DG-category and simplicial bar complex

DG-CA TEGOR Y AND SIMPLICIAL BAR COMPLEX TOMOHIDE TERASOMA Contents 1. In tro duction and con v en tions 1 1.1. In tro duction 1 1.2. Con ven tions 3 2. DG category 4 2.1. Definition of DG category and examples 4 2.2. DG complexes 6 3. T op ological motiv ation f or simplicial bar complex 7 3.1. The space of mark ed path 7 3.2. Chain complex asso ciated to the system of marked paths 8 4. Simplicial bar complex 9 4.1. Definition of simplicial bar complex 9 4.2. Augmen tations and copro duct 11 5. Comparison to Chen’s t heory 12 6. Simplicial bar complex and DG category K C A 15 6.1. Definition of como dules ov er DG coalgebras 15 6.2. Corresp ondences on ob jects 17 6.3. The functor ψ : ( B − com ) re d,b → K b C A on Morphisms 21 7. H 0 ( B ( ǫ | A | ǫ ))-como dule for Connected DGA 23 8. P atc hing of DG category 26 9. Graded case 28 10. Deligne complex 31 10.1. Definition of Deligne al gebra 31 10.2. Nilp otent v ariation of mix ed T ate Ho dge structure 32 10.3. Pro of o f Theorem 10.7 33 Reference s 36 1. Introductio n and conventions 1.1. In tro duction. In the pap er [C], Chen defined a bar complex of an as- so ciative di fferential g raded al g ebra whi ch computes the real homotopy type of a C ∞ -manifold. There he prov ed that the Hopf algebra of the dual of the nilp oten t completion of the group ring R [ π 1 ( X , p )] ˆ of the fundamen tal group 1 2 DG category and bar complex π 1 ( X , p ) is canonically isomorphic to the 0-th cohomology of the bar complex of the differen tial graded algebra A • ( X ) of smo oth differential forms on X . There are several metho ds to construct mix ed T ate motives ov er a field K . First construction is due t o Blo c h and Kriz [BK1], base on the bar complex of the differen tial graded algebra of Blo c h cycle complex. They defined the category mixed T ate motiv es as the category of como dules o v er the cohomol- ogy o f the bar complex. Another constructi o n i s due t o Hanamura [H]. In the b o ok of Kriz-Ma y [KM], they used a homotopi cal approach to define the cat- egory of mi xed T ate mot iv es. Hanam ura used some generali zat ion of complex to construct t he derived cat egory of mixed T at e motives. One can form ulate this cons truction of complex i n the setting of DG category , whic h is called DG complex in this paper. DG complexes is called t wisted complexes in a pap er of Bondal-Kaprano v [B K2]. In pap er [Ke1 ], similar noti on of perfect complexes are in tro duced. A notion of DG categories a r e also useful to study cyclic homo l ogy . (See [Ke1], [Ke2]). In this pap er, w e study tw o categories, one is the cat egory of como dules o ver the bar comp lex of a differential graded al gebra A and the other is the category of DG complexes of a DG category arising from the differential gra ded al g ebra. Roughly sp eaking, we show that these t w o categories are homotopy equiv al ent (Theorem 6.4). W e use this equiv alence t o construct a certai n coa l gebra which classifies nilp otent v ariati o n of mixed T a t e Ho dge struc tres on an al gebraic v arieti es X (Theorem 10.7). This coalgebra is isomorphic to the co ordinate ring of the T anakian category of mixed T ate Ho dge structures when X = S p ec ( C ). Bar construction is al so used to construct t he motives asso ciated to rational fundamen tal groups o f algebraic v arieti es in [DG]. They used anot her t yp e of bar construction due to Beilinson. In this pap er, we adopt this bar con- struction, call ed simpli ci al bar construction, for differen tial graded algebras. Simplicial bar complexes dep end on the ch oices of tw o augmen tati o ns of the differen tial graded al gebras. If these tw o augmentations happ en to b e equal , then the simplicial bar complex i s quasi-isomorphic to the classical reduced bar complex defined b y Chen. Let me explain in the case of the DGA of smo oth differential forms A • of a smo oth manifold X . T o a p oin t p of X , w e can asso ciate an augmen tation ǫ p : A • → R . In Chen’s reduced bar complexes, the c hoice of t he augmen tation, reflects the c hoice of t he base p oint p of the rational fundamen tal gro up. On the other hand, simplicial bar complex depends on tw o augmen tations ǫ 1 and ǫ 2 . The cohomol ogy of the simpli cial bar complex of A • with resp ect to t he t w o augmen tations a rising from tw o p oin ts p 1 and p 2 of X is iden tified with the ni lpot en t dual of the li near h ull of paths connecting the p oints p 1 and p 2 . By applying simpli cial bar construc tion with t w o augmentations o f cycle DGA arising from t w o realizat i ons, w e obtai n the dual of the space generated b y fu nctorial isomorphism b et w een t w o real ization functors. A comparison theorem giv es a path connecting these t w o realization fun ctors. Using this formalism, w e t reat the category of v ariation of mixed T ate Ho dge structures T omohide T era soma 3 o ve r smooth algebraic v arieties. In this pap er, w e show that the category of como dules o ver the bar complex of t he differen tial graded al g ebra of the Deligne complex of an algebraic v ari et y X is equiv alen t to the category of nilp oten t v ariations of mixed T a te Ho dge structures on X . 1.2. Con v entions. Let k b e a fie ld. Let C b e a k -linear ab elian category with a tensor structure. The category of complexes in C is denoted as K C . F or ob jects A = ( A • , δ A ) and B = ( B • , δ B ) in K C , w e define tensor pro duct A ⊗ B as an ob je ct in K C by the rule ( A ⊗ B ) p = ⊕ i + j = p A i ⊗ B j . The differen tial d A ⊗ B on A i ⊗ B j is defined b y d A ⊗ B = δ A ⊗ 1 B + ( − 1) i 1 A ⊗ δ B . An elemen t H om p K C ( A, B ) = Y i H om C ( A i , B i + p ) is called a homogeneous homomorphism of degree p from A to B i n K C . Let A, A ′ , B , B ′ ∈ K C and ϕ = ( ϕ i ) i ∈ H om p K C ( A, B ) and ψ = ( ψ j ) j ∈ H om q K C ( A ′ , B ′ ), we define ϕ ⊗ ψ ∈ H om p + q K C ( A ⊗ A ′ , B ⊗ B ′ ) b y setti ng ( ϕ ⊗ ψ ) i + j = ( − 1) q i ϕ i ⊗ ψ j on A i ⊗ A ′ j → B i + p ⊗ B ′ j + q . ( T o remem b er this form ula, the rule ( ϕ ⊗ ψ )( a ⊗ a ′ ) = ( − 1) deg( a ) deg( ψ ) ϕ ( a ) ⊗ ψ ( a ′ ) is useful.) An ob ject M in C is regarded as an ob ject in K C b y setting M at degree zero part. F or a complex A ∈ K C , we define the complex A [ i ] b y A [ i ] j = A i + j , where the differen tial is defined through this isomorphism. The shift k [ i ] of the unit ob ject k is defined i n this manner. The homogeneous morphism k [ i ] → k [ j ] of degree i − j , whose degree − i part k [ i ] − i = k → k [ j ] − j = k is defined b y the iden tit y map, is denoted as t j i . The degree − i element “1” of k [ i ] is denoted as e i . F or an ob ject B ∈ K C , the tensor complex B ⊗ k [ i ] is denoted as B e i . F or ob jects A, B ∈ K C and ϕ ∈ H om p K C ( A, B ), a homomorphism ϕ ⊗ t j i ∈ H om K C ( Ae i , B e j ) is a degree ( p + i − j ) homomo r phism. As a sp ecial case, for ϕ ∈ H om 0 K C ( A, B ), the map ϕ ⊗ t i − 1 ,i ∈ H om 1 K C ( Ae i , B e i − 1 ) is a degree one elemen t. It i s denoted as ϕ ⊗ t for simplicit y . The differen tial of M can b e regarded as a degree one map from M to itself. Therefore d can b e regarded as an elemen t in H om 1 K C ( M , M ). An ob ject in K K C can b e considere d as a double complex in C . Let ( · · · A • i d → A • i +1 → · · · ) b e an ob ject i n K K C . Since d is a homomorphism of complex, we ha ve d ⊗ t ∈ H om 1 K C ( A • ,j e − j , A • ,j +1 e − j − 1 ) . is a degree one elemen t in K C . Let δ ⊗ 1 b e the differen tial of A • i e − i ∈ K C . The summati on of δ ⊗ 1 for i is also denoted as δ ⊗ 1. Then the degree one map δ ⊗ 1 + d ⊗ t b ecomes a differen tial on the to t al graded ob ject. Here δ ⊗ 1 is called the inner differen tial and d ⊗ t is call ed the outer differen tial . Note that for an “elemen t” a ∈ A ij , w e hav e ( d ⊗ t )( a ⊗ e − j ) = ( − 1) i d ( a ) ⊗ e − j − 1 , whic h coincides with the standard sign conv en tion of the asso ciate simple complex of a double complex. The resulti ng complex is called the asso ciate si mpl e complex of A •• ∈ K K C and denoted as s ( A ) = s ( A • , • ). F or o b jects A = A • , • , B • , • ∈ K K C , the 4 DG ca tegory and bar c omplex tensor pro duct A ⊗ B is defi ned as an ob ject in K K C . Then w e ha v e a natural i somorphism in K C (1.1) ν : s ( A ) ⊗ s ( B ) ≃ s ( A ⊗ B ) defined by ν ( a e − j ⊗ be − j ′ ) = ( − 1) j i ′ ( a ⊗ b ) e − j − j ′ for ae − j ∈ A ij e − j , be − j ′ ∈ B i ′ j ′ e − j ′ . This isomorphism is compati ble with the natural a ssociati vit y iso- morphism. 2. DG ca te gor y 2.1. Defini t ion of D G cat egory and examples. Let k b e a field. A DG category C ov er k consists of the follo wing data (1) A cla ss of ob jects ob ( C ). (2) A complex H om • C ( A, B ) = ( H om • C ( A, B ) , ∂ ) of k v ector spaces for ev ery ob jects A and B in ob ( C ). W e sometimes imp ose t he following shift structure on C . (3) Bijective corresp ondence T : C 7→ C for ob jects in C . An ob ject T k ( A ) in C is denoted as Ae k for k ∈ Z . with t he follo wing axioms. (1) F or three ob jects A, B and C in C , t he comp osite H om • C ( B , C ) ⊗ H om • C ( A, B ) → H om • C ( A, C ) is defined as a homomorphism of complexes o v er k . (2) The ab o ve comp osite homomorphism is asso ciative. That i s, the fol- lo wing diagram of complexes commutes. H om • C ( C , D ) ⊗ H om • C ( B , C ) ⊗ H om • C ( A, B ) → H om • C ( C , D ) ⊗ H om • C ( A, C ) ↓ ↓ H om • C ( B , D ) ⊗ H om • C ( A, B ) → H om • C ( A, D ) (3) There is a degree zero closed elemen t id A in H om 0 ( A, A ) for eac h A , whic h i s a left and righ t iden t i t y under the ab ov e comp o site homo- morphism. If we assume the shift structure T , the following sign con ven t i on should b e satisfied. (4) There i s a natural isomorphism of complexes H om • C ( A, B )[ − i + j ] ≃ H om • C ( Ae i , B e j ) : ϕ 7→ ϕ ⊗ t j i satisfying the rule ( ϕ ⊗ t j i ) ◦ ( ψ ⊗ t ik ) = ( − 1) ( i − j ) deg( ψ ) ( ϕ ◦ ψ ) ⊗ t j k . (It is compati bl e with the formal comm utation rule for ψ and t j i .) Definit ion 2.1. (1) L et C b e a DG c ate gory and a, b obje cts in C . A close d morphism ϕ : a → b of de gr e e 0 (i.e. ∂ ϕ = 0 ) is c al le d an isomorphism if ther e is a close d morphi s m ψ of de gr e e zer o such that ψ ◦  = 1 a , ϕ ◦ ψ = 1 b . T omohide T era soma 5 (2) L et C 1 , C 2 b e DG c ate gori es. A DG f unctor F is a c ol le ction { F ( a ) } a of obje cts in C 2 indexe d by obj e cts in C 1 and a c ol le ction { F a,b } of ho- momorphisms of c omplexes F a,b : H om • C 1 ( a, b ) → H om • C 2 ( F ( a ) , F ( b )) indexe d by a, b ∈ C 1 , whi c h pr eserves the c omp osites, identities and de g r e e shift op er ator A 7→ A [1] . We define sub DG c ate gories and ful l sub DG c ate g or i es similarly as i n usual c ate gories. We a lso define essential ly surje ctive functors as in the usual c ate gory. A functor is e qui valent if and only if it is essential ly s urje ctive and f ul ly f aithful. (3) A DG f unctor F : C 1 → C 2 is said to b e homotopy e quivalent if and only if i t i s essential ly surje ctive, and th e induc e d map H i ( F a,b ) : H i ( H om • C 1 ( a, b )) → H i ( H om • C 2 ( F ( a ) , F ( b )) ) is a n i somorphism for al l i ∈ Z and a, b ∈ C 1 . Example 2.2. L et V ec k b e a c ate gory of k -ve ctor sp ac es. The c ate gory of c omplexes of k -ve ctor sp ac es is denote d as K V ec k . Then K V ec k b e c omes a DG c ate gory by s etting H om p K V ec k ( A, B ) = Y i H om ( A i , B i + p ) for c omplexes A = A • and B = B • . The differ ential ∂ ϕ of an element ϕ = ( ϕ i ) i ∈ H om p K V ec k ( A, B ) i s define d by the formula (2.1) ( ∂ ( ϕ )) i = d B ◦ ϕ i − ( − 1) p ϕ i +1 ◦ d A . Ther ef or e ϕ ∈ H om 0 K V ec k ( A, B ) is a homomorphi sm of c omplexes if and only if ∂ ( ϕ ) = 0 . Two homomorphisms o f c omplexes ϕ and ψ ar e homo- topic to e a c h other by the homotopy θ i f and o n ly if ϕ − ψ = ∂ ( θ ) wi th θ ∈ H om − 1 K V ec k ( A, B ) . Definit ion 2.3 (DG category asso ciated with a DGA) . L et A = A • b e a unitary asso ciative differ entia l gr ade d algebr a (denote d as DGA f o r short) over a file d k with the multiplic ation µ : A • ⊗ A • → A • . We define a DG c ate gory C A asso ciate d to A as fol lows. (1) A n obje ct of C A is a c omplex V = V • of ve ctor s p ac es ov e r k . (2) F or two obj e cts V = V • and W = W • , the s e t of morphis ms H om p C A ( V , W ) is d e fine d as H om • C A ( V , W ) = H om K V ec k ( V • , A • ⊗ W • ) . Then H om • C A ( V , W ) b e c omes a c omplex by the formula (2 . 1) and the structur e of tensor c omplex A • ⊗ W • define d in (1 . 2). (3) F or thr e e obje cts U = U • , V = V • and W = W • , we define the c omp osite µ : H om • C A ( V • , W • ) ⊗ H om • C A ( U • , V • ) → H om • C A ( U • , W • ) 6 DG ca tegory and bar c omplex by the c omp osite of the fol lowing ho momorphi sms of c omplexes: H om K V ec k ( V • , A • ⊗ W • ) ⊗ H om K V ec k ( U • , A • ⊗ V • ) ↓ H om K V ec k ( A • ⊗ V • , A • ⊗ A • ⊗ W • ) ⊗ H om K V ec k ( U • , A • ⊗ V • ) ↓ H om K V ec k ( U • , A • ⊗ A • ⊗ W • ) ↓ µ ⊗ 1 H om K V ec k ( U • , A • ⊗ W • ) Remark 2.4. L et C b e a DG c ate gory and M b e a n obje ct of C . Then E nd • C ( M ) = H om • C ( M , M ) b e c omes an (asso ciative) differ ential g r ade d al- gebr a. We have E nd • C A ( k ) ≃ ( A • ) op as DGA’s. Note that ( A • ) op is a c op y of A • as a c omplex and the mul tiplic ation rule is give n by a ◦ · b ◦ = ( − 1) deg( a ) deg( b ) ( b · a ) ◦ 2.2. DG complexes. W e intro duc e the notion of compl ex es in the sett ing of DG category , whi ch is called DG compl ex es. Definit ion 2.5. (1) L et C b e a DC c ate gory. A p air ( { M i } i ∈ Z , { d ij } i>j ) c ons isting of ( 1) a seri es of obje cts { M i } i ∈ Z in C indexe d by Z , and (2) a s eries of morphi sms d ij ∈ H om j − i +1 C ( M j , M i ) indexe d by i > j in Z is c al le d a DG c omplex in C if it satisfies the fol lowing e quality. (2.2) ∂ ( d ij ) + X i>p>j ( − 1) ( i − p )( p − j +1) d ip ◦ d pj = 0 . The unc omfortable sign in this c ondition wil l b e simplifie d by using d # ij = d ij ⊗ t − i, − j ∈ H om 1 C ( M j e − j , M i e − i ) . Then the c ondition wil l b e (2.3) ∂ ( d # ij ) + X i>p>j d # ip ◦ d # pj = 0 . (2) L et M = ( M • , d M ) and N = ( N • , d N ) b e DG c omplexes in C . We set H om i K C ( M , N ) = lim → α Y q − α ≤ r H om i + q − r C ( M q , N r ) . (3) L et i ∈ Z . F or an element ϕ ∈ H om i K C ( M , N ) , we define a map D ( ϕ ) ∈ H om i +1 K C ( M , N ) as fol lows. F or an element ϕ = ( ϕ r,q ) ∈ Y q − α ≤ r H om i + q − r C ( M q , N r ) , T omohide T era soma 7 we set ϕ # rq = ϕ rq ⊗ t − r , − q ∈ H om i C ( M q e − q , N r e − r ) and define D ( ϕ ) # r,q = ∂ ( ϕ # r,q ) + X q − α ≤ r ′ N ( A ) b e the sub double complex of B ( A ) defined by → ⊕ N <α 0 <α 1 <α 2 B α 0 ,α 1 ,α 2 ( A ) → ⊕ N <α 0 <α 1 B α 0 ,α 1 ( A ) → ⊕ N <α 0 B α 0 ( A ) → 0 Then we hav e the natural inclusion i : B >N ( A ) → B >N − 1 ( A ). W e define a map θ : B >N ( A ) → B >N − 1 ( A )[ − 1] θ : ⊕ N <α 0 < ··· <α i B α 0 , ··· ,α i ( A ) → ⊕ N − 1 <β <α 0 < ··· <α i B β ,α 0 , ··· ,α i ( A ) b y θ (1 α 0 ⊗ x 1 α 1 ⊗ · · · α n − 1 ⊗ x n α n ⊗ 1 ) = 1 N ⊗ 1 α 0 ⊗ x 1 α 1 ⊗ · · · α n − 1 ⊗ x n α n ⊗ 1 . Then w e hav e θ ◦ d + d ◦ θ = i ( x ). Therefore t he double complex (for the di f- feren tial d ) B ( A ) = li m → N B >N ( A ) is an acycli c complex . Therefore its asso ciate simple complex is also acyclic.  Remark 4.2. L et X b e a C ∞ manifold. L et A = A • ( X ) b e the DGA of C ∞ forms. Sinc e X α is i somorphic to the pr o duct of c opies of X , the nat- ur al homomorphism B α ( A ) → A • ( X α ) of c omplexes ar e quasi-i somorphism by K ¨ uneth f ormul a. The homom orphis ms B α ( A ) → B β ( A ) and A • ( X α ) → A • ( X β ) ar e c omp atible with the ab ove quasi -isomorphisms. As a c onse quenc e, the natur al ho momorphi sm B ( A ) → A • ( X • ) is a quasi - isomorphism. Pr op o- sition 3. 1 fol lows fr om Pr op ositio n 4.1. W e define left A righ t A act i on ( A - A action for short) on the compl ex B α ( A ) by A ⊗ B α ( A ) ⊗ A → B α y ⊗ ( x 0 α 0 ⊗ x 1 · · · x n α n ⊗ x n +1 ) ⊗ z → ( y x 0 ) α 0 ⊗ x 1 · · · x n α n ⊗ ( x n +1 z ) . Since the differen tials of B ( A ) are A - A homomorphisms, B ( A ) is an A - A mo dule. W e introduce a bar filtra t ion F • b on B ( A ) as follows : F − i b B ( A ) : · · · → 0 → ⊕ | α | = i B α ( A ) → · · · → ⊕ | α | =0 B α ( A ) → 0 T omohide T erasoma 11 Then we hav e the follo wing sp ectral sequence E − i,p 1 = ⊕ | α | = i B α ( H • ( A )) p ⇒ E − i + p ∞ = H − i + p ( B ( A )) . This sp ectral sequence is called t he bar sp ectral sequence. 4.2. Augmentations a n d copro duct. Let ǫ 1 , ǫ 2 b e tw o augmen tations of A = A • , i.e. DGA homomorphisms ǫ i : A → k , where k is the trivial DGA. W e define B ( ǫ 1 | A | ǫ 2 ) b y the asso ciate simple complex of the double complex B ( ǫ 1 | A | ǫ 2 ) = k ⊗ A,ǫ 1 B ( A ) ⊗ A,ǫ 2 k . Therefore the degree − n part B ( ǫ 1 | A | ǫ 2 ) n of B ( ǫ 1 | A | ǫ 2 ) is isomorphic to ⊕ | α | = n B α ( ǫ 1 | A | ǫ 2 ) where B α ( ǫ 1 | A | ǫ 2 ) = k α 0 ⊗ A α 1 ⊗ · · · α n − 1 ⊗ A α n ⊗ k and the differen tial (outer differential) is given b y d (1 α 0 ⊗ x 1 α 1 ⊗ · · · α n − 1 ⊗ x n α n ⊗ 1 ) = ǫ 1 ( x 1 ) α 1 ⊗ x 2 α 2 ⊗ · · · α n − 1 ⊗ x n α n ⊗ 1 + n − 1 X i =1 ( − 1) i 1 α 0 ⊗ x 1 · · · α i − 1 ⊗ x i x i +1 α i +1 ⊗ · · · x n α n ⊗ 1 + ( − 1) n 1 α 0 ⊗ x 1 α 1 ⊗ · · · α n − 1 ⊗ x n − 1 α n − 1 ⊗ ǫ 2 ( x n ) . As in the last subsection, w e hav e t he foll o wing bar sp ectral sequence E − i,p 1 = ⊕ | α | = i B α ( ǫ 1 | H • ( A ) | ǫ 2 ) p ⇒ E − i + p ∞ = H − i + p ( B ( ǫ 1 | A | ǫ 2 )) . W e i ntro duc e a coproduct structure o n B ( A ). Let ǫ 1 , ǫ 2 , ǫ 3 b e augmen ta- tions on A . Let α = ( α 0 < · · · < α n ) be a sequence of integers. W e define ∆ ǫ 2 (4.2) ∆ ǫ 2 : B ( A ) → ( B ( A ) ⊗ ǫ 2 k ) ⊗ ( k ⊗ ǫ 2 B ( A )) b y ∆ ǫ 2 ( x 0 α 0 ⊗ x 1 α 1 ⊗ · · · α n − 1 ⊗ x n α n ⊗ x n +1 ) = n X i =0 ( x 0 α 0 ⊗ · · · α i − 1 ⊗ x i α i ⊗ 1) ⊗ (1 α i ⊗ x i +1 α i +1 ⊗ · · · α n ⊗ x n +1 ) for an elemen t in B α ( A ). This is a morphism in the category K K V ect k . W e in tro duce an A − A mo dule structure on the rig h t hand side o f (4.2) b y y 1 · ( x 0 α 0 ⊗ · · · α i − 1 ⊗ x i α i ⊗ 1) ⊗ (1 α i ⊗ x i +1 α i +1 ⊗ · · · α n ⊗ x n +1 ) · y 2 =(( y 1 x 0 ) α 0 ⊗ · · · α i − 1 ⊗ x i α i ⊗ 1) ⊗ (1 α i ⊗ x i +1 α i +1 ⊗ · · · α n ⊗ ( x n +1 y 2 )) for y 1 , y 2 ∈ A . W e can prov e t he following by direct computation. Prop osition 4.3. (1) The homomorphism ∆ ǫ 2 is a h o momorphi sm of c omplex of K V ect k and c omp atible with the A - A action. Ther ef or e we have the f ol lowing homomorphism i n K K V ect k : ∆ ǫ 1 ,ǫ 2 ,ǫ 3 : B ( ǫ 1 | A | ǫ 3 ) → B ( ǫ 1 | A | ǫ 2 ) ⊗ B ( ǫ 2 | A | ǫ 3 ) . 12 DG ca tegory and bar c omplex Using the i somorphism (1.1), we have the fo l lowing morphi sm i n K V ect k : ∆ ǫ 1 ,ǫ 2 ,ǫ 3 : B ( ǫ 1 | A | ǫ 3 ) → B ( ǫ 1 | A | ǫ 2 ) ⊗ B ( ǫ 2 | A | ǫ 3 ) . (2) L et ǫ b e an augmentation of A . F or α = ( α 0 ) , let ǫ α 0 : B α 0 = k α 0 ⊗ k → k b e the natur al map. Then u = P α 0 ǫ α 0 defines a homomorphi s m of differ ential gr ade d c o algebr as u : B ( ǫ | A | ǫ ) → k . (3) The fol lowing c omp os ite maps ar e i dentity: B ( ǫ 1 | A | ǫ 2 ) ∆ ǫ 1 ,ǫ 1 ,ǫ 2 → B ( ǫ 1 | A | ǫ 1 ) ⊗ B ( ǫ 1 | A | ǫ 2 ) (4.3) u ⊗ 1 → B ( ǫ 1 | A | ǫ 2 ) B ( ǫ 1 | A | ǫ 2 ) ∆ ǫ 1 ,ǫ 2 ,ǫ 2 → B ( ǫ 1 | A | ǫ 2 ) ⊗ B ( ǫ 2 | A | ǫ 2 ) 1 ⊗ u → B ( ǫ 1 | A | ǫ 2 ) wher e ∆ ǫ 1 ,ǫ 2 ,ǫ 3 is d e fine d i n (1). Definit ion 4.4. (1) The map ∆ ǫ 1 ,ǫ 2 ,ǫ 3 define d i n (1) of Pr op ositi on 4.3 is c al le d the c opr o duct of B ( A ) . It is e asy to se e that the c opr o duct is c o asso ciative. Thus B ( A ) forms a DG c o algebr oid ov er the set S paug ( A ) of augmentations of A . (2) The map u : B ( ǫ | A | ǫ ) → k define d in (2) of P r op osition 4.3 is is c al le d the c ounit. In gener al, for an as so ciati v e differ ential gr ade d c o algebr a B , the map u satisfying the pr op erties as (4.3) is c al le d a c ouni t. 5. Comp arison to Chen’s theor y In this section, w e compare the simplicial bar complex defined as abov e and the reduced bar complex defined by Chen. Let A = A • b e a differen tial graded algebra ov er a field k and ǫ : A → k b e an augmen t a tion. Let I = Ker( ǫ : A → k ) b e the augmen ta t ion ideal. W e de fine the double complex B re d ( A, ǫ ) as B re d ( A, ǫ ) : · · · → I ⊗ 3 ] d B ,red → I ⊗ 2 d B ,red → I 0 → k → 0 , where t he outer differen tial d B ,red : I ⊗ i → I ⊗ ( i − 1) is defined b y (5.1) d B ,red : [ x 1 | · · · | x i ] 7→ i − 1 X p =1 ( − 1) p [ x 1 | · · · | x p x p +1 | · · · | x i ] . Here an elemen t x 1 ⊗ · · · ⊗ x i in I ⊗ i is denoted as [ x 1 | · · · | x i ]. The total complex B re d ( A, ǫ ) i s called the Chen’s reduced bar complex. W e define a degree preserving l inear map B ( ǫ | A | ǫ ) → B re d ( A, ǫ ) by (5.2) 1 α 0 ⊗ x 1 α 1 ⊗ · · · α n − 1 ⊗ x n − 1 α n ⊗ 1 7→ [ π ( x 1 ) | · · · | π ( x n − 1 )] where π ( x ) = x − ǫ ( x ) ∈ I for an elemen t x ∈ A . T omohide T erasoma 13 Lemma 5.1. It is a homomorphis m of c omplexes of K V ect k . W e ha ve the fol lo wing theorem. Theorem 5.2. The homomorphism of asso ciate simple c omplexes B ( ǫ | A | ǫ ) → B re d ( A, ǫ ) obtai ne d f orm the map ( 5.2) is a q uasi-isomorphism. Pr o of. First, we define t he bar filtratio n F − i b B re d ( A, ǫ ) on the double complex B re d ( A, ǫ ) by F − i b B re d ( A, ǫ ) : · · · → 0 → I ⊗ i → · · · → I → k → 0 . Then the bar filtration on B ( ǫ | A | ǫ ) and that on B re d ( A, ǫ ) are compatible and we ha ve the foll owing homomorphisms of sp ectral sequences: E − i,p 1 = ⊕ | α | = i B α ( ǫ | H • ( A ) | ǫ ) p ⇒ E − i + p ∞ = H − i + p ( B ( ǫ | A | ǫ )) ↓ ↓ ′ E − i,p 1 = ( H • ( I ) ⊗ i ) p ⇒ ′ E − i + p ∞ = H − i + p ( B re d ( A, ǫ )) W e prov e that the v ertical a rr ow coincides from E 2 -terms. E 1 -terms are the follo wing complex es of graded vector spaces: · · · → ⊕ α 0 <α 1 <α 2 B α 0 ,α 1 ,α 2 → ⊕ α 0 <α 1 B α 0 ,α 1 → ⊕ α 0 B α 0 → 0 ↓ ↓ ↓ · · · → H • ( I ) ⊗ 2 → H • ( I ) ⊗ 1 → k → 0 Here we used t he abbreviatio n B α 0 , ··· ,α n = k α 0 ⊗ H • ( A ) ⊗ · · · ⊗ H • ( A ) α n ⊗ k . W e sho w that the vertical homomorphism of complex es are quasi-isomorphism. W e ha v e B α 0 , ··· ,α n = k α 0 ⊗ ( H • ( I ) ⊕ k ) ⊗ · · · ⊗ ( H • ( I ) ⊕ k ) α n ⊗ k = ⊕ n k =0 ⊕ c ∈ C ( n,k ) H • ( I ) ⊗ k , where C ( n, k ) is the subset of { 1 , . . . , n } of cardinality k . Since the ima ges of H • ( I ) ⊗ H • ( A ) and H • ( A ) ⊗ H • ( I ) under the m ultipli cation is con tained in H • ( I ), the follo wing filt ration G on ( E − i, • 1 , d 1 ) i and ( ′ E − i, • 1 , d 1 ) i defines a sub compl ex es: G − l E − i, • 1 = ⊕ l k =0 ⊕ c ∈ C ( n,k ) H • ( I ) ⊗ k , G − l ′ E − i, • 1 = ( H • ( I ) ⊗ i ( i ≤ l ) 0 ( i > l ) Th us t he asso ciate g ra ded complexes of E − i, • 1 → ′ E − i, • 1 are (5.3) · · · → ⊕ | α | = l +1 , c ∈ C ( l +1 ,l ) H • ( I ) ⊗ l → ⊕ | α | = l, c ∈ C ( l,l ) H • ( I ) ⊗ l → 0 → · · · ↓ ↓ ↓ · · · → 0 → H • ( I ) ⊗ l → 0 → · · · 14 DG ca tegory and bar c omplex W e consider an al gebra k ⊕ k x suc h that x 2 = 0. W e define the complex K m,l b y (5.4) · · · → ⊕ | α | = l +2 , α ⊂ [1 ,m ] c ∈ C ( l +2 ,l ) k t α,c → ⊕ | α | = l +1 , α ⊂ [1 ,m ] c ∈ C ( l +1 ,l ) k t α,c → ⊕ | α | = l, α ⊂ [1 ,m ] c ∈ C ( l,l ) k t α,c → 0 where t α,c = 1 α 0 ⊗ u 1 α 1 ⊗ · · · α k − 1 ⊗ u k α k ⊗ 1 , with u i = ( x if i ∈ c, 1 if i / ∈ c. The differential is similar to that of simplicial bar complex. Then t here is a natural i nclusion K m,l → K m +1 ,l and a map ǫ m : K m,l → k . Prop osition 5.3. (1) The map ǫ l : K l,l → k is an isomorphis m. (2) The natur al inclusion K m,l → K m +1 ,l is a quasi-isomorphis m for m ≥ l . (3) The c omplex K m,l → k is an acyclic c omplex for l ≤ m . Pr o of. W e pro v e the prop osition b y the induction on l . The statemen t (1) is direct from the definitions. The statemen t ( 3) follows from the statemen t (1) and (2) . W e pro ve the statemen t (2 ) for l assuming the statemen t (3) for l − 1. The cokern el of the complex Cok er( K m,l → K m +1 ,l ) i s isomorphic to · · · → ⊕ | α | = k +1 , α ⊂ [1 ,m +1] , α k +1 = m +1     ⊕ c ∈ C ( k,l ) k t ′ α,c L ⊕ c ∈ C ( k,l − 1) k t ′′ α,c     d → ⊕ | α | = k , α ⊂ [1 ,m +1] , α k = m +1     ⊕ c ∈ C ( k − 1 ,l ) k t ′ α,c L ⊕ c ∈ C ( k − 1 ,l − 1) k t ′′ α,c     → · · · where t ′ α,c = 1 α 0 ⊗ u 1 α 1 ⊗ · · · α k − 1 ⊗ u k α k ⊗ 1 m +1 ⊗ 1 , t ′′ α,c = 1 α 0 ⊗ u 1 α 1 ⊗ · · · α k − 1 ⊗ u k α k ⊗ x m +1 ⊗ 1 , with u i = ( x if i ∈ c, 1 if i / ∈ c Th us it is isomorphic t o the cone of a complex homomorphism K m,l → K m,l − 1 b y considering the b oth cases for u k = 1 , x . Since K m,l → k and K m,l − 1 → k is quasi-isomorphism b y the induction h yp othesis for ( m, l ) and ( m, l − 1), w e ha v e the statemen t (2).  Pro of of Theorem 5.2. Si nce the diagram (5 .3) is obtained from (5. 4) b y tensoring H • ( I ) ⊗ l and ta k ing the inductiv e limit on m , the homomorphism of asso ciate graded complex of E − i, • 1 → ′ E − i, • 1 is a q uasi-isomorphism b y Prop osition 5.3. This prov es the theorem.  T omohide T erasoma 15 Remark 5.4. Sinc e the sp e ctr al se quenc e asso ci a te d to the filtr ation G is isomorphic f r om E 2 terms, the induc e d filtr ation on H i ( B ( ǫ | A | ǫ )) and that on H i ( B re d ( A, ǫ )) ar e e qual. 6. Simplicial bar complex and D G c a tegor y K C A 6.1. Defini t ion of como dules o v er DG coalgebras. In this section, w e pro v e that t he DG category of comodules o ver the simplicial bar complex B ( ǫ | A | ǫ ) is homotopy equiv al en t to the DG category K C A . Let B b e a coasso ciative differen tial graded coal gebra o v er k with a counit u : B → k . The com ultipl i cation B → B ⊗ B is written as ∆ B . Definit ion 6.1 (DG categro y o f B como dules) . (1) A k -c omplex M with a homomorphism ∆ M of c omplexes ∆ M : M → B ⊗ M is c al le d a ( left) B -c omo dule if the fol lowi ng pr op erties hold. (a) Coassocia tivity The fol lowing diagr am c ommutes: M ∆ M → B ⊗ M ∆ M ↓ ↓ 1 ⊗ ∆ M B ⊗ M ∆ B ⊗ 1 → B ⊗ B ⊗ M . (b) Counit arity The c omp osite homomorphis m M ∆ M → B ⊗ M u ⊗ 1 → M is the identi ty of M . (2) L et M , N b e B - c omo dules. We define the c omplex of h o momorphi s ms H om • B − com ( M , N ) b y the asso ciate si mple c omplex of Hom • B − com ( M , N ) define d by Hom • B − com ( M ) : H om K V ec k ( M , N ) d H → H om K V ec k ( M , B ⊗ N ) d H → H om K V ec k ( M , B ⊗ B ⊗ N ) d H → . . . wher e H om K V ec k ( M , B ⊗ n ⊗ N ) d H → H om K V ec k ( M , B ⊗ ( n +1) ⊗ N ) is d e fine d by d H ϕ = ( − 1) n +1 (1 B ⊗ ϕ ) ◦ ∆ M (6.1) + n X i =1 ( − 1) n − i +1 (1 ⊗ ( i − 1) B ⊗ ∆ B ⊗ 1 ⊗ ( n − i ) B ⊗ 1 N ) ◦ ϕ + (1 ⊗ n B ⊗ ∆ N ) ◦ ϕ. (3) L et L, M , N b e B -c omo dules. We define the c omp osite morphism µ : H om B − com ( M , N ) ⊗ H om B − com ( L, M ) → H om B − com ( L, N ) 16 DG ca tegory and bar c omplex by P i,j ≥ 0 µ ij , wher e the morphi s m µ ij : H om K V ec k ( M , B ⊗ i ⊗ N ) e − i ⊗ H om K V ec k ( L, B ⊗ j ⊗ M ) e − j → H om K V ec k ( L, B ⊗ ( i + j ) ⊗ M ) e − i − j is d e fine d by µ i,j ( f ⊗ g ) = (1 ⊗ j B ⊗ f ) ◦ g . Prop osition 6.2. The c ate gory of B -c omo dules ( B − com ) forms a DG- c ate gory by setti ng (1) the c omplex of h o momorphi s m by H om • B − com ( • , • ) , and (2) the c omp osite homomorphism b y µ . The c omp osite is denote d as “ ◦ ”. Pr o of. T o sho w that the m ultiplicati o n homomorphism µ is a homomorphism of complex, it is enough to consider the outer differen tials. Let f ⊗ g ∈ H om • K V ec k ( M , B ⊗ i ⊗ N ) ⊗ H om • K V ec k ( L, B ⊗ j ⊗ M ) and d H b e the outer differen tial. Then we ha ve d H ( f ◦ g ) =( − 1) n +1 (1 ⊗ ( j +1) B ⊗ f ) ◦ (1 B ⊗ g ) ◦ ∆ L + n X p =1 ( − 1) n − p +1 (1 ⊗ ( p − 1) B ⊗ ∆ B ⊗ 1 ⊗ ( n − p ) B ⊗ 1 N ) ◦ (1 ⊗ j B ⊗ f ) ◦ g + (1 ⊗ n B ⊗ ∆ N ) ◦ (1 ⊗ j B ⊗ f ) ◦ g =( − 1) i + j +1 (1 ⊗ ( j +1) B ⊗ f ) ◦ (1 B ⊗ g ) ◦ ∆ L + ( − 1) i j X p =1 ( − 1) j − p +1 (1 ⊗ ( j +1) B ⊗ f ) ◦ (1 ⊗ ( p − 1) B ⊗ ∆ B ⊗ 1 ⊗ ( j − p ) B ⊗ 1 N ) ◦ g + ( − 1) i (1 ⊗ ( j +1) B ⊗ f ) ◦ (1 ⊗ j B ⊗ ∆ M ) ◦ g + ( − 1) i +1 (1 ⊗ ( j +1) B ⊗ f ) ◦ (1 ⊗ j B ⊗ ∆ M ) ◦ g + i X q =1 ( − 1) i − q +1 (1 ⊗ ( q + j − 1) B ⊗ ∆ B ⊗ 1 ⊗ ( i − q ) B ⊗ 1 N ) ◦ (1 ⊗ j B ⊗ f ) ◦ g + (1 ⊗ ( i + j ) B ⊗ ∆ N ) ◦ (1 ⊗ j B ⊗ f ) ◦ g =( − 1) i f ◦ d H ( g ) + d H ( f ) ◦ g . The a ssociati vit y for the comp osite can b e pro ved similarly .  Let A be a DGA and ǫ : A → k b e an augmen tation. Un til the end of this subsection, let B = B ( ǫ | A | ǫ ) be the simplicial bar complex and B re d = B re d ( A, ǫ ) b e the reduced bar complex for the augmen tati on ǫ . Let π α : B → k α ⊗ k b e the pro jection. Definit ion 6.3 ( ( B − com ) re d,b ) . L et S ⊂ Z a subset of Z . A B -c omo dule M is s aid to b e supp orte d o n S if and only if the c omp osite M ∆ M → B ⊗ M π α ⊗ 1 M → k α ⊗ k ⊗ M T omohide T erasoma 17 is zer o if α / ∈ S . A mo dule M supp orte d on a finite set S is c al le d a b ounde d B c omo dule. The class of obje cts in the DG c ate gory ( B − com ) re d,b is define d by b ounde d B c omo dules and the c omplex of morphis ms fr om M to N is de fine d by H om B red − com ( M re d , N re d ) . Her e M re d and N re d ar e the B re d -c omo dules induc e d by M and N . The rest of this section is sp en t to prov e the follo wing theorem. Theorem 6.4 (Main Theorem) . DG c ate gorie s K b C A and ( B − co m ) re d,b ar e homotopy e quivalent. 6.2. Corresp ondence s on ob jects. W e construct a one to one corresp on- dence ϕ from the class of ob jects of ( K b C A ) t o that of ob jects of ( B − com ) re d,b . In t his section, w e simply denote B α and B for B α ( ǫ | A | ǫ ) and B ( ǫ | A | ǫ ), resp ectiv ely . 6.2.1. Definition of ϕ : ob ( K b C A ) → ob ( B − com ) re d,b . Let M = ( M i , d ij ) b e an ob ject of K b C A , where M i ∈ C A . W e set s ( M ) b y the graded ve ctor space ⊕ i M i e − i . The morphism d ij ∈ H om j − i +1 K V ec k ( M j , A • ⊗ M i ) defines an elemen t D j i = d ij ⊗ t − i, − j ∈ H om 1 K V ec k ( M j e − j , A • ⊗ M i e − i ) (6.2) = H om 1 K V ec k ( M j e − j , ( k j ⊗ A • i ⊗ k ) ⊗ M i e − i ) = H om 1 K V ec k ( M j e − j , B j i ⊗ M i e − i ) . W e consider the foll o wing linear map M k e − k D kj → B kj ⊗ M j e − j 1 ⊗ D j i → B kj i ⊗ M i e − i µ ⊗ 1 → B ki ⊗ M i e − i , where µ is the mu lti plication map. The condition (2.2) is equiv alent to the relation X j : k 1 . Then A is a sub DGA of A whic h is quasi-isomorphic to A and satisfies the conditions of the lemma.  Definit ion 7.2. (1) We define K 0 C A (r esp. K ≤ 0 C A , K ≥ 0 C A ) as the ful l sub c ate gory of K b C A of obje cts M = ( M i , d i,j ) wher e M i is of the form M − i,i e i (r esp. ⊕ i + j ≤ 0 M j,i e − j , ⊕ i + j ≥ 0 M j,i e − j ) for k -ve ctor sp ac es M − i,i . (2) We define an additive c ate gory H 0 K 0 C A as fol lows. The class of obje cts of H 0 K 0 C A is the same as i n K 0 C A . F or a, b ∈ K 0 C A , we define the morphi sm H om H 0 K 0 C A ( a, b ) = H 0 ( H om K b C A ( a, b )) . Definit ion 7.3. L et A • b e a differ ential gr ade d algebr a. (1) A p air ( M , ∇ ) of fr e e A 0 -mo dule M and a k -line ar map ∇ : M → M ⊗ A 0 A 1 is c al le d an A c onne ction if ∇ ( am ) = a ∇ ( m ) + da · m . A n A c onne ction is said to b e inte gr able if ∇ ◦ ∇ = 0 . (2) A n A • c onne ction ( M , ∇ ) is said to b e trivial if it is gener ate d b y horizontal s e ctions. (3) A n A • c onne ction ( M , ∇ ) is c al le d nilp otent if ther e exis ts a finite filtr ation by c o nne ctions F p M s uch that Gr p F ( M ) i s a trivial c onne c- tion. The c ate gory of inte gr able nilp otent c onne ctions is denote d as 24 DG ca tegory and bar c omplex ( I N C A ) . Morphism is a A 0 homomorphism c omp atible with c onne c- tions. (4) A homomorphism F ∈ H om I N C A ( M , N ) is homotopy e quivalent if ther e is a map h : M → N ⊗ A 0 A − 1 of A 0 -homomorphism such that F = ∇ ◦ h + h ◦ ∇ . (Se e the fol lowing di agr am.) M h → A − 1 ⊗ A 0 N ∇ M ↓ ↓ ∇ N A 1 ⊗ A 0 M h → A 0 ⊗ A 0 N By lo c alizing the homomorphisms by homotopy e quivalent, we have a c ate gory ( H I N C A ) of homotopy inte gr able nilp otent c onne cti ons. By the definition, if the c ondi tion (p) is satisfie d, then ( I N C A ) and ( H I N C A ) ar e e q ui valent. Prop osition 7.4. L et A • b e a c onne cte d DGA with an augmentation ǫ . The c ate gory ( H I N C A ) of homotopy e quivalenc e class of inte gr able nilp otent A • c onne ctions and H 0 K 0 C A ar e e quivalent. Pr o of. W e define a functor F : K 0 C A → ( I N C A ) b y t aking ( ⊕ i M − i,i ⊗ A 0 , ∇ ) for an ob ject ( M i , d i,j ) ∈ K 0 C A with M i = ⊕ i M − i,i e i . Here ∇ is defined b y ( P i,j d i,j ) ⊗ 1 + 1 ⊗ d . By restricting the ab ov e functor, we hav e a functor Z 0 K 0 C A → ( I N C A ) . By the definition of ( I N C A ), the following natural homomorphisms are isomorphisms: Z 0 H om K 0 C A ( M , N ) → H om I N C A ( F ( M ) , F ( N )) H 0 H om K 0 C A ( M , N ) → H om H I N C A ( F ( M ) , F ( N )) Th us the functor F : Z 0 K 0 C A → ( I N C A ) and F : H 0 K 0 C A → ( H I N C A ) are fully faithful functors. W e show the essen tially surjectivit y . Let ∇ : M → A 1 ⊗ M be a nilp oten t in tegrable A -connection. Let F • M b e a nilp oten t filtration of M for the connection ∇ . W e c ho ose a sp lit ting M ≃ ⊕ Gr i F ( M ), where Gr i F ( M ) = F i ( M ) /F i +1 ( M ). Let M − i,i = Gr i F ( M ) and ∇ ij : M − i,i → A 1 ⊗ M − j,j b e t he corresp onding comp onen t of ∇ . W e set M i = M − i,i e i and d ij = ∇ ij ∈ H om 1 C A ( M i e − i , M j e − j ). Then ( { M i } , d ij ) is an ob ject o f K 0 C A .  W e define the follo wing homomorphism (1) b y taking the fiber of F ( M ) at the augmen tati on ǫ : Z 0 H om K 0 C A ( M , N ) (1) − → H om k ( F ( M ) ⊗ ǫ k , F ( N ) ⊗ ǫ k ) = H om k ( ⊕ M − i,i , ⊕ N − j,j ) . Lemma 7.5. The i mage of ( 1) is identifie d with H 0 H om K 0 C A ( M , N ) . Pr o of. Since the i mage o f B 0 H om K 0 C ( M , N ) under the fu nctor (1) is zero, the functor ( 1) factors t hrough H 0 H om K 0 C ( M , N ) (2) → H om k ( ⊕ i M − i,i , ⊕ j N − j,j ) . T omohide T erasoma 25 W e sho w the injectivity of (2) by t he induction of the nilp oten t length of the corresp onding connections. Let M and N b e k -v ector spaces. Si nce A is connected, the natural homo- morphism H 0 H om K 0 C A ( M , N ) → H om K 0 C A ( ⊕ i M − i,i , ⊕ j N − j,j ) is an isomorphism. Let N ≥ i and N ≤ i b e the stupid filtration of N as DG complex in C A and the quot i en t of N b y N ≥ i . Since A is connected, w e hav e the following left exact sequences 0 → H 0 H om K 0 C A ( M , N ≥ i ) → H 0 H om K 0 C A ( M , N ) → H 0 H om K 0 C A ( M , N ≤ i ) , 0 → H 0 H om K 0 C A ( M ≤ i , N ) → H 0 H om K 0 C A ( M , N ) → H 0 H om K 0 C A ( M ≥ i , N ) . By induction on t he nilp oten t length of N and M , we hav e the l emma by 5-lemma.  Theorem 7.6. L et A b e a c onne cte d DGA. The c ate gory H 0 K 0 C A is e quiv- alent to the c ate gory of nilp o te nt H 0 ( B ( ǫ | A | ǫ )) - c omo dules. Pr o of. By Lemma 7.1, w e c ho ose a quasi-isomorphic sub DGA A ′ of A • suc h that A ′ − i = 0 for i < 0 and A ′ 0 ≃ k . In this case, t he conditio n ( p) is satisfied and the categories ( I N C A ′ ) and ( H I N C A ′ ) are equiv alent. Th us w e ha v e the follo wing commutativ e dia g ram of categories: ( I N C A ′ ) α → ( I N C A ) ↓ ↓ ( H I N C A ′ ) α ′ → ( H I N C A ) F A ′ ↓ ↓ F A ( H 0 ( B A ′ ) − como d) α ′′ → ( H 0 ( B A ) − como d) . W e know that α ′ is fully faithful and α ′′ is equiv al en t . Th us it is enough t o sho w that F A is fully faithful and F A ′ is equiv alen t. W e show the fully faithfulness of F A . Let N , M b e B -como dule cor- resp onding to the nilp oten t in tegrable A connections M and N . W e set H 0 = H 0 ( B ( ǫ | A | ǫ )). B y the induction of the nilp oten t length, w e can sho w that the natural map H i H om B − com ( M , N ) → H i H om H 0 − com ( H 0 ( M ) , H 0 ( N )) is an isomorphism for i = 0 and injective for i = 1 using 5-lemma. Therefore the functor F A is a fully faithful functor. W e introduce a univ ersal integrable nilp oten t connection on H 0 ( B ( ǫ | A ′ | ǫ )) to sho w the essen tial surjectivit y of ( H I N C A ′ ) → ( H 0 ( B A ′ ) − comod ). W e use the i somorphism H 0 ( B re d ( A ′ , ǫ )) ≃ H 0 ( B ( ǫ | A ′ | ǫ )) b etw een simplicial bar complex and Chen’s reduced bar complex. B y the conditio ns (p) and (r), 26 DG ca tegory and bar c omplex H 0 ( B re d ( A ′ , ǫ )) is iden tified with a subspace o f B re d ( A ′ , ǫ ) 0 . The copro duct on B re d ( A ′ , ǫ ) induces a connection H 0 ( B re d ( A ′ , ǫ )) → ( A ′ ) 1 ⊗ H 0 ( B re d ( A ′ , ǫ )) on H 0 = H 0 ( B re d ( A ′ , ǫ )). F or an H 0 -como dule ( M , ∆ M ), W e define M as the kernel of the follo wing map: H 0 ⊗ M ∆ H 0 ⊗ 1 M − 1 H 0 ⊗ ∆ M → H 0 ⊗ H 0 ⊗ M . Then M has a structure of A ′ -connection and F A ′ ( M ) = M . This pro v es the essen tially surjectivity .  Since the category of nilp oten t H 0 -como dules is stable under taking kernels and cokerne ls, we hav e the follo wing corollary . Corollary 7.7. (1) If A • is a c onne c te d DGA, then A = H 0 K 0 C A is an ab e li a n c ate gory. (2) L et A and A ′ b e c onne cte d DGA’s and ϕ : A → A ′ a quasi - isomorphism of DGA. Then the as so ciate d c ate go r i es ( H I N C A ) → ( H I N C A ′ ) ar e e qui valent. 8. P a tching of DG ca tegor y In this section, we consider patching of DG categor i es and their bar com- plexes. A t ypical example for patc hing a pp ears as v an Kamp en’s theorem. Let X b e a manifold, whic h i s co v ered by t w o op en sets X 1 and X 2 . Sup- p ose that X 12 = X 1 ∩ X 2 is connected. Then the fundamen tal g r o up π 1 ( X ) is isomorphic to the amal gam pro duct π 1 ( X 1 ) ∗ π 1 ( X 12 ) π 1 ( X 2 ). W e can inter- pret this isomorphism as a n equiv alence of tw o categori es. The first category Loc ( X ) is a cat ego ry of lo cal systems on X and the second is a category Loc ( X 1 ) × Loc ( X 12 ) Loc ( X 2 ) of triples ( L 1 , L 2 , ϕ ), where L 1 , L 2 are lo cal sys- tems o n X 1 and X 2 and ϕ is an isomorphism of lo cal sy stems ϕ : L 1 | X 12 → L 2 | X 12 . W e m ust b e careful in the nil potent version. Let L oc nil ( X ) b e the category of nilp otent lo cal systems. Then the natural functor Loc ( X ) nil → Loc ( X 1 ) nil × Loc ( X 12 ) nil Loc ( X 2 ) nil is not an eq ui v alen t in general. F or example, if X 12 is con tractible, then there migh t b e no filt ration on L 1 | X 12 = L 2 | X 12 whic h i ndu ces a nilp oten t filtrat ion F 1 on the lo cal system L 1 on X 1 and that on the lo cal system L 2 on X 2 . Definit ion 8.1. (1) L et A b e an ab elian c ate gory. A is said to b e nilp o- tent (a) if ther e is an obje ct 1 , and (b) for any obje ct M i n A , ther e is a filtr ation F • on M such that the ass o ciate d gr ade d quotie nt Gr i F ( M ) i s a dir e ct sum o f 1 A . This filtr ation is c al le d a nilp otent filtr ation of M . T omohide T erasoma 27 (2) L et A 1 and A 2 b e nilp otent ab elian c ate gories and ϕ : A 1 → A 2 b e an exact additive functor. The functor ϕ is said to b e nilp otent if for any nilp ote nt filtr ation F i on an obje ct M , ϕ ( F i ) is a nilp otent filtr ation on ϕ ( M ) . (3) L et A 2 , A 2 , A 12 b e a nilp otent ab elian c ate gory and F 1 : A 1 → A 12 and F 2 : A 2 → A 12 b e nilp otent functors. We define n i lp otent fib er pr o duct ( A 1 × A 12 A 2 ) nil as the ful l sub c ate gory of ( A 1 × A 12 A 2 ) c onsisti ng of obje cts ( L 1 , L 2 , ϕ ) such th at ther e exist ni lp otent filtr ati ons N 1 and N 2 on L 1 and L 2 such that ϕ ( F 1 ( N i 1 )) = F 2 ( N i 2 ) . (4) L et A 1 , A 2 , A 12 b e DGA’s and u 1 : A 1 → A 12 and u 2 : A 2 → A 12 b e homomorphisms of DGA’s . We define a fib er pr o duct e A = A 1 × A 12 A 2 as e A p = A p 1 ⊕ A p 2 ⊕ A p − 1 12 . We intr o duc e a pr o duct of elements a = ( a 1 , a 2 , a 12 ) and b = ( b 1 , b 2 , b 12 ) of e A p and e A q as a · b = ( a 1 · b 1 , a 2 · b 2 , u 1 ( a 1 ) · b 12 + ( − 1) deg b 2 a 12 · u 2 ( b 2 )) . By s etting s ( A 1 × A 12 A 2 ) = A 1 ⊕ A 2 ⊕ A 12 e − 1 , this rule c an b e written as the f ol lowing simpler rule ( a 1 + a 2 + a 12 e − 1 )( b 1 + b 2 + b 12 e − 1 ) = a 1 b 1 + a 2 b 2 + u 1 ( a 1 ) b 12 e − 1 + a 12 e − 1 u 2 ( b 2 ) . Prop osition 8.2. L et A 1 , A 2 and A 12 b e c onne cte d DGA’s, u 1 : A 1 → A 12 and u 2 : A 2 → A 12 homomorphisms of DGA’s and e A = A 1 × A 12 A 2 . Assume that ther e e xists an augmentation ǫ : A 12 → k . Then the natur al functor H 0 K 0 C ˜ A → ( H 0 K 0 C A 1 × H 0 K 0 C A 12 H 0 K 0 C A 2 ) nil is a n e quivalenc e of c ate gories . Pr o of. The fully faithfulness foll ows from the direct calculation. W e sho w the essen tially surjectivity . Since A ⋆ is connected, H 0 K 0 C A ⋆ is equiv a l en t to the category ( H I N C A ⋆ ) for ⋆ = 1 , 2 , 12. Let M = ( M 1 , M 2 , ϕ ) b e an ob ject of ( H I N C A 1 × H I N C A 12 H I N C A 2 ) nil , where M p = ( M p , ∇ p : M p → A 1 p ⊗ A 0 p M p ) ∈ H I N C A p ( p = 1 , 2 ) is a nilp oten t in tegrable connection. By the definition of nilp oten t fib er pro d- uct, there exist filtrations F • 1 and F • 2 on M 1 and M 2 and an A 0 12 -isomorphism ϕ : u 1 ( M 1 ) → u 2 ( M 2 ) suc h that (1 ) ϕ is compat i ble with the connections on u 1 ( M 1 ) and u 2 ( M 2 ), and (2) the fil t rations u 1 ( F • 1 ) and u 2 ( F • 2 ) are isomor- phic via the isomorphism ϕ . By the isomor phism, we ha ve an iden tificati on k ⊗ ǫ,A 0 1 M 1 ≃ k ⊗ ǫ,A 0 2 M 2 = M (0) . The k -vector space M (0) has a filt ration F • induced by u 1 ( F • 1 ) = u 2 ( F • 2 ). Using i den ti fication A 0 p ⊗ k M (0) ≃ M p w e ha v e a map ∇ p : M (0) → A 1 p ⊗ k M (0) for p = 1 , 2. On the other hand, the morphism ϕ defines a map ϕ : M (0) ⊂ u 1 ( M 1 ) → u 2 ( M 2 ) ≃ M (0) ⊗ A 0 12 . Th us we hav e a map M (0) ( ∇ 1 , ∇ 2 ,ϕ ) − → ( A 1 1 ⊕ A 1 2 ⊕ A 0 12 ) ⊗ M (0) . 28 DG ca tegory and bar c omplex By simple calculation, this map giv es rise to an integrable nilp oten t filtration. This nilp oten t filtration gi v e rise to an ob ject ( M 1 , M 2 , ϕ ) of ( H I N C A 1 × H I N C A 12 H I N C A 2 ) nil .  Let ǫ : A 12 → k b e an augmentation of A 12 . By comp osing t he morphism ϕ i : A i → A 12 , w e ha v e a augmentation ǫ i = ǫ ◦ ϕ 1 : A i → k for i = 1 , 2 By comp osing the natural pro jection e A → A i , w e ha v e t w o augmen t ations e i : e A → k for i = 1 , 2. W e in tro duce a comparison copath morphism p ( ǫ ) : B ( e 1 | e A | e 2 ) → k connecting e 1 and e 2 asso ciated to ǫ . Since B ( e 1 | e A | e 2 ) 0 = ⊕ | α | = k B − k α and ⊕ | α | =0 B 0 α = ⊕ α 0 k α 0 ⊗ k ⊕ | α | =1 B 1 α = ⊕ α 0 <α 1 k α 0 ⊗ e A 1 α 1 ⊗ k = ⊕ α 0 <α 1 k α 0 ⊗ ( A 1 1 ⊕ A 1 2 ⊕ A 0 12 ) α 1 ⊗ k . W e define a linear map p ( ǫ ) : B ( e 1 | e A | e 2 ) 0 → k b y the summation of (1) pro jection to the factor k α 0 ⊗ k ov er α 0 , and (2) the comp osite of ǫ and the pro jection to k α 0 ⊗ A 0 12 α 1 ⊗ k ov er α 0 < α 1 . W e can sho w that the comp osite B ( e 1 | e A | e 2 ) − 1 → B ( e 1 | e A | e 2 ) 0 → k is zero. F or example 1 α 0 ⊗ x α 1 ⊗ 1 ∈ k α 0 ⊗ A 0 1 α 1 ⊗ k g o es to ǫ 1 ( x ) α 1 ⊗ 1 + 1 α 0 ⊗ dx α 1 ⊗ 1 + 1 α 0 ⊗ ϕ 1 ( x ) α 1 ⊗ 1 ∈ B ( e 1 | e A | e 2 ) 0 , whic h is an elemen t of the k ernel of p ( ǫ ). Definit ion 8.3. The map p ( ǫ ) i s c al le d the c omp arison c op ath ass o ciate d to ǫ . 9. Graded case W e consider a graded vers ion of DGA. Let A = ⊕ k ≥ 0 A k = ⊕ k ≥ 0 A • k b e a graded DGA o ve r a field k . W e assume that the imag e of A p ⊗ A q under the m ultipli cation map is con tained in A p + q . W e in tro duce a DG category C g r A as follows. Ob jects are finite direct sum o f t he form V • ( i ) where V • is a complex of k -v ector space. F or ob jects V • ( i ) and W • ( j ), w e define H om • C g r A ( V • ( i ) , W • ( j )) = ( H om • K V ec k ( V • , A j − i ⊗ W • ) i f j ≥ i 0 if j < i. Then C g r A b ecomes a DG category . The DG category of DG complex es in C g r A is denoted as K C g r A . The category C A g r has a t ens or structure b y V ( p ) ⊗ W ( q ) = ( V ⊗ W )( p + q ). The full sub category of the b ounded DG complexes is denoted T omohide T erasoma 29 as K b C g r A . By considering k as a graded DGA in a trivial wa y , we hav e a DG- category C g r k . F or ex ample, for k -vector spaces V , W , we hav e H om C g r k ( V ( p ) , W ( q ) ) = ( H om ( V , W ) if p = q 0 otherwise . The ob ject V ( p ) is called the p -T ate twist of V and the T ate we ight of V ( p ) is defined to b e p . The category C g r k is the category of formal T ate t wist of k -v ector spaces. Definit ion 9.1. L et M = ⊕ i M • i b e a gr ade d c omplex of k -ve ctor sp ac es. We intr o duc e a homo genization M h of M by ⊕ i ( M i ⊗ k ( − i )) as an obje ct in C g r k . By thi s c orr esp ond e n c e, the c ate gory of gr ade d c omplex is e quiva lent to the c ate gory C g r k . Under this identific ation, the obje ct k ( p ) ⊗ k ( q ) i s identifie d with k ( p + q ) . Using t he ab ov e notations, for V • ( p ) , W • ( q ) ∈ C g r A , we ha v e H om C g r A ( V • ( p ) , W • ( q )) = H om C g r k ( V • ( p ) , A h ⊗ W • ( q )) . Let ǫ 0 : A 0 → k be an augmen tation of A 0 and ǫ : A → k a composit e of ǫ 0 and the natural pro jection. W e define the homogeneous bar complex B h ( ǫ | A | ǫ ) in K C k as fol l o ws. Let α = ( α 0 < · · · < α n ) b e a sequence of in tegers. W e define B h α ∈ C g r k b y k α 0 ⊗ A h α 1 ⊗ · · · α n − 1 ⊗ A h α n ⊗ k , and B h = B h ( ǫ | A | ǫ ) ∈ K C g r k b y t he complex · · · → ⊕ | α | =1 B h α → ⊕ | α | =0 B h α → 0 . The asso ciate simple ob ject in C g r k is denoted as B h = B h ( ǫ | A | ǫ ). Then the T ate weigh t w part of B h is equal to the sum o f ( k α 0 ⊗ A • p 1 α 1 ⊗ · · · α n − 1 ⊗ A • p n α n ⊗ k ) ⊗ Q ( − p 1 − · · · − p n − 1 ) where w = − ( p 1 + · · · + p n ). The weigh t w part of B h is denoted as B h ( w ). W e define the homogenized r educed bar complex B h re d = B h re d ( A, ǫ ) in the same wa y . Th us B h , B h re d are graded DG coalgebras in C k . W e intro duc e DG cat ego ry structure o n b ounded homogeneous B h como dules. Definit ion 9.2. (1) L et V • = ⊕ V • ( i ) b e an obje ct in C g r k . A homomor- phism ∆ : V • → B h ⊗ V • in C g r k (i.e. homomorphism of gr ade d c omplex) is c al le d a c opr o duct if i t satisfies the c o asso ci ativity and c ouni tar i ty define d in Definition 6.1. A n obje ct V • in C g r k e qui pp e d with a c opr o duct i s c al le d a homo gene ous B h -c omo dule. We define B h re d c omo dules i n the same way. 30 DG ca tegory and bar c omplex (2) L et V • and W • b e homo gene ous B h c omo dules. A morphism f in H om C g r k ( V • , W • ) is c al le d a B h homomorphism if the di agr am V • f → W • ∆ V ↓ ↓ ∆ W B h ⊗ V • 1 B h ⊗ f → B h ⊗ W • is c ommutative. Let V = V • , W = W • b e homo g ene ous B h como dules. W e define t he complex of homomorphism H om B h − com ( V , W ) from V to W b y the asso ciate simple complex of 0 → H om C g r k ( V , W ) d H → H om C g r k ( V , B h ⊗ W ) d H → H om C g r k ( V , B h ⊗ 2 ⊗ W ) d H → · · · where d H is defined similarly to (6.1). W e define the DG category ( B h − com ) b as the category o f b ounded homogeneous B h -como dules with H om B h − com ( • , • ) as the complex of morphisms. The following prop osition is a graded version of Theorem 6.4, and Theorem 7.6 Prop osition 9.3. (1) The DG c ate gor y K b C g r A is homotopy e quivalent to the c ate gory of b ounde d homo gene ous B h ( ǫ | A | ǫ ) c omo dules. (2) Assume that the gr ade d DGA A is c o nne cte d. Then H 0 K 0 C g r A is e qui v - alent to the c ate gory of homo gene ous H 0 ( B h ( ǫ | A | ǫ )) c omo dules. As a c o ns e quenc e, H 0 K 0 C g r A is a n a b elian c ate gory. Pr o of. (1) W e set B h = B h ( ǫ | A | ǫ ). W e give a one to one corresp ondenc e ob ( B h − comod ) b → ob ( K b C g r k ) of ob jects. Let V = ⊕ i V • ( i ) b e an ob ject in C g r k and V → B h ⊗ V b e the comultiplication of V . W e use the direct sum decomp o sition B h = ( ⊕ α 0 k α 0 ⊗ k ) ⊕ ( ⊕ α 0 <α 1 k α 0 ⊗ A h α 1 ⊗ k e ) ⊕ M | α | = i> 1 B h α e i . Let π α 0 b e the pro jectio n B h → k α 0 ⊗ k . Let p α 0 b e the comp osite of the map V ∆ V → B ⊗ V π α 0 → k α 0 ⊗ V and V α 0 b e I m ( p α 0 ) e α 0 . Since the ma p p α 0 is homogeneous, V α 0 is an ob ject in C g r k . By the assumption of b ounde dness, V α 0 = 0 except for finite n um b ers of α 0 . Let π α 0 ,α 1 b e the pro jection B h → k α 0 ⊗ A h α 0 ⊗ k e and consider the comp osite D α 0 ,α 1 b y t he comp osite V ∆ V → B ⊗ V π α 0 ,α 1 → k α 0 ⊗ A h α 0 ⊗ V e. By the asso ciativ it y condition, t he morphism D α 0 ,α 1 induces a morphism H om 1 C g r k ( V α 0 e − α 0 , A h ⊗ V α 1 e − α 1 ) = H om 1 C g r A ( V α 0 e − α 0 , V α 1 e − α 1 ) whic h i s also denoted as D α 0 ,α 1 . This defi nes a one to one correspondence ob ( B h − comod ) b → ob ( K b C g r k ). F or the construction and the pro of of the T omohide T erasoma 31 in v erse corresp ondence and the homotop y equiv alence is similar t o t hose in Section 6, so w e omit the detai led pro of. (2) Using t he follo wing lemma, the pro of of ( 2) is similar. Lemma 9.4. L et A b e a c onne cte d gr ade d DGA. Then ther e e xists a gr ade d sub algebr a A of A such that (1) the natur al inclusion A → A is a quasi - isomorphism, a nd (2)the c onditions (p) and (r) in L emma 7.1 a r e satisfie d.  10. Deligne complex 10.1. Definition of Deligne alge bra. Here we give an application of DG category to Ho dge theory . Let X b e a smo oth irreducible v ariety ov er C and X b e a smo oth compactification of X suc h that D = X − X is a simple normal crossing div isor. Let U = { U i } i ∈ I b e an affine co vering of X i ndexed b y a totally ordered set I , and U an = { U an,j } j ∈ J b e a to polo g ical simple co v ering of X indexed b y a totally o rdered set J , whic h i s a refinemen t of U ∩ X . Assume that the map J → I defining the refinemen t preserv es the ordering of I and J . Let Ω • X (log ( D )) b e t he sheaf of algebraic logarithmic de Rh am complex of X along the b oundary divisor D . Let F • b e t he Ho dge filtration of Ω • X (log ( D )) : F i : 0 → · · · → 0 → Ω i X (log ( D )) → Ω i +1 X (log( D )) → · · · . Then F • is compatible with the pro duct structure of Ω • X (log( D )). The Cec h complex ˇ C ( U , Ω • X (log( D ))) b ecomes a n asso ciativ e DGA by standard Alexander-Whitney asso ciative pro duct using the total order of I . W e define A F dR,i = ˇ C ( U , F i Ω • X (log( D ))) . Then the pro duct i nduc es a morphism of complex A F dR,i ⊗ A F dR,j → A F dR,i + j and b y t hi s mu lti plication A F dR = ⊕ i ≥ 0 A F dR,i b ecomes a graded DGA, whic h is called the algebraic de Rham DGA. W e define t he homogenized al gebraic de Rham DGA A h F dR ∈ K C g r k as ⊕ i ≥ 0 A F dR,i ( − i ). Let Ω • X an b e t he analytic de Rham complex and ˇ C ( U an , Ω • X an ) the top o- logical Cech complex of Ω • X an . Since the ma p J → I preserve the orderings, w e hav e a natural quasi-isomorphism of DGA’s: ˇ C ( U , Ω • X (log( D ))) → ˇ C ( U an , Ω • X an ) . W e set A dRan,i = ˇ C ( U an , Ω • X an ) and A dRan = ⊕ i ≥ 0 A dRan,i . Then A dRan b ecomes a graded DGA b y Alexander-Whitney a ssociati v e pro duct, whic h is called the analy t ic de Rh am DGA. W e also define A B ,i = ˇ C ( U an , (2 π i ) i Q ) and set A B = ⊕ i ≥ 0 A B ,i , which i s call ed the Betti DGA. It is a sub graded DGA of A dRan . W e define the homogenized analytic de Rham DGA A h dRan and the Betti DGA A h B as A h dRan = ⊕ i A dRan,i ( − i ) and A h B = ⊕ i A B ,i ( − i ). 32 DG ca tegory and bar c omplex Definit ion 10.1 (Deligne algebra) . We define the Deligne algebr a A Del = A Del ( X ) of X by the fib er p r o duct A F dR × A dRan A B . It is gr ade d by A Del, i = A F dR,i × A dRan,i A B ,i . (Se e Defini tion 8.1 f or the defini ti on of fib er pr o duct of DGA’s.) Example 10.2. In the c ase of X = S pec ( C ) , the c omplexes A Del and A h Del ar e e qual to A 0 Del = C M ⊕ i ≥ 0 (2 π i ) i Q , A 1 Del = ⊕ i ≥ 0 C , A h, 0 Del = C M ⊕ i ≥ 0 (2 π i ) i Q ( − i ) , A h, 1 Del = ⊕ i ≥ 0 C ( − i ) . The differ entials ar e the natur al maps. In this c ase, the b ar sp e ctr al se quenc e de g e ner ates at E 1 and Gr ( H 0 ( B ( ǫ B | A Del | ǫ B ))) is isomorphi c to the tensor algebr a gener ate d by ⊕ i> 0 C / (2 π i ) i Q over Q . Remark 10.3. (1) We c an define A dR for a smo oth variety over an ar- bitr ary subfield K of C . (2) A ctual ly, A Del ( X ) dep ends on the choi c e of the c omp actific ation X , and c overings U and U an . But the choic e of admissible pr op er mor- phisms and r e fin e ments do es not affe ct up to quasi-i somorphism. (3) We c an r eplac e the r ole Q in A B by an arbitr ary subfield F of R . The c or r esp onding Deligne algebr a i s denote d as A Del, F ( X ) . Let p dR (resp. p B ) b e a C -v alued p oint of X (resp. a p oin t in X ( C ) an ). Then w e hav e an augmen tation ǫ dR ( p ) (resp. ǫ B ( p )) of A dR (resp. A B ). The follo wing prop ositio n is a consequence of Prop osition 9.3. Prop osition 10.4. The c ate gory H 0 K 0 C g r A Del is e quivalent to the c ate gory o f homo gene ous H 0 ( B h ( ǫ B | A Del | ǫ B )) -c omo dules. 10.2. Nilp otent v ariation of mixed T ate H o dge struct ure. In this sub- section, w e pro ve that the category of nilp oten t v ari ation o f mix ed T ate Ho dge structures on a smo oth irreducible algebraic v ariet y is equiv alent to t hat of como dules ov er H 0 ( B h ( ǫ | A Del ( X ) | ǫ )). Definit ion 10.5 (VMTHS) . L et X b e a smo oth irr e ducible algebr aic variety over C . A triple ( F , V , comp ) of the fol lowing data is c al le d a variati on of mixe d T a te Ho dge structur e on X . (1) A 4ple F = ( F , ∇ , F • , W • ) i s a lo c al ly fr e e she af F on X with a c onne ction lo gari thmic c onne ction ∇ : F → Ω 1 X (log ( D )) ⊗ F and de cr e asing and i nc r e asing filtr ations F • and W • on F with the fol lowing pr op erties . ( W e ass um e that the filtr ati on W is indexe d by even i nte gers.) (a) ∇ ( F i F ) ⊂ Ω 1 X (log( D )) ⊗ F i − 1 F . (b) ∇ ( W − 2 j F ) ⊂ Ω 1 X (log ( D )) ⊗ W − 2 i F . T omohide T erasoma 33 (2) A p air V = ( V , W • ) is a lo c al system with a filtr ation W • on X ( C ) an . (3) A n isomorphism of lo c al s ystem V ⊗ C ≃ → ( F ⊗ O X ( C ) an ) ∇ =0 on X ( C ) an c omp atible with the filtr ations W • on F and V . (4) The fib er of the as so ciate d g r ade d obje ct ( Gr W − 2 i F , Gr W − 2 i V , comp ) at x is i somorphic to a sum of mixe d T ate Ho dge structur e of wei g ht − 2 i for al l x ∈ X ( C ) an Definit ion 10.6 (NVMTHS) . (1) A v ariation of mixe d T ate Ho dge struc- tur e is said to b e c onstant i f the lo c al system V i s a trivial lo c al system. (2) A c onstant var i ation of mixe d T a te Ho dg e structur e is said to b e s plit if it i s is omorphic to a s um of pur e T ate Ho dge structur es. (3) A vari ation of mixe d T ate Ho dge structur e is said to b e nilp otent (de- note d as NVMTHS for short) if the r e exi sts a filtr ation ( N • F , N • V , comp ) of ( F , V , comp ) such that the asso c i ate d gr ade d obje ct ( Gr N p F , Gr N p V , comp ) is a split c onstant v a ri ation o f mixe d T ate Ho dg e structur es o n X ( C ) an . Theorem 10.7. The c ate gory H 0 K 0 C g r A Del is e quivalent to N V M T H S ( X ) . It is also e quivalent to the c ate gory of homo gene o us H 0 ( B h ( ǫ B | A Del | ǫ B )) - c omo dules. By a pply ing the ab o v e theorem for the case X = S pec ( C ), w e hav e the follo wing corol lary . Corollary 10.8. The c ate gory o f mixe d Ho dge structur es is e quivalent to the c ate gory of homo gene ous H 0 ( B h ( ǫ B | A Del ( C ) | ǫ B )) -c omo dules. 10.3. Proof of Theorem 10.7. W e construct a nilp otent v ari a tion of mixed T ate Ho dge structure for an ob ject V = { V i } in K 0 C A Del . W e set V i = V ip e i ( p ), V = ⊕ i V i e − i and V ( p ) = ⊕ i V ip e i ( p ). Then w e ha ve V = ⊕ i,p V ip ( p ) = ⊕ p V ( p ). Let O , O an , Ω i and Ω i an denote O X , O X an , Ω i X (log D ) and Ω i X an . F or indices s, t, u ∈ J , the corresponding elemen t in I b y t he map J → I for re- finemen t is also denoted as s, t, u for short. The sum D of the differen tial d ij ∈ H om 1 C g r A Del ( V i e − i , V j e − j ) defines an elemen t in H om 0 C g r k ( V , A h, 1 Del ⊗ V ) = H om 0 C g r k ( V , ⊕ i Y s Γ( U s , F i Ω 1 )( − i ) ⊗ V ) ⊕ H om 0 C g r k ( V , ⊕ i Y s,t Γ( U s,t , F i O )( − i ) ⊗ V ) ⊕ H om 0 C g r k ( V , ⊕ i Y s,t Γ( U an,s,t , (2 π i ) i Q B ( − i )) ⊗ V ) ⊕ H om 0 C g r k ( V , ⊕ i Y s Γ( U an,s , O an )( − i ) ⊗ V ) . As in the pro of of P ro positi on 7.4, ∇ = D + 1 ⊗ d defines an in tegrable A Del - connection V → A 1 Del ⊗ V . W e write D = ( { ω s } , { f st } , { ρ st } , { ϕ s } ) according 34 DG ca tegory and bar c omplex to the ab o ve direct sum decomp osition, i.e. ρ st ∈ H om 0 C g r k ( V , ⊕ i Γ( U an,s,t , (2 π i ) i Q B ( − i )) ⊗ V ) , etc. Then as an elemen t of H om 0 C g r k ( V , A h, 2 Del ⊗ V ) = H om 0 C g r k ( V , ⊕ i Y s Γ( U s , F i Ω 2 )( − i ) ⊗ V ) ⊕ H om 0 C g r k ( V , ⊕ i Y s,t Γ( U s,t , F i Ω 1 )( − i ) ⊗ V ) ⊕ H om 0 C g r k ( V , ⊕ i Y s,t,u Γ( U s,t,u , F i O )( − i ) ⊗ V ) ⊕ H om 0 C g r k ( V , ⊕ i Y s,t,u Γ( U an,s,t,u , (2 π i ) i Q B ( − i )) ⊗ V ) ⊕ H om 0 C g r k ( V , ⊕ i Y s Γ( U an,s , Ω 1 an )( − i ) ⊗ V ) ⊕ H om 0 C g r k ( V , ⊕ i Y s Γ( U an,s,t , O an )( − i ) ⊗ V ) , w e hav e dD =( { dω s } , {− ω t + ω s + d f st } , {− f tu + f su − f st } , {− ρ tu + ρ su − ρ st } , { ω s + dϕ s } , { f st − ρ st + ϕ t − ϕ s } ) and D ◦ D =( { ω s ω s } , {− ω s f st + f st ω t } , { f st f tu } , { ρ st ρ tu } , {− ω s ϕ s } , {− f st ϕ t + ϕ s ρ st } ) . Here the pro ducts are tensor pro ducts of comp osites of endomorphisms of V and exterior pro ducts of differen tial forms. Then by the integrabilit y condi- tion, we ha ve dD = D ◦ D . By this rela tion, w e ha ve co cycle relat i ons (10.1) (1 + f su ) = (1 + f st )(1 + f tu ) , (1 + ρ su ) = (1 + ρ st )(1 + ρ tu ) , in tegrability of connections dω s = ω s ω s , and the follo wing commutativ e dia- grams: (10.2) ⊕ p V ( p ) ⊗ Γ( U an,s,t , (2 π i ) − p Q ) 1+ ρ st → ⊕ p V ( p ) ⊗ Γ( U an,s,t , (2 π i ) − p Q ) 1 + ϕ t ↓ ↓ 1 + ϕ s V ⊗ Γ( U an,s,t , O an ) 1+ f st → V ⊗ Γ( U an,s,t , O an ) (10.3) V ⊗ Γ( U s,t , O ) 1+ f st → V ⊗ Γ( U s,t , O ) ω t + 1 ⊗ d ↓ ↓ ω s + 1 ⊗ d V ⊗ Γ( U s,t , Ω 1 ) (1+ f st ) ⊗ 1 → V ⊗ Γ( U s,t , Ω 1 ) T omohide T erasoma 35 (10.4) V ⊗ Γ( U an,s , O an ) 1 ⊗ d → V ⊗ Γ( U an,s , Ω 1 an ) 1 + ϕ s ↓ ↓ 1 + ϕ s V ⊗ Γ( U an,s , O an ) ω s +1 ⊗ d → V ⊗ Γ( U an,s , Ω 1 an ) By the co cycle relations, w e ha v e a lo cal system V and lo cally free sheaf F b y patching constan t shea ves V and free shea v es V ⊗ O on U an,s and U s b y the patc hing data 1 + ρ st and 1 + f st . The connection ω s +1 ⊗ d defines an integrable connection on V ⊗ Γ( U s , O ) b y the integrabilit y condition and it is pat ched together to a global connection b y the comm utative diagram (10.3). The lo cal sheaf homomorphisms 1 + ϕ s patc hed to g ether i nto a global homomorphism comp of shea ve s V → F ⊗ O an on X an b y the comm utative diagram (10.2). Via the sheaf homomorphism comp , the lo cal system V ⊗ C is i den t i fied with the subshaef of horizontal sections o f F ⊗ O an b y the commutativ e diagram (10.4). W e introduce filtrati o ns F • and W • on U s and U an,s b y F i F = ⊕ p ≥ i V ( − p ) ⊗ O , W 2 i F = ⊕ p ≤ i V ( − p ) ⊗ O , W 2 i V = ⊕ p ≤ i V ( − p ) . The filtration F gi v es rise t o a w ell defined filtrati on on F since the patc hing data f ij for F i s con tained in Γ( U s,t , A h, 0 F dR ) = Γ( U s,t , O (0)). Since ω is contained in M p H om k ( V ( p ) , V ( p ) ⊗ Γ( U s , Ω 1 )) ⊕ M p H om k ( V ( p ) , V ( p + 1) ⊗ Γ( U s , Ω 1 )( − 1)) , the Griffiths tra nsv ersalit y condition in Definition 10.5 is sat isfied. It defines a mixed T at e Ho dge structure at an y p oint x in X ( C ) an b y the definitions of tw o filtrations. Therefore ( F , V , comp ) defines a v ariation of mi xed T ate Ho dge structure on X . W e set N i V = ⊕ i ≤ p V p . Then t hi s filtrat i on defines a filt ration on the v a ri - ation of mixed T ate H odge str ucture ( F , V , comp ) and the asso ciated graded v ariat ion is split constan t mixed T ate Ho dge structures on X . Th us w e ha v e the required nilp oten t v ari ation of mi xed T ate Ho dge structure ( F , V , comp ) on X . W e can construct an in vers e corresp onden ce b y attaching an A Del -connection to a nil potent v a riation of mixed T ate Ho dge structures as follows. W e c ho ose sufficien tly fine Zariski cov ering U = { U i } suc h t hat the restrictions of F are free on eac h U i . Using tw o filtrations F and W , F split s into ⊕ p F ( p ) as an O mo dule. W e choose a tri v ialization of F ( p ) | U i . By tak ing a refinemen t U an of U ∩ X an , we may assume t hat U an is a simple cov ering. By restricti ng U an,s , w e c ho o se a tri vialization 1 + ϕ i of the lo cal system V . At l ast we choose a nilp otent filtration N • compatible with the connection and lo cal system 36 DG ca tegory and bar c omplex and a splitt ing of N • . Using t hese data we construct an integrable nilp oten t A Del -connection D = ( { ω s } , { f st } , { ρ st } , { ϕ s } ) by the comm utati ve diagram (10.2), (10 .3), (10.4). By passing to homotop y equiv al enc e classes of morphisms, w e ha v e the required equiv alence of tw o categories. Reference s [BD] Beilinson, A.-Deligne, P . , R e al Ho dge structur e Zagier’s c onje ctur e, Proceedi ng of symp osia in pure mathemetice s 55, Amer. Math. Soc . [BK1] Bloch, S., Kri z, I., Mixe d T ate motives, Ann. of M ath. 140 (1994), 557–605. [BK2] Bondal, A.I., Kapranov, M.M. , Enh anc e d triangulate d c ate gories, Mat.Sb. 181. (1990), no. 5. 669-683, translation in Math. USSR-Sb. 70 no.1, 93107. [C] Chen, K. T., Iter ate d inte gr als of differ ential forms and loop spac e homo l o gy, Ann. of Math. (2) 97 (1973), 217–246. [De] D eligne, P ., L e gr oup e fondamental d e la dr oite proje ctive moins tr ois p oints, Galois groups o ver Q (Berkeley , CA, 1987), 79–297, Math. Sci . Res. Inst. Publ. , 16, Springer, New Y ork, 1989. [Dr] Drinfeld, V., D Gquotie nts of DGc ate gories, J. Algebra 272 (2004), no. 2, 643.691. [DM] D eligne, P .-Mi lne, J., T annakian Cate gories, In Ho dge Cycles, Motives, and Shimura V arie ties, in Lecture Notes in Math. 900, pp.101–228. [DG] Deligne, P .,Goncharo v, A., Gr oup es fondamentaux mo tiv iques de T ate mixte, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 1, 1–56. [H] Hanam ura, M. : Mixe d motives and algebr aic cycles. I, Math. Res. Lett. 2 (1995), no. 6, 811.821. [Ke1] Keller, B ., On differ ential gr ade d c ate gory, International Congress of Mathematic ians. V ol. I I, 151–190 , Eur. M ath. So c., Zurich, 2006. [Ke2] Keller, B. Deriving DG c ate gories, Ann. Sci. Ecol e Norm. Sup. (4) 27 (1994), no. 1, 63–102. [KM] K riz, I.,May , J. P ., Oper ads, algebr as, modules and motives, Asterisque No. 233 (1995), iv+145pp.