Triangulated structures induced by mutations

Ryota Iitsuka

Abstract.

In representation theory of algebras, there exist two types of mutation pairs: rigid subcategories by Iyama-Yoshino and orthogonal collections by Coelho Simões-Pauksztello. It is known that such mutation pairs induce triangulated categories, however, these facts have been proved in different ways. In this paper, we introduce the concept of “premutation triples”, which is a simultaneous generalization of two different types of mutation pairs as well as concentric twin cotorsion pairs. We present two main theorems concerning mutation triples. The first theorem is that premutation triples induce pretriangulated categories. The second one is that pretriangulated categories induced by mutation triples, which are premutation triples satisfying an additional condition (MT4), become triangulated categories.

1. Introduction

The notion of “mutation” plays important roles in representation theory and related fields. Roughly speaking, mutation is an operation to obtain new objects from old ones, usually considered in triangulated categories, exact categories, or extriangulated categories [IY08, AI12, AIR14, GNP23]. There are many studies about mutation of tilting objects (for example, APR-tilting [ASS06]), silting objects[AI12], cluster-tilting objects[IY08, BMRRT06] and support $\tau$ -tilting objects[AIR14]. They are respectively called tilting mutations, silting mutations, cluster-tilting mutations and support $\tau$ -tilting mutations. There are some mutations which are considered in more generalized situations [LZ13, ZZ18]. In many cases, we can study characters of certain objects (silting objects and so on) by mutating them [AIR14, BMRRT06, IY08]. We collected some results on mutations of rigid subcategories in Appendix LABEL:Rigid_mutation_pairs and LABEL:Triangulated_structures_induced_by_rigid_mutation_pairs.

On the other hand, we may consider not only mutations of rigid subcategories (called “rigid mutations” here) but also those of orthogonal collections (called “orthogonal mutations” here). For example, simple-minded collections [KY14] and simple-minded systems [SP20, IJ23, Sim17, Dug15, SPP22]. We also collected some results in orthogonal mutations in Appendix LABEL:Orthogonal_mutation_pairs and LABEL:Triangulated_structures_induced_by_orthogonal_mutation_pairs.

What is more interesting is that both rigid and orthogonal mutations in a triangulated category induce another smaller triangulated category [IY08, BMRRT06, AI12, SP20]. Furthermore, it is also known that some mutation-like concepts induce triangulated categories. For example, a Frobenius extriangulated category induces a triangulated category whose shift functor is exactly a cosyzygy functor, which can be seen as a special case of rigid mutations in extriangulated categories [NP19]. For another example, a concentric twin cotorsion pair [Nak18, NP19, LN19] in triangulated category induces a pretriangulated category [BR07] and induces a triangulated category with some conditions [Nak18]. We review extriangulated categories and pretriangulated categories in section 2. (For details on concentric twin cotorsion pairs, see Appendix LABEL:ccTCP.)

However, the proofs showing that they induce triangulated structures are independent of all four cases: rigid mutations, orthogonal mutations, Frobenius extriangulated categories and concentric twin cotorsion pairs [IY08, IY18, Jin23, SP20, NP19, Nak18]. So our goal is to understand these triangulated structures within the same framework. In other words, we consider a simultaneous generalization of all four cases.

In section 3, we introduce the new concept of “premutation triples”, which is the framework we wanted to explain induced triangulated structures. Then we collect elementary results of premutation triples.

In the following definition, the concept of “strongly functorially finite” is defined in Definition 2.21 and the extriangulated categories $(\mathcal{C},\mathbb{E}^{\mathcal{I}},\mathfrak{s}^{\mathcal{I}})$ and $(\mathcal{C},\mathbb{E}_{\mathcal{I}},\mathfrak{s}_{\mathcal{I}})$ are defined in Example 2.17.

Definition 1.1.

(Condition 3.1, 3.9 and Definition 3.11) Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category and $(\SS,\mathcal{Z},\mathcal{V})$ be a triplet of subcategories of $\mathcal{C}$ . $(\SS,\mathcal{Z},\mathcal{V})$ is called premutation triple if it satisfies the following conditions.

(MT1)

$\SS\cap\mathcal{Z}=\mathcal{Z}\cap\mathcal{V}$ , denoted by $\mathcal{I}$ , and $\mathcal{I}$ is strongly functorially finite in $\mathcal{Z}$ .
(MT2)
1. (i)
  
  $\mathbb{E}^{\mathcal{I}}(\SS,\mathcal{Z})=0$ and $\mathbb{E}_{\mathcal{I}}(\SS,\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{I}}}(\mathcal{I},\mathcal{Z}))=0$ .
2. (ii)
  
  $\mathbb{E}_{\mathcal{I}}(\mathcal{Z},\mathcal{V})=0$ and $\mathbb{E}^{\mathcal{I}}(\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\mathcal{I}),\mathcal{V})=0$ .
(MT3)
1. (i)
  
  $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})\subset\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\SS)$ .
2. (ii)
  
  $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})\subset\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{I}}}(\mathcal{V},\mathcal{Z})$ .
3. (iii)
  
  $\SS$ and $\mathcal{Z}$ are closed under extensions in $(\mathcal{C},\mathbb{E}^{\mathcal{I}},\mathfrak{s}^{\mathcal{I}})$ and $\mathcal{V}$ and $\mathcal{Z}$ are closed under extensions in $(\mathcal{C},\mathbb{E}_{\mathcal{I}},\mathfrak{s}_{\mathcal{I}})$ .

In the last of this section, We show the first main theorems below, which is a generalization of the results in [Nak18].

Theorem 1.2.

(Theorem LABEL:main_thm1) Let $(\SS,\mathcal{Z},\mathcal{V})$ be a premutation triple. Then $\mathcal{Z}/[\mathcal{I}]$ has a pretriangulated structure.

In section LABEL:triangulated, we collect sufficient conditions for mutation triples to induce a triangulated category. We consider two cases. The former one requires an additional condition (MT4), but it is not necessary that $\mathcal{C}$ is a triangulated category.

Theorem 1.3.

(Theorem LABEL:main_thm2, Remark LABEL:MT4) Let $(\SS,\mathcal{Z},\mathcal{V})$ be a premutation triple. $(\,\cdot\,)^{-}$ and $(\,\cdot\,)^{+}$ are defined in Proposition LABEL:prop_+_and_-. We consider the following new condition (MT4).

(MT4)

$(\mathcal{Z}\langle 1\rangle/[\mathcal{I}])^{-}=(\mathcal{Z}\langle-1\rangle/[\mathcal{I}])^{+}$

$(\SS,\mathcal{Z},\mathcal{V})$ is called a mutation triple if it satisfies (MT4). If $(\SS,\mathcal{Z},\mathcal{V})$ is a mutation triple, then $\mathcal{Z}/[\mathcal{I}]$ has a triangulated structure.

The latter one is the result in [Nak18], which can be applied to mutation triples defined by concentric twin cotorsion pairs in triangulated category with specific conditions: Hovey and heart-equivalent. We show that (MT4) follows from these conditions. Therefore, the latter case is a special case of the former one.

In section LABEL:redMT, we introduce another triplet of subcategories, named reducible triple. We consider the following different version of (MT3) and (MT4) to define reducible triples. The extriangulated category $(\mathcal{C},\mathbb{E}^{\mathcal{I}}_{\mathcal{I}},\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}})$ is also defined in Example 2.17.

(RT3)
1. (i)
  
  $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})\subset\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}(\mathcal{Z},\SS)$ .
2. (ii)
  
  $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})\subset\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}(\mathcal{V},\mathcal{Z})$ .
3. (iii)
  
  $\SS$ , $\mathcal{Z}$ and $\mathcal{V}$ are closed under extensions in $(\mathcal{C},\mathbb{E}^{\mathcal{I}}_{\mathcal{I}},\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}})$ .
(RT4)
1. (i)
  
  $\mathcal{I}$ is strongly contravariantly finite in $\SS$ .
2. (ii)
  
  $\mathcal{I}$ is strongly covariantly finite in $\mathcal{V}$ .
3. (iii)
  
  $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{I}}}(\mathcal{I},\SS)=\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{V},\mathcal{I})$ , denoted by $\mathcal{R}$ .

Then we define a reducible triple as a triplet of subcategories satisfying (MT1), (MT2), (RT3) and (RT4).

Reducible triples have the following nice property.

Theorem 1.4.

(Theorem LABEL:main_thm3) Let $(\SS,\mathcal{Z},\mathcal{V})$ be a reducible triple. Let $\mathcal{E}$ be an extension closed subcategory in $(\mathcal{C},\mathbb{E}^{\mathcal{I}}_{\mathcal{I}},\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}})$ containing $\mathcal{R}$ . Then $(\mathcal{E}\cap\mathcal{Z})/[\mathcal{I}]$ is an extension closed in $\mathcal{Z}/[\mathcal{I}]$ .

As an application of this theorem, we may consider restricting mutations to extension closed subcategory $\mathcal{E}$ in $(\mathcal{C},\mathbb{E}^{\mathcal{I}}_{\mathcal{I}},\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}})$ , which induces mutations in the extriangulated category $(\mathcal{E}\cap\mathcal{Z})/[\mathcal{I}]$ . Mutations of 2-term silting complexes [AIR14] are one of these examples.

Another advantage of introducing reducible triples is that we may define mutations of collections in $\mathcal{C}$ , which is a simultaneous generalization of cluster-tilting mutations, silting mutations, mutations of simple-minded systems and mutations of simple-minded collections.

Definition 1.5.

(Definition LABEL:defi_mu-rMT) Let $(\SS,\mathcal{Z},\mathcal{V})$ be a reducible triple and $\mathcal{R}^{\prime}$ be a collection in $\operatorname{\mathrm{Ob}}(\mathcal{C})$ whose extension closure in $(\mathcal{C},\mathbb{E}^{\mathcal{I}}_{\mathcal{I}},\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}})$ is $\mathcal{R}$ . Assume that $\mathcal{X}\subset\operatorname{\mathrm{Ob}}(\mathcal{Z})$ be a collection containing $\mathcal{R}^{\prime}$ . We denote $\mathcal{X}\setminus\mathcal{R}^{\prime}$ by $\mathcal{X}_{\mathcal{R}^{\prime}}$ .

(1)

We define right $\mathcal{R}^{\prime}$ -mutation of $\mathcal{X}$ as $\mathcal{R}^{\prime}\cup\Sigma\mathcal{X}_{\mathcal{R}^{\prime}}$ , which is denoted by $\mu^{-}_{\mathcal{R}^{\prime}}(\mathcal{X})$ .
(2)

We define left $\mathcal{R}^{\prime}$ -mutation of $\mathcal{X}$ as $\mathcal{R}^{\prime}\cup\Omega\mathcal{X}_{\mathcal{R}^{\prime}}$ , which is denoted by $\mu^{+}_{\mathcal{R}^{\prime}}(\mathcal{X})$ .

Throughout this thesis, let $k$ be a field and $\mathcal{C}$ be a skeletally small additive category, thus the isomorphism class of $\operatorname{\mathrm{Ob}}(\mathcal{C})$ is a set. If $\mathcal{C}$ is an extriangulated or triangulated, we denote the extension closure in $\mathcal{C}$ by $\langle\cdot\rangle$ . We denote the category of abelian groups (resp. sets) by $\operatorname{\mathsf{Ab}}$ (resp. $\operatorname{\mathsf{Set}}$ ).

We also assume that all subcategories are additive, full and closed under isomorphisms. We do not always assume that all subcategories are closed under direct summands, so we denote the smallest subcategory containing $\mathcal{D}$ and closed under direct summands by $\operatorname{\mathsf{add}}\mathcal{D}$ for a subcategory $\mathcal{D}$ .

We recall the concept of approximations.

Definition 1.6.

Let $\mathcal{I}$ and $\mathcal{Z}$ be subcategories of $\mathcal{C}$ and let $X\in\mathcal{C}$ .

(1)

A morphism $a\colon I_{X}\to X$ in $\mathcal{C}$ is $\mathcal{I}$ -epic if $\mathcal{C}(I,a)\colon\mathcal{C}(I,I_{X})\to\mathcal{C}(I,X)$ is surjective for any $I$ in $\mathcal{I}$ .
(2)

A morphism $b\colon I_{X}\to X$ in $\mathcal{C}$ is a right $\mathcal{I}$ -approximation of $X$ if $I_{X}\in\mathcal{I}$ and $b$ is $\mathcal{I}$ -epic.
(3)

$\mathcal{I}$ is contravariantly finite in $\mathcal{Z}$ if any $Z$ in $\mathcal{Z}$ has a right $\mathcal{I}$ -approximation.

Dually, we define $\mathcal{I}$ -monic, a left $\mathcal{I}$ -approximation of $X$ and covariantly finite in $\mathcal{Z}$ . $\mathcal{I}$ is called functorially finite in $\mathcal{Z}$ if $\mathcal{I}$ is both covariantly finite and contravariantly finite in $\mathcal{Z}$ .

Acknowledgements.

The author would like to thank H. Nakaoka and Professor Michael Wemyss for valuable suggestions to improve this paper.

2. Structures associated with additive category

2.1. Extriangulated categories

First, we start this section from the definition of extriangulated categories [NP19].

Definition 2.1.

[NP19, Definition 2.7, 2.8]

(1)

For $X,Y\in\mathcal{C}$ , we denote the collection of three-term sequences whose first-term is $X$ and third-term is $Y$ by $\widetilde{\mathcal{E}}(Y,X)$ (note the order of $X$ and $Y$ ). Then we introduce an equivalence relation $\sim$ in $\widetilde{\mathcal{E}}(Y,X)$ as follows.

For $\mathbf{E}=(X\xrightarrow{x}E\xrightarrow{y}Y),\mathbf{E}^{\prime}=(X\xrightarrow{x^{\prime}}E^{\prime}\xrightarrow{y^{\prime}}Y)$ in $\widetilde{\mathcal{E}}(Y,X)$ ,

\mathbf{E}\sim\mathbf{E}^{\prime}\iff\begin{aligned} \text{There }&\text{exists an isomorphism $e\colon E\to E^{\prime}$}\\ &\text{such that }x^{\prime}=ex\text{ and }y=y^{\prime}e.\end{aligned}

We denote $\widetilde{\mathcal{E}}(Y,X)/\sim$ by $\mathcal{E}(Y,X)$ .

(2)

For $X,Y\in\mathcal{C}$ , we denote as $0=(X\xrightarrow{\scalebox{0.6}{$\begin{bmatrix}1\\ 0\end{bmatrix}$}}X\oplus Y\xrightarrow{\scalebox{0.6}{$\begin{bmatrix}0&1\end{bmatrix}$}}Y)$ in $\mathcal{E}(Y,X)$ .
(3)

For $(X\xrightarrow{a}E\xrightarrow{b}Y)$ in $\mathcal{E}(Y,X)$ and $(X^{\prime}\xrightarrow{a^{\prime}}E^{\prime}\xrightarrow{b^{\prime}}Y^{\prime})$ in $\mathcal{E}(Y^{\prime},X^{\prime})$ , $(X\xrightarrow{a}E\xrightarrow{b}Y)\oplus(X^{\prime}\xrightarrow{a^{\prime}}E^{\prime}\xrightarrow{b^{\prime}}Y^{\prime})$ is defined by $(X\oplus X^{\prime}\xrightarrow{a\oplus a^{\prime}}E\oplus E^{\prime}\xrightarrow{b\oplus b^{\prime}}Y\oplus Y^{\prime})$ in $\mathcal{E}(Y\oplus Y^{\prime},X\oplus X^{\prime})$ .

Remark 2.2.

[NP19, Definition 2.1-2.3, Remark 2.2] Let $\mathbb{E}\colon\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\operatorname{\mathsf{Ab}}$ be an additive bifunctor and $X,X^{\prime},Y,Y^{\prime},Z\in\mathcal{C}$ .

(1)

An element $\delta\in\mathbb{E}(X,Y)$ is called $\mathbb{E}$ -extension.
(2)

Let $a\colon X\to Y$ and $b\colon Y\to Z$ be morphisms in $\mathcal{C}$ , we can define the following natural transformations.

$\mathbb{E}(b,-)\colon\mathbb{E}(Z,-)\Rightarrow\mathbb{E}(Y,-)$

$\mathbb{E}(-,a)\colon\mathbb{E}(-,X)\Rightarrow\mathbb{E}(-,Y)$

(3)

There exists the following isomorphism.

\mathbb{E}(X\oplus Y,X^{\prime}\oplus Y^{\prime})\cong\mathbb{E}(X,X^{\prime})\oplus\mathbb{E}(X,Y^{\prime})\oplus\mathbb{E}(Y,X^{\prime})\oplus\mathbb{E}(Y,Y^{\prime})

Then we define $\delta\oplus\delta^{\prime}$ in left-hand side as the element which corresponds to $(\delta,0,0,\delta^{\prime})$ in right-hand side by the above isomorphism.

(4)

Let $\delta\in\mathbb{E}(Z,X)$ . We write $\mathbb{E}(b,X)(\delta),\mathbb{E}(Z,a)(\delta)$ as $\delta b,a\delta$ respectively.

(5)

In the rest of this paper, we sometimes regard $\mathbb{E}$ -extensions as “morphisms” in $\mathcal{C}$ , that is, we interpret $\delta b$ as a “composition” of $(Y\xrightarrow{b}Z\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta$}\vss}}}X)$ and $a\delta$ as a “composition” of $(Z\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta$}\vss}}}X\xrightarrow{a}Y)$ . Then we can consider commutative diagrams with $\mathbb{E}$ -extensions by this notation.

For example, let $\delta\in\mathbb{E}(X,Y),\delta^{\prime}\in\mathbb{E}(X^{\prime},Y^{\prime}),x\in\mathcal{C}(X,X^{\prime}),y\in\mathcal{C}(Y,Y^{\prime})$ , then

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Definition 2.3.

[NP19, Definition 2.4, 2.5]

(1)

Let $\mathbb{E}\colon\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\operatorname{\mathsf{Ab}}$ be an additive bifunctor. $\mathfrak{s}$ is called a realization of $\mathbb{E}$ if $\mathfrak{s}$ satisfies the following conditions.

(i)

$\mathfrak{s}$ is a collection of correspondence $\{\mathfrak{s}_{X,Y}\colon\mathbb{E}(X,Y)\to\mathcal{E}(X,Y)\}_{X,Y\in\mathcal{C}}$ . We often denote $\mathfrak{s}_{X,Y}$ as $\mathfrak{s}$ if there is no confusion.

(ii)

For $\delta\in\mathbb{E}(Y,X),\delta^{\prime}\in\mathbb{E}(Y^{\prime},X^{\prime})$ , let $\mathfrak{s}(\delta)=(X\xrightarrow{x}E\xrightarrow{y}Y),\mathfrak{s}(\delta^{\prime})=(X^{\prime}\xrightarrow{x^{\prime}}E^{\prime}\xrightarrow{y^{\prime}}Y^{\prime})$ . Then for any commutative diagrams in $\mathcal{C}$ ,

there exists a morphism $e\colon E\to E^{\prime}$ which makes the following diagram commutative.

(2)
Let $\mathfrak{s}$ be a realization of $\mathbb{E}$ . $\mathfrak{s}$ is additive if it satisfies the following conditions.
1. (i)
  
  For any $X,Y\in\mathcal{C}$ , $\mathfrak{s}(0)=(X\xrightarrow{\scalebox{0.5}{$\begin{bmatrix}1\\ 0\\ \end{bmatrix}$}}X\oplus Y\xrightarrow{\scalebox{0.5}{$[0\ 1]$}}Y)$ , that is, $\mathfrak{s}$ maps $0$ in $\mathbb{E}(Y,X)$ to $0$ in $\mathcal{E}(Y,X)$ .
2. (ii)
  
  For any $\delta\in\mathbb{E}(X,Y),\delta^{\prime}\in\mathbb{E}(X^{\prime},Y^{\prime})$ , $\mathfrak{s}(\delta\oplus\delta^{\prime})=\mathfrak{s}(\delta)\oplus\mathfrak{s}(\delta^{\prime})$ .

Remark 2.4.

[NP19, Definition 2.15, 2.19]

(1)

Let $\delta\in\mathbb{E}(Y,X)$ and $\mathfrak{s}(\delta)=(X\xrightarrow{x}E\xrightarrow{y}Y)$ . This sequence $(X\xrightarrow{x}E\xrightarrow{y}Y)$ is called $\mathfrak{s}$ -conflation. We often call it conflation if there is no confusion.
(2)

The left morphism of a conflation is called an $\mathfrak{s}$ -inflation and the right one is called an $\mathfrak{s}$ -deflation.

(3)

A pair $(\delta,\mathfrak{s}(\delta))$ is called an $\mathfrak{s}$ -triangle and it is denoted by

X\xrightarrow{x}E\xrightarrow{y}Y\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\delta$}\vss}}}X\quad\text{or}\quad Y\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\delta$}\vss}}}X\xrightarrow{x}E\xrightarrow{y}Y.

Definition 2.5.

[NP19, Definition 2.12]

A triplet ( $\mathcal{C}$ , $\mathbb{E}$ , $\mathfrak{s}$ ) is called an extriangulated category, or ET category if the triplet satisfies the following conditions.

(ET1)

$\mathbb{E}\colon\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\operatorname{\mathsf{Ab}}$ is an additive bifunctor.
(ET2)

$\mathfrak{s}$ is an additive realization of $\mathbb{E}$ .

(ET3)

Let $\delta\in\mathbb{E}(Y,X),\delta^{\prime}\in\mathbb{E}(Y^{\prime},X^{\prime})$ . For $\mathfrak{s}(\delta)=(X\xrightarrow{x}E\xrightarrow{y}Y),\mathfrak{s}(\delta^{\prime})=(X^{\prime}\xrightarrow{x^{\prime}}E^{\prime}\xrightarrow{y^{\prime}}Y^{\prime})$ and any diagram in $\mathcal{C}$ ,

there exists a morphism $b\colon Y\to Y^{\prime}$ which makes the following diagram commutative.

(ET3)^op

Dual of $\mathrm{(ET3)}$ .

(ET4)

For $\delta\in\mathbb{E}(C,X),\epsilon\in\mathbb{E}(D,Y)$ , $\mathfrak{s}(\delta)=(X\xrightarrow{x}Y\xrightarrow{x^{\prime}}C),\mathfrak{s}(\epsilon)=(Y\xrightarrow{y}Z\xrightarrow{y^{\prime}}D)$ , there exist $\mathfrak{s}$ -triangles $X\xrightarrow{z}Z\xrightarrow{z^{\prime}}E\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\delta^{\prime}$}\vss}}}X$ and $C\xrightarrow{c}E\xrightarrow{d}D\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\epsilon^{\prime}$}\vss}}}C$ which make the following diagram commutative.

(ET4)^op

Dual of $\mathrm{(ET4)}$ .

Remark 2.6.

[NP19, Corollary 3.12] Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an ET category and $(X\xrightarrow{x}Y\xrightarrow{y}Z\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta$}\vss}}}X)$ be an $\mathfrak{s}$ -triangle. Then the following are exact sequences.

\mathcal{C}(Z,-)\xrightarrow{\scalebox{0.6}{$\mathcal{C}(y,-)$}}\mathcal{C}(Y,-)\xrightarrow{\scalebox{0.6}{$\mathcal{C}(x,-)$}}\mathcal{C}(X,-)\xrightarrow{-\circ\delta}\mathbb{E}(Z,-)\xrightarrow{\scalebox{0.6}{$\mathbb{E}(y,-)$}}\mathbb{E}(Y,-)\xrightarrow{\scalebox{0.6}{$\mathbb{E}(x,-)$}}\mathbb{E}(X,-)

\mathcal{C}(-,X)\xrightarrow{\scalebox{0.6}{$\mathcal{C}(-,x)$}}\mathcal{C}(-,Y)\xrightarrow{\scalebox{0.6}{$\mathcal{C}(-,y)$}}\mathcal{C}(-,Z)\xrightarrow{\delta\circ-}\mathbb{E}(-,X)\xrightarrow{\scalebox{0.6}{$\mathbb{E}(-,x)$}}\mathbb{E}(-,Y)\xrightarrow{\scalebox{0.6}{$\mathbb{E}(-,y)$}}\mathbb{E}(-,Z)

The following proposition is often used in this paper.

Proposition 2.7.

[NP19, Proposition 3.15] (Shifted octahedrons) Let $X_{i}\xrightarrow{x_{i}}Y_{i}\xrightarrow{y_{i}}Z\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta_{i}$}\vss}}}X_{i}$ be an $\mathfrak{s}$ -triangle for $i=1,2$ . Then there exist $\mathfrak{s}$ -triangles $X_{2}\xrightarrow{v_{1}}W\xrightarrow{w_{1}}Y_{1}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\epsilon_{1}$}\vss}}}X_{2}$ and $X_{1}\xrightarrow{v_{2}}W\xrightarrow{w_{2}}Y_{2}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\epsilon_{2}$}\vss}}}X_{1}$ which make the following diagram commutative.

Proof.

See [NP19, Proposition 3.15]. ∎

The following definitions of projective objects and injective objects are analogies of exact category.

Definition 2.8.

[NP19] Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an ET category.

(1)

We define a subcategory of $\mathcal{C}$ , $\mathrm{Proj}_{\mathbb{E}}\mathcal{C}$ as $\{X\in\mathcal{C}\mid\mathbb{E}(X,\mathcal{C})=0\}$ . An object in $\operatorname{Proj}_{\mathbb{E}}\mathcal{C}$ is called a projective object.
(2)

$\mathcal{C}$ has enough projectives if, for any $X$ in $\mathcal{C}$ , there exists a conflation $X^{\prime}\rightarrow P\rightarrow X$ with $P\in\operatorname{Proj}_{\mathbb{E}}\mathcal{C}$ .

(3)

For subcategories $\mathcal{X},\mathcal{Y}$ of $\mathcal{C}$ , we define the following three subcategories.

\mathcal{X}\ast\mathcal{Y}=\left\{E\in\mathcal{C}\ \middle|\ \begin{aligned} \text{There }&\text{exists an }\mathfrak{s}\text{-conflation }\\ &X\rightarrow E\rightarrow Y\text{ where }X\in\mathcal{X},Y\in\mathcal{Y}\end{aligned}\right\}

\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}}(\mathcal{X},\mathcal{Y})=\left\{Z\in\mathcal{C}\ \middle|\ \begin{aligned} \text{There }&\text{exists an }\mathfrak{s}\text{-conflation }\\ &X\rightarrow Y\rightarrow Z\text{ where }X\in\mathcal{X},Y\in\mathcal{Y}\end{aligned}\right\}

\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}}(\mathcal{X},\mathcal{Y})=\left\{Z^{\prime}\in\mathcal{C}\ \middle|\ \begin{aligned} \text{There }&\text{exists an }\mathfrak{s}\text{-conflation }\\ &Z^{\prime}\rightarrow X\rightarrow Y\text{ where }X\in\mathcal{X},Y\in\mathcal{Y}\end{aligned}\right\}

We denote $\operatorname{Proj}_{\mathbb{E}}\mathcal{C}$ by $\operatorname{Proj}\mathcal{C}$ when there is no confusion. Dually, we define $\operatorname{Inj}_{\mathbb{E}}\mathcal{C}$ and enough injectives. $\mathcal{C}$ is called Frobenius if $\operatorname{Proj}\mathcal{C}=\operatorname{Inj}\,\mathcal{C}$ and $\mathcal{C}$ has enough projectives and enough injectives.

Example 2.9.

[NP19, Corollary 3.18, Proposition 3.22]

(1)

An exact category is an ET category whose inflations are monomorphic and deflations are epimorphic. In this situation, $\mathfrak{s}$ -conflations are exactly conflations in the exact structure.
(2)

A triangulated category is exactly a Frobenius ET category with $\operatorname{Proj}\mathcal{C}=0$ .

There are some ways to obtain a new ET category from old one. First case is a generalized statement of Happel’s theorem [Hap88].

Proposition 2.10.

[NP19, Proposition 3.30] Let $\mathcal{C}$ be an ET category and let $\mathcal{I}\subset\operatorname{Proj}\mathcal{C}\cap\operatorname{Inj}\mathcal{C}$ . Then $\mathcal{C}/[\mathcal{I}]$ also becomes an ET category.

Proof.

See [NP19, Proposition 3.30]. ∎

Corollary 2.11.

[NP19, Corollary 7.4, Remark 7.5] Let $\mathcal{C}$ be an Frobenius ET category. Then $\mathcal{C}/[\operatorname{Proj}\mathcal{C}]$ becomes a triangulated category.

Proof.

See [NP19, Corollary 7.4, Remark 7.5]. ∎

Next way is to restrict the bifunctor $\mathbb{E}$ and the realization $\mathfrak{s}$ to an extension-closed subcategory. We start from the definition of “extension-closed” subcategories.

Definition 2.12.

[NP19, Definition 2.17] Let $\mathcal{C}$ be an ET category and $\mathcal{Z}$ be a subcategory. $\mathcal{Z}$ is called extension-closed if, for any conflation $A\rightarrow B\rightarrow C$ where $A,C$ in $\mathcal{Z}$ , then $B$ is also in $\mathcal{Z}$ .

Lemma 2.13.

[NP19, Remark 2.18] Let $\mathcal{C}$ be an ET category and $\mathcal{Z}$ is an extension-closed subcategory. Then $\mathcal{Z}$ has an ET structure defined by restricting $\mathbb{E}$ and $\mathfrak{s}$ to $\mathcal{Z}$ .

Last way is to restrict the bifunctor to a closed subfunctor. See also [DRSSK99, p649] for the following definitions in exact categories.

Definition 2.14.

[HLN21, Definition 3.7] Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an ET category.

(1)
A functor $\mathbb{F}\colon\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\operatorname{\mathsf{Set}}$ is called a subfunctor of $\mathbb{E}$ if it satisfies the following conditions.
1. (i)
  
  For any $X,Y\in\mathcal{C}$ , $\mathbb{F}(X,Y)$ is a subset of $\mathbb{E}(X,Y)$ .
2. (ii)
  
  For any morphism $x\colon X^{\prime}\to X$ and $y\colon Y\to Y^{\prime}$ , $\mathbb{F}(x,y)=\mathbb{E}(x,y)|_{\mathbb{F}(X,Y)}$ .
Then we denote $\mathbb{F}\subset\mathbb{E}$ .
(2)

A subfunctor $\mathbb{F}\subset\mathbb{E}$ is called additive if $\mathbb{F}$ is an additive bifunctor.

Definition 2.15.

[HLN21, Definition 3.8] Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an ET category and $\mathbb{F}$ be an additive subfunctor of $\mathbb{E}$ . We define $\mathfrak{s}|_{\mathbb{F}}$ by restriction of $\mathfrak{s}$ onto $\mathbb{F}$ .

Proposition 2.16.

[HLN21, Proposition 3.16] [DRSSK99, Proposition 1.4] Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an ET category and $\mathbb{F}$ be an additive subfunctor of $\mathbb{E}$ . Then the following are equivalent.

(i)

$\mathfrak{s}|_{\mathbb{F}}$ -inflations are closed under composition.
(ii)

$\mathfrak{s}|_{\mathbb{F}}$ -deflations are closed under composition.
(iii)

$(\mathcal{C},\mathbb{F},\mathfrak{s}|_{\mathbb{F}})$ satisfies (ET4).
(iv)

$(\mathcal{C},\mathbb{F},\mathfrak{s}|_{\mathbb{F}})$ satisfies (ET4)^op.
(v)

$(\mathcal{C},\mathbb{F},\mathfrak{s}|_{\mathbb{F}})$ is an ET category.

In this case, $\mathbb{F}$ is called closed.

The following are examples of closed subfunctors defined in [HLN21].

Example 2.17.

[HLN21, Definition 3.18, Proposition 3.19] Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an ET category and $\mathcal{I}$ be a subcategory of $\mathcal{C}$ .

(1)

We define a closed subfunctor $\mathbb{E}_{\mathcal{I}}$ of $\mathbb{E}$ as follows.

\mathbb{E}_{\mathcal{I}}(C,A)=\{\delta\in\mathbb{E}(C,A)\mid\delta\circ-\colon\mathcal{C}(\mathcal{I},C)\to\mathbb{E}(\mathcal{I},A);\text{zero morphism}\}.

(2)

We define a closed subfunctor $\mathbb{E}^{\mathcal{I}}$ of $\mathbb{E}$ as follows.

\mathbb{E}^{\mathcal{I}}(C,A)=\{\delta\in\mathbb{E}(C,A)\mid-\circ\delta\colon\mathcal{C}(A,\mathcal{I})\to\mathbb{E}(C,\mathcal{I});\text{zero morphism}\}.

(3)

We define a closed subfunctor $\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}$ of $\mathbb{E}$ as follows.

$\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}(C,A)=\mathbb{E}^{\mathcal{I}}(C,A)\cap\mathbb{E}_{\mathcal{I}}(C,A).$

We denote $\mathfrak{s}|_{\mathbb{E}_{\mathcal{I}}}$ (resp. $\mathfrak{s}|_{\mathbb{E}^{\mathcal{I}}}$ , $\mathfrak{s}|_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}$ ) by $\mathfrak{s}_{\mathcal{I}}$ (resp. $\mathfrak{s}^{\mathcal{I}}$ , $\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}}$ ).

Remark 2.18.

In this paper, the ET structures defined by $(\mathcal{C},$ $\mathbb{E}_{\mathcal{I}},$ $\mathfrak{s}_{\mathcal{I}})$ and $(\mathcal{C},$ $\mathbb{E}^{\mathcal{I}},$ $\mathfrak{s}^{\mathcal{I}})$ are called relative extriangulated structure, or more simply relative structure. On the other hand, in [FGPPP24, Section 2], all extriangulated substructures are called relative extriangulated structure.

Remark 2.19.

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an ET category and $\mathcal{I}$ be a subcategory of $\mathcal{C}$ . Then $\mathcal{I}\subset\operatorname{Proj}_{\mathbb{E}_{\mathcal{I}}}\mathcal{C}$ and $\mathcal{I}\subset\operatorname{Inj}_{\mathbb{E}^{\mathcal{I}}}\mathcal{C}$ . This follows from the long exact sequences in Remark 2.6.

Now, we consider the approximation theory in ET categories. In the rest of this subsection, we fix an ET category $(\mathcal{C},\mathbb{E},\mathfrak{s})$ . We start from a reformulation of extensions in relative structures by using $\mathcal{I}$ -epic and $\mathcal{I}$ -monic morphisms in ET categories for a subcategory $\mathcal{I}$ .

Lemma 2.20.

[Ara24, Proposition 3.2] Let $\mathcal{I}$ be a subcategory of $\mathcal{C}$ . Let $X\xrightarrow{x}Y\xrightarrow{y}Z\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta$}\vss}}}X$ be an $\mathfrak{s}$ -triangle.

(1)

$\delta\in\mathbb{E}_{\mathcal{I}}(Z,X)\iff y$ is $\mathcal{I}$ -epic.
(2)

$\delta\in\mathbb{E}^{\mathcal{I}}(Z,X)\iff x$ is $\mathcal{I}$ -monic.

Proof.

From Remark 2.6, this follows from definitions of $\mathcal{I}$ -epic and $\mathcal{I}$ -monic. ∎

Definition 2.21.

[ZZ18, Definition 3.21] Let $\mathcal{I},\mathcal{X}$ be subcategories of $\mathcal{C}$ where $\mathcal{I}$ is closed under direct summands.

(1)

$\mathcal{I}$ is strongly contravariantly finite in $\mathcal{X}$ with respect to $(\mathcal{C},\mathbb{E},\mathfrak{s})$ if, for any $X\in\mathcal{X}$ , there exists an $\mathfrak{s}$ -deflation $I_{X}\xrightarrow{g}X$ where $g$ is a right $\mathcal{I}$ -approximation.
(2)

$\mathcal{I}$ is strongly covariantly finite in $\mathcal{X}$ with respect to $(\mathcal{C},\mathbb{E},\mathfrak{s})$ if, for any $X\in\mathcal{X}$ , there exists an $\mathfrak{s}$ -inflation $X\xrightarrow{f}I^{X}$ where $f$ is a left $\mathcal{I}$ -approximation.
(3)

$\mathcal{I}$ is strongly functorially finite in $\mathcal{X}$ with respect to $(\mathcal{C},\mathbb{E},\mathfrak{s})$ if, $\mathcal{I}$ is both strongly covariantly finite and strongly contravariantly finite in $\mathcal{X}$ .

Remark 2.22.

We do not assume that $\mathcal{I}$ is contained in $\mathcal{X}$ in Definition 2.21.

Example 2.23.

Assume that $\mathcal{C}$ has enough projectives (resp. injectives). Then $\operatorname{Proj}\mathcal{C}$ (resp. $\operatorname{Inj}\,\mathcal{C}$ ) is strongly contravariantly (resp. covariantly) finite in $\mathcal{C}$ .

The following lemma is an ET version of Lemma LABEL:lem_bracket_rigidver.

Lemma 2.24.

[Ara24, Lemma 3.5] Let $\mathcal{I}\subset\mathcal{X}$ be subcategories of $\mathcal{C}$ .

(1)

Assume that $\mathcal{I}$ is strongly covariantly finite in $\mathcal{X}$ .

(i)

For $X\in\mathcal{X}$ , there exists an inflation $i^{X}\colon X\to I^{X}$ which is a left $\mathcal{I}$ -approximation of $X$ . Then we obtain the following $\mathfrak{s}^{\mathcal{I}}$ -triangle.

X\xrightarrow{i^{X}}I^{X}\xrightarrow{p^{X}}X\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X}$}\vss}}}X

(ii)

For a morphism $x\colon X\to X^{\prime}$ in $\mathcal{X}$ , we define $x\langle 1\rangle\colon X\langle 1\rangle\to X^{\prime}\langle 1\rangle$ as a morphism in $\mathcal{C}$ which makes the following diagram in $\mathcal{C}$ commutative.

Then $\langle 1\rangle$ induces an additive functor $\langle 1\rangle\colon\mathcal{X}/[\mathcal{I}]\to\mathcal{C}/[\mathcal{I}]$ . Moreover, $\langle 1\rangle$ is unique up to natural isomorphisms.

(2)

Assume that $\mathcal{I}$ is strongly contravariantly finite in $\mathcal{X}$ .

(i)

For $X\in\mathcal{X}$ , there exists a deflation $p_{X}\colon I_{X}\to X$ which is a right $\mathcal{I}$ -approximation of $X$ . Then we obtain the following $\mathfrak{s}_{\mathcal{I}}$ -triangle.

X\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda_{X}$}\vss}}}X\langle-1\rangle\xrightarrow{i_{X}}I_{X}\xrightarrow{p_{X}}X

(ii)

For a morphism $x\colon X\to X^{\prime}$ in $\mathcal{X}$ , we define $x\langle-1\rangle\colon X\langle-1\rangle\to X^{\prime}\langle-1\rangle$ as a morphism in $\mathcal{C}$ which makes the following diagram commutative.

Then $\langle-1\rangle$ induces an additive functor $\langle-1\rangle\colon\mathcal{X}/[\mathcal{I}]\to\mathcal{C}/[\mathcal{I}]$ . Moreover, $\langle-1\rangle$ is unique up to natural isomorphisms.

Proof.

We prove only (1). First, for each $X$ in $\mathcal{C}$ , choose an $\mathfrak{s}$ -triangle $X\xrightarrow{i^{X}}I^{X}\xrightarrow{p^{X}}X\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X}$}\vss}}}X$ where $i^{X}$ is a left $\mathcal{I}$ -approximation. For any morphism $x\colon X\to X^{\prime}$ in $\mathcal{X}$ , since $i^{X}$ is a left $\mathcal{I}$ -approximation and (ET3), there exist morphisms $y$ and $i$ which make the following diagram commutative.

Then we define $x\langle 1\rangle$ by $y$ . Assume that morphisms $y,y^{\prime}\colon X\langle 1\rangle\to X^{\prime}\langle 1\rangle$ in $\mathcal{C}$ satisfy $x\lambda^{X}=\lambda^{X^{\prime}}y=\lambda^{X^{\prime}}y^{\prime}$ . Then $\lambda^{X^{\prime}}(y-y^{\prime})=0$ . So, $y-y^{\prime}=0$ in $\mathcal{C}/[\mathcal{I}]$ and $\langle 1\rangle\colon\mathcal{X}\to\mathcal{C}/[\mathcal{I}]$ is well-defined. This induces a functor $\langle 1\rangle\colon\mathcal{X}/[\mathcal{I}]\to\mathcal{C}/[\mathcal{I}]$ because $\lambda^{X^{\prime}}x\langle 1\rangle=x\lambda^{X}=0$ for $x\in[\mathcal{I}]$ . Therefore, $y$ is uniquely determined by $x$ up to $[\mathcal{I}]$ and $\langle 1\rangle$ is an additive functor.

Uniqueness of $\langle 1\rangle$ up to natural isomorphisms follows from the diagram below and $n^{X}m^{X}=\operatorname{id}_{X\langle 1\rangle}$ in $\mathcal{C}/[\mathcal{I}]$ where $X\xrightarrow{j^{X}}J^{X}\xrightarrow{q^{X}}X\langle 1\rangle^{\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle{\lambda^{\prime}}^{X}$}\vss}}}X$ is another $\mathfrak{s}$ -triangle with a left $\mathcal{I}$ -approximation $j^{X}$ .

∎

Remark 2.25.

In the proof of Lemma 2.24, we used the following commutative diagram.

(2.1)

Then we denote $m^{X}$ in $\mathcal{C}/[\mathcal{I}]$ (that is $\underline{m^{X}}$ in Notation 3.5) by $\mu_{X}$ because $\mu$ induces natural isomorphism $\langle 1\rangle\Rightarrow\langle 1\rangle^{\prime}$ where $\langle 1\rangle,\langle 1\rangle^{\prime}\colon\mathcal{X}/[\mathcal{I}]\to\mathcal{C}/[\mathcal{I}]$ .

Notation 2.26.

Let $\mathcal{I}$ and $\mathcal{X}$ be subcategories of $\mathcal{C}$ .

(1)

We denote $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{X},\mathcal{I})$ by $\mathcal{X}\langle 1\rangle_{\mathcal{I}}$ . In particular, $\mathcal{I}\subset\mathcal{X}\langle 1\rangle_{\mathcal{I}}$ .
(2)

We denote $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{I}}}(\mathcal{I},\mathcal{X})$ by $\mathcal{X}\langle-1\rangle_{\mathcal{I}}$ . In particular, $\mathcal{I}\subset\mathcal{X}\langle-1\rangle_{\mathcal{I}}$ .

If there is no confusion, we often drop $\mathcal{I}$ of $\langle 1\rangle_{\mathcal{I}}$ and $\langle-1\rangle_{\mathcal{I}}$ .

$\{X\langle 1\rangle\mid X\in\mathcal{X}\}$ in Lemma 2.24 and $\mathcal{X}\langle 1\rangle$ are same up to isomorphisms in $\mathcal{C}/[\mathcal{I}]$ .

Note that we can define $\mathcal{X}\langle 1\rangle$ (resp. $\mathcal{X}\langle-1\rangle$ ) even if $\mathcal{I}$ is not strongly covariantly (resp. contravariantly) finite in $\mathcal{X}$ .

Lemma 2.27.

[Ara24, Proposition 3.2] Let $\mathcal{I}$ be a subcategory of $\mathcal{C}$ .

(1)

If $\mathcal{I}$ is strongly contravariantly finite in $\mathcal{C}$ , then $\mathcal{I}=\operatorname{Proj}_{\mathbb{E}_{\mathcal{I}}}\!\mathcal{C}$ and $(\mathcal{C},\mathbb{E}_{\mathcal{I}},\mathfrak{s}_{\mathcal{I}})$ has enough projectives.
(2)

If $\mathcal{I}$ is strongly covariantly finite in $\mathcal{C}$ , then $\mathcal{I}=\operatorname{Inj}_{\mathbb{E}^{\mathcal{I}}}\!\mathcal{C}$ and $(\mathcal{C},\mathbb{E}^{\mathcal{I}},\mathfrak{s}^{\mathcal{I}})$ has enough injectives.

Proof.

We only prove (2). Let $\mathcal{I}^{\prime}=\operatorname{Inj}_{\mathbb{E}^{\mathcal{I}}}\mathcal{C}$ . By definition of relative structure, $\mathcal{I}\subset\mathcal{I}^{\prime}$ . On the other hand, for any $I^{\prime}\in\mathcal{I}^{\prime}$ , there exists an $\mathfrak{s}^{\mathcal{I}}$ -triangle $I^{\prime}\rightarrow I\rightarrow Z\dashrightarrow I^{\prime}$ where $I\in\mathcal{I}$ since $\mathcal{I}$ is strongly covariantly finite in $\mathcal{C}$ . Since $I^{\prime}\in\mathcal{I}^{\prime}$ and $\mathcal{I}$ is closed under direct summands, the above $\mathfrak{s}$ -triangle splits and $I^{\prime}\in\mathcal{I}$ . ∎

ET categories are different from triangulated categories because not every morphism has a cone or cocone. However, we may sometimes replace any morphisms by inflations (resp. deflations) up to ideal quotient in the following meanings.

Lemma 2.28.

[LN19, Proposition 1.20] Take a morphism $f\colon X\to X^{\prime}$ . Let $X\xrightarrow{x}E\xrightarrow{y}Y\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\delta$}\vss}}}X$ be an $\mathfrak{s}$ -triangle and $X^{\prime}\xrightarrow{x^{\prime}}E^{\prime}\xrightarrow{y^{\prime}}Y\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle f\delta$}\vss}}}X^{\prime}$ be a realization of $f\delta$ . Then there exists a morphism $g\colon E\to E^{\prime}$ which satisfies the following two conditions.

(i)

$g$ makes the following diagram commutative.

(ii)

$X\xrightarrow{\scalebox{0.5}{$\begin{bmatrix}f\\ x\\ \end{bmatrix}$}}X^{\prime}\oplus E\xrightarrow{\scalebox{0.5}{$[-x^{\prime}\ g]$}}E^{\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\delta y^{\prime}$}\vss}}}X$ is an $\mathfrak{s}$ -triangle, in particular, $\begin{bmatrix}f\\ x\\ \end{bmatrix}$ is an inflation.

Proof.

See [LN19, Proposition 1.20]. ∎

The following statement is also useful.

Lemma 2.29.

[NP19, Proposition 3.17] Let $X\xrightarrow{a}Y\xrightarrow{b^{\prime}}Z\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta_{1}$}\vss}}}X$ , $X\xrightarrow{c}Z^{\prime}\xrightarrow{d}Z^{\prime\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta_{2}$}\vss}}}X$ and $X^{\prime}\xrightarrow{a^{\prime}}Y\xrightarrow{b}Z^{\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta_{3}$}\vss}}}X^{\prime}$ be $\mathfrak{s}$ -triangles where $c=ba$ . Then there exists an $\mathfrak{s}$ -triangle $X^{\prime}\xrightarrow{f}Z\xrightarrow{g}Z^{\prime\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta$}\vss}}}X^{\prime}$ which makes the following diagram commutative.

Proof.

This is a dual statement of [NP19, Proposition 3.17]. ∎

Corollary 2.30.

Let $\mathcal{I}$ be a strongly covariantly finite subcategory in $\mathcal{C}$ and $a\colon X\to Y$ be a morphism in $\mathcal{C}$ . Take $\mathfrak{s}^{\mathcal{I}}$ -triangles $X\xrightarrow{i^{X}}I^{X}\xrightarrow{p^{X}}X\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X}$}\vss}}}X$ and $X\xrightarrow{j^{X}}J^{X}\xrightarrow{q^{X}}X\langle 1\rangle^{\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle{\lambda^{\prime}}^{X}$}\vss}}}X\vphantom{\Big{(}}$ where $i^{X}$ and $j^{X}$ are left $\mathcal{I}$ -approximations.

There exist the following $\mathfrak{s}^{\mathcal{I}}$ -triangles from Lemma 2.28.

	$\displaystyle X\xrightarrow{\scalebox{0.5}{$\begin{bmatrix}a\\ i^{X}\\ \end{bmatrix}$}}Y\oplus I^{X}\xrightarrow{\widetilde{b}}C^{a}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\widetilde{\delta}$}\vss}}}X$
	$\displaystyle X\xrightarrow{\scalebox{0.5}{$\begin{bmatrix}a\\ j^{X}\\ \end{bmatrix}$}}Y\oplus J^{X}\xrightarrow{\widetilde{b^{\prime}}}C^{a^{\prime}}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\widetilde{\delta^{\prime}}$}\vss}}}X$
	$\displaystyle X\xrightarrow{\scalebox{0.5}{$\begin{bmatrix}a\\ i^{X}\\ j^{X}\\ \end{bmatrix}$}}Y\oplus I^{X}\oplus J^{X}\xrightarrow{\widetilde{b^{\prime\prime}}}C^{a^{\prime\prime}}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\widetilde{\delta^{\prime\prime}}$}\vss}}}X$

Then $C^{a^{\prime\prime}}\cong C^{a}\oplus J^{X}\cong C^{a^{\prime}}\oplus I^{X}$ . In particular, $C^{a}$ is uniquely determined up to isomorphisms in $\mathcal{C}/[\mathcal{I}]$ and does not depend on choices of $\mathfrak{s}^{\mathcal{I}}$ -triangle $X\xrightarrow{i^{X}}I^{X}\xrightarrow{p^{X}}X\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X}$}\vss}}}\vphantom{\Big{(}}X$ .

Proof.

From Lemma 2.29, there exists a commutative diagram.

Since $\mathbb{E}^{\mathcal{I}}(C^{a},J^{X})=0$ , $C^{a^{\prime\prime}}\cong C^{a}\oplus J^{X}$ . One can show $C^{a^{\prime\prime}}\cong C^{a^{\prime}}\oplus I^{X}$ in the same way. ∎

Corollary 2.31.

[Nak18, Corollary 3.7] Let $\mathcal{I}\subset\mathcal{Z}$ be subcategories in $\mathcal{C}$ and assume that $\mathcal{I}$ is strongly functorially finite in $\mathcal{Z}$ . If a morphism $f\colon X\to Y$ in $\mathcal{Z}$ is an isomorphism in $\mathcal{C}/[\mathcal{I}]$ , then there exist $I,J\in\mathcal{I}$ and an isomorphism $\scalebox{0.8}{$\begin{bmatrix}f&j\\ i&k\end{bmatrix}$}\colon X\oplus J\to Y\oplus I$ .

Proof.

Take an inflation $i^{X}\colon X\to I^{X}$ which is also a left $\mathcal{I}$ -approximation. Since $f$ is an isomorphism in $\mathcal{C}/[\mathcal{I}]$ , $\scalebox{0.8}{$\begin{bmatrix}f\\ i^{X}\end{bmatrix}$}\colon X\to Y\oplus I^{X}$ is both an inflation and a section. We denote $\begin{bmatrix}f\\ i^{X}\end{bmatrix}$ by $\widetilde{f}$ . Next, there exists a deflation $p\colon I_{Y}\to Y\oplus I^{X}$ which is also a right $\mathcal{I}$ -approximation. Since $\widetilde{f}$ is also an isomorphism in $\mathcal{C}/[\mathcal{I}]$ , $\scalebox{0.8}{$\begin{bmatrix}\widetilde{f}&p\end{bmatrix}$}\colon X\oplus I_{Y}\to Y\oplus I^{X}$ is both an deflation and a retraction. From Lemma 2.29, there exists the following commutative diagram in $\mathcal{C}$ .

Because $\mathcal{I}$ is closed under direct summands, $J\in\mathcal{I}$ . By taking a section $\scalebox{0.8}{$\begin{bmatrix}j\\ k\end{bmatrix}$}\colon J\to Y\oplus I^{X}$ , $\begin{bmatrix}f&j\\ i^{X}&k\end{bmatrix}$ induces an isomorphism. ∎

At the end of this subsection, we add the following lemma which is used in section LABEL:triangulated.

Lemma 2.32.

Let $\mathcal{I}\subset\mathcal{X}$ be subcategories of $\mathcal{C}$ and suppose that $\mathcal{I}$ is strongly covariantly finite in $\mathcal{X}$ . Assume that $\mathbb{E}^{\mathcal{I}}(\mathcal{I},\mathcal{X})=0$ , then $\langle 1\rangle\colon\mathcal{X}/[\mathcal{I}]\to\mathcal{X}\langle 1\rangle/[\mathcal{I}]$ is an equivalence. In particular, an $\mathfrak{s}^{\mathcal{I}}$ -triangle $X\rightarrow I^{X}\rightarrow X\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X}$}\vss}}}X\vphantom{\bigg{(}}$ is an $\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}}$ -triangle.

Proof.

Since $\langle 1\rangle\colon\mathcal{X}/[\mathcal{I}]\to\mathcal{X}\langle 1\rangle/[\mathcal{I}]$ is essentially surjective by definition and one can directly prove $\langle 1\rangle$ is full, we only show that $\langle 1\rangle$ is faithful. Let $x\colon X\to X^{\prime}$ be a morphism in $\mathcal{X}$ and take $\mathfrak{s}^{\mathcal{I}}$ -triangles $X\rightarrow I^{X}\rightarrow X\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X}$}\vss}}}X\vphantom{\Big{(}}$ and $X^{\prime}\rightarrow I^{X^{\prime}}\rightarrow X^{\prime}\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X^{\prime}}$}\vss}}}X^{\prime}\vphantom{\Big{(}}$ . Then we obtain a morphism $x\langle 1\rangle\colon X\langle 1\rangle\to X^{\prime}\langle 1\rangle$ where $x\lambda^{X}=\lambda^{X^{\prime}}x\langle 1\rangle$ . If $x\langle 1\rangle=0$ in $\mathcal{C}/[\mathcal{I}]$ , $x\lambda^{X}=0$ from $\mathbb{E}^{\mathcal{I}}(\mathcal{I},\mathcal{X})=0$ . Thus, $x$ factors through $I^{X}$ . ∎

2.2. Pretriangulated categories

In this subsection, we introduce right triangulated categories, left triangulated categories and pretriangulated categories in [BR07]. First, we define right triangles and left triangles like as distinguished triangles in triangulated categories.

Definition 2.33.

[BR07, II.1] Let $\Sigma,\Omega\colon\mathcal{C}\to\mathcal{C}$ be additive (endo)functors.

(1)

We define a category $\mathsf{RTri}$ as follows.

(i)

Objects are sequences in $\mathcal{C}$ of the form $X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}\Sigma X$ .

(ii)

Morphisms are triplets $(x,y,z)$ which make the following diagram commutative.

(2)

We define a category $\mathsf{LTri}$ as follows.

(i)

Objects are sequences in $\mathcal{C}$ of the form $\Omega Z\xrightarrow{h^{\prime}}X\xrightarrow{f}Y\xrightarrow{g}Z$ .

(ii)

Morphisms are triplets $(x,y,z)$ which satisfy the following commutative diagram.

Now, let us define right triangulated categories and left triangulated categories.

Definition 2.34.

[BR07, II.1]

(1)
A right triangulated category is a triplet $(\mathcal{C},\Sigma,\nabla)$ where
1. (i)
  
  $\Sigma\colon\mathcal{C}\to\mathcal{C}$ be an additive functor.
2. (ii)
  
  $\nabla$ is a full subcategory of $\mathsf{RTri}$ .
3. (iii)
  
  $(\mathcal{C},\Sigma,\nabla)$ satisfies all of the axioms of a triangulated category except that $\Sigma$ is not necessarily an equivalence.
(2)
A left triangulated category is a triplet $(\mathcal{C},\Omega,\Delta)$ where
1. (i)
  
  $\Omega\colon\mathcal{C}\to\mathcal{C}$ be an additive functor.
2. (ii)
  
  $\Delta$ is a full subcategory of $\mathsf{LTri}$ .
3. (iii)
  
  $(\mathcal{C},\Omega,\Delta)$ satisfies all of the axioms of a triangulated category except that $\Omega$ is not necessarily an equivalence.

Remark 2.35.

For the convenience of the reader, we list the axioms of right triangulated categories $(\mathcal{C},\Sigma,\nabla)$ below.

(rTR0)

$\nabla$ is closed under isomorphisms.
(rTR1)
1. (i)
  
  For any $X\in\mathcal{C}$ , the sequence $X\xrightarrow{\text{id}}X\rightarrow 0\rightarrow\Sigma X$ is in $\nabla$ .
2. (ii)
  
  For any morphism $f\colon X\to Y$ , there exists the sequence $X\xrightarrow{f}Y\rightarrow Z\rightarrow\Sigma X$ in $\nabla$ .
(rTR2)

Let $X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}\Sigma X$ be a sequence in $\nabla$ , then $Y\xrightarrow{g}Z\xrightarrow{h}\Sigma X\xrightarrow{-\Sigma f}\Sigma Y$ is also in $\nabla$ .

(rTR3)

Assume that there exists a commutative diagram where each row is in $\nabla$ .

Then there exists a morphism $z\colon Z_{1}\to Z_{2}$ which makes the following diagram commutative.

(rTR4)

Assume that $a^{\prime\prime}=a^{\prime}a$ and $X\xrightarrow{a}Y\xrightarrow{b}C\xrightarrow{c}\Sigma X$ , $Y\xrightarrow{a^{\prime}}Z\xrightarrow{b^{\prime}}D\xrightarrow{c^{\prime}}\Sigma Y$ , $X\xrightarrow{a^{\prime\prime}}Z\xrightarrow{b^{\prime\prime}}E\xrightarrow{c^{\prime\prime}}\Sigma X$ are in $\nabla$ . Then there exists $C\xrightarrow{s}E\xrightarrow{t}D\xrightarrow{u}\Sigma C$ in $\nabla$ which makes the following diagram commutative.

Remark 2.36.

Assume that $(\mathcal{C},\Sigma,\nabla)$ satisfies from (rTR0) to (rTR3). Then one can show the following statements like triangulated categories.

(1)

For any right triangle $X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}\Sigma X$ ,

$\mathcal{C}(-,X)\xrightarrow{f\circ-}\mathcal{C}(-,Y)\xrightarrow{g\circ-}\mathcal{C}(-,Z)$

is exact.

(2)

Assume that there exists a commutative diagram where each row is in $\nabla$ and $x,y$ are isomorphisms.

Then there exists an isomorphism $z\colon Z_{1}\to Z_{2}$ which makes the following diagram commutative.

Finally, we define pretriangulated categories.

Definition 2.37.

[BR07, II.1] $(\mathcal{C},\Sigma,\Omega,\nabla,\Delta)$ is a pretriangulated category if it satisfies the following conditions.

(i)

$(\Sigma,\Omega)$ is an adjoint pair of additive endofunctors $\Sigma,\Omega\colon\mathcal{C}\to\mathcal{C}$ .
Now, let $\alpha$ be a unit and $\beta$ be a counit.
(ii)

$(\mathcal{C},\Sigma,\nabla)$ is a right triangulated category.
(iii)

$(\mathcal{C},\Omega,\Delta)$ is a left triangulated category.

(iv)

For any commutative diagrams in $\mathcal{C}$

where $X_{1}\xrightarrow{f_{1}}Y_{1}\xrightarrow{g_{1}}Z_{1}\xrightarrow{h_{1}}\Sigma X_{1}$ is a right triangle and $\Omega Z_{2}\xrightarrow{h_{2}^{\prime}}X_{2}\xrightarrow{f_{2}}Y_{2}\xrightarrow{g_{2}}Z_{2}$ is a left triangle, then there exist morphisms $u\colon Z_{1}\to Y_{2}$ and $s^{\prime}\colon Y_{1}\to X_{2}$ which make following diagrams commutative.

Example 2.38.

[BR07, II.1]

(1)

A triangulated category $(\mathcal{C},[1],\triangle)$ is a pretriangulated category $(\mathcal{C},[1],[-1],$ $\triangle,\triangle)$ .
(2)

Assume that $\mathcal{A}$ is an abelian category. Let $\mathsf{REx}$ be the collection of right exact sequences and $\mathsf{LEx}$ be the collection of left exact sequences. Then $(\mathcal{A},$ $0,$ $0,$ $\mathsf{REx},$ $\mathsf{LEx})$ is a pretriangulated category.

3. Pretriangulated structures induced by premutation triples

We fix an ET category $(\mathcal{C},\mathbb{E},\mathfrak{s})$ in this section.

3.1. The condition (MT1) and (MT2)

In the following definition, we use notations which are compatible with previous section and [Nak18].

Condition 3.1.

Let $\SS,\mathcal{V},\mathcal{Z}$ be subcategories of $\mathcal{C}$ . We denote $\SS\cap\mathcal{Z},\mathcal{V}\cap\mathcal{Z}$ by $\mathcal{I},\mathcal{J}$ , respectively. We consider the following two conditions.

(rMT1)

$\mathcal{I}$ is strongly contravariantly finite in $\mathcal{Z}$ .
(rMT2)

$\mathbb{E}^{\mathcal{I}}(\SS,\mathcal{Z})=0$ and $\mathbb{E}_{\mathcal{I}}(\SS,\mathcal{Z}\langle-1\rangle)=0$ .

Dually, we also consider the following two conditions.

(lMT1)

$\mathcal{J}$ is strongly covariantly finite in $\mathcal{Z}$ .
(lMT2)

$\mathbb{E}_{\mathcal{J}}(\mathcal{Z},\mathcal{V})=0$ and $\mathbb{E}^{\mathcal{J}}(\mathcal{Z}\langle 1\rangle,\mathcal{V})=0$ .

For convenience, we also use the following conditions.

(MT1)

$\mathcal{I}=\mathcal{J}$ , (rMT1) and (lMT1).
(MT2)

(rMT2) and (lMT2).

Remark 3.2.

(1)

By definition of strongly contravariantly (resp. covariantly) finite, we assume that $\mathcal{I}$ (resp. $\mathcal{J}$ ) is closed under direct summands. On the other hand, recall that we do not always assume that $\SS,\mathcal{Z}$ and $\mathcal{V}$ are closed under direct summands.
(2)

$\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})$ $=$ $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})$ if (rMT2) holds. Dually, $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})$ $=$ $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})$ if (lMT2) holds.
(3)

From Lemma 2.32, $\mathbb{E}^{\mathcal{I}}(\SS,\mathcal{Z}\langle-1\rangle)=0$ and $\mathbb{E}_{\mathcal{J}}(\mathcal{Z}\langle 1\rangle,\mathcal{V})=0$ under (MT1) and (MT2). That is because $\delta\in\mathbb{E}^{\mathcal{I}}(S,Z^{\prime})$ where $S\in\SS,Z^{\prime}\in\mathcal{Z}\langle-1\rangle$ factors through $\lambda\in\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}(Z,Z^{\prime})$ where $Z\in\mathcal{Z}$ . We can show $\mathbb{E}_{\mathcal{J}}(\mathcal{Z}\langle 1\rangle,\mathcal{V})=0$ in the same way.
(4)

From Lemma 2.32, $\mathbb{E}_{\mathcal{I}}(\SS,\mathcal{Z})=0$ and $\mathbb{E}^{\mathcal{J}}(\mathcal{Z},\mathcal{V})=0$ under (MT1) and (MT2). That is because $\delta\in\mathbb{E}_{\mathcal{I}}(S,Z)$ where $S\in\SS,Z\in\mathcal{Z}$ factors through $\lambda^{Z}\in\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}(Z\langle 1\rangle,Z)$ from $\mathbb{E}_{\mathcal{I}}(\SS,\mathcal{I})\subset\mathbb{E}_{\mathcal{I}}(\SS,\mathcal{Z}\langle-1\rangle)=0$ . We can show $\mathbb{E}^{\mathcal{J}}(\mathcal{Z},\mathcal{V})=0$ in the same way.

The definitions of rigid mutation pairs and orthogonal mutation pairs are in Appendix LABEL:tri_str_by_MP. The definition of concentric twin cotorsion pairs is in Appendix LABEL:ccTCP.

Example 3.3.

(1)

[NP19] Assume that $(\mathcal{C},\mathbb{E},\mathfrak{s})$ is Frobenius with $\mathcal{P}=\operatorname{Proj}_{\mathbb{E}}\mathcal{C}$ . Then $(\mathcal{P},\mathcal{C},\mathcal{P})$ satisfies (MT1) and (MT2).
(2)

[IY08] Assume that $\mathcal{C}$ is a triangulated category. Let $\mathcal{I}$ be a functorially finite rigid subcategory of $\mathcal{C}$ and $(\mathcal{X},\mathcal{Y})$ be a rigid $\mathcal{I}$ -mutation pair. We define $\mathcal{Z}=\mathcal{X}\cap\mathcal{Y}$ . Then $(\mathcal{I},\mathcal{Z},\mathcal{I})$ satisfies (MT1) and (MT2).
(3)

[SP20] Assume that $\mathcal{C}$ is a Hom-finite Krull-Schmidt triangulated $k$ -category with a Serre functor $\mathbb{S}$ . Let $\mathcal{M}$ be a collection of $\operatorname{\mathrm{Ob}}(\mathcal{C})$ which satisfies the condition (SP1) in Condition LABEL:SP_condi1 and $(\mathcal{X},\mathcal{Y})$ be an orthogonal $\langle\mathcal{M}\rangle$ -mutation pair. We define $\mathcal{Z}=\mathcal{X}\cap\mathcal{Y}$ . Then $(\langle\mathcal{M}[1]\rangle,\mathcal{Z},\langle\mathcal{M}[-1]\rangle)$ satisfies (MT1) and (MT2).
(4)

[Nak18] Let $((\SS,\mathcal{T}),(\mathcal{U},\mathcal{V}))$ be a concentric twin cotorsion pairs. We define $\mathcal{Z}=\mathcal{T}\cap\mathcal{U}$ . Then $(\SS,\mathcal{Z},\mathcal{V})$ satisfies (MT1) and (MT2).

Proof.

(1) (MT1) follows from the definition of Frobenius ET categories. Since $\mathcal{P}$ is both projective and injective, $\mathbb{E}=\mathbb{E}^{\mathcal{P}}=\mathbb{E}_{\mathcal{P}}$ and $\mathbb{E}(\mathcal{P},-)=0,\mathbb{E}(-,\mathcal{P})=0$ . Thus, (MT2) holds.

(2) By definition of rigid $\mathcal{I}$ -mutation pairs, $\mathcal{I}$ is functorially finite in $\mathcal{Z}$ . In particular, (MT1) holds. $\mathcal{Z}\subset{}^{\perp}\mathcal{I}[1]\cap\mathcal{I}[-1]^{\perp}$ also follows from definition of rigid $\mathcal{I}$ -mutation pairs. Note that $\mathcal{Z}\langle 1\rangle\subset\mathcal{Y},\mathcal{Z}\langle-1\rangle\subset\mathcal{X}$ . Thus, $\mathbb{E}(\mathcal{I},\mathcal{Z})=0,\mathbb{E}(\mathcal{I},\mathcal{Z}\langle-1\rangle)=0,\mathbb{E}(\mathcal{Z},\mathcal{I})=0$ and $\mathbb{E}(\mathcal{Z}\langle 1\rangle,\mathcal{I})=0$ . In particular, (MT2) holds.

(3) Note that $\mathcal{I}=\mathcal{J}=0$ since $(\mathcal{X},\mathcal{Y})$ is an orthogonal mutation pair, in particular, $\mathbb{E}=\mathbb{E}^{\mathcal{I}}=\mathbb{E}_{\mathcal{I}}$ and $\langle 1\rangle=[1]$ , then (MT1) hold. (MT2) follows from $\mathcal{Z}\subset{}^{\perp}\mathcal{M}^{\perp}\cap{}^{\perp}\mathcal{M}[-1]\cap\mathcal{M}[1]^{\perp}$ .

(4) $\mathcal{I}=\mathcal{J}$ since $((\SS,\mathcal{T}),(\mathcal{U},\mathcal{V}))$ is concentric. By definition of twin cotorsion pairs, $\mathbb{E}(\SS,\mathcal{Z})=0$ and $\mathbb{E}(\mathcal{Z},\mathcal{V})=0$ . From $\mathcal{Z}\langle 1\rangle\subset\mathcal{U}$ and $\mathcal{Z}\langle-1\rangle\subset\mathcal{T}$ , $\mathbb{E}(\SS,\mathcal{Z}\langle-1\rangle)=0$ and $\mathbb{E}(\mathcal{Z}\langle 1\rangle,\mathcal{V})=0$ . Thus, we obtain (MT2). (MT1) is direct from the definition of twin cotorsion pairs (for details, see [LN19]). ∎

From Example 3.3(4), we may consider a triplet $(\SS,\mathcal{Z},\mathcal{V})$ satisfying (MT1) and (MT2) as a generalization of concentric twin cotorsion pairs. We define new subcategories of $\mathcal{C}$ , $\widetilde{\mathcal{U}}$ and $\widetilde{\mathcal{T}}$ , which are denoted by $\mathcal{C}^{-}$ and $\mathcal{C}^{+}$ in Definition LABEL:defi_plus-minus_ccTCP, respectively. We use these notations because $\mathcal{U}\subset\widetilde{\mathcal{U}}=\mathcal{C}^{-}$ and $\mathcal{T}\subset\widetilde{\mathcal{T}}=\mathcal{C}^{+}$ hold for any concentric twin cotorsion pair $((\SS,\mathcal{T}),(\mathcal{U},\mathcal{V}))$ .

Notation 3.4.

(1)

Let $(\SS,\mathcal{Z})$ be a pair of subcategories which satisfies (rMT1) and (rMT2). We define $\widetilde{\mathcal{U}}$ as $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\SS)$ .
(2)

Let $(\mathcal{Z},\mathcal{V})$ be a pair of subcategories which satisfies (lMT1) and (lMT2). We define $\widetilde{\mathcal{T}}$ as $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{J}}}(\mathcal{V},\mathcal{Z})$ .

Notation 3.5.

(1)

Let $(\SS,\mathcal{Z})$ be a pair of subcategories which satisfies (rMT1) and (rMT2). We denote $\mathcal{X}/[\mathcal{I}]$ by $\underline{\mathcal{X}}$ for a subcategory $\mathcal{X}$ containing $\mathcal{I}$ .
(2)

Let $(\mathcal{Z},\mathcal{V})$ be a pair of subcategories which satisfies (lMT1) and (lMT2). We denote $\mathcal{Y}/[\mathcal{J}]$ by $\underline{\mathcal{Y}}$ for a subcategory $\mathcal{Y}$ containing $\mathcal{J}$ .
(3)

For a morphism $x$ in $\mathcal{Z}$ , we denote $\underline{x\langle 1\rangle}$ by $\underline{x}\langle 1\rangle$ .

Lemma 3.6.

[Nak18, Fact 2.1, Definition 3.10]

(1)

Let $(\SS,\mathcal{Z})$ be a pair of subcategories which satisfies (rMT1) and (rMT2). For $U\in\widetilde{\mathcal{U}}$ , take an $\mathfrak{s}^{\mathcal{I}}$ -triangle $U\xrightarrow{h^{U}}Z^{U}\xrightarrow{g^{U}}S\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle$}\vss}}}U$ . Then $-\circ\underline{h^{U}}\colon\underline{\mathcal{Z}}(Z^{U}\!,\mathcal{Z})\to\underline{\widetilde{\mathcal{U}}}(U,\mathcal{Z})$ is a natural isomorphism. In particular, there exists an additive functor $\sigma\colon\underline{\widetilde{\mathcal{U}}}\to\underline{\mathcal{Z}}\,;U\mapsto Z^{U}$ which is a left adjoint of the inclusion functor $\iota_{\sigma}\colon\underline{\mathcal{Z}}\to\underline{\widetilde{\mathcal{U}}}$ .
(2)

Let $(\mathcal{Z},\mathcal{V})$ be a pair of subcategories which satisfies (lMT1) and (lMT2). For $T\in\widetilde{\mathcal{T}}$ , take an $\mathfrak{s}_{\mathcal{I}}$ -triangle $T\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle$}\vss}}}V\xrightarrow{}Z_{T}\xrightarrow{h_{T}}T$ . Then $\underline{h_{T}}\circ-\colon\underline{\mathcal{Z}}(\mathcal{Z},Z_{T})\to\underline{\widetilde{\mathcal{T}}}(\mathcal{Z},T)$ is a natural isomorphism. In particular, there exists an additive functor $\omega\colon\underline{\widetilde{\mathcal{T}}}\to\underline{\mathcal{Z}}\,;T\mapsto Z_{T}$ which is a right adjoint of the inclusion functor $\iota_{\omega}\colon\underline{\mathcal{Z}}\to\underline{\widetilde{\mathcal{T}}}$ .

Proof.

We only prove (1). We denote $h^{U},g^{U}$ by $h,g$ respectively. Since $-\circ\underline{h}\colon\underline{\mathcal{Z}}(Z^{U}\!,\mathcal{Z})\to\underline{\widetilde{\mathcal{U}}}(U,\mathcal{Z})$ is clearly well-defined and functorial, we only have to show that this is bijective for any $Z^{\prime}\in\mathcal{Z}$ . Take a morphism $z\colon Z^{U}\to Z^{\prime}$ with $\underline{zh}=0$ . There exists an $\mathfrak{s}_{\mathcal{I}}$ -triangle $Z^{\prime}\langle-1\rangle\xrightarrow{}I\xrightarrow{p_{Z^{\prime}}}Z^{\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda_{Z^{\prime}}$}\vss}}}Z^{\prime}\langle-1\rangle$ because $\mathcal{I}$ is strongly contravariantly finite in $\mathcal{Z}$ . We denote $p_{Z^{\prime}},\lambda_{Z^{\prime}}$ by $p,\lambda$ , respectively. Since $\lambda$ is an $\mathbb{E}_{\mathcal{I}}$ -extension and $zh\in[\mathcal{I}]$ , there exists a morphism $a\colon U\to I$ with $zh=pa$ . Since $\mathbb{E}^{\mathcal{I}}(\SS,\mathcal{Z})=0$ , there exists a morphism $a^{\prime}\colon Z^{U}\to I$ such that $a=a^{\prime}h$ . Then $0=zh-pa=(z-pa^{\prime})h$ . Thus, there exists a morphism $z^{\prime}\colon S\to Z^{\prime}$ where $z-pa^{\prime}=z^{\prime}g$ . From $\mathbb{E}_{\mathcal{I}}(\SS,\mathcal{Z}\langle-1\rangle)=0$ , then $\lambda z^{\prime}=0$ and there exists a morphism $z^{\prime\prime}\colon S\to I$ where $z^{\prime}=pz^{\prime\prime}$ . Therefore $\underline{z}=\underline{pa^{\prime}+z^{\prime}g}=\underline{p(a^{\prime}+z^{\prime\prime}g)}=0$ , that is, $-\circ\underline{h}$ is injective. On the other hand, $-\circ\underline{h}$ is surjective because $\mathbb{E}^{\mathcal{I}}(\SS,\mathcal{Z})=0$ . ∎

Remark 3.7.

(1)

From Lemma 3.6(1), for a morphism $z\colon Z^{U}\to Z$ in $\mathcal{Z}$ ,

$\underline{z}=0\text{ in }\underline{\mathcal{Z}}\iff\underline{zh^{U}}=0\text{ in }\widetilde{\underline{\mathcal{U}}}\,.$
(2)

From the above proof, for any morphism $u\colon U\to Z^{\prime}$ where $U\in\widetilde{\mathcal{U}}$ and $Z^{\prime}\in\mathcal{Z}$ , there exists a morphism $z\colon Z^{U}\to Z^{\prime}$ which satisfies $u=zh^{U}$ in $\widetilde{\mathcal{U}}$ and such $z$ is unique up to $[\mathcal{I}]$ .

(3)

We can construct $\sigma\colon\widetilde{\underline{\mathcal{U}}}\to\underline{\mathcal{Z}}$ directly like $\langle 1\rangle$ .

(i)

For $U\in\widetilde{\mathcal{U}}$ , there exists an $\mathfrak{s}^{\mathcal{I}}$ -triangle

$U\xrightarrow{h^{U}}\sigma U\xrightarrow{g^{U}}S\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\rho^{U}$}\vss}}}U$

where $\sigma U\in\mathcal{Z}$ and $S\in\SS$ .

(ii)

For a morphism $u\colon U_{1}\to U_{2}$ in $\widetilde{\mathcal{U}}$ , there exist $\mathfrak{s}^{\mathcal{I}}$ -triangles $U_{1}\xrightarrow{h^{U_{1}}}\sigma U_{1}\xrightarrow{g^{U_{1}}}S_{1}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\rho^{U_{1}}$}\vss}}}U_{1}$ and $U_{2}\xrightarrow{h^{U_{2}}}\sigma U_{2}\xrightarrow{g^{U_{2}}}S_{2}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\rho^{U_{2}}$}\vss}}}U_{2}$ where $S_{1},S_{2}\in\SS$ . We denote $\sigma U_{1},\sigma U_{2}$ by $Z_{1},Z_{2}$ , respectively. Then there exists a unique morphism $z\colon Z_{1}\to Z_{2}$ in $\mathcal{C}$ up to $[\mathcal{I}]$ which makes the following diagram commutative from (2).

We define $\sigma(u)$ as $z$ .

Then $\sigma$ induces an additive functor $\sigma\colon\underline{\widetilde{\mathcal{U}}}\to\underline{\mathcal{Z}}$ . From uniqueness of left adjoint functor up to natural isomorphisms, $\sigma$ does not depend on the choices of $\mathfrak{s}^{\mathcal{I}}$ -triangle $U\xrightarrow{h^{U}}Z^{U}\rightarrow S\dashrightarrow U$ in Lemma 3.6 up to natural isomorphisms.

Notation 3.8.

(1)

For a morphism $u$ in $\widetilde{\mathcal{U}}$ , we denote $\underline{\sigma(u)}$ by $\sigma(\underline{u})$ .
(2)

We denote the unit of the adjoint pair $(\sigma,\iota_{\sigma})$ by $\eta$ .
(3)

We denote the counit of the adjoint pair $(\iota_{\omega},\omega)$ by $\varepsilon$ .

3.2. Definition of premutation triples

Condition 3.9.

We consider the following conditions.

(1)
Assume (rMT1) and (rMT2).
- (rMT3)
  1. (i)
    
    $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})\subset\widetilde{\mathcal{U}}$ .
  2. (ii)
    
    $\SS$ and $\mathcal{Z}$ are closed under extensions in $(\mathcal{C},\mathbb{E}^{\mathcal{I}},\mathfrak{s}^{\mathcal{I}})$ .
(2)
Assume (lMT1) and (lMT2).
- (lMT3)
  1. (i)
    
    $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}_{\mathcal{J}}}(\mathcal{Z},\mathcal{Z})\subset\widetilde{\mathcal{T}}$ .
  2. (ii)
    
    $\mathcal{Z}$ and $\mathcal{V}$ are closed under extensions in $(\mathcal{C},\mathbb{E}_{\mathcal{J}},\mathfrak{s}_{\mathcal{J}})$ .
(3)
Assume (MT1) and (MT2).
- (MT3)
  
  (rMT3) and (lMT3).

Remark 3.10.

If (rMT2) holds, then $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})$ $=$ $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})$ . On the other hand, $\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}(\mathcal{Z},\SS)\subsetneq\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\SS)=\widetilde{\mathcal{U}}$ in general. In Example LABEL:ex_HoveyTCP_ARquiv(LABEL:ex_HoveyTCP_ARquiv_1), $Y\in\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\SS)=\mathcal{U}$ but $Y\notin\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}(\mathcal{Z},\SS)$ . That is because there exists a triangle $Y\xrightarrow{f}I_{1}\xrightarrow{g}I_{2}\rightarrow Y[1]$ where $f$ is $\mathcal{I}$ -monic but $g$ is not $\mathcal{I}$ -epic.

The condition $\operatorname{Cone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}}(\mathcal{Z},\mathcal{Z})\subset\operatorname{CoCone\hskip 0.5pt}_{\mathbb{E}^{\mathcal{I}}_{\mathcal{I}}}(\mathcal{Z},\SS)$ is contained in Condition LABEL:condi_rMT3.

Definition 3.11.

Let $\SS,\mathcal{Z},\mathcal{V}$ be subcategories of $\mathcal{C}$ .

(1)

$(\SS,\mathcal{Z})$ is a right mutation double if $(\SS,\mathcal{Z},\mathcal{I})$ satisfies (MT1), (rMT2) and (rMT3).
(2)

$(\mathcal{Z},\mathcal{V})$ is a left mutation double if $(\mathcal{J},\mathcal{Z},\mathcal{V})$ satisfies (MT1), (lMT2) and (lMT3).
(3)

$(\SS,\mathcal{Z},\mathcal{V})$ is a premutation triple if it satisfies from (MT1) to (MT3).

Remark 3.12.

(1)

Note that both right and left mutation doubles are required to satisfy (MT1). That is because right (resp. left) mutation doubles are required to have the additive functor $\langle 1\rangle$ (resp. $\langle-1\rangle$ ) so that we may define $\Sigma$ (resp. $\Omega$ ) in Definition 3.14.
(2)

A triplet $(\SS,\mathcal{Z},\mathcal{V})$ which satisfies (MT1), (MT2) and (MT3), that is $(\SS,\mathcal{Z})$ is a right mutation double and $(\mathcal{Z},\mathcal{V})$ is a left mutation double with $\mathcal{I}=\mathcal{J}$ , is not mutation triple. (We define mutation triples in Section LABEL:triangulated.) Later, we check that right mutation doubles (resp. left mutation doubles, premutation triples, mutation triples) induce right triangulated (resp. left triangulated, pretriangulated, triangulated) categories.

From the following examples, Frobenius ET categories, rigid mutation pairs, orthogonal mutation pairs and concentric twin cotorsion pairs are examples of premutation triples.

Example 3.13.

(1)

Assume that $\mathcal{C}$ has enough projectives and $\operatorname{Proj}\mathcal{C}$ is strongly functorially finite in $\mathcal{C}$ . Then $(\operatorname{Proj}\mathcal{C},\mathcal{C})$ is a right mutation double. Dually, assume that $\mathcal{C}$ has enough injectives and $\operatorname{Inj}\mathcal{C}$ is strongly functorially finite in $\mathcal{C}$ . Then $(\mathcal{C},\operatorname{Inj}\mathcal{C})$ is a left mutation double.
(2)

[NP19] Assume that $\mathcal{C}$ is Frobenius with $\mathcal{P}=\operatorname{Proj}\mathcal{C}$ . Then $(\mathcal{P},\mathcal{C},\mathcal{P})$ is a premutation triple. More generally, for any strongly functorially finite subcategory $\mathcal{X}$ in $\mathcal{C}$ , $(\mathcal{X},\mathcal{C},\mathcal{X})$ in $(\mathcal{C},\mathbb{E}^{\mathcal{X}}_{\mathcal{X}},\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}})$ is a premutation triple.
(3)

[IY08] In the case of Example 3.3(2), we additionally assume that $\mathcal{X}=\mathcal{Y}$ and (IY3) in Condition LABEL:IY_condi. Then $(\mathcal{I},\mathcal{Z},\mathcal{I})$ is a premutation triple.
(4)

[SP20] In the case of Example 3.3(3), we additionally assume that $\mathcal{X}=\mathcal{Y}$ and (SP3) in Condition LABEL:SP_condi2. Then $(\langle\mathcal{M}[1]\rangle,\mathcal{Z},\langle\mathcal{M}[-1]\rangle)$ is a premutation triple.
(5)

[Nak18] In the case of Example 3.3(4), $(\SS,\mathcal{Z},\mathcal{V})$ is a premutation triple.

Proof.

(1) We only prove when $\mathcal{C}$ has enough projectives. From assumptions, (MT1) holds. (rMT2) follows from $\mathbb{E}(\mathcal{P},-)=0$ . (rMT3) is also true because $\widetilde{\mathcal{U}}=\mathcal{C}$ and $\mathcal{P}$ is closed under extensions in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ . (2) is direct from (1). (3) follows from Lemma 2.20 and [IY08, Lemma 4.3]. (4) is by definition of (SP3). (5) follows from Lemma LABEL:conic and the definition of twin cotorsion pairs. ∎

Because of the condition (rMT3)1)((i) and (lMT3)2)((i), we can finally define mutation functors.

Definition 3.14.

(1)

Let $(\SS,\mathcal{Z})$ be a right mutation double. We define an additive functor $\Sigma=\sigma\circ\langle 1\rangle\colon\underline{\mathcal{Z}}\to\underline{\mathcal{Z}}$ , called a right mutation functor.
(2)

Let $(\mathcal{Z},\mathcal{V})$ be a left mutation double. We define an additive functor $\Omega=\omega\circ\langle-1\rangle\colon\underline{\mathcal{Z}}\to\underline{\mathcal{Z}}$ , called a left mutation functor.

Example 3.15.

(1)

In the case of Example 3.13(2), a right (resp. left) mutation functor is exactly a cosyzygy (resp. syzygy) functor.
(2)

In the case of Example 3.13(3), a right (resp. left) mutation functor is exactly right (resp. left) mutation functor in Lemma LABEL:lem_bracket_rigidver.
(3)

In the case of Example 3.13(4), a right (resp. left) mutation functor is exactly right (resp. left) mutation functor in Definition LABEL:sigma_and_omega_inSP.
(4)

In the case of Example 3.13(5), a right (resp. left) mutation functor is exactly right (resp. left) mutation functor in Definition LABEL:defi_sigma-omega_CTP.

In the last part of this section, we collect some lemmas we use later.

Lemma 3.16.

[Nak18, Proposition 4.3] Let $(\SS,\mathcal{Z},\mathcal{V})$ be a premutation triple. Then $(\Sigma,\Omega)$ is an adjoint pair.

Proof.

From Lemma 3.6, we only have to show that there exists a bifunctorial isomorphism $\Phi\colon\underline{\widetilde{\mathcal{U}}}(Z\langle 1\rangle,Z^{\prime})\xrightarrow{\sim}\widetilde{\underline{\mathcal{T}}}(Z,Z^{\prime}\langle-1\rangle)$ for any $Z,Z^{\prime}\in\mathcal{Z}$ . For a morphism $z\colon Z\langle 1\rangle\to Z^{\prime}$ , we take an $\mathfrak{s}^{\mathcal{I}}$ -triangle $Z\xrightarrow{i^{Z}}I^{Z}\xrightarrow{p^{Z}}Z\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{Z}$}\vss}}}Z$ and an $\mathfrak{s}_{\mathcal{I}}$ -triangle $Z^{\prime}\langle-1\rangle\xrightarrow{i_{Z^{\prime}}}I_{Z^{\prime}}\xrightarrow{p_{Z^{\prime}}}Z^{\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda_{Z^{\prime}}$}\vss}}}Z^{\prime}\langle-1\rangle$ . Then there exists a morphism $z^{\prime}\colon Z\to Z^{\prime}\langle-1\rangle$ which makes the following diagram commutative since $p_{Z^{\prime}}$ is a right $\mathcal{I}$ -approximation.

Then we define $\Phi(\underline{z})=\underline{z^{\prime}}$ . This correspondence is well-defined and injective from the commutativity of the right most square. In particular, $z^{\prime}$ is unique up to $[\mathcal{I}]$ which satisfies $z^{\prime}\lambda^{Z}=-\lambda_{Z^{\prime}}z$ . This correspondence is also surjective since $i^{Z}$ is a left $\mathcal{I}$ -approximation. Finally, this is bifunctorial because $z^{\prime}$ is uniquely determined by the commutativity of right square. ∎

Notation 3.17.

We denote the unit (resp. counit) of $(\Sigma,\Omega)$ by $\alpha$ (resp. $\beta$ ).

Remark 3.18.

From Lemma 3.6 and 3.16, we have the following isomorphisms for $Z,Z^{\prime}\in\mathcal{Z}$ .

\displaystyle\underline{\mathcal{Z}}(\Sigma Z,Z^{\prime})\xrightarrow{-\circ\underline{h^{Z\langle 1\rangle}}}\underline{\widetilde{\mathcal{U}}}(Z\langle 1\rangle,Z^{\prime})\xrightarrow{\phantom{x}\Phi\phantom{x}}\underline{\widetilde{\mathcal{T}}}(Z,Z^{\prime}\langle-1\rangle)\xleftarrow{\underline{h_{Z^{\prime}\langle-1\rangle}}\circ-}\underline{\mathcal{Z}}(Z,\Omega Z^{\prime})

(3.1)

Then for a morphism $f\colon Z\to\Omega Z^{\prime}$ in $\mathcal{Z}$ , the corresponding morphism $f^{\prime}\colon\Sigma Z\to Z^{\prime}$ in (3.1) is uniquely determined by the following commutative diagram up to $[\mathcal{I}]$ .

(3.2)

Note that $\underline{f^{\prime}}=\beta_{Z^{\prime}}\circ\Sigma(\underline{f})$ and $\underline{f}=\Omega(\underline{f^{\prime}})\circ\alpha_{Z}$ .

Remark 3.19.

We assume that $\mathcal{C}$ is a triangulated category and let $(\SS,\mathcal{Z},\mathcal{V})$ be a premutation triple induced by a concentric twin cotorsion pair. The negative sign in the proof of Lemma 3.16 comes from the following isomorphic correspondence.

\begin{array}[]{ccc}\mathbb{E}(X\langle 1\rangle,X)&\xrightarrow{\sim}&\mathcal{C}(X\langle 1\rangle,X[1])\\[3.0pt] \lambda^{X}&\mapsto&l^{X}\end{array}

\begin{array}[]{ccccc}\mathbb{E}(Y,Y\langle-1\rangle)&\xrightarrow{\sim}&\mathcal{C}(Y[-1],Y\langle-1\rangle)&\xrightarrow{[1]}&\mathcal{C}(Y,Y\langle-1\rangle[1])\\[3.0pt] \lambda_{Y}&\mapsto&-l_{Y}[-1]&\mapsto&-l_{Y}\end{array}

Then $\Phi$ in Lemma 3.16 is defined by the following commutative diagram in $\underline{\mathcal{C}}$ .

This correspondence is used in [Nak18, Definition 4.1].

Lemma 3.20.

[Nak18, Lemma 3.11] Let $(\SS,\mathcal{Z})$ be a right mutation double. Assume that there exists an $\mathfrak{s}^{\mathcal{I}}$ -triangle $U\xrightarrow{u}U^{\prime}\rightarrow S\dashrightarrow U$ where $U,U^{\prime}\in\widetilde{\mathcal{U}}$ . Then $\sigma(\underline{u})\colon\sigma U\to\sigma U^{\prime}$ is an isomorphism.

Proof.

By definition of right mutation doubles, there exists an $\mathfrak{s}^{\mathcal{I}}$ -triangle $S^{\prime}\dashrightarrow U^{\prime}\rightarrow Z^{\prime}\rightarrow S^{\prime}$ where $Z^{\prime}\in\mathcal{Z}$ and $S^{\prime}\in\SS$ . Since $\SS$ is closed under extensions in $(\mathcal{C},\mathbb{E}^{\mathcal{I}},\mathfrak{s}^{\mathcal{I}})$ , there exists the following commutative diagram in $\mathcal{C}$ where $S^{\prime\prime}\in\SS$ .

Thus, we obtain an $\mathfrak{s}^{\mathcal{I}}$ -triangle $U\rightarrow Z^{\prime}\rightarrow S^{\prime\prime}\dashrightarrow U$ . From uniqueness of $\sigma$ , $\sigma(\underline{u})$ is an isomorphism. ∎

3.3. Right triangles induced by a right mutation double

In this subsection, we consider right triangulated (resp. left triangulated, pretriangulated) structures induced by right mutation doubles (resp. left mutation doubles, premutation triples). We assume that $(\SS,\mathcal{Z})$ is a right mutation double.

First, we fix the following $\mathfrak{s}^{\mathcal{I}}$ -triangles to define $\langle 1\rangle$ and $\sigma$ .

(1)

For $X\in\mathcal{Z}$ , there exists the following $\mathfrak{s}^{\mathcal{I}}$ -triangle (and also an $\mathfrak{s}_{\mathcal{I}}$ -triangle) where $I^{X}\in\mathcal{I}$ and we fix it.

X\xrightarrow{i^{X}}I^{X}\xrightarrow{p^{X}}X\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X}$}\vss}}}X

Then we define $i^{X},p^{X},\lambda^{X}$ by the above fixed $\mathfrak{s}^{\mathcal{I}}$ -triangle.

(2)

For $U\in\widetilde{\mathcal{U}}$ , there exists the following $\mathfrak{s}^{\mathcal{I}}$ -triangle (this is not an $\mathfrak{s}_{\mathcal{I}}$ -triangle in general) where $\sigma U\in\mathcal{Z},S^{U}\in\SS$ and we fix it.

$U\xrightarrow{h^{U}}\sigma U\xrightarrow{g^{U}}S^{U}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\rho^{U}$}\vss}}}U$

Then we define $h^{U},g^{U},\rho^{U}$ by the above fixed $\mathfrak{s}^{\mathcal{I}}$ -triangle. For $Z\in\mathcal{Z}$ , we always take $\sigma Z$ and $h^{Z}$ so that $\sigma Z=Z$ and $h^{Z}=\mathrm{id}_{Z}$ .

We remind readers that $\langle 1\rangle\colon\underline{\mathcal{Z}}\to\underline{\widetilde{\mathcal{U}}}$ and $\sigma\colon\underline{\widetilde{\mathcal{U}}}\to\underline{\mathcal{Z}}$ do not depend on the choices of the above $\mathfrak{s}^{\mathcal{I}}$ -triangles up to natural isomorphisms.

Remark 3.21.

Let $U\in\widetilde{\mathcal{U}}$ . From Remark 3.7(3), $\sigma(h^{U})$ is one of the morphisms which makes the following diagram in $\mathcal{C}$ commutative. Note that $h^{\sigma U}=\mathrm{id}_{\sigma U}$ .

Then $(\sigma(h^{U})-\operatorname{id}_{\sigma U})h^{U}=0$ . From Remark 3.7(1), $\sigma(\underline{h^{U}})=\underline{\mathrm{id}_{\sigma U}}$ .

Notation 3.22.

(1)

Let $a\colon X\to Y$ be a morphism in $\mathcal{Z}$ . From Lemma 2.28, there exists the following $\mathfrak{s}^{\mathcal{I}}$ -triangle (this is also an $\mathfrak{s}_{\mathcal{I}}$ -triangle from Remark 3.2(2)) and we fix it.

X\xrightarrow{\scalebox{0.6}{$\begin{bmatrix}a\\ i^{X}\\ \end{bmatrix}$}}Y\oplus I^{X}\xrightarrow{\widetilde{b}}C^{a}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\widetilde{\delta}$}\vss}}}X

Then we define $\widetilde{b},C^{a}$ and $\widetilde{\delta}$ by the above $\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}}$ -triangle. We also define $\widetilde{a}=\scalebox{0.8}{$\begin{bmatrix}a\\ i^{X}\\ \end{bmatrix}$}$ and $b$ as the composition of $Y\xrightarrow{\scalebox{0.6}{$\begin{bmatrix}1\\ 0\end{bmatrix}$}}Y\oplus I^{X}\xrightarrow{\widetilde{b}}C^{a}$ .

(2)

From Lemma 2.29, there exists the following commutative diagram in $(\mathcal{C},\mathbb{E}^{\mathcal{I}}_{\mathcal{I}},\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}})$ and we fix it.

(3.3)

Then we define $c^{a},\gamma^{a}$ by the above diagram. We often drop “ $a$ ” if there is no confusion.

There exists the following sequence in $\underline{\mathcal{Z}}$ .

X\xrightarrow{\,\underline{\widetilde{a}}\,}Y\oplus I^{X}\xrightarrow{\,\underline{h^{C^{a}}\widetilde{b}}\,}\sigma C^{a}\xrightarrow{\sigma({\underline{c}})}\Sigma X

We show that the sequence (3.3) in Notation 3.22 does not depend on choices of morphism $c$ and $\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}}$ -triangle $X\rightarrow I^{X}\rightarrow X\langle 1\rangle\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\lambda^{X}\vphantom{X^{X^{x}}}$}\vss}}}X$ .

Lemma 3.23.

(1)

The morphism $c$ in Notation 3.22(2) is uniquely determined in $\underline{\mathcal{Z}}$ .
(2)

The sequence (3.3) in Notation 3.22 does not depend on the choices of $\langle 1\rangle$ , up to isomorphisms of sequences in $\underline{\mathcal{Z}}$ .

Proof.

(1) If morphisms $c,c^{\prime}\colon C^{a}\to X\langle 1\rangle$ satisfies $\widetilde{\delta}=\lambda c=\lambda c^{\prime}$ , then $c-c^{\prime}$ factors through $I^{X}$ .

(2) Take another $\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}}$ -triangle $X\xrightarrow{j^{X}}J^{X}\xrightarrow{q^{X}}X\langle 1\rangle^{\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle{\lambda^{\prime}}^{X}$}\vss}}}X$ where $j^{X}$ is a left $\mathcal{I}$ -approximation. Let $\mu\colon\langle 1\rangle\Rightarrow\langle 1\rangle^{\prime}$ be a natural isomorphism defined in Remark 2.25 and $\Sigma^{\prime}=\sigma\circ\langle 1\rangle^{\prime}$ . Then $\lambda^{X}={\lambda^{\prime}}^{X}m^{X}$ holds. From Corollary 2.30, there exists the following commutative diagrams where $\underline{f_{i}}$ and $\underline{g_{i}}$ are isomorphisms for $i=1,2$ .

(3.4)

Let $c\colon C^{a}\to X\langle 1\rangle$ and $c^{\prime}\colon C^{a^{\prime}}\to X\langle 1\rangle^{\prime}$ be the morphisms defined in Notation 3.22(2), then ${\lambda^{\prime}}^{X}m^{X}cg_{1}=\widetilde{\delta}g_{1}=\widetilde{\delta^{\prime}}g_{2}={\lambda^{\prime}}^{X}c^{\prime}g_{2}$ from (2.1), (3.3) and (3.4). Thus, $\underline{m^{X}c}=\underline{c^{\prime}g_{2}}\underline{{g_{1}}^{-1}}$ . Applying $\sigma$ , $\sigma(\underline{m^{X}})\sigma(\underline{c})=\sigma(\underline{c^{\prime}})\sigma(\underline{g_{2}{g_{1}}^{-1}})$ . Therefore, the following commutative diagram exists in $\underline{\mathcal{C}}$ .

∎

Lemma 3.24.

Let $a\colon X\to Y$ be a morphism in $\mathcal{Z}$ and $X\xrightarrow{a^{\prime}}Y\xrightarrow{b^{\prime}}U^{\prime}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\delta^{\prime}$}\vss}}}X$ be an $\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}}$ -triangle where $\underline{a}=\underline{a^{\prime}}$ . Then there exists a morphism $s\colon\sigma C^{a}\to\sigma U^{\prime}$ which makes the following diagram in $\underline{\mathcal{Z}}$ commutative where $\underline{s}$ is an isomorphism.

In particular, the isomorphism class of the sequence (3.3) in $\mathcal{Z}$ does not depend on the choices of $a$ up to $[\mathcal{I}]$ .

Before we show the above statement, we prove the following claim.

Claim 3.25.

Assume that there exists a commutative diagram in $\mathcal{C}$ with two $\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}}$ -triangles $X_{1}\xrightarrow{a_{1}}Y_{1}\xrightarrow{b_{1}}U_{1}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta_{1}$}\vss}}}X_{1}$ and $X_{2}\xrightarrow{a_{2}}Y_{2}\xrightarrow{b_{2}}U_{2}\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle\delta_{2}$}\vss}}}X_{2}$ where $X_{i},Y_{j}\in\mathcal{Z}$ for $1\leq i,j\leq 2$ .

This induces the following commutative diagram in $\underline{\mathcal{Z}}$ .

Proof.

First, we recall the following commutative diagrams (3.5) in $\mathcal{C}$ .

Then $\lambda^{X_{2}}x\langle 1\rangle c_{1}=x\lambda^{X_{1}}c_{1}=x\delta_{1}=\lambda^{X_{2}}c_{2}u$ . Thus, $x\langle 1\rangle c_{1}-c_{2}u$ factors through $I^{X_{2}}$ and $\underline{x}\langle 1\rangle\underline{c_{1}}=\underline{c_{2}u}$ . Applying $\sigma$ , $\Sigma(\underline{x})\sigma(\underline{c_{1}})=\sigma(\underline{c_{2}})\sigma(\underline{u})$ . Therefore, the claim holds.

(3.5)

∎

Proof of Lemma 3.24. Since $i^{X}\colon X\to I^{X}$ is a left $\mathcal{I}$ -approximation, $a-a^{\prime}$ factors through $i^{X}$ . Let $k^{X}\colon I^{X}\to Y$ be a morphism where $a-a^{\prime}=k^{X}i^{X}$ . There exists an $\mathfrak{s}$ -triangle $I^{X}\xrightarrow{\scalebox{0.6}{$\begin{bmatrix}k^{X}\\ 1\end{bmatrix}$}}Y\oplus I^{X}\xrightarrow{\scalebox{0.6}{$\begin{bmatrix}1&-k^{X}\end{bmatrix}$}}Y\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-3.0pt\hbox{$\scriptstyle 0$}\vss}}}I^{X}$ . Then we obtain the following commutative diagram from Lemma 2.29.

From the above claim, we only have to prove $\sigma(\underline{u})$ is an isomorphism. However, this is clear because $\underline{u}$ is isomorphic from the above diagram. ∎

Therefore, the sequence (3.3) does not depend on the choices of $a$ up to $[\mathcal{I}]$ . Thus, the following definition makes sense.

Definition 3.26.

Let $\underline{a}\colon X\to Y$ be a morphism in $\underline{\mathcal{Z}}$ . Then there exists the following commutative diagram in $\underline{\mathcal{Z}}$ . That is because $\underline{h^{C^{a}}\widetilde{b}}$ $=$ $\underline{h^{C^{a}}\widetilde{b}\scalebox{0.8}{$\begin{bmatrix}1&0\\ 0&0\end{bmatrix}$}}$ $=$ $\underline{h^{C^{a}}b\scalebox{0.8}{$\begin{bmatrix}1&0\end{bmatrix}$}}$ .

Then the following sequence in $\underline{\mathcal{Z}}$ is unique up to isomorphisms. It is called the standard right triangle of $\underline{a}$ .

\displaystyle X\xrightarrow{\,\underline{a}\,}Y\xrightarrow{\,\underline{h^{C^{a}}b}\,}\sigma C^{a}\xrightarrow{\sigma(\underline{c})}\Sigma X

(3.6)

We define

\nabla=\left(\begin{array}[]{ll}\text{sequences}&\text{in }\underline{\mathcal{Z}}\text{ isomorphic to one in }\\ &\{X\xrightarrow{\underline{a}}Y\xrightarrow{\underline{h^{C^{a}}b}}\sigma C^{a}\xrightarrow{\sigma(\underline{c})}\Sigma X\mid a\text{ is a morphism in }\mathcal{Z}\}\end{array}\right)

and a sequence in $\nabla$ is called a right triangle in $\underline{\mathcal{Z}}$ .

Example 3.27.

The diagram (3.3) is induced by $a=\mathrm{id}_{Z}$ for $Z\in\mathcal{Z}$ .

Thus, $Z\xrightarrow{\underline{\mathrm{id}_{Z}}}Z\xrightarrow{\underline{0}}0\xrightarrow{\underline{0}}\Sigma Z$ is a right triangle.

3.4. Right triangulated structures induced by right mutation doubles

We assume that $(\SS,\mathcal{Z})$ is a right mutation double. Now, we check the triplet $(\underline{\mathcal{Z}},\Sigma,\nabla)$ satisfies the axioms of right triangulated category.

Lemma 3.28.

The triplet $(\underline{\mathcal{Z}},\Sigma,\nabla)$ satisfies (rTR0) and (rTR1) in Remark 2.35.

Proof.

(rTR0) is by definition of $\nabla$ . (rTR1)(i) follows from Example 3.27. Since $\mathcal{I}$ is strongly covariantly finite in $\mathcal{Z}$ , (rTR1)(ii) holds from Lemma 2.28. ∎

Lemma 3.29.

$(\underline{\mathcal{Z}},\Sigma,\nabla)$ satisfies (rTR2) in Remark 2.35.

Proof.

We only have to show for the standard right triangles $X\xrightarrow{\underline{a}}Y\xrightarrow{\underline{h^{C^{a}}b}}\sigma C^{a}\xrightarrow{\sigma({\underline{c})}}\Sigma X$ . In the rest of this proof, we denote $\sigma C^{a}$ by $Z$ . Recall that both $b\colon Y\to C^{a}$ and $h^{C^{a}}\colon C^{a}\to Z$ are $\mathfrak{s}^{\mathcal{I}}$ -inflations. Thus $h^{C^{a}}b$ , now denoted by $b^{\prime}$ , is also an $\mathfrak{s}^{\mathcal{I}}$ -inflation and there exists an $\mathfrak{s}^{\mathcal{I}}$ -triangle $Y\xrightarrow{b^{\prime}}Z\xrightarrow{c^{\prime}}U\mathrel{\mathop{\dashrightarrow}\limits^{\vbox to0.0pt{\kern-5.0pt\hbox{$\scriptstyle\delta^{\prime}$}\vss}}}Y$ (this is also an $\mathfrak{s}_{\mathcal{I}}$ -triangle since $Y\in\underline{\mathcal{Z}}$ ). We define a morphism $a^{\prime}\colon U\to Y\langle 1\rangle$ by the following commutative diagram.

(3.7)

This $\mathfrak{s}^{\mathcal{I}}_{\mathcal{I}}$ -triangle induces the following right triangle.

Y\xrightarrow{\underline{b^{\prime}}}Z\xrightarrow{\underline{h^{U}c^{\prime}}}\sigma U\xrightarrow{\sigma(\underline{a^{\prime}})}\Sigma Y

It is enough to show that the above sequence is isomorphic to $Y\xrightarrow{\underline{b^{\prime}}}Z\xrightarrow{\sigma(\underline{c})}\Sigma X\xrightarrow{-\Sigma\underline{a}}\Sigma Y$ . From (ET4), we obtain a morphism $u\colon X\langle 1\rangle\to U$ which is defined by the following left commutative diagram. We also obtain the following right commutative diagram.

$\textstyle{C^{a}}$ $\textstyle{Z}$ $\textstyle{X\langle 1\rangle}$ $\textstyle{\Sigma X}$ $\textstyle{U}$ $\textstyle{\sigma U}$ $\scriptstyle{h^{C^{a}}}$ $\scriptstyle{h^{X\langle 1\rangle}}$