Homotopy theory of Spectral categories

HOMOTOPY THEOR Y OF SPECTRAL CA TEGORIES GONC ¸ ALO T ABUADA Abstract. W e construct a Quill en model structure on the categ ory of spectral categories, where th e w eak equiv alences are the s ym m etric sp ectra analogue of the not ion of equiv al ence of cate gories. Contents 1. Int ro duction 1 2. Preliminarie s 2 3. Simplicial categ ories 4 4. Levelwise quasi- e quiv alences 5 5. Stable quas i-equiv alences 11 Appendix A. Non-additive filtra tion argument 16 References 18 1. Introduction In the pas t fifteen years, the discovery of highly structured catego ries of spec tr a ( S -mo dules [12], symmetric sp ectr a [16], simplicial functors [22], o rthogonal spe c- tra [23], . . . ) has op ened the wa y for an impo rtation of more and more algebraic techn iques into stable homotopy theory [1] [10] [11]. In this pap er, we study a new ingredient in this ‘brav e new alg ebra’: Sp e ctr al c ate gories . Spec tr al ca tegories are ca teg ories enriched over the symmetric monoidal category of symmetric sp ectra. As linear categorie s can b e understo o d as rings with sever al obje cts , s pec tr al catego ries can be understoo d a s symmetric ring sp e ctr a wi th sever al obje cts . They a ppe a r now adays in s e veral (related) sub jects: On one hand, they are considere d as the ‘top olo gical’ a nalogue of differential graded (=DG) ca tegories [7] [18] [30]. The main idea is to replace the monoida l category C h ( Z ) of complexes of ab elian gro ups by the monoidal category Sp Σ of symmetric spectra, which one should imagine as ‘complexes o f abelian groups up to homotopy’. In this wa y , spectr al categ ories provide a non- a dditive framework for non-commutativ e a lgebraic geometry in the s ense of Bondal, Drinfeld, Kapranov, Kontsevic h, T o¨ en, V a n den Bergh . . . [2] [3] [7] [8] [20] [2 1] [31]. They can b e seen as non-additive de r ived categ ories of q ua si-coherent sheav es on a hypothetical non-commutativ e space. On the other hand they app ear naturally in stable homotopy theor y by the work of Dugger , Sch wede-Shipley , . . . [6] [27]. F or example, it is shown in [27, 3.3.3] Key wor ds and phr ases. Symmetric sp ectra, Sp ectral category , Quill en model structure, Bous- field’s lo calization Q -functor, Non-additive filtration. 1 2 GONC ¸ ALO T ABUADA that sta ble mo del categories with a set of compact generato rs ca n be c har acterized as mo dules ov er a s p ectr a l category . In this wa y s e veral different sub jects such as: equiv a riant homoto p y theory , stable motivic theory of sc hemes, . . . and all the classical algebr aic situations [27, 3 .4] fit in the con tex t of sp ectral ca tegories. It turns o ut that in a ll the ab ove different situations, sp ectr a l ca tegories should b e considered only up to the no tio n of st able quasi-e quivalenc e (5.1): a mix tur e b et ween stable equiv alenc e s of symmetric spectr a and categorica l equiv a lences, whic h is the correct notion of equiv alence b et ween sp ectral c a tegories. In this article, we co nstruct a Quillen model s tructure [24] on the categor y Sp Σ -Cat of sp ectra l categories, with r esp ect to the class of s ta ble quasi-equiv alences. Starting from simplicia l categ o ries [4], we constr uct in theorem 4.8 a ‘levelwise’ cofi- brantly g enerated Quillen mo del structure on Sp Σ -Cat. Then we adapt Sch wede- Shipley’s non-a dditiv e filtration arg umen t (App endix A) to our s itua tion and pr ov e our main theor em: Theorem (5.10) . The c ate gory Sp Σ -Cat admits a right pr op er Quil len mo del struc- tur e whose we ak e quivalenc es ar e the stable qu asi-e qu ivalenc es and whose c ofibr a- tions ar e those of the or em 4.8. Using theore m 5.10 and the same ge neral arguments o f [31], we ca n describ e the mapping spa c e betw een tw o sp ectra l catego r ies A a nd B in terms of the nerve of a certain c a tegory of A - B -bimo dules and pr ov e that that the homotopy category Ho ( Sp Σ -Cat) p osses s es int ernal Hom’s r elative to the der ived smash pro duct of sp ectral catego ries. Ac knowledgmen ts : It is a grea t pleasure to thank B. T o¨ en for stating this prob- lem to me and G. Granja for s everal discussions and r e ferences. I am a lso v ery grateful to S. Sc hwede for his interest a nd for k indly s ug gesting several improv e - men ts a nd r eferences. I w ould like als o to thank A. Joyal and A. E . Stancules cu for po in ting out an err or in a previo us version and the a no nymous refere e for numerous suggestions for the improv ement of this pa per . 2. Preliminaries Throughout this ar ticle the adjunctions are display ed vertically with the left, resp. right, adjoint on the le ft side, resp. right side. 2.1. Definition. L et ( C , − ⊗ − , I C ) and ( D , − ∧ − , I D ) b e t wo symmetric monoidal c ate gories. A s trong monoidal functor is a functor F : C → D e quipp e d with an isomorphi sm η : I D → F ( I C ) and natur al isomorphisms ψ X,Y : F ( X ) ∧ F ( Y ) → F ( X ⊗ Y ) , X , Y ∈ C which ar e c oher en t ly asso ciative and unital ( se e diagr ams 6 . 27 and 6 . 28 in [5] ). A strong monoida l adjunction b etwe en monoidal c ate gories is an adjunction for which the left adj oint is str ong monoidal. Let s Set , resp. s Set • , b e the (symmetric monoidal) catego ry of simplicial sets, resp. p ointed simplicial s e ts. By a simplicial c ate gory , r esp. p ointe d simplicial c ate gory , we mean a c ategory enriched o ver s Set , resp. ov er s Set • . W e de no te by s Set -Ca t, resp. s Set • -Cat, the c ategory of s ma ll simplicial categor ies, res p. HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 3 po in ted simplicial ca tegories. Obser ve that the usual adjunction [17] (on the left) s Set •   s Set • -Cat   s Set ( − ) + O O s Set -Cat ( − ) + O O is strong monoida l and so it induces the adjunction on the right. Let Sp Σ be the (symmetric mono idal) categor y of sy mmetric sp ectra of p ointed simplicial sets [16] [25]. W e de no te by ∧ its smash pro duct and by S its unit, i.e. the sphere symmetric sp ectrum [25, I- 3]. Reca ll that the pro jective level mo del structure on Sp Σ [25, I I I-1.9] a nd the pro jective stable mo del s tructure on Sp Σ [25, II I-2 .2] are monoidal with resp ect to the smash pro duct. 2.2. Lemma. The pr oje ctive level mo del st ructur e on Sp Σ satisfies the monoid axiom [26, 3.3] . Pr o of. Let Z b e a symmetric spectrum and f : X → Y a trivia l cofibra tion in the pro jective level model structure. By pro p os ition [25, I I I-1.11 ] the morphism Z ∧ f : Z ∧ X → Z ∧ Y is a trivial cofibra tion in the injectiv e level mo del s tr ucture [25, II I-1.9 ]. Since trivial cofibra tions ar e s table under c o -base change and transfinite comp osition, we conclude that ea ch map in the class ( { pro jective trivia l cofibration } ∧ Sp Σ ) − cof r eg is in par ticular a level equiv alence. This pro ves the lemma. √ 2.3. Definition. A sp ectral categ o ry A is a Sp Σ -c ate gory [5, 6.2 .1] . Recall that this means that A consists in the follo wing da ta: - a class of ob jects ob j( A ) (usually denoted b y A itself ); - for ea ch o rdered pa ir o f ob jects ( x, y ) of A , a symmetric spec tr um A ( x, y ); - for ea ch o rdered triple of ob jects ( x, y , z ) of A , a comp ositio n morphism in Sp Σ A ( x, y ) ∧ A ( y , z ) → A ( x, z ) , satisfying the usual asso ciativity condition; - for any ob ject x of A , a mor phism S → A ( x, x ) in Sp Σ , sa tisfying the usual unit co ndition with resp ect to the ab ov e comp osition. If ob j( A ) is a set we say that A is a smal l spectra l categor y . 2.4. Definition. A sp ectral functor F : A → B is a Sp Σ -functor [5, 6.2.3] . Recall that this means that F consists in the following data: - a map ob j( A ) → ob j( B ) and - for each or dered pair of ob jects ( x, y ) of A , a morphism in Sp Σ F ( x, y ) : A ( x, y ) − → B ( F x, F y ) satisfying the usual unit and a sso ciativity conditions. 2.5. Notation. W e denote by Sp Σ -Cat the ca tegory of small spectr a l categor ies. 4 GONC ¸ ALO T ABUADA Observe that the classical adjunction [25, I-2.12] (on the left) Sp Σ ( − ) 0   Sp Σ -Cat ( − ) 0   s Set • Σ ∞ O O s Set • -Cat Σ ∞ O O is strong monoida l and so it induces the adjunction on the right. 3. Simplicial ca tegories In this chapter we give a detailed pro o f o f a technical lemma concerning simplicia l categorie s, which is due to A. E. Stanculescu. 3.1. R emark. Notice that w e have a fully faithful functor: s Set -Cat − → Cat ∆ op A 7→ A ∗ given by ob j( A n ) = o b j( A ) , n ≥ 0 a nd A n ( x, x ′ ) = A ( x, x ′ ) n . Recall from [4, 1.1], that the category s Set -Cat carries a cofibrantly g enerated Quillen mo del str uc tur e whose weak equiv alences are the Dwyer-Kan (= DK) e quiv- alences, i.e. the simplicial functors F : A → B such that: - for all o b jects x, y ∈ A , the map F ( x, y ) : A ( x, y ) − → B ( F x, F y ) is a weak equiv alence o f simplicial sets and - the induced functor π 0 ( F ) : π 0 ( A ) − → π 0 ( B ) is an eq uiv a lence of ca tegories. 3.2. Notation. Let A be a n (enriched) catego ry and x ∈ ob j( A ). W e denote by x ∗ A the full (enriched) sub c ategory of A whose set of ob jects is { x } . 3.3. Lemma. (St anculescu [29, 4.7] ) L et A b e a c ofibr ant simplicial c ate gory. Then for every x ∈ obj ( A ) , the simplicial c ate gory x ∗ A is also c ofibr ant (as a simplici al monoid). Pr o of. Let O be the set of ob jects of A . Notice that if the simplicial catego ry A is cofibrant then it is a lso cofibra nt in s Set O -Cat [9, 7 .2 ]. Moreov er a simplicia l category with one ob ject (for example x ∗ A ) is cofibrant if and only if it is co fibrant as a simplicia l monoid, i.e. cofibrant in s Set { x } -Cat. Now, b y [9, 7.6] the cofibrant ob jects in s Set O -Cat can be c ha racterized as the retra c ts o f the free simplicial categories . Recall from [9, 4 .5] that a simplicial category B (i.e. a simplicial ob ject B ∗ in Cat) is fr e e if a nd o nly if: (1) for every n ≥ 0, the category B n is free on a graph G n of gener ators and (2) all degenera ncies of generator s are generators. Therefore it is enough to show the following: if A is a free simplicial categor y , then x ∗ A is also free (as a simplicia l monoid). Since for every n ≥ 0, the category A n is free o n a gra ph, lemma 3.4 implies tha t the simplicial category x ∗ A satisfies condition (1). Moreover, since the degenera ncies in A ∗ induce the ident it y ma p on HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 5 ob jects and send ge ne r ators to g e ne r ators, the simplicial ca tegory x ∗ A sa tisfies also condition (2). This prov es the lemma. √ 3.4. Lem ma. L et C b e a c ate gory which is fr e e on a gr aph G of gener ators. Then for every obje ct x ∈ obj ( C ) , the c ate gory x ∗ C is also fr e e on a gr aph e G of gener ators. Pr o of. W e s tart by defining the generator s of e G . An element of e G is a path in C from x to x such that: (i) every a r row in the path b elong s to G and (ii) the path starts in x , finishes in x and never passes throught x in an inter- mediate step. Let us now sho w that ev ery morphism in x ∗ C can b e written uniquely as a finite comp osition of elements in e G . Let f be a morphism in x ∗ C . Since x ∗ C is a full sub c ategory o f C and C is free on the graph G , the morphis m f can b e written uniquely as a finite comp osition f = g n · · · g i · · · g 2 g 1 , where g i , 1 ≤ i ≤ n b elongs to G . Now consider the partition 1 ≤ m 1 < · · · < m j < · · · < m k = n , where m j is such tha t the targ et of the mor phis m g m j is the ob ject x . If we deno te by M 1 = g m 1 · · · g 1 and b y M j = g m j · · · g m ( j − 1) +1 , j ≥ 2 the morphisms in e G , w e can factor f as f = M k · · · M j · · · M 1 . Notice that our arguments shows us als o that this facto rization is unique and so the lemma is prov en. √ 4. Level wise quasi-equiv alences In this chapter we construct a cofibra n tly g e ne r ated Quillen mo del structure on Sp Σ -Cat whose weak equiv alences a re defined as follows. 4.1. Definition. A sp e ctr al functor F : A → B is a levelwise quasi-eq uiv a lence if: L1) for al l obje cts x, y ∈ A , the morph ism of symmetric sp e ctr a F ( x, y ) : A ( x, y ) → B ( F x, F y ) is a level e quivalenc e of symmetric sp e ctr a [25, I I I-1.9] and L2) the induc e d simplici al fun ctor F 0 : A 0 → B 0 is a DK-e quivalenc e in s Set -Cat. 4.2. Notation. W e denote by W l the class of levelwise qua si-equiv alences in Sp Σ -Cat. 4.3. Rema rk. Notice that if conditio n L 1) is verified, co ndition L 2) is equiv alent to: L2’) the induced functor π 0 ( F 0 ) : π 0 ( A 0 ) − → π 0 ( B 0 ) is ess e ntially surjective. W e now define our se ts of (trivial) gener a ting c o fibrations in Sp Σ -Cat. 4.4. Definition. The set I of generating cofibratio ns c onsists in: 6 GONC ¸ ALO T ABUADA - the sp e ctr al fu n ctors obtaine d by applying the functor U (A.1) to the set of gener ating c ofibr ations of t he pr oje ctive level mo del stru ct ur e on Sp Σ [25, II I-1 .9] . Mor e pr e cisely, we c onsider the sp e ctr al functors C m,n : U ( F m ∂ ∆[ n ] + ) − → U ( F m ∆[ n ] + ) , m, n ≥ 0 , wher e F m denotes the level m fr e e symmetric sp e ctr a functor [25, I-2.12 ] . - the sp e ctr al functor C : ∅ − → S fr om t he empty sp e ctr al c ate gory ∅ (which is the initial obje ct in Sp Σ -Cat) to the sp e ctr al c ate gory S with one obje ct ∗ and endomorphi s m ring sp e ctrum S . 4.5. Definition. The set J of trivial g enerating cofibratio ns c onsists in: - the sp e ctra l functors obtaine d by applyi ng the functor U (A .1) to the set of trivial gener ating c ofibr ations of the pr oje ctive level mo del s t ructur e on Sp Σ . Mor e pr e cisely, we c onsider the s p e ctr al functors A m,k,n : U ( F m Λ[ k , n ] + ) − → U ( F m ∆[ n ] + ) , m ≥ 0 , n ≥ 1 , 0 ≤ k ≤ n . - the sp e ctra l funct ors obtaine d by applying the c omp ose d funct or Σ ∞ ( − + ) to the set ( A 2) of trivial gener ating c ofibr ations in s Set -Cat [4] . Mor e pr e cisely, we c onsider the sp e ctr al functors A H : S − → Σ ∞ ( H + ) , wher e A H sends ∗ to the obje ct x . 4.6. Notation. W e denote b y J ′ , resp. J ′′ , the subset of J consisting of the sp ectr a l functors A m,k,n , resp. A H . In this way J ′ ∪ J ′′ = J . 4.7. Rema rk. B y definition [4] the simplicial categor ies H hav e weakly contractible function co mplexes and ar e cofibra n t in s Set -Cat. By lemma 3.3, we c onclude that x ∗ H (i.e. th e full simplicial sub categor y o f H whose s et of ob jects is { x } ) is a cofibrant simplicial category . 4.8. Theorem. If we let M b e the c ate gory Sp Σ -Cat, W b e the class W l , I b e the set of s p e ctr al functors of definition 4.4 and J the set of sp e ctr al functors of defi- nition 4.5, t hen the c onditions of t he r e c o gnition the or em [15, 2.1.1 9] ar e satisfie d. Thus, the c ate gory Sp Σ -Cat admits a c ofibr antly gener ate d Quil len mo del st ructur e whose we ak e quivalenc es ar e the levelwise quasi-e quivalenc es. Pro of of Theorem 4 .8. W e sta rt by observing that the categ ory Sp Σ -Cat is com- plete and co complete and that the class W l satisfies the tw o out of thr ee axio m and is stable under r etracts. Since the domains of the (trivial) genera ting cofibr ations in Sp Σ are sequentially small, the same holds by [19] for the domains o f spectr al functors in the sets I and J . This implies that the first three conditions of the recognition theorem [1 5, 2.1.19 ] a re v erified. W e now pr ov e that J -inj ∩ W l = I -inj. F or this we introduce the following auxiliary class of spectra l functors: 4.9. Definition. L et Surj b e the class of s p e ctr al functors F : A → B such that: Sj1) for al l obje cts x, y ∈ A , the morphism of symmetric sp e ctr a F ( x, y ) : A ( x, y ) → B ( F x, F y ) is a trivial fibr ation in the pr oje ctive level mo del str u ctur e [25, I I I-1 .9] and HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 7 Sj2) the sp e ctr al functor F induc es a surje ctive map on obje cts. 4.10. Lemma. I -inj = S urj . Pr o of. Notice that a spe ctral functor satis fie s condition S j 1) if and o nly if it ha s the right lifting pro pe r t y (=R.L.P .) with resp ect to the sp ectral functors C m,n , m, n ≥ 0. Clearly a sp ectra l functor has the R.L.P . with resp ect to the sp ectral functor C if and only if it satisfies co ndition S j 2). √ 4.11. Lemma. Surj = J -inj ∩ W l . Pr o of. W e prov e fir s t the inclusio n ⊆ . Let F : A → B b e a spectra l functor which b elongs to Surj . Conditions S j 1) and S j 2) clea rly imply conditions L 1) and L 2) and so F b elongs to W l . Notice also that a sp ectral functor which satisfies condition S j 1) ha s the R.L.P . with resp ect to the trivia l gener a ting cofibra tions A m,k,n . It is then enough to show that F has the R.L.P . with resp ect to the sp ectra l functors A H . By adjunction, this is equiv alent to dema nd that the simplicial functor F 0 : A 0 → B 0 has the R.L.P . with resp ect to the set ( A 2) of trivial gener a ting cofibrations { x } → H in s Set -Cat [4]. Since F satisfies c o nditions S j 1 ) and S j 2), prop osition [4, 3.2] implies that F 0 is a trivial fibra tion in s Set -Cat and so the claim follows. W e now pr ov e the inclusio n ⊇ . Observe that a sp ectra l functor satisfies co ndition S j 1) if and only if it satisfies condition L 1) and it has moreov er the R.L.P . with resp ect to the trivial g enerating cofibra tions A m,k,n . Now, le t F : A → B b e a sp ectral functor whic h belong s to J -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 r esp ect to the trivial g e ne r ating cofibrations A H : S − → Σ ∞ ( H + ) the simplicial functor F 0 : A 0 → B 0 has the R.L.P . with resp ect to the inclusions { x } → H . This implies that F 0 is a trivia l fibra tion in s Set -Cat and so by prop osi- tion [4, 3.2 ], the simplicial functor F 0 induces a surjective map on o b jects. Since F 0 and F induce the sa me ma p on the se t o f ob jects , the sp ectra l functor F sa tisfies condition S j 2). √ W e now c haracterize the class J -inj. 4.12. Lemma. A sp e ctr al fun ct or F : A → B has the R.L.P. with r esp e ct to t he set J of trivial gener ating c ofibr ations if and only if it satisfies: F1) for al l obje cts x, y ∈ A , the morphism of symmetric sp e ctr a F ( x, y ) : A ( x, y ) → B ( F x, F y ) is a fibr ation in t he pr oje ctive level mo del st ructur e [25, I I I-1.9] and F2) the induc e d simplicial fun ctor F 0 : A 0 → B 0 is a fibr ation in t he Quil len mo del stru ctur e on s Se t -Cat. Pr o of. Observe that a sp ectral functor F sa tisfies condition F 1) if and only if it has the R.L.P . with r esp e c t to the trivial gener a ting co fibrations A m,k,n . By a djunction F has the R.L.P . with r esp e c t to the sp ectral functors A H if and o nly if the simplicia l functor F 0 has the R.L.P . with resp ect to the inclusions { x } → H . In conclusion F has the R.L.P . with resp ect to the set J if and only if it satisfies conditions F 1) and F 2) altogether. √ 8 GONC ¸ ALO T ABUADA 4.13. Lemma. J ′ -c el l ⊆ W l . Pr o of. Since the class W l is stable under transfinite comp ositions [14, 10.2.2 ] it is enough to prove the following: let m ≥ 0 , n ≥ 1 , 0 ≤ k ≤ n and R : U ( F m Λ[ k , n ] + ) → A a sp ectra l functor. Cons ider the following pushout: U ( F m Λ[ k , n ] + ) A m,k,n   R / / y A P   U ( F m ∆[ n ] + ) / / B . W e need to show that P b elongs to W l . Since the symmetric sp ectra mor phisms F m Λ[ k , n ] + − → F m ∆[ n ] + , m ≥ 0 , n ≥ 1 , 0 ≤ k ≤ n are trivial co fibrations in the pr o jective level mo del structure, lemma 2 .2 and prop o- sition A.2 imply tha t the sp ectral functor P sa tisfies co nditio n L 1). Since P induces the iden tit y map on ob jects, condition L 2 ′ ) is automatically satisfied a nd so P b e- longs to W l . √ 4.14. Prop ositi on. J ′′ -c el l ⊆ W l . Pr o of. Since the class W l is stable under tra nsfinite comp ositions , it is enough to prov e the following: let A b e a small spe c tral categor y and R : S → A a s pectr al functor. Consider the following pushout S A H   R / / y A P   Σ ∞ ( H + ) / / B . W e need to show that P b elongs to W l . W e star t by showing c o ndition L 1). F actor the sp ectral functor A H as S − → x ∗ Σ ∞ ( H + ) ֒ → Σ ∞ ( H + ) , where x ∗ Σ ∞ ( H + ) is the full sp ectra l sub category of Σ ∞ ( H + ) whose s e t of o b jects is { x } (3.2). Consider the iter a ted pushout S   ∼   R / / y A P 0   P   x ∗ Σ ∞ ( H + ) / /  _   y e A P 1   Σ ∞ ( H + ) / / B . In the low er pushout, since x ∗ Σ ∞ ( H + ) is a full spectr a l subc a tegory of Σ ∞ ( H + ), prop osition [13, 5.2] implies that e A is a full spectra l s ubca tegory of B and so P 1 satisfies condition L 1). In the upper pushout, since x ∗ Σ ∞ ( H + ) = Σ ∞ (( x ∗ H ) + ) and x ∗ H is a cofibrant simplicial c a tegory (4.7), the sp e ctral functor S / / ∼ / / x ∗ Σ ∞ ( H + ) is a trivia l cofi- bration. Now, let O denote the set o f ob jects of A (notice that if A = ∅ , then there HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 9 is no sp ectra l functor R ) and O ′ := O \ R ( ∗ ). By lemma 2.2 a nd pro po sition [28, 6.3], the categor y ( Sp Σ ) O -Cat of spectr al catego ries with a fixed set of ob jects O carries a natura l Quillen mo del structure. Notice that e A iden tifies with the follow- ing pushout in ( Sp Σ ) O -Cat ` O ′ S ∐ S   ∼   R / / y A   P 0 ∼   ` O ′ S ∐ x ∗ Σ ∞ ( H + ) / / e A . Since the left vertical ar row is a tr ivial cofibratio n so it is P 0 . In par ticula r P 0 satisfies condition L 1 ) and so we conclude that the comp osed sp ectral functor P satisfies also co ndition L 1 ). W e now sho w that P satisfies condition L 2 ′ ). Let f b e a 0- s implex in H ( x, y ). By construction [4] of the simplicial categories H , f b ecomes inv ertible in π 0 ( H ). W e consider it as a morphism in the spectra l catego ry Σ ∞ ( H + ). Notice that the sp ectral category B is obtained from A , by gluing Σ ∞ ( H + ) to the ob ject R ( ∗ ). Since f clea r ly b ecomes inv ertible in π 0 (Σ ∞ ( H + )) 0 , its image by the s pectr al functor Σ ∞ ( H + ) → B b ecomes inv er tible in π 0 ( B 0 ). This implies that the functor π 0 ( P 0 ) : π 0 ( A 0 ) − → π 0 ( B 0 ) is essentially sur jective and so P s atisfies condition L 2 ′ ). In conclusion, P sa tis fie s condition L 1) a nd L 2 ′ ) and so it belo ngs to W l . √ W e hav e shown that J -cell ⊆ W l (lemma 4.13 and prop os ition 4 .14) and that I -inj = J -inj ∩ W l (lemmas 4.1 0 and 4.11). This implies that the la st three condi- tions of the r ecognition theore m [15, 2.1.19 ] a re satisfied. This finishes the pro o f of theorem 4.8. Prop erties. 4.15. Prop osition. A sp e ctr al functor F : A → B is a fibr ation with r esp e ct to the mo del structu r e of the or em 4.8 , if and only if it satisfies c onditions F 1) and F 2) of lemma 4.12. Pr o of. This follows fr om lemma 4 .12, since by the recognition theorem [15, 2.1 .19], the set J is a set of gener ating trivial cofibra tions. √ 4.16. Co rol lary . A sp e ctr al c ate gory A is fibr ant with r esp e ct to the mo del str u ctur e of t he or em 4.8, if and only if A ( x, y ) is a levelwise Kan simplicial set for al l obje cts x, y ∈ A . Notice that b y pr op osition 4.15 we hav e a Q uillen adjunction Sp Σ -Cat ( − ) 0   s Set -Cat . Σ ∞ ( − + ) O O 4.17. Prop o sition. The Qu il len mo del struct u r e on Sp Σ -Cat of the or em 4.8 is right pr op er. 10 GONC ¸ ALO T ABUADA Pr o of. Consider the following pullback square in Sp Σ -Cat A × B C   P / / p C F     A R ∼ / / B , with R a levelwise quas i-equiv alence and F a fibration. W e need to show that P is a levelwise quasi-equiv alence . No tice that pullbacks in Sp Σ -Cat are calcula ted on ob jects and on symmetric spectra morphisms. Since the pro jective level mo del structure o n Sp Σ is right pro per [25, II I-1.9 ] and F satisfies condition F 1), the sp ectral functor P satisfies condition L 1 ). Notice that the compo sed functor Sp Σ -Cat ( − ) 0 / / s Set • -Cat / / s Set -Cat commutes with limits and that by propo sition 4.1 5, F 0 is a fibration in s Set -Ca t. Since the mode l structure on s Set -Cat is right prop er [4, 3 .5] and R 0 is a DK- equiv alence, we conclude that the spectra l functor P s atisfies also co ndition L 2). √ 4.18. Prop o sition. L et A b e a c ofibr ant sp e ctr al c ate gory (in the Qu il len m o del structur e of t he or em 4.8). The n for al l obj e cts x, y ∈ A , the symmetric sp e ctr a A ( x, y ) is c ofibr ant in the pr oje ctive level mo del structur e on Sp Σ [25, I I I-1.9] . Pr o of. The Quillen mo del structure o f theorem 4.8 is co fibrantly generated a nd so a ny c ofibrant ob ject in Sp Σ -Cat is a re tract of a I - cell c o mplex [14, 11.2.2]. Since co fibrations ar e stable under transfinite comp osition it is e nough to prove the prop osition for pushouts a long a ge ne r ating cofibration. Let A a spec tral category such that A ( x, y ) is cofibra nt for all ob jects x, y ∈ A : - consider the following pushout ∅ C   / / y A   S / / B . Notice that B is obtained from A , by simply in tro ducing a new ob ject. It is then clear that, for all ob jects x, y ∈ B , the symmetric sp ectra B ( x, y ) is cofibrant. - No w, consider the follo wing pushout U ( F m ∂ ∆[ n ] + ) / / C m,n   y A P   U ( F m ∆[ n ] + ) / / B . Notice that A and B ha ve the same set of ob jects a nd P induces the iden- tit y ma p on the set of ob jects. Since F m ∂ ∆[ n ] + → F m ∆[ n ] + is a pro jective cofibration, prop ositio n A.3 implies that the mor phism of symmetric sp ec- tra P ( x, y ) : A ( x, y ) − → B ( x, y ) HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 11 is still a pro jective cofibration. Finally , s ince the I -cell complexes in Sp Σ -Cat are built fr om ∅ (the initial ob ject), the prop o sition is proven. √ 4.19. Lemma. The functor U : Sp Σ − → Sp Σ -Cat ( A.1 ) sends pr oje ct ive c ofibr ations to c ofibr ations. Pr o of. The Quillen mo del structure of theore m 4.8 is cofibr a nt ly g enerated and so any cofibr ation in Sp Σ -Cat is a retr act of a transfinite comp osition of pushouts along the genera ting co fibrations. Since the functor U preserves retrac tio ns, co limits a nd send the g enerating pr o jective co fibrations to (generating) cofibr ations, the lemma is prov en. √ 5. St able quasi-equiv alences In this chapter we constr uct a ‘lo calized’ Quillen mo del structur e on Sp Σ -Cat. W e denote b y [ − , − ] the set of morphisms in the stable homo topy category Ho ( Sp Σ ) of s ymmetric sp ectra. F rom a sp ectra l ca tegory A o ne ca n for m a g enu ine ca tegory [ A ] by k eeping the s ame set of ob jects a nd defining the set of morphisms betw een x and y in [ A ] to b e [ S , A ( x, y )]. W e obtain in this w ay a functor [ − ] : Sp Σ -Cat − → Cat , with v alues in the categor y of s mall categor ies. 5.1. Definition. A sp e ctr al functor F : A → B is a stable quasi-equiv alence if: S1) for al l obje cts x, y ∈ A , the m orphism of symmetric sp e ct r a F ( x, y ) : A ( x, y ) → B ( F x, F y ) is a stable e quivalenc e [25, I I-4 .1] and S2) the induc e d funct or [ F ] : [ A ] − → [ B ] is an e qu ivalenc e of c ate gories. 5.2. Notation. W e denote by W s the class of stable quasi-equiv alence s . 5.3. Rema rk. Notice that if condition S 1) is verified, condition S 2) is equiv alent to: S2’) the induced functor [ F ] : [ A ] − → [ B ] is ess e ntially surjective. F unctor Q . In this subsection we construc t a functor Q : Sp Σ -Cat − → Sp Σ -Cat and a natural transfo r mation η : Id → Q , from the iden tity functor on Sp Σ -Cat to the functor Q . W e start with a few definitio ns (see the pro of of prop osition [25, I I- 4.21]). Let m ≥ 0 and λ m : F m +1 S 1 → F m S 0 the morphism of symmetric sp ectra which is adjoin t to the w e dge summand inclusion S 1 → ( F m S 0 ) m +1 = Σ + m +1 ∧ S 1 indexed by the iden tit y e le men t. The morphism λ m factors throug h the mapping cylinder as λ m = r m c m where c m : F m +1 S 1 → Z ( λ m ) is the ‘front’ mapping cylinder inclusio n and r m : Z ( λ m ) → F m S 0 is the pr o jection (which is a homotopy 12 GONC ¸ ALO T ABUADA equiv alence). Notice that c m is a trivia l cofibration [16, 3.4.10] in the pr o jective stable mo del str ucture. Define the s et K as the set of all pusho ut pro duct ma ps ∆[ n ] + ∧ F m +1 S 1 ` ∂ ∆[ n ] + ∧ F m +1 S 1 ∂ ∆[ n ] + ∧ Z ( λ m ) i n + ∧ c m   ∆[ n ] + ∧ Z ( λ m ) , where i n + : ∂ ∆[ n ] → ∆[ n ] , n ≥ 0 is the inclusion map. Let F I Λ be the set of all morphisms of symmetric sp ectra F m Λ[ k , n ] + → F m ∆[ n ] + [25, I-2 .12] induced by the horn inclus io ns for m ≥ 0 , n ≥ 1 , 0 ≤ k ≤ n . 5.4. R emark. By adjointness, a s y mmetric sp ectrum X has the R.L.P . with resp ect to the set F I Λ if and only if for all n ≥ 0, X n is a Kan simplicial set and it has the R.L.P . with re spe c t to the set K if and only if the induced map o f simplicial sets Map( c m , X ) : Map( Z ( λ m ) , X ) − → Map( F m +1 S 1 , X ) ≃ Ω X m +1 has the R.L.P . with resp ect to all inclusio ns i n , n ≥ 0, i.e. it is a trivial Kan fibr ation of s implicial sets. Since the mapping cylinder Z ( λ m ) is homotopy equiv alent to F m S 0 , Map( Z ( λ m ) , X ) is ho motopy equiv alent to Map( F m S 0 , X ) ≃ X m . So altogether, the R.L.P . with r esp ect to the union set K ∪ F I Λ implies tha t for n ≥ 0, X n is a Ka n simplicia l set and for m ≥ 0, f δ m : X m → Ω X m +1 is a w ea k equiv alence, i.e. X is an Ω- sp ectrum. Notice that the conv erse is also tr ue . Let X b e an Ω-spectr um. F o r all n ≥ 0, X n is a Ka n simplicial set, and so X ha s the R.L.P . with resp ect to the set F I Λ . Moreov er, since c m is a cofibration, the map Map( c m , X ) is a Kan fibratio n [1 7, II- 3.2]. Since for m ≥ 0, f δ m : X m → Ω X m +1 is a weak equiv alence, the map Map( c m , X ) is in fact a trivia l Kan fibration. Now consider the set U ( K ∪ F I Λ ) of sp ectr a l functors obtained b y applying the functor U (A.1) to the set K ∪ F I Λ . Since the do mains of the elements of the set K ∪ F I Λ are s equentially small in Sp Σ , the same ho lds by [19] to the domains o f the elements o f U ( K ∪ F I Λ ). Notice tha t U ( K ∪ F I Λ ) = U ( K ) ∪ J ′ (4.5). 5.5. Definitio n. L et A b e a smal l sp e ctr al c ate gory. The functor Q : Sp Σ -Cat → Sp Σ -Cat is obtaine d by applying t he s mal l obje ct ar gu m ent, using the set U ( K ) ∪ J ′ to fac tor the sp e ctr al functor A − → • , wher e • denotes the terminal obje ct in Sp Σ -Cat. 5.6. R emark. W e obtain in this w ay a functor Q a nd a natura l tra nsformation η : Id → Q . Notice also tha t Q ( A ) ha s the s a me o b jects a s A , and the R.L.P . with resp ect to the s et U ( K ) ∪ J ′ . By remar k 5.4 and [19], we get the following pr o pe r ty: Ω) for all ob jects x, y ∈ Q ( A ), the symmetric s p ectr um Q ( A )( x, y ) is an Ω- sp ectrum. 5.7. Prop ositio n. Le t A b e a smal l sp e ctr al c ate gory. The sp e ct r al functor η A : A − → Q ( A ) is a stable quasi-e qu ivalenc e. HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 13 Pr o of. The elements of the set K ∪ F I Λ are trivial cofibra tions in the pro jective stable mo del str ucture. This mo del str ucture is mono idal and satisfies the monoid axiom [16, 5.4 .1]. This implies, b y pr op osition A.2, that the sp ectral functor η A satisfies condition S 1). Since the sp ectral functor η A : A − → Q ( A ) induce the ident it y on se ts of ob jects, co ndition S 2 ′ ) is automa tically verified. √ Main theorem . 5.8. Definition. A sp e ctr al functor F : A → B is: - a Q -weak equiv alence if Q ( F ) is a levelwise qu asi-e qu ivalenc e (4.1). - a c ofibration if it is a c ofibr ation in t he mo del struct u r e of the or em 4.8. - a Q -fibra tion if it has t he R.L.P. with re sp e ct t o al l c ofibr ations which ar e Q -we ak e quivalenc es. 5.9. Lemma. A sp e ctr al functor F : A → B is a Q -we ak e quivalenc e if and only if it is a s t able quasi-e quivalenc e. Pr o of. W e hav e a t our disp osal a comm uta tiv e square A F   η A / / Q ( A ) Q ( F )   B η B / / Q ( B ) , where by pr op osition 5.7, the sp ectral functors η A and η B are stable qua si-equiv alences. Since the class W s satisfies the tw o out of three axiom, the sp ectr a l functor F is a stable quasi-eq uiv a lence if and o nly if Q ( F ) is a stable q ua si-equiv alence. The sp ectral categories Q ( A ) and Q ( B ) satisfy condition Ω) and so by lemma [1 6, 4.2 .6], Q ( F ) sa tisfies condition S 1) if and only if it satisfies condition L 1). Notice that, since Q ( A ) (and Q ( B )) satisfy condition Ω), the set [ S , Q ( A )( x, y )] can b e canonica lly identified with π 0 ( Q ( A )( x, y )) 0 and so the categ ories [ Q ( A )] and π 0 ( Q ( A )) are natura lly identified. This a llows us to conclude that Q ( F ) satisfies condition S 2 ′ ) if and only if it sa tisfies co ndition L 2 ′ ) and so the lemma is pr ov en. √ 5.10. Theorem. The c ate gory Sp Σ -Cat admi ts a right pr op er Quil len mo del struc- tur e whose we ak e quivalenc es ar e the stable quasi-e quivalenc es (5.1) and t he c ofi- br ations those of the or em 4.8. 5.11. Notation. W e denote by Ho ( Sp Σ -Cat) the ho motopy catego ry hence obtained. In order to prove theorem 5.10, we will use a slight v a riant of theor em [17, X-4.1]. Notice that in the proo f of lemma [17, X-4.4] it is only used the right prop erness assumption and in the pro o f of lemma [17 , X-4.6 ] it is o nly us ed the fo llowing condition (A3). This allows us to state the following re sult. 5.12. Theorem. [17, X-4.1] L et C b e a right pr op er Qu il len mo del structu r e, Q : C → C a functor and η : I d → Q a natur al tr ansformation su ch that the fol lowing thr e e c onditions hold: (A1) The functor Q pr eserves we ak e qu ivalenc es. (A2) The maps η Q ( A ) , Q ( η A ) : Q ( A ) → QQ ( A ) ar e we ak e quivalenc es in C . 14 GONC ¸ ALO T ABUADA (A3) Given a diagr am B p   A η A / / Q ( A ) with p a Q -fibr ation, t he 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 Quil len mo del struct u r e on C for which the we ak e quiv- alenc es ar e the Q -we ak e quivalenc es, the c ofibr ations those of C and the fi br ations the Q -fibr ations. Pro of of theorem 5.10. The pro of will consist on verifying the conditions o f theorem 5 .1 2. W e consider for C the Q uillen mo del structure of theore m 4.8 and for Q a nd η , the functor and natural transformation defined in 5.5. The Quillen mo del str ucture of theore m 4.8 is rig h t prop er (4.1 7) and by lemma 5.9 the Q -weak equiv alences a re precisely the stable qua si-equiv alences. W e now verify condition (A1), (A2) a nd (A3): (A1) Let F : A → B b e a levelwise quasi-equiv ale nce. W e hav e the following commutativ e squa re A F   η A / / Q ( A ) Q ( F )   B η B / / Q ( B ) , with η A and η B stable qua s i-equiv alences. Notice that since F sa tisfies con- dition L1), the sp ectral functor Q ( F ) satisfie s condition S1 ). The sp ectral categorie s Q ( A ) and Q ( B ) satis fy condition Ω) and so by lemma [16, 4.2 .6 ] the sp ectral functor Q ( F ) satisfies condition L1). Observe that since the spectr al functors η A and η B induce the identit y on sets o f ob jects and F satisfies condition L2’), the spectr al functor Q ( F ) satisfies also condition L2’). (A2) W e now s how that for every sp ectral ca tegory A , the spectra l functors η Q ( A ) , Q ( η A ) : Q ( A ) → QQ ( A ) are levelwise quasi-equiv alences. Since the sp ectral functors η Q ( A ) and Q ( η A ) are sta ble qua si-equiv alences betw een sp ectral ca tegories which sa t- isfy c ondition Ω), they satisfy by lemma [16, 4.2.6] co ndition L1 ). The functor Q induce the iden tit y on sets of ob jects and so the sp ectral func- tors η Q ( A ) and Q ( η A ) clear ly sa tisfy condition L2 ’). (A3) W e start by observing that if P : C → D is a Q -fibr ation, then for all x, y ∈ C the mor phism of symmetr ic spectra P ( x, y ) : C ( x, y ) − → D ( P x, P y ) is a fibration in pro jective stable model structure [25, I I I-2.2]. In fact, by prop osition 4.1 9, the functor U : Sp Σ − → Sp Σ -Cat HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 15 sends pro jective cofibrations to cofibrations . Since it sends a lso stable equiv alences to stable quasi-equiv alenc e s the claim follows. Now consider the diagram A × Q ( A ) B   / / p B P   A η A / / Q ( A ) , with P a Q -fibra tio n. The pro jective stable mo del structure o n Sp Σ is rig ht prop er a nd so, b y construction of fib er pro ducts in Sp Σ -Cat, w e conclude that the induced spe c tr al functor η A ∗ : A × Q ( A ) B − → B satisfies condition S1). Since η A induces the iden tit y o n sets of ob jects so it thus η A ∗ , and so condition S2’) is verified. 5.13. Prop osi tion. A sp e ctr al c ate gory is fibr ant with r esp e ct t o the or em 5.10 if and only if for al l obje cts x, y ∈ A , the symmetric sp e ctrum A ( x, y ) is an Ω - sp e ctrum. Pr o of. By cor o llary [1 7, X-4 .12] A is fibrant with resp ect to theorem 5 .10 if and only if it is fibrant (4.1 6), with resp ect to the mo del str ucture of theorem 4.8, and the sp ectral functor η A : A → Q ( A ) is a levelwise quasi-eq uiv a lence. Obser ve that η A is a lev elwise quasi-equiv alence if and only if for all ob jects x, y ∈ A the mor phism of symmetric sp ectra η A ( x, y ) : A ( x, y ) − → Q ( A )( x, y ) is a level equiv alence. 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 / / Ω A ( x, y ) n +1   Q ( A )( x, y ) n f δ n / / Ω Q ( A )( x, y ) n +1 , where the b ottom and vertical arrows ar e w ea k equiv alences o f pointed simplicia l sets. This implies that e δ n : A ( x, y ) n ∼ − → Ω A ( x, y ) n +1 , n ≥ 0 is a weak equiv alence of p ointed simplicial se ts a nd so we conclude that for a ll ob jects x, y ∈ A , Q ( x, y ) is an Ω-sp ectrum. √ 5.14. R emark. Notice tha t prop osition 5.7 and remark 5 .4 imply that η A : A → Q ( A ) is a functoria l fibrant replacement of A in the mo del structure of theorem 5.10. 16 GONC ¸ ALO T ABUADA Appendix A. Non-additive fil tra tion argument In this app endix, w e a da pt Sch wede-Shipley’s non-a dditive filtration a rgument [26] to a ‘s everal ob jects’ co n text. Let V b e a monoidal mo del ca tegory , with cofibrant unit I , initial ob ject 0, and which satisfies the monoid axiom [26, 3 .3]. A.1. Definition. L et U : V − → V -Cat , b e the functor which sen ds an obje ct X ∈ V t o the V -c ate gory U ( X ) , with two obje cts 1 and 2 and su ch that U ( X )(1 , 1) = U ( X )(2 , 2) = I , U ( X )(1 , 2) = X and U ( X )(2 , 1) = 0 . Comp osition is n atur al ly define d ( the initial obje ct acts as a zer o with r esp e ct t o ∧ sinc e t he bi-functor − ∧ − pr eserves c olimits in e ach of its vari- ables). In what follows, by smash pr o duct w e mea n the symmetric pr o duct − ∧ − of V . A.2. Prop osition. L et A b e a V -c ate gory, j : K → L a t rivial c ofibr ation in V and F : U ( K ) → A a morphi sm in V -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 qu ivalenc es in V . Pr o of. Notice that A and B ha ve the s ame set of ob jects and the mo rphism R induces the identit y on sets of o b jects. The descriptio n of the mo rphisms R ( x, y ) : A ( x, y ) → B ( x, y ) , x, y ∈ A in V is analogous to the one given b y Sch wede-Shipley in the pro of of lemma [26, 6.2]. The ‘ideia ’ is to think of B ( x, y ) as co ns isting of fo r mal smash pro ducts of ele ments in L with elements in A , with the relations coming fro m K a nd the comp ositio n in A . Consider the same (co nc e ptual) pr o of as the one of lemma [26, 6.2]: B ( x, y ) will app ear as the colimit in V of a seq uence A ( x, y ) = P 0 → P 1 → · · · → P n → · · · , that we now describ e. W e start by defining a n -dimensio nal cub e in V , i.e. a functor W : P ( { 1 , 2 , . . . , n } ) − → V from the p os et category of subsets o f { 1 , 2 , . . . , n } to V . If S ⊆ { 1 , 2 , . . . , n } is a subset, the vertex of the cub e at S is W ( S ) := A ( x, F (0)) ∧ C 1 ∧ A ( F (1) , F (1)) ∧ C 2 ∧ · · · ∧ C n ∧ A ( F (1) , y ) , with C i =  K if i / ∈ S L if i ∈ S . The ma ps in the cub e W a re induced fro m the map j : K → L and the identit y on the remaining factors. So at each v ertex, a total of n + 1 factors of o b jects in V , alternate with n smash factors of either K or L . The initial vertex, corr esp onding to HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 17 the empty subset has a ll its C i ’s equal to K , and the termina l vertex corr e spo nding to the whole s et has a ll it’s C i ’s equa l to L . Denote by Q n , the colimit of the punctur ed cub e, i.e. the cub e with the terminal vertex removed. Define P n via the pusho ut in V Q n   / / y A ( x, F (0)) ∧ L ∧ ( A ( F (1) , F (1)) ∧ L ) ∧ ( n − 1) ∧ A ( F (1) , y )   P n − 1 / / P n , where the left vertical map is defined as follows: for each prop er subset S of { 1 , 2 , . . . , n } , w e consider the co mp os ed map W ( S ) − → A ( x, F (0)) ∧ L ∧ A ( F (1) , F (1)) ∧ . . . ∧ L ∧ A ( F (1) , y ) | {z } | S | factors L obtained by first mapping each factor of W ( S ) equal to K to A ( F (1) , F (1)), and then comp osing in A the a djacent factors . Finally , since S is a proper subset, the right hand side b elongs to P | S | and so to P n +1 . Now the sa me (conceptual) arguments as those of lemma [26, 6.2] sho ws us that the ab ov e construction furnishes us a descriptio n o f the V - category B . W e now analyse the constructed filtration. The cub e W used in the inductive definiton of P n has n + 1 factors of ob jects in V , which map by the ide ntit y ev e r y- where. Using the sy mmetry isomo rphism of − ∧ − , w e can shuffle them all to one side and o bserve tha t the map Q n − → A ( x, F (0)) ∧ L ∧ ( A ( F (1) , F (1)) ∧ L ) ∧ ( n − 1) ∧ A ( F (1) , y ) is isomorphic to Q n ∧ Z n − → L ∧ n ∧ Z n , where Z n := A ( x, F (0)) ∧ L ∧ ( A ( F (1) , F (1)) ∧ L ) ∧ ( n − 1) ∧ A ( F (1) , y ) and Q n is the co limit of a punctured cub e ana logous to W , but with a ll the smash factors diferent from K or L deleted. By iterated applica tion o f the pushout pro duct axiom, the map Q n → L ∧ n is a trivia l co fibration a nd so by the monoid axiom, the map P n +1 → P n is a weak equiv alence in V . Since the map R ( x, y ) : A ( x, y ) = P 0 − → B ( x, y ) is the kind of map considered in the mo noid a xiom, it is also a weak equiv alence and so the propo sition is proven. √ A.3. Prop osi tion. L et A b e a V -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 V . Then in the pushout U ( N ) F / / U ( i )   y A R   U ( M ) / / B , the morphisms R ( x, y ) : A ( x, y ) − → B ( x, y ) , x, y ∈ A 18 GONC ¸ ALO T ABUADA ar e c ofibr ations in V . Pr o of. The description o f the morphisms R ( x, y ) : A ( x, y ) − → B ( x, y ) , x, y ∈ A is analog ous to the one of prop osition A.2. Since for all x, y ∈ A , A ( x, y ) is cofibra n t in V , the pushout pro duct axiom implies that in this s ituation the map Q n ∧ Z n − → L ∧ n ∧ Z n is a cofibration. Since cofibra tions are s ta ble under co-base c hang e and transfinite comp osition, we conclude that the mor phisms R ( x, y ) , x, y ∈ A are co fibrations. √ References [1] A. Bak er, A. Lazarev T op olo gic al Ho chschild c ohomolo gy and g ener alize d Morita e quiv alenc e . Algebr. Geom. T op ol. 4 (2004), 623–645 (electronic). [2] A. Bondal, M. Kaprano v, F r ame d t riangulate d c ate g ories . (Russi an) Mat. Sb. 181 ( 1990) no. 5, 669–683; transl ation in M ath. USSR-Sb. 70 no. 1, 93–107. [3] A. Bondal, M. V an den Bergh, Gener ators and R epr e sentability of F unctors in c ommutative and Nonc ommutative Ge ometry . Moscow Mathematical Journal, 3 (1), 1–37 (2003). [4] J. Bergner, A Quil len mo del st ructur e on the c ategory of simplicial c ategories . T rans. Amer. Math. So c. 3 59 (2007), 2043–2058. [5] F. Borceux, Handb o ok of c ate goric al algebr a. 2 . Encyclopedia of Mathematics and its Appli- cations, 51 , Cambridge Unive rsity Press, 1994. [6] D. Dugger, Sp e ctr al enrichments of mo del c ate gories . Homol ogy , Homotop y , Appl. 8 (2006), no. 1, 1-30. [7] V. Drinf eld, DG quotients of DG c ate gories . J. Algebra 27 2 (2004), 643–691. [8] V. Dr infeld, DG c ate gories . Universit y of Chicago Geometric Langlands Semi nar . Notes av ail- able at http://www .math.ut exas.edu/users/benzvi/GRASP/lectures/Langlands.html . [9] W. Dwy er and D. Kan, Simplicial lo c alizations of c ate gories . J. Pure Appl. Algebra 17 (1980) , no. 3, 267–284. [10] W. Dwyer, J. Greenlees, S. Iyeng ar, Duality in algebr a and top olo gy . Adv. M ath. 200 (2006), no. 2 , 357–402. [11] W. Dwyer, J. Greenlees, S. Iy engar, Finiteness in deriv e d c ate gories of lo c al rings . Comment. Math. Helv. 81 (2006), no. 2, 383–432. [12] A . Elmendorf, I. Kriz, M. Mandell, P . Ma y , Rings, Mo dules, and Algebr as in Stable Homotopy The ory, with an app endix b y M. Cole . Mathematical Surve ys and Monographs, 47 . American Mathematical So ciet y , Providence, RI, 1997. [13] R . F ritsch , D. Latc h, Homotopy inverses for nerve . Math. Z. 17 7 (1981) (2), 147-179. [14] P . Hirschhorn, Mo del c ategories and their lo c alizations . Mathematical Surveys and Mono- graphs, 99 , American M athematica l So ciet y , 2003. [15] M . Hov ey , Mo del c ate gories . Mathematical Surveys and Monographs, 63 , American Mathe- matical So ciety , 1999. [16] M . Ho vey , B. Shipley , J. Smith, Symmetric sp e ctr a . J. Amer. Math. Soc. 13 (200 0), no. 1, 149–208. [17] P . Go erss and J. Jardine, Simplicial homotopy the ory . Progress in Mathematics, 174 , 1999. [18] B. Keller, O n differ ential gr ade d c ategor ies . In ternational Congress of Mathe maticians, V ol. II, 151–190, Eur. Math. So c., Z ¨ uri c h, 2006. [19] G. M. Kelly and S. Lac k, V -Cat is lo c al ly pr e sentable or lo ca l ly b ounde d if V is so . Theory and Applications of Categories, V ol. 8 , no. 23, 2001, 555-575. [20] M . Kontsevic h, Cate gorific ation, NC Mo tives, Ge ometric L anglands and L attic e Mo dels . Univ ers i t y of Chicago Geometric Langlands Seminar. Notes av ail able at http://w ww.math. utexas.edu/users/benzvi/notes.html . [21] M . Kontsevic h, Notes on motives in finite char acteristic . Preprint arXi v:0702206. T o appear in Manin F estsc hrift. [22] M . Lydakis, Simplicial functors and stable homotopy the ory . Prepri n t (1998). Av ailable at h ttp://hopf.math.purdue.edu. HOMOTOPY THEOR Y OF SP E CTRAL CA TEGORIES 19 [23] M . A. Mandell, J. P . M a y , S. Sc hw ede, B. Shipley , Mo del ca te gories of diagr am sp e ctr a . Pro c. London Math. Soc. (3) 82 (2001), no. 2, 441–512. [24] D . Quill en, H omotopic al algebr a . Lecture Notes i n Mathematics, 43 , Spri nger-V erlag, 1967. [25] S. Sc h we de, A n untitled b o ok pr oje ct ab out symmetric sp ectr a . Av ailable at www.math .uni-bon n.de/people/schwede . [26] S. Sc hw ede, B. Shipley , Algebr as and mo dules in monoidal mo del c ate gories . Pro c. London Math. So c. (3) 80 (2000), no. 2, 491–511. [27] S. Sch wede, B. Shipley , St able mo del c ate gories ar e c ate gories of mo dules . T opology 42 (2003) , no. 1, 103–153. [28] S. Sch wede, B. Shipley , Equivalenc es of monoidal mo del ca te gories . Algebr. Geom. T opol . 3 (2003), 287–334. [29] A . E. Stanculescu, A mo del c ategory structur e on the ca te gory of simplicial multica te gories . Av ailable at arXi v:0805 . 2611. [30] G. T abuada, Th´ eorie homotopique des DG-c at ´ egories . Author’s Ph.D thesis. Av ailable at arXiv:0710 . 4303. [31] B. T o¨ en, The homotopy the ory of dg- c ate gories and derive d Morita the ory . Inv ent. Math. 167 (2007), no. 3, 615–667. Dep ar t am ento de M a tem ´ atica e CMA, F CT-UNL, Quint a da Torre, 2829-516 Ca- p arica, Por tugal E-mail addr ess : tabuada@ fct.unl. pt