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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.