AA d´evissage theorem of non-connective K -theory Satoshi Mochizuki
Contents C -homotopy and P -homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Homotopy commutative squares and diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Mapping cylinder and mapping cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 K -theory 76 K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2 Fundamental properties of non-connective K -theory . . . . . . . . . . . . . . . . . . . . . . 775.3 Homotopy additivity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.4 D´evissage theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A A d´evissage theorem for modular Waldhausen exact categories 84
Introduction
The purpose of this article is to show a version of d´evissage theorem of non-connective K -theory. As a math-ematical jargon, the term ‘d´evissage’ is introduced by Grothendieck. It is a technique for verifying assertionsabout coherent sheaves on noetherian schemes [DG61, Th´eor`eme 3.1.2] and in the mid-80’s he returned backto this theme in the context of ‘tame topology’ [Gro84, § K -groups and abstracted by Heller [BHS64], [Hel65]. In the renowned paper [Qui73],Quillen provided a general d´evissage theorem for higher K -theory of abelian categories. Namely for an es-sentially small abelian category A and a topologizing subcategory (1.1.14) B of A such that the inclusionfunctor B (cid:44) → A satisfies Quillen’s d´evissage condition (4.3.3), the inclusion functor induces a homotopyequivalence K ( B ) → K ( A ) on K -theory. It might be well-known that for an exact category which satisfies asuitable Jordan-H¨older theorem, the proof of d´evissage theorem in [Qui73] with appropriate refinements still1 a r X i v : . [ m a t h . K T ] J un orks well. See for example [Mit94, p.175, l.5]. Notice that there exists a d´evissage theorem for K -theory ofexact ∞ -categories by Barwick in [Bar13]. These approach to a d´evissage theorem crucially depends upon (ageneralization of) Quillen’s Q -construction. Especially in the situation above, we analyze the homotopy fiberof Q B (cid:44) → Q A by utilizing a (generalized) Jordan-H¨older theorem. We will illustrate this phenomena in thecontext of appropriate Waldhausen K -theory in Appendix.In [Wal84, p.188], Waldhausen wrote:‘ As to a general attack on the spaces M ( ∗ , p , n ) , the first ( and perhaps main ) step should be the search for ad´evissage theorem. ’In this scheme, he actually developed the cell filtration theorem in [Wal85, 1.7.1] (for precise statementfor non-connective K -theory, see Corollary 5.4.2). Even though its statement is apparently different fromQuillen’s d´evissage theorem, in this article, we will regard it as a version of d´evissage theorem. In line withthis thinking, there also exists similar flavoured approaches for d´evissage theorem by Yao in [Yao95] and byBlumberg and Mandell in [BM08] and by Barwick [Bar15] and recently Raptis [Rap18]. These investigationsare close to theorem of heart originated with Neeman in series of papers [Nee98], [Nee99], [Nee01a] (forrecent development about theorem of heart into K -theory of infinite categories, see [Sos17], [Fon18], [Rap18],[Bar15]). The theorem says K -theory of a triangulated category is equivalent to K -theory of its heart (for ourversion, see Corollary 5.4.4).Recall that in the celebrated paper written by Thomason (and Trobaugh) [TT90, 1.11.1], Thomason pro-posed an open problem;‘ Find a general result for Waldhausen categories that specialises to Quillen’s d´evissage theorem when appliedto the category of bounded complexes in an abelian category. ’The d´evissage theorem 5.4.1 which we will establish in this article contains Waldhausen’s cell filtration theo-rem 5.4.2, theorem of heart 5.4.4 and Quillen’s d´evissage theorem 5.4.6 as special cases. Namely, we will givean affirmative answer to Waldhausen’s and Thomason’s problems above in some sense. We will turn in thenotions of cell structures (4.1.2) and d´evissage spaces (4.3.1) in Section 4 and our d´evissage theorem statesa structure of non-connective K -theory of d´evissage spaces in terms of non-connective K -theory of heart ofcell structures. The specific feature in our proof is ‘motivic’ in the sense that properties of K -theory which wewill utilize to prove the theorem is only categorical homotopy invariance, localization and cocontinuity (seeCorollary 5.4.8). On the other hands, the analogue of the d´evissage theorem for K -theory does not hold forHochschild homology theory (see [Kel99, 1.11]). In this point of view, we could say that d´evissage theoremis not ‘motivic’ over dg-categories. To overcome this dilemma, the notion of d´evissage spaces should not beexpressed by the language of dg-categories. From Section 1 to Section 3 are devoted to the foundation of ourmodel of stable ( ∞ , ) -categories which we will play on to give a description of d´evissage spaces.In a standpoint of geometry over categories (for example, topoi theory or non-commutative geometry ofabelian categories or dg-categories), a d´evissage condition is related to nilpotent immersions (see 4.3.3). Inthe future work, the author plans to clarify the meaning of a d´evissage condition in a perspective of noncom-mutative motive theory. The conception of complicial biWaldhausen categories (closed under the formations of the canonical homo-topy push-outs and the canonical homotopy pull-backs) is introduced by Thomason in [TT90], which is aspecial class of Waldhausen categories whose underlying categories are full subcategories of categories ofchain complexes on abelian categories. The notion is abstracted and further studied by Schlichting in thesurvey article of algebraic K -theory [Sch11] under the name of complicial exact categories ( with weak equiv-alences ), whose underlying exact categories are equipped with monoidal actions of the symmetric monoidalcategory of bounded chain complexes of finitely generated free abelian groups (there exists a similar conceptwhich is called C ( k ) -model categories and discussed in [Toe11, § bicomplicial categories (or bicomplicial pairs ), we discussthe possibility of simplifying Schlichting’s axioms to establish the theory of complicial exact categories withweak equivalences in [Sch11]. In this section, following the papers [Moc10] and [Moc13b], we recall thenotion of complicial exact categories (with slightly different conventions) from Ibid. and study them further.Now we give a guide for the structure of this section. In the first subsection 1.1, we start by recalling the2otion of exact categories. In the next subsection 1.2, we will introduce the notion of complicial objects ina locally exact 2-category. Practical examples of complicial objects are normal ordinary complicial exactcategories and ordinary complicial structure will be explained in subsection 1.3. In the final subsection 1.4,we will discuss Frobenius complicial exact structure. -categories In this subsection, we recall the notion of exact categories in the sense of Quillen. In particular we study2-category
ExCat of small exact categories and the 2-category
ExCat I of I -diagrams in ExCat for a smallcategory I . The Hom categories in the both categories are equipped with the natural exact structures but theboth categories are not enriched over the category of small exact categories for the reason that the compositionfunctors H om ExCat ( x , y ) × H om ExCat ( y , z ) → H om ExCat ( x , z ) are not exact. Thus as a substitutive concept,we introduce the notion of locally exact -categories (see 1.1.5) which contains ExCat and
ExCat I for asmall category I (see 1.1.8) as typical examples. In the next subsection, this notion will make our treatmentsof complicial structures simplify. Basically, for the conventions of exact categories , we follows the notations in[Qui73]. Recall that a functor between exact categories f : E → F reflects exactness if for a sequence x → y → z in E such that f x → f y → f z is an admissible exact sequence in F , x → y → z is an admissibleexact sequence in E . For an exact category E , we say that its full subcategory F is an exact subcategory if itis an exact category and the inclusion functor F (cid:44) → E is exact and say that F is a strict exact subcategory if itis an exact subcategory and moreover the inclusion functor reflects exactness. We say that F is an extensionclosed ( full ) subcategory of E or closed under extensions in E if for any admissible exact sequence x (cid:26) y (cid:16) z in E , x and z are isomorphic to objects in F respectively, then y is isomorphic to an object in F .We will sometimes use the following lemmata. (cf. [Kel90, Step 1 in the proof of A.1].) Let E be an exact category and letx i (cid:47) (cid:47) j (cid:15) (cid:15) y q (cid:15) (cid:15) z p (cid:47) (cid:47) w (1) be a commutative square in E . If the square ( ) is a push-out ( resp. pull-back ) and the morphism i ( resp. q ) is an admissible monomorphism ( resp. epimorphism ) , then the sequence x (cid:16) i − j (cid:17) (cid:26) y ⊕ z ( q p ) (cid:16) w is an admissibleexact sequence in E . Let E be an exact category and letx (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) y (cid:15) (cid:15) (cid:15) (cid:15) z (cid:47) (cid:47) (cid:47) (cid:47) wbe a commutative square of admissible monomorphisms in E . Then the following conditions are equivalent. ( ) The induced morphism z (cid:116) x y → w is an admissible monomorphism. ( ) The induced morphism z / x → w / y is an admissible monomorphism. ( ) The induced morphism y / x → w / z is an admissible monomorphism.Proof. Assertion that ( ) implies ( ) and ( ) is proven in [Wal85, 1.1.1] and assertion that ( ) or ( ) implies ( ) is proven in [TT90, 1.7.4]. 3 .1.4. ( -categories). In this article, a 2 -category means a category enriched over the category of smallcategories. For a 2-category C and a pair of objects x and y , we write H om C ( x , y ) for the Hom categoryfrom x to y . We call an object in H om C ( x , y ) an 1 -morphism ( from x to y ) and call a morphism a : f → g in H om C ( x , y ) a 2 -morphism ( from f to g ). The compositions of 2-morphisms in the category H om C ( x , y ) iscalled the vertical compositions and denoted by the letter · and the composition of 2-morphisms comes fromthe composition functors H om C ( x , y ) × H om C ( y , z ) → H om C ( x , z ) is called the horizontal compositions anddenoted by ∗ . For example let x , y and z be a triple of objects in C and f , f , f : x → y and g , g , g : y → z be 1-morphisms in C and α : f → f , β : f → f , γ : g → g and δ : g → g be 2-morphisms in C , thenwe have the equality ( δ ∗ β ) · ( γ ∗ α ) = ( δ · γ ) ∗ ( β · α ) . (2)For 1-morphisms f , f : x → y and g , g : y → z and 2-morphisms a : f → f and b : g → g in a2-category, we denote b ∗ a by b ∗ f if f = f and a = id f and by g ∗ a if g = g and b = id g .Let x and y be a pair of objects in a 2-category C . We say that an 1-morphism f : x → y is an equivalence if there exists an 1-morphism g : y → x and a pair of 2-isomorphisms α : f g ∼ → id y and β : id x ∼ → g f (3)We say that a quadruple ( f , g , α , β ) consisting of a pair of 1-morphisms f : x → y and g : y → x and a pair of2-isomorphisms as in ( ) in a 2-category C is an adjoint equivalence from x to y if the following equalitieshold: ( α ∗ f ) · ( f ∗ β ) = id f , (4) ( g ∗ α ) · ( β ∗ g ) = id g . (5)We can show that for any equivalence f : x → y , there exists an 1-morphism g : y → x and a pair of 2-isomorphisms as in ( ) in a 2-category C such that the quadruple ( f , g , α , β ) is an adjoint equivalence.An exemplar of 2-categories is the 2-category of small categories Cat whose objects are small categories,whose 1-morphisms are functors and whose 2-morphisms are natural transformations. The horizontal com-positions and the vertical compositions of natural transformations are given in the following way. For func-tors between categories, f , f , f : X → Y and g , g : Y → Z and natural transformations a : f → f , b : f → f and c : g → g , we define the vertical composition b and a , b · a : f → f and the horizontalcomposition c and a , c ∗ a : g f → g f by the formulas ( b · a )( x ) : = b ( x ) a ( x ) , (6) ( c ∗ a )( x ) : = c ( f ( x )) g ( a ( x )) = g ( a ( x )) c ( f ( x )) (7)for any object x in X . -categories). A local exact -category is a 2-category C such that for anypair of objects x and y in C , the Hom category H om C ( x , y ) is equipped with an exact category structure whichsubjects to the condition that for any triple of objects x , y , z in C and a 1-morphisms f : x → y , the inducedfunctors H om C ( z , f ) : H om C ( z , x ) → H om C ( z , y ) and H om C ( f , z ) : H om C ( y , z ) → H om C ( x , z ) are exactfunctors.In 1.1.7 and 1.1.8, we will provide representatives of locally exact 2-categories. First we consider generalremarks of diagram categories over small exact categories. Let E be an exact category and let I be a category. We say that a sequence x f → y g → z of morphisms in E I the functor category from I to E is a level admissible exact sequence if x ( i ) f ( i ) → y ( i ) g ( i ) → z ( i ) is an admissible exact sequence in E for any object i in I . We assume that I isessentially small. Then E I with level admissible exact sequences is an exact category. Let f : E → E (cid:48) be anexact functor between exact categories. We define f I : E I → E (cid:48) I to be a functor by sending an I -diagram x in E to f x the composition with f . Then f I be an exact functor. Let θ : f → f (cid:48) be a natural transformationbetween exact functors f , f (cid:48) : E → E (cid:48) . We define θ I : f I → f (cid:48) I to be a natural transformation by setting θ I ( x )( i ) : = θ ( x ( i )) (8)for any objects x in E and i in I . Moreover we denote the full subcategory of E I consisting of exactfunctors I → E by Ex ( I , E ) . Then Ex ( I , E ) is a strict exact subcategory of E I . Notice that for any4evel admissible monomorphism f i → g in Ex ( I , E ) and for any morphism x a → y in E , the square below isadmissible in the sense that the induced morphism f ( y ) (cid:116) f ( x ) g ( x ) (cid:26) g ( y ) is an admissible monomorphism in E . (See [Moc10, § A.2].) f ( x ) i x (cid:47) (cid:47) f ( a ) (cid:15) (cid:15) g ( x ) g ( a ) (cid:15) (cid:15) f ( y ) i y (cid:47) (cid:47) g ( y ) . We write
ExCat for the category of small exact categories and exact functors and if weregard
ExCat as a 2-category where the class of 2-morphisms is the class of all natural transformations,then we denote it by
ExCat . Notice that for any pair of small exact categories E and F , the Hom category H om ( E , F ) with level admissible exact sequences is an exact category by 1.1.6 and ExCat is a locally exact2-category. -categories). Let I be a small category and let C be a 2-categoryand E and F be a pair of I -diagrams in C , namely 1-functors I → C and f and g be a pair of naturaltransformations from E to F . A modification from f to g is a family of 2-morphisms θ = { θ i : f i → g i } i ∈ Ob I in C indexed by the set of objects of I subjects to the conditions that for any morphism a : i → j in I , wehave the equality θ j ∗ E a = F a ∗ θ i . (9)We denote this situation by θ : f → g .Moreover let G be an I -diagram in C and let h : E → F and f (cid:48) , g (cid:48) : F → G be natural transformationsand let α : f → g , β : g → h and γ : f (cid:48) → g (cid:48) be modifications. Then we define β · α : f → h and γ ∗ α : f (cid:48) f → g (cid:48) g to be modifications by setting for any object i of I , ( β · α ) i : = β i · α i , (10) ( α (cid:48) ∗ α ) i = α (cid:48) i ∗ α i . (11)We call β · α and α (cid:48) ∗ α the vertical composition of β and α and the horizontal composition α (cid:48) and α respectively. We denote the 2-category of I -diagrams in C by C I whose objects are I -diagrams in C ,whose 1-morphisms are natural transformations and whose 2-morphisms are modifications.Assume that C is locally exact and let E and F be a pair of I -diagrams in C . A sequence f α → g β → h in H om C I ( E , F ) the category of natural transformations from E to F and modifications is a level admissibleexact sequence if for any object i in I , the sequence f i α i → g i β i → h i is a level admissible exact sequence of exactfunctors from E i to F i . We can show that H om C I ( E , F ) with the set of level admissible exact sequences isan exact category and C I is a locally exact 2-category. In particular for a small category I , the 2-category ExCat I of I -diagrams over ExCat is a locally exact 2-category.Next we define the suitable notion of morphisms between locally exact 2-categories. -functors). Let C and D be locally exact 2-categories. A 2-functor f : C → D is locally exact if for any pair of objects x and y in C , the functor f : H om C ( x , y ) → H om D ( f x , f y ) is exact. Let I be a small category. Assume that I is filtered , namely I satisfies the following three conditions: • I is a non-empty category. • For any pair of objects i and j in I , there exists an object k in I and a pair of morphisms i → k and j → k . • For any pair of morphisms a , b : i → j in I , there exists a morphism c : j → k such that ca = cb .Then we have the colimit I : ExCat I → ExCat . We briefly recall the construction of thisfunctor. Let E : I → ExCat be an I -diagram in ExCat . Then we have the equalitiesOb colim I E = colim I Ob E , (12)5or colim I E = colim I Mor E . (13)Namely for example, for the set of objects, we have the equalityOb colim I E = Ob colim I E : = (cid:71) i ∈ Ob I Ob E i / ∼ (14)where the equivalence relation ∼ is defined as follows. For any pair of objects x ∈ Ob E i and y ∈ Ob E j , we saythat x and y are equivalent if there exists a pair of morphisms a : i → k and b : j → k such that E a ( x ) = E b ( y ) and we denote this situation by x ∼ y . Then we can show that the relation ∼ is an equivalence relation on (cid:71) i ∈ Ob I Ob E i . We say that a sequence x → y → z in colim I E is an admissible exact sequence if it representedby an admissible exact sequence in some E i . We can show that colim I E with the set of admissible exactsequences is an exact category and we can show that 2-functor colim I : ExCat I → ExCat is locally exact2-functor.
Let { E i } i ∈ I be a family of pointed categories indexedby a set I . We denote the full subcategory of ∏ i ∈ I E i consisting of those objects x = ( x i ) i ∈ I such that { i ∈ I ; x i (cid:54) = } < + ∞ by (cid:95) i ∈ I E i . We assume that E i is an exact category for all i ∈ I . We say that asequence ( x i ) i ∈ I → ( y i ) i ∈ I → ( z i ) i ∈ I in ∏ i ∈ I E i is a level admissible exact sequence if all i ∈ I , a sequence x i → y i → z i is an admissible exact sequence in E i . ∏ i ∈ I E i with level admissible exact sequences is an exactcategory and (cid:95) i ∈ I E i is a strict exact subcategory of ∏ i ∈ I E i . One of a typical example for a locally exact 2-functor
ExCat → ExCat is the idempotent completion functor. We recall the definition and fundamental properties from[Kar68] and [TT90, § A]. An additive category A is idempotent complete if any idempotent e : x → x with e = e , arises from a splitting of x , x ∼ → Im ( e ) ⊕ Ker ( e ) . For an additive category A , its idempotent completion (cid:99) A is a category whose objects are pair ( x , e ) consisting of an object x in A and an idempotent endomorphism e of x . A morphism a : ( x , e ) → ( x (cid:48) , e (cid:48) ) in (cid:99) A is a morphism a : x → x (cid:48) in A subjects to the condition that ae = e (cid:48) a = a . Then for a pair of objects ( x , e ) and ( y , e (cid:48) ) in (cid:99) A , ( x , e ) ⊕ ( y , e (cid:48) ) ∼ → ( x ⊕ y , (cid:18) e e (cid:48) (cid:19) ) In particular (cid:99) A is an additive category. There is the fully faithful additive functor i A : A → (cid:99) A , x (cid:55)→ ( x , id x ) satisfyingthe following universal property:For any idempotent complete additive category B and any additive functor f : A → B , up to natural equiv-alence, f factor in a unique way through i A .Let E be an exact category. Then we can make its idempotent completion (cid:98) E into an exact category bydeclaring that a composable sequence in (cid:98) E is an admissible exact sequence if and only if it is a direct summandof an admissible exact sequence in E . Then the functor i E : E → (cid:98) E is exact and reflects exactness.Now we define the idempotent completion 2-functor (cid:100) ( − ) : ExCat → ExCat . For an exact functor f : E → F , we define (cid:98) f : (cid:98) E → (cid:99) F to be an exact functor by sending an object ( x , e ) in (cid:98) E to ( f ( x ) , f ( e )) . For anatural transformation θ : f → g between exact functors f , g : E → F , we define (cid:98) θ : f → g to be a naturaltransformation by setting for an object ( x , e ) in (cid:98) E , (cid:98) θ ( x , e ) : = g ( e ) θ ( x ) (15)We can check that this association is 2-functorial and this 2-functor is locally exact. Let I be a small category. Then we define ( − ) I : ExCat → ExCat to be a2-functor by sending an exact category E to E I the category of I -diagrams in E with level admissible exactstructure (See 1.1.6) and sending an exact functor f : E → E (cid:48) to f I : E I → E (cid:48) I and sending a naturaltransformation θ : f → f (cid:48) between exact functors f , f (cid:48) : E → E (cid:48) to θ I : f I → f (cid:48) I . Then we can show that ( − ) I is locally exact and we call it the I -diagram association -functor .6 .1.14. (Topologizing subcategories). Let E be an exact category and let D be a non-empty full subcategoryof E . We say that D is a topologizing subcategory of E if D is closed under finite direct sums and closedunder admissible sub- and quotient objects. The last condition means that for an admissible exact sequence x (cid:26) y (cid:16) z in E , if y is in D , then x and z are also in D . Thus in this case, D contains the zero object andclosed under isomorphisms in E . Namely for any object x in E which is isomorphic to an object in D is alsoin D . The naming of the term ‘topologizing’ comes from noncommutative geometry of abelian categoriesby Rosenberg (see [Ros08, Lecture 2 1.1]). By [Moc13a, 5,3], a topologizing subcategory of E naturallybecomes a strict exact subcategory of E . Let E be an exact category. Wesay that a full subcategory D of E is a Serre subcategory if it is an extensional closed topologizing subcategoryof E . For any full subcategory D of E , we write S √ D for intersection of all Serre subcategories which contain D and call it the Serre radical of D ( in E ).We say that a full subcategory D of E is a semi-Serre subcategory if it is closed under finite direct sumsand for an admissible sequence x (cid:26) y (cid:16) z in E , if x and y are in D , then z is also in D and if y and z are in D , then x is also in D . Let E be an exact category and let D be a semi-Serre subcategory. Then D is a strict exactsubcategory of E .Proof. We will show that ( i ) for a pair of admissible monomorphisms x (cid:26) y and y (cid:26) z with x , y and z are in D , Coker ( x (cid:26) z ) is in D , ( ii ) and in the pushout diagram in E below x (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) y (cid:15) (cid:15) z (cid:47) (cid:47) (cid:47) (cid:47) w , if x , y and z are in D , then w and Coker ( z (cid:26) w ) are also in D . Proof of ( i ) . In the admissible exact sequence x (cid:26) z (cid:16) Coker ( x (cid:26) z ) in E , since x and z are in D , Coker ( x (cid:26) z ) are also in D . Proof of ( ii ) . By 1.1.2, the sequence x (cid:26) z ⊕ y (cid:16) w is an admissible exact sequence in E with x and z ⊕ y arein D . Thus w is in D and considering the admissible exact sequence z (cid:26) w (cid:16) Coker ( z (cid:26) w ) with z and w are in D , it turns out that Coker ( z (cid:26) w ) is in D .We can show similar statements for admissible epimorphisms and hence D is a strict exact subcategoryof E . Let C be a 2-category and x an object in C . A semi-commutative unitary mag-mad on x is a quadruple ( C , ι , r , σ ) consisting of an 1-morphism C : x → x , a pair of 2-morphisms ι : id C → C and r : CC → C and a 2-isomorphism σ : CC ∼ → CC which satisfy the following conditions: (Unitary law). r · ( C ∗ ι ) = r · ( ι ∗ C ) = id C . (Semi-commutativity law). σ · ( C ∗ ι ) = ι ∗ C . (Involution law). σ · σ = id CC . C C ∗ ι (cid:47) (cid:47) id C (cid:32) (cid:32) CC r (cid:15) (cid:15) C ι ∗ C (cid:111) (cid:111) id C (cid:126) (cid:126) C , C C ∗ ι (cid:32) (cid:32) ι ∗ C (cid:126) (cid:126) CC ∼ σ (cid:47) (cid:47) CC . C is a locally exact 2-category (see Definition 1.1.5). A semi-commutative unitarymagmad ( C , ι , r , σ ) on an object x in C is exact if the 2-morphism ι : id x → C is an admissible monomorphismin the Hom category H om C ( x , x ) from x to x . When we consider an exact semi-commutative unitary magmad,we fix an admissible exact sequence id x ι (cid:26) C π (cid:16) T . (16)We call T the suspension (1 -morphism associated with C ).An exact semi-commutative unitary magmad on an object x is stable if the suspension 1-morphism T : x → x is an equivalence. When we consider a stable exact semi-commutative unitary magmad, we fix an adjointequivalence ( T , T − , α , β ) from x to x . Namely T − : x → x is an 1-morphism and a pair of 2-isomorphisms α : T T − ∼ → id x and β : id x ∼ → T − T which satisfy the equalities: ( α ∗ T ) · ( T ∗ β ) = id T , (17) ( T − ∗ α ) · ( β ∗ T − ) = id T − . (18)In the paper [Moc10], we consider C = ExCat the locally exact 2-category of small exact categories(see 1.1.7) and we call an exact semi-commutative unitary magmad on an exact category E a complicialstructure on E . But in this paper we only use stable complicial structures and for simplicity we call stablesemi-commutative unitary magmads on an object x in a locally exact 2-category complicial structures on x .Thus more precisely, a complicial structure on x is a ninefold of ( C , ι , r , σ , T , T − , π , α , β ) consisting of atriple of 1-morphisms C , T and T − : x → x and 2-morphisms ι : id x → C , r : CC → C , π : C → T and a tripleof 2-isomorphisms σ : CC ∼ → CC , α : T T − ∼ → id x and β : id x ∼ → T − T such that a sequence id x ι (cid:26) C π (cid:16) T is exact in the Hom category H om C ( x , x ) from x to x and satisfy the equalities r · ( C ∗ ι ) = r · ( ι ∗ C ) = id C , σ · σ = id CC , σ · ( C ∗ ι ) = ι ∗ C , ( T − ∗ α ) · ( β ∗ T − ) = id T − and ( α ∗ T ) · ( T ∗ β ) = id T . We call an objectequipped with a complicial structure a complicial object . In particular we call a complicial object in ExCat the 2-category of small exact categories a complicial exact category . When we denote a complicial structure,we often omit T , T − , π , α and β in the notations. Here we will illustrate an archetype of complicial exactcategories. Let E be an exact category. Now we give the complicial structure on Ch b ( E ) the category ofbounded chain complexes over E as follows. Here we use the homological notations for chain complexes.Namely boundary morphisms are of degree − C : Ch b ( E ) → Ch b ( E ) is given by sending x to Cx : = Cone id x the canonical mapping coneof the identity morphism of x . Namely the degree n part of Cx is ( Cx ) n = x n − ⊕ x n and the degree n boundarymorphism d Cxn : ( Cx ) n → ( Cx ) n − is given by d Cxn = (cid:18) − d xn − − id x n − d xn (cid:19) . For a chain morphism f : x → y , wedefine C f : Cx → Cy to be a chain morphism by setting ( C f ) n : = (cid:18) f n − f n (cid:19) for an integer n . For anycomplex x , we define ι x : x → C ( x ) , r x : CC ( x ) → C ( x ) and σ x : CC ( x ) → CC ( x ) to be chain morphisms bysetting ( ι x ) n = (cid:18) x n (cid:19) , (19) ( r x ) n = (cid:18) x n − id x n −
00 0 0 id x n (cid:19) and (20) ( σ x ) n = − id x n − x n −
00 id x n − x n . (21)Then Ch b ( E ) together with the quadruple ( C , ι , r , σ ) forms a complicial exact category.In this example, we give the functor T , T − : Ch b ( E ) → Ch b ( E ) and a natural transformation π : C → T and natural equivalences α : T T − ∼ → id Ch b ( E ) and β : id Ch b ( E ) ∼ → T − T by the following formulas for anycomplex x and integer n ; ( T x ) n = x n − , d Txn = − d xn − , (22) ( T − x ) n = x n + , d T − xn = − d xn + , (23)8 π x ) n = (cid:0) id x n − (cid:1) , (24) α = β = id Ch b ( E ) . (25) ζ : TC → CC ). Let x be a complicial object in a locally exact 2-category C . For the exact sequenceof 1-morphisms in H om C ( x , x ) C ι ∗ C (cid:26) CC π ∗ C (cid:16) TC , (26)since we have the retraction r : CC → C of ι ∗ C , there exists the section ζ : TC → CC such that we haveequalities r · ζ = , (27) ( π ∗ C ) · ζ = id TC , (28) ( ι ∗ C ) · r + ζ · ( π ∗ C ) = id CC . (29)In Example 2.3.5, we will give an explicit construction of ζ .In the standard example 1.2.2, for any chain complex x over an exact category and any integer n , themorphism ( ζ x ) n : ( TCx ) n → ( TCx ) n is given by ( ζ x ) n = id x n −
00 id x n − − id x n − . (30) -morphism). Let x be a complicial object in a locally exact 2-category C . Then we define P : x → x to be an 1-morphism and j : T − → P and q : P → id C to be 2-morphisms by setting P : = CT − and j : = ι ∗ T − and q : = α · ( π ∗ T − ) . Then we have the admissible exact sequence T − i (cid:26) P q (cid:16) id C in H om C ( x , x ) . We say that P is the path -morphism ( associated with C ).In the standard example 1.2.2, for any chain complex x over an exact category, the complex Px and thechain morphisms j x : T − x → Px and q x : Px → x are given by ( Px ) n : = x n ⊕ x n + , d Pxn : = (cid:18) d xn − id x n − d xn + (cid:19) , (31) ( j x ) n : = (cid:18) x n + (cid:19) , (32) ( q x ) n : = (cid:0) id x n (cid:1) (33)for any integer n .We will illustrate the 1-morphism P is a dual notion of the 1-morphism C in some sense. In Lemma-Definition 1.3.5, we will show that for a complicial structure ( C , ι , r , σ ) on an exact category E which satisfiesthe certain conditions, we will define a natural transformations s : P → PP and a natural equivalence τ : PP ∼ → PP such that the quadruple ( P , q , s , τ ) is a complicial structure on E op the opposite category of E . (cid:12) -products). (cf. [Moc10, 2.20].) Let x i ( i =
1, 2, 3) be objects in a 2-category C and let O i : x i → x i and f : x → x and g : x → x be 1-morphisms and let c : O f → f O and d : O g → g O be 2-morphisms.Then we define d (cid:12) O c to be a 2-morphism from O g f to g f O by setting d (cid:12) O c : = ( g ∗ c ) · ( d ∗ f ) . (34) -morphisms). A complicial -morphism between complicial objects x → x (cid:48) in a locallyexact 2-category C is a pair of an 1-morphism f : x → x (cid:48) and a 2-morphism c : C x (cid:48) f ∼ → fC x which satisfiesthe equality c · ( ι x (cid:48) ∗ f ) = f ∗ ι x (35)where ( C x , ι x , r x , σ x ) and ( C x (cid:48) , ι x (cid:48) , r x (cid:48) , σ x (cid:48) ) are complicial structures on x and x (cid:48) respectively. We often omit c in the notation. For complicial 1-morphisms x ( f , c ) → x (cid:48) ( g , d ) → x (cid:48)(cid:48) , we define composition of ( g , d ) and ( f , c ) by ( g , d )( f , c ) : = ( g f , d (cid:12) C c ) . 9e say that a complicial 1-morphism ( f , c ) : x → x (cid:48) is normal if it satisfies the following two conditions ( f ∗ r x ) · ( c ∗ C x ) · ( C x (cid:48) ∗ c ) = c · ( r x (cid:48) ∗ f ) , (36) ( f ∗ σ x ) · ( c ∗ C x ) · ( C x (cid:48) ∗ c ) = ( c ∗ C x ) · ( C x (cid:48) ∗ c ) · ( σ x (cid:48) ∗ f ) . (37) C x (cid:48) C x (cid:48) f C x (cid:48) ∗ c ∼ (cid:47) (cid:47) r x (cid:48) ∗ f (cid:15) (cid:15) C x (cid:48) fC x c ∗ C x ∼ (cid:47) (cid:47) fC x C xf ∗ r x (cid:15) (cid:15) C x (cid:48) f ∼ c (cid:47) (cid:47) fC x , C x (cid:48) C x (cid:48) f C x (cid:48) ∗ c ∼ (cid:47) (cid:47) σ x (cid:48) ∗ f (cid:111) (cid:15) (cid:15) C x (cid:48) fC x c ∗ C x ∼ (cid:47) (cid:47) fC x C xf ∗ σ x (cid:111) (cid:15) (cid:15) C x (cid:48) C x (cid:48) f C x (cid:48) ∗ c ∼ (cid:47) (cid:47) C x (cid:48) fC x c ∗ C x ∼ (cid:47) (cid:47) fC x C x . We say that a complicial 1-morphism ( f , c ) : x → x (cid:48) is strictly normal if it is normal and moreover itsatisfies the following two equalities: C x (cid:48) f = fC x , (38) c = id fC x . (39)We can show that for a pair of composable normal (resp. strictly normal) complicial 1-morphisms x ( f , c ) → x (cid:48) ( g , d ) → x (cid:48)(cid:48) , the compositions ( g f , d (cid:12) C c ) : x → x (cid:48)(cid:48) is also a normal (resp. strictly normal) complicial1-morphism. In particular we call complicial 1-morphisms in ExCat complicial exact functors . Let x be an object in a locally exact 2-category C . We say that acomplicial structure ( C , ι , r , σ ) on x is normal (resp. strictly normal ) if the pair ( C , σ ) is a normal (resp.strictly normal) complicial 1-morphism x → x . We say that a complicial object is normal (resp. strictlynormal ) if its complicial structure is normal (resp. strictly normal). We can show that the standard complicialstructure on the category of bounded chain complexes on an exact category (see 1.2.2) is strictly normal. -morphisms). A complicial -morphisms between complicial 1-morphisms ( f , c ) , ( g , d ) : x → x (cid:48) from ( f , c ) to ( g , d ) in a locally exact 2-category C is a 2-morphism φ : f → g which subjects to the conditionthat d · ( C x (cid:48) ∗ φ ) = ( φ ∗ C x ) · c . (40)In particular we call complicial 2-morphisms in ExCat complicial natural transformations . We can checkthat for any triple of complicial 2-morphisms ϕ : ( f , c ) → ( g , d ) , ϕ (cid:48) : ( g , d ) → ( h , e ) and ψ : ( f (cid:48) , c (cid:48) ) → ( g (cid:48) , d (cid:48) ) between complicial 1-morphisms ( f , c ) , ( g , d ) , ( h , e ) : x → x (cid:48) and ( f (cid:48) , c (cid:48) ) , ( g (cid:48) , d (cid:48) ) : x (cid:48) → x (cid:48)(cid:48) between complicialobjects, the vertical and the horizontal compositions ϕ (cid:48) · ϕ and ψ ∗ ϕ are again complicial 2-morphisms ( f , c ) → ( h , e ) and ( f (cid:48) f , c (cid:48) (cid:12) C c ) → ( g (cid:48) g , d (cid:48) (cid:12) C d ) respectively. ( − ) ). Let C be a locally exact 2-category. We write Comp ( C ) for the category ofcomplicial objects and complicial 1-morphisms in C and if we regard Comp ( C ) as a 2-category whose 2-morphisms are complicial 2-morphisms with the usual horizontal and vertical compositions, then we denoteit by Comp ( C ) . We write Comp nor ( C ) and Comp sn ( C ) (resp. Comp nor ( C ) and Comp sn ( C ) ) for the(2-)subcategory of Comp ( C ) (resp. Comp ( C ) ) consisting of complicial objects and normal complicial 1-morphisms and complicial objects and strictly normal complicial 1-morphisms respectively (and complicial2-morphisms).Let x and y be a pair of complicial objects in C . For a pair of complicial 1-morphisms ( f , c ) , ( g , d ) : x → y and a pair of complicial 2-morphisms φ , ψ : ( f , c ) → ( g , d ) , we consider the addition φ + ψ : f → g inHom C ( f , g ) . We can show that φ + ψ can be regarded as a complicial 2-morphism φ + ψ : ( f , c ) → ( g , d ) and by this addition, H om Comp ( C ) ( x , y ) is enriched by the category of abelian groups. We say that a sequenceof complicial 1-morphisms from x to y , ( f , c ) u → ( g , d ) v → ( h , e ) in H om Comp ( C ) ( x , y ) is an admissible exactsequence if the sequence f u → g v → h is an admissible exact sequence in H om C ( x , y ) . Then we can show that H om Comp ( C ) ( x , y ) with the class of admissible exact sequences is an exact category. For example, push-outof admissible monomorphism ( f , c ) (cid:44) → ( g , d ) by a morphism ( f , c ) → ( h , e ) in H om Comp ( C ) ( x , y ) is given10y ( h (cid:116) f g , b · ( e (cid:116) c d ) · a ) where h (cid:116) f g and e (cid:116) c d : C y h (cid:116) C y f C y g ∼ → hC x (cid:116) fC x gC x are cofiber products in thecategory H om C ( x , y ) and a : C y ( h (cid:116) f g ) ∼ → C y h (cid:116) C y f C y g and b : hC x (cid:116) fC x gC x ∼ → ( h (cid:116) f g ) C x are canonicalisomorphisms. We can show that by this exact structures, Comp ( C ) and Comp nor ( C ) are locally exact2-category.For a locally exact 2-category C and a small category I , inspection shows the following equality Comp sn ( C I ) = ( Comp sn ( C )) I . (41)We can show the following lemma. Let C and C (cid:48) be a pair of locally exact -categories and let f : C → C (cid:48) be a locally exact -functor. Then f induces a -functors Comp ( f ) : Comp ( C ) → Comp ( C (cid:48) ) and Comp sn ( f ) : Comp sn ( C ) → Comp sn ( C (cid:48) ) . By the equality ( ) and the lemma above, we obtain the following: Let C be a locally exact -category and let I be a small category. Assume that C isclosed under I -index exact colimits. Namely there exists a locally exact -functor colim I : C I → C . Then Comp sn ( C ) is also closed under I -index exact colimits. That is, there exists a locally exact -functor colim I : Comp sn ( C ) I → Comp sn ( C ) . To calculate general filtered colimits in
Comp nor ( ExCat ) , the following strification lemma is useful. Let I be a small filtered category and let E : I → Comp nor ( ExCat ) be adiagram of small complicial exact categories. Then there exists an I -diagram E (cid:48) : I → Comp sn ( ExCat ) and the natural equivalence Θ : E ∼ → E (cid:48) .Proof. For an object i in I , we define E (cid:48) i to be a category whose objects are pairs ( a : j → i , x ) consisting of amorphism a : j → i in I and an object x in E j and whose morphisms u : ( a : j → i , x ) → ( a (cid:48) : j (cid:48) → i , x (cid:48) ) is a morphism u : E a ( x ) → E (cid:48) a (cid:48) ( x ) in E i . For a morphism a : i → j in I , we define E (cid:48) a : E (cid:48) i → E (cid:48) j tobe a functor by sending an object ( α : k → i , x ) in E (cid:48) i to an object ( a α : k → j , x ) in E (cid:48) j and a morphism g : ( α : k → i , x ) → ( β : l → i , y ) in E (cid:48) i to a morphism E a ( g ) : ( a α : k → j , x ) → ( a β : l → j , y ) in E (cid:48) j . Foran object i in I , we define Θ i : E (cid:48) i → E i and Ψ i : E i → E (cid:48) i to be a pair of functors by sending an object ( α : j → i , x ) in E (cid:48) i to E a ( x ) in E i and an object y in E i to an object ( id i : i → i , y ) in E (cid:48) i respectively. More-over for an object ( α : j → i , x ) in E (cid:48) i , we set u ( α : j → i , x ) : = id E a ( x ) : ( α : j → i , x ) ∼ → ( id i : i → i , E a ( x )) .Then we can show that Θ i Ψ i = id E i and u : id E (cid:48) i ∼ → Ψ i Θ i and thus Θ i : E (cid:48) i ∼ → E i is an equivalence of cat-egories. We can make E (cid:48) into an exact category such that the functor Θ i : E (cid:48) i → E i is an exact func-tor. We define C i : E (cid:48) i → E (cid:48) i , ι (cid:48) i : id E (cid:48) i → C (cid:48) i , r (cid:48) i : C (cid:48) i C (cid:48) i → C (cid:48) i and σ (cid:48) i : C (cid:48) i C (cid:48) i ∼ → C (cid:48) i C (cid:48) i to be a functor and nat-ural transformations respectively by sending a morphism g : ( a : j → i , x ) → ( b : k → i , y ) to a morphism ( c b )( C i g )( c − a ) : ( a : j → i , C j x ) → ( b : k → i , C k y ) and by setting ι (cid:48) i ( a : j → i , x ) : = ( c a )( ι E a ( x ) ) : ( a : j → i , x ) → ( a : j → i , C j x ) , r (cid:48) i ( a : j → i , x ) : = ( c ax )( r jc ax )( C i ∗ c ax − )( c ax ∗ C j x − ) and σ (cid:48) i ( a : j → i , x ) : = σ ix : ( a : j → i , x ) ∼ → ( a : j → i , x ) for an object ( a : j → i , x ) in E i respectively where c a : C i E a ∼ → E a C j is a part of com-plicial 1-functor ( E a , c a ) : E i → E j associated to a morphism a : i → j and C i : E i → E i , ι i : id E i → C i , r i : C i C i → C i and σ i : C i C i ∼ → C i C i are parts of complicial structure on E i . Then we can show that thequadruple ( C (cid:48) i , ι (cid:48) i , r (cid:48) i , σ (cid:48) i ) is a complicial structure on E (cid:48) i and the equivalence Θ i : E (cid:48) i ∼ → E i is a complicial1-morphism and u : id E (cid:48) i ∼ → Ψ i Θ i is a complicial 2-morphism. Moreover we can regard E (cid:48) is a I -diagram on Comp sn ( ExCat ) . By Lemma 1.2.10, the idempotent completion2-functor (cid:100) ( − ) : ExCat → ExCat induces a 2-functor
Comp ( (cid:100) ( − )) : Comp ( ExCat ) → Comp ( ExCat ) . Inparticular for a complicial exact category C , (cid:98) C the idempotent completion of C has a natural complicialstructure induced from the functor Comp ( (cid:100) ( − )) and the canonical exact functor i C : C → (cid:98) C , x (cid:55)→ ( x , id x ) is astrictly normal complicial exact functor. Let I be a small category. Then the I -diagram associ-ation 2-functor ( − ) I : ExCat → ExCat (see 1.1.13) induces a 2-functor
Comp (( − ) I ) : Comp ( ExCat ) → omp ( ExCat ) by Lemma 1.2.10. In particular for a complicial exact category C , C I the category of I -diagrams in C has the complicial structure induced from the functor Comp (( − ) I ) which we call a levelcomplicial structure ( on C ). (cf. [Moc10, 2.21, 2.33, 2.35].) Let C be a locally exact -category. ( ) Let ( f , c ) : x → x (cid:48) be a complicial -morphism in C . Then there exists a -isomorphism c F : F f ∼ → f F forany F ∈ { T , T − , P } which characterized by the following equalities:c T · ( π ∗ f ) = ( f ∗ π ) · c , (42) c T − : = ( β − ∗ f T − ) · ( T − ∗ c − T ∗ T − ) · ( T − f ∗ α − ) , (43) c P : = ( c ∗ T − ) · ( C ∗ c T − ) . (44) We also have the following equalities: c P · ( j C (cid:48) ∗ f ) = ( f ∗ j C ) · c T − , (45) ( f ∗ q C ) · c P = q C (cid:48) ∗ f . (46) ( ) Let ( f , c ) , ( g , d ) : x → x (cid:48) be complicial -morphisms between complicial objects and let φ : ( f , c ) → ( g , d ) be a complicial -morphism in C . Then for any F ∈ { T , T − , P } , we have the equalityd F · ( F ∗ φ ) = ( φ ∗ F ) · c F . (47) ( ) Let ( f , c ) : x → x (cid:48) and ( g , d ) : x (cid:48) → x (cid:48)(cid:48) be complicial -morphisms between complicial objects in C . Thenfor any F ∈ { T , T − , P } , we have the equality ( d (cid:12) C c ) F = d F (cid:12) F c F . (48) ( ) For an -morphism F : x → x in F ∈ { T , T − , P } , by applying ( ) to the complicial -morphism ( C , σ ) : x → x, there exists a -isomorphism σ F : FC ∼ → CF. Then the pair ( F , σ − F ) is a complicial -morphism from x tox. Namely we have the equalities: σ − T · ( ι ∗ T ) = T ∗ ι (49) σ − T − · ( ι ∗ T − ) = T − ∗ ι (50) σ − P · ( ι ∗ P ) = P ∗ ι (51) (cf. [Moc10, 2.52, 2.54].) Let C be a locally exact 2-category and let x be acomplicial object. For any pair ( F , G ) of 1-morphisms from x to x , where F , G ∈ { C , T , T − , P } by applying1.2.15 ( ) for a complicial 1-morphism ( F , σ − F ) : x → x , we define τ F , G to be a 2-isomorphism FG ∼ → GF bythe formula τ F , G : = ( σ − G ) F , (52)where by convention, we set σ − C = σ − (= σ ) . Then we have the equality ( τ F , G ) − = τ G , F (53)for any F and G in { C , T , T − , P } . For simplicity we set τ F : = τ F , F .In the standard example 1.2.2, the natural equivalence τ F , G : FG ∼ → GF is given by the following formulasfor any chain complex x on an exact category and an integer n ; G \ F C T T − PC − id xn − xn − xn − xn (cid:32) − id xn − xn − (cid:33) (cid:18) − id xn
00 id xn + (cid:19) id xn − xn − id xn xn + T (cid:32) − id xn − xn − (cid:33) − id xn − − id xn (cid:18) id xn − − id xn (cid:19) T − (cid:18) − id xn
00 id xn + (cid:19) − id xn − id xn + (cid:32) id xn + − id xn + (cid:33) P id xn − − id xn
00 id xn xn + (cid:18) id xn − − id xn (cid:19) (cid:32) id xn + − id xn + (cid:33) id xn xn + xn + − id xn + (54)12 .2.17. Lemma. Let C be a locally exact -category and let ( f , c ) : x → x (cid:48) be a complicial -morphismbetween complicial objects in C . Then we have the equalities: α ∗ f = ( f ∗ α ) · ( c T ∗ T − ) · ( T ∗ c T − ) , (55) f ∗ β = ( c T − ∗ T ) · ( T − ∗ c T ) · ( β ∗ f ) . (56) T T − f ∼ T ∗ c T − (cid:47) (cid:47) (cid:111) α ∗ f (cid:15) (cid:15) T f T − (cid:111) c T ∗ T − (cid:15) (cid:15) f f T T − , ∼ f ∗ α (cid:111) (cid:111) f ∼ f ∗ β (cid:47) (cid:47) (cid:111) β ∗ f (cid:15) (cid:15) f T − TT − T f ∼ T − ∗ c T (cid:47) (cid:47) T − f T . (cid:111) c T − ∗ T (cid:79) (cid:79) Proof.
The equality ( ) follows from the left commutative diagram below and the equality ( ) followsfrom the right commutative diagram below: T T − f ∼ TT − f ∗ α − (cid:47) (cid:47) (cid:111) α ∗ f (cid:15) (cid:15) T T − f T T − ∼ TT − ∗ c − T ∗ T − (cid:47) (cid:47) (cid:111) α ∗ fTT − (cid:15) (cid:15) T T − T f T − (cid:111) T ∗ β − ∗ fT − (cid:15) (cid:15) f f T T − ∼ c − T ∗ T − (cid:47) (cid:47) ∼ f ∗ α − (cid:111) (cid:111) T f T − , T − T f ∼ T − ∗ c T (cid:47) (cid:47) id TT − f (cid:35) (cid:35) T − f T ∼ T − f ∗ α − ∗ T (cid:47) (cid:47) (cid:111) T − ∗ c − T (cid:15) (cid:15) T − f T T − T (cid:111) T − ∗ c − T ∗ T − T (cid:15) (cid:15) T − T f ∼ T − T f ∗ β (cid:47) (cid:47) T − T f T − Tf f ∗ β ∼ (cid:47) (cid:47) β ∗ f (cid:111) (cid:79) (cid:79) f T T − . (cid:111) β ∗ fT − T (cid:79) (cid:79) Let E be a complicial exact category and let A be a strict exactsubcategory of E . If for any object x in A , Cx and T ± x are also in A , then the restriction of the complicialstructure on E to A is a complicial structure on A . We say that A with this complicial structure is a complicial exact subcategory of E .Recall the definition of coproduct of exact categories from 1.1.11. Let { C i } i ∈ I be a family of complicial exactcategories indexed by a set I . Then we define C : ∏ i ∈ I C i → ∏ i ∈ I C i and ι : id ∏ i ∈ I C i → C , r : CC → C and σ : CC ∼ → CC to be a functor and natural transformations and a natural equivalence by setting component-wisely. Then the quadruple ( C , ι , r , σ ) is a complicial structure on ∏ i ∈ I C i and (cid:95) i ∈ I C i is a complicial exactsubcategory of ∏ i ∈ I C i . Let E be a complicial exact category and let A be a full subcategory of E . We say that A is a complicial topologizing subcategory of E if A is topologizing(see 1.1.14) and if for any object x in A , Cx and T − x are also in A . Then since A is closed under admissiblequotient objects, for any object x in A , T x is in A . Thus A is a complicial exact subcategory of E . In this subsection, we introduce the specific class of complicial structures, namely ordinary complicial struc-tures. We will associate an ordinary complicial structure on an exact category with its dual complicial structurein Lemma-Definition 1.3.5. We start by defining commutativity of semi-commutative unital magmads (recallthe definition of semi-commutative unital magmads from 1.2.1).13 .3.1. Definition (Commutative complicial structure).
We say that a semi-commutative unital magmad ( C , ι , r , σ ) on an object x in a locally exact 2-category category C is commutative if we have an equality r · σ = r . (57) CC ∼ σ (cid:47) (cid:47) r (cid:32) (cid:32) CC r (cid:126) (cid:126) C . We say that a complicial object is commutative if its complicial structure is commutative.The category of bounded chain complexes Ch b ( E ) over an exact category E equipped with the standardcomplicial structure in 1.2.2 is a commutative complicial exact category.Recall the definition of the 2-morphisms ζ : TC → T T , τ T ± : T ± T ± ∼ → T ± T ± from 1.2.3 and Lemma-Definition 1.2.16 respectively. l : TC → TC ). Let x be a commutative complicial object in a locally exact 2-category. Then thereexists an involution l of TC , which is characterized by the equality σ · ζ = ζ · l . (58) TC l (cid:15) (cid:15) ζ (cid:47) (cid:47) CC r (cid:47) (cid:47) σ (cid:15) (cid:15) CTC ζ (cid:47) (cid:47) CC r (cid:47) (cid:47) C . In the standard example 1.2.2, the chain morphism l : TC → TC is given by l = − id TC . (59) Let C be a complicial exact category. We wish to define s : P → PP to be a natural transformationwhich makes the quadruple ( P op , q op , s op , τ P op ) a complicial structure on C op the opposite category of C .There are two kinds of candidates of definition of the natural transformation s . Namely we set s : = ( C ∗ τ C , T − ∗ T − ) · ( σ ∗ T − T − ) · ( ζ ∗ T − T − ) · ( τ C , T ∗ T − T − ) · ( C ∗ τ T − , T ∗ T − ) · ( P ∗ α − ) , (60) s : = ( C ∗ τ C , T − ∗ T − ) · ( CC ∗ τ T − ) · ( ζ ∗ T − T − ) · ( TC ∗ τ T − ) · ( T ∗ τ T − , C ∗ T − ) · ( α − ∗ P ) . (61)Then the equation ( π ∗ T − T − ) · ( ζ ∗ T − T − ) = id T − T − (62)and the commutative diagrams ( ) below show the equalities ( P ∗ q ) · s = ( q ∗ P ) · s = id P . (63) CT − CT − CT ∗ π ∗ T − (cid:15) (cid:15) CCT − T − ∼ C ∗ τ C , T − ∗ T − (cid:111) (cid:111) C ∗ π ∗ T − T − (cid:15) (cid:15) CCT − T − ∼ CC ∗ τ T − (cid:47) (cid:47) ∼ σ ∗ T − T − (cid:111) (cid:111) π ∗ CT − T − (cid:15) (cid:15) CCT − T − ∼ C ∗ τ T − , C ∗ T − (cid:47) (cid:47) π ∗ CT − T − (cid:15) (cid:15) CT − CT − π ∗ CT − (cid:15) (cid:15) CT − T T − ∼ C ∗ τ T − , T ∗ T − (cid:47) (cid:47) CT T − T − ∼ τ C , T ∗ T − T − (cid:47) (cid:47) TCT − T − TCT − T − ∼ TC ∗ τ T − (cid:111) (cid:111) T T − CT − . ∼ T ∗ τ T − , C ∗ T − (cid:111) (cid:111) (64)We propose a sufficient condition which provides an equality s = s .14 .3.4. Definition (Ordinary complicial structure). A complicial structure on an object in a locally exact2-category is ordinary if it is commutative and if the following equality holds l ∗ τ T − = id TCT − T − . (65)A complicial structure on an object is strictly ordinary if it is commutative and we have the equalities ( ) and τ T = − id TT , and (66) τ T − = − id T − T − . (67)We can show that a strictly ordinary complicial structure is ordinary. We say that a complicial object is ordinary (resp. strictly ordinary ) if its complicial structure is ordinary (resp. strictly ordinary). For example,the standard complicial structure on Ch b ( E ) the category of bounded chain complexes over an exact category E (see 1.2.2) is strictly ordinary. (cf. [Moc10, 2.60].) Let C be an ordinary complicial exact category. Then ( ) We have the equality s = s (68)of natural transformations P → PP . ( ) We define s : P → PP to be a natural transformation by setting s : = s (= s ) . (69)Then the quadruple ( P , q , s , τ P ) is a commutative complicial structure on C op the opposite category of C andwe call it the dual complicial structure of ( C , ι , r , σ ) . ( ) We have the equality s · ( r ∗ T − ) = ( r ∗ T − P ) · ( C ∗ s ) . (70) Proof. ( ) By applying Lemma 1.2.17 to the complicial functor ( P , σ − P ) : C → C , we obtain the equality ( τ P , T ∗ T − ) · ( P ∗ α − ) = ( T ∗ τ T − , P ) · ( α − ∗ P ) . (71)Namely we have the equality ( τ C , T ∗ T − T − ) · ( c ∗ τ T − , T ∗ T − ) · ( P ∗ α − ) = ( TC ∗ τ T − ) · ( T ∗ τ T − , C ∗ T − ) · ( α − ∗ P ) . (72)On the other hands, by the equality ( ) , we have the equalities ( CC ∗ τ T − ) · ( ζ ∗ T − T − ) = ( ζ ∗ T − T − ) · ( TC ∗ τ T − )= ( ζ ∗ T − T − ) · ( l ∗ T − T − ) = ( σ ∗ T − T − ) · ( ζ ∗ T − T − ) . (73)Thus we obtain the equality ( ) . ( ) We have the commutative diagram below: P (cid:111) P ∗ α − (cid:15) (cid:15) P (cid:111) α − ∗ P (cid:15) (cid:15) CT − T T − ∼ C ∗ τ T − , T ∗ T − (cid:47) (cid:47) CT T − T − ∼ τ C , T ∗ T − T − (cid:47) (cid:47) TCT − T − ζ ∗ T − T − (cid:15) (cid:15) TCT − T − ∼ TC ∗ τ T − (cid:111) (cid:111) T T − CT − ∼ T ∗ τ T − , C ∗ T − (cid:111) (cid:111) CT − CT − CCT − T − ∼ C ∗ τ C , T − ∗ T − (cid:111) (cid:111) CCT − T − ∼ CC ∗ τ T − (cid:47) (cid:47) ∼ σ ∗ T − T − (cid:111) (cid:111) CCT − T − ∼ C ∗ τ T − , C ∗ T − (cid:47) (cid:47) CT − CT − PP τ P ∼ (cid:47) (cid:47) PP τ P · s = τ P . We apply the equality ( ) to ( f , c ) = ( P , σ − P ) , we obtain theequality q ∗ P = ( P ∗ q ) · τ P . (74)Thus it turns out that the quadruple ( P , q , s , τ P ) is a commutative complicial structure on the opposite category C op of C by Lemma-Definition 1.2.15 and Lemma-Definition 1.2.16. ( ) By the virtue of the equality ( ) which will be proven in Lemma 2.3.7, we have the following commu-tative diagram and it implies the equality ( ) . Notice that there is no circular reasoning in our argument. CP ∼ CP ∗ α − (cid:47) (cid:47) r ∗ T − (cid:15) (cid:15) CPT T − ∼ CC ∗ τ T − , T ∗ T − (cid:47) (cid:47) r ∗ T − TT − (cid:15) (cid:15) CCT T − T − ∼ C ∗ τ C , T ∗ T − T − (cid:47) (cid:47) r ∗ TT − T − (cid:15) (cid:15) CTCT − T − C ∗ ζ ∗ T − T − (cid:47) (cid:47) CCCT − T − ∼ CCC ∗ τ T − (cid:47) (cid:47) r ∗ PT − (cid:15) (cid:15) CCCT − T − ∼ CC ∗ τ C , T − ∗ T − (cid:47) (cid:47) r ∗ PT − (cid:15) (cid:15) CPP r ∗ T − P (cid:15) (cid:15) P ∼ P ∗ α − (cid:47) (cid:47) PT T − ∼ C ∗ τ T − , T ∗ T − (cid:47) (cid:47) CT T − T − ∼ τ C , T ∗ T − T − (cid:47) (cid:47) TCT − T − ζ ∗ T − T − (cid:47) (cid:47) CCT − T − ∼ CC ∗ τ T − (cid:47) (cid:47) CCT − T − ∼ C ∗ τ C , T − ∗ T − (cid:47) (cid:47) PP . In this subsection, we recall Frobenius exact structure on complicial exact categories. We start by recallingFrobenius categories. A Frobenius category is an exact category such that it has enough injectiveobjects and projective objects and the class of injective objects and projective objects coincided. The naming‘Frobenius’ comes from the following Frobenius theorem:
A category of modules over a group ring of a finite group over a commutative field is a Frobenius category.
We give a typical example of Frobenius categories. Let A be an additive category and let Ch b ( A ) be thecategory of bounded chain complexes on A . We regard Ch b ( A ) as an exact category by declaring degree-wised split short exact sequences to be admissible exact sequences. Then Ch b ( A ) is a Frobenius category anda chain complex in Ch b ( A ) is projective-injective if and only if it is contractible, namely chain homotopic tothe zero complex. (See [Moc10, 2.26, 2.27, 2.30, 2.40, 2.46], [Sch11, 6.5, B.16], [Moc13b,3.6].) Let E be a complicial exact category. An admissible monomorphism x i (cid:26) y in E is Frobenius if forany object u of E and any morphism x f → Cu , there exists a morphism y g → Cu such that f = gi . Similarly anadmissible epimorphism x p (cid:16) y in E is Frobenius if for any object u of E and any morphism Cu f → y , thereexists a morphism Cu g → x such that f = pg .In an admissible exact sequence x i (cid:26) y p (cid:16) z (75)in E , i is Frobenius if and only if p is Frobenius. In this case we call the sequence ( ) a Frobenius admissibleexact sequence . We denote E with the Frobenius exact structure by E frob and we can show that E frob is aFrobenius category. Moreover, we can show that E frob together with the original complicial structure is acomplicial exact category. The purpose of this section is to establish a theory of higher homotopical structures on complicial exactcategories. Many notions on a category of chain complexes on an additive category are generalized to thoseon a complicial exact category. In the first subsection 2.1, we will generalize the notion of chain homotopiesand in the next subsection 2.2, we will introduce the category of homotopy commutative diagrams on a normalordinary complicial exact category. In the last subsection 2.3, we will study mapping cone on a complicialexact category. 16 .1 C -homotopy and P -homotopy We can generalize the notion of chain homotopies in a category of chain complexes on an additive categoryto a complicial exact category. C -homotopy, P -homotopy). For a complicial exact category C , morphisms f , g : x → y in C are( C- ) homotopic if there exists a morphism H : Cx → y such that f − g = H ι x . We denote this situation by H : f ⇒ C g and we say that H is a C-homotopy from f to g . A morphism f : x → y in C is a ( C- ) homotopyequivalence if there exists a morphism g : y → x such that g f and f g are homotopic to id x and id y respectively.Then we say that x and y are ( C- ) homotopy equivalent .Similarly we can define the notion of P-homotopy . Namely a pair of morphisms f , g : x → y in C is P-homotopic if there exists a morphism H : x → Py such that q y H = f − g . In this situation, we denote by H : f ⇒ P g and call H a P-homotopy ( from f to g ).We denote the set of C -homotopies from f to g by Hom C ( f , g ) and the set of P -homotopies from f to g by Hom P ( f , g ) .Assume that C is ordinary (see Definition 1.3.4). Recall the definition of the natural transformation s : P → PP from Lemma-Definition 1.3.5. Then first we define s (cid:48) : C → PC to be a natural transformation bysetting s (cid:48) : = ( PC ∗ β − ) · ( s ∗ T ) · ( C ∗ β ) . (76)Second we define A f , g : Hom C ( f , g ) → Hom P ( f , g ) and B f , g : Hom P ( f , g ) → Hom C ( f , g ) to be maps bysending a C -homotopy H : f ⇒ C g to a P -homotopy from f to gA f , g ( H ) : = PH · s (cid:48) x · ι x (77)and sending a P -homotopy H (cid:48) : f ⇒ P g to a C -homotopy from f to gB f , g ( H (cid:48) ) : = q y · ( r ∗ T − y ) · CH (cid:48) . (78) Let Ch b ( E ) be the category of bounded chain complexes on an exact category E equippedwith the standard complicial structure in 1.2.2 and let f , g : x → y be a pair of chain morphisms in Ch b ( E ) .Recall that a chain homotopy from f to g is a family of morphisms { h n : x n → y n + } n ∈ Z in E indexed by theset of integers such that it satisfies the equality d yn + h n + h n − d xn = f n − g n (79)for any integer n . Then we define H : Cx → y and H (cid:48) : x → Py to be chain morphisms by setting H n : = (cid:0) − h n − f n − g n (cid:1) (80)and H (cid:48) n : = (cid:18) f n − g n − h n (cid:19) (81)for any integer n . We can show that H is a C -homotopy from f to g and H (cid:48) is a P -homotopy form f to g .Conversely if we give a C -homotopy or a P -homotopy from f to g , it provides a chain homotopy from f to g .Thus the notion of chain homotopies, C -homotopies and P -homotopies are equivalent in the standard exampleof the complicial exact category Ch b ( E ) .In the general situation, the relationship of C -homotopies and P -homotopies is summed up with the fol-lowing. Let f , g : x → y be a pair of morphisms in a complicial exact category C . Then ( ) f ⇒ C g if and only if f ⇒ P g. ( ) Moreover we assume that C is ordinary. Then for the maps A f , g : Hom C ( f , g ) → Hom P ( f , g ) andB f , g : Hom P ( f , g ) → Hom C ( f , g ) , we have equalities B f , g A f , g = id Hom C ( f , g ) and A f , g B f , g = id Hom P ( f , g ) . roof. ( ) If there exists a morphism H : Cx → y such that H ι x = f − g , then since q y is a Frobenius admissibleepimorphism, there exists a morphism H (cid:48) : Cx → Py such that q y H (cid:48) = H . Then q y H (cid:48) ι x = H ι x = f − g . Thus f ⇒ P g . x H (cid:48)(cid:48) (cid:47) (cid:47) ι x (cid:15) (cid:15) Py q y (cid:15) (cid:15) Cx H (cid:47) (cid:47) (cid:62) (cid:62) y . Conversely if there is a morphism H (cid:48)(cid:48) : x → Py such that q y H (cid:48)(cid:48) = f − g , then since ι x is a Frobenius admissiblemorphism, there exists a morphism H (cid:48) : Cx → Py such that H (cid:48) ι x = H (cid:48)(cid:48) . Then q y H (cid:48) ι x = q y H (cid:48)(cid:48) = f − g . Thus f ⇒ C g . ( ) Let H : f ⇒ C g and H (cid:48) : f ⇒ P g be a C -homotopy and a P -homotopy from f to g respectively. Then wehave the commutative diagrams below by virtue of Lemma 2.1.4. Cx C ∗ ι x (cid:47) (cid:47) (cid:32) (cid:32) CCx C ∗ s (cid:48) x (cid:47) (cid:47) r x (cid:15) (cid:15) CPCx r ∗ T − Cx (cid:15) (cid:15) CPH (cid:47) (cid:47)
CPy r ∗ T − y (cid:15) (cid:15) Cx s (cid:48) x (cid:47) (cid:47) id Cx (cid:34) (cid:34) Pcx PH (cid:47) (cid:47) q ∗ C x (cid:15) (cid:15) Py q y (cid:15) (cid:15) Cx H (cid:47) (cid:47) y , x H (cid:48) (cid:47) (cid:47) ι x (cid:126) (cid:126) ι x (cid:15) (cid:15) Py ι ∗ P y (cid:15) (cid:15) id Py (cid:34) (cid:34) Cx id Cx (cid:47) (cid:47) s (cid:48) x (cid:32) (cid:32) Cx CH (cid:48) (cid:47) (cid:47) CPy r ∗ T − y (cid:47) (cid:47) PyPCx
PCH (cid:48) (cid:47) (cid:47) q ∗ C x (cid:79) (cid:79) PCPy q ∗ CP y (cid:79) (cid:79) P ∗ r ∗ T − y (cid:47) (cid:47) PPy . q ∗ P y (cid:79) (cid:79) The left diagram implies B f , g A f , g ( H ) = H and the right diagram shows A f , g B f , g ( H (cid:48) ) = H (cid:48) . We complete theproof. Let C be an ordinary complicial exact category. Then we have the following equalities. ( q ∗ C ) · s (cid:48) = id C , (82) ( r ∗ T − C ) · ( C ∗ s (cid:48) ) = s (cid:48) · r . (83) Proof.
By Lemma-Definition 1.3.5, we have the following commutative diagrams and they imply the equali-ties ( ) and ( ) . C C ∗ s (cid:48) (cid:47) (cid:47) CC ∗ β (cid:32) (cid:32) r (cid:15) (cid:15) CPC
CPC ∗ β (cid:123) (cid:123) r ∗ T − C (cid:15) (cid:15) C C ∗ s ∗ T (cid:47) (cid:47) r ∗ T − T (cid:15) (cid:15) CPPT r ∗ T − PT (cid:15) (cid:15) PT s ∗ T (cid:47) (cid:47) C ∗ β − (cid:126) (cid:126) PPT P ∗ β − (cid:35) (cid:35) C s (cid:48) (cid:47) (cid:47) PC , C s (cid:48) (cid:47) (cid:47) (cid:111) C ∗ β (cid:15) (cid:15) PC q ∗ C (cid:47) (cid:47) (cid:111) PC ∗ β (cid:15) (cid:15) C C ∗ β (cid:111) (cid:15) (cid:15) PT s ∗ T (cid:47) (cid:47) id PT (cid:60) (cid:60) PPT q ∗ T (cid:47) (cid:47) PT . C -contractible). (cf. [Moc10, 2.7, 2.24, 2.44], [Sch11, B.16].) Let C be a complicial exact category.We say that an object x in C is C-contractible if it is C -homotopy equivalent to the zero object. We can showthat x is C -contractible if and only if x is a direct summand of Cu for some object u in C if and only if x is aretraction of Cu for some object u in C and if and only if x is a projective-injective object in C frob .18e can prove the following lemma. A morphism in a complicial exact category which admits both left and right C-homotopyinverses is a C-homotopy equivalence. More precisely, let f : x → y and g, h : y → x be morphisms in acomplicial exact category and let H : h f ⇒ C id x and K : f g ⇒ C id y be C-homotopies. Then − hK + HCg isa C-homotopy from h to g and − f hK + f HCg + K is a C-homotopy from f h to id y . In particular f is aC-homotopy equivalence. Let C be a complicial exact category and f : x → y a morphism in C . Then ( ) If f is a Frobenius admissible monomorphism, then C ( f ) is a split monomorphism. ( ) If f is a Frobenius admissible epimorphism, then C ( f ) is a split epimorphism.Proof. ( ) Since C ( f ) is a Frobenius admissible monomorphism and C ( x ) is an injective object in C frob by1.4.2 and by 2.1.5 respectively, C ( f ) is a split monomorphism. A proof of ( ) is similar. In this subsection we introduce concepts of homotopy commutative squares and commutative diagrams incomplicial exact categories. We will prove that the category of homotopy commutative diagrams on a normalordinary complicial exact category becomes a complicial exact category (see 2.2.10). We start by defining anotion of homotopy commutative squares.
Let C be a complicial exact category and let [ f : x → x (cid:48) ] and [ g : y → y (cid:48) ] be a pair of objects in C [ ] the morphisms category of C . A ( C- ) homotopy commutative square ( from [ f : x → x (cid:48) ] to [ g : y → y (cid:48) ] ) is a triple ( a , b , H ) C consisting of morphisms a : x → y , b : x (cid:48) → y (cid:48) and H : Cx → y (cid:48) in C such that H ι x = ga − b f . Namely H is a C -homotopy from ga to b f . We often write ( a , b , H ) for ( a , b , H ) C . Similarly a P-homotopy commutative square ( from [ f : x → x (cid:48) ] to [ g : y → y (cid:48) ] ) is atriple ( a , b , H ) P consisting of morphisms a : x → y , b : x (cid:48) → y (cid:48) and H : x → Py (cid:48) in C such that q y H = ga − b f .Notice that for a pair of morphisms a , b : x → y and for a C -homotopy H : a ⇒ C b , we can regard it as a C -homotopy commutative square [ x id x → x ] ( a , b , H ) → [ y id y → y ] . For simplicity We denote this situation by x ( a , b , H ) → y .Let [ f : x → x (cid:48) ] , [ g : y → y (cid:48) ] , [ f (cid:48) : x (cid:48) → x (cid:48)(cid:48) ] , [ g (cid:48) : y (cid:48) → y (cid:48)(cid:48) ] and [ h : z → z (cid:48) ] be objects in C [ ] and let ( a , b , H ) C , ( a (cid:48) , b (cid:48) , H (cid:48) ) C and ( b , c , K ) C be C -homotopy commutative squares from [ f : x → x (cid:48) ] to [ g : y → y (cid:48) ] , from [ g : y → y (cid:48) ] to [ h : z → z (cid:48) ] and from [ f (cid:48) : x (cid:48) → x (cid:48)(cid:48) ] to [ g (cid:48) : y (cid:48) → y (cid:48)(cid:48) ] respectively. Then we define ( a (cid:48) , b (cid:48) , H (cid:48) ) C ( a , b , H ) C tobe a homotopy commutative square from [ f : x → x (cid:48) ] to [ h : z → z (cid:48) ] by setting ( a (cid:48) , b (cid:48) , H (cid:48) ) C ( a , b , H ) C : = ( a (cid:48) a , b (cid:48) b , H (cid:48) (cid:63) C H ) C (84)where H (cid:48) (cid:63) C H is a C -homotopy from ha (cid:48) a to b (cid:48) b f given by the formula H (cid:48) (cid:63) C H : = b (cid:48) H + H (cid:48) Ca . (85)We also define K • C H to be a C -homotopy c f (cid:48) f ⇒ C g (cid:48) ga by the formula K • C H : = KC f + g (cid:48) H . (86)We often write ( a , b , H ) , H (cid:48) (cid:63) H and K • H for ( a , b , H ) C , H (cid:48) (cid:63) C H and K • C H respectively.Next let ( a , b , H ) P , ( a (cid:48) , b (cid:48) , H (cid:48) ) P and ( b , c , K ) P be P -homotopy commutative squares from from [ f : x → x (cid:48) ] to [ g : y → y (cid:48) ] , from [ g : y → y (cid:48) ] to [ h : z → z (cid:48) ] and from [ f (cid:48) : x (cid:48) → x (cid:48)(cid:48) ] to [ g (cid:48) : y (cid:48) → y (cid:48)(cid:48) ] respectively. Then wedefine ( a (cid:48) , b (cid:48) , H (cid:48) ) P ( a , b , H ) P to be a homotopy commutative square from [ f : x → x (cid:48) ] to [ h : z → z (cid:48) ] by setting ( a (cid:48) , b (cid:48) , H (cid:48) ) P ( a , b , H ) P : = ( a (cid:48) a , b (cid:48) b , H (cid:48) (cid:63) P H ) P (87)where H (cid:48) (cid:63) P H is a P -homotopy from ha (cid:48) a to b (cid:48) b f given by the formula H (cid:48) (cid:63) P H : = Pb (cid:48) H + H (cid:48) a . (88)We also define K • P H to be a P -homotopy c f (cid:48) f ⇒ P g (cid:48) ga by the formula K • P H : = K f + Pg (cid:48) H . (89)19e define C [ ] h to be a category whose objects are morphisms in C and whose morphisms are homotopycommutative squares and compositions of morphisms are give by the formula ( ) and we define C [ ] → C [ ] h to be a functor by sending an object [ f : x → x (cid:48) ] to [ f : x → x (cid:48) ] and a morphism ( a , b ) : [ f : x → x (cid:48) ] → [ g : y → y (cid:48) ] to ( a , b , ) : [ f : x → x (cid:48) ] → [ g : y → y (cid:48) ] . By this functor, we regard C [ ] as a subcategory of C [ ] h .Recall the definitions of the maps A and B from 2.1.1. We can show the following lemma. Let C be an ordinary complicial exact category. Let F be a letter in F ∈ { C , P } and let [ f : x → x (cid:48) ] , [ g : y → y (cid:48) ] , [ f (cid:48) : x (cid:48) → x (cid:48)(cid:48) ] , [ g (cid:48) : y (cid:48) → y (cid:48)(cid:48) ] and [ h : z → z (cid:48) ] be objects in C [ ] and let ( a , b , H ) F , ( a (cid:48) , b (cid:48) , H (cid:48) ) F and ( b , c , K ) F be F-homotopy commutative squares from [ f : x → x (cid:48) ] to [ g : y → y (cid:48) ] , from [ g : y → y (cid:48) ] to [ h : z → z (cid:48) ] and from [ f (cid:48) : x (cid:48) → x (cid:48)(cid:48) ] to [ g (cid:48) : y (cid:48) → y (cid:48)(cid:48) ] respectively. Then ( ) If F = C, then we have the following equalitiesA ha (cid:48) a , b (cid:48) b f ( H (cid:48) (cid:63) C H ) = A ha (cid:48) , b (cid:48) g ( H (cid:48) ) (cid:63) P A ga , b f ( H ) , (90) A g (cid:48) ga , c f (cid:48) f ( K • C H ) = A g (cid:48) b , c f (cid:48) ( K ) • P A ga , b f ( H ) . (91) ( ) If F = P, then we have the following equalitiesB ha (cid:48) a , b (cid:48) b f ( H (cid:48) (cid:63) P H ) = B ha (cid:48) , b (cid:48) g ( H (cid:48) ) (cid:63) C B ga , b f ( H ) , (92) B g (cid:48) ga , c f (cid:48) f ( K • P H ) = B g (cid:48) b , c f (cid:48) ( K ) • C B ga , b f ( H ) . (93) CC -homotopy). Let C be a complicial exact category and let f : x → y be a morphismin C and let H , H (cid:48) : f ⇒ C g be a pair of C -homotopies. A CC-homotopy ( from H to H (cid:48) ) is a morphism S : CCx → y such that L ( C ∗ ι x − ι ∗ Cx ) = H − H (cid:48) . We denote this situation by L : H ⇒ CC H (cid:48) . We say that apair of C -homotopies H , H (cid:48) : f ⇒ C g are CC-homotopic if there exists a CC -homotopy H ⇒ CC H (cid:48) . As lemmabelow, the CC -homotopic relation between C -homotopies is an equivalence relation.Let ( a , b , H ) , ( a , b , H (cid:48) ) : [ f : x → x (cid:48) ] → [ g : y → y (cid:48) ] , ( d , e , K ) , ( d , e , K (cid:48) ) : [ g : y → y (cid:48) ] → [ h : z → z (cid:48) ] and ( b , c , L ) , ( b , c , L (cid:48) ) : [ f (cid:48) : x (cid:48) → x (cid:48)(cid:48) ] → [ g (cid:48) : y (cid:48) → y (cid:48)(cid:48) ] be homotopy commutative squares and let S : H ⇒ CC H (cid:48) , T : K ⇒ CC K (cid:48) , U : L ⇒ CC L (cid:48) be a triple of CC -homotopies. Then we define T (cid:63) CC S : K (cid:63) CC H ⇒ CC K (cid:48) (cid:63) CC H (cid:48) and U • CC S : L • CC H ⇒ CC L (cid:48) • CC H (cid:48) to be CC -homotopies by setting T (cid:63) CC S : = b (cid:48) S + TCCa , (94) U • CC S : = g (cid:48) S + UCC f . (95) Let Ch b ( E ) be the category of bounded chain complexes on an exact category E equippedwith the standard complicial structure as in 1.2.2 and let f , g : x → y be a pair of chain morphisms in Ch b ( E ) and let H , L : Cx → y be a pair of C -homotopies from f to g . That is, as in 2.1.2, there are fam-ilies { h n : x n → y n + } n ∈ Z and { l n : x n → y n + } n ∈ Z of morphisms in E indexed by set of integers such that H n = (cid:0) − h n − f n − g n (cid:1) and L n = (cid:0) − l n − f n − g n (cid:1) , f n − g n = d yn + h n + h n − d xn = d yn + l n + l n − d xn for eachinteger n . In this case, for a CC -homotopy S : CCx → y from H to L , there exists a family of morphisms { s n : x n → y n + } n ∈ Z in E indexed by the set of integers such that S n = (cid:0) s n − − h n − + l n − (cid:1) , (96) H n − L n = d yn + s n − s n − d xn (97)for any integer n . Conversely the family of morphisms { s n : x n → y n + } n ∈ Z in E which satisfies the equalities ( ) for each integer n gives a CC -homotopy S : CCx → y from H to L by setting as in ( ) . Let a , b , c : x → y and let c , d , e : y → z be a sextuple of morphisms in a complicial exactcategory and let H : a ⇒ C b , K : c ⇒ C d , L : b ⇒ C c and M : d ⇒ C e be a quadruple of C -homotopies. Thenas in 2.2.6 ( ) below, in general ( M (cid:63) C L ) • C ( K (cid:63) C H ) and ( M • C K ) (cid:63) C ( L • C H ) are not equal but they areonly C -homotopic to each other. Thus to regard Ch b ( E ) the category of bounded chain complexes on anexact category E as a 2-category by declaring the class of chain homotopies to be the class of 2-morphisms,we need some arrangement of the class of chain homotopies. Namely we adopt the class of CC -homotopic20lasses of chain homotopies as the class of 2-morphisms. Then the compositions • C and (cid:63) C give the verticaland horizontal compositions of 2-morphisms respectively and we can make Ch b ( E ) into a 2-category. If weregard Ch b ( E ) as a 2-category in this way, we denote it by Ch b ( E ) . This is a motivation of defining F b , h C in 2.2.9 below. Since the exact structure and the complicial structure on Ch b ( E ) only involve 1-categoricalstructures and Ch b ( E ) does not change a matter of objects and 1-morphisms from Ch b ( E ) , Ch b ( E ) naturallybecomes a complicial exact category.We can show the following lemma. Let C be a complicial exact category. Then ( ) The CC-homotopic relation between C-homotopies is an equivalence relation. More precisely, let f ,g : x → y be a pair of morphisms in C and let H, L, M : f ⇒ C g be a triple of C-homotopies from f to g andlet S : H ⇒ CC L and T : L ⇒ CC M be a pair of CC-homotopies. Then ( a ) CCx → y is a CC-homotopy from H to H. ( b ) − S : CCx → y is a CC-homotopy from L to H. ( c ) S + T : CCx → y is a CC-homotopy from H to M. ( ) The C-homotopic relation between morphisms and the CC-homotopic relation between C-homotopies arecompatible with additions of morphisms. Namely for i = , , let f i , g i : x → y be a morphisms and let H i ,L i : f i ⇒ C g i be C-homotopies and let S i : H i ⇒ CC L i be CC-homotopies. Then H + H and L + L areC-homotopies from f + f to g + g and S + S is a CC-homotopy from H + H to L + L . ( ) C-homotopic relation between morphisms and CC-homotopic relation between C-homotopies are com-patible with the compositions of morphisms. That is, let f : x → y, g, g (cid:48) : y → z, h : z → w be a quadru-ple of morphisms in C and let H, H (cid:48) : g ⇒ C g (cid:48) be a pair of C-homotopies and let S : H ⇒ CC H (cid:48) be a CC-homotopies from H to H (cid:48) . Then HC f , H (cid:48) C f : g f ⇒ C g (cid:48) f and hH, hH (cid:48) : hg ⇒ C hg (cid:48) are C-homotopies andSCC f : Hc f ⇒ CC H (cid:48) C f and hS : hH ⇒ CC hH (cid:48) are CC-homotopies. ( ) Let ( a , b , H ) : [ u : x → x (cid:48) ] → [ v : y → y (cid:48) ] , ( d , e , K ) : [ v : y → y (cid:48) ] → [ w : z → z (cid:48) ] , ( b , c , L ) : [ u (cid:48) : x (cid:48) → x (cid:48)(cid:48) ] → [ v (cid:48) : y (cid:48) → y (cid:48)(cid:48) ] and ( e , f , M ) : [ v (cid:48) : y (cid:48) → y (cid:48)(cid:48) ] → [ w (cid:48) : z (cid:48) → z (cid:48)(cid:48) ] be a quadruple of homotopy commutative squares.Then the morphism MCH : CCx → z (cid:48)(cid:48) is a CC-homotopy from ( M • C K ) (cid:63) C ( L • C H ) to ( M (cid:63) C L ) • C ( K (cid:63) C H ) . ( ) Let ( f , d ) : C → C (cid:48) be a complicial exact functor between complicial exact categories C and C (cid:48) and letu, v : x → y be a morphism in C and let H, K : u ⇒ C v be a C-homotopy from u to v and let S : H ⇒ CC K be aCC-homotopy from H to K. Then f H · d x and f K · d x are C-homotopies from f u to f v and f S · ( d ∗ Cx ) · ( C (cid:48) ∗ d x ) is a CC-homotopy from f H · d x to f K · d x . ( ) Let f : x → y be a morphism in a complicial exact category C and let H : f ⇒ C f be a C-homotopy fromf to f . Namely we have an equality H ι x = . Assume x is C-contractible. That is, there exists a morphismU : Cx → x such that U ι x = id x . Then HCU : CCx → y is a CC-homotopy from H to . Let X be a category with a specific zero object 0. Let Z be the linearordered set of all integers with the usual linear order. We regard Z as a category in the usual way. We denote X Z the functor category from Z to X by F X . We call an object F X a filtered objects in X . For an object x in F X , we write x n and i xn for x ( n ) and x ( n ≤ n + ) respectively and illustrate x as follows: · · · i xn − → x n i xn → x n + i xn + → x n + i xn + → · · · . Let a ≤ b be a pair of integers and x an object in F X . We say that x has amplitude contained in [ a , b ] if forany k < a , x k = b ≤ k , x k = x b and i xk = id x b . In this case we write x ∞ for x b = x b + = · · · . Weset for x (cid:54) =
0, dim x : = min { k ∈ Z ; x n = x ∞ for any n ≥ k . } (98)and call it the dimension of x . By convention, we set dim x = − x =
0. Similarly for any morphism f : x → y in F X between objects which have amplitude contained in [ a , b ] , we denote f b = f b + = · · · by f ∞ .We write F [ a , b ] X for the full subcategory of F X consisting of those objects having amplitude contained in [ a , b ] . We also set F b X : = (cid:91) a < b ( a , b ) ∈ Z F [ a , b ] X , F b , ≤ b X : = (cid:91) a < ba ∈ Z F [ a , b ] X and F b , ≥ a X : = (cid:91) a < bb ∈ Z F [ a , b ] X and we callan object in F b X a bounded filtered object ( in X ). 21et u : x → y be a morphism in X . We write j ( x ) and j ( f ) : j ( x ) → j ( y ) for the object and the morphism in F [ , ] X such that j ( x ) ∞ = x and j ( f ) ∞ = f . We define ( − ) ∞ : F b X → X and j : X → F b X to be functorsby sending an object x in F b X to x ∞ in X and an object y in X to j ( y ) respectively. Obviously we have theequality ( − ) ∞ j = id X .If X is an exact category, then F X with the level admissible exact sequences (see 1.1.6) is an exactcategory and F b X and F [ a , b ] X are strict exact subcategories of F X . We also denote the full subcategory of F b X consisting of those objects x such that i xn is an admissible monomorphism for any integer n by F (cid:26) b C and we set F (cid:26) [ a , b ] X = F [ a , b ] X ∩ F (cid:26) b X and so on. By 2.2.8 below, it turns out that F (cid:26) b X and F (cid:26) [ a , b ] X arestrict exact subcategories of F b X .Moreover if X is a complicial exact category, then F X with the level complicial structure (see 1.2.14)is a complicial exact category and F b X , F [ a , b ] X , F (cid:26) b X and F (cid:26) [ a , b ] X are complicial exact subcategories of F X . Let C be an exact category and let a < b be a pair of integers. Then F (cid:26) b C and F (cid:26) [ a , b ] C arestrict exact subcategory of F b C .Proof. We will only give a proof for F (cid:26) b C and a proof for F (cid:26) [ a , b ] C is similar. Let x → y → z be a sequencein F (cid:26) b C such that it is an admissible exact sequence in F b C . Then by 1.1.3, for each integer n , the in-duced morphism x n (cid:116) x n − y n − → y n is an admissible monomorphism in C . Thus by [Wal85, 1.1.4.], theclass of admissible monomorphisms in F (cid:26) b C is closed under compositions and co-base change by arbitrarymorphisms.Next by applying the similar argument above to C op the opposite category of C , it turns out that the classof admissible epimorphisms in F (cid:26) b C is closed under compositions and base change by arbitrary morphisms.Thus F (cid:26) b C is a strict exact subcategory of F b C .The notion of homotopy commutative diagrams below is a generalization of that of homotopy commuta-tive squares 2.2.1. Let C be a complicial exact category and x and y objects in F b C . A homotopy commutative diagram ( f , H ) from x to y consisting of a family of morphisms f = { f n : x n → y n } n ∈ Z in C indexed by Z and a family of C -homotopy { H n : i yn f n ⇒ C f n + i xn } n ∈ Z indexedby Z such that for sufficiently large m , f m = f m + = · · · ( : = f ∞ ) and H m = H m + = · · · =
0. We illustrate by ( f , H ) : x → y in this situation. We say that a pair of homotopy commutative diagrams ( f , H ) , ( g , K ) : x → y are CC-homotopic if f = g and if for any integer n , H n and K n are CC -homotopic. By 2.2.6, we can show thatthe CC -homotopic relation between homotopy commutative diagrams is an equivalence relation.Next we define the compositions of homotopy commutative diagrams. Let ( f , H ) : x → y and ( g , K ) : y → z be homotopy commutative diagrams between objects in F b C . We define ( g , K )( f , H ) : x → z to be a homotopycommutative diagram by the formula ( g , K )( f , H ) : = ( g f , K (cid:63) H ) (99)where g f : = { g n f n : x n → z n } n ∈ Z and K (cid:63) H : = { K n (cid:63) C H n : i zn g n f n ⇒ C g n + f n + i xn } n ∈ Z . (For the definitionof the operation (cid:63) , see 2.2.1.) We can show that the compositions of homotopy commutative diagrams isassociative.We define F b , h C to be a category whose class of objects is Ob F b C and whose morphisms are homotopycommutative diagrams and whose compositions are given by compositions of homotopy commutative dia-grams. We also define F b , h C to be a category by setting Ob F b , h C = Ob F b C and for any pair of objects x and y , Hom F b , h C ( x , y ) : = Hom F b , h C ( x , y ) / ( CC -homotopic relation ) . (100)Then by 2.2.6 again, compositions of morphisms in F b , h C induces the compositions of morphisms in F b , h C .Similarly for any pair of integers a ≤ b , we define the category F [ a , b ] , h C , F b , ≥ a , h C , F b , ≤ b , h C , F [ a , b ] , h C , F b , ≥ a , h C and F b , ≤ b , h C . For any integer n , We can regard F [ n , n + ] C and F [ n , n + ] , h C as C [ ] and C [ ] h . Let C be a complicial exact category and let a ≤ b be a pair of integers. Then ( ) Let ( f , H ) , ( g , K ) : x → y be a pair of homotopy commutative diagrams. We define the addition of homo-topy commutative diagrams by the formula ( f , H ) + ( g , K ) : = ( f + g , H + K ) . (101)22hen F b , h C , F [ a , b ] , h C , F b , ≤ b , h C and F b , ≥ a , h C naturally become additive categories. Here for any objects x and y in F b , h C , x ⊕ y is given by ( x ⊕ y ) n = x n ⊕ y n and i x ⊕ yn = (cid:18) i xn i yn (cid:19) for any integer n . ( ) We say that a sequence x ( f , H ) (cid:26) y ( g , K ) (cid:16) z (102)in F b , h C is a level Frobenius admissible exact sequence if we have the equality ( g , K )( f , H ) = n , the sequence x n f n (cid:26) y n g n (cid:16) z n is a Frobenius admissible exact sequence in C . In this case we call ( f , H ) a level Frobenius admissible monomorphism and call ( g , K ) a level Frobenius admissible epimorphism .Then ( a ) For a homotopy commutative diagram ( f , H ) : x → y in F b , h C , ( f , H ) is a level Frobenius admissiblemonomorphism if and only if for any integer n , the morphism f n : x n → y n is a Frobenius admissible monomor-phism in C . Moreover if we assume that C is ordinary (see Definition 1.3.4), then similarly ( f , H ) is a levelFrobenius admissible epimorphism if and only if for any integer n , the morphism f n : x n → y n is a Frobeniusadmissible epimorphism in C . ( b ) If C is ordinary, then F b , h C , F [ a , b ] , h C , F b , ≤ b , h C and F b , ≥ a , h C endowed with the class of level Frobeniusadmissible exact sequences are exact categories. ( ) We define C : F b , h C → F b , h C to be a functor by setting Cx n : = C ( x n ) and i Cxn : = C ( i xn ) and C ( f , H ) n : =( f n , CH n σ x n ) for an object x in F b , h C and a homotopy commutative diagram ( f , H ) : x → y and an integer n .Moreover we assume C is normal and we also define ι : id F b , h → C , r : CC → C and σ : CC ∼ → CC to be naturaltransformations by setting component-wisely. Then the quadruple ( C , ι , r , σ ) is a normal ordinary complicialstructure on F b , h C and F [ a , b ] , h C , F b , ≤ b , h C and F b , ≥ a , h C are complicial exact subcategories of F b , h C . We callthis complicial structure the level complicial structure ( on F b , h C ) (resp. F [ a , b ] , h C , F b , ≤ b , h C , F b , ≥ a , h C ).In the proof of 2.2.10, we will use 2.3.13. Notice that there is no circular reasoning in our argument. Proof. ( ) We can show that F b , h C with the addition defined by ( ) is enriched over the category of abeliangroups. We will check the universal property of direct sums. Let x , y , z and w be objects in F b , h C . For anyhomotopy commutative diagrams ( f , H ) : x → z , ( f (cid:48) , H (cid:48) ) : y → z , ( g , K ) : w → x and ( g (cid:48) , K (cid:48) ) : w → y , thereexists unique homotopy commutative diagrams (( f f (cid:48) ) , ( H H (cid:48) )) : x ⊕ y → z and (cid:16)(cid:16) gg (cid:48) (cid:17) , (cid:16) KK (cid:48) (cid:17)(cid:17) : w → x ⊕ y which defined by the formula (cid:0) f f (cid:48) (cid:1) n = (cid:0) f n f (cid:48) n (cid:1) , (cid:0) H H (cid:48) (cid:1) n = (cid:0) H n H (cid:48) n (cid:1) , (cid:18) gg (cid:48) (cid:19) n = (cid:18) g n g (cid:48) n (cid:19) , (cid:18) KK (cid:48) (cid:19) n = (cid:18) K n K (cid:48) n (cid:19) and make the diagrams below commutative. x (cid:18) id x (cid:19) (cid:47) (cid:47) ( f , H ) (cid:33) (cid:33) x ⊕ y (cid:15) (cid:15) y (cid:18) y (cid:19) (cid:111) (cid:111) ( f (cid:48) , H (cid:48) ) (cid:125) (cid:125) z , w ( g , K ) (cid:125) (cid:125) ( g (cid:48) , K (cid:48) ) (cid:33) (cid:33) (cid:15) (cid:15) x x ⊕ y ( y ) (cid:47) (cid:47) ( id x ) (cid:111) (cid:111) y . Hence F b , h C is an additive category. ( ) ( a ) Let ( f , H ) : x → y be a homotopy commutative diagram in F b , h C . Assume that for any integer n , f n : x n → y n is a Frobenius admissible monomorphism in C . Then we set z n : = y n / x n and let g n : y n (cid:16) z n be the canonical quotient morphism. Then by Lemma 2.3.13, there exists a morphism i zn : z n → z n + and a C -homotopy K n : i zn g n ⇒ C g n + i yn such that K n (cid:63) C H n =
0. Thus the system { z n , i zn } n ∈ Z forms an object in F b , h C and the pair ( g , K ) forms a homotopy commutative diagram y → z and the sequence x ( f , H ) (cid:26) y ( g , K ) (cid:16) z is a levelFrobenius admissible exact sequence in F b , h C . 23or level Frobenius admissible epimorphisms, we shall apply the argument above to C op the oppo-site category of C and the dual complicial structure ( P , q , s , τ P ) of ( C , ι , r , σ ) and utilize Lemma 2.1.3 andLemma 2.2.2. ( b ) What we need to prove are the following assertions: ( i ) For any level Frobenius admissible exact sequence of the form ( ) , the commutative square x ( f , H ) (cid:47) (cid:47) (cid:15) (cid:15) y ( g , K ) (cid:15) (cid:15) (cid:47) (cid:47) z (103)is a biCartesian square in F b , h C . ( ii ) The class of level Frobenius admissible monomorphisms in F b , h C is closed under compositions, andpush-out along arbitrary morphisms. The class of level Frobenius admissible epimorphisms in F b , h C is closedunder compositions, and pull-back along arbitrary morphisms. ( iii ) Split short exact sequences are level Frobenius admissible exact sequences.Assertion ( iii ) is clear. Proof of ( i ) . We only prove that the commutative square ( ) is a push-out diagram. A proof of the pull-backcase is similar. For any object u and any morphism ( h , L ) : y → u in F b , h C such that ( h , L )( f , H ) = , (104)we will show that there exists a unique morphism ( ψ , M ) : z → u such that ( h , L ) = ( ϕ , M )( g , K ) . First noticethat the equality ( ) implies the following equalities h f = , (105) h n + H n + L n C f n = , (106) g n + H n + K n C f n = n . Moreover, since K n and L n are C -homotopies i zn g n ⇒ C g n + y yn and i un h n ⇒ C h n + i yn respec-tively, we have equalities i zn g n = K n ι y n + g n + i yn , (108) L n ι y n = i un h n − h n + i yn (109)for any integer n . Let us fix an integer n . First notice that there exists a morphism ϕ n : z n → u n in C such that ϕ n g n = h n by the universality of the left commutative square below x n f n (cid:47) (cid:47) (cid:15) (cid:15) y n g n (cid:15) (cid:15) h n (cid:11) (cid:11) (cid:47) (cid:47) (cid:51) (cid:51) z n ϕ n (cid:32) (cid:32) u n , Cx n C f n (cid:47) (cid:47) (cid:15) (cid:15) Cy nCg n (cid:15) (cid:15) L n − ϕ n + K n (cid:13) (cid:13) (cid:47) (cid:47) (cid:50) (cid:50) Cz n M n (cid:34) (cid:34) u n + . Next since we have the equalities ( L n − ϕ n + K n ) C f n = L n C f n − ϕ n + K n C f n = − h n + H n + ϕ n + g n + H n = , there exists a morphism M n : Cz n → u n + such that L n − ϕ n + K n = M n Cg n (110)24y the universal property of the right commutative square above. Next we will show that the equality i un ϕ n − M n ι z n = ϕ n + i zn . (111)We have the equalities ( i un ϕ n − M n ι z n ) g n = i un ϕ n g n − M n ι z n g n = i un h n − M n ι z n g n = i un h n + ( ϕ n + K n − L n ) ι y n − M n Cg n = ϕ n + K n ι y n + i un h n − L n ι y n = ϕ n + K n ι y n + h n + i yn = ϕ n + K n ι y n + ϕ n + g n + i yn = ϕ n + ( K n ι y n + g n + i yn ) = ϕ n + i zn g n . Since g n is a Frobenius admissible epimorphism, a fortiori, an epimorphism, we obtain the equality ( ) .The equality ( ) means the pair ( ϕ , M ) is a homotopy commutative diagram from z to u and the equality ( ) shows we have the equality L = M (cid:63) K . By construction the homotopy commutative diagram ( ϕ , M ) such that ( h , L ) = ( ϕ , M )( g , K ) is unique. Hence we complete the proof. Proof of ( ii ) . First notice that for a pair of level Frobenius admissible monomorphisms ( f , H ) : x (cid:26) y and ( g , K ) : y (cid:26) z , the composition ( g f , K (cid:63) H ) is a level Frobenius admissible monomorphism by ( a ) .Second, let ( f , H ) : x → y be a level Frobenius admissible monomorphism and let ( g , K ) : x → z be anarbitrary homotopy commutative diagram. We will show that there exists a pair of homotopy commutativediagrams ( f (cid:48) , H (cid:48) ) : z → u and ( g (cid:48) , K (cid:48) ) : y → u which is a push out of ( f , H ) along ( g , K ) and ( g (cid:48) , K (cid:48) ) is a levelFrobenius admissible monomorphism. x (cid:47) (cid:47) ( f , H ) (cid:47) (cid:47) ( g , K ) (cid:15) (cid:15) y ( g (cid:48) , K (cid:48) ) (cid:15) (cid:15) z ( f (cid:48) , H (cid:48) ) (cid:47) (cid:47) u . (112)Then it turns out that the homotopy commutative diagram (cid:16)(cid:16) fg (cid:17) , (cid:16) HK (cid:17)(cid:17) : x → y ⊕ z is a level Frobenius admis-sible monomorphism by 1.1.2 and ( a ) . Thus by the proof of ( a ) , there exists an object u in F b , h C and a levelFrobenius admissible epimorphism (( g (cid:48) − f (cid:48) ) , ( K (cid:48) − H (cid:48) )) : y ⊕ z → u such that the sequence x (cid:18)(cid:18) fg (cid:19) , (cid:18) HK (cid:19)(cid:19) (cid:26) y ⊕ z (( g (cid:48) − f (cid:48) ) , ( K (cid:48) − H (cid:48) )) (cid:16) u is a level Frobenius admissible exact sequence in F b , h C . Hence the commuta-tive square of the form ( ) is a coCartesian square in F b , h C and it turns out that the homotopy commutativediagram ( f (cid:48) , H (cid:48) ) : z → u is a level Frobenius admissible monomorphism by 1.1.2 and ( a ) again.Finally by applying the previous argument to C op the opposite category of C and the dual complicialstructure ( P , q , s , τ P ) of ( C , ι , r , σ ) and by utilizing Lemma 2.1.3 and Lemma 2.2.2, we obtain the results forlevel Frobenius admissible epimorphisms. We complete the proof. ( ) Almost all verifications are straightforwards. What we use normality assumption are following:For any homotopy commutative diagram ( f , H ) : x → y and any integer n , the following diagrams are com-mutative and they follow from equalities ( ) and ( ) . [ CCx n CCi xn → CCx n + ] ( r xn , r xn + , ) (cid:47) (cid:47) ( CC f n , CC f n + , CCH n C σ xn σ Cxn ) (cid:15) (cid:15) [ Cx n Ci xn → Cx n + ] ( C f n , C f n + , CH n σ xn ) (cid:15) (cid:15) [ CCy n CCi yn → CCy n + ] ( r yn , r yn + , ) (cid:47) (cid:47) [ Cy n Ci yn → Cy n + ] , CCx n CCi xn → CCx n + ] ( σ xn , σ xn + , ) (cid:47) (cid:47) ( CC f n , CC f n + , CCH n C σ xn σ Cxn ) (cid:15) (cid:15) [ CCx n CCi xn → CCx n + ] ( C f n , C f n + , CCH n C σ xn σ Cxn ) (cid:15) (cid:15) [ CCy n CCi yn → CCy n + ] ( σ yn , σ yn + , ) (cid:47) (cid:47) [ CCy n CCi yn → CCy n + ] . By 2.2.6 and 2.2.10 (and observations in 2.2.5), we obtain the following result.
Let C be a normal ordinary complicial exact category. Then we can naturally makeF b , h C into a complicial exact category such that the canonical functor F b , h C → F b , h C is a strictly normalcomplicial exact functor. Moreover for any pair of integers a < b, similar statements hold for F [ a , b ] , h C and soon. We can show the following.
Let ( f , d ) : C → C (cid:48) be a complicial exact functor between complicial exactcategories and let ( φ , H ) : x → x (cid:48) be a homotopy commutative diagram in F b , h C . Then we define f ( x ) and f ( φ , H ) to be a filtered object on C (cid:48) and a homotopy commutative diagram in F b , h C (cid:48) by setting ( f ( x )) n : = f ( x n ) , i f xn : = f ( i xn ) and ( f ( φ , H )) n : = ( f ( φ n ) , f ( H n ) · d x n ) for any integer n . We also define F b , h : F b , h C → F b , h C (cid:48) to be a functor by sending a bounded filtered object x on C to f ( x ) and a homotopy commutativediagram ( φ , H ) : x → y in F b , h C to f ( φ , H ) . The functor F b , h f sends a level Frobenius admissible exactsequence in F b , h C to a level Frobenius admissible exact sequence in F b , h C (cid:48) . In particular if C and C (cid:48) areordinary, then F b , h f is an exact functor.Moreover if we assume ( f , d ) is normal, then d induces a natural equivalence d lv : C (cid:48) lv F b , h f ∼ → F b , h fC lv defined by the formula ( d lv x ) : = ( { d x n } n ∈ Z , ) for any object x in F b , h C where C lv and C (cid:48) lv are the endofunctorson F b , h C and F b , h C (cid:48) respectively defined by level-wisely. In particular if C and C (cid:48) are normal ordinary, thenthe pair ( F b , h f , d lv ) is a complicial exact functor from F b , h C to F b , h C (cid:48) .For a complicial exact category C , we will define several operations on F b , h C or F b , h C . Let C be a complicial exact category and let n be an integer andlet x be a bounded filtered object on C and let ( f , H ) : x → y be a homotopy commutative diagram. We define σ ≤ n x and σ ≥ n x to be filtered objects on C by setting ( σ ≤ n x ) k : = (cid:40) x k if k ≤ nx n if k ≥ n , i σ ≤ n xk : = (cid:40) i xk if k ≤ n − x n if k ≥ n (113) ( σ ≥ n x ) k : = (cid:40) x k if k ≥ n k < n , i σ ≥ n xk : = (cid:40) i xk if k ≥ n k ≤ n − . (114)We also define σ ≤ n ( f , H ) : σ ≤ n x → σ ≤ n y and σ ≥ n ( f , H ) : σ ≥ n x → σ ≥ n y to be homotopy commutative dia-grams by setting ( σ ≤ n ( f , H )) k : = (cid:40) ( f k , H k ) if k < n ( f n , ) if k ≥ n , ( σ ≥ n ( f , H )) k : = (cid:40) ( f n , H n ) if k ≥ n ( , ) if k ≤ n − . (115)The associations σ ≤ n , σ ≥ n : F b , h C → F b , h C are functors which send a level Frobenius admissible exact se-quence to a level Frobenius admissible exact sequence.For any object x in F b , h C , there exists a pair of functorial natural morphism σ ≥ n x → x and σ ≤ n x → x andboth σ ≥ n : F b , h C → F b , ≥ n , h C and σ ≤ n : F b , h C → F b , ≤ n , h C are right adjoint functors of the inclusion functors F b , ≥ n , h C → F b , h C and F b , ≤ n , h C → F b , h C respectively.Moreover if C is normal, then we have the equalities σ ≤ n C lv = C lv σ ≤ n , σ ≥ n C lv = C lv σ ≥ n . (116)In particular if C is normal ordinary, then σ ≤ n and σ ≥ n are strictly normal complicial exact functors F b , h C → F b , h C . 26 .2.14. Definition (Degree shift). Let C be a complicial exact category and let k be an integer and let ( f , H ) : x → y be a homotopy commutative diagram in F b , h C . We define x [ k ] and ( f [ k ] , H [ k ]) : x [ k ] → y [ k ] tobe an object and a morphism in F b , h C by setting x [ k ] n = x k + n , i x [ k ] n = i xk + n , f [ k ] n = f k + n and H [ k ] n = H k + n .The association ( − )[ k ] : F b , h C → F b , h C , x (cid:55)→ x [ k ] gives a functor which sends a level Frobenius admissibleexact sequence to a level Frobenius admissible exact sequence. In particular if C is normal ordinary, then thefunctor ( − )[ k ] is a strictly normal complicial exact functor. Let k be an integer and let C be a complicial exact category. We define c k : F b , h C → F b , h C to be a functor by setting ( c k ( x )) n : = x n if n ≤ k − Cx n if n ≥ k , i c k ( x ) n : = i xn if n ≤ k − ι x n i xk if n = k − Ci xn if n ≥ k , c k ( f , H ) n : = ( f n , H n ) if n ≤ k − ( f k − , (cid:63) H k − ) if n = k − ( C f n , CHn · σ x n ) if n ≥ k (117)for an object x and a morphism ( f , H ) : x → y in F b , h C .For a pair of integers a < b , we also denote the restrictions of c k to F [ a , b ] , h C and F [ a , b ] , h C and so on by thesame letters c k .If C is normal ordinary, then we define d c k : C lv c k ∼ → c k C lv to be a natural equivalence by setting for anyobject x in F b , h C , ( d c k ) n = (cid:40) id Cx n if n ≤ k − , σ x n if n ≥ k . (118)Then the pair ( c k , d c k ) is a complicial exact functor F b , h C → F b , h C . We call c k the kth cone functor . Let k be an integer and let C be a complicial exact category. We define s k : F b , h C → F b , h C to be a functor by setting ( s k ( x )) n : = x n if n ≤ k − x n + if n ≥ k , i s k ( x ) n : = i xn if n ≤ k − i xk i xk − if n = k − i xk + if n ≥ k , ( s k ( f , H )) n : = ( f n , H n ) if n ≤ k − ( f k − , H k (cid:63) H k − ) if n = k − ( f n + , H n + ) if n ≥ k (119)for an object x and a morphism ( f , H ) : x → y in F b , h C .Notice that s k is not 1-functorial on F b , h C , but on F b , h C . If C is normal ordinary, then s k is a strictlynormal complicial exact. We call s k the kth skip functor .Recall the definition of F (cid:26) [ a , b ] C from Definition 2.2.7 r [ a , b ] ). Let C be a normal ordinary complicial exact category and let a < b be a pairof integers. We will define r [ a , b ] : F [ a , b ] , h C → F (cid:26) [ a , b ] C to be a complicial functor by setting in the followingway. For a pair of objects x and y in F [ a , b ] , h C and a pair of integers a ≤ s ≤ t ≤ b and a family of mor-phisms { f i : x i → y i } i ∈ [ s , t ] in C indexed by the integers i ∈ [ s , t ] , we set Cx [ s , t ] : = Cx t ⊕ Cx t − ⊕ · · · ⊕ Cx s and C f [ s , t ] : = C f t C f t − . . . . . . C f s : Cx [ s , t ] → Cy [ s , t ] and we define r [ a , b ] x to be an object in F (cid:26) [ a , b ] C bysetting ( r [ a , b ] x ) a + k : = k < x a if k = x a + k ⊕ Cx [ a , a + k − ] if 1 ≤ k < b − ax b ⊕ Cx [ a , b − ] if b − a ≤ k i r [ a , b ] xa + k : = k < (cid:32) ξ , i xa + k
00 id Cx [ a , a + k − ] (cid:33) if 0 ≤ k < b − a id x b ⊕ Cx [ a , b − ] if b − a ≤ k . (120)27or a homotopy commutative diagram ( f , H ) : x → y in F [ a , b ] , h C , we will inductively construct a morphism r [ a , b ] ( f , H ) : r [ a , b ] x → r [ a , b ] y in F (cid:26) [ a , b ] C . First we set r [ a , b ] ( f , H ) a + k : = k < f a if k = ( f a , f a + , H a ) if k = . (121)For 2 ≤ k ≤ b − a , we will set r [ a , b ] ( f , H ) a + k : = (cid:18) Cyl ( f a + k − , f a + k , H a + k − ) , k − k − , C f [ a , a + k − ] (cid:19) + (cid:18) k , H ( f , H ) k , , k − (cid:19) (122)where O s , t is the s × t th zero matrix and H ( f , H ) k : Cx [ a , a + k − ] → y a + k + ⊕ Cy [ a + , a + k ] to be a morphism in C which will be defined by inductively. By convention we set H ( f , H ) =
0. If we define the morphisms r [ a , b ] ( f , H ) a + k and r [ a , b ] ( f , H ) a + k + appropriately, then we need to have an equality i ya + k r [ a , b ] ( f , H ) a + k = r [ a , b ] ( f , H ) a + k + i xa + k . (123)Thus inspection shows that we need to have an equality H ( f , H ) k + : = (cid:32) − ξ , i ya + k H a + k − k − , (cid:32) ξ , i ya + k , k − k − , id Cy [ a + , a + k − ] (cid:33) H ( f , H ) k (cid:33) . (124)Hence we inductively define H ( f , H ) k by using the recurrence formula ( ) . For k > b − a , we set r [ a , b ] ( f , H ) a + k : = r [ a , b ] ( f , H ) b . Let ( g , K ) : y → z be another homotopy commutative diagram in F b , h C . Thenwe have an equality r [ a , b ] ( g , K ) a + k r [ a , b ] ( f , H ) a + k = r [ a , b ] ( g f , K (cid:63) H ) a + k (125)for k ≤
1. Thus by induction on k and uniqueness of construction of H ( f , H ) k , we obtain the equality r [ a , b ] ( g , K ) r [ a , b ] ( f , H ) = r [ a , b ] ( g f , K (cid:63) H ) (126)as a morphism from r [ a , b ] x to r [ a , b ] z .The natural equivalence σ : CC ∼ → CC induces a natural equivalence d r [ a , b ] : C lv r [ a , b ] ∼ → r [ a , b ] C lv . (127)and the pair ( r [ a , b ] , d r [ a , b ] ) is a complicial exact functor F [ a , b ] , h C → F (cid:26) [ a , b ] C . We write i [ a , b ] and i (cid:48) [ a , b ] for theinclusion functors F (cid:26) [ a , b ] C (cid:44) → F [ a , b ] , h C and F (cid:26) [ a , b ] C (cid:44) → F [ a , b ] C respectively. Then there are complicial naturaltransformations id F (cid:26) [ a , b ] C → r [ a , b ] i [ a , b ] and id F [ a , b ] C → i (cid:48) [ a , b ] r [ a , b ] | F [ a , b ] C . We can also generalize the notion of mapping cone and mapping cylinder of chain morphisms in a categoryof chain complexes on an additive category to a complicial exact category.
Let C be a complicial exact category. Like as categoriesof chain complexes, we will define a mapping cone and a mapping cylinder for morphisms in C as follows.First we write dom and ran for the exact functors C [ ] → C induced from the functors [ ] → [ ] definedby sending 0 to 0 and 1 respectively. We define ε : dom → ran to be a natural transformation by settingfor any object f : x → y in C [ ] , ε ( f ) : = f . We define Cone , Cyl : C [ ] → C to be exact functors calledthe mapping cone functor and the mapping cylinder functor respectively by setting Cyl : = ran ⊕ C dom andCone : = ran (cid:116) dom C dom where Cone is defined the following push-out diagram:dom (cid:47) (cid:47) ι ∗ dom (cid:47) (cid:47) ε (cid:15) (cid:15) C dom µ (cid:15) (cid:15) π ∗ dom (cid:47) (cid:47) (cid:47) (cid:47) T dom id T dom (cid:15) (cid:15) ran (cid:47) (cid:47) κ (cid:47) (cid:47) Cone ψ (cid:47) (cid:47) (cid:47) (cid:47) T dom28here ψ is induced from the universal property of Cone. We have the Frobenius admissible exact sequencedom ξ (cid:26) Cyl η (cid:16) Conewhere ξ : = (cid:16) ε − ι ∗ dom (cid:17) and η = ( κ µ ) . Moreover we define ξ : ran → Cyl, ξ : C dom → Cyl and υ : Cyl → ran to be natural transformations by setting ξ : = (cid:16) id ran (cid:17) , ξ : = (cid:16) C dom (cid:17) and υ : = ( id ran ) . Then we havethe following commutative diagram: dom (cid:47) (cid:47) ξ (cid:47) (cid:47) ε (cid:34) (cid:34) Cyl υ (cid:15) (cid:15) ran (cid:111) (cid:111) ξ (cid:111) (cid:111) id ran (cid:125) (cid:125) ran . Since the functor Cone is defined by the push-out diagram, we can choose the cofiber products as follows.For an object x in C , since the diagrams below are push-outs, we shall assume that Cone id x = Cx , κ id x = ι x , µ id x = id Cx , Cone ( x → ) = T x , µ x → = π x , Cone ( → x ) = x and κ → x = id x . x (cid:47) (cid:47) ι x (cid:47) (cid:47) id x (cid:15) (cid:15) Cx id Cx (cid:15) (cid:15) x (cid:47) (cid:47) ι x (cid:47) (cid:47) Cx , x (cid:47) (cid:47) ι x (cid:47) (cid:47) (cid:15) (cid:15) Cx π x (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) T x , (cid:47) (cid:47) ι (cid:47) (cid:47) (cid:15) (cid:15) C ( ) = (cid:15) (cid:15) x (cid:47) (cid:47) id x (cid:47) (cid:47) x . Moreover we define s , t , ∆ : C → C [ ] to be exact functors by sending an object x to [ x → ] , [ → x ] and [ x id x → x ] respectively. Then we shall assume that Cone s = T , Cone t = id C and Cone ∆ = C .For a homotopy commutative square ( a , b , H ) : [ f : x → x (cid:48) ] → [ g : y → y (cid:48) ] in C , we define a morphismCyl ( a , b , H ) : Cyl f → Cyl g in C by setting Cyl ( a , b , H ) : = (cid:18) b − H Ca (cid:19) . Then the left square below is com-mutative and there exists a morphism Cone ( a , b , H ) : Cone f → Cone g in C which makes the right squarebelow commutative: x (cid:47) (cid:47) ξ f (cid:47) (cid:47) a (cid:15) (cid:15) Cyl f Cyl ( a , b , H ) (cid:15) (cid:15) η f (cid:47) (cid:47) (cid:47) (cid:47) Cone f Cone ( a , b , H ) (cid:15) (cid:15) x (cid:48) (cid:47) (cid:47) ξ g (cid:47) (cid:47) Cyl g η g (cid:47) (cid:47) (cid:47) (cid:47) Cone g . Then we can regard Cyl and Cone as additive functors on C [ ] h and ξ : dom → Cyl, ξ : ran → Cyl and κ : ran → Cone as natural transformations between functors on C [ ] h . Since Cyl and Cone are defined bypush-out diagrams in H om ExCat ( C [ ] , C ) the (large) exact category of exact functors from C [ ] to C , Cyl andCone are exact functors from C [ ] to C . r C , Cone , r Cone , C ). Let C be a complicial exact category. We define two natural transforma-tions r C , Cone : C Cone → C ran and r Cone , C : Cone C → C ran between functors from C [ ] to C by the universal29roperties of C Cone and Cone C as in the commutative diagrams below. C dom (cid:47) (cid:47) C ∗ ι (cid:47) (cid:47) C ∗ ε (cid:15) (cid:15) CC dom r (cid:47) (cid:47) C ∗ µ (cid:15) (cid:15) C dom C ∗ ε (cid:15) (cid:15) C ran (cid:47) (cid:47) C ∗ κ (cid:47) (cid:47) id C dom (cid:42) (cid:42) C Cone r C , Cone (cid:36) (cid:36) C ran , (128) C dom (cid:47) (cid:47) ι ∗ C (cid:47) (cid:47) ε ∗ C (cid:15) (cid:15) CC dom r (cid:47) (cid:47) µ ∗ C (cid:15) (cid:15) C dom ε ∗ C (cid:15) (cid:15) C ran (cid:47) (cid:47) κ ∗ C (cid:47) (cid:47) id C dom (cid:42) (cid:42) Cone C r Cone , C (cid:36) (cid:36) C ran . (129) Let C be a complicial exact category and let f : x → y be a morphism. Then we have anequality Cone ( f , id y ) = r C , Cone f · ι Cone f . (130) Proof.
Notice that we have equalities r C , Cone f · ι Cone f · κ f = r C , Cone f · C ∗ κ f · ι y = ι y , (131) r C , Cone f · ι Cone f · µ f = r C , Cone f · C ∗ µ f · ι Cx = C f · r x · ι Cx = C f . (132)Thus it turns out that the morphism r C , Cone f ι Cone f : Cone f → Cy makes the diagram below commutative. Inparticular we obtain the equality ( ) . x (cid:47) (cid:47) (cid:16) − f ι x (cid:17) (cid:47) (cid:47) f (cid:15) (cid:15) y ⊕ Cx ( κ f µ f ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:16) id y Cf (cid:17) (cid:15) (cid:15) Cone f r C , Cone f · ι Cone f (cid:15) (cid:15) y (cid:47) (cid:47) (cid:16) − id y ι y (cid:17) (cid:47) (cid:47) y ⊕ Cy ( ι y id Cy ) (cid:47) (cid:47) (cid:47) (cid:47) Cy . Θ X ). Let C be a complicial exact category and let X be a commutative diagram ( ) . x a (cid:47) (cid:47) f (cid:15) (cid:15) x (cid:48) f (cid:48) (cid:15) (cid:15) y b (cid:47) (cid:47) h (cid:64) (cid:64) y (cid:48) . (133)Then we define Θ X : C Cone f → Cone f (cid:48) to be a C -homotopy Cone ( a , b ) ⇒ C Θ X : = Cone ( h , b ) · r C , Cone f . (134)30one f (cid:47) (cid:47) ι Cone f (cid:47) (cid:47) Cone ( a , b ) (cid:15) (cid:15) Cone ( f , id y ) (cid:37) (cid:37) C Cone f r C , Cone f (cid:15) (cid:15) Cone f (cid:48) Cy . Cone ( h , b ) (cid:111) (cid:111) In particular if x (cid:48) = y , a = f , y (cid:48) = f (cid:48) = b = h = id y , then we write Θ f for Θ X . Let C be a complicial exact category. Then the exact sequence of exact endofunctors on C [ ] h , [ → C ] ( , id C ) → [ C id C → C ] ( id C , ) → [ C → ] (135)is not split on C [ ] , but split on C [ ] h , where a retraction and a section are given by ( , id C , − r ) : [ C id C → C ] → [ → C ] and ( id C , , r ) : [ C → ] → [ C id C → C ] respectively. By the additive functor Cone : C [ ] h → C , thesequence ( ) induces a split exact sequence ( ) , and we have the equalities r = Cone ( , id C , − r ) , (136) ζ = Cone ( id C , , r ) , (137)where ζ : TC → CC is defined in 1.2.3. Let f : x → y be a morphism in a complicial exact category C . Then we have an equality Cone ( , κ f , µ f ) = id Cone f . (138) Proof.
For a C -homotopy commutative square ( , κ f , − µ f ) : [ x f → y ] → [ → Cone f ] , we have the equalityCyl ( , κ f , − µ f ) = (cid:0) κ f µ f (cid:1) : y ⊕ Cx → Cone f . (139)Thus by the following commutative diagram, we obtain the equality ( ) . x (cid:47) (cid:47) (cid:15) (cid:15) ( f − ι x ) (cid:15) (cid:15) (cid:15) (cid:15) y ⊕ Cx ( κ f µ f ) (cid:47) (cid:47) ( κ f µ f ) (cid:15) (cid:15) (cid:15) (cid:15) Cone f ⊕ id Cone f (cid:15) (cid:15) Cone f id Cone f (cid:47) (cid:47) Cone f . In a complicial exact category, we have the following equalities: ( r ∗ C ) · ( C ∗ ζ ) · ( C ∗ π ∗ C ) · ( C ∗ σ ) · ( ζ ∗ C ) = , (140) ζ · ( π ∗ C ) · σ · ( r ∗ C ) = ( r ∗ C ) · ( C ∗ ζ ) · ( C ∗ π ∗ C ) · ( C ∗ σ ) , (141) ( ζ ∗ C ) · ( T ∗ ζ ) · ( T ∗ π ∗ C ) · ( T ∗ σ ) = ( C ∗ ζ ) · ( C ∗ π ∗ C ) · ( C ∗ σ ) · ( ζ ∗ C ) , (142) ζ · τ C , T · ( r ∗ T ) = ( r ∗ C ) · ( C ∗ ζ ) · ( C ∗ τ C , T ) . (143) Proof.
First we will prove the equality ( ) . By using the equalities ( ) and ( ) , we have the equalities ( r ∗ C ) · ( C ∗ ζ ) · ( C ∗ π ∗ C ) · ( C ∗ σ ) · ( ζ ∗ C )= Cone ( , id CC , − r ∗ C ) · Cone ( ζ , ζ ) · Cone ( π ∗ C , π ∗ C ) · Cone ( σ , σ ) · Cone ( id CC , , r ∗ C )= Cone ( , , ) = . ( ) and it turns out that we have the following commutative diagram TCC (cid:47) (cid:47) ζ ∗ C (cid:47) (cid:47) (cid:111) T ∗ σ (cid:15) (cid:15) CCC r ∗ C (cid:47) (cid:47) (cid:47) (cid:47) (cid:111) C ∗ σ (cid:15) (cid:15) CC (cid:111) σ (cid:15) (cid:15) TCC T ∗ π ∗ C (cid:15) (cid:15) (cid:15) (cid:15) CCC C ∗ π ∗ C (cid:15) (cid:15) (cid:15) (cid:15) CC π ∗ C (cid:15) (cid:15) (cid:15) (cid:15) T TC (cid:15) (cid:15) T ∗ ζ (cid:15) (cid:15) CTC (cid:15) (cid:15) C ∗ ζ (cid:15) (cid:15) TC (cid:15) (cid:15) ζ (cid:15) (cid:15) TCC (cid:47) (cid:47) ζ ∗ C (cid:47) (cid:47) CCC r ∗ C (cid:47) (cid:47) (cid:47) (cid:47) CC . Namely we obtain the equalities ( ) and ( ) . Finally, the following commutative diagram and the factthat the morphism CC ∗ π : CCC → CCT is an epimorphism imply the equality ( ) . CCT r ∗ T (cid:47) (cid:47) (cid:47) (cid:47) (cid:111) C ∗ τ C , T (cid:15) (cid:15) CT (cid:111) τ C , T (cid:15) (cid:15) CCC CC ∗ π (cid:99) (cid:99) (cid:99) (cid:99) r ∗ C (cid:47) (cid:47) (cid:47) (cid:47) (cid:111) C ∗ σ (cid:15) (cid:15) CC C ∗ π (cid:61) (cid:61) (cid:61) (cid:61) (cid:111) σ (cid:15) (cid:15) CCC C ∗ π ∗ C (cid:123) (cid:123) (cid:123) (cid:123) CC π ∗ C (cid:33) (cid:33) (cid:33) (cid:33) CTC (cid:15) (cid:15) C ∗ ζ (cid:15) (cid:15) TC (cid:15) (cid:15) ζ (cid:15) (cid:15) CCC r ∗ C (cid:47) (cid:47) (cid:47) (cid:47) CC . Let x f → y g → z to be a pair of morphisms in a complicial exact categories. Then applyingCone to the admissible exact sequence [ x f → y ] (cid:16)(cid:16) id x − f (cid:17) , (cid:16) g − id y (cid:17)(cid:17) (cid:26) [ x ⊕ y (cid:16) gf
00 id y (cid:17) → z ⊕ y ] (( f id y ) , ( id z g )) (cid:16) [ y g → z ] in C [ ] , we obtain the admissible exact sequenceCone f (cid:16) Cone ( id x , g ) − Cone ( f , id y ) (cid:17) (cid:26) Cone g f ⊕ Cy ( Cone ( f , id z ) Cone ( id y , g ) ) (cid:16) Cone g (144)in C . The morphism Cone ( κ f , κ g f ) : Cone g → Cone Cone ( id x , g ) is a C -homotopy equivalence. (See [Moc10,3.34].) Let f : x → y be a morphism in a complicial exact category. Then there are sequence of(homotopy) commutative squares [ x f → y ] ( ι x , κ f ) → [ Cx µ f → Cone f ] ( , id Cone f , − µ f r x ) → [ → Cone f ] and by applyingCone to these (homotopy) squares, we obtain the equalityCone ( , id Cone f , − µ f r x ) Cone ( ι x , κ f ) = id Cone f . (145)32one f id Cone f (cid:47) (cid:47) Cone ( ι x , κ f ) (cid:36) (cid:36) Cone f Cone µ f . Cone ( , id Cone f , − µ f r x ) (cid:58) (cid:58) We can show the following.
Let C and C (cid:48) be complicial exact categories and let ( f , d ) : C → C (cid:48) be a com-plicial exact functor. Then f induces a functor f [ ] h : C [ ] h → C (cid:48) [ ] h by Lemma-Definition 2.2.12. Namely f [ ] h sends an object [ a : x → y ] in C [ ] h to an object [ f a : f x → f y ] in C (cid:48) [ ] h and sends a homotopy commuta-tive square ( ψ , φ , H ) : [ a : x → y ] → [ a (cid:48) : x (cid:48) → y (cid:48) ] in C [ ] h to a homotopy commutative square ( f ψ , f φ , f H · c x ) : [ f a : f x → f y ] → [ f a (cid:48) : f x (cid:48) → f y (cid:48) ] in C (cid:48) [ ] h . We define d Cyl : Cyl f [ ] h ∼ → f Cyl to be a natural equivalenceby setting d Cyl : = (cid:18) id f d (cid:19) . The natural equivalence d Cyl induces a natural equivalence d Cone : Cone f [ ] h ∼ → f Cone characterized by the following equalities: d Cone · ( κ ∗ f ) = f ∗ κ , (146) ( f ∗ µ ) · c = d Cone · ( µ ∗ f ) . (147) f ran (cid:47) (cid:47) κ ∗ f (cid:47) (cid:47) (cid:35) (cid:35) f ∗ κ (cid:35) (cid:35) Cone f [ ] h (cid:111) d Cone (cid:15) (cid:15)
C f dom µ ∗ f (cid:111) (cid:111) (cid:111) c (cid:15) (cid:15) f Cone fC dom . f ∗ µ (cid:111) (cid:111) We write C lv and ι lv for the endofunctor on C [ ] h and the natural transformation id C [ ] h → C lv defined bylevel-wisely. By applying the argument above to the complicial functor ( C , σ ) : C → C , we obtain naturalequivalences σ Cyl : C Cyl ∼ → Cyl C lv and σ Cone : C Cone ∼ → Cone C lv which satisfy the following equalities:Cyl ∗ ι lv = σ Cyl · ( ι ∗ Cyl ) , (148)Cone ∗ ι lv = σ Cone · ( ι ∗ Cone ) . (149)In particular, ( Cyl , σ Cyl ) , ( Cone , σ Cone ) : C [ ] → C are complicial exact functors.Recall the definitions of ψ f and Θ f from 2.3.1 and 2.3.4 respectively and recall the definition of strictlyordinary from 1.3.4. Let C be a strictly ordinary complicial exact category and let f : x → y be a morphism in C . Then a homotopy commutative square ( ψ f , , Θ f ) : [ C Cone f → ] → [ T f : T x → Ty ] induces an equality Cone ( ψ f , , Θ f ) = − σ Cone T . (150) In particular the left square in the commutative diagram ( ) below is a push-out diagram. Cone f (cid:47) (cid:47) ι Cone f (cid:47) (cid:47) ψ f (cid:15) (cid:15) C Cone f π Cone f (cid:47) (cid:47) (cid:47) (cid:47) (cid:16) − Θ fC ψ f (cid:17) (cid:15) (cid:15) T Cone f (cid:111) − σ Cone T (cid:15) (cid:15) T x (cid:47) (cid:47) (cid:16) − T f ι Tx (cid:17) (cid:47) (cid:47) Ty ⊕ CT x ( κ T f µ T f ) (cid:47) (cid:47) (cid:47) (cid:47) Cone
T f . (151) Thus we can take
Cone ψ f = Ty ⊕ CT x , (152)33 ψ f = (cid:18) − T f ι Tx (cid:19) and , (153) µ ψ f = (cid:18) − Θ f C ψ f (cid:19) . (154) Proof.
First we assume that y = x and f = id x . In this case, Cone ( ψ f , , Θ f ) = Cone ( π x , , π x r x ) = Cone ( π x , π x ) · ζ x . Since C is strictly ordinary we have an equality ζ = − σ · ζ . Thus we have equalities ( C ∗ π ) · π = − ( C ∗ π ) · σ · ζ = − τ T , C · ( π ∗ C ) · ζ = − τ T , C . For general case, the equality ( ) follows from the commu-tative diagram below.Cone f (cid:47) (cid:47) ι ∗ Cone f (cid:47) (cid:47) ψ f (cid:15) (cid:15) C Cone f π ∗ Cone f (cid:47) (cid:47) (cid:47) (cid:47) T Cone f − σ Cone T (cid:111) (cid:15) (cid:15) Cx (cid:47) (cid:47) ι ∗ Cx (cid:47) (cid:47) π x (cid:15) (cid:15) (cid:15) (cid:15) µ f (cid:98) (cid:98) CCx π Cx (cid:47) (cid:47) (cid:47) (cid:47) C ∗ µ f (cid:79) (cid:79) (cid:16) − π xrxC ∗ π x (cid:17) (cid:15) (cid:15) TCx T ∗ µ f (cid:58) (cid:58) (cid:111) − τ T , C (cid:15) (cid:15) T x (cid:47) (cid:47) (cid:16) − id Tx ι ∗ Tx (cid:17) (cid:47) (cid:47) id Tx (cid:124) (cid:124) T x ⊕ CT x (cid:16)
T f
00 id
CTx (cid:17) (cid:15) (cid:15) ( ι ∗ Tx id CTx ) (cid:47) (cid:47) (cid:47) (cid:47) CT x µ ∗ T f (cid:36) (cid:36)
T x (cid:47) (cid:47) (cid:16) − T f ι ∗ Tx (cid:17) (cid:47) (cid:47) Ty ⊕ CT x ( κ ∗ T f µ ∗ T f ) (cid:47) (cid:47) (cid:47) (cid:47) Cone
T f . Let C be a complicial exact category. Then ( ) Let ( a , b , H ) : [ x f → y ] → [ x (cid:48) g → y (cid:48) ] be a homotopy commutative square and let V : f ⇒ C f and W : g ⇒ C g be C-homotopies. Then ( i ) A triple ( a , b , − WCa + bV + H ) is a homotopy commutative square from [ x f → y ] to [ x (cid:48) g → y (cid:48) ] . ( ii ) We define a morphism
Cyl ( V , W , H ) : C ( Cyl a ) → Cyl b by setting
Cyl ( V , W , H ) : = (cid:18) W ( − WCa + bV ) r x C ( V ) σ (cid:19) . (155) Then
Cyl ( V , W , H ) is a C-homotopy from Cyl ( f , g , H ) to Cyl ( f , g , − WCa + bV + H ) and makes the leftsquare below commutative. ( iii ) There exists a morphism
Cone ( V , W , H ) : C Cone a → Cone b which makes the right square below com-mutative and it is a C-homotopy from
Cone ( f , g , H ) to Cone ( f , g , − WCa + bV + H ) .Cx (cid:47) (cid:47) C ∗ ξ a (cid:47) (cid:47) V (cid:15) (cid:15) C Cyl a C ∗ η a (cid:47) (cid:47) (cid:47) (cid:47) Cyl ( V , W , H ) (cid:15) (cid:15) C Cone a Cone ( V , W , H ) (cid:15) (cid:15) y (cid:47) (cid:47) ξ b (cid:47) (cid:47) Cyl b η b (cid:47) (cid:47) (cid:47) (cid:47) Cone b . ( ) Let ( a , b , H ) : [ x f → x (cid:48) ] → [ y g → y (cid:48) ] be a homotopy commutative square in C and let f (cid:48) : x (cid:48) → x and g : y (cid:48) → ybe C-homotopy inverse morphisms of f and g respectively. Namely there exists C-homotopies K : f (cid:48) f ⇒ C id x ,K (cid:48) : f f (cid:48) ⇒ C id x (cid:48) , L : g (cid:48) g ⇒ C id y and L (cid:48) : gg (cid:48) ⇒ C id y (cid:48) . We set H (cid:48) : = − g (cid:48) BK (cid:48) − g (cid:48) HC ( f (cid:48) ) + LC ( a f (cid:48) ) and U : = − LC ( a ) + bK + H (cid:48) (cid:63) H. Then ( i ) The triple ( b , a , H (cid:48) ) is a homotopy commutative square [ x (cid:48) f (cid:48) → x ] → [ y (cid:48) g (cid:48) → y ] . ( ii ) The morphism
Cyl ( f , g , H ) : Cyl a → Cyl b is a C-homotopy equivalence. ( iii ) The morphism
Cone ( f , g , H ) : Cone a → Cone b is a C-homotopy equivalence. roof. A proof of ( ) is given in [Moc10, 3.11] and ( ) ( i ) is straightforward. ( ii ) By ( ) , Cyl ( K , L , H (cid:48) (cid:63) H ) is a C -homotopy Cyl ( f (cid:48) f , g (cid:48) g , H (cid:48) (cid:63) H ) ⇒ C Cyl ( id x , id y , U ) and Cyl ( id x , id y , U ) = (cid:18) id y − U Cx (cid:19) is an isomorphism. Thus Cyl ( id x , id y , U ) − Cyl ( f (cid:48) , g (cid:48) , H (cid:48) ) is a left C -homotopy inverse of themorphism Cyl ( f , g , H ) . Similarly it turns out that Cyl ( f , g , H ) admits a right C -homotopy inverse. ThusCyl ( f , g , H ) is a C -homotopy equivalence by Lemma 2.1.6. A proof of ( iii ) is similar. Let C be a complicial exact category and x f (cid:26) y g (cid:16) z and x (cid:48) f (cid:48) (cid:26) y (cid:48) g (cid:48) (cid:16) z (cid:48) be Frobeniusadmissible exact sequences in C and a : x → x (cid:48) and b : y → y (cid:48) morphisms in C and H : b f ⇒ C f (cid:48) a a C-homotopy. Then there exists a morphism c : z → z (cid:48) and a retraction ρ : Cy → Cx of C f : Cx → Cy such that − g (cid:48) H ρ : Cy → z (cid:48) is a C-homotopy from cg to g (cid:48) b.x (cid:47) (cid:47) f (cid:47) (cid:47) a (cid:15) (cid:15) y g (cid:47) (cid:47) (cid:47) (cid:47) b (cid:15) (cid:15) z c (cid:15) (cid:15) x (cid:48) (cid:47) (cid:47) f (cid:48) (cid:47) (cid:47) y (cid:48) g (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) z (cid:48) . Moreover sequences
Cyl a Cyl ( f , f (cid:48) , H ) (cid:26) Cyl b Cyl ( g , g (cid:48) , − g (cid:48) H ρ ) (cid:16) Cyl c and
Cone a Cone ( f , f (cid:48) , H ) (cid:26) Cone b Cone ( g , g (cid:48) , − g (cid:48) H ρ ) (cid:16) Cone c are Frobenius admissible sequences and we have the equality ( − g (cid:48) H ρ ) (cid:63) C H = . (156) Proof.
There exists a morphism ρ : Cy → Cx which is a retraction of C f by Lemma 2.1.7. Since (cid:18) id y (cid:48) − H ρ y (cid:19) and (cid:18) f (cid:48) C f (cid:19) are Frobenius admissible monomorphisms, the composition (cid:18) f (cid:48) − H C f (cid:19) = (cid:18) id y (cid:48) − H ρ Cy (cid:19) (cid:18) f (cid:48) C f (cid:19) is also a Frobenius admissible monomorphism. Similarly it turns out that the composition (cid:18) g (cid:48) g (cid:48) H ρ Cg (cid:19) = (cid:18) g (cid:48) Cg (cid:19) (cid:18) id y (cid:48) H ρ Cy (cid:19) is a Frobenius admissible epimorphism. In the commutative diagram below, by the universality of cokernel,there exists the dotted morphism (cid:16) cu (cid:17) : z → z (cid:48) in the diagram below which makes the right square belowcommutative: x (cid:47) (cid:47) f (cid:47) (cid:47) (cid:16) a − ι x (cid:17) (cid:15) (cid:15) y g (cid:47) (cid:47) (cid:47) (cid:47) (cid:16) b − ι y (cid:17) (cid:15) (cid:15) z (cid:16) cu (cid:17) (cid:15) (cid:15) x (cid:48) ⊕ Cx (cid:47) (cid:47) (cid:16) f (cid:48) − H Cf (cid:17) (cid:47) (cid:47) (cid:36) (cid:36) (cid:16) f (cid:48) Cf (cid:17) (cid:36) (cid:36) y (cid:48) ⊕ Cy (cid:16) g (cid:48) g (cid:48) H ρ Cg (cid:17) (cid:47) (cid:47) (cid:47) (cid:47) (cid:111) ϕ (cid:15) (cid:15) z (cid:48) ⊕ Czy (cid:48) ⊕ Cy (cid:16) g (cid:48) Cg (cid:17) (cid:58) (cid:58) (cid:58) (cid:58) where the morphism ϕ is (cid:18) id y (cid:48) H ρ Cy (cid:19) . By the commutativity of the right square above, we have the equal-ities ug = − Cg ι y = − ι z g and it turns out that u = − ι z by surjectivity of g . By applying the 3 × a Cone ( f , f (cid:48) , H ) (cid:26) b Cone ( g , g (cid:48) , g (cid:48) H ρ ) (cid:16) Cone c . x (cid:47) (cid:47) f (cid:47) (cid:47) (cid:15) (cid:15) ξ a (cid:15) (cid:15) y g (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) ξ b (cid:15) (cid:15) z (cid:15) (cid:15) ξ c (cid:15) (cid:15) Cyl a (cid:47) (cid:47) Cyl ( f , f (cid:48) , H ) (cid:47) (cid:47) η a (cid:15) (cid:15) (cid:15) (cid:15) Cyl b Cyl ( g , g (cid:48) , g (cid:48) H ρ ) (cid:47) (cid:47) (cid:47) (cid:47) η b (cid:15) (cid:15) (cid:15) (cid:15) Cyl c η c (cid:15) (cid:15) (cid:15) (cid:15) Cone a (cid:47) (cid:47) Cone ( f , f (cid:48) , H ) (cid:47) (cid:47) Cone b Cone ( g , g (cid:48) , g (cid:48) H ρ ) (cid:47) (cid:47) (cid:47) (cid:47) Cone c . Finally we have equalities ( − g (cid:48) H ρ ) (cid:63) C H = g (cid:48) H − g (cid:48) H ρ C f = g (cid:48) H − g (cid:48) H = Let C be a complicial exact category. Then the functors Cyl and
Cone send a levelFrobenius admissible exact sequence in C [ ] h to a Frobenius admissible exact sequence in C . In particular if C is ordinary, then Cyl and
Cone are exact functors.Moreover if C is normal, then the pairs ( Cyl , σ Cyl ) and ( Cone , σ Cone ) are complicial exact functors from C [ ] h to C . Let C be a complicial exact category and let [ f : x → y ] ( a , b , H ) → [ f (cid:48) : x (cid:48) → y (cid:48) ] be a homotopycommutative square in C . Then we have the canonical isomorphism Cone Cone ( a , b , H ) ∼ → Cone Cone ( f , f (cid:48) , − H ) . (157) Proof. If C is normal and ordinary, it is just a consequence of 2.3.10 and 2.3.14. For general C , it followsfrom the fact that both Cone Cone ( a , b , H ) and Cone Cone ( f , f (cid:48) , − H ) are naturally isomorphic to z in thecommutative diagram below. x (cid:47) (cid:47) (cid:16) a − ι x (cid:17) (cid:47) (cid:47) (cid:15) (cid:15) (cid:16) f − ι x (cid:17) (cid:15) (cid:15) x (cid:48) ⊕ Cx ( κ a µ a ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) f (cid:48) H − ι x (cid:48) Cf − C ∗ ι x (cid:15) (cid:15) Cone a (cid:15) (cid:15) (cid:16) Cone ( f , f (cid:48) , H ) − ι Cone a (cid:17) (cid:15) (cid:15) y ⊕ Cx (cid:47) (cid:47) (cid:32) b − H Ca − ι y − C ∗ ι x (cid:33) (cid:47) (cid:47) ( κ f µ f ) (cid:15) (cid:15) (cid:15) (cid:15) y (cid:48) ⊕ Cx (cid:48) ⊕ Cy ⊕ CCx (cid:16) κ b µ b C ∗ κ a C ∗ µ a (cid:17) (cid:47) (cid:47) (cid:47) (cid:47) (cid:16) κ f (cid:48) µ f (cid:48) C ∗ κ f C ∗ µ f (cid:17) (cid:15) (cid:15) (cid:15) (cid:15) Cone b ⊕ C Cone a (cid:15) (cid:15) (cid:15) (cid:15) Cone f (cid:47) (cid:47) (cid:16) Cone ( a , b , H ) − ι Cone f (cid:17) (cid:47) (cid:47) Cone f (cid:48) ⊕ C Cone f (cid:47) (cid:47) (cid:47) (cid:47) z . Let C be a complicial exact category and let f : x → y be a morphism in C . By applying2.3.15 to the commutative square (cid:47) (cid:47) (cid:15) (cid:15) x f (cid:15) (cid:15) y id y (cid:47) (cid:47) y , we obtain the canonical isomorphism Cone κ f ∼ → Cone Cone ( , f ) . (158)36 morphism ( id x , ) : [ Cone ( , f ) : x → Cy ] → [ x → ] in C [ ] induces a C -homotopy equivalenceCone ( id x , ) : Cone Cone ( , f ) → T x (159)by 2.3.12 ( ) . Let C be a complicial exact category. Then thereare dual notions of the mapping cylinder and the mapping cone functors. We define Hfib , Pat : C [ ] → C tobe functors which called the homotopy fiber functor and the mapping path space functor respectively in thefollowing way. Pat : = dom ⊕ P ran (160)Hfib : = dom × ran P ran (161)where Hfib is defined by the following pull back diagram: T − (cid:47) (cid:47) λ (cid:47) (cid:47) id T − (cid:15) (cid:15) Hfib ν (cid:15) (cid:15) υ (cid:47) (cid:47) (cid:47) (cid:47) (cid:70) dom ε (cid:15) (cid:15) T − (cid:47) (cid:47) j (cid:47) (cid:47) P ran q ran (cid:47) (cid:47) (cid:47) (cid:47) ran (162)where λ is induced from the universal property of Hfib. Then there is an admissible exact sequenceHfib ρ (cid:26) Pat t (cid:16) ran (163)where t : = (cid:0) ε q ran (cid:1) and ρ = (cid:18) υ − ν (cid:19) . (164)Moreover we define t : Pat → dom and χ : dom → Pat to be natural transformations by setting t : = (cid:0) id dom (cid:1) and χ : = (cid:18) id dom (cid:19) . (165)Then we have the following commutative diagram: dom id dom (cid:124) (cid:124) ε (cid:34) (cid:34) χ (cid:15) (cid:15) dom Pat t (cid:47) (cid:47) (cid:47) (cid:47) t (cid:111) (cid:111) (cid:111) (cid:111) ran . (166)Since the functor Hfib is obtained by the pull-back diagram, we can choose the fiber products in the followingway. For an object x in C , since the diagrams below are pull-backs, we shall assume that Hfib id x = Px , ν id x = id Px , υ id x = q x , Hfib ( → x ) = T − x , ν → x = ι T − x , Hfib ( x → ) = x and υ x → = id x . Px q x (cid:47) (cid:47) (cid:47) (cid:47) id Px (cid:15) (cid:15) x id x (cid:15) (cid:15) Px q x (cid:47) (cid:47) (cid:47) (cid:47) x , T − x (cid:47) (cid:47) (cid:47) (cid:47) ι T − x (cid:15) (cid:15) (cid:15) (cid:15) Px q x (cid:47) (cid:47) (cid:47) (cid:47) x , x id x (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) x (cid:15) (cid:15) P ( ) = q (cid:47) (cid:47) (cid:47) (cid:47) . Recall the definitions s , t and ∆ : C → C [ ] from 2.3.1. Then we shall assume that Hfib s = id C , Hfib t = T − and Hfib ∆ = P . 37 .3.18. Lemma. Let C be a complicial exact category. Then there is a canonical natural equivalence betweenfunctors C [ ] → C . Cone ∼ → T Hfib . (167) Proof.
Let f : x → y be a morphism in C . By applying the operation Cone to the Frobenius exact sequencein C [ ] , T − x T − f (cid:15) (cid:15) T − y (cid:32) id T − x T − f (cid:33) →→ id T − y T − x ⊕ T − y (cid:16) T − y (cid:17) (cid:15) (cid:15) T − y (cid:16) − T − f id T − y (cid:17) →→ T − y (cid:15) (cid:15) , (168)we obtain the Frobenius admissible exact sequence in C Cone T − f (cid:26) x ⊕ Py ( − f q y ) (cid:16) y . (169)Comparing the sequence ( ) , we obtain the canonical isomorphism Cone T − ∼ → Hfib as the functors C [ ] → C . Let C be a complicial exact category andlet x f ← y g → z be a pair of morphisms in C . Then we set x (cid:116) hy z : = Cone ( y (cid:16) fg (cid:17) → x ⊕ z ) . (170)Then there exists a Frobenius admissible exact sequence y (cid:18) ι yfg (cid:19) (cid:26) Cy ⊕ x ⊕ z (cid:16) ψ (cid:18) fg (cid:19) ig − if (cid:17) (cid:16) x (cid:116) hy z (171)and homotopy commutative diagram y g (cid:47) (cid:47) f (cid:15) (cid:15) z i f (cid:15) (cid:15) x i g (cid:47) (cid:47) x (cid:116) hy z . (172)We call it the canonical homotopy push out ( of the diagram x f ← y g → z ) and it has the following universalproperty:For a homotopy commutative square [ f : y → x ] ( g , g (cid:48) , H ) → [ f (cid:48) : z → u ] , there exists a unique morphism a : x (cid:116) hy z → u such that f (cid:48) = ai f , g (cid:48) = ai g and H = ψ (cid:16) fg (cid:17) a .Dually there is a notion of homotopy pull-back. Let x f → y g ← z be a pair of morphism in C . Then we set x × hy z : = Hfib ( x ⊕ z ( f g ) → y ) . (173)Then there exists a Frobenius admissible exact sequence x × hy z (cid:32) uf , gpg − pf (cid:33) (cid:26) Py ⊕ x ⊕ z ( qy f g ) (cid:16) y (174)38nd P -homotopy commutative diagram x × hy z p f (cid:47) (cid:47) p g (cid:15) (cid:15) z g (cid:15) (cid:15) x f (cid:47) (cid:47) y . (175)We call it the canonical homotopy pull-back ( of the diagram x f → y g ← z ) and it has the following universalproperty:For a P -homotopy commutative square [ g (cid:48) : u → x ] ( f , f (cid:48) , H ) → [ g : z → y ] , there exists a unique morphism a : u → x × hy z such that f (cid:48) = p f a , g (cid:48) = p g a and H = u f , g a . As pointed out in [Bar15, Remark 3.6], in the view of ( ∞ , ) -category theory, a complicial exact category withweak equivalences is actually nothing more than a model for some stable ( ∞ , ) -category. In this section, wewill further develop such a theory under the name of relative complicial exact categories. In the first subsec-tion 3.1, we will study the notion of (thick) null classes which is a variant of the notion of (thick) triangulatedsubcategories of triangulated categories. In the next subsection 3.2, we will interpret the fundamental termi-nologies about relative (complicial) exact categories. A typical example of relative complicial exact categoriesis a category of chain complexes on a relative exact category which will be detailed in subsection 3.3. In thenext subsection 3.4, we will study homotopy categories of relative complicial exact categories and introducethe notion of derived equivalences. In 3.4.19, we will show the (large) relative category of relative exact cate-gories with derived equivalences is categorical homotopy equivalent to the (large) relative category of relativecomplicial exact categories with derived equivalences. This stands for that a relative exact category is also amodel for some stable infinity category. In the final subsection 3.5, following [Wal85, § t -structure on a homotopy category 3.5.3. w -closure, w -trivial objects). Let C be a category and let w be a class of morphisms in C and let S be a class of objects in C . We say that an object x is w-equivalent to an object y if there exists a zig-zagsequence of finite morphisms in w which connects x and y . We denote the class of all objects which are w -equivalent to some objects in S by S w , C or simply S w and call it the w-closure of S ( in C ). We regard S w , C as the full subcategory of C .Assume that C admits a zero object and all isomorphisms between zero objects are in w and w is closedunder finite compositions. Then we say that an object x is w-trivial if the canonical morphism 0 → x is in w .By assumption, the definition of w -triviality does not depend upon the choice of a zero object. We denote theclass of all w -trivial objects in C by C w . We regard C w as the full subcategory of C . C -homotopy closure). Let C be a complicial exact category and let A be a non-emptyfull subcategory of C and heq be a class of all C -homotopy equivalences in C . Then by the notation in 3.1.1,let A heq denote the full subcategory of C consisting of those objects x which are C -homotopy equivalent toobjects in A . We call A heq the C-homotopy closure of A ( in C ). Since C -homotopy equivalence relation istransitive, an object in C which is C -homotopy equivalent to an object in A heq is also in A heq . Let C be a complicial exact categoryand let N be a full subcategory of C . We say that N is a null class of C if N contains all C -contractibleobjects and if for any admissible exact sequence x (cid:26) y (cid:16) z in C , if two of x , y and z are in N , then the thirdone is also in N . We say that N is a prenull class if N contains a zero object and if for any morphism f : x → y in N , the objects Cone f and x ⊕ y are also in N . Since T x = Cone ( x → ) , in this case, T x is in39 for any object x in N . We say that N is a semi-null class if N is a prenull class a and for any object x in N , T − x is in N .We say that a null class (resp. prenull class, semi-null class) N is thick if it is closed under retractions.Namely for any object x in C if there exists an object y in N and a pair of morphisms f : x → y and g : y → x such that g f = id x , then x is also in N . Let C be a complicial exact category and let S be a full subcategory of C . Then thefollowing conditions are equivalent. ( ) S is closed under the operations Cone and
Hfib . ( ) S is closed under the operations Cone and T − . ( ) S is closed under the operations Hfib and T .In particular, S is a semi-null class of C if and only if it is closed under finite direct sums and satisfies oneof the conditions above.Proof. Notice that for an object x and a morphism f : x → y in C , T − x = Hfib ( x → ) , T x = Cone ( → x ) and Cone f ∼ → T Hfib f and Hfib f ∼ → T − Cone f by Lemma 2.3.18. By mimicking the definition of complicial exact functors between complicial exact cat-egories, we can define the notion to prenull classes. That is, let C and C (cid:48) be complicial exact categoriesand let N ⊂ C and N (cid:48) ⊂ C (cid:48) be prenull classes of C and C (cid:48) respectively. A complicial exact functor ( f , c ) : N → N (cid:48) from N to N (cid:48) is a pair ( f , c ) consisting of a functor N → N (cid:48) which is additive andpreserves Frobenius admissible exact sequences and a natural equivalence c : C f ∼ → fC satisfies the equality c · ( ι N (cid:48) ∗ f ) = f ∗ ι N . (176)Then we can show the similar statements of 1.2.15, 1.2.17 and 2.3.10. Let f : X → Y be a functor between categories X and Y and let Z be a full subcategory of Y and let w be a class of morphisms in Y . Then we denote the fullsubcategory of X consisting of those objects x such that f ( x ) is isomorphic to some object in Z by f − Z and we call it the pull-back of Z by f . We also write f − ( w ) for the class of morphisms in X consisting ofthose morphisms u : x → y such that f ( u ) is in w and call it the pull-back of w by f . Let ( f , c ) : C → C (cid:48) be a complicial exact functor betweencomplicial exact categories C and C (cid:48) and let N be a prenull ( resp. semi-null, null ) class of C (cid:48) . Then f − N is also a prenull ( resp. semi-null, null ) class of C .Proof. Let u : x → y be a morphism in f − N . Then by definition, there exists a pair of objects x (cid:48) and y (cid:48) in N and a pair of isomorphisms a : f x ∼ → x (cid:48) and b : f y ∼ → y (cid:48) in C (cid:48) . We set u (cid:48) : = b · f u · a − : x (cid:48) → y (cid:48) . Then by1.2.15 and 2.3.10, f ( x ⊕ y ) , f ( T ± x ) and f ( Cone u ) are isomorphic to x (cid:48) ⊕ y (cid:48) , T ± x (cid:48) and Cone u (cid:48) respectively.Thus if N is a prenull (resp. semi-null) class, then x ⊕ y , Cone u (and T − x ) are in f − N and f − N is aprenull (resp. semi-null) class.Next assume that N is a null class of C (cid:48) . Then since f sends a C -contractible object in C to a C (cid:48) -contractible object in C (cid:48) , f − N contains all C -contractible objects. For an admissible exact sequences x (cid:26) y (cid:16) z in C , if two of f ( x ) , f ( y ) and f ( z ) are in N , then the third one is also in N . Thus f − N is anull class. Let A be an additive category and let O be a full subcategory of A . We write (cid:104) O (cid:105) ⊕ for the full subcategory consisting of those objects which are isomorphic to finite directsums of some objects in O . In particular, (cid:104) O (cid:105) ⊕ contains all zero objects in A and closed under finite directsum operation and it is the smallest full subcategory which contains O and closed under finite direct sumoperation. We call it the direct sum closure of O . T -closure). Let C be a complicial exact category and let A be a full subcategory of C .We write (cid:104) A (cid:105) T for the full subcategory of C consisting of those objects which are isomorphic to T n y forsome integers n and some objects y in A . We call (cid:104) A (cid:105) T the T -closure of A in C .40 .1.10. Lemma. Let C be a complicial exact category and let A be a full subcategory of C . Assume that A is closed under finite direct sums and the operation T . Namely for any objects x and y in A , x ⊕ y and T x arein A . Then (cid:104) A (cid:105) T is also closed under finite direct sums.Proof. Let x and y be objects in (cid:104) A (cid:105) T . Then there exists objects x (cid:48) and y (cid:48) in A and isomorphisms x ∼ → T m x (cid:48) and y ∼ → T n y (cid:48) for suitable integers m and n . We set k : = min { m , n } . Then x ⊕ y ∼ → T k ( T m − k x (cid:48) ⊕ T n − k y (cid:48) ) . Since m − k , n − k ≥ A is closed under the operation T and finite direct sums, T m − k x (cid:48) ⊕ T n − k y (cid:48) is in A . Thus x ⊕ y is in (cid:104) A (cid:105) T . Let C be a complicial exact category and let D be a full addi-tive subcategory of C . For a non-negative integer n , we will inductively define P n ( D ) to be a full subcategoryof C . First we set Ob P ( D ) = Ob D and we set Ob P n + ( D ) : = { Cone f } f ∈ Mor P n ( D ) . For an object x in P n ( D ) ,since we have an equality x = Cone ( → x ) , P n + ( D ) contains P n ( D ) . Finally we set (cid:104) D (cid:105) prenull : = (cid:91) n ≥ P n ( D ) .We claim that (cid:104) D (cid:105) prenull is the smallest full subcategory of C which contains D and we call it the prenullclosure of D .For a full subcategory S of C , We write (cid:104) S (cid:105) prenull for (cid:104)(cid:104) S (cid:105) ⊕ (cid:105) prenull and also call it prenull closure of S . Proof.
First we will show that for each non-negative integer n , P n ( D ) is closed under finite direct sum opera-tions by induction on n . For n =
0, it is clear by assumption. Assume that assertion is true for n = k and wewill prove for n = k +
1. Let x and y be a pair of objects in P k + ( D ) . Then there exists a pair of morphisms f : x (cid:48) → x (cid:48)(cid:48) and g : y (cid:48) → y (cid:48)(cid:48) in P k ( D ) such that x = Cone f and y = Cone g . Then x ⊕ y = Cone (cid:18) f g (cid:19) is in P k + ( D ) .Next let f : z → w be a morphism in (cid:104) D (cid:105) prenull . Then there exists a non-negative integer n such that f is in P n ( D ) . Then Cone f is in P n + ( D ) (cid:44) → (cid:104) D (cid:105) prenull . Thus (cid:104) D (cid:105) prenull is a prenull class in C .Finally let P be prenull class in C which contains D . Then we can show for non-negative integer n , P n ( D ) is in P by proceeding induction on n . Thus it turns out that (cid:104) D (cid:105) prenull is the smallest prenull classwhich contains D . Let P be a prenull class of a complicial exact category C . Then weclaim that (cid:104) P (cid:105) T is the smallest semi-null class which contains P . Therefore we write (cid:104) P (cid:105) semi-null for (cid:104) P (cid:105) T and we call it the semi-null closure of P .For a full additive subcategory S of C , we simply write (cid:104) S (cid:105) semi-null for (cid:104)(cid:104) S (cid:105) prenull (cid:105) semi-null and also callit the semi-null closure of S . Proof.
Let f : x → y be a morphism in (cid:104) P (cid:105) semi-null . Then there exists a pair of integers n and m and a pairof objects x (cid:48) and y (cid:48) in P and a pair of isomorphisms a : T n x (cid:48) ∼ → x and b : T m y (cid:48) ∼ → y . We set k : = min { n , m } and f (cid:48) : = b − f a . Then the morphism T − k f (cid:48) : T n − k x (cid:48) → T m − k y (cid:48) is in P and T k Cone f (cid:48) and T k − T n − k x (cid:48) areisomorphic to Cone f and T − x respectively. Thus Cone f and T − x are in (cid:104) P (cid:105) T . Moreover x ⊕ y is in (cid:104) P (cid:105) T by Lemma 3.1.10.Let S be a semi-null class of C which contains P . Then for any object x in P and any integer n , T n x isin S . Thus S contains (cid:104) P (cid:105) T . Let C be a complicial exact category and let N be a full subcategory of C . Then ( ) If N is a null class of C , then ( i ) N is closed under C-homotopy equivalences. Namely an object in C which is C-homotopy equivalent toan object in N is also in N . ( ii ) For any object x in N , the objects Cx, T x and T − x are also in N . In particular N is a complicialexact subcategory of C . ( ) If N is a semi-null class of C , then C-homotopy closure of N in C is the smallest null class of C frob which contains N .Proof. A proof of ( ) ( i ) is given in [Moc10, 5.4]. ( ii ) For any object z in C , Cz is C -contractible and thus it is in N . For any object x in N , there are Frobeniusadmissible exact sequences x (cid:26) Cx (cid:16) T x and T − x (cid:26) CT − x (cid:16) x and Cx , CT − x are in N . Thus T x and T − x are also in N . 41 ) First notice that for any object x in N , Cx = Cone id x is in N and for any morphism x f → y in N ,Cyl f = y ⊕ Cx is in N .Next we we will show that for any morphism f : x → y in N heq , the objects Cone f , T − x and x ⊕ y arein N heq . Then by definition, there exists objects x (cid:48) , y (cid:48) in N and C -homotopy equivalences x (cid:48) a → x , x a (cid:48) → x (cid:48) and y b → y (cid:48) and a C -homotopy H : aa (cid:48) ⇒ C id x . Then we can check that ( a (cid:48) , b , b f H ) : [ x f → y ] → [ x (cid:48) b f a → y (cid:48) ] is a homotopy commutative square. Thus the morphism Cyl ( a (cid:48) , b , b f H ) : Cyl f → Cyl b f a is a C -homotopyequivalence by Lemma 2.3.12 ( ) ( ii ) and Cyl b f a is in N . Hence it turns out that Cone f is in N heq . By[Moc10, 2.13 ( ) ], the morphism (cid:18) a (cid:48) b (cid:19) : x ⊕ y → x (cid:48) ⊕ y (cid:48) is a C -homotopy equivalence and x (cid:48) ⊕ y (cid:48) are in N . Thus x ⊕ y is in N heq . Since T − preserves C -homotopy equivalences by [Moc10, 2.12 ( ) , 2.52], T − x T − a (cid:48) → T − x (cid:48) is a C -homotopy equivalence and T − x (cid:48) is in N . Thus T − x is in N heq and N heq is asemi-null class in C . Hence by 3.2.13 and 3.2.17 ( ) , it is a null class in C frob . Notice that there is no circularreasoning in our argument.Let N (cid:48) be a null class of C frob which contains N . Then N heq (cid:44) → N (cid:48) heq = N (cid:48) . Thus N heq is thesmallest null class of C frob which contains N . Let C be a complicial exact category. Since the class of all null classesof C is closed under intersection, for any full subcategory O of C , there exists the smallest null class of C which contains O . We denote it by (cid:104) O (cid:105) null and call it the null closure of O .We write (cid:104) O (cid:105) null , frob for ( (cid:104) O (cid:105) semi-null ) heq and by the previous lemma 3.1.13, it is the smallest null class in C frob which contains O . We call it the Frobenius null closure of O .Recall the definition of coproduct of complicial exact categories from 1.2.19. Let { C i } i ∈ I be a family of complicial exact categories indexed by a set I and let O i bea full subcategory of C i for all i ∈ I . Then we have equality (cid:42) (cid:95) i ∈ I O i (cid:43) null , frob , (cid:87) i ∈ I C i = (cid:95) i ∈ I (cid:104) O i (cid:105) null , frob , C i . (177) Let C be a category and let F be a full subcategory of C . We write F thi , C or simply F thi for the full subcategory of C consisting of those objects x such that x are retraction ofsome objects of F . We call it thick closure of F ( in C ) and then F thi is closed under retractions. Let f : C → C (cid:48) be an additive functor between additive cate-gories and let S and S (cid:48) be a pair of full subcategories of C and C (cid:48) respectively such that f ( Ob S ) ⊂ S (cid:48) and let w be a class of morphisms in C . Then ( ) We have the inclusions f ( Ob (cid:104) S (cid:105) ⊕ ) ⊂ Ob (cid:104) S (cid:48) (cid:105) ⊕ and f ( Ob S thi ) ⊂ Ob S (cid:48) thi and f ( Ob S w ) ⊂ Ob S (cid:48) f ( w ) . ( ) Moreover assume that C and C (cid:48) be a complicial exact categories and there exists a natural equivalencec : C (cid:48) f ∼ → fC such that the pair ( f , c ) is a complicial exact functor C → C (cid:48) . Then we have the inclusionf ( Ob (cid:104) S (cid:105) ) ⊂ Ob (cid:104) S (cid:48) (cid:105) where ∈ { prenull , semi-null , null } .Proof. ( ) Let N be a prenull (resp. semi-null, null) class in C (cid:48) which contains S (cid:48) . Then for = prenull(resp. semi-null, null), we have the inclusions Ob S ⊂ Ob (cid:104) S (cid:105) ⊂ Ob f − N and we obtain the inclusion f ( Ob (cid:104) S (cid:105) ) ⊂ Ob N . Thus we get the desired inclusion f ( Ob (cid:104) S (cid:105) ) ⊂ (cid:84) S (cid:48) ⊂ N Ob N = Ob (cid:104) S (cid:48) (cid:105) whereintersection taking all prenull (resp. semi-null, null) class which contains S (cid:48) .A proof of ( ) is similar. Let C be a complicial exact category and let x and y be a pairof objects in C . We say that x is a ( C -) homotopy retraction of y if there exists a pair morphisms i : x → y and p : y → x and C -homotopy H : id x ⇒ pi . We say that a full subcategory F in C is closed under C-homotopyretractions if for a pair of objects x and y in C such that y is in F and x is a C -homotopy retraction of y , then x is also in F . 42 .1.19. Proposition. Let C be an additive category and let F be an additive full subcategory. Then ( ) We have an equalities ( (cid:99) F ) thi , (cid:98) C = ( (cid:99) F ) isom , (cid:98) C , (178) F thi , C = ( (cid:99) F ) isom , (cid:98) C ∩ C , (179) where we denote the idempotent completion functor by (cid:100) ( − ) .Moreover assume C is a complicial exact category. Then ( ) If F is a prenull ( resp. semi-null ) class of C , then ( (cid:99) F ) isom , (cid:98) C is also a prenull ( resp. semi-null ) class of (cid:98) C . ( ) If F is closed under C-contractible objects and finite direct sum and retraction, then F is closed underC-homotopy retractions. In particular F is closed under C-homotopy equivalence. ( ) If F is closed under finite direct sum and contains all objects of the form Cx for some object x in C , thenfor an object y in F thi , there exists an object y (cid:48) in C such that y ⊕ y (cid:48) is in F . ( ) If F is a prenull (resp. semi-null, null) class of C , then F thi is also.Proof. ( ) First we will show the equality ( ) . What we need to prove are the following assertions: ( a ) ( (cid:99) F ) isom , (cid:98) C ∩ C is closed under retractions and contains F . ( b ) For a full subcategory M of C , if M contains F and is closed under retractions, then M contains ( (cid:99) F ) isom , (cid:98) C ∩ C . Proof of ( a ) . We regard F as a full subcategory of (cid:99) F by the canonical functor F → (cid:99) F , x (cid:55)→ ( x , id x ) . Thus F ⊂ ( (cid:99) F ) isom , (cid:98) C ∩ C .Next let x i → y p → x be a pair of morphisms such that pi = id x and y = ( y , id y ) is in ( (cid:99) F ) isom , (cid:98) C . Then bydefinition there exists an object ( z , e ) in (cid:99) F and a pair of morphisms y a → z b → y in C such that ba = id y and ab = e . We set e (cid:48) : = aipb . Then e (cid:48) = aipbaipb = aipipb = aipb = e (cid:48) . Notice that ai : ( x , id x ) → ( z , e (cid:48) ) and pb : ( z , e (cid:48) ) → ( x , id x ) give isomorphisms between ( z , e (cid:48) ) and x . Since z is in F , ( z , e (cid:48) ) is in (cid:99) F and x = ( x , id x ) is in ( (cid:99) F ) isom , (cid:98) C . Proof of ( b ) . Let ( x , id x ) be an object ( (cid:99) F ) isom , (cid:98) C ∩ C . By definition there exists an object y in F and a pairof morphisms x a → y x → in C with ba = id x . Since y is in F ⊂ M and M is closed under retractions, x is in M .Next applying the equality ( ) to (cid:98) C and (cid:99) F , we obtain the equalities ( (cid:99) F ) thi , (cid:98) C = ( (cid:99)(cid:99) F ) isom , (cid:98)(cid:98) C ∩ (cid:98) C =( (cid:99) F ) isom , (cid:98) C where we regard (cid:98)(cid:98) C = (cid:98) C and (cid:99)(cid:99) F = (cid:99) F . ( ) Let f : ( x , e ) → ( y , e (cid:48) ) be a morphism in ( (cid:99) F ) isom , (cid:98) C . By assumption there exists a pair of objects x (cid:48) and y (cid:48) and morphisms a : x → x (cid:48) , b : x (cid:48) → x , c : y → y (cid:48) and d : y (cid:48) → y such that ae = aba = a , eb = bab = b , ba = e , ce (cid:48) = cdc = c , e (cid:48) d = dcd = d , dc = e (cid:48) . Then we can show that ( x (cid:48) ⊕ y (cid:48) , (cid:18) ab cd (cid:19) ) , ( T ± x (cid:48) , T ± ab ) and ( Cone ( c f b ) , Cone ( ab , cd )) are isomorphic to ( x ⊕ y , (cid:18) e e (cid:48) (cid:19) ) , ( T ± x , T ± e ) and ( Cone f , Cone ( e , e (cid:48) )) respectively and it turns out that if F is a prenull (resp. semi-null) class in C , then ( (cid:99) F ) isom , (cid:98) C is also a prenull(resp. semi-null) class in (cid:98) C . ( ) Let x be a C -homotopy retraction of an object y in F . Namely there is a triple of morphisms i : x → y , p : y → x and H : Cx → y in C such that H ι x = id x − pi . Then since we have the equality (cid:0) H p (cid:1) (cid:18) ι i (cid:19) = id x , x is a retraction of Cx ⊕ y . By assumptions, Cx ⊕ y is in F and by assumption again, x is also in F .43 ) Let x be an object in F thi . Then there is a retraction x i → y p → x with y ∈ Ob F and pi = id x . Then byassumption y ⊕ Cx is in F and it is isomorphic to x ⊕ Cone i by the commutative diagram below. x (cid:47) (cid:47) (cid:16) i ι x (cid:17) (cid:47) (cid:47) y ⊕ Cx ( κ i − ξ i ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:111) (cid:16) p κ i − ξ i (cid:17) (cid:15) (cid:15) Cone ix (cid:47) (cid:47) (cid:16) id x (cid:17) (cid:47) (cid:47) x ⊕ Cone i ( i ) (cid:47) (cid:47) (cid:47) (cid:47) Cone i . Thus x ⊕ Cone i is also in F . ( ) Assume F is a null class in C . We will show that F thi is also a null class of C . What we need to proveare the following assertions: ( a ) F thi contains all C -contractible objects. ( b ) Let x i (cid:26) y p (cid:16) z be an admissible exact sequence in C . If two of x , y and z are in F thi , then third one isalso in F thi . Proof of ( a ) . Since F contains all C -contractible objects and F thi contains F , F thi also contains all C -contractible objects. Proof of ( b ) . Since F is closed under finite direct sums and contains all C -contractible objects, we can useassertion ( ) . Assume there exists a pair of objects x (cid:48) and z (cid:48) (resp. x (cid:48) and y (cid:48) , y (cid:48) and z (cid:48) ) such that both x ⊕ x (cid:48) and z ⊕ z (cid:48) (resp. x ⊕ x (cid:48) and y ⊕ y (cid:48) , y ⊕ y (cid:48) and z ⊕ z (cid:48) ) are in F . Then by considering an admissible exactsequence x ⊕ x (cid:48) (cid:18) i id x (cid:48) (cid:19) (cid:26) y ⊕ x (cid:48) ⊕ z (cid:48) ( p z (cid:48) ) (cid:16) z ⊕ z (cid:48) (resp. x ⊕ x (cid:48) i
00 00 id x (cid:48) (cid:26) y ⊕ y (cid:48) ⊕ x (cid:48) ⊕ x (cid:18) p y (cid:48) x (cid:19) (cid:16) z ⊕ y (cid:48) ⊕ x , x ⊕ y (cid:48) ⊕ z i y (cid:48)
00 0 id z (cid:26) y ⊕ y (cid:48) ⊕ z ⊕ z (cid:48) (cid:16) p z (cid:48) (cid:17) (cid:16) z ⊕ z (cid:48) ), it turns out that y ⊕ x (cid:48) ⊕ z (cid:48) (resp. z ⊕ y (cid:48) ⊕ x , x ⊕ y (cid:48) ⊕ z )is in F . Thus y (resp. z , x ) is in F thi .Next let us assume that F is a prenull (resp. semi-null) class in C . Then by ( ) , ( (cid:99) F ) isom , (cid:98) C is also prenull(resp. semi-null) in (cid:98) C . Then the pull-back of ( (cid:99) F ) isom , (cid:98) C by the strictly normal complicial exact functor C → (cid:98) C , x (cid:55)→ ( x , e ) is also prenull (resp. semi-null) in C by 3.1.7. Finally by the equality ( ) , it is just F thi a thick closure of F . In this subsection we study a special class of Waldhausen categories which we call relative complicial exactcategories. First we recalling the notion of relative categories in the sense of Barwick and Kan [BK12] andwe recall the notion of relative exact categories which are relative categories whose underlying categoriesare Quillen exact categories. Next we recall the notion of relative complicial exact categories from [Sch11],[Moc10] and [Moc13b] (in the references, we call them complicial exact categories with weak equivalencesor bicomplicial pairs).
A notion of relative categories introduced by Barwick and Kan in [BK12] isa model of homotopy theory of homotopy theories. We briefly review the these notions from [BK12]. A relative category X is a pair ( X , v ) consisting of a category X and a class of morphisms v in X which isclosed under finite compositions. Namely for any object x in X , the identity morphism id x is in v and for anypair of composable morphisms x f → y g → z in v , g f is also in v . Thus we can regard v as the subcategory of X .We call X the underlying category of X and denote it by C X and call v the class of weak equivalences of X and write w X for v . We say that a relative category X is small if the underlying category C X of X is small.44 relative functor between relative categories is a functor which preserves weak equivalences. Let f , g : X → Y be relative functors between relative categories. A natural weak equivalences Θ from f to g is anatural transformation Θ : f → g such that for any object x in C X , Θ x is in w Y . We say that the pair f andg are categorical homotopic or f is categorical homotopic to g if there exists a zig-zag sequence of naturalweak equivalences which connects f and g .Let f : X → Y be a relative functor between relative categories X and Y . We say that f is a categoricalhomotopy equivalence if there exists a relative functor g : Y → X such that f g and g f are categorical homo-topic to id C Y and id C X respectively. We denote the category of small relative categories and relative functorsby RelCat . Let X and Y be small relative categories. We define X × Y to be a relative category by setting C X × Y : = C X × C Y and w X × Y : = w X × w Y and call it the product of X and Y . The relative category X × Y is a categorical product of X and Y in RelCat . Next we write X Y for the relative category whose underlyingcategory is the category of relative functors from Y to X and natural transformations and whose weak equiv-alences are natural weak equivalences. Then X Y is the internal hom object in RelCat , namely for any object Z in RelCat , the functor − × Z : RelCat → RelCat admits a right adjoint ( − ) Z . Thus RelCat is Cartesianclosed.Let S be a full subcategory of C X the underlying category of a relative category X . Then the pair ( S , w X ∩ Mor S ) is a relative category and call it the restriction of X to S and denote it by X | S . We oftenwrite w | S for w X | S . Let C = ( C , w ) be a relative category and let I be a smallcategory. We say that a morphism f : x → y in C I the category of I -diagrams is a level weak equivalence iffor any object i , f ( i ) : x ( i ) → y ( i ) is in w . We denote the class of all level weak equivalences in C I by lw C I or shortly lw .Assume if C admits a zero object. Then we can consider Ch ( C ) the category of chain complexes on C and F b C the category of bounded filtered objects on C . We can regard both of them as the full subcategoryof C Z the category of Z -diagrams and we denote the restriction of lw C Z to Ch ( C ) and F b C by lw Ch ( C ) and lw F b C or simply lw . We also denote the restriction of lw to Ch ( C ) by lw or shortly lw for ∈ {± , b } .Moreover assume ( C , w ) be a relative complicial exact category. Then we similarly say that a homotopycommutative diagram ( f , H ) : x → y in F b , h C is a level weak equivalence if for any integer n , f n : x n → y n isin w and we write lw F b , h C or simply lw for the class of all level weak equivalences in F b , h C . A relative exact category E = ( E , w ) is a relative categorywhose underlying category is an exact category. A relative exact functor between relative exact categories f : E = ( E , w ) → F = ( F , v ) is a relative and exact functor. We denote the category of small relative exactcategories and relative exact functors by RelEx . We write
RelEx for the 2-category of small relative exactcategories and relative exact functors and natural weak equivalences.
Let I be a small category and let E : I → RelEx be an I -diagram of relative exact categories. Then colim I E = colim i ∈ Ob I E i is defined as follows. The underlying category C colim I E of colim I E is the colimit of the I -diagram I E → RelEx forget → ExCat of small exact categories and theclass of weak equivalences w colim I E of colim I E is the subset of the class of morphisms in C colim I E consisting ofthose morphisms which are represented by morphisms in w i for some i in Ob I . Let E be an exact category. We denote the class of all isomorphismsin E by i E or simply i . Then the pair ( E , i E ) is a relative category. In this way, we regard exact categoriesas relative exact categories. There is a functor from ExCat the category of exact categories to
RelEx thecategory of relative exact categories which sends an exact category E to the relative exact category ( E , i E ) .We often simply denote the relative category ( E , i E ) by E . We recall the notion of quasi-isomorphisms on Ch ( E ) the categoryof chain complexes on an exact category E . We say that a chain complex x is strictly acyclic if for any integer n , the boundary morphism d xn : x n → x n − factors as x n d xn (cid:47) (cid:47) p n (cid:32) (cid:32) (cid:32) (cid:32) x n − z n (cid:61) (cid:61) i n (cid:61) (cid:61) p n is a cokernel for d xn + and an admissible epimorphism and i n is a kernel for d xn − and an admissiblemonomorphism. We say that a chain complex x is acyclic if it it chain homotopy equivalent to a strictlyacyclic complex. We can show that if E is idempotent complete, then acyclic complexes on E are strictlyacyclic complexes (see [Nee90, 1.8], [Kel96, 11.2]).We say that a chain morphism f : x → y is a quasi-isomorphism if Cone f is acyclic. We can show that if E is essentially small and idempotent complete, then a morphism f : x → y in Ch ( E ) is a quasi-isomorphismif and only if it induces isomorphisms on H k ( x ) ∼ → H k ( y ) on homologies for any integer k where homologiesare taken in Lex E the category of left exact functor from E op to the category of abelian groups via Yonedaembedding E (cid:44) → Lex E (see [TT90, 1.11.8]).We denote the class of all quasi-isomorphisms in Ch ( E ) by qis Ch ( E ) or simply qis. Similarly on Ch b ( E ) , Ch + ( E ) and Ch − ( E ) the categories of bounded chain complexes, bounded below chain complexes andbounded above chain complexes on E respectively, we define the class qis of quasi-isomorphisms. The pair ( Ch ( E ) , qis ) ( ∈ { b , ± , nothing } ) are typical examples of relative exact categories. Let C be a complicial exact category. We say that a classof morphisms w in C is a class of complicial weak equivalences if it contains all C -homotopy equivalencesand satisfies the extension and the saturation axioms in [Wal85, p.327]. In this case, we call the class ofmorphisms in w a complicial weak equivalences .We say that a class of complicial weak equivalences on C is thick if it closed under retractions in themorphisms category C [ ] .A relative complicial exact category C = ( C , w ) is a relative category whose underlying category is acomplicial exact category and whose class of morphisms is a class of complicial weak equivalences. A relative complicial exact functor between relative complicial categories C = ( C , w ) → C (cid:48) = ( C (cid:48) , w (cid:48) ) is acomplicial exact functor ( f , c ) : C → C (cid:48) such that f : C → C (cid:48) is a relative functor.A complicial natural weak equivalence between relative complicial exact functors ( f , c ) , ( f (cid:48) , c (cid:48) ) : C =( C , w ) → C (cid:48) = ( C (cid:48) , w (cid:48) ) is a complicial natural transformation θ : ( f , c ) → ( f (cid:48) , c (cid:48) ) such that for any object x in C , θ ( x ) is in w (cid:48) . We denote the category of small relative complicial exact categories and relativecomplicial exact functors (resp. small thick normal ordinary relative complicial exact categories and strictlynormal relative complicial exact functors) by RelComp (resp.
RelComp sn ) and we write RelComp (resp.
RelComp sn ) for the 2-category of small relative complicial exact categories, relative complicial exact functors(resp. small thick normal ordinary relative complicial exact categories, strictly normal relative complicialexact functors) and complicial natural weak equivalences. (cf. [Moc10, 5.18].) Let ( C , w ) be a relative complicial exact category. Then the pair ( C , w ) satisfies the functorial factorization axiom, the functorial cofactorization axiom, the gluing axiom and thecogluing axiom. In particular ( C , w ) and ( C op , w op ) are Waldhausen categories. Moreover w is proper.Namely w is stable under co-base change along any admissible monomorphisms and stable under base changealong any admissible epimorphisms. Let ( C , w ) be a relative complicial exact category. Then w is homotopy proper in thefollowing sense:w is stable under canonical homotopy push-out along any morphisms and stable under canonical homotopypull-back along any morphisms.Proof. Let x f ← y g → z be a pair of morphisms in C such that f is in w . Then in the homotopy push-outdiagram ( ) the morphism i f is just a composition of the C -homotopy equivalence (cid:16) z (cid:17) : z → C ( y ) ⊕ z and f (cid:48) : C ( y ) ⊕ z → which is the co-base change of f by the admissible monomorphism (cid:16) ι y − g (cid:17) : y → Cy ⊕ z . Thusby 3.2.8, i f is in w . A proof of the homotopy pull-back case is similar. Let ( C , w ) be a relative complicial exact category and let x and z be a pair of objects in C . Then the following two conditions are equivalent. ( ) There exists a pair of morphisms x f ← y g → z in w. ( ) There exists a pair of morphisms x f → y g ← z in w.Proof. We assume the condition ( ) . Then in the canonical homotopy push-out diagram ( ) , i f is in w by3.2.9 and it turns out that i g is also in w by 2 out of 3 property of w . Hence we obtain the condition ( ) . Aproof of the converse implication is similar. 46 .2.11. (Derivable Waldhausen category). In [Cis10b, 1.1], we say that a Waldhausen category C =( C , w C ) is derivable if it satisfies the saturation axiom in [Wal85, p.327] and (non-functorial) factorizationaxiom in [Cis10b, 1.1]. Thus a relative complicial exact category is a derivable Waldhausen category byLemma 3.2.8. Let C be a complicial exact category. For a family of classes of complicial (resp. thickcomplicial) weak equivalences { w λ } λ ∈ Λ in C , (cid:92) λ ∈ Λ w λ is also a class of complicial (resp. thick complicial)weak equivalences. For any class of morphisms v in C , we write (cid:104) v (cid:105) (resp. (cid:104) v (cid:105) thick ) for (cid:84) w where w runsthrough all classes of complicial (resp. thick complicial) weak equivalences which contains v . We call (cid:104) v (cid:105) (resp (cid:104) v (cid:105) thick ) the class of complicial (resp. thick complicial ) weak equivalences spanned by v . C -homotopy equivalences). Let C be a complicial exact category. Recall wedenote the class of all C -homotopy equivalences in C by heq C or simply heq. The class heq is the class ofcomplicial weak equivalences in C frob (see [Moc13b, 3.15 ( ) ]). Let E = ( E , w ) be a relative exact category. We denote theclass of thick complicial weak equivalences in Ch ( E ) spanned by qis and lw by qw Ch ( E ) or simply qw andwe call a morphism in qw a quasi-weak equivalence . In particular if w = i E the class of all isomorphisms in E ,then the class qw is just the class of all quasi-isomorphisms in Ch ( E ) . We denote the pair ( Ch ( E ) , qw ) by Ch ( E ) . Then for any relative exact functor f : E = ( E , w ) → F = ( F , v ) , there is a strictly normal complicialexact functor Ch ( f ) : Ch ( E ) → Ch ( F ) . The association Ch : RelEx → RelComp sn is a 2-functor. (cf. [Moc13b, 3.5].)Let C be a complicial small exact category. We denote the class of all null classes (resp. thick null classes) in C by NC ( C ) (resp. NC thi ( C ) and we write CW ( C ) (resp. CW thi ( C ) ) for the class of all classes of complicialweak equivalences (resp. thick complicial weak equivalences) in C .For a null class in C , we define w N to be a class of those morphisms f : x → y in C such that Cone f is in N . Then we can show that w N is a class of complicial weak equivalences in C and moreover if N is thick,then w N is also thick. For a class of complicial weak equivalences u in C , C u the class of u -trivial objects in C (See 3.1.1) is a null class and if u is thick, then C u is also thick.We can also show the equalities C w N = N , and (180) w C u = u . (181)Thus the associations NC ( C ) → CW ( C ) , N (cid:55)→ w N and CW ( C ) → NC ( C ) , u (cid:55)→ C u give order preservingbijections between NC ( C ) and CW ( C ) , and NC thi ( C ) and CW thi ( C ) . Let C = ( C , w ) be a relative complicial exact category and let i : x → y be an admissiblemonomorphism ( resp. a Frobenius admissible monomorphism ) in C . Then ( ) The canonical morphism
Cone i → y / x is in w ( resp. a C-homotopy equivalence ) . ( ) Assume that y / x is in C w ( resp. C-contractible ) , then i is in w ( resp. a C-homotopy equivalence ) . ( ) Let f : x → x (cid:48) be a morphism and let x f (cid:47) (cid:47) (cid:15) (cid:15) i (cid:15) (cid:15) x (cid:48) (cid:15) (cid:15) i (cid:48) (cid:15) (cid:15) y f (cid:48) (cid:47) (cid:47) y (cid:48) be a co-base change of f along i. Then the morphism Cone ( i , i (cid:48) ) : Cone f → Cone f (cid:48) is in w ( resp. a C-homotopy equivalence ) .Proof. ( ) By considering the following commutative diagram of admissible exact sequences, x (cid:47) (cid:47) ξ i (cid:47) (cid:47) Cyl i (cid:47) (cid:47) (cid:47) (cid:47) υ i (cid:15) (cid:15) Cone i (cid:15) (cid:15) x (cid:47) (cid:47) i (cid:47) (cid:47) y (cid:47) (cid:47) (cid:47) (cid:47) y / x , (182)
47t turns out that the canonical morphism Cone i → y / x is in w by the gluing axiom. ( ) By ( ) , Cone i is in C w . Thus i is in w by by 3.2.15. ( ) Notice that the induced morphism f (cid:48) / f : y / x → y (cid:48) / x (cid:48) is an isomorphism and there is an admissible exactsequence Cone f Cone ( i , i (cid:48) ) (cid:26) Cone f (cid:48) (cid:16) Cone f (cid:48) / f . Since Cone f (cid:48) / f is C -contractible, we obtain the result by ( ) .Finally by applying assertions from ( ) to ( ) to the pair ( C frob , heq ) , we obtain assertions for ‘respectively’parts. Let C = ( C , w ) be a relative complicial exact category and let N be a semi-null classof C . Then ( ) Let u : x → y be a morphism in C . Then ( i ) Cu : Cx → Cy is in w. ( ii ) If u is in w, then T − u is in w. ( ) Let [ f (cid:48) : x (cid:48) → y (cid:48) ] ( u , v , H ) → [ f : x → y ] be a homotopy commutative square with v ∈ w. Then ( i ) Cyl ( u , v , H ) : Cyl f (cid:48) → Cyl f is in w. ( ii ) Moreover if u is in w, then
Cone ( u , v , H ) : Cone f (cid:48) → Cone f is in w. ( ) w is closed under C-homotopic relations. Namely for a pair of morphisms u, v : x → y in C , if u is in wand there exists a C-homotopy H : u ⇒ C v, then v is also in w. ( ) If N is closed under w-weak equivalences, then N is a null class in C .Proof. ( ) ( i ) Since Cx and Cy are C -contractible, the morphisms Cx → Cy → w and by 2 outof 3, Cu is also in w . ( ii ) By considering cogluing axiom in the commutative diagram below, since Pu and u are in w , T − u is alsoin w . T − x (cid:47) (cid:47) ι T − x (cid:47) (cid:47) T − u (cid:15) (cid:15) Px Pu (cid:15) (cid:15) q x (cid:47) (cid:47) (cid:47) (cid:47) x u (cid:15) (cid:15) T − y (cid:47) (cid:47) ι T − y (cid:47) (cid:47) Py q y (cid:47) (cid:47) (cid:47) (cid:47) y . ( ) ( i ) Since w is closed under extensions and v and Cu are in w , by considering the commutative diagrambelow, it turns out that Cyl ( u , v , H ) is in w . y (cid:48) (cid:47) (cid:47) (cid:16) y (cid:48) (cid:17) (cid:47) (cid:47) v (cid:15) (cid:15) Cx (cid:48) ⊕ y (cid:48) ( id Cx (cid:48) ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:16) Cu − H v (cid:17) (cid:15) (cid:15) Cx (cid:48) Cu (cid:15) (cid:15) y (cid:47) (cid:47) (cid:16) y (cid:17) (cid:47) (cid:47) Cx ⊕ y ( id Cx ) (cid:47) (cid:47) (cid:47) (cid:47) Cx . ( ii ) In the commutative diagram below, we obtain the result by the gluing axiom. x (cid:48) (cid:47) (cid:47) ξ , f (cid:48) (cid:47) (cid:47) u (cid:15) (cid:15) Cyl f (cid:48) η f (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) Cyl ( u , v , H ) (cid:15) (cid:15) Cone f (cid:48) Cone ( u , v , H ) (cid:15) (cid:15) x (cid:47) (cid:47) ξ , f (cid:47) (cid:47) Cyl f η f (cid:47) (cid:47) (cid:47) (cid:47) Cone f . ( ) If u is in w , then Cone u is in C w by 3.2.15. Since there exists an isomorphism Cone ( id x , id y , H ) : Cone u ∼ → Cone v , Cone v is also in C w and it turns out that v is in w by 3.2.15 again.48 ) Since w contains all C -homotopy equivalences, N contains all C -contractible objects. Let x i (cid:26) y p (cid:16) z bean admissible exact sequence in C . Since ( y ) : Cx ⊕ y → y is a C -homotopy equivalence, by applying thegluing axiom to the commutative diagram below, it turns out that Cone ( , p ) is also in w . x id x (cid:15) (cid:15) (cid:47) (cid:47) ι i (cid:47) (cid:47) Cyl i ( y ) (cid:15) (cid:15) η i (cid:47) (cid:47) (cid:47) (cid:47) Cone i Cone ( , p ) (cid:15) (cid:15) x (cid:47) (cid:47) i (cid:47) (cid:47) y p (cid:47) (cid:47) (cid:47) (cid:47) z . Thus z is in N if and only if Cone i is in N . If x and y are in N , then Cone i and z are also in N .If y (resp. x ) and z are in N , then by applying argument above to an admissible exact sequence y (cid:26) Cone i (cid:16) T x (resp. Cone i (cid:26) T x ⊕ Cy (cid:16) Ty ), T x and x (resp. Ty and y ) are in N . Let C = ( C , w ) be a relative complicial exact category. Then for F ∈ { T ± , P , C } , Fpreserves weak equivalences, namely, the pair ( F , σ − F ) : C → C is a relative complicial exact functor.Proof. Let f : x → y be a morphism in w . Then C f and
P f = CT − f are in w by 3.2.17 ( ) ( i ) . Thus T f and T − f are also in w by the left and right commutative diagrams below and the gluing and cogluing axiomsrespectively. x (cid:47) (cid:47) ι x (cid:47) (cid:47) f (cid:15) (cid:15) Cx π x (cid:47) (cid:47) (cid:47) (cid:47) C f (cid:15) (cid:15)
T x
T f (cid:15) (cid:15) y (cid:47) (cid:47) ι y (cid:47) (cid:47) Cy π y (cid:47) (cid:47) (cid:47) (cid:47) Ty , T − x (cid:47) (cid:47) j x (cid:47) (cid:47) Pf (cid:15) (cid:15) Px q x (cid:47) (cid:47) (cid:47) (cid:47) Pf (cid:15) (cid:15) x f (cid:15) (cid:15) T − y (cid:47) (cid:47) j y (cid:47) (cid:47) Py q y (cid:47) (cid:47) (cid:47) (cid:47) y . Let C = ( C , w C ) be a relative complicial exact category and let A be a full subcategoryof C . If A is closed under finite direct sums ( resp. the operation T , resp. the operation T − ) , then A w thew-closure of A is also.Proof. Let x and y be objects in A w . Then there exists a zig-zag sequences of morphisms x a ← x (cid:48) a (cid:48) → x (cid:48)(cid:48) and y b ← y (cid:48) b (cid:48) → y (cid:48)(cid:48) in w C such that x (cid:48)(cid:48) and y (cid:48)(cid:48) are in A .If A is closed under finite direct sums, then x (cid:48)(cid:48) ⊕ y (cid:48)(cid:48) is in A and the morphisms (cid:18) a b (cid:19) and (cid:18) a (cid:48) b (cid:48) (cid:19) are in w C . Thus x ⊕ y is in A w C .If A is closed under the operation T (resp. T − ), then T x (cid:48)(cid:48) (resp. T − x (cid:48)(cid:48) ) is in A and morphisms Ta and Ta (cid:48) (resp. T − a and T − a (cid:48) ) are in w C . Thus T x (resp. T − x ) is in A w C . Let C = ( C , w C ) be a relative complicial exact category and let A be a null class in C .Then A is w C -closed if and only if it contains C w C .Proof. If A is w C -closed, then it contains all objects which is w -equivalent to 0, in other words, A contains C w C . Conversely assume that A contains C w C . We have an equality A = C w A by ( ) and w A contains w C by assumption. Thus A is w A -closed and in particular w C -closed. w -closed null closure). Let C = ( C , w C ) be a relative complicial exact categoryand let A be a full subcategory of C . We set (cid:104) A (cid:105) null , w C : = (cid:104) A (cid:83) C w C (cid:105) null , frob . (cid:104) A (cid:105) null , w C is the smallest w C -closed null class which contains A in C . We call (cid:104) A (cid:105) null , w C the w C -closed null closure of A . Proof.
Since (cid:104) A (cid:105) null , w C contains C w C and it is a null closure in C frob , (cid:104) A (cid:105) null , w C is w C -closed by Lemma 3.2.20.Then it turns out that (cid:104) A (cid:105) null , w C is a null class in C by Proposition 3.2.17 ( ) . If B is a w C -closed null classwhich contains A , then B contains C w C and therefore contains (cid:104) A (cid:105) null , w C .By 3.1.7 and 3.2.15, we obtain the following result.49 .2.22. Lemma. Let ( f , c ) : C → C (cid:48) be a complicial exact functor between complicial exact categoriesand let w be a class of complicial weak equivalences in C (cid:48) . Then f − ( w ) is a class of complicial weakequivalences in C . Moreover if w is thick, then f − ( w ) is also thick. Let X be a category with a specific zero object 0 and w a class of morphisms in X . We say that a morphism f : x → y in F b X is a stable weak equivalence if f ∞ : x ∞ → y ∞ is in w . We denote the class of all stable weak equivalences in F b X by w st F b X or shortly w st .Assume X is a complicial exact category. Then we similarly say that a homotopy commutative diagram ( f , H ) : x → y in F b , h X is a stable weak equivalence if f ∞ : x ∞ → y ∞ is in w . We similarly denote the class ofall stable weak equivalences in F b , h X by w st F b , h X or shortly w st .For any integer k , the degree shift functor ( − )[ k ] : F b , h C → F b , h C preserves lw and w st .Moreover assume that the pair X : = ( X , w ) is a normal ordinary relative complicial exact category. Thensince the functor ( − ) ∞ : F b , h X → X is a strictly normal complicial exact functor and w st is just a pull-backof w in C by this functor, w st is a class of complicial weak equivalences in F b , h X by 3.2.22. We denote thenormal ordinary relative complicial exact category Moreover if w is thick, then w st is also thick. We denotethe normal ordinary relative complicial exact category ( F b , h X , w st ) by F b , h X .Recall the definition of categorical homotopic of relative functors from 3.2.1. Let ( X , w ) be a relative category and let k be an integer. Then the degree shift functor ( − )[ k ] : F b X → F b X is categorical homotopic to id F b X .Proof. There exists a natural weak equivalence ( − )[ k ] Θ k → ( − )[ k + ] on ( F b X , w st ) . Namely for an object x in F b X and an integer n , Θ k ( x ) n : = i xn + k : x n + k → x n + k + . Therefore there exists a sequence of natural weakequivalences from id F b X = ( − )[ ] to ( − )[ k ] .Notice that Θ k is natural weak equivalence with respect to w st , not to lw and if X is a complicial exactcategory, then Θ k is functorial on F b X , but not on F b , h X . Throughout this subsection, let C = ( C , w ) be a relative complicial exact category. In this subsection, wewill study compatibilities of several complicial structures and the class of complicial weak equivalences on Ch b ( C ) the category of bounded chain complexes on C . b ( C )) . There are two kind of natural complicial structures on Ch b ( C ) .The first one is the standard complicial structure illustrated in 1.2.2. We can regard Ch b ( C ) as the fullsubcategory of F C = C Z the category of functors from the totally ordered set Z to C and the second oneis the level complicial structure (see 1.2.14) induced from C Z . We denote these two complicial structuresby ( C sta , ι sta , r sta , σ sta ) and ( C lv , ι lv , r lv , σ lv ) respectively. We denote the associated suspension functor, pathfunctor, cone functor and so on by T , P , Cone and so on for ∈ { sta , lv } .We denote the forgetful functor CompEx → ExCat by F . For a small exact category E , we define j E : E → Ch b ( E ) to be an exact functor by setting j E ( x ) k is x if k = k (cid:54) =
0. Then j E is natural on E , namely j gives a natural transformation j : id ExCat → F Ch b . If E is a complicial exact category, then thefunctor j E : E → Ch b ( E ) is a strictly normal complicial exact functor with respect to ( C lv , ι lv , r lv , σ lv ) . Let x be a chain complex on C and let f : x → y be a chainmorphism on C and let n be an integer. We define σ ≥ n x and σ ≥ n f to be a subcomplex of x and a chainmorphism σ ≥ n x → σ ≥ n y by setting ( σ ≥ n x ) k = (cid:40) x k if k ≥ n , ( σ ≥ n f ) k = (cid:40) f k if k ≥ n σ ≤ n x and σ ≤ n f : σ ≤ n x → σ ≤ n y . We simply write x [ n ] and f [ n ] for ( T sta ) n ( x ) and ( T sta ) n ( f ) respectively.For an integer n ≤ m , we write Ch b , [ n , m ] ( C ) , Ch b , ≥ n ( C ) and Ch b , m ≥ ( C ) for the full subcategories of Ch b ( C ) consisting of those complexes x such that x k = k / ∈ [ n , m ] , k < n and m < k respectively.50or a complex x in Ch b ( C ) , we set min x : = min { n ∈ Z ; x n (cid:54) = } , (184)max x : = max { n ∈ Z ; x n (cid:54) = } , (185)length x : = min { b − a ; x ∈ Ob Ch [ a , b ] ( C ) } (186)and call it the length of x . Tot ). There exists an exact functor Tot : Ch b ( C ) → C and the nat-ural equivalences d sta : C Tot ∼ → Tot C sta and d lv : C Tot ∼ → Tot C lv such that the pairs ( Tot , d sta ) and ( Tot , d lv ) are complicial exact functors with resepct to the standard complicial structure and the level complicial struc-ture on Ch b ( C ) respectively.For an integer n , we have the equality Tot j C ( − )[ n ] = T n . (187)Moreover we can regard Tot is the natural transformation Tot : Ch b F → id CompEx such that the pair ( j , Tot ) gives an adjunction of the pair ( Ch b , F ) where F : CompEx → ExCat is the forgetful functor.
Proof.
We recall the construction of Tot from [Moc10, 4.1, 4.9]. First we will define Tot ≥ : Ch b , ≥ ( C ) → C , d sta : C Tot ≥ ∼ → Tot ≥ C sta and d lv : C Tot ≥ ∼ → Tot ≥ C lv to be exact functor and natural equivalencesrespectively by proceeding induction on the length of chain complexes. For an object x and a morphism f : x → y in C , we set Tot ≥ · j C ( x ) : = x and Tot ≥ j C ( f ) : = f . Notice that we have the equality Cx = Cone id x = Tot ( x id x → x ) = Tot C sta x . We set d lv j C ( x ) = d sta j C ( x ) = id Cx : C Tot ≥ j C ( x ) = Cx = Tot C lv j C ( x ) = Tot C sta j C ( x ) .For a chain complex x in Ch b , ≥ ( C ) and a chain morphism f : x → y , we defineTot ≥ ( x ) : = Cone ( Tot ≥ (( σ ≥ x )[ − ]) → x ) , (188)Tot ≥ ( f ) : = Cone ( Tot ≥ (( σ ≥ f )[ − ]) , f ) . (189)Moreover for ∈ { sta , lv } , we define d x : C Tot x ∼ → Tot C x by the compositions C Tot x = C Cone ( Tot ( σ ≥ x [ − ]) dx → x ) c Cone ∼→ Cone ( C Tot ( σ ≥ x [ − ]) Cdx → Cx ) Cone ( d σ ≥ x [ − ] , id Cx ) ∼→ Cone ( Tot ( C σ ≥ x [ − ]) Cdx → Cx )= Tot C x . Next for any positive integer n , We define Tot ≥− n : Ch b , ≥− n ( C ) → C and X − n : Tot ≥− ( n + ) | Ch b , ≥− n ( C ) ∼ → Tot ≥− n to be an exact functor and a natural equivalence by settingTot ≥− n : = ( T C ) − n · Tot ≥ · ( T sta ) n (190)and compositions of natural equivalences Tot ≥− ( n + ) =( T C ) − ( n + ) · Tot ≥ · ( T sta ) n + ∼→ ( T C ) − n · Tot ≥ · ( T sta ) − · ( T sta ) n + ∼→ ( T C ) − n · Tot ≥ · ( T sta ) n = Tot ≥− n (191)respectively. We will define Tot : Ch b ( C ) → C to be an exact functor by patching the family of functors andnatural equivalences { Tot ≥− n , X − n } n > . To do so, for any pair of integers n > m , we define X n , m : Tot ≥ n ∼ → Tot ≥ m to be a natural equivalence by setting X n , m : = X m X m + · · · X n − and for any complex x in Ch b ( C ) weset m ( x ) : = min { n ; n ≥ , x − k = k ≥ n } . (192)For a complex x in Ch b ( C ) , we set Tot ( x ) : = Tot ≥− m ( x ) ( x ) . Next for a morphism f : x → y , we take an integer n ≥ max { m ( x ) , m ( y ) } and we set Tot ( f ) : = X n , m ( y ) ( y ) · Tot ≥ n ( f ) · ( X n , m ( x ) )( x ) − . We can easily check that thedefinition of Tot ( f ) does not depend upon a choice of n and that Tot is an exact functor on Ch b ( C ) .For ∈ { sta , lv } , we will define d : C C Tot ∼ → Tot C to be a natural equivalence in the following way.For a complex x in Ch b ( C ) , let us notice that we have equalities m ( x ) = m ( C x ) and T sta C = C T sta for51 ∈ { sta , lv } . We define d x : C C Tot ( x ) ∼ → Tot C ( x ) to make the diagram below commutativeTot ( x ) Tot ∗ ι sta x (cid:15) (cid:15) ( T C ) − m ( x ) · Tot ≥ · ( T sta ) m ( x ) ( x )(( T C ) − m ( x ) · Tot ≥ · ( T sta ) m ( x ) ) ∗ ι x (cid:15) (cid:15) Tot ( x ) ι Tot ( x ) (cid:15) (cid:15) Tot · C ( x ) ( T C ) − m ( x ) · Tot ≥ · ( T sta ) m ( x ) · C ( x ) A ∼ (cid:47) (cid:47) C C · Tot ( x ) where the morphism A is compositions of natural equivalences ( T C ) − m ( x ) · Tot · ( T sta ) − m ( x ) · C ( x ) ∼→ ( T C ) − m ( x ) · C C · Tot · ( T sta ) − m ( x ) ( x ) ∼→ C C · ( T C ) − m ( x ) · Tot · ( T sta ) − m ( x ) ( x ) . For ∈ { sta , lv } , by induction on the length of chain complexes, we can show the equality d · ( ι ∗ Tot ) =
Tot ∗ ι . Let a ≤ i ≤ b be a triple of integers such that b − a ≥ f : x → y be achain morphism in Ch [ a , b ] ( C ) . Then there is a canonical chain morphism σ ≥ i x [ − ] → σ ≤ i − x induced fromthe i th boundary morphism d xi : x i → x i − and we denote it by d xi : ( σ ≥ i x )[ − ] → σ ≤ i − x . We setCone Tot d fi : = Cone (( σ ≥ i f )[ − ] , σ ≤ i − f ) : Cone Tot d xi → Cone Tot d yi . Then the association Cone Tot d i : Ch [ a , b ] C → C is an exact functor and there is a natural equivalence e : Tot ∼ → Cone Tot d i . Proof.
We will construct a natural equivalence e by proceeding induction on the length of σ ≤ i − x . By re-placing x with x [ N ] for a suitable integer N , we shall assume a =
0. Notice that if b = i =
1, then e isdefined in 2.3.15. If length σ ≤ i − x =
1, then by definition, we have the equality Tot x = Cone Tot d xi . Assumelength σ ≤ i − x >
1, then there are functorial isomorphisms below by inductive hypothesisCone Tot d xi = Cone ( Tot (( σ ≥ i x )[ − ]) → Cone ( Tot (( σ ≥ σ ≤ i − x )[ − ]) → x ) ∼ → Cone ( Cone ( Tot ( σ ≥ i x ) → Tot (( σ ≥ σ ≤ i − x )[ − ])) → x ) ∼ → Cone ( Tot (( σ ≥ x )[ − ]) → x )= Tot x . Let F be an additive full subcategory of C . We regard F as a full subcategory of Ch b ( C ) by the functors F (cid:44) → C j C → Ch b ( C ) . Then the prenull closure and semi-null closure of F in Ch b ( C ) with re-spect to the standard complicial structure are Ch b , ≥ ( F ) and Ch b ( F ) respectively. The functor Tot inducesthe complicial functors
Tot ≥ : Ch b , ≥ ( F ) → (cid:104) F (cid:105) prenull and Tot : Ch b ( F ) → (cid:104) F (cid:105) semi-null . Let A be an additive category and let B be an additive full subcategory of A . Then wehave the equality Ch b ( B thi , A ) = ( Ch b ( B )) thi , Ch b ( A ) . (193) Proof.
We regard A as an exact category with split exact sequences. Then Ch b ( A ) is a complicial exactcategory with the standard complicial structure. We regard B and B thi , A as full subcategories of Ch b ( A ) bythe functor j A : A → Ch b ( A ) and inclusions B , B thi , A (cid:44) → A .We will show that Ch b ( B thi , A ) is thick in Ch b ( A ) . Let y be a chain complex in Ch b ( B thi , A ) and let x i → y p → x be a pair of chain morphisms in Ch b ( A ) such that pi = id x . Then for each integer n , we have anequality p n i n = id x n and it turns out that x n is in B thi , A . Hence x is in Ch b ( B thi , A ) and Ch b ( B thi , A ) is thickin Ch b ( A ) . In particular we have the inclusion ( Ch b ( B )) thi , Ch b ( A ) (cid:44) → Ch b ( B thi , B ) .Next the inclusion B (cid:44) → Ch b ( B ) implies the inclusion B thi , A (cid:44) → ( Ch b ( B )) thi , Ch b ( A ) . Notice that by3.1.19 ( ) , ( Ch b ( B )) thi , Ch b ( A ) is a semi-null class in Ch b ( A ) . Thus we obtain the inclusion Ch b ( B thi , A ) = (cid:104) B thi , A (cid:105) semi-null , Ch b ( A ) (cid:44) → ( Ch b ( B )) thi , Ch b ( A ) . 52 .3.7. Lemma-Definition. We say that a morphism f : x → y in Ch ( C ) is a Frobenius quasi-isomorphism ifit is a quasi-isomorphism with respect to the Frobenius admissible exact structure of C . We denote the class ofall Frobenius quasi-isomorphisms in Ch ( C ) by q f is Ch ( C ) or simply q f is. We write q f w Ch ( C ) or simply q f w for the class of thick complicial weak equivalences spanned by q f is and lw and call it the class of Frobeniusquasi-weak equivalences ( in Ch ( C ) ). Assume that w is thick. We denote the class of all morphisms f : x → y in Ch b ( C ) such that Tot f is in w by tw and call it the class of total quasi-weak equivalences . Then there areequalities q f w = tw = qw . (194) Proof.
Since the class tw is the pull-back of w by the complicial exact functor Tot, it is a class of thickcomplicial weak equivalences in C by 3.2.22. We will show the inclusions q f w ⊂ I qw ⊂ II tw ⊂ III q f w . Since a Frobenius admissible exact sequence in C is an admissible exact sequence in C , the inclusion I isobvious. To show inclusion II , what we need to prove is the following two inclusions: IV qis ⊂ tw . V lw ⊂ tw .The inclusion IV follows form [Moc13b, 4.5]. We will show the inclusion V . Let f : x → y be a chainmorphism in Ch b ( C ) . By considering f [ N ] : x [ N ] → y [ N ] for sufficiently large integer N , without loss ofgenerality, we shall assume that f is in Ch b , ≥ ( C ) . Assume that f is in lw . Since by definition, wehave the equality Tot ( f ) = Cone (( σ ≥ f )[ − ] , f ) , we obtain the result by 3.2.17 ( ) ( ii ) and induction onmax { length ( x ) , length ( y ) } .Next we will show the inclusion III . What we need to show is the inclusion ( Ch b ( C )) tw ⊂ ( Ch b ( C w )) q f is and it follows from 3.3.8 below. (cf. [Moc13b, 4.10].) Let k be an integer. We write i : Ch b , ≤ k − ( C ) → Ch b , ≤ k ( C ) for the canonical inclusion functor. We define Q k : Ch b , ≤ k ( C ) → Ch b , ≤ k ( C ) and L k : Ch b , ≤ k ( C ) → Ch b , ≤ k − ( C ) and u k : id Ch b , ≤ k ( C ) → Q k and v k : i L k → Q k be a pair of functors and a pair of natural transformations re-spectively by setting for a chain morphism f : x → y in Ch b , ≤ k ( C ) , Q k ( x ) n : = x n if n ≥ k − d xk if n = k − Cx k if n = k n > k , d Q k ( x ) n : = d xn if n ≥ k − ( , d xk − ) if n = k − µ d xk if n = k n > k , Q k ( f ) n : = f n if n ≥ k − ( f k , f k − ) if n = k − C f k if n = k n > k , (195) L k ( x ) n : = x n if n ≥ k − d xk if n = k −
10 if n > k − , d L k ( x ) n : = d xn if n ≥ k − ( , d xk − ) if n = k −
10 if n > k − , L k ( f ) n : = f n if n ≥ k − ( f k , f k − ) if n = k −
10 if n > k − . (196) u k ( x ) n : = id x n if n ≥ k − κ d xk if n = k − ι x k if n = k n > k , v k ( x ) n : = id x n if n ≥ k − Cone d xk if n = k −
10 if n > k − . (197)Then for each complex x in Ch b , ≤ k ( C ) , u k ( x ) is a Frobenius quasi-isomorphism by 1.1.2 and v k ( x ) is a levelweak equivalence.For a complex x in Ch b , ≤ k ( C ) , we denote the zig-zag sequence of morphisms x u k ( x ) → Q k ( x ) v k ( x ) ← L k ( x ) by U kx . If for an integer n , x [ n ] is also in Ch b , ≤ k ( C ) , then we have the equality U kx [ n ] = U k − nx [ n ] . (198)For a pair of integers a < b , we write L [ a , b ] for the compositions L a + L a + · · · L b : Ch b , ≤ b ( C ) → Ch b , ≤ a ( C ) x in Ch b , ≤ b ( C ) , we write U [ a , b ] x for the composition of zig-zag sequence of morphisms U bx U b − x · · · U a + x which connects x and L [ a , b ] x . Moreover we have the equality L [ a , b ] ( x ) = j C ( T m Tot x )[ − m ] (199)where m = min x − a . Thus for a complex x in Ch b ( C ) , there exists a zig-zag sequence of morphisms in lw ∪ q f is which connects x and j C ( T m Tot ( x ))[ − m ] for a suitable integer m . For a relative complicial exact category C = ( C , w ) , we have equalitiesl ( (cid:104) w (cid:105) thi , C ) = (cid:104) lw (cid:105) thi , Ch b ( C ) , (200) qw = q ( (cid:104) w (cid:105) thi , C ) . (201) Proof.
By correspondence 3.2.15, the equality ( ) is equivalent to the equality ( Ch b ( C )) thi , Ch b ( C ) = Ch b (( C w ) thi , C ) and it follows from the equality ( ) .The inclusion w ⊂ (cid:104) w (cid:105) thi , C implies the inclusion qw ⊂ q ( (cid:104) w (cid:105) thi , C ) . On the other hand, since the class qw isthick, the inclusion lw ⊂ qw implies l ( (cid:104) w (cid:105) thi , C ) = (cid:104) lw (cid:105) thi , Ch b ( C ) ⊂ qw . Thus we obtain the converse inclusion q ( (cid:104) w (cid:105) thi , C ) ⊂ qw . We denote the forgetful functor
RelComp → RelEx by F. Then the pair ( j , Tot ) canbe regarded as natural transformations j : id RelEx → F Ch b and Tot : Ch b F → id RelComp between functors Ch b : RelEx → RelComp and F : RelComp → RelEx and it gives an adjunction of the pair ( Ch b , F ) . Let C = ( C , w C ) be a relative complicial exact category and let n be a non-negative integer and let x be a complex in Ch [ , n ] ( C ) . We define ¯ x to be a complex in Ch [ , n ] ( C ) in thefollowing way. First by convention, we set d (cid:48) xk : = (cid:40) → x n if k = n + d xn : x n → x n − if k = n . (202)For k ≥
1, we inductively set d (cid:48) xn − k : = Cone ( , d xn − k ) : Cone d (cid:48) xn − k + → x n − k − . (203) x n (cid:47) (cid:47) d (cid:48) xn (cid:15) (cid:15) (cid:15) (cid:15) x n − d xn − (cid:47) (cid:47) x n − , Cone d (cid:48) xn − k + (cid:47) (cid:47) d (cid:48) xn − k + (cid:15) (cid:15) (cid:15) (cid:15) x n − k d xn − k (cid:47) (cid:47) x n − k − . Well-definedness of d (cid:48) xn − k follows from the equality d xn − k d (cid:48) xn − k + = . (204)Then we set ¯ x k : = Cone d (cid:48) xk + and d ¯ xk = κ d (cid:48) xk d (cid:48) xk . (205)We define u x : x → ¯ x to be a chain morphism by setting u xk : = κ d (cid:48) xk + : x k → ¯ x k . (206)Notice that we can identify ¯ x = Tot x .Next let y be an object in C and let u : Tot x → y be a morphism in C . Then we define z u and g u : ¯ x → z u to be a complex and a chain morphism in Ch [ , n ] ( C ) by setting z uk : = (cid:40) C ( ¯ x k ) if k ≥ y if k = d z u k : = (cid:40) C ( d ¯ xk ) if k ≥ u µ d (cid:48) x if k = uk : = (cid:40) ι x k if k ≥ ¯ x if k = . (208)For a complex s in Ch [ , n ] ( C ) , we denote the canonical morphism Tot σ ≥ s → s induced from d s by d s .Then we have an isomorphism c Tot σ ≥ ¯ x : C ( Tot σ ≥ ¯ x ) ∼ → Tot ( C lv ( σ ≥ ¯ x )) = Tot σ ≥ z u and there is a homotopycommutative square ( , id y , − d z u r Tot σ ≥ ¯ x ( c Tot σ ≥ ¯ x ) − ) : [ Tot σ ≥ z u → y ] → [ → y ] . We set a u : = Cone ( , id y , − d z u r Tot σ ≥ ¯ x ( c Tot σ ≥ ¯ x ) − ) : Tot z u → y . (209)Then a u is in w C and we have an equality u = a u Tot ( g u u x ) . (210) Proof.
The equality ( ) follows from d xn − k d (cid:48) xn − k + = Cone ( , d xn − k d xn − k + ) = Cone ( , ) = . To show that ¯ x and z u are complexes, we check equalities d ¯ xk − d ¯ xk = κ d (cid:48) xk − d (cid:48) xk − κ d (cid:48) xk d (cid:48) xk = κ d (cid:48) xk − d xk − d xk = , d z u k − d z u k = C ( d ¯ xk − d ¯ xk ) = ( k ≥ ) , d z u d z u = u µ d (cid:48) x C ( d ¯ x ) = u µ d (cid:48) x C ( κ d (cid:48) x d (cid:48) x ) = u κ d (cid:48) x d (cid:48) x µ d (cid:48) x = u κ d (cid:48) x Cone ( , d x ) Cone ( id Cone d (cid:48) x , d (cid:48) x ) = u κ d (cid:48) x Cone ( , d x d (cid:48) x ) = . a u is in w C by Proposition 3.2.17 ( ) ( ii ) and applying Lemma 3.3.12 below to homotopy commutativesquares [ Tot σ ≥ x dx → x ] ( Tot σ ≥ u x , κ dx ) → [ Tot σ ≥ ¯ x d ¯ x → Tot x ] ( c Tot σ ≥ x ι Tot σ ≥ x , u ) → [ Tot σ ≥ z udzu → y ] ( , id y , − dzu r Tot σ ≥ x ( c Tot σ ≥ x ) − ) → [ → y ] , we obtain the equality ( ) . Let C be a complicial exact category and let f : x → y and g : Cone f → z be morphismsin C . Then the homotopy commutative squares [ f : x → y ] ( ι x , κ f ) → [ Cx µ f → Cone f ] ( , g , − g µ f r x ) → [ → z ] induces anequality g = Cone ( , g , − g µ f r x ) Cone ( ι x , κ f ) . (211) Proof.
It follows from the following commutative diagram of exact sequences. x (cid:47) (cid:47) (cid:16) − f ι x (cid:17) (cid:47) (cid:47) (cid:15) (cid:15) y ⊕ Cx ( κ f µ f ) (cid:47) (cid:47) (cid:47) (cid:47) ( g κ f g µ f ) (cid:15) (cid:15) Cone f g (cid:15) (cid:15) (cid:47) (cid:47) z id z (cid:47) (cid:47) z . In this subsection, we review the notion of homotopy categories of relative categories and that of derivedcategories of relative exact categories from [Sch11] and [Moc13b] with slightly different conventions. In thissubsection except 3.4.19, we assume that underlying categories of relative categories are small.55 .4.1. (Homotopy category of relative category). (cf. [GZ67, Chapter one 1.1, 1.2].) Let C = ( C , w ) be arelative category. A Gabriel-Zisman localization of C with respect to w is a pair ( w − C , Q w ) consisting of acategory w − C and a functor Q w : C → w − C which satisfies the following two conditions: • The functor Q w sends a morphism in w to an isomorphism in w − C . • For any small category D , Q w induces an isomorphism of categories D Q w : D w − C ∼ → D C w − inv where D C w − inv is the full subcategory of D C the category of functors from C to D consisting of those functors f : C → D such that it sends all morphisms in w to isomorphisms in D .We denote w − C by Ho ( C ) or Ho ( C ; w ) and call it the homotopy category of a relative category C . Let C = ( C , w C ) be a relative category. We say that w C satisfies the strongly saturation axiom or C is strongly saturated if for any morphism f : x → y in C , f is in w C if and only if f is an isomorphism in Ho ( C ) . We recall the conventions of triangulated categories. Basically we followsthe notations in [Kel96] and [Nee01b]. We denote a triangulated category by ( T , Σ , ∆ ) or simply ( T , Σ ) or T where T is an additive category, Σ is an additive self category equivalence on T which is said to be the suspension functor and ∆ is a class of sequences in T of the form x u → y v → z w → Σ x (212)such that vu = wv = ( u , v , w ) and call it a distinguished triangle and they satisfiesthe usual Verdier axioms from (TR ) to (TR ) . In the sequence ( ) , we sometimes write Cone u for theobject z .A triangle functor between triangulated categories from ( T , Σ ) to ( T (cid:48) , Σ (cid:48) ) is a pair ( f , a ) consisting of anadditive functor f : T → T (cid:48) and a natural equivalence a : f Σ → Σ (cid:48) f such that they preserves distinguishedtriangles. Namely for a distinguished triangle ( ) in T , the sequence f ( x ) f ( u ) → f ( y ) f ( v ) → f ( z ) a f ( w ) → Σ (cid:48) f ( x ) isa distinguished triangle in T (cid:48) . We say that a triangle functor ( f , a ) is strictly normal if f Σ = Σ (cid:48) f and a = id f Σ .A triangle natural transformation θ : ( f , a ) → ( g , b ) between triangulated functors ( f , a ) , ( g , b ) : ( T , Σ ) → ( T (cid:48) , Σ (cid:48) ) is a natural transformation θ : f → g which satisfies the equality ( Σ (cid:48) ∗ θ ) · a = b · ( θ ∗ Σ ) .We denote the category of small triangulated categories and triangle functors (resp. strictly normal trianglefunctors) by TriCat (resp.
TriCat sn ) and we write TriCat (resp.
TriCat sn ) for the 2-category of smalltriangulated categories, triangulated functors (resp. strictly normal triangle functors) and triangle naturaltransformations.Let ( T , Σ ) be a triangulated category. We say that a full subcategory D of T is a quasi-triangulatedsubcategory ( of T ) if ( D , Σ ) is a triangulated category and the inclusion functor ( ι , id Σ ) : D → T is a trian-gulated functor. We say that a quasi-triangulated subcategory D of T is a triangulated subcategory ( of T ) ifit is closed under isomorphisms. Namely an additive full subcategory D of T is a triangulated subcategory if Σ ± ( Ob D ) ⊂ Ob D and if for any distinguished triangle x → y → z → Σ x in T , if x and y are in D , then z isalso in D . Assuming the condition Σ ± ( Ob D ) ⊂ Ob D , the last condition is equivalent to the condition that iftwo of x , y and z are in D , then the other one is also in D .We say that a triangulated subcategory D of T is thick if D is closed under direct summand. Namely forany objects x and y in T , if x ⊕ y is in D , then both x and y are also in D . Let C be a complicial exact category. We write π ( C ) for the quotient category of C by C -homotopic relations. Namely we set Ob π ( C ) = Ob C and forany pair of objects x and y in Ob C , we setHom π ( C ) ( x , y ) : = Hom C ( x , y ) / ( C -homotopic relation ) . (213)Since composition of morphisms in C compatible with C -homotopic relation (see [Moc10, 2.13]), it inducesthe compositions of morphisms in π ( C ) . Moreover since the suspension functor on C preserves C -homotopicrelations, it induces an endofunctor on π ( C ) and we denote it by the same letter T . We call π ( C ) the stablecategory associated with C . Then we can make the pair ( π ( C ) , T ) into a triangulated category by declaringthat a sequence x (cid:48) → y (cid:48) → z (cid:48) → T x (cid:48) is a distinguished triangle if it is isomorphic to a sequence of the followingform x f → y κ f → Cone f ψ f → T x . (214)56e define the association from NC ( C frob ) the set of all null classes of C frob to Tri ( π ( C )) the set of alltriangulated subcategories of π ( C ) by sending a null class N to π ( N ) . It give rises to bijective correspon-dences between NC ( C frob ) and Tri ( π ( C )) , and NC thi ( C ) and Tri thi ( π ( C )) the set of all thick subcategoriesof π ( C ) (see [Moc13b, 3.15]). Let C be a normal ordinary complicial exact category and let ( f , H ) , ( f , H (cid:48) ) : x → y be apair of homotopy commutative diagrams. Assume that ( f , H ) and ( f , H (cid:48) ) are C -homotopic as morphisms in F b , h C . Then there exists a C -homotopy ( , S ) : ( f , H ) ⇒ C ( f , H (cid:48) ) . Then for each integer n , S n can be regardedas a CC -homotopy from H n to H (cid:48) n . Namely the pair ( f , H ) and ( f , H (cid:48) ) are CC -homotopic. Thus the canonicalfunctor F b , h C → F b , h C induces an equivalence of triangulated categories π ( F b , h C ) ∼ → π ( F b , h C ) . For a pairof integers a < b , similar statements hold for F [ a , b ] , h C and so on. (cf. [Moc10, 3.29].) Let C = ( C , w ) be a relative complicial exact category. Then the identity functor on C induces an equivalence ofcategories Ho ( C ) the homotopy category of C and π ( C ) / π ( C w ) the Verdier quotient of π ( C ) with respectto thick closure of the subcategory π ( C w ) . In particular we can make the pair ( Ho ( C ) , T ) into a trian-gulated category to make the equivalence above an equivalence of triangulated categories. The association Ho :
RelComp → TriCat is a -functor and it induces a -functor RelComp sn → TriCat sn . For a relative complicial exact category C = ( C , w ) , we denote the canonical functor C → Ho ( C ) by Q C .Recall the definition of thick 3.2.7 and strongly saturated 3.4.2 relative complicial exact categories. (cf. [Sch11, 3.2.18].) Let C = ( C , w C ) is a relative complicial exact category. If C is thick,then C is strongly saturated.Proof. Let f : x → y be a morphism in C such that its image in Ho ( C ) is an isomorphism. What we needto prove is that f is in w C . Then Cone f is trivial in Ho ( C ) and π ( C w C ) is thick by assumption and 3.2.15.Moreover C w C is closed under C -homotopy equivalences by 3.2.20. Thus Cone f is in C w C and it means f isin w C by ( ) . Thus we obtain the result. (cf. [HS85], [TT90, 1.9.2], [Moc13b, 3.16].) Let C = ( C , w ) be a relative complicial exactcategory. Then id C the identity functor of C induces the following equivalences of triangulated categories. Ho ( C frob ; w ) ∼ → Ho ( C frob ; (cid:104) w (cid:105) thi ) ∼ → Ho ( C ; w ) ∼ → Ho ( C ; (cid:104) w (cid:105) thi ) . (215) Let C = ( C , w ) be a relative complicial exact categoryand let F be a full subcategory of C . We regard (cid:104) F (cid:105) null , frob as a complicial exact subcategory of C frob andwrite Ho ( F in C ) for the homotopy category of the relative exact category ( (cid:104) F (cid:105) null , frob , w | (cid:104) F (cid:105) null , frob ) and callit a relative homotopy category of F ( in C . )Assume that w is thick. We say that a null class N of C frob compatible with w if for any morphism u : x → y from an object x in N to an object y in C w , there exists a pair of morphisms u (cid:48) : x → z and u (cid:48)(cid:48) : z → y with z ∈ Ob ( N ∩ C w ) and a C -homotopy H : u ⇒ C u (cid:48)(cid:48) u (cid:48) . We say that a full subcategory F iscompatible with w if the null class (cid:104) F (cid:105) null , frob of C frob is compatible with w . In this case the inclusionfunctor (cid:104) F (cid:105) null , frob → C induces a fully faithful functor Ho ( F in C ) → Ho ( C ) and the inclusion functor (cid:104) F (cid:105) null , frob (cid:44) → (cid:104) F (cid:105) null , w induces an equivalence of triangulated categoriesHo ( F in C ) ∼ → Ho ( (cid:104) F (cid:105) null , w in C ) (216)by [Kel96, 10.3]. Let E = ( E , w ) be a relative exact category. Thenwe write D b ( E ) or D b ( E ; w ) for the homotopy category of Ch b ( E ) the relative category of bounded chaincomplexes on E and we call it the ( bounded ) derived category of E . By virtue of Proposition 3.4.6, D b ( E ) naturally becomes a triangulated category. The association D b : RelEx → TriCat sn is a 2-functor.57 .4.12. Definition (Derived equivalence). We say that a relative exact functor f : E → E (cid:48) between relativeexact categories E and E (cid:48) is a derived equivalence if it induces an equivalence of triangulated categories D b ( f ) : D b ( E ) ∼ → D b ( E (cid:48) ) on bounded derived categories. We denote the class of derived equivalences in RelEx by deq
RelEx or simply deq. Since the association D b is 2-functorial and it sends a natural weakequivalence to a triangle natural equivalence, it turns out that a categorical homotopy equivalence in RelEx isa derived equivalence.By virtue of Proposition 3.4.18 below, for a relative complicial exact functor ( f , c ) : C → C (cid:48) , the followingtwo conditions are equivalent. • ( f , c ) induces an equivalence of triangulated categories Ho ( C ) ∼ → Ho ( C (cid:48) ) . • The relative functor f : C → C (cid:48) between relative exact categories is a derived equivalence in RelEx .In this case, we say that ( f , c ) is derived equivalence and we denote the class of derived equivalences in RelComp by deq
RelComp or simply deq.
Let ( f , c ) : C = ( C , w ) → C (cid:48) = ( C (cid:48) , w (cid:48) ) be a relative complicial exact functor betweenrelative complicial exact categories and let F be a full subcategory of C and let D be a T -system of C (fordefinition of T -system, see 4.1.1). Then ( ) By Theorem 1.5 in [BM11], if the following two conditions hold, then f is a derived equivalence. (App 1). For a morphism u : x → y in C , u is in w if and only if f ( u ) is in w (cid:48) . (App 2). For an object x in C and an object y in C (cid:48) and a morphism u : f ( x ) → y , there exists an object z in C and a morphism s : x → z in C and a morphism v : f ( z ) → y in w (cid:48) such that u = v f ( s ) .In (App 2) , we call the triple ( z , s , v ) a factorization of u ( along f ). ( ) In the following cases, the condition (App 1) is automatically verified. Here F is a strict exact subcat-egory of C frob or a prenull class of C .(a) The relative complicial exact functor Tot : ( Ch b ( F ) , tw ) → ( (cid:104) F (cid:105) null , frob , w | (cid:104) F (cid:105) null , frob ) . (See 3.3.3,3.3.7.)(b) The relative complicial exact functor ( − ) ∞ : ( F D | F b , h (cid:104) F (cid:105) null , frob , w st ) → ( (cid:104) F (cid:105) null , frob , w | (cid:104) F (cid:105) null , frob ) (see 3.2.23 and for the definition of the category F D | F b , h (cid:104) F (cid:105) null , frob of d´evissage filtrations, see4.2.1). ( ) In condition (App 2) , by replacing s with ξ s , z with Cyl s and u with u f ( υ s ) , we shall assume that s isan admissible monomorphism. ( ) Assume that C and C (cid:48) are thick. Then by Lemma 3.4.8, they are strongly saturated (see 3.4.2). Sincerelative complicial exact categories are derivable Waldhausen categories by 3.2.11, we can replacecondition (App 2) with the more weaker condition (App 2)’ below by [Cis10b, Th´eor`eme 2.9.]. (App 2)’ For an object x in C and an object y in C (cid:48) and a morphism u : f ( x ) → y , there exists amorphism b : y → y (cid:48) in w (cid:48) such that bu : f ( x ) → y (cid:48) admits a factorization ( z , s , v ) . ( ) We can replace condition (App 2) with the following more weaker condition (App 2)” . (App 2)” For an object x in C and an object y in C (cid:48) and a morphism u : f ( x ) → y , there exists a homotopy factorization of u . Namely there exists an object z in C and a morphism s : x → z in C and amorphism v : f ( z ) → y in w (cid:48) and a C -homotopy H : u ⇒ C v f ( s ) .In (App 2)” , we call the quadruple ( z , s , v , H ) a homotopy factorization of u ( along f ). Proof.
If we have a homotopy factorization s : x → z , v : f ( z ) → y and H : u ⇒ C v f ( s ) of u : f ( x ) → y , thenby setting z (cid:48) : = z ⊕ Cx and s (cid:48) : = (cid:18) s − ι x (cid:19) and v (cid:48) : = (cid:0) a − Hc − x (cid:1) where c − x : fCx ∼ → C f x is the canonicalisomorphism, we obtain a factorization f ( x ) f ( s (cid:48) ) → f ( z (cid:48) ) v (cid:48) → y of u .58 .4.14. Lemma. Let ( f , c ) : E = ( E , w E ) → C = ( C , w C ) be a relative complicial exact functor betweenrelative complicial exact categories and let F be an additive full subcategory of C . Assume that ( ) C is strictly ordinary and ( ) for any object x in E , f ( x ) is in (cid:104) F (cid:105) null , frob and ( ) for any object x in E and any object y in F , a morphism u : f ( x ) → y admits a factorization f ( x ) f ( v ) → f ( z ) a → y with a ∈ w C .Then ( f , c ) : E → ( (cid:104) F (cid:105) null , frob , w C | (cid:104) F (cid:105) null , frob ) satisfies (App 2)” . Proof.
Let x be an object in E and let y be an object in (cid:104) F (cid:105) null , frob and let u : f ( x ) → y be a morphism in C . We shall show that u admits a homotopy factorization. Since (cid:104) F (cid:105) null , frob is a C -homotopy equivalencesclosure of (cid:104) F (cid:105) semi-null by Lemma 3.1.13 ( ) , there exists an object y (cid:48) in (cid:104) F (cid:105) semi-null and a C -homotopyequivalences a (cid:48) : y (cid:48) → y and b (cid:48) : y (cid:48) → y and a C -homotopy H (cid:48) : id y (cid:48) ⇒ C b (cid:48) a (cid:48) . If the composition a (cid:48) u : f ( x ) → y (cid:48) admits a homotopy factorization ( z , c , a , H ) , then the quadruple ( z , c , b (cid:48) a , b (cid:48) H + H (cid:48) Ca ) gives a homotopyfactorization of u along f . Therefore we shall assume that y is in (cid:104) F (cid:105) semi-null . Moreover we shall assume that y is in (cid:104) F (cid:105) prenull by replacing f ( x ) u → y with f ( T n x ) c − Tn ∼ → T n f ( x ) T n u → T n y for suitable positive integer n . Here c T n : T n f ∼ → f T n is compositions of T n f T n − ∗ c T ∼ → T n − f T T n − ∗ c T ∗ T ∼ → · · · T ∗ c T ∗ T n − ∼ → T f T n − c T ∗ T n − ∼ → f T n . Recallthe definition of P m ( F ) from Lemma-Definition 3.1.11. There exists a non-negative integer m such that y is in P m ( F ) . If m =
0, then f ( x ) → y admits a factorization by assumption ( ) . Assume that m ≥ m . Namely, assuming that if y is in P m − ( F ) , then u admits a factorization, we willprove that if y is in P m ( F ) , then u admits a factorization. By definition of P m ( F ) , there exists a morphism d : y → y in P m − ( F ) such that y = Cone d . Applying an inductive hypothesis to the composition f ( x ) u → y ψ d → Ty , we obtain a factorization ( s , v (cid:48) , a (cid:48) ) of ψ d u : f ( x ) → Ty . We apply Lemma 3.4.16 ( ) below to thecomposition f ( Cone v (cid:48) ) Cone ( u , a (cid:48) ) → Cone ψ d ( id Ty ) → Ty , we obtain a factorization ( t , v (cid:48)(cid:48) , b (cid:48) ) of f ( Cone v (cid:48) ) → Ty by inductive hypothesis. We set z : = Cone v (cid:48)(cid:48) κ v (cid:48) and a : = σ Cone T · Cone ( a (cid:48) , b (cid:48) ) · d Cone v (cid:48)(cid:48) κ v (cid:48) : f ( z ) → Ty and v : = Cone ( v (cid:48)(cid:48) , , v (cid:48)(cid:48) µ v (cid:48) ) : T x → z . Then a is in w C by Proposition 3.2.17 ( ) ( ii ) and a · f ( v ) = Tu · ( c − T ) x byLemma 2.3.11 and Lemma 3.4.15 below. Thus we obtain a factorization of u by Lemma 3.4.16 ( ) below.We complete the proof. Let C be a strictly ordinary complicial exact category and let f : x → y, g : z → Cone f ,h : z → w and k : w → T x be morphisms in C such that ψ f g = kh. Then we have an equality − (cid:0) id Ty (cid:1) Cone ( g , k ) µ h = Θ f Cg . (217) Namely we have a commutative diagram of homotopy squares [ z → ] ( h , , µ h ) (cid:47) (cid:47) ( g , , ) (cid:15) (cid:15) [ w κ h → Cone h ] ( k , − ( id Ty ) Cone ( g , k ) , ) (cid:15) (cid:15) [ Cone f → ] ( ψ f , , Θ f ) (cid:47) (cid:47) [ T x
T f → Ty ] . (218) In particular we have an equality
Cone ( ψ f , , Θ f ) T g = Cone ( k , − (cid:0) id Ty (cid:1) Cone ( g , k )) Cone ( h , , µ h ) . (219)59 roof. By definition of Cone ( g , k ) , the following diagram is commutative and it implies the equality ( ) . z (cid:47) (cid:47) (cid:16) − h ι z (cid:17) (cid:47) (cid:47) g (cid:15) (cid:15) w ⊕ Cz (cid:16) k Cg (cid:17) (cid:15) (cid:15) ( κ h µ h ) (cid:47) (cid:47) (cid:47) (cid:47) Cone h Cone ( g , k ) (cid:15) (cid:15) Cone f (cid:47) (cid:47) (cid:16) − ψ f ι Cone f (cid:17) (cid:47) (cid:47) T x ⊕ C Cone f (cid:16) − T f − Θ f ι Tx C ψ f (cid:17) (cid:47) (cid:47) (cid:47) (cid:47) Ty ⊕ CT x . Let ( f , c ) : C = ( C , w C ) → C (cid:48) = ( C (cid:48) , w C (cid:48) ) be a relative complicial exact functor and let xand y be objects in C and C (cid:48) respectively. Then ( ) For a morphism u : f ( x ) → Ty, if the composition F ( T − x ) c − T − ∼ → T − f ( x ) T − u → T − Ty β − y ∼ → y admits afactorization ( z , s , a ) , then the triple ( T z , T s · α − x , Ta · ( c − T ) z ) is a factorization of u. ( ) For a morphism v : f ( x ) → y, if the composition f ( T x ) ( c − T ) x ∼ → T f ( x ) Tu → Ty admits a factorization ( z , s , a ) ,then the triple ( T − z , T − s · β y , β − y · T − a · ( c − T − ) z ) is a factorization of v.Proof. Assertion ( ) (resp. ( ) ) follows from the left (resp. right) commutative diagram below by Lemma 1.2.17. f ( x ) ∼ f ∗ α − x (cid:117) (cid:117) (cid:111) α − ∗ f ( x ) (cid:15) (cid:15) u (cid:47) (cid:47) Ty α − ∗ Ty (cid:111) (cid:15) (cid:15) f ( TT − x ) ∼ ( c − T ) T − x (cid:47) (cid:47) f ( Ts ) (cid:15) (cid:15) T f ( T − x ) ∼ T ( c − T − ) x (cid:47) (cid:47) T f ( s ) (cid:35) (cid:35) T − f ( x ) TT − u (cid:47) (cid:47) TT − Ty (cid:111) T ∗ β − y (cid:15) (cid:15) f ( Tz ) ∼ ( c − T ) z (cid:47) (cid:47) T f ( z ) Ta (cid:47) (cid:47) Ty , f ( x ) u (cid:47) (cid:47) ∼ f ∗ β y (cid:117) (cid:117) (cid:111) β ∗ f ( x ) (cid:15) (cid:15) y (cid:111) β y (cid:15) (cid:15) f ( T − Tx ) ∼ c − T ∗ Tx (cid:47) (cid:47) f ( T − s ) (cid:15) (cid:15) T − f ( Tx ) ∼ T − ( c − T ) x (cid:47) (cid:47) T − f ( s ) (cid:35) (cid:35) T − T f ( x ) T − Tu (cid:47) (cid:47) T − Tyf ( T − z ) ∼ ( c − T − ) z (cid:47) (cid:47) T − f ( z ) . T − a (cid:59) (cid:59) Recall the definition of the zig-zag sequence of morphisms U [ a , b ] x and the functor L [ a , b ] from 3.3.8. Let C = ( C , w ) be a relative complicial exact category and let P be a prenull class of C and let a < b be a pair of integers and let x be a complex in Ch b , ≤ b ( P ) . Then we can regard the zig-zagsequence U [ a , b ] x as an isomorphism x ∼ → L [ a , b ] x in D b ( P , w | P ) . Moreover we have the equality Tot U [ a , b ] x = id Tot x (220) as a morphism in Ho ( P in C ) .Proof. Since U [ a , b ] x is the zig-zag sequence of morphisms in lw ∪ q f is, it can be regarded as an isomorphism in D b ( P , w | P ) . To show the equality ( ) , by 3.3.4 and the equality ( ) , by replacing x with ( σ b − x )[ − b + ] , we shall assume that a = b = x is in Ch [ , ] ( C ) . In this case, the equality follows form theequality ( ) .Tot x Tot u ( x )= Cone ( ι x , κ dx ) (cid:47) (cid:47) id Tot x (cid:43) (cid:43) Tot Q ( x ) = Cone µ d x ( , idCone dx , − µ dx rx ) (cid:15) (cid:15) Tot L ( x ) = Tot x id Tot x (cid:115) (cid:115) Tot v ( x )= Cone ( , id Cone f ) (cid:111) (cid:111) Tot x . .4.18. Proposition (Comparison of homotopy and derived categories). (cf. [Moc13b, 4.15].) Let C =( C , w ) be a relative complicial exact category and let P be a full additive subcategory of C . Then ( ) If P is a prenull class, then the triangulated structures on Ho ( Ch b ( P ) ; q ( w | P )) induced from the stan-dard and the level complicial structures on Ch b ( P ) are equivalence. ( ) Assume that either condition ( i ) or ( ii ) below ( i ) P is a prenull class or ( ii ) C is strictly ordinary and P contains all C-contractible objects and P is closed under the operationsT ± .Then the functor Tot : Ch b ( P ) → (cid:104) P (cid:105) null , frob induces an equivalence of triangulated categories Ho ( Ch b ( P ) ; q ( w | P )) ∼ → Ho ( P in C ) . (221) ( ) The canonical functor j C : C → Ch b ( C ) which sends an object x to a complex j C ( x ) such that j C ( x ) k isx if k = and if k (cid:54) = induces an equivalence of triangulated categories Ho ( C ) ∼ → D b ( C ) . ( ) If P is a strict exact subcategory of C , then the identity functor of Ch b F induces an equivalence oftriangulated categories D b ( F ; w | F )(= Ho ( Ch b ( F ) ; q ( w | F ))) ∼ → Ho ( Ch b F in Ch b C ) .Proof. ( ) By 3.3.8 and the equality ( ) , the suspension functors on Ho ( Ch b ( P ) ; q ( w | P )) induced fromthe standard and the level complicial structures on Ch b ( P ) are equivalence of functors. By 3.4.9, theclass of distinguished triangles on Ho ( Ch b ( P ) ; q ( w | P )) only depends upon the Frobenius exact structureon Ch b ( P ) . Thus we obtain the result. ( ) First we assume that P is a prenull class. By 3.3.9 and 3.4.9, replacing w with (cid:104) w (cid:105) thi , C , we shallassume that w is thick. We will show that the functor Ho ( Tot ) is essentially surjective. Let x be an ob-ject in (cid:104) P (cid:105) semi-null , C , then there exists an integer m and an object y in P such that T m y is isomorphicto x . Then by the equality ( ) , Tot j P ( y )[ m ] = T m y is isomorphic to x . Since the inclusion functor π ( (cid:104) P (cid:105) semi-null , C ) → π ( (cid:104) P (cid:105) null , frob ) is an equivalence of triangulated categories, we obtain the result.Next for a pair of chain complexes x and y in Ch b ( P ) , we will show that the mapTot : Hom Ho ( Ch b ( P ) ; q ( w | P )) ( x , y ) → Hom Ho ( P in C ) ( Tot x , Tot y ) is surjective. Since the suspension functors on both Ho ( Ch b ( P ) ; q ( w | P )) and Ho ( P in C ) are equivalences,by replacing x and y with x [ N ] and y [ N ] for sufficiently large integer N respectively, we shall assume that x and y are in Ch b , ≥ ( P ) . Next by 3.3.8, replacing x and y with j P ( Tot x )[ m ] and j P ( Tot y )[ m (cid:48) ] for a suitableintegers m and m (cid:48) , we shall assume that x and y are in P . In this case, surjectivity comes from commutativeof the diagram below and the commutativity follows from 3.4.17.Hom Ho ( Ch b ( P ) ; q ( w | P )) ( j P ( T m x ) , j P ( T m (cid:48) y )) (cid:111) (cid:15) (cid:15) Hom Ho ( P in C ) ( T m x , T m (cid:48) y ) Ho ( j P ) (cid:111) (cid:111) Hom Ho ( Ch b ( P ) ; q ( w | P )) ( j P ( x )[ m ] , j P ( y )[ m (cid:48) ]) Ho ( Tot ) (cid:47) (cid:47) Hom Ho ( P in C ) ( T m x , T m (cid:48) y ) . By the equality ( ) , Ho ( Tot ) has the trivial kernel and thus Ho ( Tot ) is equivalence of triangulated categoriesby [Bal07, 3.18].Next we assume that C is strictly ordinary and P contains all C -contractible objects and P is closedunder the operations T ± . Let x be an object in Ch b ( P ) and let y be an object y in P and let u : Tot x → y be a morphism in P . We prove that u admits a factorization of u along Tot. By replacing u : Tot x → y with T m u : Tot x [ m ] → T m y for a suitable integer m , we shall assume that x is in Ch [ , n ] ( P ) for somenon-negative integer n . Then by Lemma-Definition 3.3.11, the triple ( z u , g u u x , a u ) gives a factorization of u along Tot. Notice that by construction and assumption z u is in Ch [ , n ] ( P ) . Thus by Lemma 3.4.14,Tot : ( Ch b P , q ( w | P )) → ( (cid:104) P (cid:105) null , frob , w | (cid:104) P (cid:105) null , frob ) satisfies (App 2)’ and we obtain the equivalence ( ) by Remark 3.4.13 ( ) . 61 ) By the equality ( ) , it turns out that D b ( j C ) is the right inverse functor of D b ( Tot ) which is an equiva-lence of triangulated categories D b ( C ) ∼ → Ho ( C ) by ( ) . Thus D b ( j C ) is also an equivalence of triangulatedcategories. ( ) Since Ch b F is a prenull class in Ch b C , relative complicial exact functors ( Ch b F , q ( w | F )) j Ch b F → ( Ch b Ch b F , qq ( w | F )) Tot → ( (cid:104) Ch b F (cid:105) null , frob , q ( w | (cid:104) Ch b F (cid:105) null , frob )) induce equivalences of triangulated categoriesHo ( Ch b F ; q ( w | F )) ∼ → Ho ( Ch b Ch b F ; qq ( w | F )) ∼ → Ho ( (cid:104) Ch b F (cid:105) null , frob ; q ( w | (cid:104) Ch b F (cid:105) null , frob )) by ( ) and ( ) . For the (large) relative categories ( RelEx , deq RelEx ) and ( RelComp , deq RelComp ) , theadjoint pair ( RelEx , deq RelEx ) Ch b (cid:29) F ( RelComp , deq RelComp ) with the adjunction ( j , Tot ) gives categorical ho-motopy equivalences. Let C = ( C , w ) be a relative complicial exact category and let A be an additive fullsubcategory of C . Then ( ) Assume that A contains all C-contractible objects. Then Ch b A is compatible with qw (see 3.4.10) . ( ) Assume that A is a strict exact subcategory of C . Then the inclusion functor (cid:104) Ch b A (cid:105) null , frob , Ch b C (cid:44) → (cid:104) Ch b A (cid:105) null , qw , Ch b C induces an equivalence of categories Ho ( Ch b A in Ch b C ) ∼ → Ho ( (cid:104) Ch b A (cid:105) null , qw , Ch b C in Ch b C ) . (222) Proof. ( ) First notice that A has enough objects to resolve in C in the following sense:For an object y in C , there exists an admissible epimorphism P ( y ) q y (cid:16) y with P ( y ) ∈ Ob A .Thus by [TT90, 1.9.5], for any morphism f : x → y with x ∈ Ob Ch b A and y ∈ Ob ( Ch b C ) qw , there existsan object z ∈ Ob Ch b A and an admissible monomorphism x i (cid:26) z and a quasi-isomorphism z a → y such that f = ai . Then since ( Ch b C ) qw is qis-closed by Lemma 3.2.20, z is in ( Ch b C ) qw . Thus Ch b A is compatiblewith qw . ( ) First notice that (cid:104) Ch b A (cid:105) null , frob = (cid:104) ( Ch b A ) heq (cid:105) null , frob . Thus the inclusion functor Ch b A (cid:44) → ( Ch b A ) heq induces an equivalence of triangulated categories Ho ( Ch b A in Ch b C ) ∼ → Ho (( Ch b A ) heq in Ch b C ) . In thecommutative diagram of triangulated categories Ho (( Ch b A ) heq in Ch b C ) II (cid:47) (cid:47) I (cid:15) (cid:15) Ho ( Ch b ( Ch b A ) heq in Ch b Ch b C ) III (cid:47) (cid:47) Ho ( (cid:104) Ch b ( Ch b A ) heq (cid:105) null , qqw in Ch b Ch b C ) Ho ( (cid:104) Ch b A (cid:105) null , qw in Ch b C ) II (cid:47) (cid:47) Ho ( Ch b (cid:104) Ch b A (cid:105) null , qw in Ch b Ch b C ) III (cid:47) (cid:47) Ho ( (cid:104) Ch b (cid:104) Ch b A (cid:105) null , qw (cid:105) null , qqw in Ch b Ch b C ) , the triangulated functors II and III are equivalences of triangulated categories by Proposition 3.4.18 ( ) and ( ) respectively. Thus the triangulated functor I above is also an equivalence of triangulated categories andwe complete the proof. Let E be an exact category and let A and B be a pair offull subcategory of E . We write E ( A , E , B ) for the full subcategory of F [ , ] E consisting of those objects x such that x is in A and x is in B and the sequence x i x → x i x → x is an admissible exact sequence in E . if A = B = E , then we denote E ( A , E , B ) by E ( E ) . 62 .5.2. Lemma-Definition (Homology theory of complicial exact categories). (cf. [Wal85, 1.7].) Let E bean exact category and let B be an abelian category. A homology theory on E ( with values in B ) is a family H = { H n : C → B , δ n : H n (( − ) ) → H n − (( − ) ) } n ∈ Z indexed by the set of all integers consisting of additivefunctors H n and natural transformations δ n between the functors from E ( E ) to B where ( − ) and ( − ) arefunctors which sends an object x (cid:26) y (cid:16) z in E ( E ) to x and z respectively which subjects to the followingcondition:For any admissible exact sequence x i (cid:26) y p (cid:16) z in E , the long sequence · · · δ ( i , p ) n + → H n + ( x ) H n + ( i ) → H n + ( y ) H n + ( p ) → H n + ( z ) δ ( i , p ) n → H n ( x ) H n ( i ) → H n ( y ) H n ( p ) → H n ( z ) δ ( i , p ) n − → · · · (223)is exact in E . For a homology theory H = { H n , δ n } n ∈ Z on E , we say that a morphism f : x → y is a H -quasi-isomorphism if H n ( f ) is an isomorphism for all integer n . We denote the class of all H -quasi-isomorphismsin E by w H , cE or simply w H .Moreover, we assume that E is a complicial exact category. Then for a homology theory H = { H n , δ n } n ∈ Z on E , the following conditions are equivalent: ( ) For an object x in E and an integer n , H n ( r x : CCx → Cx ) = ( ) For an object x in E and an integer n , H n ( Cx ) = ( ) For a pair of morphisms f , g : x → y in E , if f and g are C -homotopic, then H n ( f ) = H n ( g ) for any integer n . If H satisfies the conditions above, we say that H is C-homotopy invariant . For a C -homotopy invarianthomology theory H on E , we can show that the class w H of H -quasi-isomorphisms in E is a class ofcomplicial weak equivalences in E . Proof.
Assume condition ( ) . Then for an object x in E , id H n ( Cx ) = H n ( r x ι x ) = ( ) holds. Next assume condition ( ) and let f , g : x → y be a pair of morphisms in E and let H : f ⇒ C g be a C -homotopy from f to g . Then H n ( f ) − H n ( g ) = H n ( H ι x ) = n . Thuscondition ( ) holds. Finally assume condition ( ) . Then for an object x in E , r x r Cx : r x ⇒ C C -homotopyfrom r x to 0. Hence H n ( r x ) = n . Thus condition ( ) holds. t -structures). A typical example of a homology theoryon a relative exact category C comes from a t -structure on D b ( C ) the derived category of C . In this article,we use homological t-structure as in [Lur17, § t -structures from [BBD82] (but conventions are written by cohomological notions inIbid).Let ( T , Σ ) be a triangulated category and let D be a full subcategory of T . For any integer n , we write D [ n ] for the full subcategory of T consisting of those objects x [ n ] for some object x in D and we denotethe full subcategory of C consisting of those objects x such that Hom T ( x , y ) = T ( y , x ) =
0) forany object y in D by ⊥ D (resp. D ⊥ ). We say that D is a ( homological ) t-structure of T if D satisfies thefollowing two conditions. (Suspension and isomorphisms closed condition). D is closed under isomorphisms and closed under thesuspension functor, that is D [ ] ⊂ D . (Decomposition condition). For any object x in D , there exists a pair of object τ ≥ x in D and τ ≤− x in D ⊥ and a distinguished triangle τ ≥ x → x → τ ≤− x → Σ τ ≥ x . (224)In this case, we write D ≥ n and D ≤ n for D [ n ] and ( D [ n + ]) ⊥ respectively. We can show that for an integer n , there exists a left (resp. right) adjoint functor of the inclusion functor D ≤ n (cid:44) → D (resp. D ≥ n (cid:44) → D ) whichwe denote by τ ≤ n : D → D ≤ n (resp. τ ≥ n : D → D ≥ n ) and D ♥ : = D ≥ (cid:84) D ≤ is an abelian category and callit the heart of D . Moreover, the associations x (cid:55)→ τ ≥ τ ≤ x gives homological functor H D : T → D ♥ whichwe call the homological functor associated with the t-structure D .Thus for a relative complicial exact category C = ( C , w ) and a homological t -structure D on D b ( C ) thebounded derived category of C , the composition with the canonical functor C → D b ( C ) with the homologicalfunctors H D ( T n ( − )) : D b ( C ) → D ♥ gives a homotopy invariant homology theory on C .63 .5.4. Lemma-Definition (Puppe exact sequences). Let C be a complicial exact category and let B be anabelian category and let H = { H n , δ n } n ∈ Z be a C -homotopy invariant homology theory on C with valuesin B . Then there exists a family of natural transformations { δ (cid:48) n : H n ( Cone ) → H n − ( dom ) } n ∈ Z between thefunctors from C [ ] to B indexed by the set of integers such that the long sequence · · · δ (cid:48) fn + → H n + ( x ) H n + ( f ) → H n + ( y ) H n + ( κ f ) → H n + ( Cone f ) δ (cid:48) fn → H n ( x ) H n ( f ) → H n ( y ) H n ( κ f ) → H n ( Cone f ) δ (cid:48) fn − → · · · (225)induced from a morphism f : x → y in C is exact in B . We call a sequence ( ) the Puppe exact sequence ( associated with the morphism f : x → y ). Proof.
Let f : x → y be a morphism in C . Notice that Cyl f υ f → y is a C -homotopy equivalence and thus byconsidering the admissible exact sequence in the top line in the commutative diagram below, x (cid:47) (cid:47) ξ f (cid:47) (cid:47) f (cid:32) (cid:32) Cyl f υ f (cid:15) (cid:15) η f (cid:47) (cid:47) (cid:47) (cid:47) Cone fy , κ f (cid:59) (cid:59) we obtain the long exact sequence ( ) where we set δ (cid:48) fn : = δ ( ξ f , η f ) n for all integer n . The main theme of this section is to define d´evissage spaces associated with cell structures and by usingthis phraseology, we will discuss a derived version of d´evissage condition. In the first subsection 4.1, we willintroduce an idea of cell structures on relative complicial exact categories which is a resemblance of t -structureor weight structure on triangulated categories and the next subsection 4.2, we will renew the conception ofd´evissage filtrations with respect to a cell structure. In subsection 4.3, we will innovate a derived version ofd´evissage conditions and discuss relationship this conditions with Quillen’s, Raptis’ and Waldhausen’s one.The last two subsections are devoted to categorified calculation of d´evissage spaces. In subsection 4.4, wewill study a derived quasi-split sequence of relative exact categories. In the context of quasicategory theory,analogous concept is studied in [FP07]. The final section 4.5, we provide natural derived flag structures onthe category of d´evissage filtrations by utilizing results in the previous subsections. Let C be a complicial exact category and let w be a class of morphisms in C and let D = { D n } n ∈ Z be a family of full subcategories of C indexed by the set of all integers Z . We say that D is topologizing (resp. semi-Serre , Serre , prenull , semi-null , closed under finite direct sums , thick , w-closed ) iffor all integer n , D n is a topologizing (resp. semi-Serre and so on) subcategory of C . We say that D is a T -system if for all integer n , T ( Ob D n ) ⊂ Ob D n + .Let D = { D n } n ∈ Z and D (cid:48) = { D (cid:48) n } n ∈ Z be a pair of T -systems of relative complicial exact categories C =( C , w ) and C (cid:48) = ( C (cid:48) , w (cid:48) ) respectively. Assume that D (cid:48) are isomorphisms closed. Namely for each integer n , D (cid:48) n is closed under isomorphisms. A morphism of T -system ( f , d ) : D → D (cid:48) is a relative complicial exactfunctor C → C (cid:48) such that for each integer n , f ( Ob D n ) is contained in Ob D (cid:48) n . Let C be a complicial exact category. A cell structure D = { D ≤ m , D ≥ n } n , m ∈ Z of C is a family of class of full subcategories in C satisfying the following conditions: • For any integer n , D ≤ n ⊂ D ≤ n + and D ≥ n + ⊂ D ≥ n . • For any integer n , D ≥ n and D ≤ n are closed under C -homotopy equivalences. • For any integer n and an object x in C , – if x is in D ≥ n , then T x is in D ≥ n + and – if x is in D ≤ n , then T − x is in D ≤ n − . 64 For any morphism f : x → y in C , – if y is in D ≥ n − and Cone f is in D ≥ n , then x is in D ≥ n − , and – if x is in D ≤ n and Cone f is in D ≤ n , then y is in D ≤ n .We set D n : = D ≤ n ∩ D ≥ n for any integer n and set D ♥ : = { D n } n ∈ Z the family of full subcategory of indexedby the set of all integers. Then we will show that D ♥ is a T -system of C in 4.1.7. We call D ♥ the heart of D .Moreover let w be a class of complicial weak equivalences of C . We say that D is compatible with w or D isa cell structure of ( C , w ) , if C w ⊂ (cid:92) n ∈ Z D n .We say that a cell structure D is topologizing (resp. semi-Serre , Serre , prenull , semi-null , closed underfinite direct sums , thick , w-closed ) if the heart D ♥ of D is a topologizing (resp. semi-Serre and so on). Wesay that a cell structure D is strongly topologizing (resp. strongly semi-Serre , strongly Serre , strongly prenull , strongly semi-null , strongly closed under finite direct sums , strongly thick , strongly w-closed ) if for any integer n , both D ≥ n and D ≤ n are topologizing (resp. semi-null and so on). Let C = ( C , w ) be a relative complicial exact cate-gory and let D = { D ≥ n , D ≤ m } n , m ∈ Z be a cell structure of C and let F be a full subcategory of C . Thenfor each integer n , we set ( D | F ) ≥ n : = D ≥ n (cid:84) (cid:104) F (cid:105) null , frob and ( D | F ) ≤ n : = D ≤ n (cid:84) (cid:104) F (cid:105) null , frob and D | F : = { ( D | F ) ≥ n , ( D | F ) ≤ m } n , m ∈ Z . Then D | F is a cell structure of ( (cid:104) F (cid:105) null , frob , w | (cid:104) F (cid:105) null , frob ) and call it the restric-tion of D ( to F ).Recall the definition of homology theory on a complicial exact categories from 3.5.2. (cf. [Wal85, 1.7]) Let C be a complicialexact category and let H = { H n , δ n } n ∈ Z be a homology theory on C with values in an abelian category B .Then we define a family D H = { D ≤ n , D ≥ m } n , m ∈ Z of full subcategories in C by the formulaOb D ≤ n : = { x ∈ Ob X ; H k ( x ) = k > n } and (226)Ob D ≥ m : = { x ∈ Ob X ; H k ( x ) = k < m } . (227)Then the family D H is a cell structure on C by Puppe exact sequence 3.5.4. Let C = ( C , w ) be a relative complicialexact category and let A be a full subcategory of C . Then we set D A ≥ n = D A ≤ m = (cid:104) A (cid:105) null , w for any integers n and m . The family D A : = { D A ≥ n , D A ≤ m } n , m ∈ Z forms a cell structure of C . We call D A the cell structureassociated with a full subcategory A . m -connected morphisms). Let C be a complical exact category and D = { D ≥ n , D ≤ m } n , m ∈ Z a cell structure of C and let f : x → y be a morphism in C and let m be an integer. We say that f is m-connected ( with respect to D ) or ( m , D ) -connected if Cone f is in D ≥ m + . Let C be a complicial exact category and let D = { D ≤ n , D ≥ m } n , m ∈ Z be a cell structure of C and let f : x → y and g : y → z be a pair of morphisms in C and let n be an integer. Then ( ) If x is in D ≥ n and if Cone f is in D ≥ n , then y is in D ≥ n . ( ) If x is in D ≥ n − and if y is in D ≥ n , then Cone f is in D ≥ n . ( ) If y is in D ≤ n and if x is in D ≤ n − , then Cone f is in D ≤ n . ( ) If y is in D ≤ n − and if Cone f is in D ≤ n , then x is in D ≤ n − . ( ) If x is in D ≤ n , then T x is in D ≤ n + . ( ) If x is in D ≥ n , then T − x is in D ≥ n − . ( ) If Cone f is in D ≥ n and if Cone g is in D ≥ n , then Cone g f is in D ≥ n . ( ) If Cone f is in D ≥ n − and if Cone g f is in D ≥ n , then Cone g is in D ≥ n . ( ) If Cone g is in D ≥ n and if Cone g f is in D ≥ n − , then Cone f is in D ≥ n − . ( ) If Cone f is in D ≤ n and if Cone g is in D ≤ n , then Cone g f is in D ≤ n . ( ) If Cone f is in D ≤ n − and if Cone g f is in D ≤ n , then Cone g is in D ≤ n . ( ) If Cone g is in D ≤ n and if Cone g f is in D ≤ n − , then Cone f is in D ≤ n − . ( ) T : C → C induces equivalences of categories T : D ≤ n ∼ → D ≤ n + , T : D ≥ n ∼ → D ≥ n + and T : D n ∼ → D n + . In particular, D ♥ is a T -system. roof. By 2.3.16, the canonical morphism Cone κ f → T x is a C -homotopy equivalence. Thus by applying theaxiom of cell structure to the morphism κ f : y → Cone f , we obtain assertions from ( ) to ( ) . Next Applying ( ) to the morphism x → ( ) to the morphism T − x →
0, we obtain assertions ( ) and ( ) . Nextby 2.3.8, the canonical morphism Cone ( κ f , κ g f ) : Cone g → Cone Cone ( id x , g ) is a C -homotopy equivalence.Thus by applying the axiom of cell structures and assertions from ( ) to ( ) to the morphism Cone ( κ f , κ g f ) ,we obtain assertions from ( ) to ( ) . Finally ( ) follows from the axioms of cell structures and ( ) and ( ) . In 4.1.7, D is essentially determined from the pair D ≥ and D ≤ by ( ) . Thus we oftenabbreviate D as { D ≥ , D ≤ } . Let C be a complicial exact category and let D = { D ≥ , D ≤ } be a cell structure on C . For a pair of integers n and m , we set D [ n , m ] : = D ≥ n ∩ D ≤ m . We say that D is bounded if C = (cid:91) ( n , m ) ∈ Z n < m D [ n , m ] . (228) Let C = ( C , w ) be a relative complicial exact category andlet D = { D ≥ , D ≤ } be a cell structure on C . We say that D is ordinary if for any pair of integers n < m , D [ m , n ] = C w . Let C = ( C , w ) be a relative complicial exact category and let D = { D ≥ , D ≤ } be a cellstructure of C and let i : x (cid:26) y be an admissible monomorphism in C and let f : x → z be a morphism in C and let m be an integer. Assume that i is a Frobenius admissible monomorphism or D is strongly w-closed.Then if Cone i is in D ≥ m ( resp. D ≤ m ) , then Cone ( i (cid:48) : z (cid:26) z (cid:116) x y ) is also in D ≥ m ( resp. D ≤ m ) .Proof. Assume that i is a Frobenius admissible monomorphism (resp. D is strongly w -closed). Then thecanonical morphism Cone i → Cone i (cid:48) is a C -homotopy equivalence (resp. is in w ) by 3.2.16. Thus if Cone i is in D ≥ m (resp. D ≤ m ), then Cone i (cid:48) is also in D ≥ m (resp. D ≤ m ). Let C be a complicial exact category and let D = { D n } n ∈ Z be afamily of full subcategories of C indexed by the set of all integers Z . A d´evissage filtration ( with respect to thefamily D in C ) is a bounded filtered objects (see Definition 2.2.7) x on C such that for any integer n , Cone i xn isin D n + . We write F D b C (resp. F D b , h C ) for the full subcategory of F b C (resp. F b , h C ) consisting of d´evissagefiltrations with respect to the family D . We also set for any pair of integers a ≤ b , F D [ a , b ] C : = F [ a , b ] C (cid:84) F D b C and so on. Let C be a complicial exact category and let D and D be a pair of full subcategories of C . Then we set C [ ] h ( D , D ) : = F { D , D } [ , ] , h C . Namely C [ ] h ( D , D ) is a full subcategory of C [ ] h consistingof those objects [ f : x → y ] such that x is in D and Cone f is in D . Let C be a normal ordinary complicial exact category and let D = { D n } n ∈ Z be a familyof full subcategories of C indexed by the set of all integers Z . Then ( ) If D is topologizing ( resp. semi-Serre, Serre, closed under finite direct sums, prenull, semi-null, thick ) in C , then F D b , h C is also in F b , h C . ( ) Moreover let w be a class of complicial weak equivalences of C . If D is w-closed in C , then F D b , h C islw-closed in F b , h C .Proof. ( ) For a pair of objects x and y in F b , h C and an integer n , we have the canonical isomorphismCone i x ⊕ yn ∼ → Cone i xn ⊕ Cone i yn . Thus if D is closed under finite direct sum in C , then F D b , h C is also closedunder finite direct sum in F b , h C .For a level Frobenius admissible exact sequence ( ) in F b , h C and an integer n , the sequence Cone i xn (cid:26) Cone i yn (cid:16) Cone i zn is a Frobenius admissible sequence in C by 2.3.14. Thus it turns out that if D is topologizing(resp. semi-Serre, Serre), then F D b , h C is also in F b , h C .66et ( f , H ) : x → y be a homotopy commutative diagram in F b , h C . Then by 2.3.15, for each integer n ,there is a natural isomorphisms Cone i Cone ( f , H ) n ∼ → Cone Cone ( f n , f n + , H n ) . Thus if D n is closed under takingthe functor Cone for all n , then F D b , h C is also closed under taking Cone in F b , h C . If there exists a homotopycommutative diagram ( g , K ) : y → x such that ( g , K )( f , H ) = id x , then for any integer n , we have the equalityCone ( g n , g n + , K n ) Cone ( f n , f n + , H n ) = id Cone i xn . Thus if y is in F D b , h C and D is thick, then x is also in F D b , h C . ( ) Let ( f , H ) : x → y be a homotopy commutative diagram in F b , h C and assume that ( f , H ) is in lw and x (resp. y ) is in F D b , h C . Then for each integer n , Cone ( f n , f n + , H n ) : Cone i xn → Cone i yn is in w by 3.2.17 ( )( ii ) . Thus Cone i yn (resp. Cone i xn ) is in D n + and it turns out that y (resp. x ) is in F D b , h C .Recall the functors j and ( − )[ k ] from 2.2.7 and 2.2.14. Let ( C , w ) be a relative complicial exact category. For an integer k and a fullsubcategory S in C , we denote the full subcategory of F b , ≤ k , h C consisting of those objects x such that x k isin S and x i is C -contractible for any i < k by F b , h ( S [ k ] in C ) . We define j ( − )[ k ] : S → F b , h ( S [ k ] in C ) and ( − ) k : F b , h ( S [ k ] in C ) → S to be a pair of functors by sending an object x in S to j ( x )[ k ] and anobject z in F b , h ( S [ k ] in C ) to z k respectively. We have the equality ( j ( − )[ k ]) k = id S and there exists anatural weak equivalence j (( − ) k )[ k ] → id F b , h ( S [ k ] in C ) with respect to the class of level weak equivalencesof F b , h ( S [ k ] in C ) . Thus the pair of relative categories ( S , w | S ) and ( F b , h ( S [ k ] in C ) , lw ) are categoricalhomotopy equivalent.Next assume that ( C , w ) is thick normal ordinary and let a < b be a pair of integers. We set F : = F [ a , b ] , h C and G : = F [ ] h ( F [ a , b − ] , h C , F b , h ( C [ b ] in C )) . Recall the definition of the category F [ ] h ( − , − ) from 4.2.2. Wedefine G : G → F and H : F → G to be strictly normal complicial functors by sending an object [ f : x → y ] in G to y in F and y in F to [ σ ≤ b − y → y ] in G . We have the equality G H = id F and there exist a naturalweak equivalence id G → H G with respect to the class llw of level level weak equivalences of G . Thus thepair of relative categories ( G , llw ) and ( F , lw ) are categorical homotopy equivalent each other.Moreover let D = { D n } n ∈ Z be a family of full subcategories of C indexed by the set of integers Z .Assume that for any integer n , D n contains all C -contractible objects in C . Then we set F (cid:48) : = F D [ a , b ] , h C and G (cid:48) : = F (cid:48) [ ] h ( F D [ a , b − ] , h C , F b , h ( D b [ b ] , C )) . Then the restrictions of G and H to G (cid:48) and F (cid:48) induce the categoricalhomotopy equivalences ( G (cid:48) , llw ) G (cid:29) H ( F (cid:48) , lw ) and ( (cid:104) G (cid:48) (cid:105) null , frob , llw ) G (cid:29) H ( (cid:104) F (cid:48) (cid:105) null , frob , lw ) by 3.1.17. Let ( C , w ) be a relative complicial exact category and D = { D ≤ n , D ≥ m } n , m ∈ Z be a cellstructure of ( C , w ) and x a d´evissage filtration with respect to D having amplitude contained in [ a , b ] . Then ( ) x k is in D ≤ k for any k. ( ) If x b is in C w , then x k is in D k for any k.Proof. ( ) We proceed by induction on k . Since x k = k < a , x k is in D k . Assume that x k is in D ≤ k . Thenby considering an admissible sequence x k i xk → x k + κ ixk → Cone i xk (229)with x k ∈ D ≤ k and Cone i xk ∈ D k + , it turns out that x k + is in D ≤ k + . ( ) We proceed by descending induction on k . Since x k = x b for k ≥ b , x k is in D k . Assume that x k + is in D k + . Then by considering a sequence ( ) with x k + ∈ D k + and Cone i xk ∈ D k + , it turns out that x k is in D ≥ k . Thus x k is in D k .Recall the functors j , ( − )[ k ] , c k and s k from 2.2.7, 2.2.14, 2.2.15 and 2.2.16. Let ( C , w ) be a thick normal ordinary relative complicial exact category and let S be a full subcategory and let k be an integer. Then there are a pair of functors S j ( − )[ k ] (cid:29) ( − ) k F [ k , k ] , h S and theygives an isomorphism of categories.For a triple of integers a ≤ k ≤ b , we define E to be a full subcategory of F [ a , b ] , h C consisting of thoseobjects x such that x b − is C -contractible and let E (cid:48) be a subcategory of E such that Ob E (cid:48) : = Ob E andMor E (cid:48) : = { ( f , H ) : x → y ∈ Mor E ; i yb − f b − = f b i xb − , H b − = } . Then E (cid:48) is a complicial exact categorysuch that the inclusion functor E (cid:48) (cid:44) → E is a strictly normal complicial functor and reflects exactness. For an67bject x and a homotopy commutative diagram ( f , H ) : x → y in E , we define r x and r ( f , H ) : r x → r y to bean object and a morphism in E (cid:48) by setting ( r x ) n : = (cid:26) x n if n ≤ b − x b ⊕ Cx b − if n ≥ b , i r xn : = i xn if n ≤ b − (cid:32) i xb − − ι x b − (cid:33) if n = b − id x b ⊕ Cx b − if n ≥ b , ( r ( f , H )) n : = ( f n , H n ) if n ≤ b − ( f b − , ) if n = b − ( Cyl ( f b − , f b , H b ) , ) if n ≥ b . Then the association r : E → E (cid:48) is an exact functor. Moreover the natural equivalence σ : CC ∼ → CC in-duces a natural equivalence d r : C lv r ∼ → r C lv and the pair ( r , d r ) : ( E , lv ) → ( E (cid:48) , lv ) is a relative complicialexact functor. We denote the inclusion functor by i : E (cid:48) (cid:44) → E . Then by 2.2.6 ( ) , there are natural weakequivalences i r → id E and r i → id E (cid:48) with respect to level weak equivalences. Restrictions of s b − and c b induces a pair of complicial exact functors E w st s b − (cid:29) c b (cid:16) F [ a , b − ] , h C (cid:17) w st and there are natural weak equivalencesid E (cid:48) → c b s b − | E (cid:48) w st and id (cid:16) F [ a , b − ] , h C (cid:17) w st → s b − c b . Thus ( E w st , lw ) , ( E (cid:48) w st , lw ) and ( (cid:16) F [ a , b − ] , h C (cid:17) w st , lw ) arecategorical homotopy equivalences each other.We set F : = (cid:16) F [ a , b ] , h C (cid:17) w st and G : = F [ ] h ( F [ b − , b − ] , h C , F , E ) . Then there exists a pair of strictly normalcomplicial exact functors F L (cid:29) M G defined by sending an object x in F to [ x b − [ b − ] → x ] in G and an object [ x → y ] in G to y in F . We have the equality M L = id F and there exists a natural transformation id G → L M .Thus the pair of relative categories ( F , lw ) L (cid:29) M ( G , llw ) are categorical homotopy equivalence.Moreover let D = { D ≤ n , D ≥ m } n , m ∈ Z be a cell structure of ( C , w ) such that it is closed under C -homotopyequivalences. We set E (cid:48)(cid:48) : = E ∩ F D ♥ [ a , b ] , h C , F (cid:48) : = (cid:16) F D ♥ [ a , b ] , h C (cid:17) w st and G (cid:48) : = F (cid:48) [ ] h ( F [ b − , b − ] , h D b − , F (cid:48) , E (cid:48)(cid:48) ) .Then by 3.1.17 and 4.2.5, restriction of L and M induce a pair of strictly normal complicial exact functors F (cid:48) L (cid:48) (cid:29) M (cid:48) G (cid:48) and it induces a categorical homotopy equivalences ( F (cid:48) , lw ) L (cid:48) (cid:29) M (cid:48) ( G (cid:48) , llw ) and ( (cid:104) F (cid:48) (cid:105) null , frob , lw ) L (cid:48) (cid:29) M (cid:48) ( (cid:104) G (cid:48) (cid:105) null , frob , llw ) . Let C = ( C , w ) be a thick normal ordinary relative com-plicial exact category and let D be a cell structure of C . Then we write F D ♥ b , h C for the relative category ( F D ♥ b , h C , w st ) and call it the d´evissage space of C with respect to the cell structure D .We say that C satisfies derived d´evissage condition ( with respect to the cell structure D ) if the relativecomplicial exact functor ( − ) ∞ : F D ♥ b , h C → C induces an equivalence Ho ( F D ♥ b , h C in F b , h C ) ∼ → Ho ( C ) of trian-gulated categories.Moreover let F be a full subcategory of C . Then we write F D ♥ b , h C | F for the relative complicial exactcategory ( F D ♥ b , h (cid:104) F (cid:105) null , frob , w st | F D ♥ b , h (cid:104) F (cid:105) null , frob ) and call it the relative d´evissage space of F in C with respect tothe cell structure D | F . We say that F satisfies derived d´evissage condition with respect to the cell structure D in C if the relative complicial exact functor ( − ) ∞ : F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob → (cid:104) F (cid:105) null , frob induces an equivalenceHo ( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob in F b , h C ) ∼ → Ho ( F in C ) of triangulated categories. Let C be a category with cofibration and let A be a fullsubcategory of C which is closed under isomorphisms and let f : x (cid:26) y be a cofibration and let m be a non-negative integer. We write [ m ] for the linear ordered set [ m ] = { k ∈ Z ; 0 ≤ k ≤ m } with the usual ordering.A ( m , A ) -d´evissage decomposition of f or simply A -d´evissage decomposition of f is a functor z : [ m ] → C such that ( ) z ( ) = x and z ( m ) = y , ( ) for all 0 ≤ i ≤ m − z ( i ≥ i + ) : z ( i ) → z ( i + ) is a cofibration, ( ) z ( i + ) / z ( i ) is in A for all 0 ≤ i ≤ m − ) the compositions z ( m − ≤ n ) z ( m − ≤ m − ) · · · z ( ≤ ) is equal to f . Let C be a category with cofibration and let A a full subcategory of C whichis closed under isomorphisms. (Quillen’s d´evissage condition). We say that A satisfies Quillen’s d´evissage condition ( in C ) if forany object x in C , the canonical morphism 0 → x admits a A -d´evissage decomposition.Recall the definition of Serre radical S √ from 1.1.15 and assume that C is a noetherian abelian categoryand A is a topologizing subcategory (see 1.1.14) of C . Then we can show that A satisfies Quillen’sd´evissage condition in C if and only if C = S √ A . (cf. [Her97, 3.1], [Gar09, 2.2].) (Raptis’ d´evissage condition). (cf. [Rap18, 5.1].) We say that A satisfies Raptis’ d´evissage condition ( in C ) if for any cofibration f : x (cid:26) y in C , there exists A -d´evissage decomposition of f .Relationship between Quillen’s and Raptis’ d´evissage conditions for abelian categories is summed up withthe following lemma in [Rap18, 5.12, 5.13]. Let A be an abelian category and let B be a full subcategory closed under quotients.Namely for an object x in A , if there exists an epimorphism y (cid:16) x with y ∈ Ob B , then x is also in B . Thenthe following conditions are equivalent. ( ) B satisfies Quillen’s d´evissage condition in A . ( ) B satisfies Raptis’ d´evissage condition in A . ( ) Ch b ( B ) satisfies Raptis’ d´evissage condition in Ch b ( A ) . Recall the definition of cell structures from Definition 4.1.2.
Let C = ( C , w C ) be a relative complicial exactcategory and let D be a cell structure of C and let f : x (cid:26) y be an admissible monomorphism in C and let m > n be integers. A ([ m , n ] , D ) -homotopy d´evissage decomposition of f or simply D -homotopy d´evissagedecomposition of f is a pair ( z , a ) consisting of an object z in F D [ m , n ] C and a morphism a : z n = z ∞ → y in w C such that ( ) x = z m and ( ) the composition x = z m i zm → z m + i zm + → z m + i zm + → · · · i zn − → z n = z ∞ a → y (230)is equal to f .Recall the definition of cell structures associated with full subcategories from 4.1.5. Let C = ( C , w C ) be a relative complicial exact categoryand let A be a full subcategory of C . We say that A satisfies the homotopy d´evissage condition (in C ) if forany morphism f : x → y in C , there exists a D A -homotopy d´evissage decomposition of f .Relationship between Raptis’ and homotopy d´evissage conditions is summed up with the following lemma. Let C = ( C , w ) be a relative complicial exact category and let A be a full subcategory closedunder isomorphisms and let f : x → y be a morphism in C . Then ( ) If z is a ( m , A ) -d´evissage decomposition of f , then the pair ( z , id z ) is a ([ , m ] , D A ) -d´evissage decompo-sition of f . ( ) If A satisfies Raptis’ d´evissage condition in C , then A satisfies homotopy d´evissage condition in C .Proof. ( ) For each 0 ≤ i ≤ m −
1, the canonical morphism Cone z ( i ≤ i + ) → z ( i + ) / z ( i ) is in w byCorollary 3.2.16 ( ) . Thus Cone z ( i ≤ i + ) is in (cid:104) A (cid:105) null , w . ( ) It follows from ( ) . 69 .3.8. Definition (Waldhausen’s d´evissage condition). (cf. [Wal85, § C = ( C , w ) be arelative complicial exact category and let D = { D ≤ n , D ≥ m } n , m ∈ Z be a cell structure of C . We say that the pair C = ( C , w ) satisfies Waldhausen’s d´evissage condition ( with respect to a cell structure D ) if the followingtwo conditions hold. • C = (cid:91) m ∈ Z D ≥ m . • For any ( m , D ) -connected morphism f : x → y in C , there exits a ([ m , n ] , D ) -homotopy d´evissage de-composition of f for a suitable integer n ≥ m . ( ) In the second assertion in Waldhausen’s d´evissage conditions, by replacing z with r [ m , n ] z (for definition of r [ m , n ] , see 2.2.17) and a : z ∞ → y with the composition of a with the projection r [ m , n ] z ∞ → z ∞ ,we shall assume that i zj is a Frobenius admissible monomorphism for m ≤ j ≤ n − ( ) If D is a bounded (4.1.9) and ordinary (4.1.10) cell structure, then we can replace the second assertion inWaldhausen’s d´evissage conditions with the following weaker assertion:For any ( m , D ) -connected morphism f : x → y , there exists a factorization x i → z f (cid:48) → y of f such that f (cid:48) is ( m + , D ) -connected and Cone i is in D m + . ( ) The following condition implies assertion in ( ) .For any integer m and any object x in D ≥ m , there exists an object y in D ≥ m + and a morphism g : x → y in C such that Cone g is in D m + . Proof. ( ) Let f : x → y be a morphism in C . By bounded assumption of D , we shall assume that Cone f isin D [ m + , n ] for some pair of integers m + < n . Then by hypothesis, there exists a factorization z m = x i zm → z m + f (cid:48) → y with Cone i zm is in D m + and Cone f (cid:48) is ( m + , D ) -connected. Then by 4.1.7 ( ) , Cone f (cid:48) is in D [ m + , n ] . Now by proceeding by induction, we will finally obtain a factorization ( ) of f such that a is in w by ordinarily of D . ( ) Let f : x → y be a morphism in C with Cone f is in D ≥ m + . Then applying assumption to Cone f , thereexists an object z in D ≥ m + and a morphism g : Cone f → z with Cone g is in D m + . Then there exists acommutative diagram of distinguished triangles in π C below x f (cid:47) (cid:47) i (cid:48) (cid:15) (cid:15) y κ f (cid:47) (cid:47) id y (cid:15) (cid:15) Cone f (cid:47) (cid:47) g (cid:15) (cid:15) T x Ti (cid:48) (cid:15) (cid:15) u f (cid:48) (cid:47) (cid:47) κ i (cid:48) (cid:15) (cid:15) y g κ f (cid:47) (cid:47) (cid:15) (cid:15) z κ g (cid:15) (cid:15) (cid:47) (cid:47) Tu T κ i (cid:48) (cid:15) (cid:15) Cone i (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) Cone g (cid:47) (cid:47) T Cone i (cid:48) . Thus there exists a C -homotopy H : f ⇒ C f (cid:48) i (cid:48) . We shall assume that f = f (cid:48) i (cid:48) and Cone f (cid:48) is C -homotopyequivalent to z by replacing i (cid:48) with ξ , i (cid:48) : x → u ⊕ Cx and f (cid:48) with (cid:0) f (cid:48)− H (cid:1) : u ⊕ Cx → y . Hence f (cid:48) is ( m + , D ) -connected. By 3 × g is C -homotopy equivalent to T Cone i (cid:48) . Therefore Cone i (cid:48) is in D m + by4.1.7 ( ) . Let C = ( C , w ) be a normal ordinary relative complicial exact category and let A bea full subcategory of C and D be a cell structure of C . Then ( ) (cf. [Wal85, 1.7.2].) Waldhausen’s d´evissage condition with respect to D implies derived d´evissagecondition with respect to D . ( ) Homotopy d´evissage condition with respect to A implies derived d´evissage condition with respect to D A .Proof. We only give a proof of assertion ( ) . A proof of assertion ( ) is similar. We will show that ( − ) ∞ : F D b , h C → C satisfies App 2 in 3.4.13. Then by 3.4.13 ( ) ( b ) , it turns out that C satisfies derived70´evissage condition with respect to D . We proceed by induction on dim x (for definition of dim x , see 2.2.7).If dim x = −
1, then applying Waldhausen’s d´evissage condition to the morphism 0 → y , we obtain the result.For m ≥
0, by applying the inductive hypothesis to the composition σ ≤ m − x → x → y , we obtain a leveladmissible monomorphism i (cid:48) : σ ≤ m − x (cid:26) z (cid:48)(cid:48) and a morphism a (cid:48)(cid:48) : z (cid:48)(cid:48) ∞ → y in w which makes diagram belowcommutative. ( σ ≤ m − x ) ∞ (cid:47) (cid:47) (cid:15) (cid:15) i (cid:48) ∞ (cid:15) (cid:15) x ∞ f (cid:15) (cid:15) z (cid:48)(cid:48) ∞ a (cid:48)(cid:48) (cid:47) (cid:47) y Then we will show the following assertions. ( a ) For each integer i , the compositions z (cid:48)(cid:48) i i z (cid:48)(cid:48) i → z (cid:48)(cid:48) i + i z (cid:48)(cid:48) i + → · · · → z (cid:48)(cid:48) ∞ a (cid:48)(cid:48) → y is ( i − , D ) -connected. ( b ) The canonical morphism x m (cid:116) x m − z (cid:48)(cid:48) m − → y is ( m − , D ) -connected. Proof of ( a ) . We proceed by descending induction on i . For n = dim z (cid:48)(cid:48) , the morphism z (cid:48)(cid:48) n = z (cid:48)(cid:48) ∞ a (cid:48)(cid:48) → y is in w , in particular ( n − , D ) -connected. In general, applying 4.1.7 ( ) to the pair of composable morphisms z (cid:48)(cid:48) i i z (cid:48)(cid:48) i → z (cid:48)(cid:48) i + and z (cid:48)(cid:48) i + → y , it turns out that the composition z (cid:48)(cid:48) i → y is ( i − , D ) -connected by inductivehypothesis. Proof of ( b ) . By 4.1.11, Cone ( z (cid:48)(cid:48) m − (cid:26) x m (cid:116) x m − z (cid:48)(cid:48) m − ) is in D m . Now by applying 4.1.7 ( ) to the pair of com-posable monomorphisms z (cid:48)(cid:48) m − (cid:26) x m (cid:116) x m − z (cid:48)(cid:48) m − → y , it turns out that the canonical morphism x m (cid:116) x m − z (cid:48)(cid:48) m − → y is ( m − , D ) -connected.By applying Waldhausen’s d´evissage condition to the morphism x m (cid:116) x m − z (cid:48)(cid:48) m − → y , there exists an ob-ject z (cid:48) in F D b , h , ≥ m − C and a morphism a (cid:48) : z (cid:48) ∞ → y in w such that z (cid:48) m − = x m (cid:116) x m − z (cid:48)(cid:48) m − and the compo-sition z (cid:48) m − i z (cid:48) m − → z (cid:48) m i z (cid:48) m → · · · → z (cid:48) ∞ a (cid:48) → y is equal to the morphism x m (cid:116) x m − z (cid:48)(cid:48) m − → y . We write I (cid:48) for thecomposition of z (cid:48)(cid:48) m − (cid:26) x m (cid:116) x m − z (cid:48)(cid:48) m − (cid:26) z (cid:48) m . By 4.1.11, Cone ( z (cid:48)(cid:48) m − (cid:26) x m (cid:116) x m − z (cid:48)(cid:48) m − ) is in D m . Thenit turns out that Cone I (cid:48) is also in D m by applying 4.1.7 ( ) and ( ) to the pair of composable morphisms z (cid:48)(cid:48) m − (cid:26) x m (cid:116) x m − z (cid:48)(cid:48) m − (cid:26) z (cid:48) m .We define z , I : x (cid:26) z and a : z ∞ → y to be an object in F D b , h C and a level admissible monomorphism anda morphism in w such that the composition x ∞ I (cid:26) z ∞ a → y is equal to f by setting z k : = (cid:40) z (cid:48)(cid:48) k if k ≤ m − z (cid:48) k if k ≥ m , i zk : = i z (cid:48)(cid:48) k if k ≤ m − I (cid:48) if k = m − i z (cid:48) k if k ≥ m . I k : = (cid:40) i (cid:48) k if k ≤ m − x m → x m (cid:116) x m − z (cid:48)(cid:48) m − → z k if k ≥ m and a : = a (cid:48) . Now we complete the proof.
The notion of weight structures on triangulated categories is introduced byBondarko [Bon10] and [Bon12] and in this article we will use homological weight structure as in [Fon18].Let T be a triangulated category. A homological weight structure W of T is an additive full subcategory of T closed under retract in T which satisfies the following properties:We set W ≤ n : = Σ n ( W ) and W ≥ n : = Σ n − ( ⊥ W ) . (Suspension closed). W ≥ ⊂ W ≥ and W ≤− ⊂ W ≤ . (Weight decomposition). For an object x in T , there exists a distinguished triangle b → x → a → Σ b with b ∈ W ≤ and a ∈ W ≥ . 71et C = ( C , w ) be a relative complicial exact category and let W be a homological weight structure onHo ( C ) . Then we define D W ≥ n and D W ≤ m to be full subcategories of Ho ( C ) by settingOb D W ≥ n : = { x ∈ Ob C ; Q C ( x ) is in W ≥ n } , (231)Ob D W ≤ n : = { x ∈ Ob C ; Q C ( x ) is in W ≤ n } . (232)Here Q C : C → Ho ( C ) is the canonical functor. Then it turns out that a family D W : = { D W ≥ n , D W ≤ m } n , m isa cell structure of C by [Bon10, 1.3.3]. We call D W the cell structure of C associated with a homologicalweight structure W on Ho ( C ) . In Lemma 4.3.12 below, we see a relationship between weight decompositioncondition and Waldhausen’s d´evissage condition with respect to a cell structure associated with a homologicalweight structure. Let C = ( C , w ) be a relative complicial exact category and let W be a homological weightstructure of Ho ( C ) and let D W be an associated cell structure of W in C . If W is bounded below, then C satisfies Waldhausen’s d´evissage condition with respect to the cell structure D W .Proof. We shall check condition ( ) in Remark 4.3.9. Let x be an object in D W ≥ m . Then there exists an object y in D W ≥ m + and a morphism g : x → y such that the image of T − Cone g in Ho ( C ) is in W ≤ m . Then Cone g is in D W m + by Lemma 4.1.7 ( ) and ( ) . In this subsection, we give a typical example of a derived quasi-split sequence. We start by recalling thedefinition of derived exact sequences of relative exact categories from [Moc13b].
We say that a sequence of triangulated categories and triangle functors ( T , Σ ) i → ( T (cid:48) , Σ (cid:48) ) p → ( T (cid:48)(cid:48) , Σ (cid:48)(cid:48) ) (233)is exact if pi is the zero functor and i and p induce the equivalences of triangulated categories T ∼ → Ker p and T (cid:48) / Ker p ∼ → T (cid:48)(cid:48) . We say that the sequence ( ) is weakly exact if the functor i is fully faithful andif the composition pi is the zero morphism and if the induced morphism T (cid:48) / T → T (cid:48)(cid:48) is cofinal . The lastcondition means that it is fully faithful and for any object x in T (cid:48)(cid:48) , there exists a pair of objects y in T (cid:48)(cid:48) and z in T (cid:48) such that x ⊕ y is isomorphic to z .We say that a sequence of relative exact categories and relative exact functors E i → E (cid:48) p → E (cid:48)(cid:48) (234)is derived exact (resp. derived weakly exact ) if the induced sequence D b ( E ) D b ( i ) → D b ( E (cid:48) ) D b ( p ) → D b ( E (cid:48)(cid:48) ) (235)of triangulated categories and triangle functors is exact (resp. weakly exact).We can show the following. Let E i (cid:47) (cid:47) a (cid:15) (cid:15) E (cid:48) p (cid:47) (cid:47) b (cid:15) (cid:15) E (cid:48)(cid:48) c (cid:15) (cid:15) F j (cid:47) (cid:47) F (cid:48) q (cid:47) (cid:47) F (cid:48)(cid:48) be a commutative diagram of derived exact sequences of relative exact categories. If a and b ( resp. b and c ) are derived equivalences, then c ( resp. a ) is also a derived equivalence. By 3.4.18, we obtain the following result. 72 .4.3. Lemma.
The functor Ch b : RelEx → RelComp sn preserves derived exact sequences and derivedweakly exact sequences. Namely if the sequence ( ) of relative exact categories and relative exact functorsis derived exact ( resp. derived weakly exact ) , then the sequence of relative complicial exact categories andrelative complicial exact functors Ch b ( E ) Ch b ( i ) → Ch b ( E (cid:48) ) Ch b ( p ) → Ch b ( E (cid:48)(cid:48) ) (236) is also derived exact ( resp. derived weakly exact ) . (cf. [Kel96, § ( R , ρ ) : ( S , Σ ) → ( T , Σ (cid:48) ) and ( L , λ ) : T → S be twotriangle functors between triangulated categories ( S , Σ ) and ( T , Σ (cid:48) ) such that L is left adjoint to R . Let A : id T → RL and B : LR → id S be adjunction morphisms. For any objects x in T and y in S , we write µ ( x , y ) for the bijection Hom S ( Lx , y ) → Hom T ( x , Ry ) , f (cid:55)→ ( R f ) · A x . We say that ( L , λ ) (resp. ( R , ρ ) ) is left (resp. right ) triangle adjoint to ( R , ρ ) (resp. ( L , λ ) ) if for any objects x in T and y in S , the diagram belowis commutative. Hom S ( Lx , y ) Σ (cid:47) (cid:47) µ ( x , y ) (cid:15) (cid:15) Hom S ( Σ Lx , Σ y ) Hom ( λ x , Σ y ) (cid:47) (cid:47) Hom S ( L Σ (cid:48) x , Σ y ) µ ( Σ (cid:48) x , Σ y ) (cid:15) (cid:15) Hom T ( x , Ry ) Σ (cid:48) (cid:47) (cid:47) Hom T ( Σ x , Σ (cid:48) Ry ) Hom ( Σ (cid:48) x , R Σ y ) (cid:47) (cid:47) Hom T ( Σ (cid:48) x , R Σ y ) . µ ( Σ (cid:48) , Σ ) Hom ( λ , Σ ) Σ = Hom ( Σ (cid:48) , ρ − ) Σ (cid:48) µ .Let ( g , ρ (cid:48) ) : T (cid:48) → T be a triangle functor, and let f be a left (resp. right) adjoint of g with Φ : f g → id T and Ψ : id T (cid:48) → g f (resp. Φ : id T → f g and Ψ : g f → id T (cid:48) ) adjunction morphisms and λ (cid:48) : = ( Φ ∗ Σ (cid:48) f ) · ( f ∗ ρ (cid:48)− ∗ f ) · ( f Σ ∗ Ψ ) (resp. λ (cid:48) : = { ( f Σ ∗ Ψ ) · ( f ∗ ρ (cid:48) ∗ f ) · ( Φ ∗ Σ (cid:48) f ) } − ). Then the pair ( f , λ (cid:48) ) is a trianglefunctor and it is a left (resp. right) triangle adjoint of ( g , ρ (cid:48) ) (cf. [Kel96, 8.3]). (cf. [Moc13b, 6.3].) We say that a sequence ( ) is right (resp. left ) quasi-split if both i and p admit right (resp. left) adjoint functors q : T (cid:48) → T and j : T (cid:48)(cid:48) → T (cid:48) with adjunctionmorphisms iq A → id T (cid:48) , id T B → qi , id T (cid:48) C → jp and p j D → id T (cid:48)(cid:48) (resp. id T (cid:48) A → iq , qi B → id T , jp C → id T (cid:48) andid T (cid:48)(cid:48) D → p j ) respectively such that B and D are natural equivalences and if there exists a triangulated naturaltransformation jp E → Σ (cid:48) iq (resp. iq E → Σ (cid:48) jp ) such that a triangle ( A , C , E ) (resp. ( C , A , E ) ) is a distinguishedtriangle in T (cid:48) . We call a system ( j , q , A , B , C , D , E ) (resp. ( j , q , C , D , A , B , E ) ) or shortly ( j , q , E ) a right (resp. left ) splitting of a sequence ( i , p ) .We say that a sequence ( ) of relative exact categories and relative exact functors is derived right (resp. left ) quasi-split if the induced sequence ( ) of bounded derived categories is a right (resp. left) quasi-splitsequence of triangulated categories. If the sequence ( ) of triangulated categories and triangle functors is a right quasi-splitsequence with ( j , q , E ) a right quasi-splitting, then the sequence ( j , q , E ) is a left quasi-split sequence with aleft quasi-splitting ( i , p , E ) .Thus if the sequence ( ) of relative exact categories and relative exact functors is a derived right quasi-split sequence and if for a pair of relative exact functors E (cid:48) q → E and E (cid:48)(cid:48) j → E (cid:48) , induced triangle functors D b ( E (cid:48) ) D b ( q ) → D b ( E ) and D b ( E (cid:48)(cid:48) ) D b ( j ) → D b ( E (cid:48) ) give a part of a right quasi-splitting of the sequence ( ) ,then the sequence E (cid:48)(cid:48) j → E (cid:48) q → E is a derived left quasi-split sequence.For simplicity, we will only mention the cases for derived right quasi-split sequences and similar state-ments also hold for the left case.By 3.4.18, we can show the following. The functor Ch b : RelEx → RelComp sn preserves right quasi-split sequences. Namely if thesequence of relative exact categories and relative exact functors ( ) is a derived right quasi-split, then thesequence ( ) of relative complicial exact categories is also a derived right quasi-split sequence. Moreover ifthere exists q : E (cid:48) → E ( resp. j : E (cid:48)(cid:48) → E (cid:48) ) such that D b ( q ) (resp. D b ( j ) ) gives a part of a right quasi-splittingof the sequence ( ) of triangulated categories, then D b ( Ch b ( q )) ( resp. D b ( Ch b ( j )) ) also gives a part ofright quasi-splitting of the sequence D b ( Ch b ( E )) D b ( Ch b ( i )) → D b ( Ch b ( E (cid:48) )) D b ( Ch b ( p )) → D b ( Ch b ( E (cid:48)(cid:48) )) . .4.8. Lemma. (cf. [Moc13b, 6.6].) ( ) A right quasi-split sequence of triangulated categories and trianglefunctors is exact. ( ) A derived right quasi-split sequence of relative exact categories and relative exact functors is derivedexact.
Let C = ( C , w ) be a normal ordinary relative complicial exact category and let S and T be a pair of full subcategories of C . Assume that S contains the zero objects. We set C (cid:48) : = C [ ] h ( S , T heq ) .We define i : C → C [ ] h , q : C [ ] h → C , j : C → C [ ] h to be a strictly normal exact functors by sending an object x in C to [ x id x → x ] , an object in C [ ] h to and an object y in C to [ → y ] in C [ ] h respectively. For simplicity wewrite ( p , c ) for the complicial exact functor ( Cone , σ Cone ) : C [ ] h → C . We also denote the restriction of i , q to full subcategories of C and C [ ] h etc and the induced functors of i , q on homotopy categories by the sameletters i and q respectively. Moreover we denote the restriction of w and lw to full subcategories of C and C [ ] h by the same letters w and lw . In the convention 4.4.9, the sequence ( (cid:104) S (cid:105) null , frob , w ) i → ( (cid:104) C (cid:48) (cid:105) null , frob , lw ) p → ( (cid:104) T heq (cid:105) null , frob , w ) (237) is a right derived quasi-split sequence and the sequence ( (cid:104) T heq (cid:105) null , frob , w ) j → ( (cid:104) C (cid:48) (cid:105) null , frob , lw ) q → ( (cid:104) S (cid:105) null , frob , w ) (238) is a left derived quasi-split sequence.Proof. We define B : id C [ ] h → jp and C : jp → Tiq to be a pair of complicial natural transformations bysetting for an object x f → y in C [ ] h , B ( x f → y ) : = ( , κ f , − µ f ) and C ( x f → y ) : = ( , ψ f , ) . For simplicity wedenote Ho ( B ) and Ho ( C ) by the same letter B and C respectively. Moreover for an object x f → y in C [ ] h , weset A ( x f → y ) : = ( id x , f , ) : [ x id x → x ] → [ x f → y ] and for a morphism ( a , b , H ) : [ x f → y ] → [ x (cid:48) f (cid:48) → y (cid:48) ] in C [ ] h , weset A ( a , b , H ) : = ( , − H , H · c x ) : ( a , b , H )( id x , f , ) ⇒ C ( id (cid:48) x , f (cid:48) , )( a , a , ) . Then A induces a triangle naturaltransformation iq → id Ho ( C [ ] h ) on homotopy categories and we denote it by the same letter A . We can showthat there exists a functorial distinguished triangle iq A → id Ho ( C [ ] h ) B → jp C → Tiq and the equalities qi = id C , p j = id C , B ∗ j = id j , p ∗ B = id p , A ∗ i = id i and q ∗ A = id q . Thus it turns out that the sequence ( ) is aderived right quasi-split sequence. A proof of the sequence ( ) is similar. In 4.4.10, replacing C [ ] h ( S , T heq ) with C [ ] ( S , T heq ) , similar statement also holds. Thesame proof works fine. Let C = ( C , w ) be a normal ordinary relative complicial exact category and let a < b be apair of integers and let D = { D ≤ n , D ≥ m } n , m ∈ Z be a C -homotopy closed cell structure of C . Then the inclusionfunctor F [ a , b − ] , h C (cid:44) → F [ a , b ] , h C and the functors Cone i ( − ) b , ( − ) b − : F [ a , b ] , h C → C which sends an object x toCone i xb − and x b − respectively induce a right and a left derived flags below respectively by 3.4.5, 4.2.4 and4.2.6. ( (cid:104) F D ♥ [ a , b − ] , h C (cid:105) null , frob , lw ) → ( (cid:104) F D ♥ [ a , b ] , h C (cid:105) null , frob , lw ) → ( (cid:104) D b (cid:105) null , frob , lw ) , (239) ( (cid:104) ( F D ♥ [ a , b − ] , h C (cid:105) null , frob ) w st , lw ) → ( (cid:104) ( F D ♥ [ a , b ] , h C (cid:105) null , frob ) w st , lw ) → ( (cid:104) D b − (cid:105) null , frob , lw ) . (240) In this subsection, we give examples of right flags on the category of d´evissage filtrations with level weakequivalences which enable us to calculate the K -theory of d´evissage spaces in § .5.1. (Derived flag). Let T be a triangulated category and let E be a relative exact category. A right (resp. left ) flag of T is a finite sequence of fully faithful triangle functors { } = T k → T k → · · · k n − → T n = T (241)such that for each 0 ≤ i ≤ n −
1, the canonical sequence T i k i → T i + → T i + / T i of triangulated categoriesis a right (resp. left) quasi-split exact sequence (See 4.4.5) of triangulated categories.A derived right (resp. left ) flag of E is a finite sequence of fully faithful relative exact functors { } = E k → E k → · · · k n − → E n = E (242)such that it induces a right (resp. left) flag of D b ( E ) { } = D b ( E ) D b ( k ) → D b ( E ) D b ( k ) → · · · D b ( k n − ) → D b ( E n ) = D b ( E ) on bounded derived categories.A derived right (resp. left ) retractions of the derived right (resp. left) flag ( ) of E is a family of relativeexact functors R = { u i : E i → E i − } ≤ i ≤ n such that for each 1 ≤ i ≤ n , D b ( u i ) : D b ( E i ) → D b ( E i − ) gives apart of a right (resp. left) quasi-splitting of the sequence D b ( E i − ) D b ( k i ) → D b ( E i ) → D b ( E i ) / D b ( E i − ) .An associated graded relative exact categories of the derived right (resp. left) flag ( ) is a family ofrelative exact categories and relative exact functors G = { G i , t i : E i → G i } ≤ i ≤ n such that for each 1 ≤ i ≤ n ,the sequence E i − k i − → E i t i → G i is a derived right (resp. left) quasi-split sequence.By 4.4.7, we obtain the following lemma. We will only mention the case for right flags and similarstatements for left flags also hold. Let E be a relative exact category and let ( ) be a derived right flag of E . Then ( ) The sequence { } = Ch b ( E ) Ch b ( k ) → Ch b ( E ) Ch b ( k ) → · · · Ch b ( k n − ) → Ch b ( E n ) = Ch b ( E ) (243) is a derived right flag of Ch b ( E ) . ( ) If a family R = { u i : E i → E i − } ≤ i ≤ n is a derived right retractions of the derived right flag ( ) of E ,then a family Ch b ( R ) = { Ch b ( u i ) : Ch b ( E i ) → Ch b ( E i − ) } ≤ i ≤ n is a derived right retractions of the derivedright flag ( ) of Ch b ( E ) . ( ) If a family G = { F i , t i : E i → F i } ≤ i ≤ n is an associated graded relative exact categories of a derived rightflag ( ) , then the family Ch b ( G ) = { Ch b ( F i ) , Ch b ( t i ) : Ch b ( E i ) → Ch b ( F i ) } ≤ i ≤ n is an associated gradedrelative exact categories of a derived right flag ( ) of Ch b ( E ) . Let C = ( C , w ) be a normal ordinary relative complicial exact category and let D = { D ≤ n , D ≥ m } n , m ∈ Z be a cell structure of C and let a < b be a pair of integers. For each integer 1 ≤ k ≤ b − a +
1, we set E k : = (cid:104) F D ♥ [ a , a + k − ] , h C (cid:105) null , frob , G k : = (cid:104) D a + k − (cid:105) null , frob , E k = ( E k , lw | E k ) , G k = ( G k , w | G k ) and we define k i : E i → E i + , u i : E i → E i − and t i : E i → G i to be restrictions of relative complicial ex-act functors F [ a , a + i − ] , h C (cid:44) → F [ a , a + i ] , h C , σ ≤ a + i − : F [ a , a + i ] , h C → F [ a , a + i − ] C and F [ a , a + i − ] , h C → C wherethe last functor is defined by sending an object x in F [ a , a + i − ] , h C to Cone i xa + i − in C . Then the sequence { } = E → E k → · · · k b − a → E b − a + is a right derived flag of E b − a + and the families { u i : E i → E i − } ≤ i ≤ b − a + and { G i , t i : E i → G i } ≤ i ≤ b − a + are a right derived retractions and an associated graded relative exact cate-gories of this flag by 4.4.12. By 4.4.11, there exists a similar right derived flag on ( (cid:104) F D ♥ [ a , b ] , h C (cid:105) null , frob , lw ) .Next for each integer 1 ≤ k ≤ b − a +
1, we set cE (cid:48) k : = (cid:104) (cid:16) F D ♥ [ a , a + k − ] , h C (cid:17) w st (cid:105) null , frob , E (cid:48) k : = ( E (cid:48) k , lw | E (cid:48) k ) and we define k (cid:48) i : E (cid:48) i → E (cid:48) i + , u (cid:48) i : E (cid:48) i → E (cid:48) i + and t (cid:48) i : E (cid:48) i → G i − to be restrictions of complicial exactfunctors F [ a , a + i − ] , h C (cid:44) → F [ a , a + i ] , h C , s a + i − : F [ a , a + i ] , h C → F [ a , a + i − ] , h C and ( − ) a + i − : F [ a , a + i − ] , h C → C for 2 ≤ i ≤ b − a +
1. where by convention we set G : = { } . Then the sequence { } = E (cid:48) k (cid:48) → E (cid:48) k (cid:48) → · · · k (cid:48) b − a → E (cid:48) b − a + is a left derived flag of E (cid:48) b − a + and the families { u (cid:48) i : E (cid:48) i → E (cid:48) i − } ≤ i ≤ b − a + and { G i − , t (cid:48) i : E (cid:48) i → i − } ≤ i ≤ b − a + are a left derived retractions and an associated graded relative exact categories of this flagrespectively by 4.4.12 again.Since by 3.4.5, for each integer k , the canonical morphism ( F D ♥ [ a , a + k ] , h C ) w st → (cid:16) F D ♥ [ a , a + k ] , h C (cid:17) w st is a derivedequivalence, there exists a similar left derived flag on ( (cid:104) ( F D ♥ [ a , b ] , h C ) w st (cid:105) null , frob , lw ) and ( (cid:104) ( F D ♥ [ a , b ] C ) w st (cid:105) null , frob , lw ) by 4.4.11. K -theory A goal of this section is a d´evissage theorem of non-connective K -theory 5.4.1. In the first subsection 5.1 andthe second subsection 5.2, we will recall the definition and fundamental properties of non-connective K -theoryof relative exact categories. In subsection 5.3, we will study a homotopy version of additivity theorem 5.3.2which enable us to compute K -theory of d´evissage filtrations with level equivalences 5.3.8 by utilizing thecanonical flag structures introduced in 4.5.3. In the final subsection 5.4, we will discuss a d´evissage theoremof non-connective K -theory and as its special cases, we will reprove Waldhausen’s cell filtration theorem 5.4.2,theorem of heart 5.4.4, Raptis’ d´evissage theorem 5.4.5 and Quillen’s d´evissage theorem 5.4.6. In this section,we assume that underlying categories of relative categories are essentially small. K -theory K -theory). In the series of papers [Sch04], [Sch06] and [Sch11], MarcoSchlichting define and establish non-connective K -theory for exact categories, Frobenius pairs and complicialexact categories with weak equivalences respectively. We write K S , exact ( E ) for the non-connective K -theoryof an exact category E and denote the non-connective K -theory of a relative complicial exact category C =( C , w ) by K S , comp ( C ) or K S , comp ( C ; w ) . K -theory). For a relative exact category E = ( E , w ) , we set K ( E )(= K ( E ; w )) : = K S , comp ( Ch b ( E )) . (244)Moreover let C = ( C , w ) be a relative complicial category and let F be a full subcategory of C and let D = { D n } n ∈ Z be a T -system (for definition see 4.1.1) of C . Then we set K ( F in C ) : = K S , comp ( (cid:104) F (cid:105) null , frob ; w | (cid:104) F (cid:105) null , frob ) , and (245) K ( D in C ) : = hocolim T K ( D n in C ) . (246)We call K ( F in C ) and K ( D in C ) the relative K -theory of F in C and the relative K -theory of D in C .Recall the definition of morphisms of T -systems from 4.1.1. Let D and D (cid:48) be a pair of T -systems ofrelative complicial exact categories C = ( C , w ) and C (cid:48) = ( C (cid:48) , w (cid:48) ) respectively. Assume that D (cid:48) is closedunder isomorphisms and let ( f , d ) : D → D (cid:48) be a morphism of T -system. Then ( f , d ) induces a morphism ofspectra K ( D in C ) → K ( D (cid:48) in C (cid:48) ) . Let C = ( C , w ) be a relative complicial exact category and let D = { D n } n ∈ Z be a T -system of C . If we regard C as a T -system { C n } n ∈ Z by setting C n : = C for all integer n ,then the family of inclusion functors D n (cid:44) → C for all integers can be regarded as a morphism of T -systemsand it induces a morphism of spectra E D : K ( D in C ) → hocolim T K ( C ) (247)which we call the Euler characteristic morphism ( associated with the T -system D ). In 5.3.4, we will show theequality ( ) and by this equality it turns out that hocolim T K ( C ) is homotopy equivalent to K ( C ) . We denotecomposition of E D with a non-canonical homotopy equivalence hocolim T K ( C ) → K ( C ) by the same letters E D and call it Euler characteristic morphism too. For a full subcategory F of C , we similarly construct amorphism of spectra E D , F : K ( D | F in C | F ) → hocolim T K ( C | F ) . (248)76 .1.4. Proposition (Comparison theorem of non-connective K -theory). (cf. [Moc13b, 0.3]) Let E = ( E , w ) be a relative exact category. Then ( ) If w is the class of all isomorphisms in E , then there is a natural homotopy equivalence of spectra K ( E ) ∼ → K S , exact ( E ) . This equivalence is functorial in the sense that it gives a natural equivalence between the functors from thecategory of (essentially small) exact categories and exact functors to the stable category of spectra. ( ) If E is a relative complicial exact category, then there is a natural homotopy equivalence of spectra K ( E ) ∼ → K S , comp ( E ) . This equivalence is functorial in the sense that it gives a natural equivalence between the functors from thecategory of (essentially small) relative complicial exact categories and relative complicial exact functors tothe stable category of spectra. K -theory Recall the definition of categorical homotopic of relative (exact) functors from 3.2.1. (cf. [Moc13b, 3.25].)
Let f , f (cid:48) : E → E (cid:48) be a pair ofrelative exact functors between relative exact categories E and E (cid:48) . If f and f (cid:48) are categorical homotopic in RelEx , then they induce same maps K ( f ) = K ( f (cid:48) ) : K ( E ) → K ( E (cid:48) ) on K -theory. (cf. [Moc13b, 0.2].) Let ( ) be a derived weakly exact sequence ( ) of relative exact categories and relative functors. Then the induced sequence of spectra K ( E ) K ( i ) → K ( E (cid:48) ) K ( p ) → K ( E (cid:48)(cid:48) ) is a fibration sequence of spectra. In the statements of [Moc13b, 0.2, 3.25], there is an assumption that we assume that w isconsistent. In this paper we change the definition of the class of quasi-weak equivalences as in Example 3.2.14and our proof in Ibid with the new definition of the class of quasi-weak equivalences still works fine in thesetting of Theorem 5.2.1 and Theorem 5.2.2. Let f : E → F be a relative exact functor between relative exactcategories. If f induces a cofinal functor on their derived categories D b ( E ) → D b ( F ) , then f induces ahomotopy equivalence of spectra K ( E ) → K ( F ) on K -theory. Let C = ( C , w ) be a relative complicial exactcategory and let A be a strict exact subcategory of C . Then the inclusion functor j F : F → Ch b F inducesa homotopy equivalence of spectra K ( F ; w | F ) → K ( Ch b F in Ch b C ) on K -theory.Proof. It follows from Proposition 3.4.18 ( ) and Corollary 5.2.4. w -envelope invariance). Let C = ( C , w ) be a relative complicial exact category and let F be a full subcategory of C . Assume that F is compatible with w (see 3.4.10). Then the inclusion functor (cid:104) F (cid:105) null , frob (cid:44) → (cid:104) F (cid:105) null , w induces a homotopy equivalence of spectra K ( F in C ) → K ( (cid:104) F (cid:105) null , w in C ) on K -theory. Proof.
It follows from the equivalence ( ) and Corollary 5.2.4. w -envelope invariance II). Let C = ( C , w ) be a relative complicial exact category and let A be an additive full subcategory of C . Assume either condition ( i ) or condition ( ii ) below. ( i ) A contains all C-contractible objects of C , ( ii ) A is a strict exact subcategory of C .Then the inclusion functor (cid:104) Ch b A (cid:105) null , frob , Ch b C (cid:44) → (cid:104) Ch b A (cid:105) null , qw , Ch b C induces a homotopy equivalence ofspectra K ( Ch b A in Ch b C ) → K ( (cid:104) Ch b A (cid:105) null , qw , Ch b C in Ch b C ) on K -theory. roof. It follows from Corollary 3.4.20 and Corollary 5.2.6.
Let C = ( C , w ) be a thick relative complicial exact category and let P be a full additive subcategory of C . Assume that either condition ( i ) or condition ( ii ) below. ( i ) P is a prenull class or ( ii ) C is strictly ordinary and P contains all C-contractible objects and P is closed under operations T ± .Then the relative complicial exact functor Tot : Ch b ( C | P ) → ( (cid:104) P (cid:105) null , frob , w | (cid:104) P (cid:105) null , frob ) induces a homotopyequivalence of spectra K S , comp ( Ch b ( C | P )) ∼ → K ( P in C ) on K -theory.Proof. It follows from 3.4.18 and 5.2.4.
Recall the definition of derived quasi-split sequences of relative exact categories from 4.4.5. (cf. [Moc13b, 7.10].)
In the sequence ( ) of relative exact cate-gories, if there exists a relative exact functor q : E (cid:48) → E such that D b ( q ) gives a part of right splitting ofa sequence ( ) of triangulated categories, then the projection functor E (cid:48) (cid:18) qp (cid:19) → E × E (cid:48) induces a homotopyequivalence of spectra K ( E (cid:48) ) → K ( E ) ∨ K ( E (cid:48)(cid:48) ) on K -theory. By Proposition 4.4.10 and Theorem 5.3.1, we obtain the following result.
Corollary 5.3.2. Corollary (Homotopy additivity theorem).
In the convention 4.4.9, the complicial exactfunctor (cid:16) qp (cid:17) : (cid:104) C (cid:48) (cid:105) null , frob → (cid:104) S (cid:105) null , frob × (cid:104) T heq (cid:105) null , frob induces a homotopy equivalence of spectra K ( C (cid:48) in C [ ] h ) → K ( S in C ) × K ( T heq in C ) on K -theory. If we replace C [ ] h with C [ ] , then we also have similar result. Let ( f , c ) , ( g , d ) : C → C (cid:48) be a pair ofcomplicial exact functors between complicial exact categories C and C (cid:48) and let θ : ( f , c ) → ( g , d ) be a com-plicial natural transformation. Then we define Cone θ : C → C (cid:48) to be an exact functor by sending an object x in C to Cone θ x in C (cid:48) and a morphism a : x → y in C to Cone ( f ( a ) , g ( a )) : Cone θ x → Cone θ y in C (cid:48) . Thereexists a natural equivalence c Cone θ : C (cid:48) Cone θ ∼ → Cone θ C which is characterized by the following equalities c Cone θ · ( c (cid:48) ∗ κ ∗ θ ) = ( κ ∗ θ ∗ C ) · d , and (249) c Cone θ · ( c (cid:48) ∗ µ ∗ θ ) = ( µ ∗ C ) · ( C (cid:48) ∗ c ) · ( σ ∗ f ) . (250)Then the pair ( Cone θ , c Cone θ ) is a complicial exact functor. There is a complicial natural transformation κ ∗ θ : ( g , d ) → ( Cone θ , c Cone θ ) .If w and w (cid:48) are classes of complicial weak equivalences of C and C (cid:48) respectively and f and g are relativefunctors, then Cone θ is also relative exact functor. Let ( f , c ) , ( g , d ) : C → C (cid:48) be a relative complicial exact functors between relative compli-cial exact categories C and C (cid:48) and let θ : ( f , c ) → ( g , d ) be a complicial natural transformation. Then for themorphisms of spectra K ( f ) , K ( g ) , K ( Cone θ ) : K ( C ) → K ( C (cid:48) ) , we have the equality K ( g ) = K ( f ) ∨ K ( Cone θ ) . (251) If θ is an admissible monomorphisms in the category of complicial functors, then we have the equality K ( Cone θ ) = K ( Coker θ ) . (252) In particular for the suspension functor T : C → C , we have the equality K ( T ) = − id K ( C ) . (253)78 roof. By virtue of 3.4.18, replacing C and C (cid:48) with Ch b C and Ch b C (cid:48) , without loss of generality, we shallassume that C = ( C , w ) and C (cid:48) = ( C (cid:48) , w (cid:48) ) are thick normal ordinary. We define U , V : C → ( C (cid:48) [ ] h , lw ) and W : ( C (cid:48) [ ] h , lw (cid:48) ) → ( C (cid:48) , w (cid:48) ) and X : ( C (cid:48) [ ] h , lw ) → ( C (cid:48) , w (cid:48) ) × ( C (cid:48) , w (cid:48) ) to be relative complicial exact functors bysending an object x in C to [ f ( x ) θ x → g ( x )] and [ f ( x ) (cid:16) id f ( x ) (cid:17) → f ( x ) ⊕ Cone θ ( x )] in C (cid:48) [ ] h and an object [ x u → y ] in C (cid:48) to y in C (cid:48) and ( x , Cone u ) in C (cid:48) × C (cid:48) respectively. We have the equalities K ( g ) = K ( W ) K ( U ) and K ( f ) ∨ K ( Cone θ ) = K ( W ) K ( V ) and K ( X ) K ( U ) = (cid:18) K ( f ) K ( Cone θ ) (cid:19) = I (cid:18) K ( f ) K ( C ⊕ Cone θ ) (cid:19) = K ( X ) K ( V ) where theequality I follows from categorical homotopy equivalence 5.2.1. By homotopy additivity theorem 5.3.2, K ( X ) is a homotopy equivalence of spectra and thus we obtain the equality ( ) . If θ is an admissiblemonomorphism in the category of relative complicial exact functors, then Coker θ and Cone θ is a categoricalhomotopic. Thus the equality ( ) follows from 5.2.1.Finally the equality ( ) follows form the equality T = Cone ( id C → ) and the equality ( ) .Recall the functors j , σ ≥ k , ( − )[ k ] , c k from 2.2.7, 2.2.13, 2.2.14 and 2.2.15. Let C = ( C , w ) be a normal ordinary complicial relative exact category and let n be aninteger. Then we have the equality K ( c n + ) = id K ( F b , h C ) ∨ K ( σ ≥ n + T ) . (254) In particular we also have the equality. K ( c n + ( j ( − )[ n ])) = K ( j ( − )[ n ]) ∨ K ( j ( T ( − ))[ n ]) . (255) Proof.
Notice that there is a level Frobenius admissible exact sequence of relative complicial exact functorson F b , h C . id F b , h C (cid:26) c n + (cid:16) σ ≥ n + T . x n (cid:47) (cid:47) (cid:47) (cid:47) i xn (cid:15) (cid:15) x n (cid:47) (cid:47) (cid:47) (cid:47) Ci xn ι xn (cid:15) (cid:15) (cid:15) (cid:15) x n + (cid:47) (cid:47) ι xn + (cid:47) (cid:47) i xn + (cid:15) (cid:15) Cx n + π xn + (cid:47) (cid:47) (cid:47) (cid:47) Ci xn + (cid:15) (cid:15) T x n + Ti xn + (cid:15) (cid:15) ... ... ... . Thus we obtain the equalities ( ) and ( ) by 5.3.4.From 5.3.1, proceeding by induction, we obtain the result for K -theory of derived flags below. We willonly state the case for right derived flags and the case for left derived flags also holds. K -theory of derived flags). Let E be a relative exact category and let ( ) be a rightderived flag of E with a derived right retractions R = { u i : E i → E i − } ≤ i ≤ n and an associated graded rela-tive exact categories G = { G i , t i : E i → G i } ≤ i ≤ n . Then the relative complicial exact functor E tntn − untn − un − un...t u ··· un − un → G n × G n − × · · · × G induces a homotopy equivalence of spectra K ( E ) ∼ → (cid:87) ni = K ( G i ) . Recall the functors j , ( − )[ k ] , c k from 2.2.7, 2.2.14 and 2.2.15. Let ( C , w ) be a thick normal ordinary relative complicial exact category and let a < b bea pair of integers. For a positive integer m , we write C m for C × · · · × C (cid:124) (cid:123)(cid:122) (cid:125) m the m -times products of C . We de-fine A [ a , b ] : C b − a + → F [ a , b ] , h C and B [ a , b ] : C b − a → ( F [ a , b ] , h C ) w st C to be complicial exact functors by setting A [ a , b ] : = j ( − )[ a ] ⊕ j ( − )[ a + ] ⊕ · · · ⊕ j ( − )[ b ] and B [ a , b ] : = c a + j ( − )[ a ] ⊕ c a + j ( − )[ a + ] ⊕ · · · ⊕ c b j ( − )[ b − ] C [ a , b ] : F [ a , b ] , h C → C b − a + and D [ a , b ] : ( F [ a , b ] , h C ) w st C → C b − a to be complicial exact functorsby sending an object x in F [ a , b ] , h C to ( x a , Cone i xa , Cone i xa + , · · · , Cone i xb − ) in C b − a + and an object x in ( F [ a , b ] , h C ) w st to ( x a , x a + , · · · , x b − ) in C b − a and we also define E [ a , b ] = ( e i j ) : C b − a → C b − a + to be a com-plicial exact functor by setting e i j = id C if i = jT if i = j +
10 otherwise . We also write the same letters A [ a , b ] , B [ a , b ] , C [ a , b ] , D [ a , b ] and E [ a , b ] for the restrictions of A [ a , b ] , B [ a , b ] , C [ a , b ] , D [ a , b ] and E [ a , b ] to subcategories of C b − a + , C b − a , F [ a , b ] , h C , ( F [ a , b ] , h C ) w st and C b − a . Since D is a cell structure, these functors are well-defined. Let C = ( C , w ) be a thick normal ordinary relative complicial exact category and let a < bbe a pair of integers and let D be a C-homotopy invariance closed cell structure of C . Then C [ a , b ] and D [ a , b ] induce the homotopy equivalence of spectra K ( F D ♥ b C in ( F b C , lw )) → ∏ k ∈ [ a , b ] K ( D k in C ) , K ( F D ♥ b , h C in ( F b , h C , lw )) → ∏ k ∈ [ a , b ] K ( D k in C ) (256) K (( F D ♥ b C ) w st in ( F b C , lw )) → ∏ k ∈ [ a , b − ] K ( D k in C ) , and K (( F D ♥ b , h C ) w st in ( F b , h C , lw )) → ∏ k ∈ [ a , b − ] K ( D k in C ) (257) and A [ a , b ] and B [ a , b ] induce the inverse morphisms of K ( C [ a , b ] ) and K ( D [ a , b ] ) respectively. Moreover thefollowing diagram is commutative ∏ k ∈ [ a , b − ] K ( D k in C ) K ( B [ a , b ] ) (cid:47) (cid:47) K ( E [ a , b ] ) (cid:15) (cid:15) K (( F D ♥ b , h C ) w st in ( F b , h C , lw )) (cid:15) (cid:15) ∏ k ∈ [ a , b ] K ( D k in C ) K ( A [ a , b ] ) (cid:47) (cid:47) K ( F D ♥ b , h C in ( F b , h C , lw )) . (258) Proof.
The first assertion follows from 3.4.5, 4.5.3, 5.2.4 and 5.3.6. We have an equality C [ a , b ] A [ a , b ] = id C b − a + and there is a natural weak equivalence D [ a , b ] B [ a , b ] → id C b − a . Thus K ( A [ a , b ] ) and K ( B [ a , b ] ) are the inversemorphisms of K ( C [ a , b ] ) and K ( D [ a , b ] ) respectively. Commutativity of the diagram ( ) follows form theequality ( ) . Let ( C , w ) be a thick normal ordinary relative complicial exact category and let D be aC-homotopy equivalent cell structure of ( C , w ) . Then the inclusion functors induces homotopy equivalencesof spectra K ( F D ♥ b C in ( F b C , lw )) → K ( F D ♥ b , h C in ( F b , h C , lw )) , (259) K (( F D ♥ b C ) w st in ( F b C , lw )) → K ( F D ♥ b , h C in ( F b , h C , lw )) and (260) K ( F D ♥ b C in F b C ) → K ( F D ♥ b , h C in F b , h C ) . (261) Proof.
Let a < b be a pair of integers. Then there are the following commutative diagrams of K -theory. K ( F D ♥ b C in ( F b C , lw )) (cid:47) (cid:47) K ( C [ a , b ] ) (cid:39) (cid:39) K ( F D ♥ b , h C in ( F b , h C , lw )) K ( C [ a , b ] ) (cid:119) (cid:119) ∏ k ∈ [ a , b ] K ( D k in C ) , K (( F D ♥ b C ) w st in ( F b C , lw )) (cid:47) (cid:47) K ( D [ a , b ] ) (cid:40) (cid:40) K (( F D ♥ b , h C ) w st in ( F b , h C , lw )) K ( D [ a , b ] ) (cid:118) (cid:118) ∏ k ∈ [ a , b − ] K ( D k in C ) .
80y 5.3.8, horizontal morphisms in the diagrams are homotopy equivalence of spectra. By taking colimits, weobtain the homotopy equivalences ( ) and ( ) .The last homotopy equivalence of spectra ( ) follows from the diagram of fibration sequences below. K (( F D ♥ b C ) w st in ( F b C , lw )) (cid:47) (cid:47) (cid:15) (cid:15) K ( F D ♥ b C in ( F b C , lw )) (cid:47) (cid:47) (cid:15) (cid:15) K ( F D ♥ b C in F b C ) (cid:15) (cid:15) K (( F D ♥ b , h C ) w st in ( F b , h C , lw )) (cid:47) (cid:47) K ( F D ♥ b , h C in ( F b , h C , lw )) (cid:47) (cid:47) K ( F D ♥ b , h C in F b , h C ) . Let C = ( C , w ) be a normal ordinary relative complicial exact category and let D be aC-homotopy closed cell structure of C . Then the relative complicial exact functor T ( − )[ ] : ( F D ♥ b , h C , w st ) → ( F D ♥ b , h C , w st ) which sends an object x to an object T ( x )[ ] induces − id K ( F D ♥ b , h C ) on K -theory.Proof. We will show T ( − )[ ] : ( F D ♥ [ a , b ] C , w st ) → ( F D ♥ [ a , b ] C , w st ) induces − id on K -theory for any pair of inte-gers a < b . Then by taking direct limit as a goes to − ∞ and b goes to + ∞ and by 5.3.9, we will obtain the result.For m > b , since there are stable weak equivalences σ ≥ n ( − )[ ] → id [ ] ← id and σ ≥ n C → σ ≥ n ( − )[ ] (cid:26) σ ≥ n C ( − )[ ] (cid:16) σ ≥ n T ( − )[ ] ,we have equalities by 5.2.1 and 5.3.4 K ( T [ ]) = K ( σ ≥ n T [ ]) = − K ( σ ≥ n ( − )[ ]) = − id . K -theory of d´evissage spaces). Let C = ( C , w ) be a normalordinary relative complicial exact category and let D = { D ≥ n , D ≥ m } n , m ∈ Z be a cell structure of C . Then foreach integer n , the commutative diagram below D n j ( − )[ n ] (cid:47) (cid:47) T (cid:15) (cid:15) F D ♥ b , h C T ( − )[ ] (cid:15) (cid:15) D n + j ( − )[ n + ] (cid:47) (cid:47) F D ♥ b , h C induces a morphism E D : K ( D in C ) → hocolim T ( − )[ ] K ( F D ♥ b , h C ) (262)which we call the Euler characteristic morphism ( in K -theory of d´evissage space associated with D ♥ ). As avariant, for a full subcategory F in C , we also define the relative Euler characteristic morphismE D , F : K (( D | F ) in C | F ) → hocolim T ( − )[ ] K ( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob in F D ♥ b , h C ) . (263)For simplicity, we set E : = K ( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob in F D ♥ b , h C ) . Then by 5.3.10, hocolim T ( − )[ ] E is homotopy equiv-alent to E . We denote the composition of E D , F with a non-canonical homotopy equivalence between thesetwo spectra by the same letters E D , F and call it relative Euler characteristic morphism too.For each integer i , we set C i : = C and we regard T : C → C as the functor C i → C i + for each integer i . Then it induces a relative complicial exact functor T : C ∨ Z → C ∨ Z where we denote the pair (cid:32) (cid:95) i ∈ Z C i , lw (cid:33) by C ∨ Z . For each integer n , we write D [ − ∞ , n ] for the full subcategory of (cid:95) i ∈ Z D i consisting of those ob-jects ( x i ) i ∈ Z such that x i = i > n . There is a relative complicial exact functor T : D [ − ∞ , n ] → D [ − ∞ , n + ] which sends an object ( x i ) i ∈ Z to ( T x i ) i ∈ Z where we regard T x i as an object D i + for each integer i . Thus81e can regard the family { D [ − ∞ , n ] } n ∈ Z as a T -system of C ∨ Z . Then we write K ( D [ − ∞ , ∞ ] | F in C ∨ Z ) for K ( { D [ − ∞ , n ] } n ∈ Z in C ∨ Z ) and for each integer n , the functors A [ a , b ] and B [ a , b ] in 5.3.7 induces the func-tors A [ − ∞ , n ] : D [ − ∞ , n ] | F → (cid:18) F D | F ♥ b , h , ≤ n (cid:104) F (cid:105) null , frob (cid:19) w st and B [ − ∞ , n ] : D [ − ∞ , n ] | F → F D | F ♥ b , h , ≤ n (cid:104) F (cid:105) null , frob . We define ∇ [ − ∞ , n ] : D [ − ∞ , n ] | F → D n | F to be a functor by sending an object ( x i ) i ≤ n in D [ − ∞ , n ] | F . Then they induce atriple of morphisms of spectra K ( A [ − ∞ , ∞ ] ) : K ( D [ − ∞ , ∞ ] | F in C ∨ Z ) → hocolim T ( − )[ ] K (( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob ) w st in ( F D ♥ b , h C , lw )) , (264) K ( B [ − ∞ , ∞ ] ) : K ( D [ − ∞ , ∞ ] | F in C ∨ Z ) → hocolim T ( − )[ ] K ( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob in ( F D ♥ b , h C , lw )) and (265) K ( ∇ [ − ∞ , ∞ ] ) : K ( D [ − ∞ , ∞ ] | F in C ∨ Z ) → K ( D ♥ | F in C | F ) (266)respectively.We can show the following lemma. In the convention 5.3.11, the following diagrams of spectra are commutative. K ( D | F in C | F ) E D , F (cid:47) (cid:47) E D , F (cid:41) (cid:41) hocolim T ( − )[ ] K ( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob in F D ♥ b , h C ) ( − ) ∞ (cid:15) (cid:15) hocolim T K ( C | F ) , (267) K ( D [ − ∞ , ∞ ] | F in C ∨ Z ) K ( B [ − ∞ , n ] ) (cid:47) (cid:47) K ( ∇ [ − ∞ , n ] ) (cid:15) (cid:15) hocolim T ( − )[ ] K ( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob in ( F D ♥ b , h C , lw )) (cid:15) (cid:15) K ( D | F in C | F ) E D , F (cid:47) (cid:47) hocolim T ( − )[ ] K ( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob in F D ♥ b , h C ) . (268) The goal of this subsection is to give d´evissage theorems for relative complicial exact categories. The spe-cific feature in our proof is ‘motivic’ in the sense that the properties of K -theory we will utilize to provethe theorems are only localization (additivity and derived invariance), categorical homotopy invariance andcocontinuity. Namely our proof works to any other localizing theory on relative complicial exact categories(for the precise statement, see Corollary 5.4.8). Recall the definition of the Euler characteristic morphismsfrom 5.3.11. K -theory of D´evissage spaces). Let C = ( C , w ) be a normal ordinary relative complicialexact category and let D be a cell structure of C and let F be a full subcategory of C . Then the Euler char-acteristic morphism E D , F : K ( D ♥ | F in C | F ) → hocolim T ( − )[ ] K ( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob in F D ♥ b , h C ) is a homotopyequivalence of spectra. In particular if F satisfies relative derived d´evissage condition with respect to thecell structure D in C , then E D , F : K ( D | F in C | F ) → K ( F in C ) is a homotopy equivalence of spectra. roof. By commutative diagrams ( ) and ( ) , there exists a commutative diagram of fibration sequencesof spectra K ( D [ − ∞ , ∞ ] | F in C ∨ Z ) K ( B [ − ∞ , ∞ ] ) (cid:47) (cid:47) K ( E [ − ∞ , ∞ ] ) (cid:15) (cid:15) hocolim T ( − )[ ] K (( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob ) w st in ( F D ♥ b , h C , lw )) (cid:15) (cid:15) K ( D [ − ∞ , ∞ ] | F in C ∨ Z ) K ( A [ − ∞ , ∞ ] ) (cid:47) (cid:47) K ( ∇ [ − ∞ , ∞ ] ) (cid:15) (cid:15) hocolim T ( − )[ ] K (( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob ) in ( F D ♥ b , h C , lw )) (cid:15) (cid:15) K ( D | F in C | F ) E D , F (cid:47) (cid:47) hocolim T ( − )[ ] K (( F ( D | F ) ♥ b , h (cid:104) F (cid:105) null , frob ) in F D ♥ b , h C ) . By Corollary 5.3.8, K ( A [ − ∞ , ∞ ] ) and K ( B [ − ∞ , ∞ ] ) are homotopy equivalence of spectra. Thus we obtain theresult.By Proposition 4.3.10 and Theorem 5.4.1, we obtain the following Corollary 5.4.2 and Corollary 5.4.3. K -theory). (cf. [Wal85, 1.7.1].) Let C = ( C , w ) be a thick normal ordinary relative complicial exact category and let D be a cell structureof C . Assume that D satisfies Waldhausen’s d´evissage condition. Then the Euler characteristic morphismE D : K ( D ♥ in C ) → K ( C ) is a homotopy equivalence of spectra. Let C = ( C , w ) be a thick normal ordinary relativecomplicial exact category and let A be a full subcategory of C . Assume that A satisfies homotopy d´evissagecondition in C . Then the Euler characteristic morphism E D : K ( (cid:104) A (cid:105) null , w in C ) → K ( C ) is a homotopyequivalence of spectra. By Proposition 4.3.10, Lemma 4.3.12 and Corollary 5.4.2, we obtain the following.
Let C = ( C , w ) be a normal ordinary relative complicial exact cate-gory and let W be a bounded below homological weight structure on Ho ( C ) . Then the Euler characteristicmorphism E D W : K (( D W ) ♥ in C ) → K ( C ) is a homotopy equivalence of spectra. By Lemma 4.3.7 and Corollary 5.4.3, we obtain the following. K -theory). Let C = ( C , w ) be a thicknormal ordinary relative complicial exact category and let A be a full subcategory of C . Assume that A sat-isfies Raptis’ d´evissage condition in C . Then the inclusion functor A (cid:44) → C induces a homotopy equivalenceof spectra K ( (cid:104) A (cid:105) null , w in C ) → K ( C ) on K -theory. K -theory). Let A be an abelian cate-gory and let B be a topologizing subcategory of A . Assume that B satisfies Quillen’s d´evissage conditionin A . Then the inclusion functor B (cid:44) → A induces a homotopy equivalence of spectra K ( B ) → K ( A ) on K -theory.Proof. By Lemma 4.3.4, Ch b ( B ) satisfies Raptis’ d´evissage condition in Ch b A . Thus by Corollary 5.4.5,the inclusion functor Ch b B (cid:44) → Ch b A induces a homotopy equivalence of spectra K ( (cid:104) Ch b B (cid:105) null , qis in ( Ch b A , qis )) → K ( Ch b A ; qis ) on K -theory. Thus by Corollary 5.2.5 and Corollary 5.2.7, we obtain the result. We define the full subcategory of
RelComp [ ] consisting of thoseobjects x such that the sequence x ( ) → x ( ) → x ( ) are derived weakly exact sequences (4.4.1) of relativecomplicial exact categories and relative complicial exact functors by EX ( RelComp ) . The functor [ ] → [ ] ,0 (cid:55)→ k induces functor q k : EX ( RelComp ) → RelComp for 0 ≤ k ≤ ( T , Σ ) be a triangulated category. A localization theory on RelComp to T is a pair ( L , ∂ ) consistingof functor L : EX ( RelComp ) → T and a natural transformation ∂ : L q → Σ L q such that they satisfiesthe following conditions: (Categorical homotopy invariance). Let f , f (cid:48) : C → C (cid:48) be a pair of relative complicial exact func-tors between relative complicial exact categories C and C (cid:48) . If f and f (cid:48) are categorical homotopic in RelComp , then they induce same maps L ( f ) = L ( f (cid:48) ) : L ( C ) → L ( C (cid:48) ) . (Localization). Let ( ) be a derived weakly exact sequence (4.4.1) of relative complicial exactcategories and relative complicial exact functors. Then the induced sequence L ( E ) L ( i ) → L ( E (cid:48) ) L ( p ) → L ( E (cid:48)(cid:48) ) is a distinguished triangle in T . (Cocontinuity). L sends filtered colimits to homotopy colimits in T .As alluded at the outset of this subsection, we obtain the following result. For a localization theory ( L , ∂ ) from RelComp to ( T , Σ ) , similar statements of 5.2.4, 5.2.5, 5.2.6, 5.2.7, 5.2.8, 5.4.1, 5.4.2, 5.4.3, 5.4.4, 5.4.5and 5.4.6 hold. A A d´evissage theorem for modular Waldhausen exact categories
In this section, we recall Quillen’s original d´evissage theorem in [Qui73] with an adequate generalization.Throughout this section, let E be a small exact category. A.0.1. (Modular exact categories, modular Waldhausen categories).
We say that E is modular if it satis-fies the following two conditions: ( Mod1 ) . For any admissible monomorphisms x (cid:26) z , y (cid:26) z in E , there exists a fiber product x × z y and thecanonical morphisms x × z y → x , x × z y → y and x (cid:116) x × z y y → z are admissible monomorphism. ( Mod2 ) . In the following commutative diagram of admissible monomorphisms, x (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) y (cid:15) (cid:15) (cid:15) (cid:15) z (cid:47) (cid:47) (cid:47) (cid:47) w the canonical induced morphism x → z × w y is also an admissible monomorphisms.For a modular exact category E and admissible monomorphisms x (cid:26) z and y (cid:26) z , we set x ∩ z y : = x × z y and x ∪ z y : = x (cid:116) x ∩ z y y . For simplicity we often write x ∩ y and x ∪ y for x ∩ z y and x ∪ z y respectively. Then by[Kel90, p.406 step 1], the commutative diagram of admissible monomorphisms x ∩ y (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) x (cid:15) (cid:15) (cid:15) (cid:15) y (cid:47) (cid:47) (cid:47) (cid:47) x ∪ y . is a biCartesian square. Hence we have an admissible exact sequence x ∩ y (cid:26) x ⊕ y (cid:16) x ∪ y . (269)84oreover for any sequence of admissible monomorphisms x (cid:26) z (cid:26) a (cid:27) u (cid:27) y in E , we can producethe commutative diagram below: x ∩ y (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) x ⊕ y (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) x ∪ y (cid:15) (cid:15) (cid:15) (cid:15) z ∩ w (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) z ⊕ u (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) z ∪ w (cid:15) (cid:15) (cid:15) (cid:15) z ∩ w / x ∩ y (cid:47) (cid:47) z / x ⊕ u / y (cid:47) (cid:47) z ∪ w / x ∪ y . Here by the sequence ( ) , the first and the second horizontal lines are admissible short exact sequences andwe can check that the left, the middle and the right vertical lines are also admissible short exact sequences.Thus by 3 × z ∩ w / x ∩ y (cid:26) z / x ⊕ u / y (cid:16) z ∪ w / x ∪ y (270)is also an admissible short exact sequence.We say that a pair ( E , w ) is a modular Waldhausen exact category if the pair ( E , w ) and the pair of theopposite categories ( E op , w op ) are Waldhausen categories and E is modular and moreover w satisfies thefollowing strong cogluing axiom :In the commutative diagram below x (cid:47) (cid:47) (cid:47) (cid:47) a (cid:15) (cid:15) y b (cid:15) (cid:15) z (cid:111) (cid:111) (cid:111) (cid:111) c (cid:15) (cid:15) x (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) y (cid:48) z (cid:48) (cid:111) (cid:111) (cid:111) (cid:111) if a , b and c are in w , then the induced canonical morphism a × b c : x × y z → x (cid:48) × y (cid:48) z (cid:48) is also in w .In the rest of this section, we assume that the pair ( E , w ) is a modular Waldhausen exact category. A.0.2. Remark.
To explain where the naming of modularity comes from, we need to prepare some notations.Let L be a lattice, namely a partially ordered set closed under sup and inf of any finite sets. For any elements a and b in L , we write a ∪ b and a ∩ b for sup { a , b } and inf { a , b } respectively. Recall that a lattice L is modular if for any triple of elements a ≤ b and c in L , the following equality called modular law holds: a ∪ ( b ∩ c ) = ( a ∪ b ) ∩ c . (271)For any object e , we denote the isomorphism class of admissible subobjects of e by P ( e ) . Then we will showthat P ( e ) is a modular lattice. Indeed, for any triple of admissible subobjects x (cid:26) y and z of e , by applyingthe admissible short exact sequence ( ) to x , y , z and u = y , it turns out that the conditions x ∩ y = z ∩ y and x ∪ y = z ∪ y imply x ∼ → z . Thus by Dedekind’s criterion of modularity (see for example [MY14, 1.5]), P ( e ) is a modular lattice. A.0.3.
For any non-negative integer n , we define E ( n , w ) to be the full subcategory of E [ ] the functor categoryfrom the ordinal [ n ] = { , , · · · , n } to E consisting of those objects x such that for any pair i ≤ j of elementsof [ n ] , the morphism x ( i ≤ j ) from x ( i ) to x ( j ) is in w .We natural make E ( n , w ) into an exact category where a sequence x f → y g → z of morphisms in E ( n , w ) isan admissible exact sequence if x ( i ) f ( i ) → y ( i ) g ( i ) → z ( i ) is an admissible exact sequence in E for any i ∈ [ n ] .The association [ n ] (cid:55)→ E ( n , w ) , gives a simplicial exact category(= a simplicial object in the category ofexact categories and exact functors). A.0.4. Lemma.
Let ( E , w ) be a modular Waldhausen exact category. Then for any non-negative integer n,the exact category E ( n , w ) is a modular exact category. roof. Recall that an admissible monomorphism in E ( n , w ) is a term-wise admissible monomorphisms in E (see A.0.3). Only non-trivial assertion is existence of a fiber product x × z y for any pair of admissiblemonomorphisms x (cid:26) z and y (cid:26) z in E ( n , w ) . Since E is modular it exists term-wisely, and therefore it existsin E [ n ] the functor category from [ n ] to E . By virtue of strong cogluing axiom, it is actually in E ( n , w ) . A.0.5. (Relative d´evissage condition).
Let D be a topologizing subcategory of E (see 1.1.14). Then D naturally becomes a strict exact subcategory of E (see the definition of strict exact subcategories for 1.1.1 andsee [Moc13a, 5.3] for the proof of this fact).For a non-negative integer n , we can check that D ( n , w ) is again a topologizing subcategory of E ( n , w ) byutilizing strong gluing axiom. For simplicity we also write the same letter w for w ∩ Mor D . We say that theinclusion functor ( D , w ) (cid:44) → ( E , w ) satisfies the relative d´evissage condition if for any non-negative integer n ,the inclusion functor D ( n , w ) (cid:44) → E ( n , w ) satisfies the d´evissage condition (see 4.3.3). A.0.6. Theorem (D´evissage).
If the inclusion functor ( D , w ) (cid:44) → ( E , w ) satisfies the relative d´evissage con-dition, then it induces a homotopy equivalence wS · D → wS · E on K-theory.Proof. We fix a non-negative integer n and we apply Quillen’s d´evissage theorem in [Qui73] to the inclusionfunctor D ( n , w ) (cid:44) → E ( n , w ) . In [Qui73], we need to assume that E ( n , w ) is abelian. But this hypothesis isonly using to prove the fact that for any object a in E ( n , w ) , the isomorphism class of subobjects of a becomesa (modular) lattice and the fact that for any sequence of admissible monomorphisms z (cid:26) x (cid:26) a (cid:27) y (cid:27) u in E ( n , w ) , there exists an admissible short exact sequence ( ) in E ( n , w ) . These facts are proven inA.0.1, A.0.4 and A.0.5. Thus the proof in [Qui73] works fine for our situation. Hence it turns out that theinclusion functor Q D ( n , w ) (cid:44) → Q E ( n , w ) is a homotopy equivalence. Here the letter Q means the Quillen’s Q -construction. Then by virtue of Waldhausen’s Q = s theorem in [Wal85, 1.9], the inclusion functor D (cid:44) → E induces a homotopy equivalence wS n D → wS n E for any non-negative integer n . Finally by the realizationlemma in [Seg74, Appendix A] or [Wal78, 5.1], we obtain the result. Acknowledgement.
The author wishes to express his deep gratitude to Masana Harada, Kei Hagihara, ShoSaito, Seidai Yasuda, Toshiro Hiranouchi and Gonc¸alo Tabuada for useful conversations in the early stage ofthis work.
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DEPARTMENT OF MATHEMATICS, CHUO UNIVERSITY, BUNKYO-KU, TOKYO, JAPAN. e-mail: [email protected]@gug.math.chuo-u.ac.jp