Locally well generated homotopy categories of complexes

LOCALL Y WELL GENERA TED HOMOTOPY CA TEGORIES OF COMPLEXE S JAN ˇ S ˇ TO V ´ I ˇ CEK Abstra ct. W e sho w that the h omotop y category of complexes K ( B ) o ver any finitely accessible additive category B is locally well generated. That i s, any lo calizing sub category L in K ( B ) whic h is generated by a set is w ell generated in th e sense of Neeman. W e also show th at K ( B ) itself b eing w ell generated is equ iv al ent to B being p ure semisimple, a concept which natu rally generalizes right pure semisimplicit y of a ring R for B = Mo d- R . Introduction The main motiv ation for this pap er is to stu dy wh en the h omotop y cate- gory of complexes K ( B ) ov er an additiv e catego ry B is compactly generated or, more generally , well generated. In the last f ew d ecades, the theory of compactly generated triangulated catego ries has b ecome an imp ortan t to ol unify in g concepts from v arious fields of mathematics. Stand ard examples are the u n b ound ed d eriv ed cate- gory of a rin g or the stable homotop y category of sp ectra. The ke y prop erty of suc h a category T is the Bro wn Representa bilit y Th eorem, cf. [30, 25], originally du e to Bro w n [9]: An y con trav arian t cohomolo gical fun ctor F : T → Ab wh ic h sends copro ducts to pro du cts is represent able. This theorem is an imp ortan t tool and has b een used in sev eral places. W e men tion Neeman’s pro of of the Grothendiec k Dualit y Theorem [30], Krause’s w ork on the T elescope Conjecture [28, 24], or Keller’s r epresen tation theorem for algebraic compactly generated triangulated categ ories [23]. Recen tly , there has b een a gro wing int erest in giving criteria for cer- tain homotop y categories K ( B ) to b e compactly generated, [15, 20, 29, 31]. Here, B t yp ically w as a suitable sub category of a mo dule catego ry . The main reason f or studying su c h homotop y categories were results concern- ing the Grothendiec k Dualit y Theorem [17, 31] and relativ e homological algebra [19]. There is, ho w ever, a conceptual r eason, to o. Namely , ev ery Date : May 14, 2010. 2000 Mathematics Subje ct Cl assific ation. 18G35 (Primary) 18E30, 18E35, 16D90 (Secondary). Key wor ds and phr ases. Compactly and wel l generated triangulated categories, com- plexes, pure semisimplic it y . The author was supp orted by the Research Council of Norw a y through the Storforsk- pro ject “Homological and geometric meth ods in algebra” and also by th e grant GAUK 301-10/25 2216. 1 2 JAN ˇ S ˇ TOV ´ I ˇ CEK algebraic triangulated ca tegory is triangle equiv alen t to a fu ll sub category of some homotop y category , [25, § 7.5]. It turned out when stud ying the h omotop y category of complexes of pro- jectiv e mo du les o v er a ring R in [31] that it is useful to consider well gen- erated triangulated categories in this con text. More precisely , K (Pro j- R ) is alw ays well generated, but m a y not b e compactly generated. W ell gener- ated c atego ries ha v e b een d efined by Ne eman [3 2] in a natural a ttempt to extend r esults such as the Bro wn Represen tabilit y from compact ly generated triangulated categories to a wider class of triangulated categories. Although one has already kno wn for some time that there exist rather nat- ural triangulated catego ries, such as the homotop y catego ry of complexes of ab elian group s, which are not ev en well generated, on e has t ypically viewe d those as rare and exceptional cases. W e will giv e some argum en ts to sh ow that this interpretation is n ot very accurate. First, the catego ries K (Mo d- R ) for a rin g R are rarely w ell gener- ated. It happ ens if and only if R is right pu re semisimple, wh ich establishes the con ve rse of [15, § 4 (3), p. 17]. Moreo ver, w e generalize this result to the homotop y categ ories K ( B ) with B additiv e fi n itely accessible. This wa y , w e obtain a fairly complete answer regarding wh en K (Flat- R ) is compactly or w ell generated, see [15, Question 4.2]. W e also g iv e a partial remedy for the t ypical failure of K ( B ) to b e well generated. Roughly sp eaking, the main problem with K ( B ), where B is finitely accessible, is that it ma y not h a ve an y set of generators at all. But if we take a lo calizing sub catego ry L generated by an y set of ob jects, it will automatica lly b e w ell generated. W e will call a triangulated ca tegory with this prop ert y locally we ll generated. W e will also giv e b asic prop erties of lo cally w ell generated catego ries and see that some of the usual resu lts regarding lo caliza tion hold in the new setting. F or example, an y localizing sub category generated b y a set of ob- jects is realized as the k ernel of a lo calization end ofunctor. Th is v ers ion of a Bousfield lo calizati on theorem generalizes [26, § 7.2] and [2, 5.7]. Ho we v er , one has to b e m ore careful. T he Bro wn Repr esen tabilit y theorem as stated ab o v e do es not work for lo cally w ell generated categories in general, and there are lo calizing sub categories which are not asso ciated to any lo caliza- tion endofun ctor. W e illustrate this in Example 3.7. Ac knowledgemen ts. Th e author w ould lik e to thank Hennin g Krause for sev eral helpf u l discussions and suggestions, as we ll as for his hospitalit y during the au th or’s visits in P aderb orn. 1. Preliminaries Let T b e a triangulated categ ory . A triangulated su b category S ⊆ T is called thick if, whenev er X ∐ Y ∈ S , then also X ∈ S . F rom no w on, w e will assume that T has arb itrary (set-indexed) copro ducts. A full triangulated su b category L ⊆ T is called lo c alizing if it is closed under forming coprod ucts. Note that b y [32, 1.6.8], T has splitting idemp otents and any lo calizing sub category L ⊆ T is thick. LOCALL Y WELL GENERA T E D HOMOTOPY CA TEGORIES OF COMPLEXES 3 If S is an y class of ob jects of T , we d enote b y Lo c S the smallest localizing sub categ ory of T wh ic h con tains S . I n other w ords, Lo c S is the closure of S und er sh ifts, copro du cts and triangle completions. Giv en T and a localizing sub category L ⊆ T , o ne can construct the so- called V er dier quotient T / L b y formally inv erting in T all morph isms in the class Σ( L ) defin ed as Σ( L ) = { f | ∃ triangle X f → Y → Z → X [1] in T suc h that Z ∈ L} . It is a well known fact that the V erd ier qu otien t alw a ys has copro ducts, ad- mits a natural triangulated structure, and the canonical lo calization fun ctor Q : T → T / L is exact and preserv es copro ducts, [32, Chapter 2]. Ho w- ev er, one has to b e careful, since T / L m ight not b e a usu al category in the sense that the homomorph ism spaces might b e prop er classes rather than s ets. Th is fact, although often inessen tial and n eglecte d, as T / L has a very straightfo rw ard and constructiv e description, ma y nev ertheless ha ve imp ortant consequences in some cases; s ee eg. [6]. Let L : T → T b e an exact end ofunctor of T . Then L is called a lo c alization functor if there exists a natural transformation η : Id T → L suc h that Lη X = η LX and η LX : LX → L 2 X is an isomorph ism f or eac h X ∈ T . It is easy to c h ec k that the full sub catego ry K er L of T giv en by Ker L = { X ∈ T | LX = 0 } is alw a ys localizing [2, 1.2]. Moreov er, there is a canonical triangle equiv a- lence b et w een T / Ker L and Im L , the essential image of L ; see [32, 9.1.16] or [26, 4.9.1] . T his among other things implies that all morphism sp aces in T / Ker L are sets. Note that although Im L has copro ducts as a category , it migh t not b e closed und er copro du cts in T . This t yp e of localization, coming from a lo calization functor, is often referred to as Bousfield lo c aliza- tion . Ho wev er, not ev ery lo calizing su b category L is realized as the k ernel of a lo calization functor, [6 , 1. 3]. Namely , L is of th e form Ker L for some lo calizat ion functor if and only if the inclusion L → T has a righ t adjoin t, [2, 1.6]. A cen tral concept in this pap er is that of a w ell generated triangulated catego ry . Let κ b e a regular cardinal n u m b er. An o b ject Y in a ca tegory with arbitrary copro ducts is called κ –smal l provided th at e v ery morphism of the form Y − → a i ∈ I X i factorizes through a sub copro duct ` i ∈ J X i with | J | < κ . Definition 1.1. Let T b e a triangulated cat egory w ith arbitrary copro ducts and κ b e a regular cardinal. Then T is called κ –wel l gener ate d provided there is a set S of ob jects of T satisfying the follo wing conditions: (1) If X ∈ T suc h that T ( Y , X ) = 0 for eac h Y ∈ S , then X = 0; (2) Eac h ob ject Y ∈ S is κ –small; (3) F or an y morphism in T of the form f : Y → ` i ∈ I X i with Y ∈ S , there exists a family of morph isms f i : Y i → X i suc h that Y i ∈ S for 4 JAN ˇ S ˇ TOV ´ I ˇ CEK eac h i ∈ I and f factorizes as Y − − − → a i ∈ I Y i ` f i − − − → a i ∈ I X i . The catego ry T is called wel l g e ner ate d if it is κ –w ell generated for some regular cardinal κ . This definition d iffers to some exten t from Neeman’s original definition in [32, 8.1.7]. Th e equiv alence b et ween th e tw o follo w s from [27, Theorem A] and [27, Lemmas 4 and 5]. Note that if κ = ℵ 0 , then condition (3) is v acuous and ℵ 0 –w ell generated triangulated categories are precisely the c omp actly gener ate d triangulated categories in the usu al sense. The k ey prop ert y of w ell generated categories is that the Bro wn Repre- sen tabilit y Theorem holds: Prop osition 1.2. [32, 8.3.3] L et T b e a wel l gener ate d triangulate d c ate gory. Then: (1) Any c ontr avariant c ohomolo gic al functor F : T → Ab which takes c opr o ducts to pr o ducts i s, up to isomorp hism, of the form T ( − , X ) for some X ∈ T . (2) If S is a set of obje c ts of T which me ets assumptions (1) , (2) and (3) of Definition 1.1 for some c ar dinal κ , then T = Lo c S . Next w e turn our atten tion to catego ries of complexes. Let B b e an addi- tiv e cat egory . Using a sta ndard notation, w e denote by C ( B ) the category of c h ain complexes X : · · · → X n − 1 d n − 1 → X n d n → X n +1 → . . . , of ob jects of B . By K ( B ), we denote the factor-category of C ( B ) mo d - ulo the ideal of null-homoto pic c hain complex morp h isms. It is well known that K ( B ) has a triangulated stru cture where triangle completions are con- structed usin g mapp ing cones (see for example [14, Chapter I]). Moreo v er, if B h as arbitrary copro ducts, so hav e them b oth C ( B ) and K ( B ), and the canonical fun ctor C ( B ) → K ( B ) preserves copr o ducts. W e will often tak e f or B m o dule categories or their s u b categories. In this case, R w ill denote an asso ciativ e u nital ring and Mo d- R th e category of all (unital) right R –mo dules. By Pro j- R and Flat- R we denote, r esp ectiv ely , the full s ub categories of pr o jectiv e and flat R –mo du les. In fact, our considerations will us u ally w ork in a more general setting. Let A b e a sk eleta lly small additive cate gory and Mo d- A b e the category of all con trav arian t add itiv e functors A → Ab. W e will call such fun ctors right A –mo dules . Th en Mo d - A shares many formal prop erties with usual mo dule catego ries. W e r efer to [18, App endix B] for more details. Corresp ondingly , w e denote b y P r o j- A the full sub category of pro jectiv e functors and b y Flat- A the category of flat functors. W e discuss the ca tegories of the form Flat- A more in detail in Section 4 since those are, up to equ iv alence, pr ecisely the so called additiv e fi nitely accessible categories. Many natural ab elian catego ries are of this form. Finally , we sp end a few words on set-theoretic considerations. All our pro ofs wo rk in ZF C with an extra tec hnical assumption: the axiom of choice LOCALL Y WELL GENERA T E D HOMOTOPY CA TEGORIES OF COMPLEXES 5 for p rop er classes. The latter assumption has no algebraic significance, it is only used to k eep argumen ts simple in the follo w ing case: Let F : C → D b e a c o v ariant additive fun ctor. If w e kno w, for example b y the Brown Repr esen tability Theorem, th at the comp osition of fun ctors C F − − − − → D D ( − ,X ) − − − − − → Ab is r ep resen table for eac h X ∈ D , w e w ould lik e to conclude that F h as a righ t adjoin t G : D → C . In order to do that, w e m ust f or eac h Y ∈ C cho ose one particular v alue for GY fr om a class of m utually isomorphic candidates. 2. Pure semisiplicity A relativ ely straightfo rw ard bu t crucial obstacle causing a h omotop y cat- egory of complexes K ( B ) n ot to b e w ell generated is th at the ad d itiv e base catego ry B is not pur e semisimple. Here, we use the f ollo wing v ery general definition: Definition 2.1. An additiv e category B with arbitrary copro du cts is called pur e semisimple if it has an additiv e g enerator. That is, th er e is an ob ject X ∈ B such that B = Add X , where Add X stand s for th e full su b category formed b y all ob jects wh ich are summand s in (p ossib ly infi n ite) copro du cts of copies of X . The term is inspir ed by the case B = Mo d - R , where w e ha ve the follo wing prop osition: Prop osition 2.2. A ring R is right pur e semisimple (that is, e ach pur e monomor phism b etwe en right R –mo dules splits) if and only i f Mo d- R is pur e semisimple in the sense of Definition 2.1. Pr o of. If every pur e monomorphism in Mod- R splits, then also ev ery pure epimorphism splits. That is, ev ery mod ule is pure pro jectiv e, or equ iv alently a su mmand in a direct sum of fin itely presented m o dules. By a th eorem of Kaplansky , [21, Theorem 1], it f ollo ws that ev ery mo dule is a d irect sum of coun tably generated mo dules. Hence, Mo d- R is pu re semisimp le according to our defin ition. In fact, one can sh o w more in th is case: Ev ery mo d ule is ev en a d ir ect su m of finitely present ed mo dules; see for example [16] or [18, App. B]. Let u s conv ersely assu me that Mo d - R is a pu re semisimp le add itiv e c at- egory . Using [3, Th eorem 26.1], w hic h is a v ariation of [21, Th eorem 1] for higher cardinalities, w e see that if Mo d- R = Add X for some κ –generated mo dule X , then eac h mo dule in Mod - R is a d irect sum of λ –generated mo d- ules where λ = max( κ, ℵ 0 ). This fact implies that every mo dule is Σ–pure injectiv e, [12 ]. In particular, eac h p ure monomorphism in Mo d- R splits and R is r ight pure semisimp le.  If R is an artin algebra, then the co nditions o f Pr op osition 2.2 are well- kno wn to b e fur th er equiv alen t to R b eing of finite repr esen tatio n t yp e; see [4, Theorem A] . F or more details and references on this topic, w e also refer to [16]. It tur ns out that the pure semisimplicit y condition has a nice in terpretation for finitely acce ssible additiv e catego ries as well. W e w ill discuss this m ore in detail in Section 4. 6 JAN ˇ S ˇ TOV ´ I ˇ CEK F or giving a connection b et ween pu re semisimp licit y of B and prop erties of K ( B ), w e recall a structure result f or the so-calle d con tractible complexes in C ( B ). A complex Y ∈ C ( B ) is c ontr actible if it is mapp ed to a zero ob ject under C ( B ) → K ( B ). It is clear that the complexes of the form I X,n : · · · → 0 → 0 → X = X → 0 → 0 → . . . , suc h that the first X is in degree n , are con tractible. Moreo v er, all other con tractible complexes are obtained in the f ollo wing wa y : Lemma 2.3. L et B b e an ad ditive c ate gory with splitting idemp otents and Y ∈ C ( B ) . Then th e fol lowing ar e e quivalent: (1) Y is c ontr actible; (2) Y is isomorphic in C ( B ) to a c omplex of the for m ` n ∈ Z I X n ,n . Pr o of. (2) = ⇒ (1). This is trivial giv en the fact that the fu nctor C ( B ) → K ( B ) preserves those comp onent wise copro du cts o f complexes whic h exist in C ( B ). (1) = ⇒ (2). Let us fix a contrac tible complex in K ( B ): Y : . . . d n − 2 − − − → Y n − 1 d n − 1 − − − → Y n d n − − − → Y n +1 d n +1 − − − → . . . . By definition, the ident it y morphism of Y is h omotopy equ iv alen t to the zero morphism in C ( B ), so there are morphisms s n : Y n → Y n − 1 in B su c h that 1 Y n = d n − 1 s n + s n +1 d n . When comp osing with d n , w e g et d n = d n s n +1 d n , so s n +1 d n : Y n → Y n is idemp otent in B for eac h n ∈ Z . Hence there are morp h isms p n : Y n → X n and j n : X n → Y n in B such that p n j n = 1 X n and j n p n = s n +1 d n . Let us denote b y f n : X n − 1 ∐ X n → Y n and g n : Y n → X n − 1 ∐ X n the morphisms defined as f ollo ws: f n = ( d n − 1 j n − 1 , j n ) , a nd g n =  p n − 1 s n p n  . Using the identitie s ab o ve, it is easy to c hec k that f n g n = 1 Y n and g n f n is an isomorphism in B f or eac h n . Therefore, b oth f n and g n are isomorphisms and g n f n is the iden tit y morphism. Finally , it is straigh tforw ard to c hec k that the f amily of morph isms ( f n | n ∈ Z ) induces an (iso)morphism f : ` n ∈ Z I X n ,n → Y in C ( B ).  It is not diffi cu lt to see that the condition of B h a vin g splitting idem- p oten ts is really necessary in Lemma 2.3. Ho we v er, there is a standard construction wh ic h allo ws us to a mend B with the missing summands if B do es not ha ve sp litting idemp oten ts. Definition 2.4. Let B b e an additive category . Then an additiv e category ¯ B is called an idemp otent c ompletion of B if (1) ¯ B has sp litting idemp oten ts; (2) B is a full sub category of ¯ B ; (3) Ev ery ob ject in ¯ B is a direct s u mmand of an ob ject in B . LOCALL Y WELL GENERA T E D HOMOTOPY CA TEGORIES OF COMPLEXES 7 It is a classical result that idemp oten t completions alwa ys exist. W e refer for example to [5, § 1] for a particular construction. Moreo v er, it is we ll- kno wn that if B h as arbitrary copr o ducts, then also ¯ B has them and they are compatible with copro ducts in B . No w w e can state the main result of the sect ion sho w in g that for K ( B ) b eing generated by a set (and, in particular, for K ( B ) b eing well generated), the catego ry B is necessarily p ure semisimple. Theorem 2.5. L et B b e an additive c ate gory with arbitr ary c opr o ducts and assume that ther e is a set of obje cts S ⊆ K ( B ) such that K ( B ) = Lo c S . Then B is pur e semisimple. Pr o of. Note that we can replace S by a singleton { Y } ; tak e for instance Y = ` Z ∈S Z . Let us denote by X ∈ B th e copro duct ` n ∈ Z Y n of all comp onent s of Y . W e will sho w th at B = Add X . First, w e claim that K (Add X ) is a dense sub catego ry of K ( B ), that is, eac h ob ject in K ( B ) is isomorphic to one in K (Add X ). I ndeed, Y ∈ K (Add X ) and one easily c h ecks that the closure of K (Add X ) un der taking isomorphic ob jects in K ( B ) is a lo calizing su b category . Hence K (Ad d X ) is dense in K ( B ) and the claim is p ro v ed. Supp ose for the moment th at B h as splitting idemp oten ts. If w e identify B with t he f ull su b category of K ( B ) formed b y complexes concen trated in degree zero, we ha ve prov ed that eac h ob ject Z ∈ B is isomorphic to a complex Q ∈ K (Ad d X ). Th at is, there is a chain complex homomo rphism f : Z → Q such that Q ∈ C (Add X ) and f b ecomes an isomorphism in K ( B ). In p articular, the mapping cone C f of f is con tr actible: C f : . . . − → Q − 3 d − 3 − → Q − 2 ( d − 2 0 ) − → Q − 1 ∐ Z ( d − 1 ,f 0 ) − → Q 0 d 0 − → Q 1 − → . . . Here, f 0 is the degree 0 co mp onent of f . Con s equen tly , Lemma 2.3 yields the follo wing comm utativ e d iagram in B with isomorph isms in columns: Q − 2 ( d − 2 0 ) − − − − → Q − 1 ∐ Z ( d − 1 ,f 0 ) − − − − − → Q 0 ∼ =   y ∼ =   y ∼ =   y U ∐ V ( 0 1 0 0 ) − − − − → V ∐ W ( 0 1 0 0 ) − − − − → W ∐ Z It follo ws that V , W and also Q − 1 ∐ Z and Z are in Ad d X . Hence B = Add X . Finally , let B b e a general additiv e catego ry with copro d ucts and ¯ B b e its idemp otent completio n. F rom th e fact that K ( B ) h as splitting idemp oten ts, [32, 1.6.8], one easily sees that the full em b edding K ( B ) → K ( ¯ B ) is d ense. W e already kno w that if K ( B ) = Lo c S f or a set S , then ¯ B = Add X for some X ∈ ¯ B . In fact, we can tak e X ∈ B b y the ab o v e construction. But then clearly B = Ad d X when the additive closure is tak en in B . Hence B is pure semisimple.  R emark. When studying well generated triangulated categories, an imp or- tan t role is pla y ed b y s o-calle d κ –localizing sub categories, see [3 2, 26]. W e recall that giv en a cardinal n um b er κ , a κ –c opr o duct is a copro duct with few er than κ su mmands. If T is a triangulated category w ith arb itrary κ – copro ducts, a th ic k sub catego ry L ⊆ T is cal led κ – lo c alizing if it is closed 8 JAN ˇ S ˇ TOV ´ I ˇ CEK under taking κ –copro du cts. In this cont ext, one can state the follo wing “b ound ed ” ve rsion of T heorem 2.5: Let κ b e an u ncount able r egular card inal and B b e an add itive catego ry with κ –copro ducts. If K ( B ) is generated as a κ –lo calizing sub category by a set S of fewer than κ ob jects, then th ere is X ∈ B such that eve ry ob ject of B is a summand in a κ –copro du ct of copies of X . Note that Theorem 2.5 giv es immediately a wide ran ge of examples of catego ries w hic h are not w ell generated. F or instance, K (Mod - R ) is n ot w ell generated for any ring R which is not righ t pure semisimple. O ne can tak e R = Z or R = k ( · ⇒ · ), the Kr on eck er algebra o v er a field k . The fact that K (Ab) is n ot w ell generated w as first observ ed by Neeman, [32, E .3.2], using differen t argumen ts. In fact, w e can state the follo wing pr op osition, whic h w e later generalize in Section 5: Prop osition 2.6. L et R b e a ring. Then th e fol lowing ar e e qu ivalent: (1) K (Mo d- R ) is wel l gener ate d; (2) K (Mo d- R ) is c omp actly gener ate d; (3) R is right pur e semisimp le. If R is an artin a lgebr a, the c onditions ar e furth er e qui v alent to: (4) R is of finite r epr esentation typ e. Pr o of. (2) = ⇒ (1) is clear, as compactly generated is the same as ℵ 0 – w ell ge nerated. (1) = ⇒ (3) follo ws by Th eorem 2.5 and Prop osition 2 .2. (3) = ⇒ (2) has b een pro ve d b y Holm and Jørgensen, [15, § 4 (3), p. 17]. Finally , the equiv alence b etw een (3) and (4) is due to Auslander, [4 , T heorem A].  3. Locall y we ll gene ra t ed triangul a te d ca t egories W e hav e seen in the last section that a triangulated catego ry of the form K (Mo d - R ) is o ften not w ell generated. One migh t get an impression that handling such categories is hop eless, but the main problem h ere actually is that th e category is v ery b ig in the sense that it is not generated by any set. Otherwise, it has a very reasonable structure. W e shall see that it is lo cally w ell generated in the follo wing sense: Definition 3.1. A triangulated category T with arbitrary copro du cts is called lo c al ly wel l gener ate d if Lo c S is wel l generated for an y set S of ob jects of T . In f act, we prov e that K (Mo d- A ) is lo cally well generated for any sk eletally small additiv e category A . T o this end, w e firs t need to b e able to measure the size of m o dules and complexes. Definition 3.2. Let A b e a sk eleta lly small additiv e category and M ∈ Mo d- A . Reca ll that M is a cont ra v ariant additiv e functor A → Ab by definition. Th en the c ar dinality of M , d enoted b y | M | , is defin ed as | M | = X A ∈S | M ( A ) | , LOCALL Y WELL GENERA T E D HOMOTOPY CA TEGORIES OF COMPLEXES 9 where | M ( A ) | is j ust the usual cardinalit y of the group M ( A ) and S is a fixed represent ativ e set for isomorphism classes of ob jects f rom A . The c ar dinality of a c omplex Y = ( Y n , d n ) ∈ K (Mo d - A ) is defined as | Y | = X n ∈ Z | Y n | . It is not so difficult to see that the category of all co mplexes whose car- dinalities are b ounded b y a giv en regular cardinal alwa ys giv es r ise to a w ell-ge nerated sub category of K (Mod - A ): Lemma 3.3. L et A b e a skeletal ly smal l additive c ate gory and κ b e an infinite c ar dinal. Then the fu l l sub c ate gory S κ forme d by al l c omplexes of c ar dinality less than κ me ets c onditions (2) a nd (3) of Definition 1.1. In p articular, T κ = Lo c S κ is a κ –wel l gener ate d sub c ate gory of K (Mod - A ) for any r e gular c ar dinal κ . Pr o of. Let Y ∈ K (Mo d- A ) su c h that | Y | < κ . If ( Z i | i ∈ I ) is an arbitrary family of complexes in K (Mod - A ), we can construct their copro duct as a comp onent wise co pro du ct in C (Mo d- A ). Then whenev er f : Y → ` i ∈ I Z i is a m orphism in C (Mo d- A ), it is str aightforw ard to see that f factorizes through ` i ∈ J Z i for some J ⊆ I of cardinalit y less th an κ . Hence Y is κ –small in K (Mo d - A ). Regarding p art (3) of Definition 1.1, consid er a m orp hism f : Y → ` i ∈ I Z i . W e ha v e the follo w ing factorization in the ab elian category of complexes C (Mo d- A ): Y ( f i ) − → a i ∈ I Im f i j − → a i ∈ I Z i . Here, f i : Y → Z i are the comp ositions of f with th e canonical pro jections π i : ` i ′ ∈ I Z i ′ → Z i , and j stands for the ob vious in clusion. It is easy to see that | Im f i | < κ for ea c h i ∈ I and that the morphism j is a copro duct of the inclusions Im f i → Z i . Hence (3) is satisfied. F or the seco nd part, let κ b e regular and T κ = Lo c S κ . Let us denote b y S ′ a representa tiv e set of ob jects in S κ . It only r emains to pro ve that S ′ satisfies condition (1) of Defin ition 1.1, w h ic h is r ather easy . Namely , let X ∈ T κ suc h th at T κ ( Y , X ) = 0 for eac h Y ∈ S ′ . Th en T ′ = { Y ∈ T κ | T κ ( Y , X ) = 0 } defines a localizing sub category of T κ con taining S κ . Hence, T ′ = T κ and X = 0.  W e will also need (a simplified v ersion of ) an imp ortant result, whic h is essen tially conta ined already in [32]. It says that the pr op ert y of b eing we ll generated is preserv ed when passing to any lo calizing s u b category gener- ated b y a set. In p articular, every well generated categ ory is lo cally w ell generated. Prop osition 3.4. [26, Theorem 7.2 .1] L et T b e a wel l gener ate d triangulate d c ate g ory and S ⊆ T b e a set of obje cts. Then Lo c S is a wel l gener ate d triangulate d c ate gory, to o. No w, we are in a p osition to state a theorem whic h giv es u s a ma jor source of examples of lo cally we ll generated triangulated categories. 10 JAN ˇ S ˇ TOV ´ I ˇ CEK Theorem 3.5. L et A b e a skeletal ly smal l additive c ate gory. Then the triangulate d c ate gory K (Mod- A ) is lo c al ly wel l gener ate d. Pr o of. As in Lemma 3.3, w e denote by S κ the full sub category of K (Mo d- A ) formed by complexes of cardinalit y less than κ and put T κ = Lo c S κ , the lo calizing class generated by S κ in K (Mo d- A ). Th en T κ is ( κ –)w ell generated for eac h regular card in al κ by Lemma 3.3 and clearly K (Mo d - A ) = [ κ regular S κ = [ κ regular T κ . No w, if S ⊆ K (Mo d- A ) is a set of ob jects, then S ⊆ T κ for some κ . Hence also Lo c S ⊆ T κ and Lo c S is w ell generated by Prop osition 3. 4. It follo ws that K (Mo d- A ) is locally we ll generated.  Ha ving obtained a large class of examples of lo cally wel l generated triangu- lated categories, one might ask for some basic pr op erties of suc h categories. W e will p ro v e a version of the s o-called Bousfield L o calizat ion Theorem here: Prop osition 3.6. L et T b e a lo c al ly wel l gener ate d triangulate d c ate gory and S ⊆ T b e a set of obje cts. Then T / Lo c S is a Bousfield lo c alization; that is, ther e is a lo c alization functor L : T → T such that Ker L = Lo c S . In p articular, we have Im L = { X ∈ T | T ( Y , X ) = 0 for e ach Y ∈ S } , ther e is a c anonic al triangle e quiv alenc e b etwe en T / Loc S and Im L giv e n by the c omp osition Im L ⊆ − → T Q − → T / Lo c S , and al l morp hism sp ac es in T / Lo c S ar e sets. Pr o of. The pro of is rather stand ard. Lo c S is we ll generated, s o it satisfies the Bro wn Repr esen tabilit y Theorem (see Prop osition 1.2). Hence the in- clusion i : Lo c S → T has a righ t adjoint by [32, 8.4.4]. The comp osition of this right adjoin t with i giv es a so-called colo calizatio n fun ctor Γ : T → T whose essen tial image is equal to Loc S . The definition of a col o calizati on functor is fo rmally dual to the one of a lo calization functor; see [26, § 4 .12] for details. A well-kno wn construction then yields a lo calization f unctor L : T → T suc h that Ker L = Lo c S . W e refer to [32, 9.1.14 ] or [26, 4.12.1] for details. The rest follo w s from [32, 9.1.16] or [26, 4.9.1].  R emark. Prop osition 3.6 has b een pro ved b efore for w ell generated triangu- lated categories. Th is is implicitly conta ined for example in [26 , § 7.2]. It also generalizes more classical resu lts, suc h as a corresp onding s tatement for the derive d catego ry D ( B ) of a Grothendieck abelian category B , [2 , 5.7]. T o see th is, one only needs to observ e that D ( B ) is well generated, see [26, Example 7.7]. An ob vious question is wh ether the Bro wn Represen tabilit y Theorem also holds for lo cally w ell generated categories, as this wa s the crucial feature of w ell generated categ ories. Unfortun ately , this is not the case in general, as the follo wing example s uggested by Henn ing Krause sh o ws . LOCALL Y WELL GENERA T E D HOMOTOPY CA TEGORIES OF COMPLEXES 11 Example 3.7 . According to [10 , Exercise 1, p. 131], one can construct an ab elian catego ry B with some Ext-spaces b eing p rop er classes. Namely , le t U b e the class o f all cardinals, and let B = Mo d- Z h U i , th e category o f all “mo dules ov er the free ring o n t he prop er class o f generators U .” That is, an ob ject X of B is an ab elian group such th at eac h κ ∈ U has a Z -linear action on X and this ac tion is trivial for all but a set o f cardinals. Such a catego ry ad m its a v alid set-theoretical d escription in ZF C . If w e denote by Z th e ob ject of B whose un derlying group is free of rank 1 and κ · Z = 0 for eac h κ ∈ U , then Ext 1 B ( Z , Z ) is a prop er class (see also [26, 4.15] or [6, 1.1]). Giv en the ab o v e description of ob jects of B , one can easily adjus t the pro of of T heorem 3.5 to see that K ( B ) is lo cally we ll generated. Let K ac ( B ) stand for the full sub category of all acyclic complexes in K ( B ). T hen K ac ( B ) is clearly a lo calizing sub category of K ( B ), h ence lo cally w ell-g enerated. It has b een sh own in [6] that K ac ( B ) do es n ot satisfy th e Bro wn Repr e- sen tabilit y T heorem. In fact, one pr o ved eve n more: K ac ( B ) is lo calizing in K ( B ), but it is not a k ernel of an y localization functor L : K ( B ) → K ( B ). More sp ecifically , the composition of functors, the second of whic h is con- tra v ariant, K ac ( B ) ⊆ − − − − → K ( B ) K ( B )( − , Z ) − − − − − − − → Ab is not rep r esen table b y an y ob ject of K ac ( B ). Y et another natural question is what other triangulated categ ories are lo cally w ell generated. A deep er analysis of this pr oblem is left for future researc h, bu t w e w ill see in Section 4 th at K ( B ) is lo cally well generated for an y finitely accessible add itiv e category B . F or no w, we w ill prov e that the class of locally w ell generated triangulated catego ries is closed under some natural constructions. Let us start with a general lemma, w h ic h h olds even if morphism sp aces in the qu otien t T / L are prop er classes: Lemma 3.8. L et T b e a triangulate d c ate gory and L ⊆ L ′ b e two lo c alizing sub c ate gories of T . Then L ′ / L is a lo c alizing sub c ate gory of T / L . Pr o of. It is easy to see that L ′ / L is a full sub category of T / L whic h is closed under taking isomorp hic ob jects, see [33, Th´ eor ` e me 4-2] or [22, Prop osition 1.6.5]. T he rest follo w s directly from th e constr u ction of T / L .  No w we can sho w th at taking lo calizing sub catego ries and lo calizing with resp ect to a set of ob jects preserves the lo cally wel l generated p rop erty . Prop osition 3.9. L et T b e a lo c al ly wel l gener ate d triangulate d c ate gory. (1) Any lo c alizing sub c ate gory L of T is itself lo c al ly wel l gener ate d. (2) The V er dier quotient T / Loc S is lo c al ly wel l gener ate d for any set S of obje cts in T . Pr o of. (1) is trivial. F or (2), put L = Lo c S and consider a set C of ob jects in T / L . W e h av e to pro v e that the lo calizing s u b category generated b y C in T / L is w ell generated. Since the ob jects of T and T / L coincide by definition, w e can co nsider a lo calizing sub category L ′ ⊆ T defin ed by L ′ = Lo c ( S ∪ C ). One easily sees using Lemma 3.8 that L ′ / L = Lo c C in T / L . Since b oth L and L ′ are wel l generated b y definition, s o is L ′ / L by [26, 7.2.1]. Hence T / L is lo cally well generated.  12 JAN ˇ S ˇ TOV ´ I ˇ CEK W e conclude this section with an immediate consequence of Theorem 3.5 and Prop osition 3.9, whic h will b e u seful in th e next section: Corollary 3.10. L et A b e a smal l additive c ate gory and B b e a ful l su b c at- e gory of Mo d- A which is c lose d under arbitr ary c opr o ducts. Then K ( B ) is lo c al ly wel l g e ner ate d. 4. Finitel y accessibl e additive ca tegories There is a natural generalization of mo dule catego ries, namely the ad- ditiv e version of finitely accessible catego ries in the terminology of [1 ]. As w e ha ve seen, there is quite a lot of freedom to c ho ose B in the ab ov e Corollary 3.10. W e will us e this fact and a standard tric k to (seemingly) generalize Theorem 3.5 from mod ule cat egories to finitely accessible add itiv e catego ries. W e start with a defin ition. Definition 4.1. Let B be an a dditiv e category which adm its arbitrary fil- tered colimits. Then: • An ob ject X ∈ B is called finitely pr esentable if the repr esen table functor B ( X, − ) : B → Ab pr eserves filtered colimits. • The category B is ca lled finitely ac c essible if there is a set A of finitely present able ob jects from B such that eve ry o b ject in B is a fi ltered colimit of ob jects from A . Note that if B is finitely accessible, the full sub catego ry fp( B ) of B formed b y all finitely pr esen table ob jects in B is skelet ally small, [1, 2.2]. Sev- eral other general pr op erties of finitely accessible categories will follo w from Prop osition 4.2. Finitely accessible categories o ccur at man y o ccasions. The simplest and most n atural example is the m o dule category Mo d- R o v er an asso ciativ e unital ring. It is w ell-kno wn that finitely p resen table ob j ects in Mod - R coincide w ith fi nitely presen ted R –mo dules in the usu al sense. T he s ame holds for Mo d- A , the category of mo d ules o ver a small additive category A . Motiv ated by represent ation th eory , finitely accessible categories w ere studied b y Cra w ley-Boeve y [8] under the name lo cally finitely p r esen ted catego ries; see [8, § 5] for further examples. The term fr om [8], h ow ev er, ma y cause some confusion in th e ligh t of other definitions. Namely , Gabriel an d Ulmer [11] h a ve d efi ned the concept of a lo c al ly finitely pr esentable category whic h is, in o ur terminology , a co complete finitely accessible categ ory . As the latt er concept has b een u sed quite su bstan tially in one of our main references, [26], w e stic k to the term in ology of [1]. The crucial fact ab out fi nitely accessible add itiv e categories is the f ollo w- ing representa tion theorem: Prop osition 4.2. The a ssignments A 7→ Flat- A and B 7→ fp( B ) form a bije ctive c orr esp ondenc e b etwe en (1) e quivalenc e classes of skeletal ly smal l additive c ate gories A with split- ting idemp otents, and (2) e quivalenc e classes of a dditive finitely ac c essible c ate gories B . LOCALL Y WELL GENERA T E D HOMOTOPY CA TEGORIES OF COMPLEXES 13 Pr o of. See [8, § 1.4].  R emark. The corresp ond ence from Pr op osition 4.2 restricts, using [8, § 2.2], to a bijection b et ween equiv alence classes of s keleta lly small add itiv e cat- egories with finite colimits (equiv alen tly , with cok er n els) and equiv alence classes of locally finitely present able categories in the sense o f Gabr iel and Ulmer [11]. One of the main results of this pap er has no w b ecome a mere corollary of preceding results: Theorem 4.3. L et B b e a finitely ac c essib le additive c ate gory. Then K ( B ) is lo c al ly wel l gener ate d. Pr o of. Let us put A = fp( B ), the f ull su b category of B formed b y all finitely present able ob j ects. Using Prop osition 4.2 , we see that B is equiv alen t to the category Flat- A . The category K (Flat- A ) is lo cally we ll generated b y Corollary 3.10, and so m ust b e K ( B ).  The remaining question when K ( B ) is κ –w ell generated and whic h car- dinals κ can o ccur will b e ans w ered in the next section. F or now, we kno w b y Th eorem 2.5 th at a necessary condition is that B b e pure semisimple. In f act, we will sho w that th is is also sufficient, bu t at the moment we will only give a b etter description of pu re semisimple finitely accessible add itiv e catego ries. Prop osition 4.4. L et B b e a finitely ac c essible additive c ate gory. Then the fol lowing ar e e quivalent: (1) B is pur e semisimple in th e sense of Definition 2.1; (2) Each obje ct in B is a c opr o duct of (inde c omp osable) finitely pr e- sentable obje cts; (3) Each flat right A –mo dule is pr oje ctiv e , wher e A = fp( B ) . Pr o of. F or th e whole argument, we put A = fp( B ) and without loss of generalit y assume that B = Flat- A . (1) = ⇒ (3). Assu me that Flat- A is pure semisimple. As in the pro of for P r op osition 2.2, w e can use a generalization [3, T heorem 26.1] of Ka- plansky’s theorem, to deduce that there is a cardinal num b er λ suc h th at eac h fl at A –mo dule is a direct s u m of at most λ –generated flat A –mo dules. The ke y step is then con tained in [13, Corollary 3.6] w hic h sa ys th at under the latter condition A is a r igh t p er f ect categ ory . Th at is, it satisfies the equiv alen t conditions of Bass’ theorem [18, B.12] (or more p recisely , its v er- sion for con tra v ariant functors A → Ab). One of the equiv alent cond itions is condition (3). (3) = ⇒ (2). Th is is a consequence of Bass’ theorem; see [18, B.13]. (2) = ⇒ (1). T rivial, B = Add X where X = L Y ∈A Y .  F or further reference, we men tion one more condition w hic h one migh t imp ose on a finitely accessible additiv e categ ory . Namely , it is we ll k n o w n that f or a ring R , the category Flat- R is closed under pro du cts if and only if R is left coheren t. This generalizes in a natural wa y for finitely accessible additiv e catego ries. Let u s recall that an add itiv e category A is said to ha ve 14 JAN ˇ S ˇ TOV ´ I ˇ CEK we ak c okernels if for eac h morp hism X → Y there is a morp hism Y → Z suc h that A ( Z , W ) → A ( Y , W ) → A ( X, W ) is exact for all W ∈ A . Lemma 4.5. L et B b e a finitely ac c essible additive c ate gory and A = fp ( B ) . Then the fol lowing ar e e qu i valent: (1) B has pr o ducts. (2) Flat- A is close d u nder pr o ducts in Mo d- A . (3) A has we ak c okernels. Pr o of. See [8, § 2.1].  R emark. If B has pro du cts, one can give a more classical p ro of for Prop osi- tion 4.4. Namely , one can th en replace the argument by Guil Asensio, Izur- diaga and T orr ecillas [13] by an older and simpler argument b y Chase [7, Theorem 3.1]. 5. When is the homotopy ca tegor y we ll gene ra t ed? In this final section, we hav e develo p ed enough to ols to answer the ques- tion when exactly is th e homotop y categ ory of complexes K ( B ) well gener- ated if B is a finitely accessible additiv e catego ry . This wa y , we will generalize Prop osition 2.6 and also giv e a rather complete answ er to [15, Question 4.2] ask ed b y Holm an d Jørgensen. Finally , w e will give another crite rion f or a triangulated category to b e (or not to b e) wel l generated and this w a y con- struct other classes of examples of categ ories whic h are not w ell generated. First, we recall a crucial result d u e to Neeman: Lemma 5.1. L et A b e a skeletal ly smal l additive c ate gory. Then the ho- motopy c ate gory K (Pro j - A ) i s ℵ 1 –wel l gener ate d. If, mor e over, A has we ak c okernels, then K (Pro j- A ) is c omp actly gener ate d. Pr o of. Neeman has p ro v ed in [31 , Theorem 1.1] that, giv en a ring R , the catego ry K (Pro j- R ) is ℵ 1 –w ell generated, and if R is left coherent then K (Pro j- R ) is ev en c ompactly generated. The act ual argumen ts, con tained in [31, §§ 4–7], immed iately generalize to the setting of pro jectiv e mo dules o ver small categ ories. The role of finitely generated fr ee mo dules o ver R is tak en by repr esen table fu nctors, and instead of the dualit y b et ween the cate- gories of left and r igh t pro jectiv e finitely generated mo d u les we consider the dualit y b et ween the idemp oten t completio ns of the cate gories of co v ariant and con tra v ariant rep r esen table functors.  W e already know th at K ( B ) is alw a ys locally w ell generated. When em- plo yin g Lemma 5.1, w e can sho w the follo wing statemen t, whic h is one of the main r esults of this pap er: Theorem 5.2. L et B b e a finitely ac c essible additive c ate g ory. Then the fol lowing ar e e quivalent: (1) K ( B ) is wel l g e ner ate d; (2) K ( B ) is ℵ 1 –wel l gener ate d; (3) B is pur e semisimple. If, mor e over, B has pr o ducts, then the c onditions ar e further e quivalent to (4) K ( B ) is c omp actly gener ate d. LOCALL Y WELL GENERA T E D HOMOTOPY CA TEGORIES OF COMPLEXES 15 Pr o of. (1) = ⇒ (3). If K ( B ) is w ell generated, it is in p articular generated b y a s et of ob jects as a lo calizing s ub category of itself; see Prop osition 1.2. Hence B is p ure semisimp le by Theorem 2.5. (3) = ⇒ (2) and (4). If B is pure semisimple and A = fp( B ), t hen B is equiv alen t to Flat- A by Proposition 4.2, and Flat - A = Pro j- A b y Pr op osi- tion 4.4. The conclusion follo ws by Lemm as 5.1 and 4.5. (2) or (4) = ⇒ (1). This is obvious.  R emark. (1) Neeman prov ed in [31] m ore than stated in Lemma 5.1. He describ ed a particular set of generators for K (Pro j- A ) satisfying conditions of Definition 1.1. Namely , K (Pro j- A ) is alw ays ℵ 1 –w ell generated by a rep- resen tativ e set of b ound ed b elo w complexes of finitely generated pro jectiv es. Moreo ve r, he ga v e a n explicit d escription of compact ob jects in K (Pr o j- A ) in [31, 7.12]. (2) An exact c haracterizat ion of when K ( B ) is compactly generated and thereb y a complete answe r to [15, Q uestion 4.2] d o es n ot seem to b e kno wn. W e ha v e s ho w n that this red uces to the pr oblem when K (Pro j- A ) is com- pactly generated. A sufficien t condition is give n in Lemma 5.1, but it is pr ob- ably n ot n ecessary . On the other hand, if R = k [ x 1 , x 2 , x 3 , . . . ] / ( x i x j ; i, j ∈ N ) w h ere k is a field, then K (Flat- R ) coincides with K (Pr o j- R ), but the latter is not a compactly generated triangulated category; see [31, 7.16] for details. Example 5.3 . The ab o v e theorem adds other lo cally wel l generated b ut not well generated triangulated catego ries to our rep ertoire. F or example K ( T F ), where T F stands for the catego ry of all torsion-free abelian group s, has this pr op ert y . W e finish the p ap er with some examples of triangulated categories wher e the fact th at they are not generated b y a s et is less obvious. F or this purp ose, w e will use the follo wing criterion: Prop osition 5.4. L et T b e a lo c al ly wel l gener ate d triangulate d c ate gory and L b e a lo c alizing sub c ate gory. Consider th e diagr am L ⊆ − − − − → T Q − − − − → T / L . If two of the c ate gories L , T and T / L ar e wel l gener ate d, so is the thir d. Pr o of. If L = Lo c S and T / L = Lo c C for some sets S , C , let L ′ b e the lo calizing sub category of T generated b y the set of ob jects S ∪ C . Lemma 3.8 yields the equ ality T / L = L ′ / L . Hence also T = L ′ , so T is generated by a set, and consequently T is w ell generated. If L and T are we ll generated, so is T / L by [26, 7.2.1]. Finally , one kn o w s that X ∈ T b elongs to L if a nd only if QX = 0; see [32, 2.1.33 and 1.6.8]. Therefore, if T and T / L are w ell generated, so is L by [26 , 7.4.1].  R emark. W e stress here that b y sa ying that T / L is w ell generated, we in particular mean th at T / L is a usual category in the sense that all morphism spaces are sets and not prop er classes. No w we can conclude b y showing that some homotop y categories of acyclic complexes are n ot we ll generated. 16 JAN ˇ S ˇ TOV ´ I ˇ CEK Example 5.5 . Let R b e a r ing, K ac (Mod - R ) b e the fu ll su b category of K (Mo d - R ) formed by a ll acyclic complexes, and L = Lo c { R } . It is w ell- kno wn but also an easy consequence of Prop osition 3.6 that the comp osition K ac (Mod - R ) ⊆ − → K (Mod- R ) Q − → K (Mo d- R ) / L is a triangle equiv alence b etw een K (Mo d- R ) / L and K ac (Mod - R ). By Prop osition 2.6, K (Mo d - R ) is we ll generated if and only if R is right pure semisimple. Therefore, K ac (Mod - R ) is w ell generated if and only if R is right pure semisimple by Prop osition 5.4. In fact, K ac (Mod - R ) is not generated by any set of ob jects if R is not right pure semisimple. As particular examples, w e ma y tak e R = Z or R = k ( · ⇒ · ) for any field k . Example 5.6 . Let B b e a finitely accessible categ ory . Recall that B is equiv- alen t to Flat- A for A = fp( B ). Then the natural e xact structure on Flat- A coming from Mod- A is nothing else than the w ell-kno wn exact stru cture giv en b y pu re exact short sequences in B (see eg. [8 ]). W e d enote b y K pac (Flat- A ) th e full sub catego ry of K (Flat- A ) formed b y all complexes exact with r esp ect to this exact stru cture, and call s u c h complexes pur e acyclic . More explicitly , X ∈ K (Flat- A ) is pure acyclic if and only if X is acyclic in Mod- A and all the cycles Z i ( X ) are fl at. Note that K pac (Flat- A ) is closed under taking copro du cts in K (Flat- A ). Neeman pro v ed in [31, T heorem 8.6] that X ∈ K (Flat- A ) is pure acycl ic if and only if there are no non-zero homomorph isms from an y Y ∈ K (Pro j- A ) to X . Then either by com bining Prop osition 3.6 with Lemma 5.1 or b y using [31, 8.1 and 8.2], one sh o ws that the comp osition K pac (Flat- A ) ⊆ − → K (Flat- A ) Q − → K (Flat - A ) / K (Pro j- A ) is a triangle equiv alence. No w aga in, Prop osition 5.4 implies that K pac (Flat- A ) is well generated if and only if B is pur e semisimple. If B is of the form Flat- R for a rin g R , this p recisely means that R is right p erfect. As a p articular example, K pac ( T F ) is lo cally well generated bu t not we ll generated, wh ere T F stands f or the class of all torsion-free ab elian groups. Referen ces [1] J. Ad´ amek and J. Rosic k ´ y, L o c al ly Pr esentab le and A c c essible Cate gories , London Math. So c. Lect. Note S er., V ol. 189, Cam bridge Univ. Press, Cambridge, 1994. [2] L. Alonso T arr ´ ıo, A . 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Charles University in Prague, F acul ty of Ma thema tics and Physics, De- p ar tment of Algebra, Sok olo vska 83, 186 75 Praha, C zech Republic E-mail addr ess : stovicek@ka rlin.mff. cuni.cz