Categories of Fractions Revisited

CA TEGORIES OF FRA CTIONS REVISITED TOBIAS FRITZ Abstract. The theory of categories of fractions as o riginally deve loped by Gabriel and Zisman [ 1 ] is review ed in a p edagogical manner g iving detailed proof s of all statemen ts. A w eak en ing of the category of fractions axioms used b y Higson [ 4 ] is di scussed and shown to be equiv alen t to the or iginal axioms. Contents 1. Int ro duction 1 Summary . 2 Notation and terminology . 2 2. Lo calization of categories 2 3. Categorie s of fractions 4 4. W eakening the requirements 10 5. Additiv e categorie s of fractions 11 References 14 1. Introduction In ca tegory theor y , the co ncept of lo calizatio n is a too l for constructing a new category from a g iv en one. The idea is a s follows: a categor y may have a certain class of morphisms whic h are not all in vertible, although morally they “should” b e inv ertible. As an example, one may cons ider weak homotopy equiv a le nc e s in the homotopy categor y of top ological spaces: some weak homotopy e quiv alences are homotopy equiv alences, and hence isomorphisms, but not a ll o f them a r e [ 3 ]; on the other ha nd, tw o weakly ho motopy equiv alent spaces b ehave in absolutely the same wa y concer ning the prop erties pro bed by maps fr om or to suita bly nice spaces, a nd hence should morally be iso morphic. Given such a class of mo rphisms in a categ ory , one ca n fo r m a lo c alization of the original c a tegory , which is a new category which g uarantees all “morally in v ertible” morphisms to b e in v ertible, while approximating the original catego ry as closely as p ossible. This idea ca n b e made precise in terms of a univ ersal pro per t y; see section 2 . Lo calizations exist no t o nly for categ ories, but a ls o for other kinds of a lgebraic structures. F or example for rings: adjoining for mal in verses for a cer tain class of ring elements yields a new r ing f rom a given one. Under certa in conditions on the class W of elements to b e inv erted—the so-ca lle d Or e conditions—there is a particularly nice wa y to describ e the elemen ts of the lo c a lized ring in terms of a n equiv alence class of formal fractions, where a forma l fra ction is defined to hav e a n element of the original ring in the n umera tor and an elemen t of W in the denominator. 1 2 TOBIAS FRITZ It turns out that pretty muc h the s ame technique that works for r ing s can a lso applied to catego ries. Under ce r tain conditions, th e localizatio n o f a category with resp ect to a class of mor phis ms can b e describ ed in terms o f “formal fractions”. If this co nstruction is p ossible, the resulting lo c a lization is a c ate gory of fr actions . In some cases, such an a bstract construction can be more useful than a concrete (in the category-theo retical se ns e!) description of the lo caliza tion. F urthermore, categorie s of fractions can b e relev a nt for o ther gener al categ o rical constructions; the theory of V erdier lo calization in the context o f triangulated categories is an example. Due to the metamathematical nature of category theory , the ob jectives in cate- gory theory a re quite different fr om those in ring theory: thinking of a categor y as representing the collection of mo dels of a ma thematical theory , taking a categor y of fractions is a tool to construct a new mathematic al the ory from a given one. Summary . In s e ction 2 , the co ncept o f lo calization of a category is int ro duced and compared to taking a quotien t category . Section 3 then gives a detailed account of the catego ry of fraction axioms and their c o nsequences; in pa rticular, all pro ofs are pres e nted in complete detail. Sectio n 4 go es on to study a weak ening of the category of fra ction axio ms which was originally intro duced by Higson [ 4 ] in the context of biv a riant K -theor y o f C ∗ -algebra s. It is shown that this weakening is equiv alent to the usual set of axioms . This is the only new r esult of the present work. Finally , section 5 shows that a categor y of fractions is additive in case the original category is additiv e. Notation and termi nology . In all commutativ e diagra ms, the ob jects are simply denoted by fat dots “ • ”. Unles s noted other wise, all diagr a ms comm ute. Idenitity morphisms are pictured as double lines “ ”. The w or ds “iso morphism” and “monomor phis m” are abbrevia ted resp ectively a s “is o ” a nd “mono”. A split mono is a morphism which has a le ft inv er s e; it automatica lly is a mono. Domain and co domain of a morphism f are written as dom( f ) and cod( f ), resp ectively . This ar ticle is a revised version of par t of the a uthor’s Master’s thesis written at the Univ ersity of M ¨ unster in 2007. 2. Localiza tion of ca tegories In some contexts it may happ en that we ha ve a category C which is – in a sense depe nding on the situation – not well-b e hav ed. F or e x ample, it might b e that it is to o hard to do c o ncrete calculations , o r it mig h t b e that C do es not have some desired formal pro per t y . Then one can try to find a s e cond category b C whic h has the sa me ob jects as C together with a functor C → b C which is the identit y o n ob jects, such that b C is b etter-b ehav ed and a pproximates C in some appropria te sense also dep ending on the s ituation. Then instead of working in C directly , one can transp or t the morphisms from C t o b C via the functor C → b C and prov e theor e ms ab out the mo rphisms in the well-b ehav ed catego ry b C . The price one ha s to pay is that in general some information about the structure of C is lost on the w ay . Now there a re at lea st tw o concrete wa y s to make this precise. The first one is the notion of a q uotient category . Supp ose we are giv e n an equiv alence relation ∼ on every morphism set C ( A, B ) which is preserved under comp ositio n, meaning that ( f 1 ∼ f 2 ) = ⇒ ( f 1 g ∼ f 2 g ) ∧ ( hf 1 ∼ hf 2 ) ∀ f 1 , f 2 , g, h ∈ C (1) CA TEGORIES OF FRAC TIONS REVISITED 3 whenever t hese comp ositions a r e defined. Then comp osition of eq uiv alence classes is well-defined and defines the quotient catego ry C / ∼ toge ther with the canonica l pro jection functor C → C / ∼ . An y kind of ho mo topy theory serves as a go o d example. The second way is a conce pt ca lled lo cali zation . It may b e familia r from ring theory . Suppo s e w e are given a ca teg ory C and a subcla s s of morphisms called W , which “morally ” ought to b e isos, but in C not necessarily all of them a r e; using the letter W is supp osed to s uggest a reading like “weak equiv a lence” [ 5 ]. W e try to turn all the morphisms in W in to isos by adjoining formal inv erses for them. More precisely , we ar e lo oking for a catego ry b C = C [ W − 1 ] eq uipped with a lo calization functor Loc : C → C [ W − 1 ] which has the following univ er sal proper t y: (a) Loc ( w ) is an iso for all w ∈ W , (b) If F : C → D is any functor whic h maps W to isos , then F factors uniquely ov er Loc as in the diagram C Loc / / F   > > > > > > > > C [ W − 1 ] ∃ ! { { x x x x x D (2) In ca se such a functor exis ts , the categor y C [ W − 1 ] is called the “lo calizatio n o f C with respe c t to W ”. It serves a s the desired approximation b C to C . Since C [ W − 1 ] is defined via a universal prop erty , it is ce r tainly unique (up to a unique iso). Proving ex istence is the non trivial part. 2.1. Theorem. C [ W − 1 ] and Loc always exist. Pr o of. (from [ 2 , I I I.2.2] and [ 1 , 1 .1]). The catego ry C [ W − 1 ] can b e cons tr ucted in tw o steps: s tart with the category of paths—call it P ( C , W − 1 )—whic h has a s ob jects the o b jects of C , and as morphisms finit e strings h l 1 , . . . , l n i of comp osable literals, where a literal l k is either a morphis m of C (including W ) or a formal inv erse of a morphism in W . Comp o sition of these morphisms is defined as concatena tion of strings. F or every ob ject A ∈ C , w e also have the e mpty s tring hi A which starts and ends at A and is the identit y morphism of A in P ( C , W − 1 ). This whole definition can b e summar ized by saying that P ( C , W − 1 ) is th e free ca tegory genera ted b y the graph C ∪ W − 1 . There is a ca nonical ma p C → P ( C , W − 1 ) which is the iden tit y o n ob jects and maps every mo rphism f ∈ C to the corr esp onding sing le-literal string h f i . This map already has the desired universal prop erty (b) . How ev er, neither is this map a functor nor does it map W to is os. W e can easily fi x both of these issues by taking a quotient ca tegory of P ( C , W − 1 ) in whic h these pr o per ties are enforced. T o this end, w e in tro duce the equiv alence relation ∼ on string s generated b y closure under comp osition together with the elemen ta ry equiv alences (a) h i A ∼ h id A i ∀ A ∈ Ob j( C ), (b) h g , f i ∼ h g f i ∀ f , g ∈ C fo r which the composition g f ex ists, (c) h w, w − 1 i ∼ h i co d( w ) , h w − 1 , w i ∼ h i dom( w ) ∀ w ∈ W . Then it is clear that the induced map Loc : C → P ( C , W − 1 ) / ∼ is a functor a nd maps W to isos. As for universality , supp ose we are given some functor F : C → D mapping W to is os. It induces a unique functor P ( C , W − 1 ) → D . This functor maps the 4 TOBIAS FRITZ ab ov e ele men tary eq uiv alences to equalities, th us uniquely factors over the q uotient category P ( C , W − 1 ) / ∼ .  2.2. Remark. (a) F or lo ca lly s ma ll C , the lo ca lization C [ W − 1 ] need not b e lo cally small. Even under the conditions to b e discussed in the nex t section, it may well happ en that the lo calization has prop er cla sses as the colle ctions of morphisms betw een some pairs o f ob jects. Showing that this do es not happ en in a co nc r ete case seems to b e a hard problem; one case where loc a l smallness is known is for mo del categories and localizing with resp ect to the class of weak equiv a lences (see [ 5 , p.7 and 1.2.10]). (b) The ca nonical functor to a quo tien t ca tegory C → C / ∼ is full b y defini- tion of C / ∼ . How ever, this is usually not true for a lo caliza tion functor Loc : C → C [ W − 1 ]. 3. Ca tegories of fractions In all diagra ms dea ling with c ategories of fractions, a wig g ly arr ow / / /o /o /o denotes a morphism in W , while a straight ar row / / is an y morphism of C . In certain s itua tions, the lo calization C [ W − 1 ] ca n b e descr ibed muc h more ex- plicitly , which implies a lar ge gain of control ov er the structure of this category . W e s ay that the pair W ⊆ C allows a calculus of left fractions , if the following conditions are satisfied: (L0) W contains all identit y morphisms a nd is closed under comp osition. In other words, W ⊆ C is a sub categor y con ta ining all ob jects. (L1) Giv en any w ∈ W a nd a n arbitra r y mor phism f with dom( f ) = dom( w ), we can find w ′ ∈ W with dom( w ′ ) = co d( f ) and some morphism f ′ with co d( f ′ ) = co d( w ′ ), such t hat the diagram • w / / /o /o /o f   • f ′   • w ′ / / /o /o /o • commutes. (L2) Giv en w ∈ W and par a llel morphisms f 1 , f 2 such that f 1 w = f 2 w , there exists w ′ ∈ W such tha t w ′ f 1 = w ′ f 2 . • w / / /o /o /o • f 1 ( ( f 2 6 6 • w ′ / / /o /o /o • These co nditions ar e exact analogues of the Ore conditio ns in the theory of (not necessarily commutativ e) rings [ 6 , p. 3]. 3.1. Remark. Condition (L0) is not an essen tial restriction: if (L1) and (L2) hold for some clas s of morphisms W , then b oth also hold for the C -sub category g enerated by W ∪ { id A , A ∈ Ob j( C ) } . Hence W ca n be replac e d by th is sub category . Pr o of. W e assume that W satisfies (L1) and (L2) , but not ne c e ssarily (L0) . Then W ∪ { id A , A ∈ O b j( C ) } certainly also sa tisfies (L1 ) and (L2) , so it is eno ug h to s how that closing W under co mpo sition gives a morphism class c W which a lso satisfies (L1) and (L2) . CA TEGORIES OF FRAC TIONS REVISITED 5 Let w 1 , w 2 ∈ W be comp osable to w = w 1 w 2 . Given a n y f with dom( f ) = dom( w 1 ), applying (L1) twice s hows that we ca n find w ′ 1 , w ′ 2 ∈ W and f ′ , f ′′ ∈ C such that the diagram • w 1 / / /o /o /o f   • w 2 / / /o /o /o f ′   • f ′′   • w ′ 1 / / /o /o /o • w ′ 2 / / /o /o /o • commutes. No w w ′ = w ′ 1 w ′ 2 ∈ c W and f ′′ hav e the required prop erties with re- sp ect to w = w 1 w 2 and f . Applying this arg umen t inductively prov e s the claim ab out (L1) . Concerning (L2) , w e simila r ly c onsider the situation f 1 w 2 w 1 = f 2 w 2 w 1 , and obtain • w 1 / / /o /o /o • w 2 / / /o /o /o • f 1 ( ( f 2 6 6 • w ′ 1 / / /o /o /o • w ′ 2 / / /o /o /o • where, thanks to (L2) , we could choos e w ′ 1 such that w ′ 1 f 1 w 2 = w ′ 1 f 2 w 2 , and then w ′ 2 such that w ′ 2 w ′ 1 f 1 = w ′ 2 w ′ 1 f 2 , as desired.  3.2. Definitio n. A r o of ( f , w ) b etwe en t wo obje cts dom( f ) and dom( w ) is a dia- gr am of the form • • f ? ?        • w _ _ _ _ _ _ F rom now o n, we assume that W ⊆ C s atisfies (L0) , (L1) and (L2) , a nd der ive some consequences from this assumption. The wa y to think of a ro of ( f , w ) is a s b eing a for ma l “left fractio n” w − 1 f , defining a formal morphism from the lo wer left ob ject to the low er right ob ject. Then (L1) intuitiv ely s tates that it is p ossible to turn any formal “ r ight fraction” f w − 1 int o a left fra ction w ′− 1 f ′ , since w ′ f = f ′ w together with inv ertibilit y of w and w ′ implies f w − 1 = w ′− 1 f ′ . 3.3. Definition. Two r o ofs ( f 1 , w 1 ) and ( f 2 , w 2 ) ar e e quivalent if t her e ar e mor- phisms g and h forming a thir d r o of ( g f 1 , g w 1 ) = ( hf 2 , hw 2 ) as in the diagr am • • g ? ?        • h _ _ @ @ @ @ @ @ @ • f 1 ? ?        f 2 4 4 j j j j j j j j j j j j j j j j j j j j • w 1 j j j* j* j* j* j* j* j* j* j* j* j* j* j* w 2 _ _ _ _ hw 2 = gw 1 P Q R T V X [ o o c# f& h( j* l, m- o/ Note that it is not r equired tha t g or h b e an e le men t o f W , only the c omp osition g w 1 = hw 2 has to be in W . The equality of ( g f 1 , g w 1 ) = ( hf 2 , hw 2 ) is expres sed by commutativit y of the t wo squares in the diagram. The g oal o f this section is to e stablish that equiv a lence cla sses o f ro o fs form a category under the appropria te comp osition o per ation, and that this categ ory is the lo calization C [ W − 1 ]. This will b e done in a sequence o f small steps. In tuitiv ely , the first step is to show that the r o of ( f ′ , w ′ ) o ne obtains fro m using (L1) to turn 6 TOBIAS FRITZ a formal right fractio n f w − 1 int o a formal left fra ction w ′− 1 f ′ is unique up to equiv alence. This will let us define compositio n o f equiv alence classes o f ro o fs later on. 3.4. Lemma. A ny two ways to cho ose f ′ and w ′ in (L1) define e qu ivalent r o ofs. Pr o of. Imag ine tw o p oss ible choices ( f ′ 1 , w ′ 1 ) and ( f ′ 2 , w ′ 2 ) as in the partia lly co m- m utative diag ram • • b w O O O O O • g ? ?        • e w _ _ _ _ _ _ • f ′ 1 ? ?        f ′ 2 4 4 j j j j j j j j j j j j j j j j j j j j • w ′ 1 j j j* j* j* j* j* j* j* j* j* j* j* j* j* w ′ 2 _ _ _ _ _ _ • w g g g' g' g' g' g' g' g' g' g' f 7 7 o o o o o o o o o o o o o o By (L1) , g and e w w er e c hosen suc h that g w ′ 1 = e w w ′ 2 . This is not yet an equiv alence of ro ofs, since, in genera l, g f ′ 1 6 = e w f ′ 2 . How ev er, we do know that gf ′ 1 w = e wf ′ 2 w , so by (L2) we ca n c ho ose b w such that b wg f ′ 1 = b w e w f ′ 2 . This makes ( f ′ 1 , w ′ 1 ) and ( f ′ 2 , w ′ 2 ) equiv alent v ia b wg a nd b w e w .  3.5. Lemm a. The e quivalenc e of r o ofs fr om definition 3.3 is an e quivalenc e r elation. Pr o of. Reflexivity a nd symmetry are obvious. F or transitivity , supp ose w e a re given an e quiv alence be tw een ( f 1 , w 1 ) and ( f 2 , w 2 ), and one b etw een ( f 2 , w 2 ) and ( f 3 , w 3 ), as in the partially comm utativ e diagram • • b w O O O O O • k 7 7 p p p p p p p p p p p p p • e w g g g' g' g' g' g' g' g' g' • g 7 7 p p p p p p p p p p p p p • h g g N N N N N N N N N N N N N g ′ 7 7 p p p p p p p p p p p p p • h ′ g g N N N N N N N N N N N N N • f 1 _ _ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? f 2 ? ?                 f 3 4 4 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j • w 1 j j j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* j* w 2 _ _ _ _ _ _ _ _ _ _ _ _ w 3 ? ? ? ? ? ? ? ? ? ? ? ? gw 1 = hw 2 r2 q1 q1 p0 p0 o/ n. m- l, k+ j* i) h( f& e% c# a! _ ] [ Y X W V U T S P P R Q P g ′ w 2 = h ′ w 3 O O O O O O O O O O O O O Here, the equiv alence b etw een ( f 1 , w 1 ) and ( f 2 , w 2 ) is assumed to b e implemented by g and h , while the one b etw een ( f 2 , w 2 ) and ( f 3 , w 3 ) is implemented by g ′ and h ′ . The commut ativity co nditions for the t w o equiv a lences are g f 1 = hf 2 , g w 1 = hw 2 ; g ′ f 2 = h ′ f 3 , g ′ w 2 = h ′ w 3 (3) CA TEGORIES OF FRAC TIONS REVISITED 7 In the upp er par t of the diag ram, k and e w were obtained by applying (L1) to the t wo wiggly arr ows g w 1 = hw 2 and g ′ w 2 = h ′ w 3 . The c o rresp onding co mm utativit y assertion o f (L1) then is k hw 2 = e wg ′ w 2 . By virtue o f (L2) , we can then find the drawn b w such that b w k h = b w e w g ′ . T ogether with the relations ( 3 ), this means that the comp ositions b w k g and b w e w h ′ of the morphisms which go up along the sides implemen t an equiv alence betw een ( f 1 , w 1 ) and ( f 3 , w 3 ).  Under a closer lo o k, this argument is actually a special ca se of the arg umen t used to prov e lemma 3.4 . In fact, we co uld also have applied lemma 3.4 directly to the tw o ro ofs ( h, hw 2 ) a nd ( g ′ , g ′ w 2 ), since bo th are (L1) -co mplemen ts of the formal right fraction id dom( w 2 ) w − 1 2 . 3.6. Rem ark. One ca n also take a 2- categorica l p oint o f view whic h gives some more intuitiv e insight on the no tion o f equiv alence of ro ofs. W e ge t so mething resembling a 2-catego r y as follo ws: on the ob jects of C we define a 1-mor phism to be a ro of in C with resp ect to W . F or a ro of ( f , w ), w e define dom(( f , w )) = do m( f ) and cod(( f , w )) = dom( w ). A 2-morphism from a ro o f ( f 1 , w 1 ) to a par allel ro o f ( f 2 , w 2 ) is then defined to be a co mmutative diagra m • % % • • f 1 ? ?        f 2 5 5 j j j j j j j j j j j j j j j j j j j • w 1 i i i) i) i) i) i) i) i) i) i) i) i) i) w 2 _ _ _ _ _ _ The existence of such a 2- mo rphism makes ( f 1 , w 1 ) and ( f 2 , w 2 ) eq uiv alent; we call such a 2-morphism an elementar y equiv alence . A 2-morphism from ( f 1 , w 1 ) to ( f 2 , w 2 ) can b e comp osed with a 2-morphism from ( f 2 , w 2 ) to ( f 3 , w 3 ). This resem- bles the vertical co mpo sition in a 2-catego ry . Now the observ ation is that t w o ro ofs are equiv alent if and only if they can b e connected by a finite path o f 2-morphisms, where ea c h 2-mor phism is either traversed from its domain to its co domain or in the reverse directio n. T o see this, note tha t the third ro of ( g f 1 , hw 2 ) in the dia gram of definition 3.3 is connected to eac h of the other t wo ro ofs b y a 2-mor phism. The other implication dir e ction follows fro m the transitivity statement of lemma 3.5 and the fact that tw o par a llel ro ofs connected by a single 2-morphism are equiv alent. Hence le mma 3.5 can also b e r einterpreted as a connectivity sta temen t ab out the category of parallel roo fs betw ee n some pair of ob jects. In what follows, we will define a (w eakly asso ciative) comp osition of 1-mo rphisms. A horizontal comp osition of 2-mo rphisms do es not seem to exist in genera l, although it seems related to the upc o ming pro o f that the compo sition of 1-morphisms is w ell- defined up to equiv alence. W e end this remark by p ointing out ag ain that this 2-categor ical picture is a non-rigo rous intuition. Lemma 3.4 a lso allows the definition o f comp osition for e q uiv alence cla sses o f ro ofs: 3.7. Definition. Given two r o ofs ( f 1 , w 1 ) and ( f 2 , w 2 ) whi ch ar e c omp osable in the sense that dom( w 1 ) = dom( f 2 ) , we define their c omp osition as ( f 2 , w 2 ) ◦ ( f 1 , w 1 ) ≡ ( e f f 1 , e w w 2 ) 8 TOBIAS FRITZ wher e e f and e w in • • e f ? ?        • e w _ _ _ _ _ _ • f 1 ? ?        • w 1 _ _ _ _ _ _ f 2 ? ?        • w 2 _ _ _ _ _ _ wer e obtaine d by me ans of (L1) . Thanks to lemma 3.4 , the eq uiv alence class of ( e f , e w ) is unique. Ther efore, so is the equiv alence class of ( e f f 1 , e w w 2 ). 3.8. Lemma. This c omp osition do es not dep end on the e quivalenc e class of either of the two r o ofs. Pr o of. F or both pa irs of ro ofs, it is sufficient to consider the case that they are connected b y an elemen tar y equiv a lence as des crib ed in remark 3.6 . Th us suppose we ar e g iven the low er half of the dia gram • • g 1 / / • e f ? ?        • e w _ _ _ _ _ _ • g 2 o o • f ′ 1 ? ?        f 1 5 5 j j j j j j j j j j j j j j j j j j j • w 1 _ _ _ _ _ _ w ′ 1 i i i) i) i) i) i) i) i) i) i) i) i) i) f 2 ? ?        f ′ 2 5 5 j j j j j j j j j j j j j j j j j j j • w ′ 2 _ _ _ _ _ _ w 2 i i i) i) i) i) i) i) i) i) i) i) i) i) which represents t wo pairs of elementarily equiv alent ro o fs. After po ssible re na m- ings ( f 1 , w 1 ) ↔ ( f ′ 1 , w ′ 1 ) and ( f 2 , w 2 ) ↔ ( f ′ 2 , w ′ 2 ), we can assume that g 1 go es from co d( f ′ 1 ) to cod( f 1 ), while g 2 similarly p oints from co d( f ′ 2 ) to cod( f 2 ). Applying (L1) to the pa ir w 1 , f 2 yields e f and e w . Then ( e f f 1 , e w w 2 ) is a po ssible ro of representing the comp ositio n ( f 2 , w 2 ) ◦ ( f 1 , w 1 ). Similar ly , ( e f g 1 f ′ 1 , e w g 2 w ′ 2 ) is a possible ro of representing the comp osition ( f ′ 2 , w ′ 2 ) ◦ ( f ′ 1 , w ′ 1 ). By commutativit y , these ro ofs coincide, so in particular they are equiv alen t.  3.9. Theorem . If W ⊆ C admits a c alculus of left fr actions, then the c ate gory C [ W − 1 ] c an b e describ e d as the c ate gory with the same obje cts as C , morphisms e quivalenc e classes of r o ofs, and c omp osition as defin e d ab ove. The lo c alization functor Loc : C → C [ W − 1 ] is given by f 7→ ( f , id) . Pr o of. Asso cia tivit y of composition follows from t he symbolic diagram • • ? ?        • _ _ _ _ _ _ • ? ?        • _ _ _ _ _ _ ? ?        • _ _ _ _ _ _ • ? ?        • _ _ _ _ _ _ ? ?        • _ _ _ _ _ _ ? ?        • _ _ _ _ _ _ CA TEGORIES OF FRAC TIONS REVISITED 9 where the three low er ro ofs ar e tho s e to b e co mpos ed; the rest of the diag ram is o btained by thr e e a pplications of (L1) . Then the la r ge ro o f from the left to the r ight for med by co mpos ing the morphisms along the sides is a representativ e for the compo sition of the three low er ro ofs in both po s sible wa ys of bracketing. This shows ass o ciativity . F ur ther more, the equiv alence c lasses of the r o ofs (id , id) obviously function as iden tit y morphisms . Therefore, taking eq uiv alence classes o f ro ofs as morphisms on Ob j( C ) gives a well-defined category C [ W − 1 ]. Concerning functoriality , L oc pre s erves iden tities b y definition, and preserves comp osition b y the diagram • • g ? ?        • @ @ @ @ @ @ @ @ @ @ @ @ @ @ • f ? ?        • @ @ @ @ @ @ @ @ @ @ @ @ @ @ g ? ?        • @ @ @ @ @ @ @ @ @ @ @ @ @ @ which says that the r o of ( g f , id) is a repr esentativ e for the equiv alence class o f ( g , id) ◦ ( f , id). Under Loc , the ima ge of some w ∈ W is ( w , id ), and this image has a s its in v erse element the c la ss of (id , w ) since ( w , w ) is a repres en tative of b oth (id , w ) ◦ ( w , id ) and ( w, id) ◦ (id , w ), and there is a n obvious equiv alence ( w , w ) ∼ (id , id). In particular, Loc maps W to isos. It r e ma ins to chec k univ ersality . Supp ose we hav e some functor F : C → D which ma ps W to isos . First w e need to show that F uniquely extends to ro o fs. By the desired commutativit y of ( 2 ), any suc h extension has to map the ro o f ( f , id) to F ( f ). Similarly , since the class [(id , w )] is the inv erse of the class [( w , id)], any such extension maps (id , w ) to F ( w ) − 1 . B ut now ( f , w ) is a r epresentativ e of the comp osition (id , w ) ◦ ( f , id), so we ne e d ( f , w ) 7→ F ( w ) − 1 F ( f ). W e still hav e to c hec k that this assignment is w ell-defined on equiv a lence clas ses and that it is functor ial. Co nsider an elementary equiv alence of ro ofs a s in re- mark 3.6 , • g % % • • f 1 ? ?        f 2 5 5 j j j j j j j j j j j j j j j j j j j • w 1 i i i) i) i) i) i) i) i) i) i) i) i) i) w 2 _ _ _ _ _ _ Then in D we hav e F ( w 2 ) = F ( g ) F ( w 1 ), so F ( w 1 ) − 1 = F ( w 2 ) − 1 F ( g ). Then the calculation F ( w 1 ) − 1 F ( f 1 ) = F ( w 2 ) − 1 F ( g ) F ( f 1 ) = F ( w 2 ) − 1 F ( f 2 ) (4) shows that the equiv ale n t roofs get mapp ed to identical mo rphisms in D . F unctoria lit y fo llows b y very similar reaso ning. Given a pair of co mpos able ro o fs together with their compo sition as in definition 3.7 , it holds that F ( e f ) F ( w 1 ) = F ( e w ) F ( f 2 ) (5) so that w e g et F ( e w ) − 1 F ( e f ) = F ( f 2 ) F ( w 1 ) − 1 (6) 10 TOBIAS FRITZ Applying first the functor and compo sing the roofs afterwards yields  F ( w 2 ) − 1 F ( f 2 )  ◦  F ( w 1 ) − 1 F ( f 1 )  (7) while for the o ther direction we end up with F ( e w w 2 ) − 1 F ( e f f 1 ), which coincides with ( 7 ) b y ( 6 ) and functorialit y of F .  If W ⊆ C satisfies (L0) (whic h is self-dua l) and a dditionally the conditions (R1) and (R2), which ar e defined to b e the category-theo retic duals of (L1) and (L2) , then we say that W allows a calculus of right fractions . In this case, the dual theorem holds: C [ W − 1 ] can be describ ed in terms o f equiv alence classes of ro ofs ( w, f ) which now repr esent rig h t frac tions f w − 1 . If all five of the (L ∗ ) and (R ∗ ) conditions hold, w e say that W a dmits a calculus of left and righ t fractions . 4. Weakening the requirements In [ 4 ], a notion of categor y of fractions is introduced which, on first sight, is seemingly w ea ker in its premises than the one discussed in the previous section. While keeping (L0 ) and (L1) , the axiom (L2) gets replaced by the conditio n (L2’) Denote b y W L the class of morphisms in C genera ted by W a nd all split monos in C . Then g iv en w ∈ W and par allel mo rphisms f 1 , f 2 such that f 1 w = f 2 w , there exists w ′ ∈ W L such that w ′ f 1 = w ′ f 2 . • w / / /o /o /o • f 1 ( ( f 2 6 6 • w ′ / / /o /o /o • 4.1. Propos ition. Given w ′ ∈ W L , we c an find k ∈ C s u ch that k w ′ ∈ W . Pr o of. Let us cons ider the cases how w ′ might lo ok like, one by o ne and in increa sing order of difficult y . If alr e ady w ′ ∈ W , we are done since we ca n take k = id co d( w ′ ) . If w ′ = m b w , where m is a s plit mono and b w ∈ W , w e can take k to b e a left-in verse of m , so w e are done as well. The only non- trivial type o f situatio n o ccurs w ′ is a co mpos ition of morphisms in W a nd split monos such that morphisms of W come after split monos. The prototype for this situation is a morphism lik e w ′ = b w m , with b w ∈ W and m a split mono. By as sumption, m has a left inv erse e , which means em = id. Now apply (L1) to the pair b w , e , • b w / / /o /o /o e   • k   • e w / / /o /o /o m A A w ′ ? ? ? ? ? ? • which gives the morphis m k and so me mor phism e w ∈ W . The commutativit y assertion of (L1) states in this case e w e = k b w , so after comp osing with m on the right we ha v e e w = e w e m = k b w m = k w ′ ∈ W . Now for the gener a l case. By definition of W L and (L0) , our w ′ is of the form w ′ = w n m n · · · w 1 m 1 (8) where the m j are s plit mo no s a nd w j ∈ W . Starting fro m the left, we can iteratively apply the previous a rgument and use (L0) to compo se the mo rphisms in W to a single morphism in W , un til w e ha ve only a single morphism in W left.  CA TEGORIES OF FRAC TIONS REVISITED 11 4.2. Corollary . (a) Given (L0) and (L1) , t he assertions (L2) and (L2’) ar e e quivalent. (b) Two r o ofs ( f 1 , w 1 ) and ( f 2 , w 2 ) ar e e quivalent if and only if ther e is a dia- gr am • • g ? ?        • h _ _ @ @ @ @ @ @ @ • f 1 ? ?        f 2 4 4 j j j j j j j j j j j j j j j j j j j j • w 1 j j j* j* j* j* j* j* j* j* j* j* j* j* j* w 2 _ _ _ _ _ _ wher e now we only demand c ommut ativity and hw 2 ∈ W L (inste ad of hw 2 ∈ W ). Pr o of. These are b oth immediate conseque nc e s of the previous prop osition.  4.3. Rem ark. As alrea dy noticed in [ 4 , 1.2 .4], when (L0) holds the axiom (L1) is in fact equiv alent to the v ar iant where w ∈ W L : (L1’) Giv en any w ∈ W L and an arbitra ry morphism f with dom( f ) = dom( w ), we can find w ′ ∈ W and so me mo r phism f ′ with co d( f ′ ) = cod( w ′ ), such that the diagram • w / / /o /o /o f   • f ′   • w ′ / / /o /o /o • commutes. Clearly , (L1) is trivially implied b y this. F or the other imp lication direction, by remark 3.1 it is s ufficien t to show that (L1’) holds if w is any split mono as in the diagram • w / / f   • f e   e } } • • with e some left-inv er s e of w . Then by (L0) we ha ve w ′ ≡ id co d( f ) ∈ W , so toge ther with f ′ ≡ f e this do es the job; co mm uta vity f ew = f holds since e is left-inv ers e to w . 5. Additive ca tegories of fracti ons Often, the w o rking mathematician de a ls with additiv e catego r ies. In particular, they ma y w ant to do lo calization completely within the framew ork of additiv e c at- egories. In o ther words, g iven an additive categ ory C and a clas s of “mora l isomor- phisms” W in C , is there an additiv e categor y C [ W − 1 ] and an additive lo calization functor Loc : C → C [ W − 1 ] which ma ps W to isos and is the universal additiv e functor with this prope rty? And if yes, ho w can this lo calization b e constructed? F or simplicit y , we consider only the catego ry of fra ctions ca se. Then, in fact, the loc alization constructed in theorem 3.9 alr eady is a dditive. Inituitiv ely , the reason is that one can find a “co mmo n denominator” for pairs of roofs repres en ting 12 TOBIAS FRITZ parallel morphisms in C [ W − 1 ]. The purp o se of this section is to turn th is in tuitiv e explanation in to a formal pro of. In the following, C is an additive categ o ry , a nd W ⊆ C is a class of mor phisms satisfying (L0) , (L1) and (L2) (or the alternativ es (L1’) and (L2 ’) discussed in the previous section). W e s tart b y constructing “common denominators” and using them to define an addition oper ation on equiv alence classes of ro ofs. Given parallel r o ofs ( f 1 , w 1 ) and ( f 2 , w 2 ), w e apply (L1) to the pair w 1 , w 2 and obtain a diagram • • g ? ?        • e w _ _ _ _ _ _ • f 1 ? ?        f 2 4 4 j j j j j j j j j j j j j j j j j j j j • w 2 _ _ _ _ _ _ w 1 j j j* j* j* j* j* j* j* j* j* j* j* j* j* (9) which commutes o nly in the sens e that g w 1 = e ww 2 . There is an equiv alence ( f 1 , w 1 ) ∼ ( gf 1 , e w w 2 ), and similarly ( f 2 , w 2 ) ∼ ( e w f 2 , e w w 2 ). Th us w e have iden- tified e w w 2 as a “common denominator” . Now w e can define the sum of ( f 1 , w 1 ) and ( f 2 , w 2 ) as ( f 1 , w 1 ) + ( f 2 , w 2 ) ≡ ( g f 1 + e w f 2 , e w w 2 ) . (10) It needs to b e c heck ed that the cla ss of ( g f 1 + e w f 2 , e w w 2 ) do es no t dep end o n the particular choice of g and e w . Thanks to lemma 3.4 , the c lass [( g , e w )] is w ell-defined by w 1 and w 2 . Now if ( g ′ , e w ′ ) is a nother c hoice connected to ( g , e w ) b y a n element ary equiv alence h , then w e hav e the diagram • h % % • • g ? ?        g ′ 5 5 j j j j j j j j j j j j j j j j j j j • e w i i i) i) i) i) i) i) i) i) i) i) i) i) e w ′ _ _ _ _ _ _ • f 1 ? ?        f 2 2 2 e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e • w 2 _ _ _ _ _ _ w 1 l l l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, which commutes only in the sense that g w 1 = e w w 2 , g ′ w 1 = e w ′ w 2 , hg = g ′ and h e w = e w ′ . The equation g ′ f 1 + e w ′ f 2 = h ( g f 1 + e w f 2 ) shows that h likewise implements an equiv alence ( g ′ f 1 + e w ′ f 2 , e w ′ w 2 ) ∼ ( g f 1 + e w f 2 , e w w 2 ) , as w as to be shown. While it has b een prov en that the definition ( 10 ) pro duce s a well-defined class of roofs from every pair of ro ofs, it is still unclear whether the sum depends on the particular representativ es of the summands or only on their classes. 5.1. Lemma. The class of the sum only dep ends on the classes of the summ ands and not on the p articular r epr esentatives. CA TEGORIES OF FRAC TIONS REVISITED 13 Pr o of. Still using the same no tation, it is sufficient to consider an elementary equiv- alence betw een ( f 1 , w 1 ) and some ( f ′ 1 , w ′ 1 ): • h % % • • f ′ 1 ? ?        f 1 5 5 j j j j j j j j j j j j j j j j j j j • w ′ 1 i i i) i) i) i) i) i) i) i) i) i) i) i) w 1 _ _ _ _ _ _ Then taking the common denomina tor of ( f 1 , w 1 ) and ( f 2 , w 2 ) as ab ov e yields the partially comm utative diag ram • • h % % • g ? ?        • e w _ _ _ _ _ _ • f ′ 1 O O f 1 7 7 p p p p p p p p p p p p p f 2 3 3 g g g g g g g g g g g g g g g g g g g g g g g g g g • w ′ 1 k k k+ k+ k+ k+ k+ k+ k+ k+ k+ k+ k+ k+ k+ k+ k+ k+ k+ w 1 g g g' g' g' g' g' g' g' g' g' w 2 O O O O O Now the sum of ( f 1 , w 1 ) and ( f 2 , w 2 ) is the class of ( g f 1 + e w f 2 , e w w 2 ) (11) while the sum of ( f ′ 1 , w ′ 1 ) and ( f 2 , w 2 ) is the class of ( g hf ′ 1 + e w f 2 , e w w 2 ) (12) which coinides with ( 11 ) by c omm utativity of the diag ram.  5.2. Theorem. Supp ose C is add itive and al lows a c alculus of left fr actions with r esp e ct to W . Then the c ate gory of fr actions C [ W − 1 ] is additive. Pr o of. A category is additive if it is preadditive, has a zero ob ject, and has a bipro duct for ev ery pair of ob jects. It was alrea dy shown how to add equiv a lence classes of ro ofs and that this o per - ation is well-defined. Its asso cia tivit y can b e seen from a diagra m of the sy mbo lic form • • ? ?        • _ _ _ _ _ _ • ? ?        • _ _ _ _ _ _ ? ?        ^ ^ ^ ^ ^ ^ ^ • ? ?        4 4 j j j j j j j j j j j j j j j j j j j j 2 2 e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e • ^ ^ ^ ^ ^ ^ ^ i i i) i) i) i) i) i) i) i) i) i) i) i) l l l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, Its commutativit y is eviden t fro m ( 9 ) by r ealizing that g and e w play identical rˆ oles in ( 9 ): it is no t relev ant that e w ∈ W , but only that e w w 2 ∈ W . Neutral elements of th e addition opera tion a re giv en by the eq uiv alence classes [(0 , id)]. An additiv e inv ers e o f [( f , w )] is [( − f , w )]. Hence the catego ry of fr a ctions is preadditive. The lo caliza tion functor w as defined as Loc : f 7→ ( f , id). F or pa r allel morphisms f , g ∈ C , adding ro ofs giv es [( f , id)] + [( g , id)] = [( f + g , id)]; in other words, Loc is additive. In par ticular, L oc maps bipro duct diagr ams to bipr o duct diagra ms. Then 14 TOBIAS FRITZ since the functor is s ur jective on ob jects, C [ W − 1 ] has bipro ducts. Any n ull ob ject of C also is a null ob ject in C [ W − 1 ]. All in all, this makes C [ W − 1 ] additive.  5.3. Remark. In an additive category , one can obviously r e place the a xiom (L2) by the slightly simpler requirement (L2”) Given w ∈ W and a mor phism f suc h that f w = 0, there exists w ′ ∈ W (or w ′ ∈ W L ) such t hat w ′ f = 0. • w / / /o /o /o • f / / • w ′ / / /o /o /o • References [1] P . Gabriel and M . Zisman. Calculus of fr actions and homotop y the ory . Ergeb nisse der Math- ematik und ihrer Gr enzgebiet e, Band 35. Springer- V erl ag New Y ork, Inc., New Y ork, 1967. [2] Sergei I. Gelfand and Y uri I. Manin. Metho ds of homolo gic al algebr a . Spri nger Monographs in Mathematics. Springer-V erlag, Ber lin, second edition, 2003. [3] All en Hatc her. Algebr aic top olo gy . Cambridge Unive rsity Press, Cambridge, 2002. [4] Nigel Higson. Categories of fr actions and excision in K K - theory . J. Pur e Appl. Algebr a , 65(2):119– 138, 199 0. [5] Mark Hov ey . Mo del c ate gories , v olume 63 of Mathematic al Survey s and M ono gr aphs . Am erican Mathematical Society , Pro vidence, RI, 1999. [6] A. V . Jategaonk ar . L o c alization in No etherian rings , v olume 98 of L ondon Mathema tic al So- ciety L e ctur e Note Series . Cambridge Univ ersity Press, Camb ridge, 1986. [7] Saunders Mac Lane. Ca te gories for the working mathematician , volume 5 of Gr aduate T exts in Ma thematics . Springer-V erlag, New Y ork, second edition, 1998. E-mail addr ess : tobias.fritz@ icfo.es ICF O – Institut de Ci ` encies Fo t ` oniques, Mediterranean Technology P ark, 0886 0 Castelldefels (Barcelona), Sp ain