Differential graded versus Simplicial categories

DIFFERENTIAL GRADED VERSUS SIMPLICIAL CA TEGORIES GONC ¸ ALO T ABUADA Abstract. W e construct a z ig-zag of Quillen adjunctions b et w een the ho- motop y theories of differential graded and si m plicial categories. In an i n- termediate step we generalize Shipley-Sch w ede’s work [21] on connect ive DG algebras by extending the Dold-Kan corresp ondence to a Quil l en equiv alence betw een categories enriche d ov er positive graded cha in complexes and sim- plicial k -mo dules. As an application we obtain a conceptual explanation of Simpson’s homotop y fiber construction [22]. Contents 1. Int ro duction 1 2. Ac knowledgmen ts 2 3. Preliminarie s 2 4. Homotopy theory o f p ositive g raded DG c ategories 3 4.1. Pro of of Theor em 4.7 4 4.2. The truncation functor 6 5. Extended Dold-Ka n equiv alenc e 7 5.1. Left adjoint 10 5.2. Path ob ject 13 5.3. Quillen equiv alence 18 6. The global pictur e 19 References 21 1. Introduction A differe ntial graded (=dg ) ca tegory is a ca teg ory enriched in the catego ry of complexes o f mo dules ov er some c o mm utative base ring k . Dg categ o ries provide a framework for ‘homolo gical g eometry’ and for ‘non commutativ e algebr aic ge - ometry’ in the sense of Drinfeld and Kontsevic h [4] [5] [1 4] [15] [16]. In [23] the homotopy theory of dg catego ries was cons tructed. This theory w as allowed several developmen ts such as: the cre a tion by T o¨ en of a derived Morita theory [24]; the construction of a catego ry of non commut ative motives [23]; the fir st c onceptual characterization o f Q uillen-W aldha us en’s K -theory [23] . . . . On the o ther hand a s implicia l catego r y is a categor y enr iched ov er the category of s implicia l sets. Simplicial c ategories (and their close cousins: quasi-categ ories) provide a framework for ‘homo topy theories’ and for ‘higher category theory’ in the sense of Joy al, Lurie, Rezk, T o¨ en . . . [11][17][20][25]. In [1] Berg ne r constructed a Key wor ds and phr ases. Dg category , Simplicial catego ry , Dol d-Kan corresp ondence, Quillen model structure, Eilenberg-MacLane’s sh uffle map. 1 2 GONC ¸ ALO T ABUADA homotopy theory of simplicial c a tegories by fixing an e r ror in a previo us version of [6]. This theo ry can b e consider ed as one of the four Quillen mo dels for the theory of ( ∞ , 1)-categories, see [2] for a s ur vey . W e observe that the homotopy theories of differential gra ded and simplicia l cat- egories are for mally similar and s o a ‘bridge ’ b et ween the t wo should be developed. In this paper we establish the fir st connexion b etw een these theories by constructing a zig-za g o f Quillen a djunctions relating the t w o: In first pla c e, we co nstruct a Quillen mo del structure on p ositive graded dg categorie s by ‘trunca ting’ the mo de l structure of [23], see theo r em 4.7. Secondly we generaliz e Shipley- Sc hw e de ’s work [21] on connective DG algebr as by extending the Dold-Ka n cor resp ondence to a Quillen equiv alence b etw een ca te- gories enriched ov er p ositive gra ded chain co mplexes and simplicial k -mo dules, see theorem 5.19. Finally we extend the k -lineariz a tion functor to a Quillen adjunction b etw een simplicial catego ries and simplicial k - linear categ ories. As a n applicatio n, the zig- zag of Quillen a djunctions obta ined allow us to give a conceptual ex planation of Simpson’s homotopy fiber construction [22] used in his nonab elian mixed Ho dge theory . 2. Ackno wledgments I am deeply gr ateful to Gustav o Granja for se veral useful disc ussions and for his kindness. 3. Preliminaries In what follows, k will denote a comm utativ e ring with unit. The tensor pro duct ⊗ will deno te the tensor pr o duct ov er k . Let Ch denote the catego ry of complexe s ov er k and Ch ≥ 0 the full sub categ ory of p ositive gra de d c o mplexes. Throughout this a rticle w e cons ide r homologic a l notation (the different ial decreases the degree). Observe that Ch ≥ 0 is a full symmetric monoidal sub c ategory of Ch and that the inclusion Ch ≥ 0 ֒ → Ch commutes with limits and colimits. W e denote by Ch ≥ 0 ( − , − ) the internal Ho m-functor in Ch ≥ 0 with resp ect to ⊗ . By a dg c ate gory , r e sp. p ositiv e gr ade d dg c ate gory , we mea n a catego ry enriched ov er the sy mmetr ic mo noidal ca tegory Ch , resp. Ch ≥ 0 , see [4] [1 2] [13] [23]. W e denote by dgcat , resp. dgcat ≥ 0 , the categor y of small dg catego ries, resp. small po sitive gr a ded dg categories . Notice that dgcat ≥ 0 is a full sub categor y o f d gcat and the inclusion dgcat ≥ 0 ֒ → dgcat commutes with limits and colimits. Let s Set b e the symmetric mono idal ca tegory of simplicial sets and s Mo d the category of simplicia l k -mo dules. W e denote by ∧ the levelwise tensor pr o duct of simplicial k -mo dules. The ca teg ory ( s M o d , − ∧ − ) is a closed sy mmetr ic monoidal category . W e deno te by s Mo d ( − , − ) its internal Ho m-functor. By a simplicial c ate gory , res p. simplic ial k -line ar c ate gory , we mean a ca tegory enriched over s Set , resp. s Mo d , see [1]. DG VERSUS SIMPLICIAL CA TEGORIES 3 W e denote by s Se t -Cat, resp. s Mo d -Cat, the categ ory o f small s implicial ca te- gories, res p. simplicia l k -linear ca tegories. Let ( C , − ⊗ − , I C ) and ( D , − ∧ − , I D ) b e tw o symmetric monoidal categ ories. A lax monoidal functor is a functor F : C → D equippe d with: - a mor phis m η : I D → F ( I C ) and - natural morphisms ψ X,Y : F ( X ) ∧ F ( Y ) → F ( X ⊗ Y ) , X , Y ∈ C which a re coherently asso ciative a nd unital (see diagr ams 6 . 27 a nd 6 . 28 in [3]). A lax mono idal functor is str ong monoidal if the mor phisms η and ψ X,Y are iso- morphisms. Throughout this article the adjunctions ar e display ed vertically with the left, resp. right, a djoin t on the left side, r e s p. right side. 4. Homotopy theor y of positive graded DG ca tegories In this section we will construct a Quillen mo del structure on dgcat ≥ 0 . F or this we will adapt to our situa tion the Quillen mo del structure on dg cat c onstructed in chapter 1 of [23]. R emark 4 .1 . Chapter 1 o f [2 3] (and the whole thesis) is wr itten using cohomo lo gical notation. Througho ut this article we a re alwa ys using homolog ical notation. W e now define the weak equiv alence s in dgcat ≥ 0 . Definition 4 .2. A dg functor F : A → B in dg cat ≥ 0 is a quasi-eq uiv alence if: (i) F ( x, y ) : A ( x, y ) → B ( x, y ) is a quasi-isomorphism in Ch ≥ 0 for al l obje cts x, y ∈ A and (ii) The induc e d functor H 0 ( F ) : H 0 ( A ) → H 0 ( B ) is essential ly surje ctive. Notation 4 .3 . W e denote by Q qe the class of quas i-equiv a le nces in d gcat ≥ 0 . R emark 4.4 . Notice that the class Q qe consist exactly of those qua si-equiv a lences in dgcat , see [23, 1.6], which b elong to dgcat ≥ 0 . In order to build a Quillen mo del structure on dg cat ≥ 0 we consider the gener - ating (trivial) c o fibrations in d gcat which b elong to dgcat ≥ 0 and intro duce a new generating co fibration. Let us now reca ll these constructions, s ee s ection 1 . 3 in [23]. Definition 4.5. F ol lowing D rinfeld [4, 3 .7.1] we define K to b e the dg c ate gory that has two obje cts 1 , 2 and whose morphisms ar e gener ate d by f ∈ K (1 , 2 ) 0 , g ∈ K (2 , 1) 0 , r 1 ∈ K (1 , 1) 1 , r 2 ∈ K (2 , 2) 1 and r 12 ∈ K (1 , 2) 2 subje ct to t he r elations d ( f ) = d ( g ) = 0 , d ( r 1 ) = g f − 1 1 , d ( r 2 ) = f g − 1 2 and d ( r 12 ) = f r 1 − r 2 f . 1 r 1 : : f ( ( r 12   2 r 2 d d g h h L et k b e the dg c ate gory with one obje ct 3 , s uch that k (3 , 3) = k . L et F b e the dg functor fr om k to K that sends 3 t o 1 . L et B b e the dg c ate gory with two obje cts 4 and 5 such that B (4 , 4) = k , B (5 , 5) = k , B (4 , 5) = 0 and B (5 , 4) = 0 . L et n ≥ 1 , S n − 1 the c omplex k [ n − 1] and let D n b e the mapping c one on the identity 4 GONC ¸ ALO T ABUADA of S n − 1 . W e denote by P ( n ) the dg c ate gory with two obje cts 6 and 7 su ch that P ( n )(6 , 6 ) = k , P ( n )(7 , 7) = k , P ( n )(7 , 6) = 0 , P ( n )(6 , 7) = D n and whose c omp osition given by mult iplic ation. L et R ( n ) b e the dg funct or fr om B to P ( n ) that sends 4 to 6 and 5 to 7 . L et C ( n ) b e t he dg c ate gory with two obje cts 8 et 9 such that C ( n )(8 , 8) = k , C ( n )(9 , 9 ) = k , C ( n )(9 , 8) = 0 , C ( n )(8 , 9) = S n − 1 and whose c omp osition given by multiplic ation. L et S ( n ) b e the dg fun ctor fr om C ( n ) to P ( n ) that sends 8 to 6 , 9 t o 7 and S n − 1 to D n by the identity on k in de gr e e n − 1 . L et Q b e the dg functor fr om the empty dg c ate gory ∅ , which is the initial obje ct in dgcat ≥ 0 , to k . Final ly let N b e the dg functor fr om B t o C (1) that sends 4 t o 8 and 5 to 9 . Let us now recall the following standard recognitio n theor em: Theorem 4.6. [8, 2.1.19] L et M b e a c omplete and c o c omplete c ate gory, W a class of maps in M and I and J sets of maps in M such that: 1) The class W satisfies t he two out of thr e e axiom and is stable u nder r etr acts. 2) The domains of the element s of I ar e smal l r elative to I -c el l. 3) The domains of the element s of J ar e smal l r elative to J -c el l. 4) J − c el l ⊆ W ∩ I − c of. 5) I − inj ⊆ W ∩ J − inj. 6) W ∩ I − c of ⊆ J − c of or W ∩ J − inj ⊆ I − inj. Then t her e is a c ofibr ant ly gener ate d mo del c ate gory structur e on M in which W is the class of we ak e quivalenc es, I is a set of gener ating c ofibr ations, and J is a set of gener ating t rivial c ofibr ations. Theorem 4.7. If we let M b e the c ate gory dgcat ≥ 0 , W b e the class Q qe , J b e the set of dg fun ctors F and R ( n ) , n ≥ 1 , and I t he set of dg functors Q , N and S ( n ) , n ≥ 1 , then the c onditions of the r e c o gnition the or em 4.6 ar e s atisfie d. Thus, the c ate gory dgcat ≥ 0 admits a c ofibr antly gener ate d Qu il len mo del struct u r e whose we ak e quivalenc es ar e t he qu asi-e quivalenc es. 4.1. Proo f of Theorem 4.7. W e start b y obser ving that the catego ry dgcat ≥ 0 is complete a nd co co mplete and that the class Q qe satisfies the t wo out o f three axiom and that it is sta ble under r etracts. W e obser ve a lso that the domains and co domains of the morphisms in I and J a r e s mall in the c a tegory dgcat ≥ 0 . This implies that the first thr e e conditions of the reco gnition theorem 4.6 are verified. Lemma 4 .8. J − c el l ⊆ Q qe . Pr o of. Since the inclus io n dgcat ≥ 0 ֒ → dgcat preserves co limits and the class Q qe consist exa ctly of those quas i-equiv ale nc e s in dgcat which b elong to dg cat ≥ 0 , the pro of follows from lemma 1 . 10 in [23]. √ W e now prov e that J − inj ∩ Q qe = I − inj. F or this we introduce the following auxiliary class o f dg functors: Definition 4. 9. L et Surj ≥ 0 b e t he class of dg funct ors G : H → I in dgcat ≥ 0 such that: - G ( x, y ) : H ( x, y ) → I ( Gx, Gy ) is a surje ct ive quasi-isomorphism for al l obje cts x, y ∈ H and - G induc es a surje ct ive map on obje cts. DG VERSUS SIMPLICIAL CA TEGORIES 5 R emark 4.10 . Notice that the clas s Surj ≥ 0 consist exactly o f thos e dg functors in Surj , see se c tion 1 . 3 . 1 in [23], which belo ng to dgcat ≥ 0 . Lemma 4 .11. I − inj = Surj ≥ 0 . Pr o of. W e prove first the inclusion ⊇ . Let G : H → I b e a dg functor in Surj ≥ 0 . By rema r k 4.10, G belo ngs to Surj a nd so lemma 1 . 11 in [23] implies that G has the right lifting prop erty with r esp ect to the dg functors Q and S ( n ) , n ≥ 1 . Since the morphis m o f complexes G ( x, y ) : H ( x, y ) → I ( Gx, Gy ) , x, y ∈ H is surjective o n the deg ree zer o co mp onent, the dg functor G als o has the right lifting prop er t y with resp ect to N . This prov es the inclusion ⊇ . W e now prove the inclusio n ⊆ . Let R : C → D b e a dg functor in I − inj. Lemma 1 . 11 in [2 3] implies that: - R induces a sur jectiv e map on ob jects and - for all ob jects x, y ∈ C : - R ( x, y ) : C ( x, y ) → D ( Rx, Ry ) is a sur jectiv e quasi- isomorphism for n ≥ 1 and - H 0 R ( x, y ) : H 0 C ( x, y ) → H 0 D ( R x, Ry ) is an injective map. Since R b elongs to I − inj it has the rig h t lifting prop erty with resp ect to N and so the mor phism o f complexes R ( x, y ) is also s urjective on the deg ree zer o component. This clearly implies that R b elongs to Surj ≥ 0 and proves the inclus ion ⊆ . √ W e now consider the following ‘dia gram chasing’ lemma: Lemma 4 .12. L et f : M • → N • b e a morphism in Ch ≥ 0 such that: - f n : M n → N n is surje ctive map for n ≥ 1 and - H n ( M • ) → H n ( N • ) is an isomorphism for n ≥ 0 . Then f 0 : M 0 → N 0 is also a surje ctive map. Pr o of. It’s a simple dia gram chasing argument. √ Lemma 4 .13. J − inj ∩ Q qe = Surj ≥ 0 . Pr o of. The inclus ion ⊇ follows from remark 4.10 and from the inclusion ⊇ in lemma 1 . 12 of [2 3]. W e now prov e the inc lus ion ⊆ . Let R : C → D b e a dg functor in J − inj ∩ Q qe . Since R b elongs to Q qe and it has the right lifting pr op erty with resp ect to the dg functors R ( n ) , n ≥ 1 the mo rphism of complexes R ( x, y ) : C ( x, y ) → D ( R x, Ry ) , x, y ∈ C satisfies the conditions of lemma 4.12 and so R ( x, y ) is a sur jective quasi-isomor phism. Finally the fa c t that R induces a surjective map on ob jects follows fr o m lemma 1 . 12 in [23]. This proves the lemma. √ Lemma 4 .14. J − c el l ⊆ I − c of. Pr o of. Observe that the morphisms in J − cell hav e the left lifting pro p er ty with resp ect to the class J − inj. By lemmas 4 .11 and 4.13 I − inj = J − inj ∩ Q qe and so the morphisms in J − cell hav e also the left lifting prop erty with r esp ect to the class I − inj, i.e. J − cell ⊆ I − cof. √ 6 GONC ¸ ALO T ABUADA W e hav e shown tha t J − cell ⊆ Q qe ∩ I − cof (lemmas 4 .8 a nd 4.14) and that I − inj = J − inj ∩ Q qe (lemmas 4.11 and 4.1 3). This implies that the las t thr ee conditions of the recognition theore m 4.6 are satisfied. This finishes the pro of o f theorem 4.7. R emark 4.15 . Since every ob ject in dgc at is fibra nt, s e e remark 1 . 14 in [23], a nd the set J of g enerating trivia l cofibr ations in dgcat ≥ 0 is a subse t of the g e ne r ating trivial co fibrations in dg cat we conclude that ev ery ob ject in dg cat ≥ 0 is also fibrant. 4.2. The truncation functor. In this subsection we construct a functorial path ob ject in the Quillen mo del categor y d gcat ≥ 0 . Consider the following adjunction: Ch τ ≥ 0   Ch ≥ 0 ?  i O O where τ ≥ 0 denotes the ‘intelligent’ tr uncation functor: to a complex M • : · · · ← M − 2 d − 1 ← M − 1 d 0 ← M 0 d 1 ← M 1 ← · · · it ass o c ia tes the complex τ ≥ 0 M • : · · · ← 0 ← 0 ← K er ( d 0 ) d 1 ← M 1 ← · · · . The truncation functor τ ≥ 0 is a lax mo noidal functor. In particular we have natural morphisms τ ≥ 0 M • ⊗ τ ≥ 0 N • − → τ ≥ 0 ( M • ⊗ N • ) , M • , N • ∈ Ch which s a tisfy the asso ciativity conditions. Observe that the truncation functor τ ≥ 0 preserve the unit · · · ← 0 ← k ← 0 ← · · · of b oth symmetric monoidal structure s. Definition 4. 16. L et A b e a s mal l dg c ate gory. The trunca tion τ ≥ 0 A of A is the p ositive gr ad e d dg c ate gory with the same obje cts as A and whose c omplexes of morphisms ar e define d as τ ≥ 0 A ( x, y ) := τ ≥ 0 A ( x, y ) , x, y ∈ A . F or x, y and z obje ct s in τ ≥ 0 A the c omp osition is define d as τ ≥ 0 A ( x, y ) ⊗ τ ≥ 0 A ( y , z ) − → τ ≥ 0 ( A ( x, y ) ⊗ A ( y , z )) τ ≥ 0 ( c ) − → τ ≥ 0 A ( x, z ) , wher e c denotes t he c omp osition op er ation in A . The units in τ ≥ 0 A ar e the same as those of A . Observe that we hav e a natural adjunction dgcat τ ≥ 0   dgcat ≥ 0 . ?  i O O R emark 4 .17 . Notice that b oth functors i and τ ≥ 0 preserve qua si-equiv a lences. DG VERSUS SIMPLICIAL CA TEGORIES 7 Prop ositio n 4.18. The adjunction ( i, τ ≥ 0 ) is a Quil len adjunction. Pr o of. Clearly , by remar k 4.4 the functor i preserves weak equiv alences. W e now show that it also pr e s erves cofibrations. The Quillen mo del str uc tur e of theorem 4.7 is c o fibrantly genera ted and s o by prop osition 11 . 2 . 1 in [7] the c la ss of cofibrations equals the class of r etracts of relative I − cell complexe s. Since the functor i preser ves colimits it is then eno ugh to pr ov e that it sends the generating cofibrations in dgcat ≥ 0 to cofibrations in dg cat . This is clear , by definition, for the generating cofibrations Q a nd S ( n ) , n ≥ 1. W e now obs erve that i ( N ) = N is also a cofibratio n in dgcat . In fact N can b e obtained by the following push-out C (0)   S (0)   P / / y B   N   P (0) / / C (1) , where S (0) is a generating cofibratio n in d gcat , see section 1 . 3 in [2 3], and P se nds 8 to 4 a nd 9 to 5. This prov es the lemma. √ R emark 4.19 . Recall from [23, 4.1] the cons tr uction o f a path ob ject P ( A ) for each dg catego ry A ∈ dgcat . Lemma 4 . 20. L et A b e a p ositive gr ade d dg c ate gory. Then τ ≥ 0 P ( A ) is a p ath obje ct of A in dgcat ≥ 0 . Pr o of. Consider the diago nal dg functor A ∆ − → A × A in dgcat . W e have, as in [23, 4.1], a factorizatio n A ∆ / / I ∼ " " E E E E E E E E E A × A P ( A ) , P : : : : t t t t t t t t t where I is a quasi-e q uiv alence a nd P a fibra tion. By r emark 4.1 7 and lemma 4.18 the functor τ ≥ 0 preserves qua s i-equiv a lences and fibra tions. Since the functor τ ≥ 0 also preserves limits we o btain the following factoriza tion A ∆ / / ∼ τ ≥ 0 ( I ) # # H H H H H H H H H A × A τ ≥ 0 P ( A ) τ ≥ 0 ( P ) 9 9 9 9 s s s s s s s s s s . This prov es the lemma. √ 5. Extended Do ld-Kan equ iv alence In this section we will first cons tr uct a Quillen mo del s tr ucture on s Mo d -Cat and then show that it is Quillen equiv alen t to the mo del structure on dg cat ≥ 0 of theorem 4.7. 8 GONC ¸ ALO T ABUADA Recall from [9, I I I-2.3] the Dold-Ka n e quiv alence betw een simplicial k -mo dules and p ositive g r aded complexes s Mo d N   Ch ≥ 0 , Γ O O where N is the nor malization functor and Γ its inverse. The normalization functor N is a lax monoidal functor, see [21, 2.3], via the Eilenber g-MacLane shuffle map, see [18, VI I I-8.8 ] ∇ : N A ⊗ N B − → N ( A ∧ B ) , A, B ∈ s Mo d . Observe that the nor malization functor N preserves the unit of the tw o symmetric monoidal structures. As it is shown in [2 1, 2.3] the la x monoidal str ucture on N , given by the s h uffle map ∇ , induces a lax c o monoida l structure on Γ: e ψ : Γ( M ⊗ M ′ ) − → Γ( M ) ∧ Γ( M ′ ) , M , M ′ ∈ Ch ≥ 0 . Now, let I be a set. Notation 5 .1 . W e denote by Ch I ≥ 0 -Gr, r esp. by Ch I ≥ 0 -Cat, the catego r y o f Ch ≥ 0 - graphs with a fixed set o f o b jects I , resp. the ca teg ory of categor ies e nr iched over Ch ≥ 0 which hav e a fixed set of ob jects I . The morphisms in Ch I ≥ 0 -Gr a nd Ch I ≥ 0 -Cat induce the identit y map o n the ob jects. W e hav e a natural adjunction Ch I ≥ 0 -Cat U   Ch I ≥ 0 -Gr , T I O O where U is the forge tful functor and T I is defined as T I ( A )( x, y ) :=    k ⊕ L x,x 1 ,...,x n ,y A ( x, x 1 ) ⊗ . . . ⊗ A ( x n , y ) if x = y L x,x 1 ,...,x n ,y A ( x, x 1 ) ⊗ . . . ⊗ A ( x n , y ) if x 6 = y Comp osition is given by conc a tenation and the unit corr esp onds to 1 ∈ k . R emark 5 .2 . - Notice that the categ o ries Ch I ≥ 0 -Gr and Ch I ≥ 0 -Cat admit stan- dard Quillen mo del structures whose weak equiv alences (r esp. fibrations ) are the morphisms F : A → B s uc h that F ( x, y ) : A ( x, y ) − → B ( x, y ) , x, y ∈ I is a weak equiv alence (resp. fibration) in Ch ≥ 0 . In fact the pro jective Quillen mo del str ucture on Ch ≥ 0 , see [9 , I I I-2], na turally induces a mo del structure on Ch I ≥ 0 -Gr which c a n be lifted along the functor T I using theorem 11 . 3 . 2 in [7]. - If the set I has a unique element, then the previous adjunction co rresp onds to the (Quillen) adjunction betw een connective dg alg ebras and po sitive graded co mplexes, see [10]. DG VERSUS SIMPLICIAL CA TEGORIES 9 Notation 5.3 . W e denote by s Mo d I -Gr, resp. by s M o d I -Cat, the categ ory of s Mo d -gra phs with a fixed set of ob jects I , r esp. the c a tegory of ca teg ories enriched ov er s Mo d which have a fixed set o f ob jects I . The morphisms in s M o d I -Gr and s Mo d I -Cat induce the identit y map on the o b jects. In an a nalogous wa y we hav e a n adjunction s Mo d I -Cat U   s Mo d I -Gr , T I O O where U is the forge tful functor and T I is defined as T I ( B )( x, y ) :=    k ∆ 0 ⊕ L x,x 1 ,...,x n ,y B ( x, x 1 ) ∧ . . . ∧ B ( x n , y ) if x = y L x,x 1 ,...,x n ,y B ( x, x 1 ) ∧ . . . ∧ B ( x n , y ) if x 6 = y Comp osition is given by conc a tenation and the unit corr esp onds to 1 ∈ k ∆ 0 . R emark 5.4 . If the set I has a n unique element, then the previous a djunction corres p onds to the classical adjunction betw een simplicial k -algebr a s and simplicial k -mo dules, see [9 , I I- 5.2]. Clearly the Dold-Kan equiv alence induces an equiv a le nce of categor ies s Mo d I -Gr N   Ch I ≥ 0 -Gr Γ O O that we still denote by N and Γ. Since the functor N : s Mo d → Ch ≥ 0 is lax monoidal it induces, as in [21, 3 .3], a normaliz a tion functor s Mo d I -Cat N I   Ch I ≥ 0 -Cat . In fact, let A ∈ s Mo d I -Cat and x, y and z o b jects in A . Then N I ( A ) has the same ob jects a s A , the complexes of mor phisms are given b y N I ( A )( x, y ) := N A ( x, y ) , x, y ∈ A and the co mpos ition is defined by N A ( x, y ) ⊗ N A ( y , z ) ∇ − → N ( A ( x , y ) ∧ A ( y , z )) N ( c ) − → N A ( x, z ) , where c denotes the c o mpo sition op eration in A . The units in N I ( A ) are induced by those o f A under the normaliz ation functor N . As it is shown in section 3 . 3 of [21] the functor N I admits a left adjoint L I . 10 GONC ¸ ALO T ABUADA Let A ∈ Ch I ≥ 0 -Cat. The v alue of the left adjoint L I on A is defined as the co equalizer of tw o morphisms in s Mo d I -Cat T I Γ U T I U ( A ) ψ 1 / / ψ 2 / / T I Γ U ( A ) / / L I ( A ) . The morphism ψ 1 is obta ined from the unit of the adjunction T I U A − → A by applying the comp osite functor T I Γ U ; the morphism ψ 2 is the unique mor phism in s Mo d I -Cat induced b y the s Mo d I -Gr morphism Γ U T I U ( A ) − → U T I Γ U ( A ) whose v alue a t Γ U T I U ( A )( x, y ) , x, y ∈ I is L x,x 1 ,...,x n ,y Γ( A ( x, x 1 ) ⊗ . . . ⊗ A ( x n , y )) ˜ ψ   L x,x 1 ,...,x n ,y Γ A ( x, x 1 ) ∧ . . . ∧ Γ A ( x n , y ) , where ˜ ψ is the lax c o monoidal structure on Γ induced by the lax monoidal structure on N , see section 3 . 3 o f [21]. 5.1. Left adjoint. Notice that the normalization functor N I : s Mo d I -Cat → Ch I ≥ 0 -Cat, o f the previous subsection, ca n b e na tur ally defined for every se t I a nd so it induces a ‘globa l’ normalization functor s Mo d -Cat N   dgcat ≥ 0 . In this subsec tio n we will construct the left adjoint of N . Let A ∈ dgcat ≥ 0 and denote by I its set of ob jects. Define L ( A ) a s the simplicial k -linear catego ry L I ( A ). Now, le t F : A → A ′ be a dg functor. W e deno te by I ′ the s et of ob jects of A ′ . The dg functor F induces the following dia g ram in s Mo d -Cat: T I Γ U T I U ( A ) ψ 1 / / ψ 2 / /   T I Γ U ( A )   / / L I ( A ) =: L ( A ) T I ′ Γ U T I ′ U ( A ′ ) ψ 1 / / ψ 2 / / T I ′ Γ U ( A ′ ) / / L I ′ ( A ′ ) = : L ( A ′ ) . Notice that the sq ua re who se hor izontal ar rows are ψ 1 (resp. ψ 2 ) is co mm utative. Since the inclusio ns s Mo d I -Cat ֒ → s Mo d -Cat and s Mo d I ′ -Cat ֒ → s Mo d -Cat DG VERSUS SIMPLICIAL CA TEGORIES 11 clearly preser ve co equalizers the pr evious diagra m in s Mo d -Cat induces a simplicial k -linear functor L ( F ) : L ( A ) − → L ( A ′ ) . W e hav e constructed a functor L : dgcat ≥ 0 − → s Mo d -Cat . Prop ositio n 5.5. The functor L is left adjoi nt to N . Pr o of. Let A ∈ dgcat ≥ 0 and B ∈ s Mo d -Cat. Let us deno te by I the set of ob jects of A . W e will co nstruct tw o natur al maps s Mo d -Cat( L ( A ) , B ) φ / / dgcat ≥ 0 ( A , N ( B )) η o o and then show that they are inv erse o f each other. Let G : L ( A ) → B b e a simplicial k -linear functor. W e denote by B ′ the full sub c ategory o f B who se o b jects are those which b elong to the image of G . W e hav e a natural factorization L ( A ) G / / G ′ " " D D D D D D D D B B ′ /  ? ? ~ ~ ~ ~ ~ ~ ~ ~ . Now, let e B be the simplicial k -linear categ ory whose set o f ob jects is ob j( e B ) := { ( a, b ) | a ∈ L ( A ) , b ∈ B ′ and G ′ ( a ) = b } and whose simplicia l k -mo dule o f morphisms is defined as e B (( a, b ) , ( a ′ , b ′ )) := B ′ ( b, b ′ ) . The comp ositio n is given by the comp ositio n in B ′ . No w co nsider the simplicia l k -linear functor e G : L ( A ) − → e B which maps a to ( a, G ′ ( a )) and the simplicial k - linear functor P : e B − → B ′ which maps ( a, b ) to b . The ab ove constructions a llow us to fa c tor G a s the following comp o sition L ( A ) e G / / e B P / / B ′   / / B . Notice that e G induces a bijectio n o n ob jects a nd s o it b elongs to s M o d I -Cat. Finally define φ ( G ) as the following comp osition φ ( G ) : A e G ♯ / / N e B N P / / N B ′   / / N B , where e G ♯ denotes the mor phism in Ch I ≥ 0 -Cat which co rresp onds to e G under the adjunction ( L I , N I ). 12 GONC ¸ ALO T ABUADA W e now co nstruct in a simila r wa y the map η . Let F : A → N B b e a dg functor and ( N B ) ′ be the full sub catego ry of N B whose ob jects are those which belo ng to the image of F . W e hav e a natura l factorizatio n A F / / F ′ " " E E E E E E E E N B ( N B ) ′ ,  ; ; v v v v v v v v v . Now, let g N B b e the p ositive g raded dg ca tegory whose s e t of ob jects is ob j( g N B ) := { ( a, b ) | a ∈ A , b ∈ N B ′ and F ′ ( a ) = b } and whose p ositive graded complex of morphisms is defined as g N B (( a, b ) , ( a ′ , b ′ )) := ( N B ) ′ ( b, b ′ ) . The comp osition is given by the comp ositio n in ( N B ) ′ . Consider the dg functor e F : A − → g N B which maps a to ( a, F ′ ( a )) and the dg functor P : g N B − → ( N B ) ′ which maps ( a, b ) to b . The ab ove constructions a llow us to fa c tor F as the following co mpo sition A e F / / g N B P / / ( N B ) ′   / / N B . Notice that e F induces a bijection on ob jects and so belo ngs to Ch I ≥ 0 -Cat. Since the normalizatio n functor N preserve the set of ob jects, the ab ov e constructio n g N B P / / ( N B ) ′   / / N B can b e natur ally lifted to the categor y s Mo d -Cat. W e hav e the folowing diagr a m s Mo d -Cat N   e B P / / _   B ′ _     / / B _   dgcat ≥ 0 g N B P / / ( N B ) ′   / / N B . W e can now define η ( F ) as the following comp osition η ( F ) : L ( A ) e F ♮ / / e B P / / B ′   / / B , where e F ♮ denotes the morphism in s Mo d I -Cat, which co rresp onds to e F under the adjunction ( L I , N I ). The maps η and φ are clea rly inv erse of each o ther and so the prop osition is prov en. √ DG VERSUS SIMPLICIAL CA TEGORIES 13 5.2. P ath ob ject. In this s ubsection w e lift the Quillen mo del structure on dgcat ≥ 0 , see theorem 4 .7, along the adjunction s Mo d -Cat N   dgcat ≥ 0 L O O of the previous subs e c tion. F or this we will use theor em 5 . 1 2 and pr op osition 5 . 1 3 of [23]. Definition 5 .6. A simplicial k -line ar fun ctor G : A → B is: - a w eak equiv alence if N G is a quasi-e quival enc e in dgc at ≥ 0 . - a fibration if N G is a fi br ation in dgcat ≥ 0 . - a cofibration if it has the left lifting pr op erty with r esp e ct to al l trivial fi- br ations in s Mo d -Cat. Definition 5.7. L et A b e a smal l simplicial k - line ar c ate gory. The homotop y category π 0 ( A ) of A is the c ate gory which has the same obje cts as A and whose morphisms ar e define d as π 0 ( A )( x, y ) := π 0 ( A ( x, y )) , x, y ∈ A . Lemma 5.8. L et x f → y b e a 0 - simplex morphism in A . Then π 0 ( f ) is invertible in π 0 ( A ) iff H 0 ( N f ) is invertible in H 0 ( N A ) . Pr o of. W e start by obser ving that if we restric t ourselves to the 0-simplex mor- phisms in A and to the degree zero mo rphisms in N A we have the same category . In fact the degree zero comp onent of the s h uffle map ∇ , used in the definition of N A , is the iden tit y map, see [18, VI I I-8.8]. Now supp ose that π 0 ( f ) is invertible. Then ther e exis ts a 0- simplex morphism g : y → x and 1 -simplex morphisms h 1 ∈ A ( x, x ) and h 2 ∈ A ( y , y ) such that d 0 ( h 1 ) = 1 X , d 1 ( h 1 ) = g f , d 0 ( h 2 ) = 1 Y and d 1 ( h 2 ) = f g . Observe that the image of h 1 , r esp. h 2 , by the normaliza tion functor N is a de g ree 1 morphism in N A ( x, x ), resp. in N A ( y , y ), whose differential is g f − 1 X (resp. f g − 1 Y ). This implies that H 0 ( N f ) is also inv ertible in H 0 ( N A ). T o pr ov e the co n verse we c o nsider an a nalogous argument. √ Prop ositio n 5.9. A simplicial k -line ar funct or G : A → B is a we ak e quivalenc e iff: (1) G ( x, y ) : A ( x, y ) → B ( Gx, Gy ) induc es an isomorphism on π i for al l i ≥ 0 and for al l obje cts x, y ∈ A , and (2) π 0 ( G ) : π 0 ( A ) → π 0 ( B ) is essential ly surje ctive. Pr o of. W e show that condition (1), re sp. condition (2), is equiv alent to condition ( i ), resp. co ndition ( ii ), of definition 4.2. By the Dold-Ka n equiv alence, we hav e the following co mm utative diagr am π i A ( x, y ) G / / ∼   π i B ( Gx, Gy ) ∼   H i N A ( x, y ) N G / / H i N B ( Gx, Gy ) 14 GONC ¸ ALO T ABUADA where the vertical a rrows ar e iso morphisms. This implies that condition (1) is equiv ale nt to condition ( i ) of definition 4.2. Concerning condition (2), we star t by suppos ing that π 0 ( G ) is essentially surjec- tive. Consider the functor H 0 ( N G ) : H 0 ( N A ) → H 0 ( N B ) and let z b e a n ob ject in H 0 ( N B ). Since π 0 ( B ) and H 0 ( N B ) have the same ob- jects w e can consider z as a n o b ject in π 0 ( B ). By hypothesis, π 0 ( G ) is essentially surjective and so there exists an ob ject w ∈ π 0 ( A ) and a 0-simplex mor phism Gw f → z which beco mes invertible in π 0 ( B ). Now lemma 5.8 implies that N f is inv ertible in H 0 ( N B ) and so we conclude that the functor H 0 ( N G ) is essentially surjective. This shows that condition (2) implies condition ( ii ) of definitio n 4.2. T o prov e the conv erse we co nsider an ana logous argument. √ Theorem 5.10. The c ate gory s Mo d -Cat when endowe d with the notions of we ak e quivalenc e, fibr ation and c ofibr ation as in definition 5.6 , b e c omes a c ofibr antly gen- er ate d Quil len mo del c ate gory and the adjunction ( L, N ) b e c omes a Quil len adjunc- tion. The pro of will consist on verifying the conditions of theo rem 5 . 12 and prop osition 5 . 13 in [23]. Since the Quillen mo del s tructure on dgcat ≥ 0 is cofibra nt ly genera ted, see theor em 4.7; every ob ject in dg cat ≥ 0 is fibran t, see remar k 4.15; and the functor N clearly commutes with filtered co limits it is enough to show that: - for each s implicia l k -linea r category A , w e hav e a facto rization A ∆ / / ∼ I A " " D D D D D D D D A × A P ( A ) , P 0 × P 1 : : : : u u u u u u u u u with I A is a weak equiv a lence and P 0 × P 1 is a fibra tio n in s Mo d -Cat. F or this we need a few lemmas. W e start with the following definitio n. Definition 5.11. L et us define P ( A ) as the simplicial k -line ar c ate gory whose obje cts ar e t he 0 -simplex morphisms f : x → y in A which b e c ome invertible in π 0 ( A ) . We define the simplicial k -mo dule of morphisms P ( A )( x f → y , x ′ f ′ → y ′ ) , f , f ′ ∈ P ( A ) as the homotopy pul l- b ack in s Mo d of the diagr am A ( y , y ′ ) f ∗   A ( x, x ′ ) f ′ ∗ / / A ( x, y ′ ) , by which we me an t he simplicia l k -mo dule A ( x, x ′ ) × A ( x,y ′ ) s Mo d ( k ∆[1] , A ( x, y ′ )) × A ( x,y ′ ) A ( y , y ′ ) . DG VERSUS SIMPLICIAL CA TEGORIES 15 We denote the simplexes in A ( x, x ′ ) and A ( y , y ′ ) la teral morphisms and the sim- plexes in s Mo d ( k ∆[1] , A ( x, y ′ )) homoto pies . The c omp osition op er ation P ( A )( f , f ′ ) ∧ P ( A )( f ′ , f ′′ ) − → P ( A )( f , f ′′ ) , f , f ′ , f ′′ ∈ P ( A ) de c omp oses on: - a c omp osition of later al morphisms, which is induc e d by the c omp osition on A and - a c omp ositio n of homotopies, which is given by the map s Mo d ( k ∆[1] , A ( x, y ′ )) ∧ A ( y ′ , y ′′ ) × s Mo d ( k ∆[0] ,A ( x ,y ′′ )) A ( x, x ′ ) ∧ s Mo d ( k ∆[1] , A ( x ′ , x ′′ )) c omp osition   s Mo d ( k ∆[1] L k ∆[0] k ∆[1] , A ( x, y ′′ ))   s Mo d ( k ∆[1] , A ( x, y ′′ )) , wher e the last map is induc e d by the diago nal map in k ∆[1] . R emark 5.12 . Notice that a 0-simplex morphism α : f → f ′ in P ( A ) is o f the form α = ( m x , h, m y ), with m x : x → x ′ and m y : y → y ′ 0-simplex mo rphisms in A and h is a 1-simplex mor phism in A ( x, y ′ ) such that d 0 ( h ) = m y f and d 1 ( h ) = f ′ m x . W e hav e a natural commutativ e diag ram in s Mo d -Cat (1) A ∆ / / I A ! ! D D D D D D D D A × A P ( A ) P 0 × P 1 : : v v v v v v v v v where I A is the simplicial k -linear functor that asso ciates to an ob ject x ∈ A the 0-simplex mo rphism x I d → x a nd P 0 , resp. P 1 , is the simplicial k -linea r functor that sends a mor phism x f → y in P ( A ) to x , resp. y . Notice that by applying the normalization functor N to the ab ov e diag ram and lemma 4.20 to the dg ca tegory N A we obtain tw o factorizations N A ∆ / / % % K K K K K K K K K K τ ≥ 0 ( I ) $ $ N A × N A N P ( A ) 7 7 o o o o o o o o o o o τ ≥ 0 P ( N A ) τ ≥ 0 ( P ) H H of the diago na l dg functor. By lemma 4 .20 τ ≥ 0 P ( N A ) is a path ob ject of N A in dgcat ≥ 0 . W e will show in prop ositio n 5.17 tha t N P ( A ) is also a path ob ject of N A . Lemma 5 .13. L et A, B ∈ s Mo d . The shuffle map ∇ induc es a n atur al surje ctive chain homotopy e quiva lenc e N ( s Mo d ( A, B )) ∇ ♯ − → Ch ≥ 0 ( N A, N B ) , 16 GONC ¸ ALO T ABUADA which has a natura l se ction induc e d by the Alexander-Whitney map. Pr o of. First note that if ( L, R ) and ( L ′ , R ′ ) are a djoin t pair s o f functors , a natura l transformatio n ζ : L → L ′ induces a natural trans fo rmation ζ ♯ : R ′ → R which is a natural equiv a lence iff ζ is a lso. Fixing a chain complex N A ∈ Ch ≥ 0 let L, L ′ : Ch ≥ 0 → Ch ≥ 0 be defined by L ( C ) := C ⊗ N A, L ′ ( C ) := N (Γ C ∧ A ) . Using the Dold-Ka n equiv alence in the case of L ′ , w e see that these functors have right adjoints R ( C ) = Ch ≥ 0 ( N A, C ) , R ′ ( C ) = N ( s Mo d ( A, Γ C )) resp ectively . The shuffle map determines a natura l inclusion ∇ : L → L ′ which has a rig h t inv erse given by the Alexander-Whitney map AW , see [2 1, 2.7]. It follows that ∇ ♯ : R ′ → R is a natural sur jection with a s e c tion given by AW ♯ . The fact that ∇ ♯ is a natural transfo rmation of bi-functors is clear. Since ∇ is a chain homotopy equiv alence, in order to finish the pr o of it is now enough to show that the functors L, L ′ , R, R ′ send chain homotopic maps to chain homotopic maps (for ( L, R ) a nd ( L ′ , R ′ ) will then induce adjunctions on the ho- motopy category Ho ( Ch ≥ 0 ) and ∇ : L → L ′ will b e a natural isomor phism b et ween endo-functors of Ho ( Ch ≥ 0 )). The functors L and R clearly preserve the chain homotopy relatio n. F or the same reason, L ′ and R ′ preserve the relation on Ch ≥ 0 ( C, D ) defined by the cylinder ob ject C / / & & M M M M M M M M M M M M N (Γ C ∧ k ∆[1 ])   C o o x x q q q q q q q q q q q q C . Since the Alexa nder-Whitney and s h uffle maps give ma ps b et ween this cylinder ob ject a nd the usual one, we see that this r elation is the usual chain ho mo topy relation. This concludes the pro of. √ W e now define a map φ relating the Ch ≥ 0 -graphs asso ciated with the dg cate- gories N P ( A ) and τ ≥ 0 P ( N A ). Observe that: - By lemma 4.2 0, N P ( A ) and τ ≥ 0 P ( N A ) hav e exactly the same ob jects and - F or ea ch pair of ob jects x f → y , x ′ f ′ → y ′ in N P ( A ), the map of lemma 5.13 (with A = k ∆[1]) induces a surjective quas i-isomorphism φ f ,f ′ N A ( x, x ′ ) × N A ( x,y ′ ) N s Mo d ( k ∆[1] , A ( x, y ′ )) × N A ( x,y ′ ) N A ( y , y ′ ) 1 ×∇ ♯ × 1 ∼     N A ( x, x ′ ) × N A ( x,y ′ ) Ch ≥ 0 ( N k ∆[1] , N A ( x, y ′ )) × N A ( x,y ′ ) N A ( y , y ′ ) in Ch ≥ 0 . Notation 5 .14 . W e deno te by φ : N P ( A ) / / _ _ _ τ ≥ 0 P ( N A ) DG VERSUS SIMPLICIAL CA TEGORIES 17 the map of Ch ≥ 0 -graphs which is the identit y on ob jects and φ f ,f ′ on the complexes of morphisms. R emark 5.15 . Notice that b y definition of P ( A ) and r e ma rk 5.12 the map φ preserve ident ities and the comp osition of deg ree zero morphisms. W e now establish a ‘homotopy eq uiv alence lifting prop erty’ of φ . Prop ositio n 5.16. L et α b e a de gr e e zer o morphism in τ ≥ 0 P ( N A ) that b e c omes invertible in H 0 ( τ ≥ 0 P ( N A )) . Then ther e exists a de gr e e zer o morphism α in N P ( A ) which b e c omes invertible in H 0 ( N P ( A )) and φ ( α ) = α . Pr o of. Let α : ( x f → y ) − → ( x ′ f ′ → y ′ ) be a degr e e zero morphism in τ ≥ 0 P ( N A ). Notice that α is of the form ( m x , h, m y ) with m x : x → x ′ and m y : y → y ′ degree zero morphisms in N A a nd h : x → y a degree 1 morphism in N A . Now, by definition o f P ( A ) we ca n c ho ose a represe n- tative h ∈ A ( x, y ) 1 of h and so we obtain a degree zer o morphism α = ( m x , h, m y ) in N P ( A ) such that φ f ,f ′ : N P ( A )( f , f ′ ) → τ ≥ 0 P ( N A )( f , f ′ ) α = ( m x , h, m y ) 7→ ( m x , h, m y ) . Now supp ose that α is inv ertible in H 0 ( τ ≥ 0 P ( N A )). Then ther e exist morphisms β of degre e 0 a nd r 1 and r 2 of degre e 1 such that d ( r 1 ) = β α − 1 and d ( r 2 ) = αβ − 1 . As ab ov e, we can lift β to a mor phism β in N P ( A ). Since the map φ pr e serve the ident ities a nd the co mp os itio n of degree zero morphisms it ma ps αβ to αβ and β α to β α . Finally s ince the maps φ f ,f ′ are surjective quasi-is omorphisms we can lift r 1 to r 1 , r esp. r 2 to r 2 , in N P ( A ) by applying the lemma [8 , 2.3 .5] to the couple ( r 1 , 1 ), res p. ( r 2 , 1 ). This implies tha t α is also inv ertible in H 0 ( N P ( A )). √ Prop ositio n 5.17. In the fol lowing c ommu tative diagr am in dgcat ≥ 0 N A ∆ / / N ( I A ) $ $ I I I I I I I I I N A × N A N P ( A ) , N ( P 0 ) × N ( P 1 ) 8 8 q q q q q q q q q q q obtaine d by applying the normalization functor N to the diagr am (1) in s Mo d -Cat, the dg functor N ( I A ) is a quasi-e quivale nc e and N ( P 0 ) × N ( P 1 ) is a fi br ation. Pr o of. W e first pr ov e that N ( I A ) is a qua si-equiv a lence. By definition of P ( A ) the dg functor I A clearly sa tisfies co ndition (1) of pro po s ition 5.9. W e now prov e that N ( I A ) s atisfies condition ( ii ) o f definition 4.2. Let f b e an ob ject in N P ( A ). The dg categories N P ( A ) and τ ≥ 0 P ( N A ) hav e the same ob jects and so we can consider f as an ob ject in τ ≥ 0 P ( N A ). Since the dg functor τ ≥ 0 ( I ) : N A − → τ ≥ 0 P ( N A ) is a quasi-equiv alence, see lemma 4.20, there exists an ob ject x in N A a nd a homo- topy equiv alence α in τ ≥ 0 P ( N A ) betw een I ( x ) and f . By pr o po sition 5.16 we can lift α to a homo topy equiv alenc e α in N P ( A ) a nd so the dg functor N ( I A ) : N A − → N P ( A ) 18 GONC ¸ ALO T ABUADA satisfies condition ( ii ) of definition 4.2. This prov es that N ( I ) is a quasi- e q uiv alence. W e now prove that N ( P 0 ) × N ( P 1 ) is a fibra tion. By definition o f P ( A ) the dg functor N ( P 0 ) × N ( P 1 ) is c le arly surjective on the complexes of morphisms. W e now prov e that it ha s the right lifting pro p er ty with resp ect to the g enerating trivial cofibra tion F , s ee definition 4 .5. Let x f → y b e a n ob ject in N P ( A ) and γ : ( x, y ) → ( x ′ , y ′ ) a ho mo topy equiv alence in N A × N A . Since the dg functor τ ≥ 0 ( P ) : τ ≥ 0 P ( N A ) / / / / N A × N A is a fibra tion there exists a ho motopy equiv alence α : f → f ′ in τ ≥ 0 P ( N A ) such that τ ≥ 0 ( P )( α ) = γ . Now, by prop osition 5.1 6 we can lift α to a ho motopy equiv alence α : f → f ′ in N ( P A ) such that N ( P 0 ) × N ( P 1 )( α ) = γ . This proves the prop ositio n. √ Notice that the previo us pr op osition implies theorem 5.10. R emark 5.18 . Since every o b ject in d gcat ≥ 0 is fibrant, see re ma rk 4.15, all simplicial k -linear catego ries will b e fibrant with resp ect to this Q uillen mo del structure. 5.3. Quillen equiv alence. In this subse c tion we prove that the Quillen adjunction constructed in the previo us subs e ction s Mo d -Cat N   dgcat ≥ 0 L O O is in fact a Quillen eq uiv alence. Theorem 5.19. The Quil le n adjunction ( L, N ) is a Quil len e quivalenc e. Pr o of. Let A ∈ dg cat ≥ 0 be a cofibra n t dg c ategory and B a simplicia l k -linea r category . Recall fro m remark 5.18 that every ob ject in s Mo d -Cat is fibrant. W e need to s how that a simplicial k -linea r functor F : L ( A ) − → B is a weak equiv a lence in s M o d -Cat iff the corresp onding dg functor F ♯ : A − → N B is a quas i-equiv ale nce in dgcat ≥ 0 . W e hav e the folowing commutativ e diagram A F ♯ / / η   N B N L ( A ) N F : : v v v v v v v v v , where η is the co unit of the adjunction ( L, N ). Since, by definition, F is a weak equiv ale nc e in s Mo d -Cat iff N F is a qua si-equiv a lence it is enough to s how that η is a qua si-equiv a lence. The dg functor η is the identit y map on ob jects and so it is enough to show that η ( x, y ) : A ( x, y ) − → N L ( A )( x, y ) , x, y ∈ A DG VERSUS SIMPLICIAL CA TEGORIES 19 is a quasi-iso morphism. Now, let I be the s et of ob jects o f A . Since A is co fibrant in dgcat ≥ 0 it clea rly stays cofibrant whe n consider e d as an ob ject of the Quillen mo del structure on Ch I ≥ 0 -Cat, see remark 5.2. By prop osition 6 . 4 of [21] the a djunction morphism in Ch I ≥ 0 -Gr Γ U ( A ) − → L I ( A ) is such that Γ U ( A )( x, y ) − → L I ( A )( x, y ) induces an isomo rphism in π i for i ≥ 0 and for all o b jects x, y ∈ Γ U ( A ). This implies by the Dold-Ka n equiv alence that A ( x, y ) = N (Γ U ( A )( x , y )) ∼ − → N ( L I ( A )( x, y )) , x, y ∈ A is a quas i-isomorphism and so η ( x, y ) : A ( x, y ) − → N L A ( x, y ) , x, y ∈ A is a quas i-isomorphism. This pr oves the theorem. √ R emark 5.20 . Notice that the ob jects in dgcat ≥ 0 , resp. in s Mo d -Cat, with only one ob ject consis t ex a ctly on the connective dg algebras , s ee [21, 1.1], r esp. simplicia l k -algebr as. W e hav e the fo llowing commutativ e diagra m s Alg N     / / s Mo d -Cat N   DGA ≥ 0 L O O   / / dgcat ≥ 0 , L O O where DGA ≥ 0 denotes the c a tegory of connec tive dg algebras and s Al g the cate- gory of simplicial k -algebr as. Observe that if we restr ic t the Quillen mo del struc- tures to these full s ubca tegories we obtain Shipley- Schw e de ’s Quillen eq uiv alence [21, 1.1] s Alg N   DGA ≥ 0 . L O O W e hav e then extended Shipley- Sc hw e de ’s work to a ‘several ob jects’ con text: the notion of weak equiv a le nce in s Mo d -Cat a nd dgcat ≥ 0 (see definition 4.2 and pr op o- sition 5.6) is now a mixture betw een quasi-iso morphisms and categorica l equiv a- lences. 6. The gl obal picture Recall from [9, I I I] that w e hav e a n adjunction s Mo d U   s Set , k ( − ) O O 20 GONC ¸ ALO T ABUADA where U is the forge tful functor and k ( − ) the k -linear ization functor. The functor k ( − ) is lax strong monoida l and so we hav e the natural a djunction s Mo d -Cat U   s Set -Cat . k ( − ) O O Recall from [1, 1.1] that the catego ry s Set -Cat is endow ed with a Quillen mo del structure whose weak equiv alences are the Dwyer-Kan (=DK) equiv a lences. Let us recall this no tion. Definition 6 .1. A simplicial functor F : A → B is a Dwyer-Kan equiv alence if: - for any obje cts x and y in A , the map F ( x, y ) : A ( x, y ) − → B ( F x, F y ) is a we ak e quivalenc e of simplicia l sets and - the induc e d functor π 0 ( F ) : π 0 ( A ) − → π 0 ( B ) is essential ly surje ctive. Prop ositio n 6. 2. The adjunction ( k ( − ) , U ) is a Q uil len adjunction, when we c on- sider on s Mo d -Cat the Q uil len mo del structur e of the or em 5.10. Pr o of. W e fir s t observe that b y prop osition 5.9 the functor U : s Mo d -Cat − → s Set -Cat preserves weak equiv a le nc e s. W e now show that it also preserves fibrations. Let G : A → B be a simplicial k -linear functor such tha t N G : N A → N B is a fibratio n in dgcat ≥ 0 . W e need to show that U G is a fibration in s Set -Cat. Recall from [1] that U G is a fibra tion iff: (F1) fo r any o b ject x and y in U A , the ma p U G ( x, y ) : U A ( x, y ) − → U B ( Gx, Gy ) is a fibration in s Set and (F2) fo r any ob ject x ∈ U A , y ∈ U B and homotopy equiv alence f : Gx → y in U B (= f b e c omes inv ertible in π 0 ( U B )), there is an ob ject z ∈ A and a homotopy eq uiv alence h : x → z in U A such that U G ( h ) = f . Since by hypothesis N G : N A → N B is a fibratio n in dgcat ≥ 0 , the dg functors R ( n ) , n ≥ 1 (which belo ng to the set J of genera ting trivia l cofibratio ns) allow us to conclude that the morphis ms N G ( x, y ) n : N A ( x, y ) n − → N B ( Gx, Gy ) n , x, y ∈ A are surjective for n ≥ 1. Now, by [9, II I-2.1 1], U G ( x, y ) is a fibration in s Set iff the morphisms N G ( x, y ) n are surjective for n ≥ 1. This implies that condition ( F 1) is verified. Concerning condition (F2), let x ∈ U A , y ∈ U B and f : Gx → y b e a ho- motopy equiv alence in U B . This mea ns that f is invertible in π 0 ( B ) and so by lemma 5.8 N ( f ) is also inv ertible in H 0 ( N B ). This data allow us to construct, DG VERSUS SIMPLICIAL CA TEGORIES 21 using prop ositio n 1 . 7 in [23], the following (so lid) commutativ e square k / /   F ∼   N A N G     K / / = = N B . Since N G is a fibra tio n in dgcat ≥ 0 we can lift N f to a mo rphism h : x → z in N A which is inv ertible in H 0 ( N A ). Since the 0-simplex morphisms in A a nd the degree zero mor phisms in N A ar e exactly the same, lemma 5.8 implies that h : x → z , when considered a s a mor phis m in U A , sa tisfies condition (F2). This proves the prop ositio n. √ W e hav e obtained the following z ig-zag of Quillen adjunctions rela ting the ho- motopy theories o f differential graded and s implicial categor ies: s Set -Cat k ( − )   s Mo d -Cat U O O N   dgcat ≥ 0 L O O _    dgcat . τ ≥ 0 O O R emark 6 .3 . Since the a djunction ( L, N ) is a Q uille n equiv a lence - the comp osed functor L ( N ◦ k ( − )) : Ho ( s Set -Ca t) − → Ho ( dgcat ) prese rves homotopy co limits and - the comp osed functor R ( U ◦ L ◦ τ ≥ 0 ) : Ho ( dgcat ) − → Ho ( s Set -Cat) preser ves homotopy limits. The following result was prov ed by Simpson in a n adho c way in [22, 5.1]. Corollary 6.4. 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