Model structure arising from one hereditary cotorsion pair on extriangulated categories

Jiangsheng Hu$^{a}$, Dongdong Zhang$^{b}$[Corresponding author. Jiangsheng Hu is supported by the NSF of China (Grant No. 12171206) and Jiangsu 333 Project. Panyue Zhou is supported by the National Natural Science Foundation of China (Grant No. 12371034) and by the Hunan Provincial Natural Science Foundation of China (Grant No. 2023JJ30008).] and Panyue Zhou$^{c}$

$^{a}$School of Mathematics, Hangzhou Normal University, Hangzhou 311121, P. R. China

$^b$Department of Mathematics, Zhejiang Normal University,
Jinhua 321004, P. R. China

$^c$School of Mathematics and Statistics, Changsha University of Science and Technology,

Changsha 410114, P. R. China

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>
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Let $\mathcal{C}$ be a weakly idempotent complete extriangulated category. In contrast with the Hovey correspondence of admissible model structures on weakly idempotent complete exact categories from two
complete cotorsion pairs, we give a construction of model structures on
$\mathcal{C}$ from only one complete cotorsion pair. Our main result not only generalizes the work by Beligiannis-Reiten and Cui-Lu-Zhang, but also provides methods to construct model structures from silting objects of $\mathcal{C}$ and co-$t$-structures in triangulated categories.

Keywords: extriangulated category; cotorsion pair; model structure; silting object; co-$t$-structure

2020 Mathematics Subject Classification: 18N40; 16D90; 16E30; 16E65

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§ INTRODUCTION

The Hovey correspondence [17, 18, 14, 26] is a tool for constructing model structures on abelian categories, exact categories or triangulated categories. It is inspired by the somewhat canonical model structure on a
Frobenius category, but with two cotorsion pairs mimicking the role played by the projectives
and the injectives. Extriangulated category, introduced in [22], can be viewed as a simultaneous generalization of exact categories and triangulated categories. Hovey's correspondence has been generalized by Nakaoka and Palu in [22] as a bijective correspondence between admissible model structures and some pairs of cotorsion pairs which they called Hovey twin cotorsion pairs on a weakly idempotent complete extriangulated category.

A similar, but weaker correspondence, is obtained by Beligiannis and  Reiten in [8]. More precisely, they give a construction of weakly projective model structures on abelian categories $\mathcal{A}$ from only one complete cotorsion pair. Recenty, it has been extended by Cui, Lu and Zhang [11] to weakly idempotent complete exact categories $\mathcal{E}$, by adding the condition of hereditary of the complete cotorsion pair. These weakly projective model structures are not necessarily exact. The
two approaches get the same result if and only if $\mathcal{E}$ has enough projective objects and the model
structure is projective, i.e., it is exact and each object is fibrant (see <cit.>).

The aim of this paper is to extend the results of Cui, Lu and Zhang to weakly idempotent
complete extriangulated categories. But our proof method is far different from the exact category because we cannot use the properties of monomorphism and epimorphism.
To state our results, we first introduce some notation and definitions.

Assume that $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$ is an extriangulated category, where $\mathcal{C}$ is an additive category, $\mathbb{E}: \mathcal{C}^{\rm op}\times \mathcal{C} \rightarrow {\rm Ab}$ is an additive bifunctor and $\mathfrak{s}$ assigns to each $\delta\in \mathbb{E}(C,A)$ a class of $3$-term sequences with end terms $A$ and $C$ as in <cit.>. If there is no confusion, we also write $\mathcal{C}$ $:=$ $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$.

A sequence<EMAIL_ADDRESS>\ar[r]^{x} & B \ar[r]^{y} & C }$ in $\mathcal{C}$ is called a conflation if it realizes some $\mathbb{E}$-extension $\delta\in\mathbb{E}(C, A)$. In this case, $x$ is called an inflation, $y$ is called a deflation, and we write it as

For an inflation $f\in\mathcal{C}(A, B)$, take a conflation<EMAIL_ADDRESS>and denote this $C$ by ${\rm Cone}(f)$.
Similarly, for a deflation $f\in\mathcal{C}(A, B)$, we take a conflation<EMAIL_ADDRESS>and denote this $L$ by ${\rm CoCone}(f)$.

Let $\mathcal{X}$ and $\mathcal{Y}$ be full subcategories of $\mathcal{C}$
which are closed under direct summands and isomorphisms. Put $\omega:=\mathcal{X}\cap \mathcal{Y}$.

$\bullet$ Denote by ${\rm CoFib}_{\omega}$ the class of inflations $f$ with ${\rm Cone}(f)\in\mathcal{X}$.

$\bullet$ Denote by ${\rm Fib}_{\omega}$ the class of morphisms $f:A\rightarrow B$ such that $f$ is $\omega$-epic, that is, $\mathcal{C}(W, f): \mathcal{C}(W, A)\rightarrow \mathcal{C}(W, B)$ is surjective for any $W\in\omega$.

$\bullet$ Denote by ${\rm TCoFib}_{\omega}$ the class of splitting inflations $f$ with ${\rm Cone}(f)\in\omega$.

$\bullet$ Denote by ${\rm TFib}_{\omega}$ the class of deflations $f$ with ${\rm CoCone}(f)\in\mathcal{Y}$.

$\bullet$ Denote by ${\rm Weq}_{\omega}$ the class of morphisms $f: A\rightarrow B$ such that there is a deflation $(f, t): A\oplus W\rightarrow B$ with $W\in\omega$ and ${\rm CoCone}(f, t)\in\mathcal{Y}$. Therefore, a morphism $f: A\rightarrow B$ is in ${\rm Weq}_\omega$ if and only if there is a commutative diagram

$$\xymatrix{A\ar[rr]^f\ar[dr]_{\tiny\begin{pmatrix}1\\0\end{pmatrix}}&&B\\&A\oplus W\ar[ur]_{(f,~ t)}&}$$

such that $W\in\omega, (f, t)$ is a deflation, and ${\rm CoCone}(f, t)\in\mathcal{Y}$. Since ${\tiny\begin{pmatrix}1\\0\end{pmatrix}}\in {\rm TCofib}_\omega$, ${\rm Weq}_\omega$ can be reformulated as

$${\rm Weq}_\omega=\{gf|g\in{\rm TFib}_\omega, f\in{\rm TCoFib}_\omega\}.$$

Now, our main result can be stated as follows.
Keep the notations above. Then $({\rm CoFib}_\omega, {\rm Fib}_\omega, {\rm Weq}_\omega)$ is a model structure if and only if
$(\mathcal{X, Y})$ is a hereditary cotorsion pair in $\mathcal{C}$, and $\omega$ is contravariantly finite in $\mathcal{C}$. In this case, the class $\mathcal{C}_\omega$ of cofibrant objects is $\mathcal{X}$, the class $\mathcal{F}_\omega$ of fibrant objects is $\mathcal{C}$, the class $\mathcal{W}_\omega$ of trivial objects is $\mathcal{Y}$; and the homotopy category ${\rm Ho}(\mathcal{C})$ is $\mathcal{X}/\mathcal{W}$.

A few comments on Theorem <ref> are in order. First, the originality of Theorem <ref> is due to Beligiannis and Reiten [8] for abelian categories.
If $\mathcal{C}$ is a weakly idempotent complete exact category, Theorem <ref> also recovers the weakly projective model structure established by Cui, Lu and Zhang in <cit.>.

Second, thanks to the the bijection between bounded hereditary cotorsion pairs
and silting subcategories in extriangulated categories established by Adachi and Tsukamoto in [2], a direct consequence of Theorem <ref> yields that any silting object in a weakly idempotent complete extriangulated category induces a model structure (see Corollary <ref>). We provided some examples (see Examples <ref>-<ref>) to illustrate this result.

Last, assume that $\mathcal{C}$ is a triangulated category (it can be regarded as an extriangulated category). Specializing $\mathcal{C}$ to a triangulated category, we give a construction of model structures from co-t-structures under some certain conditions (see Corollary <ref>).

The contents of this paper are as follows. In Section 2, we fix notations and recall some
definitions and basic facts. Section 3 contributes to showing Theorem <ref>. Some applications of Theorem <ref> are given in the final section.

§ PRELIMINARIES

Let us briefly recall some definitions and basic properties of extriangulated categories from [22]. Throughout this paper, we assume that $\mathcal{C}$ is an additive category.

§.§ 2.1 Extriangulated categories
<cit.> Suppose that $\mathcal{C}$ is equipped with an additive bifunctor
$$\mathbb{E}: \mathcal{C}^{op}\times \mathcal{C}\rightarrow {\rm Ab},$$
where ${\rm Ab}$ is the category of abelian groups. For any objects $A, C\in\mathcal{C}$, an element $\delta\in \mathbb{E}(C,A)$ is called an $\mathbb{E}$-extension. Thus formally, an $\mathbb{E}$-extension is a triple $(A,\delta, C)$. For any $A, C\in\mathcal{C}$, the zero element $0\in\mathbb{E}(C, A)$ is called the split $\mathbb{E}$-extension.

Let $\delta\in \mathbb{E}(C, A)$ be any $\mathbb{E}$-extension. By the functoriality, for any $a\in\mathcal{C}(A, A')$ and $c\in\mathcal{C}(C', C)$, we have $\mathbb{E}$-extensions

$\mathbb{E}(C, a)(\delta)\in\mathbb{E}(C, A')$  and  $\mathbb{E}(c, A)(\delta)\in\mathbb{E}(C', A)$.

We simply denote them by $a_*\delta$ and $c^*\delta$  respectively. In this terminology, we have
$$\mathbb{E}(c, a)(\delta)=c^*a_*\delta=a_*c^*\delta$$
in $\mathbb{E}(C', A')$.
In the following, we always assume that the  bifunctor $\mathbb{E}$ exists.

<cit.> Let $\delta\in\mathbb{E}(C, A)$ and $\delta'\in\mathbb{E}(C', A')$ be any pair of $\mathbb{E}$-extensions. A morphism $(a, c): \delta\rightarrow \delta'$ of $\mathbb{E}$-extensions is a pair of morphisms $a\in\mathcal{C}(A, A')$ and $c\in\mathcal{C}(C, C')$ in $\mathcal{C}$ satisfying the equality

<cit.> Let $\delta=(A, \delta, C)$ and $\delta'=(A', \delta', C')$ be any pair of $\mathbb{E}$-extensions. Let

$\xymatrix{C\ar[r]^{\iota_C\ \ \ }&C\oplus C'&C'\ar[l]_{\qquad\iota_{C'}}}$
$\xymatrix{A&A\oplus A'\ar[l]_{ p_A\;\;}\ar[r]^{\quad p_{A'} }&A'}$

be coproduct and product in $\mathcal{C}$, respectively. Remark that, by the additivity of $\mathbb{E}$, we have a natural isomorphism

$\mathbb{E}(C\oplus C', A\oplus A')\simeq\mathbb{E}(C, A)\oplus\mathbb{E}(C, A')
\oplus\mathbb{E}(C', A)\oplus\mathbb{E}(C', A')$.

Let $\delta\oplus \delta'\in\mathbb{E}(C\oplus C', A\oplus A')$ be the element corresponding to $(\delta, 0, 0, \delta')$ through this isomorphism. This is the unique element which satisfies

$\mathbb{E}(\iota_C, p_A)(\delta\oplus \delta')=\delta,\ \mathbb{E}(\iota_C, p_{A'})(\delta\oplus \delta')=0, \ \mathbb{E}(\iota_{C'}, p_A)(\delta\oplus \delta')=0,\ \mathbb{E}(\iota_{C'}, p_{A'})(\delta\oplus \delta')=\delta'$.

<cit.> Let $A, C\in\mathcal{C}$ be any pair of objects. Two sequences of morphisms in $\mathcal{C}$

$\xymatrix{A\ar[r]^x&B\ar[r]^y&C}$ and $\xymatrix{A\ar[r]^{x'}&B'\ar[r]^{y'}&C}$

are said to be equivalent if there exists an isomorphism $b\in\mathcal{C}(B, B')$ which makes the following diagram commutative.

We denote the equivalence class of $\xymatrix{A\ar[r]^x&B\ar[r]^y&C}$ by $\xymatrix{[A\ar[r]^x&B\ar[r]^y&C].}$

<cit.> (1) For any $A, C\in\mathcal{C}$, we denote as

$$0=[A\xrightarrow{~\tiny\begin{bmatrix}1\\0\end{bmatrix}~}A\oplus C\xrightarrow{\tiny\begin{bmatrix}0&1\end{bmatrix}}C].$$

(2) For any $\xymatrix{[A\ar[r]^x&B\ar[r]^y&C]}$ and $\xymatrix{[A'\ar[r]^{x'}&B'\ar[r]^{y'}&C']}$, we denote as

$\xymatrix{[A\ar[r]^x&B\ar[r]^y&C]}\oplus$$\xymatrix{[A'\ar[r]^{x'}&B'\ar[r]^{y'}&C']}=$$\xymatrix{[A\oplus A'\ar[r]^{x\oplus x'}&B\oplus B'\ar[r]^{y\oplus y'}&C\oplus C'].}$

<cit.> Let $\mathfrak{s}$ be a correspondence which associates an equivalence class
to any $\mathbb{E}$-extension $\delta\in\mathbb{E}(C, A)$. This $\mathfrak{s}$ is called a realization of $\mathbb{E}$, if it satisfies
the following condition $(\star)$. In this case, we say that the sequence $\xymatrix{A\ar[r]^x&B\ar[r]^y&C}$ realizes $\delta$, whenever it satisfies $\mathfrak{s}(\delta)=$$\xymatrix{[A\ar[r]^x &B\ar[r]^y&C].}$

$(\star)$ Let $\delta\in\mathbb{E}(C, A)$ and $\delta'\in\mathbb{E}(C', A')$ be any pair of $\mathbb{E}$-extensions, with

$\mathfrak{s}(\delta)=$$\xymatrix{[A\ar[r]^x &B\ar[r]^y&C]}$ and $\mathfrak{s}(\delta')=$$\xymatrix{[A'\ar[r]^{x'} &B'\ar[r]^{y'}&C'].}$
Then, for any morphism $(a, c): \delta\rightarrow \delta'$, there exists $b\in\mathcal{C}(B, B')$ which makes the following diagram commutative

In the above situation, we say that the triplet $(a, b, c)$ realizes $(a, c)$.

<cit.> Let $\mathcal{C}, \mathbb{E}$ be as above. A realization of $\mathbb{E}$ is said to be additive, if it satisfies the following conditions.

(i) For any $A, C\in\mathcal{C}$, the split $\mathbb{E}$-extension $0\in\mathbb{E}(C, A)$ satisfies $\mathfrak{s}(0)=0.$

(ii) For any pair of $\mathbb{E}$-extensions $\delta\in\mathbb{E}(C, A)$ and $\delta'\in\mathbb{E}(C', A')$, we have

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

(ET1) $\mathbb{E}: \mathcal{C}^{\rm op}\times \mathcal{C}\rightarrow \rm{Ab}$ is an additive bifunctor.

(ET2) $\mathfrak{s}$ is an additive realization of $\mathbb{E}$.

(ET3) Let $\delta\in\mathbb{E}(C, A)$ and $\delta'\in\mathbb{E}(C', A')$ be any pair of $\mathbb{E}$-extensions, realized as

$\mathfrak{s}(\delta)=$$\xymatrix{[A\ar[r]^x &B\ar[r]^y&C]}$ and $\mathfrak{s}(\delta')=$$\xymatrix{[A'\ar[r]^{x'} &B'\ar[r]^{y'}&C'].}$
For any commutative square

in $\mathcal{C}$, there exists a  morphism $(a, c): \delta\rightarrow \delta'$ satisfying $cy=y'b$.

(ET3)$^{\rm op}$ Dual of (ET3).

(ET4) Let $\delta\in\mathbb{E}(D,A)$ and $\delta'\in\mathbb{E}(F, B)$ be $\mathbb{E}$-extensions realized by

$\xymatrix{A\ar[r]^f&B\ar[r]^{f'}&D}$ and $\xymatrix{B\ar[r]^g&C\ar[r]^{g'}&F}$

respectively. Then there exist an object $E\in\mathcal{C}$, a commutative diagram

in $\mathcal{C}$, and an $\mathbb{E}$-extension $\delta^{''}\in\mathbb{E}(E, A)$ realized by $\xymatrix{A\ar[r]^h&C\ar[r]^{h'}&E,}$
which satisfy the following compatibilities.

(i) $\xymatrix{D\ar[r]^d&E\ar[r]^{e}&F}$ realizes $f'_*\delta'$,

(ii) $d^*\delta^{''}=\delta$,

(iii) $f_*\delta^{''}=e^*\delta'$.

(ET4)$^{\rm op}$ Dual of (ET4).

We will use the following terminology in the sequel.

<cit.> Let $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$ be an extriangulated category.

(1) A sequence $\xymatrix@C=1cm{A\ar[r]^x&B\ar[r]^{y}&C}$ is called a conflation if it realizes some $\mathbb{E}$-extension $\delta\in\mathbb{E}(C, A)$.
In this case, $x$ is called an inflation and $y$ is called a deflation.

(2) If a conflation<EMAIL_ADDRESS>realizes $\delta\in\mathbb{E}(C, A)$, we call the pair
<EMAIL_ADDRESS>\delta)$ an $\mathbb{E}$-triangle, and write it in the following way.

We usually do not write this “$\delta$" if it is not used in the argument.

(3) Let $\xymatrix{A\ar[r]^x&B\ar[r]^{y}&C\ar@{-->}[r]^{\delta}&}$ and $\xymatrix{A'\ar[r]^{x'}&B'\ar[r]^{y'}&C'\ar@{-->}[r]^{\delta'}&}$
be any pair of $\mathbb{E}$-triangles. If a triplet $(a, b, c)$ realizes $(a, c): \delta\rightarrow \delta'$, then we write it as

and call $(a, b, c)$ a morphism of $\mathbb{E}$-triangles.

(4) (i) For an inflation $f\in\mathcal{C}(A, B)$, take a conflation $\xymatrix@C=0.6cm{A\ar[r]^f&B\ar[r]&C,}$ and denote this $C$ by ${\rm Cone}(f)$.

(ii) For a deflation $f\in\mathcal{C}(A, B)$, take a conflation $\xymatrix@C=0.6cm{L\ar[r]&A\ar[r]^f&B,}$ and denote this $L$ by ${\rm CoCone}(f)$.

Assume that $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$ is an extriangulated category. By Yoneda's Lemma, any $\mathbb{E}$-extension $\delta\in \mathbb{E}(C, A)$ induces  natural transformations

$\delta_\sharp: \mathcal{C}(-, C)\Rightarrow \mathbb{E}(-, A)$ and $\delta^\sharp: \mathcal{C}(A, -)\Rightarrow \mathbb{E}(C, -)$.

For any $X\in\mathcal{C}$, these $(\delta_\sharp)_X$ and $\delta^\sharp_X$ are given as follows:

(1) $(\delta_\sharp)_X: \mathcal{C}(X, C)\Rightarrow \mathbb{E}(X, A); ~f\mapsto f^*\delta.$

(2) $\delta^\sharp_X: \mathcal{C}(A, X)\Rightarrow \mathbb{E}(C, X); ~g\mapsto g_*\delta.$
<cit.>  Let $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$ be an extriangulated category, and
an $\mathbb{E}$-triangle. Then we have the following long exact sequences:

$\xymatrix@C=1cm{\mathcal{C}(C, -)\ar[r]^{\mathcal{C}(y, -)}&\mathcal{C}(B, -)\ar[r]^{\mathcal{C}(x, -)}&\mathcal{C}(A, -)\ar[r]^{\delta^\sharp}&\mathbb{E}(C, -)\ar[r]^{\mathbb{E}(y, -)}&\mathbb{E}(B, -)\ar[r]^{\mathbb{E}(x, -)}&\mathbb{E}(A, -);}$

$\xymatrix@C=1cm{\mathcal{C}(-, A)\ar[r]^{\mathcal{C}(-, x)}&\mathcal{C}(-, B)\ar[r]^{\mathcal{C}(-, y)}&\mathcal{C}(-, C)\ar[r]^{\delta_\sharp}&\mathbb{E}(-, A)\ar[r]^{\mathbb{E}(-, x)}&\mathbb{E}(-, B)\ar[r]^{\mathbb{E}(-, y)}&\mathbb{E}(-, C).}$

By (ET4) and (ET)$^{\rm op}$, we can directly obtain the following results.

Let $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ be an extriangulated category. Then the following statements hold.

(1) If $f$ and $g$ are inflations, then $gf$ is an inflation, and there exists a conflation

<EMAIL_ADDRESS>Cone}(f)\ar[r]&{\rm Cone}(gf)\ar[r]&{\rm Cone}(g)}$ in $\mathcal{C}$.

(2) If $f$ and $g$ are deflations, then $gf$ is a deflation, and there exists a conflation

<EMAIL_ADDRESS>CoCone}(f)\ar[r]&{\rm CoCone}(gf)\ar[r]&{\rm CoCone}(g)}$ in $\mathcal{C}$.

Let $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ be an extriangulated category and $f: A\rightarrow B,  g: B\rightarrow C$ are morphisms in $\mathcal{C}$. Then the following statements hold.

(1) If $f$ is an inflation,  $gf$ is a deflation, then $g$ is a deflation, and there exists a conflation

<EMAIL_ADDRESS>CoCone}(gf)\ar[r]&{\rm CoCone}(g)\ar[r]&{\rm Cone}(f)}$ in $\mathcal{C}$.

(2) If $g$ is a deflation, $gf$ is an inflation, then $f$ is an inflation, and there exists a conflation

<EMAIL_ADDRESS>CoCone}(g)\ar[r]&{\rm Cone}(f)\ar[r]&{\rm Cone}(gf)}$ in $\mathcal{C}$.

By duality, we only prove (1). Assume there are $\mathbb{E}$-triangles<EMAIL_ADDRESS>Cone}(f)\ar@{-->}[r]^{\quad\delta_f}&}$ and $\xymatrix{{\rm CoCone}(gf)\ar[r]^{\quad h}&A\ar[r]^{gf}&C\ar@{-->}[r]^{\delta_h}&}$. So there exist morphisms of $\mathbb{E}$-triangles which make the following diagram commutative

$$\xymatrix{{\rm CoCone}(gf)\ar[r]^h\ar@{=}[d]&A\ar[r]^{gf}\ar[d]_f&C\ar[d]^d\ar@{-->}[r]^{\delta_h}&\\
{\rm CoCone}(gf)\ar[r]^k&B\ar[r]^{k'}\ar[d]_{f'}&E\ar[d]^e\ar@{-->}[r]^{\delta_k}&\\
&{\rm Cone}(f)\ar@{-->}[d]^{\delta_f}\ar@{=}[r]&{\rm Cone}(f)\ar@{-->}[d]^{\delta_d}&\\

by (ET4). It follows from <cit.> that the upper right square of above diagram is a weak pushout, then there exists a commutative diagram

Hence $t$ is a splitting deflation. Since $k'$ is a deflation, $g=tk'$ is a deflation.

Assume that $\xymatrix{{\rm CoCone}(g)\ar[r]^{g'}&B\ar[r]^{g\quad}&C\ar@{-->}[r]^{\quad\delta_{g'}}&}$ is an $\mathbb{E}$-triangle. Then there exist morphisms of $\mathbb{E}$-triangles which make the following diagram commutative

$$\xymatrix{{\rm CoCone}(gf)\ar[r]^h\ar[d]_x&A\ar[r]^{gf}\ar[d]_f&C\ar@{=}[d]\ar@{-->}[r]^{\delta_h}&\\
{\rm CoCone}(g)\ar[r]^{g'}\ar[d]_{x'}&B\ar[r]^{g}\ar[d]_{f'}&C\ar@{-->}[r]^{\delta_{g'}}&\\
{\rm Cone}(f)\ar@{=}[r]\ar@{-->}[d]_{\delta_x}&{\rm Cone}(f)\ar@{-->}[d]^{\delta_f}&&\\

by <cit.>. So we have a conflation<EMAIL_ADDRESS>CoCone}(gf)\ar[r]&{\rm CoCone}(g)\ar[r]&{\rm Cone}(f)}$ in $\mathcal{C}$.

(Extension-Lifting Lemma) The Extension-Lifting Lemma will play important roles. It has been proved for abelian categories and for exact categories.

Let $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ be an extriangulated category and $X, Y\in\mathcal{C}$. Then $\mathbb{E}(X, Y)=0$ if and only for any commutative diagram with $\mathbb{E}$-triangles

there exists a morphism $h:B\rightarrow C$ such that $f=hi$ and $g=ph$.

“$\Leftarrow$". Let $\delta\in\mathbb{E}(X, Y)$. Then there exists a conflation $\xymatrix{Y\ar[r]^c&C\ar[r]^p&X}$ which realizes $\delta$, so we have a commutative diagram with $\mathbb{E}$-triangles

Therefore, there exists a morphism $h: X\rightarrow C$ such that $ph=1_X$, which implies $\delta=0$, and so $\mathbb{E}(X, Y)=0$.

“$\Rightarrow$". Assume that $\mathbb{E}(X, Y)=0$. For any commutative diagram with $\mathbb{E}$-triangles

There exists a commutative diagram of $\mathbb{E}$-triangles

such that the right square is a weak pullback by the dual of <cit.>.
There exists a morphism $t:A\rightarrow K$ such that $f=kt$ and $i=yt$ since $pf=gi$.  Note that $i$ is an inflation and $y$ is a deflation. It follows from Lemma <ref>(2) that $t$ is an inflation. Assume that $\xymatrix{A\ar[r]^t&K\ar[r]^{t'}&L\ar@{-->}[r]^{\theta}&}$ is an $\mathbb{E}$-triangle. Since $i=yt$, there exists a commutative diagram with $\mathbb{E}$-triangles

with $\theta=s^*\eta$ and $d^*\rho=g^*\delta$ by the dual of <cit.>. Hence $\rho=0$ because $\mathbb{E}(X, Y)=0$, which implies $g^*\delta=0$. So there exists a morphism $r: B\rightarrow K$ such that $yr=1_B$. Then $p(f-kri)=0$, there exists a morphism $m:A\rightarrow Y$ such that $f-kri=cm$. By exactness of the sequence $\xymatrix{\mathcal{C}(B, Y)\ar[r]^{\mathcal{C}(i, Y)}&\mathcal{C}(A, Y)\ar[r]&\mathbb{E}(X, Y)=0,}$ there is a morphism $n: B\rightarrow Y$ such that $m=ni$. Put $h=cn+kr$, then $f=hi$ and $ph=g$, as desired.

<cit.> An extriangulated category $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ satisfies Condition (WIC) if the following two conditions hold:

(a) If $f\in \mathcal{C}(A, B)$ and $g\in\mathcal{C}(B, C)$ is a pair of composable morphisms in $\mathcal{C}$ and $gf$ is an inflation, then so is $f$.

(b) If $f\in \mathcal{C}(A, B)$ and $g\in\mathcal{C}(B, C)$ is a pair of composable morphisms in $\mathcal{C}$ and $gf$ is a deflation, then so is $g$.

(i) If the category $\mathcal{C}$ is exact, then $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ satisfies  Condition (WIC) if and only if $\mathcal{C}$ is weakly idempotent complete.

(ii) If $\mathcal{C}$ is a triangulated category, then Condition (WIC) is automatically satisfied.

It is worth nothing that an extrianglated category $\mathcal{C}$ satisfies Condition (WIC) if and only if $\mathcal{C}$ is weakly idempotent complete, see <cit.>.
In this paper, we always assume that any extrianglated category is weakly idempotent complete.

§.§ 2.2 Cotorsion pairs in extriangulated categories

Let $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ be an extriangulated category and $\mathcal{W}$ a class of objects in $\mathcal{C}$.

(1) $\mathcal{W}$ is closed under cocones of deflations if for any $\mathbb{E}$-triangle $\xymatrix{A\ar[r]^x&B\ar[r]^y&C\ar@{-->}[r]^\delta&}$ which satisfies $B, C\in\mathcal{W}$, we have $A\in\mathcal{W}$.

(2) $\mathcal{W}$ is closed under cones of inflations if for any $\mathbb{E}$-triangle $\xymatrix{A\ar[r]^x&B\ar[r]^y&C\ar@{-->}[r]^\delta&}$ which satisfies $A, B\in\mathcal{W}$, we have $C\in\mathcal{W}$.

(3) $\mathcal{W}$ is closed under extensions if for any $\mathbb{E}$-triangle $\xymatrix{A\ar[r]^x&B\ar[r]^y&C\ar@{-->}[r]^\delta&}$ which satisfies $A, C\in\mathcal{W}$, we have $B\in\mathcal{W}$.

<cit.> Assume that $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ is an extriangulated category and $\mathcal{X, Y}$ are two classes of objects in $\mathcal{C}$. We call $(\mathcal{X}, \mathcal{Y})$ a cotorsion pair if it satisfies the following conditions:

(a) $\mathbb{E}(\mathcal{X}, \mathcal{Y})=0$.

(b) For any $C\in\mathcal{C}$, there are $\mathbb{E}$-triangles

$\xymatrix{Y_C\ar[r]&X_C\ar[r]&C\ar@{-->}[r]&}$ and $\xymatrix{C\ar[r]&Y^C\ar[r]&X^C\ar@{-->}[r]&}$

satisfying $X_C, X^C\in\mathcal{X}$ and  $Y_C, Y^C\in\mathcal{Y}$.

We say that a cotorsion pair $(\mathcal{X}, \mathcal{Y})$ is hereditary if $\mathcal{X}$ is closed under cocones of deflations and $\mathcal{Y}$ is closed under cones of inflations. The following result gives several equivalent conditions of a cotorsion pair being hereditary.

Let $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ be an extriangulated category and $(\mathcal{X}, \mathcal{Y})$ a cotorsion pair. Consider the following conditions:

(1)   $\mathbb{E}^{i}(X, Y)=0$ for each $i\geqslant2$ and all $X\in{\mathcal{X}}$ and $Y\in{\mathcal{Y}}$.

(2)  $\mathcal{Y}$ is closed under cones of inflations.

(3)  $\mathcal{X}$ is closed under cocones of deflations.

(4)   $(\mathcal{X}, \mathcal{Y})$ is hereditary.

Then $(1)\Rightarrow(2)\Leftrightarrow(3)\Leftrightarrow(4)$. The converses hold if $\mathcal{C}$ has enough projectives and injectives.

$(1)\Rightarrow(2)$ is trivial.

$(2)\Rightarrow(3)$. Assume that $\xymatrix{A\ar[r]^x&X_1\ar[r]^y&X_2\ar@{-->}[r]&}$ is an $\mathbb{E}$-triangle with $X_1, X_2\in\mathcal{X}$. Next we claim that any morphism $f: A\rightarrow Y$ with $Y\in\mathcal{Y}$ can factor through $\omega$.
Since $\mathbb{E}(X_2, Y)=0$, it is easy to see that there is a morphism $g: X_1\rightarrow Y$ such that $f=gx$. As $(\mathcal{X}, \mathcal{Y})$ is a cotorsion pair, there is an $\mathbb{E}$-triangle $\xymatrix{X_1\ar[r]&Y'\ar[r]&X\ar@{-->}[r]&}$ with $X\in\mathcal{X}$, and $Y'\in\omega$ because $\mathcal{X}$ is closed under extensions. So $g$ can factor through $Y'$ as $\mathbb{E}(X, Y)=0$, hence $f$ factors through $\omega$.

Since $(\mathcal{X}, \mathcal{Y})$ is a cotorsion pair, there are $\mathbb{E}$-triangles

with $X_3, X_4\in\mathcal{X}$, $Y_1\in\mathcal{Y}$, and $W\in\omega$ because $\mathcal{X}$ is closed under extensions. Hence there is a commutative diagram with $\mathbb{E}$-triangles

by (ET4). Note that $Y_1, W\in\mathcal{Y}$, we have $Y''\in\mathcal{Y}$ since $\mathcal{Y}$ is closed under cones of inflations. Then $f$ factors through $\omega$ by above argument, it is easy to check that $f$ factors through $W$. Therefore, $1_{Y_1}$ factors through $g$ by <cit.>, which implies that $A$ is isomorphism a direct summand of $X_3$. So we have $A\in\mathcal{X}$.

$(3)\Rightarrow(2)$. The proof is similar to that of $(2)\Rightarrow(3)$.

$(3)\Rightarrow(4)$ is clear because of the equivalence of (2) and (3).

$(4)\Rightarrow(1)$ follows from  <cit.>.

It should be noted that the notion of hereditary cotorsion pairs introduced here is different from that of <cit.>, where the authors call a cotorsion pair $(\mathcal{X}, \mathcal{Y})$ hereditary if it satisfies the condition (1) of Proposition <ref>.

§.§ 2.3 Model structures

A closed model structure on a category $\mathcal{M}$ is a triple $({\rm CoFib, Fib, Weq})$ of classes of morphisms, where the morphisms in the three classes are respectively called cofibration, fibrations, and weak equivalences, satisfying the following axioms:

(Two out of three axiom) Let  $f: A\rightarrow B$ and $g: B\rightarrow C$ be a pair of composable morphisms in $\mathcal{M}$. If two of the morphisms $f, g, gf$ are weak equivalences, then so is the third one.

(Retract axiom) If $g$ is a retract of $f$, and $f$ is a cofibration (a fibration, a weak equivalence, respectively), then so is $g$.

(Lifting axiom) Cofibrations have the left lifting property with respect to all morphisms in ${\rm Fib}\cap{\rm Weq}$, and fibrations have the right lifting property with respect to all morphisms in ${\rm CoFib}\cap{\rm Weq}$. That is, given a commutative diagram

If either $i\in{\rm CoFib}$ and $p\in{\rm Fib}\cap{\rm Weq}$,  or $i\in{\rm CoFib}\cap{\rm Weq}$ and $p\in{\rm Fib}$, then there exists a morphism $h: B\rightarrow X$ such that $a=hi, b=ph$.

(Factorization axiom) Any morphism $f: A\rightarrow B$ admits factorizations $f=pi$ and $f=qj$, where $i\in{\rm CoFib}\cap{\rm Weq}, p\in{\rm Fib}, j\in{\rm CoFib}$ and $q\in{\rm Fib}\cap{\rm Weq}$.

The morphisms in  ${\rm CoFib}\cap{\rm Weq}$ (respectively,  ${\rm Fib}\cap{\rm Weq}$) are called trivial cofibrations (respectively, trivial fibrations). Put ${\rm TCoFib}:={\rm CoFib}\cap{\rm Weq}$ and  ${\rm TFib}:={\rm Fib}\cap{\rm Weq}$.

For a model structure $({\rm CoFib, Fib, Weq})$ on category $\mathcal{M}$ with zero object, an object $X$ is trivial if $0\rightarrow X$ is a weak equivalence, or, $X\rightarrow 0$ is a weak equivalence. It is cofibrant if $0\rightarrow X$ is a cofibration, and it is  fibrant if $X\rightarrow 0$ is a fibration. An object is trivial cofibrant (respectively, trivial fibrant) if it is both trivial and cofibrant (respectively, fibrant).

§ PROOF OF THEOREM <REF>

Throughtout this section, we always assume that $\mathcal{C}:=(\mathcal{C}, \mathbb{E},\mathfrak{s})$ is a weakly idempotent complete extriangulated category.

§.§ Model structures induces by a hereditary cotorsion pair

The aim of this subsection is to prove the “if" part of Theorem <ref>. Thus, we must check that these structures in Theorem <ref> satisfy the factorization, two-out-of-three, retract and lifting axioms for a model structure. We first make some preliminary observations. First of all, we give a description of trivial cofibrations ${\rm TCoFib}_\omega$ and trivial fibrations ${\rm TFib}_\omega$. Since $1_X\in {\rm TFib}_\omega$ and $1_X\in {\rm TCoFib}_\omega$ for any object $X\in\mathcal{C}$, we have ${\rm TCoFib}_\omega\subseteq {\rm Weq}_\omega$ and ${\rm TFib}_\omega\subseteq {\rm Weq}_\omega$.

We begin the proof with the descriptions of ${\rm CoFib}_\omega\cap{\rm Weq}_\omega$ and ${\rm Fib}_\omega\cap{\rm Weq}_\omega$.

Let $\mathcal{X}, \mathcal{Y}$ be full subcategories of $\mathcal{C}$ which are closed under direct summands and isomorphisms, and $\omega:=\mathcal{X}\cap \mathcal{Y}$. If $\mathbb{E}(\mathcal{X}, \mathcal{Y})=0$, then we have

$${\rm TCoFib}_\omega={\rm CoFib}_\omega\cap{\rm Weq}_\omega,\quad{\rm TFib}_\omega={\rm Fib}_\omega\cap{\rm Weq}_\omega.$$

We first prove ${\rm TCoFib}_\omega={\rm CoFib}_\omega\cap{\rm Weq}_\omega$. It is to see that ${\rm TCoFib}_\omega\subseteq {\rm CoFib}_\omega\cap{\rm Weq}_\omega$. Conversely, if $f: A\rightarrow B\in {\rm CoFib}_\omega\cap{\rm Weq}_\omega$, then $f$ is an inflation with ${\rm Cone}(f)\in\mathcal{X}$ and there is an $\mathbb{E}$-triangle $\xymatrix{Y\ar[r]^x&A\oplus W\ar[r]^{(f, t)}&B\ar@{-->}[r]^{\delta}&}$ with $W\in\omega$ and $Y\in\mathcal{Y}$. Suppose $\xymatrix{A\ar[r]^f&B\ar[r]^{g}&{\rm Cone}(f)\ar@{-->}[r]^{\eta}&}$ is an $\mathbb{E}$-triangle with ${\rm Cone}(f)\in\mathcal{X}$. Since $f=(f, t){\tiny\begin{pmatrix}1\\0\end{pmatrix}}$, there exists a commutative diagram of $\mathbb{E}$-triangles

A\ar[r]^{\tiny\begin{pmatrix}1\\0\end{pmatrix}}\ar@{=}[d]&A\oplus W\ar[r]^{(0, 1)}\ar[d]_{(f,t)}&W\ar[d]^s\ar@{-->}[r]^0&\\
A\ar[r]^f&B\ar[r]^{g}\ar@{-->}[d]_{\delta}&{\rm Cone}(f)\ar@{-->}[d]^{\rho}\ar@{-->}[r]^{\eta}&\\

with $\delta=g^*\rho$ by the dual of <cit.>. Since $\mathbb{E}(\mathcal{X}, \mathcal{Y})=0$, we have $\rho=0$, which implies that ${\rm Cone}(f)\in\omega$ as ${\rm Cone}(f)$ is a direct summand of $W$. Meanwhile, we have $\delta=g^*\rho=0$. It is easy to check that $\mathcal{C}({\rm Cone}(f), g)$ is surjective by applying the functor $\mathcal{C}({\rm Cone}(f), -)$ to above commutative diagram. Therefore, $f$ is a splitting inflation, and so $f\in{\rm TCoFib}_\omega$ by definition. This proves ${\rm TCoFib}_\omega={\rm CoFib}_\omega\cap{\rm Weq}_\omega$.

Next we prove ${\rm TFib}_\omega={\rm Fib}_\omega\cap{\rm Weq}_\omega$. It is easy to check that ${\rm TFib}_\omega\subseteq{\rm Fib}_\omega$, and so ${\rm TFib}_\omega\subseteq{\rm Fib}_\omega\cap{\rm Weq}_\omega$ since ${\rm TFib}_\omega\subseteq{\rm Weq}_\omega$. Conversely, if $f\in {\rm Fib}_\omega\cap{\rm Weq}_\omega$, there is an $\mathbb{E}$-triangle $\xymatrix{{\rm CoCone}(f,t)\ar[r]^{\quad x}&A\oplus W\ar[r]^{\quad(f,~ t)}&B\ar@{-->}[r]^{\delta}&}$ with $W\in\omega$ and ${\rm CoCone}(f,t)\in\mathcal{Y}$.
Since $f$ is $\omega$-epic, there is a morphism $s:W\rightarrow A$ such that $t=fs$. Then $(f, t)=f(1, s)$. Note that $(\mathcal{C}, \mathbb{E},\mathfrak{s})$ is an extriangulated category satisfies Condition (WIC) and $(f, t)$ is a deflation. It follows that $f$ is a deflation. So we have a commutative diagram with $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}(f)\ar[r]^r\ar@{-->}[d]_g&A\ar[r]^{f}\ar[d]_{\tiny\begin{pmatrix}1\\0\end{pmatrix}}&B\ar@{=}[d]\ar@{-->}[r]^{\eta}&\\
{\rm CoCone}(f,t)\ar[r]^x\ar@{-->}[d]_h&A\oplus W\ar[r]^{(f, t)}\ar[d]_{(1, s)}&B\ar@{=}[d]\ar@{-->}[r]^{\delta}&\\
{\rm CoCone}(f)\ar[r]^r&A\ar[r]^{f}&B\ar@{-->}[r]^{\eta}&}$$

by (ET3)$^{\rm op}$. So we have a commutative diagram with $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}(f)\ar[r]^{\quad r}\ar@{-->}[d]_{hg}&A\ar[r]^{f}\ar@{=}[d]&B\ar@{=}[d]\ar@{-->}[r]^{\eta}&\\
{\rm CoCone}(f)\ar[r]^{\quad<EMAIL_ADDRESS>
Hence, $hg$ is an isomorphism, which implies that ${\rm CoCone}(f)$ is a direct summand of ${\rm CoCone}(f, t)$, thus ${\rm CoCone}(f)\in\mathcal{Y}$. So $f\in {\rm TFib}_\omega$. This completes the proof.

The following is the Factorization axiom.
Let $(\mathcal{X}, \mathcal{Y})$ be a hereditary cotorsion pair, and $\omega:=\mathcal{X}\cap \mathcal{Y}$. Then we have ${\rm Mor}(\mathcal{C})= {\rm TFib}_\omega\circ{\rm CoFib}_\omega={\rm Fib}_\omega\circ {\rm TCoFib}_\omega$.

To prove Proposition <ref>, we need the following lemma.

Let $(\mathcal{X}, \mathcal{Y})$ be a hereditary cotorsion pair, and $\omega:=\mathcal{X}\cap \mathcal{Y}$. Then the following classes of morphisms

$${\rm CoFib}_\omega, ~{\rm Fib}_\omega,~ {\rm TCoFib}_\omega, ~{\rm TFib}_\omega$$
are closed under compositions.

For ${\rm TCoFib}_\omega$, this follows from  splitting inflations are closed under compositions, the extension-closedness of $\omega$ and Lemma <ref>. Similarly for the others.

Proof of Proposition <ref>. We can prove ${\rm Mor}(\mathcal{C})={\rm Fib}_\omega\circ {\rm TCoFib}_\omega$ by a similar proof of <cit.>.
Next we prove ${\rm Mor}(\mathcal{C})= {\rm TFib}_\omega\circ{\rm CoFib}_\omega$. For any morphism $f: A\rightarrow B$, there exists an $\mathbb{E}$-triangle $\xymatrix{Y_B\ar[r]&X_B\ar[r]^{t}&B\ar@{-->}[r]&}$ with $X_B\in\mathcal{X}$ and $Y_B\in\mathcal{Y}$. Hence there is a deflation $(f,~ t): A\oplus X_B\rightarrow B$ by the dual of <cit.>. Assume that $\xymatrix{K\ar[r]&A\oplus X_B\ar[r]^{\quad(f,~ t)}&B\ar@{-->}[r]&}$ is an $\mathbb{E}$-triangle. There exists an  $\mathbb{E}$-triangle $\xymatrix{K\ar[r]&Y^K\ar[r]&X^K\ar@{-->}[r]&}$ with $X^K\in\mathcal{X}$ and $Y^K\in\mathcal{Y}$. It follows from <cit.> that there is a commutative diagram of $\mathbb{E}$-triangles

$$\xymatrix{K\ar[r]\ar[d]&A\oplus X_B\ar[r]^{(f, t)}\ar[d]_{i}&B\ar@{=}[d]\ar@{-->}[r]&\\

Then $i\in {\rm CoFib}_\omega$ and $h\in{\rm TFib}_\omega$ by definition.

Note that $\xymatrix{{\tiny \begin{pmatrix}1\\0\end{pmatrix}}: A\rightarrow A\oplus X_B\in{\rm CoFib}_\omega}$, one has $i\circ{\tiny \begin{pmatrix}1\\0\end{pmatrix}}\in{\rm CoFib}_\omega$ by Lemma <ref>. Thus $f=h(i\circ{\tiny \begin{pmatrix}1\\0\end{pmatrix}})\in  {\rm TFib}_\omega\circ{\rm CoFib}_\omega$, as desired.  $\Box$

Let $(\mathcal{X}, \mathcal{Y})$ be a hereditary cotorsion pair, and $\omega:=\mathcal{X}\cap \mathcal{Y}$. Then we have
$${\rm TCoFib}_\omega\circ{\rm TFib}_\omega\subseteq{\rm Weq}_\omega.$$
Moreover, the class ${\rm Weq}_\omega$ is closed under compositions.

Let $a\in{\rm TFib}_\omega$ and $x'\in{\rm TCoFib}_\omega$. Hence there are some $x\in{\rm CoFib}_\omega$ and $b\in{\rm TFib}_\omega$ such that $x'a=bx$ by Proposition <ref>. It suffices to show that $x\in{\rm TCoFib}_\omega$. There are $\mathbb{E}$-triangles $\xymatrix{A'\ar[r]^{x'}&B'\ar[r]^{y'}&W\ar@{-->}[r]^{\delta'}&}$, $\xymatrix{A\ar[r]^{x}&B\ar[r]^{y}&X\ar@{-->}[r]^\delta&}$, $\xymatrix{Y\ar[r]^{k}&A\ar[r]^{a}&A'\ar@{-->}[r]^\eta&}$ and $\xymatrix{Y'\ar[r]^{k'}&B'\ar[r]^{b}&B'\ar@{-->}[r]^\theta&}$ with $Y, Y'\in\mathcal{Y}, X\in\mathcal{X}, W\in\omega$ and $x'$ is a splitting inflation by definition. Therefore there is a commutative diagram with $\mathbb{E}$-triangles

\xymatrix{
Y\ar@{}[dr]|{\circlearrowleft} \ar[d]_{k}\ar[r]^m  & Y' \ar@{}[dr]|{\circlearrowleft}\ar[d]_{k'}\ar[r]^{m'}  & Y'' \ar[d]_{k''}\ar@{-->}[r]^{\delta''}  & \\
A\ar@{}[dr]|{\circlearrowleft} \ar[d]_{a} \ar[r]^{x} & B \ar@{}[dr]|{\circlearrowleft}\ar[d]_{b} \ar[r]^{y} &X \ar[d]_{c} \ar@{-->}[r]^{\delta} &  \\
A' \ar@{-->}[d]_{\eta} \ar[r]^{x'} & B' \ar@{-->}[d]_{\theta} \ar[r]^{y'} & W \ar@{-->}[d]_{\rho} \ar@{-->}[r]^{\delta'} &  \\
&  & &  }

such that $(k, k', k''), (m, x, x')$ and $(m', y, y')$ are morphisms of $\mathbb{E}$-triangles by <cit.>. Note that $Y, Y'\in\mathcal{Y}$ and $\mathcal{Y}$ is closed under cones of inflations. Then we have $Y''\in\mathcal{Y}$. It is easy to see that $X\in\omega$ as $\mathcal{Y}$ is closed under extensions. Since $\mathbb{E}(\mathcal{X}, \mathcal{Y})=0$, one has $\rho=0$, which implies $Y''\in\omega$ because $Y''$ is a direct summand of $X$. So we have $\delta''=0$ as $\delta''\in\mathbb{E}(\mathcal{X}, \mathcal{Y})$.  Applying the functor $\mathcal{C}(X, -)$ to above commutative diagram, it is easy to check that $\mathcal{C}(X, y): \mathcal{C}(X, B)\rightarrow \mathcal{C}(X, X)$ is surjective by Snake Lemma. Therefore, $x$ is a splitting inflation, and one has $x\in{\rm TCoFib}_\omega$.

It is easy to check that the class ${\rm Weq}_\omega$ is closed under compositions by Lemma <ref>.

Let $(\mathcal{X}, \mathcal{Y})$ be a hereditary cotorsion pair, and $\omega:=\mathcal{X}\cap \mathcal{Y}$. If $f:A\rightarrow B$ is a morphism in ${\rm Weq}_\omega$, then the morphism $(f, t): A\oplus W\rightarrow B$ is in ${\rm TFib}_\omega$ for an arbitrary right $\omega$-approximation morphism $t: W\rightarrow B$.

By the assumption, there is a morphism $(f, t'): A\oplus W'\rightarrow B$ in ${\rm TFib}_\omega$ with $W'\in\omega$ and ${\rm CoCone}(f, t')\in\mathcal{Y}$. Let $t: W\rightarrow B$ be any right $\omega$-approximation. Then there is a morphism $s: W'\rightarrow W$ such that $t'=ts$. Since $(f, t')$ is a deflation and $(f, t')=(f, t){\tiny \begin{pmatrix}1&0\\0&s\end{pmatrix}}$, $(f, t): A\oplus W\rightarrow B$ is also a deflation. It remains to prove that ${\rm CoCone}(f, t)\in\mathcal{Y}$.

Since ${\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}: A\oplus W'\rightarrow A\oplus W'\oplus W$ is a splitting inflation with ${\rm Cone}{\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}=W\in\omega$. Note that ${\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}$ is an inflation and $(f, t')=(f, t', t){\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}$ is a deflation, it follows from Lemma <ref>(2) that $(f, t', t)$ is a deflation. Therefore, there is a commutative diagram with $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}(f, t')\ar[r]^i\ar[d]&A\oplus W'\ar[r]^{(f, t')}\ar[d]_{\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}&B\ar@{=}[d]\ar@{-->}[r]&\\
{\rm CoCone}(f, t', t)\ar[r]^{i'}\ar[d]&A\oplus W'\oplus W\ar[r]^{\qquad (f, t', t)}\ar[d]&B\ar@{-->}[r]&\\

by <cit.>. Hence ${\rm CoCone}(f, t', t)\in\mathcal{Y}$ because $\mathcal{Y}$ is closed under extensions. Since $(f, t)=(f, t', t){\tiny \begin{pmatrix}1&0\\0&0\\0&1\end{pmatrix}}$ and $(f, t', t)=(f, t){\tiny \begin{pmatrix}1&0&0\\0&s&1\end{pmatrix}}$, there is a  commutative diagram of $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}(f, t)\ar[r]\ar@{-->}[d]_g&A\oplus W\ar[r]^{(f, t)}\ar[d]_{\tiny\begin{pmatrix}1&0\\0&0\\0&1\end{pmatrix}}&B\ar@{=}[d]\ar@{-->}[r]&\\
{\rm CoCone}(f, t', t)\ar[r]\ar@{-->}[d]_h&A\oplus W'\oplus W\ar[r]^{\qquad(f, t', t)}\ar[d]_{\tiny\begin{pmatrix}1&0&0\\0&s&1\end{pmatrix}}&B\ar@{=}[d]\ar@{-->}[r]&\\
{\rm CoCone}(f,t)\ar[r]&A\oplus W\ar[r]^{(f, t)}&B\ar@{-->}[r]&}$$

by (ET3)$^{\rm op}$. Note that ${\tiny\begin{pmatrix}1&0&0\\0&s&1\end{pmatrix}}
{\tiny\begin{pmatrix}1&0\\0&0\\0&1\end{pmatrix}}={\tiny\begin{pmatrix}1&0\\0&1\end{pmatrix}}$, so we have a commutative diagram of $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}(f,t)\ar[r]\ar@{-->}[d]_{hg}&A\oplus W\ar[r]^{(f, t)}\ar@{=}[d]&B\ar@{=}[d]\ar@{-->}[r]&\\
{\rm CoCone}(f,t)\ar[r]&A\ar[r]^{(f, t)}&B\ar@{-->}[r]&}$$

Therefore, $hg$ is an isomorphism, which implies that ${\rm CoCone}(f,t)$ is a direct summand of ${\rm CoCone}(f, t', t)$, thus ${\rm CoCone}(f,t)\in\mathcal{Y}$.  This completes the proof.

Let $f:A\rightarrow B$ and $g:B\rightarrow C$ be morphisms in $\mathcal{C}$ with $f\in{\rm TFib}_\omega$ and $gf\in{\rm Weq}_\omega$. Then $g\in{\rm Weq}_\omega$.

Let $t:W\rightarrow C$ be a right $\omega$-approximation of $C$. It follows from Lemma <ref> that $(gf, t): A\oplus W\rightarrow C$ is in ${\rm TFib}_\omega$ as $gf\in{\rm Weq}_\omega$, that is, $(gf, t)$ is a deflation with ${\rm CoCone}(gf, t)\in\mathcal{Y}$. Note that $(gf, t)=(g, t){\tiny\begin{pmatrix}f&0\\0&1\end{pmatrix}}$, which implies that $(g, t):B\oplus W\rightarrow C$ is a deflation. Then there is a commutative diagram of $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}{\tiny\begin{pmatrix}f&0\\0&1\end{pmatrix}}\ar[d]\ar@{=}[r]&{\rm CoCone}{\tiny\begin{pmatrix}f&0\\0&1\end{pmatrix}}\ar[d]&&\\
{\rm CoCone}(gf,  t)\ar[r]\ar[d]&A\oplus W\ar[r]^{\qquad(gf,t)}\ar[d]_{\tiny\begin{pmatrix}f&0\\0&1\end{pmatrix}}&C\ar@{=}[d]\ar@{-->}[r]&\\
{\rm CoCone}(g,t)\ar[r]\ar@{-->}[d]&B\oplus W\ar[r]^{(g, t)}\ar@{-->}[d]&C\ar@{-->}[r]&\\

by <cit.>. Since ${\rm CoCone}{\tiny\begin{pmatrix}f&0\\0&1\end{pmatrix}}={\rm CoCone}(f)\in\mathcal{Y}$, which implies that ${\rm CoCone}(g, t)\in\mathcal{Y}$ because $\mathcal{Y}$ is closed under cones of inflations. So $(g, t)\in{\rm TFib}_\omega$, and we have $g\in{\rm Weq}_\omega$ by definition.

Let $f:A\rightarrow B$ and $g:B\rightarrow C$ be morphisms in $\mathcal{C}$ such that $f, gf\in{\rm Weq}_\omega$. Then $g\in{\rm Weq}_\omega$.

Since $f\in{\rm Weq}_\omega$, there is a morphism $(f, t): A\oplus W\rightarrow B$ with $W\in\omega$ such that $(f, t)\in{\rm TFib}_\omega$. To prove $g\in {\rm Weq}_\omega$, it suffices to prove $(gf, gt)\in{\rm Weq}_\omega$ by the the commutative diagram

$$\xymatrix{A\oplus W\ar[rr]^{(gf, gt)}\ar[rd]_{(f, t)}&&C\\

and Lemma <ref>.

Let $t':W'\rightarrow C$ be a right $\omega$-approximation of $C$. It follows from Lemma <ref> that $(gf, t'): A\oplus W'\rightarrow C$ is in ${\rm TFib}_\omega$ as $gf\in{\rm Weq}_\omega$, that is, $(gf, t')$ is a deflation with ${\rm CoCone}(gf, t')\in\mathcal{Y}$. Since ${\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}: A\oplus W'\rightarrow A\oplus W'\oplus W$ is a splitting inflation with ${\rm Cone}{\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}=W\in\omega$. Note that ${\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}$ is an inflation and $(gf, t')=(gf, t', gt){\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}$ is a deflation, where $(gf, t', gt): A\oplus W'\oplus W\rightarrow C$. It follows from Lemma <ref>(2) that $(gf, t', gt)$ is a deflation. Therefore, there is a commutative diagram with $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}(gf, t')\ar[r]^i\ar[d]&A\oplus W'\ar[r]^{(gf, t')}\ar[d]_{\tiny \begin{pmatrix}1&0\\0&1\\0&0\end{pmatrix}}&C\ar@{=}[d]\ar@{-->}[r]&\\
{\rm CoCone}(gf, t', gt)\ar[r]^{i'}\ar[d]&A\oplus W'\oplus W\ar[r]^{\qquad (gf, t', gt)}\ar[d]&C\ar@{-->}[r]&\\

by <cit.>. Since ${\rm CoCone}(gf, t')\in\mathcal{Y}$ and $W\in\mathcal{Y}$, which implies ${\rm CoCone}(gf, t', gt)\in\mathcal{Y}$, and hence $(gf, t', gt)\in{\rm TFib}_\omega$. Therefore, $(gf, gt)\in{\rm Weq}_\omega$ by the commutative diagram

$$\xymatrix{A\oplus W\ar[rr]^{(gf, gt)}\ar[rd]_{\tiny \begin{pmatrix}1&0\\0&0\\0&1\end{pmatrix}}&&C\\
&A\oplus W'\oplus W\ar[ur]_{(gf, t', gt)}&}$$

This completes the proof.

Let  Let $f: A\rightarrow B$ and $g:B\rightarrow C$ be morphisms in $\mathcal{C}$ with $gf\in{\rm Weq}_\omega$, $f\in{\rm CoFib}_\omega$ and $g\in{\rm TFib}_\omega$. Then $f\in{\rm Weq}_\omega$.

We first show that $f$ is a splitting inflation. Since $f\in{\rm CoFib}_\omega$, $f$ is an inflation with ${\rm Cone}(f)\in\mathcal{X}$. Since $h:=gf\in{\rm Weq}_\omega$, there is a deflation $(h, t): A\oplus W\rightarrow C$ with $W\in\omega$ and ${\rm CoCone}(h,t)\in\mathcal{Y}$. Consider the commutative diagram with $\mathbb{E}$-triangles

$$\xymatrix{&A\ar[r]^f\ar[d]_{\tiny\begin{pmatrix}1\\0\end{pmatrix}}&B\ar[r]\ar[d]^g&{\rm Cone}(f)\ar@{-->}[r]&\\
{\rm CoCone}(h, t)\ar[r]&A\oplus W\ar[r]^{(h,t)}&C\ar@{-->}[r]&&}$$

Since $\mathbb{E}({\rm Cone}(f), {\rm CoCone}(h, t))=0$, there exists a morphism ${\tiny\begin{pmatrix}s_1\\s_2\end{pmatrix}}: B\rightarrow A\oplus W$ such that $1_A=sf$ by Lemma <ref>, which implies that $f$ is a splitting inflation.
It remains to prove that ${\rm Cone}(f)\in\mathcal{Y}$.

Since $f$ is a splitting inflation, one can write $h=pi$ as

$$\xymatrix{A\ar[rr]^{h}\ar[rd]_{i={\tiny \begin{pmatrix}1\\0\end{pmatrix}}}&&C\\
&A\oplus X\ar[ur]_{p=(h, t')}&}$$

where $X:={\rm Cone}(f)\in\mathcal{X}$ and $p=(h, t')\in{\rm TFib}_\omega$.

Note that $(h, t')=(h, t, t'){\tiny \begin{pmatrix}1&0\\0&0\\0&1\end{pmatrix}}$ is a deflation where $(h, t, t'): A\oplus W\oplus X\rightarrow C$, and ${\tiny \begin{pmatrix}1&0\\0&0\\0&1\end{pmatrix}}: A\oplus X\rightarrow A\oplus W\oplus X$ is an inflation. It follows from Lemma <ref>(2) that $(h, t, t')$ is a deflation. Therefore, there is a commutative diagram with $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}(h, t')\ar[r]\ar[d]&A\oplus X\ar[r]^{(h, t')}\ar[d]_{\tiny \begin{pmatrix}1&0\\0&0\\0&1\end{pmatrix}}&C\ar@{=}[d]\ar@{-->}[r]&\\
{\rm CoCone}(h, t, t')\ar[r]^{\tiny \begin{pmatrix}k_1\\k_2\\k_3\end{pmatrix}}\ar[d]&A\oplus W\oplus X\ar[r]^{\qquad (h, t, t')}\ar[d]&C\ar@{-->}[r]&\\

by <cit.>. Since ${\rm CoCone}(h, t')\in\mathcal{Y}$ and $W\in\omega$, we have ${\rm CoCone}(h, t, t')\in\mathcal{Y}$. The commutative square

$$\xymatrix{{\rm CoCone}(h, t, t')\ar[r]^{\qquad k_3}\ar[d]_{\tiny \begin{pmatrix}k_1\\k_2\end{pmatrix}}&X\ar[d]^{-t'}\\
A\oplus W\ar[r]^{(h, t)}&C}$$

is a homotopy cartesian in <cit.> because there is an $\mathbb{E}$-triangle

$$\xymatrix{{\rm CoCone}(h, t, t')\ar[r]^{\tiny \begin{pmatrix}k_1\\k_2\\k_3\end{pmatrix}}&A\oplus W\oplus X\ar[r]^{\qquad (h, t, t')}&C\ar@{-->}[r]&}$$

Hence there is a commutative diagram of $\mathbb{E}$-triangles

$$\xymatrix{{\rm CoCone}(h,t)\ar@{=}[d]\ar[r]&{\rm CoCone}(h, t, t')\ar[r]^{\qquad k_3}\ar[d]_{\tiny \begin{pmatrix}k_1\\k_2\end{pmatrix}}&X\ar[d]^{-t'}\ar@{-->}[r]&\\
{\rm CoCone}(h,t)\ar[r]& A\oplus W\ar[r]^{(h, t)}&C\ar@{-->}[r]&}$$

by the dual of <cit.>. Since ${\rm CoCone}(h,t), {\rm CoCone}(h,t, t')\in\mathcal{Y}$, we have $X\in\mathcal{Y}$ as $(\mathcal{X}, \mathcal{Y})$ is a hereditary cotorsion pair. This completes the proof.

Let $f: A\rightarrow B$ and $g:B\rightarrow C$ be morphisms in $\mathcal{C}$ with $gf, g\in{\rm Weq}_\omega$. Then $f\in{\rm Weq}_\omega$.

The proof is similar to <cit.>.

Now, Lemmas <ref>, <ref> and <ref> yield the Two out of three axiom.

Let $f: A\rightarrow B$ and $g:B\rightarrow C$ be morphisms in $\mathcal{C}$. If two of three morphisms $f, g, gf$ in ${\rm Weq}_\omega$, then so is the third.

In the following, we prove that ${\rm CoFib}_\omega, {\rm Fib}_\omega$ and ${\rm Weq}_\omega$ are closed under retracts. Suppose that $g: A\rightarrow B$ is a retract of $f: C\rightarrow D$, i.e., one has a commutative diagram of morphisms

with $ca=1_{A}$ and $db=1_{B}$.

${\rm CoFib}_\omega$,  ${\rm TCoFib}_\omega$, and ${\rm TFib}_\omega$ are closed under retracts.

Let $f\in{\rm CoFib}_\omega$, i.e., $f$ is an inflation with ${\rm Cone}(f)\in\mathcal{X}$. Since $bg=fa$ is an inflation, $g$ is an inflation. So there exist morphisms $m$ and $n$ which make the following diagram commutative

{\rm Cone}(g)\ar@{-->}[r]^{m}\ar@{-->}[d]&{\rm Cone}(f)\ar@{-->}[r]^{n}\ar@{-->}[d]&{\rm Cone}(g)\ar@{-->}[d]&\\&&}$$

where all columns are $\mathbb{E}$-triangles by (ET3). Hence there is a commutative diagram with $\mathbb{E}$-triangles

$$\xymatrix{A\ar[r]^g\ar@{=}[d]&B\ar[r]^{p}\ar@{=}[d]&{\rm Cone}(g)\ar[d]^{nm}\ar@{-->}[r]&\\
A\ar[r]^g&B\ar[r]^{p}&{\rm Cone}(g)\ar@{-->}[r]&}$$

Therefore $nm$ is an isomorphism, which implies that $m$ is a splitting inflation. So ${\rm Cone}(g)$ is a direct summand of ${\rm Cone}(f)$. Hence ${\rm Cone}(g)\in\mathcal{X}$, and one has $g\in{\rm CoFib}_\omega$.

Similarly for the others.

${\rm Fib}_\omega$  is closed under retracts.

The proof is similar to  that of ${\rm Fib}_\omega$ is closed under retracts in [11].

${\rm Weq}_\omega$  is closed under retracts.

Let $g=g_1\oplus g_2: A_1\oplus A_2\rightarrow B_1\oplus B_2$ and suppose that $g\in{\rm Weq}_\omega$. Then $g_i$ factor as $g_n=p_ni_n$, $i_n\in{\rm CoFib}_\omega, p_n\in{\rm TFib}_\omega$ for $n=1, 2$ by Proposition <ref>. Therefore, $g=g_1\oplus g_2=(p_1\oplus p_2)\circ (i_1\oplus i_2)$. Since $p_1, p_2\in{\rm TFib}_\omega$, one has $p_1\oplus p_2\in{\rm TFib}_\omega\subseteq{\rm Weq}_\omega$. So we have $i_1\oplus i_2\in{\rm Weq}_\omega$ by Lemma <ref>. Since $i_1, i_2\in{\rm CoFib}_\omega$, we have $i_1\oplus i_2\in{\rm CoFib}_\omega$. Therefore, $i_1\oplus i_2\in{\rm CoFib}_\omega\cap{\rm Weq}_\omega={\rm TCoFib}_\omega$ by Proposition <ref>, which implies that $i_1, i_2\in{\rm TCoFib}_\omega$ by Lemma <ref> for $n=1, 2$. Therefore, we have $g_1, g_2\in{\rm Weq}_\omega$, i.e. ${\rm Weq}_\omega$  is closed under retracts.

Proof of the Retract axiom. Now, the Retract axiom follows from Lemmas <ref>-<ref>.  $\Box$

Finally, we have the following Lifting axiom and its proof is similar to that of Lifting axiom in <cit.>.

We have the following.

(1) ${\rm TCoFib}_\omega$ satisfies the left lifting property against ${\rm Fib}_\omega$.

(2) ${\rm TFib}_\omega$ satisfies the left lifting property against ${\rm CoFib}_\omega$.

Up to now the “if" part of Theorem <ref> is proved.

§.§ Hereditary cotorsion pair arising from a model structure

The aim of this subsection is to prove the “only if" part of Theorem <ref> as follows.

Let $\mathcal{A}$ be a weakly idempotent complete extriangulated category, $\mathcal{X}, \mathcal{Y}$ full subcategories of $\mathcal{C}$ which are closed under direct summands and isomorphisms, and $\omega=\mathcal{X}\cap \mathcal{Y}$. If $({\rm CoFib}_\omega, {\rm Fib}_\omega, {\rm Weq}_\omega)$ is a model structure, then
$(\mathcal{X, Y})$ is a hereditary cotorsion pair in $\mathcal{C}$, and $\omega$ is contravariantly finite in $\mathcal{C}$; and the class $\mathcal{C}_\omega$ of cofibrant objects is $\mathcal{X}$, the class $\mathcal{F}_\omega$ of fibrant objects is $\mathcal{C}$, the class $\mathcal{W}_\omega$ of trivial objects is $\mathcal{Y}$; and the homotopy category ${\rm Ho}(\mathcal{C})$ is $\mathcal{X}/\mathcal{W}$.

To prove Theorem <ref>, we need some preparations. Remark that, a model category $\mathcal{M}$ induces an equivalence

$$\mathcal{M}_{cf}/{\sim}\s{\simeq}\longrightarrow{\rm Ho}(\mathcal{M}).$$

Here, the right hand side is the localization $\mathcal{M}[\textrm{Weq}^{-1}]$. The left hand side is the category of cofibrant-fibrant objects modulo homotopies. We refer
to [25, 16] for a detailed discussion on this matter.

The following lemma is essentially taken from <cit.>, where a variation of it
appears. The proof given there carries over to the present situation.

Keep the notations as in Theorem <ref>. Assume that $({\rm CoFib}_\omega, {\rm Fib}_\omega, {\rm Weq}_\omega)$ is a model structure on $\mathcal{C}$. For any $X,Y\in{\mathcal{C}}$, morphisms $f,g\in{\mathcal{C}(X, Y)}$ satisfy $f\sim g$ if and only if $f-g$ factors through some object $W\in{\omega}$.

Proof of Theorem <ref>. By definition one easily sees that $\mathcal{C}_\omega=\mathcal{X}$ and $\mathcal{F}_\omega=\mathcal{C}$. Next we claim that $\mathcal{W}_\omega=\mathcal{Y}$. If $Y\in\mathcal{Y}$, since $(0, 0): Y\oplus 0\rightarrow 0$ is a deflation with $0\in\omega$ and ${\rm CoCone}(0,0)=Y\in\mathcal{Y}$, hence $0: Y\rightarrow 0$ is a weak equivalence by definition, i.e., $Y\in\mathcal{W}_\omega$. Conversely, if $Y\in\mathcal{W}_\omega$, then $0: Y\rightarrow 0$ is a weak equivalence. Hence there is a deflation $(0, 0): Y\oplus W\rightarrow 0$ with ${\rm CoCone}(0, 0)=Y\oplus W\in \mathcal{Y}$, which implies $Y\in \mathcal{Y}$. So we have $\mathcal{W}_\omega=\mathcal{Y}$. Thus ${\rm T}\mathcal{C}_\omega=\mathcal{C}_\omega\cap \mathcal{Y}=\omega$ and ${\rm T}\mathcal{F}_\omega=\mathcal{F}_\omega\cap\mathcal{W}_\omega=\mathcal{Y}$. Thanks to Lemma <ref>, we have  ${\rm Ho}(\mathcal{C})\cong\mathcal{X}/\mathcal{W}$.

Next we prove that $(\mathcal{X, Y})$ is a hereditary cotorsion pair. Let $\delta\in\mathbb{E}(X, Y)$ with $X\in\mathcal{X}$ and $Y\in\mathcal{Y}$. Realize it as an $\mathbb{E}$-triangle $\xymatrix{Y\ar[r]^x&A\ar[r]^y&X\ar@{-->}[r]^\delta&}$. Since $Y$ is in $\mathcal{Y}$, it follows that $y$ is a trivial fibration. Note that $X\in\mathcal{X}=\mathcal{C}_\omega$. Then the morphism $1_X: X\rightarrow X$ factors through $y$ by <cit.>. Thus $\delta$ splits by <cit.>, and so $\mathbb{E}(X, Y)=0$.

Let $C\in\mathcal{C}$ be any object, and let $\xymatrix@=2em{0\ar[r]&X_C\ar[r]^{f}&C}$ be a factorization of  $0: 0\rightarrow C$ into a cofibration followed by a trivial fibration. Then $X_C\in \mathcal{C}_\omega=\mathcal{X}$ by definition.
Since $f$ is a trivial fibration, there exists an $\mathbb{E}$-triangle $\xymatrix{Y_C\ar[r]&X_C\ar[r]^{f}&C\ar@{-->}[r]&}$ with $Y_C\in\mathcal{Y}$. Dually, there exists an $\mathbb{E}$-triangle $\xymatrix{C\ar[r]&Y^C\ar[r]&X^C\ar@{-->}[r]&}$ with $X^C\in\mathcal{X}$ and $Y^C\in\mathcal{Y}$.

We can prove that $\mathcal{Y}$ is closed under cones of inflations by a similar argument of <cit.>, hence  $(\mathcal{X, Y})$ is a hereditary cotorsion pair by  Proposition <ref>.

Finally, we claim that $\omega$ is contravariantly finite. For any object $C\in\mathcal{C}$, there exists an $\mathbb{E}$-triangle $\xymatrix{Y_C\ar[r]&X_C\ar[r]^{f}&C\ar@{-->}[r]&}$ with $X_C\in\mathcal{X}$ and $Y_C\in\mathcal{Y}$. Hence $f$ is a trivial fibration by definition. Let $\xymatrix@=2em{0\ar[r]&F_C\ar[r]^{g}&X_C}$ be a factorization of $0: 0\rightarrow X_C$ into a trivial cofibration followed by a fibration. Then $F_C\in{\rm T}\mathcal{C}_\omega=\omega$. Next claim that $fg: F_C\rightarrow C$ is a right $\omega$-approximation of $C$. Since fibrations are closed under compositions, $fg$ is a fibration. Let $t: W\rightarrow C$ be a morphism with $W\in \omega$. Consider the following diagram

Since $0\rightarrow W$ is a trivial cofibration, by the Lifting axiom, $t$ factors through $fg$. Hence $f g: F_C\rightarrow C$ is a right $\omega$-approximation of $C$. $\Box$

§ APPLICATIONS

We begin this section with the following definitions.
Let $\mathcal C$ be an extriangulated category and $\X$ a subcategory of $\C$.
For each $n\geq 0$, we inductively define subcategories
$\mathcal X_n^{\wedge}$ and $\mathcal X_n^{\vee}$ of $\mathcal C$
as $\mathcal X_n^{\wedge}:={\rm Cone}(\mathcal X_{n-1}^{\wedge},\mathcal X)$
and $\mathcal X_n^{\vee}:={\rm CoCone}(\mathcal X,\mathcal X_{n-1}^{\vee})$,
where $\mathcal X_{-1}^{\wedge}:=\{0\}$
and $\mathcal X_{-1}^{\vee}:=\{0\}$. Put

$$\mathcal X^{\wedge}:=\bigcup\limits_{n\geq 0}\mathcal X_n^{\wedge}~~\mbox{and}~~
\mathcal X^{\vee}:=\bigcup\limits_{n\geq 0}\mathcal X_n^{\vee}.$$

Let $\mathcal{X}$ be a subcategory of $\C$. Recall from [28] that $\mathcal{X}$ is called a thick subcategory of $\C$ if it is closed under extensions, cones of inflations, cocones of deflations and direct summands. Denote by ${\rm add} \mathcal{X}$ the smallest subcategory of
$\C$ containing $\mathcal{X}$ and closed under finite direct sums and direct summands.

Let $\mathcal C$ be an extriangulated category.
A subcategory $\mathcal M$ of $\C$ is called a silting subcategory if
it satisfies the following conditions.

(a) $\mathcal M$ is a presilting subcategory, that is, $\mathbb{E}^k(\mathcal M,\mathcal M)=0$ for all $k\geq 1$.

(b)  $\C$ is the smallest thick subcategory containing
$\mathcal M$.

We say that an object $M\in\mathcal C$ is silting if so is ${\rm add} M$.

We give some examples of silting subcategories.

This example comes from <cit.>.

(1) Let $A$ be a ring. Then $A$ regarded as a stalk complex is a tilting object
in $K^b({\rm proj}\emph{-}A)$. More generally, any tilting complex of $A$
is a tilting object in $K^b({\rm proj}\emph{-}A)$.
By definition, any tilting object is silting object in triangulated categories.

(2)  Let $A$ be a differential graded ring and let $D(A)$ be the derived category of $A$.
If $H^i(A)=0$ for any $i>0$,
then the thick subcategory ${\rm thick}(A)$ of $D(A)$ generated by $A$ has a silting
object $A$.

Let $\Lambda$ be an artin algebra. We denote by $\mathcal P^{\infty}(\Lambda)$
the category of finitely
generated right $\Lambda$-modules of finite projective dimension. Since $\mathcal P^{\infty}(\Lambda)$
is closed under extensions, it is an extriangulated category. By <cit.>,
we know that ${\rm add}A$ is a silting
subcategory of $\mathcal P^{\infty}(\Lambda)$.
Moreover, ${\rm add}A$ is functorially finite.

(1) Let $\Lambda$ be a finite dimensional $k$-algebra. If ${\rm gl. dim}\Lambda<\infty$, then any tilting object in ${\rm mod}\Lambda$ is also a silting object.

(2) Let $Q$ be the following infinite quiver:

$$\xymatrix<EMAIL_ADDRESS>\ar[r]^{x_{-5}} &-4 \ar[r]^{x_{-4}} &-3 \ar[r]^{x_{-3}} &-2 \ar[r]^{x_{-2}} &-1 \ar[r]^{x_{-1}} &0 \ar[r]^{x_{0}} &1 \ar[r]^{x_1} &2 \ar[r]^{x_2} &3 \ar[r]^{x_3} &4 \ar[r]^{x_4} &\cdots}$$

Let $\Lambda=kQ/[x_ix_{i+1}x_{i+2},x_{4k}x_{4k+1}](i\neq 4k)$. Then the Auslander-Reiten quiver of ${\rm mod}\Lambda$ is the following.

&\bigstar \ar[dr] &&&&\bigstar\ar[dr] &&\bigstar \ar[dr] &&\bigstar \ar[dr] &&&&\bigstar \ar[dr] &&\bigstar \ar[dr] &&\bigstar \ar[dr] &&&&\bigstar \ar[dr]\\
\cdots \ar[ur] \ar[dr] &&\bigstar \ar[dr] &&\circ \ar[ur] \ar[dr] &&\circ \ar[ur] \ar[dr] &&\circ \ar[ur] \ar[dr] &&\bigstar  \ar[dr] &&\bigstar \ar[ur] \ar[dr] &&\circ \ar[ur] \ar[dr] &&\circ \ar[ur] \ar[dr] &&\circ  \ar[dr] &&\bigstar \ar[ur] \ar[dr] &&\cdots\\
&\circ \ar[ur] &&\circ \ar[ur] &&\circ\ar[ur] &&\circ \ar[ur] &&\circ \ar[ur] &&\bigstar \ar[ur] &&\circ \ar[ur] &&\circ\ar[ur] &&\circ \ar[ur]  &&\circ \ar[ur] &&\circ \ar[ur]

The indecomposable objects denoted by $\bigstar$ consist a silting subcategory in ${\rm mod}\Lambda$.

Adachi and Tsukamoto [2] establish a bijection between bounded hereditary cotorsion
pairs and silting subcategories in extriangulated categories.

Let $\mathcal C$ be an extriangulated category. Then there exists a
bijection between the following sets:

(1) The set of bounded cotorsion pairs $(\mathcal X, \mathcal Y)$ satisfying $\mathbb{E}^{i}(X, Y)=0$ for each $i\geqslant2$ and all $X\in{\mathcal{X}}$ and $Y\in{\mathcal{Y}}$.

(2) The set of silting subcategories of $\mathcal C$.

In particular, if $\mathcal M$ is a silting subcategory of $\mathcal C$,
then $(\mathcal M^{\vee},\mathcal M^{\wedge})$ is a bounded hereditary cotorsion pair
in $\mathcal C$ satisfying $\mathcal M=\mathcal M^{\vee}\cap\mathcal M^{\wedge}$.

A direct consequence of Theorem <ref> and Lemma <ref> yields the following corollary.
Let $\mathcal C$ be a weakly idempotent complete extriangulated category. Then any silting object $M$ of $\mathcal C$ induces the model structure $({\rm CoFib}_{\mathcal M}, {\rm Fib}_{\mathcal M}, {\rm Weq}_{\mathcal M})$ on $\mathcal{C}$, where $\mathcal M={\rm add}M$ is a silting subcategory of $\mathcal C$.

We introduce the following notion of co-$t$-structures, which was independently introduced by Bondarko [9] and Pauksztello [24] as an analog of $t$-structures defined in [5].

Let $\mathcal{C}$ be a triangulated category with the shift functor [1]. A co-t-structure on $\mathcal{T}$ is a pair $(\mathcal{A},\mathcal{B})$ of subcategories of $\mathcal{C}$ such that

* $\mathcal{A}[-1]\subseteq \mathcal{A}$ and $\mathcal{B}[1]\subseteq \mathcal{B}$.

* $\mathrm{Hom}_{\mathcal{C}}(\mathcal{A}[-1],\mathcal{B})=0$.

* Any object $T\in{\mathcal{T}}$ has a distinguished triangle
in $\mathcal{C}$ with $X\in{\mathcal{A}[-1]}$ and $Y\in{\mathcal{B}}$.

Note that hereditary cotorsion pairs on $\mathcal{C}$ are exactly co-t-structures. In combination this with Theorem <ref>, we have the following result.

Let $\mathcal{C}$ be a triangulated category, and $\mathcal{X}$ and $\mathcal{Y}$ additive full subcategories of $\mathcal{C}$ which are closed under direct summands and isomorphisms, and $\omega=\mathcal{X}\cap\mathcal{Y}$. Then  $({\rm CoFib}_\omega, {\rm Fib}_\omega, {\rm Weq}_\omega)$ is a model structure if and only if $(\mathcal{X, Y})$ is a co-t-structure in $\mathcal{C}$, and $\omega$ is contravariantly finite in $\mathcal{C}$. In this case, the class $\mathcal{C}_\omega$ of cofibrant objects is $\mathcal{X}$, the class $\mathcal{F}_\omega$ of fibrant objects is $\mathcal{C}$, the class $\mathcal{W}_\omega$ of trivial objects is $\mathcal{Y}$; and the homotopy category ${\rm Ho}(\mathcal{C})$ is $\mathcal{X}/\mathcal{W}$.

Both authors acknowledge the inspiration they received from reading the paper [11].

[1]
T. Aihara, O. Iyama,
Silting mutation in triangulated categories,
J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668.

[2] T. Adachi, M. Tsukamoto, Hereditary cotorsion pairs and silting subcategories in extriangulated categories, J. Algebra 594 (2022) 109-137.

[5] A. Beilinson, J. Bernstein, P. Deligne, Faisceaux Pervers, Ast$\mathrm{\acute{e}}$risque 100 (1982) 5-171.

[8] A. Beligiannis, I. Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. vol. 188, 2007.
[9] M.V. Bondarko, Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6(3) (2010) 387-504.

[11] J. Cui, X.S. Lu, P. Zhang, Model structure from one hereditary complete cotorsion pair, arXiv: 2401.08078v1.

[14] J. Gillespie, Model structures on exact categories, J. Pure Appl. Algebra 215 (2011) 2892-2902.
[15] J. He, C.B. Xie, P.X. Zhou, Homotopy cartesian squares in extriangulated categories, Open Math. 21 (2023) 20220570.

[16] M. Hovey, Model categories, Math. Surveys and Monographs 63, Amer. Math. Soc., Providence, 1999.
[17] M. Hovey, Cotorsion pairs, model category structures, and representation theory,
Math. Z. 241 (2002) 553-592.
[18] M. Hovey, Cotorsion pairs and model categories,
in Interactions between homotopy theory and algebra, Contemp. Math.
436. Amer. Math. Soc., Providence, RI, 2007, 277-296.
[19] Y. Liu, P.Y. Zhou, Hereditary cotorsion pairs on extriangulated subcategories, arXiv:2012.06997.

[21] C. Klapproth, $n$-extension closed subcategories of $n$-exangulated categories, arXiv: 2209.01128v3, 2023.

[22]  H. Nakaoka, Y. Palu,  Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cahiers de Topologie et Geometrie Differentielle Categoriques, Volume LX-2 (2019) 117-193.

[24] D. Pauksztello, Compact corigid objects in triangulated categories and co-t-structures, Cent. Eur. J. Math. 6(1) (2008) 25-42.
[25] D. Quillen, Homotopical algebra, Lecture Notes in Math.43, Springer-Verlag, 1967.

[26]   X.Y. Yang, Model structures on triangulated categories. Glasg. Math. J. 57(2) (2015) 263-284.

[28]  P. Zhou, Hovey triples arising from two cotorsion pairs of extriangulated categories.
Colloq. Math. 171(2) (2023),  211-230.