math.AG 2010-07-13 0

Tensor Structure on Smooth Motives

Grothendieck first defined the notion of a "motif" as a way of finding a universal cohomology theory for algebraic varieties. Although this program has not been realized, Voevodsky has constructed a triangulated category of geometric motives over a p…

Authors: 논문은 **저자 미상**(학위 논문)으로, 본문에 “my thesis” 라는 표현이 사용되어 저자가 직접 연구를 수행했음을 알 수 있다. ###

T ensor Structur e on Smo oth Motiv es Anandam Banerjee Abstra ct. Grothendiec k first defi ned the n otion of a “motif ” as a w a y of finding a univers al co homology theory for algebraic v arieties. Al- though this program has not b een realized, V oevodsk y has constructed a triangulated category of geometric motives ov er a p erfect fi eld, which has many of the p rop erties ex p ected of the derived category of the con- jectural abelian category of motiv es. The construction of t he triangu- lated category of motiv es has been exten ded b y Cisinski-D´ eglise to a triangulated category of motives ov er a base-scheme S . Recently , Bon- darko constructed a DG category of motives, whose homotopy category is eq u iv alent to V o evodsky’s category of eff ective geometric motives, as- suming resolution of singularities. So on after, Levine extended th is idea to construct a D G category whose h omotopy category is equiva len t to the full sub category of motives ov er a base-scheme S generated by the motives of smo oth pro jective S -schemes, assuming that S is itself smo oth o ver a p erfect field. In b oth construct ions, the tensor structure requires Q -coefficients. In my th esis, I show how to provide a tensor structure on the homotopy category mentioned ab ov e, when S is semi-lo cal and essen tially smo oth o ve r a fi eld of c haracteristic zero. This is done by defining a p seudo-tensor structure on t he DG category of motives con- structed by Levine, which indu ces a tensor structure on its homotop y category . Ac kno wledgemen ts I would lik e to thank my advisor Marc Levine for enlight ening me with his infin ite kno wledge, helping me un conditionally wh en ev er I n eeded and b eing very supp ortiv e and motiv ating. I am v ery grateful to the Departmen t of Mathematics of North eastern Univ er s it y for pr o vidin g me with financial supp ort du ring my gradu ate stud - ies. I w ould also lik e to thank my fello w graduate studen ts and friends for alw ays b eing helpfu l and encouraging. 3 Dedication Dedicated to my b eloved grandma Jut hik a Ganguly who taught me to b e pat ient and p erseverant. It is also dedicat ed to my pa rents Prana b and Jay ati Banerjee whose pa tient supp ort and constant encouragement has b een with me all the w a y s ince the b eginning of my studies. 5 Con ten ts Ac knowledgeme n ts 3 Dedicatio n 5 T able of C on tents 7 In tro duction 9 Chapter 1. DG categories of motiv es 11 1.1. Preliminaries 11 1.1.1. DG catego ries 11 1.1.2. The category Pre-T r( C ) of DG complexes 13 1.1.3. DG catego ries and triangulated categories 14 1.1.4. Sheafification of DG categories 16 1.2. Cubical enric hment s in DG categorie s 17 1.2.1. Cubical categories 17 1.2.2. Cub es and complexes 17 1.2.3. Multiplica tions and co-m ultiplications 20 1.2.4. Extended cub es 22 1.2.5. DG catego ries asso cia ted with cub ical categories 22 1.3. More on DG categories and cubical categories 23 1.3.1. Homot op y in v ariance 23 1.3.2. Multi-c ub es 25 1.3.3. The extended DG category 26 1.3.4. Extended multi- cubical complexes 28 1.4. DG categories of motiv es 29 Chapter 2. Pseudo-tensor stru cture 33 2.1. Pseudo-tensor structur e on DG categories 33 2.2. Pseudo-tensor structur e on DG complexes 39 2.3. Pseudo-tensor structur e on th e category C pretr 45 2.4. Pseudo-tensor structur e in the category dg e C 49 Chapter 3. T ensor Structur e on S mo oth Motiv es 55 3.1. Main Theorem 55 3.2. F uture directions 56 Bibliograph y 57 7 In tro duction Although this program has n ot b een realized, V oevodsky h as constr u cted a triangulated category of geometric motiv es ov er a p erf ect fi eld, whic h has man y of the p rop erties exp ected of the deriv ed catego ry of the conjectural ab elian category of motive s. T h e construction of th e triangulated category of motiv es h as b een extended by Cisinski-D ´ eglise ([ CD07 ]) to a triangulated catego ry of motiv es o v er a base-sc heme S . Hanamura h as also constructed a triangulate d categ ory of motiv es o v er a field, using the idea of a “higher corresp ondence”, with morphisms built out of Blo ch’s cycle complex. Re- cen tly , Bondark o (in [ Bon09 ]) has r efi ned Hanam ur a’s idea and limited it to smo oth pro jectiv e v arieties to constru ct a DG catego ry of m otiv es. Assum- ing resolution of singularities, the homotop y category of this DG category is equiv alent to V o ev o d sky’s catego ry of effectiv e geometric motiv es. So on after, Levine (in [ Lev09 ]) extended this idea to construct a DG category of “smo oth motiv es” ov er a base-sc h eme S generated by the motive s of smo oth pro jectiv e S -sc hemes, where S is itself smo oth o v er a p erfect field. Its ho- motop y category is equiv alen t to the full sub category of Cisinsk i-D´ eglise catego ry of effecti v e motive s o v er S generated by the smo oth pro jectiv e S -sc hemes. Both these constructions lac k a tensor structure in general. Ho wev er, passing to Q -co efficien ts, Levine replaced th e cu b ical construction with alternating cub es, whic h yields a tensor structure on his DG catego ry . In [ BV08 ], Beilinson and V olog o dsky constructs a “homotop y tensor structure” on the DG catgego ry of V oevodsky’s effectiv e geometric motiv es. That is, they constructed a “pseudo-tensor stru ctur e” (see Defin ition 2.1.2 ) on the DG category D M (defined in [ BV08 ], § 6.1), suc h that the corre- sp ond in g pseudo-tensor structure on its homotop y categ ory is actuall y a tensor structur e. In the follo win g, w e construct a p seudo-tensor stru cture on a DG cate- gory dg e C or S whic h in d uces a tensor str ucture on the homotop y category of DG complexes, s uc h that, in case S is semi-lo cal and essentia lly smo oth o ver a field of c haracteristic zero, it ind uces a tensor stru cture on the category of smo oth motive s o v er S . Thus, we prov e Theorem 1. Su pp ose S is semi-lo c al and essential ly smo oth over a field of char acteristic zer o. Then, ther e is a tensor structur e on the c ate gory S mM ot eff g m ( S ) of smo oth effe ctive ge ometric motives over S making it into a tensor tria ngulate d c ate gory. 9 See T heorem 3.1.1 and corollary 3.1.2 for a more p recise statement . The p s eudo-tensor structur e is constru cted in t w o main steps: • W e sho w that a ps eudo-tensor structure on a DG category C in d uces a pseudo-tensor structure on the ca tgeory Pre-T r( C ) (defined in 1.1.2 ). Then w e pro v e that if the ps eudo-tensor str ucture on C induces a tensor structur e on the h omotop y catego ry H 0 C , then w e hav e an ind uced tensor structure on K b ( C ). • F or a tensor category C , with a cubical ob ject with com u ltiplica- tion, Levine constructs a DG categ ory dg C (see 1.2.5 ). W e pr o duce a pseudo-tensor stru cture on dg e C that induces a tensor structure on its homotopy category , assuming some add itional tec hn ical con- ditions. Und er these conditions, the DG categories dg C and dg e C are quasi-equiv alen t, so th e homotop y categories of complexes are equiv alen t triangulated categories. In Chapter 1, we go o v er some basic definitions and recall Levine’s con- struction of the DG category of smo oth effectiv e geo metric motiv es ov er S . In Chapter 2, we define a p seudo-tensor stru cture on a DG category and sho w when it ind u ces a tensor structure on the homoto p y cat egory . Then, w e show h o w it induces a pseudo-tensor s tructure on the catego ry Pr e-T r( C ) and on C b dg ( C ) (defin ed in page 15 ). Next, we sho w that if the pseud o-tensor structure on C in duces a tensor structure on the homotop y category H 0 C , then the indu ced p seudo-tensor s tructure on C b dg ( C ) giv es a tensor stru cture on its homotop y category K b ( C ). W e also pr o ve the existence of a homotop y tensor structure on dg e C for a tensor category C , with a cubical ob ject with com ultiplication. Finally , in C hapter 3, we give a p ro of of Th eorem 1 . 10 CHAPTER 1 DG categories of motiv es 1.1. Preliminaries 1.1.1. DG categories. W e b egin b y recalling some basic facts about DG categ ories. F or a complex of ab elia n groups C ∈ C ( Ab ), we ha v e the group of cycles in degree n , Z n C and cohomology H n C . F or complexes X, Y , we ha v e the Hom-complex H om C ( Ab ) ( X, Y ) n := Y p Hom Ab ( X p , Y n + p ) and differenti al d X,Y ( f ) := d Y ◦ f − ( − 1) deg f f ◦ d X . Assigning the morphism s to b e the group of maps of complexes Hom C ( Ab ) ( X, Y ) := Z 0 H om C ( Ab ) ( X, Y ) ∗ . giv es us the additiv e category C ( Ab ). C ( Ab ) h as the shift fu nctor X 7→ X [1] with X [1] n := X n +1 and dif- feren tial d n X [1] := − d n +1 X , and for a morph ism f = { f n : X n → Y n } , f [1] : X [1] → Y [1] is defined as f [1] n := f n +1 . F or a morphism f : X → Y , we ha ve the cone complex Cone( f ) with Cone( f ) n := Y n ⊕ X n +1 and differen tial d :=  d Y f 0 d A [1]  . and the cone sequence X f → Y i → Cone( f ) p → X [1] . (1.1.1) T en s or pro du ct of complexes X ∗ ⊗ Y ∗ is defin ed b y ( X ∗ ⊗ Y ∗ ) n := ⊕ i + j = n X i ⊗ Y j , with differen tial give n b y th e Leibniz rule d ( a ⊗ b ) = da ⊗ b + ( − 1) deg a a ⊗ db. The comm utativit y constrain t τ X,Y : X ∗ ⊗ Y ∗ → Y ∗ ⊗ X ∗ is giv en by τ X,Y ( a ⊗ b ) = ( − 1) deg a · de g b b ⊗ a. This mak es C ( Ab ) into a tensor category . A pre-DG category C is a catego ry in whic h, for ob jects X , Y , one has the Hom complex H om C ( X, Y ) ∗ ∈ C ( Ab ) and for ob jects X , Y , Z in C , one has th e comp ositio n la w ◦ X,Y ,Z : H om C ( Y , Z ) ∗ ⊗ H om C ( X, Y ) ∗ → H om C ( X, Z ) ∗ . 11 1.1. Pr eliminaries The map ◦ X,Y ,Z is a map of complexes, that is, we ha v e th e Leibniz rule, d ( f ◦ g ) = d f ◦ g + ( − 1) deg f f ◦ dg . Also, the map ◦ X,Y ,Z is associativ e and there is an iden tit y elemen t id X ∈ Z 0 H om C ( X, X ) 0 . A DG category is a pre-DG category admitting finite direct sums. If C is a (p r e-)DG category , w e ha ve the opp osite (pre-)DG catego ry C op with H om C op ( X, Y ) n := H om C ( Y , X ) − n . Letting f op : X → Y b e the morphism in C op corresp ondin g to a morph ism f : Y → X in C , the differen tial d op is given by d op ( f op ) = ( d f ) op and the comp osition la w is f op ◦ g op := ( − 1) deg f · deg g ( g ◦ f ) op . W e h a ve the DG cate gory C dg ( Ab ) with Hom-co mplexes as d efined ab o v e and comp osition la w ind u ced by the comp osition la w in Ab . A DG fu nctor F : A → B is a fu nctor of the underlying pre-additive catego ries suc h that the map F : H om A ( X, Y ) ∗ → H om B ( F ( X ) , F ( Y )) ∗ is a map of complexes for eac h pair of ob jects X , Y in A . Supp ose that A and B are small DG categories. A degree n natural transformation of DG functors F , G : A → B , ϑ : F → G, is a collectio n of elemen ts ϑ ( X ) ∈ H om B ( F ( X ) , G ( X )) n ; X ∈ A suc h that, for eac h f ∈ H om A ( X, Y ) m , one has G ( f ) ◦ ϑ ( X ) = ( − 1) mn ϑ ( Y ) ◦ F ( f ) . F or ϑ : F → G a degree n natural transf orm ation, w e hav e the d egree n + 1 natural transform ation dϑ : F → G with dϑ ( X ) := d ( ϑ ( X )); X ∈ A , giving us the complex N at ( F , G ) ∗ := H om D GF un ( A , B ) ( F , G ) ∗ of natural tr ansformations. T he comp ositi on in B induces a comp osition la w N at ( G, H ) ∗ ⊗ N at ( F, G ) ∗ → N at ( F, H ) ∗ giving us the DG category of DG fun ctors D GF un ( A , B ). F or a DG category C , one has the additiv e categ ories Z 0 C and H 0 C with the same ob jects as C but with m orphisms Hom Z 0 C ( X, Y ) := Z 0 H om C ( X, Y ) ∗ and Hom H 0 C ( X, Y ) := H 0 H om C ( X, Y ) ∗ . 12 1.1. Pr eliminaries A DG functor F : A → B induces fun ctors of additiv e categ ories Z 0 F : Z 0 A → Z 0 B and H 0 F : H 0 A → H 0 B . This mak es Z 0 and H 0 in to fun ctors from DG categories to add itive catego ries. 1.1.2. The category Pre-T r ( C ) of DG complexes. Definition 1.1.1. Let C b e a DG cate gory . The DG category Pr e-T r( C ) has ob jects E consisting of the follo win g d ata: (1) A finite collection of ob jects of C , { E i , N ≤ i ≤ M } ( N and M dep end ing on E ). (2) Morph isms e ij : E j → E i in C of degree j − i + 1, N ≤ i, j ≤ M , satisfying ( − 1) i de ij + X k e ik e k j = 0 . F or ob jects E := { E i , e ij } , F := { F i , f ij } , a morphism ϕ : E → F of degree n is a col lection of morphisms ϕ ij : E j → E i of degree n + j − i in C . Comp osition of morphisms ϕ : E → F , ψ : F → G is defined by ( ψ ◦ ϕ ) ij := X k ψ ik ◦ ϕ k j . Giv en a morphism ϕ : E → F of degree n , d efine ∂ F , E ( ϕ ) ∈ Hom Pre-T r( C ) ( E , F ) n +1 as the collectio n ∂ F , E ( ϕ ) ij := ( − 1) i d ( ϕ i j ) + X k f ik ϕ k j − ( − 1) n X k ϕ ik e k j . F or an ob ject E in Pre-T r( C ) and n ∈ Z , we defin e an ob ject E [ n ] b y ( E [ n ]) i := E n + i , e [ n ] ij := ( − 1) n e n + i,n + j : ( E [ n ]) j → ( E [ n ]) i . Let E , F b e ob jects in Pre-T r( C ) and ψ ∈ Z 0 H om Pre-T r( C ) ( E , F ). W e define an ob ject C one ( ψ ) in Pre-T r( C ) as C one ( ψ ) i := F i ⊕ E [1] i cone ( ψ ) ij :=  f ij ψ i,j +1 0 e [1] ij  : C on e ( ψ ) j → C one ( ψ ) i W e ha v e the cone sequence of morphisms in Z 0 Pre-T r ( C ): E ψ − → F i − → C one ( ψ ) p − → E [1] with i and p the eviden t inclusion and pro jection. Prop osition 1.1.1 ([ Bon09 ], Pr op osition 2.2.3 ) . L et T r ( C ) := H 0 Pr e-T r ( C ) b e the homot opy c ate gory of Pr e- T r ( C ) . Then, T r ( C ) is a triangulate d c ate- gory, with tr anslation functor induc e d by the tr anslation define d ab ove, and with distinguishe d triangles, those triangles which ar e isomorphic to the i m- age of a c one se quenc e. W e hav e an inclusion i 0 : C ֒ → Pre-T r( C ) giv en by i 0 ( E ) = { E 0 = E ; e 00 = 0 } . Let C pretr b e the smallest full sub cat egory of Pre-T r ( C ) conta ining i 0 ( C ) and 13 1.1. Pr eliminaries closed under taking translatio ns and cones. This giv es a fully faithful em- b eddin g i : C → C pretr , since H om C pretr ( i ( E ) , i ( F )) ∗ = H om C ( E , F ) ∗ Clearly H 0 C pretr is a full triangulated sub categ ory of H 0 Pre-T r ( C ). 1.1.3. DG categories and tria ngulated categories. In this section, w e recall some basic constructions in volving DG categories and triangulated catego ries. Let C b e a DG category . Definition 1.1.2. F or a DG category C , we define a C -mo d ule to b e a DG functor M : C → C dg ( Ab ). W e d en ote b y C dg ( C ) the DG ca tegory of C op -mo dules: C dg ( C ) := D GF un ( C op , C dg ( Ab )) . The op erations of trans lation and cones in C dg ( Ab ) ind u ce these op er- ations in the fu nctor category C dg ( C ): M [1]( E ) := M ( E )[1] and for ψ : M → N a morp hism in Z 0 C dg ( C ), C one ( ψ )( E ) := C one ( ψ ( E )) Similarly , we ha v e the cone sequ ence M ψ − → N i − → C one ( ψ ) j − → M [1] with v alue at E ∈ C the cone sequence M ( E ) ψ ( E ) − − − → N ( E ) i − → C one ( ψ ( E )) j − → M ( E )[1] in Z 0 C dg ( Ab ). Let K ( C ) := H 0 C dg ( C ) b e the homotopy category . With translation functor ind uced from that of C dg ( C ) and distinguished triangles those se- quences isomorph ic to the image of a cone sequence, K ( C ) b ecomes a trian- gulated category (see [ Kel94 ], § 2.2 ). Let AcK ( C ) ⊂ K ( C ) b e the full su b category with ob jects the fu nctors M such that M ( A ) is acyclic for all A ∈ C , that is, H p ( M ( A )) = 0 for all p . It is easy to see that AcK ( C ) is a thick sub ca tegory of K ( C ). Definition 1.1.3 ([ Kel94 ], § 4.1) . Th e d eriv ed catego ry D ( C ) is the local- ization of K ( C ) with resp ect to qu asi-isomorphisms, that is, D ( C ) = K ( C ) / AcK ( C ) . A represen table C op -mo dule is a functor A ∨ := H om C ( , A ) for some ob ject A ∈ C . S ending A to A ∨ defines a fu lly faithful embed d ing of DG catego ries C → C dg ( C ) . 14 1.1. Pr eliminaries Let C b dg ( C ) b e the smallest full DG su b category of C dg ( C ), con taining all the ob jects A ∨ and closed u nder translations and cones. Let K b ( C ) := H 0 C b dg ( C ) b e its homotop y catego ry , giving us a fu ll triangulated sub ca tegory of K ( C ). W e let K b ( C ) ess ⊂ K ( C ) b e the full su b category of K ( C ) w ith ob ject s those ob jects of K ( C ) which are isomorph ic to an ob ject of K b ( C ). Clearly K b ( C ) ess is a full triangulated s ub category of K b ( C ) and the inclusion K b ( C ) → K b ( C ) ess is an equiv alence of triangulated categories. W e h av e a fu lly faithful em b edding i pretr : C pretr → C dg ( C ) giv en by E 7→ H om C pretr ( i ( ) , E ) ∗ The comp osition i pretr ◦ i is th e fu nctor A 7→ A ∨ . Prop osition 1.1.2 ([ Lev09 ], Prop. 2.12) . 1. The e mb e dding i pretr induc es an e quivalenc e of DG c ate gories ϕ : C pretr − → C b dg ( C ) and an e q u ivalenc e of triangulate d c ate gories H 0 ϕ : H 0 C pretr − → K b dg ( C ) 2. The evident functor K b dg ( C ) → D ( C ) is a ful ly faithful emb e dding. Recall that a quasi-equiv alence of DG categories is a DG fu nctor F : A → B whic h is surjectiv e on isomorphism classes and for eac h pair of ob jects X , Y ∈ A , the map F A,B : H om A ( X, Y ) ∗ → H om B ( F ( X ) , F ( Y )) is a quasi-isomorphism. Clearly , a quasi-equiv alence indu ces an equiv alence on the h omotop y ca tegories H 0 F : H 0 A → H 0 B . Prop osition 1.1.3 ([ T o¨ e07 , prop osition 3.2] see also [ Lev09 , theorem 2.14]) . A quasi-e qu ivalenc e F : A → B induc es an e qui valenc e of triangulate d c ate- gories K b dg ( F ) : K b dg ( A ) → K b dg ( B ) . 15 1.1. Pr eliminaries 1.1.4. Sheafification of DG categories. Let n 7→ A n , n 7→ B n b e cosimplicial ab elian groups, giving us the diagonal cosimplicial ab elian group n 7→ A n ⊗ B n , which we denote as n 7→ ( A ⊗ B ) n . The Alexander-Whitney map is a qu asi-isomorphism of complexes AW : ( A ∗ , d ) ⊗ ( B ∗ , d ) → (( A ⊗ B ) ∗ , d ) and is defined as follo w s: for eac h pair ( p, q ), let δ 1 p,p + q : [ p ] → [ p + q ] and δ 2 q ,p + q : [ q ] → [ p + q ] b e the maps δ 1 p,p + q ( i ) = i, δ 2 q ,p + q ( j ) = p + j. The sum of the maps A ( δ 1 p,p + q ) ⊗ B ( δ 2 q ,p + q ) : A p ⊗ B q → A p + q ⊗ B p + q is easily seen to giv e the desired m ap of complexes AW . W e will not n eed the fact that AW is a quasi-isomorphism. See [ W ei94 ], § 8.5.4 for more details. If no w n 7→ ( A ∗ ) n is a cosimplicial ob ject in C ( Ab ) (a cosimplicial complex), th e cosimplicial structure give s a double complex A ∗∗ with second differen tial coming from the cosimplicial s tructure, and the asso ciated total complex T ot( A ∗∗ ). Giv en t wo cosimplicial complexes n 7→ ( A ∗ ) n , n 7→ ( B ∗ ) n , th e A W map giv es a map AW A,B : T ot( A ∗∗ ) ⊗ T ot( B ∗∗ ) → T ot(( A ⊗ B ) ∗ ) ∗ ) , (1.1.2) ho wev er, one needs to in tro duce a sign: f or a ∈ ( A p ) n , b ∈ ( B q ) m AW A,B ( a ⊗ b ) = ( − 1) q n AW A p ,B q ( a ⊗ b ) . F ollo w ing [ Lev09 ], w e use the Go d emen t resolution to giv e a go o d global mo del for a sheaf of DG categories. In particular, for a complex of sh eav es F ∗ on a top ological space X , w e h a ve the cosimplicial Go dement resolution n 7→ G ∗ ( F ∗ ) and the asso ciated total complex of shea v es on X T ot( G ∗ ( F ∗ )). Giv en shea ves F 1 , F 2 on X , the naturalit y of the Go d emen t resolution giv es us a canonical map of cosimplicial sh eav es n 7→ ∪ n ( F 1 , F 2 ) : G n ( F 1 ) ⊗ G n ( F 2 ) → G n ( F 1 ⊗ F 2 ) . F or complexes of s hea ves F ∗ 1 , F ∗ 2 , one extends this to a map of cosimplicial complexes by in tro ducing the appr opriate signs: ∪ n ( F ∗ 1 , F ∗ 2 ) a,b : G n ( F a 1 ) ⊗ G n ( F b 2 ) → G n ( F a 1 ⊗ F b 2 ) ∪ n ( F ∗ 1 , F ∗ 2 ) a,b := ( − 1) na ∪ n ( F a 1 , F b 2 ) Comp osing with the signed Alexa nder-Whitney map ( 1.1.2 ) giv es us the map of complexes AW : T ot( G ∗ ( F ∗ 1 )) ⊗ T ot ( G ∗ ( F ∗ 2 )) → T ot( G ∗ ( F ∗ 1 ⊗ F ∗ 2 )) . (1.1.3) 16 1.2. C u bical enric hment s in DG categorie s 1.2. Cubical enric hmen ts in DG categories 1.2.1. Cubical categories. W e first d efine the “cubical category” Cub e . It is a sub categ ory of Set s with ob jects n := { 0 , 1 } n , n ≥ 0 an integer, and morphisms generated by (1) In clusions: η n,i,ǫ : n → n + 1 , ǫ ∈ { 0 , 1 } , 1 ≤ i ≤ n + 1, η n,i,ǫ ( y 1 , . . . , y n ) = ( y 1 , . . . , y i − 1 , ǫ, y i , . . . , y n ) (2) Pr o jectio ns: p n,i : n → n − 1 , 1 ≤ i ≤ n , p n,i ( y 1 , . . . , y n ) = ( y 1 , . . . , y i − 1 , y i +1 , . . . , y n ) (3) Perm utation of factors: ( y 1 , . . . , y n ) 7→ ( y σ (1) , . . . , y σ (1) ) for σ ∈ S n . (4) Inv olutions: τ n,i : n → n exc hanging 0 and 1 in the i th factor. Definition 1.2.1. F or a category C , we call a functor F : Cub e op → C a cubical ob ject of C and a f unctor G : C ub e → C a co-cubical ob ject of C . Remark 1.2.2. 1. Defining a morp hism of (co)-cubical ob jects to b e a n at- ural transformation give s us the category of (co) cubical ob jects in a (small) catego ry C . 2. Replacing Cub e with the k -fold pro d uct catego ry Cub e k giv es us the catego ries of k -cubical ob jects in C and the category of k -co-cubical ob jects in C . 3. The pr o duct of sets mak es Sets a symmetric monoidal category of whic h Cub e is a symmetric monoidal sub categ ory . 1.2.2. Cube s and complexes. Let A b e a pseud o-ab elian category and A : Cub e op → A a cubical ob ject. Let ( A ∗ , d ) b e the complex with A n := A ( n ) and d n := n X i =1 ( − 1) i ( η ∗ n,i, 1 − η ∗ n,i, 0 ) : A n +1 → A n . F or ǫ ∈ { 0 , 1 } , d efine π ǫ n,i := p ∗ n,i ◦ η ∗ n,i,ǫ : A ( n ) → A ( n ) and let π n,m := ( id − π 1 n,m ) ◦ · · · ◦ ( id − π 1 n, 1 ) Note that π ǫ n,i are comm uting idemp otent s, and that ( id − π ǫ n,i )( A ( n )) ⊂ k er η ∗ n − 1 ,i,ǫ since p n,i ◦ η n − 1 ,i,ǫ = id . Let I = ( i 1 , . . . , i m ) b e an m tup le of inte gers 1 ≤ i 1 < . . . < i m ≤ n , and let p n,I : n → m b e the corresp ond ing pro jection. Let η n,I : m → n 17 1.2. C u bical enric hment s in DG categorie s b e the inclusion defined by η m,I ( ǫ 1 , . . . , ǫ m ) = ( η m,I ( ǫ ∗ ) 1 , . . . , η m,I ( ǫ ∗ ) n ) η m,I ( ǫ ∗ ) i = ( ǫ j if i = i j for some j 0 else. The collectio n of maps { p n,I } for fixed n form an n -cub e of m aps, which is compatibly split by the maps { η n,I } , in th e sense of [ GL01 , § 5.6]. The follo wing result follo w s d irectly fr om [ GL01 , prop osition 5.7]. Lemma 1.2.1. L et A : Cub e op → A b e a cubi c al obje ct in a pseudo-ab elian c ate gory A . 1. F or n ≥ 1 and 1 ≤ m ≤ n , ther e ar e wel l wel l-define d obje c ts A 0 n,m := ∩ m i =1 k er η ∗ n − 1 ,i, 1 ⊂ A ( n ) , A degn n,m := m X i =1 p ∗ n,i ( A ( n − 1)) ⊂ A ( n ) . We set A 0 n := A 0 n,n , A degn n := A degn n,n . 2. F or e ach n, m , π n,m maps A ( n ) to A 0 n,m and defines a splitting A ( n ) = A degn n,m ⊕ A 0 n,m . 3. d n ( A degn n +1 ,m ) = 0 , d n ( A 0 n +1 ) ⊂ A 0 n Definition 1.2.3. F or a cubical ob ject A : C ub e op → A in a pseudo-ab elian catego ry A , define th e complex ( A ∗ , d ) to b e A ∗ := A ∗ /A degn ∗ . Lemma 1.2.1 sho ws th at A ∗ is w ell-defined and is isomorphic to the sub complex ( A 0 ∗ , d ) of ( A ∗ , d ). Note that d n = P n i =1 ( − ) i − 1 η ∗ n,i, 0 on A n +1 . W e h av e the map of complexes λ : A 0 → A ∗ viewing A 0 as a complex concen trated in degree 0. W e no w tak e A to b e the category of complexes in an ab elian ca tegory A 0 (with some b oundedness condition b, + , − , ∅ . App lyin g the total complex functor to the map λ giv es us the map λ : A 0 → T ot( A ∗ ) . (1.2.1) Lemma 1.2.2. Supp ose that for e ach n , the map i n : 0 → n with image 0 n induc es a quasi-isomorph ism A ( i n ) : A n → A 0 Then the map ( 1.2.1 ) is a quasi-i somorphism. 18 1.2. C u bical enric hment s in DG categorie s Pr oof. F or eac h m , let A ( m ) ( n ) := A ( n ) / X I =( i 1 ,...,i m ) p ∗ n,I ( A ( m )) . This defin es A ( m ) : Cub e → A with A ( m ) ( n ) = 0 for n ≤ m and with ( A ( m ) ) n = A n for n > m . F or m = 0, th e unique p ro jection p n : n → 0 is split (by any inclus ion i n : 0 → n ), so w e ha v e the dir ect sum d ecomp osition of cubical ob jects A = A 0 ⊕ A (0) where we consider A 0 as a constan t cubical ob ject. Since p ∗ n : A 0 → A ( n ) is a quasi-isomorphism, it thus follo ws that A (0) ( n ) is acyclic for all n . W e no w sh ow by indu ction on m that A ( m ) ( n ) is acyclic for all n and m . Indeed, by construction, the sum P I =( i 1 ,...,i m ) p ∗ n,I ( A ( m − 1 ) ( m )) in A ( m − 1 ) ( n ) is a direct su m of copies of A ( m − 1 ) ( m ). As A ( m − 1 ) ( m ) and A ( m − 1 ) ( n ) are acyclic by the in d uction assump tion, the induction goes thr ough. In particular, the complex A n := A ( n − 1) ( n ) is acyclic for n > 0, and thus the total complex of the double complex A ∗ , ∗ ≥ 1, is acyclic. As this latter complex is q u asi-isomorphic to the cone of the map λ : A 0 → T ot( A ∗ ), the map λ is a quasi-isomorphism, as claimed.  If w e hav e tw o cub ical ob jects A , B : Cub e op → C in a tensor cat egory C , w e can defin e a diagonal ob ject A ⊗ B as A ⊗ B ( n ) := A ( n ) ⊗ B ( n ) , and on morphism s by A ⊗ B ( f ) := A ( f ) ⊗ B ( f ) . Let p 1 n,m : m + n → n and p 2 n,m : m + n → m b e the pro jections on to the first n and last m f actors resp ectiv ely . Let ∪ n,m A ,B : A ( n ) ⊗ B ( m ) → A ( n + m ) ⊗ B ( n + m ) b e the m ap A ( p 1 n,m ) ⊗ B ( p 2 n,m ). T aking direct sum of ∪ n,m A ,B o ver n, m yields a map of complexes ∪ A,B : A ∗ ⊗ B ∗ → A ⊗ B ∗ (1.2.2) with an asso ciativit y prop ert y ∪ A ⊗ B ,C ◦ ( ∪ A,B ⊗ id C ∗ ) = ∪ A,B ⊗ C ◦ ( id A ∗ ⊗ ∪ B ,C ) 19 1.2. C u bical enric hment s in DG categorie s 1.2.3. Multiplications and co-m ultiplications. In this s ection, w e fix a co-cubical ob ject n 7→  n (denoted  ∗ ) in a tens or category ( C , ⊗ ), suc h that  0 is the unit ob ject with resp ect to ⊗ . Definition 1.2.4. A multiplic ation µ on  ∗ is a collectio n of morp hisms µ n,m :  n ⊗  m →  n + m whic h are (1) b i-natural: Let f : n → p b e a morphism in C ub e , giving the morphism f × id : n + m → p + m . Then the diagram  n ⊗  m µ n,m / / f ⊗ id    n + m f × id    p ⊗  m µ p,m / /  p + m comm utes. (2) asso ciativ e: The diagram  p ⊗  n ⊗  m µ p,n ⊗ id / / id ⊗ µ n,m    p + n ⊗  m µ p + n,m    p ⊗  n + m µ p,n + m / /  p + n + m comm utes. (3) commutati v e: Let τ n,m : n + m → n + m b e the m orphism in Cub e defined as the comp osition n + m = n × m σ − → m × n = n + m where σ is the symmetry isomorph ism in S ets . Let t n,m :  n ⊗  m →  m ⊗  n b e the s y m metry isomorph ism in the tensor cate- gory C . Th en the diagram  n ⊗  m t n,m   µ n,m / /  n + m τ n,m    m ⊗  m µ m,n / /  n + m comm utes. (4) u nital: Let µ n :  0 ⊗  n →  n b e the iden tit y isomorphism in C . Then the comp osition  n µ − 1 n − − →  0 ⊗  n µ 0 ,n − − →  n is th e identi t y . 20 1.2. C u bical enric hment s in DG categorie s Definition 1.2 .5. Let  ∗ ⊗  ∗ b e th e diagonal co-cubical ob ject n 7→  n ⊗  n . A co-m ultiplication δ ∗ on  ∗ is a morphism of co-cubical ob jects δ ∗ :  ∗ →  ∗ ⊗  ∗ suc h that δ ∗ is (1) co-asso ciativ e: T he diagram  ∗ δ ∗ / / δ ∗    ∗ ⊗  ∗ id ⊗ δ ∗    ∗ ⊗  ∗ δ ∗ ⊗ id / /  ∗ ⊗  ∗ ⊗  ∗ (2) co-comm utativ e: Let t b e the commuta tivit y constrain t in ( C , ⊗ ). Then the comp osition  ∗ δ ∗ − →  ∗ ⊗  ∗ t  ∗ ,  ∗ − − − − →  ∗ ⊗  ∗ is th e identi t y . (3) co-unital: Let p n : n → 0 := { 0 } b e the pro jection. The comp osi- tion  n δ n − →  n ⊗  n p n ⊗ id − − − − →  0 ⊗  n µ n − →  n is th e identi t y . Let p 1 n,m : n + m → n, p 2 n,m : n + m → m b e the pr o jectio ns on the first n (resp. the last m ) factors. Given a co- m u ltiplication δ ∗ on  ∗ , we ha ve the maps δ n,m :  n + m →  n ⊗  m defined as the comp ositio n  n + m δ n + m − − − →  n + m ⊗  n + m p 1 n,m ⊗ p 2 n,m − − − − − − − →  n ⊗  m . Definition 1.2.6. A b i-m u ltiplicatio n on  ∗ is a m u ltiplication µ ∗∗ and a co-m ultiplication δ ∗ on  ∗ suc h th at µ n,m and δ n,m are in v erse isomorph isms, for all n, m ≥ 0. Remark 1.2.7. Clea rly a co-cubical ob ject  ∗ with a bi-m ultiplication ( µ ∗∗ , δ ∗ ) is canonically isomorph ic, as a co-cubical ob ject with bi-multiplica tion, to the co-cubical ob ject n 7→ (  1 ) ⊗ n with a b i-m u ltiplication of the form (id , ˜ δ ∗ ). 21 1.2. C u bical enric hment s in DG categorie s 1.2.4. Extended cube s. Definition 1.2.8. Let ECub e b e the smallest symmetric monoidal s u b- catego ry of Sets ha vin g the same ob jects as Cub e , con taining Cub e and con taining th e morphism m : 2 → 1 defined by m ultiplication of inte gers: m ((1 , 1 )) = 1; m (( a, b )) = 0 for ( a, b ) 6 = (1 , 1) . An extended cubical ob ject in a category C is a fu n ctor F : ECub e op → C , and an extended co-cubical ob ject in C is a functor F : ECub e → C . If  ∗ is an extended co-cubical ob ject in a tensor catego ry C , a m ulti- plication, resp. c o-m ultiplication, r esp . b i-m u ltiplication on  ∗ is defined as for a co-cubical ob ject in C , with all fun ctorialities and naturalities ex- tending to ECub e . Concretely , a co-m ultiplicatio n δ ∗ :  ∗ →  ∗ ⊗  ∗ is required to b e a morp hism of extended co-cubical ob jects and a m ultiplica- tion is required to s atisfy the bi-n atur alit y condition of defin ition 1.2.4 (1) with resp ect to all m orphisms in ECub e . 1.2.5. DG categories asso ciated with cubical categories. Th e catego ry of cub ical ab elian groups Ab Cube op carries the s tr ucture of a symmetric monoidal catego ry in the follo wing wa y: If w e ha v e tw o cubi- cal ab elian group s n 7→ A ( n ) , n 7→ B ( n ), the tensor pr o duct A ⊗ B is the cubical ab elia n group n 7→ A ( n ) ⊗ B ( n ), with morphisms acting by g ( a ⊗ b ) = g ( a ) ⊗ g ( b ) . A cubical category is a catego ry C enric h ed with cubical ab elian groups. Explicitly , for ob jects X , Y in C , we hav e cub ical ab elian groups H om ( X, Y , ) : Cub e op → Ab n 7→ H om C ( X, Y , n ) with the follo wing p rop erty: F or eac h ob ject X in C , w e ha v e an elemen t id X ∈ H om C ( X, X , 0) and we ha ve an asso ciativ e comp osition law, for ob jects X, Y , Z , ◦ X,Y ,Z : H om ( Y , Z, ) ⊗ H om ( X , Y , ) → H om ( X, Z, ) with f ◦ X,X ,Z id X = f and id Z ◦ X,Z ,Z g = g . There is a fun ctor C 7→ C 0 from cubical cate gories to pre-additive cate- gories, where C 0 has the same ob jects as C and H om C 0 ( X, Y ) := H om C ( X, Y , 0) . A cu bical enric h men t of a pre-additiv e category C is a cubical category ˜ C with an isomorphism C ≃ ˜ C 0 . W e can associate a DG cate gory to a cub ical categ ory C . F or ob jects X, Y in C , let H om dg C ( X, Y ) ∗ b e the non-degenerate complex H om C ( X, Y ) ∗ /H om C ( X, Y ) ∗ degn 22 1.3. More on DG categories and cubical categories asso ciated to the cubical ab elian group n 7→ H om C ( X, Y , n ). W e hav e the comp osition law ◦ X,Y ,Z : H om dg C ( Y , Z ) ∗ ⊗ H om dg C ( X, Y ) ∗ → H om dg C ( X, Z ) ∗ induced by ◦ X,Y ,Z and the pro d uct ( 1.2.2 ). It is easy to c hec k that the complexes H om dg C ( X, Y ) ∗ together with the ab o v e comp osition la w defin es a DG catego ry dg C with the s ame ob jects as C . W e no w sho w ho w to construct a cubical category and hence a DG catego ry from a tensor category with a co-cubical ob ject. Let  ∗ b e a co-cubical ob ject with a co-multiplic ation δ . Defining H om ( X, Y , n ) := H om C ( X ⊗  ∗ , Y ) giv es a cubical abelian group n 7→ H om ( X, Y , n ). Let H om ( X, Y ) ∗ b e the asso ciated complex. The co-m ultiplication gives a map ◦ X,Y ,Z : H om C ( Y , Z, ∗ ) ⊗ H om C ( X, Y , ∗ ) → H om C ( X, Z, ∗ ) b y sending f ⊗ g ∈ H om C ( Y , Z, n ) ⊗ H om C ( X, Y , n ) to the morphism X ⊗  n id X ⊗ δ n r ed − → X ⊗  n ⊗  n g ⊗ id  n − → Y ⊗  n f → Z Prop osition 1.2.3. L et ( C , ⊗ ) b e a tensor c ate gory with a c o-cubic al ob- je ct  ∗ and a c o- multiplic ation δ on  ∗ . Then, the cubic al ab elian gr oup H om ( X, Y , ) with the c omp osition law ◦ X,Y ,Z define d ab ove describ es a cu- bic al enrichment of C . Th us, follo wing the m etho d d escrib ed ab o ve, we get a DG catego ry dg C = ( C , ⊗ ,  ∗ , δ ∗ ). 1.3. More on DG categories and cubical categories In this section, we give some v ariations on the theme of cubical categories and DG cate gories discu s sed in § 1.2.5 . Throughout this section, we fix an tensor category C with a co-cubical ob jects  ∗ ha vin g a co-m ultiplication δ ∗ 1.3.1. Homotop y in v aria nce. Let p 1 :  1 →  0 b e the map induced b y 1 → 0 in Cub e . F or X ∈ M , let p X : X ⊗  1 → X b e th e comp osition X ⊗  1 id X ⊗ p 1 − − − − − → X ⊗  0 µ X − − → X where µ X : X ⊗  0 → X is the u nit isomorph ism in C . Prop osition 1.3.1. Supp ose that  ∗ is an extende d c o-cubic al obje c t of C and that δ ∗ extends to a bi -multiplic ation ( µ ∗∗ , δ ∗ ) on the extende d c o-cub e  ∗ . Then the map p ∗ X : H om dg C ( X, Y ) ∗ → H om dg C ( X ⊗  1 , Y ) ∗ is a ho motopy e quiv alenc e. 23 1.3. More on DG categories and cubical categories Pr oof. W e ha v e the map i 0 := η 0 , 1 , 0 : 0 → 1, sending 0 = 0 to 0 ∈ 1. W e h a ve the map i X := id X ⊗ i 0 : X → X ⊗  1 . Clearly p X ◦ i X = id X , hence i ∗ X ◦ p ∗ X = id. T o complete the pro of, it suffices to sh ow that p ∗ X ◦ i ∗ X is homotopic to the identit y . F or this, recall th e multiplicat ion map m : 2 → 1 m (1 , 1) = 1, m ( a, b ) = 0 if ( a, b ) 6 = (1 , 1). Consider the map q n :  1 ⊗  n +1 →  1 ⊗  n defined as the comp ositio n  1 ⊗  n +1 id ⊗ δ 1 ,n − − − − →  1 ⊗  1 ⊗  n µ 1 , 1 ⊗ id − − − − − →  2 ⊗  n m ⊗ id − − − →  1 ⊗  n . W e th us h a ve the map h n := q ∗ n : H om dg C ( X ⊗  1 , Y ) n → H om dg C ( X ⊗  1 , Y ) n +1 , whic h we claim giv es a homotop y b etw een the identit y and p ∗ X ◦ i ∗ X . T o pr o ve this, w e n ote the follo wing iden tities (w e ident ify  0 ⊗  a and  a ⊗  0 with  a via the un it isomorphism) (1) Let f : n → m b e a morph ism in Cub e and let f 1 := id 1 × f . Th en q m ◦ (id ⊗ f 1 ) = (id ⊗ f ) ◦ q n . In p articular, for i ≥ 2, n ≥ 1 and ǫ ∈ { 0 , 1 } , w e ha v e q n ◦ (id ⊗ η n,i,ǫ ) = (id ⊗ η n − 1 ,i − 1 ,ǫ ) ◦ q n − 1 . (2) q n ◦ (id ⊗ η n, 1 , 1 ) = id (3) q n ◦ (id ⊗ η n, 1 , 0 ) = ( i 0 ◦ p 1 ) ⊗ id In the additive categ ory generated by Cub e (whic h is a tensor category with pr o duct × ), th is giv es the iden tit y q n ◦ (id ⊗ n X i =1 ( − 1) i ( η n,i, 1 − η n,i, 1 ) + (id × n − 1 X i =1 ( − 1) i ( η n − 1 ,i, 1 − η n − 1 ,i, 1 ) ◦ q n − 1 = q n ◦ (id × η n, 1 , 0 − id ⊗ η n, 1 , 1 ) + q n ◦ (id × n X i =2 ( − 1) i ( η n,i, 1 − η n,i, 1 )) + (id × n − 1 X i =1 ( − 1) i ( η n − 1 ,i, 1 − η n − 1 ,i, 1 ) ◦ q n − 1 = q n ◦ (id × η n, 1 , 0 − id × η n, 1 , 1 ) + (id × n − 1 X i =1 ( − 1) i +1 ( η n − 1 ,i, 1 − η n − 1 ,i, 1 )) ◦ q n − 1 + (id × n − 1 X i =1 ( − 1) i ( η n − 1 ,i, 1 − η n − 1 ,i, 1 )) ◦ q n − 1 = ( i 0 ◦ p 1 − id 1 ) × id n Therefore, the maps h n giv es th e desired homotop y .  24 1.3. More on DG categories and cubical categories Remark 1.3.1. Let i X 0 : X → X ⊗  1 b e the map in d uced by i 0 := η 0 , 1 , 0 :  0 →  1 with image 0. As p X ◦ i X 0 = id X , it f ollo ws from pr op osition 1.3.1 that i ∗ X 0 : H om dg C ( X ⊗  1 , Y ) ∗ → H om dg C ( X, Y ) ∗ is a homotop y equiv alence, assuming that we ha ve a bi-multiplic ation on the extended cub e  ∗ . 1.3.2. Multi-cubes. Next, we see wh at happ ens when we replace a cub e with a multi-cub e. Let C b e a tensor category w ith a co-cubical ob ject  ∗ . F orm the k -cubical ob ject ( a 1 , . . . , a k ) 7→ H om C ( X ⊗  a 1 ⊗ . . . ⊗  a k , Y ) T aking the qu otient b y the degenerate elemen ts with resp ect to eac h v ariable giv es us the k -dimensional complex H om C ( X, Y ) ∗ 1 ,..., ∗ k and then the total complex H om C ( X, Y ) ∗ k := T ot H om C ( X, Y ) ∗ ,..., ∗ T ake an in teger k ′ with 1 ≤ k ′ < k . Ident ifying H om C ( X, Y ) ∗ 1 ,..., ∗ k ′ with the k ′ -dimension su b complex H om C ( X, Y ) ∗ 1 ,..., ∗ k ′ , 0 ,..., 0 of H om C ( X, Y ) ∗ 1 ,..., ∗ k induces the inclusion of total complexes λ k ′ ,k : H om C ( X, Y ) ∗ k ′ → H om C ( X, Y ) ∗ k . (1.3.1) Lemma 1.3.2. Supp ose that  ∗ is an extende d c o-cubic al obje ct with a bi- multiplic ation. Then for 1 ≤ k ′ < k , the map ( 1.3.1 ) is a quasi-isomorphism for al l X and Y in C . Pr oof. W e pr o ceed b y ind uction on k . F or k = 1 there is nothin g to pro v e and it suffices to pro v e the case k ′ = k − 1. Note that H om C ( X, Y ) ∗ k is isomorphic to the non-degenerate complex total complex of the complex asso ciated to the cubical ob ject n 7→ H om C ( X ⊗  n , Y ) ∗ k − 1 : H om C ( X, Y ) ∗ k ∼ = T ot [ H om C ( X ⊗  ∗ , Y ) ∗ k − 1 ] . In addition, via this identit y , the map λ k − 1 ,k is trans formed to the map ( 1.2.1 ) λ : H om C ( X, Y ) ∗ k − 1 → T ot[ H om C ( X ⊗  ∗ , Y ) ∗ k − 1 ] . By our indu ction hypothesis the map λ 1 ,k − 1 : H om C ( X ⊗  n , Y ) ∗ 1 → H om C ( X ⊗  n , Y ) ∗ k − 1 is a quasi-isomorphism for all X , Y and n . Also, H om C ( X ⊗  n , Y ) ∗ 1 = H om dg C ( X ⊗  n , Y ) ∗ , so by p rop osition 1.3.1 , the map p ∗ n : H om C ( X, Y ) ∗ 1 → H om C ( X ⊗  n , Y ) ∗ 1 25 1.3. More on DG categories and cubical categories is a quasi-isomorphism for all n . Our induction hyp othesis thus imp lies that p ∗ n : H om C ( X, Y ) ∗ k − 1 → H om C ( X ⊗  n , Y ) ∗ k − 1 is a quasi-isomorphism for all n . By lemma 1.2.2 the map λ is a qu asi- isomorphism, hence λ k − 1 ,k is a quasi-isomorphism.  1.3.3. The ext ended DG category. Next, w e lo ok at what hap p ens when w e add cub es to the target. Let C b e an additiv e cat egory with a co- cubical ob ject n 7→  n . W e define a n ew DG category dg e C , with the same ob jects as C . The Hom complexes are defin ed as follo ws: F or eac h m , w e hav e the non-degenerate complex H om dg C ( X, Y ⊗  m ) ∗ ; let H om C ( X, Y ⊗  m ) ∗ 0 b e the sub complex consisting of f suc h that p m,i ◦ f = 0 ∈ H om C ( X, Y ⊗  m − 1 ) ∗ ; i = 1 , . . . , m. Let H om dg e C ( X, Y ) p := Y m − n = p H om dg C ( X, Y ⊗  m ) − n 0 ; for f := ( f n ∈ H om C ( X, Y ⊗  p + n ) − n 0 ), d efi ne d f = (( d f ) n ∈ H om C ( X, Y ⊗  n + p +1 ) − n 0 ) w ith ( d f ) n := n X i =1 ( − 1) i f n +1 ◦ ( η n,i, 1 − η n,i, 0 ) − ( − 1) p n + p X i =1 ( − 1) i ( η n + p,i, 1 − η n + p,i, 0 ) ◦ f n , whic h we write as ( d f ) n = f n +1 ◦ d X − ( − 1) p d Y ◦ f n ; d f := f ◦ d X − ( − 1) p d Y ◦ f . The comp ositi on ( g m ) ◦ ( f n ) w ith f n ∈ H om dg C ( X, Y ⊗  n + p ) − n 0 ⊂ H om C ( X ⊗  n , Y ⊗  n + p ) g m ∈ H om dg C ( Y , Z ⊗  m + q ) − m 0 ⊂ H om C ( Y ⊗  m , , Z ⊗  m + q ) is the s equ ence ( g n + p ◦ f n ), wh ere w e use the comp osition in C . One chec ks that this do es indeed d efine a DG category , whic h w e d en ote by dg e C . Remark 1.3.2. Note that contrary to the DG category dg C , w e did not require a co-m ultiplication to define the comp osition la w . No w assume th at  ∗ has a co-m ultiplication δ ∗ . W e d efine a DG fun ctor F : dg C → dg e C as follo ws : Supp ose w e are giv en f : X ⊗  n → Y . Define F ( f ) := ( F ( f ) m ), wh ere F ( f ) m : X ⊗  n + m → Y ⊗  m is the map d efined by the comp ositio n X ⊗  n + m δ n + m − − − → X ⊗  n + m ⊗  n + m p 1 n,m ⊗ p 2 n,m − − − − − − − → X ⊗  n ⊗  m f ⊗ id  m − − − − − → Y ⊗  m . One chec k s th at send ing f to F ( f ) defines a map of complexes F X,Y : H om dg C ( X, Y ) ∗ → H om dg e C ( X, Y ) ∗ 26 1.3. More on DG categories and cubical categories and is compatible with comp ositio n, giving us the DG fun ctor F : dg C → dg e C . (1.3.2) In many situations, th e functor F is a quasi-equiv alence of DG categories, that is, th e indu ced map F X,Y on the Hom-complexes is a quasi-isomorphism (in general, one also sup p oses th at F is a su r jection on isomorph ism classes, but as F is a b ij ection on ob jects, this is immediate). W e fi r st require a defin ition Definition 1.3.3. Let  ∗ b e a co-cubical ob ject in an additiv e cate gory C , w ith a co-m ultiplication δ , giving us the DG category dg C . Call (  ∗ , δ ) homotop y in v ariant if for al l n and all X , Y in C , the morp hism i n : 0 → n with image 0 n induces a quasi-isomorphism i n ∗ : H om dg C ( X, Y ) ∗ ∼ = H om dg C ( X, Y ⊗  0 ) ∗ → H om dg C ( X, Y ⊗  n ) ∗ Prop osition 1.3.3. Supp ose that  ∗ is homotopy invariant. Then the func- tor ( 1.3.2 ) F : dg C → dg e C is a quasi-e quivalenc e. Pr oof. Let π − n : H om dg e C ( X, Y ) − n → H om dg C ( X, Y ) − n b e the pr o jectio n on the H om C ( X ⊗  n , Y )-comp onen t; this giv es us the map of complexes π X,Y : H om dg e C ( X, Y ) ∗ → H om dg C ( X, Y ) ∗ with π X,Y ◦ F X,Y = id. T o show that F X,Y is a quasi-equiv alence, it su ffices to sho w th at, for a given elemen t g = ( g n ) ∈ H om dg e C ( X, Y ) p with dg = 0, w e can fi nd an h ∈ H om dg e C ( X, Y ) p − 1 with g − F ( π ( g )) = dh . F or this, w e note that g ′ := g − F ( π ( g )) has π ( g ′ ) = 0, that is, the H om C ( X ⊗  − p , Y )-comp onent of g ′ is zero. Let n > 0 b e th e minimal in teger su c h that the H om C ( X ⊗  n − p , Y ⊗  n )-comp onen t g ′ n of g ′ is non- zero. T hen the “ X -differen tial” d X g ′ n of g ′ n in H om C ( X ⊗  n − p − 1 , Y ⊗  n ) is zero, hence g ′ n defines a cohomology class in H n − p ( H om dg C ( X, Y ⊗  n ) ∗ ). Also, by definition, p n, 1 ( g ′ n ) = 0, bu t since  ∗ is h omotop y in v ariant, the map on cohomology induced b y p n, 1 is an isomorph ism. Thus, there is an elemen t h n ∈ H om dg C ( X, Y ⊗  n ) n − p − 1 with d X h n = g ′ n . Using the splittings η n − 1 ,i, 1 to p n,i , we can assume that h n is in the subgroup H om dg C ( X, Y ⊗  n ) n − p − 1 0 of of H om dg C ( X, Y ⊗  n ) n − p − 1 . W e view h n as an element of H om dg e C ( X, Y ) p − 1 b y taking all other comp onents to b e zero. No w we can replace with g ′ with g ′′ := g ′ − dh n giving u s a new elemen t suc h that the minimal m f or w hic h g ′′ has a non-zero H om C ( X ⊗  m − p , Y ⊗  m )-comp onen t has m > n . Rep eating, w e construct a sequence of elements h n ∈ H om C ( X, Y ⊗  n )) n − p − 1 0 with d ( h n ) = ( g ′ n ) in H om dg e C ( X, Y ) p , as desired.  27 1.3. More on DG categories and cubical categories 1.3.4. Extended multi-cubic al complexes. F or late r use, w e com- bine the extended total complex constr u ction with mult-cub es in th e source. Let H om C ( X, Y ⊗  m ) n k , 0 ⊂ H om C ( X, Y ⊗  m − 1 ) n k b e the int ersection of the k ernels of the maps p m,i : H om C ( X, Y ⊗  m ) n k → H om C ( X, Y ⊗  m − 1 ) n k i = 1 , . . . , m , and let H om C ( X, Y ) p k , ext := Y n + m = p H om C ( X, Y ⊗  m ) n k . W e note th at H om dg e C ( X, Y ) p = H om C ( X, Y ) p 1 , ext ; just as in the case k = 1, the differen tial d X in H om C ( X ⊗  m , Y ) ∗ k and the differen tial d Y formed using the co-cubical structur e m 7→ H om C ( X ⊗  m , Y ) ∗ k : F or f = ( f n ) ∈ H om C ( X, Y ) p k , ext , w ith f n ∈ H om C ( X, Y ⊗  p + n ) − n k , set ( d f ) n := f n +1 ◦ d X − ( − 1) p d Y ◦ f n . The maps ( 1.3.1 ) give rise to maps λ k ′ ,k , ext : H om C ( X, Y ) ∗ k ′ , ext → H om C ( X, Y ) ∗ k , ext (1.3.3) for 1 ≤ k ′ < k . Prop osition 1.3.4. Supp ose that  ∗ is an extende d c o- c ubic al obje ct with bi-multiplic ation. Supp ose further that  ∗ is homotopy invariant. Then the map ( 1.3.3 ) is a quasi-i somorphism. Pr oof. By our assump tion that  ∗ is homotop y inv ariant, together with lemma 1.3.2 , th e map p n ∗ : H om C ( X, Y ×  n ) ∗ k → H om C ( X, Y ) ∗ k is a quasi-isomorphism for all X , Y , k and n . The same pro of as w e used in prop osition 1.3.3 sho ws that the pro jection on the H om C ( X, Y ) ∗ k -factor giv es a quasi-isomorphism π k : H om C ( X, Y ) ∗ k , ext → H om C ( X, Y ) ∗ k . In add ition, the diagram H om C ( X, Y ) ∗ k ′ , ext λ k ′ ,k, ext   π k ′ / / H om C ( X, Y ) ∗ k ′ λ k ′ ,k   H om C ( X, Y ) ∗ k , ext π k / / H om C ( X, Y ) ∗ k comm utes; as the map s π k , π k ′ and λ k ′ ,k are quasi-isomorphisms, the map λ k ′ ,k , ext is a quasi-isomorphism as we ll.  28 1.4. DG categories of motiv es 1.4. DG categories of motiv es W e b r iefly recall Levin e’s construction of the DG category of s m o oth motiv es o v er a base. Let S b e a fi xed regular s c heme of fin ite Krull di- mension. Let S m /S b e the category of smo oth S -sc h emes of finite type and Pro j /S ⊂ S m /S b e the fu ll sub cate gory of S m /S consisting smooth pro jectiv e S -sc h emes. Definition 1.4.1. F or X , Y ∈ S m /S , the group of finite corresp ondences, C or S ( X, Y ), is d efined to b e the free ab elian group on the inte gral closed subschemes W ⊂ X × S Y suc h that th e pro jection W → X is fin ite and surjectiv e onto an irr educible comp onen t of X . The category C or S consists of the ob jects of S m /S and h as morph isms Hom C or S ( X, Y ) := C or S ( X, Y ) where the comp osition of corresp ondences ◦ : C or S ( X, Y ) ⊗ C or S ( Y , Z ) → C or S ( X, Z ) is defin ed as W ◦ W ′ := p X Z ∗ ( p ∗ X Y ( W ) · X Y Z p ∗ Y Z ( W ′ )) where · X Y Z is the in tersection pr o duct of cycles on X × S Y × S Z and p X Y , p Y Z , p X Z are the resp ectiv e pro jections. The pro du ct × S extends to fi nite corresp ondences, making C or S a tensor catego ry . Assigning  n S = A n S giv es a co-cubical ob ject in C or S . In fact,  ∗ S extends to a f unctor  n S : ECub e → S m /S sending m : 2 → 1 to the usual multiplicati on m S :  2 S →  1 S ; µ S ( x, y ) = xy Since the tensor pro duct in C or S arises f r om the pro duct in S m /S , we ha ve the iden tit y  n S = (  1 S ) ⊗ n ; the collectio n of iden tit y maps th us giv es a m u ltiplication µ ∗∗ /S on the ex- tended co-cubical ob ject  ∗ S . The diagonal δ  n S :  n S →  n S × S  n S =  n S ⊗  n S giv es the co-m ultiplication δ on  ∗ . It is easy to chec k that ( µ ∗∗ , δ ∗ ) defin es a bi-multi plication on th e extended co-cubical ob ject  ∗ . Definition 1.4.2. W e define the category dg C or S = ( C or S , ⊗ ,  ∗ S , δ ∗ ). Let dg P r C or S b e the full sub category of dg C or S with ob jects in Pro j /S . 29 1.4. DG categories of motiv es Prop osition 1.4.1. 1. C or S is a tensor c ate g ory 2. The identity multiplic ation µ ∗∗ and diagonal c o-multiplic ation δ ∗ define a bi- multiplic ation on the c o-cubic al obje ct  ∗ S . 3. The c o-cu b ic al obje ct  ∗ S is homotopy invariant. Pr oof. W e ha v e alrea dy remark ed on (1) and (2). Th e pr o of of (3) follo ws the pro of of the homotop y inv ariance of the s implicial Su slin complex giv en in [ FV00 , lemma 4.1], whic h w e recall for the conv enience of the reader. Since C or S is an additive category with disjoint union of schemes indu c- ing th e d irect sum , we ma y assume that X and S are irred ucible. W e need to show that, for i n : 0 → n the inclusion with im age 0 n , th e map i n ∗ : H om dg C or S ( X, Y ) ∗ → H om dg C or S ( X, Y ⊗  n S ) ∗ is a quasi-isomorphism. Since  n +1 S =  n S ⊗  1 S , we need only show that for the 0-section i : Y → Y × A 1 = Y × S  1 S , th e induced map i ∗ : H om dg C or S ( X, Y ) ∗ → H om dg C or S ( X, Y ⊗  1 S ) ∗ = H om dg C or S ( X, Y × S  1 S ) ∗ is a quasi-isomorphism. W e sho w in fact that i ∗ is a homotop y equiv alence with homotopy in v erse the map p ∗ : H om dg C or S ( X, Y × S  1 S ) ∗ → H om dg C or S ( X, Y ) ∗ . As p ∗ ◦ i ∗ = id, we need to show that i ∗ ◦ p ∗ is homotopic to the identit y on H om dg C or S ( X, Y × S  1 S ) ∗ . F or this, recall that H om C or S ( X × S  n S , Y × S  1 ) n is the free ab elian group on the in tegral closed subs chemes W ⊂ X × S  n S × Y × S  1 S suc h that the pro jection W → X ×  n S is fi nite and surjectiv e. Let p :  n +1 S →  n S b e the pro jection on the first n factors. W e hav e the m ultiplication m ap m S :  1 S × S  1 S →  1 S giving us the map ( p 1 , m S ) :  1 S × S  1 S →  1 S × S  1 S sending ( x, y ) to ( x, xy ). F or a cycle Z ∈ H om C or S ( X × S  n S , Y × S  1 ), w e asso ciate the cycle p ∗ ( Z ) ∈ H om C or S ( X × S  n +1 S , Y × S  1 ). Next, we apply th e map q n : X × S  n +1 S × S Y × S  1 → X × S  n +1 S × S Y × S  1 30 1.4. DG categories of motiv es defined as the comp ositio n X ×  n +1 S × Y ×  1 S = X ×  n S × S  1 S × S Y × S  1 S τ − → X ×  n S × S Y × S  1 S × S  1 S id × S ( p 1 ,m S ) − − − − − − − − → X ×  n S × Y × S  1 S × S  1 S τ − 1 − − → X ×  n S × S  1 S × S Y × S  1 S = X ×  n +1 S × Y ×  1 S . W e wo uld lik e to form the cycle q n ∗ ( p ∗ ( Z )); the p roblem is that q n is n ot a prop er m orphism. How ev er one sh o ws that the restriction of q n to a closed subset W of X ×  n +1 S × Y ×  1 S whic h is finite ov er X ×  n +1 S is prop er, whic h suffices. T o see th is, we note that q n is a m orp hism o ver X ×  n S × S Y , w hic h reduces u s to s h o w in g th at the map ( p 1 , m S ) is prop er when restricted to a closed subsc heme W ⊂  1 S × S  1 S whic h is fi n ite o v er  1 S via the fi rst pro jection. W e m ay enlarge W , and thus we may assu me that W is giv en b y a monic equation of th e form f ( X 1 , X 2 ) := X n 2 + n − 1 X i =0 a i ( X 1 ) X i 2 = 0; ] a i ( X 1 ) ∈ k [ X 1 ] . The map ( p 1 , m S ) r estricted to W is then giv en by the map ( p 1 , m S ) ∗ : k [ T 1 , T 2 ] → k [ X 1 , X 2 ] / ( f ) sending T 1 to X 1 and T 2 to X 1 X 2 , so it is clear th at k [ X 1 , X 2 ] / ( f ) is a fi nite k [ T 1 , T 2 ]-mo dule. W e th er efore hav e th e well-defined map h n := q n ∗ ◦ p ∗ : H om C or S ( X × S  n S , Y × S  1 ) → H om C or S ( X × S  n +1 S , Y × S  1 ) The maps h n satisfy th e follo wing relations: (1) Let f : n → m b e a map in Cub e , giving the map f × id : n + 1 → m + 1 , and maps f ∗ : H om C or S ( X × S  n S , Y × S  1 ) → H om C or S ( X × S  m S , Y × S  1 , ( f × id ) ∗ : H om C or S ( X × S  n +1 S , Y × S  1 ) → H om C or S ( X × S  m +1 S , Y × S  1 Then h m ◦ f ∗ = ( f × id) ∗ ◦ h n . (2) η ∗ n,n +1 , 1 ◦ h n = id, η ∗ n,n +1 , 0 ◦ h n = i ∗ ◦ p ∗ . The relat ion (1) applied to the pro jections p n,i sho w s that the maps h n descend to maps on the non-degenerate quotien ts ¯ h n : H om dg C or S ( X, Y × S  1 S ) − n → H om dg C or S ( X, Y × S  1 S ) − n − 1 . The relations (1) and (2) sho w that the collection of maps ( − 1) n +1 ¯ h n giv e a homotop y b etw een id and i ∗ ◦ p ∗ .  31 1.4. DG categories of motiv es Let S b e a fixed r egular sc heme of finite Kr u ll dimension. Fix a Grothendiec k top ology τ on some full sub categ ory O p n τ S of Sch S . S u pp ose w e ha ve a p re- sheaf of DG categories U → C ( U ) on S τ . F or ob jects X and Y in C ( S ), we ha ve the presheaf [ f : U → S ] 7→ H om C ( U ) ( f ∗ X, f ∗ Y ) , whic h we denote by H om τ C ( X, Y ). Let R Γ( S, C ) b e th e DG category with the same ob jects as C ( S ) an d with Hom-complex H om R Γ( S, C ) ( X, Y ) ∗ = T ot( G ∗ ( H om τ C ( X, Y )) ∗ ( S )) . The comp ositio n law is d efined using the Alexander-Whitney map ( 1.1.3 ) comp osed with the comp osit ion la w on the p resheaf of DG catego ries U → C ( U ). F or a pro of that the comp ositio n of morp hism is an asso ciativ e map of complexes, see [ Lev09 , § 3.1] (but note that w e hav e introd uced a sign- correction in b oth the Go demen t r esolution and the Alexander-Whitney map, wh ich w as missin g in [ Lev09 , lo c . cit ]). Definition 1.4.3. W e den ote b y dg P r C or S the Zariski presheaf of DG catego ries U 7→ dg P r C or U The DG category of smo oth effectiv e motiv es ov er S is defined to b e dg S mM ot eff S := C b dg ( R Γ( S, dg P r C or S )) . taking τ to b e the Zarisk i top olo gy on the category O pn Z ar S of Zariski op en subsets of S . The triangulated cate gory , S mM ot eff g m ( S ) of smo oth effectiv e geometric motiv es ov er S is defined as the idemp otent completion of the homotop y category H 0 dg S mM ot eff S . The triangulated category of smo oth motiv es o ve r S , S mM ot g m ( S ) is the triangulated category formed by in v erting ⊗ L on S mM ot eff g m ( S ). Remark 1.4.4. Su p p ose that S = S p ec O X,v for X a smo oth sc heme of finite t yp e o v er k , and v a finite s et of p oin ts of X . Supp ose that the field k h as c haracteristic zero. Then by [ Lev09 , corollary 5.6] together with [ FV00 , theorem 8.1], the natur al fu nctor Γ( S, dgP rC or S ) → R Γ( S, dg P r C or S ) is a quasi-equiv alence of DG categories. Th us, w e ha v e th e q u asi-equiv alence of DG catego ries C b dg (Γ( S, dgP rC or S )) → C b dg ( R Γ( S, dg P r C or S )) = dg S mM ot eff S and therefore the idemp ote n t completion of H 0 C b dg (Γ( S, dgP rC or S )) is equiv- alen t to S mM ot g m ( S ). 32 CHAPTER 2 Pseudo-tensor structure 2.1. Pseudo-tensor structure on DG categories Definition 2.1.1 ([ BD04 ], pg. 1 1–14) . A p seudo-tensor str ucture on an additiv e category A is the follo wing d atum: (1) F or any finite n on-empt y set I , an I -family of ob jects X i ∈ A , i ∈ I , and an ob ject Y ∈ A , we ha ve an ab elia n group P A I ( { X i } i ∈ I , Y ). [W e den ote P A n ( { X i } n i =1 , Y ) := P A { 1 ,...,n } ( { X i } i ∈ I , Y ).] (2) Given an y surjectiv e map of fi nite sets π : J ։ I , w e hav e the comp osition m ap P A I ( { Y i } , Z ) ⊗ Y i ∈ I P A J i ( { X j } j ∈ J i , Y i ) − → P A J ( { X j } , Z ) , ( f , ( g i )) 7→ f ( g i ) where J i := π − 1 ( i ). The follo win g prop erties must hold: (1) T h e comp ositio n is asso ciativ e: for another su r jectiv e map K ։ J , { W k } an K -family of ob jects, and h j ∈ P A K j ( { W k } k ∈ K j , X j ), w e ha ve f ( g i ( h j )) = ( f ( g i ))( h j ) ∈ P A K ( { W k } , Z ). (2) F or an y ob ject E ∈ A , there is an elemen t id E ∈ P A 1 ( { E } , E ) with ∂ id E = 0 suc h that for an y f ∈ P A I ( { X i } , Y ), w e ha ve id Y ( f ) = f ( id X i ) = f . No w, let C b e an additiv e DG catego ry . W e define the notion of a pseudo-tensor str ucture on C . Definition 2.1.2. A pseud o-tensor structure on C is the follo wing d atum: (1) F or any finite n on-empt y set I , an I -family of ob ject s X i ∈ C , i ∈ I , and an ob ject Y ∈ C , w e h a ve a complex of ab elian groups ( P C I ( { X i } i ∈ I , Y ) ∗ , ∂ ). [W e den ote P C n ( { X i } n i =1 , Y ) ∗ := P C { 1 ,...,n } ( { X i } i ∈ I , Y ) ∗ .] (2) Given an y surjectiv e map of fi nite sets π : J ։ I , w e hav e the comp osition m ap P C I ( { Y i } , Z ) m ⊗ O i ∈ I P C J i ( { X j } j ∈ J i , Y i ) n i − → P C J ( { X j } , Z ) m + P n i , ( f , ( g i )) 7→ f ( g i ) where J i := π − 1 ( i ). The follo win g prop erties must hold: 33 2.1. Pseudo-tensor structur e on DG categories (1) T h e comp osition is a map of complexes: ∂ ( f ( g i )) = ( ∂ f )( g i ) + p X k =1 ( − 1) m + P k − 1 i =1 n i f ( g 1 , . . . , g k − 1 , ∂ g k , g k +1 , . . . , g p ) taking I = { 1 , . . . , p } . (2) T h e comp ositio n is asso ciativ e: for another su r jectiv e map K ։ J , { W k } an K -family of ob jects, an d h j ∈ P C K j ( { W k } k ∈ K j , X j ) p j , we ha ve in P C K ( { W k } , Z ) m + P n i + P p j , f ( g i ( h j )) = ( − 1) P p i =1 ( P i − 1 l =1 ( P j ∈ J l p j )) n i ( f ( g i ))( h j ) . (3) F or any ob ject E ∈ C , there is an element id E ∈ P C 1 ( { E } , E ) 0 with ∂ id E = 0 such that for an y f ∈ P C I ( { X i } , Y ) n , w e ha ve id Y ( f ) = f ( id X i ) = f . Remark 2.1.3. The condition for asso ciativit y stated ab o v e is precisely the comm utativit y of the f ollo wing d iagram. P I ( { Y i } , Z ) ⊗ O i ∈ I  P J i ( { X j } , Y i ) ⊗ O j ∈ J i P K j ( { W k } , X j )  * * V V V V V V V V V V V V V V V V V θ   P I ( { Y i } , Z ) ⊗ O i ∈ I  P K i ( { W k } , X j )    P I ( { Y i } , Z ) ⊗ O i ∈ I P J i ( { X j } , Y i )  ⊗ O j ∈ J P K j ( { W k } , X j ) * * V V V V V V V V V V V V V V V V V E P J ( { X j } , Z ) ⊗ O j ∈ J P K j ( { W k } , X j ) O O The sign that comes in to the asso ciativit y condition is to make sur e that θ is a map of complexes. Lemma 2.1.1. If P C I is a pseudo-tensor structur e on a DG c ate gory C , then it induc es a pseudo-tensor structur e on the homo topy c ate gory H 0 C . Pr oof. Let P H 0 C I ( { X i } , Y ) := H 0 P C I ( { X i } , Y ) ∗ . W e wan t to sho w that P H 0 C I is a pseudo-tensor structure on H 0 C . Since composition map on P C is a map of complexes, w e hav e th e map P H 0 C I ( { Y i } , Z ) ⊗ Y i ∈ I P H 0 C J i ( { X j } j ∈ J i , Y i ) − → P H 0 C J ( { X j } , Z ) 34 2.1. Pseudo-tensor structur e on DG categories Asso ciativit y of comp osition is clear and id X ∈ Z 0 P C 1 ( { X } , X ) implies that its image in H 0 P C 1 ( { X } , X ) has the requir ed pr op erties.  Lemma 2.1.2. A pseudo-tensor structur e P C I ( { X i } , Y ) on a DG c ate gory C is functorial in e ach of the X i ’s. The same is true for a pseudo-tensor structur e on an add itive c ate gory. Pr oof. Let f : X ′ k → X k b e a m orphism of degree m for any k ∈ I = { 1 , . . . , n } . Th en , we ha ve a map P C n ( { X i } n i =1 , Y ) f ∗ − → P C n ( { X 1 , . . . , X k − 1 , X ′ k , X k +1 , . . . , X n } , Y ) giv en b y g 7→ g ( id X 1 , . . . , id X k − 1 , f , id X k +1 , . . . , id X n ). Not e that f ∗ is a map of degree m in C dg ( Ab ). Clea rly , id ∗ X k = id and it follo ws from the asso ciativit y of comp ositi on that ( f 1 ◦ f 2 ) ∗ = f ∗ 2 ◦ f ∗ 1 .  Corollary 2.1.3. The functor P C I is additive in e ach c omp onent. Lemma 2.1.4. If π : J ≃ I is a bije ction and the obje cts { X j } j ∈ J is a p ermutation of the obje cts { Y i } i I such that X π − 1 ( i ) = Y i , then P C I ( { Y i } , ) ∗ ∼ − → P C J ( { X j } , ) ∗ Pr oof. S ince π is a b ijection, for an y j ∈ J , j = π − 1 ( i ) for i = π ( j ) ∈ I . Then, X j = X π − 1 ( i ) = Y i = Y π ( j ) . Th us w e ha ve g i = id Y i ∈ P C 1 ( { X π − 1 ( i ) } , Y i ) 0 and h j = id X j ∈ P C 1 ( { Y π ( j ) , X j ) 0 . This give s us maps σ : P C I ( { Y i } , Z ) ∗ − → P C J ( { X j } , Z ) ∗ τ : P C J ( { X j } , Z ) ∗ − → P C I ( { Y i } , Z ) ∗ giv en b y σ ( f ) = f ( g i ) and τ ( f ′ ) = f ′ ( h j ). Then τ ◦ σ ( f ) = ( f ( g i ))( h j ) = f ( g i ( h j )) = f ( g i ◦ h π − 1 ( i ) ) = f ( id Y i ) = f and similarly , σ ◦ τ = id . Hence, σ is an isomorphism w ith inv erse τ .  If we h a ve a pseudo-tensor structure on a DG category C , then eac h collect ion of ob jects { X i } i ∈ I in C giv es rise to the C -mo dule P C I ( { X i } , ) ∗ : C → C ( Ab ) , whic h we ma y co nsider as an ob ject of D ( C op ). Similarly , eac h ob ject X of C giv es us the ob ject H om C ( X, ) ∗ of D ( C op ), lying in th e full su b category K b ( C op ). Definition 2.1.4. A pseudo-tensor structure is called representa ble, if for an y collection of ob jects { X i } i ∈ I in C , there exists an ob ject ⊗ i ∈ I X i in C and an isomorphism in D ( C op ) λ { X i } i ∈ I : P C I ( { X i } , ) ∗ ∼ − → H om C ( ⊗ i ∈ I X i , ) ∗ suc h that for | I | = 1, the ob ject corresp onding to { X 1 } is X 1 and λ { X 1 } is the identit y . 35 2.1. Pseudo-tensor structur e on DG categories Remark 2.1.5. 1. F or an additive catego ry C , a pseudo-tensor structure is represent able if and only if for any collectio n of ob jects { X i } i ∈ I in C , there exists an ob ject ⊗ i ∈ I X i in C and an isomorph ism of f unctors from C to Ab λ { X i } i ∈ I : P C I ( { X i } , ) → H om C ( ⊗ i ∈ I X i , ) . 2. Supp ose a giv en pseud o-tensor structure on C is repr esen table, via ob jects X 1 ⊗ . . . ⊗ X n and isomorph isms λ { X i } i ∈ I for eac h collectio n X 1 , . . . , X n of ob jects of C . Supp ose w e h a ve another c h oice of ob jects ( X 1 ⊗ . . . ⊗ X n ) ′ and isomorphisms λ ′ { X i } i ∈ I . Then we ha v e the isomorphism in D ( C op ) H om C (( ⊗ i ∈ I X i ) ′ , ) ∗ ∼ = H om C ( ⊗ i ∈ I X i , ) ∗ ; in particular, the r esp ectiv e iden tit y maps give us the morp hisms in H 0 C f : X 1 ⊗ . . . ⊗ X n → ( X 1 ⊗ . . . ⊗ X n ) ′ g : ( X 1 ⊗ . . . ⊗ X n ) ′ → X 1 ⊗ . . . ⊗ X n whic h are inv erse isomorph isms in H 0 C , and we hav e λ ′ { X i } i ∈ I = λ { X i } i ∈ I ◦ g . Th us, the data ( X 1 ⊗ . . . ⊗ X n , λ { X i } i ∈ I ) is determined u p to canonical iso- morphism in H 0 C . If the P C I are repr esen table, the comp ositio n map giv es for a surjection π : J ։ I , H om C ( O i ∈ I ( O j ∈ J i X j ) , O i ∈ I ( O j ∈ J i X j )) m ⊗ Y i ∈ I H om C ( O j ∈ J i X j , O j ∈ J i X j ) n i ≀   P C I ( { O j ∈ J i X j } , O i ∈ I ( O j ∈ J i X j )) m ⊗ Y i ∈ I P C J i ( { X j } j ∈ J i , O j ∈ J i X j ) n i   P C J ( { X j } , O i ∈ I ( O j ∈ J i X j )) m + P n i ≀   H om C ( O j ∈ J X j , O i ∈ I ( O j ∈ J i X j )) m + P n i The image of ( id ⊗ i ∈ I ( ⊗ j ∈ J i X j ) , ( id ⊗ j ∈ J i X j ) i ) u nder this map is denoted ǫ π : ⊗ j ∈ J X j − → ⊗ i ∈ I ( ⊗ j ∈ J i X j ) Note that ǫ π ∈ Z 0 H om C ( ⊗ j ∈ J X j , ⊗ i ∈ I ( ⊗ j ∈ J i X j )). 36 2.1. Pseudo-tensor structur e on DG categories If f : X → Y and f ′ : X ′ → Y ′ are morphisms in C , w e define a morphism f ⊗ f ′ as follo ws. Consider the map H om ( Y ⊗ Y ′ , ) ⊗ H om ( X, Y ) ⊗ H om ( X ′ , Y ′ ) ≀   P 2 ( { Y , Y ′ } , ) ⊗ P 1 ( { X } , Y ) ⊗ P 1 ( { X ′ } , Y ′ )   P 2 ( { X, X ′ } , ) ≀   H om ( X ⊗ X ′ , ) The image of ( id Y ⊗ Y ′ , f , f ′ ) und er this map is denoted f ⊗ f ′ ∈ H om ( X ⊗ X ′ , Y ⊗ Y ′ ). Lemma 2.1.5. If K π ′ ։ J π ։ I ar e surje ctive maps and the pseudo-tensor structur e is r epr esentable, then the fol lowing diagr am c ommutes. ⊗ K X k ǫ π ′ / / ǫ ππ ′ & & M M M M M M M M M M M ⊗ J ( ⊗ K j X k ) ǫ π / / ⊗ I ( ⊗ J i ( ⊗ K j X k )) ⊗ I ( ⊗ K i X k ) ⊗ i ∈ I ǫ π ′ i 6 6 m m m m m m m m m m m m m wher e J i = π − 1 ( i ) , K j = π ′− 1 ( j ) , K i = ( π π ′ ) − 1 ( i ) and π ′ i := π ′ | K i : K i ։ J i . Pr oof. Commutativit y of the ab o v e d iagram is the same as th e com- m u tativit y of the f ollo wing d iagram P J ( {⊗ k ∈ K j X k } , ) ǫ ∗ π ′ / / P K ( { X k } , ) P I ( {⊗ j ∈ J i ( ⊗ k ∈ K j X k ) } , ) ( ⊗ I ǫ π ′ i ) ∗ / / ǫ ∗ π O O P I ( {⊗ k ∈ K i X k } , ) ǫ ∗ ππ ′ O O 37 2.1. Pseudo-tensor structur e on DG categories Let f ∈ P I ( {⊗ j ∈ J i ( ⊗ k ∈ K j X k ) } , ), then ǫ ∗ π ′ ( ǫ ∗ π ( f )) = ( f ◦ (( λ − 1 {⊗ K j X k } j ∈ J i ( id ⊗ J i ( ⊗ K j X k ) )) i ∈ I )) ◦ (( λ − 1 { X k } k ∈ K j ( id ⊗ K j X k )) j ∈ J ) = f ◦ (( λ − 1 {⊗ K j X k } j ∈ J i ( id ⊗ J i ( ⊗ K j X k ) )) ◦ (( λ − 1 { X k } k ∈ K j ( id ⊗ K j X k )) j ∈ J i )) i ∈ I = f ◦ ( λ − 1 { X k } k ∈ K i ( ǫ π ′ i )) i ∈ I [By d efi nition of ǫ π ′ i ] = f ◦ ( ǫ π ′ i ◦ ( λ − 1 { X k } k ∈ K i ( id ⊗ K i X k ))) i ∈ I [Since λ { X k } k ∈ K i ( ǫ π ′ i ◦ ( λ − 1 { X k } k ∈ K i ( id ⊗ K i X k ))) = ǫ π ′ i ◦ ( id ⊗ K i X k ) = ǫ π ′ i b y n aturalit y and λ is an isomorphism.] = ( f ◦ ( ǫ π ′ i ) i ∈ I ) ◦ ( λ − 1 { X k } k ∈ K i ( id ⊗ K i X k )) i ∈ I = ǫ ∗ π π ′ (( ⊗ I ǫ π ′ i ) ∗ ( f ))  Lemma 2.1.6. Supp ose that the pseudo-tensor structur e P C ∗ is r epr esentable. Then ǫ π is an isomorphism for al l su rje ctions π if and only if the pseudo- tensor structur e P C I gives a tensor structur e on C . Pr oof. Clearly if th e pseudo-tensor structure giv es a tensor stru cture, then the maps ǫ π are all isomorphisms. Con v ersely , if w e ha ve morphisms X f → Y g → Z and X ′ f ′ → Y ′ g ′ → Z ′ , then we ha v e that g f ⊗ g ′ f ′ = λ { X,X ′ } ( λ − 1 { Z,Z ′ } ( id Z ⊗ Z ′ )( g f , g ′ f ′ )) = λ { X,X ′ } (( λ − 1 { Z,Z ′ } ( id Z ⊗ Z ′ )( g , g ′ ))( f , f ′ )) = λ { X,X ′ } (( g ⊗ g ′ ) ∗ λ − 1 { Y , Y ′ } ( id Y ⊗ Y ′ )( f , f ′ )) = ( g ⊗ g ′ ) ∗ λ { X,X ′ } ( λ − 1 { Y , Y ′ } ( id Y ⊗ Y ′ )( f , f ′ )) = ( g ⊗ g ′ ) ◦ ( f ⊗ f ′ ) . (2.1.1) since, by naturalit y , λ { Y , Y ′ } (( g ⊗ g ′ ) ∗ λ − 1 { Y , Y ′ } ( id Y ⊗ Y ′ )) = ( g ⊗ g ′ ) ∗ λ { Y , Y ′ } ( λ − 1 { Y , Y ′ } ( id Y ⊗ Y ′ )) = g ⊗ g ′ = λ { Y , Y ′ } ( λ − 1 { Z,Z ′ } ( id Z ⊗ Z ′ )( g , g ′ )) and λ { Y , Y ′ } is an isomorphism. The asso ciativit y isomo rphisms are giv en b y the diagrams X ⊗ ( Y ⊗ Z ) α / / ( X ⊗ Y ) ⊗ Z X ⊗ Y ⊗ Z ∼ ǫ 1 , 23 h h P P P P P P P P P P P P ∼ ǫ 12 , 3 6 6 n n n n n n n n n n n n 38 2.2. Ps eu do-tensor structur e on DG complexes W e need to c hec k that the follo wing p en tagonal d iagram commutes: ( X ⊗ Y ) ⊗ ( Z ⊗ W ) α * * U U U U U U U U U U U U U U U U X ⊗ ( Y ⊗ ( Z ⊗ W )) α 4 4 i i i i i i i i i i i i i i i i id X ⊗ α   (( X ⊗ Y ) ⊗ Z ) ⊗ W X ⊗ Y ⊗ Z ⊗ W ǫ 12 , 34 O O id X ⊗ ǫ 2 , 34 j j U U U U U U U U U U U U U U U U ǫ 12 , 3 ⊗ id W 4 4 i i i i i i i i i i i i i i i i id X ⊗ ǫ 23 , 4 t t i i i i i i i i i i i i i i i i ǫ 1 , 23 ⊗ id W * * U U U U U U U U U U U U U U U U X ⊗ (( Y ⊗ Z ) ⊗ W ) α / / ( X ⊗ ( Y ⊗ Z )) ⊗ W α ⊗ id W O O The upp er left triangle comm u tes since id X ⊗ ǫ 2 , 34 = ǫ 1 , 23 ◦ ǫ 1 , 2 , 34 and ǫ 12 , 34 = ǫ 12 , 3 ◦ ǫ 1 , 2 , 34 b y Lemma 2.1.5 . Similarly , the up p er right triangle commutes to o. Again, u sing Lemma 2.1.5 , the lo wer triangle comm utes since id X ⊗ ǫ 23 , 4 = ǫ 1 , 23 ◦ ǫ 1 , 23 , 4 and ǫ 1 , 23 ⊗ id W = ǫ 12 , 3 ◦ ǫ 1 , 23 , 4 . The other t w o triangles comm ute by d efinition of α and ( 2.1.1 ). The p en tagon comm u tes since all the inner triangles commute. By Lemma 2.1.4 , w e hav e the isomorph ism P 2 ( { X 2 , X 1 } , ) ∼ → P 2 ( { X 1 , X 2 } , ) whic h along w ith repr esen tabilit y , giv es the commutativi t y constrain t τ X 1 ,X 2 : X 1 ⊗ X 2 ∼ → X 2 ⊗ X 1 . W e also need to v erify th e commutat ivit y of the fol- lo wing diagram. X ⊗ ( Y ⊗ Z ) α / / id X ⊗ τ Y ,Z   ( X ⊗ Y ) ⊗ Z τ ( X ⊗ Y ) ,Z / / Z ⊗ ( X ⊗ Y ) α   X ⊗ Y ⊗ Z ∼ h h P P P P P P P P P P P P ∼ 6 6 n n n n n n n n n n n n ∼ ( ( Q Q Q Q Q Q Q Q Q Q Q Q ∼ / / Z ⊗ X ⊗ Y ∼ 6 6 n n n n n n n n n n n n ∼ ! ! B B B B B B B B B B B B B B B B B B B B X ⊗ Z ⊗ Y ∼ 6 6 m m m m m m m m m m m m ≀   ∼ s s f f f f f f f f f f f f f f f f f f f f f f f f X ⊗ ( Z ⊗ Y ) α / / ( X ⊗ Z ) ⊗ Y τ X,Z ⊗ id Y / / ( Z ⊗ X ) ⊗ Y The thr ee inn er triangles comm u te b y defin ition of α . But all the inner quadrilaterals commute b y Lemma 2.1.5 , implying the comm utativit y of the outer hexagon. Hence, ⊗ giv es a tensor structure on C .  2.2. Pseudo-tensor structure on DG complexes The go al in this sectio n is to sho w that a pseudo-tensor str ucture on a DG category C ind u ces a canonical pseudo-tensor structur e on the catego ry Pre-T r ( C ) (see Definition 1.1.1 ). In section 2.3 , we sho w that, if the original pseudo-tensor structure is represen table an d defines a tensor stru cture on 39 2.2. Ps eu do-tensor structur e on DG complexes H 0 C , th en the pseudo-tensor structur e on Pre-T r( C ) is represen table and induces a tensor structure on the category K b ( C ). W e b egin by defining the in duced pseudo-tensor structure on Pre-T r( C ). Let E i = { E i j , e i j k : E i k → E i j } N i ≤ j,k ≤ M i , i ∈ I := { 1 , . . . , m } and F = { F k , f j k : F k → F j } N ′ ≤ j,k ≤ M ′ b e ob jects in P r e-T r( C ). W e d efi ne P pretr m ( {E i } i ∈ I , F ) n = M N i ≤ j i ≤ M i N ′ ≤ k ≤ M ′ P C m ( { E i j i } i ∈ I , F k ) n − k + P i ∈ I j i (2.2.1) Let ϕ = ( ϕ k , { j i } ) ∈ P pretr m ( {E i } i ∈ I , F ) n . Then we define ( ∂ ϕ ) k , { j i } =( − 1) k dϕ k , { j i } + X l f k l ( ϕ l, { j i } ) (2.2.2) − ( − 1) n m X i =1 X N i ≤ l ≤ M i ( − 1) s ( i,j i ,l ) ϕ k ,J ( i,j i ,l ) ( id E 1 j 1 , . . . , id E i − 1 j i − 1 , e i lj i , id E i +1 j i +1 , . . . , id E m j m ) where s ( i, j i , l ) := j 1 + · · · + j i − 1 + ( j i − l + 1)( j i +1 + · · · + j m ) and J ( i, j i , l ) := { j 1 , . . . , j i − 1 , l, j i +1 , . . . , j m } . W e chec k that ∂ 2 = 0. F or ϕ as ab o v e, ( ∂ 2 ϕ ) k , { j i } = ( − 1) k d ( ∂ ϕ ) k , { j i } (2.2.3) + X l f k l  ( ∂ ϕ ) l, { j i }  (2.2.4) − ( − 1) n +1 m X i =1 X N i ≤ l ≤ M i ( − 1) s ( i,j i ,l ) ( ∂ ϕ ) k ,J ( i,j i ,l ) ( id E 1 j 1 , . . . , id E i − 1 j i − 1 , e i lj i , id E i +1 j i +1 , . . . , id E m j m ) (2.2.5) W e h av e, ( 2.2.3 ) = ( − 1) k X l X q ( − 1) k +1 f k q f q l ( ϕ l, { j i } ) + X l ( − 1) l − k +1 f k l ( dϕ l, { j i } ) − ( − 1) n m X i =1 X N i ≤ l ≤ M i ( − 1) s ( i,j i ,l )  dϕ k ,J ( i,j i ,l ) ( id E 1 j 1 , . . . , id E i − 1 j i − 1 , e i lj i , id E i +1 j i +1 , . . . , id E m j m ) +( − 1) l − j i + P j i − k +1 ϕ k ,J ( i,j i ,l ) ( id E 1 j 1 , . . . , X p ( − 1) l +1 e i lp e i pj i , . . . , id E m j m )  ! ( 2.2.4 ) = X l ( − 1) l f k l ( dϕ l, { j i } ) + X l X m f k l f lm ( ϕ m, { j i } ) − ( − 1) n X q X i X l ( − 1) s ( i,j i ,l ) f k q ( ϕ q ,J ( i,j i ,l ) ( id E 1 j 1 , . . . , id E i − 1 j i − 1 , e i lj i , . . . , id E m j m )) 40 2.2. Ps eu do-tensor structur e on DG complexes And, ( 2.2.5 ) = − ( − 1) n +1 m X i =1 X l ( − 1) s ( i,j i ,l ) dϕ k ,J ( i,j i ,l ) ( id E 1 j 1 , . . . , id E i − 1 j i − 1 , e i lj i , id E i +1 j i +1 , . . . , id E m j m ) − ( − 1) n +1 X q X i X l ( − 1) s ( i,j i ,l ) f k q ( ϕ q ,J ( i,j i ,l ) ( id E 1 j 1 , . . . , id E i − 1 j i − 1 , e i lj i , . . . , id E m j m )) − ( − 1) n +1+ n +1  m X i =1 X N i ≤ l,p ≤ M i ( − 1) s ( i, { j i } ,l )+ s ( i,J ( i, { j i } ,l ) ,p ) ϕ k ,J ( i, { j i } ,p ) ( id E 1 j 1 , . . . , e i pl e i lj i , . . . , id E m j m ) m X i,q =1 i 6 = q M i X l = N i M q X p = N q ( − 1) s ( i, { j i } ,l )+ s ( q ,J ( i, { j i } ,l ) ,p ) ( ϕ k ,J ( q ,J ( i, { j i } ,l ) ,p ) ( id E 1 j 1 , . . . , e q pj q , . . . , id E m j m ))( id E 1 j 1 , . . . , e i lj i , . . . , id E m j m )  W e hav e s ( i, { j i } , l ) + s ( i, J ( i, { j i } , l ) , p ) = j 1 + · · · + j i − 1 + ( j i − l + 1)( j i +1 + · · · + j m ) + j 1 + · · · + j i − 1 + ( l − p + 1)( j i +1 + · · · + j m ) = ( j i − l i )( j i +1 + · · · + j m ) + 2( − j i + P j i ) = s ( i, { j i } , p ) − j i + P j i . No w, the last term in the expansions of ( 2.2.3 ) is: ( − 1) k + n +1 m X i =1 X l ( − 1) s ( i,j i ,l ) (( − 1) l − j i + P j i − k + n ϕ k ,J ( i,j i ,l ) ( id E 1 j 1 , . . . , X p ( − 1) l +1 e i lp e i pj i , . . . , id E m j m )) Since the s ign of the term ϕ k ,J ( i, { j i } ,p ) ( id E 1 j 1 , . . . , e i pl e i lj i , . . . , id E m j m ) is ( − 1) s ( i, { j i } ,p ) − j i + P j i , w e can see that th e last term in the expansion of ( 2.2.3 ) and the p enultimat e term of ( 2.2.5 ) cancel eac h other. F or the last term of ( 2.2.5 ), note that for i ≤ q , ( ϕ k ,J ( q ,J ( i, { j i } ,l ) ,p ) ( id E 1 j 1 , . . . , e q pj q , . . . , id E m j m ))( id E 1 j 1 , . . . , e i lj i , . . . , id E m j m ) = ( − 1) ( j i − l +1)( j q − p +1) ( ϕ k ,J ( i,J ( q , { j i } ,p ) ,l ) ( id E 1 j 1 , . . . , e i lj i , . . . , id E m j m ))( id E 1 j 1 , . . . , e q pj q , . . . , id E m j m ) . Also, s ( i, { j i } , l ) + s ( q , J ( i, { j i } , l ) , p ) − ( s ( q , { j i } , p ) + s ( i, J ( q , { j i } , p ) , l )) ≡ ( j i − l + 1)( j q − p + 1) + 1 mo d 2 implying that the last su m P 1 ≤ i 6 = q ≤ m P N i ≤ l ≤ M i P N q ≤ p ≤ M q in the expan- sion of ( 2.2.5 ) is 0. Clearly , all the other terms also cancel eac h other, giving ∂ 2 ϕ = 0. 41 2.2. Ps eu do-tensor structur e on DG complexes The comp ositi on map is defined as follo w s P pretr I ( {E i } , F ) m ⊗ Q i ∈ I P pretr J i ( {H j } , E i ) n i L j i ,k,k j ,l i P C I ( { E i j i } , F k ) m − k + P j i ⊗ Q i ∈ I ( { H j k j } , E j l i ) n i − l i + P j ∈ J i k j   L j i ,k,k j P C I ( { E i j i } , F k ) m − k + P j i ⊗ Q i ∈ I ( { H j k j } , E j j i ) n i − j i + P j ∈ J i k j   L j i L k j ,k P C J ( { H j k j } , F k ) m + P n i − k + P j ∈ J k j L j i P pretr J ( {H j } , F ) m + P n i f P   P pretr J ( {H j } , F ) m + P n i (2.2.6) where f P is defined as, for ϕ ∈ P pretr I ( {E i } , F ) m and ψ i ∈ P pretr J i ( {H j } , E i ) n i , ( ϕ ( ψ i )) k , { k j } = X { j i } ( − 1) S ( { j i } , { k j } , { n i } ) ϕ k , { j i } ( ψ i j i , { k j } ) . where S ( { j i } , { k j } , { n i } ) = P d p =2 ( P p − 1 i =1 j p ( n i − j i + P J i k j ) + ( P J i k j ) n p ). T o c hec k that the comp osition defined ab o v e is a map of complexes, note that ( ∂ ( ϕ ( ψ i ))) k , { k j } = ( − 1) k X { j i } ( − 1) S ( { j i } , { k j } , { n i } ) d ( ϕ k , { j i } ( ψ i j i , { k j } )) (2.2.7) + X l X { j i } ( − 1) S ( { j i } , { k j } , { n i } ) f k l ( ϕ l, { j i } ( ψ i j i , { k j } )) (2.2.8) + − ( − 1) m + P n i e i X j =1 X l X { j i } ( − 1) s ( j, { k j } ,l )+ S ( { j i } ,J ( j, { k j } ,l ) , { n i } ) ( ϕ k , { j i } ( ψ i j i ,J ( j, { k j } ,l ) ))( id H 1 k 1 , . . . , h j lk j , . . . , id H p i k p i ) (2.2.9) 42 2.2. Ps eu do-tensor structur e on DG complexes whereas (( ∂ ϕ )( ψ i )) k , { k j } = X { j i } ( − 1) S ( { j i } , { k j } , { n i } )  ( − 1) k dϕ k , { j i } ( ψ i j i , { k j } ) (2.2.10 ) + X l f k l ϕ l, { j i } ( ψ i j i , { k j } ) (2.2.11 ) − ( − 1) m d X p =1 X l ( − 1) s ( p, { j i } ,l )+( j p − l p +1)( P i

p ( n i − j i + P J i k j )( P j ∈ J p ( k j − l +1)) ( ϕ k , { j i } ( ψ p j p ,J ( j, { k j } ,l ) ))( id H 1 k 1 , . . . , h j lk j , . . . , id H p i k p i ) (2.2.16 ) Comparing signs of the like term s and using the definitions of s and S , it follo ws b y easy computation that ∂ ( ϕ ( ψ i )) = ( ∂ ϕ ) ( ψ i ) + d X p =1 ( − 1) m + P p − 1 i =1 n i ϕ ( ψ 1 , . . . , ∂ ψ p , . . . , ψ d ) Next, to c h ec k that the comp osition defi n ed ab o v e is asso ciativ e, note that, for sur jectiv e maps K ։ J ։ I , and ϕ ∈ P pretr I ( {E i } , F ) m , ψ i ∈ P pretr J i ( {H j } , E i ) n i 43 2.2. Ps eu do-tensor structur e on DG complexes and ρ j ∈ P pretr K j ( {G k } , H j ) p j , (( ϕ ( ψ i ))( ρ j )) k , { q l } = X { k j } ( − 1) S ( { k j } , { q l } , { p j } ) ( ϕ ( ψ i )) k , { k j } ( ρ j k j , { q l } ) = X { k j } X { j i } ( − 1) S ( { k j } , { q l } , { p j } )+ S ( { j i } , { k j } , { n i } ) ( ϕ k { j i } ( ψ i j i { k j } ))( ρ j k j , { q l } ) = X { k j } X { j i } ( − 1) S ( { k j } , { q l } , { p j } )+ S ( { j i } , { k j } , { n i } )+ P d p =1 P i

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