Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories
aa r X i v : . [ m a t h . C T ] S e p MUTATION VIA HOVEY TWIN COTORSION PAIRS ANDMODEL STRUCTURES IN EXTRIANGULATED CATEGORIES.
HIROYUKI NAKAOKA AND YANN PALU
Abstract.
We give a simultaneous generalization of exact categories and tri-angulated categories, which is suitable for considering cotorsion pairs, andwhich we call extriangulated categories. Extension-closed, full subcategoriesof triangulated categories are examples of extriangulated categories. We givea bijective correspondence between some pairs of cotorsion pairs which wecall Hovey twin cotorsion pairs, and admissible model structures. As a con-sequence, these model structures relate certain localizations with certain idealquotients, via the homotopy category which can be given a triangulated struc-ture. This gives a natural framework to formulate reduction and mutation ofcotorsion pairs, applicable to both exact categories and triangulated categories.These results can be thought of as arguments towards the view that extrian-gulated categories are a convenient setup for writing down proofs which applyto both exact categories and (extension-closed subcategories of) triangulatedcategories.
Contents
1. Introduction and Preliminaries 22. Extriangulated category 32.1. E -extensions 32.2. Realization of E -extensions 42.3. Definition of extriangulated category 62.4. Terminology in an extriangulated category 83. Fundamental properties 93.1. Associated exact sequence 93.2. Shifted octahedrons 173.3. Relation with triangulated categories 243.4. Projectives and injectives 254. Cotorsion pairs 294.1. Cotorsion pairs 294.2. Associated adjoint functors 304.3. Concentric twin cotorsion pairs 325. Bijective correspondence with model structures 345.1. Hovey twin cotorsion pair 345.2. From admissible model structure to Hovey twin cotorsion pair 355.3. From Hovey twin cotorsion pair to admissible model structure 37 Both authors would like to thank Professor Osamu Iyama and Professor Ivo Dell’Ambrogio forinspiring comments on a previous version of the paper.The first author wishes to thank Professor Frederik Marks and Professor Jorge Vit´oria for theirinterests, and for giving him several opportunities.The first author is supported by JSPS KAKENHI Grant Numbers 25800022.
6. Triangulation of the homotopy category 486.1. Shift functor 486.2. Connecting morphism 516.3. Triangulation 567. Reduction and mutation via localization 637.1. Happel and Iyama-Yoshino’s construction 637.2. Mutable cotorsion pairs 63References 681.
Introduction and Preliminaries
Cotorsion pairs, first introduced in [Sal], are defined on an exact category or atriangulated category, and are related to several homological structures, such as: t -structures [BBD], cluster tilting subcategories [KR, KZ], co- t -structures [Pau],functorially finite rigid subcategories. A careful look reveals that what is neces-sary to define a cotorsion pair on a category is the existence of an Ext bifunctorwith appropriate properties. In this article, we formalize the notion of an extri-angulated category by extracting those properties of Ext on exact categories andon triangulated categories that seem relevant from the point-of-view of cotorsionpairs.The class of extriangulated categories not only contains exact categories andextension-closed subcategories of triangulated categories as examples, but it is alsoclosed under taking some ideal quotients (Proposition 3.30). This will allow us toconstruct an extriangulated category which is not exact nor triangulated. Moreover,this axiomatization rams down the problem of the non-existence of a canonicalchoice of the middle arrow in the axiom (TR3) to the ambiguity of a representativeof realizing sequences (Section 2.2) and the exactness of the associated sequencesof natural transformations (Proposition 3.3).Let us motivate a bit more the use of extriangulated categories. Many resultswhich are homological in nature apply (after suitable adaptation) to both setups:exact categories and triangulated categories. In order to transfer a result knownfor triangulated categories to a result that applies to exact categories, the usualstrategy is the following (non-chronological):(1) Specify to the case of stable categories of Frobenius exact categories.(2) Lift all definitions and statements from the stable category to the Frobeniuscategory.(3) Adapt the proof so that it applies to any exact categories (with suitableassumptions).Conversely, a result known for exact categories might have an analog for triangu-lated categories proven as follows:(1) Specify to the case of a Frobenius exact category.(2) Descend all definitions and statements to the stable category.(3) Adapt the proof so that it applies to any triangulated categories (withsuitable assumptions).Even though step (2) might be far from trivial, the main difficulty often lies instep (3) for both cases. The use of extriangulated categories somehow removes that OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 3 difficulty. It is not that this difficulty has vanished into thin air, but that it hasalready been taken care of in the first results on extriangulated categories obtainedin Section 3.The term “extriangulated” stands for externally triangulated by means of a bi-functor. It can also be viewed as the mixing of exact and triangulated , or as anabbreviation of Ext -triangulated . The precise definition will be given in Section 2.Fundamental properties including several analogs of the octahedron axiom in anextriangulated category, will be given in Section 3.On an extriangulated category, we can define the notion of a cotorsion pair,which generalizes that on exact categories [Ho1, Ho2, Liu, S] and on triangulatedcategories [AN, Na1]. Basic properties will be stated in Section 4.In Section 5, we give a bijective correspondence between
Hovey twin cotorsionpairs and admissible model structures . This result is inspired from [Ho1, Ho2, G],where the case of exact categories is studied in more details. We note that ananalog of Hovey’s result in [Ho1, Ho2] has been proven for triangulated categoriesin [Y]. The use of extriangulated categories allows for a uniform proof.As a result, we can realize the associated homotopy category by a certain idealquotient, on which we can give a triangulated structure as in Section 6. Thistriangulation can be regarded as a simultaneous generalization of those given byHappel’s theorem on stable categories of Frobenius exact categories and of Iyama-Yoshino reductions, and gives a link to the one given by the Verdier quotient. Asa consequence, the homotopy category of any exact model structure on a weaklyidempotent complete exact category is triangulated. This result was previouslyknown in the case of hereditary exact model structures [G, Proposition 5.2].With this view, in Section 7, we propose a natural framework to formulatereduction and mutation of cotorsion pairs, applicable to both exact categories andtriangulated categories. Indeed, we establish a bijective correspondence betweenthe class of mutable cotorsion pairs associated with a Hovey twin cotorsion pairand the class of all cotorsion pairs on the triangulated homotopy category.2.
Extriangulated category
In this section, we abstract the properties of extension-closed subcategory oftriangulated or exact category, to formulate it in an internal way by means of anExt functor. This gives a simultaneous generalization of triangulated categoriesand exact categories, suitable for dealing with cotorsion pairs.2.1. E -extensions. Throughout this paper, let C be an additive category. Definition 2.1.
Suppose C is equipped with a biadditive functor E : C op × C → Ab .For any pair of objects A, C ∈ C , an element δ ∈ E ( C, A ) is called an E -extension .Thus formally, an E -extension is a triplet ( A, δ, C ). Remark 2.2.
Let ( A, δ, C ) be any E -extension. Since E is a bifunctor, for any a ∈ C ( A, A ′ ) and c ∈ C ( C ′ , C ) , we have E -extensions E ( C, a )( δ ) ∈ E ( C, A ′ ) and E ( c, A )( δ ) ∈ E ( C ′ , A ) . We abbreviately denote them by a ∗ δ and c ∗ δ . In this terminology, we have E ( c, a )( δ ) = c ∗ a ∗ δ = a ∗ c ∗ δ in E ( C ′ , A ′ ) . HIROYUKI NAKAOKA AND YANN PALU
Definition 2.3.
Let (
A, δ, C ) , ( A ′ , δ ′ , C ′ ) be any pair of E -extensions. A morphism ( a, c ) : ( A, δ, C ) → ( A ′ , δ ′ , C ′ ) of E -extensions is a pair of morphisms a ∈ C ( A, A ′ )and c ∈ C ( C, C ′ ) in C , satisfying the equality a ∗ δ = c ∗ δ ′ . Simply we denote it as ( a, c ) : δ → δ ′ .We obtain the category E -Ext( C ) of E -extensions, with composition and identi-ties naturally induced from those in C . Remark 2.4.
Let ( A, δ, C ) be any E -extension. We have the following. (1) Any morphism a ∈ C ( A, A ′ ) gives rise to a morphism of E -extensions ( a, id C ) : δ → a ∗ δ. (2) Any morphism c ∈ C ( C ′ , C ) gives rise to a morphism of E -extensions (id A , c ) : c ∗ δ → δ. Definition 2.5.
For any
A, C ∈ C , the zero element 0 ∈ E ( C, A ) is called the split E -extension . Definition 2.6.
Let δ = ( A, δ, C ) , δ ′ = ( A ′ , δ ′ , C ′ ) be any pair of E -extensions. Let C ι C −→ C ⊕ C ′ ι C ′ ←− C ′ and A p A ←− A ⊕ A ′ p A ′ −→ A ′ be coproduct and product in C , respectively. Remark that, by the biadditivity of E , we have a natural isomorphism E ( C ⊕ C ′ , A ⊕ A ′ ) ∼ = E ( C, A ) ⊕ E ( C, A ′ ) ⊕ E ( C ′ , A ) ⊕ E ( C ′ , A ′ ) . Let δ ⊕ δ ′ ∈ E ( C ⊕ C ′ , A ⊕ A ′ ) be the element corresponding to ( δ, , , δ ′ ) throughthis isomorphism. This is the unique element which satisfies E ( ι C , p A )( δ ⊕ δ ′ ) = δ , E ( ι C , p A ′ )( δ ⊕ δ ′ ) = 0 , E ( ι C ′ , p A )( δ ⊕ δ ′ ) = 0 , E ( ι C ′ , p A ′ )( δ ⊕ δ ′ ) = δ ′ . If A = A ′ and C = C ′ , then the sum δ + δ ′ ∈ E ( C, A ) of δ, δ ′ ∈ E ( C, A ) isobtained by δ + δ ′ = E (∆ C , ∇ A )( δ ⊕ δ ′ ) , where ∆ C = h i : C → C ⊕ C , ∇ A = [1 1] : A ⊕ A → A .2.2. Realization of E -extensions. Let C , E be as before. Definition 2.7.
Let
A, C ∈ C be any pair of objects. Sequences of morphisms in C A x −→ B y −→ C and A x ′ −→ B ′ y ′ −→ C are said to be equivalent if there exists an isomorphism b ∈ C ( B, B ′ ) which makesthe following diagram commutative. A BB ′ C x ♦♦♦♦♦♦♦♦ y ' ' ❖❖❖❖❖❖❖❖ x ′ ' ' ❖❖❖❖❖❖❖ y ′ ♦♦♦♦♦♦♦ b ∼ = (cid:15) (cid:15) (cid:8) (cid:8) OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 5
We denote the equivalence class of A x −→ B y −→ C by [ A x −→ B y −→ C ]. Definition 2.8. (1) For any
A, C ∈ C , we denote as0 = [ A h i −→ A ⊕ C [0 1] −→ C ] . (2) For any [ A x −→ B y −→ C ] and [ A ′ x ′ −→ B ′ y ′ −→ C ′ ], we denote as[ A x −→ B y −→ C ] ⊕ [ A ′ x ′ −→ B ′ y ′ −→ C ′ ] = [ A ⊕ A ′ x ⊕ x ′ −→ B ⊕ B ′ y ⊕ y ′ −→ C ⊕ C ′ ] . Definition 2.9.
Let s be a correspondence which associates an equivalence class s ( δ ) = [ A x −→ B y −→ C ] to any E -extension δ ∈ E ( C, A ). This s is called a realization of E , if it satisfies the following condition ( ∗ ). In this case, we say thatsequence A x −→ B y −→ C realizes δ , whenever it satisfies s ( δ ) = [ A x −→ B y −→ C ].( ∗ ) Let δ ∈ E ( C, A ) and δ ′ ∈ E ( C ′ , A ′ ) be any pair of E -extensions, with s ( δ ) = [ A x −→ B y −→ C ] , s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ]. Then, for anymorphism ( a, c ) ∈ E -Ext( C )( δ, δ ′ ), there exists b ∈ C ( B, B ′ ) which makesthe following diagram commutative.(1) A B CA ′ B ′ C ′ x / / y / / a (cid:15) (cid:15) b (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / (cid:8) (cid:8) Remark that this condition does not depend on the choices of the representativesof the equivalence classes. In the above situation, we say that (1) (or the triplet( a, b, c )) realizes ( a, c ). Definition 2.10.
Let C , E be as above. A realization of E is said to be additive ,if it satisfies the following conditions.(i) For any A, C ∈ C , the split E -extension 0 ∈ E ( C, A ) satisfies s (0) = 0 . (ii) For any pair of E -extensions δ = ( A, δ, C ) and δ ′ = ( A ′ , δ ′ , C ′ ), s ( δ ⊕ δ ′ ) = s ( δ ) ⊕ s ( δ ′ )holds. Remark 2.11. If s is an additive realization of E , then the following holds. (1) For any
A, C ∈ C , if ∈ E ( C, A ) is realized by A x −→ B y −→ C , then thereexist a retraction r ∈ C ( B, A ) of x and a section s ∈ C ( C, B ) of y whichgive an isomorphism h ry i : B ∼ = −→ A ⊕ C . (2) For any f ∈ C ( A, B ) , the sequence A h − f i −→ A ⊕ B [ f −→ B realizes the split E -extension ∈ E ( B, A ) . HIROYUKI NAKAOKA AND YANN PALU
Definition of extriangulated category.Definition 2.12.
We call the pair ( E , s ) an external triangulation of C if it satisfiesthe following conditions. In this case, we call s an E - triangulation of C , and callthe triplet ( C , E , s ) an externally triangulated category , or for short, extriangulatedcategory .(ET1) E : C op × C → Ab is a biadditive functor.(ET2) s is an additive realization of E .(ET3) Let δ ∈ E ( C, A ) and δ ′ ∈ E ( C ′ , A ′ ) be any pair of E -extensions, realized as s ( δ ) = [ A x −→ B y −→ C ] , s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ] . For any commutative square(2)
A B CA ′ B ′ C ′ x / / y / / a (cid:15) (cid:15) b (cid:15) (cid:15) x ′ / / y ′ / / (cid:8) in C , there exists a morphism ( a, c ) : δ → δ ′ which is realized by ( a, b, c ).(ET3) op Let δ ∈ E ( C, A ) and δ ′ ∈ E ( C ′ , A ′ ) be any pair of E -extensions, realized by A x −→ B y −→ C and A ′ x ′ −→ B ′ y ′ −→ C ′ respectively. For any commutative square A B CA ′ B ′ C ′ x / / y / / b (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / (cid:8) in C , there exists a morphism ( a, c ) : δ → δ ′ which is realized by ( a, b, c ).(ET4) Let ( A, δ, D ) and (
B, δ ′ , F ) be E -extensions realized by A f −→ B f ′ −→ D and B g −→ C g ′ −→ F respectively. Then there exist an object E ∈ C , a commutative diagram(3) A B DA C EF F f / / f ′ / / g (cid:15) (cid:15) d (cid:15) (cid:15) h / / h ′ / / g ′ (cid:15) (cid:15) e (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) in C , and an E -extension δ ′′ ∈ E ( E, A ) realized by A h −→ C h ′ −→ E , whichsatisfy the following compatibilities.(i) D d −→ E e −→ F realizes E ( F, f ′ )( δ ′ ),(ii) E ( d, A )( δ ′′ ) = δ ,(iii) E ( E, f )( δ ′′ ) = E ( e, B )( δ ′ ). OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 7
By (iii), ( f, e ) : δ ′′ → δ ′ is a morphism of E -extensions, realized by( f, id C , e ) : [ A h −→ C h ′ −→ E ] → [ B g −→ C g ′ −→ F ] . (ET4) op Dual of (ET4) (see Remark 2.22).
Example 2.13.
Exact categories (with a condition concerning the smallness) andtriangulated categories are examples of extriangulated categories. See also Re-mark 2.18.We briefly show how an exact category can be viewed as an extriangulatedcategory. As for triangulated categories, see the construction in Proposition 3.22.For the definition and basic properties of an exact category, see [Bu] and [Ke].Let
A, C ∈ C be any pair of objects. Remark that, as shown in [Bu, Corollary 3.2],for any morphism of short exact sequences (= conflations in [Ke]) of the form A B CA B ′ C x / / y / / b (cid:15) (cid:15) x ′ / / y ′ / / (cid:8) (cid:8) , the morphism b in the middle automatically becomes an isomorphism. Considerthe same equivalence relation as in Definition 2.7, and define Ext ( C, A ) to be thecollection of all equivalence classes of short exact sequences of the form A x −→ B y −→ C . We denote the equivalence class by [ A x −→ B y −→ C ] as before.This becomes a small set, for example in the following cases. • C is skeletally small. • C has enough projectives or injectives.In such a case, we obtain a biadditive functor Ext : C op × C → Ab , as stated in[S, Definitions 5.1]. Its structure is given as follows.- For any δ = [ A x −→ B y −→ C ] ∈ Ext ( C, A ) and any a ∈ C ( A, A ′ ), take apushout in C , to obtain a morphism of short exact sequences A B C PO A ′ M C x / / y / / a (cid:15) (cid:15) (cid:15) (cid:15) m / / e / / (cid:8) . This gives Ext ( C, a )( δ ) = a ∗ δ = [ A ′ m −→ M e −→ C ].- Dually, for any c ∈ C ( C ′ , C ), the map Ext ( c, A ) = c ∗ : Ext ( C, A ) → Ext ( C ′ , A ) is defined by using pullbacks.- The zero element in Ext ( C, A ) is given by the split short exact sequence0 = [ A h i −→ A ⊕ C [0 1] −→ C ] . For any pair δ = [ A x −→ B y −→ C ] , δ = [ A x −→ B y −→ C ] ∈ Ext ( C, A ),its sum δ + δ is given by the Baer sum∆ ∗ C ( ∇ A ) ∗ ( δ ⊕ δ ) = ∆ ∗ C ( ∇ A ) ∗ ([ A ⊕ A x ⊕ x −→ B ⊕ B y ⊕ y −→ C ⊕ C ]) . This shows (ET1). Define the realization s ( δ ) of δ = [ A x −→ B y −→ C ] to be δ itself.Then (ET2) is trivially satisfied. For (ET3) and (ET4), the following fact is useful. HIROYUKI NAKAOKA AND YANN PALU
Fact 2.14. ( [Bu, Proposition 3.1] ) For any morphism of short exact sequences A B CA ′ B ′ C ′ x / / y / / a (cid:15) (cid:15) b (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / (cid:8) (cid:8) in C , there exists a commutative diagram A B CA ′ ∃
M CA ′ B ′ C ′ PO PB x / / y / / a (cid:15) (cid:15) (cid:15) (cid:15) m / / e / / (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / (cid:8)(cid:8) whose middle row is also a short exact sequence, the upper-left square is a pushout,and the down-right square is a pullback. Remark that this means a ∗ [ A x −→ B y −→ C ] = c ∗ [ A ′ x ′ −→ B ′ y ′ −→ C ′ ] . By Fact 2.14, (ET3) follows immediately from the universality of cokernel. Sim-ilarly, (ET4) follows from [Bu, Lemma 3.5]. Dually for (ET3) op and (ET4) op .2.4. Terminology in an extriangulated category.
To allow an argument withfamiliar terms, we introduce terminology from both exact categories and triangu-lated categories (cf. [Ke, Bu, Ne]).
Definition 2.15.
Let ( C , E , s ) be a triplet satisfying (ET1) and (ET2).(1) A sequence A x −→ B y −→ C is called a conflation if it realizes some E -extension δ ∈ E ( C, A ). For the ambiguity of such an E -extension, seeCorollary 3.8.(2) A morphism f ∈ C ( A, B ) is called an inflation if it admits some conflation A f −→ B → C . For the ambiguity of such a conflation, see Remark 3.10.(3) A morphism f ∈ C ( A, B ) is called a deflation if it admits some conflation K → A f −→ B . Remark 2.16.
Condition (ET4) implies that inflations are closed under composi-tion. Dually, (ET4) op implies the composition-closedness of deflations. Definition 2.17.
Let
D ⊆ C be a full additive subcategory, closed under isomor-phisms. We say D is extension-closed if it satisfies the following condition. • If a conflation A → B → C satisfies A, C ∈ D , then B ∈ D .The following can be checked in a straightforward way. Remark 2.18.
Let ( C , E , s ) be an extriangulated category, and let D ⊆ C be anextension-closed subcategory. If we define E D to be the restriction of E onto D op ×D ,and define s D by restricting s , then ( D , E D , s D ) becomes an extriangulated category. Definition 2.19.
Let ( C , E , s ) be a triplet satisfying (ET1) and (ET2). OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 9 (1) If a conflation A x −→ B y −→ C realizes δ ∈ E ( C, A ), we call the pair( A x −→ B y −→ C, δ ) an E -triangle , and write it in the following way.(4) A x −→ B y −→ C δ (2) Let A x −→ B y −→ C δ and A ′ x ′ −→ B ′ y ′ −→ C ′ δ ′ be any pair of E -triangles. If a triplet ( a, b, c ) realizes ( a, c ) : δ → δ ′ as in (1), then we writeit as A B CA ′ B ′ C ′ x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ (cid:8) (cid:8) and call ( a, b, c ) a morphism of E -triangles . Caution 2.20.
Although the abbreviated expression (4) looks superficially asym-metric, we remark that the definition of an extriangulated category is completelyself-dual.
Remark 2.21. (ET3) means that any commutative square (2) bridging E -trianglescan be completed into a morphism of E -triangles. Dually for (ET3) op .Condition ( ∗ ) in Definition 2.9 means that any morphism of E -extensions canbe realized by a morphism of E -triangles. In the above terminology, condition (ET4) op can be stated as follows. Remark 2.22. ( Paraphrase of (ET4) op ) Let D f ′ −→ A f −→ B δ and F g ′ −→ B g −→ C δ ′ be E -triangles. Then there exist an E -triangle E h ′ −→ A h −→ C δ ′′ and acommutative diagram D E FD A BC C d / / e / / h ′ (cid:15) (cid:15) g ′ (cid:15) (cid:15) f ′ / / f / / h (cid:15) (cid:15) g (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) in C , satisfying the following compatibilities. (i) D d −→ E e −→ F g ′∗ δ is an E -triangle, (ii) δ ′ = e ∗ δ ′′ , (iii) d ∗ δ = g ∗ δ ′′ . Fundamental properties
Associated exact sequence.
In this section, we will associate exact se-quences of natural transformations C ( C, − ) C ( y, − ) = ⇒ C ( B, − ) C ( x, − ) = ⇒ C ( A, − ) δ ♯ = ⇒ E ( C, − ) E ( y, − ) = ⇒ E ( B, − ) E ( x, − ) = ⇒ E ( A, − ) , C ( − , A ) C ( − ,x ) = ⇒ C ( − , B ) C ( − ,y ) = ⇒ C ( − , C ) δ ♯ = ⇒ E ( − , A ) E ( − ,x ) = ⇒ E ( − , B ) E ( − ,y ) = ⇒ E ( − , C ) to any E -triangle A x −→ B y −→ C δ in an extriangulated category ( C , E , s )(Corollary 3.12).Here, δ ♯ and δ ♯ are defined in the following. Definition 3.1.
Assume C and E satisfy (ET1). By Yoneda’s lemma, any E -extension δ ∈ E ( C, A ) induces natural transformations δ ♯ : C ( − , C ) ⇒ E ( − , A ) and δ ♯ : C ( A, − ) ⇒ E ( C, − ) . For any X ∈ C , these ( δ ♯ ) X and δ ♯X are given as follows.(1) ( δ ♯ ) X : C ( X, C ) → E ( X, A ) ; f f ∗ δ .(2) δ ♯X : C ( A, X ) → E ( C, X ) ; g g ∗ δ .We abbreviately denote ( δ ♯ ) X ( f ) and δ ♯X ( g ) by δ ♯ f and δ ♯ g , when there is noconfusion. Lemma 3.2.
Assume ( C , E , s ) satisfies (ET1),(ET2),(ET3),(ET3) op . Then for any E -triangle A x −→ B y −→ C δ , the following hold.(1) y ◦ x = 0.(2) x ∗ δ (= δ ♯ x ) = 0.(3) y ∗ δ (= δ ♯ y ) = 0. Proof. (1) By (ET2), the conflation A id −→ A → ∈ E (0 , A ). Applying(ET3) to A A A B C id A / / / / id A (cid:15) (cid:15) x (cid:15) (cid:15) x / / y / / (cid:8) , we obtain a morphism of E -triangles (id A , x, y ◦ x = 0.(2) Similarly, applying (ET3) to A B CB B x / / y / / x (cid:15) (cid:15) id B (cid:15) (cid:15) id B / / / / (cid:8) , we obtain a morphism of E -extensions ( x,
0) : δ →
0. Especially we have x ∗ δ = 0.(3) is dual to (2). (cid:3) Proposition 3.3.
Assume ( C , E , s ) satisfies (ET1),(ET2). Then the following areequivalent.(1) ( C , E , s ) satisfies (ET3) and (ET3) op .(2) For any E -triangle A x −→ B y −→ C δ , the following sequences of naturaltransformations are exact.(i) C ( C, − ) C ( y, − ) = ⇒ C ( B, − ) C ( x, − ) = ⇒ C ( A, − ) δ ♯ = ⇒ E ( C, − ) E ( y, − ) = ⇒ E ( B, − )in Mod( C ). Here Mod( C ) denotes the abelian category of additivefunctors from C to Ab .(ii) C ( − , A ) C ( − ,x ) = ⇒ C ( − , B ) C ( − ,y ) = ⇒ C ( − , C ) δ ♯ = ⇒ E ( − , A ) E ( − ,x ) = ⇒ E ( − , B )in Mod( C op ). OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 11
Remark 3.4.
In the above (i) , the category
Mod( C ) is not locally small in general.The “exactness” of the sequence in (i) simply means that C ( C, X ) C ( y,X ) −→ C ( B, X ) C ( x,X ) −→ C ( A, X ) δ ♯X −→ E ( C, X ) E ( y,X ) −→ E ( B, X ) is exact in Ab for any X ∈ C . Similarly for (ii) .Proof of Proposition 3.3. First we assume (1). We only show the exactness of (i),since (ii) can be shown dually. By Lemma 3.2, composition of any consecutivemorphisms in (i) is equal to 0. Let us show the exactness of C ( C, X ) C ( y,X ) −→ C ( B, X ) C ( x,X ) −→ C ( A, X ) δ ♯X −→ E ( C, X ) E ( y,X ) −→ E ( B, X )for any X ∈ C .Exactness at C ( B, X )Let b ∈ C ( B, X ) be any morphism satisfying C ( x, X )( b ) = b ◦ x = 0. Applying(ET3) to A B C X X x / / y / / (cid:15) (cid:15) b (cid:15) (cid:15) / / id X / / (cid:8) , we obtain a morphism c ∈ C ( C, X ) satisfying b = c ◦ y = C ( y, X )( c ).Exactness at C ( A, X )Let a ∈ C ( A, X ) be any morphism satisfying δ ♯X ( a ) = a ∗ δ = 0. This meansthat ( a,
0) : δ → E -extensions. Since s realizes E , there exists b ∈ C ( B, X ) which gives the following morphism of E -triangles. A B CX X x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) (cid:15) (cid:15) id X / / / / / / ❴❴❴ (cid:8) (cid:8) Especially we have a = b ◦ x = C ( x, X )( b ).Exactness at E ( C, X )Let θ ∈ E ( C, X ) be any E -extension satisfying E ( y, X )( θ ) = y ∗ θ = 0. Realizethem as E -triangles X f −→ Y g −→ C θ and X m −→ Z e −→ B y ∗ θ . Then the morphism (id X , y ) : y ∗ θ → θ can be realized by X Z BX Y C m / / e / / y ∗ θ / / ❴❴❴ e ′ (cid:15) (cid:15) y (cid:15) (cid:15) f / / g / / θ / / ❴❴❴ (cid:8) (cid:8) with some e ′ ∈ C ( Z, Y ). Since y ∗ θ splits by assumption, e has a section s . Applying(ET3) op to A B CX Y C x / / y / / δ / / ❴❴❴ e ′ ◦ s (cid:15) (cid:15) f / / g / / θ / / ❴❴❴ (cid:8) , we obtain a ∈ C ( A, X ) which gives a morphism ( a, id C ) : δ → θ . This means θ = a ∗ δ = δ ♯ a .Conversely, let us assume (2) and show (ET3). Let A x −→ B y −→ C δ and A ′ x ′ −→ B ′ y ′ −→ C ′ δ ′ be any pair of E -triangles. Suppose that we are given acommutative diagram A B CA ′ B ′ C ′ x / / y / / a (cid:15) (cid:15) b (cid:15) (cid:15) x ′ / / y ′ / / (cid:8) in C . Remark that E ( − , x ) ◦ δ ♯ = 0 is equivalent to x ∗ δ = 0 by Yoneda’s lemma.Similarly, we have y ′∗ δ ′ = 0.By the exactness of C ( C, C ′ ) ( δ ′ ♯ ) C −→ E ( C, A ′ ) E ( C,x ′ ) −→ E ( C, B ′ )and the equality E ( C, x ′ )( a ∗ δ ) = x ′∗ a ∗ δ = b ∗ x ∗ δ = 0 , there is c ′ ∈ C ( C, C ′ ) satisfying a ∗ δ = δ ′ ♯ c ′ = c ′∗ δ ′ . Thus ( a, c ) : δ → δ ′ is amorphism of E -extensions. Take its realization as follows. A B CA ′ B ′ C ′ x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b ′ (cid:15) (cid:15) c ′ (cid:15) (cid:15) δ ′ / / ❴❴❴ x ′ / / y ′ / / (cid:8) (cid:8) Then by the exactness of C ( C, B ′ ) C ( y,B ′ ) −→ C ( B, B ′ ) C ( x,B ′ ) −→ C ( A, B ′ )and the equality ( b − b ′ ) ◦ x = x ′ ◦ a − x ′ ◦ a = 0 , there exists c ′′ ∈ C ( C, B ′ ) satisfying c ′′ ◦ y = b − b ′ . If we put c = c ′ + y ′ ◦ c ′′ , thissatisfies c ◦ y = c ′ ◦ y + y ′ ◦ c ′′ ◦ y = y ′ ◦ b ′ + ( y ′ ◦ b − y ′ ◦ b ′ ) = y ′ ◦ b,c ∗ δ ′ = c ′∗ δ ′ + c ′′∗ y ′∗ δ ′ = a ∗ δ. Dually, (2) implies (ET3) op . (cid:3) OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 13
Corollary 3.5.
Assume ( C , E , s ) satisfies (ET1),(ET2),(ET3),(ET3) op . Let(5) A B CA ′ B ′ C ′ x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ (cid:8) (cid:8) be any morphism of E -triangles. Then the following are equivalent.(1) a factors through x .(2) a ∗ δ = c ∗ δ ′ = 0.(3) c factors through y ′ .In particular, in the case δ = δ ′ and ( a, b, c ) = (id , id , id), we obtain x has a retraction ⇔ δ splits ⇔ y has a section . Proof.
By the definition of δ ♯ , it satisfies δ ♯A ′ ( a ) = a ∗ δ . Thus (1) ⇔ (2) followsfrom the exactness of C ( B, A ′ ) C ( x,A ′ ) −→ C ( A, A ′ ) δ ♯A ′ −→ E ( C, A ′ ) . (2) ⇔ (3) can be shown dually. (cid:3) Corollary 3.6.
Assume ( C , E , s ) satisfies (ET1),(ET2),(ET3),(ET3) op . Then thefollowing holds for any morphism of E -triangles.(1) If a and c are isomorphisms in C (equivalently, if ( a, c ) is an isomorphismin E -Ext( C ) in Definition 2.3), then so is b .(2) If a and b are isomorphisms in C , then so is c .(3) If b and c are isomorphisms in C , then so is a . Proof. (1) By Proposition 3.3, we have morphisms of exact sequences of abeliangroups(6) C ( B ′ , A ) C ( B ′ , B ) C ( B ′ , C ) E ( B ′ , A ) C ( B ′ , A ′ ) C ( B ′ , B ′ ) C ( B ′ , C ′ ) E ( B ′ , A ′ ) C ( B ′ ,x ) / / C ( B ′ ,y ) / / ( δ ♯ ) B ′ / / ∼ = C ( B ′ ,a ) (cid:15) (cid:15) C ( B ′ ,b ) (cid:15) (cid:15) ∼ = C ( B ′ ,c ) (cid:15) (cid:15) ∼ = E ( B ′ ,a ) (cid:15) (cid:15) C ( B ′ ,x ′ ) / / C ( B ′ ,y ′ ) / / ( δ ′ ♯ ) B ′ / / (cid:8) (cid:8) (cid:8) and(7) C ( C ′ , B ) C ( B ′ , B ) C ( A ′ , B ) E ( C ′ , B ) C ( C, B ) C ( B, B ) C ( A, B ) E ( C, B ) C ( y ′ ,B ) / / C ( x ′ ,B ) / / δ ′ ♯B / / ∼ = C ( c,B ) (cid:15) (cid:15) C ( b,B ) (cid:15) (cid:15) ∼ = C ( a,B ) (cid:15) (cid:15) ∼ = E ( c,B ) (cid:15) (cid:15) C ( y,B ) / / C ( x,B ) / / δ ♯B / / (cid:8) (cid:8) (cid:8) . Diagram (6) shows the surjectivity of C ( B ′ , b ). Thus b has a right inverse in C .Dually, (7) shows the surjectivity of C ( b, B ), and thus b also has a left inverse. (2) By Yoneda’s lemma, this follows from the five lemma applied to C ( − , A ) C ( − , B ) C ( − , C ) E ( − , A ) E ( − , B ) C ( − , A ′ ) C ( − , B ′ ) C ( − , C ′ ) E ( − , A ′ ) E ( − , B ′ ) + + + + ∼ = (cid:11) (cid:19) ∼ = (cid:11) (cid:19) C ( − ,c ) (cid:11) (cid:19) ∼ = (cid:11) (cid:19) ∼ = (cid:11) (cid:19) + + + + (cid:8) (cid:8) (cid:8) (cid:8) . (3) is dual to (2). (cid:3) Proposition 3.7.
Assume ( C , E , s ) satisfies (ET1),(ET2),(ET3),(ET3) op . Let A x −→ B y −→ C δ be any E -triangle. If a ∈ C ( A, A ′ ) and c ∈ C ( C ′ , C ) areisomorphisms, then A ′ x ◦ a − −→ B c − ◦ y −→ C ′ a ∗ c ∗ δ becomes again an E -triangle. Proof.
Put s ( a ∗ c ∗ δ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ]. Since( c − ) ∗ (cid:0) a ∗ c ∗ δ (cid:1) = ( c ◦ c − ) ∗ a ∗ δ = a ∗ δ, we see that ( a, c − ) : δ → c ∗ a ∗ δ is a morphism of E -extensions. Take a morphism of E -triangles ( a, b, c − ) realizing ( a, c − ). Then b is an isomorphism by Corollary 3.6.Since A ′ BB ′ C ′ x ◦ a − ♦♦♦♦♦♦♦ c − ◦ y ' ' ❖❖❖❖❖❖❖ x ′ ' ' ❖❖❖❖❖❖❖ y ′ ♦♦♦♦♦♦♦ b ∼ = (cid:15) (cid:15) (cid:8) (cid:8) is commutative, it follows that [ A ′ x ′ −→ B ′ y ′ −→ C ′ ] = [ A ′ x ◦ a − −→ B c − ◦ y −→ C ′ ]. (cid:3) Corollary 3.8.
Assume ( C , E , s ) satisfies (ET1),(ET2),(ET3),(ET3) op . Let A x −→ B y −→ C δ be any E -triangle. Then for any δ ′ ∈ E ( C, A ), the following areequivalent.(1) s ( δ ) = s ( δ ′ ).(2) δ ′ = a ∗ δ for some automorphism a ∈ C ( A, A ) satisfying x ◦ a = x .(3) δ ′ = c ∗ δ for some automorphism c ∈ C ( C, C ) satisfying c ◦ y = y .(4) δ ′ = a ∗ c ∗ δ for some pair of automorphisms a ∈ C ( A, A ) , c ∈ C ( C, C )satisfying x ◦ a = x and c ◦ y = y . Proof. (2) ⇒ (4) is trivial. Similarly for (3) ⇒ (4). Proposition 3.7 shows (4) ⇒ (1).Let us show (1) ⇒ (2). Suppose δ ′ satisfies s ( δ ′ ) = s ( δ ) = [ A x −→ B y −→ C ].Applying (ET3) op to A B CA B C x / / y / / δ / / ❴❴❴ x / / y / / δ ′ / / ❴❴❴ (cid:8) , we obtain a ∈ C ( A, A ) with which ( a, id B , id C ) gives a morphism between the above E -triangles. By Corollary 3.6, this a is an isomorphism. (1) ⇒ (3) can be shown ina similar way. (cid:3) For simplicity, we use the following notations.
OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 15
Definition 3.9.
Assume ( C , E , s ) satisfies (ET1),(ET2),(ET3),(ET3) op .(1) For an inflation f ∈ C ( A, B ), take a conflation A f −→ B → C , and denotethis C by Cone( f ).(2) For a deflation f ∈ C ( A, B ), take a conflation K → A f −→ B . We denotethis K by CoCone( f ).Cone( f ) is determined uniquely up to isomorphism by the following remark.They are not functorial in general, as the case of triangulated category suggests.Dually for CoCone( f ). Remark 3.10.
Let f ∈ C ( A, B ) be an inflation, and suppose A f −→ B g −→ C δ , A f −→ B g ′ −→ C ′ δ ′ are E -triangles. Then by (ET3) applied to A B CA B C ′ f / / g / / δ / / ❴❴❴ f / / g ′ / / δ ′ / / ❴❴❴ (cid:8) , there exists c ∈ C ( C, C ′ ) which gives a morphism of E -triangles (id A , id B , c ) . ByCorollary 3.6, this c is an isomorphism. Proposition 3.11.
Assume ( C , E , s ) satisfies (ET1),(ET2),(ET3),(ET3) op . Let A x −→ B y −→ C δ be any E -triangle. Then, we have the following.(1) If ( C , E , s ) satisfies (ET4), then E ( − , A ) E ( − ,x ) = ⇒ E ( − , B ) E ( − ,y ) = ⇒ E ( − , C )is exact.(2) If ( C , E , s ) satisfies (ET4) op , then E ( C, − ) E ( y, − ) = ⇒ E ( B, − ) E ( x, − ) = ⇒ E ( A, − )is exact. Proof. (1) E ( − , y ) ◦ E ( − , x ) = 0 follows from Lemma 3.2. Let X ∈ C be any object.Let θ ∈ E ( X, B ) be any E -extension, realized by an E -triangle B f −→ Y g −→ X θ .By (ET4), there exist E ∈ C , θ ′ ∈ E ( E, A ) and a commutative diagram
A B CA Y EX X x / / y / / f (cid:15) (cid:15) d (cid:15) (cid:15) h / / h ′ / / g (cid:15) (cid:15) e (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) which satisfy s ( y ∗ θ ) = [ C d −→ E e −→ X ] , s ( θ ′ ) = [ A h −→ Y h ′ −→ E ] ,x ∗ θ ′ = e ∗ θ. Thus if E ( X, y )( θ ) = y ∗ θ = 0, then e has a section s ∈ C ( X, E ). If we put ρ = s ∗ θ ′ ,then this satisfies E ( x, X )( ρ ) = x ∗ s ∗ θ ′ = s ∗ x ∗ θ ′ = s ∗ e ∗ θ = θ. (2) is dual to (1). (cid:3) Corollary 3.12.
Let ( C , E , s ) be an extriangulated category. For any E -triangle A x −→ B y −→ C δ , the following sequences of natural transformations are exact. C ( C, − ) C ( y, − ) = ⇒ C ( B, − ) C ( x, − ) = ⇒ C ( A, − ) δ ♯ = ⇒ E ( C, − ) E ( y, − ) = ⇒ E ( B, − ) E ( x, − ) = ⇒ E ( A, − ) , C ( − , A ) C ( − ,x ) = ⇒ C ( − , B ) C ( − ,y ) = ⇒ C ( − , C ) δ ♯ = ⇒ E ( − , A ) E ( − ,x ) = ⇒ E ( − , B ) E ( − ,y ) = ⇒ E ( − , C ) . Proof.
This immediately follows from Propositions 3.3 and 3.11. (cid:3)
The following lemma shows that the upper-right square of Diagram (3) obtainedby (ET4) is a weak pushout.
Lemma 3.13.
Let (3) be a commutative diagram in C , where A f −→ B f ′ −→ D d ∗ δ ′′ , B g −→ C g ′ −→ F δ ′ A h −→ C h ′ −→ E δ ′′ , D d −→ E e −→ F f ′∗ δ ′ are E -triangles, which satisfy e ∗ δ ′ = f ∗ δ ′′ .Suppose we are given a commutative square B DC Y f ′ / / g (cid:15) (cid:15) y (cid:15) (cid:15) x / / (cid:8) in C . Then there exists z ∈ C ( E, Y ) which makes the following diagram commu-tative.
B DC E Y f ′ / / g (cid:15) (cid:15) d (cid:15) (cid:15) h ′ / / z (cid:31) (cid:31) ❄❄❄❄❄❄ y (cid:17) (cid:17) x - - (cid:8) (cid:8) (cid:8) Proof.
By ( f ′∗ δ ′ ) ♯ ( y ) = y ∗ f ′∗ δ ′ = x ∗ g ∗ δ ′ = 0 and the exactness of C ( E, Y ) C ( d,Y ) −→ C ( D, Y ) ( f ′∗ δ ′ ) ♯Y −→ E ( F, Y ) → , there exists z ∈ C ( E, Y ) satisfying z ◦ d = y. Then by( x − z ◦ h ′ ) ◦ g = y ◦ f ′ − z ◦ d ◦ f ′ = 0 OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 17 and the exactness of C ( F, Y ) C ( g ′ ,Y ) −→ C ( C, Y ) C ( g,Y ) −→ C ( B, Y ) , there exists z ∈ C ( F, Y ) satisfying z ◦ g ′ = x − z ◦ h ′ . If we put z = z + z ◦ e ,this satisfies the desired commutativity. (cid:3) Shifted octahedrons.
Condition (ET4) in Definition 2.12 is an analog of theoctahedron axiom (TR4) for a triangulated category. As in the case of a triangu-lated category, we can make it slightly more rigid as follows.
Lemma 3.14.
Let ( C , E , s ) be an extriangulated category. Let A f −→ B f ′ −→ D δ f ,B g −→ C g ′ −→ F δ g ,A h −→ C h −→ E δ h be any triplet of E -triangles satisfying h = g ◦ f . Then there are morphisms d , e in C which make the diagram(8) A B DA C E F F f / / f ′ / / g (cid:15) (cid:15) d (cid:15) (cid:15) h / / h / / g ′ (cid:15) (cid:15) e (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) commutative, and satisfy the following compatibilities.(i) D d −→ E e −→ F f ′∗ ( δ g ) is an E -triangle,(ii) d ∗ ( δ h ) = δ f ,(iii) f ∗ ( δ h ) = e ∗ ( δ g ). Proof.
By (ET4), there exist an object E ∈ C , a commutative diagram (3) in C ,and an E -triangle A h −→ C h ′ −→ E δ ′′ , which satisfy the following compatibilities.(i ′ ) D d −→ E e −→ F f ′∗ ( δ g ) is an E -triangle,(ii ′ ) d ∗ ( δ ′′ ) = δ f ,(iii ′ ) f ∗ ( δ ′′ ) = e ∗ ( δ g ).By Remark 3.10, we obtain a morphism of E -triangles A C EA C E h / / h ′ / / δ ′′ / / ❴❴❴ u (cid:15) (cid:15) h / / h / / δ h / / ❴❴❴ (cid:8) (cid:8) in which u is an isomorphism. In particular we have δ ′′ = u ∗ ( δ h ). If we put d = u ◦ d and e = e ◦ u − , then the commutativity of (8) follows from that of (3).By the definition of the equivalence relation, we have [ D d −→ E e −→ F ] = [ D d −→ E e −→ F ]. It is straightforward to check that (i ′ ),(ii ′ ),(iii ′ ) imply (i),(ii),(iii). (cid:3) Proposition 3.15.
Let ( C , E , s ) be an extriangulated category. Then the followingholds.(1) Let C be any object, and let A x −→ B y −→ C δ , A x −→ B y −→ C δ be any pair of E -triangles. Then there is a commutative diagram in C (9) A A A M B A B C m (cid:15) (cid:15) x (cid:15) (cid:15) m / / e / / e (cid:15) (cid:15) y (cid:15) (cid:15) x / / y / / (cid:8)(cid:8) (cid:8) which satisfies s ( y ∗ δ ) = [ A m −→ M e −→ B ] , s ( y ∗ δ ) = [ A m −→ M e −→ B ] ,m ∗ δ + m ∗ δ = 0 . (2) Dual of (1). Proof.
By the additivity of s , we have s ( δ ⊕ δ ) = [ A ⊕ A x ⊕ x −→ B ⊕ B y ⊕ y −→ C ⊕ C ] . Let A A ⊕ A A ι / / p o o o o ι / / p be a biproduct in C . Put µ = (∆ C ) ∗ ( δ ⊕ δ ) and take s ( µ ) = [ A ⊕ A j −→ M k −→ C ]. Then µ satisfies(10) p ∗ µ = δ and p ∗ µ = δ . Applying (ET4) to s (0) = [ A ι −→ A ⊕ A p −→ A ] and s ( µ ) = [ A ⊕ A j −→ M k −→ C ], we obtain B ′ ∈ C , θ ∈ E ( B ′ , A ) and a commutative diagram A A ⊕ A A A M B ′ C C ι / / p / / j (cid:15) (cid:15) x ′ (cid:15) (cid:15) m / / e ′ / / k (cid:15) (cid:15) y ′ (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) which satisfy(i) s ( p ∗ µ ) = [ A x ′ −→ B y ′ −→ C ],(ii) x ∗ θ = 0,(iii) s ( θ ) = [ A m −→ M e ′ −→ B ′ ], OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 19 (iv) ( ι , id M , y ′ ) is a morphism of E -triangles. Especially, we have y ′∗ µ = ι ∗ θ .In particular, we have[ A x ′ −→ B y ′ −→ C ] = s ( δ ) = [ A x −→ B y −→ C ] . Thus there is an isomorphism b ∈ C ( B , B ′ ) satisfying b ◦ x = x ′ and y ′ ◦ b = y .If we put e = b − ◦ e ′ , then s ( b ∗ θ ) = [ A m −→ M e −→ B ]by Proposition 3.7. Thus we obtain a commutative diagram A A ⊕ A A A M B C C ι / / p / / j (cid:15) (cid:15) x (cid:15) (cid:15) m / / e / / k (cid:15) (cid:15) y (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) which satisfies y ∗ δ = b ∗ y ′∗ p ∗ µ = b ∗ p ∗ y ′∗ µ = b ∗ p ∗ ι ∗ θ = b ∗ θ . Thus we obtain s ( y ∗ δ ) = [ A m −→ M e −→ B ]. Similarly, from s (0) = [ A ι −→ A ⊕ A p −→ A ] and s ( µ ) = [ A ⊕ A j −→ M k −→ C ], we obtain a commutativediagram A A ⊕ A A A M B C C ι / / p / / j (cid:15) (cid:15) x (cid:15) (cid:15) m / / e / / k (cid:15) (cid:15) y (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) which satisfies s ( y ∗ δ ) = [ A m −→ M e −→ B ]. Since e ◦ m = e ◦ j ◦ ι = x ◦ p ◦ ι = x ,e ◦ m = e ◦ j ◦ ι = x ◦ p ◦ ι = x ,y ◦ e = k = y ◦ e , diagram (9) is commutative. Moreover, we have m ∗ δ + m ∗ δ = j ∗ (cid:0) ι ∗ δ + ι ∗ δ (cid:1) = j ∗ (cid:0) ( ι ◦ p ) ∗ + ( ι ◦ p ) ∗ (cid:1) ( µ ) = j ∗ µ = 0by (10) and Lemma 3.2. (cid:3) Corollary 3.16.
Let x ∈ C ( A, B ) , f ∈ C ( A, D ) be any pair of morphisms. If x isan inflation, then so is h fx i ∈ C ( A, D ⊕ B ). Dually for deflations. Proof.
Let A x −→ B y −→ C δ be an E -triangle. Realize f ∗ δ by an E -triangle D d −→ E e −→ C f ∗ δ . By Proposition 3.15 (1), we obtain a commutative diagram made of E -triangles A AD M BD E C m (cid:15) (cid:15) x (cid:15) (cid:15) k / / ℓ / / y ∗ f ∗ δ / / ❴❴❴ e (cid:15) (cid:15) y (cid:15) (cid:15) d / / e / / f ∗ δ / / ❴❴❴ e ∗ δ (cid:15) (cid:15) ✤✤✤✤ δ (cid:15) (cid:15) ✤✤✤✤ (cid:8)(cid:8) (cid:8) satisfying m ∗ δ + k ∗ f ∗ δ = 0. Since y ∗ f ∗ δ = f ∗ y ∗ δ = 0, we may assume M = D ⊕ B, k = h i , ℓ = [0 1], and take p ∈ C ( M, D ) , i ∈ C ( B, M ) which make
D M B k / / p o o o o i / / ℓ a biproduct. By the exactness of C ( B, M ) C ( x,M ) −→ C ( A, M ) δ ♯ −→ E ( C, M )and the equality δ ♯ ( m + k ◦ f ) = m ∗ δ + k ∗ f ∗ δ = 0, there exists b ∈ C ( B, M )satisfying b ◦ x = m + k ◦ f .Modifying A m −→ M e −→ E by the automorphism n = (cid:20) − p ◦ b (cid:21) = (cid:20) − (cid:21) ◦ (id M − k ◦ p ◦ b ◦ ℓ ) : M ∼ = −→ M, we obtain a conflation A n ◦ m −→ D ⊕ B e ◦ n − −→ E. Then, since p ◦ ( n ◦ m ) = − p ◦ (id M − k ◦ p ◦ b ◦ ℓ ) ◦ m = p ◦ k ◦ p ◦ b ◦ ℓ ◦ m − p ◦ m = p ◦ b ◦ x − p ◦ m = p ◦ k ◦ f = f and ℓ ◦ ( n ◦ m ) = ℓ ◦ (id M − k ◦ p ◦ b ◦ ℓ ) ◦ m = ℓ ◦ m = x, we have n ◦ m = h fx i . (cid:3) Proposition 3.17.
Suppose we are given E -triangles D f −→ A f ′ −→ C δ f ,A g −→ B g ′ −→ F δ g ,E h −→ B h ′ −→ C δ h OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 21 satisfying h ′ ◦ g = f ′ . Then there is an E -triangle D d −→ E e −→ F θ which makes(11) D A CE B CF F f / / f ′ / / d (cid:15) (cid:15) g (cid:15) (cid:15) h / / h ′ / / e (cid:15) (cid:15) g ′ (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) commutative in C , and satisfy the following equalities.(i) d ∗ ( δ f ) = δ h .(ii) f ∗ ( θ ) = δ g .(iii) g ′∗ ( θ ) + h ′∗ ( δ f ) = 0. Proof.
By (ET4), we have E -triangles D g ◦ f −→ B a −→ G µ and C b −→ G c −→ F ν which make the following diagram commutative in C ,(12) D A CD B GF F f / / f ′ / / g (cid:15) (cid:15) b (cid:15) (cid:15) g ◦ f / / a / / g ′ (cid:15) (cid:15) c (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) and satisfy f ′∗ ( δ g ) = ν, b ∗ µ = δ f , c ∗ ( δ g ) = f ∗ µ. It follows from Lemma 3.2 that ν = f ′∗ ( δ g ) = h ′∗ g ∗ ( δ g ) = 0. Thus, up to equivalence,we may assume G = C ⊕ F, b = h i , c = [0 1]from the start. Then a = h a a i : B → G = C ⊕ F should satisfy a ◦ g = f ′ , a = g ′ by the commutativity of (12). Since h ′ − a ∈ C ( B, C ) satisfies ( h ′ − a ) ◦ g = f ′ − f ′ = 0, there exists z ∈ C ( F, C ) satisfying z ◦ g ′ = h ′ − a . Put z ′ = h − z i . Applying the dual of Lemma 3.14 to the following diagram madeof E -triangles, E FD B GC C h (cid:15) (cid:15) z ′ (cid:15) (cid:15) g ◦ f / / a / / µ / / ❴❴❴ h ′ (cid:15) (cid:15) [1 z ] (cid:15) (cid:15) δ h (cid:15) (cid:15) ✤✤✤ (cid:15) (cid:15) ✤✤✤ (cid:8) we obtain an E -triangle D d −→ E e −→ F θ which makes the following diagram commutative, D E FD B GC C d / / e / / h (cid:15) (cid:15) z ′ (cid:15) (cid:15) g ◦ f / / a / / h ′ (cid:15) (cid:15) [1 z ] (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) and satisfies θ = z ′∗ µ, d ∗ µ = [1 z ] ∗ ( δ h ) . Then the commutativity of (11) can be checked in a straightforward way. Let usshow the equalities (i),(ii),(iii).(i) follows from d ∗ ( δ f ) = d ∗ b ∗ µ = b ∗ [1 z ] ∗ ( δ h ) = ([1 z ] ◦ b ) ∗ ( δ h ) = δ h . (ii) follows from the injectivity of E ( c, A ) = c ∗ and c ∗ f ∗ ( θ ) = c ∗ f ∗ z ′∗ µ = f ∗ ( z ′ ◦ c ) ∗ µ = f ∗ (cid:20) − z (cid:21) ∗ µ = f ∗ (1 − b ◦ [1 z ]) ∗ µ = f ∗ µ − f ∗ [1 z ] ∗ ( δ f ) = f ∗ µ − [1 z ] ∗ f ∗ ( δ f ) = f ∗ µ = c ∗ ( δ g ) . (iii) follows from g ′∗ ( θ ) + h ′∗ ( δ f ) = g ′∗ z ′∗ µ + h ′∗ b ∗ µ = (cid:16)h − z ◦ g ′ g ′ i + h h ′ i(cid:17) ∗ µ = h a g ′ i ∗ µ = a ∗ µ = 0 . (cid:3) As in Example 2.13, an exact category (with some smallness assumption) canbe regarded as an extriangulated category, whose inflations are monomorphic andwhose deflations are epimorphic. Conversely, the following holds.
OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 23
Corollary 3.18.
Let ( C , E , s ) be an extriangulated category, in which any inflationis monomorphic, and any deflation is epimorphic. If we let S be the class ofconflations given by the E -triangles (see Definition 2.15), then ( C , S ) is an exactcategory in the sense of [Bu]. Proof.
By the exact sequences obtained in Proposition 3.3, for any conflation A x −→ B y −→ C , the pair ( A, x ) gives a weak kernel of y . Since x is monomorphic byassumption, it is a kernel of y . Dually ( C, y ) gives a cokernel of x , and A x −→ B y −→ C becomes a kernel-cokernel pair.Thus S consists of some kernel-cokernel pairs. Moreover, it is closed underisomorphisms. Indeed, let A x −→ B y −→ C δ be any E -triangle, let A ′ x ′ −→ B ′ y ′ −→ C ′ be a kernel-cokernel pair, and suppose there are isomorphisms a ∈ C ( A, A ′ ) , b ∈ C ( B, B ′ ) , c ∈ C ( C, C ′ ) satisfying x ′ ◦ a = b ◦ x and y ′ ◦ b = c ◦ y . By Proposition 3.7,we obtain an E -triangle A ′ x ◦ a − −→ B c ◦ y −→ C ′ ( c − ) ∗ a ∗ δ . Since A ′ BB ′ C ′ x ◦ a − ♦♦♦♦♦♦♦ c ◦ y ' ' ❖❖❖❖❖❖❖ x ′ ' ' ❖❖❖❖❖❖❖ y ′ ♦♦♦♦♦♦♦ b ∼ = (cid:15) (cid:15) (cid:8) (cid:8) is commutative in C , this gives s ( δ ) = [ A ′ x ◦ a − −→ B c ◦ y −→ C ′ ] = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ],which means that A ′ x ′ −→ B ′ y ′ −→ C ′ belongs to S .Let us confirm conditions [E0],[E1],[E2] in [Bu, Definition 2.1]. Since our assump-tions are self-dual, the other conditions [E0 op ],[E1 op ],[E2 op ] can be shown dually.[E0] For any object A ∈ C , the split sequence A id A −→ A → S by(ET2).[E1] The class of inflations (= admissible monics) is closed under compositionby (ET4), as in Remark 2.16.[E2] Let A x −→ B y −→ C δ be any E -triangle, and let a ∈ C ( A, A ′ ) be anymorphism. By Corollary 3.16, there exists a conflation A s −→ B ⊕ A ′ ∃ [ b x ′ ] −→ ∃ B ′ , where s = h x − a i . Since it becomes a kernel-cokernel pair by the above argument,it follows that A BA ′ B ′ x / / a (cid:15) (cid:15) b (cid:15) (cid:15) x ′ / / (cid:8) is a pushout square. By the dual of Proposition 3.17, we obtain the followingcommutative diagram made of conflations, A ′ A ′ A B ⊕ A ′ B ′ A B C h i (cid:15) (cid:15) x ′ (cid:15) (cid:15) s / / [ b x ′ ] / / [1 0] (cid:15) (cid:15) ∃ y ′ (cid:15) (cid:15) x / / y / / (cid:8)(cid:8) (cid:8) which shows that x ′ is an inflation. (cid:3) Relation with triangulated categories.
In this section, let C be an addi-tive category equipped with an equivalence [1] : C ≃ −→ C , and let E : C op × C → Ab be the bifunctor defined by E = Ext ( − , − ) = C ( − , − [1]). Remark 3.19.
As usual, we use notations like X [1] and f [1] for objects X andmorphisms f in C . The n -times composition of [1] is denoted by [ n ] . We will show that, to give a triangulation of C with shift functor [1], is equivalentto give an E -triangulation of C (Proposition 3.22). Remark 3.20.
Let C , [1] , E be as above. Then for any δ ∈ E ( C, A ) = C ( C, A [1]) ,we have the following. (1) δ ♯ = C ( − , δ ) : C ( − , C ) ⇒ C ( − , A [1]) . (2) δ ♯ is given by δ ♯X : C ( A, X ) → C ( C, X [1]) ; f ( f [1]) ◦ δ for any X ∈ C . Lemma 3.21.
Let C , [1] , E be as above. Suppose that s is an E -triangulation of C . Then for any A ∈ C , the E -extension = id A [1] ∈ E ( A [1] , A ) = C ( A [1] , A [1])can be realized as s ( ) = [ A → → A [1]] . Namely, A → → A [1] is an E -triangle. Proof.
Put s ( ) = [ A x −→ X y −→ A [1]]. By Proposition 3.3, C ( − , A ) C ( − ,x ) = ⇒ C ( − , X ) C ( − ,y ) = ⇒ C ( − , A [1]) ♯ =id = ⇒ C ( − , A [1]) C ( − ,x [1]) = ⇒ C ( − , X [1])is exact. In particular id X ∈ C ( X, X ) satisfies y = ( ♯ ) X ◦ C ( X, y )(id X ) = 0.Similarly x [1] = 0 implies x = 0. Thus0 ⇒ C ( − , X ) ⇒ X = 0. (cid:3) Proposition 3.22.
As before, let C be an additive category equipped with anauto-equivalence [1], and put E = C ( − , − [1]). Then we have the following. OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 25 (1) Suppose C is a triangulated category with shift functor [1]. For any δ ∈ E ( C, A ) = C ( C, A [1]), take a distinguished triangle A x −→ B y −→ C δ −→ A [1]and define as s ( δ ) = [ A x −→ B y −→ C ]. Remark that this s ( δ ) does not de-pend on the choice of the distinguished triangle above. With this definition,( C , E , s ) becomes an extriangulated category.(2) Suppose we are given an E -triangulation s of C . Define that A x −→ B y −→ C δ −→ A [1] is a distinguished triangle if and only if it satisfies s ( δ ) =[ A x −→ B y −→ C ]. With this class of distinguished triangles, C becomes atriangulated category.By construction, distinguished triangles correspond to E -triangles by the above(1) and (2). Proof. (1) is straightforward. For (2), all the axioms except for (TR2) are easilyconfirmed. Let us show (TR2).Let A x −→ B y −→ C δ be any E -triangle. Applying Proposition 3.15 (2) to A → → A [1] and δ , we obtain A B C ∃ M CA [1] A [1] x / / y / / (cid:15) (cid:15) m ′ (cid:15) (cid:15) / / e / / (cid:15) (cid:15) e ′ (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) which satisfies(i) [0 → M e −→ C ] = 0 ∗ δ = 0,(ii) s ( x [1]) = s ( x ∗ ) = [ B m ′ −→ M e ′ −→ A [1]],(iii) e ∗ δ + e ′∗ = 0.(i) shows that e is an isomorphism, by Remark 2.11 (1). (iii) means δ ◦ e + e ′ = 0in C ( M, A [1]), namely e ′ ◦ e − = − δ . Thus we have s ( x [1]) = [ B y −→ C − δ −→ A [1]]by (ii), which means B y −→ C − δ −→ A [1] x [1] −→ B [1] is a distinguished triangle. Thisis isomorphic to B y −→ C δ −→ A [1] − x [1] −→ B [1]. (cid:3) Projectives and injectives.
If ( C , E , s ) has enough “ projectives ”, then thebifunctor E can be described in terms of them. Throughout this section, let ( C , E , s )be an extriangulated category. Duals of the results in this section hold true for“ injectives ”. Definition 3.23.
An object P ∈ C is called projective if it satisfies the followingcondition. • For any E -triangle A x −→ B y −→ C δ and any morphism c ∈ C ( P, C ),there exists b ∈ C ( P, B ) satisfying y ◦ b = c . We denote the full subcategory of projective objects in C by Proj( C ). Dually, thefull subcategory of injective objects in C is denoted by Inj( C ). Proposition 3.24.
An object P ∈ C is projective if and only if it satisfies E ( P, A ) = 0 for any A ∈ C . Proof.
Suppose P satisfies E ( P, A ) = 0 for any A ∈ C . Then for any E -triangle A x −→ B y −→ C δ , C ( P, B ) C ( P,y ) −→ C ( P, C ) → P is projective.Conversely, suppose P is projective. Let A ∈ C be any object, and let δ ∈ E ( P, A ) be any element, with s ( δ ) = [ A x −→ M y −→ P ]. Since P is projective, thereexists m ∈ C ( P, M ) which makes the following diagram commutative.0
P PA M P / / id P / / m (cid:15) (cid:15) id P (cid:15) (cid:15) x / / y / / (cid:8) By (ET3) op , the triplet (0 , m, id P ) realizes the morphism (0 , id P ) : 0 → δ . Espe-cially we have δ = E ( P, (cid:3) Definition 3.25.
Let ( C , E , s ) be an extriangulated category, as before. We say it has enough projectives , if it satisfies the following condition. • For any object C ∈ C , there exists an E -triangle A x −→ P y −→ C δ satisfying P ∈ Proj( C ). Example 3.26. (1) If ( C , E , s ) is an exact category, then these agree with theusual definitions.(2) If ( C , E , s ) is a triangulated category as in the previous section, thenProj( C ) consists of zero objects. Moreover it always has enough projectives.(3) If ( C , E , s ) is a triangulated category with a rigid subcategory R (i.e. forall R , R ∈ R , Ext ( R , R ) = 0), let D be its full subcategory whoseobjects are those objects X that satisfy Ext ( R, X ) = 0 for all R ∈ R .Then D is an additive and extension-closed subcategory of C , which isthus extriangulated by Remark 2.18. We then have:(a) R ⊆ Proj( D );(b) Proj( D ) = R and D has enough projectives if and only if R is con-travariantly finite. Corollary 3.27.
Assume that ( C , E , s ) has enough projectives. For any object C ∈ C and any E -triangle A x −→ P y −→ C δ ( P ∈ Proj( C )) , the sequence C ( P, − ) C ( x, − ) = ⇒ C ( A, − ) δ ♯ = ⇒ E ( C, − ) ⇒ E ( C, − ) ∼ = Cok (cid:0) C ( x, − ) (cid:1) . OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 27
Proof.
This immediately follows from Propositions 3.3 and 3.24. (cid:3)
The isomorphism (13) in Corollary 3.27 is natural in C , in the following sense. Remark 3.28.
Assume that ( C , E , s ) has enough projectives. Let c ∈ C ( C, C ′ ) beany morphism, and let A x −→ P y −→ C δ , A ′ x ′ −→ P ′ y ′ −→ C ′ δ ′ be any pair of E -triangles satisfying P, P ′ ∈ Proj( C ) . By the projectivity of P and (ET3) op , we obtain a morphism of E -triangles A P CA ′ P ′ C ′ x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) p (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ (cid:8) (cid:8) . This gives the following morphism of exact sequences. C ( P, − ) C ( A, − ) E ( C, − ) 0 C ( P ′ , − ) C ( A ′ , − ) E ( C ′ , − ) 0 C ( x, − ) + δ ♯ + + C ( p, − ) K S C ( a, − ) K S E ( c, − ) K S C ( x ′ , − ) + δ ′ ♯ + + (cid:8) (cid:8) Lemma 3.29.
Let A x −→ B y −→ C δ be any E -triangle, and let i ∈ C ( A, I )be any morphism to I ∈ Inj( C ). Then for the projection p C : C ⊕ I → C , the E -extension p ∗ C δ is realized by an E -triangle of the form(14) A x I −→ B ⊕ I y I −→ C ⊕ I p ∗ C δ , ( x I = h xi i , y I = (cid:20) y ∗∗ ∗ (cid:21) ) . Proof.
By Corollary 3.16, we have an E -triangle A x I −→ B ⊕ I d −→ D ν . By thedual of Proposition 3.17, we obtain the following commutative diagram made of E -triangles I IA B ⊕ I DA B C (cid:15) (cid:15) h i e (cid:15) (cid:15) x I / / d / / ν / / ❴❴❴❴ [1 0] (cid:15) (cid:15) f (cid:15) (cid:15) x / / y / / δ / / ❴❴❴❴ (cid:15) (cid:15) ✤✤✤✤ ∃ θ (cid:15) (cid:15) ✤✤✤✤ (cid:8)(cid:8) (cid:8) satisfying f ∗ δ = ν . Since I ∈ Inj( C ), we have θ = 0. Thus there is some iso-morphism n : C ⊕ I ∼ = −→ D satisfying n ◦ h i = e and f ◦ n = [1 0]. Then for p C = [1 0] : C ⊕ I → C , A B ⊕ I C ⊕ IA B C x I / / n − ◦ d / / n ∗ ν / / ❴❴❴ [1 0] (cid:15) (cid:15) [1 0]= p C (cid:15) (cid:15) x / / y / / δ / / ❴❴❴❴ (cid:8) (cid:8) is a morphism of E -triangles. Then n − ◦ d satisfies p C ◦ n − ◦ d ◦ h i = y ◦ [1 0] ◦ h i = y , and thus is of the form (cid:20) y ∗∗ ∗ (cid:21) . (cid:3) The following construction gives extriangulated categories which are not exactnor triangulated in general.
Proposition 3.30.
Let
I ⊆ C be a full additive subcategory, closed under iso-morphisms. If I satisfies I ⊆
Proj( C ) ∩ Inj( C ), then the ideal quotient C / I hasthe structure of an extriangulated category, induced from that of C . In particular,we can associate a “ reduced ” extriangulated category C ′ = C / (Proj( C ) ∩ Inj( C ))satisfying Proj( C ′ ) ∩ Inj( C ′ ) = 0, to any extriangulated category ( C , E , s ). Proof.
Put C = C / I . Let us confirm conditions (ET1),(ET2),(ET3),(ET4). Theother conditions (ET3) op ,(ET4) op can be shown dually.(ET1) Since E ( I , C ) = E ( C , I ) = 0, one can define the biadditive functor E : C op × C → Ab given by • E ( C, A ) = E ( C, A ) ( ∀ A, C ∈ C ), • E ( c, a ) = E ( c, a ) ( ∀ a ∈ C ( A, A ′ ) , c ∈ C ( C, C ′ )). Here, a and c denote theimages of a and c in C / I .(ET2) For any E -extension δ ∈ E ( C, A ) = E ( C, A ), define s ( δ ) = s ( δ ) = [ A x −→ B y −→ C ] , using s ( δ ) = [ A x −→ B y −→ C ]. Let us show that s is an additive realization of E .Let ( a, c ) : δ = ( A, δ, C ) → δ ′ = ( A ′ , δ ′ , C ′ ) be any morphism of E -extensions. Bydefinition, this is equivalent to that ( a, c ) : δ → δ ′ is a morphism of E -extensions.Put s ( δ ) = [ A x −→ B y −→ C ], s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ]. Since the condition inDefinition 2.9 does not depend on the representatives of the equivalence classes ofsequences in C , we may assume s ( δ ) = [ A x −→ B y −→ C ], s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ]. Then there is b ∈ C ( B, B ′ ) with which ( a, b, c ) realizes ( a, c ). It follows that( a, b, c ) realizes ( a, c ).As for the additivity, the equality s (0) = 0 is trivially satisfied. Since s ( δ ) ⊕ s ( δ ′ )only depends on the equivalence classes s ( δ ) and s ( δ ′ ), the equality s ( δ ⊕ δ ′ ) = s ( δ ) ⊕ s ( δ ′ ) follows from s ( δ ⊕ δ ′ ) = s ( δ ) ⊕ s ( δ ′ ).(ET3) Suppose we are given s ( δ ) = [ A x −→ B y −→ C ], s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ],and morphisms a ∈ C ( A, A ′ ) , b ∈ C ( B, B ′ ) satisfying x ′ ◦ a = b ◦ x . As in the proofof (ET2), we may assume s ( δ ) = [ A x −→ B y −→ C ], s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ]. By x ′ ◦ a = b ◦ x , there exist I ∈ I , i ∈ C ( A, I ) , j ∈ C ( I, B ′ ) which satisfy x ′ ◦ a = b ◦ x + j ◦ i . By Lemma 3.29, we obtain an E -triangle (14). This gives the following OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 29 isomorphism of E -triangles.(15) A B ⊕ I C ⊕ IA B C x I / / y I / / p ∗ C δ / / ❴❴❴ p B (cid:15) (cid:15) p C (cid:15) (cid:15) x / / y / / δ / / ❴❴❴❴ (cid:8) (cid:8) On the other hand, by (ET3) for ( C , E , s ), we have a morphism of E -triangles asfollows.(16) A B ⊕ I C ⊕ IA ′ B ′ C ′ x I / / y I / / p ∗ C δ / / ❴❴❴ [ b j ] (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴❴ (cid:8) (cid:8) From (15) and (16), we obtain a morphism of E -extensions ( a, c ◦ p C − ) : δ → δ ′ which satisfies ( c ◦ p C − ) ◦ y = y ′ ◦ b .(ET4) Let A f −→ B f ′ −→ D δ and B g −→ C g ′ −→ F δ ′ be E -triangles. As inthe above arguments, we may assume A f −→ B f ′ −→ D δ and B g −→ C g ′ −→ F δ ′ are E -triangles. Then by (ET4) for ( C , E , s ), we obtain a commutative diagram(3) made of conflations, satisfying s ( f ′∗ δ ′ ) = [ D d −→ E e −→ F ], d ∗ δ ′′ = δ and f ∗ δ ′′ = e ∗ δ ′ . The image of this diagram in C shows (ET4) for ( C , E , s ). (cid:3) Remark 3.31.
Proposition 3.30 applied to an exact category, together with Corol-lary 3.18, gives an another proof of [DI, Theorem 3.5] . Corollary 3.32.
Let
I ⊆ C be a full additive subcategory closed under isomor-phisms, satisfying E ( I , I ) = 0. Let Z ⊆ C be the full subcategory of those Z ∈ C satisfying E ( Z, I ) = E ( I , Z ) = 0. Then, Z / I is extriangulated. Proof.
This follows from Remark 2.18 and Proposition 3.30. (cid:3) Cotorsion pairs
In the rest of this article, let ( C , E , s ) be an extriangulated category.4.1. Cotorsion pairs.Definition 4.1.
Let U , V ⊆ C be a pair of full additive subcategories, closed underisomorphisms and direct summands. The pair ( U , V ) is called a cotorsion pair on C if it satisfies the following conditions.(1) E ( U , V ) = 0.(2) For any C ∈ C , there exists a conflation V C → U C → C satisfying U C ∈ U , V C ∈ V .(3) For any C ∈ C , there exists a conflation C → V C → U C satisfying U C ∈ U , V C ∈ V . The first author wishes to thank Professor Osamu Iyama for informing this to him.
Definition 4.2.
Let X , Y ⊆ C be any pair of full subcategories closed underisomorphisms. Define full subcategories Cone( X , Y ) and CoCone( X , Y ) of C asfollows. These are closed under isomorphisms.(i) C ∈ C belongs to Cone( X , Y ) if and only if it admits a conflation X → Y → C satisfying X ∈ X , Y ∈ Y .(ii) C ∈ C belongs to CoCone( X , Y ) if and only if it admits a conflation C → X → Y satisfying X ∈ X , Y ∈ Y .If X and Y are additive subcategories of C , then so are Cone( X , Y ) and CoCone( X , Y ),by condition (ET2). Remark 4.3.
In the case of exact categories, cotorsion pairs satisfying (2) and (3)are often called complete cotorsion pairs. Since all cotorsion pairs considered inthis article are complete, this adjective is omitted. Also remark that completenessis equivalent to the equalities C = Cone( V , U ) = CoCone( V , U ) , in the notation ofDefinition 4.2. Remark 4.4.
Let ( U , V ) be a cotorsion pair on ( C , E , s ) . By Remark 2.11 (1) , thefollowing holds for any C ∈ C . (1) C ∈ U ⇔ E ( C, V ) = 0 . (2) C ∈ V ⇔ E ( U , C ) = 0 . Corollary 4.5.
Let ( U , V ) be a cotorsion pair on ( C , E , s ). Let C ∈ C , U ∈ U beany pair of objects. If there exists a section C → U or a retraction U → C , then C also belongs to U . Similarly for V . Proof.
In either case, there are s ∈ C ( C, U ) and r ∈ C ( U, C ) satisfying r ◦ s = id C .This gives the following commutative diagram of natural transformations. E ( C, − ) E ( U, − ) E ( C, − ) E ( r, − ) ? ✈✈✈✈✈ ✈✈✈✈✈ E ( s, − ) (cid:31) ' ❍❍❍❍❍ ❍❍❍❍❍ E (id C , − )=id / (cid:8) Thus E ( U, V ) = 0 implies E ( C, V ) = 0, and thus C ∈ U by Remark 4.4. (cid:3) Remark 4.6.
Let ( U , V ) be a cotorsion pair on C . By Proposition 3.11, the sub-categories U and V are extension-closed in C . Remark 4.7.
By Proposition 3.24 and Remark 4.4, the following are equivalent. (1) ( X , C ) is a cotorsion pair for some subcategory X ⊆ C . (2) (Proj( C ) , C ) is a cotorsion pair. (3) C has enough projectives.Dually, the following are equivalent. (1) ( C , X ) is a cotorsion pair for some subcategory X ⊆ C . (2) ( C , Inj( C )) is a cotorsion pair. (3) C has enough injectives. Associated adjoint functors.Definition 4.8.
For a cotorsion pair ( U , V ) on C , put I = U ∩ V , and call it the core of ( U , V ). For any full additive subcategory X ⊆ C containing I , let X / I denote the ideal quotient. The image of a morphism f in the ideal quotient isdenoted by f . OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 31
Lemma 4.9.
For any cotorsion pair ( U , V ), we have ( C / I )( U / I , V / I ) = 0. Namely,for any U ∈ U and V ∈ V , any morphism f ∈ C ( U, V ) factors through some I ∈ I . Proof.
Resolve V by an E -triangle V ′ x −→ U ′ y −→ V λ satisfying U ′ ∈ U , V ′ ∈ V . Since V is extension-closed, it follows that U ′ ∈ U ∩ V = I . By the exactness of C ( U, U ′ ) C ( U,y ) −→ C ( U, V ) ( λ ♯ ) U −→ E ( U, V ′ ) = 0obtained in Proposition 3.3, any morphism f ∈ C ( U, V ) factors through U ′ ∈ I . (cid:3) Proposition 4.10.
Let C ∈ C be any object, and let λ be an E -extension with s ( λ ) = [ V C v C −→ U C u C −→ C ] ( U C ∈ U , V C ∈ V ) . Then the morphism u C has the following property. • For any U ∈ U , the map(17) u C ◦ − : ( C / I )( U, U C ) → ( C / I )( U, C )is bijective.
Proof.
By the exactness of C ( U, U C ) C ( U,u C ) −→ C ( U, C ) ( λ ♯ ) U −→ E ( U, V ) = 0 , the map C ( U, U C ) → C ( U, C ) is surjective. This implies the surjectivity of (17).Let us show the injectivity of (17). Let g ∈ C ( U, U C ) be any morphism whichsatisfies u C ◦ g = u C ◦ g = 0. By definition, there exist I ∈ I , i ∈ C ( U, I ) and i ∈ C ( I, C ) which makes the following diagram commutative.
U IV C U C C i / / g (cid:15) (cid:15) i (cid:15) (cid:15) v C / / u C / / (cid:8) Since E ( I, V C ) = 0, C ( I, V C ) C ( I,v C ) −→ C ( I, U C ) C ( I,u C ) −→ C ( I, C ) → j ∈ C ( I, U C ) satisfying u C ◦ j = i . Then by u C ◦ ( g − j ◦ i ) = 0, we obtain h ∈ C ( U, V C ) satisfying v C ◦ h = g − j ◦ i .By Lemma 4.9, this h factors through some I ′ ∈ I . It follows that g = v C ◦ h + j ◦ i factors through I ⊕ I ′ ∈ I . (cid:3) Proposition 4.10 means that ( U C , u C ) is coreflection of C ∈ C / I along theinclusion functor U / I ֒ → C / I ([Bo, Definition 3.1.1]). As a corollary, we have thefollowing. Corollary 4.11.
The inclusion functor U / I ֒ → C / I has a right adjoint ω U : C / I →U / I , which assigns ω U ( C ) = U C for any C ∈ C / I , where V C v C −→ U C u C −→ C ( U C ∈ U , V C ∈ V )is a conflation. Moreover ε U = { u C } C ∈ Ob( C / I ) gives the counit of this adjoint pair. Concentric twin cotorsion pairs.Definition 4.12.
Let ( S , T ) and ( U , V ) be cotorsion pairs on C . Then the pair P = (( S , T ) , ( U , V )) is called a twin cotorsion pair if it satisfies E ( S , V ) = 0. (Pairsof cotorsion pairs are considered in abelian/exact categories in [Ho1, Ho2], and intriangulated categories in [Na2, Na3].)If moreover it satisfies S ∩ T = U ∩ V (= I ), then P is called a concentric twincotorsion pair similarly as in the triangulated case [Na4]. In this case, we put Z = T ∩ U . Remark 4.13.
Let P = (( S , T ) , ( U , V )) be a concentric twin cotorsion pair. ByCorollary 4.11, the inclusion U / I ֒ → C / I has a right adjoint ω U . Dually theinclusion T / I ֒ → C / I has a left adjoint σ T .These restrict to yield the following. - Left adjoint σ of the inclusion Z / I ֒ → U / I . - Right adjoint ω of the inclusion Z / I ֒ → T / I . Definition 4.14.
Let P = (( S , T ) , ( U , V )) be a twin cotorsion pair on C . Wedefine full subcategories N i , N f of C as follows.(1) N i = Cone( V , S ).(2) N f = CoCone( V , S ). Remark 4.15.
The notation N i , N f is motivated by Section 5. Morally, an object X belongs to N i if and only if the morphism → X from the initial object is aweak equivalence. For a more precise statement, see Proposition 5.7. Remark 4.16. If P is concentric, then for any C ∈ C , we have (1) C ∈ N i ⇔ ω U ( C ) ∈ S / I , (2) C ∈ N f ⇔ σ T ( C ) ∈ V / I . Remark 4.17.
Let P = (( S , T ) , ( U , V )) be a twin cotorsion pair. Then the follow-ing holds. (1) S ⊆ N i , V ⊆ N f . (2) U ∩ N i = S , T ∩ N f = V . (3) If P is concentric, S ⊆ N f , V ⊆ N i . Lemma 4.18.
Let P = (( S , T ) , ( U , V )) be a concentric twin cotorsion pair. Thenthe following holds.(1) Cone( V , N i ) ⊆ N i .(2) CoCone( N f , S ) ⊆ N f . Proof. (1) By definition, C ∈ Cone( V , N i ) admits a conflation V → N → C ( V ∈ V , N ∈ N i ) . Resolve N by a conflation V N → S N → N ( S N ∈ S , V N ∈ V ) . OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 33
By (ET4) op , we obtain a commutative diagram in C V N V N ∃ E S N CV N C (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:8)(cid:8) (cid:8) in which V N → E → V and E → S N → C are conflations. Since V ⊆ C isextension-closed, it follows that E ∈ V . This means C ∈ N i . (2) is dual to (1). (cid:3) Lemma 4.19.
Let P = (( S , T ) , ( U , V )) be a concentric twin cotorsion pair. Let U ∈ U be any object. Assume there is a conflation M → U → S satisfying M ∈ N f and S ∈ S . Then U belongs to N f .Dually, if T ∈ T appears in a conflation V → T → N satisfying V ∈ V , N ∈ N i ,then T belongs to N i . Proof.
By definition, M admits a conflation M → V M → S M ( V M ∈ V , S M ∈ S ) . By Proposition 3.15 (2), we obtain a commutative diagram in C M U SV M ∃ X SS M S M / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) consisting of conflations. Since U is extension-closed, it follows that X ∈ U . Since E ( S, V M ) = 0, the E -extension realized by V M → X → S splits. Especially V M is adirect summand of X , and thus it follows that V M ∈ U ∩ V = I . By the extension-closedness of S , we obtain X ∈ S . Thus U ∈ N f follows from Remark 4.17 (3) andLemma 4.18 (2). (cid:3) Lemma 4.20.
Let P be as in Lemma 4.19. Let T ∈ T , M ∈ N f be any pair ofobjects. If there is a section T → M or a retraction M → T , then T belongs to V . Proof.
In either case, we have morphisms s ∈ C ( T, M ) and r ∈ C ( M, T ) satisfying r ◦ s = id. By definition, M admits a conflation M v −→ V → S. By E ( S, T ) = 0, the morphism r factors through v . Then v ◦ s ∈ C ( T, V ) becomesa section, and thus it follows from Corollary 4.5 that T ∈ V . (cid:3) Bijective correspondence with model structures
In the rest, let ( C , E , s ) be an extriangulated category.In this section, we give a bijective correspondence between Hovey twin cotorsionpairs and admissible model structures which we will soon define. This gives aunification of the following preceding works. • For an abelian category, Hovey has shown their correspondence in [Ho1,Ho2] ( abelian model structure ). This has been generalized to an exact cate-gory by Gillespie [G] ( exact model structure ), and investigated by ˇSˇtov´ıˇcek[S]. • For a triangulated category, Yang [Y] has introduced an analogous notion of triangulated model structure and showed its correspondence with cotorsionpairs.5.1.
Hovey twin cotorsion pair.
We recall that N i (resp. N f ) is the collectionof all objects X ∈ C which are part of a conflation V −→ S −→ X (resp. X −→ V −→ S ), for some V ∈ V and S ∈ S . Definition 5.1.
Let P = (( S , T ) , ( U , V )) be a twin cotorsion pair. We call P a Hovey twin cotorsion pair if it satisfies N f = N i . We denote this subcategory by N . Remark 5.2.
Any Hovey twin cotorsion pair is concentric. In fact, we have
U ∩V = U ∩ ( N f ∩ T ) = ( U ∩ N i ) ∩ T = S ∩ T by Remark 4.17 (2) . For any Hovey twin cotorsion pair, the subcategory
N ⊆ C is extension-closed.More strongly, it satisfies the following. Proposition 5.3.
Let P = (( S , T ) , ( U , V )) be a Hovey twin cotorsion pair. Forany conflation A x −→ B y −→ C , if two out of A, B, C belong to N , then so does thethird. Namely, we have the following.(1) A, C ∈ N ⇒ B ∈ N .(2) A, B ∈ N ⇒ C ∈ N .(3) B, C ∈ N ⇒ A ∈ N . Proof. (1) Resolve
A, C by conflations A → V A → S A and V C → S C → C ( S A , S C ∈ S , V A , V C ∈ V )respectively. By Proposition 3.15 (1),(2), we obtain commutative diagrams V C V C A ∃ X S C A B C (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:8)(cid:8) (cid:8) , A X S C V A ∃ Y S C S A S A / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) which are made of conflations. Since E ( S C , V A ) = 0, the conflation V A → Y → S C realizes the split E -extension. It follows that Y ∼ = V A ⊕ S C ∈ N . We obtain X ∈ N by Lemma 4.18 (2), and thus B ∈ N by Lemma 4.18 (1).(2) Resolve A by a conflation A → V A → S A ( S A ∈ S , V A ∈ V ) . OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 35
Then by Proposition 3.15 (2), we have a commutative diagram
A B CV A ∃ G CS A S A / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) made of conflations. Since B, S A ∈ N , we have G ∈ N by (1). From Lemma 4.18(1), it follows that C ∈ N .(3) is dual to (2). (cid:3) Remark 5.4.
As a corollary, N becomes an extriangulated category by Remark 2.18.Almost by definition of N , the pair ( S , V ) gives a cotorsion pair on N . From admissible model structure to Hovey twin cotorsion pair.
Through-out this section, let M = ( Fib , Cof , W ) be a model structure on C , where Fib , Cof , W are the classes of fibrations, cofibrations, and weak equivalences. Let w Fib = Fib ∩ W and w Cof = Cof ∩ W denote the classes of acyclic fibrations and acycliccofibrations, respectively. Associate full subcategories S , T , U , V ⊆ C as follows. C ∈ S ⇔ (0 → C ) ∈ w Cof ,C ∈ T ⇔ ( C → ∈ Fib ,C ∈ U ⇔ (0 → C ) ∈ Cof ,C ∈ V ⇔ ( C → ∈ w Fib . Remark that these are full additive subcategories of C , closed under isomorphismsand direct summands. In particular, the definition below makes sense. Definition 5.5. M is called an admissible model structure if it satisfies the follow-ing conditions for any morphism f ∈ C ( A, B ).(1) f ∈ w Cof if and only if it is an inflation with Cone( f ) ∈ S .(2) f ∈ Fib if and only if it is a deflation with CoCone( f ) ∈ T .(3) f ∈ Cof if and only if it is an inflation with Cone( f ) ∈ U .(4) f ∈ w Fib if and only if it is a deflation with CoCone( f ) ∈ V .We note that the model structures which might appear in [Pal] are not admissible. Proposition 5.6.
Let M be an admissible model structure. Then P = (( S , T ) , ( U , V ))is a twin cotorsion pair on ( C , E , s ). Proof.
S ⊆ U is obvious from the definition. Since a similar argument works for( S , T ), we show that ( U , V ) is a cotorsion pair. Let us confirm the conditions(1),(2),(3) in Definition 4.1.(1) Let U ∈ U , V ∈ V be any pair of objects, and let δ ∈ E ( U, V ) be any element.Realize it as an E -triangle V v −→ B u −→ U δ . Since U ∈ U and u ∈ w Fib , there exists a section s ∈ C ( U, B ) of u . Thus δ splitsby Corollary 3.5. (2) Let C ∈ C be any object. Factorize the zero morphism 0 : 0 → C as follows.0 CD / / i (cid:26) (cid:26) ✻✻✻✻✻✻✻ D D u ✟✟✟✟✟✟✟ (cid:8) i ∈ Cof ,u ∈ w Fib
Since M is admissible, we have conflations0 i −→ D j −→ U, and V → D u −→ C with U ∈ U , V ∈ V . This shows that j is an isomorphism, and thus we obtain aconflation V → U → C .(3) is dual to (2). (cid:3) Proposition 5.7.
Let M be an admissible model structure as above. Then the as-sociated twin cotorsion pair P obtained in Proposition 5.6 is a Hovey twin cotorsionpair.Indeed, if we let N i , N f ⊆ C be as in Definition 4.14, then the following areequivalent for any object N ∈ C .(1) N ∈ N i .(2) (0 → N ) ∈ W .(3) ( N → ∈ W .(4) N ∈ N f . Proof. (1) ⇒ (2) If N ∈ N i , there is a conflation V → S s −→ N ( V ∈ V , S ∈ S )by definition. Thus 0 → N can be factorized as follows.0 NS / / (cid:26) (cid:26) ✻✻✻✻✻✻✻ D D s ✟✟✟✟✟✟✟ (cid:8) s ∈ w Fib , (0 → S ) ∈ w Cof
It follows that (0 → N ) ∈ w Fib ◦ w Cof = W .(2) ⇒ (1) Factorize 0 → N as follows.0 ND / / i (cid:26) (cid:26) ✻✻✻✻✻✻✻ D D u ✟✟✟✟✟✟✟ (cid:8) i ∈ w Cof ,u ∈ w Fib
A similar argument as in the proof (2) of Proposition 5.6 gives a conflation V → S → N .(2) ⇔ (3) follows from the 2-out-of-3 property of W .(3) ⇔ (4) is dual to (1) ⇔ (2). (cid:3) OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 37
From Hovey twin cotorsion pair to admissible model structure.
Through-out this section, let P = (( S , T ) , ( U , V )) be a Hovey twin cotorsion pair on ( C , E , s ).In addition, we assume the following condition, analogous to the weak idempotentcompleteness ([Bu, Proposition 7.6]). Condition 5.8 (WIC) . Let ( C , E , s ) be an extriangulated category. Consider thefollowing conditions. (1) Let f ∈ C ( A, B ) , g ∈ C ( B, C ) be any composable pair of morphisms. If g ◦ f is an inflation, then so is f . (2) Let f ∈ C ( A, B ) , g ∈ C ( B, C ) be any composable pair of morphisms. If g ◦ f is a deflation, then so is g . With the assumption of Condition (WIC), we have the following analog of thenine lemma.
Lemma 5.9.
Assume ( C , E , s ) is an extriangulated category satisfying Condi-tion (WIC). Let K K ′ A B CA ′ B ′ C ′ k (cid:15) (cid:15) k ′ (cid:15) (cid:15) x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ κ (cid:15) (cid:15) ✤✤✤✤ κ ′ (cid:15) (cid:15) ✤✤✤✤ (cid:8) be a diagram made of E -triangles. Then for some X ∈ C , we obtain E -triangles K m −→ K ′ n −→ X ν and X i −→ C c −→ C ′ τ which make the following diagram commutative,(18) K K ′ XA B CA ′ B ′ C ′ m / / n / / ν / / ❴❴❴ k (cid:15) (cid:15) k ′ (cid:15) (cid:15) i (cid:15) (cid:15) x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ κ (cid:15) (cid:15) ✤✤✤✤ κ ′ (cid:15) (cid:15) ✤✤✤✤ τ (cid:15) (cid:15) ✤✤✤✤ (cid:8) (cid:8)(cid:8) (cid:8) in which, those ( k, k ′ , i ) , ( a, b, c ) , ( m, x, x ′ ) , ( n, y, y ′ ) are morphisms of E -triangles. Proof.
By (ET4) op , we obtain an E -triangle E f −→ B ′ y ′ ◦ b −→ C ′ θ and a commutative diagram K ′ E A ′ K ′ B B ′ C ′ C ′ d / / e / / f (cid:15) (cid:15) x ′ (cid:15) (cid:15) k ′ / / b / / y ′ ◦ b (cid:15) (cid:15) y ′ (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) in C , satisfying the following compatibilities.(i) K ′ d −→ E e −→ A ′ x ′∗ κ ′ is an E -triangle,(ii) δ ′ = e ∗ θ ,(iii) d ∗ κ ′ = y ′∗ θ .By the dual of Lemma 3.13, the upper-right square E BA ′ B ′ f / / e (cid:15) (cid:15) b (cid:15) (cid:15) x ′ / / (cid:8) is a weak pullback. Thus there exists a morphism g ∈ C ( A, E ) which makes thefollowing diagram commutative.
A E BA ′ B ′ f / / e (cid:15) (cid:15) b (cid:15) (cid:15) x ′ / / g (cid:31) (cid:31) ❄❄❄❄❄❄ x " " a (cid:28) (cid:28) (cid:8)(cid:8)(cid:8) By Condition (WIC), this g becomes an inflation. Complete it into an E -triangle A g −→ E h −→ X µ . By Lemma 3.14, we obtain a commutative diagram
A E XA B CC ′ C ′ g / / h / / f (cid:15) (cid:15) i (cid:15) (cid:15) x / / y / / y ′ ◦ b (cid:15) (cid:15) c (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) made of conflations, which satisfy(iv) X i −→ C c −→ C ′ h ∗ θ is an E -triangle,(v) µ = i ∗ δ ,(vi) g ∗ δ = c ∗ θ . OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 39
By Proposition 3.17, we obtain an E -triangle K m −→ K ′ n −→ X ν −→ which makes the diagram K A A ′ K ′ E A ′ X X k / / a / / m (cid:15) (cid:15) g (cid:15) (cid:15) d / / e / / n (cid:15) (cid:15) h (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) commutative in C , and satisfies(vii) m ∗ κ = x ′∗ κ ′ ,(viii) µ = k ∗ ν ,(ix) h ∗ ν + e ∗ κ = 0.Put τ = h ∗ θ . It is straightforward to show that the diagram (18) is indeed com-mutative. Moreover,- (v) and (viii) show k ∗ ν = i ∗ δ ,- (ii) and (vi) show a ∗ δ = c ∗ δ ′ ,- (vii) shows m ∗ κ = x ′∗ κ ′ ,- (iii) shows n ∗ κ ′ = y ′∗ τ . (cid:3) Remark 5.10.
In the proof of Lemma 5.9, we have obtained an extra compatibility (ix) . This can be interpreted by the following analog of
Ext -group.Let A, D ∈ C be any pair of objects. We denote triplet of X ∈ C , σ ∈ E ( D, X ) , τ ∈ E ( X, A ) by ( σ, X, τ ) . For any pair of such triplets ( σ, X, τ ) and ( σ ′ , X ′ , τ ′ ) , we writeas ( σ, X, τ ) x ( σ ′ , X ′ , τ ′ ) ( or simply ( σ, X, τ ) ( σ ′ , X ′ , τ ′ )) if and only if there exists x ∈ C ( X, X ′ ) satisfying x ∗ σ = σ ′ and τ = x ∗ τ ′ .Let ∼ be the equivalence relation generated by , and denote the equivalenceclass of ( σ, X, τ ) by τ ◦ X σ . Let us denote their collection by E ( D, A ) = (cid:18) a X ∈ C E ( D, X ) × E ( X, A ) (cid:19) / ∼ . The proof of Lemma 5.9 shows ( δ ′ , A ′ , − κ ) e ( θ, A ′ , h ∗ ν ) h ( τ, X, ν ) and thus ( − κ ) ◦ A ′ δ ′ = ν ◦ X τ holds in E ( C ′ , K ) . Definition 5.11.
Define classes of morphisms
Fib , w
Fib , Cof , w
Cof and W in C as follows.(1) f ∈ Fib if it is a deflation with CoCone( f ) ∈ T .(2) f ∈ w Fib if it is a deflation with CoCone( f ) ∈ V .(3) f ∈ Cof if it is a inflation with Cone( f ) ∈ U .(4) f ∈ w Cof if it is a inflation with Cone( f ) ∈ S . (5) W = w Fib ◦ w Cof . Claim 5.12. (1)
If a conflation A f −→ B → N satisfies N ∈ N , then f belongs to W . (2) If a conflation N → A f −→ B satisfies N ∈ N , then f belongs to W .Proof. This follows from Proposition 3.15. (cid:3)
Proposition 5.13.
Fib , w
Fib , Cof , w
Cof are closed under composition.
Proof.
For
Fib , this follows from (ET4) and the extension-closedness of T . Similarlyfor the others. (cid:3) Proposition 5.14.
We have the following.(1) w Cof satisfies the left lifting property against
Fib .(2) w Fib satisfies the right lifting property against
Cof . Proof. (1) Suppose we are given a commutative square(19)
A CB D a / / f (cid:15) (cid:15) g (cid:15) (cid:15) b / / (cid:8) in C , satisfying f ∈ w Cof and g ∈ Fib . By definition, there are E -triangles A f −→ B s −→ S δ ,T t −→ C g −→ D κ . By Corollary 3.12, C ( B, T ) C ( f,T ) −→ C ( A, T ) → → E ( B, T ) E ( f,T ) −→ E ( A, T ) , (20) C ( A, T ) C ( A,t ) −→ C ( A, C ) C ( A,g ) −→ C ( A, D ) , (21) C ( B, C ) C ( B,g ) −→ C ( B, D ) ( κ ♯ ) B −→ E ( B, T )(22)are exact.By the commutativity of (19), we have E ( f, T )( b ∗ κ ) = f ∗ b ∗ κ = a ∗ g ∗ κ = 0by Lemma 3.2. Exactness of (20) shows κ ♯ b = b ∗ κ = 0 . Thus by the exactness of (22), there exists c ∈ C ( B, C ) satisfying g ◦ c = b . Then a − c ◦ f ∈ C ( A, C ) satisfies g ◦ ( a − c ◦ f ) = g ◦ a − b ◦ f = 0 . By the exactness of (21), there is c ′ ∈ C ( A, T ) satisfying t ◦ c ′ = a − c ◦ f .By the exactness of (20), there is c ′′ ∈ C ( A, T ) satisfying c ′′ ◦ f = c ′ . If we put h = c + t ◦ c ′′ ∈ C ( B, C ), it satisfies h ◦ f = c ◦ f + t ◦ c ′′ ◦ f = c ◦ f + t ◦ c ′ = a and g ◦ h = g ◦ c + g ◦ t ◦ c ′′ = b .(2) is dual to (1). (cid:3) OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 41
Proposition 5.15.
Mor( C ) = w Fib ◦ Cof = Fib ◦ w Cof . Proof.
We only show Mor( C ) = w Fib ◦ Cof . Let f ∈ C ( A, B ) be any morphism.Resolve A by a conflation A v A −→ V A u A −→ U A ( U A ∈ U , V A ∈ V ) , and put f ′ = h fv A i : A → B ⊕ V A . By Corollary 3.16, it admits some conflation A f ′ −→ B ⊕ V A → C. Resolve C by a conflation V C → U C → C ( U ∈ U , V ∈ V ) . Then by Proposition 3.15 (1), we obtain a diagram made of conflations as follows. V C V C A M U C A B ⊕ V A C (cid:15) (cid:15) (cid:15) (cid:15) m / / / / e (cid:15) (cid:15) (cid:15) (cid:15) f ′ / / / / (cid:8)(cid:8) (cid:8) We have m ∈ Cof . Moreover, for the projection p B = [1 0] ∈ C ( B ⊕ V A , B ),we have p B ◦ e ∈ w Fib ◦ w Fib = w Fib . Thus f = ( p B ◦ e ) ◦ m gives the desiredfactorization. (cid:3) Proposition 5.16.
Fib , w
Fib , Cof , w
Cof are closed under retraction.
Proof.
We only show the result for
Fib . Suppose we are given a commutativediagram
A C AB D B a - - ❩❩❩❩❩❩ c ❞❞❞❞❞❞ id ' ' f (cid:15) (cid:15) g (cid:15) (cid:15) f (cid:15) (cid:15) b ❞❞❞❞❞❞ d - - ❩❩❩❩❩❩ id (cid:8) (cid:8)(cid:8)(cid:8) in C , satisfying g ∈ Fib . By definition, there is an E -triangle T t −→ C g −→ D θ ( T ∈ T ) . By Condition (WIC), d ◦ b = id implies that d is a deflation. Thus d ◦ g becomesa deflation by (ET4) op . Again by Condition (WIC), it follows that f is a deflation.Thus there exists an E -triangle X x −→ A f −→ B δ . By (ET3) op , we obtain thefollowing morphisms of E -triangles. X A BT C D x / / f / / δ / / ❴❴❴ k (cid:15) (cid:15) a (cid:15) (cid:15) b (cid:15) (cid:15) t / / g / / θ / / ❴❴❴ (cid:8) (cid:8) , T C DX A B t / / g / / θ / / ❴❴❴ ℓ (cid:15) (cid:15) c (cid:15) (cid:15) d (cid:15) (cid:15) x / / f / / δ / / ❴❴❴ (cid:8) (cid:8) Composing them, we obtain a morphism
X A BX A B x / / f / / δ / / ❴❴❴ ℓ ◦ k (cid:15) (cid:15) id (cid:15) (cid:15) id (cid:15) (cid:15) x / / f / / δ / / ❴❴❴ (cid:8) (cid:8) of E -triangles. By Corollary 3.6, it follows that ℓ ◦ k is an isomorphism. Especially k is a section, and thus X ∈ T . This means that f belongs to Fib . (cid:3) Lemma 5.17.
Suppose that a commutative diagram in C A B C f @ @ ✂✂✂✂✂ g (cid:30) (cid:30) ❁❁❁❁❁ h / / (cid:8) satisfies f ∈ w Cof , g ∈ Fib and h ∈ w Cof . Then g belongs to w Fib . Proof.
By assumption, there are conflations A f −→ B s −→ S ( S ∈ S ) ,T t −→ B g −→ C ( T ∈ T ) ,A h −→ C s −→ S ( S ∈ S ) . By the dual of Lemma 3.17, we obtain the following commutative diagram madeof conflations.
T TA B S A C S t (cid:15) (cid:15) (cid:15) (cid:15) f / / s / / g (cid:15) (cid:15) (cid:15) (cid:15) h / / s / / (cid:8)(cid:8) (cid:8) By Lemma 4.18 (2) and Remark 4.17 (2), we obtain T ∈ T ∩ N = V . This means g ∈ w Fib . (cid:3) Proposition 5.18. W is closed under composition. Proof.
It suffices to show that w Cof ◦ w Fib ⊆ W . Let a ∈ w Fib and x ′ ∈ w Cof . ByProposition 5.15, there are some x ∈ w Cof and b ∈ Fib such that b ◦ x = x ′ ◦ a . It isthus enough to show that b belongs to w Fib . By definition, there is a commutative
OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 43 diagram of E -triangles: V TA B SA ′ B ′ S ′ k (cid:15) (cid:15) k ′ (cid:15) (cid:15) x / / y / / δ / / ❴❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ κ (cid:15) (cid:15) ✤✤✤✤ κ ′ (cid:15) (cid:15) ✤✤✤✤ (cid:8) with V ∈ V , T ∈ T and S, S ′ ∈ S . Applying Lemma 5.9 gives some X ∈ C andtwo conflations X i −→ S c −→ S ′ and V m −→ T n −→ X . The existence of the firstconflation (and Lemma 4.18(2)) shows that X belongs to N ; that of the latterconflation and the dual of Lemma 4.19 imply that T belongs to V , and thereforethat b ∈ w Fib . (cid:3) Lemma 5.19.
Suppose that a commutative diagram in C A B C f @ @ ✂✂✂✂✂ g (cid:30) (cid:30) ❁❁❁❁❁ h / / (cid:8) satisfies f ∈ w Cof , g ∈ Fib and h ∈ w Fib . Then g belongs to w Fib . Proof.
By assumption, there are conflations A f −→ B s −→ S ( S ∈ S ) ,T t −→ B g −→ C ( T ∈ T ) ,V v −→ A h −→ C ( V ∈ V ) . By Proposition 3.17, we obtain a conflation V → T → S . Thus from Proposition 5.3(1) and Remark 4.17 (2), it follows T ∈ T ∩ N = V . This means g ∈ w Fib . (cid:3) Lemma 5.20.
Suppose that a commutative diagram in C A B C f @ @ ✂✂✂✂✂ g (cid:30) (cid:30) ❁❁❁❁❁ h / / (cid:8) satisfies f ∈ w Fib , g ∈ Fib and h ∈ w Fib . Then g belongs to w Fib . Proof.
By assumption, there are conflations V f → A f −→ B ( V f ∈ V ) ,T → B g −→ C ( T ∈ T ) ,V h → A h −→ C ( V h ∈ V ) . By (ET4) op , we obtain a conflation V f → V h → T . Thus from Lemma 4.18 (1) andRemark 4.17 (2), it follows that T ∈ T ∩ N = V . This means g ∈ w Fib . (cid:3) Proposition 5.21.
Suppose that a commutative diagram in C A B C f @ @ ✂✂✂✂✂ g (cid:30) (cid:30) ❁❁❁❁❁ h / / (cid:8) satisfies f, h ∈ W . Then g also belongs to W . Proof.
By definition and the dual of Proposition 5.15, the morphisms f, g, h can befactorized as
A BX ff / / f (cid:26) (cid:26) ✻✻✻✻✻✻✻ D D f ✟✟✟✟✟✟✟ (cid:8) , B CX gg / / g (cid:26) (cid:26) ✻✻✻✻✻✻✻ D D g ✟✟✟✟✟✟✟ (cid:8) , A CX hh / / h (cid:26) (cid:26) ✻✻✻✻✻✻✻ D D h ✟✟✟✟✟✟✟ (cid:8) , with f , g , h ∈ w Cof , f , h ∈ w Fib and g ∈ Fib .By Proposition 5.18, the morphism g ◦ f ∈ W can be factorized as follows. X f X g X g ◦ f / / w (cid:26) (cid:26) ✻✻✻✻✻✻✻ D D w ✟✟✟✟✟✟✟ (cid:8) ( w ∈ w Cof )( w ∈ w Fib )By Proposition 5.14, there is k ∈ C ( X h , X ) which makes A XX h C w ◦ f / / h (cid:15) (cid:15) g ◦ w (cid:15) (cid:15) h / / k ⑧⑧⑧ ? ? ⑧⑧⑧⑧ (cid:8) (cid:8) commutative in C . By Proposition 5.15, we can factorize k as follows X h XX kk / / k (cid:26) (cid:26) ✻✻✻✻✻✻✻ D D k ✟✟✟✟✟✟✟ (cid:8) ( k ∈ w Cof )( k ∈ Fib )Thus we obtain the following commutative diagram.
A X X g X k X h C w ◦ f / / w % % ▲▲▲ h (cid:15) (cid:15) k ; ; ✈✈✈✈ g (cid:15) (cid:15) k ; ; ✈✈✈ h / / (cid:8) (cid:8) ( k , h , w ◦ f ∈ w Cof )( w , h ∈ w Fib )( k , g ∈ Fib )Lemma 5.17 shows k ∈ w Fib . On the other hand, Lemma 5.19 shows g ◦ w ◦ k ∈ w Fib . Thus Lemma 5.20 shows g ∈ w Fib . (cid:3) Corollary 5.22.
The class W satisfies the 2-out-of-3 condition. OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 45
Proof.
This follows from Proposition 5.18, Proposition 5.21 and its dual. (cid:3)
When a category has enough pull-backs or enough push-outs, the fact that weakequivalences are stable under retracts follows from the other axioms (e.g. [Joy,Proposition E.1.3], attributed to Joyal–Tierney). However, that proof does notcarry over to the setup of extriangulated categories. The following lemma will thusbe used for proving that the class W is closed under retracts. Lemma 5.23.
Let A x −→ B y −→ C δ and A x ′ −→ B ′ y ′ −→ C ′ δ ′ be E -triangles.Suppose that b ∈ C ( B, B ′ ) belongs to W and satisfies b ◦ x = x ′ . Then there is c ∈ C ( C, C ′ ) which belongs to W and gives a morphism of E -triangles as follows. A B CA B ′ C ′ x / / y / / δ / / ❴❴❴ b (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ (cid:8) (cid:8) Proof.
By definition, b can be factorized as b = v ◦ s , using E -triangles B s −→ P → S θ and V → P v −→ B ′ τ with S ∈ S , V ∈ V . By (ET4), and then by thedual of Proposition 3.17, we obtain the following commutative diagrams made of E -triangles, A B CA P ∃ QS S x / / y / / δ / / ❴❴❴ s (cid:15) (cid:15) ∃ c (cid:15) (cid:15) s ◦ x / / ∃ p / / ∃ ν / / ❴❴❴ (cid:15) (cid:15) (cid:15) (cid:15) θ (cid:15) (cid:15) ✤✤✤✤ y ∗ θ (cid:15) (cid:15) ✤✤✤✤ (cid:8) (cid:8)(cid:8) , A AV P B ′ V Q C ′ s ◦ x (cid:15) (cid:15) x ′ (cid:15) (cid:15) / / v / / τ / / ❴❴❴ p (cid:15) (cid:15) y ′ (cid:15) (cid:15) / / ∃ c / / / / ❴❴❴ ν (cid:15) (cid:15) ✤✤✤ δ ′ (cid:15) (cid:15) ✤✤✤✤ (cid:8)(cid:8) (cid:8) in which c ∗ ν = δ holds. Then c = c ◦ c belongs to wCof ◦ wFib ⊆ W byProposition 5.18, satisfies c ◦ y = c ◦ p ◦ s = y ′ ◦ v ◦ s = y ′ ◦ b and c ∗ δ ′ = c ∗ c ∗ δ ′ = c ∗ ν = δ . (cid:3) Proposition 5.24.
The class W is closed under retracts. Proof.
Suppose we are given a commutative diagram in C A C AB D B id $ $ a / / c / / f (cid:15) (cid:15) g (cid:15) (cid:15) f (cid:15) (cid:15) id : : b / / d / / (cid:8)(cid:8) (cid:8)(cid:8) in which g ∈ W . Let us show that f ∈ W . If we decompose f and g as A BM f / / i (cid:27) (cid:27) ✻✻✻✻✻ C C x ✟✟✟✟✟ (cid:8) , C DN g / / j (cid:27) (cid:27) ✻✻✻✻✻ C C y ✟✟✟✟✟ (cid:8) (cid:18) i ∈ Cof , j ∈ wCof ,x, y ∈ w Fib (cid:19) by Proposition 5.15, then there exist morphisms m, n which make the followingdiagram commutative by Proposition 5.14 .
A C AM N MB D B a / / c / / i (cid:15) (cid:15) j (cid:15) (cid:15) i (cid:15) (cid:15) m / / n / / x (cid:15) (cid:15) y (cid:15) (cid:15) x (cid:15) (cid:15) b / / d / / (cid:8) (cid:8)(cid:8) (cid:8) By Corollary 5.22 applied to the lower half, it follows n ◦ m ∈ W . By definition of Cof and wCof , there are E -triangles A i −→ M p −→ U ρ , C j −→ N q −→ S τ with U ∈ U , S ∈ S . It suffices to show U ∈ S .Realize c ∗ τ by an E -triangle A j ′ −→ N ′ q ′ −→ S c ∗ τ . Put c ′ = h − cj i : C → A ⊕ N .Then by an argument similar to that of the proof of Corollary 3.16, we can find amorphism of E -triangles C N SA N ′ S j / / q / / τ / / ❴❴❴ c (cid:15) (cid:15) n (cid:15) (cid:15) j ′ / / q ′ / / c ∗ / / ❴❴❴ (cid:8) (cid:8) which gives an E -triangle C c ′ −→ A ⊕ N [ j ′ n ] −→ N ′ q ′∗ τ (cf. [LN, Proposition 1.20]).Since we have [ i n ] ◦ c ′ = n ◦ j − i ◦ c = 0, there is n ′ ∈ C ( N ′ , M ) which satisfies n ′ ◦ [ j ′ n ] = [ i n ], namely n ′ ◦ j ′ = i and n ′ ◦ n = n .Put m ′ = n ◦ m . This satisfies n ′ ◦ m ′ = n ◦ m ∈ W and m ′ ◦ i = j ′ . Resolve N ′ by an E -triangle T ′ → S ′ s ′ −→ N ′ θ ( S ′ ∈ S , T ′ ∈ T ) . Then by the dual of Corollary 3.16, the morphism [ m ′ s ′ ] : M ⊕ S ′ → N ′ canbe completed into an E -triangle ∃ L → M ⊕ S ′ [ m ′ s ′ ] −→ N ′ . By the dual of OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 47
Proposition 3.17, we obtain the following commutative diagram made of conflations.(23)
A AL M ⊕ S ′ N ′ L U ⊕ S ′ S (cid:15) (cid:15) h i i j ′ (cid:15) (cid:15) / / [ m ′ s ′ ] / / p ⊕ id (cid:15) (cid:15) q ′ (cid:15) (cid:15) ∃ ℓ / / ∃ k / / (cid:8)(cid:8) (cid:8) If we put m = n ′ ◦ [ m ′ s ′ ], then since M ⊕ S ′ M M (cid:0) (cid:0) ✂✂✂✂✂ h i m / / n ′ ◦ m ′ (cid:30) (cid:30) ❁❁❁❁❁ (cid:8) is commutative, m ∈ W follows from h i ∈ wCof and n ′ ◦ m ′ ∈ W , by Corol-lary 5.22.Applying Lemma 5.23 to A M ⊕ S ′ U ⊕ S ′ A M U / / h i i p ⊕ id / / / / ❴❴❴❴ m (cid:15) (cid:15) i / / p / / / / ❴❴❴❴❴ (cid:8) we obtain u ∈ C ( U ⊕ S ′ , U ) which belongs to W satisfying u ◦ ( p ⊕ id S ′ ) = p ◦ m .Then since u ◦ ℓ = 0, we see that u factors through S , in the bottom E -trianglein (23). Thus if we apply the functor σ : U / I → Z / I , it follows σ ( u ) = 0. Onthe other hand, it can be easily seen that u ∈ W implies that u can be written ascomposition of U ⊕ S ′ u ′ −→ U ⊕ I [1 0] −→ U , with u ′ ∈ wCof and I ∈ I . By (ET4),we can show that σ ( u ′ ) is an isomorphism, and thus σ ( u ) is an isomorphism. Thismeans σ ( U ) = 0 in Z / I , which shows U ∈ U ∩ CoCone( I , S ) ⊆ U ∩ N = S . (cid:3) By the argument so far, admissible model structures and Hovey twin cotorsionpairs on ( C , E , s ) correspond bijectively. Remark that, a model structure inducesan equivalence C cf / ∼ ≃ −→ C [ W − ] . Here, the right hand side is the localization ℓ : C → C [ W − ]. The left hand side isthe category of fibrant-cofibrant objects modulo homotopies. Let us describe it interms of the corresponding Hovey twin cotorsion pair P = (( S , T ) , ( U , V )).- X ∈ C is fibrant if and only if ( X → ∈ Fib , if and only if X ∈ T .Dually, X is cofibrant if and only if X ∈ U . Thus the full subcategory offibrant-cofibrant objects in C agrees with Z ⊆ C .- For any X, Y ∈ Z , morphisms f, g ∈ Z ( X, Y ) satisfy f ∼ g if and only if f − g factors through some object I ∈ I .Thus we have C cf / ∼ = Z / I . In summary, we obtain the following. This gives anexplanation for the equivalence in [Na4, Proposition 6.10] and [IYa, Theorem 4.1]. Corollary 5.25.
Let ( C , E , s ) be an extriangulated category, and let P = (( S , T ) , ( U , V ))be a Hovey twin cotorsion pair. Then for W = w Fib ◦ w Cof defined as above, wehave an equivalence Z / I ≃ −→ C [ W − ] which makes the following diagram commu-tative up to natural isomorphism. Z C Z / I C [ W − ] inclusion / / idealquotient (cid:15) (cid:15) ℓ (cid:15) (cid:15) ≃ / / (cid:8) In particular, the map( Z / I )( X, Y ) → C [ W − ]( X, Y ) ; f ℓ ( f )is an isomorphism for any X, Y ∈ Z . Remark 5.26.
By the generality of a model structure, we can also deduce that ( C / I )( U, T ) → C [ W − ]( U, T ) ; f ℓ ( f ) is an isomorphism for any U ∈ U and T ∈ T . (This also follows from the adjointproperty given in Remark 4.13.) Remark 5.27. If C is abelian, then we can show easily that W agrees with the classof morphisms f satisfying Ker( f ) ∈ N and Cok( f ) ∈ N . Remark that N ⊆ C becomes a Serre subcategory only when N = C . Indeed, if N ⊆ C is a Serresubcategory, then the localization ℓ : C → C [ W − ] becomes an exact functor betweenabelian categories. Since any C ∈ C admits an inflation C → V to some V ∈ V ,this shows that C = 0 holds in C [ W − ] , which means C ∈ N . Triangulation of the homotopy category
In this section, we assume (( S , T ) , ( U , V )) to be a Hovey twin cotorsion pair.Put e C = C [ W − ] and let ℓ : C → e C be the localization functor. We will show that e C is triangulated (Theorem 6.20). Remark 6.1.
We note that the category C is usually not complete and cocomplete,so that the model structure is not stable. However, axiom (ET4) gives specificchoices of weak bicartesian squares which will compensate for the lack of stability. Shift functor.
We first aim at defining a shift functor on the category e C . Definition 6.2.
Let us fix a choice, for any object A ∈ C , of an E -triangle A v A −→ V A u A −→ U A ρ A , with V A ∈ V and U A ∈ U . The functor [1] : C → e C is defined onobjects by A [1] = U A , and on morphisms as follows: Let A f −→ B be a morphismin C . Then there exists u f ∈ C ( U A , U B ) which gives a morphism of E -extensions( f, u f ) : ρ A → ρ B . Indeed, since E ( U A , V B ) = 0, one shows, by using the longexact sequence of Proposition 3.3 that there is a morphism V A v f −→ V B such that v f ◦ v A = v B ◦ f . There is an induced morphism of E -triangles as follows. A V A U A B V B U Bv A / / u A / / ρ A / / ❴❴❴ f (cid:15) (cid:15) v f (cid:15) (cid:15) u f (cid:15) (cid:15) v B / / u B / / ρ B / / ❴❴❴ (cid:8) (cid:8) OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 49
Define f [1] to be the image ℓ ( u f ) of u f in e C . Since f ∗ ρ A = u ∗ f ρ B , the morphism u f is uniquely defined in C / V by Corollary 3.5. This implies that f [1] = ℓ ( u f ) iswell-defined. Claim 6.3.
The functor [1] does not essentially depend on the choices made. Moreprecisely, fix any other choice of E -triangles A v ′ A −→ V ′ A u ′ A −→ U ′ A ρ ′ A and let { } bethe functor defined as above by means of these E -triangles. Then [1] and { } arenaturally isomorphic.Proof. The identity on A induces two morphisms of E -extensions(id , ∃ t A ) : ρ A → ρ ′ A and (id , ∃ t ′ A ) : ρ ′ A → ρ A . Then, since both (id , t ′ A ◦ t A ) , (id , id) : ρ A → ρ A are morphisms of E -extensions, we have ℓ ( t ′ A ◦ t A ) = ℓ (id) = id as in the argument inDefinition 6.2. Similarly we have ℓ ( t A ◦ t ′ A ) = id, and thus ℓ ( t A ) is an isomorphism.Put τ A = ℓ ( t A ), and let us show the naturality of τ = { τ A ∈ e C ( A [1] , A { } ) } A ∈ C . Let f ∈ C ( A, B ) be any morphism. By definition, f [1] = ℓ ( u ) and f { } = ℓ ( u ′ ) aregiven by morphisms of E -extensions( f, u ) : ρ A → ρ B and ( f ′ , u ′ ) : ρ ′ A → ρ ′ B . Then, since both ( f, t B ◦ u ) and ( f, u ′ ◦ t A ) are morphisms ρ A → ρ ′ B , we obtain ℓ ( t B ◦ u ) = ℓ ( u ′ ◦ t A ), namely τ B ◦ f [1] = f { } ◦ τ A . (cid:3) We would like to show that [1] : C → e C induces an endofunctor of e C . For this,it is enough to show that [1] sends weak equivalences to isomorphisms. Since weakequivalences are compositions of a morphism in w Cof followed by a morphism in w Fib , it is enough to show that [1] inverts all morphisms in w Cof and in w Fib . Lemma 6.4.
Let j ∈ w Cof . Then j [1] is an isomorphism in e C . Proof.
Let A j −→ B be a morphism in w Cof . There is an E -triangle A j −→ B → S , with S ∈ S . Then (ET4) gives morphisms of E -triangles as follows. A B SA V B CU B U Bj / / / / / / ❴❴❴ v B (cid:15) (cid:15) (cid:15) (cid:15) v ′ A / / u ′ A / / ρ ′ / / ❴❴❴ u B (cid:15) (cid:15) w (cid:15) (cid:15) ρ B (cid:15) (cid:15) ✤✤✤ (cid:15) (cid:15) ✤✤✤ (cid:8) (cid:8)(cid:8) Since S and U B belong to U , so does C . Moreover, w is a weak equivalence byClaim 5.12 (2). Claim 6.3 allows to conclude that j [1] is an isomorphism in e C , since A V B CB V B U Bv ′ A / / u ′ A / / ρ ′ A / / ❴❴❴ j (cid:15) (cid:15) w (cid:15) (cid:15) v B / / u B / / ρ B / / ❴❴❴ (cid:8) (cid:8) is a morphism of E -triangles. (cid:3) Lemma 6.5.
Let q ∈ w Fib . Then q [1] is an isomorphism in e C . Proof.
Let X q −→ Y be a morphism in w Fib . It induces a morphism of E -triangles: X V X U X Y V Y U Yv X / / u X / / ρ X / / ❴❴❴ q (cid:15) (cid:15) n (cid:15) (cid:15) q [1] (cid:15) (cid:15) v Y / / u Y / / ρ Y / / ❴❴❴ (cid:8) (cid:8) Since V X → V Y → V X n −→ V Y is a weakequivalence. Factor n as an acyclic cofibration j followed by an acyclic fibration p .Then (ET4) gives a diagram of E -triangles: X V X U X X B CS S v X / / u X / / j (cid:15) (cid:15) (cid:15) (cid:15) j ◦ v X / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) where S ∈ S . We have B ∈ N (since V X , S ∈ N ) and C ∈ U (since U X , S ∈ U ).The nine Lemma 5.9 gives morphisms of E -triangles: V V ′ DX B CY V Y U Y / / / / / / ❴❴❴ (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) j ◦ v X / / / / / / ❴❴❴ q (cid:15) (cid:15) p (cid:15) (cid:15) q ′ (cid:15) (cid:15) v Y / / u Y / / ρ Y / / ❴❴❴ (cid:15) (cid:15) ✤✤✤✤ (cid:15) (cid:15) ✤✤✤ (cid:15) (cid:15) ✤✤✤ (cid:8) (cid:8)(cid:8) (cid:8) with V, V ′ ∈ V , and thus B ∈ V . This implies D ∈ N and thus q ′ is a weakequivalence by Claim 5.12 (2). We conclude that q [1] is an isomorphism in e C inthe same manner as in the end of the proof of Lemma 6.4. (cid:3) Corollary 6.6.
The functor [1] : C → e C induces an endofunctor of e C , which wedenote by the same symbol [1]. Thus f [1] for a morphism f in C will be denotedalso by ℓ ( f )[1] in the rest. OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 51
Proof.
By Lemma 6.4 and Lemma 6.5, the functor [1] : C → e C inverts all weakequivalences. By the universal property of the localization of a category, it inducesa functor [1] : e C → e C . (cid:3) Definition 6.7.
By using E -triangles T C t C −→ S C s C −→ C λ C one can define duallya functor [ −
1] : C → e C with C [ −
1] = T C , which induces an endofunctor [ −
1] of e C .6.2. Connecting morphism.
Define the bifunctor e E : e C op × e C → e C by e E = e C ( − , − [1]). In this section, we will construct a homomorphism e ℓ = e ℓ C,A : E ( C, A ) → e E ( C, A )for each pair
A, C ∈ C . For any A ∈ C , we continue to use the E -triangle(24) A v A −→ V A u A −→ U A ρ A chosen to define the shift functor [1]. Lemma 6.8.
Let
A, C ∈ C be any pair of objects. If f , f ∈ C ( X, U A ) satisfy f ∗ ρ A = f ∗ ρ A , then ℓ ( f ) = ℓ ( f ) holds. Proof.
This immediately follows from the exactness of C ( X, V A ) u A ◦− −→ C ( X, U A ) ( ρ A ) ♯ −→ E ( X, A )shown in Proposition 3.3. (cid:3)
Definition 6.9.
For any E -extension δ ∈ E ( C, A ), define e ℓ ( δ ) ∈ e E ( C, A ) by thefollowing. • Take a span of morphisms ( C w ←− D d −→ U A ) from some D ∈ C , whichsatisfy(25) w ∈ w Fib and w ∗ δ = d ∗ ρ A . Then, define as e ℓ ( δ ) = ℓ ( d ) ◦ ℓ ( w ) − . Claim 6.10.
For any δ ∈ E ( C, A ) , the morphism e ℓ ( δ ) in Definition 6.9 is well-defined. More precisely, the following holds. (1) Take any D ∈ C , w ∈ C ( D, C ) . If both d , d ∈ C ( D, U A ) satisfy w ∗ δ = d ∗ i ρ A ( i = 1 , , then ℓ ( d ) = ℓ ( d ) holds. (2) If both spans ( C w ←− D d −→ U A ) and ( C w ←− D d −→ U A ) satisfy (25) ,then ℓ ( d ) ◦ ℓ ( w ) − = ℓ ( d ) ◦ ℓ ( w ) − holds. (3) There exists at least one span ( C w ←− D d −→ U A ) satisfying (25) . Proof. (1) This immediately follows from Lemma 6.8.(2) Let V i v i −→ D i w i −→ C ( i = 1 ,
2) be conflations. By Proposition 3.15 (1), wehave a commutative diagram made of conflations as follows. V V V ∃ D D V D C m (cid:15) (cid:15) v (cid:15) (cid:15) m / / e / / e (cid:15) (cid:15) w (cid:15) (cid:15) v / / w / / (cid:8)(cid:8) (cid:8) If we put w = w ◦ e = w ◦ e , then we have w ∈ w Fib ◦ w Fib = w Fib . If we put k = d ◦ e and k = d ◦ e , then they give ℓ ( k ) ◦ ℓ ( w ) − = ℓ ( d ◦ e ) ◦ ℓ ( w ◦ e ) − = ℓ ( d ) ◦ ℓ ( w ) − ,ℓ ( k ) ◦ ℓ ( w ) − = ℓ ( d ◦ e ) ◦ ℓ ( w ◦ e ) − = ℓ ( d ) ◦ ℓ ( w ) − . Since both k , k satisfy k ∗ δ = e ∗ d ∗ δ = e ∗ w ∗ δ = w ∗ δ,k ∗ δ = e ∗ d ∗ δ = e ∗ w ∗ δ = w ∗ δ, we obtain ℓ ( k ) = ℓ ( k ) by (1). Thus it follows that ℓ ( d ) ◦ ℓ ( w ) − = ℓ ( d ) ◦ ℓ ( w ) − .(3) Realize δ by an E -triangle A x −→ B y −→ C δ . Then Proposition 3.15 (2) gives a commutative diagram made of E -triangles A B CV A ∃ D CU A U Ax / / y / / δ / / ❴❴❴ v A (cid:15) (cid:15) (cid:15) (cid:15) / / w / / ( v A ) ∗ δ / / ❴❴❴ u A (cid:15) (cid:15) e (cid:15) (cid:15) ρ A (cid:15) (cid:15) ✤✤✤ x ∗ ρ A (cid:15) (cid:15) ✤✤✤ (cid:8) (cid:8)(cid:8) satisfying w ∗ δ + e ∗ ρ A = 0. Thus the span ( C w ←− D − e −→ U A ) satisfies (25). (cid:3) Proposition 6.11.
For any
A, C ∈ C , the map e ℓ : E ( C, A ) → e E ( C, A ) is an additivehomomorphism.
Proof.
Let δ , δ ∈ E ( C, A ) be any pair of elements. By (3) of Claim 6.10, we canfind spans ( C w ←− D d −→ U A ) , ( C w ←− D d −→ U A ) which give e ℓ ( δ ) , e ℓ ( δ ). Asin the proof of (2) in Claim 6.10, replacing ( w i , D i ) by a common ( w, D ), we mayassume D = D = D and w = w = w OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 53 from the start. Then by w ∗ δ = d ∗ ρ A and w ∗ δ = d ∗ ρ A , we have w ∗ ( δ + δ ) = ( d + d ) ∗ ρ A . Thus the span ( C w ←− D d + d −→ U A ) satisfies (25) for δ + δ . This shows e ℓ ( δ + δ ) = ℓ ( d + d ) ◦ ℓ ( w ) − = ℓ ( d ) ◦ ℓ ( w ) − + ℓ ( d ) ◦ ℓ ( w ) − = e ℓ ( δ ) + e ℓ ( δ ) . (cid:3) Lemma 6.12.
For any U ∈ U and T ∈ T , the map e ℓ : E ( U, T ) → e E ( U, T )is monomorphic.
Proof.
Let δ ∈ E ( U, T ) be any E -extension. Realize it by an E -triangle T x −→ A y −→ U δ . Let T v T −→ V T u T −→ U T ρ T be the chosen E -triangle, as before. By Proposition 3.15(2), we obtain a diagram made of E -triangles T A UV T M UU T U Tx / / y / / δ / / ❴❴❴ v T (cid:15) (cid:15) m (cid:15) (cid:15) m ′ / / e ′ / / ( v T ) ∗ δ / / ❴❴❴ u T (cid:15) (cid:15) e (cid:15) (cid:15) ρ T (cid:15) (cid:15) ✤✤✤ x ∗ ( ρ T ) (cid:15) (cid:15) ✤✤✤ (cid:8) (cid:8)(cid:8) satisfying e ∗ ( ρ T ) + e ′∗ δ = 0. As the proof of (3) of Claim 6.10 suggests, we have e ℓ ( δ ) = − ℓ ( e ) ◦ ℓ ( e ′ ) − . By E ( U, V T ) = 0, we have ( v T ) ∗ δ = 0. Thus, replacing M by an isomorphicobject, we may assume M = V T ⊕ U, m ′ = ι V T : V T → M, e ′ = p U : M → U, where V T V T ⊕ U U ι VT / / p VT o o o o ι U / / p U is a biproduct. Put q = − e ◦ ι U : U → U T . Then we have e = u T ◦ p V T − q ◦ e ′ .Since ℓ ( p V T ) = 0, this gives ℓ ( e ) = − ℓ ( q ) ◦ ℓ ( e ′ ), namely(26) e ℓ ( δ ) = ℓ ( q ) . On the other hand, since the morphism e ′∗ is a monomorphism, the equality e ′∗ δ = − e ∗ ( ρ T ) = − p ∗ V T u ∗ T ( ρ T ) + e ′∗ q ∗ ( ρ T ) = e ′∗ q ∗ ( ρ T )implies(27) δ = q ∗ ( ρ T ) . By (26) and (27), it suffices to show ℓ ( q ) = 0 ⇒ q ∗ ( ρ T ) = 0 . Assume ℓ ( q ) = 0. Take E -triangles U T z −→ Z → S ( Z ∈ Z , S ∈ S ) ,T ′ → I i −→ Z ( I ∈ I , T ′ ∈ T ) . Then by ℓ ( z ◦ q ) = ℓ ( z ) ◦ z ◦ q ∈ C ( U, Z ) factors through i byRemark 5.26. Namely, there exists k ∈ C ( U, I ) which makes the following diagramcommutative.
U U T I Z q / / k (cid:15) (cid:15) z (cid:15) (cid:15) i / / (cid:8) By (ET4) op , we obtain a diagram made of conflations T ′ ∃ N U T T ′ I ZS S / / ∃ p / / ∃ n (cid:15) (cid:15) z (cid:15) (cid:15) / / i / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) in which, N belongs to N by Lemma 4.18 (2).By the dual of Lemma 3.13, we obtain j ∈ C ( U, N ) which makes the followingdiagram commutative.
U N U T I Z p / / n (cid:15) (cid:15) z (cid:15) (cid:15) i / / j (cid:31) (cid:31) ❄❄❄❄❄❄ q " " k (cid:29) (cid:29) (cid:8)(cid:8)(cid:8) By N ∈ N = Cone( V , S ) and E ( U, V ) = 0, this j factors through some S ∈ S , asfollows. US N U T (cid:11) (cid:11) ✖✖✖✖✖✖✖ j (cid:31) (cid:31) ❄❄❄❄❄❄ ∃ r ❥❥❥❥❥❥ p / / q " " (cid:8) (cid:8) By E ( S , T ) = 0, the morphism p ◦ r factors through u T . This implies q ∗ ( ρ T ) = 0by Lemma 3.2. (cid:3) Lemma 6.13.
Let (
A, δ, C ) , ( A ′ , δ ′ , C ′ ) be E -extensions, and let ( a, c ) : δ → δ ′ bea morphism of E -triangles. Then( ℓ ( a ) , ℓ ( c )) : e ℓ ( δ ) → e ℓ ( δ ′ ) OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 55 is a morphism of e E -extensions. Namely, C U A = A [1] C ′ U A ′ = A ′ [1] e ℓ ( δ ) / / ℓ ( c ) (cid:15) (cid:15) ℓ ( a )[1] (cid:15) (cid:15) e ℓ ( δ ′ ) / / (cid:8) is commutative in e C . Proof.
Take spans ( C w ←− D d −→ U A ) and ( C ′ w ′ ←− D ′ d ′ −→ U A ′ ) satisfying w, w ′ ∈ w Fib , w ∗ δ = d ∗ ρ A , w ′∗ δ ′ = d ′∗ ρ A ′ . By definition, we have e ℓ ( δ ) = ℓ ( d ) ◦ ℓ ( w ) − , e ℓ ( δ ′ ) = ℓ ( d ′ ) ◦ ℓ ( w ′ ) − . Remark that ℓ ( a )[1] = ℓ ( u ) is given by a morphism of E -triangles A V A U A A ′ V A ′ U A ′ v A / / u A / / ρ A / / ❴❴❴ a (cid:15) (cid:15) v (cid:15) (cid:15) u (cid:15) (cid:15) v A ′ / / u A ′ / / ρ A ′ / / ❴❴❴ (cid:8) (cid:8) . Since w ′ ∈ w Fib , there exists an E -triangle V ′ → D ′ w ′ −→ C ′ ν ′ ( V ′ ∈ V ) . By realizing c ∗ ν ′ , we obtain a morphism of E -triangles V ′ ∃ D ′′ CV ′ D ′ C ′ / / ∃ w ′′ / / c ∗ ν ′ / / ❴❴❴ ∃ f (cid:15) (cid:15) c (cid:15) (cid:15) / / w ′ / / ν ′ / / ❴❴❴ (cid:8) (cid:8) . Then both spans ( C w ←− D u ◦ d −→ U A ′ ) and ( C w ′′ ←− D ′′ d ′ ◦ f −→ U A ′ ) satisfy w ∗ ( c ∗ δ ′ ) = w ∗ a ∗ δ = a ∗ w ∗ δ = a ∗ d ∗ ρ A = d ∗ u ∗ ρ A ′ = ( u ◦ d ) ∗ ρ A ′ ,w ′′∗ ( c ∗ δ ′ ) = f ∗ w ′∗ δ ′ = f ∗ d ′∗ ρ A ′ = ( d ′ ◦ f ) ∗ ρ A ′ . Thus by Claim 6.10 (2), we obtain ℓ ( u ◦ d ) ◦ ℓ ( w ) − = ℓ ( d ′ ◦ f ) ◦ ℓ ( w ′′ ) − = ℓ ( d ′ ) ◦ ( ℓ ( f ) ◦ ℓ ( w ′′ ) − ) = ℓ ( d ′ ) ◦ ( ℓ ( w ′ ) − ◦ ℓ ( c )) , which means ℓ ( u ) ◦ e ℓ ( δ ) = e ℓ ( δ ′ ) ◦ ℓ ( c ). (cid:3) Proposition 6.14.
The functor [1] : e C → e C is an auto-equivalence, with quasi-inverse [ − Proof.
By the definitions of [ −
1] and [1], there are E -triangles C [ − → S C → C λ C ( S C ∈ S ) ,C [ − → V C [ − → ( C [ − ρ C [ − ( V C [ − ∈ V )for each C ∈ C . Then by Proposition 3.15 (2), we have a commutative diagrammade of conflations C [ − S C CV C [ − ∃ D C ( C [ − C [ − / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / w / / (cid:15) (cid:15) e (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) which gives e ℓ ( λ C ) = − ℓ ( e ) ◦ ℓ ( w ) − as in the proof of (3) in Claim 6.10. Since e ∈ W , it follows that e ℓ ( λ C ) is an isomorphism in e C . Let us show the naturality of { e ℓ ( λ C ) : C → ( C [ − } C ∈ e C . For this purpose, it suffices to show the naturality with respect to the mor-phisms in C . For any morphism f ∈ C ( C, C ′ ), the morphism ℓ ( f )[ −
1] = ℓ ( g ) ∈ e C ( C [ − , C ′ [ − E -extensions ( g, f ) : λ C → λ C ′ , duallyto Definition 6.2. Thus by Lemma 6.13, C ( C [ − C ′ ( C ′ [ − e ℓ ( λ C ) / / ℓ ( f ) (cid:15) (cid:15) ℓ ( g )[1]=( ℓ ( f )[ − (cid:15) (cid:15) e ℓ ( λ C ′ ) / / (cid:8) becomes commutative. This shows [1] ◦ [ − ∼ = Id. The isomorphism [ − ◦ [1] ∼ = Idcan be shown dually. (cid:3) Triangulation.Definition 6.15.
For an E -triangle A x −→ B y −→ C δ , its associated standardtriangle in e C is defined to be A ℓ ( x ) −→ B ℓ ( y ) −→ C e ℓ ( δ ) −→ A [1] . A distinguished triangle in e C is a triangle isomorphic to some standard triangle. Proposition 6.16.
Any morphism of E -triangles A B CA ′ B ′ C ′ x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) c (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ (cid:8) (cid:8) OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 57 gives the following morphism between standard triangles.
A B C A [1] A ′ B ′ C ′ A ′ [1] ℓ ( x ) / / ℓ ( y ) / / ℓ ( δ ) / / ℓ ( a ) (cid:15) (cid:15) ℓ ( b ) (cid:15) (cid:15) ℓ ( c ) (cid:15) (cid:15) ℓ ( a )[1] (cid:15) (cid:15) ℓ ( x ′ ) / / ℓ ( y ′ ) / / ℓ ( δ ′ ) / / (cid:8) (cid:8) (cid:8) Proof.
This immediately follows from Lemma 6.13. (cid:3)
This gives a cofibrant replacement of a standard triangle, as follows.
Corollary 6.17.
Assume ( C , E , s ) satisfies Condition (WIC) as before. Any stan-dard triangle is isomorphic to a standard triangle associated to an E -triangle U → U ′ → U ′′ whose terms satisfy U, U ′ , U ′′ ∈ U . Proof.
Let A x −→ B y −→ C δ be any E -triangle. Resolve A by an E -triangle V v −→ U a −→ A λ satisfying U ∈ U and V ∈ V . By Proposition 5.15, we have x ◦ a ∈ w Fib ◦ Cof . Namely, there are E -triangles U x ′ −→ B ′ y ′ −→ U δ ′ and V v ′ −→ B ′ b −→ B satisfying U ∈ U , V ∈ V and x ◦ a = b ◦ x ′ .Since U is extension-closed, it follows that B ′ ∈ U . Moreover, by Lemma 5.9 andLemma 4.18 (1), we obtain a morphism of E -triangles U B ′ U A B C x ′ / / y ′ / / δ ′ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) c (cid:15) (cid:15) x / / y / / δ / / ❴❴❴ (cid:8) (cid:8) which admits an E -triangle N → U c −→ C with some N ∈ N . By Claim 5.12, it follows that c ∈ W .By Proposition 6.16, we obtain an isomorphism of standard triangles U B ′ U U [1] A B C A [1] ℓ ( x ′ ) / / ℓ ( y ′ ) / / e ℓ ( δ ′ ) / / ℓ ( a ) ∼ = (cid:15) (cid:15) ℓ ( b ) ∼ = (cid:15) (cid:15) ℓ ( c ) ∼ = (cid:15) (cid:15) ℓ ( a )[1] ∼ = (cid:15) (cid:15) ℓ ( x ) / / ℓ ( y ) / / e ℓ ( δ ) / / (cid:8) (cid:8) (cid:8) in e C . (cid:3) Remark 6.18.
Similarly, for any E -triangle A x −→ B y −→ C δ , we can constructa morphism of E -triangles A B CT A T B T Cx / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) c (cid:15) (cid:15) / / / / / / ❴❴❴ (cid:8) (cid:8) which satisfies T A , T B , T C ∈ T and a, b, c ∈ W . Lemma 6.19.
Let(28) A x −→ B y −→ C δ be any E -triangle. From the E -triangle A v A −→ V A u A −→ U A ρ A , we obtain an E -triangle(29) A h xv A i −→ B ⊕ V A → X θ by Corollary 3.16. Then the standard triangles associated to (28) , (29) are isomor-phic. Also remark that we have V A ∈ I if A ∈ U . Proof.
By the dual of Proposition 3.17, we obtain a commutative diagram made of E -triangles, as follows. V A V A A B ⊕ V A XA B C (cid:15) (cid:15) (cid:15) (cid:15) / / h i xv A / / θ / / ❴❴❴❴ [1 0] (cid:15) (cid:15) ∃ e (cid:15) (cid:15) x / / y / / δ / / ❴❴❴❴ (cid:15) (cid:15) ✤✤✤✤ (cid:15) (cid:15) ✤✤✤✤ (cid:8)(cid:8) (cid:8) Remark that ℓ ([1 0]) : B → B ⊕ V A and ℓ ( e ) : C → Z are isomorphisms in e C . ThusLemma 6.19 follows from Proposition 6.16. (cid:3) Theorem 6.20.
The shift functor in Definition 6.2 and the class of distinguishedtriangles in Definition 6.15 give a triangulation of e C .Proof. (TR1) By definition, the class of distinguished triangles is closed underisomorphisms. From the E -triangle A id A −→ A → , we obtain a distinguishedtriangle A id A −→ A → → A [1].Let α ∈ e C ( A, B ) be any morphism, and let us show the existence of a distin-guished triangle of the form A α −→ B → C → A [1] . Up to isomorphism in e C , we may assume A ∈ U , B ∈ T from the start. As inRemark 5.26, then there is a morphism f ∈ C ( A, B ) satisfying ℓ ( f ) = α .By Corollary 3.16, we have an E -triangle A h fv A i −→ B ⊕ V A g −→ C δ , OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 59 which gives a standard triangle compatible with ℓ ( f ) as follows. A B ⊕ V A C A [1] B / / h i ℓ ( f ) ℓ ( v A ) ℓ ( g ) / / e ℓ ( δ ) / / ℓ ( f ) % % ❑❑❑❑❑❑❑❑❑❑ ∼ = (cid:15) (cid:15) (cid:8) (TR2) It suffices to show this axiom for standard triangles. Let A x −→ B y −→ C δ be an E -triangle, and let A ℓ ( x ) −→ B ℓ ( y ) −→ C e ℓ ( δ ) −→ A [1] be its associated standardtriangle. Let us show that B ℓ ( y ) −→ C e ℓ ( δ ) −→ A [1] − ℓ ( x )[1] −→ B [1]is distinguished. By Proposition 3.15 (2), we obtain a commutative diagram madeof E -triangles A B CV A ∃ D CU A U Ax / / y / / δ / / ❴❴❴ v A (cid:15) (cid:15) m (cid:15) (cid:15) / / w / / ( v A ) ∗ δ / / ❴❴❴ u A (cid:15) (cid:15) e (cid:15) (cid:15) ρ A (cid:15) (cid:15) ✤✤✤ x ∗ ρ A (cid:15) (cid:15) ✤✤✤ (cid:8) (cid:8)(cid:8) satisfying w ∗ δ + e ∗ ρ A = 0. In particular, we obtain a distinguished triangle B ℓ ( m ) −→ D ℓ ( e ) −→ A [1] e ℓ ( x ∗ ρ A ) −→ B [1] . Remark that we have e ℓ ( δ ) = − ℓ ( e ) ◦ ℓ ( w ) − , as the proof of (3) of Claim 6.10suggests. Thus it remains to show the commutativity of the right-most square of B D A [1] B [1] B C A [1] B [1] ℓ ( m ) / / ℓ ( e ) / / e ℓ ( x ∗ ρ A ) / / ℓ ( w ) ∼ = (cid:15) (cid:15) − ∼ = (cid:15) (cid:15) ℓ ( y ) / / e ℓ ( δ ) / / − ℓ ( x )[1] / / (cid:8) (cid:8) (cid:8) in e C . Let A V A U A B V B U Bv A / / u A / / ρ A / / ❴❴❴ x (cid:15) (cid:15) v (cid:15) (cid:15) u (cid:15) (cid:15) v B / / u B / / ρ B / / ❴❴❴ (cid:8) (cid:8) be a morphism of E -triangles, which gives ℓ ( x )[1] = ℓ ( u ). Then, since the span( U A id ←− U A u −→ U B ) satisfies id ∗ ( x ∗ ρ A ) = x ∗ ρ A = u ∗ ρ B , it follows that e ℓ ( x ∗ ρ A ) = ℓ ( u ) = ℓ ( x )[1]. (TR3) Up to isomorphism of triangles, it suffices to show this axiom for standardtriangles. Let A x −→ B y −→ C δ , A ′ x ′ −→ B ′ y ′ −→ C ′ δ ′ be E -triangles, and suppose we are given a commutative diagram(30) A BA ′ B ′ ℓ ( x ) / / α (cid:15) (cid:15) β (cid:15) (cid:15) ℓ ( x ′ ) / / (cid:8) in e C . By Corollary 6.17 and Remark 6.18, we may assume A, B, C ∈ U and A ′ , B ′ , C ′ ∈ T from the start. In that case, α and β can be written as α = ℓ ( a ) , β = ℓ ( b ) for some a ∈ C ( A, A ′ ) and b ∈ C ( B, B ′ ) by Remark 5.26. Moreover,the commutativity of (30) means that b ◦ x − x ′ ◦ a factors through some I ∈ I . Bythe exactness of C ( V A , I ) −◦ v A −→ C ( A, I ) → E ( U A , I ) = 0 , this shows that there exists j ∈ C ( V A , B ′ ) which makes A B ⊕ V A A ′ B ′ x / / a (cid:15) (cid:15) [ b j ] (cid:15) (cid:15) x ′ / / (cid:8) commutative in C , where x = h xv A i . By Lemma 6.19, replacing A x −→ B by A x −→ B ⊕ V A , we may assume A B CA ′ B ′ C ′ x / / y / / δ / / ❴❴❴ a (cid:15) (cid:15) b (cid:15) (cid:15) x ′ / / y ′ / / δ ′ / / ❴❴❴ (cid:8) is commutative, from the start. Now (TR3) follows from (ET3) and Proposi-tion 6.16.(TR4) Let A α −→ B → D → A [1] ,B β −→ C → F → B [1] ,A γ −→ C → E → A [1]be any distinguished triangles in e C satisfying β ◦ α = γ . In a similar way as inthe proof of (TR3), we may assume that they are standard triangles associated to E -triangles A f −→ B f ′ −→ D δ f ,B g −→ C g ′ −→ F δ g ,A h −→ C h ′ −→ E δ h OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 61 satisfying g ◦ f = h . By Lemma 3.14, we obtain a commutative diagram made of E -triangles as follows. A B DA C EF F f / / f ′ / / δ f / / ❴❴❴ g (cid:15) (cid:15) ∃ d (cid:15) (cid:15) h / / h ′ / / δ h / / ❴❴❴ g ′ (cid:15) (cid:15) ∃ e (cid:15) (cid:15) δ g (cid:15) (cid:15) ✤✤✤✤ f ′∗ δ g (cid:15) (cid:15) ✤✤✤✤ (cid:8) (cid:8)(cid:8) Thus by Proposition 6.16, we obtain the following diagram made of distinguishedtriangles, as desired.
A B D A [1]
A C E A [1]
F F B [1] B [1] D [1] ℓ ( f ) / / ℓ ( f ′ ) / / e ℓ ( δ f ) / / ℓ ( g ) (cid:15) (cid:15) ℓ ( d ) (cid:15) (cid:15) ℓ ( h ) / / ℓ ( h ′ ) / / e ℓ ( δ h ) / / ℓ ( g ′ ) (cid:15) (cid:15) ℓ ( e ) (cid:15) (cid:15) ℓ ( f )[1] (cid:15) (cid:15) e ℓ ( δ g ) / / e ℓ ( δ g ) (cid:15) (cid:15) e ℓ ( f ′∗ δ g ) (cid:15) (cid:15) ℓ ( x ′ )[1] / / (cid:8) (cid:8) (cid:8)(cid:8) (cid:8)(cid:8) (cid:3) The following argument ensures the dual arguments concerning distinguishedtriangles, in the following sections. Recall that the functor [ −
1] : e C → e C inducedfrom the chosen E -triangle T C t C −→ S C s C −→ C λ C ( S C ∈ S , T C ∈ T , T C = C [ − C , gives a quasi-inverse of [1] by Proposition 6.14. Its proof shows thatthe isomorphisms e ℓ ( λ C ) : C → T C [1]give a natural isomorphism Id ∼ = = ⇒ [1] ◦ [ − Definition 6.21.
For any E -triangle A x −→ B y −→ C δ , take a cospan of mor-phisms(31) ( T C m −→ E n ←− A )to some T ∈ C satisfying(32) n ∈ w Cof and n ∗ δ = m ∗ λ C . Then, ℓ † ( δ ) = ℓ ( n ) − ◦ ℓ ( m ) ∈ e C ( C [ − , A ) is well-defined.With this definition, we can give a triangulation of e C by requiring(33) T C ℓ † ( δ ) −→ A ℓ ( x ) −→ B ℓ ( y ) −→ C to be a left triangle. The following proposition (and its dual) shows that theresulting triangulation is the same as that defined in Definition 6.15. Proposition 6.22.
For any E -triangle A x −→ B y −→ C δ ,(34) T C ℓ † ( δ ) −→ A ℓ ( x ) −→ B e ℓ ( λ C ) ◦ ℓ ( y ) −→ T C [1]becomes a distinguished triangle in e C , with respect to the triangulation given inDefinition 6.15. Proof.
Take the standard triangle A ℓ ( x ) −→ B ℓ ( y ) −→ C e ℓ ( δ ) −→ A [1]. Since e C is trian-gulated, by the converse of (TR2), it suffices to show the commutativity of thefollowing diagram. A B T C [1] A [1] C ℓ ( x ) / / e ℓ ( λ C ) ◦ ℓ ( y ) / / − ℓ † ( δ )[1] / / ℓ ( y ) ❍❍❍❍❍❍❍❍❍❍❍❍ ∼ = e ℓ ( λ C ) O O e ℓ ( δ ) ; ; ✈✈✈✈✈✈✈✈✈✈✈✈ (cid:8) (cid:8) As ℓ † ( δ ) does not depend on the choice of a cospan (31), we may take it in thefollowing way.By Proposition 3.15 (1), we obtain a commutative diagram made of E -triangles T C T C A ∃ E S C A B C k (cid:15) (cid:15) t C (cid:15) (cid:15) n / / / / ( s C ) ∗ δ / / ❴❴❴ (cid:15) (cid:15) s C (cid:15) (cid:15) x / / y / / δ / / ❴❴❴ y ∗ λ C (cid:15) (cid:15) ✤✤✤✤ λ C (cid:15) (cid:15) ✤✤✤✤ (cid:8)(cid:8) (cid:8) satisfying n ∗ δ + k ∗ λ C = 0. Then the cospan ( T C − k −→ E n ←− A ) satisfies the desiredproperty (32), and thus gives ℓ † ( δ ) = − ℓ ( n ) − ◦ ℓ ( k ). If we put θ = n ∗ δ = − k ∗ λ C ,then ( n, id C ) : δ → θ and ( − k, id C ) : λ C → θ are morphisms of E -extensions. Thus C C CT C [1] E [1] A [1] e ℓ ( λ C ) (cid:15) (cid:15) e ℓ ( θ ) (cid:15) (cid:15) e ℓ ( δ ) (cid:15) (cid:15) − ℓ ( k )[1] / / ℓ ( n )[1] o o (cid:8) (cid:8) becomes commutative by Lemma 6.13. This shows( ℓ † ( δ )[1]) ◦ e ℓ ( λ C ) = ( − ℓ ( n ) − [1] ◦ ℓ ( k )[1]) ◦ e ℓ ( λ C ) = e ℓ ( δ ) . OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 63 (cid:3) Reduction and mutation via localization
Happel and Iyama-Yoshino’s construction.Definition 7.1.
An extriangulated category ( C , E , s ) is said to be Frobenius if itsatisfies the following conditions.(1) ( C , E , s ) has enough injectives and enough projectives.(2) Proj( C ) = Inj( C ). Example 7.2. (1) If ( C , E , s ) is an exact category, then this agrees with theusual definition ([Ha, section I.2]).(2) Suppose that T is a triangulated category and ( Z , Z ) is an I -mutation pairin the sense of [IYo, Definition 2.5]. Then Z becomes a Frobenius extrian-gulated category, with the extriangulated structure given in Remark 2.18. Remark 7.3.
Let ( C , E , s ) be an extriangulated category, as before. By Remark 4.7,the following are equivalent. (1) (( I , C ) , ( C , I )) is a twin cotorsion pair for some subcategory I ⊆ C . (2) ( C , E , s ) is Frobenius.Moreover I in (1) should satisfy I = Proj( C ) = Inj( C ) . The following can be regarded as a generalization of the constructions by Happel[Ha] and Iyama-Yoshino [IYo]. See also [Li] for the triangulated case.
Corollary 7.4.
Let ( F , E , s ) be a Frobenius extriangulated category satisfyingCondition (WIC), with I = Inj( F ). Then its stable category, namely the idealquotient F / I , becomes triangulated. Proof.
Since (( I , F ) , ( F , I )) becomes a Hovey twin cotorsion pair with Cone( I , I ) =CoCone( I , I ) = I , this follows from Corollary 5.25 and Theorem 6.20. (cid:3) Remark 7.5.
A direct proof for Corollary 7.4 is not difficult either, by imitatingthe proofs by [Ha] or [IYo] , even without assuming Condition (WIC). Corollary 7.6.
For any category C , the following are equivalent.(1) ( C , E , s ) is triangulated, as in Proposition 3.22.(2) ( C , E , s ) is a Frobenius extriangulated category, with Proj( C ) = Inj( C ) =0. Proof. (1) ⇒ (2) is trivial. (2) ⇒ (1) follows from Corollary 7.4. (cid:3) Mutable cotorsion pairs.Lemma 7.7. (1) For any weak equivalence f ∈ C ( U, U ′ ) between U, U ′ ∈ U , there exist I ∈ I and i ∈ C ( U, I ), with which h fi i : U → U ′ ⊕ I becomes an acyclic cofibration. (2) Dually, for any weak equivalence g ∈ C ( T, T ′ ) between T, T ′ ∈ T , thereexist J ∈ I and j ∈ C ( J, T ′ ), with which[ g j ] : T ⊕ J → T ′ becomes an acyclic fibration. Proof.
We only show (1). Since f ∈ W = w Fib ◦ w Cof , there are E -triangles U m −→ E → S , (35) V → E e −→ U ′ δ (36)satisfying S ∈ S , V ∈ V and e ◦ m = f . By E ( U ′ , V ) = 0, we have δ = 0. Thus wemay assume E = U ′ ⊕ V and e = [1 0]in (36), from the start.By the extension-closedness of U ⊆ C , the E -triangle (35) gives U ′ ⊕ V = E ∈ U ,which implies V ∈ I . Moreover by e ◦ m = f , the acyclic cofibration m : U → U ′ ⊕ V should be of the form m = h fi i , with some i ∈ C ( U, V ). (cid:3) The following is an immediate consequence of the existence of the model struc-ture.
Remark 7.8.
For any morphism f ∈ C ( A, B ) in C , the following are equivalent. (1) f ∈ W . (2) ℓ ( f ) is an isomorphism in e C . For any extriangulated category ( C , E , s ), let CP ( C ) denote the class of cotorsionpairs on C . Since e C is triangulated as shown in Theorem 6.20, we may use theusual notation Ext e C for e E . Definition 7.9.
Let P = (( S , T ) , ( U , V )) be a Hovey twin cotorsion pair on C andlet ℓ : C → e C be the associated localization functor as before. Define the class of mutable cotorsion pairs on C with respect to P by M P = (cid:26) ( A , B ) ∈ CP ( C ) (cid:12)(cid:12)(cid:12)(cid:12) S ⊆ A ⊆ UV ⊆ B ⊆ T , Ext e C ( ℓ ( A ) , ℓ ( B )) = 0 (cid:27) . Here, ℓ ( A ) , ℓ ( B ) ⊆ e C denote the essential images of A , B under ℓ . Remarkthat S ⊆ A is equivalent to
B ⊆ T , and
A ⊆ U is equivalent to
V ⊆ B , for any( A , B ) ∈ CP ( C ). Theorem 7.10.
For any Hovey twin cotorsion pair P = (( S , T ) , ( U , V )) on C , wehave mutually inverse bijective correspondences R = R P : M P → CP ( e C ) , I = I P : CP ( e C ) → M P given by R (( A , B )) = ( ℓ ( A ) , ℓ ( B )) , I (( L , R )) = ( U ∩ ℓ − ( L ) , T ∩ ℓ − ( R )) . Proof.
It suffices to show the following.(1) For any ( A , B ) ∈ M P , we have R (( A , B )) ∈ CP ( e C ). OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 65 (2) For any ( L , R ) ∈ CP ( e C ), we have I (( L , R )) ∈ M P .(3) I ◦ R = id.(4) R ◦ I = id.To distinguish, in this proof, let ℓ ( X ) ∈ e C denote the image under ℓ of an object X ∈ C .(1) Since ℓ ( A ) is the essential image of A under ℓ , it is closed under isomor-phisms and finite direct sums. C = Cone( B , A ) implies e C = ℓ ( A ) ∗ ℓ ( B )[1], byDefinition 6.15. Ext e C ( ℓ ( A ) , ℓ ( B )) = 0 follows from the definition of M P .It remains to show that ℓ ( A ) , ℓ ( B ) ⊆ e C are closed under direct summands. Toshow that ℓ ( A ) ⊆ e C is closed under direct summands, it suffices to show ℓ ( A ) = ⊥ ℓ ( B )[1]. Take any X ∈ C , and suppose it satisfies Ext e C ( ℓ ( X ) , ℓ ( B )) = 0.Let us show ℓ ( X ) ∈ ℓ ( A ). By a cofibrant replacement, we may assume X belongsto U . Resolve X by an E -triangle in C B → A → X δ ( A ∈ A , B ∈ B ) . Since ℓ ( δ ) = 0 by assumption, we obtain δ = 0 by Lemma 6.12. Thus X is a directsummand of A , which implies that X itself belongs to A . Similarly for ℓ ( B ) ⊆ e C .(2) Put A = U ∩ ℓ − ( L ), B = T ∩ ℓ − ( R ). Since both U and ℓ − ( L ) areclosed under isomorphisms, finite direct sums and direct summands, so is theirintersection A . Similarly for B . By ℓ ( S ) ⊆ ℓ ( N ) = 0, we have S ⊆ A ⊆ U .By Lemma 6.12, Ext e C ( ℓ ( A ) , ℓ ( B )) = 0 implies E ( A , B ) = 0. It remains to show C = Cone( B , A ) = CoCone( B , A ). Let us show C = Cone( B , A ).Let X ∈ C be any object. By the assumption, there exist R ∈ ℓ − ( R ) , L ∈ ℓ − ( L ) and a distinguished triangle ℓ ( R ) → ℓ ( L ) → ℓ ( X ) → ℓ ( R )[1]in e C . By definition, it is isomorphic to the standard triangle associated to an E -triangle, which we may assume to be of the form(37) R x −→ L y −→ Z δ satisfying R , L , Z ∈ Z , by a fibrant-cofibrant replacement (Corollary 6.17 andRemark 6.18). Thus we have an E -triangle (37) satisfying R ∈ Z ∩ ℓ − ( R ) , L ∈Z ∩ ℓ − ( L ) and Z ∈ Z , together with an isomorphism ζ : ℓ ( Z ) ∼ = −→ ℓ ( X ) in e C .Resolve X by an E -triangle X t X −→ T X s X −→ S X ρ X ( T X ∈ T , S X ∈ S ) . Then there exists a morphism z ∈ C ( Z, T X ) which satisfies ζ = ℓ ( t X ) − ◦ ℓ ( z ).Since ζ is an isomorphism, it follows that z ∈ W . By Lemma 7.7 (2), there existsan E -triangle V → Z ⊕ I [ z i ] −→ T X ( V ∈ V , I ∈ I ) . On the other hand by (ET2), we have an E -triangle R x −→ L ⊕ I y −→ Z ⊕ I from (37), where x = h x i , y = y ⊕ id I . Thus by (ET4) op , we obtain a diagram R ∃ E VR L ⊕ I Z ⊕ IT X T X ∃ r / / / / ∃ e (cid:15) (cid:15) (cid:15) (cid:15) x / / y / / d (cid:15) (cid:15) [ z i ] (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) ( d = [ z i ] ◦ y = [ z ◦ y i ])made of conflations. Since ℓ ( r ) is an isomorphism in e C , we have E ∈ ℓ − ( R ).Besides, R , V ∈ T implies E ∈ T .By (ET4) op , we obtain a diagram E ∃ F XE L ⊕ I T X S X S X / / / / ∃ f (cid:15) (cid:15) t X (cid:15) (cid:15) e / / d / / (cid:15) (cid:15) s X (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) made of conflations. Since ℓ ( f ) is an isomorphism in e C , this shows F ∈ ℓ − ( L ).Resolve F by an E -triangle V F → U F → F ( U F ∈ U , V F ∈ V ) . By (ET4) op , we obtain a diagram V F ∃ G EV F U F FX X / / / / (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:8) (cid:8)(cid:8) made of conflations. Then in the E -triangle G → U F → X , we have U F ∈U ∩ ℓ − ( L ) = A and G ∈ T ∩ ℓ − ( R ) = B .(3) For any ( A , B ) ∈ M P , we have I ◦ R (( A , B )) = ( U ∩ ℓ − ( ℓ ( A )) , T ∩ ℓ − ( ℓ ( B ))) . Obviously,
A ⊆ U ∩ ℓ − ( ℓ ( A )) and B ⊆ T ∩ ℓ − ( ℓ ( B )) hold. Since both ( A , B ) and I ◦ R (( A , B )) are cotorsion pairs, this means ( A , B ) = I ◦ R (( A , B )).(4) For any ( L , R ) ∈ CP ( e C ), we have R ◦ I (( L , R )) = ( ℓ ( U ∩ ℓ − ( L )) , ℓ ( T ∩ ℓ − ( R ))) , OTORSION PAIRS IN EXTRIANGULATED CATEGORIES. 67 which obviously satisfies ℓ ( U ∩ ℓ − ( L )) ⊆ L and ℓ ( T ∩ ℓ − ( R )) ⊆ R . Similarly asin (3), it follows that ( L , R ) = R ◦ I (( L , R )). (cid:3) Claim 7.11.
For any ( A , B ) ∈ CP ( C ) satisfying S ⊆ A ⊆ U (or equivalently
V ⊆ B ⊆ T ), we have
U ∩ ℓ − ℓ ( A ) = U ∩
CoCone( A , S ) , T ∩ ℓ − ℓ ( B ) = T ∩
Cone( V , B ) . Proof.
We only show
U ∩ ℓ − ℓ ( A ) = U ∩
CoCone( A , S ). U ∩
CoCone( A , S ) ⊆ U ∩ ℓ − ℓ ( A ) is obvious. For the converse, let U ∈ U be anyobject satisfying ℓ ( U ) ∼ = ℓ ( A ) in e C for some A ∈ A . Resolve U by an E -triangle U → Z → S ( Z ∈ Z , S ∈ S ) . Then ℓ ( A ) ∼ = ℓ ( Z ) holds in e C . Since A ∈ U , Z ∈ T , there is a morphism f ∈ C ( A, Z )which gives the isomorphism ℓ ( f ) : ℓ ( A ) → ℓ ( Z ) by Remark 5.26. Factorize this f ∈ W as f = h ◦ g ( g ∈ w Cof , h ∈ w Fib ) . By definition, we have E -triangles V → E h −→ Z ( V ∈ V ) ,A g −→ E → S ( S ∈ S ) . Since E ( Z, V ) = 0, it follows that V ⊕ Z ∼ = E ∈ A , which implies Z ∈ A . (cid:3) The class M P can be rewritten as follows. Corollary 7.12.
Let P be a Hovey twin cotorsion pair on C . For any ( A , B ) ∈ CP ( C ) satisfying S ⊆ A ⊆ U (or equivalently
V ⊆ B ⊆ T ), the following areequivalent.(1) ( A , B ) ∈ M P i.e., it satisfies Ext e C ( ℓ ( A ) , ℓ ( B )) = 0(2) U ∩ ℓ − ℓ ( A ) = A .(2) ′ U ∩ ℓ − ℓ ( A ) ⊆ A .(3) T ∩ ℓ − ℓ ( B ) = B .(3) ′ T ∩ ℓ − ℓ ( B ) ⊆ B .Thus by Claim 7.11, we have M P = { ( A , B ) ∈ CP ( C ) | S ⊆ A ⊆ U , U ∩
CoCone( A , S ) ⊆ A} = { ( A , B ) ∈ CP ( C ) | V ⊆ B ⊆ T , T ∩
Cone( V , B ) ⊆ B} . Proof.
We only show (1) ⇔ (2) ⇔ (2) ′ .(1) ⇒ (2) follows from Theorem 7.10.(2) ⇒ (2) ′ is obvious. It remains to show (2) ′ ⇒ (1).Suppose (2) ′ is satisfied. Let us show Ext e C ( ℓ ( A ) , ℓ ( B )) = 0 for any pair ofobjects A ∈ A , B ∈ B . Resolve A by an E -triangle T A t −→ S A → A ( T A ∈ T , S A ∈ S )and T A by V T → Z T z −→ T A ( Z T ∈ Z , V T ∈ V ) . Then we have ℓ ( A )[ − ∼ = ℓ ( T A ) ∼ = ℓ ( Z T ), and thusExt e C ( ℓ ( A ) , ℓ ( B )) ∼ = e C ( ℓ ( Z T ) , ℓ ( B )) ∼ = ( C / I )( Z T , B ) by Remark 5.26. Let us show ( C / I )( Z T , B ) = 0. Factorize t ◦ z as t ◦ z = h ◦ g ( g ∈ Cof , h ∈ w Fib ) , to obtain a diagram V T ∃ V Z T ∃ E ∃ U T A S A A (cid:15) (cid:15) (cid:15) (cid:15) g / / / / z (cid:15) (cid:15) h (cid:15) (cid:15) t / / / / (cid:8) ( U ∈ U , V ∈ V )made of conflations. Since E ( S A , V ) = 0, we have E ∼ = S A ⊕ V . Besides, by theextension-closedness of U ⊆ C , we have E ∈ U , which shows V ∈ V ∩ U = I . Thusit follows that E ∈ S .By Lemma 5.9, we obtain X ∈ C and conflations V T → V → X and X → U u −→ A. By Lemma 4.18 (1), we have X ∈ N , and thus u ∈ W by Claim 5.12. This shows ℓ ( U ) ∼ = ℓ ( A ), which means U ∈ U ∩ ℓ − ℓ ( A ) ⊆ A by assumption. Thus we obtainexact sequence C ( E, B ) C ( g,B ) −→ C ( Z T , B ) → E ( U , B ) = 0 . This shows that any morphism f ∈ C ( Z T , B ) factors through E ∈ S , and thus f = 0 holds in ( C / I )( Z T , B ). (cid:3) The above arguments allows us to define the mutation of cotorsion pairs asfollows.
Definition 7.13.
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