Generalized Eilenberg Theorem I: Local Varieties of Languages

0.36 MB·English

by  Jiri Adamek

Generalized Eilenberg Theorem I:
Local Varieties of Languages
Jiˇr´ıAda´mek1,StefanMilius2,RobertS.R.Myers1andHenningUrbat1
1 Institutfu¨rTheoretischeInformatik
5
TechnischeUniversita¨tBraunschweig,Germany
1 2 Lehrstuhlfu¨rTheoretischeInformatik
0
Friedrich-Alexander-Universita¨tErlangen-Nu¨rnberg,Germany
2

n
a DedicatedtoManuelaSobral.
J

2
1 Abstract. Weinvestigatethedualitybetweenalgebraicandcoalgebraicrecog-
nitionoflanguagestoderiveageneralizationofthelocalversionofEilenberg’s

] theorem.Thistheoremstatesthatthelatticeofallbooleanalgebrasofregularlan-
L
guagesover analphabetΣ closedunderderivativesisisomorphictothelattice
F ofallpseudovarietiesofΣ-generatedmonoids. Byapplyingourmethodtodif-
.
s ferentcategories,weobtainthreerelatedresults:one,duetoGehrke,Grigorieff
c andPin,weakensbooleanalgebrastodistributivelattices,oneweakensthemto
[ join-semilattices,andthelastoneconsidersvectorspacesoverZ2.

1
v
4 1 Introduction
3
8 Regularlanguagesarepreciselythebehavioursoffiniteautomata.Amachine-indepen-
2
dentcharacterizationofregularityisthestartingpointofalgebraicautomatatheory(see
0
e.g.[16]):onedefinesrecognitionviapreimagesofmonoidmorphismsf ∶ Σ∗ → M,
.
1
where M is a finite monoid,and everyregularlanguageis recognizedin this way by
0
itssyntacticmonoid.Thissuggeststoinvestigatehowoperationsonregularlanguages
5
1 relate to operationson monoids. Recall that a pseudovariety of monoids is a class of
: finite monoidsclosed underfinite products,submonoidsandquotients(homomorphic
v
images),andavarietyofregularlanguagesisaclassofregularlanguagesclosedunder
i
X thebooleanoperations(union,intersectionandcomplement),leftandrightderivatives1
r and preimages of monoid morphisms Σ∗ → Γ∗. Eilenberg’s variety theorem [9], a
a
cornerstoneofautomatatheory,establishesalatticeisomorphism
varietiesofregularlanguages ≅ pseudovarietiesofmonoids.
Numerous variations of this correspondence are known, e.g. weakening the closure
propertiesinthedefinitionofavariety[15,18],orreplacingregularlanguagesbyfor-
mal powerseries [20]. Recently Gehrke,GrigorieffandPin [11, 12] proveda “local”
version of Eilenberg’stheorem: for every fixed alphabet Σ, there is a lattice isomor-
phismbetweenlocalvarietiesofregularlanguages(sets ofregularlanguagesoverΣ
1Theleft and right derivatives of a language L ⊆ Σ∗ arew−1L = {u ∈ Σ∗ ∶ wu ∈ L} and
Lw−1={u∈Σ∗∶uw∈L}forw∈Σ∗,respectively.
2 J.Ada´mek,S.Milius,R.S.R.MyersandH.Urbat
closedunderbooleanoperationsandderivatives)andlocalpseudovarietiesofmonoids
(setsofΣ-generatedfinitemonoidsclosedunderquotientsandsubdirectproducts).At
theheartofthisresultliestheuseofStonedualitytorelatethebooleanalgebraofreg-
ularlanguagesoverΣ,equippedwithleftandrightderivatives,tothefreeΣ-generated
profinitemonoid.
InthispaperwegeneralizethelocalEilenbergtheoremtothelevelofanabstractdu-
ality.Ourapproachstartswiththeobservationthatallconceptsinvolvedinthistheorem
areinherentlycategorical:
(1) The boolean algebra Reg of all regular languages over Σ naturally carries the
Σ
structure of a deterministic automaton whose transitions L Ð→a a−1L for a ∈ Σ
are given by left derivation and whose final states are the languages containing
the empty word. In other words, Reg is a coalgebrafor the functorT Q = 2×
Σ Σ
QΣ on the categoryof boolean algebras, where 2 = {0,1} is the two-chain. The
coalgebraReg canbecapturedabstractlyastherationalfixpoint̺T ofT ,i.e.,
Σ Σ Σ
theterminallocallyfiniteT -coalgebra[14].
Σ
(2) Monoids are precisely the monoid objects in the category of sets, viewed as a
monoidalcategoryw.r.t.thecartesianproduct.
(3) The categories of boolean algebras and sets occurring in (1) and (2) are locally
finitevarietiesofalgebras(thatis, theirfinitelygeneratedalgebrasarefinite),and
thefullsubcategoriesoffinitebooleanalgebrasandfinitesetsareduallyequivalent
viaStoneduality.
Inspiredby(3),wecalltwolocallyfinitevarietiesC undD of(possiblyordered)alge-
braspredualiftherespectivefullsubcategoriesC andD offinitealgebrasaredually
f f
equivalent.OuraimistoprovealocalEilenbergtheoremforanabstractpairofpredual
varietiesC andD,theclassicalcasebeingcoveredbytakingC =booleanalgebrasand
D =sets.Inthissettingdeterministicautomataaremodeledbothascoalgebrasforthe
functor
T ∶C →C, T Q=2×QΣ,
Σ Σ
andasalgebrasforthefunctor
LΣ ∶D→D, LΣA=1+∐A,
Σ
where2isatwo-elementalgebrainCand1isitsdualfinitealgebrainD.Thesefunctors
arepredualinthesensethattheirrestrictionsT ∶C →C andL ∶D →D tofinite
Σ f f Σ f f
algebrasare dual, and therefore the categoriesof finite T -coalgebrasand finite L -
Σ Σ
algebrasareduallyequivalent.AsafirstapproximationtothelocalEilenbergtheorem,
weconsidertherationalfixpoint̺T forT –whichisalwayscarriedbytheautomaton
Σ Σ
Reg ofregularlanguages–andtheinitialalgebraµL forL andestablishalattice
Σ Σ Σ
isomorphism
subcoalgebrasof̺T ≅ idealcompletionoftheposetoffinitequotientalgebrasofµL .
Σ Σ
Thisis“almost”thedesiredgenerallocalEilenbergtheorem.Fortheclassicalcase(C =
booleanalgebrasand D = sets) one has ̺T = Reg and µL = Σ∗, and the above
Σ Σ Σ
GeneralizedEilenbergTheoremI:LocalVarietiesofLanguages 3
isomorphismstatesthatthebooleansubalgebrasofReg closedunderleftderivatives
Σ
correspondtosetsoffinitequotientalgebrasofΣ∗closedunderquotientsandsubdirect
products.Whatismissingistheclosureunderright derivativesonthecoalgebraside,
andquotientalgebrasofΣ∗whicharemonoidsonthealgebraside.
The final step is to provethatthe aboveisomorphismrestricts to one between lo-
calvarieties of regularlanguagesin C (i.e., subcoalgebrasof ̺T closed underright
Σ
derivatives)andlocalpseudovarietiesofD-monoids.HereaD-monoidisamonoidob-
jectinthemonoidalcategory(D,⊗,Ψ1)where⊗isthetensorproductofalgebrasand
Ψ1 is the free algebraon onegenerator.In moreelementaryterms, a D-monoidis an
algebraAin D equippedwitha “bilinear”monoidmultiplicationA×A Ð→○ A, which
meansthatthemapsa○−and−○aareD-morphismsforalla ∈ A. Forexample,D-
monoidsinD =sets,posets,join-semilatticesandvectorspacesoverZ2 aremonoids,
orderedmonoids,idempotentsemiringsandZ2-algebras(inthesenseofalgebrasover
afield),respectively.Inalltheseexamplesthemonoidalcategory(D,⊗,Ψ1)isclosed:
thesetD(A,B)ofhomomorphismsbetweentwoalgebrasAandB isanalgebrainD
withthepointwisealgebraicstructure.Ourmainresultisthe
GeneralLocalEilenberg Theorem.LetC andD be preduallocallyfinite varieties
ofalgebras,wherethealgebrasinD arepossiblyordered.SupposefurtherthatD is
monoidalclosedw.r.t.tensorproduct,epimorphismsinDaresurjective,andthefree
algebrainD ononegeneratorisdualtoatwo-elementalgebrainC.Thenthereisa
latticeisomorphism
localvarietiesofregularlanguagesinC ≅ localpseudovarietiesofD-monoids.
ByapplyingthistoStoneduality(C =booleanalgebrasandD=sets)werecoverthe
“classical”localEilenbergtheorem.Birkhoffduality(C =distributivelatticesandD =
posets)givesanotherresultofGehrkeet.al,namelyalatticeisomorphismbetweenlocal
latticevarietiesofregularlanguages(subsetsofReg closedunderunion,intersection
Σ
and derivatives)and local pseudovarietiesof orderedmonoids. Finally,by taking C =
D =join-semilatticesandC = D =vectorspacesoverZ2,respectively,weobtaintwo
newlocalEilenbergtheorems.Thefirstoneestablishesalatticeisomorphismbetween
local semilattice varieties of regular languages (subsets of Reg closed under union
Σ
andderivatives)andlocalpseudovarietiesofidempotentsemirings,andthesecondone
givesan isomorphismbetween locallinear varieties of regularlanguages(subsets of
Reg closedundersymmetricdifferenceandderivatives)andlocalpseudovarietiesof
Σ
Z2-algebras.
AsaconsequenceoftheGeneralLocalEilenbergTheoremwealsogainageneral-
izedviewofprofinitemonoids.ThedualequivalencebetweenC andD liftstoadual
f f
ˆ
equivalencebetweenC andacategoryDarisingasaprofinitecompletionofD .Inthe
f
ˆ
classical case we have C = booleanalgebras,D = sets and D = Stonespaces, andthe
ˆ ˆ
dualequivalencebetweenC andDisgivenbyStoneduality.ThentheD-objectdualto
therationalfixpoint̺T ∈C canbeequippedwithamonoidstructurethatmakesitthe
Σ
freeprofiniteD-monoidonΣ.
4 J.Ada´mek,S.Milius,R.S.R.MyersandH.Urbat
Theorem.UndertheassumptionsoftheGeneralLocalEilenbergTheorem,thefree
profiniteD-monoidonΣ isdualtotherationalfixpoint̺T .
Σ
ThisextendsthecorrespondingresultsofGehrke,GrigorieffandPin[11]forD =sets
andD=posets.
Thepresentpaperisarevisedandextendedversionoftheconferencepaper[2],provid-
ingfullproofsandmoredetailedexamples.Incomparisontoloc.cit.weworkwitha
slightlymodifiedcategoricalframeworkinordertosimplifythepresentation,seeSec-
tion6.
Relatedwork. OurpaperisinspiredbytheworkofGehrke,GrigorieffandPin[11,12]
whoshowedthatthealgebraicoperationofthefreeprofinitemonoidonΣ dualizesto
thederivativeoperationsonthebooleanalgebraofregularlanguages(andsimilarlyfor
the free orderedprofinite monoid on Σ). Previously,the duality between the boolean
algebraofregularlanguagesandtheStonespaceofprofinitewordsappeared(implic-
itly)inworkbyAlmeida[5]andwasformulatedbyPippenger[17]intermsofStone
duality.
Acategoricalapproachtothedualitytheoryofregularlanguageshasbeensuggested
by Rhodesand Steinberg[21]. They introducethe notionof a booleanbialgebra,and
provethe equivalenceof bialgebrasandprofinite semigroups.The precise connection
totheirworkisyettobeinvestigated.
Another related work is Pola´k [18] and Reutenauer [20]. They consider what we
treat as the example of join-semilattices and vector spaces, respectively, and obtain a
(non-local)Eilenbergtheoremforthese cases. To the bestof ourknowledgethe local
versionwe provedoesnotfollow fromthe globalversion,and so we believethatour
resultisnew.
Theoriginofalltheaboveworkis, ofcourse,Eilenberg’stheorem[9].LaterRei-
terman[19]provedanothercharacterizationofpseudovarietiesofmonoidsinthespirit
ofBirkhoff’sclassicalvarietytheorem.Reiterman’stheoremstatesthatanypseudova-
rietyofmonoidscanbecharacterizedbyprofiniteequations(i.e.,pairsofelementsofa
freeprofinitemonoid).Wedonottreatprofiniteequationsinthepresentpaper.
2 The Rational Fixpoint
Theaimof thissection istorecalltherationalfixpointofa functor,whichprovidesa
anabstractcoalgebraicviewofthesetofregularlanguages.Asaprerequisite,weneed
acategoricalnotionof“finiteautomaton”,andsowewillworkwithcategorieswhere
enough“finite”objectsexist–viz.locallyfinitelypresentablecategories[4].
Definition2.1. (a) An object X of a category C is finitely presentable if the hom-
functorC(X,−)∶C →Setisfinitary(i.e.,preservesfilteredcolimits).LetC denote
f
thefullsubcategoryofallfinitelypresentableobjectsofC.
(b) C islocallyfinitelypresentableifitiscocomplete,C issmalluptoisomorphism
f
andeveryobjectofC isafilteredcolimitoffinitelypresentableobjects.
GeneralizedEilenbergTheoremI:LocalVarietiesofLanguages 5
Example2.2. Let Γ be a finitary signature, that is, a set of operation symbols with
finitearity.
(1) DenotebyAlgΓ thecategoryofΓ-algebrasandΓ-homomorphisms.Avarietyof
algebras is a full subcategory of AlgΓ closed under products, subalgebras (rep-
resentedbyinjectivehomomorphisms)andhomomorphicimages(representedby
surjective homomorphisms).Equivalently,by Birkhoff’stheorem [7], varieties of
algebras are precisely the classes of algebras definable by equations of the form
s = t, wheres andt areΓ-terms.Everyvarietyofalgebrasislocallyfinitelypre-
sentable[4,Corollary3.7].
(2) Similarly, let Alg Γ be the category of ordered Γ-algebras. Its objects are Γ-
≤
algebrascarryingaposetstructureforwhicheveryΓ-operationismonotone,and
its morphisms are monotone Γ-homomorphisms. A variety of ordered algebras
is a full subcategory of Alg Γ closed under products, subalgebras and homo-
≤
morphic images. Here subalgebras are represented by embeddings (injective Γ-
homomorphismsthat are both monotoneand order-reflecting),and homomorphic
imagesarerepresentedbysurjectiveΣ-homomorphismsthataremonotonebutnot
necessarilyorder-reflecting.
Bloom[8]provedanorderedanalogueofBirkhoff’stheorem:varietiesofordered
algebrasarepreciselytheclassesoforderedalgebrasdefinablebyinequalitiess≤t
betweenΓ-terms.Fromthisitiseasytoseethateveryvarietyoforderedalgebrasis
finitarymonadicoverthelocallyfinitelypresentablecategoryofposets,andhence
locallyfinitelypresentable[4,TheoremandRemark2.78].
Inourapplicationswewillworkwiththevarietiesinthetablebelow.Observethatall
these varietiesare locally finite, that is, their finitely presentableobjects are precisely
thefinitealgebras.
C objects morphisms
Set sets functions
BA booleanalgebras booleanmorphisms
DL01 distributivelatticeswith0and1 latticemorphismspreserving0and1
JSL0 join-semilatticeswith0 semilatticemorphismspreserving0
VectZ2 vectorspacesoverthefieldZ2 linearmaps
Pos partiallyorderedsets monotonefunctions
Remark2.3. For the rest of this paper the term “variety” refers to both varieties of
algebrasandvarietiesoforderedalgebras.
Notation2.4. Fix a locally finitely presentable categoryC and a finitary endofunctor
T ∶C →C.
Definition2.5. A T-coalgebra is a pair (Q,γ) of a C-object Q and a C-morphism
γ ∶Q→TQ.Ahomomorphism
h∶(Q,γ)→(Q′,γ′)
6 J.Ada´mek,S.Milius,R.S.R.MyersandH.Urbat
of T-coalgebras is a C-morphism h ∶ Q → Q′ with γ′ ⋅h = Th⋅γ. We denote by
CoalgT thecategoryofallT-coalgebrasandtheirhomomorphisms,andbyCoalg T
f
thefullsubcategoryofT-coalgebras(Q,γ)withfinitelypresentablecarrierQ(inthe
casewhereC isalocallyfinitevariety,thesearepreciselythefinitecoalgebras).
Example2.6. GivenafinitealphabetΣ andanobject2inC,theendofunctor
T =2×IdΣ =2×Id×Id×...×Id
Σ
ofC isfinitarysinceinanylocallyfinitelypresentablecategoryfilteredcolimitscom-
mutewithfiniteproducts.IfC isalocallyfinitevarietyand2isatwo-elementalgebra
inC,thenT -coalgebrasaredeterministicautomata,seee.g.[22].Indeed,bytheuni-
Σ
versalpropertyoftheproduct,togiveacoalgebraQÐ→γ T Q=2×QΣmeansprecisely
Σ
to give an algebra Q (of states), morphismsγ ∶ Q → Q for every a ∈ Σ (represent-
a
inga-transitions)anda morphismf ∶ Q → 2(representingfinalstates). Herearetwo
specialcases:
(a) Theusualconceptofadeterministicautomaton(withoutinitialstates)iscaptured
as a coalgebraforT where C = Set and 2 = {0,1}.An importantexampleof a
Σ
T -coalgebraistheautomatonReg ofregularlanguages.Itsstatesaretheregular
Σ Σ
languagesoverΣ,itstransitionsare
γ (L)=a−1L forallL∈Reg anda∈Σ,
a Σ
andthefinalstatesarepreciselythelanguagescontainingtheemptywordε.
(b) Analogously,considerT asanendofunctorofC = BAwith2 = {0,1}(thetwo-
Σ
element boolean algebra). A coalgebra for T is a deterministic automaton with
Σ
a booleanalgebrastructureonthe state set Q. Moreover,the transitionmapsγ ∶
a
Q → Q are boolean homomorphisms, and the final states (given by the inverse
image of 1 under f ∶ Q → 2) form an ultrafilter. The above automaton Reg is
Σ
alsoaT -coalgebrainBA:thesetofregularlanguagesisabooleanalgebraw.r.t.
Σ
theusualset-theoreticoperations,leftderivativespreservetheseoperations,andthe
languagescontainingεformaprincipalultrafilter.
Definition2.7. (a) A coalgebra is called locally finitely presentable if it is a filtered
colimitofcoalgebraswithfinitelypresentablecarrier.ThefullsubcategoryofCoalgT
ofalllocallyfinitelypresentablecoalgebrasisdenotedCoalg T.
lfp
(b) TherationalfixpointofT isthefilteredcolimit
r∶̺T →T(̺T)
of all coalgebras with finitely presentable carrier, i.e., the colimit of the diagram
Coalg T ↣CoalgT.
f
Theterm“rationalfixpoint”isjustifiedbyitem(a)inthetheorembelow.
Theorem2.8 (see[14]).
(a) risanisomorphism.
GeneralizedEilenbergTheoremI:LocalVarietiesofLanguages 7
(b) ̺T istheterminallocallyfinitelypresentableT-coalgebra,i.e.,everylocallyfinitely
presentableT-coalgebrahasauniquecoalgebrahomomorphisminto̺T.
Example2.9. TherationalfixpointofT ∶Set→Setistheautomaton̺T =Reg of
Σ Σ Σ
Example2.6(a),see [3].ForanylocallyfinitelypresentableT -coalgebra(Q,γ),the
Σ
uniquehomomorphism(Q,γ)→̺T mapseachstateq∈Qtoitsacceptedlanguage
Σ
Lq ={a1...an∈Σ∗ ∶ qÐa→1 q1Ða→2 q2→⋯Ða→n qnforsomefinalstateqn}.
Thisexamplecanbegeneralized:
Theorem2.10. SupposethatCisalocallyfinitevarietyandT liftsafinitaryfunctorT0
onSet,thatis,thefollowingdiagram(whereU denotestheforgetfulfunctor)commutes:
C T //C
U U
(cid:15)(cid:15) (cid:15)(cid:15)
Set //Set
T0
ThenthefunctorU∶CoalgT →CoalgT0givenby
QÐ→γ TQ ↦ UQÐUÐ→γ UTQ=T0UQ
preservestherationalfixpoint,i.e.,
U(̺T)≅̺T0.
Proof. ThefunctorUisfinitary(sincefilteredcolimitsofT-coalgebrasareformedon
the level of C and hence on the level of Set) and restricts to finite coalgebras, so we
haveacommutativesquare
CoalgT U //CoalgT0
OO OO
I I0
OO OO
CoalgfT V // CoalgfT0
whereI andI0 aretheinclusionfunctors.WewillprovebelowthatViscofinal,from
whichtheclaimfollows:
U(̺T)=U(colimI) def.̺T
≅colim(UI) Ufinitary
≅colim(I0V) UI =I0V
≅colim(I0) Vcofinal
=̺T0 def.̺T0
ThecofinalityofVamountstoprovingthat
8 J.Ada´mek,S.Milius,R.S.R.MyersandH.Urbat
(1) for every finite T0-coalgebra (Q,γ) there exists a T0-coalgebra homomorphism
(Q,γ)→V(Q′,γ′)forsomefiniteT-coalgebra(Q′,γ′),and
(2) anytwosuchcoalgebrahomomorphismsareconnectedbyazig-zag.
Proofof(1).LetΦ ∶ Set → C betheleftadjointoftheforgetfulfunctorU ∶ C → Set,
anddenotetheunitandcounitoftheadjunctionbyηandε,respectively.Givenafinite
T0-coalgebraQÐ→γ T0Qformthe“free”T-coalgebra
ΦQÐΦ→γ ΦT0QÐΦÐTÐ0ηÐQ→ΦT0UΦQ=ΦUTΦQÐεÐTΦÐQ→TΦQ.
NotethatΦQisfinitebecauseC islocallyfinite.Then
ηQ∶(Q,γ)→V(ΦQ,εTΦQ⋅ΦT0ηQ⋅Φγ)
isacoalgebrahomomorphism.Indeed,thediagrambelowcommutesbythenaturality
ofηandthetriangleidentityUε⋅ηU =id:
Q γ // T0Q
ηT0Q vv♥♥♥T♥0♥η♥Q♥♥♥♥♥♥♥
ηQ T0UΦQ T0ηQ
(cid:15)(cid:15) ww uu❧❧❧η❧T❧0❧U❧Φ❧Q❧❧❧❧❧❧ PPPPPPPPPPPPPPPPPPPPPPPP (cid:15)(cid:15)
UΦQ //UΦT0Q // UΦT0UΦQ=UΦUTΦQ //UTΦQ=T0UΦQ
UΦγ UΦT0ηQ UεTΦQ
Proofof(2).Givenanycoalgebrahomomorphismh∶(Q,γ)→V(Q′,γ′)thereexistsa
uniqueD-morphismh∶ΦQ→Q′withUh⋅η =hbytheuniversalpropertyofη.We
Q
claimthathisacoalgebrahomomorphism
h∶(ΦQ,εTΦQ⋅ΦT0ηQ⋅Φγ)→(Q′,γ′).
Indeed,thelowersquareinthediagrambelowcommuteswhenprecomposedwithη ,
Q
fromwhichtheequationγ′⋅h=Th○εTΦQ⋅ΦT0ηQ⋅Φγ follows.
Q γ //T0Q
ηQ T0ηQ
(cid:15)(cid:15) (cid:15)(cid:15)
h UΦQ // T0UΦQ T0h
U(εTΦQ⋅ΦT0ηQ⋅Φγ)
Uh T0Uh
$$ (cid:15)(cid:15) (cid:15)(cid:15) zz
UQ′ //T0UQ′
Uγ′
Now giventwo coalgebrahomomorphismsh ∶ (Q,γ) → V(Q′,γ′) and k ∶ (Q,γ) →
V(Q′′,γ′′),thedesiredzig-zaginCoalg T is Q′oo h ΦQ k //Q′′. ⊓⊔
f
GeneralizedEilenbergTheoremI:LocalVarietiesofLanguages 9
Corollary2.11. Let C be a locally finite variety with a two-element algebra 2. Then
the rational fixpoint of T = 2×IdΣ ∶ C → C is carried by the automaton Reg of
Σ Σ
Example2.6(a).For any locally finitelypresentableT -coalgebra(Q,γ), the unique
Σ
homomorphism(Q,γ)→̺T mapseachstateq∈Qtoitsacceptedlanguage.
Σ
Proof. ApplyTheorem2.10to T = TΣ andT0 = {0,1}×IdΣ.Since ̺T0 = RegΣ by
Example2.9,theclaimfollows. ⊓⊔
NextwewillshowthatthelocallyfinitelypresentableT-coalgebrasariseasa“free
completion”ofthecoalgebraswithfinitelypresentablecarrier(Theorem2.14below).
Remark2.12. (a) Recall that the free completion under filtered colimits of a small
category A is a full embedding A ↪ IndA such that IndA has filtered colimits
andeveryfunctorF ∶A→BintoacategoryB withfilteredcolimitshasafinitary
extensionF ∶IndA→B,uniqueuptonaturalisomorphim:
A // //IndA
❊❊❊F❊❊❊❊❊❊"" (cid:15)(cid:15)✤✤✤F
B
This determines IndA up to equivalence. If A has finite colimits then IndA is
locally finitely presentable and (IndA) ≅ A. Conversely, every locally finitely
f
presentablecategoryC arisesinthisway:C ≅Ind(C ).
f
(b) IfAisajoin-semilatticethenIndAisitsidealcompletion,seeRemark3.15.
Lemma2.13. LetBbeacocompletecategoryandJ ∶A↣Bbeasmallfullsubcate-
goryoffinitelypresentableobjectsclosedunderfinitecolimits.Thentheuniquefinitary
extensionJ∗∶Ind(A)→Bformsafullcoreflectivesubcategory.
Proof. By the theoreminSectionVI.1.8ofJohnstone[13],we knowthatJ∗ isa full
embeddingsothatInd(A)canbeidentifiedwiththefullsubcategoryofBgivenbyall
filteredcolimitsofobjectsfromA.Wewillshowthatthisfullsubcategoryiscoreflec-
tive.LetB beanobjectofBanddefineB tobethecolimitofthefiltereddiagram
A/B //A (cid:31)(cid:127) //B,
wherethefirstarrowisthecanonicalprojectionfunctorandthesecondonetheinclusion
functor.Wedenotethecorrespondingcolimitinjectionsby
in ∶A→B foreveryf ∶A→B inA/B.
f
Clearly, the objects in A/B form a cocone on the above diagram and so we have a
uniquemorphismb∶B →B suchthat
b⋅in =f foreveryf ∶A→BinA/B.
f
Wewillnowprovethismorphismbtobecouniversal.Tothisend,letAbeanobjectof
Ind(A),i.e.,
A=colimA
i
i∈I
10 J.Ada´mek,S.Milius,R.S.R.MyersandH.Urbat
isafilteredcolimitinB ofobjectsfromA withcolimitinjectionsa ∶ A → A, i ∈ I.
i i
Givenamorphismf ∶A→B inB,themorphismf ⋅a ∶A →B isanobjectofA/B,
i i
andforeachconnectingmorphisma ∶A →A wehave
i,j i j
(f ⋅a )⋅a =f ⋅a .
j i,j i
Thus,a isamorphisminA/B andso
i,j
in ⋅a =in ,
f⋅aj i,j f⋅ai
i.e.,themorphismsin ∶A →Bformacocone.Sowegetauniquef ∶A→B such
f⋅ai i
that
f ⋅a =in foreveryi∈I.
i f⋅ai
Nowthefollowingdiagramcommutes:
B b //B
?? OO ??
inf⑦⋅a⑦⑦i⑦⑦⑦⑦⑦ f(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)f(cid:0)(cid:0)(cid:0)
A // A
i ai
Indeed, the outside and left-hand triangle commute, and so the right-hand one com-
muteswhenprecomposedwitheverya ,i∈I,whencethattrianglecommutessincethe
i
colimitinjectionsa formajointlyepimorphicfamily.
i
Westillneedtoshowthatf isuniquesuchthatb⋅f =f.Soassumethatf ∶A→B
is anysuch morphism.Fix i ∈ I. Then,since B is a filtered colimitandA is finitely
i
presentable,itfollowsthatthereexistssomeg ∶ A′ →B inA/B andsomemorphism
i
f′∶A →A′ suchthatthesquarebelowcommutes:
i i
in
A′ g // B
OOi OO
′
f f
A // A
i ai
Itfollowsthatf′ isaconnectingmorphisminA/B fromf ⋅a tog:
i
g⋅f′=b⋅in ⋅f′=b⋅f ⋅a =f ⋅a .
g i i
Thereforewegetin ⋅f′=in sothat
g f⋅ai
f ⋅a =in ⋅f′=in ,
i g f⋅ai
whichdeterminesf uniquely.Thiscompletestheproof. ⊓⊔
Theorem2.14. Coalg T istheInd-completionofCoalg T andformsacoreflective
lfp f
subcategoryofCoalgT.