Localizations, colocalizations and non additive *-objects

LOCALIZA TIONS, COLOCALIZA TIONS AND NON ADDITIVE ∗ -OBJECTS GEORGE CIPRIAN MODO I Abstra ct. Giv en a pair of adjoin t fun ctors b etw een tw o arbitrary cat- egories it induces mutually in verse equiv a lences b etw een the full sub- categories of the initial ones, consisting of ob jects for whic h the arro ws of adjunction are isomorphisms. W e inves tigate some cases in which these sub categories may b e b etter characterize d. One application is the construction of cellular approximatio ns. Other is th e definition and the chara cterization of (weak) ∗ -ob jects in the non additive case. Introduction In mathematics the concept of localizatio n has a long history . The origin of the concept is the stud y of some prop erties of maps around a p oin t of a top ological s p ace. In the algebraic sense, the lo calization provi des a metho d to in v e rt some morphisms in a category . Making abstraction of some tec h- nical set theoretic problems, giv en a class of morphisms Σ in a category A , there is a categ ory A [Σ − 1 ] and a functor A → A [Σ − 1 ] univ ersal with the prop erty that it s en ds an y morphism in Σ to an isomorphism. This functor will b e called a lo c a lization , if it h as a right adjoint, whic h will b e frequen tly fully faithful. Dually this functor is called a c olo c alization pro vided that it has a left adjoin t. One of the starting p oin t of this pap er is the observ ation that the con- sequences of the dualit y b etw een lo calizatio n and col o calizati on we re not exhausted. F or example the concept, b orrow ed fr om top ology , of cell ular appro ximation in arb itrary category is a p articular case of a colo calizatio n, fact remark ed for example in [4]. Some results concerning the cellular ap- pro ximation may b e d ed uced in a formal, categorical wa y b y stressing this dualit y . On th e other hand the same formal tec hn iques are us eful in the study of s o called ∗ -mo d u les, defined as in [3]. No w let us pr esen t the organization and the m ain results of the pap er. In the first section we set the notations, w e define the main notions used throughout of the pap er and w e record some easy prop ertie s concerning these notions. 2000 Mathematics Subje ct Classific ation. 18A40, 20M50, 18G50. Key wor ds and phr ases. adjoin t pair; localization; colocalization; cellular approxima- tion; ∗ -act ov er a monoid. The aut hor w as supp orted by the gran t PN2CD-ID- 489. 1 2 GEORG E C IPRIAN M ODOI In Section 2 are s tated the formal results, on whic h it is based the rest of th e p ap er. There are three main results here: First T heorem 2.4 w here are giv en necessary and sufficient conditions for a pair of ad j oin t fu nctors to induce an equiv alence b et w een the full sub categories consisting of colo cal resp ectiv ely lo cal ob jects with resp ect to these fun ctors (for the definition of a (co)local ob ject see Section 1). S econd and thir d Prop ositio n 2.10 and Theorem 2.11 wh ic h represent the formal c h aracterizat ion of a non additiv e (w eak) ∗ -ob ject. Section 3 con tains a non add itiv e v ersion of a theorem of Menin i and Orsatti in [7]. Consider an ob ject (or a set of ob jects) A of a categ ory A , and the category of all con tra v ariant fun ctors [ E op , S et ] where E is the full sub ca tegory of A conta ining the ob ject(s) A , situation whic h is less general but more comprehensiv e that the h yp otheses of Section 3. Under appropriate assump tions, mutually inv erse equiv alences b et w een tw o full sub categories of A and [ E op , S et ] are r epresen ted by A in the sense that they are realized by r estrictions of the representable f u nctor H A = A ( A, − ) and of its left adjoint (see Theorem 3.2). Pro v id ed that A is a co complete, w ell cop o wered, balanced category with epimorphic image s, and A i s a se t of o b jects o f A , it is sho wn in Secti on 4 that the inclusion of the sub ca tegory of H A -colocal ob jects h as a r igh t adjoin t (see Th eorem 4.2). Consequent ly fixin g an ob ject A in suc h a category , ev ery ob ject X will h a ve an A -cellular approximat ion. In Section 5 we defin e and c h aracterize the n otions of a (w eak) ∗ -act o ver a monoid, in Pr op osition 5.2 and Theorem 5.3, pro viding in this wa y a translation of the notion of (wea k) ∗ -mo dule in th ese new settings. It is in teresting to note that our appr oac h may b e conti n ued b y dev eloping a theory analogo us with s o called tilting theory for m o dules. The Morita theory for the category of acts ov er m onoids is a consequence of our results. 1. N o t a tions and p r eliminaries All sub ca tegories wh ic h we consider are full and closed un der isomor- phisms, so if we sp eak ab out a class of ob jects in a category w e unders tand also the resp ect iv e sub categ ory . F or a categ ory A w e denote b y A → the cat- egory of all morp hisms in A . W e denote b y A ( − , − ) the b ifunctor assigning to any t w o ob jects of A the set of all morphism s b et we en them. Consider a functor H : A → B . The (essent ial) image of H is the sub - catego ry Im H of B consisting of all ob jects Y ∈ B satisfying Y ∼ = H ( X ) for s ome X ∈ A . In con trast we shall den ote by im α th e categ orical notion of image of a morphism α ∈ A → . A morphism α ∈ A → is called an H - e quivalenc e , p ro v id ed that H ( α ) is an isomorphism. W e denote by E q ( H ) the sub categ ory of A → consisting of all H -equiv alences. An ob ject X ∈ A is called H - lo c al ( H -c olo c al ) if, f or an y H -equiv alence ǫ , the induced map ǫ ∗ = A ( ǫ, X ) (resp ectiv ely , ǫ ∗ = A ( X , ǫ )) is b ijectiv e, that means it is an iso- morphism in the category S et of all sets. W e denote by C H and C H the full LOCALIZA TIONS, COLOCALIZA TIONS AND NON ADDITIVE ∗ -OBJECTS 3 sub categories of A consisting of all H -lo cal, resp ect iv ely H -colocal ob jects. F or ob jects X ′ , X ∈ A , w e sa y that X ′ is a r etr act of X if there are maps α : X ′ → X and β : X → X ′ in A su c h that β α = 1 X ′ . W e record without pro of the follo wing prop erties relativ e to the ab o v e considered n otions: Lemma 1.1. The fol lowing hold: a) Eq( H ) is close d under r etr acts in A → . b) Eq( H ) satisfies the ‘two out of thr e e’ pr op erty, namely if α, β ∈ A → ar e c omp osable morphisms, then if two of the morphisms α, β , β α ar e H -e quivalenc es, then so is the thir d. c) The sub c ate gory C H (r esp e ctively C H ) is close d under limits (r e sp e c- tively c olimits) and r etr acts in A . Moreo v er if ev ery ob ject of A has a left (righ t) app ro ximation with an H -lo cal (colo cal) ob ject, in a sen s e b ecoming p recise in th e hypothesis of the Lemma b ello w, then w e are in the situation of a localization (colo calizat ion) functor, as it ma y b e seen fr om: Lemma 1.2. If for every X ∈ A ther e is an H -e qu ivalenc e X → X H with X H ∈ C H (r esp e ctively, X H → X with X H ∈ C H ), then the assignment X 7→ X H ( X 7→ X H ) is functorial and defines a left (right) adjoint of the inclusion functor C H → A ( C H → A ). Mor e over the left (right) adjoint of the inclusion functor sends every map α ∈ Eq( H ) into an isomorphism and it is universal r elative to this pr op e rty. Pr o of. Straightfo rw ard. (The first statemen t w as also notic ed in [5, 1.6 ]).  In the sequel we consider a pair of adjoint functors H : A → B at the righ t and T : B → A at the left, where A and B are arbitrary categories. W e sh all sym b olize this s ituation b y T ⊣ H . Cons ider also the arrows of adjunction δ : T ◦ H → 1 A and η : 1 B → H ◦ T . Note th at, for all X ∈ A and all Y ∈ B w e obtain the comm utativ e diagrams in B and A resp ectiv ely: (1) H ( X ) η H ( X ) / / 1 H ( X ) O O O O O O O O O O O O O O O O O O O O O O ( H ◦ T ◦ H )( X ) H ( δ X )   H ( X ) and T ( Y ) T ( η Y ) / / 1 T ( Y ) N N N N N N N N N N N N N N N N N N N N N N ( T ◦ H ◦ T )( Y ) δ T ( Y )   T ( Y ) sho wing that H ( X ) and T ( Y ) are retracts of ( H ◦ T ◦ H )( X ), r esp ectiv ely ( T ◦ H ◦ T )( Y ). Corresp ond in g to the a djoin t pair considered ab ov e, we define the f ollo wing full s u b categories of A and B : S H = { X ∈ A | δ X : ( T ◦ H )( X ) → X is an isomorphism } , and resp ectiv ely S T = { Y ∈ B | η Y : Y → ( H ◦ T )( Y ) is an isomorphism } . 4 GEORG E C IPRIAN M ODOI The ob jects in S H and S T are called δ -r eflexive , r esp ectiv ely η - r eflexiv e . Note that H and T r estrict to m utually in v erse equiv alences of categories b et w een S H and S T and these su b categories are the largest of A and B resp ectiv ely , enjo ying this p rop erty . 2. A n equiv alence ind u ced by adjoint functors In this s ection we fix a pair of adjoin t f u nctors T ⊣ H b et wee n t w o arbitrary categories A and B , as in Section 1. Lemma 2.1. The fol lowing inclusions hold: a) S H ⊆ Im T ⊆ C H ⊆ A . b) S T ⊆ Im H ⊆ C T ⊆ B . Pr o of. a) The first in clusion is ob vious. F or th e second in clusion observ e that for all ǫ ∈ Eq( H ) and all Y ∈ B the isomorp h ism in S et → ǫ ∗ = A ( T ( Y ) , ǫ ) ∼ = B ( Y , H ( ǫ )) sho ws that ǫ ∗ is b ij ective . The in clusions from b ) follo w by dualit y .  Lemma 2.2. L et C b e a sub c ate gory of A su ch that the inclusion functor I : C → A has a right adjoint R : A → C and the arr ow of the adjunction µ X : ( I ◦ R )( X ) → X is an H -e quivalenc e for al l X ∈ A . Then µ X is an isomorph ism for al l X ∈ C H , and c onse quently C H ⊆ C . Pr o of. Let X ∈ C H . Since µ X ∈ Eq( H ), we deduce that the in duced map µ ∗ X : A ( X, ( I ◦ R )( X )) → A ( X , X ) is b ij ectiv e, consequently there is a morphism µ ′ X : X → ( I ◦ R )( X ) su ch that µ X µ ′ X = 1 X . Since R ◦ I ∼ = 1 C naturally , and µ is also n atural, w e obtain a commutat iv e diagram ( I ◦ R )( X ) ( I ◦ R )( µ ′ X ) / / µ X   U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U ( I ◦ R ◦ I ◦ R )( X ) µ ( I ◦ R )( X )   X µ ′ X / / ( I ◦ R )( X ) sho wing that µ ′ X µ X = 1 ( I ◦ R )( X ) , hence µ X is an isomorphism.  Lemma 2.3. If S T = Im H then S H = C H . Dual ly if S H = Im T then S T = C T . Pr o of. Consider an arb itrary ob ject X ∈ A . By h yp othesis H ( X ) ∈ S T , so η H ( X ) is an isomorp h ism. T ogether with diagrams (1), this implies H ( δ X ) is an isomorphism. Thus δ X : ( T ◦ H )( X ) → X is an H -equiv alence, and w e kno w ( T ◦ H )( X ) ∈ Im T ⊆ C H . As w e learned from Lemm a 1.2, this means that the assignment X 7→ ( T ◦ H )( X ) defines a r ight adjoin t of the inclusion of C H in A . If in add ition X ∈ C H , then δ X is an isomorphism by LOCALIZA TIONS, COLOCALIZA TIONS AND NON ADDITIVE ∗ -OBJECTS 5 Lemma 2.2, pro ving the inclusion C H ⊆ S H . Since the conv erse inclusion is alw ays true, the conclusion holds.  R emark 2.4 . F r om the pro of of Lemma 2.3 w e can see that the condition S T = Im H implies that the fun ctor A → C H , X 7→ ( T ◦ H )( X ) is the r igh t adjoin t of the inclusion fun ctor of S H = C H in to A . Dually if S H = Im T , then the functor B → C T , Y 7→ ( H ◦ T )( Y ) is the left adjoin t of the inclusion functor of S T = C T in to B . Theorem 2.5. The fol lowing ar e e quivalent: (i) S T = C T . (ii) S T = Im H . (iii) S H = C H . (iv) S H = Im T . (v) The functors H and T induc e mutual ly inverse e quivalenc e of c ate- gories b etwe en C H and C T . Pr o of. The equiv alence of the conditions (i)–(iv) follo ws by Lemmas 2.1 and 2.3. Finally the equiv alen t conditions (i)–(ii) are also equiv alen t to (v), b ecause S H and S T are the largest sub ca tegories of A and B for wh ic h H and T restrict to m utually inv erse equiv alences.  Corollary 2.6. The adjoint functors T ⊣ H i nduc e mutual ly inverse e quiv- alenc es C H ⇄ B if and o nly if T is ful ly faithful. Dual ly the ad joint p air induc es e quivalenc es A ⇄ C T if and only if H is ful ly faithful. Pr o of. The functor T is fully faithful exactly if the unit of the adjunction η : 1 B → ( H ◦ T ) is an isomorph ism , or equiv alen tly , S T = B . No w, Theorem 2.5 applies.  Theorem 2.5 and Corollary 2.6 generalize [1, Th eorem 1.6 and Corollary 1.7], where the work is done in the setting of ab eli an categories, and the pro of stresses th e ab elian structur e. These r esults m a y b e also compared with [9, Theorem 1.18], wh ere the framew ork is also that of ab elia n categories. W e consider n ext other t w o su b categories of A and B resp ectiv ely: G H = { X ∈ A | δ X : ( T ◦ H )( X ) → X is an epimorph ism } , G T = { Y ∈ B | η Y : Y → ( H ◦ T )( Y ) is a monomorp hism } . The du al c haracter of all considerations in the pr esen t S ection contin ues to hold for G H and G T . Lemma 2.7. The fol lowing statements hold: a) The sub c ate gory G H (r esp e ctively G T ) is close d under quotient obje cts (sub obje cts). b) Im T ⊆ G H (r esp e ctively Im H ⊆ G T ). Pr o of. a) Let α : X ′ → X b e an epimorphism in A with X ′ ∈ G H . Sin ce δ is n atural, w e obtain the equalit y αδ X ′ = δ X ( T ◦ H )( α ), sh o win g that δ X is an epimorphism together with αδ X ′ . 6 GEORG E C IPRIAN M ODOI b) F rom the diagrams (1), w e see that δ T ( Y ) is right inv er tib le, s o it is an epimorphism for any Y ∈ B . Th us Im T ⊆ G H .  The sub categ ory G H of A is more inte resting in the case w hen A has e pi- morphic images , what m eans that it has images and the factoriza tion of a morphism through its image is a comp ositio n of an epimorphism follo wed by a monomorphism (for example, A has epimorphic images, provided that it has equ alizators and images, by [8, C hapter 1, Pr op osition 10.1]). Supp ose also that A is b alanc e d , that is ev ery morphism w hic h is b oth epimorphism and monomorphism is an isomorphism. Thus eve ry factorization of a m or- phism as a comp osition of an epimorphism follo wed b y a monomorphism is a factorizatio n through image , b y [8, Chapter 1, Prop ositio n 10.2]. With these h yp otheses it is not hard to see that the factorizatio n of a morp hism through its image is functorial, that means the assignmen t α 7→ im α defines a functor A → → A . Prop osition 2.8. If A is a b alanc e d c ate g ory with epimorphic images, then the functor A → G H , X 7→ im δ X is a right adjoint of the inclusion f unctor G H → A . Pr o of. By hypothesis im δ X is a quotien t of ( H ◦ T )( Y ) and H ( T ( Y )) ∈ G H , so the fu nctor A → G H , X 7→ im δ X is wel l d efined, b y Lemma 2.7. Let no w α : X ′ → X in A → , w here X ′ ∈ G H and X ∈ A . Since δ X ′ is an epimorphism, it follo ws im α = im ( αδ X ′ ) = im ( δ X ( T ◦ H )( α )) ⊆ im δ X , so α factors through im δ X . This means that the map A ( X ′ , im δ X ) → A ( X ′ , X ) is sur jectiv e. But it is also injectiv e since the f unctor A ( X ′ , − ) pr eserves monomorphisms, and the conclusion follo ws.  Corollary 2.9. If A is a b alanc e d c ate gory with epimorphic images, then the morphism im δ X → X is an H -e qu ivalenc e and C H ⊆ G H . Pr o of. The second s tatement of the conclusion follo w s from the fi rst one by using Prop osition 2.8 and Lemma 2.2. But H carries the monomorphism im δ X → X into a monomorphism in B , b ecause H is a righ t adjoin t. Mo re- o ver, s ince H ( δ X ) is right inv ertible, th e same is tru e f or the m orp hism H (im δ X ) → H ( X ), as we ma y see from the commuta tiv e diagram ( H ◦ T ◦ H )( X ) H ( δ X ) / / ( ( P P P P P P P P P P P P H ( X ) H (im δ X ) 9 9 s s s s s s s s s s .  LOCALIZA TIONS, COLOCALIZA TIONS AND NON ADDITIVE ∗ -OBJECTS 7 Prop osition 2.10. Supp ose b oth A and B ar e b alanc e d c ate gories with epi- morphic images. The fol low ing ar e e qu ivalent: (i) The p air of adjoint functors T ⊣ H induc es mutual ly i nverse e quiv- alenc es C H ⇄ G T . (ii) η Y : Y → ( H ◦ T )( Y ) is an epimorphism for al l Y ∈ B . Pr o of. (i) ⇒ (ii). Denote Y ′ = im η Y . Then the u nit η Y of adjunction facto rs as Y → Y ′ → ( H ◦ T )( Y ), where the epimorp hism Y → Y ′ is a T -equiv alence b y the du al of C orollary 2.9, and Y ′ → ( H ◦ T )( Y ) a monomorp hism. Since ( H ◦ T )( Y ) ∈ Im H ⊆ G T and G T is closed under sub ob jects, w e deduce Y ′ ∈ G T . No w (i) imp lies that η Y ′ is an isomorph ism , so the diagram Y / / η Y   Y ′ η Y ′   ( H ◦ T )( Y ) ∼ = / / ( H ◦ T )( Y ′ ) pro v es (ii). (ii) ⇒ (i). Condition (ii) implies that T ( η Y ) is an epimorphism , for eve ry Y ∈ B , since T pr eserv es epimorph isms. But it is also left in vertible b y diagrams 1. Th us it is inv ertible, with the inv erse δ T ( Y ) . W e ha ve just s h o w n that S H = Im T , h ence Th eorem 2.5 tells us that C H and C T are equiv alen t via H and T . Finally , since B is balanced, clearly G T = S T = C T .  Com b ining Prop ositio n 2.10 and its d ual we obtain: Theorem 2.11. Supp ose b oth A and B ar e b alanc e d c ate gories with epimor- phic images. The fol low ing ar e e quivalent: (i) The p air of adjoint functors T ⊣ H induc es mutual ly i nverse e quiv- alenc es G H ⇄ G T . (ii) δ X : ( T ◦ H )( X ) → X is a monomor phism for al l X ∈ A and η Y : Y → ( H ◦ T )( Y ) is an epimorphism for al l Y ∈ B . Remark that [3, Pr op osition 2.2.4 and Th eorem 2.3.8] p ro vide charac ter- izations of (weak) ∗ -mo dules whic h are analogous to Prop osition 2.10 and Theorem 2.11 ab o v e. These r esults will b e used in Section 5 , for defining the corresp ond ing notions in a non add itiv e situation. 3. R eprese nt ab l e equiv al ences Ov erall in this section A is a co complete cate gory and E is small category . Denote by [ E op , S et ] the category of all con tra v ariant functors from E into S et . Then we view E as a su b categ ory of [ E op , S et ], via th e Y oneda emb ed - ding E → [ E op , S et ], e 7→ E ( − , e ). F or simplicit y , we shall write [ Y ′ , Y ] for [ E op , S et ]( Y ′ , Y ), where Y ′ , Y ∈ [ E op , S et ]. F or ev er y Y ∈ [ E op , S et ] denote b y E ↓ Y the comma category whose ob jects are of the form ( e, y ) with e ∈ E and y ∈ Y ( e ) and whose m orp hisms are ( E ↓ Y )(( e ′ , y ′ ) , ( e, y )) = { α ∈ E ( e ′ , e ) | Y ( α )( y ′ ) = y } . 8 GEORG E C IPRIAN M ODOI The pro jection fun ctor E ↓ Y → E is giv en by ( e, y ) 7→ e and α 7→ α for all ( e, y ) ∈ E ↓ Y and all α ∈ ( E ↓ Y )(( e ′ , y ′ ) , ( e, y )). Ob serv e then that the sub category E is dense in [ E op , S et ], wh at means, for ev ery Y ∈ [ E op , S et ] it holds Y ∼ = colim(( E ↓ Y ) → E → [ E op , S et ]) = colim ( e,y ) ∈E ↓ Y E ( − , e ) , where the last notation is a shorthand f or the previous colimit. F or a fu nctor A : E → A , consider th e left K an extension of A along the Y oneda em b edding: T A : [ E op , S et ] → A , T A ( Y ) = colim ( e,y ) ∈E ↓ Y A ( e ) , whic h ma y b e c h aracterized as th e uniqu e, up to a natural isomo rphism, colimit pr eserving fun ctor [ E op , S et ] → A , mapping E ( − , e ) in to A ( e ) for all e ∈ E . Th e functor T A has a r igh t adjoint , namely the fun ctor H A : A → [ E op , S et ] , H A ( X ) = A ( A ( − ) , X ) . In order to use the results of Section 1 , w e remaind the n otations made there, namely let δ : T A ◦ H A → 1 A and η : 1 B → H A ◦ T A b e the arrows of adjunction. F or simplicit y w e shall r eplace in the next considerations the subscript H A and the sup erscript T A with A . So ob jects in C A , C A , G A and G A will b e called A -c olo c al , A -lo c al , A -gener ate d , resp ective ly A -c o gener ate d . W e consider o verall in this s ection t wo su b categories C ⊆ A and C ′ ⊆ [ E op , S et ], such th at C is closed u nder taking colimits and retracts and C ′ is closed und er taking limits an d retracts. Lemma 3.1. L et H : C ⇄ C ′ : T b e a p air of adjoint fu nc tors T ⊣ H , and denote A : E → A the functor given by A ( e ) = T ( E ( − , e )) . If E ( − , e ) ∈ C ′ for al l e ∈ E then H is natur al ly isomorphic to the r estriction of H A and T is natur al ly isomorphic to the r estriction of T A , I m H A ⊆ C ′ and Im T A ⊆ C . If mor e over T i s ful ly faithful, then arr ow δ X : ( T A ◦ H A )( X ) → X is an H A -e quivalenc e for al l X ∈ A . Pr o of. Using Y oneda lemma, w e ha ve the natural isomorph ism s for ev ery X ∈ C , and ev ery e ∈ E : H ( X )( e ) ∼ = [ E ( − , e ) , H ( X )] ∼ = A ( T ( E ( − , e )) , X ) = A ( A ( e ) , X ) = H A ( X )( e ) , th us H ( X ) ∼ = H A ( X ) naturally . Since A ( e ) = T ( E ( − , e )) ∈ C for all e ∈ E , the closure of C un der colimits and the formula T A ( Y ) = colim ( e,y ) ∈E ↓ Y A ( e ) , v alid for all Y ∈ [ E op , S et ], sho w that Im T A ⊆ C . Therefore th e assignment Y 7→ T A ( Y ) defin es a functor C ′ → C , which is naturally isomorphic to T , as righ t adjoints of H . F urth er Im T A ⊆ C implies Im H A ⊆ C ′ , since for all X ∈ A , w e hav e ( H A ◦ T A ◦ H A )( X ) ∼ = H (( T A ◦ H A )( X )) ∈ C ′ , LOCALIZA TIONS, COLOCALIZA TIONS AND NON ADDITIVE ∗ -OBJECTS 9 H A ( X ) is a retract of ( H A ◦ T A ◦ H A )( X ) and C ′ is closed un der retracts. No w, th e fu lly faithfulness of T is equiv alen t to the fact that H ◦ T ∼ = 1 C ′ naturally , th us Im H A ∈ C implies ( H A ◦ T A ◦ H A )( X ) ∼ = ( H ◦ T )( H A ( X )) ∼ = H A ( X ) , what means that δ X is an H A -equiv alence.  Theorem 3.2. L et H : C ⇄ C ′ : T b e mutual ly inverse e quivalenc es of c ate gories, and denote A : E → A the functor given by A ( e ) = T ( E ( − , e )) . If E ( − , e ) ∈ C ′ for al l e ∈ E then H is natur al ly isomorphic to the r estriction of H A , T i s natur al ly isomorphic to the r estriction of T A , C = C A and C ′ = C A . Pr o of. The conclusions concerning H and T follo w by Lemma 3.1. F or th e rest, w e hav e for all X ∈ C : X ∼ = ( T ◦ H )( X ) ∼ = ( T A ◦ H A )( X ) ∈ Im T A ⊆ C A as w e hav e seen in Lemma 2.1. Thus C ⊆ C A , and d u ally C ′ ⊆ C A . The fun ctor A → C give n by X 7→ ( T A ◦ H A )( X ) is a right adjoin t of the inclusion fu nctor C → A . Ind eed, for all X ′ ∈ C and all X ∈ A , w e obtain X ′ ∼ = ( T ◦ H )( X ′ ) ∼ = ( T A ◦ H A )( X ′ ) and H A ( X ) ∼ = ( H A ◦ T A ◦ H A )( X ) sin ce the coun it δ X of adj unction is an H A -equiv alence, as w e observed in Lemma 3.1. No w th e natur al isomorp hisms A ( X ′ , ( T A ◦ H A )( X )) ∼ = A (( T A ◦ H A )( X ′ ) , ( T A ◦ H A )( X )) ∼ = [ H A ( X ′ ) , ( H A ◦ T A ◦ H A )( X )] ∼ = [ H A ( X ′ ) , H A ( X )] ∼ = A (( T A ◦ H A )( X ′ ) , X ) ∼ = A ( X ′ , X ) pro v e our claim. Using again the fact that δ X : ( T A ◦ H A )( X ) → X is an H A - equiv alence, Lemma 2.2 tells us that C A ⊆ C . The functor [ E op , S et ] → C ′ giv en by Y 7→ ( H A ◦ T A )( Y ) is also w ell defined . In a d ual manner w e sho w that it is a left adjoin t o f the in clusion C ′ → [ E op , S et ], and follo ws C A ⊆ C ′ .  The equiv alences H : C ⇄ C ′ : T are called r epr esente d b y A : E → A pro vided that H ∼ = H A and T ∼ = T A as in the Theorem 3.2. In the w ork [7] of Menini and Or satti (see also [3]), it is giv en an additiv e v ers ion of T heorem 3.2. There, our category E is preadditiv e with a single ob ject (that m eans it is a ring), A is an ob ject in A with endomorph ism ring E (therefore A : E → A is a fully faithful functor), and [ E op , S et ] is replaced with Mo d( E ). 4. The existenc e of cell u lar covers In this S ection consider as in the p revious one a co complete catego ry A , a fu n ctor A : E → A , wher e E is a small category and construct its left Kan extension T A : [ E op , S et ] → A along the Y oneda em b edding E → [ E op , S et ] whic h has the right adjoin t H A : A → [ E op , S et ]. I n addition supp ose that A is fully faithful. Note that, this additional assumption means that the 10 GEORG E C IPRIAN M ODOI catego ry E ma y b e identified with a (small) sub category of A and A w ith the inclusion fu nctor. F or example, if E has a single ob ject, th en A may b e iden tified with an ob ject of A . Lemma 4.1. If A is a c o c omplete, b alanc e d c ate gory with e pimorphic images and A : E → A is ful ly faithful, then i t holds: a) A ( e ) ∈ S A for al l e ∈ E . b) An obje ct X ∈ A is A -gener ate d exactly if ther e is an epimorphism A ′ → X with A ′ a c opr o duct of obje c ts of the form A ( e ) with e ∈ E . Pr o of. a) S ince A is fully faithful, we ha v e the n atural isomorph isms: ( T A ◦ H A )( A ( e )) = T A ( A ( A ( − ) , A ( e )) ∼ = T A ( E ( − , e )) ∼ = A ( e ) , for ev ery e ∈ E . b) Let A ′ = ` A ( e i ) ∈ A b e a copro duct of ob jects of the f orm A ( e ). By the result in a) we ded uce A ′ = a A ( e i ) ∼ = a ( T A ◦ H A )( A ( e i )) ∼ = T A ( a H A ( A ( e i ))) ∈ Im T A , so A ′ ∈ G A , s in ce Im T A ⊆ G A , in clusion established in Lemm a 2.7. If X ∈ A suc h th at there is an epimorph ism A ′ → X , then X ∈ G A , again by Lemma 2.7. Con ve rsely , for ev ery X ∈ A , the ob ject H A ( X ) of [ E op , S et ] ma y b e written as H A ( X ) ∼ = colim ( e,x ) ∈E ↓ H A ( X ) E ( − , e ) = colim ( e,x ) ∈ A ↓ X E ( − , e ) , where the co mma cate gory A ↓ X has as ob jects pairs of the form ( e, x ) with e ∈ E a nd x ∈ A ( A ( e ) , X ). T h us ( T A ◦ H A )( X ) ∼ = colim ( e,x ) ∈ A ↓ X T A ( E ( − , e )) ∼ = colim ( e,x ) ∈ A ↓ X A ( e ) , so there is an epimorp hism from ` ( e,x ) ∈ A ↓ X A ( e ) to ( T A ◦ H A )( X ). F urther the morphism δ X : ( T A ◦ H A )( X ) → X is an epimorph ism to o, for X ∈ G A . Comp osing them w e obtain the desired epimorph ism.  Recall that a category A is called wel l (c o)p ower e d if f or ev ery ob ject the class of sub ob jects (resp ecti v ely qu otient ob jects) is actually a set. Theorem 4.2. If A is a c o c omplete, wel l c op ower e d, b alanc e d c ate gory with epimorphic images, and A : E → A is a ful ly faithful functor, then the inclusion functor C A → A has a right adjoint, or e quivalently, e very obje ct in A has a left C A -appr oximation. Pr o of. Combining Lemma 4.1 and Corollary 2.9, we deduce th at ev ery ob ject in C is a quotien t ob ject of a direct sum of ob jects of the form A ( e ), so { A ( e ) | e ∈ E } ⊆ C A is a generating set for C A , by [8, Chapter I I, Prop osition 15.2]. The closure of C A under colimits implies that the inclusion functor C A → A preserves colimits, and the category C A inherits from A the p rop erty to b e we ll cop o w ered. Thus th e conclusion follo ws b y F reyd’s Sp ecial Adjoint F un ctor Theorem (see [8, Chapter V, Corollary 3.2]).  LOCALIZA TIONS, COLOCALIZA TIONS AND NON ADDITIVE ∗ -OBJECTS 11 If E has a single ob ject and A ia a fully faithful functor (i.e . A is an ob ject of A ), then A -colocal ob jects are sometimes called A -c e l lular , and an H A -equiv alence is called then simp ly an A -e quiv alenc e . Our Theorem 4.2 sho ws th at, under reasonable hyp otheses (t hat means A is a co complete, w ell cop o w ered, balanced cat egory with epimorph ic images), eve ry ob ject X has an A -c el lular appr oximation , what means an A -equiv alence C → X with C b eing A -cellular. Hence it is generalized in this wa y [4, S ection 2.C], where is constru cted an A -cellular approximati on for ev ery group. The same pr o of th at giv en [4, Lemma 2.6] f or th e case of the category of groups works for th e follo wing consequence of the existence of an A -cellular appro ximation for every ob ject X ∈ A : Corollary 4.3. L et A : E → A b e a ful ly faithful functor, wher e A is a c o c omplete, wel l c op ower e d, b alanc e d c ate gory with epimorphic images and E is smal l c ate gory. The fol lowing ar e e q uivalent for a morp hism α : C → X in A → : (i) α is a left C A -appr oximation of X . (ii) α is an H A -e quivalenc e and it is initial among al l H A -e quivalenc es ending in X (iii) C ∈ C A and α is terminal among al l morphisms fr om an A -c olo c al obje ct to X . Consequent ly w e ma y u s e the follo wing more of less tautolog ical formulas for determining the left C A -appro ximation of an ob ject (see [5, Sections 7.1 and 7.2]): Corollary 4.4. L et A : E → A b e a ful ly faithful functor, wher e A is a c o c omplete, wel l c op ower e d, b alanc e d c ate gory with epimorphic images and E is smal l c ate gory. If α : C → X is the left C A -appr oximation X , then it holds: a) C = lim X ′ → X X ′ , wher e X ′ → X runs over al l H A -e quivalenc es. b) C = colim X ′ → X X ′ , wher e X ′ runs over al l A -c olo c al obje cts. 5. ∗ -acts over mon o ids W e see a monoid M as a category with one ob ject whose endomorphism set is M . Thus we consider th e category [ M op , S et ] of all con trav arian t functors fr om this category to the category of sets, and we call it the catego ry of (right) acts o ver M , or simply M -acts . Clearly an M -act is a set X together with a an action X × M → X , ( x, m ) 7→ xm suc h that ( xm ) m ′ = x ( mm ′ ) and x 1 = x for all x ∈ X and all m, m ′ ∈ M . Left acts are co v arian t functors M → S et , that is sets X together with an action M × X → X , satisfying the corresp ondin g axio ms. F or the general th eory of acts o ver monoids and un defined notions concerning this sub ject w e refer to [6]. W e should men tion her e that in con trast with [6] we allo w the empt y act to b e an ob ject in our category of acts, for the sak e of (co)completness. Note 12 GEORG E C IPRIAN M ODOI that the category of M -acts is balanced and has epimorph ic im ages, b y [6, Prop osition 1.6.15 and Theorem 1.4.21]. Fix a monoid M and an ob ject A ∈ [ M op , S et ]. In ord er to use the results of the preceding S ections, we ident ify A w ith a fully faithful f unctor E → [ M op , S et ] where E is the endomorph ism monoid of A . T h us A is canonically a E − M -biact (see [6, Definition 1.4.24]), so we obtain tw o functors H A : [ M op , S et ] → [ E op , S et ] , H A ( X ) = [ A, X ] and T A : [ E op , S et ] → [ M op , S et ] , T A ( Y ) = Y ⊗ E A the second one b eing the left adjoin t of the first (see [6, Definition 2.5.1 and Prop osition 2.5.19]). Clearly these fun ctors agree w ith the functors defined at the b eginning of the Section 3. W e sa y that A is a (we ak) ∗ -act if the ab o v e adjoin t pair induces m utually in v erse equ iv alences H A : G A ⇄ G A : T A (resp ectiv e H A : C A ⇄ G A : T A ). Note that our d efinitions for sub cate gories G A and G A agree with the c har- acterizat ions of all A -generated resp ectiv ely A ∗ -cogenerate d mo du les giv en in [2, Lemma 2 .1.2]. As we may see from Pr op osition 2. 10, our sub cate- gory C A seems to b e the non–additiv e counte rpart of the su b category of all A -presen ted mo dules (compare with [2, Prop ositi on 2.2.4]). In w hat follo w s , we need m ore d efinitions relativ e to an M -act A . First A is called de c omp osable if there exists tw o non empty subacts B , C ⊆ A suc h that A = B ∪ C and B ∩ C = ∅ (see [6 , Definition 1.5.7]) . In this case A = B ⊔ C , since copro du cts in the category of acts is the disjoin t union, by [6, Prop osition 2.1.8]. If A is not d ecomp osable, then it is called inde c omp osable . Second, A is say to b e we ak self–pr oje ctive provided that ( H A ◦ T A )( g ) is an epimorp hism wh enev er g : U → Y is an epimorphism in [ E op , S et ] with U ∈ S A . More explicitly , if g : U → Y is an epimorp hism in [ E op , S et ], then T A ( g ) is an epimorphism in [ M op , S et ] and ou r defin ition requires that A is p ro jectiv e relativ e to su ch epimorph ism s for which U ∈ S A . T hird A is called (self–)smal l provided that the fun ctor H A preserve s copro ducts (of copies of A ). Lemma 5.1. With the notations ab ove, the fol lowing ar e e quivalent: (i) A is smal l. (ii) A is self–smal l. (iii) E ( I ) is η -r eflexive for any set I , wher e E ( I ) denotes th e c opr o duct indexe d over I of c opies of E . (iv) E ⊔ E is η - r eflexive. (v) A is inde c omp osable. Pr o of. (i) ⇒ (ii) is ob v ious . LOCALIZA TIONS, COLOCALIZA TIONS AND NON ADDITIVE ∗ -OBJECTS 13 (ii) ⇒ (iii). If H A comm u tes with copro ducts of copies of A then E ( I ) = a I [ A, A ] ∼ = " A, a I A # ∼ = " A, a I ( E ⊗ E A ) # ∼ = h A,  E ( I )  ⊗ E A i ∼ = ( H A ◦ T A )  E ( I )  . (iii) ⇒ (iv) is ob vious . (iii) ⇒ (iv). If A is decomp osable, that is A = B ⊔ C with B 6 = ∅ and C 6 = ∅ , then let i B : B → A and i C : C → A the canonical inj ections of th is copro d uct. Denote also by j 1 , j 2 : A → A ⊔ A the co rresp ond ing canonical injections. The homomorphisms of M -acts j 1 i B : B → A ⊔ A and j 2 i C : C → A ⊔ A indu ce a unique homomorphism f : A = B ⊔ C → A ⊔ A . Ob viously f ∈ ( H A ◦ T A )( E ⊔ E ) but f / ∈ [ A, A ] ⊔ [ A, A ] = E ⊔ E . (iv) ⇒ (i) is [6, Lemma 1.5.37] .  Prop osition 5.2. The fol low ing statements hold: a) If A is a we ak ∗ -act then A is we ak self–pr oje ctive. b) If A is we ak self–pr oje ctive and inde c omp osable, then A is a we ak ∗ -act. Pr o of. a) Let A b e a w eak ∗ -act and let g : U → Y b e an epimorphism in [ E op , S et ] with U ∈ S A . W e know b y Prop osition 2.10 that η Y is epic, and by the naturalness of η that ( H A ◦ T A )( g ) η U = η Y g . Since η U is an isomorphism and η Y g is an epimorphism we deduce that ( H A ◦ T A )( g ) is an epimorphism to o. b) As we h a ve already noticed H A preserve s copro ducts, pro vided that A is indecomp osable. Thus S A is closed u n der arbitrary copro ducts in the catego ry of E -acts. F or a fixed Y ∈ [ E op , S et ] there is an epimorphism g : E ( I ) → Y . Ho w E is η -reflexiv e the same is also true for E ( I ) . But ( H A ◦ T A )( g ) is an epimorph ism, s in ce A is weak self–pro jectiv e. F rom the equalit y ( H A ◦ T A )( g ) η E ( I ) = η Y g follo ws that η Y is an epimorph ism to o. The conclusion follo ws b y Prop osit ion 2.10.  Theorem 5.3. The fol lowing statements hold: a) If A is a ∗ -act then A is we ak self–pr oje ctive and C A = G A . b) If A is inde c omp osable, we ak self–pr oje ctiv e and C A = G A , then A is a ∗ -act. Pr o of. The b oth implications follo w at once from Pr op osition 5.2.  R emark 5 .4 . Prop ositions 2.10 and 5.2 and T heorems 2.11 an d 5.3 provi de a non additiv e ve rsion of [2, Pr op osition 2.2.4] r esp ectiv ely [2 , Theorem2.3.8]. In cont rast with the case of mo dules, where the fun ctors are additive , for acts it is n ot clear that a w eak star ob ject m ust me indecomposable (t he non add itiv e version of self–smallness as w e may seen from Lemma 5.1). T he 14 GEORG E C IPRIAN M ODOI main obstacle for deducing this im p lication in the new s etting comes fr om the fact that non add itiv e functors d o not hav e to p reserv es fin ite copr o ducts. Using the characte rization of so called tilting mo d ules giv en in [2, Th e- orem 2.4.5], w e may d efine a tilting M -act to b e a ∗ - act A suc h that the injectiv e env elop e of M b elo ngs to G A . Note that injective en v elop es exist in [ M op , S et ] b y [6, Corollary 3.1.23]. As a sub ject f or a fu ture researc h w e ma y ask ourselves w hic h from the man y b ea utiful results wh ic h are known for tilting mo d ules do h a ve corresp on d en ts for acts. Our next aim is to in f er fr om our results th e Morita–t yp e charac terizatio n of an equiv alence b et w een categories of acts (see [6, Section 5.3]). In order to p erform it we n eed a couple of lemmas. Lemma 5.5. If the M -act A is a gener ator in [ M op , S et ] then C A = G A = [ M op , S et ] . Pr o of. F or a generator A of [ M op , S et ] the equalit y G A = [ M op , S et ] follo ws b y Lemma 4.1. Moreo ver M is a r etract of A by [6, Theorem 2.3.16], there- fore M ∈ C A , since C A is closed under retracts. Th us a morphism ǫ : U → V in [ M op , S et ] is an A -equiv alence if and only if it is an isomorp hism, ther efore C A = [ M op , S et ].  Recall that the left E -act A is said to b e p ul l b ack flat if the functor T A = ( − ⊗ E A ) comm utes with pull backs (see [6, Definition 3.9.1]). Lemma 5.6. If the right M -act A is inde c omp osable, we ak self pr oje ctive and the left E -act A i s pul l b ack flat, then G A = [ E op , S et ] . Pr o of. First observe th at A is a w eak ∗ -act by Prop ositio n 5.2. Hence G A = C A = S A , and this s u b category h as to b e closed u nder su b acts and limits. Moreo v er E ( I ) is η -reflexiv e for an y set I according to L emm a 5.1. F or a fixed Y ∈ [ E op , S et ] there is an epimorphism g : E ( I ) → Y . T ak e the k ernel pair of g , that is construct the pull b ac k K k 1 / / k 2   E ( I ) g   E ( I ) g / / Y . The fu nctors T A and H A preserve pull bac ks, the first one b y h yp ot hesis and the second one au tomatically . Moreo v er K is a subact of E ( I ) × E ( I ) and the closure prop erties of S A imply K ∼ = ( T A ◦ H A )( K ). App lying the functor H A ◦ T A to th e ab o v e diagram and ha ving in the min d the p revious LOCALIZA TIONS, COLOCALIZA TIONS AND NON ADDITIVE ∗ -OBJECTS 15 observ ations we obtain a pull bac k diagram K k 1 / / k 2   E ( I ) = ( H A ◦ T A )  E ( I )  ( H A ◦ T A )( g )   E ( I ) = ( H A ◦ T A )  E ( I )  ( H A ◦ T A )( g ) / / ( H A ◦ T A )( Y ) . Note that ( H A ◦ T A )( g ) is an epimorph ism b y hyp othesis. T hen w e know by [6, Theorem 2.2.44] th at b ot h g and ( H A ◦ T A )( g ) are co equalizers for the pair ( k 1 , k 2 ). Thus we d educe Y ∼ = ( H A ◦ T A )( Y ) canonically , so Y ∈ S A . Th us G A = S A = [ E op , S et ].  No w we are in p ositi on to pro v e the desired Morita–t yp e result: Theorem 5.7. L et M and E b e two monoids. Then the c ate gories [ M op S et ] and [ E op , S et ] ar e e quivalent via the mutual ly inverse e quivalenc e functors H and T if and only i f ther e is a cyclic, pr oje ctiv e ge ner ator A of [ M op , S et ] such that E is the endomorphism monoid of A , c ase in which H = H A and T = T A . Pr o of. First note that a pr o jecti v e act is indecomp osable if and only if it is cyclic in virtue of [6, Prop ositions 1.5.8 and 3.17.7]. If H : [ M op , S et ] ⇄ [ E op , S et ] : T are mutually in verse equiv alences, then H ∼ = H A and T ∼ = T A , where A = T ( E ) according to Theorem 3.2. Moreo v er the endomorphism monoid of A is E , and A h as to b e pro jectiv e, indecomp osable and generator together w ith E . Con v ersely if A is in decomp osable and p ro jectiv e in [ M op , S et ] then it is a we ak ∗ -act by P r op osition 5.2. Since A is in addition a generator, Lemma 5.5 tell us that A is a ∗ -act and C A = G A = [ M op , S et ] and Theorem 5.3 imp lies that A is a ∗ -act. Finally the left E -act A is pro jectiv e by [6 , Corollary 3.18.17], so it is strongly flat by [6, Prop ositio n 3.15.5], that means T A comm u tes b oth with pull bac ks and equ alizers. Thus G A = [ E op , S et ], according to Lemma 5.6.  Referen ces [1] F. Casta˜ no Iglesias, J. G´ omez–T orrecillas, R. Wisbauer, Adjoint functors and equiv- alences, Bul l . sci. math. 127 (2003), 379– 395. [2] R. Colpi, K . F uller, Equivalenc e and Duali ty for Mo dule Cate gories , Cam b ridge Uni- versi ty Press, Cam bridge, 2004. [3] R. Colpi, K. F uller, Tilting ob jects in ab elian categories and quasitilted rings, T r ans. Amer . Math. So c. , 359 (2007), 741–765. [4] E. D ror F arjoun, R. G¨ ob el, Y . Segev, Cellular cov er of groups, J. Pur e Appl. Algebr a , 208 (2007 ), 61–76 . [5] W. Dwy er, Localizations in Axiomatic, Enriche d and Motivic Homotopy The ory , Proceedings of the NA TO A SI (ed. J.P .C. Greenlees), Kluw er, 2004. [6] M. Kilp, U. Knauer, A. Mikhalev, Monoids, A cts and Cate gories De Gruy t er Ex p o- sitions in Mathematics, 29, W alter de Gruyter, Berlin, New–Y ork, 2000. 16 GEORG E C IPRIAN M ODOI [7] C. Menini, A. Orsatti, Representable equiva lences b et ween categories of mo dules and applications, R end. Sem. Math. Univ. Padova , 82 (1989 ), 203–231. [8] B. Mitc hell, The ory of Cate gories , Academic Press, New Y ork and London, 1965 . [9] C. Mo doi, Equiv alences indu ced by adjoin t functors, Comm . Alg. 31(5) (2003), 2327– 2355. [10] N. Popescu, L. Popescu, The ory of Cate gories , Editura Academiei, Bucure¸ sti and Sijthoff & No ordhoff Internatio nal Pub lishers, 197 9. ”Babes ¸ -Bol y ai” Universi ty, RO-40008 4, Cluj-Napoca, Romania E-mail addr ess , George Ciprian Mo doi: cmodoi@math .ubbcluj.r o