β.
ApplyingLemma1.2.21tothecategorySet(JCPos)wegetthatUϕ◦α=β◦ftogether
withsurjectivityoffimplyfinalityof(f,ϕ).
NowwearegoingtoprovethattheaforesaidrequirementsarealsonecessaryforX(A)
beingtopologicaloverX×A.
Proposition1.2.22.Suppose1–2hold.IfX(A)istopologicaloverX×A,then
hold.5–3hProof.3:LetT(X,A)bethefibreof(X,A).ThenthemapX(X,UA)✲T(X,A):
α→αisbijective.Sincethefollowingareequivalent:
;βα(i)(ii)|(X,α,A)|(idX,idA✲)|(X,β,A)|isanX(A)-morphism;
(iii)(X,α,A)(X,β,A);
and,moreover,(T(X,A),)isacompletelattice,then(X(X,UA),)mustbealso.
αfiX(X,UA).Weshowthat(αi◦f)(αi)◦f.SinceX(A)istopological,thesource
4:LetY✲XbeanX-morphism.ConsiderasubsetS=(X✲UA)i∈Iof
i∈Ii∈I
S=((X,αi,A)(idX,idA✲)(X,αi,A))i∈Iisinitial.Since
I∈iiii|(Y,(α◦f),A)|(f,idA✲)|(X,α,A)(idX,idA✲)(X,α,A)|
i∈Ii∈I
isanX(A)-morphismforeveryi∈I,then|(Y,(αi◦f),A)|A✲|(X,αi,A)|must
(f,id)
bealsoandtherefore(αi◦f)(αi)◦f.Theconverseinclusionfollowsimmediately
i∈Ii∈I
i∈Ii∈I
fromLemma1.1.2.
5:LetAϕ✲BbeanA-morphism.Then(UB,A)(idUB,ϕ✲)|(UB,idUB,B)|isan
X×A-morphismandthereforehasaninitiallift(UB,ϕ∗,A)(idUB,ϕ✲)(UB,idUB,B)in
X(A).Considersomepropertiesofϕ∗.
(i)Since(idUB,ϕ)isanX(A)-morphism,thenUϕ◦ϕ∗idUB.

16

(ii)Since
|(UA,id,A)|(Uϕ,idA✲)|(UB,ϕ∗,A)(idUB,ϕ✲)(B,id,B)|
UBUA(Uϕ,idA)
isanX(A)-morphism,then|(UA,idUA,A)|✲|(UB,ϕ∗,A)|mustbealsoand
thereforeidUAϕ∗◦Uϕ.
α(iii)TakeX✲✲UBwithαβ.Since
β|(X,ϕ∗◦α,A)|(β,idA✲)|(B,ϕ∗,A)(idUB,ϕ✲)(UB,idUB,B)|
isanX(A)-morphism,then|(X,ϕ∗◦α,A)|(β,idA✲)|(B,ϕ∗,A)|mustbealsoand
thereforeϕ∗◦αϕ∗◦β.
∗∗DefineA(−)✲Xop:Aϕ✲B→UBϕ✲UA.Lemma1.2.2impliesthat(−)∗isa
r.otfuncProposition1.2.23.Suppose1–2hold.IfX(A)istopologicaloverX×A,then
hold.8–6)−(Proof.6:Considertherelationalisomorphism(X(X,UA),)✲(Ob(A(FX,A)),),
whereϕψiffϕ=τ⇒ψ.Proposition1.2.22impliesthat(X(X,UA),)isacompletelattice
andtherefore(Ob(A(FX,A)),)mustbealso.Lemma1.2.9impliesthatA(FX,A)isa
asilattice.quletecomp7:LetAϕ✲BbeanA-morphism.ConsiderasubsetS=(FXαi✲A)i∈Iof
τA(FX,A)-objects.Weshowthatϕ◦(αi)=⇒(ϕ◦αi).Proposition1.2.22implies
i∈Ii∈I
thatthesourceS=((X,αi,A)(idX,idA✲)(X,αi,A))i∈Iisfinal.Since
I∈i|(X,αi,A)XA✲(X,αi,A)|X✲|(X,(Uϕ◦αi),B)|
(id,id)(id,ϕ)
i∈Ii∈I
isanX(A)-morphismforeveryi∈I,then|(X,αi,A)|(idX,ϕ✲)|(X,(Uϕ◦αi),B)|
mustbealsoandthereforeUϕ◦(αi)(Uϕ◦αi).Thenϕ◦(αi)=ϕ◦αi=
i∈Ii∈I
Uϕ◦(αi)=τ⇒(Uϕ◦αi)=Uϕ◦αi=ϕ◦αi.Theconverseinclusionfollows
i∈Ii∈Ii∈Ii∈I
i∈Ii∈Ii∈Ii∈I
immediatelyfromDefinition1.2.8.
8:LetXf✲YbeanX-morphism.Then|(X,ηX,FX)|(f,idFX✲)(Y,FX)isan
)dif,(X×A-morphismandthereforehasafinallift(X,ηX,FX)FX✲(Y,ηˆX,FX)inX(A).
ηˆThenFYX✲FXisanA-morphism.ConsidersomepropertiesofηˆX.
τ(i)Since(f,idFX)isanX(A)-morphism,thenηXηˆX◦fandthereforeidFX=⇒
ηˆX◦Ff.

17

(ii)Since
|(X,ηX,FX)(f,idFX✲)(Y,ηˆX,FX)|(idY,Ff✲)|(Y,ηY,FY)|
isanX(A)-morphism,then|(Y,ηˆX,FX)|(idY,Ff✲)|(Y,ηY,FY)|mustbealsoand
thereforeUFf◦ηˆXηYthatimpliesFf◦ηˆX=τ⇒idFY.
(iii)LetXf✲YbeanX-morphismandletAbeanA-object.Considerasubset
S=(FXαi✲A)i∈IofA(FX,A)-objects.Weshowthat(αi)◦ηˆX=τ⇒(αi◦ηˆX).
i∈Ii∈I
Foreveryj∈I,itfollowsthatαjUαj◦ηˆX◦f(Uαi◦ηˆX◦f)(Uαi◦ηˆX)◦f
andthereforei∈Iαii∈I(Uαi◦ηˆX)◦f.Since
i∈Ii∈I
|(X,ηX,FX)(f,idFX✲)(Y,ηˆX,FX)|i∈I✲|(Y,(Uαi◦ηˆX),A)|
(idY,αi)
I∈i(idY,αi)
isanX(A)-morphism,then|(Y,ηˆX,FX)|i∈I✲|(Y,(Uαi◦ηˆX),A)|mustbe
I∈ialsoandthereforeU(αi)◦ηˆX(Uαi◦ηˆX).Thus(αi)◦ηˆX=τ⇒(αi◦ηˆX).
Theconverseinclusioni∈Ifollowsimmie∈dIiatelyfromDefinitioni∈I1.2.8.i∈I
DefineX(−)✲◦Aop:Xf✲Y→FYηˆX✲FX.Lemma1.2.13impliesthat(−)◦isa
r.otfuncThemainresultofthesectionisasfollows.
Proposition1.2.24.Suppose1–2hold.Thefollowingareequivalent:
(i)X(A)istopologicaloverX×A;
hold;5–3(ii)hold.8–6(iii)Proof.FollowsimmediatelyfromPropositions1.2.7,1.2.17,1.2.22,1.2.23.
AnimmediateconsequenceofProposition1.2.24isthefollowingresult.
Proposition1.2.25.ThecategoriesSet(JCPos)andSet(SetRel)aretopological.None
ofthecategoriesSet(Top),Prost(Top),Set(Cat),Set(Grp)istopological.
Proof.ForthecategoriesSet(JCPos)andSet(SetRel)considerExamples1.2.3and1.2.4.
Allothercategoriesdonotsatisfyrequirement6.

18

1.3Arelationbetweenthefunctors(−)∗and(−)◦
◦∗IntheprevioussectionweintroducedtwofunctorsA(−)✲XopandX(−)✲Aop.By
Proposition1.2.24weknowthattheexistenceofoneofthemimpliestheexistenceofboth.
Inthissectionwearegoingtoinvestigatesomewhatdeepertherelationbetweenthem.
InthefollowingweassumethatthecategoryX(A)istopologicaloverX×A.
∗◦Proposition1.3.1.X(−)✲AopcanbedefinedthroughA(−)✲Xopasfollows:
∗Xf✲Y→FYFX◦F((Ff)◦ηY✲)FX.
•fProof.LetFY✲FXbeasdefined.Considersomepropertiesoff•.
(i)SinceUFf◦(Ff)∗idUFYimpliesUFf◦(Ff)∗◦ηYηY,thenFf◦f•=Ff◦FX◦
F((Ff)∗◦ηY)=FY◦F(UFf◦(Ff)∗◦ηY)=UFf◦(Ff)∗◦ηY=τ⇒ηY=idFY.
(ii)SinceidUFX(Ff)∗◦UFfimpliesηX(Ff)∗◦UFf◦ηX=(Ff)∗◦ηY◦f,then
τidFX=ηX=⇒(Ff)∗◦ηY◦f=FX◦F((Ff)∗◦ηY◦f)=FX◦F((Ff)∗◦ηY)◦Ff=
•.Ff◦fLemma1.2.13impliesthat(−)•isafunctorandtherefore(−)•=(−)◦byLemma1.2.12.

Thenextpropositiongivesanotherpropertyofbothfunctors.
Proposition1.3.2.Thediagram
F✲AX(−)◦(−)∗
❄❄Aopop✲Xop
Us.temmuocProof.LetXf✲YbeanX-morphism.SinceFf◦f◦=τ⇒idFYimpliesU(Ff◦f◦)
UidFY,thenUf◦(Ff)∗◦UFf◦Uf◦(Ff)∗.Conversely,idFX=τ⇒f◦◦Ffimplies
UidFXU(f◦◦Ff)andtherefore(Ff)∗Uf◦◦UFf◦(Ff)∗Uf◦.
Propositions1.3.1and1.3.2implythefollowingresult.
fProposition1.3.3.IfX✲YisanX-morphism,then(Ff)∗=U(FX◦F((Ff)∗◦ηY)).
ByProposition1.3.1weknowthat(−)◦canbedefinedthrough(−)∗.Whataboutthe
converserelation,namely,isitpossibletodefine(−)∗through(−)◦.Thenextproposition
answersthequestion.

19

◦∗Proposition1.3.4.A(−)✲XopcanbedefinedthroughX(−)✲Aopas
Aϕ✲B→UBU(A◦(Uϕ)◦)◦ηUB✲UA
iffthefollowingholds:
α(A)foreveryX✲✲UB∈Fequivalentare
βτ(i)B◦Fα=⇒B◦Fβ;
ϕ(ii)A◦(Uϕ)◦◦Fα=υ⇒A◦(Uϕ)◦◦FβforeveryA-morphismA✲B.
ϕProof.Thesufficiency:LetUB✲UAbeasdefined.Considersomepropertiesofϕ.
(i)SinceFUϕ◦(Uϕ)◦=τ⇒idFUBimpliesUϕ◦ϕ=B◦F(Uϕ◦ϕ)=ϕ◦A◦Fϕ=
ϕ◦A◦F(U(A◦(Uϕ)◦)◦ηUB)=ϕ◦A◦FUA◦F(U(Uϕ)◦◦ηUB)=ϕ◦A◦(Uϕ)◦◦
υFUB◦FηUB=ϕ◦A◦(Uϕ)◦=B◦FUϕ◦(Uϕ)◦=⇒B=idUB,thenUϕ◦ϕidUB.
(ii)SinceidFUA=τ⇒(Uϕ)◦◦FUϕimpliesidUA=A=υ⇒A◦(Uϕ)◦◦FUϕ=A◦(Uϕ)◦◦
FUB◦F(ηUB◦Uϕ)=A◦FUA◦F(U(Uϕ)◦◦ηUB◦Uϕ)=A◦F(U(A◦(Uϕ)◦)◦
ηUB◦Uϕ)=A◦F(ϕ◦Uϕ)=ϕ◦Uϕ,thenidUAϕ◦Uϕ.
αϕ✲(iii)LetX✲UBwithαβandletA✲BbeanA-morphism.Sinceαβimplies
βB◦Fα=τ⇒B◦Fβ,thenϕ◦α=A◦(Uϕ)◦◦Fα=υ⇒A◦(Uϕ)◦◦Fβ=ϕ◦β
andthereforeϕ◦αϕ◦β.
Lemma1.2.2impliesthat(−)isafunctorandtherefore(−)=(−)∗byLemma1.2.1.
Thenecessity:
(i)⇒(ii)Ifαβ,thenϕ∗◦αϕ∗◦βimpliesA◦(Uϕ)◦◦Fα=ϕ∗◦α=τ⇒ϕ∗◦β=
A◦(Uϕ)◦◦Fβ.
(ii)⇒(i)Considerϕ=idB.

Proposition1.3.4saysthatundertheassumptionof(A)onecandefine(−)∗through
(−)◦.Wearegoingtoshowasimplernecessaryconditionforthepossibilityofsuchdefini-
.tionStartwithapreliminarylemma.
ϕLemma1.3.5.IfAψ✲✲BareA-morphismsandϕ=τ⇒ψ,thenψ∗ϕ∗.

20

τ∗∗ψPr∗oof.ϕ∗Si◦ncUψeϕ◦ψ=∗⇒ψϕ∗.impliesUϕUψ,thenidUAϕ◦Uϕϕ◦Uψandtherefore

Corollary∗1.3.6.LetXbeanX-object.IfϕisaninitialobjectofA(FX,FX),then
idUFXϕ.

Forshortnesssakeintroducethefollowingproperty

(B)ForeveryX-objectX,idUFXU(FX◦(Uϕ)◦)◦ηUFX,whereϕisaninitialobject
ofA(FX,FX).
Proposition1.3.7.If(−)∗canbedefinedthrough(−)◦asinProposition1.3.4,then(B)
holds.

Onecaneasilyseethat(B)holdsneitherinthecategorySet(JCPos)norinthecate-
gorySet(SetRel).

2Chapter

OncategoryaSetgeneralization(L)ofGoguen’s

InthischapterwefixanA-objectAandconsiderthesubcategoryX(A)ofthecategory
X(A)(see,e.g.,[57]).RecallthatX(A)canbeconsideredasageneralizationofthe
Goguen’scategorySet(L)ofL-fuzzysets[18].Weshowthenecessaryandsufficientcondi-
tionsforX(A)tobeaquasitopos(Proposition2.6.1).Itfollowsthatassuchthecategory
hasanadditionalrichinnerstructure(Definitions2.6.8–2.6.16).

2.1ThecategoryX(A)anditstopologicalproperties

WebeginbyrecallingthedefinitionofthecategoryX(A)fromthepreviouschapter.
Definition2.1.1.LetAbeanA-object.DefinethecategoryX(A)asfollows:
Objects:(X,α)whereXα✲UAisastructuredarrow.
Morphisms:X-morphisms(X,α)f✲(Y,β)suchthatαβ◦f.

WeconsiderthecategoryX(A)asaconcretecategoryoverXinthefollowingway.
Definition2.1.2.DefinetheforgetfulfunctorX(A)|−|✲Xasfollows:
|(X,α)f✲(Y,β)|=Xf✲Y.

Inthepreviouschapterwefoundthenecessaryandsufficientconditionsforthecate-
goryX(A)tobetopologicaloverX×A(seeProposition1.2.24).Itwascrucialtohave

21

22

thefunctors(−)∗and(−)◦.IncaseofthecategoryX(A)thesituationismuchsimpler.
Introducethefollowingweakversionsofrequirements3–4.

3ForeachX-objectX,(X(X,UA),)isacompletelattice.
4ForeveryX-morphismYf✲XitfollowsthatthemapX(X,UA)−◦f✲X(Y,UA)
ismeet-preserving.
Proposition2.1.3.Suppose1–2hold.Thefollowingareequivalent:
(i)X(A)istopologicaloverX;
(ii)3–4hold.
Proof.(i)=⇒(ii):3–4:ProceedasinProposition1.2.22.
ithasaninitiallift,defineα=[αi◦fi]andproceedasinProposition1.2.7.
(ii)=⇒(i):LetS=(Xfi✲|(Xi,αi)|)i∈Ibea|−|-structuredsource.Toshowthat
Propositions1.2.24and2.1.3implythefollowingresult.
Proposition2.1.4.Suppose1,2hold.Equivalentare:
(i)X(A)istopologicaloverX×A;
(ii)everysubcategoryX(A)ofthecategoryX(A)istopologicaloverXand5holds.

FromnowoneweassumethatthecategoryX(A)(andthusbyProposition2.1.4every
subcategoryoftheformX(A))istopological.
ffLemma2.1.5.LetX✲YbeanX-morphism.Then|(X,α)|✲|(Y,β)|isan
X(A)-morphismiff|(X,α,A)|(f,idA✲)|(Y,β,A)|isanX(A)-morphism.

WiththehelpofLemma2.1.5andProposition1.2.17onecanprovethefollowing.
Proposition2.1.6.LetS=(|(Xi,αi)|fi✲X)i∈Ibea|−|-costructuredsink.Define
α=[αi◦fi◦].ThenSˆ=((Xi,αi)fi✲(X,α))i∈IisafinalliftofSinX(A).
I∈iidXCorollary2.1.7.LetS=(|(X,αi)|✲X)i∈Ibea|−|-costructuredsink.ThenSˆ=
((X,αi)idX✲(X,αi))isafinalliftofS.
I∈iCorollary2.1.7canbeproveddirectlyincaseofX(A)isnottopological.

23

2.2Anexampleofanon-topologicalcategoryX(A)

Theprevioussectionimpliesthefollowingresult:givenaJCPos-objectA,itfollowsthat
thecategorySet(A)istopologicaloverSet.Example1.1.15introducesageneralization
ofthecategorySet(JCPos)intheformofthecategoryCat(Qtlds).Isittruethat
theaforesaidstatementholdsinthelattercategoryaswell?Unfortunatelytheanswer
isnegative.Startbyclarifyingthenatureoftherelation””onobjectsofthecategory
H✲Cat(Qtlds).GivenanaturaltransformationX⇓η✲UQinCatonecanformallydefinea
GH✲naturaltransformationFX⇓η✲QinQtldsbyputtingηX=ηX(recallthatbothXand
GFXhavethesameobjects).Thequestionarises:doesηreallyisanaturaltransformation?
HηLemma2.2.1.LetX✲✲UAbeCat(A)-objects.ThenH✲Gisanaturaltransfor-
GmationiffHη✲Gisanaturaltransformation.
Proof.Thenecessity:ifXS✲YisanFX-morphism,thenηY◦HS=ηY◦(H[S])=
f(ηY◦Hs)=(Gs◦ηX)=GS◦ηX.Thesufficiency:ifX✲YisanX-morphism,then
s∈Ss∈S
}f{X✲YisanFX-morphismandthereforeηY◦Hf=ηY◦H{f}=G{f}◦ηX=Gf◦ηX.
Lemma2.2.2.LetQ,Sbequantaloids.Thenthehom-setQtlds(Q,S)isaquantaloid.
H✲ηProof.GivenafamilyofnaturaltransformationsQ⇓ηi✲Swithi∈I,defineH✲Gby
GηX=ηiX.
I∈iLemmas2.2.1and2.2.2yieldthefollowingresult.
Lemma2.2.3.SupposeXisasmallcategoryandQisaquantaloid.Thenthefunctor
)−(HomCat(X,UQ)✲UHomQtlds(FX,Q)isanisomorphism.
Inotherwords,theadjointsituation(η,):FU:Qtlds✲CatofExam-
ple1.1.15isactuallya2-adjunction(see,e.g.,Chapter7in[8]).ByLemma2.2.3itfollows
ηthatHGinCat(X,UQ)iffthereexistsanaturaltransformationH✲G.Onecan
easilyseethattherelation””isnotalwaysantisymmetricandthereforedoesnotinduce
acompletelatticestructure.Thusonecaneasilyconstructanexampleofanon-topological
categoryCat(Q).Undersomerestrictions,however,onecanshowthat(Cat(X,UQ),)

24

stillhassomegoodproperties.Recallthataprelatticeisapreorderedsetsuppliedwiththe
structureofalattice.Weusethenthetermsprejoins(meets)insteadoftheusualjoinsand
meetsanddenotethemby””and””respectively.
Lemma2.2.4.Let(X,Q)beaCat×Qtlds-object.IfQhasproducts,thenthefollowing
hold:(i)(Cat(X,UQ),)isacompleteprelattice;
H−◦H(ii)ifY✲XisaCat-morphism,thenthemapCat(X,UQ)✲Cat(Y,UQ)is
premeet-preserving.
Proof.ThefirstassumptionfollowsfromTheorem18.22in[20]andthefactthatlimitsin
functorcategoriesarecomputed”pointwise”.Thesecondassumptionthenisstraightfor-
wardsincepremeetsarejustproducts.

AsanimmediateconsequenceofLemma2.2.4onegetsthefollowingcorollary.
Corollary2.2.5.LetQbeaquantaloidthathasproducts.Thenevery|−|-structured
sourceinCat(Q)hasa|−|-initiallift.
Proof.ConsidertheproofofProposition2.1.3.Noticethattheliftneednotbeunique.

Thelastcorollaryyieldsthefollowingresult.
Lemma2.2.6.IfQisaquantaloidthathasproducts,thenthefunctorCat(Q)|−|✲Cat
is(Generating,InitialSource)-factorizable.
Proof.Givena|−|-structuredsourceS=(XHi✲|(Xi,Γi)|)i∈I,setΓ=Γiandgeta(Ge-
nerating,InitialSource)-factorizationXHi✲|(Xi,Γi)|=XidX✲|(X,Γ)||Hi✲||(Xi,Γi)|
fori∈I.

AsanimmediateconsequencesofLemma2.2.6onegetsthefollowingcorollary(see
Definition25.1in[2]).
Corollary2.2.7.IfQisaquantaloidthathasproducts,thenthecategory(Cat(Q),|−|)
istopologicallyalgebraic.

ThelastcorollarytogetherwithProposition25.12andCorollary25.15in[2]implythe
followingresult.
Corollary2.2.8.IfQisaquantaloidthathasproducts,thenthefollowinghold:
(i)thefunctorCat(Q)|−|✲Catisadjoint;

25

(ii)Cat(Q)isstronglycompleteandcocomplete.
NoticethatifweconsiderthefullsubcategoryQtldsofQtldsconsistingofallthose
quantaloidswhosesetofobjectsisnonempty,theaboveresultscanbegeneralizedtothe
categorySet(Qtlds).Startwiththefollowinglemma.
Lemma2.2.9.Let(X,Γ,Q),(Y,Δ,S)beCat(Qtlds)-objects.TheneveryCat×Qtlds-
Ψ)H,(morphism|(X,Γ,Q)|✲|(Y,Δ,S)|isaCat(Qtlds)-morphism.
Proof.ForeveryX-objectXlet(UΨ◦Γ)XτX✲(Δ◦H)Xbethebottomelementofthe
frespectivecompletelattice.GivenanX-morphismX✲Y,itfollowsthatτY◦(UΨ◦Γ)f=
τ(∅)◦(UΨ◦Γ)f=(Δ◦H)f◦(∅)=(Δ◦H)f◦τXandthereforeUΨ◦Γ✲Δ◦H
isanaturaltransformation.
Asaconsequenceoneimmediatelygetsthefollowingcorollary.
Corollary2.2.10.InthecategoryCat(Qtlds)everysourceisinitial.
NoticethatuptonowwedonotusethecategoryQtlds.Thenextlemmahowever
needsthelattercategory.
|−|Lemma2.2.11.ThefunctorCat(Qtlds)✲Cat×Qtldsis(Generating,Initial
Source)-factorizable.
)Ψ,H(Proof.LetS=((X,Q)ii✲|(Xi,Γi,Qi)|)i∈Ibea|−|-structuredsource.SinceQ
isnotemptythereexistssomeQ-objectQandthereforeonehastheconstantfunctor
GQGQ,idX
X✲Q:f→idQ.ThusonehastheuniquefunctorX✲UQ×Xmakingthe
amagrid

XidGQGQ,idXX✲
❄✛UQΠ✛UQUQ×XΠX✲X
commute.DefineXΓ✲UQ=XΠUQ◦GQ,idX✲UQ.ByLemma2.2.9andCorollary2.2.10
(X,Q)(Hi,Ψi✲)|(Xi,Γi,Qi)|=(X,Q)(idX,idQ✲)|(X,Γ,Q)||(Hi,Ψi✲)||(Xi,Γi,Qi)|isa(Gen-
erating,InitialSource)-factorizationofS.
FromLemma2.2.11oneimmediatelygetsthefollowingcorollary.
Corollary2.2.12.Thecategory(Cat(Qtlds),|−|)istopologicallyalgebraic.

26

GoingbacknowtothecategoryCat(Q)withQbeingaQtlds-objectonecaneasily
getthefollowing.
Corollary2.2.13.EverysubcategoryCat(Q)ofthecategoryCat(Qtlds)istopologically
algebraic.

2.3OnconcretecartesianclosednessofX(A)
InthissectionweshownecessaryandsufficientconditionsforthecategoryX(A)tobe
concretelycartesianclosed.
Startwiththefollowingrequirement.
9Xiscartesianclosed.
SinceXhasfiniteproductsandX(A)istopologicaloverX,thefollowingholds.
Proposition2.3.1.ThecategoryX(A)hasfiniteproductswhichcanbeconstructedas
follows.LetS=((Xi,αi))i∈IbeafinitefamilyofX(A)-objects.Thesource
πX((Xi,∗αi)i✲(Xi,αi))i∈Iwith∗αi=αi◦πXi
i∈Ii∈Ii∈Ii∈I
isaproductofS.
Introduceadditionalrequirements.
10ForeveryX(A)-object(X,α)andeveryX-objectYitfollowsthatthemap
∗−αX(Y,UA)✲X(X×Y,UA)

isjoin-preserving.
11ForeveryX(A)-object(X,α)thefunctorX(A)(X,α)×−✲X(A)preservesfinalmor-
phisms.Forconveniencesakeintroducethefollowingnotion.
Definition2.3.2.DefineapartialmapOb(X(A))×Mor(X)−−✲Ob(X(A))asfol-
flows:givenanX(A)-object(X,α)andanX-morphismX✲Ysetαf=β,where
ff(X,α)✲(Y,β)isafinalliftofthe|−|-structuredarrow|(X,α)|✲Y.

27

Intermsof−−11canbestatedasfollows:foreveryX(A)-objects(X,α),(Y,β)
andeveryX-morphismYf✲Zitfollowsthatα∗(βf)=(α∗β)(idX×f).
Considertwoexamples.
Example2.3.3.AsubcategorySet(A)ofthecategorySet(JCPos)satisfies10iffAis
(a∧s).11followsfrom10.
aframe(orcompleteHeytingalgebra),i.e.,itsatisfiesthedistributivelaw:a∧(S)=
S∈sExample2.3.4.Sinceeachpower-setisaframe,everysubcategorySet(A)ofthecategory
Set(SetRel)satisfies10–11.
Wearegoingtoshowthat9–11guaranteeconcretecartesianclosednessofX(A).
Proposition2.3.5.If9–11hold,thenX(A)isconcretelycartesianclosed.
Proof.FixafunctorX(A)(X,α)×−✲X(A)andtakeanX(A)-object(Y,β).By9thereexists
theevaluationmorphismX×YXev✲YinX.Defineγ={YXγi✲UA|α∗γiβ◦ev}.
evThenα∗γ=α∗(γi)=(α∗γi)β◦evimpliesthat|(X×YX,α∗γ)|✲|(Y,β)|is
iianX(A)-morphism.ShowthatevisanevaluationmorphisminX(A).
Let(X,α)×(Z,δ)g✲(Y,β)beanX(A)-morphism.ThenthereexistsanX-morphism
gˆZ✲YXsuchthatev◦(idX×gˆ)=g.Since
|(X×Z,α∗δ)(idX×gˆ✲)(X×YX,(α∗δ)(idX×gˆ))|ev✲|(Y,β)|
isanX(A)-morphism,then|(X×YX,(α∗δ)(idX×gˆ))|ev✲|(Y,β)|mustbealso.Since
α∗(δgˆ)=(α∗δ)(idX×gˆ)β◦evimpliesδgˆγ,thenδ(δˆg)◦gˆγ◦gˆand
gˆtherefore|(Z,δ)|✲|(YX,γ)|isanX(A)-morphism.
Nowshowthatrequirements9–11arealsonecessaryforconcretecartesianclosedness
).A(XofProposition2.3.6.IfX(A)isconcretelycartesianclosed,then9–11hold.
f.ooPr9:Followsimmediatelyfromthedefinitionofconcretecartesianclosedness.
11:ByProposition27.15in[2]itfollowsthatforeveryX(A)-object(X,α)thefunctor
X(A)(X,α)×−✲X(A)preservesfinalsinksandthereforeitpreservesfinalmorphisms.
β10:ConsiderafamilyS=(Yi✲UA)i∈I.ByCorollary2.1.7itfollowsthatS=
idY((Y,βi)✲(Y,βi))i∈IisafinalsinkinX(A)andthereforethesink(X,α)×S=
((X×Y,α∗βi)idX×id✲Y(X×Y,α∗(βi)))i∈Imustbealso.Thusα∗(βi)=(α∗βi).
i∈I
i∈Ii∈Ii∈I

28

Propositions2.3.5and2.3.6implythefollowingresult.
Proposition2.3.7.Thefollowingareequivalent:
(i)X(A)isconcretelycartesianclosed;
hold.11–9(ii)FromExamples2.3.3and2.3.4oneimmediatelygetsthefollowingconsequenceofPropo-
sition2.3.7(cf.Proposition71.4in[60]).
Proposition2.3.8.AsubcategorySet(A)ofthecategorySet(JCPos)isconcretelycarte-
sianclosediffAisaframe.IncaseofthecategorySet(SetRel),everysubcategoryofthe
formSet(A)isconcretelycartesianclosed.

2.4X(A)isnotatopos
InthissectionweshowthatX(A)is(almostalways)notatopos.(Foranintroductioninto
thetheoryoftoposessee,e.g.,[19,10].Someaspectsrelatedtothetheoryoffuzzysetscan
befoundin[60]).
SincethecategoryX(A)istopologicalonecanintroducethefollowingnotations.
finalDefinit)stiruconture2.4.1.onXLetw.rX.t.bethaeneXmp-obtyjsouect.rceBy(resp(r.espsin.k⊥).)wedenotetheinitial(resp.the
Proposition2.4.2.SupposeXisatopos.Thefollowingareequivalent:
(i)X(A)isatopos;
(ii)UAisaterminalobjectinX.
Proof.(i)=⇒(ii):IfX(A)isatopos,thenithasasubobjectclassifier(T,)t✲(Ω,θ),
whereTisaterminalobjectsinX.ForeveryX-objectXthedigram
)X,(idXidX✲✲(X,⊥)idX✲(X,)
✲❄pullback❄
(T,)t✲(Ω,θ)
commutesandtherefore(X,)=(X,⊥)thatimpliesX(X,UA)={α}.
(ii)=⇒(i):IfUAisaterminalobjectinX,thentheforgetfulfunctorX(A)|−|✲X
isanisomorphismandthereforeX(A)isatopos.

29

AsaconsequenceofProposition2.4.2oneimmediatelygetsthefollowingresult.
Proposition2.4.3.AsubcategorySet(A)ofthecategorySet(JCPos)isatoposiffA=1.
AsubcategorySet(A)ofthecategorySet(SetRel)isatoposiffA=∅.

2.5OnrepresentabilityofpartialmorphismsinX(A)
InthissectionweshowthenecessaryandsufficientconditionsforX(A)tohaverepresentable
Minit-partialmorphisms.
StartwiththedefinitionoftheclassMinit.
Definition2.5.1.LetMbeaclassofmorphismsinX.DefineMinittobetheclassofall
initialX(A)-morphismsmwith|m|∈M.
IntroduceapropertyofX-morphisms.
mDefinition2.5.2.WesaythatanX-morphismX✲Ysatisfiesproperty(B)provided
thatforeveryX(A)-object(Z,γ),everypullbackdiagramoftheform
Wmˆ✲|(Z,γ)|
fˆ❄❄f(2.1)
✲YXminXimpliesγ◦Fmˆ◦fˆ◦=γ◦f◦◦Fm.
Consideranexample.
Example2.5.3.IfSet(A)isasubcategoryofSet(JCPos)(resp.Set(SetRel)),then
everySet-maphasproperty(B).
InfactthecategorySetsatisfiesastrongerpropertyconsideredinthefollowingdefini-
.tionDefinition2.5.4.LetXbeacategory.Xissaidtohavepullbacks,whichsatisfythe
Beck-ChevalleyProperty(BCP)providedthatforeverypullbackdiagram
mˆ✲ZWˆff❄❄✲YXminXitfollowsthatFmˆ◦fˆ◦=f◦◦Fm.

30

Lemma2.5.5.IfacategoryXhaspullbacks,whichsatisfy(BCP),theneveryX-morphism
.)B(satisfiesIntroducethefollowingrequirements.
12XhasrepresentableM-partialmorphisms.
13Everym∈Msatisfies(B).
Proposition2.5.6.LetMbeaclassofX-morphisms.Thefollowingareequivalent:
hold;13,12(i)(ii)X(A)hasrepresentableMinit-partialmorphisms.
Moreover,|−|preservesrepresentations.
Proof.(i)=⇒(ii):ByTheorem28.15in[2]itwillbeenoughtoshowthatfinalsinksin
X(A)arestableunderpullbacksalongMinit.
mfiLetS=((Yi,βi)✲(Y,β))i∈IbeafinalsinkinX(A),let(X,α)✲(Y,β)bean
ˆfithatα=(αi◦fˆi◦).
elementofMinitandletSˆ=((Xi,αi)✲(X,α))i∈IbeapullbackofSalongm.Show
I∈iSinceα=β◦m,thenα=β◦Fm=(βi◦fi◦)◦Fm=(βi◦fi◦◦Fm).Pullbackstability
i∈Ii∈I
βi◦fi◦◦Fm.Thusα=(αi◦fˆi◦).
ofinitialmorphismsinX(A)impliesαi=βi◦miandthereforeαi◦fˆ◦=βi◦Fmi◦fˆi◦=
I∈i(ii)=⇒(i):12:FollowsfromProposition28.12in[2].
13:Takem∈Mandconsiderdiagram(2.1).SinceX(A)istopologicaloverX,there
existsafinallift(Z,γ)f✲(Y,β)inX(A)ofthe|−|-costructuredsink(|(Z,γ)|f✲Y).
mSimilarly,thereexistsaninitiallift(X,α)✲(Y,β)inX(A)ofthe|−|-structuredsource
m(X✲|(Y,β)|).Finally,thereexistsastructureδonWsuchthat(2.1)becomesa
pullbackinX(A).ByTheorem28.15in[2]finalsinksinX(A)arestableunderpullbacks
alongMinit.Thenγ◦f◦◦Fm=β◦Fm=α=δ◦fˆ◦=γ◦Fmˆ◦fˆ◦.
Introduceastrongerversionof13.
13Xhaspullbacks,whichsatisfy(BCP).
ThefollowingpropositionfollowsimmediatelyfromLemma2.5.5andProposition2.5.6.
Proposition2.5.7.Suppose13holds.Thefollowingareequivalent:
holds;12(i)

31

(ii)X(A)hasrepresentableMinit-partialmorphisms.
Corollary2.5.8.Suppose13holds.IfXhasrepresentableextremalpartialmorphisms,
thenX(A)hasrepresentableextremalpartialmorphisms.
Proof.MinitistheclassofextremalmonomorphismsinX(A).
AsaconsequenceofCorollary2.5.8oneimmediatelygetsthefollowing.
Proposition2.5.9.EverysubcategorySet(A)ofthecategoriesSet(JCPos)aswellas
Set(SetRel)hasrepresentableextremalpartialmorphismsand|−|preservestheserepre-
sentations.LetXhaverepresentableM-partialmorphismsandlet(X,α)beanX(A)-object.Then
thereexistssomeXmX✲X∗thatrepresentsM-partialmorphismsintoX.ByProposi-
mXtion2.5.6wehavesome(X,α)✲(X∗,α∗)thatrepresentsMinit-partialmorphismsinto
(X,α).Thenextpropositionshowsthenatureofα∗.
mXProposition2.5.10.Suppose12,13hold.Let(X,α)beanX(A)-object,letX✲X∗
representM-partialmorphismsintoX.Defineα∗={X∗αi✲UA|α=αi◦mX}.Then
(X,α)mX✲(X∗,α∗)representsMinit-partialmorphismsinto(X,α).
Proof.Letαbeasdefined.SincemX∈Minit,thenα=α∗◦mXandthereforeα∗α.
idmConversely,letα=αi◦mX.Since((X∗,αi)✛X(X,α)X✲(X,α))isanMinit-partial
morphismfrom(X∗,αi)to(X,α),thereexistsapullback
(X,α)m✲X(X∗,αi)
∗ididXX❄❄(X,α)✲(X∗,α∗)
mXinX(A).Since|−|preservesrepresentations,thenid∗X=idX∗andthereforeαiα∗.
Thusαα∗.
Considerthefollowingexample(cf.Proposition71.3in[60]).
Set(SetRel)andlet(X,α)beaSet(A)-object.SetX∗=X{∞}andgettheobviousin-
Example2.5.11.LetSet(A)beasubcategoryofoneofthecategoriesSet(JCPos)or
mclusionXX✲X∗thatrepresents(extremal)partialmorphismsintoXinthecategorySet.
∗LetX∗α✲UAbeasfollows:α∗(x)=α(x)andα(∞)=.Then(X,α)mX✲(X∗,α∗)
representsextremalpartialmorphismsinto(X,α)inthecategoryX(A).

2.6X(A)isaquasitopos

32

Proposition2.4.2showsthatforthecategoriesoftheformX(A)theconceptofatoposis
notinteresting.However,theconceptofaquasitoposisfruitful.Inthissectionweshow
thenecessaryandsufficientconditionsforX(A)tobeaquasitopos.
Startwiththefollowingrequirement.
14Xisaquasitopos.
Nowthemainproposition.
Proposition2.6.1.Suppose13holds.Thefollowingareequivalent.
(i)10,11,14hold;
(ii)(X(A),|−|)isaconcretequasitopos,i.e.,X(A)isaquasitoposand|−|preserves
powerobjectsandrepresentationsofextremalpartialmorphisms.
Proof.FollowsimmediatelyfromPropositions2.3.7and2.5.7.
AsaconsequenceofProposition2.6.1oneimmediatelygetsthefollowing.
Proposition2.6.2.AsubcategorySet(A)ofthecategorySet(JCPos)isaquasitoposiff
Aisaframe.IncaseofthecategorySet(SetRel),everysubcategoryoftheformSet(A)
isaquasitopos.

2.6.1TheinnerstructureofX(A)
Supposerequirements10,11,13,14hold.ByProposition2.6.1X(A)isaquasitopos.
InthissubsectionweconsidertheinnerstructureofX(A).
Startwiththestandardone(cf.[2]ChapterVII).RecallthatX(X,UA)−◦✲fX(Y,UA)
preservesby4.
Proposition2.6.3.LetTt✲Ωbeanextremal-subobjectclassifier(ESC)inX.Then
(T,)t✲(Ω,)isanESCinX(A).
Proof.Propositions2.6.1and2.5.10implythat(T,)t✲(Ω,θ)representsextremal
partialmorphismsinto(T,),whereθ={Ωαi✲UA|=αi◦t}.By=◦tit
followsthatθ=.
Considerthefollowinglemma.
Lemma2.6.4.Let(X,α),(Y,)beX(A)-objects.Then(Y,)(X,α)=∼(YX,).

33

Proof.ByProposition2.3.5(Y,)(X,α)=∼(YX,α),whereα={YXγi✲UA|α∗γi
◦ev}=.
Proposition2.6.5.Let(X,α)beanX(A)-objectandletYεX✲X×ΩXbeanelement-
hood-morphism(EM)inX.Then(Y,α◦πX◦εX)εX✲(X×ΩX,α◦πX)isanEMin
.)A(XεProof.ByLemma2.6.4(Y,β)X✲(X×ΩX,α∗)isanEMinX(A),whereα∗=
(α◦πX)∧(◦πΩX)=α◦πX.SinceεXisanextremalmonomorphisminX(A),then
β=(α∗)◦εX=α◦πX◦εX.
Consideranexample.
Example2.6.6.LetSet(A)beasubcategoryofthecategorySet(JCPos),let(X,α)bea
Set(A)-object.AnEMfor(X,α)isanextremalsubobject((Y,β),εX)of(X×2X,α◦πX)
tthahsuc(i)Yistheusualrelation∈⊆X×2X;
(ii)givenx∈S⊆Xitfollowsthatβ(x,S)=α(x);
(iii)εXistheinclusion.
ThenextpropositionfollowsimmediatelyfromLemma2.6.4.
fProposition2.6.7.Let(X,α)✲(Y,β)beanX(A)-morphism.
(i)IfXσ✲ΩXisasingleton-morphism(SM)inX,then(X,α)σX✲(ΩX,)isa
SMinX(A).
∗∗(ii)IfYf✲ΩXisapreimage-morphism(PM)inX,then(Y,β)f✲(ΩX,)isaPM
inX(A).
ffΩΩ(iii)IfΩY✲ΩXisapower-set-morphism(PSM)inX,then(ΩY,)✲(ΩX,)is
aPSMinX(A).
NowconsideranadditionalinnerstructureofX(A).
Definition2.6.8.Since(UA,idUA)isanX(A)-objectthereexistsanextremalmonomor-
tAphism(UA,idUA)✲(Δ,δ)inX(A)thatrepresentsextremalpartialmorphismsinto
(UA,idUA).Thenforeveryextremalmonomorphism(X,α)m✲(Y,β)inX(A)there
χexistsauniqueX(A)-morphism(Y,β)m✲(Δ,δ)suchthatthediagram
m(X,α)✲(Y,β)
χαm❄❄(UA,idUA)tA✲(Δ,δ)

34

isapullback.
CalltAaweakextremal-subobjectclassifier(WESC).
Consideranexample.
Example2.6.9.LetSet(A)beasubcategoryofthecategorySet(JCPos).Example2.5.11
timpliesthefollowingdefinitionofaWESC(UA,idUA)A✲(Δ,δ):
(i)Δ=UA{∞};
UA∈xx,(ii)δ(x)=,x=∞;
(iii)tAistheinclusion.
Definition2.6.10.ForeachX(A)-object(X,α)thereisanX(A)-morphismε(ΔX,α),defined
bythepullback
Δ(Y,β)ε(X,α)✲(X,α)×(Δ,δ)(X,α)
evevˆ❄❄(UA,idUA)tA✲(Δ,δ)
ΔCall(Y,β)ε(X,α✲)(X×ΔX,α∗δα)aweakelementhood-morphism(WEM).
Noticethatsinceε(ΔX,α)isanextremalX(A)-monomorphism,β=(α∗δα)◦ε(ΔX,α).
Consideranexample.
Example2.6.11.LetSet(A)beasubcategoryofthecategorySet(JCPos),let(X,α)
beaSet(A)-object.AWEMfor(X,α)isanextremalsubobject((Y,β),ε(ΔX,α))of(X×
ΔX,α∗δα)suchthat
(i)Y={(a,(x,f))|a∈Aandf(x)=a};
β{b∈A|α(x)∧bδ◦f(x)forx∈X};
(ii)themapY✲UAisasfollows:(a,(x,f))→α(x)∧(α→δ◦f)withα→δ◦f=
(iii)ε(ΔX,α)istheprojection.
Definition2.6.12.ForeachX(A)-object(X,α)thereisanX(A)-morphismσ(ΔX,α),defined
bythediagram
(X,α)<idX,idX>✲(X,α)×(X,α)idX×σ(ΔX,α✲)(X,α)×(Δ,δ)(X,α)
α❄pullback❄✛ev
(UA,idUA)tA✲(Δ,δ)

35

ΔσCall(X,α)(X,α✲)(ΔX,δα)aweaksingleton-morphism(WSM).
Consideranexample.
Example2.6.13.LetSet(A)beaΔsubcategoryofthecategorySet(JCPos),let(X,α)be
σ)αX,(aSet(A)-object.AWSM(X,α)✲(ΔX,δα)canbedefinedasfollows:
Δα(x),y=x
σ(X,α)(x):X✲Δ:y→∞,y=x.
fDefinition2.6.14.ForeachX(A)-morphism(X,α)✲(Y,β)thereisanX(A)-morp-
hismfΔ∗,definedbythediagram
<id,f>id×f∗
(X,α)X✲(X,α)×(Y,β)X✲Δ(X,α)×(Δ,δ)(X,α)
α❄pullback❄ev
✛(UA,idUA)tA✲(Δ,δ)
∗fCall(Y,β)Δ✲(ΔX,δα)aweakpreimage-morphism(WPM).
Consideranexample.
Example2.6.15.ConsiderasubcategorySet(A)ofthecategorySet(JCPos).Takea
∗ffSet(A)-morphism(X,α)✲(Y,β).AWPM(Y,β)Δ✲(ΔX,δα)canbedefinedas
follows:
fΔ∗(y):X✲Δ:x→α(x),f(x)=y
∞,f(x)=y.
Definition2.6.16.ForeachX(A)-morphism(X,α)f✲(Y,β)thereisanX(A)-morp-
hism(Δ,δ)f,definedbythediagram
(X,α)×(Δ,δ)(Y,β)f×idΔY✲(Y,β)×(Δ,δ)(Y,β)
idX×(Δ,δ)fev
❄❄(X,α)×(Δ,δ)(X,α)✲Δ
evfCall(ΔY,δβ)(Δ,δ)✲(ΔX,δα)aweakpower-set-morphism(WPSM).
Consideranexample.
Example2.6.17.ConsiderasubcategorySet(A)ofthecategorySet(JCPos).Takea
Set(A)-morphism(X,α)f✲(Y,β).AWPSM(ΔY,δβ)(Δ,δ)✲f(ΔX,δα)canbedefined
asfollows:(Δ,δ)f(h)=h◦f.

36

2.6.2ArelationbetweenthestructuresgeneratedbyΩandΔ
InthelastsubsectionweintroducedX-objectsΩandΔwhichgavetherespectiveinner
structuresofX(A).ThissubsectionisdevotedtorelationsbetweenΩandΔand,asa
consequence,betweentheinnerstructuresgeneratedbythem.
Startwiththeobservationthat(Ω,)canbeembeddedinto(Δ,δ).
sProposition2.6.18.ThereexistX(A)-morphisms(Ω,)✛✲(Δ,δ)withr◦s=idΩ.
rProof.Pastetogethertwopullbacks
ttA(T,)✲(Ω,)(UA,idUA)✲(Δ,δ)
r!s❄❄❄❄(UA,idUA)✲(Δ,δ)(T,)✲(Ω,)
ttAandnoticethat!◦=idT.
ThenextpropositionfollowsimmediatelyfromthefactthatbothtAandsareextremal
monomorphismsinX(A).
stProposition2.6.19.ForX(A)-morphisms(UA,idUA)A✲(Δ,δ)✛(Ω,)itfollows
thatidUA=δ◦tAand=δ◦s.
mSuppose(X,α)✲(Y,β)isanextremalmonomorphisminX(A).Sincetheleftpart
ofthediagram

(X,α)m✲(Y,β)
❄αpullbackχmΔ❄
χ!(UA,idUA)tA✲(Δ,δ)mΩ
!pullbackr✲✲
(T,✛✛)✲(Ω,)
tcommutes,soistheright.Thusthefollowingpropositionholds.
PropΔositionΩ2.6.20.Let(X,α)m✲(Y,β)beanextremalmonomorphisminX(A).Then
r◦χm=χm.
ςProposition2.6.21.Thereexistnaturaltransformations(Ω,)(−)✛ρ✲(Δ,δ)(−)such
thatρ◦ς=id(Ω,)(−).

37

Proof.ForeveryX(A)-object(X,α)thereexistsanX(A)-morphisms(X,α),definedbythe
amagrid)αX,((X,α)×(Ω,)(X,α)idX×s✲(X,α)×(Δ,δ)(X,α)
evev❄❄(Ω,)✲(Δ,δ)
sLetς(X,α)=s(X,α).Similarletρ(X,α)=r(X,α).
Proposition2.6.22.ForeveryX(A)-object(X,α)thereexistsanX(A)-sectionsˆsuch
thatthediagram
ε(ΩX,α)
(Y,β)Ω✲(X,α)×(Ω,)(X,α)
sˆidX×ς(X,α)
❄❄(Y,β)Δ✲(X,α)×(Δ,δ)(X,α)
ε(ΔX,α)

s.temmuocProof.Straightforwardcomputationsshowthatthediagram
Ω(Y,β)Ωε(X,α)✲(X,α)×(Ω,)(X,α)
❄❄ev◦(idX×ς(X,α))
(UA,idUA)tA✲(Δ,δ)
commutes.Nowusethedefinitionofε(ΔX,α).
Straightforwardcomputationsprovethenexttwopropositions.
Proposition2.6.23.ForeveryX(A)-object(X,α)itfollowsthatρ(X,α)◦σ(ΔX,α)=σ(ΩX,α).
fProposition2.6.24.ForeveryX(A)-morphism(X,α)✲(Y,β)itfollowsthatρ(X,α)◦
f∗Δ=fΩ∗.
2.7Someremarksonrepresentabilityofpartialmorphisms
)A(XinInthissectionwegeneralizeProposition2.5.6tothecategoryX(A).
Startbyintroducingthefollowingrequirements.

38

(2.2)

(2.3)

15XhasrepresentableMX-partialmorphisms.
16Foreverym∈MXandeverypullbackdiagram
mˆ✲ZWˆ(2.2)ff❄❄✲YXminX,itfollowsthatf◦◦Fm=Fmˆ◦fˆ◦.
17AhasrepresentableMA-partialmorphisms.
18Foreveryϕ∈MAandeverypullbackdiagram
ϕˆ✲CDˆ(2.3)ψψ❄❄✲BAϕinA,itfollowsthatϕ∗◦FUψ=ψˆ◦ϕˆ∗.
),ψf(19Forevery|−|-co-structuredsinkS=(|(Yi,βi,Bi)|ii✲(Y,B))i∈Iandevery
(X,A)(m,ϕ✲)(Y,B)∈MX×MA,itfollowsthatϕ∗◦FUB◦F(U(τ◦Fm)◦ηX)=
ϕ∗◦F(U(B◦τ◦Fm)◦ηX),whereτ=(FUψi◦Fβi◦fi◦).
I∈iProposition2.7.1.Thefollowingareequivalent:
hold;19–15(i)(ii)X(A)hasrepresentableMinit-partialmorphisms,whereMinitistheclassofallinitial
X(A)-morphisms(m,ϕ)with|(m,ϕ)|∈MX×MA.
Proof.(i)=⇒(ii):ByTheorem28.15in[2]itwillbeenoughtoshowthatfinalsinksare
stableunderpullbacksalongMinit.
LetS=((Yi,βi,Bi)(fi,ψ✲i)(Y,β,B))beafinalsinkinX(A),let(X,α,A)(m,ϕ✲)(Y,β,B)
ˆˆbeanelementofMinitandletSˆ=((Xi,αi,Ai)(fi,ψi✲)(X,α,A))beapullbackofSalong
(m,ϕ).Showthatα=ψˆi◦αi◦fˆi◦.
I∈iSinceinitialmorphismsarepullbackstableinX(A),itfollowsthatαi=ϕi∗◦F(βi◦mi).
Thenψˆi◦αi◦fˆi◦=ψˆi◦ϕi∗◦Fβi◦Fmi◦fˆi◦=ϕ∗◦FUψi◦Fβi◦fi◦◦Fmandψˆi◦αi◦fˆi◦=
I∈i

39

ϕ∗◦(FUψi◦Fβi◦fi◦)◦Fm=ϕ∗◦τ◦Fm=ϕ∗◦FUB◦F(U(τ◦Fm)◦ηX).Ontheother
I∈iI∈ihand,α=ϕ∗◦Fβ◦Fm=ϕ∗◦F(Uβ◦ηY)◦Fm=ϕ∗◦F(U(ψi◦βi◦fi◦)◦ηY)◦Fm=
ϕ∗◦F(U(B◦τ◦Fm)◦ηX).By19itfollowsthatα=ψˆi◦αi◦fˆi◦.
I∈i(ii)=⇒(i):15,17:FollowimmediatelyfromProposition28.12in[2].
16:Takem∈MXandconsiderdiagram(2.2).Let((Z,ηZ,FZ)(f,idFZ✲)(Y,β,FZ))
beafinalliftinX(A)ofthe|−|-co-structuredsink(|(Z,ηZ,FZ)|(f,idZ✲)(Y,FZ)).Then
β=f◦.Let((X,α,FZ)(m,idFZ✲)(Y,β,FZ))beaninitialliftinX(A)ofthe|−|-structured
source((X,FZ)(m,idFZ✲)|(Y,β,FZ)|).Thenα=β◦m.Finally,thereexistsastructureδ
on(W,FZ)suc◦hthat(2.◦2)becomesapullbackinX(A◦).ByTheorem◦28.15in[2]◦itfollows
thatα=δ◦fˆ.Thenf◦Fm=β◦Fm=α=δ◦fˆ=ηZ◦Fmˆ◦fˆ(=idFUCmˆ,ψ)◦fˆ.
18:Takeϕ∈MAandconsiderdiagram(2.3).Let((UC,idUC,C)✲(UC,β,B))
beafinalliftinX(A)ofthe|−|-co-structuredsink(|(UC,idUC,C)|(idUC,ψ✲)(UC,B)).
Thenβ=ψ◦C.Let((UC,α,A)(idUC,ϕ✲)(UC,β,B))beaninitialliftinX(A)ofthe|−|-
structuredsource((UC,A)(idUC,ϕ✲)|(UC,β,B)|).Thenα=ϕ∗◦β.Finally,thereexistsa
structureδon(UC,D)suchthat(2.3)becomesapullbackinX(A).ByTheorem28.15in[2]
itfollowsthatα=ψˆ◦δ.Thenψˆ◦ϕˆ∗=ψˆ◦δ=α=ϕ∗◦Fβ=ϕ∗◦F(Uβ◦ηUC)=ϕ∗◦FUψ.
19:Considera|−|-co-structuredsinkS=(|(Yi,βi,Bi)|(fi,ψi✲)(Y,B))i∈Iandan
element(X,A)(m,ϕ✲)(Y,B)ofMX×MA.LetSˆ=((Yi,βi,Bi)(fi,ψi✲)(Y,β,B))i∈Ibe
afinalliftofSinX(A).Let((X,α,A)(m,ϕ✲)(Y,β,B))beaninitialliftofthe|−|-
structuredsource((X,A)(m,ϕ✲)|(Y,β,B)|)inX(A).ByProposition28.3in[2]letSˆ=
((Xi,αi,Ai)(fˆi,ψˆi✲)(X,α,A))beapullbackofSalong(m,ϕ).Similartotheimplication
(i)=⇒(ii)oneshowsthatϕ∗◦FUB◦F(U(τ◦Fm)◦ηX)=ψˆi◦αi◦fˆi◦=α=
I∈iϕ∗◦F(U(B◦τ◦Fm)◦ηX).
Theaforesaidrequirementsimplythefollowing.
Lemma2.7.2.Suppose15,16hold.Thenm◦◦Fm=idFXforeveryXm✲Y∈MX.
Proof.ByProposition28.3in[2]misamonomorphisminXandtherefore
XidˆX✲X
ˆmidX❄❄✲YXmisapullbackdiagraminX.Thenm◦◦Fm=FidX◦id◦X=idFX.
Lemma2.7.3.Suppose17,18hold.Thenϕ∗◦Uϕ=idUAforeveryAϕ✲B∈MA.

40

Proof.SimilartotheproofofLemma2.7.2.
ConsideranexampleoftheaforesaidrequirementsinthecategorySet(JCPos).
Example2.7.4(JCPoshasrepresentableMJCPos-partialmorphisms).LetMJCPosbe
theclassofallJCPos-monomorphismsAϕ✲Bsuchthatϕ[A]isanidealinB.Then
extensionA→A{∞}witha<∞foralla∈ArepresentsMJCPos-partialmorphisms
JCPoshasrepresentableMJCPos-partialmorphisms.ForeachobjectAtheone-point
.AtoinExample2.7.5(MJCPosdoesnotsatisfy18).Let2ϕ✲3betheinclusion.Thenϕis
ψanelementofMJCPos.Let2✲3:1→2.Thenthepullbackdiagram
i✲121ψi2❄❄✲32ϕdoesnotsatisfy18sinceϕ∗◦FUψ(2)=1=0=i2◦i1∗(2).
Example2.7.6(Set(JCPos)doesnotsatisfy19).LetMSetbetheclassofallinjective
maps.ThenSetsatisfies15.
Definea|−|-co-structuredsink(|(Y1,β1,B1)|(f1,ψ1✲)(Y,B))asfollows:
(i)B={⊥,a,b,c,}witha,b,cunrelatedandψ1istheinclusionofthesublattice
B1={⊥,a,b,};
(ii)f1istheuniquemapfromY1={y1,y2}toY=1;
(iii)β1(y1)=aandβ1(y2)=b.
Define(X,A)(m,ϕ✲)(Y,B)∈MSet×MJCPosasfollows:X=1=Y,A=2withϕ(1)=c.
Sinceϕ∗◦FUB◦F(U(τ◦Fm)◦ηX)({0})=0=1=ϕ∗◦F(U(B◦τ◦Fm)◦ηX)({0}),
thenSet(JCPos)doesnotsatisfy19fortheaforesaidMSetandMJCPos.
Example2.7.7(Set(JCPos)doesnothaverepresentableMinit-partialmorphismsforthe
aforesaidMSetandMJCPos).Examples2.7.4-2.7.6implythatSet(JCPos)satisfies15–
17anddoesnotsatisfy18–19.

3Chapter

Aspectsofcommacategories

InthischapterweconsiderthesubcategoryX∗(A)ofthecategoryX(A)(see,e.g.,[49]).
WeshowthenecessaryandsufficientconditionsforX∗(A)tobealgebraic(coalgebraic)and
monadic(Propositions3.1.14,3.3.9and3.5.3)aswellasconsiderfactorizationstructures
onX∗(A).

3.1DefinitionofthecommacategoryX∗(A)anditsalgebraic
ertiesoprp

WebeginbyrecallingthedefinitionofthecategoryX∗(A)fromthefirstchapter.
Definition3.1.1.LetAU✲Xbeafunctor.ThecategoryX∗(A)isdefinedasfollows:
Objects:(X,α,A)whereXα✲UAisanU-structuredarrow.
Morphisms:X×A-morphisms(X,α,A)(f,ϕ✲)(Y,β,B)suchthatUϕ◦α=β◦f.
ConsiderthecategoryX∗(A)asaconcretecategoryoverX×Ainthefollowingway.
|−|Definition3.1.2.DefinetheforgetfulfunctorX∗(A)✲X×Aasfollows:
|(X,α,A)(f,ϕ✲)(Y,β,B)|=(X,A)(f,ϕ✲)(Y,B).

InthischapterweconsiderthecategoryX∗(A)asanindependentcategoryorrather
asthecommacategory(idX↓U).Asaresultweforgetallrequirementsexcept2and
introducethenewones.

41

42

ConsiderthenecessaryandsufficientconditionsforthecategoryX∗(A)tobealgebraic.
Introducethefollowingrequirements.

20ThecategoryAhasthefollowingproperties:
(i)Ais(Epi,Mono-Source)-factorizable;
(ii)ForeveryX×A-object(X,A)thereexistsacoproductofA,FXinA.
21ThecategoryXis(Epi,Mono-Source)-factorizable.
Remark3.1.3.Requirement2impliesthatAU✲Xpreservesmono-sources.Require-
ments20.(i)and21implythatAandXare(ExtrEpi,Mono-Source)-categories.
Recallthedefinitionofanalgebraiccategory(cf.[2]):
Definition3.1.4.Aconcretecategory(C,V)iscalledessentiallyalgebraicprovidedthat
thefollowingholds:
(i)Vcreatesisomorphisms;
(ii)Vis(Generating,Mono-Source)-factorizable.
Anessentiallyalgebraiccategory(C,V)iscalledalgebraicprovidedthatVpreservesex-
tremalepimorphisms.
Wearegoingtoshowthattheaforesaidrequirementsguaranteethatthecategory
(X∗(A),|−|)isalgebraic.
Forthefirstpropertyonedoesnotneedanyrequirements.
Proposition3.1.5.ThefunctorX∗(A)|−|✲X×Acreatesisomorphisms.
Proof.Let(X,A)(f,ϕ)✲|(Y,β,B)|beanX×A-isomorphism.DefineXα✲UA=
XUϕ−1◦β◦✲fUA.
Nowthesecondproperty.Startbyconsidering|−|-generatingarrows.Noticethatthere
existsthefunctorAU,idA✲X×Adefinedbythediagram
AidUU,idAA✲
❄✛X✛πXX×AπA✲A

43

Lemma3.1.6.Let(X,A)(e,ϕ✲)|(Y,β,B)|bea|−|-structuredarrow.Ifeisanepimorphism
inXandtheU,idA-structuredarrow(Y,A)(β,ϕ✲)U,idA(B)isU,idA-generating,then
((e,ϕ),(Y,β,B))is|−|-generating.
Proof.Let(Y,β,B)(r,τ✲✲)(Z,γ,C)beX∗(A)-morphismssuchthat|(r,τ)|◦(e,ϕ)=|(s,ψ)|◦
)s,ψ((e,ϕ).Thenr◦e=s◦eimpliesr=s.Sinceτ◦ϕ=ψ◦ϕandUτ◦β=γ◦r=γ◦s=Uψ◦β
implyU,idA(τ)◦(β,ϕ)=U,idA(ψ)◦(β,ϕ),thenτ=ψ.
Proposition3.1.7.Suppose2,20,21hold.ThenX∗(A)|−|✲X×Ais(Generating,
Mono-Source)-factorizable.
Proof.LetS=((X,A)(fi,ϕ✲i)|(Xi,αi,Ai)|)i∈Ibea|−|-structuredsource.LetXfi✲Xi=
Xe✲Ymi✲Xibean(Epi,Mono-Source)-factorizationof(fi)i∈IinX.Theuniversal
arrowYηY✲UFYgivesasource(FYαi◦m✲iAi)i∈IinA.Withthehelpofacoproduct
((μA,μFY),AFY)ofAandFYinAgetasourceT=(AFY[ϕi,αi◦m✲i]Ai)i∈IinA.
Finally,letAFY[ϕi,αi◦mi✲]Ai=AFYψ✲Bτi✲Aibean(Epi,Mono-Source)-
factorizationofTinA.ByLemma3.1.6itfollowsthat(X,A)(fi,ϕi✲)|(Xi,αi,Ai)|=
(X,A)(e,ψ◦μA✲)|(Y,U(ψ◦μFY)◦ηY,B)||(mi,τi)✲||(Xi,αi,Ai)|isa(Generating,Mono-
Source)-factorizationofS(seethediagrambelow).
Xfi✲Xi
✲eim✲Y.....ηY.....❄
YUF....UμFY....❄αi
U(AFY)Uα
✲Uψ.....i◦m
i..❄UUμA✲UB[ϕi,α
i◦m]iU(ψ◦μA)Uτi✲✲✲❄
UAUϕi✲UAi

Propositions3.1.5and3.1.7implythefollowing.

44

Proposition3.1.8.Suppose2,20,21hold.ThenthecategoryX∗(A)isessentially
algebraic.
Nowthelastproperty.
puLemmashoutfore3.1.9.ach2-Susoupprosecewi20thcoholdsdomaiandn(leA,tF(X,X)A.)beanX×A-object.ThenAhasa
Proof.Followsfrom20bythecanonicalconstructionofpushoutsthroughcoproductsand
coequalizers.
Lemma3.1.10.Suppose2,20hold.AnX∗(A)-morphism(X,α,A)(e,ϕ✲)(Y,β,B)is
anepimorphismiffbotheandϕareepimorphismsinXandArespectively.
Proof.Trhesufficiencyisclear.ForthenecessityshowthateisanepimorphisminXfirst.
LetY✲✲ZbeX-morphismssuchthatr◦e=s◦e.ByLemma3.1.9thereexistsa
sutpusho

FXF(r◦e✲)FZ
β◦e❄❄p
✲CBqinA.By20.(i)thereexistsamultiplecoequalizer(c,C)ofp◦ηZ◦s,q◦βandp◦ηZ◦r
(r,c◦q)
inA.Thefactthat|(Y,β,B)|✲✲|(Z,U(c◦p)◦ηZ,C)|areX∗(A)-morphismssuchthat
)q◦s,c((r,c◦q)◦(e,ϕ)=(s,c◦q)◦(e,ϕ)impliesr=s.
NowshowthatϕisanepimorphisminA.LetBτ✲✲CbeA-morphismssuchthat
ψτ◦ϕ=ψ◦ϕ.Then(Uτ◦β)◦e=Uτ◦Uϕ(id◦Yα,τ)=Uψ◦Uϕ◦α=(Uψ◦β)◦eimplies
Uτ◦β=Uψ◦β.Thefactthat|(Y,β,B)|✲✲|(Y,Uτ◦β,C)|areX∗(A)-morphisms
),ψid(Ywith(idY,τ)◦(e,ϕ)=(idY,ψ)◦(e,ϕ)impliesτ=ψ.
|−|Proposition3.1.11.Suppose2,20,21hold.ThenX∗(A)✲X×Apreserves
extremalepimorphisms.
Proof.Let(X,α,A)(e,ϕ✲)(Y,β,B)beanextremalX∗(A)-epimorphism.ByLemma3.1.10
itisenoughtoshowtheextremalcondition.LetXe✲Y=Xf✲Zm✲Ybea
factorizationofeinXwi(the,ϕ)amonomorphismm.T(hf,eϕn)β◦m◦f=β◦(m,eid=B)Uϕ◦αgives
afactorization(X,α,A)✲(Y,β,B)=(X,α,A)✲(Z,β◦m,B)✲(Y,β,B)of
(e,ϕ)inX∗(A)withamonomorphism(m,idB)thatimpliesmisanX-isomorphism.Let

45

Aϕ✲B=Aτ✲Cψ✲BbeafactorizationofϕinAwithamonomorphismψ.By
Remark3.1.3thereexistsanX-morphismYd✲UCsuchthatthediagram
e✲YXdUτ◦α❄✛❄β
✲UBUCUψcommutes.Thenthereexistsafactorization
(X,α,A)(e,ϕ✲)(Y,β,B)=(X,α,A)(e,τ✲)(Y,d,C)(idY,ψ✲)(Y,β,B)
of(e,ϕ)inX∗(A)withamonomorphism(idY,ψ)thatimpliesψisanA-isomorphism.
Nowthepromisedresult.
Proposition3.1.12.Suppose2,20,21hold.Thenthecategory(X∗(A),|−|)is
algebraic.
Proof.FollowsfromPropositions3.1.8and3.1.11.
Nowshowthatrequirements20,21arealsonecessaryforthecategoryX∗(A)tobe
algebraicoverX×A.
Proposition3.1.13.Suppose2holds.If(X∗(A),|−|)isalgebraic,then20,21hold.
Proof.20.(i):LetS=(Aϕi✲Ai)i∈IbeasourceinA.By2getasourceSˆ=
((UFUA,UA,A)(UFUϕi,ϕi✲)(UFUAi,UAi,Ai))i∈IinX∗(A).SinceX∗(A)isalgebraic
thereexistsan(ExtrEpi,Mono-Source)-factorizationofSˆinX∗(A).Apply|−|andget
an(ExtrEpi,Mono-Source)-factorizationofSinA.
20.(ii):Let(X,A)beanX×A-object.SinceX∗(A)isalgebraicthefunctor|−|isad-
jointandthenthereexistsauniversalarrow(X,A)(e,ϕ✲)|(Y,β,B)|.Defineτ=β◦e.Show
thatthesinkS=((ϕ,τ),B)isacoproductofA,FXinA.LetT=(Aψ1✲C✛ψ2FX)
beasinkinA.Then(X,A)(Uψ2◦ηX,ψ1✲)|(UC,idUC,C)|isa|−|-structuredarrowand
thereforethereexistsauniqueX∗(A)-morphism(Y,β,B)(g,ψ✲)(UC,idUC,C)suchthat
amiagrdeth(X,A)(e,ϕ)✲|(Y,β,B)|
(Uψ2◦η|(g,ψ)|
❄X,ψ1)✲|(UC,idUC,C)|

46

commutes.ThenU(ψ◦τ)◦ηX=Uψ◦β◦e=g◦e=Uψ(2Uψ◦η◦Xβ,ψ)impliesψ◦τ=ψ2and
Xth∗er(Aef)or-emoψrphi◦Ssm.=TS.inceSuppUψose◦βψ◦e◦S==UψT.T◦ηhe,tnh|(enY,|β(,UψB)|◦β,ψ)|◦✲(e,|(ϕUC)=,i(dUψUC,C◦η)|is,ψan)
1X2X2andtherefore(Uψ◦β,ψ)=(g,ψ)impliesψ=ψ.
fi21:LetS=((fiX,Ffi)✲Xi)i∈IbeasourceinX.Withthehelpof2getasource
Sˆ=((X,ηX,FX)✲(Xi,ηXi,FXi))i∈IinX∗(A).Nowproceedasin20.(i).
Propositions3.1.12and3.1.13togetherimplythefollowing.
Proposition3.1.14.Suppose2holds.Thefollowingareequivalent:
(i)Thecategory(X∗(A),|−|)isalgebraicoverX×A;
hold.21,20(ii)Considersomeexamples.ByCorollary15.17in[2]eachofthecategoriesJCPos,Top,
Prost,Cat,Grpis(Epi,Mono-Source)-factorizable.TogetherwithProposition3.1.14
thatimpliesthefollowingresult.
Proposition3.1.15.ThecategoriesSet∗(JCPos),Set∗(Top),Prost∗(Top),Set∗(Cat)
andSet∗(Grp)arealgebraic.
NoticethatthecategorySet∗(SetRel)isnotalgebraicasshowsthefollowingremark.
Remark3.1.16.ThecategorySetReldoesnothave(Epi,Mono-Source)-factorizations.
Thiscanbeseenasfollows.By[48]SetRelhascoproductsbutisnotcocomplete.The
existenceof(Epi,Mono-Source)-factorizationsimpliestheexistenceofcoequalizersand
thereforecocompletenessofSetRel.
Proposition3.1.7andTheorem17.10in[2]implythefollowingresult.
Proposition3.1.17.Suppose2,20,21hold.ThenX∗(A)|−|✲X×Aisan(Ex-
tremallyGenerating,Mono-Source)-functor.

Thenextlemmaclarifiesthenatureof(extremally)generating|−|-structuredarrows.
)e,ϕ(struLemmacturedarr3.1.18.ow.TSuheppfoloselowi2,ngar20e,equ21ivaleholdnt:andlet(X,A)✲|(Y,β,B)|bea|−|-
(i)((e,ϕ),(Y,β,B))is(extremally)|−|-generating;
e(ii)X✲(β,Yϕ)isan(extremal)epimorphisminXandtheU,idA-structuredarrow
(Y,A)✲U,idA(B)is(extremally)U,idA-generating.

47

Proof.(i)=⇒(ii):ThefactthatXe✲Yisan(extremal)epimorphisminXcanbe
shownwiththehelpofproofsofLemma3.1.10andProposition3.1.11.Showthesecond
property.LetBτ✲✲CbeA-morphismssuchthatU,idA(τ)◦(β,ϕ)=U,idA(ψ)◦
ψ),τid((β,ϕ).Thefactthat|(Y,β,B)|Y✲✲|(Y,Uτ◦β,C)|areX∗(A)-morphismssuchthat
),ψid(Y|(idY,τ()β,|◦ϕ)(e,ϕ)=|(idY,ψ)|◦(e,(f,ϕτ))impliesτ=ψU,.idN(oψw)theextremalcondition.Let
(Y,A)✲U,idA(B)=(Y,A)✲U,idA(C)A✲U,idA(B)beafactorization
dwithanA-monomorphismψ.By21thereexistsanX-morphismY✲UCsuchthat
amiagrdethe✲YXde◦f❄✛❄β
✲UBUCUψcommutes.Thenthereexistsafactorization
(X,A)(e,ϕ✲)|(Y,β,B)|=(X,A)(e,τ✲)|(Y,d,C)||(idY,ψ✲)||(Y,β,B)|
withanX∗(A)-monomorphism(idY,ψ)thatimpliesψisanA-isomorphism.
(ii)=⇒(i):Thefactthat(X,A)(e,ϕ)✲|(Y,β,B)|is|−|-generatingfollowsfrom
Lemma3.1.6.Showtheextremalcondition.Let
(X,A)(e,ϕ✲)|(Y,β,B)|=(X,A)(f,τ✲)|(Z,γ,C)||(m,ψ✲)||(Y,β,B)|
beafactorizationwithanX∗(A)-monomorphism(m,ψ).ByProposition3.1.14thefunctor
|−|isadjointandthereforebothmandψaremonomorphismsinrespectivecategories.
ThefactthatXe✲Y=Xf✲Zm✲YisafactorizationofeinXimpliesthatmis
anX-isomorphism.Thatgivesafactorization
1−(Y,A)(β,ϕ✲)U,idA(B)=(Y,A)(γ◦(m),τ✲)U,idA(C)U,idA(ψ✲)U,idA(B)
andthereforeψisanA-isomorphism.
3.2AfactorizationstructureforsourcesonX∗(A)
InthissectionweshowthesufficientconditionsforX∗(A)tobean(Epi,ExtrMono-Source)-
.ycategorIntroducethefollowingstrongerversionsofrequirements20,21.
20ThecategoryAsatisfies20where(i)ischangedto

48

(i)Aisan(Epi,ExtrMono-Source)-category.
21ThecategoryXisan(Epi,ExtrMono-Source)-category.
Weshowthatrequirements20,21togetherwith2guaranteethatX∗(A)isan
(Epi,ExtrMono-Source)-category.Startwiththefollowinglemmas.
),ϕf(Lemma3.2.1.LetS=((X,α,A)i✲i(Xi,αi,Ai))i∈IbeasourceinX∗(A).If(Uϕi)i∈I
isamono-sourceinX,thenthesourceSisinitial.
)ψg,(Proof.Let|(Y,β,B)|✲|(X,α,A)|beanX×A-morphismsuchthat|S|◦(g,ψ)=|T|
(hi,ψi)
forasourceT=((Y,β,B)✲(Xi,αi,Ai))i∈IinX∗(A).ThenUϕi◦α◦g=Uϕi◦Uψ◦β
impliesα◦g=Uψ◦β.
Corollary3.2.2.Suppose2holdsandletS=((X,α,A)(fi,ϕ✲i)(Xi,αi,Ai))i∈Ibeasource
inX∗(A).If(ϕi)i∈Iisamono-sourceinA,thenthesourceSisinitial.
Lemma3.2.3.Suppose2,20holdandletS=((X,α,A)(fi,ϕi✲)(Xi,αi,Ai))i∈Ibea
sourceinX∗(A).If(fi)i∈Iand(ϕi)i∈Iareextremalmono-sourcesinXandArespectively,
thenSisanextremalmono-sourceinX∗(A).
Proof.FollowsfromLemma3.1.10.
NowshowthatthecategoryX∗(A)has(Epi,ExtrMono-Source)-factorizations.
Proposition3.2.4.Suppose2,20,21hold.ThenthecategoryX∗(A)is(Epi,
ExtrMono-Source)-factorizable.
Proof.LetS=((X,α,A)(fi,ϕi✲)(Xi,αi,Ai))i∈IbeasourceinX∗(A).Proposition3.1.7
givesafactorization
|(X,α,A)||(fi,ϕi✲)||(Xi,αi,Ai)|=|(X,α,A)|(f,ϕ✲)|(Y,β,B)||(mi,τi✲)||(Xi,αi,Ai)|
of|S|,wherethefirstcomponentis|−|-generatingandthesecondoneisanextremalmono-
sourceinX×A.ByCorollary3.2.2(f,ϕ)isanX∗(A)-morphismandthereforeanX∗(A)-
epimorphism.ByLemma3.2.3((mi,τi))i∈Iisanextremalmono-sourceinX∗(A).
ShowtheconverseofLemma3.2.3.
Lemma3.2.5.Suppose2,20,21holdandletS=((X,α,A)(fi,ϕi✲)(Xi,αi,Ai))i∈I
beasourceinX∗(A).ThenSisanextremalmono-sourceinX∗(A)iffboth(fi)i∈Iand
(ϕi)i∈Iareextremalmono-sourcesinXandArespectively.

49

Proof.InviewofLemma3.2.3itwillbeenoughtoshowthenecessity.
LetS=((X,α,A)(fi,ϕi✲)(Xi,αi,Ai))i∈Ibeanextremalmono-sourceinX∗(A).By
Proposition3.1.14both(fi)i∈Iand(ϕi)i∈Iaremono-sourcesinrespectivecategories.Show
thatthemono-source(fi)i∈Iisextremal.LetXfi✲Xi=Xe✲Ygi✲Xibeafactor-
izationinXwithanepimorphisme.Proposition3.1.7givesa(Generating,Mono-Source)-
factorization(Y,A)(gi,ϕi✲)|(Xi,αi,Ai)|=(Y,A)(e,ϕ✲)|(Z,γ,C)||(mi,τi✲)||(Xi,αi,Ai)|.By
Corollary3.2.2|(X,α,A)|(e◦e,ϕ✲)|(Z,γ,C)|isanX∗(A)-morphismandthereforeanX∗(A)-
epimorphism.ThenS=((mi,τi))i∈I◦(e◦e,ϕ)impliesthateisanX-isomorphism.
Nowshowthatthemono-source(ϕi)i∈Iisextremal.LetAϕi✲Ai=Aϕ✲Bψi✲Ai
beafactorizationinAwithanepimorphismϕ.Proposition3.2.4givesan(Epi,Mono-
Source)-factorization
(X,Uϕ◦α,B)(fi,ψi✲)(Xi,αi,Ai)=(X,Uϕ◦α,B)(f,ϕ✲)(Z,γ,C)(mi,τi✲)(Xi,αi,Ai)
inX(A).Thefactsthat(f,ϕ◦ϕ)isanX∗(A)-epimorphismandS=((mi,τi))i∈I◦(f,ϕ◦ϕ)
implythatϕisanX∗(A)-isomorphism.
Nowthemainresult.
Proposition3.2.6.Suppose2,20,21hold.ThenthecategoryX∗(A)isan(Epi,
ExtrMono-Source)-category.
Proof.Inview(g,ofψ)Proposition3(f,.2.4ϕ)itwillbesufficienttoshowthediagonalizationproperty.
Let(Z,γ,C)✛(X,α,A)✲(Y,β,B)beX∗(A)-morphismswithanepimorphism
(f,ϕ),letM=((Z,γ,C)(fi,ϕi✲)(Zi,γi,Ci))i∈I,S=((Y,β,B)(gi,ψi✲)(Zi,γi,Ci))i∈Ibe
sourcesinX∗(A)withanextremalmono-sourceMsuchthatM◦(g,ψ)=S◦(f,ϕ).By
Lemma3.2.5both(fi)i∈Iand(ϕi)i∈Iareextremalmono-sourcesinrespectivecategories.
SinceX×Aisan(Epi,ExtrMono-Source)-categorythereexistsauniqueX×A-morphism
(Y,B)(d,d✲)(Z,C)suchthatthediagram
(X,A)(f,ϕ)✲(Y,B)
(g,ψ)(d,d)(gi,ψi)
❄✛❄(Z,C)(fi,ϕi)✲(Zi,Ci)
commutes.ByCorollary3.2.2(Y,β,B)(d,d✲)(Z,γ,C)istherequireddiagonal.
ByCorollary15.17in[2]eachofthecategoriesJCPos,Top,Prost,Cat,Grpis
an(Epi,ExtrMono-Source)-category.TogetherwithProposition3.2.6thatimpliesthe
followingresult.
Proposition3.2.7.ThecategoriesSet∗(JCPos),Set∗(Top),Prost∗(Top),Set∗(Cat)
andSet∗(Grp)are(Epi,ExtrMono-Source)-categories.

50

3.3CoalgebraicpropertiesofX∗(A)
InthissectionweconsiderthenecessaryandsufficientconditionsforthecategoryX∗(A)
tobecoalgebraic.
Introducethefollowingrequirements.

22ThecategoryAis(Epi-Sink,Mono)-factorizable.
23ThecategoryXhasthefollowingproperties:
(i)Xis(Epi-Sink,Mono)-factorizable;
(ii)ForeveryX×A-object(X,A)thereexistsaproductofX,UAinX.

Wearegoingtoshowthattheaforesaidrequirementsguaranteethatthecategory
(X∗(A),|−|)iscoalgebraic(cf.thedualofDefinition3.1.4).Sincethefirstpropertyis
provedinProposition3.1.5beginwiththesecondone.Consider|−|-cogeneratingarrows.
Lemma3.3.1.Let|(X,α,A)|(f,ϕ✲)(Y,B)bea|−|-costructuredarrow.Ifϕisamonomor-
phisminA,(X,(f,α))isamono-sourceinX,then((X,α,A),(f,ϕ))is|−|-cogenerating.
)τr,(Proof.Let(Z,γ,C)✲✲(X,α,A)beX∗(A)-morphismssuchthat(f,ϕ)◦|(r,τ)|=(f,ϕ)◦
)s,ψ(|(s,ψ)|.Thenϕ◦τ=ϕ◦ψimpliesτ=ψ.Further,f◦r=f◦sandα◦r=Uτ◦γ=
Uψ◦γ=α◦simplyr=s.
Proposition3.3.2.Suppose22,23hold.ThenX∗(A)|−|✲X×Ais(Epi-Sink,
Cogenerating)-factorizable.
Proof.LetS=(|(Xi,αi,Ai)|(fi,ϕ✲i)(X,A))i∈Ibea|−|-costructuredsink.LetAiϕi✲A=
Aiτi✲Bψ✲Abean(Epi-Sink,Mono)-factorizationof(ϕi)i∈IinA.Withthehelpof
aproduct(UB×X,(πUB,πX))ofUB,XinXgetasourceT=(XiUτi◦αi,fi✲UB×
X)i∈IinX.Finally,letXiUτi◦αi,fi✲UB×X=Xiei✲Ym✲UB×Xbean
(Epi-Sink,Mono)-factorizationofTinX.ByLemma3.3.1|(Xi,αi,Ai)|(fi,ϕi✲)(X,A)=

51

|(e,τ)|(π◦m,ψ)
|(Xi,αi,Ai)|ii✲|(Y,πUB◦m,B)|X✲(X,A)isan(Epi-Sink,Cogenerating)-
factorizationofS(seethediagrambelow).
Xifi✲X
✲✲eiπX◦m
✲Uτi◦αi,fY..πX
im....
✲....❄
αiUB.×X
..πUB....
..❄UB✲UψiUτ✲❄✲UAUAiUϕi

Propositions3.1.5and3.3.2implythefollowing.
Proposition3.3.3.Suppose22,23hold.ThenthecategoryX∗(A)isessentiallycoal-
c.iagebrNowthelastproperty.
Lemma3.3.4.Suppose23holdandlet(X,A)beanX×A-object.ThenXhasapullback
foreach2-sinkwithdomain(X,UA).
Proof.Followsfrom23bythecanonicalconstructionofpullbacksthroughproductsand
equalizers.
Lemma3.3.5.Suppose23holdandletUpreservemonomorphisms.AnX∗(A)-morp-
hism(X,α,A)(m,ϕ✲)(Y,β,B)isamonomorphismiffbothmandψaremonomorphismsin
XandArespectively.
Proof.Thesufficiencyisclear.ForthenecessityshowthatϕisamonomorphisminAfirst.
τLetC✲✲AbeA-morphismssuchthatϕ◦τ=ϕ◦ψ.ByLemma3.3.4thereexistsa
ψkacllbup

q✲XZα◦Uϕp❄❄✲UBUCU(ϕ◦τ)

52

inX.By23.(i)thereexistsamultipleequalizer(Z,e)ofUτ◦p,α◦qandUψ◦pinX.The
)e,τ◦q(factthat|(Z,p◦e,C)|✲✲|(X,α,A)|areX∗(A)-morphismssuchthat(m,ϕ)◦(q◦e,ψ)=
)e,ψ◦q((m,ϕ)◦(q◦e,τ)impliesψ=τ.
NowshowthatmisamonomorphisminX.LetZr✲✲XbeX-morphismssuchthat
sm◦r=m◦s.SinceUpreservesmonomorphisms,Uϕ◦α◦r=β◦m◦r=β◦m◦s=Uϕ◦α◦s
impliesα◦r=α◦s.Thefactthat|(Z,α◦r,A)|(r,idA✲✲)|(X,α,A)|areX∗(A)-morphisms
)ds,i(Awith(m,ϕ)◦(r,idA)=(m,ϕ)◦(s,idA)impliesr=s.
Proposition3.3.6.Suppose2,22,23hold.ThenX∗(A)|−|✲X×Apreserves
extremalmonomorphisms.
Proof.Let(X,α,A)(m,ϕ✲)(Y,β,B)beanextremalmonomorphisminX∗(A).Inviewof
Lemma3.3.5itisenoughtoshowtheextremalcondition.LetAϕ✲B=Aψ✲Cτ✲B
beafactorizationofϕinXwithanepimorphismψ.ThenUτ◦Uψ◦α=Uϕ◦α=β◦mgives
afactorization(X,α,A)(m,ϕ✲)(Y,β,B)=(X,α,A)(idX,ψ✲)(X,Uψ◦α,C)(m,τ✲)(Y,β,B)
of(m,ϕ)inX∗(A)withanepimorphism(idX,ψ)thatimpliesψisanA-isomorphism.Let
Xm✲Y=Xe✲Zf✲YbeafactorizationofminXwithanepimorphisme.By
22thereexistsanA-morphismFYd✲Asuchthatthediagram
Fe✲FZFXdf◦βαA❄✛✲B❄
ϕcommutes.Thenthereexistsafactorization
(X,α,A)(m,ϕ✲)(Y,β,B)=(X,α,A)(e,idA✲)(Z,Ud◦ηZ,A)(f,ϕ✲)(Y,β,B)
of(m,ϕ)inX∗(A)withanepimorphism(e,idA)thatimplieseisanX-isomorphism.
Nowthepromisedresult.
Proposition3.3.7.Suppose2,22,23hold.Thenthecategory(X∗(A),|−|)iscoal-
c.iagebrProof.FollowsfromPropositions3.3.3,3.3.6.
Nowshowthatrequirements22,23arealsonecessaryforthecategoryX∗(A)tobe
coalgebraicoverX×A.
Proposition3.3.8.Suppose2holds.If(X∗(A),|−|)iscoalgebraic,then22,23hold.

53

Proof.Requirements22and23.(i)canbeshownasinProposition3.1.13.
23.(ii):Let(X,A)beanX×A-object.SinceX∗(A)iscoalgebraicthefunctor|−|
isco-adjointandthenthereexistsaco-universalarrow|(Y,β,B)|(e,ϕ✲)(X,A).Define
k=Uϕ◦β.ShowthatthesourceS=(Y,(e,k))isanX-productofX,UA.LetT=
(X✛f1Zf2✲UA)beanX-source.Then|(Z,ηZ,FZ)|(f1,f✲2)(X,A)isa|−|-costructured
arrowandthusthereexistsauniqueX∗(A)-morphism(Z,ηZ,FZ)(g,ψ✲)(Y,β,B)suchthat
amiagrdeth|(Z,ηZ,FZ)|
|(g,ψ)|(f1,f2)
✲❄|(Y,β,B)|(e,ϕ)✲(X,A)
commutes.ThenUϕ◦β◦g=Uϕ◦Uψ◦ηZ=f2impliesT=S◦g.SupposeT=S◦g.
Then|(Z,ηZ,FZ)|(g,β◦g✲)|(Y,β,B)|isanX∗(A)-morphism.Sinceϕ◦β◦h=f2,then
(e,ϕ)◦|(g,β◦g)|=(f1,f2)andtherefore(g,β◦g)=(g,ψ)impliesg=g.
Propositions3.3.7and3.3.8togetherimplythefollowing.
Proposition3.3.9.Suppose2holds.Thefollowingareequivalent:
(i)Thecategory(X∗(A),|−|)iscoalgebraicoverX×A;
hold.23,22(ii)Considersomeexamples.BythedualofCorollary15.17in[2]eachofthecategories
JCPos,Top,Prost,Cat,Grpis(Epi-Sink,Mono)-factorizable.TogetherwithProposi-
tion3.3.9thatimpliesthefollowingresult.
Proposition3.3.10.ThecategoriesSet∗(JCPos),Set∗(Top),Prost∗(Top),Set∗(Cat)
andSet∗(Grp)arecoalgebraic.
NoticethatthecategorySet∗(SetRel)isnotcoalgebraicasshowsthefollowingremark.
Remark3.3.11.ThecategorySetReldoesnothave(Epi-Sink,Mono)-factorizations.
Thisfollowsfromthefactthatthefunctor
SetRelF✲SetRelop:Aρ✲B→Bρ−✲1A
isanisomorphism(cf.[6])andRemark3.1.16.
Proposition3.3.2andthedualofTheorem17.10in[2]implythefollowingresult.
|−|Proposition3.3.12.Suppose22,23hold.ThenX∗(A)✲X×Aisan(Epi-Sink,
ExtremallyCogenerating)-functor.

54

Thenextlemmaclearsthenatureof(extremally)cogenerating|−|-structuredarrows.
Lemma3.3.13.Suppose22,23holdandlet|(Y,β,B)|(e,ϕ✲)(X,A)bea|−|-costructured
arrow.Thefollowingareequivalent:
(i)((Y,β,B),(e,ϕ))is(extremally)|−|-cogenerating;
(ii)Bϕ✲Aisan(extremal)A-monomorphism,thesourceS=(UB✛βYe✲X)
inXisan(extremal)mono-source.
ϕ✲bPreooshof.w(ni)=wit⇒ht(iihe):helTphoeffaproctotfshaotfALemma3B.3.is5aannd(eProxtrepomasitl)ionmo3.3no.6.moShorphismwtheinseAcondcan
rproperty.LetZs✲✲YbeX-morphismssuchthatS◦r=S◦s.Thefactthat|(Z,β◦
r,B)|(r,idB✲✲)|(Y,β,B)|areX∗(A)-morphismssuchthat(e,ϕ)◦|(r,idB)|=(e,ϕ)◦|(s,idB)|
)ds,i(Bimpliesr=s.Nowtheextremalcondition.LetT=(UB✛f1Zf2✲X)beasourcein
Xsuchthat(e,ϕS)✲=T◦gwithanX-epimo|(g,irphidB)✲|smg.Thenthe(fre2,ϕt✲)hereexistsafactoriz∗ation
|(Y,β,B)|(X,A)=|(Y,β,B)||(Z,f1,B)|(X,A)withanX(A)-
epimorphism(g,idB)thatimpliesgisanX-isomorphism.
(ii)=⇒(i):Thefactthat|(Y,β,B)|(e,ϕ✲)(X,A)is|−|-cogeneratingfollowsfrom
Lemma3.3.1.Showtheextremalcondition.Let
|(Y,β,B)|(e,ϕ✲)(X,A)=|(Y,β,B)||(g,τ✲)||(Z,γ,C)|(f,ψ✲)(X,A)
beafactorizationwithanX∗(A)-epimorphism(g,τ).ByProposition3.3.3thefunctor|−|
isco-adjointanϕdthereforebτothganψdτareepimorphismsinrespectivecategories.The
factthatB✲A=B✲C✲AisafactorizationofϕinAimpliesτisanA-
isomorphism.ThatgivesasourceT=(UBU✛(τ−1)◦γZf✲X)inXsuchthatS=T◦g
thatimpliesgisanX-isomorphism.

3.4AfactorizationstructureforsinksonX∗(A)

InthissectionweshowthesufficientconditionsforX∗(A)tobean(ExtrEpi-Sink,Mono)-
.ycategorIntroducethefollowingweakerversionofrequirement2aswellasstrongerversionsof
requirements22,23.
U2ThefunctorA✲Xpreservesmonomorphisms.

22ThecategoryAisan(ExtrEpi-Sink,Mono)-category.

55

23ThecategoryXsatisfies23where(i)ischangedto
(i)Xisan(ExtrEpi-Sink,Mono)-category.
Weshowthatrequirements22,23togetherwith2guaranteethatthecategory
X∗(A)isan(ExtrEpi-Sink,Mono)-category.Startwiththefollowinglemmas.
Lemma3.4.1.LetS=((Xi,αi,Ai)(fi,ϕi✲)(X,α,A))i∈IbeasinkinX∗(A).If(fi)i∈Iis
anepi-sinkinX,thenthesinkSisfinal.
Proof.ConsiderthedualoftheproofofLemma3.2.1.
Lemma3.4.2.Suppose2,23hold.LetS=((Xi,αi,Ai)(fi,ϕi✲)(X,α,A))i∈Ibeasink
inX∗(A).If(fi)i∈Iand(ϕi)i∈Iareextremalepi-sinksinXandArespectively,thenSis
anextremalepi-sinkinX∗(A).
Proof.FollowsfromLemma3.3.5.
NowshowthatthecategoryX∗(A)has(ExtrEpi-Sink,Mono)-factorizations.
Proposition3.4.3.Suppose2,22,23hold.ThenthecategoryX∗(A)is(ExtrEpi-
Sink,Mono)-factorizable.
Proof.ConsidertheproofofProposition3.2.4.
ShowtheconverseofLemma3.4.2.
Lemma3.4.4.Suppose2,22,23hold.AsinkS=((Xi,αi,Ai)(fi,ϕi✲)(X,α,A))i∈I
inX∗(A)isanextremalepi-sinkiffboth(fi)i∈Iand(ϕi)i∈Iareextremalepi-sinksinX
andArespectively.
Proof.InviewofLemma3.4.2itwillbeenoughtoshowthenecessity.
LetS=((Xi,αi,Ai)(fi,ϕi✲)(X,α,A))i∈Ibeanextremalepi-sinkinX∗(A).ByPropo-
sition3.3.3both(fi)i∈Iand(ϕi)i∈Iafreepi-sinksingrespectimvecategories.Showthatthe
epi-sink(fi)i∈Iisextremal.LetXii✲X=Xii✲Y✲XbeafactorizationinX
withamonomorphismm.Proposition3.4.3givesan(ExtrEpi-Sink,Mono)-factorization
(Xi,αi,Ai)(gi,ϕi✲)(Y,α◦m,A)=(Xi,αi,Ai)(ei,τi✲)(Z,γ,C)(m,ϕ✲)(Y,α◦m,A).Thefacts
that(m◦m,ϕ)isanX∗(A)-monomorphismandS=(m◦m,ϕ)◦((ei,τi))i∈Iimplythat
misanX-isomorphism.
Nowshowthattheepi-sink(ϕi)i∈Iisextremal.LetAiϕi✲A=Aiψi✲Bϕ✲Abea
factorizationinAwithamonomorphismϕ.ByProposition3.3.2wehavean(ExtrEpi-Sink,
Cogenerating)-factorization
|(Xi,αi,Ai)|(fi,ψi✲)(X,B)=|(Xi,αi,Ai)||(ei,τi✲)||(Z,γ,C)|(m,ψ✲)(X,B)
inX(A).Lemma3.4.1impliesthat(m,ϕ◦ψ)isanX∗(A)-morphismandthereforeanX-
monomorphism.ThenS=(m,ϕ◦ψ)◦((ei,τi))i∈IimpliesthatϕisanA-isomorphism.

56

Nowthemainresult.
Proposition3.4.5.Suppose2,22and23hold.ThenthecategoryX∗(A)isan
(ExtrEpi-Sink,Mono)-category.
Proof.ConsidertheproofofProposition3.2.6.
BythedualofCorollary15.17in[2]eachofthecategoriesJCPos,Top,Prost,Cat,
Grpisan(ExtrEpi-Sink,Mono)-category.TogetherwithProposition3.4.5thatimplies
thefollowingresult.
Proposition3.4.6.ThecategoriesSet∗(JCPos),Set∗(Top),Prost∗(Top),Set∗(Cat)
andSet∗(Grp)are(ExtrEpi-Sink,Mono)-categories.

3.5MonadicpropertiesofX∗(A)
InthissectionweconsiderthenecessaryandsufficientconditionsforthecategoryX∗(A)
tobemonadicoverX×A.
Introducethefollowingmodificationofrequirement20:
24ForeveryX×A-object(X,A)thereexistsacoproductofA,FXinA.
Wearegoingtoshowthattheaforesaidrequirementguaranteesthatthecategory
(X∗(A),|−|)ismonadicoverX×A.
Proposition3.5.1.Suppose2holds.ThenthefunctorX∗(A)|−|✲X×Aisadjointiff
holds.24FXA(PrX,ooAf.)bFeorantheXn×ecesAs-obityject.consBidyer24thethperroeofexiofstsPraopcoosproitionduc3tS.1.13.=((Fμor,μthes)u,AfficienFXcyl)oetf
AandFXinA.Showthatthe|−|-structuredarrow
(X,A)(e,ψ✲)|(X,ˆˆα,Aˆ)|=(X,A)(idX,μA✲)|(X,UμFX◦ηX,AFX)|.
isuniversal.Let(X,A)(f,ϕ✲)|(Y,β,B)|beanother|−|-structuredarrow.Thenthereexists
auniqueA-morphismAˆτ✲Bsuchthatthediagram
β✛FYBϕ✲✻τ✻Ff
AμA✲Aˆ✛μFXFX

57

commutes.ThenUτ◦αˆ=Uτ◦UμFX◦ηX=Uβ◦UFf◦ηX=Uβ◦ηY◦f=β◦fimplies
(X,ˆˆα,Aˆ)(f,τ✲)(Y,β,B)isanX∗(A)-morphismsuchthat|(f,τ)|◦(e,ψ)=(f,ϕ).Suppose
(X,ˆˆα,Aˆ)(f,τ✲)(Y,β,B)isanX∗(A)-morphismsuchthat|(f,τ)|◦(e,ψ)=(f,ϕ).Then
f=fandthereforeU(τ◦μFX)◦ηX=Uτ◦αˆ=β◦f=Uβ◦ηY◦f=U(β◦Ff)◦ηX
impliesτ◦μFX=β◦Ff.Thenτ◦S=τ◦Sandthereforeτ=τ.
Proposition3.5.2.ThefunctorX∗(A)|−|✲X×Acreatescoequalizers.
Proof.Let(X,α,A)(r,τ✲✲)(Y,β,B)beX∗(A)-morphismsandlet(Y,B)(e,ϕ✲)(Z,C)bea
)s,ψ(coequalizerof|(r,τ)|and|(s,ψ)|inX×A.Then(e,Z)isacoequalizerofr,sinXand(ϕ,C)
isacoequalizerofτ,ψinA.ThenUϕ◦(β◦r)=Uϕ◦Uτ◦α=Uϕ◦Uψ◦α=Uϕ◦(β◦s)implies
theexistenceofauniqueX-morphismZγ✲UCsuchthat(Y,β,B)(e,ϕ✲)(Z,γ,C)isan
X∗(A)-morphism.Showthat((e,ϕ),(Z,γ,C))isacoequalizerof(r,τ),(s,ψ)inX∗(A).Let
(Y,β,B)(g,✲)(Z,γ,C)beanX∗(A)-morphismsuchthat(g,)◦(r,τ)=(g,)◦(s,ψ).
ThenthereexistsauniqueX×A-morphism(Z,C)(ˆg,ˆ✲)(Z,C)suchthatthediagram
|(Y,β,B)||(e,ϕ)|✲|(Z,γ,C)|
|(g,)|✲❄(ˆg,ˆ)
|(Z,γ,C)|
commutes.ByLemma3.4.1itfollowsthat|(Z,γ,C)|(ˆg,ˆ✲)|(Z,γ,C)|isanX∗(A)-
sm.rphimoNowthemainresult.
Proposition3.5.3.Suppose2holds.Thefollowingareequivalent:
(i)Thecategory(X∗(A),|−|)ismonadicoverX×A;
holds.24(ii)Proof.FollowsfromPropositions3.5.1,3.5.2andTheorem20.17in[2].
Theorem20.17in[2]impliesthattherequiredmonadT=(T,ˆη,ˆμˆ)onthecategory
X×Acanbedefinedasfollows:
ˆ(i)thefunctorX×AT✲X×Aisgivenby
(X,A)(f,ϕ✲)(Y,B)→(X,AFX)(f,ϕFf✲)(Y,BFY);

58

(ii)thenaturaltransformationidX×Aηˆ✲Tˆisgivenby
(X,A)ηˆ(X,A)=(idX,μA✲)(X,AFX);
(iii)thenaturaltransformationTˆ2μˆ✲Tˆisgivenby
(X,(AFX)FX)μˆ(X,A)=(idX,[idAFX,μFX✲])(X,AFX).
Thendefinetheconcreteisomorphism(X∗(A),|−|)H✲((X×A)T,UT)asfollows
(X,α,A)(f,ϕ✲)(Y,β,B)→((X,A),(idX,[idA,α]))(f,ϕ✲)((Y,B),(idY,[idB,β])).
ThefactthateachofthecategoriesJCPos,SetRel,Top,Prost,Cat,Grphas
coproductstogetherwithProposition3.5.3implythefollowingresult.
Proposition3.5.4.ThecategoriesSet∗(JCPos),Set∗(SetRel),Set∗(Top),Set∗(Cat),
Prost∗(Top)andSet∗(Grp)aremonadic.
Remark3.5.5.CoproductsinSetRelcanbeconstructedasinSetusingthegraphsof
thecanonicalinjections.

3.6SomeremarksonthemonadT
InthelastsectionweconstructedamonadTonthecategoryX×A.Thissectionisdevoted
tosomeofitsproperties.Startwiththefollowingpropositionwhichwillhavealotofuseful
corollaries.
Proposition3.6.1.ThefunctorTˆpreservescolimits.
Proof.SupposeID✲X×AisafunctorwithD(i)=(Xi,Ai)foreveryi∈Ob(I).
(fi,ϕi)
LetC=((Xi,Ai)✲(X,A))i∈Ob(I)beacolimitofD.ShowthatthesinkTˆC=
((Xi,AiFXi)(fi,ϕiFfi✲)(X,AFX))i∈Ob(I)isacolimitofTˆ◦D.SupposeCˆ=
((Xi,AiFXi)(fˆi,ϕˆi✲)(X,ˆAˆ))i∈Ob(I)isanaturalsinkforTˆ◦D.SincethesinksπX(C)
andπA(C)arecolimitsofthefunctorsπX◦DandπA◦Drespectively,onehastheunique
θψfX-morphismX✲XˆaswellastheuniqueA-morphismsA✲AˆandFX✲Aˆ
suchthatforeveryi∈Ob(I)thefollowingdiagramscommute:
XˆAψ✲Aˆ✛θFX
fˆi✲✻fϕ✻✻ϕˆi✻Ff
iiXif✲XAiμ✲AiFXiμ✛FXi
FXAiii

59

ϕThelasttwomorphismsgivetheuniqueA-morphismAFX✲Aˆsuchthatthediagram
ˆA✛✲✻θψϕAμ✲AFXμ✛FX
FXAcommutes.Straightforwardcomputationsshowthat(X,AFX)(f,ϕ✲)(X,ˆAˆ)isthere-
quireduniqueX×A-morphismsuchthatCˆ=(f,ϕ)◦TˆC.
Nowthecorollaries.
Corollary3.6.2.ThemonadThasfiniterank.
Corollary3.6.3.IfthecategoriesX,Aarebothcompleteandcocomplete,thenthecategory
(X×A)Tisbothcompleteandcocomplete.
Proof.SeeProposition4.3.6in[9].
Corollary3.6.4.ThecategoriesSet∗(JCPos),Set∗(Cat),Set∗(Grp),Set∗(Top)and
Prost∗(Top)arebothcompleteandcocomplete.
Proof.FollowsfromProposition3.5.4andCorollary3.6.3.
Corollary3.6.5.IfthecategoriesX,Aarelocallypresentable,thenthecategory(X×A)T
islocallypresentable.
Proof.SeeTheorem2.78in[3].
Corollary3.6.6.ThecategoriesSet∗(Cat)andSet∗(Grp)arelocallyfinitelypresentable.
Proof.ByExamples5.2.2.(a)and5.2.2.(f)in[9]bothGrpandCatarelocallyfinitely
presentable.TheresultthenfollowsfromCorollary3.6.5.

4Chapter

Ontopologicalfuzzificationstructuresofalgebraicand

In[45,11,32]differentfuzzystructuressuchasfuzzytopologicalspacesandfuzzygroups
areconsidered.AllofthemuseimplicitlyGoguen’scategorySet(L)ofL-fuzzysets[18].
InthesecondchapterageneralizationofSet(L)intheformofX(A)wasintroduced.The
nextstepwouldbetointroduceageneralizedfuzzificationprocedureoverthecategory
X(A)(see,e.g.,[51]).Inthischapterweintroducesuchaprocedureusingthecategories
Alg(T)andSpa(T)of[2].

4.1Fuzzificationmachineryforalgebraicstructures

Inthissectionweconsiderafuzzificationschemeforalgebraicstructuresoverthecategory
X(A).Inordernottorestrictourselvestoaparticularclassofalgebraswewillusethe
objectsofthecategoryAlg(T)(see,e.g.,[2])asasufficientgeneralizationofthenotionof
abstractalgebra.Startwiththefollowingdefinition.
Definition4.1.1.LetXT✲XbeafunctorandletBbeasubcategoryofthecategory
Alg(T).ForafixedAlg(T)-object(UA,τ)definethecategoryBτ(A)asfollows:
Objects:Triples(X,x,α)suchthat
(i)(X,x)isaB-object;
(ii)(X,α)isanX(A)-object;
(iii)τ◦Tαα◦x.
Morphisms:X-morphisms(X,x,α)f✲(Y,y,β)suchthat

60

f(i)(X,x)✲(Y,y)isaB-morphism;
f(ii)(X,α)✲(Y,β)isanX(A)-morphism.
WeconsiderthecategoryBτ(A)asaconcretecategoryoverBinthefollowingway.
|−|Definition4.1.2.DefinetheforgetfulfunctorBτ(A)✲Basfollows:
|(X,x,α)f✲(Y,y,β)|=(X,x)f✲(Y,y).

61

WearegoingtoshowasufficientconditionforBτ(A)tobetopologicaloverB.
FromnowonweassumethatthecategoryX(A)istopological.Introducethefollowing
requirement:
25ForeveryX-objectXthemapX(X,UA)τ◦T(−✲)X(TX,UA)isorder-preserving.
Proposition4.1.3.Suppose25holds.ThenBτ(A)istopologicaloverB.
fi(αi◦fi).Itwillbeenoughtoshowthat(X,x,α)isaBτ(A)-object.
Proof.LetS=((X,x)✲|(Xi,xi,αi)|)i∈Ibea|−|-structuredsource.Defineα=
I∈iτ◦Tα=τ◦T((αi◦fi))τ◦T(αi◦fi)(αi◦fi◦x)=(αi◦fi)◦x=α◦x.
Thefactthatτ◦T(αi◦fi)=τ◦Tαi◦Tfiαi◦xi◦Tfi=αi◦fi◦ximplies
i∈Ii∈Ii∈Ii∈I
WearegoingtoconsideraparticularcategoryBτ(A)andthereforeintroducethefol-
lowingrequirement.
26ThecategoryXhascoproductsandfiniteproducts.
RecallthefollowingnotionfromChapter2.GivenafamilyofX(A)-objects(Xi,αi)
onecanformanX(A)-object(Xi,∗αi)with∗αi=αi◦πXi.
i∈Ii∈Ii∈Ii∈I
fifamiliesofX(A)-objectsandX-morphismsrespectively.Then(∗αi)◦(fi)=∗(αi◦fi).
Lemma4.1.4.Suppose26holds.Let((Xi,αi))i∈Iand(Xi✲Xi)i∈Ibeset-indexed
i∈Ii∈Ii∈I
Proof.(∗αi)◦(fi)=(αi◦πXi)◦(fi)=(αi◦πXi◦(fi))=(αi◦fi◦πX)=
i∈Ii∈Ii∈Ii∈Ii∈Ii∈Ii∈Ii

i∈∗I(αi◦fi).
Lemma4.1.5.Suppose26holds.Let((Xi,αi))i∈Iand((Xi,βi))i∈Ibeset-indexedfami-
liesofX(A)-objects.Ifαiβifori∈I,then∗αi∗βi.
i∈Ii∈I

62Wearegoingtoconsiderthefollowingexample.
Example4.1.6.Suppose26holds.LetΩn=(ni)i∈Ibeafamilyofnatunralnumbers
indexedbyasetISn.Fiori∈IdefineXSi✲X:Xf✲Y→Xnifi✲Yni.Set
]id∗[XT✲X=Xi∈I✲XandtakeTUAτ✲UA=TUAniUA✲UA.
FromnowoneweareworkingwithExample4.1.6.Introducethefollowingdefinitions.
Definition4.1.7.LetS=(Xiαi✲UA)i∈IandT=(Xiβi✲UA)i∈IbesinksinX.
ThenSTprovidedthatαiβifori∈I.
Definition4.1.8.AsinkSinXissaidtobea-epi-sinkprovidedthatα◦Sβ◦S
impliesαβ.
Lemma4.1.9.IfX(A)istopological,thenevery-epi-sinkisanepi-sink.
Proposition4.1.10.Suppose26holds.IfcoproductsinXare-epi-sinks,thenτA◦T(−)
isorder-preserving.
Proof.TakeXα✲✲UAwithαβandletSbeacoproductof(Xni)i∈IinX.Then
βτ◦Tα=[n∗iidUA]◦(i∈Iαni)=[(n∗iidUA)◦αni]=[n∗iα]andsimilarforβ.Thefactthat
n∗iα∗niβfori∈Iimplies[n∗iα]◦S[n∗iβ]◦Sandtherefore[n∗iα][n∗iβ].
WearegoingtoshowthenecessaryandsufficientconditionforasinkinXtobea-
epi-sink.Startwiththefollowing:givenasinkS=(Xifi✲X)i∈IinXsetFS◦S◦=
◦i∈IFfi◦fi.
Proposition4.1.11.LetX(A)betopologicaloverX×AandletS=(Xifi✲X)i∈Ibe
asinkinX.Thefollowingareequivalent:
(i)Sisa-epi-sink;
(ii)FS◦S◦=idFX.
Proof.(i)=⇒(ii):SinceFfi◦fi◦=⇒idFXfori∈I,thenFS◦S◦=⇒idFX.Forthe
◦◦Ufconj◦v◦erηsXe:◦ηfjX◦fj(=UFUFffji◦◦ηUfXij◦)◦ηXUF◦ffjj◦U=((fj◦UFf(jFf)◦iη◦Xfji◦))=◦ηUFX◦fjf◦jUfjU◦(UFfFfj◦iη◦Xfji◦)=◦ηUFX◦ffjj◦.
i∈Ii∈Ii∈I
I∈iThenηXU(Ffi◦fi◦)◦ηXandthereforeidFX=⇒FS◦S◦.
α(ii)=⇒(i):LetXβ✲✲UAbesuchthatα◦Sβ◦S.ThenA◦Fα=A◦Fα◦
(FS◦S◦)=A◦Fα◦FS◦S◦=⇒A◦Fβ◦FS◦S◦=A◦Fβandtherefore
.βα

63

Asaconcreteexampleconsiderthefollowingproposition.
Proposition4.1.12.ConsiderthecategorySet(JCPos)(resp.Set(SetRel)).Forasink
S=(Xifi✲X)i∈IinSetthefollowingareequivalent:
(i)Sisanepi-sink;
(ii)Sisan-epi-sink.
ThecategorySet(JCPos):IfS∈FX,thenFS◦S◦(S)=fi◦fi−1[S]=Sand
Proof.InviewofLemma4.1.9show(i)=⇒(ii).
I∈ithereforeSisa-epi-sinkbyProposition4.1.11.
xforsomexi∈Xi}andthereforeFS◦S◦=ρfi◦ρf−i1={(x,x)|x∈X}=idFX.
ThecategorySet(SetRel):Ifi∈I,thenFfi◦fi◦=ρfi◦ρf−i1={(x,x)|fi(xi)=
I∈iInviewofProposition4.1.11introducethefollowingdefinition.
asinkinX.WesaythatSis-compatibleprovidedthatFS◦S◦=idFX.
Definition4.1.13.LetX(A)betopologicaloverX×AandletS=(Xifi✲X)i∈Ibe
Proposition4.1.14.Suppose26holds.LetX(A)betopologicalwith-compatibleco-
products,letAbeanA-objectandletT,τbesuchasdefinedinExample4.1.6.IfBisa
subcategoryofAlg(T),thenthecategoryBτ(A)istopologicaloverB.
Proof.FollowsfromPropositions4.1.3,4.1.10,4.1.11.
Corollary4.1.15.LetSet(A)betopologicalwith-compatiblecoproductsandletAbean
A-object.IfBisafinitaryquasivariety,thenBτ(A)istopologicaloverB.
Corollary4.1.15impliesthatincaseofthecategorySet(JCPos)(resp.Set(SetRel))
eachofthecategoriesVec(A),R-Mod(A),Ab(A),Grp(A),Mon(A),Sgr(A),Rng(A),
Lat(A)istopological(see,e.g.,[2]).
Considerthefollowingexample.
Example4.1.16(ContinuationofExample4.1.6).LetBbeasubcategoryofAlg(T).If
(X,x,α)isaBτ(A)-object,then[n∗iα]α◦xandthereforen∗iα=[n∗iα]◦μiα◦x◦μifor
i∈I.Considertwocases.
(i)Ifni=0,thenα◦x◦μi=.
(ii)Ifni=1,thenαα◦x◦μi.Suppose(x◦μi)◦(x◦μi)=idX.Thenα◦x◦μi
α◦x◦μi◦x◦μi=αimpliesα=α◦x◦μi.
Example4.1.16generalizesthewell-knownresultsforfuzzygroups.

64

4.2Fuzzificationmachineryfortopologicalstructures

Inthissectionweconsiderafuzzificationschemefortopologicalstructuresoverthecategory
X(A).Inordernottorestrictourselvestoaparticularclassofstructureswewillusethe
objectsofthefunctor-structuredcategorySpa(T)(see,e.g.,[2])asasufficientgeneralization
ofthenotionofabstracttopologicalstructure.Startwiththefollowingdefinition.
TDefinition4.2.1.LetX✲SetbeafunctorandletBbeasubcategoryofthecategory
Spa(T).ForafixedSpa(T)-object(UA,P)definethecategoryBP(A)asfollows:
Objects:Triples(X,S,α)suchthat
(i)(X,S)isaB-object;
(ii)(X,α)isanX(A)-object;
α(iii)(X,S)✲(UA,P)isaSpa(T)-morphism.
fMorphisms:X-morphisms(X,S,α)✲(Y,Q,β)suchthat
f(i)(X,S)✲(Y,Q)isaB-morphism;
f(ii)(X,α)✲(Y,β)isanX(A)-morphism.
WeconsiderthecategoryBP(A)asaconcretecategoryoverBinthefollowingway.
|−|Definition4.2.2.DefinetheforgetfulfunctorBP(A)✲Basfollows:
ff|(X,S,α)✲(Y,Q,β)|=(X,S)✲(Y,Q).

AsintheprevioussectionwearegoingtoshowasufficientconditionforBP(A)tobe
topologicaloverB.
FromnowoneweassumethatthecategoryX(A)istopological.Introducethefollowing
requirement.
27ForeveryB-object(X,S)itfollowsthatSpa(T)((X,S),(UA,P))isclosedunderthe
formationofmeets.
Proposition4.2.3.Suppose27holds.ThenBP(A)istopologicaloverB.

65

Proof.LetS=((X,S)fi✲|(Xi,Si,αi)|)i∈Ibea|−|-structuredsource.Defineα=
(αi◦fi).Itwillbeenoughtoshowthat(X,S,α)isaBP(A)-object.
I∈if◦αiiThefactthat(X,S)✲(UA,P)isaSpa(T)-morphismfori∈Iand27implythat
(X,S)α✲(UA,P)isaSpa(T)-morphism.
WearegoingtoconsiderthecategoryBP(A)inducedbythefollowingexample.
Example4.2.4.Suppose26holds.LetΩ=(ni)i∈Ibeafamilyofnaturalnum-
bersinndexedbyasetIandlet(nXi)i∈IbeafamilyofX-objects.Forin∈Idefine
TiXSi✲X:Xf✲Y→Xnifi✲YniandsetXTi✲Set=Xhom(Xi,Si(−✲))Set.
DefineXT✲Set=Xi∈I✲Set.
FromnowoneweareworkingwithExample4.2.4.LetP⊆TUAbeclosedunderthe
formationofmeetsinsuchawaythat
hj(A)Foreveryi0∈Iandeverysubfamily(Xi0✲(UA)ni0)j∈J⊆Pitfollowsthat
(πUA◦hj)
Xi0j∈J✲(UA)ni0∈P.
Proposition4.2.5.(A)implies27.
αjαProof.Let((X,S)✲(UA,P))j∈JbeafamilyofSpa(T)-morphisms.Weshowthat
j(X,S)j∈J✲(UA,P)isaSpa(T)-morphism.
hj∈J.SinceπUA◦(αj)ni◦h=(αj)◦πX◦h=(αj◦πX◦h)=πUA◦(αj◦πX◦h),
Ifh∈S,thenXi✲XniisanX-morphismforsomei∈Iandαjni◦h=Tαj(h)∈Pfor
nthenT(αj)(h)=(αj)ni◦h=(αj◦πY◦h)=(πUA◦αji◦h)∈Pby(A).
j∈Jj∈Jj∈Jj∈J
j∈Jj∈Jj∈Jj∈J
ConsideraspecialcaseofExample4.2.4.
Example4.2.6.LetX(A)beSet(JCPos)andletXi=1fori∈I.ThenTi(X)=
hom(Xi,Xni)=∼Xni.OnecaneasilyseethatSpa(T)isconcretelyisomorphictothe
categoryRel(Ω)definedasfollows:
Objects:Pairs(X,(ρi)i∈I),whereXisasetandρi⊆Xni.
fMorphisms:Maps(X,(ρi)i∈I)✲(Y,(σi)i∈I),wherefni[ρi]⊆σifori∈I.
LetBbeasubcategoryofSpa(T)(forexampleconsiderthecategoryofsetsandrela-
tionsRel,thecategoryofpreorderedsetsProstorthecategoryofposetsPos)andlet
(UA,(ρi)i∈I)beaSpa(T)-objectsuchthatρiisclosedundertheformationofmeetsfor
i∈I,i.e.,ajρibjforj∈Jimply(aj)ρi(bj).ThenBP(A)istopologicaloverBby
j∈Jj∈J
.2.5.4ositionopPr

66

ForafunctorXopT✲SetonegetsacategoryBPop(A).Onecaneasilyrestate
Proposition4.2.3forthecaseofthecategoryBPop(A).Consideranexample.
Example4.2.7.LetSetQ✲Setbethecontravariantpower-setfunctor.By22.7in[2]
Topisafullsubcategoryof(Spa(Q))op.LetAbeaJCPos-objectandlet(UA,O(UA))
bea(Spa(Q))op-objectsuchthat
(B)EveryV∈O(UA)hasthefollowingproperties(cf.Scottopensetsin[16]):
(i)↓V=V;
(ii)S∈VimpliesSV=∅foreverynonemptysubsetS⊆A.
AsanexampleconsideracompletechainA.Theneverysubsetoftheform↓x\{x}
satisfiestheabove-mentionedproperties.
Proposition4.2.8.TopPop(A)istopologicaloverTop.
Proof.Let(X,O(X))beaTop-objectandlet((X,O(X))αi✲(UA,O(UA)))i∈Ibea
αfamilyof(Spa(Q))op-morphisms.Setα=αi.Showthat(X,O(X))✲(UA,O(UA))
I∈iisa(Spa(Q))op-morphism.IfI=∅,thenα≡andthereforeforeveryV∈O(UA)
α−1[V]=X,∈V
∅,∈V.
SupposeI=∅andtakeV∈O(UA).Showthatα−1[V]=αi−1[V].Ifx∈α−1[V],then
I∈iαi(x)∈Vandthereforethereexistsi0∈Isuchthatαi0(x)∈Vby(B).(ii).Onthe
I∈iotherhand,x∈αi−1[V]givesi0∈Isuchthatαi0(x)∈Vandthereforeαi(x)∈Vby
i∈Ii∈I
).(i).B(

Noticethattheabove-mentionedfuzzificationdiffersfromtheonementionedin[11].

5Chapter

motaleQuandules

Givenaconcretecategory(A,U)overX,inthepreviouschaptersweconsideredthecate-
goryX(A)ofA-valuedobjectsofthecategoryX.Oneofthemostimportantrequirements
onthecategoryAwasadjointnessoftheforgetfulfunctorU.Consideringdifferentexam-
plesofsuchcategorieswearrivedatthecategoryQ-Modofquantalemodules(see,e.g.,
[1,29,35,37,38,46,47,52])motivatedbyconstructionsandresultsfrommodulesovera
ring.Thischapterpresentsthecollectionofresultsweobtainedwhilestudyingthecategory
Q-Mod.Inparticular,weshowthatthecategoryQ-Modismonadic(Proposition5.3.6),
consideritsrelationtothecategoryQ-TopofQ-topologicalspaces(Proposition5.2.6)and
generalizeamethodofcompletionofpartiallyorderedsets(Proposition5.10.14).Notice
thatsomeofthecontentsofSection5.7canbefoundin[29].Theimportanttopicofrela-
tionbetweenquantalemodulesandtopologicalspaceswillbecontinuedinourforthcoming
papers(see,e.g.,[50])wherewewilluse(toavoidsomedeficiencieswhicharoseinthischap-
ter)thenotionofquantalealgebra(see,e.g.,[53]).Itarisesfromreplacingcompletelattices
byquantalesinthedefinitionofquantalemodulesandaddinganadditionalrequirement
(see,e.g.,[4,26]fortheringanalogy).
AsaresultofourinvestigationswegetsomepropertiesofthecategorySet(Q-Mod)
(Proposition5.9.9).

5.1DefinitionofthecategoryQ-ModofQ-modules

InthissectionweintroducethecategoryQ-Mod.Startbyrecallingthedefinitionof
quantale(see,e.g.,[46,47]).

67

68

Definition5.1.1.Aquantaleisatriple(Q,,·)suchthat
(i)(Q,)isacompletelattice;
(ii)(Q,·)isasemigroup;
(iii)q·(S)=(q·s)and(S)·q=(s·q)foreveryq∈QandeveryS⊆Q.
s∈Ss∈S
Definition5.1.2.AquantaleQiscalledunitalprovidedthatthereexistsanelemente∈Q
suchthat(Q,·,e)isamonoid.
Fromnowonwithoutfurtherreferencesallquantalesaresupposedtobeunital.The
followingareexamplesofquantales:
(i)(2,,∧,1)where2={0,1};
(ii)([0,1],,∧,1)where[0,1]istheunitinterval;
(iii)([0,1],,·,1)where·istheusualmultiplication;
·(iv)thechain3withtheusualorderandthemap3×3✲3givenbythetable:
201·000020112022

Noticethat=e.
NowdefinethecategoryQ-Modofquantalemodulesorinotherwordsthecategoryof
enrichedcompletelatticesoverthecategoryJCPos.
Definition5.1.3.GivenaquantaleQ,definethecategoryQ-Modasfollows:
∗(i)Theobjectsaretriples(A,,∗)where(A,)isacompletelatticeandQ×A✲A
isamapsuchthat
(a)q∗(S)=(q∗s)foreveryq∈Q,S⊆A;
S∈s(b)(S)∗a=(s∗a)foreverya∈A,S⊆Q;
S∈s(c)q1∗(q2∗a)=(q1·q2)∗aforeveryq1,q2∈Q,a∈A;
(d)e∗a=aforeverya∈A.

69

(ii)Themorphismsaremaps(A,,∗)f✲(B,,∗)suchthat
(a)f(S)=f(S)foreveryS⊆A;
(b)f(q∗a)=q∗f(a)foreverya∈A,q∈Q.
WeconsiderthecategoryQ-ModasaconcretecategoryoverSetinthefollowingway.
Definition5.1.4.DefinetheforgetfulfunctorQ-ModU✲Setasfollows:
U((A,,∗)f✲(B,,∗))=Af✲B.

Definitions5.1.3and5.1.4giveaconstruct(Q-Mod,U).Onecaneasilyseethat2-Mod
isconcretelyisomorphictoJCPos(comparewithintegersZincaseofthecategoryR-Mod
ofmodulesovertheringR).ThuswhileconsideringthecategoryQ-Modwestudythe
categoryJCPosaswell.
Remark5.1.5.ForQ=1itfollowsthatA∈Ob(Q-Mod)iffA=∼1,i.e.,1-Modis
equivalenttotheterminalcategory.

5.2Fromquantalemodulestotopologicalspaces
Aninterestingexampleofquantalemodulesgivesthegeneralizationofthefunctor
TopO✲(JCPos)op:Xf✲Y→O(Y)O(f✲)O(X):S→f−1[S]
whereO(X)standsforthetopologyonX.Considerthefollowingdefinition.
Definition5.2.1.GivenaquantaleQ,definethecategoryQ-TopofQ-topologicalspaces
asfollows(cf.,e.g.,[24]):
(i)Theobjectsarepairs(X,OQ(X))withXasetandOQ(X)asubsetofQXsuchthat
OQ(X)=1fortheemptyXandotherwisethefollowingconditionsarefulfilled:
(a)foreveryq∈QtheconstantmapwithvalueqisanelementofOQ(X);
(b)givenh,h∈OQ(X),itfollowsthath·h∈OQ(X);
(c)givenasubset{hs|s∈S}⊆OQ(X),itfollowsthaths∈OQ(X).
S∈s(ii)Themorphismsaremaps(X,OQ(X))f✲(Y,OQ(Y))suchthath∈OQ(Y)implies
h◦f∈OQ(X).

70

Remark5.2.2.GivenaQ-topologicalspace(X,OQ(X)),wesaythatOQ(X)isaQ-to-
pologyonX.
WiththehelpofDefinition5.2.1definethefollowinggeneralizationofthefunctorO:
Q-TopOQ✲(Q-Mod)op:Xf✲Y→OQ(Y)OQ(f✲)OQ(X):h→h◦f.
ByanalogywithTheorem1.4inChapterIIof[28]weshowthatthefunctorOQhasaright
t.inojadStartwithtwosimplelemmas(see,e.g.,Remark3.2andProposition3.3in[24]).
Lemmas5.2.3.LetXbeasetandlet{OQs(X)|s∈S}beafamilyofQ-topologiesonX.
Thens∈SOQ(X)isaQ-topologyonX.
Definition5.2.4.LetXbeasetandletS⊆QX.Define
OQ(X)={OQ(X)|OQ(X)isaQ-topologyonXandS⊆OQ(X)}.
ThenSiscalledasubbaseofOQ(X)(denotedbyOQ(X)=S).
Lemma5.2.5.Let(X,OQ(X))f✲(Y,OQ(Y))beamapandletOQ(Y)=S.The
followingareequivalent:
(i)fisaQ-Top-morphism;
(ii){h◦f|h∈S}⊆OQ(X).
Nowthemainresult.
Proposition5.2.6.ThefunctorQ-TopOQ✲(Q-Mod)ophasarightadjoint.
Proof.LetAbeaQ-Mod-object.Setpt(A)=Q-Mod(A,Q).Fora∈Aletpt(A)a✲Q:
h→h(a)andsetS={a|a∈A}.LetOQ(pt(A))=SanddefineAA✲OQ(pt(A)):
a→a.ThenAisaQ-Mod-morphismwiththepropertythatforeveryQ-Mod-morphism
Af✲OQ(X)thereexistsauniqueQ-Top-morphismXf✲pt(A)definedbyf(x)(a)=
f(a)(x)suchthatOQ(f)◦A=f.
Remark5.2.7.ByProposition5.2.6thereexistsanadjointsituation
(η,):OQpt:(Q-Mod)op✲Q-Top
whereXηX✲pt(OQ(X))with(ηX(x))(h)=h(x).

71

WiththehelpofRemark5.2.7onecanintroducethefollowingnotions(see,e.g.,Defi-
nition2.5in[44]).
Definition5.2.8.AQ-topologicalspaceXiscalled
(i)Q-T0providedthatηXisinjective,i.e.,foreveryx,y∈X,x=ythereexistsh∈
OQ(X)suchthath(x)=h(y);
(ii)Q-soberprovidedthatηXisbijective.

Byanalogywith[28]onecouldtrytoshowthatpt(A)isQ-soberforeveryQ-Mod-
objectA.Unfortunatelythefollowinglemmashowsthattherearenonon-trivialQ-sober
.acespsLemma5.2.9.AQ-topologicalspaceXisQ-soberiffbothXandQcontainexactlyone
ent.elemProof.ThesufficiencyisclearbythefactthatQ=∼1impliespt(OQ(X))=∼1.Forthe
necessityletQ=∼1.SinceX=∅,leth∈OQ(X)betheconstantmapwithvalue.
Let⊥∈pt(OQ(X))betheconstantmapwithvalue⊥.Foreveryx∈Xitfollowsthat
(ηX(x))(h)=andtherefore⊥∈ηX[X],i.e.,ηXisnotsurjective.ThusQ=∼1and
thereforeX=∼1.

Lemma5.2.9showsthattheconceptofQ-sobrietyshouldbedefinedinadifferentway.
Thefirstconcept,however,ismorefruitfulasshowthefollowingresults.
Definition5.2.10.LetQ-Top0bethecategoryofQ-T0Q-topologicalspaces.LetSbe
thespace(Q,idQ).Accordingto[58]callittheQ-Sierpinskispace.
Lemma5.2.11.SisacoseparatorforQ-Top0.
WiththehelpofthenextdefinitiononecanshowthateveryQ-T0spaceisactuallya
subspaceofsomepowerSIofS.
Definition5.2.12.Let((Xi,OQ(Xi)))i∈IbeafamilyofQ-topologicalspaces.Itsproduct
isthesetXiwithOQ(Xi)=SjforSj={Xi✲Q|h∈OQ(Xj)}.
h◦πj
i∈Ii∈Ij∈Ii∈I
Proposition5.2.13.LetXbeaQ-Top-object.Thefollowingareequivalent:
(i)XisQ-T0;
(ii)XcanbeembeddedintosomepowerSIofS.
Proof.Straightforwardcomputationssimilartothestandardcase.

72

SimilarlyonecanshowthateveryQ-topologicalspaceisactuallyasubspaceofsome
powerEIofthespaceEdefinedasfollows.
Definition5.2.14.LetE={∞}Qandlet
EhE✲Q:x→⊥,x=∞
x,otherwise.
DefineEtobebethespace(E,hE).
Proposition5.2.15.EveryQ-topologicalspaceXcanbeembeddedintosomepowerEIof
thespaceE.
Remark5.2.16.Proposition5.2.15generalizesthefactthateverytopologicalspacecan
beembeddedintosomepowerofthespace(3,{∅,{2},3}).
LastlyconsiderageneralizationofExample1.1.10.
Example5.2.17.ForeveryQ-topologicalspaceXtherelationxydefinedby”h(x)
h(y)foreveryh∈OQ(X)”isapreorderonXwhichisanorderiffXisQ-T0.Thus,there
existsafunctor
ffVQ-Top✲Prost:(X,OQ(X))✲(Y,OQ(Y))→(X,)✲(Y,)
idXwhichisadjointsince(X,)✲V(X,OQ(X))withOQ(X)=Prost(X,Q)isaV-
universalarrowfor(X,).
InthesubsequentsectionsweconsidersomepropertiesofthecategoryQ-Mod.

5.3Q-Modisamonadicconstruct

InthissectionweprovethatQ-Modisamonadicconstruct.Thiswillprovidealotofnice
propertiesforthecategoryQ-Mod.
FirstofallshowthatthefunctorUisadjoint.
Proposition5.3.1.ThecategoryQ-Modhasfreeobjects.
Proof.ForasetXletQXbethesetofmapsXh✲Qwiththepoinwisestructure.Let
η{x}e,x=x
XX✲UQX:x→X✲Qwhere{x}:X✲Q:x→
⊥,x=x.
fTheneveryelementh∈QXcanbewrittenash(x)∗{x}andeverymapX✲UA
X∈xhasauniquefactorizationf=Uf◦ηXwheref(h)=h(x)∗f(x).
X∈x

73

Corollary5.3.2.Thereexistsanadjointsituation(η,):FU:Q-Mod✲Set.
Proof.LetSetF✲Q-Mod:Xf✲Y→QXf✲QYwherefisdefinedbythediagram
XηX✲UQX
fUf❄❄YηY✲UQY
andthereforeFf(h)=h(x)∗{f(x)}.ForeveryQ-Mod-objectAletFUAA✲Abe
definedbythediagramx∈X

UAηUA✲U(FUA)
idUA✲❄UA
UA

andthereforeA(h)=h(a)∗a.
A∈aUFRemark5.3.3.ThefunctorSet✲SetcanbeconsideredastheQ-valuedpower-set
functoranddenotedbyPQ.
TheadjointsituationfromCorollary5.3.2givesrisetoamonadT=(T,η,μ)onSet
definedbyT=UFandμ=UF.
Remark5.3.4.ThemonadTcanbeconsideredastheQ-valuedpower-setmonadonthe
.SetycategorKConsiderthecomparisonfunctorQ-Mod✲SetT.WearegoingtoshowthatKisa
concreteisomorphism.ForQ=1thestatementistrivial,therefore,assumethatQhasat
leasttwoelements.Startwiththeobservationthate=⊥impliesP(X)=∼{⊥,e}X⊆TX
andthereforewecanusetheusualnotationsforordinarysetsbearingtheaforesaidinclusion
inmind.Wealsodonotdistinguishbetweenasetanditscharacteristicfunction.Consider
onesimplelemma.
Lemma5.3.5.LetTXα✲Xbeamap.Then(X,α)isaSetT-objectiffthefollowing
conditionsarefulfilled:
(i)α({x})=xforeveryx∈X;
(ii)α((h)∗h)=α((h)∗{α(h)})forevery∈TTX.
h∈TXh∈TX


74

Condition(ii)impliesthefollowingconsequences:
(iii)α(A)=α({α(A)|A∈A})foreveryA⊆P(X);
(iv)α((h)∗{α(h)})=α({α((h)∗h)})forevery∈TTX.
h∈TXh∈TX
Proof.(i)and(ii)followfromthefactthat(X,α)isaSetT-objectiffα◦ηX=idXand
(h)∗hcanberewrittenas(h)∗hwhere
α◦μX=α◦Tα.(iii)istrivial.For(iv)noticethatforevery∈TTXitfollowsthat
h∈TXh∈TX
✲e,h=(h)∗hforsomeh∈TX
:TXQ:h→
⊥,otherwise.
Thenα((h)∗h)=α((h)∗{α(h)})=α({α((h)∗h)}).
h∈TXh∈TXh∈TX
Proposition5.3.6.ThecategoryQ-Modisamonadicconstruct.
Proof.ShowtheexistenceofaconcretefunctorSetTG✲Q-ModinversetoK.Let(X,α)
beaSetT-object.Forx,y∈Xletxyiffα({x,y})=y.ForS⊆XletS=α(S).Let
Q-Mod-object,forexample,ifq∈QandS⊆X,thenq∗(S)=α(q∗{α(S)})=α(q∗S)=
∗:Q×X✲X:(q,x)→α(q∗{x}).WithLemma5.3.5oneshowsthat(X,,∗)isa
α(q∗{s})=α({α(q∗{s})})=q∗s.Let(X,α)f✲(Y,β)beaSetT-morphism.
s∈Ss∈Ss∈S
Thenf◦α=β◦Tfandoneshowsthat(X,,∗)f✲(Y,,∗)isaQ-Mod-morphism,for
example,ifq∈Qandx∈X,thenf(q∗x)=f◦α(q∗{x})=β◦Tf(q∗{x})=β(q∗{f(x)})=
ffGq∗f(x).DefineSetT✲Q-Mod:(X,α)✲(Y,β)→(X,,∗)✲(Y,,∗).
StraightforwardcomputationsshowthatGistherequiredfunctor.
Corollary5.3.7.ThecategoryQ-Modiscomplete,cocomplete,wellpowered,extremally
co-wellpowered,andhasregularfactorizations.
Remark5.3.8.ThecategoryQ-Modhasconcretelimitsbutcolimitsfailtobeconcrete
(seeProposition5.5.1).
Attheendofthesectionletusconsiderthefollowingquestion.WitheverymonadT
onecannaturallyassociatetwoconcretecategories,namely,theEilenberg-Moorecategory
andtheKleislicategory(see,e.g.,[33]).ConsidertheQ-valuedpower-setmonadTon
Set.Proposition5.3.6givestheEilenberg-MoorecategoryforT.Wearegoingtoshowthe
Kleislicategoryaswell.Considerthefollowingdefinition(cf.thecategoryR(L)in[18]).
Definition5.3.9.LetQbeaquantale.DefinethecategorySetRel(Q)asfollows:
Objects:Sets.

75

Morphisms:Q-valuedrelationsXR✲Y,i.e.,mapsX×YR✲Q.
Composition:IfXS✲YandYR✲ZareQ-valuedrelations,thecompositionR◦S
R◦S(x,z)=S(x,y)·R(y,z).
isdefinedby
Y∈yIdentities:ForasetXdefinetheidentityonXtobetheidentityQ-valuedrelation,i.e.,
XEX✲XwithEX(x,y)=e,x=y
⊥,otherwise.

Proposition5.3.10.ThecategorySetRel(Q)isisomorphictotheKleislicategoryofthe
Q-valuedpower-setmonadonSet.
Proof.LetSetTbetheaforesaidKleislicategory.Thenthefunctor
SetTF✲SetRel(Q):XTfT✲YT→XRf✲Y:(x,y)→(f(x))(y)
isanisomorphism.
Corollary5.3.11.ThecategorySetRel(Q)isconcretizableoverSet.
Proof.DefinetheforgetfulfunctorSetRel(Q)U✲Setasfollows(cf.Example1.1.8)
XR✲Y→PQ(X)fR✲PQ(Y):h(x)∗{x}→h(x)∗R(x,−).
x∈Xx∈X

5.4OnsomespecialmorphismsinQ-Mod

InthissectionweconsiderspecialmorphismsinthecategoryQ-Mod.Startwithasimple
observation.
Lemma5.4.1.TheforgetfulfunctorQ-ModU✲SetisrepresentablebyQ.
Proof.ByProposition5.3.1Qisfreeoverthesingletonset1.
Corollary5.4.2.ThemonomorphismsinthecategoryQ-Modarepreciselythemorphisms
withinjectiveunderlyingfunctions.
Proof.FollowsfromLemma5.4.1andCorollary7.38in[2].

76Unlikemonomorphisms,thecaseofepimorphismsisnotsoobvious.Itisthoroughly
studiedin[36]wheretheauthorshowsthatepimorphismsareontoinQ-Mod.Thecrucial
stepistoshowthatQ-Modsatisfiestheso-calledamalgamationproperty.Unfortunately
theproofofthisfactisrathercomplicatedandinvolvesatransfiniteconstruction,therefore,
wedecidedtopresenthereourownshorterversionofit.Startwiththefollowingdefinition
from[36].
Definition5.4.3.AcategoryCissaidtosatisfytheamalgamationpropertyprovidedthat
mforeverymonomorphismA✲B,wheneverthesquare
m✲BApm❄❄✲CBqisapushout,thenitisalsoapullbackandp,qaremonomorphisms.
Theorem5.4.4.ThecategoryQ-Modsatisfiestheamalgamationproperty.
mProof.LetA✲BbeaQ-Mod-monomorphism.ByCorollary5.4.2onecanassumethat
AisasubmoduleofBandmistheinclusionmap.Let((μ1,μ2),B⊕B)beacoproductof
(B,B).ThenB⊕B=B×BwithBμ1✲B⊕B:b→(b,⊥)andBμ2✲B⊕B:b→(⊥,b)
(seeProposition5.5.1).Let(c,C)beacoequalizerof(μ1◦m,μ2◦m).Bythedualof
Theorem21.3(canonicalconstructionofpullbacks)in[20]thediagram
m✲BAmc◦μ1(5.1)
❄❄✲Bc◦μ2C
isapushout.Itwillbeenoughtoshowthat(5.1)isapullback.
LetS={((a,⊥),(⊥,a))|a∈A}andletTbethesmallestcongruencerelationonB⊕B
thatcontainsS(seeDefinition5.7.10).ThenC=(B⊕B)/T={[(x,y)]|(x,y)∈B⊕B}
candB⊕B✲C:(x,y)→[(x,y)].WearegoingtodescribeTexplicitly.
DefineR={((x∨a,y∨b),(x∨a,y∨b))|x,y,x,y∈B;a,b,a,b∈A;x∨b=
x∨bandy∨a=y∨a}.ThenRisabinaryrelationonB⊕B.WeshowthatR=T.
OnecaneasilyseethatRisanequivalencerelationonB⊕B.Forexample,toprovethatRis
reflexive,given(x0,y0)∈B⊕B,set(x,y)=(x0,y0)=(x,y)and(a,b)=(⊥,⊥)=(a,b).
ShowthatRisacongruencerelationonB⊕B.

yi∨aifori∈I.Noticethat(xi∨ai,yi∨bi)=((xi)∨(ai),(yi)∨(bi))
Suppose(xi∨ai,yi∨bi)R(xi∨ai,yi∨bi)fori∈I.Thenxi∨bi=xi∨biandyi∨ai=
i∈Ii∈Ii∈Ii∈Ii∈I

77

withi∈Iai,i∈Ibi∈A.Since(i∈Ixi)∨(i∈Ibi)=(i,j)∈I×I(xi∨bj)=i∈I(xi∨bi)=i∈I(xi∨

bi)=(xi)∨(bi)andthesecondequalityofjoinsfollowssimilarly,((xi∨ai,yi∨
i∈Ii∈Ii∈I
bi))R((xi∨ai,yi∨bi)).
I∈iSuppose(x∨a,y∨b)R(x∨a,y∨b)andletq∈Q.Noticethatq∗(x∨a,y∨b)=
((q∗x)∨(q∗a),(q∗y)∨(q∗b))withq∗a,q∗b∈A.Sincex∨b=x∨bimplies
(q∗x)∨(q∗b)=(q∗x)∨(q∗b)andthesecondequalityofjoinsfollowssimilarly,
(q∗(x∨a,y∨b))R(q∗(x∨a,y∨b)).ThusRisacongruencerelationonB⊕B.
ShowthatS⊆R.Givena0∈A,define(x,y)=(a0,⊥)=(a,b)and(x,y)=(⊥,a0)=
(a,b).Then((a0,⊥)=(x∨a,y∨b))R((x∨a,y∨b)=(⊥,a0)).SupposeRisanother
congruencerelationonB⊕BwithS⊆R.ShowthatR⊆R.
Suppose(x∨a,y∨b)R(x∨a,y∨b).Firstofallwegetthat(a,b)R(b,a)since
(a,⊥)R(⊥,a),(⊥,b)R(⊥,b)imply(a,b)R(⊥,a∨b)and(b,⊥)R(⊥,b),(⊥,a)R(⊥,a)imply
(b,a)R(⊥,b∨a).Similarlyoneshowsthat(b,a)R(a,b).Since(x,y)R(x,y)implies
(x∨a,y∨b)R(x∨b,y∨a)and(x,y)R(x,y)implies(x∨b,y∨a)R(x∨a,y∨b),it
followsthat(x∨a,y∨b)R(x∨a,y∨b).
TheaforesaidresultsshowthatR=T.Supposeb1,b2∈Baresuchthatc◦μ1(b1)=
c◦μ2(b2).Then(b1,⊥)R(⊥,b2).BythedefinitionofRitfollowsthat(b1,⊥)=(x∨a,y∨b)
andthereforey=⊥=b.Moreover,(⊥,b2)=(x∨a,y∨b)andthereforex=⊥=
a.Lastlyx∨b=x∨bimpliesx=bandy∨a=y∨aimpliesa=y.Thus
b1=x∨a=b∨a=a∨b=y∨b=b2andb1(=b2)liesinA.Itfollowsthat
A=∼{(a,a)∈B⊕B|a∈A}={(b1,b2)∈B⊕B|c◦μ1(b1)=c◦μ2(b2)}andtherefore
(5.1)isapullback(noticethatpullbacksinQ-ModareconstructedasinSet).
Itremainstoshowthatbothc◦μ1andc◦μ2areinjective.Supposeb1,b2∈Bare
suchthatc◦μ1(b1)=c◦μ1(b2).Then(b1,⊥)R(b2,⊥)andtherefore(b1,⊥)=(x∨a,y∨b)
impliesy=⊥=band(b2,⊥)=(x∨a,y∨b)impliesy=⊥=b.Lastlyx∨b=x∨b
impliesx=xandy∨a=y∨aimpliesa=a.Thusb1=x∨a=x∨a=b2.Similarly
followsinjectivityofthesecondmap.
Theorem5.4.4impliesthefollowingproposition.
Proposition5.4.5.TheregularmonomorphismsinthecategoryQ-Modarepreciselythe
sms.monomorphiProof.Itwillbeenoughtoshowthesufficiency.LetAf✲BbeaQ-Mod-monomorphism.
ByCorollary5.3.7Q-Modiscocompleteandthereforethereexistsapushout
m✲BApm❄❄✲CBqwhichisalsoapullbackbyTheorem5.4.4.Thelatterresultimmediatelyimpliesthat(A,m)
isanequalizerof(p,q)inQ-Mod(see,e.g.,thedualofProposition21.16in[20]).

78

Corollary5.4.6.TheepimorphismsinthecategoryQ-Modarepreciselythemorphisms
withsurjectiveunderlyingfunctions.
fProof.Itwillbeenoughtoshowthenecessity.LetA✲BbeaQ-Mod-epimorphism.
OnehastheobviousfactorizationinQ-Mod
f✲BA✲em✲]A[fwhereAe✲f[A]:a→f(a)andf[A]m✲B:b→b.SincemisaQ-Mod-
monomorphism,itisaregularmonomorphismbyProposition5.4.5.Moreover,sincef
isanepimorphism,mshouldbealsoandthereforemisaQ-Mod-isomorphism.Thus
f[A]=Bandthereforefissurjective.
Nowconsiderregularepimorphisms.
Proposition5.4.7.TheregularepimorphismsinthecategoryQ-Modarepreciselythe
epimorphisms.
fProof.Itwillbeenoughtoshowthesufficiency.LetA✲BbeaQ-Mod-epimorphism.
LetC={(a1,a2)∈A×A|f(a1)=f(a2)}andletπi:C✲A:(a1,a2)→aifori=1,2.
DefinetherequiredstructureonCpointwise.Then(f,B)isacoequalizerofπ1andπ2.
SincethecategoryQ-Modispointed(foreverytwoobjectsAandBthemorphism
A0AB✲B:a→⊥isthezeromorphism)onecanconsidernormalmonomorphisms(or
kernels)inQ-Mod.ForbetterresultsweconsiderasubcategoryofthecategoryQ-Mod.
Definition5.4.8.LetAbeaQ-Mod-object.Aissaidtohavenozerodivisorsprovided
that(q,a)∈Q×Aandq∗a=⊥implyq=⊥ora=⊥.
SinceQisaQ-ModobjectthedefinitionisapplicabletoQaswell.
Definition5.4.9.ThecategoryQ-ModzisthefullsubcategoryofthecategoryQ-Mod
consistingofallmodulesAwithoutzerodivisors.
Remark5.4.10.LetQhavenozerodivisors.ThenQXhasnozerodivisorsforeveryset
XandthereforeQ-Modzhasfreeobjects(cf.Proposition5.3.1).
fProposition5.4.11.LetQhavenozerodivisorsandletA✲BbeaQ-Modz-
monomorphism.Thefollowingareequivalent:
(i)fisanormalmonomorphism;

79

(ii)f[A]hasthefollowingproperties:
(a)↓f[A]=f[A];
(b)(q,b)∈Q×Bandq∗b∈f[A]implyq=⊥orb∈f[A].
Proof.(i)=⇒(ii):Byassumption(A,f)isanequalizerofQ-Modz-morphismsBg✲✲C.
0BCByRemark5.4.10f[A]=∼{b∈B|g(b)=⊥}andtherefore↓f[A]=f[A].Suppose
q∗b∈f[A].Thenq∗g(b)=g(q∗b)=⊥andthereforeq=⊥org(b)=⊥.
(ii)=⇒(i):Considerthelattice2.Define
⊥=,q⊥∗:Q×2✲2:(q,a)→a,otherwise.
Then(2,∗)isaQ-Modz-object.Define
g:B✲2:b→⊥,b∈f[A]
,otherwise.
StraightforwardcomputationsshowthatgisaQ-Modz-morphismand(A,f)isanequalizer
ofgand0B2.
Remark5.4.12.Theimplication(i)=⇒(ii).(a)inProposition5.4.11istrueinthecate-
goryQ-ModforanyquantaleQ.

5.5Q-Modisnotanabeliancategory

ItisknownthatgivenaringRthecategoryR-Modisabelian.Inthissectionweshow
thatthefactcanbepartlytransferredtothecategoryQ-Mod.
Proposition5.5.1.ThecategoryQ-Modhasbiproducts.
Proof.Let(Ai)i∈Ibeaset-indexedfamilyofQ-Mod-objectsandletP=(Ai,(πi)i∈I)
I∈ibeaproductoftheirunderlyingsets.Definetherequiredstructurepointwiseandgeta
productof(Ai)i∈IinQ-Mod.Forj∈Idefine
j=ia,μj:Aj✲Ai:a→(ai)i∈Iwithai=
i∈I⊥,i=j
andgetasinkS=((μi)i∈I,Ai)inQ-Mod.ThenSisacoproductoftheaforesaid
I∈ifamilysinceforanyothersinkT=(Aifi✲B)i∈IinQ-Modthemapf:Ai✲B:
I∈i(ai)i∈I→fi(ai)istheuniqueQ-Mod-morphismsuchthatf◦S=T.
I∈i

80

ByPropositions5.5.1and40.12inf[20]thereexistsauniquesemiadditivestructure+
onQ-Moddefinedasfollows:ifA✲✲B,thenAf+g✲B=AΔ✲A⊕A[f,g✲]B.It
gcanbeshownthat+isthejoinoperation.
ThenextpropositionshowsthatingeneralthecategoryQ-Modisnotabelian.
Proposition5.5.2.LetQhavenozerodivisorsandletQ=1.ThenQ-Modisnot
abelian.
Proof.Considertheembedding
JCPos⊂E✲Q-Mod:Af✲B→(A,∗)f✲(B,∗)
erwhe∗:Q×A✲A:(q,a)→⊥,q=⊥
a,otherwise.
Let2f✲3betheJCPos-morphismwithf(1)=2.ThenE(2f✲3)isaQ-Mod-
monomorphismwhichisnotnormalbyRemark5.4.12.
Remark5.5.3.Themodulestructure∗consideredinProposition5.5.2iscalledthetrivial
modulestructure.

5.6Quantalemodulesdonotformatopos

InthissectionweshowthatingeneralthecategoryQ-Modisnotatopos,however,itdoes
haverepresentableM-partialmorphismswhereMisnottheclassofallisomorphisms.The
firststatementfollowsfromthenextpropositionwhichisaconsequenceofProposition5.5.1.
Proposition5.6.1.IfQ=∼Q×Q,thenthecategoryQ-Modisnotcartesinclosed.
Proof.ConsiderthefunctorQ-ModQ×−✲Q-Mod.SinceQ×(1⊕1)=∼Q=∼Q×Q=∼
(Q×1)⊕(Q×1),Q×−doesnotpreservecolimitsandthereforeisnotco-adjoint.
Remark5.6.2.Since2=∼2×2,JCPosisnotcartesianclosed.
Nowthesecondstatement.
Definition5.6.3.DefineMtobetheclassofallQ-Mod-morphismsAf✲Bsuchthat
(i)misaQ-Mod-monomorphism;
(ii)↓m[A]=m[A];
(iii)foreveryq∈Qandeveryb∈B,ifq∗b∈m[A],thenq=⊥orb∈m[A].

81

WeshowthatthecategoryQ-ModhasrepresentableM-partialmorphisms.Startwith
twolemmas.
Lemma5.6.4.TheclassMispullback-stable.
Proof.Straightforwardcomputations.
Lemma5.6.5.LetCbeaQ-Mod-objectwiththetopelementidempotentw.r.t.∗,i.e.,
q∗=foreveryq=⊥.ThenCisM-injective.
Proof.LetAm✲BbeinMandletAf✲CbeaQ-Mod-morphism.Define
g:B✲C:b→f(a),b=m(a)forsomea∈A
,otherwise.
ThengisaQ-Mod-morphismsuchthatg◦m=f.

Nowthepromisedresult.
Proposition5.6.6.LetQhavenozerodivisors.ThenQ-ModhasrepresentableM-partial
morphisms.Proof.LetBbeaQ-Mod-object.LetB∗=B{∞}withtheQ-Mod-structuredefined
asfollows:b<∞foreveryb∈Bandq∗∞=∞foreveryq=⊥.SinceQhasno
zerodivisors,B∗isaQ-Mod-object.Straightforwardverificationshowsthattheinclusion
B⊂✲B∗representsM-partialmorphismsintoB.
Remark5.6.7.Proposition5.6.6generalizesExample2.7.4.

5.7Tensorproductofquantalemodules

Thebinarytensorproductinthecategoryofmodulesoveracommutativeringiswidely
knowninalgebra(see,e.g.,[26,31]).Asaresult,foreverycommutativeringRonegetsa
monoidalcategoryR-Mod,⊗,R(cf.[33]).Asappearsin[5]thenotioncanbeconsidered
inanysuitablecategory.
Definition5.7.1.AtensormultiplicationforafunctorH:Aop×A✲Aisafunctor
T:A×A✲A,essentiallyuniqueifitexists,forwhichthereisanaturalequivalence
A(T(A,B),C)✲A(A,H(B,C)).

WearegoingtousetheideaforthecategoryQ-Mod.LetQbeacommutativequantale.
ThefirstlemmaisalightversionofTheorem4.8in[26].

82

Lemma5.7.2.LetA,A,B,BbeQ-Mod-objects.
(i)hom(A,B)isaQ-Mod-object.
(ii)IfBf✲BisaQ-Mod-morphism,thenhom(A,B)hom(A,f✲)hom(A,B)isa
Q-Mod-morphism.
(iii)IfAg✲AisaQ-Mod-morphism,thenhom(A,B)hom(g,B✲)hom(A,B)isa
Q-Mod-morphism.
Proof.DefinetherequiredstructurepointwiseandusecommutativityofQ.
Lemma5.7.2givesaninternalhom-functorH:(Q-Mod)op×Q-Mod✲Q-Mod
withUH(A,B)=hom(A,B).
Proposition5.7.3.LetAandBbeQ-Mod-objects.Thenthereexistsanembedding
H(A,B)iAB✲BUAsuchthatπa◦iAB(h)=h(a)foreverya∈Aandeveryh∈H(A,B)
whereπaistheprojectionfora.
Proof.DefineiAB:H(A,B)✲BUA:h→(h(a))a∈A.
Nowconsiderthefollowingdefinitions.
fDefinition5.7.4.LetA,B,CbeQ-Mod-objects.AmapUA×UB✲UCiscalleda
bimorphismprovidedthat
(i)Bf(a,−✲)CisaQ-Mod-morphismforeverya∈A;
(ii)Af(−,b✲)CisaQ-Mod-morphismforeveryb∈B.
Noticethatthenotionofbimorphismisusedinaquitedifferentsensein[2,20].One
cansuggesttouseanothertermsuchasbilinearorsemicontinuous.However,wedecided
tousetheoriginalterminologyof[5].
Definition5.7.5.AbimorphismUA×UBf✲UCiscalleduniversal(forAandB)iff
anybimorphismUA×UBg✲UDhasafactorizationg=Uh◦fforauniqueCh✲D.
WeshowthateverypairofQ-Mod-objectshasauniversalbimorphism.Proposition4
in[5]givesthefollowing.
Proposition5.7.6.IfacategoryChascoproducts,(Epi,Mono-Source)-factorizations,
anditsmono-sourcesarepreservedbytheunderlyingsetfunctor,thenChasuniversal
bimorphisms.

83

Corollary5.7.7.ThecategoryQ-Modhasuniversalbimorphisms.
Proof.FollowsfromProposition5.7.6,5.3.1andCorollary5.3.7.
Corollary5.7.7motivatesthenextdefinition.
Definition5.7.8.DefinethefunctorofuniversalbimorphismstobethefunctorT:
Q-Mod×Q-Mod✲Q-Modgivenbythediagram
UA×UBβAB✲UT(A,B)
Uf×UgUT(f,g)
❄❄UA×UBβ✲UT(A,B)
BAwhereβABandβABareuniversalbimorphisms.
Proposition5.7.9.Thefollowingareequivalent:
(i)TisatensormultiplicationforH;
(ii)Tisafunctorofuniversalbimorphisms.
Proof.Usepropositions1and3in[5]aswellasProposition5.7.3.
ByProposition5.7.9thecategoryQ-Modhasatensormultiplication,onehasonly
tochooseafunctorofuniversalbimorphisms.Weshowanexampleofsuchfunctor.The
constructionisaslightmodificationofthetensorproductinthecategoryofmodulesover
acommutativeringconsideredin[26].
LetAandBbeQ-Mod-objects.AccordingtoProposition5.3.1thereexistsamap
UA×UBηUA×UB✲U(QUA×UB)forthesakeofbrevitydenotedbyη.Considerthefollowing
definition(cf.ChapterII§4in[7]).
Definition5.7.10.LetAbeaQ-Mod-object.AcongruencerelationonAisanequiva-
lencerelation∼onUAsuchthat
(i)ForeverysetI,ai∼bifori∈Iimpliesai∼bi.
i∈Ii∈I
(ii)Foreveryq∈Q,a∼bimpliesq∗a∼q∗b.
fExample5.7.11.EveryQ-Mod-morphismA✲Bgivesrisetoacongruencerelation
onAdefinedthrougha∼fbifff(a)=f(b).
Remark5.7.12.ForeveryQ-Mod-objectAthesetofcongruencerelationsonAisa
completelatticew.r.t.usualinclusionofsets.

84

Remark5.7.13.Everycongruencerelation∼onAdefinesaQ-Mod-objectA/∼,where
U(A/∼)isthesetofequivalenceclasses{[a]|a∈A}equippedwiththefollowingstructure
(i)Foreverysubset{[ai]|i∈I}⊆A/∼seti∈I[ai]=[i∈Iai].
(ii)Forevery[a]∈A/∼andeveryq∈Qsetq∗[a]=[q∗a].
Let∼bethesmallestcongruencerelationonQUA×UBsuchthatforeveryS⊆A,
S⊆B,a∈A,b∈B,q∈Qthefollowingconditionsarefulfilled
(i)η(S,b)∼η(s,b)andη(a,S)∼η(a,s);
s∈Ss∈S
(ii)η(q∗a,b)∼q∗η(a,b)andη(a,q∗b)∼q∗η(a,b).
ByRemark5.7.13wegetaquotientmapQUA×UBq✲QUA×UB/∼.LetA⊗B=
QUA×UB/∼andletp=Uq◦η.
Lemma5.7.14.UA×UBp✲U(A⊗B)isauniversalbimorphismforAandB.
Proof.Straightforwardcomputationsshowthatpisabimorphism.LetUA×UBf✲UC
beanotherbimorphism.SinceQUA×UBisfreeoverUA×UBthereexistsauniqueQ-Mod-
morphismQUA×UBf✲CsuchthatUf◦η=f.Bythefactthat∼⊆∼fthereexistsa
uniqueQ-Mod-morphismA⊗Bfˆ✲Csuchthatfˆ◦q=f.Thecommutativediagram
UA×UBη✲U(QUA×UB)Uq✲U(A⊗B)
f✲❄UfUfˆ
✛UC

impliesthedesiredresult.
Nowthepromisedresult.
Proposition5.7.15.ThetripleQ-Mod,⊗,Qisamonoidalcategory.
Remark5.7.16.WiththehelpofProposition5.7.15onecandefinethenotionofaQ-
algebra(see,e.g.,[33]).
NoticethatthecategoryJCPosallowsasimplerwayofconstructingtensorproduct.
Onecanproceedasfollows.
fifff(S,S)=f[S×S]foreveryS⊆A,S⊆B.
Lemma5.7.17.LetA,B,CbeJCPos-objects.AmapUA×UB✲UCisabimorphism

85

Proof.Necessity:f(S,S)=f(s,S)=(f(s,s))=f[S×S].Suffi-

ciency:Ifa∈A,S⊆B,thenf(a,S)=f({a},S)=f[{a}×S]=f(a,s).
s∈Ss∈Ss∈S
S∈sFixJCPos-objectsAandB.ForA⊥=A\{⊥},B⊥=B\{⊥}define
A⊗B={S⊆A⊥×B⊥|S=↓Sand(P1,P2)∈SforeveryP1×P2⊆S}
where↓SistakeninA⊥×B⊥.SinceA⊗Bisclosedunderarbitraryintersections,itis
acompletelatticew.r.t.usualinclusionofsets.Definep:UA×UB✲U(A⊗B):
(a,b)→↓(a,b).
Proposition5.7.18.UA×UBp✲U(A⊗B)isauniversalbimorphismforAandB.
Proof.ByLemma5.7.17pisabimorphism.ForanyotherbimorphismUA×UBf✲UC
Sincegisorder-preserving,itisenoughtoshowthatg(Si)g(Si).ConsiderSi.
showthatg:A⊗B✲C:S→f[S]isaJCPos-morphism.Let{Si|i∈I}⊆A⊗B.
i∈Ii∈Ii∈I
Bytransfiniterecursiondefine
S0=Si
Sξ={(a,b)∈A⊥×B⊥|thereexists∅=P1×P2⊆Sτwith(a,b)(P1,P2)}
i∈I
ξτ<foreveryordinalξ.ThenSi=Sξ0forξ0withthepropertySξ0=Sξ0+1.Showthatfor
everyξandevery(a,b)∈Sξ,f(a,b)g(Si).Theng(Si)g(Si).If(a,b)∈S0,
i∈I
then(a,b)∈Si0forsomei0∈Iandthereforef(a,b)g(Si).For(a,b)∈Sξthereexists
i∈Ii∈Ii∈I
∅=P1×P2⊆Sτwith(a,b)(P1,P2)andthereforef(a,b)f(P1,P2)=
i∈I
ξτ<f[P1×P2].Sincef(p1,p2)g(Si)forevery(p1,p2)∈P1×P2,f(a,b)g(Si).
i∈Ii∈I
StraightforwardcomputationsshowthatUg◦p=fistherequiredfactorization.
TheaforesaidconstructionofA⊗Bwassuggestedtotheauthorby[27].

5.8Quantale-valuedpower-setfunctors

Inthissectionweconsiderbrieflygeneralizationsofusualpower-setfunctors.Recallthatin
Remark5.3.3weintroducedthecovariantQ-valuedpower-setfunctorSetPQ✲Setdefined

86

asfollows:
PQ(Xf✲Y)=QXPQ✲fQYwith(PQf(h))(y)=h(x).
y)=x(fWewanttointroducethecontravariantQ-valuedpower-setfunctorSetQQ✲Setopaswell.
Considerthefollowingfunctor(cf.Example1.2.3):
Q-Mod(−)✲∗Setop:Aϕ✲B→UBϕ∗✲UA:b→ϕ−1[↓b].
Recallourrequirement5onthecategoryX(A)atthebeginning
Proposition5.8.1.ThefunctorQ-Mod(−)✲∗Setopsatisfiesrequirement5.
Propositions5.8.1and1.3.1implytheexistenceofthefunctorSet(−)◦✲Q-Modop
ybdfinedeXf✲Y→FYFX◦F((Ff)∗◦ηY✲)FX.
LetQQ=Uop◦(−)◦.
Lemma5.8.2.LetXf✲Ybeamap.ThenU(FX◦F((Ff)∗◦ηY))(h)=h◦f.
Proof.Leth∈UFY.ThenUFηY(h)=h(y)∗{{y}}=andUF(Ff)∗()=h(y)∗
{{y}◦f}=andUFX()=y∈Yh(y)∗({y}◦f)=h◦f.
y∈Yy∈Y
ByLemma5.8.2SetQQ✲Setopisdefinedasfollows
QQ(Xf✲Y)=QXQ✛QfQYwithQQf(h)=h◦f.
Remark5.8.3.TheaforesaidQ-valuedpower-setfunctorscoincidewiththeusualgener-
alizations(cf.,e.g.,[43]).

5.9FactorizationstructuresonQ-Mod
InthissectionweconsiderfactorizationstructuresonthecategoryQ-Mod.
Proposition5.9.1.ThecategoryQ-Modisan(Epi,Mono-Source)-category.
Proof.FollowsfromCorollary5.3.7andProposition5.4.7.
ThefactthatthecategoryQ-ModisbalancedandProposition5.9.1implythefollowing.

87

Proposition5.9.2.ThecategoryQ-Modisan(Epi,ExtrMono-Source)-category.
WeshowthatQ-Modisan(Epi-Sink,Mono)-category.InviewofProposition5.4.5it
willbesufficienttoshowthatQ-Modhas(Epi-Sink,Mono)-factorizationsforsinks.
Westartwithpropositionsclarifyingthenatureofepi-sinksinQ-Mod.
Definition5.9.3.IfSisasubsetofamoduleAoveraquantaleQ,thentheintersection
ofallsubmodulesofAcontainingSiscalledthesubmodulegeneratedbySanddenotedby
.SS={qi∗ai|Iisaset,ai∈S,qi∈Q}.
Lemma5.9.4.LetAbeaQ-Mod-objectandletSbeanonemptysubsetofA.Then
I∈iProposition5.9.5.AsinkS=(Aifi✲A)i∈IinQ-Modisanepi-sinkiffA=
i∈Ifi[Ai].
gProof.Sufficiency:LetA✲✲BbeQ-Mod-morphismswithg◦S=h◦S.Takea∈A.By
hj∈Ji∈Ij∈Jj∈J
Lemma5.9.4a=qj∗aj,aj∈fi[Ai]andg(a)=qj∗g(aj)=qj∗h(aj)=h(a).
Necessity:Usethefactthattheinclusionfi[Ai]⊂μ✲Aisanepimorphismand
Proposition5.4.5.i∈I
Nowthepromisedresult.
Proposition5.9.6.ThecategoryQ-Modis(Epi-Sink,Mono)-factorizable.
Proof.LetS=(Aifi✲A)i∈IbeasinkinQ-Mod.Considerthenaturalfactorization
ˆAifi✲A=Aifi✲fi[Ai]⊂μ✲A.
I∈iCorollary5.9.7.ThecategoryQ-Modisan(Epi-Sink,Mono)-category.
Corollary5.9.8.ThecategoryQ-Modisan(ExtrEpi-Sink,Mono)-category.
TheaforesaidresultsshowthatthecategoryQ-Modsatisfiestherequirementsimposed
onthecategoryAaspartofthecategoryX(A).Asaconsequenceonegetsthecategory
Set(Q-Mod)withthefollowingproperties.
Proposition5.9.9.LetQbeaquantale.Thenthefollowinghold:
(i)thecategorySet(Q-Mod)istopological;
(ii)thecategorySet∗(Q-Mod)is

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(a)algebraicandcoalgebraic;
(b)an(Epi,ExtrMono-Source)-categoryandan(ExtrEpi-Sink,Mono)-category;
(c)monadic.
Remark5.9.10.AsonecouldexpectProposition5.9.9generalizestheresultsforthe
categoryJCPos.

5.10Completionofpartiallyorderedsets

Inthissectionwegeneralizeastandardmethodofcompletionofpartiallyorderedsetsor
justposetsforshort(see,e.g.,[56]).LetPosbethecategoryofposetsandorder-preserving
maps.Thefollowingpropositionisdueto[2].
Proposition5.10.1.ThecategoryJCPosisreflectiveinPos.
Proof.GivenaposetAonehasthecompletelatticeBAofalllower-setsofAandthe
embeddingA⊂✲BA:a→↓awhichisthereflectionarrowforA.
WearegoingtogeneralizetheresultforthecategoryQ-Mod.Thefirstapproachisas
follows.
EProposition5.10.2.LetQbeaquantale.TheobviousforgetfulfunctorQ-Mod✲Pos
nt.oiadjsiProof.GivenaposetA,theE-universalarrowcanbeconstructedasfollows.DefineBA=
r{h∈QA|abimpliesh(b)h(a)}andletA✲BA:a→↓awhere
abe,↓a:A✲Q:b→
⊥,otherwise.
fGivenaQ-Mod-objectBandaPos-morphismA✲B,thereexistsauniqueQ-Mod-
fmorphismBA✲Bsuchthatf◦r=f,i.e.,f:BA✲B:h→h(a)∗f(a).
A∈aAnotherapproachismoresophisticated.Startwiththefollowingdefinition.
Definition5.10.3.GivenaquantaleQ,definethecategoryQ-Posasfollows:
(i)Theobjectsaretriples(A,,∗)where(A,)isaposetandQ×A∗✲Aisamap
tthahsuc∗q(a)themapA✲Aisorder-preservingforeveryq∈Q;
(b)themapQ∗✲aAisorder-preservingforeverya∈A;

89

(c)q1∗(q2∗a)=(q1·q2)∗aforeveryq1,q2∈Q,a∈A;
(d)e∗a=aforeverya∈A.
f(ii)Themorphismsaremaps(A,,∗)✲(B,,∗)suchthat
(a)fisorder-preserving;
(b)q∗f(a)f(q∗a)foreverya∈A,q∈Q.
TheobjectsofthecategoryQ-PoswillbereferredtoasQ-posets.Onecanconsider
thecategoryQ-PosasaconcretecategoryoverSetinthefollowingway.
UDefinition5.10.4.DefinetheforgetfulfunctorQ-Pos✲Setasfollows:
ffU((A,,∗)✲(B,,∗))=A✲B.

Definitions5.10.3and5.10.4giveaconstruct(Q-Pos,U).Onecaneasilyseethat2-Pos
isconcretelyisomorphictoPos(foraposetAlet∗:2×A✲A:(q,a)→a).Thus
whileconsideringthecategoryQ-PoswestudythecategoryPosaswell.
EveryQ-Pos-objectAhasthefollowingmap
→:A×A✲Q:(a,b)→{q∈Q|q∗ab}.
Considerapropertyoftheaforesaidmap.Startbyrecallingsomepreliminarynotions(cf.
Chapter0-3in[16]).
Definition5.10.5.LetCbeanorderedcategory(i.e.,hom-setsarepartiallyorderedand
compositiononbothsidesisorder-preserving).ApairofC-morphismsA✛g✲Biscalled
danadjunctionbetweenAandBprovidedthatidBg◦dandd◦gidA.
ForeveryquantaleQonehastheorderedcategoryQ-Pos.
Definition5.10.6.LetAbeaQ-posetandletq∈Q.DefineAqA✲A:a→q∗a.
Remark5.10.7.GivenaQ-Pos-morphismAf✲B,itfollowsthatqB◦ff◦qAfor
everyq∈Q.
Definition5.10.6givesthefollowingcharacterizationofadjunctionsinQ-Pos.
Lemma5.10.8.LetA✛g✲BbeQ-Pos-morphisms.Thefollowingareequivalent:
d

90

(i)(g,d)isanadjunctionbetweenAandB;
(ii)qBg◦qA◦dandd◦qB◦gqAforeveryq∈Q.
SimilarlytoTheorem0-3.2in[16]oneshowsthefollowing.
Theorem5.10.9.LetA✛g✲BbemapsbetweenQ-posets.Thefollowingareequivalent:
d(i)(g,d)isanadjunctionbetweenAandB;
(ii)(a)disaQ-Pos-morphism;
(b)g(a)=maxd−1[↓a]foreverya∈A;
(c)d◦qB◦gqAforeveryq∈Q.
Nowthepromisedproperty.
Corollary5.10.10.LetAbeaQ-posetandleta∈A.Thefollowingareequivalent:
(i)(a→,∗a)isanadjunctionbetweenAandQ(andthusa→(S)=(a→s)
S∈sforeveryS⊆AsuchthatSexistsinA);
(ii)(a→b)∗abforeveryb∈A.
Definition5.10.11.GivenaQ-posetA,saythatitsatisfiescondition(A)providedthat
(a→b)∗abforeverya,b∈A.
Remark5.10.12.LetQbeaquantale.TheneveryQ-modulesatisfiescondition(A).The
situationwithQ-posets,however,isdifferentasshowsthefollowingexample.Let2be
orderedbyequality.DefineQ×2∗✲2:(q,a)→a.Then2isaQ-posetwhichdoesnot
).A(tisfysaLemma5.10.13.LetAbeaQ-Pos-object.Then(b→c)·(a→b)(a→c)forevery
a,b,c∈A.
Nowthemainproposition.
Proposition5.10.14.LetQbeaquantale.ThenthecategoryQ-Modisreflectivein
.sPo-QProof.GivenaQ-posetA,thereflectionarrowcanbeconstructedasfollows.DefineBA=
underarbitrarymeets)andh∈BAiffh(a)=h(b)·(a→b)foreverya∈A.Let
{h∈QA|h(b)·(a→b)h(a)foreverya,b∈A}.ThenBAisasubmoduleofQA(closed
A∈bAr✲BA:a→→a.ByLemma5.10.13themapriscorrect.Fortherestsee
Proposition5.10.2.

BelowaresomepropertiesofthereflectionarrowAr✲BA.

Lemma5.10.15.LetQbecompletelydistributive.Thenrpreservesallexistingmeets.

91

(a→s).Seta→(S)=Tand(a→s)=Ts.Itwillbeenoughtoshow
Proof.LetS⊆AbesuchthatSexistsinAandleta∈A.Showthata→(S)=
thatTsT.Bytheassumption,Ts=f(s)whereFisthesetof
s∈Ss∈Ss∈S
choicefunctionsdefinedonS.Sincef(s)∈Tforeveryf∈F,theresultfollows.
s∈Ss∈Sf∈Fs∈S
S∈sLemma5.10.16.LetAsatisfycondition(A).Thenrisinjectiveandpreservesallexisting
s.teme

Proof.SincethesecondstatementfollowsfromCorollary5.10.10weshowthatrisinjective.
Leta,b∈Awithr(a)=r(b).Thenea→a=(r(a))(a)=(r(b))(a)=a→bimplies
a=e∗a(a→b)∗ab.Similarlyba.

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