Higher K-theory via universal invariants

HIGHER K -THEOR Y VIA UNIVERSAL INV ARIANTS GONC ¸ ALO T ABUADA Abstract. Using the formalism of Grothend iec k’s deriv ato rs, we construct ‘the univ ersal lo calizing inv ariant ’ of dg categories. By this, we mean a mor - phism U l from the p ointe d deriv ator H O ( dg cat ) asso ciated with the Morita homotop y t heory of dg categories to a triangulated strong deriv ator M loc dg suc h that U l comm utes with ﬁltered homotopy colimits, preserv es the p oint, sends eac h exact sequence of dg cate gories to a triangle and is unive rsal for these properties. Similary , we construct the ‘unive r sal additive in v ariant’ of dg categories, i.e. the universal morphism of deriv ators U a from HO ( dgcat ) to a strong triangu- lated deriv ator M add dg which satisﬁes the ﬁrst t wo prop erties but the third one only for split exact s eque nces. W e prov e that W aldhausen’s K -theory b ecomes co-represen table in the target of the unive rsal additive inv ariant. This is the ﬁrst conceptual characterization of Quil len-W aldhausen’s K -theory s i nce its deﬁnition in the earl y 70’s. As an application we obtain for free the higher Chern characters fr om K -theory to cyclic homology . Contents 1. Int ro duction 2 2. Ac knowledgmen ts 5 3. Preliminarie s 5 4. Derived Ka n extens io ns 11 5. Lo calization: mo del catego ries versus deriv ator s 12 6. Filtered ho motop y co limits 17 7. Poin ted deriv ator s 23 8. Small weak generato rs 25 9. Stabilization 26 10. DG quo tien ts 33 11. The universal lo calizing inv ar ian t 37 12. A Quillen model in terms of pr eshea ves of sp ectra 42 13. Upper triangular DG categories 43 14. Split sho rt exact sequenc e s 47 15. Quasi-Additivity 48 16. The universal additive inv ar ian t 53 17. Higher Chern c haracters 57 18. Concluding r emarks 59 References 59 Key wor ds and phr ases. Quillen-W aldhausen’s K -theory , Non-commut ative algebraic geome- try , Higher Chern characters, Grothendiec k deri v ator, Quil len mo del structure, Bousﬁeld lo cal- ization, Simpli cial presheav es, Stabilization, Dr infeld’s dg quotient , Cyclic homology , Additivity , Preshea ves of sp ectra, Upp er triangular dg category . Supported by FCT-P ortugal, scholarship SFRH/BD/14 035/2003 . 1 2 GONC ¸ ALO T ABUAD A 1. In troduction Diﬀerent ial gra ded categor ies (=dg catego r ies) e nha nce our understanding of triangulated categ o ries a ppearing in algebra a nd geometry , see [1 9]. They ar e co nsidered a s non-co mm utative schemes b y Drinfeld [9] [10] and Kont- sevich [23] [24] in their pro gram of non-commutativ e alge braic geometry , i.e. the study of dg c ategories and their homologica l inv a riant s. In this article, using the formalis m of Grothendieck’s deriv ator s, we construct ‘the universal lo calizing in v ariant’ o f dg categories cf. [21]. By this, we mean a morphism U l from the p ointed deriv ator HO ( dgcat ) a ssoc ia ted with the Morita ho- motopy theo r y of dg categor ies, see [37] [38] [3 9], to a tria ng ulated strong der iv a tor M loc dg such tha t U l commutes with ﬁltered homotopy co limits, preserves the point, sends each ex act sequence of dg c ategories to a triangle and is univ er sal for these prop erties. Becaus e of its univ er salit y pr operty reminiscent of motives, see section 4 . 1 of Ko n tsevich’s preprint [25], we call M loc dg the (stable) lo c alizing motivator of dg catego ries. Similary , we cons tr uct ‘the universal additive inv a riant ’ of dg categ ories, i.e. the universal morphism of deriv ators U a from HO ( dg cat ) to a strong triangulated deriv ator M add dg which satisﬁes the ﬁrst tw o pro perties but the third one only fo r split exa ct se quences. W e call M add dg the additive motivator of dg categ ories. W e prov e that W aldhausen’s K -theory sp ectrum a ppears as a sp ectrum of mor- phisms in the base ca tegory M add dg ( e ) o f the additive motiv ator. This shows us that W aldhausen’s K -theory is completely characterized by its additive pro perty a nd ‘int uitiv ely’ it is the universal co ns truction with v a lues in a stable co n text which satisﬁes additivity . T o the best of the author’s knowledge, this is the ﬁrst conceptual characteriza- tion of Quillen-W aldhausen’s K -theo ry [34] [44] since its deﬁnition in the early 70’s. This result giv es us a co mpletely new way to think ab out algebraic K -theory a nd furnishes us for free the higher Chern characters from K -theory to cyclic homol- ogy [26]. The co-repr esen ta tio n of K -theo ry a s a sp ectrum of mor phisms extends our results in [3 7] [38], where we co-r epresented K 0 using functors with v alues in addi tive c ate gories rather than morphisms of deriv ators with v alues in str ong t riangulate d derivators . F or example, the mixed complex cons truction [22], from which a ll v aria n ts of cyclic ho mo logy can b e deduced, and the non-co nnec tiv e alg ebraic K-theory [35] are lo calizing inv a rian ts and factor thro ug h U l and thr ough U a . The connective a l- gebraic K -theory [44] is an ex a mple o f an additive in v a riant whic h is not a lo calizing one. W e pro ve that it becomes co-representable in M add dg , see theorem 16.10. Our co nstruction is similar in spir it to those o f Meyer-Nest [31], Co rti˜ nas-Thom [8] a nd Garkusha [12]. It splits into several gener al steps and also oﬀers s o me insight into the relationship b etw een the theory o f deriv a to rs [14] [13] [20] [29] [3] and the c la ssical theory of Quillen mo del catego ries [33]. Der iv ator s allow us to state and prov e precise universal prop erties and to disp ense with many of the technical problems one faces in us ing mo del categories. In chapter 3 we recall the notion of Gr othendiec k der iv ator and p oin t o ut its connexion with that o f small homotop y theo ry in the sense of Heller [14]. In chap- ter 4, we recall C is inski’s theory o f derived K an extensions [4 ] and in chapter 5, HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 3 we develop his ideas on the Bousﬁe ld lo calization of der iv a tors [7]. In par ticu- lar, we characteriz e the deriv ator asso ciated with a left Bousﬁeld lo calization of a Quillen mo del category by a univ er sal prop erty , see theor em 5.4. T his is based on a constr uctiv e descr iption o f the lo cal weak equiv alences. In chapter 6, starting from a Quillen mo del category M satisfying so me com- pactness co nditio ns, we construct a mo r phism o f prederiv ator s HO ( M ) R h − → L Σ Hot M f which commutes with ﬁltered homotopy co limits, has a deriv ator a s targe t and is universal for these pro perties. In chapter 7 we study morphisms o f p oin ted deriv ators and in chapter 8 we prov e a general result which g arantees that small weak ge ne r ators are preserved under le ft Bousﬁeld lo calizations. In chapter 9, we recall Heller’s stabilization construction [1 4] and we prove that this construction takes ‘ﬁnitely generated’ unstable theories to co mpa ctly gener ated s table ones. W e establish the connection b et ween Heller’s s tabilization and Hovey/Sc hw e de’s stabi- lization [17] [3 6] by proving that if we start with a pointed Quillen model catego r y which satisﬁes so me mild ‘generation’ hypo theses, then the tw o stabilization pr o - cedures yield equiv alent results. This allows us to characterize Hov ey/Sch wede’s construction by a universal pr oper ty a nd in par ticular to g iv e a very simple charac- terisation of the classica l catego ry of sp ectra in the sense of Bous ﬁeld-F r iedlander [2]. In chapter 10, by applying the genera l arg umen ts of the pr e vious chapters to the Morita homotopy theory of dg catego ries [37] [38] [39], we construct the univ er s al morphism of deriv ator s U t : HO ( dgc at ) − → St ( L Σ ,P Hot dgcat f ) which commutes with ﬁltered ho motop y co limits, preserves the p oint and has a strong triangulated deriv ator as ta r get. F or every inclusion A K ֒ → B o f a full dg sub c ategory , we have a n induced morphism S K : co ne ( U t ( A K ֒ → B )) → U t ( B / A ) , where B / A denotes Drinfeld’s dg quotient. By a pplying the lo caliza tion techniques of section 5 and using the fact that the der iv ator St ( L Σ ,P Hot dgcat f ) admits a sta - ble Quillen mo del, we in vert the morphisms S K and obtain ﬁnally the universal lo calizing inv a riant of dg categories U l : HO ( dgc at ) − → M loc dg . W e establish a connection b et ween the triang ulated categ ory M loc dg ( e ) and W ald- hausen’s K - theory by showing that W aldhausen’s S • -construction corresp onds to the susp ension functor in M loc dg ( e ). In s ection 12, we pr o ve that the deriv ator M loc dg admits a stable Quille n mo del given by a left Bo us ﬁeld lo calization of a categ ory of presheav es o f sp ectra. In section 13, w e int ro duce the concept of upper triangula r dg category and construct a Quillen mo del str ucture on this cla ss of dg c a tegories, which sa tisﬁes strong compactness conditions. In section 14, we establish the con- nection be tween upper triang ular dg ca tegories a nd split sho rt exact sequences and use the Quillen mo del structure of sectio n 13 to prov e an ‘approximation result’, see prop osition 14.2. In s e c tion 15, by a pplying the techniques of section 5, we construct the univ e rsal mo rphism of der iv ator s U u : HO ( dgcat ) − → M unst dg 4 GONC ¸ ALO T ABUAD A which comm utes with ﬁltered homotopy colimits, preser v es the p oin t and se nds each split shor t exact sequence to a homotopy coﬁb er sequence. W e pr o ve that W aldhausen’s K - theory space construction appe ars as a ﬁbra n t ob ject in M unst dg . This allow us to obtain the w eak equiv ale nce of simplicial sets Map ( U u ( k ) , S 1 ∧ U u ( A )) ∼ − → | N .w S • A f | and the isomorphisms π i +1 Map ( U u ( k ) , S 1 ∧ U u ( A )) ∼ − → K i ( A ) , ∀ i ≥ 0 , see prop osition 15 .12 . In s ection 16 we stabilize the deriv ator M unst dg , using the fact that it admits a Quillen mo del a nd obta in ﬁnally the univ er sal additive inv ar ian t of dg categor ies U a : HO ( dgcat ) − → M add dg . Connective algebraic K -theory is additive a nd so factor s through U a . W e prov e that for a small dg ca teg ory A its connectiv e alg ebraic K -theor y co rresp onds to a ﬁbrant r e solution of U a ( A )[1], see theo rem 16.9. Using the fact that the Quillen mo del for M add dg is enr ic hed ov er sp ectra, we prov e our main co-r epresentabilit y theorem. Let A and B b e small dg catego ries with A ∈ d gcat f . Theorem 1. 1 (16.10) . We have the following we ak e quivalenc e of sp e ctr a Hom Sp N ( U a ( A ) , U a ( B )[1]) ∼ − → K c ( rep mor ( A , B )) , wher e K c ( rep mor ( A , B )) denotes Waldhausen ’s c onne ctive K -the ory sp e ctru m of rep mor ( A , B ) . In the triangulated bas e category M add dg ( e ) of the additive motiv ator we hav e: Prop osition 1. 2 (17.1) . We have the fol lowing isomorphisms of ab elian gr oups Hom M add dg ( e ) ( U a ( A ) , U a ( B )[ − n ]) ∼ − → K n ( rep mor ( A , B )) , ∀ n ≥ 0 . R emark 1.3 . Notice that if in the ab o ve theo rem (resp. pr opo s ition), w e co nsider A = k , we hav e Hom Sp N ( U a ( k ) , U a ( B )[1]) ∼ − → K c ( B ) , r esp . Hom M add dg ( e ) ( U a ( k ) , U a ( B )[ − n ]) ∼ − → K n ( B ) , ∀ n ≥ 0 . This shows that W aldhause n’s connective K -theory spec tr um (resp. g r oups) be- comes co- representable in M add dg , r e s p. in M add dg ( e ). In section 17, we show that our co -representabilit y theorem furnishes us fo r free the higher Che r n characters fr o m K - theo ry to cyclic homolo gy . Theorem 1.4 (17.3) . The c o-r epr esentability the or em furn ishes us t he high er Chern char acters ch n,r : K n ( − ) − → H C n +2 r ( − ) , n, r ≥ 0 . In sectio n 1 8, w e p oin t out so me ques tions that deserve further inv estigation. HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 5 2. Acknowledgments This ar ticle is part o f my Ph.D. thesis under the sup e rvision of Pro f. B. Keller. It is a gr eat plea s ure to thank him for sug gesting this problem to me and for countless useful discuss ions. I hav e learned a lot from discussio ns with D.-C. Cisinski. I deeply thank him for shar ing so ge ne r ously his ideas on L o c alisation des D ´ erivateurs . I am also v ery grateful to B. T o¨ en for important remarks a nd for his interest. 3. P rel iminaries In this section we r ecall the no tio n o f Grothendieck der iv ator following [3]. Notation 3.1 . W e denote by C AT , r e sp. C at the 2-catego ry o f catego ries, r esp. small catego ries. The empty category will b e wr itten ∅ , and the 1-p oint ca tegory (i.e. the ca tegory with one o b ject and one identit y mo rphism) will b e written e . If X is a s ma ll ca tegory , X op is the opposite category a sso c ia ted to X . If u : X → Y is a functor, and if y is an ob ject of Y , one deﬁnes the ca tegory X/y as follows: the ob jects a re the couples ( x, f ), where x is an ob ject of X , and f is a map in Y from u ( x ) to y ; a map from ( x, f ) to ( x ′ , f ′ ) in X/y is a map ξ : x → x ′ in X such that f ′ u ( ξ ) = f . The comp osition law in X/y is induced by the comp osition law in X . Dually , one de ﬁnes y \ X by the formula y \ X = ( X op /y ) op . W e then hav e the canonical functors X/y → X a nd y \ X → X deﬁned by pro jection ( x, f ) 7→ x . One can eas ily c heck that one gets the follo wing pullback squa res o f categories X/y / / u/y   X u   y \ X y \ u   / / X u   Y /y / / Y y \ Y / / Y . If X is any ca tegory we let p X : X → e be the canonical pro jection functor. Given any ob ject x of X we will write x : e → X for the unique functor which se nds the ob ject o f e to x . The ob jects of X , or e quiv alently the functors e → X , will b e called the p oints of X . If X and Y are tw o categories we denote b y F un ( X , Y ) the catego ry of functors from X to Y . If C is a 2-categ ory one wr ites C op for its dual 2- category: C op has the same ob jects as C and, for an y tw o ob jects X and Y , the categ o ry Fun C op ( X, Y ) of 1-arr o w s fr om X to Y in C op is Fun C ( Y , X ) op . Deﬁnition 3.2. A preder iv ator is a strict 2 - functor fr om C at op to the 2 -c ate gory of c ate gories D : C at op − → C AT . Mor e explicity: for any smal l c ate gory X ∈ C at one has a c ate gory D ( X ) . F or any functor u : X → Y in C at one gets a functor u ∗ = D ( u ) : D ( Y ) − → D ( X ) . 6 GONC ¸ ALO T ABUAD A F or any morphism of functors X u ( ( v 6 6 ⇓ α Y , one has a morph ism of functors D ( X ) α ∗ ⇑ D ( Y ) . v ∗ j j u ∗ t t Of c ourse, al l these data have to verify some c oher enc e c onditions, namely: (a) F or any c omp osable maps in C at , X u → Y v → Z , ( v u ) ∗ = u ∗ v ∗ and 1 ∗ X = 1 D ( X ) . (b) F or any c omp osable 2 -c el ls in C at X ⇓ α ⇓ β / / u ' ' w 7 7 Y we have ( β α ) ∗ = α ∗ β ∗ and 1 ∗ u = 1 u ∗ . (c) F or any 2 -diagr am in C a t X u ' ' u ′ 7 7 ⇓ α Y v ( ( v ′ 6 6 ⇓ β Z , we have ( β α ) ∗ = α ∗ β ∗ . Example 3. 3. L et M b e a smal l c ate gory. The pr e derivator M natur al ly asso ciate d with M is deﬁne d as X 7→ M ( X ) wher e M ( X ) = Fun ( X op , M ) is the c ate gory of pr eshe aves over X with values in M . Example 3. 4. L et M b e a c ate gory endowe d with a class of maps c al le d we ak e quivalenc es (for example M c an b e the c ate gory of b ounde d c omplexes in a given ab elian c ate gory, and the we ak e quivalenc es c an b e the quasi-isomorphi sms). F or any smal l c ate gory X , we deﬁne the we ak e quivalenc es in M ( X ) to b e the morphisms of pr eshe aves which ar e termwise we ak e quivalenc es in M . We c an then deﬁn e D M ( X ) as the lo c alization of M ( X ) by the we ak e quivalenc es. It is cle ar that, for any functor u : X → Y , the inverse image functor M ( Y ) → M ( X ) F 7→ u ∗ ( F ) = F ◦ u r esp e cts we ak e quivalenc es, so that it induc es a wel l deﬁne d functor u ∗ : D M ( X ) − → D M ( X ) . The 2 -functoriality of lo c alization implies that we have a pr e derivator D M . HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 7 Let X b e a s mall categ ory and let x b e an o b ject o f X . Giv e n an ob ject F ∈ D ( X ) we will write F x = x ∗ ( F ). The ob ject F x will be called the ﬁb er of F at the p oint x . F or a prederiv ator D , deﬁne its op osite to b e the prederiv ato r D op given by the formula D op ( X ) = D ( X op ) op for all small categorie s X . Deﬁnition 3.5. L et D b e a pr e derivator. A map u : X → Y i n C at has a coho- mological direct imag e functor (re sp. a homolog ic a l dir ect image functor ) in D if the inverse image functor u ∗ : D ( Y ) − → D ( X ) has a right adjo int (r esp. a left adjo int) u ∗ : D ( X ) − → D ( Y ) ( r esp. u ! : D ( X ) → D ( Y )) , c al le d the cohomolog ical direct image functor (r esp. ho mological direct ima ge func- tor ) asso ciate d to u . Notation 3.6 . Let X b e a small catego ry a nd p = p X : X → e . If p has a cohomo- logical direct imag e functor in D , o ne deﬁnes for an y ob ject F of D ( X ) the ob ject of globa l sections of F as Γ ∗ ( X, F ) = p ∗ ( F ) . Dually , if p has a homo logical direct image in D then, fo r any ob ject F of D ( X ), one sets Γ ! ( X, F ) = p ! ( F ) . Notation 3.7 . L e t D b e a prederiv ator, and let X ′ v / / u ′   X u   ⇓ α Y ′ w / / Y be a 2-dia gram in C at . By 2 -functoriality one obtains the following 2-diag r am D ( X ′ ) D ( X ) v ∗ o o D ( Y ′ ) u ′∗ O O ⇑ α ∗ D ( Y ) . w ∗ o o u ∗ O O If w e assume that the functors u and u ′ bo th ha ve cohomolo gical direct images in D one ca n deﬁne the b ase change morphi sm induced by α β : w ∗ u ∗ → u ′ ∗ v ∗ D ( X ′ ) u ′ ∗   D ( X ) v ∗ o o u ∗   D ( Y ′ ) D ( Y ) w ∗ o o ⇑ β as follows. The counit u ∗ u ∗ → 1 D ( X ) induces a mo rphism v ∗ u ∗ u ∗ → v ∗ , a nd by comp osition with α ∗ u ∗ , a morphism u ′∗ w ∗ u ∗ → v ∗ . This giv e s β by adjunction. This construction will b e used in the following s ituation: le t u : X → Y be a map in C at and let y be a po in t of Y . Accor ding to no tation 3.1 we have a functor 8 GONC ¸ ALO T ABUAD A j : X/y → X , deﬁned b y the fo r m ula j ( x, f ) = x , wher e f : u ( x ) → y is a morphism in Y . If p : X/ y → e is the canonica l map one obtains the 2 -diagram b elow, w her e α denotes the 2-cell deﬁned by the form ula α ( x,f ) = f X/y p   j / / X u   ⇓ α e y / / Y . Using nota tio n 3 .6, the ass ocia ted base change morphism g iv es rise to a ca nonical morphism u ∗ ( F ) y → Γ ∗ ( X/y , F /y ) for a n y ob ject F ∈ D ( X ), wher e F /y = j ∗ ( F ). Dually one has canonical mor phisms Γ ! ( y \ X , y \ F ) → u ! ( F ) y where y \ F = k ∗ ( F ) and k denotes the c a nonical functor from y \ X to X . Notation 3.8 . Let X a nd Y b e tw o small ca tegories. Using the 2 - functorialit y of D , one deﬁnes a functor d X,Y : D ( X × Y ) → Fun ( X op , D ( Y )) as follows. Setting X ′ = X × Y , w e ha ve a canonica l functor F un ( Y , X ′ ) op − → Fun ( D ( X ′ ) , D ( Y )) which deﬁnes a functor F un ( Y , X ′ ) op × D ( X ′ ) − → D ( Y ) and then a functor D ( X ′ ) − → Fun ( F un ( Y , X ′ ) op , D ( Y )) . Using the canonical functor X → F un ( Y , X × Y ) , x 7→ ( y 7→ ( x, y )) , this gives the desir ed functor . In par ticular, for an y small c ategory X , o ne g ets a functor d X = d X,e : D ( X ) → Fun ( X op , D ( e )) . If F is an ob ject of D ( X ), then d X ( F ) is the pres he a f on X with v alues in D ( e ) deﬁned by x 7→ F x . Deﬁnition 3. 9. A deriv a tor is a pr e derivator D with the fol lowing pr op erties. Der1 (Non-t rivia lity axiom). F or any ﬁnite set I and any family { X i , i ∈ I } of smal l c ate gories, the c anonic al fun ctor D ( a i ∈ I X i ) − → Y i ∈ I D ( X i ) is an e quivalenc e of c ate gories. HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 9 Der2 (Conservativity ax iom). F or any smal l c ate gory X , the family of functors x ∗ : D ( X ) → D ( e ) F 7→ x ∗ ( F ) = F x c orr esp onding to the p oints x of X is c onservative. I n other wor ds: if ϕ : F → G is a morphism in D ( X ) , such that for any p oint x of X the map ϕ x : F x → G x is an isomophism in D ( e ) , then ϕ is an isomorphi sm in D ( X ) . Der3 (Dir e ct image axiom). Any functor in C at has a c ohomolo gic al dir e ct image functor and a homolo gic al dir e ct image functor in D (se e deﬁnition 3.5). Der4 (Base change axiom). F or any functor u : X → Y in C at , any p oint y of Y and any obje ct F in D ( X ) , the c anonic al b ase change morphisms (se e notation 3.7) u ∗ ( F ) y → Γ ∗ ( X/y , F /y ) and Γ ! ( y \ X , y \ F ) → u ! ( F ) y ar e isomorphi sms in D ( e ) . Der5 (Essential surje ct ivi ty axiom). L et I b e the c ate gory c orr esp onding to the gr aph 0 ← 1 . F or any smal l c ate gory X , t he functor d I ,X : D ( I × X ) → Fun ( I op , D ( X )) (se e notation 3.8) is ful l and essential ly s u rje ctive. Example 3.1 0. By [5] , any Quil len mo del c ate gory M gives rise to a derivator denote d HO ( M ) . W e denote by Ho ( M ) the ho motop y category of M . By deﬁnition, it equals HO ( M )( e ). Deﬁnition 3 . 11. A derivator D is stro ng if for every ﬁnite fr e e c ate gory X and every smal l c ate gory B Y , the natur al functor D ( X × Y ) − → Fun ( X op , D ( Y )) , se e notation 3.8, is ful l and essential ly surje ct ive. Notice that a strong deriv a to r is the s a me thing as a s ma ll homotopy theory in the sense of Heller [14]. Notice also that by prop osition 2 . 15 in [6], HO ( M ) is a strong deriv ato r . Deﬁnition 3.12. A derivator D is r e gular if in D , se quent ial homotopy c olimits c ommute with ﬁnite pr o ducts and ho motopy pul lb acks. Notation 3 .13 . Let X be a categor y . Remember that a s ieve (or a crible) in X is a full sub category U of X such that, for a ny ob ject x of X , if there exists a mo rphism x → u with u in U then x is in U . Dually a c osieve (or a co c rible) in X is a full sub c ategory Z of X s uc h that, for an y mor phism z → x in X , if z is in Z then so is x . A functor j : U → X is an op en immersion if it is injectiv e o n ob jects, fully faithful, and if j ( U ) is a s iev e in X . Dually a functor i : Z → X is a close d immersion if it is injectiv e o n ob jects, fully faithful, a nd if i ( Z ) is a cosie ve in X . One can easily show that op en immersions and closed immersio ns are stable by comp osition and pullback. 10 GONC ¸ ALO T ABUAD A Deﬁnition 3. 14. A derivator D is po in ted if it satisﬁes the fol lowing pr op erty. Der6 (Exc eptional axiom). F or any close d immersion i : Z → X in C at the c ohomolo gic al dir e ct image functor i ∗ : D ( Z ) − → D ( X ) has a right adjo int i ! : D ( X ) − → D ( Z ) c al le d the exceptional in verse image functor asso ciate d to i . D ual ly, for any op en immersion j : U → X the homolo gic al dir e ct image functor j ! : D ( U ) − → D ( X ) has a left adjoi n t j ? : D ( X ) − → D ( U ) c al le d the co exceptional in verse ima ge functor asso ciate d to j . Let  be the catego ry given by the comm utative squar e (0 , 0) (0 , 1) o o (1 , 0) O O (1 , 1) . o o O O W e ar e interested in tw o of its sub catego r ies. The sub category is (0 , 1) (1 , 0) (1 , 1) . o o O O and is the subcatego ry (0 , 0) (0 , 1) o o (1 , 0) O O . W e thus have tw o inclusio n functors σ : →  and τ : →  ( σ is an open immersion and τ a closed immersion). A glob al c ommu tative squar e in D is an ob ject o f D (  ). A glo bal comm utative square C in D is th us lo cally of shap e C 0 , 0 / /   C 0 , 1   C 1 , 0 / / C 1 , 1 in D ( e ). A g lobal commutativ e squar e C in D is c artesian (or a homotopy pul lb ack squar e ) if, for an y global commutativ e s q uare B in D , the ca nonical ma p Hom D (  ) ( B , C ) − → Hom D ( y ) ( σ ∗ ( B ) , σ ∗ ( C )) HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 11 is bijective. Dually , a global commutativ e squa r e B in D is c o c artesian (or a homo- topy pushout squar e ) if, for any globa l comm uta tiv e sq ua re C in D , the canonical map Hom D (  ) ( B , C ) − → Hom D ( p ) ( τ ∗ ( B ) , τ ∗ ( C )) is bijective. As  is isomor phic to its oppo s ite  op , one ca n see tha t a global commut ative square in D is ca rtesian (re s p. c o car tesian) if a nd o nly if is co car tesian (resp. cartesian) as a global c o mm utative square in D op . Deﬁnition 3.15. A derivator D is triang ulated or s ta ble if it is p ointe d and sat- isﬁes the following axiom: Der7 (St ability axiom). A glob al c ommutative squar e in D is c artesian if and only if it is c o c artesian. Theorem 3.16 ([30]) . F or any t riangulate d derivator D and smal l c ate gory X the c ate gory D ( X ) has a c anonic al t riangulate d stru ctur e. Let D and D ′ be deriv ato rs. W e deno te by Hom ( D , D ′ ) the category o f a ll mor - phisms of deriv ato rs, by Hom ! ( D , D ′ ) the categ ory of morphisms of deriv ato rs which commute with ho motop y colimits [4, 3 .25] and by Hom f lt ( D , D ′ ) the category of morphisms o f deriv ator s which commute with ﬁltered homotopy colimits, s e e [3, 5 ]. 4. Derived Kan extensions Let A b e a small categor y and Fun ( A op , S se t ) the Quille n mo del catego ry of simplicial pre-sheaves on A , endow ed with the pro jective mo del structur e , see [40]. W e have a t our dispo sal the functor A h − → F un ( A op , S se t ) X 7− → Hom A (? , X ) , where Hom A (? , X ) is considered as a c onstan t simplicial set. The functor h gives r ise to a morphism of pre-deriv ators A h − → HO ( Fun ( A op , S se t )) . Using the notation of [4], w e denote by Hot A the deriv a tor HO ( Fun ( A op , S se t )). The following results a re proven in [4 ]. Let D be a deriv ator . Theorem 4. 1. The morphism h induc es an e qu ival enc e of c ate gories Hom ( A, D ) ϕ   Hom ! ( Hot A , D ) . h ∗ O O Pr o of. This theorem is equiv alent to corolla r y 3 . 26 in [4], since w e have Hom ( A, D ) ≃ D ( A op ) . √ 12 GONC ¸ ALO T ABUAD A Lemma 4. 2. W e have an adjunction Hom ( Hot A , D ) Ψ   Hom ! ( Hot A , D ) ?  inc O O wher e Ψ( F ) := ϕ ( F ◦ h ) . Pr o of. W e cons tr uct a univ er sal 2- morphism of functors ǫ : inc ◦ Ψ − → I d . Let F b e a morphism of deriv ators b e longing to Hom ( Hot A , D ). Let L be a small category and X an ob ject of Hot A ( L ). Reca ll fro m [4] that w e ha ve the diagram ∇ R X π { { w w w w w w w w  " " F F F F F F F F L op A . Now, let p b e the functor π op and q the functor  op . By the dual of prop osition 1 . 15 in [4], we hav e the following functoria l is o morphism p ! q ∗ ( h ) ∼ − → X . Finally let ǫ L ( X ) b e the comp osed morphism ǫ L ( X ) : Ψ( F )( X ) = p ! q ∗ F ( h ) = p ! F ( q ∗ h ) → F ( p ! q ∗ h ) ∼ → F ( X ) and notice, using theorem 4 .1, that ǫ induces an a djunction. √ 5. Lo caliza tion: model ca tegories versus deriv a tors Let M be a left proper , cellula r Quillen mo del categor y , see [15]. W e start by ﬁxing a frame on M , s ee deﬁnition 16 . 6 . 21 in [15]. Let D b e a small category and F a functor from D to M . W e denote by ho colim F the ob ject of M , as in deﬁnition 19 . 1 . 2 of [15]. Let S b e a set of morphisms in M and denote by L S M the left Bousﬁeld lo calizatio n of M by S . Notice that the Quillen adjunction M I d   L S M , I d O O induces a mo rphism of der iv ator s γ : HO ( M ) L I d − → HO ( L S M ) which commutes with homotopy co limits. Prop osition 5.1. Le t W S b e the smal lest class of morphisms in M satisfying the fol lowing pr op erties: a) Every element in S b elongs to W S . b) Every we ak e qu ival enc e of M b elongs to W S . HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 13 c) If in a c ommut ative triangle, two out of thr e e morphisms b elong to W S , then so do es t he t hir d one. The class W S is stable under r etr actions. d) L et D b e a smal l c ate gory and F and G funct ors fr om D t o M . If η is a morphism of funct ors fr om F to G such that for every obje ct d in D , F ( d ) and G ( d ) ar e c oﬁbr ant obje cts and the morphi s m η ( d ) b elongs to W S , then so do es the morphism ho c olim F − → ho c olim G . Then the class W S e quals the class of S -lo c al e qu iva lenc es in M , se e [15] . Pr o of. The cla ss of S -lo cal equiv alences satisﬁes prop erties a ), b ), c ) and d ): Prop- erties a ) and b ) are satisﬁed by deﬁnition, prop ositions 3 . 2 . 3 and 3 . 2 . 4 in [1 5] imply prop erty c ) and propo sition 3 . 2 . 5 in [15] implies prop ert y d ). Let us now show that conv ersely , ea ch S -lo cal equiv alence is in W S . Let X g → Y be an S -lo cal equiv a le nc e in M . Without loss o f generality , we can s upp ose that X is coﬁbra n t. Indeed, let Q ( X ) b e a coﬁbrant resolution of X and consider the diagram Q ( X ) π ∼   g ◦ π " " F F F F F F F F X g / / Y . Notice that since π is a w ea k equiv alence, g is an S - loca l eq uiv a le nce if and only if g ◦ π is one . By theorem 4 . 3 . 6 in [1 5], g is a n S -lo cal equiv alence if and only if the mo rphism L S ( g ) app earing in the diagr am X g / / j ( X )   Y j ( Y )   L S X L S ( g ) / / L S Y is a weak eq uiv alence in M . This shows that it is eno ugh to prove that j ( X ) and j ( Y ) belo ng to W S . Apply the small o b ject a rgument to the mor phism X − → ∗ using the set ] Λ( S ), see prop osition 4 . 2 . 5 in [15]. W e ha ve the factorization X / / j ( X ) " " F F F F F F F F ∗ L S ( X ) < < y y y y y y y y y , where j ( X ) is a relative ] Λ( S )- c e ll co mplex. W e will no w prov e tw o stabilit y conditions concerning the class W S : 14 GONC ¸ ALO T ABUAD A S1) Cons ide r the following push- out W 0 / /   f   y W 2 f ∗   W 1 / / W 3 , where W 0 , W 1 and W 2 are coﬁbrant ob jects in M and f is a coﬁbratio n which belongs to W S . Obse r v e that f ∗ corres p onds to the colimit o f the morphism of diagrams W 0   f   W 0 / / W 2 W 1 W 0 o o f o o / / W 2 . Now, prop osition 1 9 . 9 . 4 in [15] and prop erty d ) imply that f ∗ belo ngs to W S . S2) Cons ide r the following dia gram X : X 0 / / f 0 / / X 1 / / f 1 / / X 2 / / f 2 / / X 3 / / / / . . . in M , wher e the ob jects ar e coﬁbra n t and the morphisms are coﬁbrations which b elong to the class W S . Obs e rv e that the trans ﬁnite compositio n o f X cor resp o nds to the colimit o f the morphis m of diagrams X 0 X 0   f 0   X 0   f 1 ◦ f 0   . . . X 0 / / f 0 / / X 1 / / f 1 / / X 2 / / / / . . . . Now, since X is a Reedy coﬁbrant diagram on category with ﬁbrant con- stants, see deﬁnition 15 . 10 . 1 in [1 5], theorem 19 . 9 . 1 fr om [1 5] and pro perty d ) imply tha t the transﬁnite comp osition of X b elongs to W S . Notice that the ab o ve argument can be immediatly generalized to a transﬁnite com- po sition of a λ -se quence, wher e λ denotes an ordinal, see section 1 0 . 2 in [15]. Now, the construction of the morphism j ( X ) and the stability co nditio ns S 1 ) and S 2) shows us that it is enough to prove tha t the elemen ts o f ] Λ( S ) belo ng to W S . By propo sition 4 . 2 . 5 in [15], it is suﬃcien t to show that the set Λ( S ) = { ˜ A ⊗ ∆[ n ] ∐ ˜ A ⊗ ∂ ∆[ n ] ˜ B ⊗ ∆[ n ] Λ( g ) − → ˜ B ⊗ ∆[ n ] | ( A g → B ) ∈ S, n ≥ 0 } , of hor ns in S is contained in W S . Re c a ll from deﬁnition 4 . 2 . 1 in [1 5] that ˜ g : ˜ A → ˜ B denotes a cos implicial res olution of g : A → B a nd ˜ g is a Reedy co ﬁbration. W e HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 15 hav e the diagram ˜ A ⊗ ∂ ∆[ n ] 1 ⊗ i   ˜ g ⊗ 1 / / ˜ B ⊗ ∂ ∆[ n ] 1 ⊗ i   ˜ A ⊗ ∆[ n ] ˜ g ⊗ 1 / / ˜ B ⊗ ∆[ n ] . Observe that the morphism ˜ A ⊗ ∆[ n ] ˜ g ⊗ 1 − → ˜ B ⊗ ∆[ n ] ident iﬁes with ˜ A n ˜ g n − → ˜ B n , and so belongs to W S . Now, the mor phism ˜ A ⊗ ∂ ∆[ n ] ˜ g ⊗ 1 ∂ ∆[ n ] − → ˜ B ⊗ ∂ ∆[ n ] corres p onds to the induced map of latc hing ob jects L n ˜ A − → L n ˜ B , which is a coﬁbration in M by prop osition 15 . 3 . 11 in [15 ]. Now by pro positio ns 15 . 10 . 4 , 16 . 3 . 12 and theorem 19 . 9 . 1 of [1 5], we hav e the following commutativ e diagram ho colim ∂ ( → ∆ ↓ [ n ]) ˜ A / / ∼   ho colim ∂ ( → ∆ ↓ [ n ]) ˜ B ∼   L n ˜ A / / L n ˜ B , where the vertiv al ar rows ar e weak equiv alences and ∂ ( → ∆ ↓ [ n ]) denotes the categor y of strictly increa s ing maps with tar get [ n ]. By prop erty d ) of the clas s W S , we conclude that ˜ g ⊗ 1 ∂ ∆[ n ] belo ngs to W S . W e have the follo wing dia gram ˜ A ⊗ ∆[ n ] ∐ ˜ A ⊗ ∂ ∆[ n ] ˜ B ⊗ ∆[ n ] Λ( g ) * * U U U U U U U U U U U U U U U U ˜ A ⊗ ∆[ n ] I O O ˜ g ⊗ 1 / / ˜ B ⊗ ∆[ n ] . Notice that the mo r phism I b elongs to W S by the stability co ndition S 1) applied to the morphism ˜ A ⊗ ∂ ∆[ n ] ˜ g ⊗ 1 − → ˜ B ⊗ ∂ ∆[ n ] , which is a coﬁbration and b elongs to W S . Since the morphism ˜ g ⊗ 1 b elongs to W S so do es Λ( g ). This pr o ves the prop o sition. √ Let D be deriv ator and S a class of mo rphisms in D ( e ). 16 GONC ¸ ALO T ABUAD A Deﬁnition 5.2 (Cisinski [7 ]) . The derivator D admits a left Bo usﬁeld lo calization by S if ther e exists a morphism of derivators γ : D → L S D , which c ommutes with homotop y c olimits, sends the elements of S to isomorphi sms in L S D ( e ) and satisﬁes the u niversal pr op ert y: for every derivator D ′ the morphism γ induc es an e quivalenc e of c ate gories Hom ! ( L S D , D ′ ) γ ∗ − → Hom ! ,S ( D , D ′ ) , wher e Ho m ! ,S ( D , D ′ ) denotes the c ate gory of morphisms of derivators which c om- mute with homotopy c olimits and send the elements of S to isomorphisms in D ′ ( e ) . Lemma 5 .3. Su pp ose that D is a triangulate d derivator, S is st able under the lo op sp ac e functor Ω( e ) : D ( e ) → D ( e ) , se e [3] , and D admits a left Bousﬁeld lo c alization L S D by S . Then L S D is also a triangulate d derivator. Pr o of. Recall from [3] that since D is a tria ng ulated deriv ator, we hav e the fo llo wing equiv alence D Ω   D Σ O O Notice that bo th morphisms of deriv ators, Σ and Ω, commute with ho motop y col- imits. Since S is stable under the functor Ω( e ) : D ( e ) → D ( e ) a nd D admits a left Bousﬁeld lo calizatio n L S D by S , we have a n induced morphism Ω : L S D → L S D . Let s b e an elemen t of S . W e now show that the ima ge of s b y the functor γ ◦ Σ is an isomorphism in L S D ( e ). F or this co nsider the category , see section 3, and the functors (0 , 0) : e → and p : → e . Now reca ll from section 7 fro m [14] that Ω( e ) := p ! ◦ (0 , 0) ∗ . This descriptio n shows us that the image of s under the functor γ ◦ Σ is an iso mo r- phism in L S D ( e ) b ecause γ commutes with homo top y co limits. In conclusion, we hav e an induced adjunction L S D Ω   L S D Σ O O which is clearly an equiv alence. This pro ves the lemma. √ Theorem 5. 4 (Cisinski [7]) . The morphism of derivators γ : HO ( M ) L I d − → HO ( L S M ) is a left Bousﬁeld lo c alization of H O ( M ) by t he image of the set S in Ho ( M ) . HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 17 Pr o of. Let D be a der iv ato r . The mor phism γ admits a fully faithful right a djoin t σ : HO ( L S M ) − → HO ( M ) . Therefore, the induced functor γ ∗ : Hom ! ( HO ( L S M ) , D ) − → Ho m ! ,S ( HO ( M ) , D ) , admits a left adjoint σ ∗ and σ ∗ γ ∗ = ( γ σ ) ∗ is is o morphic to the identit y . Therefor e γ ∗ is fully faithful. W e now show that γ ∗ is essentially surjective. Let F b e an ob ject of Ho m ! ,S ( HO ( M ) , D ). Notice that since D satis ﬁe s the conserv ativity a xiom, it is suﬃcient to sho w that the functor F ( e ) : Ho ( M ) → D ( e ) sends the imag es in Ho ( M ) of S - local equiv alences of M to isomorphisms in D ( e ). The morphism F then b ecomes na turally a mor phism o f deriv ators F : HO ( L S M ) → D such that γ ∗ ( F ) = F . Now, s ince F commutes with homotopy colimits, the functor M → H o ( M ) → D ( e ) sends the elements of W S to isomorphis ms. This proves the theorem since by prop osition 5.1 the class W S equals the class of S -lo cal equiv alence s in M . √ 6. Fil tered h o motopy colimits Let M b e a cellula r Quillen mo del ca teg ory , with I the set of genera ting coﬁbr a- tions. Supp ose tha t the do mains and co domains o f the elements o f I are coﬁbra n t, ℵ 0 -compact, ℵ 0 -small a nd homoto pic a lly ﬁnitely presented, see deﬁnition 2 . 1 . 1 in [41]. Example 6 . 1. Consider the quasi-e quivalent, r esp. quasi-e quic onic, r esp. Morita, Quil len mo del structu r e on dgcat c onstructe d in [37] [38] [39] . R e c al l that a dg funct or F : C → E is a quasi-e qu ival enc e, r esp. quasi-e quic onic, r esp. a Morita dg functor, if it satisﬁes one of the fol lowing c onditions C 1) or C 2) : C1) The dg c ate gory C is empty and al l the obje cts of E ar e c ontr actible. C2) F or every obje ct c 1 , c 2 ∈ C , t he morphism of c omplexes fr om Hom C ( c 1 , c 2 ) to Hom E ( F ( c 1 )) , F ( c 2 )) is a qu asi-isomo rphism and the functor H 0 ( F ) , r esp. H 0 ( pr e-tr ( F )) , r esp. H 0 ( pr e-tr ( F )) ∋ , is essential ly surje ct ive. Observe that the domains and c o domains of t he set I of gener ating c oﬁbr ations in dgcat satisfy the c onditions ab ove for al l the Quil len mo del structu r es. The following prop osition is a simpliﬁcation of prop osition 2 . 2 in [41]. Prop osition 6.2. L et M b e a Qu il len mo del c ate gory which satisﬁes the c onditions ab ove. Then 1) A ﬁ lt er e d c olimit of trivial ﬁbr ations is a trivial ﬁbr ation. 2) F or any ﬁlter e d diagr am X i in M , the natur al morphism ho c olim i ∈ I X i − → c olim i ∈ I X i is an isomorphi s m in Ho ( M ) . 3) Any obje ct X in M is e quivalent to a ﬁlter e d c olimit of strict ﬁ n ite I -c el l obje cts. 18 GONC ¸ ALO T ABUAD A 4) An obje ct X in M is homotopic al ly ﬁ nitely pr esent e d if and only if it is e quivalent to a r e ctr act of a strict ﬁnite I -c el l obje ct. Pr o of. The pro of of 1), 2) and 3) is exactly the sa me as that of prop osition 2 . 2 in [41]. The pro of of 4) is a ls o the sa me once we observe that the domains and co domains of the elements of the set I are alrea dy homoto pically ﬁnitely prese nted by hypothesis. √ In everything that fo llo ws , we ﬁx: - A co-simplicial re solution functor (Γ( − ) : M → M ∆ , i ) in the mo del ca tegory M , see deﬁnition 16 . 1 . 8 in [15]. This mea ns that for every ob ject X in M , Γ( X ) is coﬁbra n t in the Reedy mo del str ucture on M ∆ and i ( X ) : Γ( X ) ∼ − → c ∗ ( X ) is a weak equiv a lence on M ∆ , where c ∗ ( X ) denotes the co ns tan t co- simplicial ob ject asso ciated with X . - A ﬁbrant resolutio n functor (( − ) f : M → M , ǫ ) in the model category M , see [15]. Deﬁnition 6. 3. L et M f b e the smal lest ful l sub c ate gory of M such that - M f c ontains (a re pr esentative of t he isomorphism class of ) e ach st r ictly ﬁnite I -c el l obje ct of M and - the c ate gory M f is stable under the functors ( − ) f and Γ( − ) n , n ≥ 0 . R emark 6.4 . No tice that M f is a small categor y and that every ob ject in M f is weakly equiv alent to a strict ﬁnite I -cell. W e have the inclusion M f I ֒ → M . Deﬁnition 6.5 . L et S b e t he set of pr e-images of the we ak e quivalenc es in M u nder the functor i . Lemma 6. 6. The induc e d functor M f [ S − 1 ] Ho ( I ) − → Ho ( M ) is ful ly faithful, wher e M f [ S − 1 ] denotes the lo c alization of M by the set S . Pr o of. Let X , Y be ob jects of M f . Notice that ( Y ) f is a ﬁbr an t r esolution of Y in M whic h b elongs to M f and Γ( X ) 0 ` Γ( X ) 0 d 0 ‘ d 1   / / Γ( X ) 0 Γ( X ) 1 s 0 7 7 p p p p p p p p p p p is a cylinder ob ject for Γ( X ) 0 , see prop osition 16 . 1 . 6 . from [15]. Since M f is also stable under the functor s Γ( − ) n , n ≥ 0, this cy linder ob ject also belo ng s to M f . This implies tha t if in the constr uction of the homotopy categor y Ho ( M ), a s in HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 19 theorem 8 . 3 5 of [15], we restrict our selv es to M f we recov er M f [ S − 1 ] as a full sub c ategory of Ho ( M ). This implies the lemma. √ W e denote by Fun ( M op f , S se t ) the Q uillen mo del categor y of simplicial pr e- sheav es on M f endow ed with the pro jectiv e mo del structure, see section 4. Let Σ be the image in Fun ( M op f , S se t ) b y the functor h , see section 4, o f the set S in M f . Since the categor y Fun ( M op f , S se t ) is cellular and left pr oper, its left Bousﬁeld lo calization b y the set Σ exis ts, see [15]. W e deno te it b y L Σ F un ( M op f , S se t ). W e hav e a co mposed functor that w e still denote by h h : M f → Fun ( M op f , S se t ) I d → L Σ F un ( M op f , S se t ) . Now, consider the functor h : M − → F un ( M op f , S se t ) X 7− → Hom (Γ( − ) , X ) |M f . W e also ha ve a comp osed functor that we s till denote b y h h : M → Fun ( M op f , S se t ) I d → L Σ F un ( M op f , S se t ) . Now, obser v e that the natural equiv ale nc e i ( − ) : Γ( − ) − → c ∗ ( − ) , induces, for ev er y o b ject X in M f , a morphism Ψ( X ) in L Σ F un ( M op f , S se t ) Ψ( X ) : h ( X ) = Hom ( c ∗ ( − ) , X ) − → Hom (Γ( − ) , X ) =: ( h ◦ I )( X ) , which is functorial in X . Lemma 6. 7. The fu n ctor h pr eserves we ak e quivalenc es b etwe en ﬁbr ant obje cts. Pr o of. Let X b e a ﬁbrant o b ject in M . W e ha ve a n equiv alence Hom (Γ( Y ) , X ) ∼ − → Map M ( Y , X ) , see [15]. This implies the lemma. √ R emark 6.8 . The previous lemma implies that the functor h admits a right der iv e d functor R h : Ho ( M ) − → Ho ( L Σ F un ( M op f , S se t )) X 7− → Hom (Γ( − ) , X f ) |M f . Since the functor h : M f → L Σ F un ( M op f , S se t ) , sends, by deﬁnition, the elements of S to weak equiv alence s , we have an induced morphism Ho ( h ) : M f [ S − 1 ] → Ho ( L Σ F un ( M op f , S se t )) . R emark 6.9 . Notice that lemma 4 . 2 . 2 from [40] implies that for every X in M f , the mor phism Ψ ( X ) Ψ( X ) : Ho ( h )( X ) − → ( R h ◦ Ho ( I ))( X ) is an is o morphism in Ho ( L Σ F un ( M op f , S se t )). 20 GONC ¸ ALO T ABUAD A This s hows that the functors Ho ( h ) , R h ◦ Ho ( I ) : M f [ S − 1 ] → Ho ( L Σ F un ( M op f , S se t ) are canonica lly isomor phic a nd so w e ha ve the follo wing dia gram M f [ S − 1 ] Ho ( I ) / / Ho ( h )   Ho ( M ) , R h t t i i i i i i i i i i i i i i i i i Ho ( L Σ F un ( M o f , S se t )) which is comm utative up to iso morphism. Lemma 6. 10. The functor R h c ommutes with ﬁlter e d homotopy c olimits. Pr o of. Let { Y i } i ∈ I be a ﬁlter ed diagram in M . W e can suppos e, without loss of generality , that Y i is ﬁbr an t in M . By prop osition 6 .2, the natural mor phism ho colim i ∈ I Y i − → colim i ∈ I Y i is an is o morphism in Ho ( M ) a nd co lim i ∈ I Y i is a lso ﬁbra n t. Since the functor Ho ( F un ( M op f , S se t )) L I d − → Ho ( L Σ F un ( M op f , S se t )) , commutes with homotopy co limits and in Ho ( Fun ( M op f , S se t )) they are calcula ted ob ject wise, it is suﬃcient to show that the morphism ho colim i ∈ I R h ( Y i )( X ) − → R h (co lim i ∈ I Y i )( X ) is an isomorphism in H o ( S set ), fo r every ob ject X in M f . Now, since every ob ject X in M f is homo topically ﬁnitely pres e n ted, see prop osition 6.2, we hav e the following equiv alences: R h (colim i ∈ I Y i )( X ) = Hom (Γ( X ) , co lim i ∈ I Y i ) ≃ Map (Γ( X ) , colim i ∈ I Y i ) ≃ colim i ∈ I Map ( X , Y i ) ≃ ho colim i ∈ I R h ( Y i )( X ) This proves the lemma. √ W e now denote by L Σ Hot M f the deriv ator asso ciated with L Σ F un ( M op f , S se t ) and by M f [ S − 1 ] the pre-deriv ato r M f lo calized a t the set S , see exa mples 3.3 a nd 3.4. Observe that the morphism of functors Ψ : h − → h ◦ I induces a 2 -morphism of der iv ator s Ψ : Ho ( h ) − → R h ◦ Ho ( I ) . Lemma 6. 11. The 2 -morphism Ψ is an isomorp hism. Pr o of. F or the terminal ca tegory e , the 2-mor phism Ψ coincides with the morphism of functors of remark 6.9 . Since this one is a n isomor phism, s o is Ψ by conser v ativity . This proves the lemma. √ HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 21 As b efore, we have the following diagram M f [ S − 1 ] Ho ( I ) / / Ho ( h )   HO ( M ) , R h v v l l l l l l l l l l l l l l L Σ Hot M f which is commutativ e up to isomorphism in the 2-catego ry of pr e-deriv ators. Notice that by lemma 6.1 0, R h commutes with ﬁltered homotopy co limits. Let D be a deriv ator . Lemma 6. 12. The morphism of pr e-derivators M f [ S − 1 ] Ho ( h ) − → L Σ Hot M f , induc es an e qu iva lenc e of c ate gories Hom ! ( L Σ Hot M f , D ) Ho ( h ) ∗ − → Hom ( M f [ S − 1 ] , D ) . Pr o of. The category Hom ! ( L Σ Hot M f , D ) is equiv ale n t, by theo rem 5 .4, to the cat- egory Ho m ! , Σ ( Hot M f , D ). This last catego ry identiﬁes, under the equiv alence Hom ! ( Hot M f , D ) → Hom ( M f , D ) given b y theor e m 4.1, with the full sub category of Hom ( M f , D ) co nsisting of the morphisms o f pr e-deriv ators which send the elements o f S to isomor phisms in D ( e ). Now obse r v e tha t this last ca teg ory identiﬁes with Ho m ( M f [ S − 1 ] , D ), by deﬁnition of the localized pre-deriv ator M f [ S − 1 ]. This prov es the lemma. √ Recall fr o m section 9 . 5 in [11] tha t the co-s implicial r esolution functor Γ( − ) that we hav e ﬁxed in the b eginning o f this section a llo ws us to construct a Q uillen adjunction: M h = sing   F un ( M op f , S se t ) . Re O O Since the functor Re s ends the elemen ts of Σ to w eak equiv ale nc e s in M , we have the following Quillen adjunction M h   L Σ F un ( M op f , S se t ) , Re O O and a na tural weak e q uiv alence η : R e ◦ h ∼ − → I , see [11]. 22 GONC ¸ ALO T ABUAD A This implies that we hav e the follo wing diagr am M f [ S − 1 ] Ho ( I ) / / Ho ( h )   Ho ( M ) , R h t t i i i i i i i i i i i i i i i i i Ho ( L Σ F un ( M op f , S se t )) L Re 4 4 i i i i i i i i i i i i i i i i i which is comm utative up to iso morphism. W e now claim that L Re ◦ R h is natura lly isomor phic to the ident it y . Indeed, by prop osition 6 .2, each ob ject of M is is o morphic in Ho ( M ), to a ﬁltered colimit of strict ﬁnite I -cell ob jects. Since R h and L Re comm ute with ﬁlter e d homotopy colimits and L Re ◦ Ho ( h ) ≃ Id, w e conclude that L R e ◦ R h is na turally iso morphic to the iden tity . This implies that the morphism R h is fully faithful. Now, obser v e that the natural weak equiv alence η induces a 2 -isomorphism and so we obtain the follo wing diagr am M f [ S − 1 ] Ho ( I ) / / Ho ( h )   HO ( M ) , R h v v l l l l l l l l l l l l l l L Σ Hot M f L Re 6 6 l l l l l l l l l l l l l l which is commutativ e up to isomorphism in the 2-catego ry of pr e-deriv ators. Notice that L Re ◦ R h is natura lly isomorphic to the ident it y (by co nserv a tivit y) and so the morphism of deriv ator s R h is fully faithful. Let D be a deriv ator . Theorem 6. 13. The morphism of deriva tors HO ( M ) R h − → L Σ Hot M f , induc es an e qu iva lenc e of c ate gories Hom ! ( L Σ Hot M f , D ) R h ∗ − → H om f lt ( HO ( M ) , D ) , wher e Hom f lt ( HO ( M ) , D ) denotes the c ate gory of morphisms of derivators which c ommute with ﬁlter e d homotopy c olimits. Pr o of. W e hav e the following a djunction Hom ( HO ( M ) , D ) L Re ∗   Hom ( L Σ Hot M f , D ) , R h ∗ O O with R h ∗ a fully faithful functor. Now notice that the adjunction o f lemma 4.2 induces naturally a n adjunction Hom ( L Σ Hot M f , D ) Ψ   Hom ! ( L Σ Hot M f , D ) . ?  O O HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 23 This implies that the c o mposed functor R h ∗ : Ho m ! ( L Σ Hot M f , D ) − → Ho m f lt ( HO ( M ) , D ) is fully faithful. W e now show that this functor is essentially sur jectiv e. Let F b e a n ob ject of Hom f lt ( HO ( M ) , D ). Consider the morphism L Re ∗ ( F ) := F ◦ L R e . Notice that this morphism does not necessarily co mm ute with homotop y co limits. Now, by the ab ove adjunction, w e ha ve a universal 2-mor phism ϕ : Ψ ( L Re ∗ ( F )) − → L R e ∗ ( F ) . Consider the 2-morphism R h ∗ : R h ∗ ((Ψ ◦ L Re ∗ )( F )) − → ( R h ∗ ◦ L Re ∗ )( F ) ≃ F . Now, we will sho w that this 2-morphism is a 2-isomo rphism. By c o nserv ativity , it is suﬃcient to show this for the case of the termina l ca tegory e . F or this, observe that R h ∗ ( ϕ ) induces an isomo rphism Ψ( L Re ∗ ( F )) ◦ R h ◦ Ho ( I ) − → F ◦ Ho ( I ) . Now each ob ject of M is iso morphic, in Ho ( M ), to a ﬁltered colimit of strict ﬁnite I -cell ob jects. Since F and Ψ( L Re ∗ ( F )) commute with ﬁltered homoto p y colimits, R h ∗ ( ϕ ) induces an is omorphism. This shows that the functor R h ∗ is essentially surjective. This pr o ves the theorem. √ 7. Pointed deriv a tors Recall from the pr evious s ection that we hav e co ns tructed a de r iv ato r L Σ Hot M f asso ciated with a Quillen mo del catego ry M satisfying suitable compactness a s- sumptions. Now supp ose that Ho ( M ) is p ointed, i.e. that the morphism ∅ − → ∗ , in M , where ∅ denotes the initial ob ject and ∗ the terminal one, is a weak equiv a- lence. Consider the morphism P : e ∅ − → h ( ∅ ) , where e ∅ denotes the initial o b ject in L Σ F un ( M op f , S se t ). Observe that, since R h admits a left adjo int, h ( ∅ ) identiﬁes with the terminal ob ject in Ho ( L Σ F un ( M op f , S se t )) , bec ause h ( ∅ ) = Ho ( h )( ∅ ) ∼ → R h ◦ Ho ( I )( ∅ ) ∼ → R h ( ∗ ) . W e denote b y L Σ ,P F un ( M op f , S se t ) , the left Bousﬁeld lo calization of L Σ F un ( M op f , S se t ) at the mor phis m P . Notice that the categ ory Ho ( L Σ ,P F un ( M op f , S se t )) , 24 GONC ¸ ALO T ABUAD A is now a pointed one. W e have the follo wing mo r phisms of der iv ator s Ho ( M ) R h   L Σ Hot M f L Re O O Φ   L Σ ,P Hot M f . By constr uction, we hav e a po in ted mo rphism of der iv ator s HO ( M ) Φ ◦ R h − → L Σ ,P Hot M f , which commutes with ﬁltered ho mo top y co limits and preser v es the po in t. Let D be a p oin ted deriv a tor. Prop osition 7.1. The morphism of derivators Φ ◦ R h induc es an e quivalenc e of c ate gories Hom ! ( L Σ ,P Hot M f , D ) (Φ ◦ R h ) ∗ − → Hom f lt,p ( HO ( M ) , D ) , wher e Hom f lt,p ( HO ( M ) , D ) denotes the c ate gory of morphisms of derivators which c ommute with ﬁlter e d homotopy c olimits and pr eserve the p oint. Pr o of. By theorem 5.4, we have an eq uiv alence of categor ies Hom ! ( L Σ ,P Hot M f , D ) Φ ∗ − → H om ! ,P ( L Σ Hot M f , D ) . By theorem 6.13, we have an equiv alence of catego ries Hom ! ( L Σ Hot M f , D ) R h ∗ − → H om f lt ( HO ( M ) , D ) . W e now show that under this last equiv alence, the category Hom ! ,P ( L Σ Hot M f , D ) ident iﬁes with Hom f lt,p ( HO ( M ) , D ). Let F b e an ob ject of Hom ! ,P ( L Σ Hot M f , D ). Since F comm utes with homotopy c o limits, it preser v es the initial ob ject. This implies that F ◦ R h belo ngs to Hom f lt,p ( HO ( M , D ) . Let now G b e an ob ject of Hom f lt,p ( HO ( M ) , D ). Co nsider, as in the pro of of theorem 6.1 3 , the mor phism Ψ( L Re ∗ ( G )) : L Σ Hot M f − → D . Since Ψ ( L Re ∗ ( G )) commutes with homotopy co limits, b y construction, it s ends e ∅ to the po in t of D . Obse rv e a ls o that h ( ∅ ) is also sent to the point of D b ecause Ψ( L Re ∗ ( G ))( h ( ∅ )) ≃ G ( ∅ ) . This pr o ves the prop o sition. √ HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 25 8. Small weak genera tors Let N b e a p ointed, left prop er, compa ctly gener ated Quillen mo del c a tegory a s in deﬁnition 2 . 1 of [41]. Observe that in particular this implies that N is ﬁnitely generated, a s in section 7 . 4 in [16]. W e denote by G the set of coﬁb ers of the generating coﬁbra tio ns I in N . By coro llary 7 . 4 . 4 in [1 6], the set G is a set of small weak generators for Ho ( N ), see deﬁnitions 7 . 2 . 1 and 7 . 2 . 2 in [16]. Le t S b e a s e t of mo rphisms in N b et ween ob jects which are ho motopically ﬁnitely presented, see [41], and L S N the left Bousﬁeld lo calization o f N by S . W e have an a djunction Ho ( N ) L I d   Ho ( L S N ) . R I d O O Lemma 8.1. The image of t he set G under the functor L I d is a set of smal l we ak gener ators in Ho ( L S N ) . Pr o of. The previous adjunction is eq uiv alent to Ho ( N ) ( − ) f   Ho ( N ) S  ? O O where Ho ( N ) S denotes the full sub category of Ho ( M ) formed by the S -lo cal ob jects of N and ( − ) f denotes a ﬁbrant resolution functor in L S N , see [15]. Clearly , this implies that the image of the set G under the functor ( − ) f is a set of weak gener a tors in Ho ( L S N ). W e now s ho w that the S -lo cal ob jects in N are stable under ﬁltered homotopy colimits. Le t { X i } i ∈ I be a ﬁltered diag ram of S -lo cal ob jects. By prop osition 6.2, we hav e a n isomorphis m ho colim i ∈ I X i ∼ − → colim i ∈ I X i in Ho ( N ). W e now show that colim i ∈ I X i is an S -lo cal ob ject. Let g : A → B b e an element of S . W e hav e a t our dispo sal the following commutativ e dia gram Map ( B , colim i ∈ I X i ) g ∗ / / Map ( A, colim i ∈ I X i ) colim i ∈ I Map ( B , X i ) ∼ O O colim i ∈ I g ∗ i / / colim i ∈ I Map ( A, X i ) . ∼ O O Now obser v e that since A and B are homotopica lly ﬁnitely presented ob jects, the vertical arrows in the diag ram are iso morphisms in Ho ( S set ). Since each o b ject X i is S -lo cal, the morphism g ∗ i is an isomorphism in Ho ( S s et ) a nd s o is c olim i ∈ I g ∗ i . This implies that colim i ∈ I X i is an S -lo cal ob ject. This shows that the inclusion Ho ( N ) S ֒ → Ho ( N ) , 26 GONC ¸ ALO T ABUAD A commutes with ﬁltered homotopy colimits and so the image of the set G under the functor ( − ) f consists o f small ob jects in Ho ( L S N ). This pr o ves the lemma. √ Recall fro m the previous chapter that we hav e constr ucted a p oin ted deriv ator L Σ ,P Hot M f . W e will now construct a strictly po in ted Quillen mo del ca tegory whose asso ciated deriv ato r is equiv alent to L Σ ,P Hot M f . Consider the p ointed Q uillen mo del ca tegory ∗ ↓ Fun ( M op f , S se t ) = F un ( M op f , S se t • ) . W e have the following Q uillen adjunction F un ( M op f , S se t • ) U   F un ( M op f , S se t ) , ( − ) + O O where U denotes the forgetful functor. W e denote b y L Σ ,P F un ( M op f , S se t • ) the left B o usﬁeld loca lization of Fun ( M op f , S se t • ) by the image of the set Σ ∪ { P } under the functor ( − ) + . W e denote by L Σ ,P Hot M f • the deriv a tor as socia ted with L Σ ,P F un ( M op f , S se t • ). R emark 8.2 . Since the deriv ator s asso ciated with L Σ ,P F un ( M op f , S se t ) and L Σ ,P F un ( M op f , S se t • ) are cara cterized by the same universal prop erty we hav e a ca nonical equiv alence of po in ted der iv ator s L Σ ,P Hot M f ∼ − → L Σ ,P Hot M f • Notice also that the categ o ry Fun ( M op f , S se t • ) endow ed with the pro jective mo del structure is p ointed, left pr oper , compactly gener ated and that the domains and co domains of the elements of the set (Σ ∪ { P } ) + are homotopically ﬁnitely pr esen ted ob jects. The r efore by lemma 8.1, the s et G = { F X ∆[ n ] + /∂ ∆[ n ] + | X ∈ M f , n ≥ 0 } , of coﬁb ers of the gener ating co ﬁbr ations in F u n ( M op f , S se t • ) is a s e t of small weak generator s in Ho ( L Σ ,P F un ( M op f , S se t • )). 9. St abiliza tion Let D be a regula r p ointed strong deriv ator. In [1 4], Heller c o nstructs a universal morphism to a triang ulated s trong deriv a to r D stab − → St ( D ) , which commutes with homotopy co limits. This co nstruction is done in tw o steps . First consider the follo wing o rdered set V := { ( i, j ) | | i − j | ≤ 1 } ⊂ Z × Z naturally as a s mall catego ry . W e denote by · V := { ( i, j ) | | i − j | = 1 } ⊂ V , the full subcateg ory o f ‘p oints on the boundar y’. HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 27 Now, let Sp ec ( D ) b e the full sub deriv ator of D V , se e deﬁnition 3 . 4 in [3 ], formed by the ob jects X in D V ( L ), who s e ima ge under the functor D V ( L ) = D ( V × L ) − → Fun ( V op , D ( L )) is of the form ∗ / / X (1 , 1) · · · ∗ / / X (0 , 0) / / O O ∗ O O · · · X ( − 1 , − 1) O O / / ∗ O O , see section 8 in [1 4]. W e hav e an ev aluation functor e v (0 , 0) : Sp ec ( D ) → D , which admits a left adjoint L [0 , 0]. Finally , let St ( D ) b e the full reﬂex iv e sub deriv ator of Sp ec ( D ) consisting of the Ω-sp ectra, as deﬁned in [14]. W e have the follo wing a djunctions D L [0 , 0]   stab & & Spe c ( D ) ev (0 , 0) O O loc   St ( D )  ? O O in the 2 -category of de r iv ator s. Let T b e a triang ulated strong der iv ator . The following theorem is prov ed in [14]. Theorem 9. 1. The morphism stab induc es an e quivalenc e of c ate gories Hom ! ( St ( D ) , T ) stab ∗ − → Hom ! ( D , T ) . Lemma 9.2. L et G b e a set of obje cts in D ( e ) which satisﬁes the fol lowing c ondi- tions: A1) If, for e ach g in G , we have Hom D ( e ) ( g , X ) = {∗} , then X is isomorphic to ∗ , wher e ∗ denotes the terminal and initial obje ct in D ( e ) . A2) F or every set K and e ach g in G the c anonic al map c olim S ⊆ K S f inite Hom D ( e ) ( g , a α ∈ S X α ) ∼ → Hom D ( e ) ( g , a α ∈ S X α ) is bije ctive. Then the set { Σ n stab ( g ) | g ∈ G , n ∈ Z } 28 GONC ¸ ALO T ABUAD A of obje cts in St ( D )( e ) , wher e Σ denotes the susp ension functor in St ( D )( e ) , satisﬁes c onditions A 1) and A 2) . Pr o of. Let X be an ob ject of St ( D )( e ). Supp ose tha t for each g in G and n in Z , we hav e Hom St ( D )( e ) (Σ n stab ( g ) , X ) = { ∗} . Then by the following isomorphis ms Hom St ( D )( e ) (Σ n stab ( g ) , X ) ≃ Hom St ( D )( e ) ( stab ( g ) , Ω n X ) ≃ Hom D ( e ) ( g , e v (0 , 0) Ω n X ) ≃ Hom D ( e ) ( g , e v ( n,n ) X ) , we conclude that, for all n in Z , we hav e ev ( n,n ) X = ∗ . By the conserv ativity a xiom, X is isomorphic to ∗ in St ( D )( e ). T his shows co ndition A 1). Now observe that condition A 2) follo ws fr om the following iso morphisms Hom St ( D )( e ) (Σ n stab ( g ) , L α ∈ K X α ) ≃ Hom St ( D )( e ) ( stab ( g ) , Ω n L α ∈ K X α ) ≃ Hom St ( D )( e ) ( stab ( g ) , L α ∈ K Ω n X α ) ≃ Hom D ( e ) ( g , e v (0 , 0) ` α ∈ K Ω n X α ) ≃ Hom D ( e ) ( g , ` α ∈ K ev (0 , 0) Ω n X α ) ≃ colim S ⊆ K S f inite Hom D ( e ) ( g , ` α ∈ S ev (0 , 0) Ω n X α ) ≃ colim S ⊆ K S f inite Hom D ( e ) ( g , e v (0 , 0) ` α ∈ S Ω n X α ) ≃ colim S ⊆ K S f inite Hom St ( D )( e ) ( stab ( g ) , L α ∈ S Ω n X α ) ≃ colim S ⊆ K S f inite Hom St ( D )( e ) (Σ n stab ( g ) , L α ∈ S X α ) √ Lemma 9.3. Le t T b e a triangulate d derivator and G a set of obje ct s in T ( e ) which satisﬁes c onditions A 1 ) and A 2) of lemma 9. 2. Then for every smal l c ate gory L and every p oint x : e → L in L , the set { x ! ( g ) | g ∈ G , x : e → L } satisﬁes c onditions A 1 ) and A 2) in t he c ate gory T ( L ) . Pr o of. Suppo se tha t Hom T ( L ) ( x ! ( g ) , M ) = {∗} , for every g ∈ G and every p oin t x in L . Then by adjunction x ∗ M is isomorphic to ∗ in T ( e ) and so b y the co nserv a tivit y axiom, M is isomo rphic to ∗ in T ( L ). This HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 29 shows condition A 1 ). Condition A 2) follows from the following is omorphisms Hom T ( L ) ( x ! ( g ) , L α ∈ K M α ) ≃ Hom T ( e ) ( g , x ∗ L α ∈ K M α ) ≃ Hom T ( e ) ( g , L α ∈ K x ∗ M α ) ≃ L α ∈ K Hom T ( e ) ( g , x ∗ M α ) ≃ L α ∈ K Hom T ( L ) ( x ! ( g ) , M α ) . √ R emark 9.4 . Notice that if D is a re gular p ointed strong deriv ator and we have at o ur dispo sal of a se t G of ob jects in D ( e ) which satisﬁes conditions A 1) a nd A 2), then le mma 8 .1 and lemma 9.3 imply that St ( D )( L ) is a co mpactly gener ated triangulated categ o ry , for ev ery small category L . Relation with Hov ey/Sc hw ede’s stabili zation. W e will now relate Heller’s construction with the constr uction of spe c tra as it is done by Hov ey in [17] a nd Sch wede in [3 6]. Let M b e a p ointed, simplicial, left pro per, cellular, almost ﬁnitely g enerated Quillen mo del category , se e deﬁnition 4 . 1 in [17], w her e sequential colimits commute with ﬁnite pro ducts and homotopy pullbac k s . This implies in particula r that the asso ciated deriv ator HO ( M ) will b e re gular. Example 9.5. Consider the c ate gory L Σ ,P F un ( M op f , S se t • ) deﬁne d in se ction 8. Notic e that t he c ate gory of p ointe d simplicial pr e-she aves F un ( M op f , S se t • ) is p ointe d, simplicia l, left pr op er, c el lular and even ﬁnitely gener ate d, se e deﬁnition 4 . 1 in [17] . Sinc e limits and c olimits in Fun ( M op f , S se t • ) ar e c alculate d obje ctwise, we c onclude that se quential c olimits c ommute with ﬁn ite pr o ducts. Now, by t he or em 4 . 1 . 1 in [15] the c ate gory L Σ ,P F un ( M op f , S se t • ) is also p ointe d, simplicial, left pr op er and c el lular. Now observe that the domains and c o domains of e ach morphism in Λ((Σ ∪ { P } ) + ) , se e deﬁnition 4 . 2 . 1 in [1 5] , ar e ﬁnitely pr esente d, sinc e t he for getful functor F un ( M op f , S se t • ) → Fun ( M op , S se t ) c ommutes with ﬁlter e d c olimits and homotopy pul lb acks. N ow, by pr op osition 4 . 2 . 4 in [15] , we c onclude t hat a morphism A f → B in L Σ ,P F un ( M op f , S se t • ) , with B a lo c al obje ct, is a lo c al ﬁbr ation if and only if it has t he right lifting pr op erty with r esp e ct to the set J ∪ Λ((Σ ∪ { P } ) + ) , wher e J denotes the set of gener ating acyclic c oﬁbr ations in Fun ( M op f , S se t • ) . This shows that L Σ ,P F un ( M op f , S se t • ) is almost ﬁnitely gener ate d. Recall from section 1 . 2 in [36] that since M is a p o in ted, simplicial mo del cate- gory , we have a Quillen adjunction M Σ( − )   M , Ω( − ) O O 30 GONC ¸ ALO T ABUAD A where Σ( X ) denotes the susp ension of an ob ject X , i.e. the pus hout of the diagr am X ⊗ ∂ ∆ 1 / /   X ⊗ ∆ 1 ∗ . Recall also that in [1 7] and [36] the a uthors co nstruct a stable Quillen mo del ca te- gory Sp N ( M ) o f sp ectra ass ocia ted with M and with the left Q uillen functor Σ ( − ). W e have the following Q uillen adjunction, see [1 7], M Σ ∞   Sp N ( M ) ev 0 O O and thus a mor phism to a stro ng tr iangulated deriv ator HO ( M ) L Σ ∞ − → HO ( Sp N ( M )) which commutes with homotopy co limits. By theo rem 9.1, we hav e at our disp osal a dia gram HO ( M ) stab   L Σ ∞ ' ' P P P P P P P P P P P P St ( HO ( M )) ϕ / / HO ( Sp N ( M )) , which is comm utative up to iso morphism in the 2-ca tegory of deriv ato rs. Now s upp ose als o tha t w e have a set G of s mall weak generator s in Ho ( M ), as in deﬁnitions 7 . 2 . 1 and 7 . 2 . 2 in [1 7]. Suppo se also that ea ch ob ject of G co nsidered in M is co ﬁbran t, ﬁnitely presented, ho motop y ﬁnitely presented and has a ﬁnitely presented cylinder ob ject. Example 9. 6 . Observe that the c ate gory F u n ( M op f , S se t • ) is p ointe d and ﬁn it ely gener ate d. By c or ol lary 7 . 4 . 4 in [16] , t he set G = { F X ∆[ n ] + /∂ ∆[ n ] + | X ∈ M f , n ≥ 0 } , is a set of s m al l we ak gener ators in Ho ( F un ( M op f , S se t • )) . Sinc e the domains and c o domains of the set (Σ ∪ { P } ) + ar e homotopic al ly ﬁ nitely pr esent e d obje cts, lemma 8.1 implies that G is a set of smal l we ak gener ators in Ho ( L Σ ,P F un ( M op f , S se t • )) . Cle arly the elements of G ar e c oﬁbr ant, ﬁn itely pr esent e d and have a ﬁnitely pr esent e d cylinder obje ct. They ar e also homotop ic al ly ﬁnitely pr esente d. Under the hypotheses a bov e o n the catego ry M , we have the following compa r - ison theor e m Theorem 9. 7. The induc e d morphism of triangulate d derivators ϕ : St ( HO ( M )) − → HO ( Sp N ( M )) is an e quivalenc e. HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 31 The pr oof of theorem 9.7 will consist in verifying the conditions of the following general pro position. Prop osition 9. 8 . L et F : T 1 → T 2 b e a morphism of str ong triangulate d deriva- tors. Supp ose t hat the t r iangulate d c ate gories T 1 ( e ) and T 2 ( e ) ar e c omp actly gener- ate d and that ther e is a set G ⊂ T 1 ( e ) of c omp act gener ators, which is stable under susp ensions and satisﬁes the fol lowi ng c onditions : a) F ( e ) induc es bije ctions Hom T 1 ( e ) ( g 1 , g 2 ) → Ho m T 2 ( e ) ( F g 1 , F g 2 ) , ∀ g 1 , g 2 ∈ G and b) t he set of obje cts { F g | g ∈ G } is a set of c omp act gener ators in T 2 ( e ) . Then the morphi sm F is an e quivalenc e of derivators. Pr o of. Conditions a ) and b ) imply that F ( e ) is an equiv alence o f triangulated cat- egories, see [32]. Now, let L b e a small categor y . W e show that conditions a ) and b ) a re also veriﬁed by F ( L ), T 1 ( L ) a nd T 2 ( L ). By lemma 9.3 the sets { x ! ( g ) | g ∈ G , x : e → L } and { x ! ( F g ) | g ∈ G , x : e → L } consist of compact generator s for T 1 ( L ), res p. T 2 ( L ), which a re stable under sus- pens ions. Since F commutes with homoto p y colimits F ( x ! ( g )) = x ! ( F g ) and so the following isomor phisms Hom T 1 ( L ) ( x ! ( g 1 ) , x ! ( g 2 )) ≃ Hom T 1 ( e ) ( g 1 , x ∗ x ! ( g 2 )) ≃ Hom T 2 ( e ) ( F ( g 1 ) , F ( x ∗ x ! ( g 2 ))) ≃ Hom T 2 ( e ) ( F g 1 , x ∗ F ( x ! ( g 2 ))) ≃ Hom T 2 ( L ) ( x ! F ( g 1 ) , x ! F ( g 2 )) imply the prop osition. √ Let us now prov e theor em 9.7: Pr o of. Let us ﬁrst prov e co ndition b ) of prop osition 9.8. Since the set G of small generator s in Ho ( M ) satisﬁes the conditions of lemma 9.2, we have a set { Σ n stab ( g ) | g ∈ G , n ∈ Z } of compact ge ner ators in St ( HO ( M ))( e ), which is stable under susp ensions. W e now show that the set { Σ n L Σ ∞ ( g ) | g ∈ G , n ∈ Z } is a set of compact gener a tors in Ho ( Sp N ( M )). These ob jects are compact b ecause the functor R ev 0 in the adjunction Ho ( M ) L Σ ∞   Ho ( Sp N ( M )) R ev 0 O O commutes with ﬁltered homoto p y co limits. W e now show tha t they form a set of generator s. Let Y b e an ob ject in Ho ( Sp N ( M )), that we ca n supp ose, without los s of genera lit y , to be an Ω-sp ectrum, se e [17]. Supp ose that Hom (Σ n L Σ ∞ ( g i ) , Y ) ≃ co lim m Hom ( g i , Ω m Y m + p ) = { ∗} , n ≥ 0 . 32 GONC ¸ ALO T ABUAD A Since Y is an Ω-sp ectrum, we hav e Y p = ∗ , ∀ p ≥ 0 . This implies that Y is isomorphic to ∗ in Ho ( Sp N ( M )). W e now show condition a ). L et g 1 and g 2 be ob jects in G . Observe that we have the following isomorphisms, see [14] Hom St ( HO ( M ))( e ) ( stab ( g 1 ) , sta b ( g 2 )) ≃ Hom Ho ( M ) ( g 1 , ( ev (0 , 0) ◦ l oc ◦ L [0 , 0])( g 2 )) ≃ Hom Ho ( M ) ( g 1 , ev (0 , 0) (ho colim ( L [0 , 0]( g 2 ) → Ω σ L [0 , 0]( g 2 ) → . . . ))) ≃ Hom Ho ( M ) ( g 1 , ho colim ev (0 , 0) ( L [0 , 0]( g 2 ) → Ω σ L [0 , 0 ]( g 2 ) → · · · )) ≃ colim j Hom Ho ( M ) ( g 1 , Ω j Σ j ( g 2 )) . Now, by co rollary 4 . 13 in [17], we have Hom Ho ( Sp N ( M )) ( L Σ ∞ ( g 1 ) , L Σ ∞ ( g 2 )) ≃ Hom Ho ( Sp N ( M )) (Σ ∞ ( g 1 ) , (Σ ∞ ( g 2 )) f ) ≃ colim m Hom Ho ( M ) ( g 1 , Ω m (Σ m ( g 2 )) f ) , where (Σ ∞ ( g 2 )) f denotes a lev elwise ﬁbrant resolution o f Σ ∞ ( g 2 ) in the category Sp N ( M ). Now, notice that since g 2 is coﬁbra n t, s o is Σ m ( g 2 ) and so we hav e the following isomorphism Ω m (Σ m ( g 2 )) f ∼ − → ( R Ω) m ◦ ( L Σ) m ( g 2 ) in Ho ( SP N ( M )). This implies that for j ≥ 0, w e have an isomorphism Ω j Σ j ( g 2 ) ∼ − → Ω j (Σ j ( g 2 )) f in Ho ( Sp N ( M )) and so Hom St ( HO ( M ))( e ) ( stab ( g 1 ) , sta b ( g 2 )) = Ho m Ho ( Sp N ( M )) ( L Σ ∞ ( g 1 ) , L Σ ∞ ( g 2 )) . Let now p be an integer. Notice that Hom St ( HO ( M ))( e ) ( stab ( g 1 ) , Σ p stab ( g 2 )) = co lim j Hom Ho ( M ) ( g 1 , Ω j Σ j + p ( g 2 )) and that Hom Ho ( Sp N ( M )) ( L Σ ∞ ( g 1 ) , Σ p L Σ ∞ ( g 2 )) = co lim m Hom Ho ( M ) ( g 1 , Ω m (Σ m + p ( g 2 )) f ) . This pr o ves condition a ) and so the theor em is prov en. √ R emark 9.9 . If we consider for M the catego ry L Σ ,P F un ( M op f , S se t • ), we hav e equiv alences o f deriv ato rs ϕ : St ( L Σ ,P Hot M f • ) ∼ − → HO ( Sp N ( L Σ ,P F un ( M op f , S se t • ))) ∼ ← St ( L Σ ,P Hot M f ) . Let D be a strong tr iangulated deriv ator . Now, by theorem 9.1 and prop osition 7.1, we hav e the following pro positio n Prop osition 9. 1 0. We hav e an e quivalenc e of c ate gories Hom ! ( St ( L Σ ,P Hot dgcat f ) , D ) ( stab ◦ Φ ◦ R h ) ∗ − → Hom f lt,p ( HO ( dgcat ) , D ) . Since the categor y S set • satisﬁes all the conditions of theorem 9.7, we hav e the following characterization of the classical categor y of spectr a, a fter Bousﬁeld- F riedlander [2], b y a universal pro perty . HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 33 Prop osition 9. 1 1. We hav e an e quivalenc e of c ate gories Hom ! ( HO ( Sp N ( S se t • )) , D ) ∼ − → D ( e ) . Pr o of. By theorems 9.7 and 4 .1, w e hav e the following equiv alences Hom ! ( HO ( Sp N ( S se t • )) , D ) ≃ Ho m ! ( HO ( S set • ) , D ) = Hom ! ( Hot • , D ) ≃ Hom ! ( Hot , D ) ≃ D ( e ) . This proves the pro positio n. √ R emark 9.1 2 . An analoguos caracteriz a tion of the category of spec tr a, but in the context of s table ∞ -categ ories is pr o ved in [27, 17 .6]. 10. DG quo tients Recall from [37] [3 8] that we hav e at our disp o sal a Morita Quille n mo del struc- ture on the categor y of sma ll dg catego ries dgca t , see example 6 .1. As s ho wn in [37] [38] the ho motop y categor y Ho ( dgcat ) is p ointed. In the following, we will be considering this Quillen mo del structure. W e denote by I its s e t of generating coﬁbrations. Notation 10.1 . W e denote b y E the set of inclusio ns of full dg sub categories G ֒ → H , where H is a strict ﬁnite I -cell. Recall that w e hav e a mo rphism o f deriv ator s U t := stab ◦ Φ ◦ R h : HO ( dgcat ) − → St ( L Σ ,P Hot dgcat f ) which commutes with ﬁltered ho mo top y co limits and preser v es the po in t. Let us now make so me ge neral a rgumen ts. Let D be a p oin ted deriv ator . W e denote by M the ca teg ory asso ciated to the graph 0 ← 1 . Consider the functor t = 1 : e → M . Since the functor t is an op en immers ion, see notation 3.8 a nd the der iv ato r D is p oint ed, the functor t ! : D ( e ) → D ( M ) has a left adjoint t ? : D ( M ) → D ( e ) , see [3]. W e denote it b y cone : D ( M ) → D ( e ) . Let F : D → D ′ be a morphism o f p oin ted deriv ator s . Notice that we hav e a natur a l transformatio n of functors . S : cone ◦ F ( M ) → F ( e ) ◦ con e . 34 GONC ¸ ALO T ABUAD A Prop osition 10. 2. L et A R ֒ → B b e an inclusion of a ful l dg sub c ate gory and R A   R / /   B , 0 the asso ciate d obje ct in HO ( dgcat )( ) , wher e 0 denotes the terminal obje ct in Ho ( dgc at ) . Then t her e ex ists a ﬁ lter e d c ate gory J and an obj e ct D R in HO ( dgcat )( × J ) , su ch that p ! ( D R ) ∼ − → R , wher e p : × J → denotes the pr oje ction fun ct or. Mor e over, for every p oint j : e → J in J t he obje ct (1 × j ) ∗ in HO ( dgcat )( ) is of the form 0 ← Y j L j → X j , wher e Y j L j → X j , b elongs to the set E . Pr o of. Apply the small ob ject a rgumen t to the morphism ∅ − → B using the set o f genera ting co ﬁbrations I a nd obtain the factorization ∅ / / ! ! i ! ! B B B B B B B B B Q ( B ) ∼ p > > > > } } } } } } } } } , where i is a n I -cell. Now consider the following ﬁb er pr oduct p − 1 ( A ) ∼       / / p Q ( B ) p ∼     A   J / / B . Notice that p − 1 ( A ) is a full dg subc a tegory of Q ( B ). Now, by prop osition 6.2, w e have an isomorphism colim j ∈ J X j ∼ − → Q ( B ) , where J is the ﬁlter ed category of inclusions of strict ﬁnite sub- I - c e lls X j int o Q ( B ). F or each j ∈ J , consider the ﬁber pro duct Y j     / / p X j   p − 1 ( A ) / /   / / Q ( B ) . In this way , we obtain a morphism o f diag rams { Y j } j ∈ J ֒ → { X j } j ∈ J , such that for each j in J , the inclusion Y j ֒ → X j HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 35 belo ngs to the set E and J is ﬁltered. Consider now the diagram D I { 0 ← Y j ֒ → X j } j ∈ J in the ca tegory Fun ( × J, dgcat ). Now, notice that we have the isomorphism colim j ∈ J { 0 ← Y j ֒ → X j } ∼ − → { 0 ← p − 1 ( A ) ֒ → Q ( B ) } in Fun ( , dgcat ) and the weak equiv a lence { 0 p − 1 ( A ) o o ∼     / / Q ( B ) } ∼   { 0 A o o   / / B } in Fun ( , dgcat ), when endow ed with the pro jective mo del structure, see [1 5]. Since F un ( , dgca t ) is clearly also compactly genera ted, we hav e an isomorphism ho colim j ∈ J (0 ← Y j → X j ) ∼ − → c o lim j ∈ J (0 ← Y j → X j ) . Finally , notice that D R is an ob ject of HO ( dgcat )( × J ) and tha t p ! ( D R ), where p : × J → J denotes the pro jection functor , identiﬁes with ho colim i ∈ J (0 ← Y j → X j ) . This proves the pro positio n. √ Notation 10.3 . W e denote by E st the set of morhisms S L , where L b elongs to the set E . Let D be a strong tr iangulated deriv ator . Theorem 10 . 4. If G : St ( L Σ ,P Hot dgcat f ) → D is a morphism of triangulate d derivators c ommuting with arbitr ary homotopy c olim- its and s uch that G ( e )( S L ) is invertible for e ach L in E , t hen G ( e )( S K ) is invertible for e ach inclus ion K : A ֒ → B of a ful l dg sub c ate gory. Pr o of. Let A K ֒ → B b e an inclusion o f a full dg sub category . Consider the morphism ϕ K := ϕ ( K ) : ( i ! ◦ U T )( K ) − → ( U T ◦ i ! )( K ) in St ( L Σ ,P Hot dgcat f )(  ). Let D K be the ob ject of HO ( dgcat )( × J ) co nstructed in propos ition 1 0.2. In particular p ′ ! ( D K ) ∼ → K , wher e p ′ : × J → deno tes the pro jection functor. The inclusio n i : ֒ →  , induces a comm utative squar e HO ( dgcat )(  × J ) ( i × 1) ∗   U T (  × J ) / / St ( L Σ ,P Hot dgcat f )(  × J ) ( i × 1) ∗   HO ( dgcat )( × J ) U T (  × J ) / / St ( L Σ ,P Hot dgcat f )( × J ) and a mo rphism Ψ : (( i × 1) ! ◦ U T ( × J ))( D K ) − → ( U T (  × J ) ◦ ( i × 1) ! )( D K ) . 36 GONC ¸ ALO T ABUAD A W e will now show that p ! Ψ ∼ − → ϕ K , where p :  × J →  , deno tes the pro jection functor . The fact that we hav e the follo wing comm utative square   × J p o o i O O × J i × 1 O O p ′ o o and that the morphis m of deriv a to rs U T commutes with ﬁlter e d homotopy co limits implies the follo wing equiv a lences p ! Ψ = p ! ◦ ( i × 1) ! ◦ U T ( × j )( D K ) → p ! ◦ U T (  × J ) ◦ ( i × 1) ! ( D k ) ≃ i ! ◦ p ′ ! ◦ U T ( × J )( D k ) → U T (  × J ) ◦ p ! ◦ ( i × 1 ) ! ( D K ) ≃ i ! ◦ U T ( ) ◦ p ′ ! ( D K ) → U T (  ) ◦ i ! ◦ p ′ ! ( D K ) ≃ ( i ! ◦ U T ( ))( K ) → ( U T (  ) ◦ i ! )( K ) = ϕ J This shows that p ! (Ψ) ∼ − → ϕ K . W e now show that Ψ is an isomorphis m. F or this, by conserv ativity , it is e no ugh to show that for ev ery ob ject j : e → J in J , the morphism (1 × j ) ∗ (Ψ) , is an isomorphism in St ( L Σ ,P Hot dgcat f )(  ). Recall from propo sition 10.2 tha t (1 × j ) ∗ ( D K ) identiﬁes with { 0 ← Y j L j ֒ → X j } , where L j belo ngs to E . W e now show that (1 × j ) ∗ (Ψ) identiﬁes with ϕ L j , which by hypotheses, is a n isomor phism. Now, the follo wing comm utative diagr am  1 × j / /  × J i O O 1 × j / / × J i × i O O and the dual o f prop osition 2 . 8 in [4] imply that we hav e the following equiv alences (1 × j ) ∗ Ψ = ((1 × j ) ∗ ◦ ( i × 1) ! ◦ U T ( × J ))( D K ) → ((1 × j ) ∗ ◦ U T (  × J ) ◦ ( i × 1) ! )( D K ) ≃ ( i ! ◦ (1 × j ) ∗ ◦ U T ( × J ))( D K ) → ( U T (  × J ) ◦ (1 × j ) ∗ ◦ ( i × 1) ! )( D K ) ≃ ( i ! ◦ U T ( ) ◦ (1 × j ) ∗ )( D K ) → ( U T (  ) ◦ i ! ◦ (1 × j ) ∗ )( D K ) ≃ i ! ◦ U T ( )( L j ) → U T (  ) ◦ i ! ( L j ) = ϕ L j Since b y h yp o theses ϕ L j is an isomor phism and the morphism G comm utes with homotopy colimits the theorem is pr o ven. √ HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 37 11. The universal lo cal izing inv ariant Recall from theorem 9.7 and remark 9.9 that if w e c o nsider for the category M the ca tegory L Σ ,P F un ( dgcat op f , S se t • ), see example 9.6, we hav e a n equiv alence of triangulated deriv ato r s ϕ : St ( L Σ ,P Hot dgcat f ) ∼ − → HO ( Sp N ( L Σ ,P F un ( dgcat op f , S se t • ))) . Now, stabilize the set E st deﬁned in the previous section under the functor lo op space and choose for each element o f this stabilized s et a r epresentativ e in the category Sp N ( L Σ ,P F un ( dgcat op f , S se t • )). W e denote the set of these re pr esen tatives by f E st . Since Sp N ( L Σ ,P F un ( dgcat op f , S se t • )) is a left prop er, cellular Q uillen mo del category , see [17], its le ft Bo usﬁeld lo caliza tion by f E st exists. W e denote it by L f E st Sp N ( L Σ ,P F un ( dgcat op f , S se t • )). By lemma 5.3 it is a stable Quillen mo del cate- gory . R emark 11.1 . Since the lo calization morphism γ : St ( L Σ ,P Hot dgcat f ) L I d − → H O ( L f E st Sp N ( L Σ ,P F un ( dgcat op f , S se t • ))) commutes with homotopy co limits and inverts the set of mor phis ms E st , theo- rem 10.4 allows us to co nclude that it in verts all morphisms S K for each inclusio n A ֒ → B of a full dg subca teg ory . Deﬁnition 11. 2. - The Lo calizing Motiv a tor of dg c ate gories M loc dg is the triangulate d derivator asso ciate d with the stable Quil len mo del c ate gory L f E st Sp N ( L Σ ,P F un ( dgcat op f , S se t • )) . - The Universal lo calizing inv ar ian t of dg c ate gories is t he c anonic al mor- phism of derivators U l : HO ( dgcat ) → M loc dg . W e sum up the construction of M loc dg in the following diag r am dgcat f [ S − 1 ] / / Ho ( h )   HO ( dgcat ) R h w w n n n n n n n n n n n n n U l | | L Σ Hot dgcat f Φ   L Re 7 7 n n n n n n n n n n n n n L Σ ,P Hot dgcat f stab   St ( L Σ ,P Hot dgcat f ) γ   M loc dg Observe that the mor phism of der iv ator s U l is p oin ted, co mm utes with ﬁltered homotopy colimits and satisﬁes the following condition: 38 GONC ¸ ALO T ABUAD A Dr) F or every inclus ion A K ֒ → B of a full dg sub categor y the canonic a l morphism S K : co ne ( U l ( A K ֒ → B )) → U l ( B / A ) is invertible in M loc dg ( e ). W e now give a conceptual characterization of condition D r ). Let us now deno te by I be the categ ory ass o ciated with the gr aph 0 ← 1. Lemma 11.3. The isomorphism classes in HO ( dgcat )( I ) asso ciate d with t he inclu- sions A K ֒ → B of ful l dg s u b c ate gories c oincide with t he classe of homotopy monomor- phims in dgcat , se e se ction 2 in [42] . Pr o of. Recall from section 2 in [42] that in a mo del categor y M a morphis m X f → Y is a homotopy monomorphism if for ev e ry ob ject Z in M , the induced morphism of simplicial sets Map ( Z, X ) f ∗ → Map ( Z, Y ) induces an injection on π 0 and isomo rphisms on a ll π i for i > 0 (for a ll ba se p oints). Now, by lemma 2 . 4 of [42] a dg functor A F → B is an homotopy monomorphis m on the quasi-equiv a len t Quillen mo del ca tegory in d gcat if and o nly if it is quasi- ful ly faithful , i.e. for any tw o ob jects X and Y in A the morphism o f complexes Hom A ( X, Y ) → Hom B ( F X , F Y ) is a qua si-isomorphism. Recall that by corolla ry 5 . 10 of [38] the mapping s pace functor Map ( A , B ) in the Morita Quillen mo del catego ry iden tiﬁes with the mapping space Map ( A , B f ) in the q ua si-equiv alent Quillen mo del category , where B f denotes a Morita ﬁbrant resolution of B . This implies that a dg functor A F → B is a homotopy monomor phism if and only if A f F f → B f is a quasi-fully faithful dg functor. Now, notice that an inclusion A ֒ → B of a full dg subc ategory is a homotopy monomorphism. Co n versely , let A F → B b e a homotop y monomorphism. C o nsider the diag ram f A f   / / B f A f π O O F f / / B f A ∼ O O F / / B , ∼ O O where f A f denotes the full dg s ubcatego ry o f B f whose ob jects a re those in the image by the dg functor F f . Since F f is a q ua si-fully faithful dg functor , the dg functor π is a quas i- equiv alenc e . This prov es the lemma. √ R emark 11.4 . Lemma 11 .3 shows that condition D r ) is equiv alent to Dr’) F or every homo to p y monomorphism A F → B in H O ( dgcat )( I ) the c anonical morphism cone ( U l ( A F → B )) → U l ( cone ( F )) is invertible in M loc dg ( e ). HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 39 Let D be a strong tr iangulated deriv ator . Theorem 11 . 5. The morphism U l induc es an e quivalenc e of c ate gories Hom ! ( M loc dg , D ) U ∗ l − → H om f lt, D r, p ( HO ( dgcat ) , D ) , wher e Hom f lt, D r ,p ( HO ( dgcat ) , D ) denotes t he c ate gory of morphisms of derivators which c ommute with ﬁlter e d homotopy c olimits, satisfy c ondition D r) and pr eserve the p oint. Pr o of. By theorem 5.4, we have the follo wing eq uiv alence of catego r ies Hom ! ( M loc dg D ) γ ∗ − → H om ! , f E st ( St ( L Σ ,P F un ( dgcat op f , S se t • )) , D ) . W e now show that w e have the following equiv alence o f categories Hom ! , f E st ( St ( L Σ ,P F un ( dgcat op f , S se t • )) , D ) ∼ → Hom ! , E st ( St ( L Σ ,P F un ( dgcat op f , S se t • )) , D ) . Let G be an element of Ho m ! , E st ( St ( L Σ ,P F un ( dgcat op f , S se t • )) , D ) and s an element of E st . W e show that the ima ge of s under the functor G ( e ) ◦ Ω( e ) is a n isomor phism in D ( e ). Reca ll from the pro of o f lemma 5.3 that the functor G ( e ) commutes with Σ( e ). Since the susp ension and lo op space functor s in D ( e ) a r e inv e rse of each other we co nclude that the image of s under the functor G ( e ) ◦ Ω( e ) is a n isomorphism in D ( e ). Now, simply observe that the catego ry o n the right hand side o f the ab ov e equiv alence identiﬁes with Hom f lt, D r ,p ( HO ( dgcat ) , D ) under the eq uiv a le nce Hom ! ( St ( L Σ ,P Hot dgcat f ) , D ) ( stab ◦ Φ ◦ R h ) ∗ − → Hom f lt,p ( HO ( dgcat ) , D ) , of prop osition 9 .10. This pr o ves the theorem. √ Notation 1 1.6 . W e call an ob ject of the right hand side category o f theorem 11.5 a lo c alizing invariant o f dg categories. W e now present so me ex amples. Ho c hschild and cyclic homology. Let A b e a small k -ﬂat k -categ ory . The Ho chschild chai n c omplex of A is the c omplex concentrated in ho mological deg r ees p ≥ 0 whose p th co mponent is the sum of the A ( X p , X 0 ) ⊗ A ( X p , X p − 1 ) ⊗ A ( X p − 1 , X p − 2 ) ⊗ · · · ⊗ A ( X 0 , X 1 ) , where X 0 , . . . , X p range throug h the ob jects of A , endowed with the diﬀerential d ( f p ⊗ . . . ⊗ f 0 ) = f p − 1 ⊗ · · · ⊗ f 0 f p + p X i =1 ( − 1) i f p ⊗ · · · ⊗ f i f i − 1 ⊗ · · · ⊗ f 0 . Via the cyclic p ermutations t p ( f p − 1 ⊗ · · · ⊗ f 0 ) = ( − 1) p f 0 ⊗ f p − 1 ⊗ · · · ⊗ f 1 this complex b ecomes a precyc lic chain complex and th us gives rise to a mixe d c omplex C ( A ), i.e. a dg mo dule o ver the dg alg ebra Λ = k [ B ] / ( B 2 ), wher e B is of degree − 1 and dB = 0 . All v a rian ts of cyclic homolo gy only depend on C ( A ) considered in D (Λ). F or ex ample, the cyclic homo logy of A is the homolog y of the complex C ( A ) L ⊗ Λ k , cf. [1 8]. If A is a k -ﬂat diﬀerential graded category , its mixed complex is the sum-total complex o f the bicomplex obtained as the natural re - in terpr etation of the ab o ve 40 GONC ¸ ALO T ABUAD A complex. If A is a n a rbitrary dg k -categor y , its Ho chsc hild chain co mplex is deﬁned as the o ne of a k -ﬂat (e.g. a coﬁbrant) r esolution of A . The following theor em is prov ed in [22]. Theorem 11 . 7. The map A 7→ C ( A ) yields a morphism of derivators HO ( dgcat ) → HO (Λ − M od ) , which c ommutes with ﬁ lter e d homotopy c olimits, pr eserves the p oint and satisﬁes c ondition Dr). R emark 11 .8 . By theor em 11 .5 the mor phism o f der iv ator s C factors throug h U l and so giv es r ise to a morphism of deriv a tors C : M loc dg → HO (Λ − M od ) . Non-connectiv e K -theory. Le t A b e a sma ll dg catego ry . Its non-connective K -theory spectr um K ( A ) is deﬁned by applying Schlic hting’s co nstruction [35] to the F ro benius pair ass ocia ted with the category of coﬁbrant p erfect A -mo dules (to the empty dg categor y we as socia te 0). Reca ll that the conﬂations in the F ro benius category of coﬁbrant perfect A -modules a re the short exact sequences which split in the ca tegory of graded A -mo dules. Theorem 11 . 9. The map A 7→ K ( A ) yields a morphism of derivators HO ( dgcat ) → HO ( S p t ) , to the derivator asso ciate d with the c ate gory of sp e ctr a, which c ommutes with ﬁlter e d homotopy c olimits, pr eserves the p oint and satisﬁes c ondition D r). Pr o of. Prop osition 11 . 15 in [3 5], which is an adaptio n of theorem 1 . 9 . 8 in [43], implies that w e ha ve a well deﬁned mor phism of deriv a to rs HO ( dgcat ) → HO ( S p t ) . Lemma 6 . 3 in [35] implies that this mo rphism commutes with ﬁltered homotopy colimits and theorem 11 . 1 0 in [3 5] implies that condition Dr) is satisﬁed. √ R emark 1 1.10 . By theor em 11.5, the mo rphism of deriv ato r s K factors thr ough U l and so giv es r ise to a morphism of deriv a tors K : M loc dg → HO ( S pt ) . W e now establish a connection b et ween W aldhausen’s S • -construction, s e e [44] and the susp ension functor in the triangulated categor y M loc dg ( e ). Let A b e a Morita ﬁbrant dg c a tegory , see [37] [38]. Notice that Z 0 ( A ) carries a natural exa ct category structure obtained b y pulling back the gar ded-split structure on C dg ( A ) along the Y oneda functor h : Z 0 ( A ) − → C dg ( A ) A 7→ Hom • (? , A ) . Notation 1 1.11 . Remark that the simplicial categ ory S • A , obtained by applying W aldhausen’s S • -construction to Z 0 ( A ), admits a natura l enric hissement ov er the complexes. W e denote by S • A this simplicial Morita ﬁbra n t dg catego ry obtained. Recall that ∆ denotes the simplicia l categ ory and p : ∆ → e the pro jection functor. HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 41 Prop osition 11. 12. Ther e is a c anonic al isomorphism in M loc dg ( e ) p ! U l ( S • A ) ∼ → U l ( A )[1] . Pr o of. As in [2 8, 3.3], we consider the sequence in HO ( dgcat )(∆) 0 → A • → P S • A → S • A → 0 , where A • denotes the constant simplicial dg category with v alue A and P S • A the path ob ject of S • A . F or each p oint n : e → ∆, the n th comp o nen t of the ab ov e sequence is the following s hort exact se q uence in Ho ( dgcat ) 0 → A I → P S n A = S n +1 A Q → S n A → 0 , where I ma ps A ∈ A to the constant sequence 0 → A I d → A I d → · · · I d → A and Q ma ps a se q uence 0 → A 0 → A 1 → · · · → A n to A 1 / A 0 → · · · → A n / A 0 . Since the mo r phism of deriv ato rs U l satisﬁes condition Dr), the conser v ativity axiom implies that w e obtain a tria ngle U l ( A • ) → U l ( P S • ) → U l ( S • ) → U l ( A • )[1] in M loc dg (∆). B y applying the functor p ! , we obtain the following triang le p ! U l ( A • ) → p ! U l ( P S • A ) → p ! U l ( S • A ) → p ! U l ( A • )[1] in M loc dg ( e ). W e no w sho w that w e have natural isomorphisms p ! U l ( A • ) ∼ → U l ( A ) and p ! U l ( P S • A ) ∼ → 0 , in M loc dg ( e ), where 0 denotes the z ero ob ject in the triangulated ca tegory M loc dg ( e ). This clear ly implies the pr opositio n. Since the morphisms of deriv ators Φ, stab and γ co mm ute with homotopy colimits it is enough to show that we hav e isomor phisms p ! R h ( A • ) ∼ → R h ( A ) and p ! R h ( P S • A ) ∼ → ∗ in Hot dgcat f ( e ), where ∗ denotes the termina l ob ject in Hot dgcat f ( e ). Notice that since A and P S n A , n ≥ 0 ar e Mo rita ﬁbrant dg catego ries, we ha ve natura l isomorphisms h ( A • ) ∼ → R h ( A • ) and h ( P S • A ) ∼ → R h ( P S • A ) in Hot dgcat f (∆). Now, since homo top y colimits in F un ( dg cat op f , S se t ) are calculated ob jectwise a nd since h ( A • ) is a co ns tan t simplicial ob ject in Fun ( dgcat op f , S se t ), co rollary 18 . 7 . 7 in [15] implies that we hav e an iso morphism p ! R h ( A • ) ∼ → R h ( A ) 42 GONC ¸ ALO T ABUAD A in M loc dg ( e ). Notice also that since P S • A is a contractible simplicia l ob ject, see [28], so is h ( P S • A ). Since ho motop y colimits in Fun ( dgcat op f , S se t ) a re calculated ob ject wise, we hav e a n isomorphis m p ! R h ( P S • A ) ∼ → ∗ in Hot dgcat f ( e ). This pr o ves the prop o sition. √ 12. A Quillen model in terms of preshea ves of spectra In this s ection, w e c o nstruct another Quillen mo del ca tegory whose asso ciated deriv ator is the lo calizing motiv ator of dg categor ies M loc dg . Consider the Quillen adjunction F un ( dgcat op f , S se t • ) Σ ∞   Sp N ( F u n ( dgcat op f , S se t • )) . ev 0 O O Recall from sectio n 8 that we have a set of morphis ms (Σ ∪ { P } ) + in the ca tegory F un ( dgcat op f , S se t • ). No w s ta bilize the image o f this set by the der ived functor L Σ ∞ , under the functor lo op s pace in H o ( Sp N ( F u n ( dgcat op f , S se t • ))). F or each one of the morphims th us obtained, c ho ose a representativ e in the mo del ca tegory Sp N ( F u n ( dgcat op f , S se t • )). Notation 1 2.1 . Let us denote this set by G a nd by L G Sp N ( F u n ( dgcat op f , S se t • )) the asso ciated left Bo us ﬁeld lo caliza tio n. Prop osition 12. 2. We have an e quivalenc e of triangulate d str ong deriv ators HO ( Sp N ( L Σ ,P F un ( M op f , S se t • ))) ∼ − → HO ( L G Sp N ( F u n ( M op f , S se t • ))) . Pr o of. Observe that theorems 5.4 and 9.7 imply that b oth der iv ator s ha ve the same universal pro perty . This prov es the pro position. √ R emark 12.3 . Notice that the stable Quillen mo del categ ory Sp N ( F u n ( M op f , S se t • )) ident iﬁes with F un ( M op f , Sp N ( S se t • )) endow ed with the pro jectiv e model structure. The a b ov e co nsiderations imply the following prop osition. Prop osition 12. 4. We have an e quivalenc e of derivators HO ( L f E st ,G F un ( dgcat op f , Sp N ( S se t • ))) ∼ − → M loc dg . HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 43 13. Upper triangular DG ca tegories In this section we study upper tr ia ngular dg categ o ries using the formalism of Quillen’s homotopica l alg e br a. In the nex t section, we will relate this imp ortant class of dg categorie s with split shor t ex a ct s e q uences in Ho ( dgcat ). Deﬁnition 13.1. An upp er triang ula r dg c ate gory B is given by an upp er triangular matrix B :=  A X 0 C  , wher e A and C ar e smal l dg c ate gories and X is a A - C - bimo dule. A morphism F : B → B ′ of upp er t riangular dg c ate gories is given by a triple F := ( F A , F C , F X ) , wher e F A , r esp. F C , is a dg functor fr om A to A ′ , r esp. fr om C to C ′ , and F X is a morphism of A - C - bimo dules fr om X to X ′ (we c onsider X ′ endowe d with the action induc e d by F A and F C ). The c omp osition is t he natur al one. Notation 13.2 . W e denote by dgcat tr the ca tegory of upp er triang ular dg categor ies. Let B ∈ dg cat tr . Deﬁnition 13.3. L et |B | b e the to talization of B , i.e. the smal l dg c ate gory whose set of obje cts is the disjoint union of the set of obje cts of A and C and whose morphisms ar e given by Hom |B | ( x, x ′ ) :=        Hom A ( x, x ′ ) if x, x ′ ∈ A Hom C ( x, x ′ ) if x, x ′ ∈ C X ( x, x ′ ) if x ∈ A , x ′ ∈ C 0 if x ∈ C , x ′ ∈ A . W e have the follo wing a djunction dgcat tr |−|   dgcat , I O O where I ( B ′ ) :=  B ′ Hom B ′ ( − , − ) 0 B ′  . Lemma 13. 4. The c ate gory dgcat tr is c omplete and c o c omplete. Pr o of. Let { B j } j ∈ J be a diag ram in dgcat tr . Obs e rv e that the upp er triangular dg category   colim A j j ∈ J colim j ∈ J |B j | ( − , − ) 0 colim j ∈ J C j   , where colim j ∈ J |B j | ( − , − ) is the colim j ∈ J A j -colim j ∈ J C j -bimo dule na turally a ssoc ia ted with the dg category colim j ∈ J |B j | , c orresp onds to colim j ∈ J B j . Observe also that the upp er triangular dg category   lim j ∈ J A j lim j ∈ J X j 0 lim j ∈ J C j   , 44 GONC ¸ ALO T ABUAD A corres p onds to lim j ∈ J B j . This pro ves the lemma. √ Notation 13.5 . Let p 1 ( B ) := A and p 2 ( B ) := C . W e have at our dispo sal the following adjunction dgcat tr p 1 × p 2   dgcat × dgcat , E O O where E ( B ′ , B ′′ ) :=  B ′ 0 0 B ′′  . Recall from [3 7][38][39] that dgcat admits a structure o f coﬁbra n tly generated Quillen mo del category whose w ea k equiv alences a re the Morita dg functors. This structure clearly induces a co mponent wise mo del structur e o n dg cat × dgcat which is also coﬁbrantly generated. Prop osition 13.6. The c ate gory dg cat tr admits a structu r e of c oﬁbr antly gener ate d Quil len mo del c ate gory whose we ak e quivalenc es, r esp. ﬁbr ation, ar e the morphisms F : B → B ′ such that ( p 1 × p 2 )( F ) ar e quasi-e quivalenc es, r esp. ﬁbr ations, in dgcat × dgcat . Pr o of. W e show that the previous adjunction ( E , p 1 × p 2 ) veriﬁes conditions (1) and (2) of theo rem 11 . 3 . 2 fro m [15]. (1) Since the functor E is als o a right adjoint to p 1 × p 2 , the functor p 1 × p 2 commutes with colimits and so condition (1) is v er iﬁed. (2) Let J , r esp. J × J , b e the set of genera ting trivia l coﬁbr a tions in dg cat , resp. in dgcat × dgcat . Since the functor p 1 × p 2 commutes with ﬁltered colimits it is enoug h to prove the fo llowing: let G : B ′ → B ′′ be an element of the set E ( J × J ), B an ob ject in dgcat tr and B ′ → B a mo r phism in dgcat tr . Cons ider the following push-out in dgcat tr : B ′ / / G   y B G ∗   B ′′ / / B ′′ ` B ′ B . W e now prov e that ( p 1 × p 2 )( G ∗ ) is a weak-equiv alence in dgcat × dgca t . Observe that the ima g e of the pr e v ious push-o ut under the functors p 1 and p 2 corres p ond to the following tw o push-outs in dg cat : A ′ / /   G A ′ ∼   y A G A ′ ∗   C ′ / /   G C ′ ∼   y C G C ′ ∗   A ′′ / / A ′′ ` A ′ A C ′′ / / C ′′ ` C ′ C . Since G A ′ ∗ and G C ′ ∗ belo ng to J the morphism ( p 1 × p 2 )( G ∗ ) = ( G A ′ ∗ , G C ′ ∗ ) HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 45 is a w ea k-equiv alence in dg cat × dgcat . This prov e s co ndition (2 ). The prop osition is then prov en. √ Let B , B ′ ∈ dg cat tr . Deﬁnition 13.7. A morphism F : B → B ′ is a total Morita dg functor if F A and F C ar e Morita dg functors, se e [3 7][38 ][39] , and F X is a quasi-isomorphism of A - C -bimo dules. R emark 1 3.8 . Notice that if F is a to tal Morita dg functor then | F | is a Morita dg functor in dgcat but the conv erse is not true. Theorem 13.9 . The c ate gory dgcat tr admits a structur e of c oﬁbr antly gener ate d Quil len mo del c ate gory whose we ak e quivalenc es W ar e the total Morita dg functors and whose ﬁ br ations ar e the morphisms F : B → B ′ such that F A and F C ar e Morita ﬁbr at ions, se e [37] [38] , and F X is a c omp onentwise surje ct ive morphism of bimo dules. Pr o of. The pr oof is ba sed on enlarging the set E ( I × I ), resp. E ( J × J ), of genera ting coﬁbrations, resp. gener a ting trivial coﬁbra tions, of the Quillen model structure of prop osition 13.6. Let ˜ I b e the set of morphisms in d gcat tr  k S n − 1 0 k  ֒ →  k D n 0 k  , n ∈ Z , where S n − 1 is the complex k [ n − 1] and D n the ma pping cone on the identit y of S n − 1 . The k - k -bimodule S n − 1 is sent to D n by the iden tity on k in degre e n − 1. Consider a lso the set ˜ J of morphis ms in dgcat tr  k 0 0 k  ֒ →  k D n 0 k  , n ∈ Z . Observe that a morphism F : B → B ′ in dgcat tr has the right lifting pro perty (=R.L.P .) with resp ect to the set ˜ J , resp. ˜ I , if and only if F X is a compo nen twise surjective mor phism, r esp. surjective qua s i-isomorphism, o f A - C - bimodules. Deﬁne I := E ( I × I ) ∪ ˜ I as the se t of gener ating c oﬁbr ations in dg cat tr and J := E ( J × J ) ∪ ˜ J as the set of gener ating t rivia l c oﬁbr ations . W e no w prov e that conditions (1)-(6) of theorem 2 . 1 . 19 from [1 6] ar e sa tisﬁed. This is clea rly the ca se for conditions (1)-(3). (4) W e now prov e that J - cel l ⊂ W , se e [15]. Since by prop osition 1 3.6 we hav e E ( J × J )- cel l ⊂ W ′ , wher e W ′ denotes the weak equiv alences of prop osition 13 .6, it is enough to prove that pushouts with r espect to a n y morphism in ˜ J belong to W . Let n be a n in teger and B an ob ject in d gcat tr . Consider the following push-out in dgcat tr :  k 0 0 k  T / /  _   y B R    k D n 0 k  / / B ′ . 46 GONC ¸ ALO T ABUAD A Notice that the morphism T corres p onds to sp ecifying a n ob ject A in A a nd an ob ject C in C . The upp er triangular dg catego ry B ′ is then obtained from B by g luing a new morphism o f degree n from A to C . O bserve that R A and R C are the identit y dg functors and that R X is a quas i- isomorphism of bimo dules. This shows that R belo ngs to W a nd s o condition (4) is prov ed. (5-6) W e now show that R.L.P .( I )= R.L.P .( J ) ∩ W . The pro of o f pr o positio n 13.6 implies that R.L.P .( E ( I × I ))=R.L.P .( E ( J × J )) ∩ W ′ . Let F : B → B ′ be a morphism in R.L.P .( ˜ I ). Clear ly F b elongs to R.L.P .( ˜ J ) and F X is a quasi-isomo rphism o f bimo dules. T his shows that R.L.P .( I ) ⊂ R.L.P .( J ) ∩ W . Let now F : B → B ′ be a morphism in R.L.P .( ˜ J ) ∩ W . Clearly F belo ngs to R.L.P .( ˜ I ) and so R.L.P .( J ) ∩ W ⊂ R.L.P( I ). This pr o ves conditions (5) and (6). This proves the theor em. √ R emark 13.1 0 . Notice that the Quillen mo del structure of theor em 1 3.9 is ce llula r, see [15] and that the domains and codo mains o f I (the set of g e nerating coﬁbr a- tions) ar e coﬁbra n t, ℵ 0 -compact, ℵ 0 -small and ho motopically ﬁnitely prese n ted, s ee deﬁnition 2 . 1 . 1 . from [41]. This implies that w e ar e in the conditions of pro posi- tion 6.2 and so a n y ob ject B in dg cat tr is weakly equiv alent to a ﬁlter ed colimit of strict ﬁnite I -cell ob jects. Prop osition 13. 11. If B b e a strict ﬁn ite I -c el l obje ct in dgcat tr , then p 1 ( B ) , p 2 ( B ) and |B| ar e st rict ﬁnite I - c el l obje cts in d gcat . Pr o of. W e cons ider the following inductive ar gumen t: - notice that the initial ob ject in dg cat tr is  ∅ 0 0 ∅  and it is sen t to ∅ (the initial ob ject in dg cat ) by the functors p 1 , p 2 and | − | . - supp ose that B is an upp er triangular dg catego ry such that p 1 ( B ), p 2 ( B ) and |B | are strict ﬁnite I -cell ob jects in dg cat . Let G : B ′ → B ′′ be an element of the set I in dg cat tr , see the pro of of theorem 13.9, and B ′ → B a mo rphism. Consider the following push-out in dgcat tr : B ′ / / G   y B G ∗   B ′′ / / PO . W e now prove that p 1 ( PO ), p 2 ( PO ) and | PO | are strict ﬁnite I - cell o b jects in dg cat . W e consider the following tw o cases : 1) G belo ngs to E ( I × I ): observe that p 1 ( PO ), p 2 ( PO ) and | PO | corre- sp ond ex actly to the following push-outs in dgcat : A ′ / /   G A ′ ∼   y A   C ′ / /   G C ′ ∼   y C   A ′ ` C ′ / / y G A ′ ∐ G C ′   |B |   A ′′ / / p 1 ( PO ) C ′′ / / p 2 ( PO ) A ′′ ` C ′′ / / | PO | . HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 47 Since G A ′ and G C ′ belo ng to I this ca se is prov e d. 2) G belo ngs to ˜ I : obse r v e that p 1 ( PO ) ident iﬁes with A , p 2 ( PO ) identiﬁes with C a nd | PO | corresp onds to the follo wing push-out in dgcat : C ( n ) / / S ( n )   y |B |   P ( n ) / / | PO | , where S ( n ) is a g enerating c o ﬁbration in dgcat , see section 2 in [39]. This prov es this case. The prop osition is prov en. √ 14. Split shor t exact sequences In this section, we esta blish the c onnexion be tw een split short exa ct sequences of dg ca tegories and upp er tr iangular dg categ ories. Deﬁnition 14. 1. A split short exact se quenc e of dg c ate gories is a short exact se quenc e of dg c ate gories, se e [19] , which is Morita e quivalent to one of the form 0 / / A i A / / B R o o P / / C i C o o / / 0 , wher e we have P ◦ i A = 0 , R is a dg functor righ t adjoint to i A , i C is a dg functor right adjoint to P and we have P ◦ i C = I d C and R ◦ i A = I d A via the adjunction morphisms. T o a split sho rt e x act sequence, we can naturally as s ocia te the upp er triangula r dg catego ry B :=  A Hom B ( i C ( − ) , i A ( − )) 0 C  . Conv ers e ly to an upp er tria ngular dg catego ry B such that C a dmits a zer o ob ject (for instance if C is Mor ita ﬁbr an t), we can asso ciate a split shor t exa ct seq uence 0 / / A i A / / |B | R o o P / / C i C o o / / 0 , where P and R are the pro jection dg functors. Mor eo ver, this cons truction is functorial in B and sends total Mor ita equiv alences to Morita e quiv alent split s hort exact seque nce s. Notice also that by lemma 13.4 this functor pr eserves colimits. Prop osition 14. 2. Every split short ex act se quenc e of dg c ate gories is we akly e quiv- alent t o a ﬁlter e d homotopy c olimit of split short exact se quenc es whose c omp onent s ar e strict ﬁnite I -c el l obje cts in dgcat . Pr o of. Let 0 / / A i A / / B R o o P / / C i C o o / / 0 , be a split short ex a ct sequence of dg ca tegories. W e can sup ose that A , B and C are Mor ita ﬁbrant dg categ ories, see [37] [38]. Co nsider the upper tr iangular dg category B :=  A Hom B ( i C ( − ) , i A ( − )) 0 C  . 48 GONC ¸ ALO T ABUAD A Now by rema rk 13 .10, B is equiv a le nt to a ﬁltered colimit of strict ﬁnite I -cell ob jects in dgcat tr . Consider the imag e of this diag ram by the functor, desc ribed ab o ve, which sends an upp er tria ngular dg categor y to a split short exact sequence. By prop osition 13 .11 the comp onents of each split short ex a ct seq uence o f this diagram are strict ﬁnite I -cell o b jects in dg cat . Since the categ ory d gcat sa tisﬁes the conditions of prop o sition 6.2, ﬁltered homoto p y co limits are equiv alent to ﬁlter ed colimits and so the prop osition is pr o ven. √ 15. Quasi-Additivity Recall from section 11 that w e hav e a t our dispos a l the Quillen model category L Σ ,P F un ( dgcat op f , S se t ) which is homotopic al ly p ointe d , i.e. the morphism ∅ → ∗ , from the initial ob ject ∅ to the ter minal one ∗ , is a weak e quiv alence. W e now consider a s trictly p oint ed Quillen model. Prop osition 15. 1. We have a Q uil len e quivalenc e ∗ ↓ L Σ ,P F un ( M op f , S se t ) U   L Σ ,P F un ( M op f , S se t ) , ( − ) + O O wher e U denotes the for getful functor. This follows fr om the fact that the catego ry L Σ ,P F un ( dgcat op f , S se t ) is homotopi- cally p ointed and from the following ge neral a rgumen t. Prop osition 15.2 . L et M b e a homotopic al ly p ointe d Quil len m o del c ate gory. We have a Quil len e quivalenc e. ∗ ↓ M U   M , ( − ) + O O wher e U denotes the for getful functor. Pr o of. Clearly the functor U preserves coﬁbrations , ﬁbra tions and weak equiv a- lences, b y construction. L et now N ∈ M a nd M ∈ ∗ ↓ M . Consider the fo llo wing commutativ e diagr a m in M N ≃ ∅ ` N f / / ∼ i ∐ 1 & & L L L L L L L L L L U ( M ) ∗ ` N f ♯ : : v v v v v v v v v , where f ♯ is the mor phism which corre s ponds to f , co nsidered as a morphism in M , under the adjunction and i : ∅ ∼ → ∗ . Since the mor phism i ∐ 1 corresp onds to the homotopy colimit of i and 1, w hich are bo th weak equiv alences, prop osi- tion 5.1 implies that i ∐ 1 is a weak equiv ale nc e . Now, by the ‘2 out o f 3’ pro perty , the morphism f is a weak equiv a lence if a nd o nly if f ♯ is one. This pr o ves the prop osition. √ HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 49 Notation 15.3 . Let A and B b e small dg categ ories. W e denote by rep mor ( A , B ) the full dg sub category of C dg ( A op c ⊗ B ), wher e A c denotes a coﬁbr an t reso lutio n of A , whose ob jects are the bimodules X such that X (? , A ) is a co mpact ob ject in D ( B ) for all A ∈ A c and which are co ﬁbran t as bimo dules. W e denote b y w A the category of homotopy equiv alences o f A and by N .w A its nerve. Now, consider the morphism Ho ( dgcat ) → Ho ( ∗ ↓ L Σ ,P F un ( M op f , S se t )) A 7→    Hom dgcat (Γ(?) , A f ) + ≃ Map dgcat (? , A ) + ≃ N .w rep mor (? , A ) + which by sections 6, 7 and prop osition 15.1 cor respo nds to the comp onent (Φ ◦ R h )( e ) of the morphism of deriv a tors Φ ◦ R h : HO ( dgcat ) − → L Σ ,P Hot dgcat f , see prop osition 7.1. Observe that the simplicial pr e sheaf N .w rep mor (? , A ) is already canonically p ointed. Prop osition 15. 4. The c anonic al morphism Ψ : N .w rep mor (? , A ) + → N .w rep mor (? , A ) is a we ak e quivalenc e in ∗ ↓ L Σ ,P F un ( M op f , S se t ) . Pr o of. Observe that N .w rep mor (? , A ) is a ﬁbrant ob ject in ∗ ↓ L Σ ,P F un ( M op f , S se t ) and that the cano nical morphism Ψ corres p onds to the co-unit of the adjunction of prop osition 15.1. Since this adjunction is a Quillen equiv ale nc e , the prop osition is prov ed. √ Recall now from remar k 8.2 that we have a ca nonical equiv alenc e of p oint ed deriv ators L Σ ,P Hot dgcat f ∼ − → L Σ ,P Hot dgcat f • , where L Σ ,P Hot dgcat f • is the deriv a tor a sso ciated with the Quillen mo del ca tegory L Σ ,P F un ( dgcat op f , S se t • ). F r om no w on, we will consider this Q uillen mo del. W e hav e the follo wing morphism of der iv a tors Φ ◦ R h : HO ( dg cat ) − → L Σ ,P Hot dgcat f • , which commutes with ﬁltered ho mo top y co limits and preser v es the po in t. Notation 15.5 . - W e deno te by E s the set of retr actions o f dg categorie s G i G / / H , R o o where G and H are strict ﬁnite I -cell ob jects in dgcat , i G is a fully faithful dg functor , R is a right adjoint to i G and R ◦ i G = I d G . - W e denote by E s un the set of morphisms S L in L Σ ,P Hot dgcat f • ( e ), s ee sec- tion 1 0, where L belongs to the set E s . Now choose for each element of the set E s un a repres en tative in the catego ry L Σ ,P F un ( dgcat op f , S se t • ). W e denote this set of re pr esen tatives by g E s un . Since L Σ ,P F un ( dgcat op f , S se t • ) is a left pr oper , cellular Quillen mo del categor y , see [15], its left Bo us ﬁeld lo calization by g E s un exists. W e denote it b y L g E s un L Σ ,P F un ( dgcat op f , S se t • ) 50 GONC ¸ ALO T ABUAD A and by L g E s un L Σ ,P Hot dgcat f • the a sso c iated deriv a tor. W e have the following mor- phism of deriv ato rs Ψ : L Σ ,P F unHot dgcat f • → L g E s un L Σ ,P F unHot dgcat f • . R emark 15.6 . - No tice that by construction the domains and co domains of the set g E s un are homotopically ﬁnitely pr esen ted o b jects. Ther efore by lemma 8 .1 the set G = { F X ∆[ n ] + /∂ ∆[ n ] + | X ∈ M f , n ≥ 0 } , of coﬁb ers of the generating coﬁbra tions in Fun ( M op f , S se t • ) is a set of sma ll weak g e nerators in Ho ( L g E s un L Σ ,P F un ( M op f , S se t • )). - Notice also that prop osition 14.2 implies that v ar ian ts of propo sition 10.2 and theor em 10.4 are a lso veriﬁed: simply consider the set E s instead o f E and a r etraction o f dg categor ies instea d of a n inclusion o f a full dg sub c ategory . The pr oo fs are exactly the same. Deﬁnition 15. 7. - The Unsta ble Motiv ator of dg c ate gories M unst dg is the derivator asso ciate d with the Quil len mo del c ate gory L g E s un L Σ ,P F un ( dgcat o f , S se t • ) . - The Universal unstable inv aria n t of dg c ate gories is t he c anonic al morphism of derivators U u : HO ( dgcat ) → M unst dg . Let M b e a left prop er cellular mo del ca tegory , S a set o f maps in M a nd L S M the left Bousﬁeld lo calization of M with respec t to S , see [15]. Reca ll from [15, 4.1.1 .] that an o b ject X in L S M is ﬁbrant if X is ﬁbr a n t in M and for every elemen t f : A → B of S the induced map o f homotopy function co mplexes f ∗ : M ap ( B, X ) → Map ( A, X ) is a weak equiv alence. Prop osition 15 . 8. A n obje ct F ∈ L g E s un L Σ ,P F un ( dgcat op f , S se t • ) is ﬁbr ant if and only if it satisﬁes the following c onditions 1) F ( B ) ∈ S s e t • is ﬁbr ant, for al l B ∈ dgca t f . 2) F ( ∅ ) ∈ S set • is c ontr actible. 3) F or every Morita e quivalenc e B ∼ → B ′ in dgcat f the morphism F ( B ′ ) ∼ → F ( B ) is a we ak e quivalenc e in S set • . 4) Every split short ex act se qu en c e 0 / / B ′ i B ′ / / B R o o P / / B ′′ i B ′′ o o / / 0 in dgcat f induc es a homotopy ﬁb er se quenc e F ( B ′′ ) → F ( B ) → F ( B ′ ) in S se t . Pr o of. Clearly condition 1) cor respo nds to the fa ct that F is ﬁbrant in Fun ( dgcat op f , S se t • ). Now observe that F un ( dg cat op f , S se t • ) is a simplicial Quillen mo del categ o ry with the simplicia l action given by S se t × Fun ( dgcat op f , S se t • ) → F un ( dg cat op f , S se t • ) ( K, F ) 7→ K + ∧ F , HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 51 where K + ∧ F denotes the co mponent wise s mash pro duct. This simplicial s tructure and the construction of the loca lized Quillen mo del category L g E s un L Σ ,P F un ( dgcat op f , S se t • ), see section 8, allow us to recover conditions 2) a nd 3). Condition 4) follows from the c o nstruction of the set g E s un and from the fa ct that the functor Map (? , F ) : Fun ( dgcat op f , S se t • ) op → S set transforms homo top y coﬁb er se q uences int o homotopy ﬁb er seq uences. This pr o ves the prop o sition. √ Let A b e a Mor ita ﬁbrant dg catego r y . Recall from no tation 11 .1 1 that S • A de- notes the s implicia l Mo rita ﬁbr a n t dg categor y obtained by applying W aldha usen’s S • -construction to the exact category Z 0 ( A ) and remembering the enrichmen t in complexes. Notation 15.9 . W e denote b y K ( A ) ∈ Fun ( dgcat op f , S se t • ) the simplicial presheaf B 7→ | N .wS • rep mor ( B , A ) | , where | − | denotes the ﬁbra n t r ealization functor of bisimplicial sets. Prop osition 15.10. The simplicial pr eshe af K ( A ) is ﬁ br ant in L g E s un L Σ ,P F un ( dgcat op f , S se t • ) . Pr o of. Observe that K ( A ) satisﬁes conditions (1)-(3 ) of prop osition 1 5.8. W e now prov e that W a ldha usen’s ﬁbration theorem [44, 1 .6.4] implies conditio n (4). Apply the contrav aria n t functor rep mor (? , A ) to the split short exac t sequence 0 / / B ′ i B ′ / / B R o o P / / B ′′ i B ′′ o o / / 0 and obtain a split shor t exa ct se quence 0 / / rep mor ( B ′′ , A ) / / rep mor ( B , A ) o o / / rep mor ( B ′ , A ) o o / / 0 . Now c o nsider the W aldhause n category v rep mor ( B , A ) := Z 0 ( rep mor ( B , A )), where the weak equiv alences are the morphisms f such that cone ( f ) is contractible. Co n- sider a lso the W aldhaus e n catego ry w rep mor ( B , A ), which has the sa me coﬁbra tions as v rep mor ( B , A ) but the weak equiv alences are the morphisms f such tha t co ne ( f ) belo ngs to rep mor ( B ′′ , A ). Obser v e that we have the inclus ion v rep mor ( B , A ) ⊂ w rep mor ( B , A ) and an equiv alence rep mor ( B , A ) w ≃ Z 0 ( rep mor ( B ′ , A )), see s ection 1 . 6 fr om [44]. The co nditions of theorem 1 . 6 . 4 from [4 4] are satisﬁed and so we hav e a ho motop y ﬁber sequence | N .w S • rep mor ( B ′′ , A ) | → | N .wS • rep mor ( B , A ) | → | N .wS • rep mor ( B ′ , A ) | in S set . This pr o ves the pr o positio n. √ Let p : ∆ → e be the pro jection functor . Prop osition 15. 11. The obje cts S 1 ∧ N .w rep mor (? , A ) and | N .wS • rep mor (? , A ) | = K ( A ) ar e c anonic al ly isomorphic in Ho ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )) . 52 GONC ¸ ALO T ABUAD A Pr o of. As in [2 8, 3.3], we consider the sequence in HO ( dgcat )(∆) 0 / / A • I / / P S • A Q / / S • A / / 0 , where A • denotes the constant simplicial dg category with v alue A and P S • A the path ob ject of S • A . B y applying the mor phis m of deriv ator s U u to this sequence, we obtain the canonical morphis m S I : cone( U u ( A • I → P S • A )) → U u ( S • A ) in M unst dg (∆). W e no w pr ove that for each po in t n : e → ∆, the n th compo nent of S I is an isomor phism in L g E s un L Σ ,P F un ( dgcat op f , S se t • ). F or each point n : e → ∆, we hav e a split short exact sequence in Ho ( dgcat ): 0 / / A I n / / P S n A = S n +1 A R n o o Q n / / S n A S n o o / / 0 , where I n maps A ∈ A to the constant sequence 0 → A I d → A I d → · · · I d → A , Q maps a sequence 0 → A 0 → A 1 → · · · → A n to A 1 / A 0 → · · · → A n / A 0 , S n maps a se quence 0 → A 0 → A 1 → · · · → A n − 1 to 0 → 0 → A 0 → · · · → A n − 1 and R n maps a sequence 0 → A 0 → A 1 → · · · → A n − 1 to A 0 . No w, by construction of L g E s un L Σ ,P F un ( dgcat op f , S se t • ) the ca nonical mor- phisms S I n : cone( U u ( A I n → P S n A )) → U u ( S n A ) , n ∈ N are isomo r phisms in M unst dg ( e ). Since homotopy colimits in L g E s un L Σ ,P F un ( dgcat op f , S se t • ) are calculated ob jectwise the n th co mponent of S I ident iﬁes with S I n and so b y the conserv a tiv ity axiom S I is a n iso morphism in M unst dg (∆). This implies that we obtain the homotopy c ocar tesian sq ua re p ! U u ( A • )   / / p ! ( U u ( P S • A ))   ∗ / / p ! ( U u ( S • A )) . As in the pro of of prop osition 11.12, we show that p ! U u ( A • ) identiﬁes with N .w rep mor (? , A ) = U u ( A ) a nd that p ! ( U u ( P S • A )) is con tractible. Since w e ha ve the equiv alence p ! ( U u ( S • A )) = p ! ( N .w rep mor (? , S • A )) ∼ → | N .w S • rep mor (? , A ) | and N .w rep mor (? , A ) is coﬁbra nt in L g E s un L Σ ,P F un ( dgcat op f , S se t • ) the prop osition is prov en. √ HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 53 Prop osition 15. 12. We have the fol lowing we ak e quivalenc e of simplicial sets Map ( U u ( k ) , S 1 ∧ U u ( A )) ∼ → | N .w S • A f | in L g E s un L Σ ,P F un ( dgcat op f , S se t • ) . In p articular, we have t he fol lowing isomorphi s ms π i +1 Map ( U u ( k ) , S 1 ∧ U u ( A )) ∼ → K i ( A ) , ∀ i ≥ 0 . Pr o of. This follows fr om pr o positio ns 1 5 .10, 15 .11 and from the following weak equiv alences Map ( U u ( k ) , S 1 ∧ U u ( A )) ≃ Map ( R h ( k ) , K ( A )) ≃ ( K ( A ))( k ) ≃ | N .w S • A f | . √ 16. The universal additive inv ariant Consider the Q uillen mo del categ ory L g E s un L Σ ,P F un ( dgcat op f , S se t • ) co nstructed in the previous section. The deﬁnition of the set g E s un and the same ar gumen ts a s those of example 9.5 and example 9.6 allows us to conclude that L g E s un L Σ ,P F un ( dgcat op f , S se t • ) satisﬁes the conditions of theo rem 9.7. In particular we hav e a n equiv ale nce o f tri- angulated deriv ator s St ( M unst dg ) ∼ → HO ( Sp N ( L g E s un L Σ ,P F un ( dgcat op f , S se t • ))) . Deﬁnition 16. 1. - The Additive motiv ator of dg c ate gories M add dg is t he triangulate d derivator asso ciate d with the stable Quil len mo del c ate gory Sp N ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )) . - The Universal additive inv aria n t of dg c ate gories is the c anonic al morphism of derivators U a : HO ( dgcat ) → M add dg . R emark 16.2 . Obser v e that rema rk 15 .6 a nd remark 9.4 imply that M add dg is a compactly g e ne r ated tria ng ulated deriv ator. W e sum up the construction of M add dg in the following diagra m 54 GONC ¸ ALO T ABUAD A dgcat f [ S − 1 ] / / Ho ( h )   HO ( dgcat ) R h w w o o o o o o o o o o o o U a y y U u x x L Σ Hot dgcat f Φ   L Re 7 7 o o o o o o o o o o o o L Σ ,P Hot dgcat f • Ψ   M unst dg ϕ   M add dg Observe that the mor phism o f der iv ato r s U a is p oin ted, commutes with ﬁlter ed homotopy colimits and satisﬁes the following condition : A) F or ev ery split short exact sequence 0 / / A i A / / B R o o P / / C i C o o / / 0 in Ho ( dgcat ), we hav e a split tr ia ngle U a ( A ) i A / / U a ( B ) R o o P / / U a ( C ) i C o o / / U a ( A )[1] in M add dg ( e ). (This implies that the dg functors i A and i C induce an iso - morphism U a ( A ) ⊕ U a ( C ) ∼ → U a ( B ) in M add dg ( e )). R emark 16 .3 . Since the dg ca teg ory B in the ab ov e split shor t exact sequence is Morita equiv alent to the dg categor y E ( A , B , C ), s ee sectio n 1 . 1 fro m [44], condition A ) is eq uiv alent to the additivit y pr operty sta ted by W aldhausen in [44, 1 .3.2]. Let D be a strong tr iangulated deriv ator . Theorem 16 . 4. The morphism U a induc es an e qu iva lenc e of c ate gories Hom ! ( M add dg , D ) U ∗ a − → Hom f lt, A ) , p ( HO ( dgcat ) , D ) , wher e Hom f lt, A ) , p ( HO ( dgcat ) , D ) denotes the c ate gory of morphisms of derivators which c ommute with ﬁ lter e d homotopy c olimits, satisfy c ondition A ) and pr eserve the p oint. Pr o of. By theorem 9.1, we have an eq uiv alence of categor ies Hom ! ( M add dg , D ) ∼ − → Hom ! ( M unst dg , D ) . By theorem 5.4, we have a n equiv alence o f ca tegories Hom ! ( M unst dg , D ) ∼ − → Hom ! , E s un ( L Σ ,P Hot dgcat f • , D ) . HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 55 Now, we obs erv e that since D is a strong tria ngulated deriv ator , the ca tegory Hom ! , E s un ( L Σ ,P Hot dgcat f • , D ) identiﬁes Hom f lt, A ) , p ( HO ( dgcat ) , D ). This proves the theorem. √ Notation 16 .5 . W e call an ob ject of the r igh t hand side category of theo rem 16.4 an additive invariant o f dg categ ories. Example 1 6 .6. - The Ho chschild and cyclic homolo gy and the non-c onne ctive K -the ory deﬁne d in se ction 11 ar e ex amples of addi tive invariants. - Another example is given by the classic al Waldhausen ’s c onne ct ive K -the ory sp e ctrum K c : HO ( dgcat ) → HO ( S pt ) , se e [44] . R emark 1 6.7 . By theo rem 16 .4 , the mo rphism of der iv ato r s K c factors through U a and so giv es r ise to a morphism of deriv a tors K c : M add dg → HO ( S pt ) . W e now will pr ove tha t this morphis m of deriv ators is co-r epresentable in M add dg . Let A be a small dg categor y . Notation 16.8 . W e denote by K ( A ) c ∈ Sp N ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )) the sp ec- trum such that K ( A ) c n := | N .wS ( n +1) • rep mor (? , A ) | , n ≥ 0 , endow ed with the natural str ucture mo rphisms β n : S 1 ∧ | N .wS ( n +1) • rep mor (? , A ) | ∼ − → | N .wS ( n +2) • rep mor (? , A ) | , n ≥ 0 , see [44]. Notice that U a ( A ) identiﬁes in Ho ( M add dg ) with the susp ension spectr um g iv en by (Σ ∞ | N .w rep mor (? , A ) | ) n := S n ∧ | N .w rep mor (? , A ) | . Now pr opos ition 15.11 and the fact that the mor phism of deriv ato rs ϕ commutes with homoto p y colimits implies that w e ha ve an isomor phism U a ( A )[1] ∼ → p ! U a ( S • A ) . In particular , we hav e a natural morphism η : U a ( A )[1] → K ( A ) c in Sp N ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )) induced b y the identit y in degree 0. Theorem 16 . 9. The morphism η is a ﬁbr ant r esolution of U a ( A )[1] . Pr o of. W e prov e ﬁrst that K ( A ) c is a ﬁbrant ob ject in Sp N ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )). By [16][36] we need to show tha t K ( A ) c is an Ω- spectrum, i.e. that K ( A ) c n is a ﬁbrant ob ject in L g E s un L Σ ,P F un ( dgcat op f , S se t • ) and that the induced map K ( A ) c n → Ω K ( A ) c n +1 is a weak eq uiv alence. By W aldhausen’s a dditivit y theorem, see [4 4], we have weak equiv alences K ( A ) c n ∼ → Ω K ( A ) c n +1 56 GONC ¸ ALO T ABUAD A in F un ( dgcat op f , S se t • ). No w o bserve that fo r every integer n , K c ( A ) n satisﬁes conditions (1)-(3) of prop osition 15 .10. Condition (4) follows from W aldhause n’s ﬁ- bration theorem, a s in the pro of o f prop osition 15.11, applied to the S. - construction. This shows that K ( A ) c is a n Ω-s pectrum. W e now prov e that η is a (comp o nen twise) weak equiv alence in Sp N ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )). F or this, we prov e ﬁrst that the structura l mor- phisms β n : S 1 ∧ N .wS ( n +1) • rep mor (? , A ) ∼ − → | N .w S ( n +2) • rep mor (? , A ) | , n ≥ 0 , see nota tion 16.8, are weak equiv alences in L g E s un L Σ ,P F un ( dgcat op f , S se t • ). By co n- sidering the s ame ar gumen t as in the pr oo f o f prop osition 15.1 1 , using S ( n +1) • A instead of A , we obta in the following homoto p y co cartesia n squar e K ( A ) c n   / / p ! ( U u ( P S ( n +2) • A ))   ∗ / / K ( A ) c n +1 in L g E s un L Σ ,P F un ( dgcat op f , S se t • ) with p ! ( U u ( P S ( n +2) • A )) contractible. Since K ( A ) c n +1 is ﬁbrant , prop osition 1 . 5 . 3 in [44], implies that the previous square is a ls o homotopy cartesian and so the ca no nical morphism β ♯ n : K ( A ) c n → Ω K ( A ) c n +1 is a weak equiv alence in L g E s un L Σ ,P F un ( dgcat op f , S se t • ). W e now show that the struc- ture morphism β n , which cor respo nds to β ♯ n by a djunction, see [44 ], is als o a w eak equiv alence. The derived adjunction ( S 1 ∧ − , R Ω( − )) induces the following com- m utative diagr am S 1 ∧ K ( A ) c n S 1 ∧ L β ♯ n ( ( Q Q Q Q Q Q Q Q Q Q Q Q / / K ( A ) c n +1 S 1 ∧ L Ω K ( A ) c n +1 ∼ O O in Ho ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )) wher e the vertical a rrow is an isomor phism since the pr evious square is homotopy bicartesian. This shows that the induced morphism S 1 ∧ K ( A ) c n − → K ( A ) c n +1 is a n isomor phism in Ho ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )) and so β n is a w e ak equiv a- lence. Now to prov e that η is a compo nen twise weak e q uiv alence, we pro ceed b y in- duction : observe that the zero co mponent of the morphism η is the iden tity . Now suppo se that the n -c o mponent of η is a weak equiv alence . The n + 1-comp onent of η is the comp osition of β n +1 , which is a w e ak equiv alence, w ith the susp e nsion of the n -c o mponent of η , which by prop osition 5.1 is also a w eak equiv a le nc e . This pr o ves the theorem. √ Let A a nd B b e small dg catego ries with A ∈ dg cat f . W e denote by H om Sp N ( − , − ) the sp ectrum of morphisms in Sp N ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )). HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 57 Theorem 16 . 10. We have the fol lowing we ak e quivalenc e of sp e ctr a Hom Sp N ( U a ( A ) , U a ( B )[1]) ∼ → K c ( rep mor ( A , B )) , wher e K c ( rep mor ( A , B )) denotes Waldhausen ’s c onne ctive K -the ory sp e ctrum of rep mor ( A , B ) . In p articular, we have the fol lowing we ak e quivalenc e of simplicial sets Map ( U a ( A ) , U a ( B )[1]) ∼ → | N .w S • rep mor ( A , B ) | and so the isomorp hisms π i +1 Map ( U a ( A ) , U a ( B )[1]) ∼ → K i ( rep mor ( A , B )) , ∀ i ≥ 0 . Pr o of. Notice that U a ( A ) identiﬁes with the susp ension sp ectrum Σ ∞ | N .w rep mor (? , A ) | which is coﬁbrant in Sp N ( L g E s un L Σ ,P F un ( dgcat o f , S se t • )). By theor em 16.9 we hav e the following equiv alences Hom Sp N ( U a ( A ) , U a ( B )[1]) ≃ Hom Sp N ( U a ( A ) , K c ( B )) ≃ K c ( B )( A ) ≃ K c ( rep mor ( A , B )) . This proves the theor em. √ R emark 16.1 1 . Notice that if in the ab ov e theor em, we consider A = k , we hav e Hom Sp N ( U a ( k ) , U a ( B )[1]) ∼ → K c ( B ) . This shows that W aldha usen’s connective K -theory sp ectrum b ecomes co -representable in M add dg . T o the b est of the author ’s knowledge, this is the ﬁrst co nc e ptual charac- terization o f Q uille n- W aldha usen K - theory [3 4] [44] since its deﬁnition in the early 70’s. This r esult g iv es us a co mpletely new wa y to think ab out alge br aic K -theory . 17. Higher Chern characte rs In this chapter we apply o ur ma in co -representabilit y theorem 16.1 0 in the co n- struction of the higher Cher n characters [2 6]. Let A and B b e small dg catego ries with A ∈ d gcat f . Prop osition 17. 1. We have the fo l lowing isomorphisms of ab elian gr oups Hom M add dg ( e ) ( U a ( A ) , U a ( B )[ − n ]) ∼ − → K n ( rep mor ( A , B )) , ∀ n ≥ 0 . Pr o of. In ﬁrs t place , notice that the abe lia n gro up Hom M add dg ( e ) ( U a ( A ) , U a ( B )[ − n ]) ident iﬁes with π 0 Map ( U a ( A ) , U a ( B )[ − n ]) , where M ap denotes the mapping space in Sp N ( L g E s un L Σ ,P F un ( dgcat op f , S se t • )). By theorem 16 .9 the morphism η : U a ( B )[1] − → K ( B ) c is a ﬁbrant reso lution o f U a ( B )[1]. This implies that in M add dg ( e ), U a ( B )[ − n ] iden- tiﬁes with the s pectrum Ω n +1 ( K c ( B )). Since U a ( A ) is coﬁbrant and Ω n +1 ( K c ( B )) 0 = Ω n +1 | N .w S • rep mor (? , B ) | 58 GONC ¸ ALO T ABUAD A we conclude that π 0 Map ( U a ( A ) , U a ( B )[ − n ]) ≃ π 0 Ω n +1 | N .w S • rep mor ( A , B ) | . Finally notice that π 0 Ω n +1 | N .w S • rep mor ( A , B ) | ≃ π n +1 | N .w S • rep mor ( A , B ) | ≃ K n ( rep mor ( A , B )) . This proves the pro positio n. √ R emark 17.2 . Notice that if in the ab ov e prop osition, w e co nsider A = k , we hav e the iso morphisms Hom M add dg ( e ) ( U a ( k ) , U a ( B )[ − n ]) ∼ − → K n ( B ) , ∀ n ≥ 0 . This s hows that the algebr aic K -theory gr oups K n ( − ) , n ≥ 0 are co-r epresentable in the tr iangulated catego r y M add dg ( e ). Now let K n ( − ) : Ho ( dgcat ) − → Mo d- Z , n ≥ 0 be the n th K -theory g r oup functor, see theorem 11 .7 , and H C j ( − ) : Ho ( dgca t ) − → Mo d- Z , j ≥ 0 the j th cyclic homology group functor, see theorem 11.9. Theorem 1 7 .3. The c o-r epr esentability the or em 16.1 0 furnishes us the hi gher Chern char acters [26] ch n,r : K n ( − ) − → H C n +2 r ( − ) , n, r ≥ 0 . Pr o of. By theorem 11.9 the morphism of der iv ator s C : HO ( dgcat ) − → H O (Λ-Mod) is an additive inv a riant a nd so descends to M add dg and induces a functor (still denoted by C ) C : M add dg ( e ) − → D (Λ) . By [18] the cyclic homolo gy functor H C j ( − ) , j ≥ 0 is o btained by compo sing C with the functor H − j ( k L ⊗ Λ − ) : D (Λ) − → Mo d- Z , j ≥ 0 Now, by pr opos ition 1 7.1 a nd remark 17.2 the functor K n ( − ) : M add dg ( e ) − → Mo d- Z is co-r epresented by U a ( k )[ n ]. This implies, by the Y o neda lemma, tha t Nat ( K n ( − ) , H C j ( − )) ≃ H C j ( U a ( k )[ n ]) . Since we hav e the follo wing isomo r phisms H C j ( U a ( k )[ n ]) ≃ H − j ( k L ⊗ Λ C ( U a ( k )[ n ])) ≃ H − j ( k L ⊗ Λ C ( k )[ n ]) ≃ H − j + n ( k L ⊗ Λ C ( k )) ≃ H C j − n ( k ) . and since H C ∗ ( k ) ≃ k [ u ] , | u | = 2 HIGHER K -THEOR Y VIA UNIVE RSAL INV ARIANTS 59 we conclude that H C j ( U a ( k )[ n ]) =  k if j = n + 2 r, r ≥ 0 0 otherwise . Finally no tice that the canonical element 1 ∈ k furnishes us the higher Chern characters and s o the theor em is prov en. √ 18. Concluding remarks By the universal pro p erties of U u , U a and U l , we obtain the following diag r am: HO ( dgcat ) U u / / U a % % K K K K K K K K K K U l   8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 M unst dg   M add dg Φ   M loc dg . Notice that W aldhausen’s connective K -theo ry is an example of a n additive in- v a riant which is NO T a der ived one, see [1 9]. W aldhausen’s connective K - theory bec omes co-repr esen ta ble in M add dg by theo rem 16.10. 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