Topological Hochschild and cyclic homology for Differential graded categories

TOPOLOGICAL HOCHSCHILD AND CYCL IC HOMOLOGY F OR DIFFERENTIAL GRADED CA TEGORIES GONC ¸ ALO T ABUADA Abstract. W e deﬁne a topological H ochsc hild ( T H H ) and cyclic ( T C ) ho- mology theory for diﬀerent ial graded (dg) catego ries and construct sev eral non-trivial natural transformations from a lgebraic K -theory to T H H ( − ). In an int ermediate step, we pro v e that the homotop y theory of dg catego ries is Q uillen equiv alen t, through a four step zig-zag of Quillen equiv alence s, to the homotop y theory of Eilenberg-Mac Lane spectral categories. Finally , w e show that ov er the rationals Q tw o dg categories are top ological equiv alent if and only if they ar e quasi-equiv alen t. Contents 1. Int ro duction 1 2. Ac knowledgemen ts 3 3. Preliminarie s 3 4. Homotopy theory of ge ne r al Sp ectral ca tegories 6 5. A Quillen equiv alence criter ion 21 6. Examples of Quillen equiv alences 23 7. General Sp ectra l algebr a 24 8. Eilenberg-Ma c Lane sp e c tral algebra 33 9. Global picture 35 10. T H H and T C for DG ca tegories 35 11. F rom K -theor y to T H H 38 12. T op olo gical equiv alence theory 40 Appendix A. Adjunctions 41 Appendix B. Bo usﬁeld lo caliza tion techn iques 44 Appendix C. Non-a dditive ﬁltration argument 45 References 46 1. Intr oduction In the pa st tw o deca des, t op ologic a l Ho chschild homolog y ( T H H ) and top o- logical cyclic homolog y ( T C ) hav e re volutionized algebraic K -theor y co mputations [10] [11] [12]. Roughly , the T H H of a ring sp ectrum is obtained b y substituting the tenso r pro duct in the c lassical Ho chsc hild complex by the smash pro duct. The Key wor ds and phr ases. Dg category , top ological Ho chsc hi l d homology , top ol ogical cyclic ho- mology , Eil en berg-M ac Lane sp ectral algebra, symmetric spectra, Quillen mo del structure, Bous- ﬁeld l ocali zation, non-commutativ e algebraic geometry . 1 2 GONC ¸ ALO T ABUADA T H H sp ectr um c o mes with a ‘cyclotomic’ structure, a nd for ea ch pr ime p , T C is deﬁned as a certa in homotopy limit ov er the ﬁxed po int sp ectra . The a im of this a rticle is to tw o fold: (1) to deﬁned these new homology theo ries in the context of diﬀer ential gr ade d (dg) ca teg ories (ov er a base ring R ) [6] [18] [27]; (2) to relate these new theor ies with algebra ic K - theory . Our motiv ation c o mes from Drinfeld-Kont sevich’s progra m in non-commutativ e alge br aic ge ometry [7] [19] [20], where dg categ ories are considered as the cor rect mode ls for non-co mmut ative spaces. F o r exa mple, one can asso c ia te to a smo oth pro jective alg ebraic v ar iety X a dg model D b dg (coh( X )), i.e. a dg categor y well deﬁned up to quasi- equiv a lence (see 3.8), whose triangulated categor y H 0 ( D b dg (coh( X ))) (obtained by applying the zero co homology group functor in each Hom-complex) is equiv alent to the b ounded derived catego ry of q ua si-coher ent complex e s on X . F or D b dg (coh( X )), one c o uld take the dg ca tegory of left b o unded complexes of injective O X -mo dules who se cohomolog y is b o unded and coherent. As the ab ove example shows, these theories would furnishes us automatically a well-deﬁned to po logical Ho chsc hild and cyclic homolo gy theory for alg ebraic v ar i- eties. In order to deﬁne these new homo logy theories for dg ca tegories , we introduce in chapter 8 the notion of Eilenb er g-Mac L ane sp e ctr al c ate gory (i.e. a catego ry enriched in mo dules ov er the E ilenberg-Mac Lane ring spec trum H R ), and prove (b y inspiring ourselves in Shipley’s work [26]) our ﬁrs t main theore m. Theorem 1.1. (se e chapter 9) The homotopy the ory of diﬀer ential gr ade d c ate- gories is Quil le n e quivalent, thr ough a four step zig-zag of Quil len e quivalenc es, t o the homotopy the ory of Eilenb er g-Mac L ane sp e ctr al c ate gories. The nov elty in thes e homoto py theories is that in c o ntrast to dg algebras or enriched c ategorie s with a ﬁxed set of o b jects, the cor r ect notion o f equiv alence (as the ab ov e exa mple shows) is not deﬁned by simply forgetting the m ultiplicative structure. It is an h ybrid notion which in volv es weak e q uiv alence s and ca tegorica l equiv ale nc e s (see deﬁnition 3.8), making not only all the ar guments very diﬀerent but also muc h more evolved. In order to prov e theo rem 1.1, w e had to develop several new genera l homo to pical algebra to ols (see theor ems 4.39, 4.45, 5.1 and 7.25) which are of indepe ndent interest. By c o nsidering the restriction functor H R -Cat − → Sp Σ -Cat from Eilenber g-Mac Lane to s pe c tr al categ ories (i.e. categ o ries e nriched over sym- metric sp ectra ), w e o btain a well-deﬁned functor Ho ( dgcat ) ∼ → Ho ( H R -Cat) → Ho ( Sp Σ -Cat) on the homo topy catego ries. Finally , b y comp osing the ab ove functor with the top ologica l and cy clic homology theories deﬁned by Blumber g and Mandell [21] for sp ectral ca tegories, we obtain our searched homolo gy theories T H H , T C : Ho ( dgcat ) − → Ho ( Sp ) . Calculations in algebr aic K -theory are very r a re and hard to get by . O ur second main theorem establishes a relationship b etw een algebraic K -theory and topolo g ical Ho chsc hild homo logy . THH AND TC FOR DG CA TEGORIES 3 Theorem 1.2 (see theore m 11.2) . L et R b e the ring of inte gers Z . Then we have non-trivial natur al tr ansformatio ns γ n : K n ( − ) ⇒ T H H n ( − ) , n ≥ 0 γ n,r : K n ( − ) ⇒ T H H n +2 r − 1 ( − ) , n ≥ 0 , r ≥ 1 . fr om the algebr aic K -the ory gr oups to the top olo gic al H o chschild homolo gy ones. When X is a quasi- compact and quasi-separa ted scheme, the T H H of X is equiv ale nt [2 1, 1.3] to the to po logical Ho chsc hild homolo gy o f schemes as deﬁned by Geisser and Hesselho lt in [9]. In pa rticular, theor e m 1.2 a llows us to use a ll the calculations of T H H developed in [9], to infer re s ults on algebra ic K -theor y . Finally , we recall in c hapter 12 Dugg er-Shipley’s no tion of topo logical equiv- alence [4] a nd sho w in prop o sition 12.6 that, w hen R is the ﬁeld of rationals nu mbers Q , tw o dg catego ries are top o logical equiv alen t if and only if they are quasi-equiv alen t. 2. Acknowledgements It is a g reat pleasure to t hank Stefan Sc h wede for suggesting me this problem and Gustav o Granja for helpful conv ersations. I would like also to thank the Institute for Adv anced Study at Princeton for excellent working conditions, w he r e pa rt o f this work was carr ied out. 3. P reliminaries Throughout this article we wil l b e w orking (to simpli fy notation) ov er the integers Z . How ever all our resul ts are v alid if we replace Z b y an y unital comm utativ e ring R . Let C h denote the catego ry of complexe s and C h ≥ 0 the full subc a tegory of p osi- tive g raded complexes. W e co nsider homological notation (the diﬀerential decreases the degr ee). W e deno te by s Ab the c a tegory of simplicial a b elian gr o ups. Notation 3.1 . Le t ( C , − ⊗ C − , 1 C ) b e a s ymmetric mono idal categ ory [1, 6.1.2]. W e denote by C -Cat the catego ry of sma ll ca tegories enr iched over C , see [1, 6.2 .1]. An ob ject in C -Cat w ill b e called a C -c ate gory and a morphism a C -functor . Notation 3.2 . Let D be a Quillen mo del category , see [22]. W e denote by W D the class of weak equiv alences in D a nd by [ − , − ] D the set of mor phisms in the homotopy catego r y H o ( D ) o f D . Mo reov er, if D is coﬁbrantly gener ated, we de no te by I D , res p. b y J D , the set of g enerating coﬁbra tio ns, resp. generating trivial coﬁbrations. Deﬁnition 3 . 3. L et ( C , − ⊗ − , 1 C ) and ( D , − ∧ − , 1 D ) b e t wo symmetric monoidal c ate gories. A la x monoida l functor is a functor F : C → D e quipp e d with: - a morphi sm η : 1 D → F ( 1 C ) and - natu r al morphisms ψ X,Y : F ( X ) ∧ F ( Y ) → F ( X ⊗ Y ) , X , Y ∈ C which ar e c oh er ent ly asso ciative and unital (se e diagr ams 6 . 2 7 and 6 . 28 in [1] ). A lax monoidal functor is stro ng mono idal if the morphisms η and ψ X,Y ar e iso- morphisms. 4 GONC ¸ ALO T ABUADA Now, supp ose that we hav e an adjunction C N   D , L O O with N a lax monoida l functor. Then, r ecall from [25, 3.3], that the left adjoint L is endow ed with a lax c o monoidal structure. γ : L ( 1 D ) − → 1 C φ : L ( X ∧ Y ) − → L ( X ) ⊗ L ( Y ) . Deﬁnition 3.4. A weak monoidal Quillen pa ir C N   D , L O O b etwe en monoidal m o del c ate gories C and D (s e e [15, 4.2.6] ) c onsists of a Q uil len p air, with a lax monoidal st r u ctur e on the right adjoint, such that the fol l owing t wo c onditions hold: (i) for al l c oﬁbr ant obje cts X and Y in D , the c o monoidal map φ : L ( X ∧ Y ) − → L ( X ) ⊗ L ( Y ) b elongs to W C and (ii) for some (henc e any) c oﬁbr ant r esolution 1 c D ∼ → 1 D , the c omp osite map L ( 1 c D ) → L ( 1 D ) γ → 1 C b elongs to W C . A stro ng monoidal Quillen pair is a we ak monoidal Quil len p ai r for which the co monoidal maps γ and φ ar e isomorphi sms. R emark 3.5 . If 1 D is coﬁbr ant a nd L is stro ng monoidal, then N is lax mo no idal and the Quillen pair is a strong mo noidal Quillen pair. Deﬁnition 3.6. A we ak (r esp. str ong) monoidal Quil len p air is a weak mo noidal Quillen equiv alence (r esp e ct ively strong mono idal Quillen eq uiv alence ) if t he under- lying p air is a Q uil len e qu ivalenc e. Let C a symmetric monoidal model category [15, 4.2.6]. W e consider the following natural notion of equiv alence in C -Cat: R emark 3.7 . W e have a functor [ − ] : C -Cat − → Cat , with v alues in the category o f small catego ries, obtained by base change along the monoidal comp osite functor C − → Ho ( C ) [ 1 C , − ] − → Set . Deﬁnition 3.8. A C -functor F : A → B is a weak equiv alence if: THH AND TC FOR DG CA TEGORIES 5 WE1) for al l obje cts x, y ∈ A , the m orphism F ( x, y ) : A ( x, y ) → B ( F x, F y ) b elongs to W C and WE2) t he induc e d functor [ F ] : [ A ] − → [ B ] is an e quivalenc e of c ate gories. R emark 3.9 . If condition W E 1) is veriﬁed, co ndition W E 2) is equiv alen t to: WE2’) the induced functor [ F ] : [ A ] − → [ B ] is es sentially surjective. Deﬁnition 3.10. L et A ∈ C -Cat. A r ight A -mo dule is a C -functor fr om A to C . R emark 3.11 . If the mo del categ ory C is coﬁbr antly generated, the categor y A -Mod of right A -mo dules is natur a lly endowed with a (coﬁbra ntly g enerated) Q uillen mo del structure, see for example [1 3, 11 ]. W e deno te by Ho ( A -Mo d) its homotopy triangulated catego ry . W e ﬁnish this chapter with s o me useful criterions: Prop ositi on 3. 1 2. [3, A.3 ] L et C U   D F O O b e an adjunction b etwe en Quil len mo del c ate gories. If the right adjo int functor U pr eserves trivial ﬁbr ations and ﬁbr ations b etwe en ﬁbr ant obje cts, then the adjunction ( F, U ) is a Quil len adjunction. Prop ositi on 3. 1 3. [15, 1.3 .16] L et C U   D F O O b e a Quil len adjunction b etwe en mo del c ate gories. The adjunction ( F, U ) is a Quil le n e quivalenc e if and only if: a) the right adjoint U r eﬂe cts we ak e qu ivalenc es b etwe en ﬁbr ant obje cts and b) for every c oﬁbr ant obje ct X ∈ D , the c omp ose d morphism X " " F F F F F F F F F ∼ / / U F ( X ) f U F ( X ) U ( i ) 9 9 s s s s s s s s s s , is a we ak e quivalenc e in D , wher e i : F ( X ) → F ( X ) f is a ﬁbr ant re solution in C . Throughout this a rticle the adjunctions a re displayed vertically with the left, resp. right, adjoint on the left side, resp. rig ht side. 6 GONC ¸ ALO T ABUADA 4. H omotopy theor y of general Spectral ca tegories Hov ey dev elop ed in [14] the homotopy theo ry of symmetric sp ectra in g eneral Quillen mo del ca tegories. Starting from a ‘well-behav ed’ mono idal mo del catego ry C and a coﬁbr a nt ob ject K ∈ C , Hovey constructed the stable monoidal mo del category Sp Σ ( C , K ) of K -symmetric sp ectr a ov er C , see [14, 7.2]. In this chapter, we cons tr uct the ho motopy theory of gener al Sp e ctra l c ate gories , i.e. small categories enriched over Sp Σ ( C , K ) (see theorem 4.3 9). F or this, we impo se the existence of a ‘w ell-b ehav ed’ Quillen mo del structur e on C -Cat and certain co nditio ns on C a nd K . Let ( C , − ⊗ C − , 1 C ) be a coﬁbrantly genera ted monoidal mo del ca tegory and K an ob ject of C . 4.1. Conditi ons o n C and K : C1) The mo del s tructure on C is prop er ([13 , 13.1 .1]) and cellular ([1 3, 12 .1.1]). C2) The domains of the morphisms of the sets I C (generating coﬁbrations ) and J C (generating trivial coﬁbr ations) are c oﬁbrant and seq uentially small. C3) The mo del structure on C satisﬁes the monoid axiom ([25, 3.3]). C4) Ther e exists a pro duct preserv ing functor | − | : C → Set s uch that a morphism f b elongs to W C (see 3 .2) if and only if | f | is a n is o morphism in Set . C5) The ob ject K ∈ C is coﬁbr a nt and seq uentially small. C6) The stable mo del structure on Sp Σ ( C , K ) (see [14, 8.7]) is right prop er and satisﬁes the monoid a xiom. C7) The identit y 1 C is a domain or co domain of an element of I C . R emark 4.1 . - Notice that since C is r ight pr op er, so it is the level mo del structure on Sp Σ ( C , K ) (see [14, 8.3]). - Notice also, that since the ev aluation functors ([1 4, 7.3]) E v n : Sp Σ ( C , K ) − → C , n ≥ 0 preserve ﬁltered colimits, the domains of the g enerating (trivial) coﬁbra- tions of the level mo del structure on Sp Σ ( C , K ) are also s equentially small. - Obs erve that the functor | − | a dmits a natural extension: | − | T : Sp Σ ( C , K ) − → Q n ≥ 0 Set ( X n ) n ≥ 0 7→ ( | X n | ) n ≥ 0 . In this way a morphism f in Sp Σ ( C , K ) is a level equiv ale nce if and only if | f | T is a n isomor phism. Example 4.2. 1) If we c onsider for C the pr oje ctive mo del st ructur e on the c ate gory C h (se e [1 5, 2 .3.11] ) and for K the chain c omplex Z [1] , which c ont ains a single c opy of Z in dimension 1 , then c onditio ns C 1) - C 7) ar e satisﬁe d: c onditions C 1) , C 2) , C 5) and C 7) ar e veriﬁe d by c onstru ction. F or c ondition C 4) , c onsider t he functor, which asso ciates t o a chain c om- plex M the set Q n ∈ Z H n ( M ) . Final ly c ondition C 3) is pr ove d in [26, 3.1] and c ondition C 6) is pr ove d in [26, 2 .9] . THH AND TC FOR DG CA TEGORIES 7 2) If we c onsider for C the pr oje ctive mo del structur e on C h ≥ 0 (se e [17, I I I] ) and for K the chain c omplex Z [1] , then as in t he pr evious example, c ondi- tions C 1) - C 7) ar e satisﬁe d. 3) If we c onsider for C the mo del stru ctur e on s Ab deﬁne d in [17, I I I-2.8] and for K t he simplicial ab elian gr oup e Z ( S 1 ) , wher e S 1 = ∆[1] /∂ ∆[1] and e Z ( S 1 ) n is the fr e e ab elian gr oup on the non-b asep oint n -simple c es in S 1 , then c onditions C 1) - C 7) ar e s at isﬁe d: c onditions C 1) , C 2) , C 5) and C 7) ar e veriﬁe d by c onst ruction. F or c ondition C 4) , c onsider the functor which asso ciates to a simplicia l ab elian gr oup M , t he set Q n ∈ N π n ( M ) . Final ly c on- ditions C 3) and C 6) ar e pr ove d in [26, 3.4 ] . 4.2. Conditi ons on C -Cat: W e imp ose the existen c e of a (ﬁxe d) set J ′′ of C - functors, such that: the ca tegory C -Cat c arries a coﬁbra nt ly generated Quillen mo del, who se weak equiv alence s a re those deﬁned in 3.8 a nd the s ets o f g enerating (trivial) coﬁbr a tions are a s follows: Deﬁnition 4.3. The set I of genera ting c o ﬁbrations on C -Cat c onsist of: - the C -functors obtaine d by applying the fun ctor U (se e C.1) to t he set I C of gener ating c oﬁbr ations on C and - the C -functor ∅ − → 1 C fr om the empty C -c ate gory ∅ ( which is the initial obje ct in C -Cat) to the C -c ate gory 1 C with one obje ct ∗ and endomorphi sm ring 1 C . Deﬁnition 4.4. The set J := J ′ ∪ J ′′ of ge ner ating trivia l coﬁbr a tions on C -Cat c onsist of: - the set J ′ of C -functors obtaine d by applying the functor U to the set J C of gener ating trivial c oﬁbr ations on C and - the ﬁxe d set J ′′ of C -functors. Moreov er, we imp ose that: H1) the C -functors in J ′′ are of the form G V − → D , with G a sequentially sma ll C -category with only one ob ject { 1 } and D a C -categor y with t wo ob- jects { 1 , 2 } , such that the map G (1) / / V (1) ∼ / / D (1 ) is a trivia l co ﬁbration in C { 1 } -Cat, see remar k A.2. H2) for every C -categ ory A , the (solid) diagr am B 1 / / P ′′   A B 2 φ ? ? ~ ~ ~ ~ admits a ‘lift’ φ , where P ′′ is a C -functor in J ′′ . H3) the Quillen mo del s tr ucture obtained is r ight prop er. R emark 4.5 . The class I -inj, i.e. the class of trivia l ﬁbrations on C -Cat co ns ist of the C -functors F : A → B such that: - for all ob jects x, y ∈ A , the morphism F ( x, y ) : A ( x, y ) → B ( F x, F y ) 8 GONC ¸ ALO T ABUADA is a trivial ﬁbr ation in C and - the C -functor F induces a surjective map on o b jects. Notice als o that the class J - inj, i.e. the class of ﬁbrations on C - Cat co nsist of the C -functors F : A → B such that: - for all ob jects x, y ∈ A , the morphism F ( x, y ) : A ( x, y ) → B ( F x, F y ) is a ﬁbration in C and - have the right lifting prop erty (=R.L.P .) with res p ec t to the set J ′′ . In particula r, b y co nditio n H 2), a C -category A is ﬁbrant if and only if for all ob jects x, y ∈ A , the ob ject A ( x, y ) is ﬁbrant in C . Example 4.6. 1) If in the ﬁrst example of 4.2, we c onsider for C -Cat the Quil le n mo del structur e of the or em [27, 1.8] , wher e the set J ′′ c onsists of the dg functor F : A → K ( s e e se ction 1 . 3 of [27] ), then c onditions H 1) - H 3) ar e satisﬁe d: c ondi tion H 1) is satisﬁe d by c onst ruction and c onditions H 2) and H 3) fol lo w fr om r emark [27, 1.1 4 ] and c or ol lary [13, 13 .1.3] . 2) If in the se c ond example of 4.2, we c ons ider for C -Cat the Q u il len mo del structur e of the or em [30, 4.7 ] , wher e the set J ′′ c onsists of the dg fun ctor F : A → K , then c onditions H 1) - H 3) ar e satisﬁe d: c onditio n H 1) is also satisﬁe d by c onstruction (se e se ction 4 of [3 0] ) and c onditions H 2 ) and H 3) ar e analo gous to the pr evious example. 3) If in the thir d ex ample of 4.2, we c onsider for C -Cat the Qu il len mo del structur e of t he or em [30, 5.10 ] , wher e the s et J ′′ c onsists of the simplicial ab elian functor L ( F ) : L ( A ) → L ( K ) (se e se ction 5 . 2 of [3 0] ), then c ondi- tions H 1) - H 3) ar e satisﬁe d: sinc e the normalization functor (se e se ct ion 5 . 2 of [30] ) N : s Ab -Cat − → dgca t ≥ 0 pr eserves ﬁlter e d c olimits, admits a left Quil len adjoi nt and every obje ct is ﬁbr ant in s Ab -Cat ( s e e r emark [30, 5.18] ) c onditions H 1) - H 3) ar e satisﬁe d. F ro m now on unt il the end of this chapter, we supp ose that C , K and C -Cat satisfy conditions C 1)- C 7) and H 1)- H 3). 4.3. Lev elwise quasi-equiv alences . In this section we consider the level mo del structure on Sp Σ ( C , K ), see [1 4, 8.3]. Recall from [14, 7.2] that the ca tegory Sp Σ ( C , K ) o f K -symmetric sp ectra ov er C is endow ed with a symmetric monoidal structure − ∧ − whos e identit y is the K -sp ectrum 1 Sp Σ ( C ,K ) = ( 1 C , K , K ⊗ K, . . . , K ⊗ n , . . . ) , where the p ermutation g roup Σ n acts on K ⊗ n by p ermutation. Moreover, a s it is shown in [1 4, 7.3], we hav e an adjunction Sp Σ ( C , K ) E v 0   C , F 0 O O THH AND TC FOR DG CA TEGORIES 9 where b oth a djoints are strong monoidal (3.3). This natura lly induces, b y re- mark A.5, the following a djunction Sp Σ ( C , K )-Cat E v 0   C -Cat . F 0 O O R emark 4 .7 . By remark 4.1, t he r ight adjoin t functor E v 0 preserves ﬁltered co limits. W e now construct a Quillen model structure on Sp Σ ( C , K )-Cat, whose weak equiv ale nc e are deﬁned as follows. Deﬁnition 4.8. A Sp Σ ( C , K ) -functor F : A → B is a levelwise quasi- equiv ale nce if: L1) for al l obje cts x, y ∈ A , the morphism F ( x, y ) : A ( x, y ) → B ( F x, F y ) is a level e quivalenc e in Sp Σ ( C , K ) ( s e e [1 4, 8.1] ) and L2) the induc e d C -functor E v 0 ( F ) : E v 0 ( A ) → E v 0 ( B ) is a we ak e quivalenc e (3.8) in C -Cat. Notation 4.9 . W e denote by W L the clas s of levelwise quasi- equiv a le nces. R emark 4.1 0 . If condition L 1 ) is veriﬁed, co ndition L 2) is equiv a lent to: L2’) the induced functor (3.7) [ E v 0 ( F )] : [ E v 0 ( A )] − → [ E v 0 ( B )] is es sentially surjective. W e now deﬁne our sets of generating (trivia l) coﬁbratio ns . Deﬁnition 4.11. The set I L of generating co ﬁbrations in Sp Σ ( C , K ) -Cat c onsists of: - the set I ′ L of Sp Σ ( C , K ) -functors obtaine d by applying t he fun ctor U (se e C.1) to the set I Sp Σ ( C ,K ) of gener ating c oﬁbr ations for the level mo del struc- tur e on Sp Σ ( C , K ) (se e [1 4, 8.2] ) and - the Sp Σ ( C , K ) -functor ∅ − → 1 Sp Σ ( C ,K ) fr om t he empty Sp Σ ( C , K ) -c ate gory ∅ (which is the initial o bje ct in Sp Σ ( C , K ) -Cat) to the Sp Σ ( C , K ) -c ate gory 1 Sp Σ ( C ,K ) with one obje ct ∗ and endomorphism ring 1 Sp Σ ( C ,K ) . Deﬁnition 4.12. The set J L := J ′ L ∪ J ′′ L of genera ting trivial coﬁbr ations in Sp Σ ( C , K ) -Cat c onsists of: - the set J ′ L of Sp Σ ( C , K ) -functors obtaine d by applying the functor U to the set J Sp Σ ( C ,K ) of trivial gener ating c oﬁbr ations for the level mo del stru ctur e on Sp Σ ( C , K ) and 10 G ONC ¸ ALO T ABUADA - the set J ′′ L of Sp Σ ( C , K ) -functors obtaine d by applying the functor F 0 to the set J ′′ of trivial gener ating c oﬁbr ations (4.4) in C -Cat. Theorem 4.13. If we let M b e the c ate gory Sp Σ ( C , K ) -Cat, W b e the class W L , I b e the set I L of deﬁnition 4.11 and J t he set J L of deﬁnition 4.12, then the c onditions of the r e c o gnition the or em [1 4, 2.1.19] ar e satisﬁ e d. Thus, the c ate gory Sp Σ ( C , K ) -Cat admits a c oﬁbr ant ly gener ate d Q u il len mo del structu re whose we ak e quivalenc es ar e the levelwise quasi-e quivalenc es. 4.4. Pro of of Theorem 4.13. Observe that the categor y Sp Σ ( C , K )-Cat is com- plete and co co mplete and that the class W L satisﬁes the tw o out of three axio m and is stable under retracts. Since the domains of the generating (trivia l) coﬁbrations in Sp Σ ( C , K ) a r e sequentially s ma ll (see 4.1) the same ho lds by remark C.2, for the domains of Sp Σ ( C , K )-functors in the s ets I ′ L and J ′ L . By rema rk 4.7, this also holds for the r emaining Sp Σ ( C , K )-functors of the sets I L and J L . This implies that the ﬁrst three c onditions of the r ecognition theore m [14, 2.1 .19] are veriﬁed. W e now prove that J L -inj ∩ W L = I L -inj. F or this we intro duce the following auxiliary clas s of Sp Σ ( C , K )-functors : Deﬁnition 4. 14. L et Surj b e the class of Sp Σ ( C , K ) -functors F : A → B su ch that: Sj1) for al l obje cts x, y ∈ A , the morphism F ( x, y ) : A ( x, y ) → B ( F x, F y ) is a trivial ﬁbr ation in Sp Σ ( C , K ) and Sj2) the Sp Σ ( C , K ) -functor F induc es a surje ctive m ap on obje cts. Lemma 4.15. I L -inj = Surj . Pr o of. Notice that by adjunction (see rema rk C.2), a Sp Σ ( C , K )-functor satisﬁes condition S j 1) if and only if it has the R.L.P . with resp ect to the s et I ′ L of ge n- erating coﬁbra tions. Clearly a Sp Σ ( C , K )-functor has the R.L.P . with resp ect to ∅ → 1 Sp Σ ( C ,K ) if and only if it sa tisﬁes condition S j 2). √ Lemma 4.16. Surj = J L -inj ∩ W L . Pr o of. W e prov e ﬁrst the inclusion ⊆ . Let F : A → B b e a Sp Σ ( C , K )-functor which belo ngs to Surj . Clear ly condition S j 1) and S j 2) imply conditions L 1) a nd L 2) a nd so F b elong s to W L . Notice also that a Sp Σ ( C , K )-functor which s atisﬁes condition S j 1 ) has the R.L.P . with resp ect to the set J ′ L of ge ner ating trivial co ﬁbrations. It is then enough to show that F has the R.L.P . with respe ct to the set J ′′ L of generating trivia l coﬁbra tions. By adjunction, this is equiv alent to demand tha t the C -functor E v 0 ( F ) : E v 0 ( A ) → E v 0 ( B ) has the R.L.P . with res p e ct to t he set J ′′ of genera ting tr ivial coﬁbrations in C -Cat. Since F sa tisﬁes co nditio ns S j 1) a nd S j 2), remark 4 .5 implies that E v 0 ( F ) is a triv ial ﬁbratio n in C -Cat and so the claim follows. W e now prov e the inclusion ⊇ . Start by o bserving that a Sp Σ ( C , K )-functor satisﬁes condition S j 1 ) if a nd only if it satisﬁes condition L 1) and it has the R.L.P . with resp ect to the set J ′ L of generating trivial coﬁbrations . No w, let F : A → B be a Sp Σ ( C , K )-functor which b elo ngs to J L -inj ∩ W L . It is then enoug h to show that it satisﬁes co ndition S j 2). Since F has the R.L.P . with resp ect to the set J L of THH AND TC FOR DG CA TEGORIES 11 generating trivial coﬁbra tio ns, the C -functor E v 0 ( F ) : E v 0 ( A ) → E v 0 ( B ) has the R.L.P . with r esp ect to the se t J of genera ting triv ial coﬁbra tio ns in C -Cat. These remarks imply that E v 0 ( F ) is a trivial ﬁbratio n in C -Cat and so by r emark 4.5, the C -functor E v 0 ( F ) induces a surjective map on ob jects. Since E v 0 ( F ) and F induce the same map o n the set o f ob jects, the Sp Σ ( C , K )-functor F satis ﬁe s condition S j 2 ). √ W e now characterize the class J L -inj. Lemma 4.17. A Sp Σ ( C , K ) -functor F : A → B has the R.L.P. with r esp e ct to the set J L of trivial gener ating c oﬁbr ations if and only if it satisﬁes: F1) for al l obje cts x, y ∈ A , the morphism F ( x, y ) : A ( x, y ) → B ( F x, F y ) is a ﬁbr ation in t he level mo del stru ctur e on Sp Σ ( C , K ) and F2) the induc e d C -functor E v 0 ( F ) : E v 0 ( A ) → E v 0 ( B ) is a ﬁbr ation (4.5) in the Qu il len mo del structure on C -C at. Pr o of. Obs erve that a Sp Σ ( C , K )-functor F sa tisﬁes condition F 1) if and only if it ha s the R.L.P . with resp ect to the set J ′ L of generating trivia l coﬁbr ations. By adjunction, F has the R.L.P . with resp ect to the set J ′′ L if a nd only if the C - functor E v 0 ( F ) ha s the R.L.P . with r esp ect to the set J ′′ . In co nclusion F has the R.L.P . with resp ect to the set J L if and only if it satisﬁes conditions F 1 ) a nd F 2) altogether. √ Lemma 4.18. J ′ L -c el l ⊆ W L . Pr o of. Since the class W L is stable under transﬁnite co mpo sitions ([13, 1 0.2.2]) it is enough to prove the following: cons ide r the fo llowing pushout: U ( X ) U ( j )   R / / y A P   U ( Y ) / / B , where j b elongs to the s e t J Sp Σ ( C ,K ) of ge nerating triv ial c o ﬁbrations on Sp Σ ( C , K ). W e need to show that P b elo ngs to W L . Since the morphism j : X − → Y is a trivial coﬁbration in Sp Σ ( C , K ), pr op osition C.3 and pr op osition [14 , 8.3] imply that the sp ectral functor P satisﬁes condition L 1). Since P induces the identit y map o n ob jects, co ndition L 2 ′ ) is automatically sa tisﬁed a nd so P b elongs to W L . √ Prop ositi on 4. 1 9. J ′′ L -c el l ⊆ W L . Pr o of. Since the class W L is sta ble under transﬁnite co mpo sitions, it is enough to prov e the following: consider the following push-o ut F 0 ( G ) F 0 ( V )   R / / y A P   F 0 ( D ) / / B , 12 G ONC ¸ ALO T ABUADA where G V → D belong s to the s e t J ′′ of generating trivia l coﬁbratio ns in C -Cat. W e need to s how that P b elongs to W L . W e start by condition L 1 ). F actor the Sp Σ ( C , K )-functor F 0 ( V ) as F 0 ( G ) → F 0 ( D )(1) ֒ → F 0 ( D ) , where F 0 ( D )(1) is the full Sp Σ ( C , K )-sub categ ory of F 0 ( D ) whos e s e t of ob jects is { 1 } , see c o ndition H 1 ). Consider the iterated pus ho ut: F 0 ( G )   ∼   R / / y A P 0   P   F 0 ( D )(1) / /  _   y e A P 1   F 0 ( D ) / / B . In the lower pusho ut, since F 0 ( D )(1) is a full Sp Σ ( C , K )-sub categ ory o f F 0 ( D ), prop osition [8, 5.2] implies that e A is a full Sp Σ ( C , K )-sub categ ory of B and so P 1 satisﬁes condition L 1). In the upper pushout, by condition H 1 ) the map G (1) / / ∼ / / D (1 ) is a trivial coﬁbration s o it is F 0 ( G ) / / ∼ / / F 0 ( D )(1) . Now, let O deno te the set of ob jects of A and let O ′ := O\ R (1). Obser ve that e A is identiﬁed with the following pushout in Sp Σ ( C , K ) O -Cat (see remar k A.2) ` O ′ F 0 ( G ) ∐ F 0 ( G )   ∼   R / / y A P 0   ` O ′ F 0 ( G ) ∐ F 0 ( D )(1) / / e A . Since the left vertical arrow is a tr ivial coﬁbration so it is P 0 . In particular P 0 satisﬁes co nditio n L 1 ) and so the compos ed dg functor P satisﬁes also condition L 1 ). W e now show that P satisﬁes condition L 2 ′ ). Co nsider the following commuta- tive squar e in Cat [ G ] [ E v 0 ( R )] / / [ V ]   [ E v 0 ( A )] [ E v 0 ( P )]   [ D ] / / [ E v 0 ( B )] . W e need to show that the functor [ E v 0 ( P )] is essentially surjective. Notice that if we restrict o urselves to the ob jects of each categ o ry (ob j( − )) in the previous THH AND TC FOR DG CA TEGORIES 13 diagram, we obtain the following co -cartesie n squa re ob j [ G ] ob j [ V ]   ob j [ E v 0 ( R )] / / y ob j [ E v 0 ( A )] ob j [ E v 0 ( P )]   ob j [ D ] / / ob j [ E v 0 ( B )] in Set . Since V be longs to J ′′ , the functor [ V ] is ess entially surjective. These facts imply , b y a simple diagr am chasing argument, that [ E v 0 ( P )] is also essen tially surjective and so the Sp Σ ( C , K )-functor P satisﬁes condition L 2 ′ ). In conc lus ion, P satisﬁes co nditio n L 1) a nd L 2 ′ ) and so it b elongs to W L . √ W e have s hown that J L -cell ⊆ W L (lemma 4.18 and pro p osition 4.19) and that I L -inj = J L -inj ∩ W L (lemmas 4.15 and 4.16). This implies that the last three conditions of the reco gnition theorem [14, 2 .1.19] are satisﬁed. This ﬁnishes the pro of of theorem 4.13. 4.5. Prop erti e s I. Prop ositi on 4.20. A Sp Σ ( C , K ) -functor F : A → B is a ﬁbr ation with r esp e ct to the mo del structu r e of the or em 4.13, if and only if it satisﬁes c onditions F 1) and F 2 ) of lemma 4.17 . Pr o of. This follows fro m lemma 4 .17, sinc e by the reco gnition theorem [14, 2.1.19 ], the set J L is a se t of g enerating trivial coﬁbr ations. √ Corollary 4.21. A Sp Σ ( C , K ) -c ate gory A is ﬁbr ant with r esp e ct t o the mo del struc- tur e of the or em 4.13, if and only if for al l obje cts x, y ∈ C , the K -sp e ctrum A ( x, y ) is levelwise ﬁbr ant in Sp Σ ( C , K ) (se e [14, 8.2] ). R emark 4.2 2 . No tice that by prop osition 4.20 and r emark 4.5 we have a Quillen adjunction Sp Σ ( C , K )-Cat E v 0   C -Cat . F 0 O O Prop ositi on 4. 23. The Qu il len mo del st ructur e on Sp Σ ( C , K ) -Cat of the or em 4.13 is right pr op er. Pr o of. Cons ider the following pullback square in Sp Σ ( C , K )-Cat A × B C   P / / p C F     A R ∼ / / B with R a levelwise quasi- equiv a lence a nd F a ﬁbration. W e now show that P is a levelwise quasi-e q uiv alence . Notice that pullbacks in Sp Σ ( C , K )-Cat ar e calculated on ob jects and on mo rphisms. Since the level mo del structure on Sp Σ ( C , K ) is right prop er (see 4.1) and F satisﬁes condition F 1), the sp ectral functor P sa tisﬁes condition L 1 ). 14 G ONC ¸ ALO T ABUADA Notice that if we apply the functor E v 0 to the previous diag ram, w e obtain a pullback squa re in C -Cat E v 0 ( A ) × E v 0 ( B ) E v 0 ( C )   E v 0 ( P ) / / p C E v 0 ( F )     E v 0 ( A ) E v 0 ( R ) / / E v 0 ( B ) with E v 0 ( R ) a w eak equiv alence and E v 0 ( F ) a ﬁbration. Now, condition C 3) allow us to conclude that P s atisﬁes co ndition L 2). √ Prop ositi on 4 . 24. L et A b e a c oﬁbr ant Sp Σ ( C , K ) -c ate gory (in t he Quil len mo del structur e of the or em 4.13). Then for al l obje cts x, y ∈ A , the K - sp e ctrum A ( x, y ) is c oﬁbr ant in t he level mo del structure on Sp Σ ( C , K ) . Pr o of. The mo del structure of theorem 4.13 is coﬁbr a ntly g e nerated and so a ny coﬁbrant ob ject in Sp Σ ( C , K )-Cat is a re tract of a I L -cell complex, see coro llary [13, 11.2.2]. Since coﬁbra tions are stable under ﬁltered colimits it is enough to prov e the pr op osition for pushouts along a generating coﬁbration. Let A b e a Sp Σ ( C , K )-categ ory such that A ( x, y ) is coﬁbr ant for all ob jects x, y ∈ A : - Cons ider the following pushout ∅   / / y A   1 Sp Σ ( C ,K ) / / B . Notice that B is obtained from A , by simply introducing a new ob ject. It is then clear that, for all ob jects x, y ∈ B , the K -spectrum B ( x, y ) is coﬁbrant in the level mo del structure on Sp Σ ( C , K ). - Now, conside r the following pushout U ( X ) / / i   y A P   U ( Y ) / / B , where i : X → Y b elongs to the set I Sp Σ ( C ,K ) of generating coﬁbr ations of Sp Σ ( C , K ). Notice that A and B have the same set o f ob jects and P induces the identit y map on the set of o b jects. Since i : X → Y is a coﬁbratio n, prop osition C.4 and prop osition [14, 8.3] imply that the morphis m P ( x, y ) : A ( x, y ) − → B ( x, y ) is s till a co ﬁbration. Since I -ce ll co mplex es in Sp Σ ( C , K )-Cat are built of ∅ (the initial ob ject), the pr op osition is prov en. √ Lemma 4.25. The functor U : Sp Σ ( C , K ) − → Sp Σ ( C , K ) -Cat ( se e C.1 ) sends c oﬁbr ations to c oﬁbr ations. THH AND TC FOR DG CA TEGORIES 15 Pr o of. The mo del structure of theorem 4.13 is coﬁbr a ntly g e nerated and so a ny coﬁbrant o b ject in Sp Σ ( C , K )-Cat is a r ectract o f a (p ossibly inﬁnite) comp os ition of pushouts along the generating coﬁbratio ns. Since the functor U preserves re- tractions, colimits and send the gener a ting coﬁbra tions to (generating) coﬁbrations (see 4.11) the lemma is proven. √ 4.6. Stable quasi-equiv alences. In this section w e consider the sta ble mode l structure on Sp Σ ( C , K ), see [1 4, 8.7]. Reca ll from [14, 7 .2] that since C is a c losed symmetric monoidal category , the category Sp Σ ( C , K ) is tensored, co-tenso red a nd enriched over C . In w ha t follows, we denote by Ho m ( − , − ) this C -enrichmen t a nd by Hom C ( − , − ) the in ternal Hom-o b ject in C . W e will now construct a Quillen mo del str uc tur e on Sp Σ ( C , K )-Cat, whos e weak equiv ale nc e s are as follows. Deﬁnition 4.26. A Sp Σ ( C , K ) -functor F is a stable quas i- equiv a lence if it is a we ak e quivalenc e in the sense of deﬁnition 3.8. Notation 4.2 7 . W e denote b y W S the clas s of sta ble quasi- equiv a lences. 4.7. Q -functor. W e now construct a functor Q : Sp Σ ( C , K )-Cat − → Sp Σ ( C , K )-Cat and a na tural transfor mation η : Id → Q , from the identit y functor on Sp Σ ( C , K )-Cat to Q . Recall from [14, 8.7] the following set of morphis ms in Sp Σ ( C , K ): S := { F n +1 ( C ⊗ K ) ζ C n − → F n C | n ≥ 0 } , where C r uns through the domains and co do mains of the ge ne r ating coﬁbratio ns of C and ζ C n is the adjoint to the map C ⊗ K − → E v n +1 F n ( C ) = Σ n +1 × ( C ⊗ K ) corres p o nding to the ident ity o f Σ n +1 . F or each e le ment of S , we co nsider a facto r- ization F n +1 ( C ⊗ K ) ζ C n / / % % f ζ C n % % L L L L L L L L L L L F n ( C ) Z C n ∼ < < < < x x x x x x x x in the level mo del structure on Sp Σ ( C , K ). Recall fro m [14, 8.7], that the stable mo del structure o n Sp Σ ( C , K ) is the left B ousﬁeld lo caliz a tion of the level mo del structure with resp ect to the morphisms ζ C n (or with resp ect to the morphisms f ζ C n ). Deﬁnition 4.28. L et e S b e the fol lowing set of morphisms in Sp Σ ( C , K ) : i ⊗ f ζ C n : B ⊗ F n +1 ( C ⊗ K ) a A ⊗ F n +1 ( C ⊗ K ) A ⊗ Z C n − → B ⊗ Z C n , wher e i : A → B is a gener ating c oﬁbr ation in C . R emark 4.29 . Since Sp Σ ( C , K ) is a monoidal mo del category , the elements o f the set e S a re co ﬁbrations with coﬁbrant (and sequentially sma ll) doma ins . 16 G ONC ¸ ALO T ABUADA Deﬁnition 4.30. L et A b e a Sp Σ ( C , K ) -c ate gory. The functor Q : Sp Σ ( C , K ) -Cat → Sp Σ ( C , K ) -Cat is obtaine d by applying the smal l obje ct ar gument, using the set J ′ L ∪ U ( e S ) (se e C.1) to factorize t he Sp Σ ( C , K ) -functor A − → • , wher e • denotes the terminal obje ct. R emark 4.31 . W e obtain in this way a functor Q and a na tural transformatio n η : Id → Q . No tice also that Q ( A ) has the sa me o b jects as A , and the R.L.P . with resp ect to the set J ′ L ∪ U ( e S ). Prop ositi on 4.32 . Le t A b e a Sp Σ ( C , K ) -c ate gory which has the right lifting pr op - erty with r esp e ct to the set J ′ L ∪ U ( e S ) . Then it satisﬁes t he fol lowing c onditio n: Ω) for al l obje cts x, y ∈ A , the K -sp e ctrum A ( x, y ) is an Ω -sp e ctrum (se e [14, 8.6] ). Pr o of. By corolla r y 4.2 1, A has the R.L.P . with resp ect to J ′ L if and only if for all ob jects x, y ∈ A , the K -sp ectrum A ( x, y ) is levelwise ﬁbra nt in Sp Σ ( C , K ). Now, suppo se that A has the R.L.P . with resp ect to U ( e S ). Then b y adjunction, for all ob jects x, y ∈ A , the induced mo r phism Hom ( Z C n , A ( x, y ) ) − → Hom( F n +1 ( C ⊗ K ) , A ( x, y )) is a tr ivial ﬁbra tion in C . Notice also that we have the fo llowing w eak equiv alences Hom ( Z C n , A ( x, y ) ) ≃ Hom( F n ( C ) , A ( x, y )) ≃ Hom C ( C, A ( x, y ) n ) and Hom ( F n +1 ( C ⊗ K ) , A ( x, y )) ≃ Hom C ( C ⊗ K , A ( x, y ) n +1 ) ≃ Hom C ( C, A ( x, y ) K n +1 ) . This implies that the induced mo r phisms in C Hom C ( C, A ( x, y ) n ) ∼ − → Hom C ( C, A ( x, y ) K n +1 ) , are weak equiv alences , where C runs thro ugh the domains and co domains o f the generating coﬁbr ations of C . By co ndition C 7), we obta in the weak equiv a lence A ( x, y ) n ≃ Ho m C ( 1 C , A ( x, y ) n ) ∼ − → Hom C ( 1 C , A ( x, y ) K n +1 ) ≃ A ( x, y ) K n +1 . In conclusio n, for all ob jects x, y ∈ A , the K -spectrum A ( x, y ) is an Ω-spe ctrum. √ Prop ositi on 4. 3 3. L et A b e a Sp Σ ( C , K ) -c ate gory. The Sp Σ ( C , K ) -functor η A : A − → Q ( A ) is a stable qu asi-e quivalenc e (4.26). Pr o of. Notice that the e le ment s of the set J Sp Σ ( C ,K ) ∪ e S are trivial coﬁbrations in the sta ble mo del structure on Sp Σ ( C , K ) (see [14, 8.8]). Notice also that the stable mo del str ucture is mono idal ([14, 8.11 ]) and that b y co ndition C 6), it satisﬁes the monoid a xiom. This implies, b y pr op osition C.3, that η A satisﬁes condition W E 1). Since the Sp Σ ( C , K )-categ ories Q ( A ) and A hav e the same set of ob jects condition W E 2 ′ ) is automatically veriﬁed. √ THH AND TC FOR DG CA TEGORIES 17 4.8. Main the o rem. Deﬁnition 4.34. A Sp Σ ( C , K ) -functor F : A → B is: - a Q -weak equiv alence if Q ( F ) is a levelwise quasi-e quivalenc e (4.8). - a c oﬁbration if it is a c oﬁbr ation in the mo del stru ctur e of t he or em 4.13. - a Q - ﬁbr ation if it has the R.L.P. with r esp e ct to al l c oﬁbr ations which ar e Q -we ak e quivalenc es. Lemma 4.35. L et A b e a Sp Σ ( C , K ) -c ate gory which satisﬁ es c ondi tion Ω) of pr op o- sition 4.32. Then the c ate gory [ A ] (se e 3.7) is n atur al ly identiﬁe d with the c ate gory [ E v 0 ( A )] . Pr o of. Reca ll fro m [14, 8.8 ], that a n ob ject in Sp Σ ( C , K ) is stably ﬁbrant if and only if it is a n Ω-s p e ctrum. Since the adjunction Sp Σ ( C , K ) E v 0   C F 0 O O is strong mo no idal, we hav e the following identiﬁcations [ 1 Sp Σ ( C ,K ) , A ( x, y ) ] ≃ [ 1 C , E v 0 ( A ( x, y ))] , x, y ∈ A , which imply the lemma. √ Corollary 4. 3 6. L et F : A → B b e a Sp Σ ( C , K ) -functor b etwe en Sp Σ ( C , K ) - c ate gories which satisfy c ondition Ω) . Th en F satisﬁes c ondition W E 2) if and only if E v 0 ( F ) satisﬁes c onditio n W E 2) . Prop ositi on 4.37. L et F : A → B b e st able quasi-e qu ivalenc e b etwe en Sp Σ ( C , K ) - c ate gories which satisfy the c ondition Ω) . Then F i s a levelwise quasi-e quivalenc e (4.8). Pr o of. Since F satisﬁes condition W E 1) and A and B satisfy condition Ω ), lem- mas [1 3, 4.3.5-4 .3.6] imply that F sa tisﬁes c o ndition L 1). By coro llary 4.36, the Sp Σ ( C , K )-functor F satisﬁes condition W E 2) if and only if it satisﬁes condition L 2). This prov es the pro p osition. √ Lemma 4.38. A Sp Σ ( C , K ) -functor F : A → B is a Q -we ak e quiva lenc e if and only if it is a stable qu asi-e quivalenc e. Pr o of. W e have at o ur disp osal a commutativ e square A F   η A / / Q ( A ) Q ( F )   B η B / / Q ( B ) where the Sp Σ ( C , K )-functors η A and η B are sta ble quasi-equiv alences by prop os i- tion 4.33. Since the class W S satisﬁes the t w o o ut of three axiom, the Sp Σ ( C , K )- functor F is a stable quasi-equiv alence if and only if Q ( F ) is a stable quasi- equiv ale nc e . The Sp Σ ( C , K )-categ ories Q ( A ) and Q ( B ) satisfy condition Ω) and so by propo sition 4.37, Q ( F ) is a levelwise q uasi-equiv alence. This prov es the lemma. √ 18 G ONC ¸ ALO T ABUADA Theorem 4.39. The c ate gory Sp Σ ( C , K ) -Cat admits a right pr op er Quil len mo del structur e whose we ak e qu ivalenc es ar e the stable quasi-e quivalenc es (4.26) and t he c oﬁbr ations those of the or em 4.13. Pr o of. The pr o of will consist on verifying the conditions o f theor em B.2. W e con- sider for M the Quillen mo del structure o f theorem 4 .13 and for Q and η , the functor a nd natural transformation deﬁned in 4.3 0. The Quillen mo del structure of theorem 4.13 is right pr op er (see 4.23) and by lemma 4.38 the Q -weak equiv alences are precisely the stable qua s i-equiv a lences. W e now v erify conditions (A1), (A2) and (A3): (A1) Le t F : A → B b e a levelwise quasi-equiv a lence. W e hav e the following commutativ e sq uare A F   η A / / Q ( A ) Q ( F )   B η B / / Q ( B ) with η A and η B stable q uasi-equiv alences. Notice that s inc e F is a levelwise quasi-equiv alence it is in par ticular a stable quas i- equiv a lence. The class W S satisﬁes the thre e out of three axiom a nd so Q ( F ) is a stable quasi- equiv ale nc e . Since the Q ( A ) a nd Q ( B ) satis fy condition Ω), propo s ition 4.37 implies that Q ( F ) is in fact a levelwise quasi- equiv a lence. (A2) W e now show that for every Sp Σ ( C , K )-categ ory A , the Sp Σ ( C , K )-functors η Q ( A ) , Q ( η A ) : Q ( A ) → QQ ( A ) are levelwise quasi- e quiv alenc e s . Since the Sp Σ ( C , K )-functors η Q ( A ) and Q ( η A ) are stable q uasi-equiv alences b e t ween Sp Σ ( C , K )-categ ories which satisfy co nditio n Ω), prop os ition 4.37 implies that they ar e levelwise quas i- equiv ale nc e s . (A3) W e star t b y observing that if P : C → D is a Q -ﬁbratio n, then for all ob jects x, y ∈ C , the mor phism P ( x, y ) : C ( x, y ) − → D ( P x, P y ) is a Q -ﬁbratio n in Sp Σ ( C , K ). In fact, by pro p osition 4.2 5, the functor U : Sp Σ ( C , K ) − → Sp Σ ( C , K )-Cat sends coﬁbr ations to coﬁbra tions. Since clear ly it sends stable eq uiv alences in Sp Σ ( C , K ) to stable quasi-equiv alences in Sp Σ ( C , K )-Cat, the claim fol- lows. Now consider the dia gram A × Q ( A ) B   / / p B P   A η A / / Q ( A ) , THH AND TC FOR DG CA TEGORIES 19 with P a Q -ﬁbration. The stable mo del structure o n Sp Σ ( C , K ) is right prop er by condition C 6), and so we conclude, by construction of ﬁb er pr o d- ucts in Sp Σ ( C , K )-Cat, that the induced Sp Σ ( C , K )-functor η A ∗ : A × Q ( A ) B − → B satisﬁes condition W E 1). Since η A induces the identit y ma p on ob jects so th us η A ∗ , and so condition W E 2 ′ ) is pr ov en. √ 4.9. Prop erti e s I I. Prop ositi on 4.40 . A Sp Σ ( C , K ) -c ate gory A is ﬁbr ant with r esp e ct to the or em 4.39 if and only if for al l obje cts x, y ∈ A , the K -sp e ctrum A ( x, y ) is an Ω -sp e ctru m in Sp Σ ( C , K ) . Notation 4.4 1 . W e denote these ﬁbr a nt Sp Σ ( C , K )-categ ories by Q -ﬁ br ant . Pr o of. By cor o llary B.4, A is ﬁbra nt with resp ect to theo rem 4.3 9 if and only if it is ﬁbrant fo r the structure of theorem 4.13 (see cor ollary 4.21) and the Sp Σ ( C , K )- functor η A : A → Q ( A ) is a levelwise quas i-equiv a lence. Observe that η A is a levelwise quasi-e q uiv alence if and only if fo r all ob jects x, y ∈ A the morphism η A ( x, y ) : A ( x, y ) − → Q ( A )( x, y ) is a level eq uiv alence in Sp Σ ( C , K ). Since Q ( A )( x, y ) is an Ω-sp ectrum we hav e the following commutativ e diagra ms (for all n ≥ 0) A ( x, y ) n   f δ n / / A ( x, y ) K n +1   Q ( A )( x, y ) n f δ n / / Q ( A )( x, y ) K n +1 , where the b ottom and vertical arr ows a re weak equiv a lences in C . This implies that e δ n : A ( x, y ) n − → A ( x, y ) K n +1 , n ≥ 0 is a w eak equiv alenc e in C and so, fo r a ll x, y ∈ A , the K - sp ectrum A ( x, y ) is a n Ω-sp ectrum in Sp Σ ( C , K ). √ R emark 4.4 2 . Remark 4.31 and prop os itions 4.32 and 4.3 3 imply that η A : A → Q ( A ) is a (functor ial) ﬁbrant repla cement of A in the mo del structur e of theo- rem 4.39. In pa rticular η A induces the identit y map on the set of ob jects. Notice also, that since the co ﬁbrations (and so the trivial ﬁbrations) of the mode l structure of theorem 4.39 a re the s ame as those of theor em 4.1 3, prop os itio n 4.24 and lemma 4.25 stay v a lid in the Quillen mo de l structure o f theorem 4 .39. Prop ositi on 4.43. L et F : A → B is a Sp Σ ( C , K ) -functor b et we en Q -ﬁbr ant Sp Σ ( C , K ) -c ate gories. Then F is a Q -ﬁbr ation if and only if it is a ﬁ br ation. 20 G ONC ¸ ALO T ABUADA Pr o of. By theorem B .3, if F is a Q - ﬁbration then it is a ﬁbr ation. Let us now prov e the conv erse. Supp ose that F : A → B is a ﬁbration. Consider the commutativ e square A F   η A / / Q ( A ) Q ( F )   B η B / / Q ( B ) . Since A and B satisfy the condition Ω), prop os itio n 4.37 implies that η A and η B are levelwise quasi-eq uiv alences . Since the mo del str ucture of theorem 4.13 is right prop er (see 4.23), the prev ious square is homotopy ca rtesian and so b y theorem B.3, F is in fact a Q - ﬁbration. √ R emark 4 .44 . Since a Q -ﬁbra tion is a ﬁbration, the a djunction of remark 4.2 2 induces a na tural Quillen adjunction Sp Σ ( C , K )-Cat E v 0   C -Cat F 0 O O with res pe c t to the mo del str ucture of theorem 4 .3 9. 4.10. Ho m otop y Idemp otence. In [1 4, 9.1], Ho vey pr ov ed the following idem- po tence prop er t y: S) If the functor − ⊗ K : C − → C is a Quillen equiv alence . Then the Q uillen a djunction Sp Σ ( C , K ) E v 0   C , F 0 O O is a Quillen equiv alence, wher e Sp Σ ( C , K ) is endow ed with the stable mo del struc- ture. Theorem 4.45. If c onditon S ) is veriﬁe d then the Qu il len adjunction of r emark 4 .44 Sp Σ ( C , K ) -Cat E v 0   C -Ca t F 0 O O is a Qu il len e quivalenc e. Pr o of. The pro of will co nsist on verifying co nditions a ) and b ) of prop osition 3.13. a) Let F : A → B be a Sp Σ ( C , K )-functor b etw een Q -ﬁbrant Sp Σ ( C , K )- categorie s, such that E v 0 ( A ) : E v 0 ( A ) → E v 0 ( B ) is a weak equiv alence in THH AND TC FOR DG CA TEGORIES 21 C -Cat. Since A and B satisfy condition Ω), condition a ) of prop os ition 3 .13 applied to the Quillen equiv alence E v 0 : Sp Σ ( C , K ) − → C implies that F satis ﬁe s condition W E 1). Notice also that since E v 0 ( F ) is a weak equiv alence in C -Cat, corolla ry 4.36 implies that F satisﬁes condition W E 2). In conclusion F is a stable qua si-equiv alence. b) Let A b e a coﬁbrant C -ca teg ory a nd let us denote by I its set of ob jects. Notice, that in particular A is coﬁbr ant in C I -Cat, with r esp ect to the mo del str ucture of remark A.2. Now, since the adjunction Sp Σ ( C , K ) E v 0   C , F 0 O O is a stro ng monoidal Quillen equiv a lence (3.6), remark A.3 implies that condition b ) of prop osition 3.13 is v eriﬁed for the adjunction ( F I 0 , E v I 0 ). By remark 4.42, the Sp Σ ( C , K )-functor η A : A → Q ( A ) is a ﬁbr ant reso lution in Sp Σ ( C , K ) I and so we co nclude that condition b ) is veriﬁed. √ R emark 4.46 . Clea rly the ﬁrst exa mple of 4.2 sa tisﬁes condition S ), since the func- tor − ⊗ Z [1] : C h → C h is the classica l susp ens io n pro cedure. This implies, by theorem 4.4 5, that we hav e a Quillen eq uiv alence Sp Σ ( C h, [ 1])-Cat E v 0   dgcat . F 0 O O 5. A Quill en equ iv al ence criterion Let C and D b e tw o monoidal mo del categor ies with K an ob ject of C and K ′ an o b ject of D . In this chapter w e esta blish a general criterion for a Quillen equiv ale nc e b etw e e n Sp Σ ( C , K ) and Sp Σ ( D , K ′ ) to induce a Quillen equiv alence betw een Sp Σ ( C , K )-Cat and Sp Σ ( D , K ′ )-Cat (see theo r em 5 .1). Suppo se that both C , K , C -Cat and D , K ′ , D -Cat satisfy conditions C 1)- C 7) and H 1)- H 3) fr o m the previous chapter. Supp ose moreov er, that we hav e a weak monoidal Quillen equiv a le nce (3.6) Sp Σ ( C , K ) N   Sp Σ ( D , K ′ ) L O O with resp ec t to the stable mo del structures (see [14, 8.7 ]), whic h preser ves ﬁbra- tions and lev elwise weak eq uiv alences. By pro po sition A.4, w e have the following 22 G ONC ¸ ALO T ABUADA adjunction Sp Σ ( C , K )-Cat N   Sp Σ ( D , K ′ )-Cat . L cat O O Theorem 5.1. If ther e exists a fun ct or e N : C -Cat − → D - Cat which pr eserves ﬁbr ations and we ak e quivalenc es and makes t he fol lowing diagr am C -Cat e N   Sp Σ ( C , K ) -Cat E v 0 o o N   D - Cat Sp Σ ( D , K ′ ) -Cat E v 0 o o c ommu te, then the adjunction ( L cat , N ) is a Quil len adjunction ( with r esp e ct to the mo del st ructur e of the or em 4.39). Mor e over, if e N satisﬁes the fol lo wing ‘r eﬂexion ’ c ondition: R) If the D - functor e N ( F ) : e N ( A ) → e N ( B ) satisﬁes c ondition W E 2) , so do es the C -functor F : A → B . Then the Qu il len adjunction ( L cat , N ) is a Quil len e qu ivalenc e. Pr o of. W e start by proving tha t the adjunction ( L cat , N ) is a Quillen adjunction with resp ec t to the Quillen mo de l structure of theor em 4.1 3. Obser ve that since the functor N : Sp Σ ( C , K ) − → Sp Σ ( D , K ′ ) preserves ﬁbrations and levelwise weak equiv ale nc e s , pro po sition 4.20, deﬁnition 4.8 and the ab ov e commutativ e sq uare imply that N : Sp Σ ( C , K )-Cat − → Sp Σ ( D , K ′ )-Cat preserves ﬁbratio ns and levelwise quasi-eq uiv alences . W e now show that this adjunction induces a Quillen adjunction on the mo del structure of theorem 4.39. F o r this we use pr op osition 3.12. The functor N c le arly preserves trivial ﬁbra tions since they are the s ame in the mo del structures of theo- rems 4.1 3 and 4.39. Now, let F : A → B b e a Q -ﬁbratio n be t ween Q-ﬁbra nt ob jects (see 4.4 0). B y prop osition 4.43, F is a ﬁbra tion a n so thus N ( F ). Since the co ndition Ω) is clear ly preserved b y the functor N , prop osition 4.43 implies that the Sp Σ ( D , K ′ )-functor N ( F ) is in fact a Q -ﬁbration. W e now show that if the ‘r eﬂexion’ condition R ) is satisﬁed, the adjunction ( L cat , N ) is in fact a Quillen eq uiv alence. F or this, we verify conditions a ) and b ) of prop os itio n 3.13. a) Let F : A → B b e a Sp Σ ( C , K )-functor b etw een Q -ﬁbrant ob jects such that N ( F ) is a stable quasi-equiv alence in Sp Σ ( D , K ′ )-Cat. Since condition Ω) is clearly preser ved by the functor N , pr op osition 4.37 implies that N ( F ) is in fact a levelwise quasi-equiv alence. Since A a nd B are Q -ﬁbr ant, condition a ) of prop os itio n 3 .13 a pplied to the Quillen equiv alence N : Sp Σ ( C , K ) − → Sp Σ ( D , K ′ ) THH AND TC FOR DG CA TEGORIES 23 implies tha t F satisﬁes condition W E 1). Now, co ndition R ) and the a b ove commutativ e square imply that it s atisﬁes condition L 2 ′ ). By prop osi- tion 4 .37, we conclude that F is a s ta ble quasi-e quiv alence . b) Let A b e a c oﬁbrant Sp Σ ( D , K ′ )-categor y a nd let us denote by I its s et of ob jects. Notice, that in pa rticular A is coﬁbrant in Sp Σ ( D , K ′ ) I -Cat, with resp ect to the module structure of remar k A.2. Since the a djunction ( L, N ) is a stro ng monoidal Quillen equiv a lence (3.6), remark A.3 implies that condition b ) of prop ositio n 3.13 is veriﬁed for the adjunction ( L I , N I ). By remark 4.42, the Sp Σ ( C , K )-functor η A : A → Q ( A ) is a ﬁbr ant reso lution in Sp Σ ( C , K ) I and so we co nclude that condition b ) is veriﬁed. √ 6. E xamples of Quillen equiv alences In this chapter we describ e tw o situations, where the conditions o f theorem 5.1 are satisﬁed. First situation: W e consider the relations hip betw een the ex a mples 1) and 2) describ ed in 4.2 and 4.6: reca ll from [26, 4.9], tha t the inclusion i : C h ≥ 0 → C h induces a s tr ong mono idal Quillen e q uiv alence Sp Σ ( C h, Z [1]) τ ≥ 0   Sp Σ ( C h ≥ 0 , Z [1 ]) , i O O with resp ect to the stable mo del structur e s. Moreov er, the tr uncation functor τ ≥ 0 preserves ﬁbr a tions and lev elwise weak equiv alences. By remark A.5, we obtain a adjunction Sp Σ ( C h, Z [1])-Ca t τ ≥ 0   Sp Σ ( C h ≥ 0 , Z [1 ])-Cat . i O O R emark 6.1 . Notice that we hav e the following commutativ e sq uare dgcat τ ≥ 0   Sp Σ ( C h, Z [1])-Ca t τ ≥ o   E v 0 o o dgcat ≥ 0 Sp Σ ( C h ≥ 0 , Z [1 ])-Cat . E v 0 o o Finally , since the functor τ ≥ 0 : dgcat − → dgcat ≥ 0 satisﬁes the ‘reﬂexion’ co ndition R ) o f theore m 5.1, the conditions of theo rem 5.1 are satisﬁed a nd so the pr evious adjunction is a Quillen equiv alence. Second situation: W e co nsider the relationship b etw een the examples 2) and 3) descr ib ed in 4 .2 and 4 .6: recall from [26, 4.4], that we have a weak mo noidal 24 G ONC ¸ ALO T ABUADA Quillen equiv alence Sp Σ ( s Ab , e Z ( S 1 )) φ ∗ N   Sp Σ ( C h ≥ 0 , Z [1 ]) , L O O with r esp ect to the stable mo del structures. Moreov er the functor φ ∗ N preserves ﬁbrations a nd levelwise weak equiv alences (see the pro of of [26, 4.4]). By pro po si- tion A.4, we hav e the adjunction Sp Σ ( s Ab , e Z ( S 1 ))-Cat φ ∗ N   Sp Σ ( C h ≥ 0 , Z [1 ])-Cat . L cat O O R emark 6.2 . Recall from [2 6, 4.4 ] that the r ing map φ : Sym( Z [1]) − → N is the identit y in degr ee z ero. This implies that we hav e the following commutativ e square s Ab -Cat N   Sp Σ ( s Ab , e Z ( S 1 ))-Cat E v 0 o o φ ∗ N   dgcat ≥ 0 Sp Σ ( ch ≥ 0 , Z [1 ])-Cat . E v 0 o o Finally , since the norma liz a tion functor (see section 5 . 2 of [3 0]) N : s Ab -Cat − → dgcat ≥ 0 satisﬁes the ‘reﬂex ion’ condition R ) o f theorem 5.1, b o th conditions of theo rem 5.1 are satisﬁed a nd so the pr evious adjunction ( L cat , φ ∗ N ) is a Quillen equiv alence . 7. General Spectral al gebra Let R b e a co mmut ative symmetric ring sp ectrum of p ointed simplicial sets (see [24, I-1.3]). In this c hapter, we construct a Q uille n mo del structure on the ca tegory of small ca tegories enr iched ov er R -mo dules (see theorem 7.25). Notation 7.1 . W e denote by ( R -Mo d , − ∧ R − , R ) the symmetric mo noidal category of right R -mo dules (see [24, I- 1.5]). Let us now reca ll some cla ssical results of sp ectr al alg e bra. Theorem 7 .2. [24, I I I-3.2 ] The c ate gory R -Mo d c arri es the fol lowi ng mo del struc- tur es: - The pro jective level mo del structure , in which the we ak e quivalenc es (r esp. ﬁbr atio ns) ar e those morphisms of R -mo dules which ar e pr oje ctive level e quivalenc es (r esp. pr oje ctive level ﬁbr ations) on the u nderlying symmetric sp e ctr a. THH AND TC FOR DG CA TEGORIES 25 - The pro jectiv e s table model structure , in which the we ak e quival enc es (r esp. ﬁbr atio ns) ar e those morphisms of R -mo dules which ar e pr oje ctive stable e quivalenc es (r esp. pr oje ctive stable ﬁbr ations) on the u nderlying symmet ric sp e ctr a. Mor e over these mo del structu r es ar e pr op er, s implicia l, c oﬁbr antly gener ate d, monoidal (with r esp e ct to the smash pr o duct over R ) and satisfy the monoid axiom. Now, le t f : P → R b e a morphism ([24, I-1.4]) o f commutativ e symmetric ring sp ectra. R emark 7.3 . W e have at o ur disp osa l a re striction/extensio n o f scalar s adjunction R -Mo d f ∗   P - Mo d . f ! O O Given a R -mo dule M , we deﬁne a P - mo dule f ∗ ( M ) as the same underlying symmetric sp ectrum as M a nd with the P -action given by the comp osite ( f ∗ M ) ∧ P = M ∧ P I d ∧ f − → M ∧ R α − → M , where α denotes the R -a c tion o n M . The left a djoint functor f ! takes an P - mo dule N to the R - mo dule N ∧ P R with the R -action given by f ! N ∧ R = N ∧ P R ∧ R I d ∧ µ − → N ∧ P R , where µ deno tes the multiplication o f R . Theorem 7.4. [24, I I I-3.4] The pr evio us adjuncton ( f ! , f ∗ ) is a Q uil len adjunction with r esp e ct to the mo del structu r es of the or em 7.2. R emark 7.5 . In the a djunction ( f ! , f ∗ ), the functor f ! is stro ng mono idal a nd the functor f ∗ is lax mo noidal (see deﬁnition 3.3). W e are now interested in the par ticula r case wher e P is the sphere s ymmetric ring sp ectrum S (see [24, I-2 .1 ]) and f = i the unique mo rphism of ring spec tra. W e hav e the restriction/ extension of sca lars adjunction R -Mo d i ∗   Sp Σ , i ! O O where Sp Σ denotes the categor y o f symmetric sp ectra [1 6]. Notation 7.6 . In what follows and to simply the notation, we denote by R -Cat the category ( R -Mo d)-Cat (see notation 3.1). In particular a ( R -Mo d)-catego ry will b e denoted by R -ca tegory and a ( R -Mo d)-functor b y R -functor. Notice that by remar k A.5, the pr evious a djunction ( i ! , i ∗ ) induces the adjunction R -Cat i ∗   Sp Σ -Cat , i ! O O 26 G ONC ¸ ALO T ABUADA where Sp Σ -Cat denotes the category of spe c tral categ ories (see [2 9, 3.3 ]). 7.1. Lev elwise quasi -equiv alences. In this section, we will construct a c o ﬁ- brantly genera ted Quillen mo del structure on R -Cat, who se weak equiv alences ar e as follows: Deﬁnition 7.7. A R -functor F : A → B is a levelwise quasi-equiv alence if t he r est ricte d sp e ctr al fun ctor i ∗ ( F ) : i ∗ ( A ) → i ∗ ( B ) is a levelwise qu asi-e quivalenc e on Sp Σ -Cat (se e [29, 4 .1] ). Notation 7.8 . W e denote by W L the class of le velwise quasi-equiv alences on R -Cat. R emark 7.9 . Rec all fro m [29, 4.8] that the category Sp Σ -Cat carr ies a coﬁbrantly generated Quillen mo del structure whos e weak equiv a lences a re prec isely the level- wise quasi- equiv a le nc e s. Theorem 7.10. If we let M b e the c ate gory Sp Σ -Cat, endowe d with the mo del structur e of the or em [2 9, 4.8 ] , and N b e the c ate gory R -Cat, then the c onditio ns of the lifting the or em [13, 11.3 .2] ar e satisﬁe d. Thus, the c ate gory R -Cat admits a c oﬁbr antly gener ate d Quil len mo del structur e whose we ak e quivalenc es ar e the levelwise quasi-e quival enc es. R emark 7.1 1 . Since the Quillen mo del structure on Sp Σ -Cat is right prop er (se e [29, 4.19]), the s ame holds for the mo del str ucture on R -Cat given by theorem 7 .10. R emark 7.1 2 . B y remar k 7.11 and c o rollar y [29, 4.16], a R - categor y A is ﬁbrant if a nd o nly if for all o b jects x, y ∈ A a nd n ≥ 0, the simplicial set i ∗ ( A )( x, y ) n is ﬁbrant. 7.2. Pro of of theorem 7.10. W e start b y obs e rving that the catego ry R -Cat is complete and co complete. Since the restr ic tion functor i ∗ : R -Cat − → Sp Σ -Cat preserves ﬁltered colimits a nd the domains a nd c o domains of the e le ments of the generating (trivial) coﬁbratio ns in Sp Σ -Cat are small (see sections 4 and 5 of [2 9]), condition (1) of the lifting theore m [13, 1 1.3.2] is veriﬁed. W e now prove condition (2). Recall fro m [29, 4.5], tha t the set J of g e nerating trivial co ﬁbrations in Sp Σ -Cat consis ts of a (disjoint) union J ′ ∪ J ′′ . Lemma 7.13. i ! ( J ′ ) -c el l ⊆ W L . Pr o of. Since the cla ss W L is stable under transﬁnite comp ositions it is enough to prov e the following: let S : i ! U ( F m Λ[ k , n ] + ) − → A be a R -functor and consider the following pushout in R -Cat i ! U ( F m Λ[ k , n ] + ) S / / i ! ( A m,k,n )   y A P   i ! U ( F m ∆[ n ] + ) / / B . W e need to show that P b elongs to W L . Observe that the R -functor i ! ( A m,k,n ) ident iﬁes with the R - functor U ( i ! ( F m Λ[ k , n ] + )) − → U ( i ! ( F m ∆[ n ] + )) . THH AND TC FOR DG CA TEGORIES 27 Since the mo rphisms i ! ( F m Λ[ k , n ] + ) − → i ! ( F m ∆[ n ] + ) are triv ial c o ﬁbrations in the pr o jectiv e level mo del s tructure of theo rem 7.2, pro p o - sition C.3 and theorem 7 .2 imply that the sp ectra l functor i ∗ ( P ) satisﬁes c o ndition L 1). Since P induces the identit y map o n ob jects, the spectr al functor i ∗ ( P ) sat- isﬁes condition L 2 ′ ) and so P belo ngs to W L . √ Lemma 7.14. i ! ( J ′′ ) -c el l ⊆ W L . Pr o of. Since the class W L is sta ble under transﬁnite co mpo sitions, it is enough to prov e the following: let S : i ! S = R − → A be a R -functor (wher e R is the category with one ob ject a nd endomorphism r ing R ) and consider the following pushout in R -cat R S / / i ! ( A H )   y A P   i ! Σ ∞ ( H + ) / / B . W e need to s how that P belong s to W L . W e start by showing that the spectr a l functor i ∗ ( P ) satisﬁes condition L 1). O bserve that the pro o f of condition L 1) is ent irely analo gous to the pro o f of c o ndition L 1 ) in prop osition 4.1 9: simply replace the sets of Sp Σ ( C , K )-functors J ′ and J ′′ by the sp ectr al functor s i ! ( A m,n,k ) and i ! ( A H ) and the set I (of gener a ting coﬁbr ations in Sp Σ ( C , K )-Cat) by the set i ! ( I ). W e now show that the s pe c tr al functor i ∗ ( P ) satisﬁes condition L 2 ′ ). Notice that the ex istence o f the sp ec tr al functor Σ ∞ ( H + ) − → i ∗ i ! Σ ∞ ( H + ) , given by the unit of the adjunction ( i ! , i ∗ ), implies that the le ft vertical spectr al functor in the commutativ e squar e i ∗ R i ∗ S / / i ∗ i ! ( A H )   i ∗ ( A ) i ∗ ( P )   i ∗ i ! Σ ∞ ( H + ) / / i ∗ ( B ) , satisﬁes condition L 2 ′ ). Notice also , that if we r estrict ourselves to the ob jects of each categor y (ob j( − )) in the previous diag ram, we obtain a co-cartes ia n s quare ob j ( i ∗ R ) / /   y ob j( i ∗ ( A )   ob j ( i ∗ i ! Σ ∞ ( H + )) / / ob j ( i ∗ ( B )) in Set . These facts clearly imply tha t i ∗ ( P ) satisﬁes condition L 2 ′ ) and so w e conclude that P b elo ngs to W L . √ Lemmas 7 .13 and 7.1 4 imply co ndition (2) of theorem [13, 11 .3 .2] and so theo- rem 7.10 is proven. 28 G ONC ¸ ALO T ABUADA Lemma 7.15. The functor U : R -Mo d − → R -Cat ( se e C.1 ) sends pr oje ctive c oﬁbr ations t o c oﬁbr ations. Pr o of. The mo del s tr ucture of theo r em 7 .10 is coﬁbrantly gene r ated and so any coﬁ- brant ob ject in R -Cat is a r ectract of a (p ossibly inﬁnite) comp os ition o f pushouts along the g e nerating coﬁbra tions. Since the functor U preserves retractions , co lim- its and send the ge nerating pro jective coﬁbr ations to (genera ting) coﬁbra tions the lemma is prov en. √ 7.3. Stable quasi-equiv alences. In this s ection, we consider the pr o jectiv e stable mo del structure on R - Mo d (see 7.2). W e will co nstruct a Quillen mo del s tructure on R -Cat, who se weak equiv a lences ar e a s follows: Deﬁnition 7.16. A R -functor F : A → B is a s ta ble qua si-equiv alence if it is an we ak e quivalenc e in the sense of deﬁnition 3.8. Notation 7.1 7 . W e denote b y W S the clas s of sta ble quasi- equiv a lences. Prop ositi on 7.18 . A R -fun ctor F : A → B is a stable quasi-e quivalenc e if and only if the r estricte d sp e ctr al funct or i ∗ ( F ) : i ∗ ( A ) → i ∗ ( B ) is a st able quasi-e qu ivalenc e in Sp Σ -Cat (se e [29, 5.1] ). Pr o of. W e start by no ticing that the deﬁnition o f stable equiv alence in R -Mo d (see 7 .2 ), implies that a R -functor F satisﬁes condition W E 1) if a nd only if the sp ectral functor i ∗ ( F ) sa tisﬁes condition S 1) of deﬁnition [29, 5.1]. Now, let A b e a R - categor y . Obse rve that, since the functor i ! : Sp Σ − → R -Mo d is str ong monoidal, the categ ory [ A ] (see 3.7) naturally identiﬁes with [ i ∗ ( A )] and so F satisﬁes co ndition W E 2) if a nd only if i ∗ ( F ) sa tisﬁes condition S 2). This pr ov es the pro p osition. √ 7.4. Q -functor. W e now construct a functor Q : R -Cat − → R - Cat and a natural trans fo rmation η : I d → Q , from the iden tit y functor on R -Cat to Q . R emark 7.19 . Recall fro m [29, 5.5] that we hav e at our dispo sal a set { A m,k,n } ∪ U ( K ) of sp ectr al functors used in the constr uction of a Q -functor in Sp Σ -Cat. Deﬁnition 7.20. L et A b e a R -c ate gory. The functor Q : R -Cat − → R -Cat is obtaine d by applying the smal l obje ct ar gument, using the set { i ! ( A m,k,n ) } ∪ i ! U ( K ) , to factorize t he R -functor A − → • , wher e • denotes the terminal obje ct in R -Cat. R emark 7.21 . W e obtain in this way a functor Q and a na tural transformatio n η : I d → Q . Notice that the set { i ! ( A m,k,n ) } ∪ i ! U ( K ) ident iﬁes natura lly with the set { U ( i ! ( A m,n,k )) } ∪ U ( i ! ( K )). Notice also that Q ( A ) ha s the same ob jects as A , and the R.L.P with r esp ect to the set { U ( i ! ( A m,n,k )) } ∪ U ( i ! ( K )). By adjunction and rema rk [29, 5 .7] this is equiv alent to the following fact: the spe c tral catego ry i ∗ Q ( A ) satisﬁes the following conditio n THH AND TC FOR DG CA TEGORIES 29 Ω) for all ob jects x, y ∈ i ∗ Q ( A ) the symmetric sp ectra ( i ∗ Q )( x, y ) is an Ω - sp ectrum. Prop ositi on 7. 2 2. L et A b e a R -c ate gory. The R -functor η A : A − → Q ( A ) is a stable qu asi-e quivalenc e (7.16). Pr o of. W e star t by observing that since the elements of the set { A m,n,k } ∪ K a re trivial coﬁbr ations in the pr o jectiv e stable model structure on Sp Σ , the elemen ts of the set { i ! ( A m,n,k ) } ∪ i ! ( K ) ar e trivial co ﬁbrations in the stable mo del structure on R -Mo d (see 7 .2). By theor em 7.2 this mo del str ucture is monoidal and satis ﬁes the monoid ax iom. This implies, by prop osition C.3, that the sp ectra l functor η A satisﬁes co ndition W E 1). Since Q ( A ) a nd A hav e the sa me set of o b jects condition W E 2 ′ ) is automatically veriﬁed and so we conclude that η A belo ngs to W S . √ 7.5. Main the o rem. Deﬁnition 7.23. A R -funct or F : A → B is: - a Q -weak equiv alence if Q ( F ) is a levelwise quasi-e quivalenc e (se e 7.7). - a c oﬁbration if it is a c oﬁbr ation in the mo del stru ctur e of t he or em 7.10 . - a Q - ﬁbr ation if it has the R.L.P. with r esp e ct to al l c oﬁbr ations which ar e Q -we ak e quivalenc es. Lemma 7. 24. A R -fun ctor F : A → B is a Q -we ak e quivalenc e if and only if it is a stable quasi-e quivalenc e (7.16). Pr o of. W e have at o ur disp osal a commutativ e square A η A / / F   Q ( A ) Q ( F )   B η B / / Q ( B ) where the R -functors η A and η B are sta ble quasi-eq uiv alence s by prop ositio n 7.22. If we apply the restriction functor i ∗ to the above squar e , we obta in i ∗ A i ∗ ( F )   i ∗ ( η A ) / / i ∗ Q ( A ) i ∗ Q ( F )   i ∗ B i ∗ ( η B ) / / i ∗ Q ( B ) , where the sp ectra l functor s i ∗ ( η A ) and i ∗ ( η B ) are stable quasi-equiv ale nces in Sp Σ -Cat (s e e 7.18) and the sp ectral categorie s i ∗ Q ( A ) and i ∗ Q ( B ) satisfy condition Ω) (see 7 .21). Now, we co nclude: F is a weak equiv alence if and only if i ∗ ( F ) is a stable quasi- equiv a lence (7.18); if and only if i ∗ Q ( F ) is a stable q uasi-equiv alence; if a nd o nly if i ∗ Q ( F ) is a le velwise q uasi-equiv alence (theorem [29, 5.1 1 -(A1)]); if and only if F is a Q -weak equiv alence. √ Theorem 7.2 5. The c ate gory R -Cat admits a right pr op er Quil len mo del structur e whose we ak e quivalenc es ar e the stable quasi-e quiva lenc es (7.16) and the c oﬁbr ations those of the or em 7.10. 30 G ONC ¸ ALO T ABUADA Pr o of. The pr o of will consist on verifying the conditions o f theor em B.2. W e con- sider for M the Quillen mo del structure o f theorem 7 .10 and for Q and η , the functor and natur a l transformation deﬁned in 7.20. The Q uille n mo del structure of theorem 7.1 0 is right prop er (7.11) and b y lemma 7.24, the Q -weak equiv alences are precisely the stable qua s i-equiv a lences. W e now v erify conditions (A1), (A2) and (A3): (A1) Le t F : A → B b e a levelwise quasi-equiv a lence. W e hav e the following commutativ e sq uare A F   η A / / Q ( A ) Q ( F )   B η B / / Q ( B ) with η A and η B stable quasi-equiv alences. If we apply the restriction functor i ∗ to the above squar e , we obta in i ∗ A i ∗ ( F )   i ∗ ( η A ) / / i ∗ Q ( A ) i ∗ Q ( F )   i ∗ B i ∗ ( η B ) / / i ∗ Q ( B ) , where i ∗ ( F ) is a le velwise quas i-equiv a lence (7.7), i ∗ ( η A ) and i ∗ ( η B ) ar e stable quas i-equiv a lences (7.18) a nd i ∗ Q ( A ) and i ∗ Q ( B ) satisfying the con- dition Ω) (see 7.21). Now, condition (A1) o f theor em [29, 5 .1 1], implies condition (A1). (A2) W e now show that for every R -catego r y A , the R -functors η Q ( A ) , Q ( η A ) : Q ( A ) − → Q Q ( A ) are stable quasi-equiv alences. If we apply the res triction functor i ∗ to the ab ov e R -functors, w e o bta in stable qua si-equiv alences be t ween s pe ctral cat- egories which satisfy condition Ω ). Now, co ndition (A2) o f theorem [29, 5.11] implies condition (A2). (A3) W e star t b y o bserving that if P : A − → D is a Q - ﬁbration, then for a ll ob jects x, y ∈ C , the mor phism P ( x, y ) : C ( x, y ) − → D ( P x, P y ) is a ﬁbration in the pro jective stable mo del structur e on R -Mo d (see 7.2). In fact, by pro p o sition 7.1 5, the functor U : R -Mo d − → R -Cat sends pr o jectiv e coﬁbrations to coﬁbr a tions. Since clea rly it sends stable equiv ale nc e s to stable quasi- e quiv alence s the claim follows. THH AND TC FOR DG CA TEGORIES 31 Now consider the following diagra m in R -Cat A × Q ( A ) B p / /   B P   A η A / / Q ( A ) , with P a Q -ﬁbr ation. If we apply the restrictio n functor i ∗ to the previous commutativ e sq uare, we obtain i ∗ A × i ∗ Q ( A ) i ∗ B p / /   i ∗ ( B ) i ∗ ( P )   i ∗ A i ∗ ( η A ) / / i ∗ Q ( A ) a car tesian square, with i ∗ ( η A ) a sta ble quasi-equiv alence. Notice also that i ∗ ( P ) is such that for all ob jects x, y ∈ i ∗ ( B ) i ∗ ( P )( x, y ) : i ∗ ( B )( x, y ) − → i ∗ Q ( A ) is a ﬁbr a tion in the pr o jectiv e stable mo del structure on Sp Σ . Now, condi- tion (A3) of theorem [29, 5.1 1] implies condition (A3). The theorem is now prov ed. √ Prop ositi on 7. 26. A R -c ate gory A is ﬁbr ant with r esp e ct to the mo del structure of the or em 7.25 if and only if for al l obje cts x, y ∈ A , the symmetric sp e ctrum ( i ∗ A )( x, y ) is an Ω -sp e ctrum. Notation 7.2 7 . W e denote these ﬁbr a nt R - c ategorie s by Q -ﬁbrant. Pr o of. By cor ollary A. 5, A is ﬁbrant, with resp ect to theorem 7.25 if and only if it is ﬁbrant for the structure of theore m 7.10 (see r emark 7 .12) and the R -functor η A : A → Q ( A ) is a levelwise quasi-eq uiv alence. Observe that i ∗ ( η A ) : i ∗ ( A ) → i ∗ Q ( A ) is a levelwise quasi- equiv ale nc e if and only if for a ll ob jects x, y ∈ A , the morphism of symmetric s pe c tra i ∗ ( η A ( x, y )) : i ∗ A ( x, y ) − → i ∗ Q ( A )( x, y ) is a level equiv alence. Since for all ob jects x, y ∈ A , the symmetric sp ectrum i ∗ Q ( A )( x, y ) is a n Ω-sp ectr um (se e 7.21) we have the following commutativ e dia- grams (for all n ≥ 0) i ∗ A ( x, y ) n   f δ n / / Ω( i ∗ A ( x, y ) n +1 )   i ∗ Q ( A )( x, y ) n f δ n / / Ω( i ∗ Q ( A )( x, y ) n +1 ) , where the b ottom and vertical arrows are w eak equiv alences of p ointed simplicial sets. This implies that e δ n : i ∗ A ( x, y ) n − → Ω( i ∗ A ( x, y ) n +1 ) , n ≥ 0 32 G ONC ¸ ALO T ABUADA is a weak equiv alence o f p ointed simplicial se ts . In co nclusion, for all ob jects x, y ∈ A , the symmetric sp ectr um i ∗ A ( x, y ) is an Ω-sp ectrum. √ R emark 7.28 . Re ma rk 7.21 and prop os ition 7.26 imply that η A : A → Q ( A ) is a (functorial) ﬁbr ant replacement of A in the mo del structure of theor em 7.2 5. In particular η A induces the identit y map on the set of o b jects. Prop ositi on 7.29. L et F : A → B b e a ﬁbr ation (in the mo del structu r e of the o- r em 7.10 ) b etwe en Q - ﬁbr ant obje cts (se e 7.26). Then F is a Q -ﬁbr ation. Pr o of. W e need to show (by theor em B.3) that the following co mmutative squar e A F   η A / / Q ( A ) Q ( F )   B η B / / Q ( B ) is homotopy c a rtesian (in the mo del structure of theorem 7.1 0). By theo r em 7.1 0 this is equiv alent to the fact that the following commutativ e squar e in Sp Σ -Cat i ∗ ( A ) i ∗ ( F )   i ∗ ( η A ) / / Q ( A ) i ∗ Q ( F )   i ∗ ( B ) i ∗ ( η B ) / / i ∗ Q ( B ) is ho motopy cartesia n. Notice that since A and B are Q -ﬁbrant ob jects, the sp ec- tral functors i ∗ ( η A ) and i ∗ ( η B ) ar e levelwise quasi- e quiv alenc e s. Since i ∗ ( F ) is a ﬁbration, the prop ositio n is proven. √ Let f : P → R b e a mo rphism of co mm utative sy mmetric ring spe ctra. No- tice that the r estriction/ex tension of sca lars adjunction (7.3) induces the natura l adjunction R -Cat f ∗   P - Cat . f ! O O Prop ositi on 7. 30. The pr evious adjunction ( f ! , f ∗ ) is a Quil le n adjunction with r esp e ct t o the mo del structur es of the or ems 7.10 and 7.25. Pr o of. W e start by consider ing the mo del structure of theorem 7 .10. Notice that the commutativ e dia gram P f / / R S i P _ _ ? ? ? ? ? ? ? i R ? ?        THH AND TC FOR DG CA TEGORIES 33 induces a st rictly commutativ e diagr a m o f categ ories P - Cat i ∗ P % % K K K K K K K K K R -Cat f ∗ o o i ∗ R y y s s s s s s s s s Sp Σ -Cat . By theor em 7.1 0, w e conclude tha t f ∗ preserves (and r e ﬂects) levelwise q ua si- equiv ale nc e s and ﬁbra tio ns. Now, to show the propos itio n with resp ect to the mo del structure of theo- rem 7.25, we use prop osition 3.1 2. a) Clear ly f ∗ preserves tr ivial ﬁbra tions, since these are the same in the Quillen mo del str uctures of theo rems 7.1 0 and 7.25. b) Let F : A → B be a Q - ﬁbr ation in R -Cat b etw een ﬁbrant ob jects (see 7 .26). By theor em B.3, F is a ﬁbration and so b y a ) so it is f ∗ ( F ). Notice that prop osition 7.2 6 a nd the ab ove strictly commutativ e dia gram of ca tegories imply that f ∗ preserves Q -ﬁbr ant o b jects. Now, we co nclude b y prop osi- tion 7 .29 that f ∗ ( F ) is in fact a Q -ﬁbratio n. The prop os itio n is proven. √ 8. E ilenber g-Ma c Lane spectral algebra In this chapter we prov e that in the ca se of the E ilenberg-Ma c Lane ring sp ectr um H Z , the mo del structure of theorem 7.25 is Quille n equiv a lent to the o ne descr ibe d in chapter 6 for the case of simplicial Z -mo dules (se e pr o p osition 8.1). Recall from [26, 4.3] tha t we have the following strong monoidal Quillen equiv alence Sp Σ ( s Ab , e Z ( S 1 )) U   H Z -Mo d , Z O O where U is the for getful functor. By rema rk A.5, it induce s the following a djunction Sp Σ ( s Ab , e Z ( S 1 ))-Cat U   H Z -Cat , Z O O denoted by the s a me functors. Prop ositi on 8.1. The pr evi ous adjunction is a Q uil len e quivalenc e, when t he c at- e gory Sp Σ ( s Ab , e Z ( S 1 )) -Cat is endowe d with the m o del struct ur e of t he or em 4.39 and H Z -Cat is endowe d with the mo del structu r e of the or em 7.25. Pr o of. W e start b y showing that the previo us adjunction is a Quillen a djunction, when Sp Σ ( s Ab )-Cat is endow ed with the mo del structure of theorem 4.13 and H Z -Cat is endow ed with the mo del structure of theo rem 7.10. Notice that we have 34 G ONC ¸ ALO T ABUADA the following commutativ e diag ram Sp Σ ( s Ab , e Z ( S 1 ))-Cat U   E v 0 / / s Ab -Cat U   H Z -Cat / / Sp Σ -Cat E v 0 / / s Set -Cat . Since the forgetful functor U : Sp Σ ( s Ab , e Z ( S 1 )) → H Z -Ca t preserves ﬁbrations and levelwise weak equiv alences the cla im follows. W e now show that it is a Quillen a djunction with resp ect to the mo del structures describ ed in the pro p o sition. Clear ly U preserves trivial ﬁbrations. Since ﬁbrations betw een Q -ﬁbra nt ob jects are Q -ﬁbra tions, see le mmas 7.29 and 4.4 3, these a re also preserved by U . By prop o sition 3.12, we conclude that it is a Quillen adjunction. Now, we show that U is a Quillen eq uiv alence. F or this we verify conditions a ) and b ) of prop os ition 3.13. a) Let R : A → B b e a Sp Σ ( s Ab , e Z ( S 1 ))-functor b etw een Q -ﬁbrant o b jects such that U ( R ) : U ( A ) → U ( B ) is a stable quasi-equiv alence. Since A and B are Q -ﬁbrant, condition a ) of prop ositio n 3.1 3 applied to the Quillen equiv ale nc e U : Sp Σ ( s Ab , e Z ( S 1 )) − → H Z - Mo d implies that R satisﬁes conditio n W E 1). Now, since the functor U : s Ab -Cat → s Set -Cat satisﬁes the ‘reﬂexion’ condition R ) of theor em 5.1, the Sp Σ ( s Ab , e Z ( S 1 ))- functor R satisﬁes condition L 2 ′ ) and so we conclude that it is a stable quasi-equiv alence. b) Let A b e a coﬁbra nt H Z -category and let us denote by I its set of ob jects. Notice, that in par ticula r A is coﬁbra nt in ( H Z -Mo d) I -Cat, with resp ect to the module structur e of r emark A.2. Now, s ince the adjunction ( L, F ) is a strong mo noidal Quillen equiv alence (3.6), remar k 7.28 implies that condition b ) of prop osition 3.13 is veriﬁed for the adjunction ( L I , F I ). By remark A.3, the Sp Σ ( C , K )-functor η A : A → Q ( A ) is a ﬁbrant reso lution in ( H Z -Mo d) I -Cat and s o we c o nclude tha t condition b ) is veriﬁed. √ THH AND TC FOR DG CA TEGORIES 35 9. Gl obal picture Notice that rema rks 4 .46, 6 .1, 6.2 and pro p osition 8.1 furnishes us the following (four steps) zig-zag o f Quillen equiv alences : H Z -Cat Z   Sp Σ ( s Ab , e Z ( S 1 ))-Cat U O O φ ∗ N   Sp Σ ( C h ≥ 0 , Z [1 ])-Cat L cat O O i   Sp Σ ( C h, Z [1])-Ca t τ ≥ 0 O O E v 0   dgcat . F 0 O O Notation 9.1 . W e denote by H : Ho ( dgcat ) ∼ − → Ho ( H Z -Cat) and by Θ : Ho ( H Z -Ca t) ∼ − → Ho ( dg cat ) the comp os ed derived equiv alences. R emark 9.2 . If, in the previo us diagr am, we restrict ourselves to enriched catego ries with only o ne ob ject, we recover the zig-za g of Quillen equiv ale nces cons tr ucted by Shipley in [2 6, 2.1 0] and [2 6, 4.9 ]. 10. T H H and T C for DG ca tegories In this chapter, we use the equiv a lence of the pr evious chapter to deﬁne a to p o - logical Ho chsc hild and cyclic homolog y theory for dg categ ories. Let us s ta rt by r ecalling from se ction 3 of [2 1], the construction of top olo gical Ho chsc hild homolo gy ( T H H ) and top ologica l cyclic homolo gy ( T C ) in the context of sp ectral categories . Construction: Let I be the category who se o b jects are the ﬁnite sets n = { 1 , . . . , n } (including 0 = {} ), a nd whos e mo rphisms ar e the injectiv e maps. F or a symmetric sp ectrum T (of p ointed simplicial sets), we wr ite T n for the n -th p ointed simplicial set. The ass o ciation n 7→ Ω n | T n | extends to a functor fr om I to spaces, where | − | denotes the geometric r ealization. More genera lly , given symmetric sp ectra T 0 , . . . , T q and a s pace X , we o btain a functor from I q +1 to spa ces, which sends ( n 0 , . . . , n q ) to Ω n 0 + ··· + n q ( | T q n q ∧ · · · ∧ T 0 n 0 | ∧ X ) . Now, let A b e a s pe ctral catego ry . Let V ( A , X ) n 0 ,...,n q be the functor from I q +1 to spaces deﬁned by Ω n 0 + ··· + n q ( _ |A ( c q − 1 , c q ) n q ∧ · · · ∧ A ( c 0 , c 1 ) n 1 ∧ A ( c q , c 0 ) n 0 | ∧ X ) , 36 G ONC ¸ ALO T ABUADA where the sum W is taken ov er the ( q + 1 )- tuples ( q 0 , · · · , q n ) of o b jects of A . Now, deﬁne T H H q ( A )( X ) := ho colim I q +1 V ( A , X ) n 0 , ··· ,n q . This co nstruction a s sembles into a simplicial space a nd we wr ite T H H ( A )( X ) for its geometric r ealization. W e obtain in this wa y a functor T H H ( A )( X ) in the v ariable X . If we restrict ourselves to the spheres S n , we obta in ﬁnally the sy mmetric sp ectrum of top olo gic al Ho chschild homolo gy of A denoted by T H H ( A ). F or the deﬁnition of top olo gical c y clic ho mology inv olving the ‘cyclo tomic’ s truc- ture o n T H H ( A ), the author is invited to consult section 3 of [21]. W e deno te by T C ( A ) the sp ectrum of t op olo gic al cyclic homolo gy of A . R emark 10 .1 . By theorem [21, 4.9] (a nd pro p o sition [21, 3 .8]), the functors T H H , T C : Sp Σ -Cat − → Sp , send q uasi-equiv alences to weak equiv ale nc e s a nd so des cend to the homotopy ca t- egories. Now, let i : S → H Z b e the uniq ue morphism of commutativ e symmetric ring sp ectra. By prop os ition 7.30, we have the restric tio n/extension of scalars Quillen adjunction H Z -Cat i ∗   Sp Σ -Cat . i ! O O Notice that the equiv alence of the previo us chapter H : Ho ( dgcat ) ∼ − → Ho ( H Z -Cat) and the previo us adjunction furnishes us well-deﬁned topo logical Hochsch ild and cyclic homolo gy theories T H H , T C : Ho ( dgcat ) H ∼ / / , , X X X X X X X X X X X X X X Ho ( H Z -Cat) i ∗ / / Ho ( Sp Σ -Cat) T H H,T C   Ho ( Sp ) for dg categ ories (up to q uasi-equiv alence). Since the T H H and T C homolog y carries muc h more torsion info r mation than its Ho chsc hild and cyc lic homo lo gy , we obtain in this way a m uc h richer in v ar iant of dg catego r ies a nd of (algebraic) v ar ieties. Example 10.2. L et X b e a smo oth pr oje ctive algebr aic variety. We asso ciate to X a dg mo del, i.e. a dg c ate gory D b dg ( c oh ( X )) , wel l deﬁne d up t o isomorphism in H o ( dgcat ) , such that the triangulate d c ate gory H 0 ( D b dg ( c oh ( X ))) is e quiva lent the b ounde d derive d c ate gory of quasi-c ohe r ent she aves on X . F or example, for D b dg ( c oh ( X )) , we c an t ake the dg c ate gory of left b ounde d c omplexes of inje ctive O X -mo du les whose homolo gy is b ounde d and c oher ent. We then deﬁne the T H H and T C of X as the T H H and T C of the sp e ctr al c ate gory i ∗ ( H ( D b dg ( c oh ( X )))) . THH AND TC FOR DG CA TEGORIES 37 10.1. T H H and T C as additiv e in v arian ts. In this section, we use freely the language of deriv ators, see chapter 1 of [2] for the main notio ns. Notice that the constructions of the pr evious chapters furnishes us morphisms of deriv ators T H H , T C : HO ( dgcat ) QE − → HO ( Sp ) , from the der iv ato r HO ( dgcat ) QE asso ciated with the Quillen mo del str uctur e [27, 1.8] to the deriv ator ass o ciated to the category o f spectra . W e now s how tha t T H H and T C descent to the deriv ator H O ( dgcat ) a sso ciated with the Morita mo del structure [27, 2.27 ]. F or this, we sta rt by r ecalling some pr op erties o f T H H and T C in the context of sp ectr a l catego ries. Deﬁnition 10.3 . (se e se ction 4 fr om [21] ) A sp e ctr al funct or F : C → D is a Morita equiv alence if: - for al l obje cts x, y ∈ C , t he morphism F ( x, y ) : C ( x, y ) − → D ( F x, F y ) is a stable we ak e quivalenc e in Sp Σ and - the smal lest thick (i.e. close d under dir e ct factors) triangulate d sub c ate gory of Ho ( D - Mo d ) (se e 3.11 ) gener ate d by the image of F is e quivalent to the ful l sub c ate gory of c omp act obje ct s in Ho ( D - Mo d ) . R emark 10.4 . Obser ve that theorem [21, 4.1 2] and prop osition [21, 3.8] imply that if a sp ectral functor F is a Mor ita equiv alence, then T H H ( F ) and T C ( F ) a re w eak equiv ale nc e s . Now, let us r eturn to the context of dg catego r ies. Prop ositi on 1 0.5. L et F : A → B b e a Morita dg functor (se e se ction 2 . 5 of [27] ). Then the asso ciate d sp e ctr al functor i ∗ H ( F ) is a Morita e qu ivalenc e. Pr o of. By pro po sitions [27, 2.35 ] and [27, 2.1 4], F is a Morita dg functor if and only if F f A F / / j A   B j B   A f F f / / B f is a q ua si-equiv alence, w her e ( − ) f denotes a functorial ﬁbrant res olution functor in the mo del ca teg ory [27, 2 .27]. Notice that if we a pply the co mpo sed functor i ∗ H to the ab ove diagra m, we obtain a diagr a m o f sp ectral c a tegories i ∗ H ( A ) i ∗ H ( F ) / / i ∗ H ( j A )   i ∗ H ( B ) i ∗ H ( j B )   i ∗ H ( A f ) i ∗ H ( F f ) / / i ∗ H ( B f ) , with i ∗ H ( F f ) a quasi- equiv ale nce (and so a Mor ita equiv ale nce ). Moreover, since j A and j B are fully faithful Mor ita dg functors with v alues in Morita ﬁbrant dg categorie s (see [27, 2.34]), pr op osition [2 1, 4 .8] implies that i ∗ H ( j A ) and i ∗ H ( j B ) are Morita equiv a lences. In conclusion, by the tw o out o f three prop erty , the sp ectral functor i ∗ H ( F ) is a lso a Morita equiv alence. √ 38 G ONC ¸ ALO T ABUADA R emark 10.6 . By re ma rk 10 .4 and prop os itio n 10.5, we o btain well-deﬁned mor- phisms of deriv a tors T H H , T C : HO ( dgcat ) − → H O ( Sp ) . Prop ositi on 10.7. Th e morphisms of derivators T H H and T C ar e additive in- variants of dg c ate gories (se e [2 7, 3.86] ). Pr o of. The pro o f will consist on verifying that T H H and T C s atisfy all the con- ditions of [27, 3.86]. Clearly T H H a nd T C preserve the p oint. Moreov er, by construction, the morphisms of deriv ators i ∗ : HO ( H Z - C a t) − → HO ( Sp Σ -Cat) and T H H , T C : HO ( Sp Σ -Cat) − → HO ( Sp ) commute with ﬁltered homotopy colimits. It remains to s how that T H H and T C satisfy the additivity condition A). Le t 0 / / A i A / / B R o o P / / C i C o o / / 0 , be a split shor t exact sequence of dg categories [27, 3.68 ]. Observe that b y applying the functor i ∗ H ( − ), we obtain a split short exact sequence of sp ectral categor ies (see section 6 of [21]) 0 / / i ∗ H ( A ) / / i ∗ H ( B ) o o / / i ∗ H ( C ) o o / / 0 . By the lo calization theore m [21, 6.1 ], we o bta in split triangles T H H ( A ) / / T H H ( B ) o o / / T H H ( C ) o o / / T H H ( A )[1] , T C ( A ) / / T C ( B ) o o / / T C ( C ) o o / / T C ( A )[1] in Ho ( Sp ). In conclusion, the dg functors i A and i C induce iso morphisms T H H ( A ) ⊕ T H H ( C ) ∼ − → T H H ( B ) , T C ( A ) ⊕ T C ( C ) ∼ − → T C ( B ) in Ho ( Sp ) and so the pro of is ﬁnished. √ 11. Fr om K -theor y to T H H In this chapter, we construct non- tr ivial natura l transfor mations from the al- gebraic K -theor y groups functors to the top o logical Ho chsc hild homology gr oups functors. Notation 11 .1 . W e denote b y K n ( − ) : Ho ( dgcat ) − → Mo d- Z , n ≥ 0 be the n th K -theo ry g roup functor [23] [31] and by T H H j ( − ) : Ho ( dgcat ) − → Mod- Z , j ≥ 0 the j th top olo gical Ho chsc hild homology gr oup functor. THH AND TC FOR DG CA TEGORIES 39 Theorem 11.2. Supp ose that we ar e working over the ring of int e gers Z . Then we have non- t rivial natur al t r ansfo rmations γ n : K n ( − ) ⇒ T H H n ( − ) , n ≥ 0 γ n,r : K n ( − ) ⇒ T H H n +2 r − 1 ( − ) , n ≥ 0 , r ≥ 1 fr om the algebr aic K -the ory gr oups to the top olo gic al H o chschild homolo gy ones. Pr o of. By prop osition 10.7, the mo rphism of deriv ators T H H : HO ( dgcat ) − → HO ( Sp ) is an additive in v aria nt and so by theorem [27, 3.85 ] it descends to the additive motiv ator of dg categ o ries M add dg (see [27, 3 .82]) and induces a triangula ted functor (still denoted b y T H H ) T H H : M add dg ( e ) − → Ho ( Sp ) . Recall that the topo logical Ho chsc hild ho mo logy functor T H H j ( − ) , j ≥ 0 is ob- tained by comp osing T H H with the functor π s j : Ho ( Sp ) − → Mo d- Z , j ≥ 0 . Now, by [2 8, 17 .2 ], the functor K n ( − ) : M add dg ( e ) − → Mo d- Z is co-r epresented by U a ( Z )[ n ]. This implies, by the Y oneda lemma, that Nat ( K n ( − ) , T H H j ( − )) ≃ T H H j ( U a ( Z )[ n ]) , where Nat ( − , − ) deno tes the ab elia n group of natur a l transforma tio ns. Notice that we hav e the following isomorphisms T H H j ( U a ( Z )[ n ]) ≃ T H H j ( U a ( Z ))[ n ] ≃ T H H j − n ( U a ( Z )) ≃ T H H j − n ( Z ) . By B¨ okstedt, we hav e the following calculation (see [5, 0.2.3]) T H H j ( Z ) =    Z if j = 0 Z /r Z if j = 2 r − 1 , r ≥ 1 0 otherwise . In co nclusion, the canonica l g enerator s o f Z a nd Z /r Z furnishes us, b y the Y oneda lemma, non-trivia l natural tra nsformations. This ﬁnishes the pr o of. √ Theorem 11.3. L et p b e a prime n umb er. If we work over t he ring Z /p Z we have non-trivial natur al tr ansformatio ns γ n : K n ( − ) ⇒ T H H n +2 r ( − ) , n , r ≥ 0 . Pr o of. The pro o f is the sa me as the one of theorem 11.2, but ov er Z /p Z we hav e the following calculation (see [5, 0.2.3]) T H H j ( Z /p Z ) =  Z /p Z if j is ev en 0 if j is odd . √ R emark 11 .4 . When X is a quasi-co mpact a nd quasi-s eparated scheme, the T H H of X is equiv a lent [21, 1.3 ] to the to p o logical Ho chsc hild homolo gy o f s chemes as deﬁned by Geisser a nd Hesselho lt in [9]. In par ticular, theorems 1 1 .2 and 11.3 allows us to use all the calculations of T H H dev elop ed in [9], to infer r esults o n algebraic K -theor y . 40 G ONC ¸ ALO T ABUADA 12. Topological equiv alence theor y In this chapter we recall Dugger- Shipley’s notion of to po logical equiv alence and show that, when R is the ﬁeld of rationals num b er s Q , tw o dg categ o ries a re top o- logical equiv alen t if and only if they are quasi-equiv alent (see pro po sition 12 .6). Let i : S → H Z b e the unique morphism of c ommutativ e symmetric r ing s p ec tr a. By prop os ition 7 .30, we hav e the restriction/ extension o f scalars Q uillen adjunction H Z -Cat i ∗   Sp Σ -Cat . i ! O O Deﬁnition 12 .1. L et A and B b e two dg c ate gories. We say t hat A and B ar e top o- logical equiv alent if the sp e ctr al c ate gories i ∗ ( H ( A )) and i ∗ ( H ( B )) ar e isomorphic in Ho ( Sp Σ -Cat ) . R emark 12.2 . If in the pre v ious deﬁnition, A a nd B have only one ob ject, we recover Dugger-Shipley’s notio n of topo logical equiv alence for DG algebra s, see [4]. R emark 12.3 . T he zig-zag o f Q uillen equiv alences descr ib ed in section 9 and def- inition 7.16 imply that: if A and B are isomorphic in Ho ( dgcat ), then they are top ologica l equiv alent . Now, supp ose we work over the rationa ls Q . As b efo r e, w e hav e the restr ic - tion/extension of scalar s Quillen adjunction H Q -Cat i ∗   Sp Σ -Cat . i ! O O Notation 12.4 . Let A b e a sp ec tr al categ o ry . W e say that A satisﬁes condition R H ) if the following condition is veriﬁed: R H ) for a ll ob jects x, y ∈ A , the symmetric sp ectrum A ( x, y ) has ra tio nal ho- motopy . Lemma 12 .5. L et F : A → B b e a stable quasi-e quivalenc e b etwe en sp e ctr al c at- e gories which satisfy c ondition R H ) . Then i ! ( F ) : i ! ( A ) → i ! ( B ) is a s t able quasi- e quivalenc e (7.16). Pr o of. Notice that the pr o of o f pr op osition [4, 1.7] implies that, for all ob jects x, y ∈ A , the morphism i ! ( A )( x, y ) = A ( x, y ) ∧ S H Q − → B ( F x, F y ) ∧ S H Q = i ! ( B )( F x, F y ) is a pr o jectiv e stable equiv alence in H Q -Mod (see 4.3 9). Now, consider the co mmu tative squa re A F   ǫ A / / i ∗ i ! ( A ) i ∗ i ! ( F )   B ǫ B / / i ∗ i ! ( B ) , THH AND TC FOR DG CA TEGORIES 41 where ǫ A and ǫ B are the sp ectra l functors induced by the adjunction ( i ! , i ∗ ). Since A and B satisfy condition R H ), the pr o of of pr op osition [4, 1.7] implies that for all ob jects x, y ∈ A , the morphism A ( x, y ) − → A ( x, y ) ∧ S H Q = i ∗ i ! ( A )( x, y ) is a pr o jectiv e sta ble eq uiv alence in H Q -Mod . Since ǫ A and ǫ B induce the identit y map on ob jects, we conclude that ǫ A and ǫ B are stable quasi-equiv alences and so by prop os ition 7.18, that i ! ( P ) is a stable q uasi-equiv alence. √ Prop ositi on 12.6. L et A and B b e two dg c ate gories over Q . Then A and B ar e top olo gic al e quivalent if and only if they ar e isomorphic in Ho ( dgcat Q ) . Pr o of. By remark 12.3, it is enough to prove one of the implicatio ns. Supp os e that A and B are to po logical e quiv alent. Then there exists a z ig -zag o f stable quasi-equiv alences in Sp Σ -Cat i ∗ ( A ) ∼ ← Z 1 ∼ → . . . ∼ → Z n ∼ ← i ∗ ( B ) relating i ∗ ( A ) and i ∗ ( B ). By applying the functor i ! , we obtain by lemma 1 2.5, the following zig-za g of stable quasi-eq uiv alences i ! i ∗ ( A ) ∼ ← i ! ( Z 1 ) ∼ → . . . ∼ → i ! ( Z n ) ∼ ← i ! i ∗ ( B ) . W e now show that the H Q -functor s η A : i ! i ∗ ( A ) → A η B : i ! i ∗ ( B ) → B , induced b y the ab ov e adjunction ( i ! , i ∗ ) are in fact stable quasi- equiv a lences (7.16). Notice that the pro of of prop osition [4, 1 .7] implies that, for a ll ob jects x, y ∈ A , the mor phism i ! i ∗ ( A )( x, y ) − → A ( x, y ) is a pr o jectiv e stable equiv alence in H Q -Mo d. Since η A induces the identit y map on ob jects, it s atisﬁes conditio n W E 2 ′ ) and so it is a stable quas i- equiv a lence. An analogo us res ult holds for B and so the pro of is ﬁnished. √ Appendix A. Adjunction s Let ( C , − ⊗ − , 1 C ) and ( D , − ∧ − , 1 D ) be tw o symmetric monoidal catego ries and C N   D L O O an adjunction, with N a lax mono idal functor (3.3). Notation A.1 . If I is a set, w e denote b y C I -Gr, r esp. b y C I -Cat, the catego ry of C -graphs with a ﬁx e d set of ob jects I , resp. the categ ory of categories enr iched ov er C which hav e a ﬁx e d set of ob jects I . The morphisms in C I -Gr and C I -Cat induce the identit y map o n the o b jects. W e hav e a natura l adjunction C I -Cat U   C I -Gr , T I O O 42 G ONC ¸ ALO T ABUADA where U is the for g etful functor and T I is deﬁned as T I ( A )( x, y ) :=    1 C ∐ ` x,x 1 ,...,x n ,y A ( x, x 1 ) ⊗ . . . ⊗ A ( x n , y ) if x = y ` x,x 1 ,...,x n ,y A ( x, x 1 ) ⊗ . . . ⊗ A ( x n , y ) if x 6 = y Comp osition is given by concatenation and the unit corr esp onds to 1 C . R emark A.2 . If the categ ory C ca r ries a coﬁbrantly generated Quillen mo de l struc- ture, the categ ories C I -Gr and C I -Cat admit standar d mo del structures. The weak equiv ale nc e s (resp. ﬁbrations) a re the morphisms F : A → B s uch that F ( x, y ) : A ( x, y ) − → B ( x , y ) , x, y ∈ I is a weak equiv alence (resp. ﬁbratio n) in C . In fact, the Quillen mo del str ucture on C naturally induces a mo de l structure on C I -Gr, which ca n b e lifted along the functor T I using theorem [13, 11.3.2]. Clearly the adjunction ( L, N ), induces the following a djunction C I -Gr N   D I -Gr L O O still denoted b y ( L, N ). Since the functor N : C → D is la x mo noidal it induces, a s in [25, 3.3], a functor C I -Cat N I   D I -Cat . More precisely , let B ∈ C I -Cat and x, y and z ob jects of B . Then N I ( B ) has the same ob jects as B , the s paces of morphisms ar e given by N I ( B )( x, y ) := N B ( x, y ) , x, y ∈ B and the comp o sition is deﬁned by N B ( x, y ) ∧ N B ( y , z ) − → N ( B ( x, y ) ⊗ B ( y , z )) N ( c ) − → N B ( x, z ) , where c denotes the comp o sition op eration in B . As it is shown in s e c tion 3 . 3 of [25] the functor N I admits a left adjoint L I : Let A ∈ D I -Cat. The v a lue of the left adjoint L I on A is deﬁned as the co eq ua l- izer of tw o morphis ms in C I -Cat T I LU T I U ( A ) ψ 1 / / ψ 2 / / T I LU ( A ) / / L I ( A ) . The morphism ψ 1 is o btained from the unit of the adjunction T I U A − → A , by applying the co mpo site functor T I LU ; the morphism ψ 2 is the unique morphism in C I -Cat induced by the C I -Gr mor phis m LU T I U ( A ) − → U T I LU ( A ) THH AND TC FOR DG CA TEGORIES 43 whose v alue at LU T I U ( A )( x, y ) , x, y ∈ I is given by ` x,x 1 ,...,x n ,y L ( A ( x, x 1 ) ⊗ . . . ⊗ A ( x n , y )) φ   ` x,x 1 ,...,x n ,y L A ( x, x 1 ) ⊗ . . . ⊗ L A ( x n , y ) , where φ is the lax c o monoidal structure on L , induce d by the lax monoidal s tructure on N , see section 3 . 3 of [2 5]. R emark A.3 . Notice that if C and D ar e Quillen mo del catego ries and the a djunction ( L, N ) is a weak monoida l Quillen equiv a lence (see 3 .6), prop o sition [25, 6.4] implies that we obtain a Quillen eq uiv alence C I -Cat N I   D I -Cat L I O O with res pe ct to the mo del struc tur e o f remark A.2. Moreover, if L is strong mono idal (3.3), the left adjoint L I is given by the original functor L . Left adjoint Notice that the functor N I : C I -Cat → D I -Cat, of the pr evious section, ca n b e na turally deﬁned for every set I and so it induces a ‘global’ functor C -Cat N   D - Cat . In this s ection we will construct the left adjoint of N : Let A ∈ D -Cat and denote by I its set of ob jects. Deﬁne L cat ( A ) a s the C - category L I ( A ). Now, let F : A → A ′ be a D - functor. W e denote by I ′ the set of ob jects o f A ′ . The D -functor F induce s the following dia gram in C -Cat: T I LU T I U ( A ) ψ 1 / / ψ 2 / /   T I LU ( A )   / / L I ( A ) = : L cat ( A ) T I ′ LU T I ′ U ( A ′ ) ψ 1 / / ψ 2 / / T I ′ LU ( A ′ ) / / L I ′ ( A ′ ) =: L cat ( A ′ ) . Notice that the square whose horizo ntal ar rows are ψ 1 (resp. ψ 2 ) is commutativ e. Since the inclus ions C I -Cat ֒ → C -Cat and C I ′ -Cat ֒ → C -Cat clearly preser ve co equalize r s the pre v ious diagra m in C -Cat induces a C -functor L cat ( F ) : L cat ( A ) − → L cat ( A ′ ) . W e hav e cons tructed a functor L cat : D -Ca t − → C -Cat . 44 G ONC ¸ ALO T ABUADA Prop ositi on A.4. [30, 5 .5 ] The funct or L cat is left adjoi nt to N . R emark A.5 . [25, 3 .3] Notice that if, in the initial adjunction ( L, N ), the left adjoint L is s tr ong mono idal (see 3 .3), remar k A.3 implies that the left adjoint functor L cat : D -Ca t − → C -Cat is given by the original left adjoint L . Appendix B. Bo usfield l ocaliza tion techniques In this app endix we recall and gener a lize some results concer ning the co nstruc- tion and lo calization of Quillen mo del structures. W e start by stating a weak er form of the Bousﬁeld lo calization theorem [17, X-4.1]. Deﬁnition B . 1. L et M b e a Q uil len m o del c ate gory, Q : M → M a funct or and η : Id → Q a natur al tr ansformation b etwe en the identity functor and Q . A morphism f : A → B in M is: - a Q -weak equiv alence if Q ( f ) is a we ak e quivalenc e in M . - a c oﬁbration if it is a c oﬁbr ation in M . - a Q - ﬁbr ation if it has the R.L.P. with r esp e ct to al l c oﬁbr ations which ar e Q -we ak e quivalenc es. An immediate analy sis of the pro of of theorem a llows us to state the following general theorem B.2. Notice that in the entire pro of, we o nly use the right prop er- ness of M a nd in the pro of of lemma [1 7, X-4.6] and theorem [17, X-4.8], we only use condition (A3). Theorem B.2. [1 7, X-4.1] L et M b e a right pr op er Quil len mo del structur e, Q : M → M a fun ctor and η : Id → Q a natur al tr ansformatio n s uch that the fol lo wing thr e e c onditions hold: (A1) The functor Q pr eserves we ak e quivalenc es. (A2) The maps η Q ( A ) , Q ( η A ) : Q ( A ) → QQ ( A ) ar e we ak e quivalenc es in M . (A3) Given a diagr am B p   A η A / / Q ( A ) with p a Q -ﬁbr ation, the induc e d map η A ∗ : A × Q ( A ) B → B is a Q -we ak e quivalenc e. Then t her e is a right pr op er Q u il len mo del stru ct ur e on M for which the we ak e quiv- alenc es ar e the Q - we ak e quivalenc es, t he c oﬁbr ations those of M and the ﬁ br ations the Q -ﬁbr ations. Theorem B. 3. [17, X-4 .8] Supp ose that the right pr op er Quil l en mo del c ate gory M and the fun ctor Q t o gether satisfy the c onditions for t he or em B.2. Then a map f : A → B is a Q -ﬁbr ation if and only if it is a ﬁ br ation in M and the squar e A f   η A / / Q ( A ) Q ( f )   B η B / / Q ( B ) THH AND TC FOR DG CA TEGORIES 45 is homotopy c artesian in M . Corollary B. 4. [17, X-4.12] Supp ose t hat the right pr op er Quil len m o del c ate gory M and t he functor Q to gether satisfy the c onditions for t he or em B.2. Then an obje ct A of M is Q -ﬁbr ant if and only if it is ﬁ br ant in M and the map η A : A → Q ( A ) is a we ak e quivalenc e in M . Appendix C. N on-additive fil tra tio n argument Let D b e a closed symmetric mo noidal catego ry . W e denote by − ∧ − its sym- metric monoidal pro duct and by 1 its unit. W e start with some gener a l a rguments. Deﬁnition C.1. Consider the functor U : D − → D - Cat , which sends an obje ct X ∈ D to the D -c ate gory U ( X ) , with t wo obje cts 1 and 2 and such t hat U ( X )(1 , 1) = U ( X )(2 , 2 ) = 1 , U ( X )(1 , 2 ) = X and U ( X )(2 , 1) = 0 , wher e 0 denotes the initial obje ct in D . Comp osition is n atur al ly deﬁne d (notic e that 0 acts as a zer o with r esp e ct to ∧ sinc e the bi-functor − ∧ − pr eserves c olimi ts in e ach of its variables). R emark C.2 . F or an ob ject X ∈ D , the ob ject U ( X ) ∈ D -Cat co-repr esents the following functor D - Cat − → Set A 7→ ` ( x,y ) ∈A×A Hom D ( X, A ( x, y )) . This shows us, in particular , that the functor U preserves seq uent ially small ob jects. Now, supp ose that D is a mo noidal mo del ca tegory , with coﬁbra nt unit, which satisﬁes the mono id axiom [25, 3.3]. In what follows, by smash pr o duct w e mean the symmetric pro duct − ∧ − of D . Prop ositi on C.3. [29, B.3] Le t A b e a D -c ate gory, j : K → L a trivial c oﬁbr ation in D and U ( K ) F → A a morphism in D -Cat. Then, in the pushout U ( K ) F / / U ( j )   y A R   U ( L ) / / B the morphisms R ( x, y ) : A ( x, y ) − → B ( x, y ) , x, y ∈ A ar e we ak e quivalenc es in V . Prop ositi on C.4. [29, B.4] L et A b e a D -c ate gory such that A ( x, y ) is c oﬁbr ant in V for al l x, y ∈ A and i : N → M a c oﬁbr ation in D . Then, in a pushout in D -Cat U ( N ) F / / U ( i )   y A R   U ( M ) / / B 46 G ONC ¸ ALO T ABUADA the morphisms R ( x, y ) : A ( x, y ) − → B ( x, y ) , x, y ∈ A ar e al l c oﬁbr ations in D . References [1] F. 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