Abelian quotients arising from extriangulated categories via morphism categories
aa r X i v : . [ m a t h . R T ] J u l ABELIAN QUOTIENTS ARISING FROM EXTRIANGULATEDCATEGORIES VIA MORPHISM CATEGORIES
ZENGQIANG LIN
Abstract.
We investigate abelian quotients arising from extriangulated cate-gories via morphism categories, which is a unified treatment for both exact cat-egories and triangulated categories. Let ( C , E , s ) be an extriangulated categorywith enough projectives P and M be a full subcategory of C containing P . Weshow that certain quotient category of s -def( M ), the category of s -deflations f : M → M with M , M ∈ M , is abelian. Our main theorem has two appli-cations. If M = C , we obtain that certain ideal quotient category s -tri( C ) / R is equivalent to the category of finitely presented modules mod- C / [ P ], where s -tri( C ) is the category of all s -triangles. If M is a rigid subcategory, we showthat M L / [ M ] ∼ = mod-( M / [ P ]) and M L / [Ω M ] ∼ = (mod-( M / [ P ]) op ) op , where M L (resp. Ω M ) is the full subcategory of C of objects X admitting an s -triangle X / / M / / M / / ❴❴❴ (resp. X / / P / / M / / ❴❴❴ )with M , M ∈ M (resp. M ∈ M and P ∈ P ). In particular, we have C / [ M ] ∼ = mod-( M / [ P ]) and C / [Ω M ] ∼ = (mod-( M / [ P ]) op ) op provided that M is a cluster-tilting subcategory. introduction In representation theory, there are a few quotient categories admitting naturalabelian structures. For both triangulated categories and exact categories, cluster-tilting subcategories provide a way to construct abelian quotient categories. Let C be a triangulated category and T be a cluster-tilting subcategory of C , then thequotient C / [ T ] is abelian; related works see [1, 8, 9]. The version of exact categoriessee [2]. Submodule categories provide another way to construct abelian quotientcategories. Certain quotients of submodule categories are realized as categories offinitely presented modules over stable Auslander algebras [16, 3]. More generally,some quotients of categories of short exact sequences in exact categories are abelian,related works see [4, 10]. For triangulated version, certain quotients of categoriesof triangles are abelian [14].Recently, Nakaoka and Palu introduced the notion of extriangulated categories[15], which is a simultaneous generalization of exact categories and triangulatedcategories. They pointed out that the notion is a convenient setup for writingdown proofs which apply to both exact categories and triangulated categories. Forrecent developments on extriangulated categories we refer to [6, 11, 12, 13, 17] etc.In this paper, we focus our attention onto the abelian quotients arising fromextriangulated categories via morphism categories, which is a unified treatment of Mathematics Subject Classification.
Key words and phrases. extriangulated categories; abelian categories.This work was supported by the national natural science foundation of China (Grants No.11871014 and No. 11871259). abelian quotients for both exact categories and triangulated categories. Our ap-proach to abelian quotients is based on identifying quotients of morphism categoriesas certain module categories.Let ( C , E , s ) be an extriangulated category and M be a full subcategory of C . Wedenote by Mor( M ) the morphism category of M and by s -def( M ) (resp. s -inf( M ))the full subcategory of Mor( M ) consisting of s -deflations (resp. s -inflations). Thefull subcategory of s -def( M ) consisting of split epimorphisms (resp. split monomor-phisms) is denoted by s-epi( M ) (resp. s-mono( M )). We denote by sp-epi( M ) (resp.si-mono( M )) the full subcategory of s -def( M ) consisting of ( M −→ M ) ⊕ ( P → M ′ )(resp. ( M −→ M ) ⊕ ( M ′ → I )) with P ∈ P (resp. I ∈ I ).Our main theorem is the following (Theorem 3.2), which generalizes [10, Theorem3.9]. Theorem 1.1.
Let C be an extriangulated category and M be a full subcategory of C . (1) If C has enough projectives P and M contains P , then s -def( M ) / [s-epi( M )] ∼ =mod- M / [ P ] and s -def( M ) / [sp-epi( M )] ∼ = (mod-( M / [ P ]) op ) op . (2) If C has enough injectives I and M contains I , then s -inf( M ) / [s-mono( M )] ∼ =(mod-( M / [ I ]) op ) op and s -inf( M ) / [si-mono( M )] ∼ = mod- M / [ I ] . Theorem 1.1 has two interesting applications. We will investigate two specialcases when M = C and when M is rigid , that is, E ( M, M ′ ) = 0 for any M, M ′ ∈ M .For the first case, we denote by s -tri( C ) the category of all s -triangles, wherethe objects are the s -triangles X • = ( X f / / X f / / X δ / / ❴❴❴ ) and the mor-phisms from X • to Y • are the triples ϕ • = ( ϕ , ϕ , ϕ ) such that the followingdiagram X f / / ϕ (cid:15) (cid:15) X f / / ϕ (cid:15) (cid:15) X δ / / ❴❴❴ ϕ (cid:15) (cid:15) Y g / / Y g / / Y δ ′ / / ❴❴❴ is a morphism of s -triangles. Let X • and Y • be two s -triangles, we denote by R ( X • , Y • ) the class of morphisms ϕ • : X • → Y • such that ϕ factors through g .It is easy to see that R is an ideal of s -tri( C ), moreover, the following three quotientcategories s -tri( C ) / R , s -def( C ) / [s-epi( C )] and s -inf( C ) / [s-mono( C )] are equivalent.Given an s -triangle δ = ( X f / / X f / / X ρ / / ❴❴❴ ), we define the con-travariant defect δ ∗ and the covariant defect δ ∗ by the following exact sequence offunctors C ( − , X ) C ( − ,f ) −−−−−→ C ( − , X ) C ( − ,f ) −−−−−→ C ( − , X ) → δ ∗ → , C ( X , − ) C ( f , − ) −−−−−→ C ( X , − ) C ( f , − ) −−−−−→ C ( X , − ) → δ ∗ → . Our first application of Theorem 1.1 is the following (Theorem 4.1, Proposition4.3 and Theorem 4.8), which generalizes [10, Theorem 4.1, Theorem 4.8, Theorem5.1].
Theorem 1.2.
Let C be an extriangulated category. (1) The quotient s -tri ( C ) / R is abelian. (2) If C has enough projectives P , then we have the following equivalence F : s -tri( C ) / R ∼ = mod- C / [ P ] , δ δ ∗ . BELIAN QUOTIENTS VIA MORPHISM CATEGORIES 3 (3) If C has enough injectives I , then we have the following equivalence G : s -tri( C ) / R ∼ = (mod-( C / [ I ]) op ) op , δ δ ∗ . We point out that the abelian quotient s -tri( C ) / R admits nice properties. Wedescribe the projectives, injectives and simple objects in s -tri( C ) / R ; see Propo-sition 4.4 and Proposition 4.6. In particular, if C has enough projectives P andenough injectives I , then there is a duality between mod- C / [ P ] and mod-( C / [ I ]) op ,which is used to derive Auslander-Reiten duality and defect formula for extriangu-lated categories; see Proposition 4.9.For describing the second application, we first give some notations. Let C ′ and C ′′ be two full subcategories of C . We denote by Cocone( C ′ , C ′′ ) (resp. Cone( C ′ , C ′′ ))the subcategory of objects X admitting an s -triangle X / / C ′ / / C ′′ / / ❴❴❴ (resp. C ′ / / C ′′ / / X / / ❴❴❴ ) with C ′ ∈ C ′ and C ′′ ∈ C ′′ .Let M be a rigid subcategory of C . For convenience we let M L = Cocone( M , M )and M R = Cone( M , M ). If C has enough projectives P and enough injectives I , then we let Ω M =Cocone( P , M ) and Σ M =Cone( M , I ). It turns out that thequotient categories s -def( M ) / [s-epi( M )] and s -def( M ) / [sp-epi( M )] can be realizedas subquotient categories of C .Our second application of Theorem 1.1 is the following (Theorem 5.4), whichgeneralizes [2, Theorem 3.2, Theorem 3.4] and [7, Proposition 6.2]. Theorem 1.3.
Let C be an extriangulated category and M be a rigid subcategoryof C . (1) If C has enough projectives P and M contains P , then M L / [ M ] ∼ = mod-( M / [ P ]) and M L / [Ω M ] ∼ = (mod-( M / [ P ]) op ) op . (2) If C has enough injectives I and M contains I , then M R / [ M ] ∼ = (mod-( M / [ I ]) op ) op and M R / [Σ M ] ∼ = mod-( M / [ I ]) . In particular, if M is a cluster tilting subcategory of C , then M L = M R = C .Thus we have the following result (Corollary 5.7). Corollary 1.4.
Let C be an extriangulated category with enough projectives P andenough injectives I . If M is a cluster tilting subcategory of C , then (1) C / [ M ] ∼ = mod-( M / [ P ]) ∼ = (mod-( M / [ I ]) op ) op . (2) C / [Ω M ] ∼ = (mod-( M / [ P ]) op ) op . (3) C / [Σ M ] ∼ = mod-( M / [ I ]) . This paper is organized as follows. In Section 2 we make some preliminaries onmorphism categories and extriangulated categories. In Section 3 we prove Theorem1.1. In Section 4 we provide the first application. In Section 5 we provide the secondapplication. 2. definitions and preliminaries
In this section, we first give some facts on morphism categories, then recall thedefinitions and basic properties on extriangulated categories from [15], [11] and [6].2.1.
Morphism categories.
Let C be an additive category. The morphism cat-egory of C is the category Mor( C ) defined by the following data. The objects ofMor( C ) are all the morphisms f : X → Y in C . The morphisms from f : X → Y to f ′ : X ′ → Y ′ are pairs ( a, b ) where a : X → X ′ and b : Y → Y ′ such that bf = f ′ a .The composition of morphisms is componentwise. For two objects f : X → Y ZENGQIANG LIN and f ′ : X ′ → Y ′ in Mor( C ), we define R ( f, f ′ ) (resp. R ′ ( f, f ′ )) to be the set ofmorphisms ( a, b ) such that there is some morphism p : Y → X ′ such that f ′ p = b (resp. pf = a ). Then R and R ′ are ideals of Mor( C ). We denote by s-epi( C ) (resp.s-mono( C )) the full subcategory of Mor( C ) consisting of split epimorphisms (resp.split monomorphisms).Recall that a right C -module is a contravariantly additive functor F : C → Ab ,where Ab is the category of abelian groups. A C -module F is called finitely presented if there exists an exact sequence C ( − , X ) → C ( − , Y ) → F →
0. We denote by mod- C the category of finitely presented C -modules, and by proj- C (resp. inj- C ) the fullsubcategory of mod- C consisting of projectives (resp. injectives).The following result was proved in [10, Lemma 3.1] and [10, Proposition 3.3]. Lemma 2.1.
Let C be an additive category, then (1) Mor( C ) / R ∼ = Mor( C ) / [s-epi( C )] ∼ = mod- C . (2) Mor( C ) / R ′ ∼ = Mor( C ) / [s-mono( C )] ∼ = (mod- C op ) op . Extriangulated categories.
Let C be an additive category equipped withan additive bifunctor E : C op × C → Ab . For any pair of objects A, C ∈ C , anobject δ ∈ E ( C, A ) is called an E - extension . For any morphism a ∈ C ( A, A ′ ) and c ∈ C ( C ′ , C ), we denote the E -extension E ( C, a )( δ ) ∈ E ( C, A ′ ) by a ∗ δ and denotethe E -extension E ( c, A )( δ ) ∈ E ( C ′ , A ) by c ∗ δ . Let δ ∈ E ( C, A ) and δ ′ ∈ E ( C ′ , A ′ )be two E -extensions. A morphism ( a, c ) : δ → δ ′ of E -extensions is a pair ofmorphisms a ∈ C ( A, A ′ ) and c ∈ C ( C, C ′ ) such that a ∗ δ = c ∗ δ ′ .Let A, C ∈ C be any pair of objects. Two sequences of morphisms in C A x −→ B y −→ C and A x ′ −→ B ′ y ′ −→ C are equivalent if there exists an isomorphism b ∈ C ( B, B ′ ) such that the followingdiagram is commutative. A x / / B y / / b ≃ (cid:15) (cid:15) CA x ′ / / B ′ y ′ / / C We denote the equivalence class of A x −→ B y −→ C by [ A x −→ B y −→ C ]. Definition 2.2.
Let E : C op ×C → Ab be an additive bifunctor. A correspondence s is called a realization of E if it associates an equivalence class s ( δ ) = [ A x −→ B y −→ C ]to any E -extension δ ∈ E ( C, A ) and associates a commutative diagram A x / / a (cid:15) (cid:15) B y / / b (cid:15) (cid:15) C c (cid:15) (cid:15) A x ′ / / B ′ y ′ / / C to any morphism ( a, c ) : δ → δ ′ of E -extensions, where s ( δ ) = [ A x −→ B y −→ C ] and s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ]. In the above situation, we say the sequence A x −→ B y −→ Crealizes δ and the triple ( a, b, c ) realizes ( a, c ). Definition 2.3.
A realization s of E is said to be additive if it satisfies the followingtwo conditions. BELIAN QUOTIENTS VIA MORPHISM CATEGORIES 5 (1) Assume that 0 ∈ E ( C, A ) is the zero element, then s (0) = [ A ( ) −−→ A ⊕ C (0 , −−−→ C ].(2) Assume that s ( δ ) = [ A x −→ B y −→ C ] and s ( δ ′ ) = [ A ′ x ′ −→ B ′ y ′ −→ C ′ ], then s ( δ ⊕ δ ′ ) = [ A ⊕ A ′ x ⊕ x ′ −−−→ B ⊕ B ′ y ⊕ y ′ −−−→ C ⊕ C ′ ], where δ ⊕ δ ′ ∈ E ( C ⊕ C ′ , A ⊕ A ′ ) isthe element corresponding to ( δ, , , δ ′ ) under the isomorphism E ( C ⊕ C ′ , A ⊕ A ′ ) ∼ = E ( C, A ) ⊕ E ( C, A ′ ) ⊕ E ( C ′ , A ) ⊕ E ( C ′ , A ′ ).Let s be an additive realization of E . If s ( δ ) = [ A x −→ B y −→ C ], then the sequence A x −→ B y −→ C is called an s - conf lation , the morphism x is called an s - inf lation and y is called an s - def lation . In this case, we say A x / / B y / / C δ / / ❴❴❴ isan s - triangle . Let A x / / B y / / C δ / / ❴❴❴ and A ′ x ′ / / B ′ y ′ / / C ′ δ ′ / / ❴❴❴ be any pair of s -triangles. Let ( a, c ) : δ → δ ′ be a morphism of E -extensions. If atriple ( a, b, c ) realizes ( a, c ), then we say ( a, b, c ) is a morphism of s -triangles. Definition 2.4. ([15, Definition 2.12]) A triple ( C , E , s ) is an extriangulated category if the following conditions are satisfied.(ET1) E : C op × C → Ab is an additive bifunctor.(ET2) s is an additive realization of E .(ET3) Each commutative diagram A x / / a (cid:15) (cid:15) B y / / b (cid:15) (cid:15) C δ / / ❴❴❴ A ′ x ′ / / B ′ y ′ / / C ′ δ ′ / / ❴❴❴ whose rows are s -triangles can be completed to a morphism of s -triangles.(ET3) op Each commutative diagram A x / / B y / / b (cid:15) (cid:15) C c (cid:15) (cid:15) δ / / ❴❴❴ A ′ x ′ / / B ′ y ′ / / C ′ δ ′ / / ❴❴❴ whose rows are s -triangles can be completed to a morphism of s -triangles.(ET4) Let A f / / B f ′ / / D δ / / ❴❴❴ and B g / / C g ′ / / F δ ′ / / ❴❴❴ be s -triangles. There exists a commutative diagram A f / / B f ′ / / g (cid:15) (cid:15) D δ / / ❴❴❴ d (cid:15) (cid:15) A h / / C g ′ (cid:15) (cid:15) h ′ / / E e (cid:15) (cid:15) δ ′′ / / ❴❴❴ F δ ′ (cid:15) (cid:15) ✤✤✤ F f ′∗ δ ′ (cid:15) (cid:15) ✤✤✤ ZENGQIANG LIN such that the second row and the third column are s -triangles, moreover, δ = d ∗ δ ′′ and f ∗ δ ′′ = e ∗ δ ′ .(ET4) op Let D f ′ / / A f / / B δ / / ❴❴❴ and F g ′ / / B g / / C δ ′ / / ❴❴❴ be s -triangles. There exists a commutative diagram D d / / E e / / h ′ (cid:15) (cid:15) F g ′∗ δ / / ❴❴❴ g ′ (cid:15) (cid:15) D f ′ / / A h (cid:15) (cid:15) f / / B g (cid:15) (cid:15) δ / / ❴❴❴ C δ ′′ (cid:15) (cid:15) ✤✤✤ C δ ′ (cid:15) (cid:15) ✤✤✤ such that the first row and the second column are s -triangles, moreover, δ ′ = e ∗ δ ′′ and d ∗ δ = g ∗ δ ′′ . Definition 2.5.
Let ( C , E , s ) be an extriangulated category.(1) An object P ∈ C is called projective if for any s -deflation y : B → C and anymorphism c : P → C , there exists a morphism b : P → B such that yb = c . Thefull subcategory of projectives is denoted by P .(2) We say that C has enough projectives if for any object C ∈ C there exists an s -triangle A x / / P y / / C δ / / ❴❴❴ with P ∈ P . Example 2.6. [15, Example 3.26] (1) Let C be an exact category, then C is anextriangulated category with E ( − , − ) = Ext C ( − , − ). In particular, if C is an exactcategory with enough projectives, then C is an extriangulated category with enoughprojectives.(2) Let C be a triangulated category with shift functor [1], then C is an extrian-gulated category with E ( − , − ) = C ( − , − [1]). Moreover, C has enough projectives.In this case, P consists of zero objects.The following lemmas will be used frequently later. Lemma 2.7. ( [15, Corollary 3.5] ) Let C be an extriangulated category. Assumethat the diagram A x / / a (cid:15) (cid:15) B y / / b (cid:15) (cid:15) C c (cid:15) (cid:15) δ / / ❴❴❴ A ′ x ′ / / B ′ y ′ / / C ′ δ ′ / / ❴❴❴ is a morphism of s -triangles. Then the following statements are equivalent. (1) a factors through x . (2) a ∗ δ = c ∗ δ ′ = 0 . (3) c factors through y ′ . Lemma 2.8. ( [11, Proposition 1.20] ) Let C be an extriangulated category. Assumethat A x / / B y / / C δ / / ❴❴❴ is an s -triangle, f : A → D is a morphism and BELIAN QUOTIENTS VIA MORPHISM CATEGORIES 7 D d / / E e / / C f ∗ δ / / ❴❴❴ is an s -triangle, then there is a morphism g : B → E which gives a morphism of s -triangles A x / / f (cid:15) (cid:15) B y / / g (cid:15) (cid:15) ✤✤✤ C δ / / ❴❴❴ D d / / E e / / C f ∗ δ / / ❴❴❴ and moreover, A ( fx ) / / D ⊕ B ( d, − g ) / / E e ∗ δ / / ❴❴❴ is an s -triangle. Lemma 2.9. ( [15, Corollary 3.12] ) Let C be an extriangulated category. Then forany s -triangle A x / / B y / / C δ / / ❴❴❴ , the following two sequences are exact. C ( − , A ) → C ( − , B ) → C ( − , C ) → E ( − , A ) → E ( − , B ) → E ( − , C ) , C ( C, − ) → C ( B, − ) → C ( A, − ) → E ( C, − ) → E ( B, − ) → E ( A, − ) . Let C be an extriangulated category with enough projectives P and enoughinjectives I . Let X be any object in C . It admits an s -triangle X / / I / / Σ X δ X / / ❴❴❴ (resp. Ω X / / P / / X δ X / / ❴❴❴ )with I ∈ I (resp. P ∈ P ). We can get s -trianglesΣ i X / / I i / / Σ i +1 X δ Σ iX / / ❴❴❴ (resp. Ω i +1 X / / P i / / Ω i X δ Ω iX / / ❴❴❴ )with I i ∈ I (resp. P i ∈ P ) for i > E ( X, Σ i Y ) ∼ = E (Ω i X, Y ) by E i ( X, Y ), where theequivalence follows from [11, Lemma 5.1].The following result extends the exact sequences appeared in Lemma 2.9.
Lemma 2.10. ( [11, Proposition 5.2] ) Let C be an extriangulated category withenough projectives P and enough injectives I . Then for any s -triangle A x / / B y / / C δ / / ❴❴❴ , the following two sequences are exact. C ( − , A ) → C ( − , B ) → C ( − , C ) → E ( − , A ) → E ( − , B ) → E ( − , C ) → E ( − , A ) → E ( − , B ) → E ( − , C ) → · · · → E i ( − , A ) → E i ( − , B ) → E i ( − , C ) → · · · , C ( C, − ) → C ( B, − ) → C ( A, − ) → E ( C, − ) → E ( B, − ) → E ( A, − ) → E ( C, − ) → E ( B, − ) → E ( A, − ) → · · · → E i ( C, − ) → E i ( B, − ) → E i ( A, − ) → · · · . Lemma 2.11.
Let C be an extriangulated category with enough projectives P and f : X → Y be a morphism in C . (1) If π : P → Y is an s -deflation with P ∈ P , then ( f, − π ) : X ⊕ P → Y is an s -deflation and ( f, − π ) ∼ = f in Mor- C / [ P ] . (2) If h : X → Z is a morphism in C and g : Y → Z is an s -deflation suchthat gf = h in Mor- C / [ P ] , then there exists an object P ∈ P and two morphisms u : X → P and v : P → Y such that g ( f − vu ) = h . ZENGQIANG LIN
Proof. (1) The first assertion follows from [15, Corollary 3.16] or the dual of Lemma2.8. The second assertion is clear.(2) Since gf = h , there is an object P ∈ P and two morphisms u : X → P and w : P → Z such that gf − h = wu . Since g : Y → Z is an s -deflation, there existsa morphism v : P → Y such that w = gv . Therefore, g ( f − vu ) = h . (cid:3) Proof of Theorem 1.1
Throughout this paper, we assume that ( C , E , s ) is an extriangulated category.Let M be a full subcategory of C . We denote by s -def( M ) (resp. s -inf( M )) thefull subcategory of Mor( M ) consisting of s -deflations (resp. s -inflations). Recallthat the full subcategory of s -def( M ) consisting of split epimorphisms (resp. splitmonomorphisms) is denoted by s-epi( M ) (resp. s-mono( M )). We denote by sp-epi( M ) (resp. si-mono( M )) the full subcategory of s -def( M ) consisting of ( M −→ M ) ⊕ ( P → M ′ ) (resp. ( M −→ M ) ⊕ ( M ′ → I )) with P ∈ P (resp. I ∈ I ). Lemma 3.1.
Let C be an extriangulated category with enough projectives P and M be a full subcategory of C containing P . Assume that the following X k / / g (cid:15) (cid:15) M f / / a (cid:15) (cid:15) M δ / / ❴❴❴ b (cid:15) (cid:15) X ′ k ′ / / M ′ f ′ / / M ′ δ ′ / / ❴❴❴ is a morphism of s -triangles with M i , M ′ i ∈ M . Then (1) The following statements are equivalent. (a)
The morphism b factors through f ′ in M / [ P ] . (b) The morphism b factors through f ′ . (c) The morphism ( a, b ) factors through some object in s-epi( M ) . (2) The following statements are equivalent. (a)
The morphism a factors through f in M / [ P ] . (b) The morphism ( a, b ) factors through some object in sp-epi( M ) .Proof. (1) Since (b) ⇔ (c) follows from Lemma 2.1 and (b) ⇒ (a) is clear, we onlyprove (a) ⇒ (b). Suppose that there is a morphism p : M → M ′ such that f ′ p = b .By Lemma 2.11, there exists an object P ∈ P and two morphisms u : M → P and v : P → M ′ such that f ′ ( p − vu ) = b . Thus b factors through f ′ .(2) (a) ⇒ (b). Suppose that there is a morphism p : M → M ′ such that pf = a .Since C has enough projectives, there is an s -deflation a : P → M ′ with P ∈ P .It is easy to see that a − pf factors through a . We assume that a − pf = a a where a : M → P . Since ( b − f ′ p ) f = f ′ a − f ′ pf = f ′ a a , we have the followingcommutative diagram. M f / / a (cid:15) (cid:15) (cid:16) fa (cid:17) ❍❍❍❍❍❍❍❍❍ M b (cid:15) (cid:15) (cid:16) b − f ′ p (cid:17) $ $ ■■■■■■■■■ M ⊕ P (cid:16) f ′ a (cid:17) / / ( p,a ) { { ✇✇✇✇✇✇✇✇✇ M ⊕ M ′ f ′ p, z z ✉✉✉✉✉✉✉✉✉ M ′ f ′ / / M ′ In other words, ( a, b ) factors through ( M ⊕ P (cid:16) f ′ a (cid:17) −−−−−−→ M ⊕ M ′ ) ∈ sp-epi( M ).(b) ⇒ (a). Assume that the morphism ( a, b ) factors through ( M ⊕ P ( π ) −−−−→ M ⊕ M ′ ) ∈ sp-epi( M ). Suppose that the following diagram M f / / a (cid:15) (cid:15) (cid:16) a a ′ (cid:17) ❍❍❍❍❍❍❍❍❍ M b (cid:15) (cid:15) (cid:18) b b ′ (cid:19) $ $ ■■■■■■■■■ M ⊕ P ( π ) / / ( a ,a ′ ) { { ✇✇✇✇✇✇✇✇✇ M ⊕ M ′ ( b ,b ′ ) z z ✈✈✈✈✈✈✈✈✈ M ′ f ′ / / M ′ is commutative. Let p = a b : M → M ′ , then pf = a b f = a a , thus a = a a = pf . (cid:3) Theorem 3.2.
Let C be an extriangulated category and M be a full subcategory of C . (1) If C has enough projectives P and M contains P , then s -def( M ) / [s-epi( M )] ∼ =mod- M / [ P ] and s -def( M ) / [sp-epi( M )] ∼ = (mod-( M / [ P ]) op ) op . (2) If C has enough injectives I and M contains I , then s -inf( M ) / [s-mono( M )] ∼ =(mod-( M / [ I ]) op ) op and s -inf( M ) / [si-mono( M )] ∼ = mod- M / [ I ] .Proof. Since (2) is dual to (1), we only prove (1).Define a functor F : s -def( M ) → Mor( M / [ P ]) , ( M f −→ M ) ( M f −→ M ) . For any object f : M → M in Mor( M / [ P ]), by Lemma 2.11 there is an object P ∈ P and an s -deflation ( f, − π ) : M ⊕ P → M such that ( f, − π ) ∼ = f . Therefore, F ( f, − π ) ∼ = f and F is dense.Assume that f : M → M and f ′ : M ′ → M ′ are objects in s -def( M ) and ( a, b )is a morphism in Mor( M / [ P ]) from f to f ′ . Then bf = f ′ a . By Lemma 2.11, thereexists an object Q ∈ P and two morphisms u : M → Q and v : Q → M ′ such that f ′ ( a − vu ) = bf . Thus, F ( a − vu, b ) = ( a, b ) and the functor F is full.The functor F induces a full and dense functor e F : s -def( M ) → Mor( M / [ P ]) / R .By Lemma 3.1(1), we have s -def( M ) / [s-epi( C )] ∼ = Mor( M / [ P ]) / R . It follows that s -def( M ) / [s-epi( M )] ∼ = mod- M / [ P ] by Lemma 2.1(1).The functor F induces a full and dense functor b F : s -def( M ) → Mor( M / [ P ]) / R ′ .By Lemma 3.1(2), we have s -def( M ) / [sp-epi( C )] ∼ = Mor( M / [ P ]) / R ′ . It follows that s -def( M ) / [sp-epi( M )] ∼ = (mod-( M / [ P ]) op ) op by Lemma 2.1(2). (cid:3) Application to category of s -triangles In this section, we will investigate the first application of Theorem 3.2 in thecase when M = C .We denote by s -tri( C ) the category of s -triangles in C , where the objects are the s -triangles X • = ( X f / / X f / / X δ / / ❴❴❴ ) and the morphisms from X • to Y • are the triples ϕ • = ( ϕ , ϕ , ϕ ) such that the following diagram is commutative X f / / ϕ (cid:15) (cid:15) X f / / ϕ (cid:15) (cid:15) X δ / / ❴❴❴ ϕ (cid:15) (cid:15) Y g / / Y g / / Y δ ′ / / ❴❴❴ and ϕ ∗ δ = ϕ ∗ δ ′ . Let X • and Y • be two s -triangles, we denote by R ( X • , Y • ) (resp. R ′ ( X • , Y • )) the class of morphisms ϕ • : X • → Y • such that ϕ factors through g (resp. ϕ factors through f ). It is easy to see that R and R ′ are ideals of s -tri( C ). Theorem 4.1.
Let C be an extriangulated category. (1) If C has enough projectives P , then s -tri( C ) / R ∼ = mod- C / [ P ] . (2) If C has enough injectives I , then s -tri( C ) / R ∼ = (mod-( C / [ I ]) op ) op .Proof. (1) We have s -tri( C ) / R ∼ = s -def( C ) / [s-epi( C )] by Lemma 3.1. Thus s -tri( C ) / R ∼ =mod- C / [ P ] follows from Theorem 3.2(1).(2) We note that R = R ′ by Lemma 2.7. Thus s -tri( C ) / R = s -tri( C ) / R ′ ∼ = s -inf( C ) / [s-mono( C )] ∼ = (mod-( C / [ I ]) op ) op , where the last equivalence follows fromTheorem 3.2(2). (cid:3) Lemma 4.2.
Let C be an extriangulated category. Assume that the following X • ϕ • (cid:15) (cid:15) X f / / ϕ (cid:15) (cid:15) X f / / ϕ (cid:15) (cid:15) X δ / / ❴❴❴ ϕ (cid:15) (cid:15) Y • Y g / / Y g / / Y δ ′ / / ❴❴❴ is a morphism of s -triangles. Then (1) The following statements are equivalent. (a) ϕ • = 0 in s -tri( C ) / R . (b) ϕ factors through f . (c) ϕ factors through g . (2) The following statements are equivalent. (a) ϕ • is a monomorphism in s -tri( C ) / R . (b) (cid:0) f ϕ (cid:1) : X → X ⊕ Y is a section.Proof. (1) It follows from Lemma 2.7.(2) The proof is similar to [10, Lemma 4.7]. (cid:3) If C has enough projectives P , then s -tri( C ) / R ∼ = mod- C / [ P ] is abelian byTheorem 4.1. The following result implies that s -tri( C ) / R is always abelian forgeneral case. Proposition 4.3.
Let C be an extriangulated category. Then s -tri( C ) / R is anabelian category.Proof. The proof is an adaption of [10, Theorem 4.8]. Assume that the following X • ϕ • (cid:15) (cid:15) X f / / ϕ (cid:15) (cid:15) X f / / ϕ (cid:15) (cid:15) X δ / / ❴❴❴ ϕ (cid:15) (cid:15) Y • Y g / / Y g / / Y δ ′ / / ❴❴❴ BELIAN QUOTIENTS VIA MORPHISM CATEGORIES 11 is a morphism of s -triangles. Thus ϕ ∗ δ = ϕ ∗ δ ′ by definition. By Lemma 2.8 andits dual, we have the following morphisms of s -triangles. K ( ϕ • ) k • (cid:15) (cid:15) X (cid:16) f ϕ (cid:17) / / X ⊕ Y a − h ) / / (1 , (cid:15) (cid:15) Z h ∗ δ / / ❴❴❴ h (cid:15) (cid:15) X • π • (cid:15) (cid:15) X f / / ϕ (cid:15) (cid:15) X f / / a (cid:15) (cid:15) X δ / / ❴❴❴ I ( ϕ • ) i • (cid:15) (cid:15) Y h / / Z h / / a (cid:15) (cid:15) X ϕ ∗ δ / / ❴❴❴ ϕ (cid:15) (cid:15) Y • c • (cid:15) (cid:15) Y g / / h (cid:15) (cid:15) Y g / / ( ) (cid:15) (cid:15) Y δ ′ / / ❴❴❴ C ( ϕ • ) Z (cid:16) h a (cid:17) / / X ⊕ Y − ϕ ,g ) / / Y h ∗ δ ′ / / ❴❴❴ Moreover, we have ϕ • = i • π • in s -tri( C ) / R . It is routine to check that k • : K ( ϕ • ) → X • is a kernel of ϕ • , c • : Y • → C ( ϕ • ) is a cokernel of ϕ • andCoker(Ker( ϕ • )) ∼ = I ( ϕ • ) ∼ = Ker(Coker( ϕ • )) . (cid:3) Proposition 4.4.
Let C be an extriangulated category. Then an s -triangle P X =( Ω X f / / P f / / X ρ / / ❴❴❴ ) with P ∈ P is a projective object in s -tri( C ) / R .Moreover, if C has enough projectives, then each projective object in s -tri( C ) / R isof the form P X .Proof. The proof is an adaption of [10, Proposition 4.11]. We omit it. (cid:3)
Definition 4.5. ([6, 17]) An s -triangle X f / / X f / / X δ / / ❴❴❴ is called Auslander-Reiten s -triangle if the following holds:(1) δ ∈ E ( C, A ) is non-split.(2) If g : X → Y is not a section, then g factors through f .(3) If h : Z → X is not a retraction, then h factors through f . Proposition 4.6.
Let C be a Krull-Smidt extriangulated category. Assume that X • : X f / / X f / / X δ / / ❴❴❴ is a non-split s -triangle such that X and X are indecomposable. Then X • is a simple object in s -tri( C ) / R if and only if X • isan Auslander-Reiten s -triangle in C .Proof. The proof is an adaption of [10, Theorem 4.20 (a)]. (cid:3)
From now on to the end of this section we assume that C is an extriangulatedcategory with enough projectives P and enough injectives I .Given an s -triangle δ = ( X f / / X f / / X ρ / / ❴❴❴ ), we define the con-travariant defect δ ∗ and the covariant defect δ ∗ by the following exact sequence offunctors C ( − , X ) C ( − ,f ) −−−−−→ C ( − , X ) C ( − ,f ) −−−−−→ C ( − , X ) → δ ∗ → , C ( X , − ) C ( f , − ) −−−−−→ C ( X , − ) C ( f , − ) −−−−−→ C ( X , − ) → δ ∗ → . Example 4.7. (1) Let δ = P X = ( Ω X f / / P f / / X ρ / / ❴❴❴ ) with P ∈ P .Then δ ∗ = C / [ P ]( − , X ) and δ ∗ = E ( X, − ).(2) Let δ = I X = ( X f / / I f / / Σ X ρ / / ❴❴❴ ) with I ∈ I . Then δ ∗ = E ( − , X )and δ ∗ = C / [ I ]( X, − ).The following result gives an explanation of [6, Theorem 4.1]. Theorem 4.8.
Let C be an extriangulated category with enough projectives P andenough injectives I . (1) We have the following equivalences s -tri( C ) / R ∼ = mod- C / [ P ] ∼ = (mod-( C / [ I ]) op ) op . Moreover, the equivalence F : s -tri( C ) / R ∼ = mod- C / [ P ] is given by δ δ ∗ and theequivalence G : s -tri( C ) / R ∼ = (mod-( C / [ I ]) op ) op is given by δ δ ∗ . (2) The abelian category mod- C / [ P ] has enough projectives and enough injectives.Moreover, each projective object is of the form C / [ P ]( − , X ) , and each injectiveobject is of the form E ( − , X ) . (3) The abelian category mod-( C / [ I ]) op has enough projectives and enough injec-tives. Moreover, each projective object is of the form C / [ I ]( X, − ) , and each injectiveobject is of the form E ( X, − ) .Proof. (1) The first assertion follows from Theorem 4.1. Assume that δ = ( X f / / X f / / X ρ / / ❴❴❴ )is an s -triangle. Recall that F ( δ ) = Coker( C / [ P ]( − , f )) and δ ∗ = Coker( C ( − , f )).Since δ ∗ ( P ) = 0, we can view δ ∗ as a finitely presented C / [ P ]-module. Thus F ( δ ) = δ ∗ . Similarly, we have G ( δ ) = δ ∗ .(2) and (3) follows from (1), Proposition 4.4 and Example 4.7. (cid:3) Corollary 4.9.
Let C be an extriangulated category with enough projectives P andenough injectives I . Then there is a duality Φ : mod- C / [ P ] → mod-( C / [ I ]) op , δ ∗ δ ∗ . Moreover, by restrictions, we obtain the following two dualities
Φ : proj- C / [ P ] → inj-( C / [ I ]) op , C / [ P ]( − , X ) E ( X, − ) . Φ : inj- C / [ P ] → proj-( C / [ I ]) op , E ( − , X )
7→ C / [ I ]( X, − ) . Proof.
It is a direct consequence of Theorem 4.8. (cid:3)
Corollary 4.10. ( [6, Proposition 4.9] ) Let C be an extriangulated category withenough projectives P and enough injectives I . (1) There is an isomorphism between C / [ P ]( Y, X ) and the group of natural trans-formations from E ( X, − ) to E ( Y, − ) . (2) There is an isomorphism between C / [ I ]( X, Y ) and the group of natural trans-formations from E ( − , X ) to E ( − , Y ) . Now we have the following Auslander-Reiten duality and defect formula for ex-triangulated categories.
BELIAN QUOTIENTS VIA MORPHISM CATEGORIES 13
Proposition 4.11.
Let C be an Ext-finite extriangulated category with enough pro-jectives P and enough injectives I . Assume that either C / [ P ] or C / [ I ] is a dualizing k -variety. Then there is an equivalence τ : C / [ P ] ∼ = C / [ I ] satisfying the followingproperties: (1) D E ( − , X ) ∼ = C / [ P ]( τ − X, − ) , D E ( X, − ) ∼ = C / [ I ]( − , τ X ) . (2) Dδ ∗ = δ ∗ τ − , Dδ ∗ = δ ∗ τ for each s -triangle δ .Proof. Without loss of generality, we assume that C / [ I ] is a dualizing k -variety.The composition of Φ : mod- C / [ P ] → mod-( C / [ I ]) op and D : mod-( C / [ I ]) op → mod- C / [ I ] defines an equivalenceΘ : mod- C / [ P ] Φ −→ mod-( C / [ I ]) op D −→ mod- C / [ I ] . It follows that Θ( C / [ P ]( − , X )) = D E ( X, − ) ∼ = C / [ I ]( − , Y ) for some Y ∈ C . There-fore, there is an equivalence τ : C / [ P ] ∼ = C / [ I ] mapping X to Y . The equivalence τ induces an equivalence τ − ∗ : mod- C / [ P ] ∼ = mod- C / [ I ] , F F τ − , such that D Φ = τ − ∗ . Assume that δ is an s -triangle, then D Φ( δ ∗ ) = Dδ ∗ . On the otherhand, τ − ∗ ( δ ∗ ) = δ ∗ τ − . Hence, we have Dδ ∗ = δ ∗ τ − . It follows that Dδ ∗ = δ ∗ τ .If δ = I X , then by Example 4.7 we have δ ∗ = E ( − , X ) and δ ∗ = C / I ( X, − ). There-fore, we have D E ( − , X ) ∼ = C / [ I ]( X, τ − ) ∼ = C / [ P ]( τ − X, − ) since Dδ ∗ = δ ∗ τ . (cid:3) Application to rigid subcategories
In this section, we will investigate the second application of Theorem 3.2 in thecase when M is a rigid subcategory of C , that is, E ( M, M ′ ) = 0 for any objects M, M ′ ∈ M .Let C ′ and C ′′ be two full subcategories of C . We denote by Cocone( C ′ , C ′′ ) the fullsubcategory of C of objects X admitting an s -triangle X / / C ′ / / C ′′ / / ❴❴❴ with C ′ ∈ C ′ and C ′′ ∈ C ′′ . We denote by Cone( C ′ , C ′′ ) the full subcategory ofobjects X admitting an s -triangle C ′ / / C ′′ / / X / / ❴❴❴ with C ′ ∈ C ′ and C ′′ ∈ C ′′ .For convenience we let M L = Cocone( M , M ) and M R = Cone( M , M ). If C hasenough projectives P , then we let Ω M =Cocone( P , M ). If C has enough injectives I , then we let Σ M =Cone( M , I ). Lemma 5.1.
Let C be an extriangulated category and M be a rigid subcategory of C . If X k / / M f / / M δ / / ❴❴❴ is an s -triangle with M i ∈ M , then k is a left M -approximation of X .Proof. For any M ∈ M , by Lemma 2.9 we have the following exact sequence C ( M , M ) → C ( M , M ) → C ( X, M ) → E ( M , M ) = 0 . Hence, k is a left M -approximation of X . (cid:3) Lemma 5.2.
Let C be an extriangulated category with enough projectives P and M be a rigid subcategory of C containing P . Assume that the following diagram X k / / g (cid:15) (cid:15) M f / / a (cid:15) (cid:15) M δ / / ❴❴❴ b (cid:15) (cid:15) X ′ k ′ / / M ′ f ′ / / M ′ δ ′ / / ❴❴❴ is a morphism of s -triangles with M i , M ′ i ∈ M . Then (1) The following statements are equivalent. (a)
The morphism b factors through f ′ . (b) The morphism ( a, b ) factors through some object in s-epi( M ) . (c) The morphism g factors through some object in M . (2) The following statements are equivalent. (a)
The morphism a factors through f in M / [ P ] . (b) The morphism ( a, b ) factors through some object in sp-epi( M ) . (c) The morphism g factors through some object in Ω M .Proof. (1) We note that (a) ⇔ (b) follows from Lemma 3.1(1).(a) ⇒ (c). Assume that b factors through f ′ . It follows that g factors through k by Lemma 2.7, which implies that g factors through M ∈ M .(c) ⇒ (a). Suppose that g has a factorization X g −→ M g −→ X ′ with M ∈ M . ByLemma 5.1 we have g factors through k . Thus g factors through k . It follows that b factors through f ′ by Lemma 2.7.(2) We note that (a) ⇔ (b) follows from Lemma 3.1(2).(a) ⇒ (c). Suppose that there is a morphism p : M → M ′ such that pf = a . Since C has enough projectives, there is an s -triangle Ω M ′ a ′ / / P a / / M ′ δ ′′ / / ❴❴❴ with P ∈ P . It is easy to see that pf − a factors through a . Assume that pf − a = a a where a : M → P . By (ET4) op , we have the following diagramΩ M ′ d ′ / / Y d / / c (cid:15) (cid:15) X ′ k ′∗ δ ′′ / / ❴❴❴ k ′ (cid:15) (cid:15) Ω M ′ a ′ / / P a / / c ′ (cid:15) (cid:15) M ′ δ ′′ / / ❴❴❴ f ′ (cid:15) (cid:15) M ′ ρ (cid:15) (cid:15) ✤✤✤ M ′ δ ′ (cid:15) (cid:15) ✤✤✤ where the first row and the second column are s -triangles. It follows that Y ∈ Ω M .Since the upper-right square of the above diagram obtained by (ET4) op is a weakpullback and ( k ′ , a ) (cid:0) ga k (cid:1) = ak + a a k = pf k = 0, there exists a morphism h : X → Y such that dh = g and ch = a k . Therefore, g factors through Y ∈ Ω M .(c) ⇒ (a). Suppose that g has a factorization X g −→ Ω M g −→ X ′ with Ω M ∈ Ω M .Then by Lemma 5.1 and (ET3) we complete the following morphism of s -triangles X k / / g (cid:15) (cid:15) M f / / a (cid:15) (cid:15) ✤✤✤ M δ / / ❴❴❴ b (cid:15) (cid:15) ✤✤✤ Ω M i / / g (cid:15) (cid:15) P π / / a (cid:15) (cid:15) ✤✤✤ M δ ′′ / / ❴❴❴ b (cid:15) (cid:15) ✤✤✤ X ′ k ′ / / M ′ f ′ / / M ′ δ ′ / / ❴❴❴ BELIAN QUOTIENTS VIA MORPHISM CATEGORIES 15 where P ∈ P . Since ( a − a a ) k = k ′ ( g − g g ) = 0, there exists a morphism p : M → M ′ such that a − a a = pf . Therefore, a = pf . (cid:3) Lemma 5.3.
Let C be an extriangulated category with enough projectives P and M be a rigid subcategory of C containing P . Then (1) s -def( M ) / [s-epi( M )] ∼ = M L / [ M ] . (2) s -def( M ) / [sp-epi( M )] ∼ = M L / [Ω M ] .Proof. (1) For any object f : M → M in s -def( M ), there exists an s -triangle X k / / M f / / M δ / / ❴❴❴ where X is unique under isomorphism. By (ET3) op ,the following commutative diagram X k / / g (cid:15) (cid:15) ✤✤✤ M f / / a (cid:15) (cid:15) M δ / / ❴❴❴ b (cid:15) (cid:15) X ′ k ′ / / M ′ f ′ / / M ′ δ ′ / / ❴❴❴ whose rows are s -triangles can be completed to a morphism of s -triangles. Themorphism g : X → X ′ is not unique in general. Assume that g ′ : X → X ′ is anothermorphism such that ( g ′ , a, b ) is a morphism of s -triangles. Then ( g − g ′ , ,
0) is alsoa morphism of s -triangles. Lemma 2.7 implies that g − g ′ factors through k , thatis, g − g ′ factors through M ∈ M . It follows that g = g ′ in M L / [ M ].Hence the assignment ( M f −→ M ) X, ( a, b ) g defines a well-defined functor F : s -def( M ) → M L / [ M ]. It is clear that F is dense. The functor F is fullby Lemma 5.1 and (ET3). Lemma 5.2(1) implies that F induces an equivalence s -def( M ) / R ∼ = M L / [ M ], that is, s -def( M ) / [s-epi( M )] ∼ = M L / [ M ].(2) For any X ∈ M L , we fix an s -triangle X k / / M f / / M δ / / ❴❴❴ with M , M ∈ M . Assume that g : X → X ′ is a morphism in M L . Then by Lemma5.1 and (ET3) there exist two morphisms a : M → M ′ and b : M → M ′ suchthat the following is a morphism of s -triangles X k / / g (cid:15) (cid:15) M f / / a (cid:15) (cid:15) M δ / / ❴❴❴ b (cid:15) (cid:15) X ′ k ′ / / M ′ f ′ / / M ′ δ ′ / / ❴❴❴ where M ′ , M ′ ∈ M . Suppose that the morphisms a ′ : M → M ′ and b ′ : M → M ′ satisfying ( g, a ′ , b ′ ) is also a morphism of s -triangles. Then (0 , a − a ′ , b − b ′ ) is amorphism of s -triangles. It follows that ( a − a ′ , b − b ′ ) factors through some objectin sp-epi( M ) by Lemma 5.2(2). We have ( a, b ) = ( a ′ , b ′ ) in s -def( M ) / [sp-epi( M )].Therefore, the assignment X ( M f −→ M ) , g ( a, b ) defines a well-definedfunctor G : M L → s -def( M ) / [sp-epi( M )].It is easy to see that G is full and dense. Lemma 5.2(2) implies that G inducesan equivalence M L / [Ω M ] ∼ = s -def( M ) / [sp-epi( M )]. (cid:3) Theorem 5.4.
Let C be an extriangulated category and M be a rigid subcategoryof C . (1) If C has enough projectives P and M contains P , then M L / [ M ] ∼ = mod-( M / [ P ]) and M L / [Ω M ] ∼ = (mod-( M / [ P ]) op ) op . (2) If C has enough injectives I and M contains I , then M R / [ M ] ∼ = (mod-( M / [ I ]) op ) op and M R / [Σ M ] ∼ = mod-( M / [ I ]) .Proof. We only prove (1). It follows from Theorem 4.1 and Lemma 5.3. (cid:3)
Definition 5.5. ([11, Definition 5.3]) Let C be an extriangulated category withenough projectives and enough injectives. A full subcategory M is called n -clustertilting for some integer n ≥
2, if it satisfies the following conditions.(1) M is functorially finite in C .(2) X ∈ M if and only if E i ( X, M ) = 0 for i ∈ { , , · · · , n − } .(3) X ∈ M if and only if E i ( M , X ) = 0 for i ∈ { , , · · · , n − } .In particular, a 2-cluster tilting subcategory of C is simply called cluster tiltingsubcategory .Assume that M is an n -cluster tilting subcategory. Define ⊥ n − M = { X ∈ C| E i ( X, M ) = 0 , i ∈ { , , · · · , n − }} and M ⊥ n − = { X ∈ C| E i ( M , X ) = 0 , i ∈ { , , · · · , n − }} . Proposition 5.6.
Let M be an n -cluster tilting subcategory of an extriangulatedcategory C , then M L = ⊥ n − M and M R = M ⊥ n − .Proof. We only prove the first equation. Let X ∈ ⊥ n − M and X x / / I y / / C δ / / ❴❴❴ be an s -triangle with I ∈ I . Assume that f : X → M is a left M -approximationof X . By Lemma 2.8, we have the following morphism of s -triangles X x / / f (cid:15) (cid:15) I y / / g (cid:15) (cid:15) C δ / / ❴❴❴ M x ′ / / C ′ y ′ / / C f ∗ δ / / ❴❴❴ such that X ( fx ) / / M ⊕ I ( x ′ , − g ) / / C ′ y ′∗ δ / / ❴❴❴ is an s -triangle. We claim that C ′ ∈ M ,thus X ∈ M L . In fact, for each M ′ ∈ M , by Lemma 2.10 we have the followingexact sequence C ( C ′ , M ′ ) → C ( M ⊕ I, M ′ ) → C ( X, M ′ ) → E ( C ′ , M ′ ) → E ( M ⊕ I, M ′ ) → E ( X, M ′ ) → · · · → E i ( M ⊕ I, M ′ ) → E i ( X, M ′ ) → E i +1 ( C ′ , M ′ ) → E i +1 ( M ⊕ I, M ′ ) → · · · . Since E i ( M ⊕ I, M ′ ) = E i +1 ( M ⊕ I, M ′ ) = 0, we get E i ( X, M ′ ) ∼ = E i +1 ( C ′ , M ′ ) for i ∈ { , , · · · , n − } . Thus, as X ∈ ⊥ n − M , E i ( C ′ , M ′ ) = 0 if 2 ≤ i ≤ n −
1. Notingthat f : X → M is a left M -approximation, we have E ( C ′ , M ′ ) = 0. Therefore C ′ ∈ M .Suppose that X ∈ M L . There exists an s -triangle X f / / M f / / M δ / / ❴❴❴ with M , M ∈ M . For any M ∈ M , we have an exact sequence0 = E i ( M , M ) → E i ( X, M ) → E i +1 ( M , M ) = 0for i ∈ { , , · · · , n − } . Therefore, X ∈ ⊥ n − M . (cid:3) Corollary 5.7.
Let C be an extriangulated category with enough projectives P andenough injectives I . If M is a cluster tilting subcategory of C , then (1) C / [ M ] ∼ = mod-( M / [ P ]) ∼ = (mod-( M / [ I ]) op ) op . (2) C / [Ω M ] ∼ = (mod-( M / [ P ]) op ) op . (3) C / [Σ M ] ∼ = mod-( M / [ I ]) . BELIAN QUOTIENTS VIA MORPHISM CATEGORIES 17
Remark 5.8.
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