# Some applications of extriangulated categories

Yann Palu  LAMFA, Université Picardie Jules Verne, 80039 Amiens, France
<EMAIL_ADDRESS>
###### Abstract.

Extriangulated categories axiomatize extension-closed subcategories of
triangulated categories and generalise both exact categories and triangulated
categories. This survey article presents three applications of extriangulated
categories to homotopical algebra, algebraic combinatorics and representation
theory. The first shows that, via some generalised Hovey’s correspondence,
extriangulated categories easily give rise to model category structures with
triangulated homotopy categories. As a second application, extriangulated
structures play a fondamental role in the construction of polytopal
realisations of $g$-vector fans. This allows for a generalisation of ABHY’s
construction appearing in the study of scattering amplitudes in theoretical
physics. Lastly, extriangulated categories provide a convenient framework for
studying mutations in representation theory and flips in algebraic
combinatorics. In nice enough hereditary extriangulated categories, there is a
well-behaved theory of mutation for silting objects, which encompass cluster
tilting, two-term silting, relative tilting, mutation of maximal almost-rigid
modules, flip of dissections and mutation of intermediate co-$t$-structures.

Most diagrams were made using the amazing https://q.uiver.app/. This work was
partially supported by the French ANR grants SC3A (15-CE40-0004-01) and CHARMS
(19-CE40-0017)

###### Contents

  1. 1 The definition of an extriangulated category
    1. 1.1 Prototypical example
    2. 1.2 Definitions
    3. 1.3 First properties
  2. 2 Examples of extriangulated categories
    1. 2.1 Some examples arising in representation theory
      1. 2.1.1 Exact categories
      2. 2.1.2 Triangulated categories
      3. 2.1.3 Extension-closed subcategories
      4. 2.1.4 Relative structures
      5. 2.1.5 Quotients by projectives-injectives
      6. 2.1.6 Categories of Cohen–Macaulay dg modules
      7. 2.1.7 Higgs categories
    2. 2.2 Enhancements
      1. 2.2.1 Topological extriangulated categories
      2. 2.2.2 Algebraic extriangulated categories
    3. 2.3 Localisation of extriangulated categories
  3. 3 Hovey’s correspondence
    1. 3.1 Exact model structures
    2. 3.2 Hovey’s correspondence for extriangulated categories
    3. 3.3 Triangulated homotopy categories
  4. 4 Polytopal realisations of g-vector fans
    1. 4.1 ABHY’s construction
    2. 4.2 Representation theoretic interpretation
    3. 4.3 Reformulation using McMullen’s type cone
      1. 4.3.1 Polyhedral cones
      2. 4.3.2 Polyhedral fans
      3. 4.3.3 Polytopes and their normal fans
      4. 4.3.4 McMullen’s type cone
      5. 4.3.5 From the type cone to ABHY’s construction
    4. 4.4 Cluster categories come into play
  5. 5 Mutation
    1. 5.1 0-Auslander extriangulated categories
    2. 5.2 Mutation of maximal rigid objects

## Introduction

In a similar way that Quillen exact categories axiomatise extension-closed
full subcategories of abelian categories, extriangulated categories,
introduced in [NP19], axiomatise extension-closed full subcategories of
triangulated categories. They also serve as a tool for unifying proofs in both
exact categories and triangulated categories. One strength of extriangulated
categories is that many constructions that are often performed on exact or
triangulated categories may leave the realm of exact or triangulated
categories: This is the case, for example, of quotients by projective-
injective objects in non-Frobenius exact categories, of taking extension-
closed subcategories or relative structures of triangulated categories.
However, in all three examples, the resulting categories remain
extriangulated.

In this survey article, we illustrate the use of extriangulated structures
with three applications.

_Hovey’s correspondence_ [Hov07, Gil11] is a tool for constructing model
category structures (see Definition 3.7) out of very few information, namely
that of fibrant and of cofibrant objects, or alternatively, that of
(co)fibrant and trivially (co)fibrant objects. In [NP19], we generalised
Hovey’s correspondence from the case of abelian and exact categories to that
of extriangulated categories (see Theorem 3.13). Moreover, we proved that the
homotopy categories of thus constructed model structures are always
triangulated (see Theorem 3.18). This gives many examples of localisations of
triangulated categories that are both triangulated and equivalent to some
ideal subquotient.

_Polytopal realisations of $g$-vector fans_. Inspired by [BMCLD+18], we
revisit in [PPPP19], Arkani-Hamed–Bai–He–Yan’s realisation [AHBHY18] of the
associahedron, that was motivated by the theory of scattering amplitudes in
theoretical physics. We do so by relating ABHY’s construction to the so-called
type cone of Peter McMullen (Definition 4.5). This allows us to generalise
ABHY’s construction and to give all polytopal realisations of $g$-vector fans
for cluster algebras of finite type (Theorem 4.10). Our approach only relies
on two purely combinatorial properties of the type cone (Theorem 4.7) and we
only use representation theory, and extriangulated structures, in order to
prove that those two properties are satisfied in the context of cluster
algebras. Working with extriangulated categories permits to apply this
strategy to other $g$-vector fans: those of (brick and 2-acyclic) gentle
algebras.

_Mutation_. Initially motivated by the additive categorification of cluster
algebras, many instances of mutation have been studied in representation
theory. Classes of objects that enjoy a nice theory of mutation include
cluster-tilting objects, two-term silting objects, relative tilting objects,
maximal almost rigid objects. It turns out that, in all those examples, some
nice hereditary extriangulated structures arise. In [GNP23] (see Section 5),
we introduced the notion of a 0-Auslander extriangulated category and proved
there is a well-behaved theory of mutation for their maximal rigid objects.
Moreover, the following are examples of 0-Auslander extriangulated categories:

  * •

Cluster categories (more generally, 2-Calabi–Yau triangulated categories) with
some cluster tilting object $T$ and endowed with the maximal extriangulated
substructure making $T$ projective. The maximal rigid objects are the cluster
tilting objects.

  * •

The homotopy category of two-term complexes of projectives over a finite-
dimensional algebra. Its maximal rigid objects are the two-term silting
objects.

  * •

The category of representations of a quiver of Dynkin type $A_{n}$ over some
field, endowed with the relative exact structure given by the Auslander–Reiten
sequences with non-indecomposable middle terms. Its maximal rigid objects are
the maximal almost-rirgid objects.

  * •

More examples are given in Examples 5.4 and 5.10.

We thus recover many notions of mutation, and we hope that 0-Auslander
extriangulated structures will be useful when studying new mutation theories
in the future.

## Acknowledgements

I would like to warmly thank the organizers of the Abel Symposium 2022, and
more specifically Petter Andreas Bergh for his encouragements in writing this
survey. The applications described in Sections 3, 4 and 5 were the result of
teamwork; I thank my collaborators for the fun we had in inventing and proving
those results together. Specific thanks to the other three P’s for useful
comments on a previous version of these notes. My deepest and friendliest
thanks go to Hiroyuki Nakaoka, without whom this whole story wouldn’t have
started.

##  1\. The definition of an extriangulated category

###  1.1. Prototypical example

We are interested in the similarities between the following two results:

######  Theorem 1.1 (Happel [Hap88]).

The stable category of a Frobenius exact category is triangulated.

######  Theorem 1.2 (Iyama–Yoshino [IY08]).

Let $\mathcal{R}$ be a functorially finite, rigid subcategory in a
2-Calabi–Yau triangulated category. Then, the subquotient
$\mathcal{R}^{\perp_{1}}/[\mathcal{R}]$ is triangulated.

Here, $\mathcal{R}^{\perp_{1}}$ is the full subcategory whose objects $X$
satisfy $\operatorname{Ext}^{1}(R,X)=0$ or, equivalently by the 2-Calabi–Yau
property, $\operatorname{Ext}^{1}(X,R)=0$. The proof of Theorem 1.2 resembles
that of Theorem 1.1 where exact categories (that are morally full extension-
closed subcategories of abelian categories) are replaced by
$\mathcal{R}^{\perp_{1}}$, which is a full extension-closed subcategory of a
triangulated category; where conflations are replaced by triangles; and where
the use of push-outs or pull-backs is replaced by that of the octahedral
axiom.

One of the motivations for the introduction of extriangulated categories was
to give a framework that would formalise the analogy above. Hence, the class
of extriangulated categories has to contain both exact categories and
extension-closed full subcategories of triangulated categories. Extriangulated
categories should thus come endowed with some class of diagrams that
generalise both conflations $X\rightarrowtail Y\twoheadrightarrow Z$ of exact
categories and triangles $X\to Y\to Z\xrightarrow{\delta}\Sigma X$. However,
for $X,Y,Z$ in some extension-closed subcategory of a triangulated category,
the shift $\Sigma X$ generally does not belong to the subcategory. One might
be tempted to only consider the part $X\to Y\to Z$ of the triangle above, but
this would be a mistake: this complex does not retain enough information to
know which commutative diagrams are morphisms of triangles. We should thus
keep track of the morphism $\delta$, while not remembering the object $\Sigma
X$. This leads us to considering diagrams of the form $X\rightarrowtail
Y\twoheadrightarrow Z\overset{\delta}{\dashrightarrow}$, with

  * •

$\delta\in\mathbb{E}(Z,X)=\operatorname{Ext}^{1}(Z,X)$ where $\delta$
classifies the conflation, if the category is exact, and

  * •

$\delta\in\mathbb{E}(Z,X)=\operatorname{Hom}(Z,\Sigma X)$ is the connecting
morphism in the triangle, if the category is extension-closed in a
triangulated category.

With these considerations in mind, we can now introduce the formal
definitions.

###  1.2. Definitions

Fix an additive category $\mathscr{C}$, and an additive bifunctor
$\mathbb{E}:\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathit{Ab}$, where
$\mathit{Ab}$ is the category of abelian groups.

######  Definition 1.3.

For any $X,Z\in\mathscr{C}$, an element $\delta\in\mathbb{E}(Z,X)$ is called
an $\mathbb{E}$-extension. A split $\mathbb{E}$-extension is a zero element
$0\in\mathbb{E}(Z,X)$, for some objects $X,Z\in\mathscr{C}$. By additivity of
$\mathscr{C}$ and $\mathbb{E}$ we can define, for any two
$\mathbb{E}$-extensions $\delta\in\mathbb{E}(Z,X)$ and
$\delta^{\prime}\in\mathbb{E}(Z^{\prime},X^{\prime})$, the
$\mathbb{E}$-extension

$\delta\oplus\delta^{\prime}\in\mathbb{E}(Z\oplus Z^{\prime},X\oplus
X^{\prime}),$

as being the image of $(\delta,0,0,\delta^{\prime})$ under the natural
isomorphism
$\mathbb{E}(Z,X)\oplus\mathbb{E}(Z^{\prime},X)\oplus\mathbb{E}(Z,X^{\prime})\oplus\mathbb{E}(Z^{\prime},X^{\prime})\cong\mathbb{E}(Z\oplus
Z^{\prime},X\oplus X^{\prime})$.

######  Remark 1.4.

Let $\delta\in\mathbb{E}(Z,X)$ be an $\mathbb{E}$-extension. By functoriality,
any morphisms $f\in\mathscr{C}(X,X^{\prime})$ and
$h\in\mathscr{C}(Z^{\prime},Z)$ induce $\mathbb{E}$-extensions
$f_{\ast}\delta=\mathbb{E}(Z,f)(\delta)$ in $\mathbb{E}(Z,X^{\prime})$ and
$h^{\ast}\delta=\mathbb{E}(h,X)(\delta)$ in $\mathbb{E}(Z^{\prime},X)$. Using
those notations, we have
$\mathbb{E}(h,f)(\delta)=h^{\ast}f_{\ast}\delta=f_{\ast}h^{\ast}\delta$ in
$\mathbb{E}(Z^{\prime},X^{\prime})$.

######  Definition 1.5.

A morphism $(f,h):\delta\to\delta^{\prime}$ of $\mathbb{E}$-extensions from
$\delta\in\mathbb{E}(Z,X)$ to
$\delta^{\prime}\in\mathbb{E}(Z^{\prime},X^{\prime})$ is a pair of morphisms
$f\in\mathscr{C}(X,X^{\prime})$ and $h\in\mathscr{C}(Z,Z^{\prime})$ in
$\mathscr{C}$, such that $f_{\ast}\delta=h^{\ast}\delta^{\prime}$.

######  Example 1.6.

If $\mathscr{C}$ is a triangulated category, and
$\mathbb{E}=\mathscr{C}(-,\Sigma-)$, then
$h^{\ast}\delta^{\prime}=\delta^{\prime}\circ h$, $f_{\ast}\delta=(\Sigma
f)\circ\delta$ and a morphism of extensions is a commutative square:

${Z}$${{\Sigma X}}$${{Z^{\prime}}}$${{\Sigma
X^{\prime}}}$$\scriptstyle{\delta}$$\scriptstyle{h}$$\scriptstyle{\Sigma
f}$$\scriptstyle{\delta^{\prime}}$

which induces a morphism of triangles. If $\mathscr{C}$ is an exact category
then the condition $f_{\ast}\delta=h^{\ast}\delta^{\prime}$ equivalently says
that $(f,h)$ can be completed to a morphism $(f,g,h)$ of conflations [Büh10,
Proposition 3.1].

######  Definition 1.7.

Let $X,Z\in\mathscr{C}$ be any two objects. Two sequences of morphisms in
$\mathscr{C}$

$X\overset{x}{\longrightarrow}Y\overset{y}{\longrightarrow}Z\text{ and
}X\overset{x^{\prime}}{\longrightarrow}Y^{\prime}\overset{y^{\prime}}{\longrightarrow}Z$

are said to be equivalent if there exists an isomorphism
$g\in\mathscr{C}(Y,Y^{\prime})$ such that the following diagram commutes.

$\textstyle{X}$$\textstyle{Y}$$\textstyle{Y^{\prime}}$$\textstyle{Z}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{x^{\prime}}$$\scriptstyle{y^{\prime}}$$\scriptstyle{g}$$\scriptstyle{\cong}$

The equivalence class of $X\xrightarrow{x}Y\xrightarrow{y}Z$ is denoted by
$[X\xrightarrow{x}Y\xrightarrow{y}Z]$.

######  Notation 1.8.

For any $X,Y,Z,A,B,C\in\mathscr{C}$, and any
$[X\xrightarrow{x}Y\xrightarrow{y}Z]$, $[A\xrightarrow{a}B\xrightarrow{b}C]$,
we let

$0=[X\overset{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]}{\longrightarrow}X\oplus
Y\overset{\left[\begin{smallmatrix}0\;1\end{smallmatrix}\right]}{\longrightarrow}Y]$

and

$[X\xrightarrow{x}Y\xrightarrow{y}Z]\oplus[A\xrightarrow{a}B\xrightarrow{b}C]=[X\oplus
A\overset{\left[\begin{smallmatrix}x\;0\\\
0\;a\end{smallmatrix}\right]}{\longrightarrow}Y\oplus
B\overset{\left[\begin{smallmatrix}y\;0\\\
0\;b\end{smallmatrix}\right]}{\longrightarrow}Z\oplus C].$

######  Definition 1.9.

A realization $\mathfrak{s}$ is a correspondence associating, with any
$\mathbb{E}$-extension ${\delta\in\mathbb{E}(Z,X)}$, an equivalence class
$\mathfrak{s}(\delta)=[X\xrightarrow{x}Y\xrightarrow{y}Z]$ and satisfying the
following condition:

  * $(\ast)$

Let $\delta\in\mathbb{E}(Z,X)$ and
$\delta^{\prime}\in\mathbb{E}(Z^{\prime},X^{\prime})$ be any pair of
$\mathbb{E}$-extensions, with

$\mathfrak{s}(\delta)=[X\xrightarrow{x}Y\xrightarrow{y}Z]\text{ and
}\mathfrak{s}(\delta^{\prime})=[X^{\prime}\xrightarrow{x^{\prime}}Y^{\prime}\xrightarrow{y^{\prime}}Z^{\prime}].$

Then, for any morphism $(f,h):\delta\to\delta^{\prime}$, there exists
$g\in\mathscr{C}(Y,Y^{\prime})$ such that the following diagram commutes:

$\textstyle{X}$$\textstyle{Y}$$\textstyle{Z}$$\textstyle{X^{\prime}}$$\textstyle{Y^{\prime}}$$\textstyle{Z^{\prime}}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{x^{\prime}}$$\scriptstyle{y^{\prime}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

The sequence $X\xrightarrow{x}Y\xrightarrow{y}Z$ is said to realize $\delta$
if $\mathfrak{s}(\delta)=[X\xrightarrow{x}Y\xrightarrow{y}Z]$, and the triple
$(f,g,h)$ is said to realize $(f,h)$ if the previous diagram commutes.

######  Definition 1.10.

A realization of $\mathbb{E}$ is called an additive realization if moreover:

  1. (1)

For any $X,Z\in\mathscr{C}$, the realization of the split
$\mathbb{E}$-extension $0\in\mathbb{E}(Z,X)$ is given by $\mathfrak{s}(0)=0$.

  2. (2)

For any two $\mathbb{E}$-extensions $\delta\in\mathbb{E}(Z,X)$ and
$\delta^{\prime}\in\mathbb{E}(Z^{\prime},X^{\prime})$, the realization of
$\delta\oplus\delta^{\prime}$ is given by
$\mathfrak{s}(\delta\oplus\delta^{\prime})=\mathfrak{s}(\delta)\oplus\mathfrak{s}(\delta^{\prime})$.

######  Definition 1.11.

([NP19, Def. 2.12]) A triple $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is called
an extriangulated category if the following holds:

  * (ET1)

$\mathbb{E}:\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathit{Ab}$ is an
additive bifunctor;

  * (ET2)

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

  * (ET3)

Let $\delta\in\mathbb{E}(Z,X)$ and
$\delta^{\prime}\in\mathbb{E}(Z^{\prime},X^{\prime})$ be
$\mathbb{E}$-extensions, respectively realized by
$X\xrightarrow{x}Y\xrightarrow{y}Z$ and
$X^{\prime}\xrightarrow{x^{\prime}}Y^{\prime}\xrightarrow{y^{\prime}}Z^{\prime}$.
Then, for any commutative square

$\textstyle{X}$$\textstyle{Y}$$\textstyle{Z}$$\textstyle{X^{\prime}}$$\textstyle{Y^{\prime}}$$\textstyle{Z^{\prime}}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{x^{\prime}}$$\scriptstyle{y^{\prime}}$$\scriptstyle{\circlearrowright}$

in $\mathscr{C}$, there exists a morphism $(f,h):\delta\to\delta^{\prime}$
satisfying $h\circ y=y^{\prime}\circ g$.

  * (ET3)op

Dual of (ET3).

  * (ET4)

Let $\delta\in\mathbb{E}(Z^{\prime},X)$ and
$\delta^{\prime}\in\mathbb{E}(X^{\prime},Y)$ be $\mathbb{E}$-extensions
realized respectively by

$X\overset{f}{\longrightarrow}Y\overset{f^{\prime}}{\longrightarrow}Z^{\prime}\quad\text{and}\quad
Y\overset{g}{\longrightarrow}Z\overset{g^{\prime}}{\longrightarrow}X^{\prime}.$

Then there exist an object $Y^{\prime}\in\mathscr{C}$, a commutative diagram
in $\mathscr{C}$

${X}$${Y}$${{Z^{\prime}}}$${X}$${Z}$${{\color[rgb]{0.2,0.23921568627451,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.23921568627451,1}Y^{\prime}}}$${{X^{\prime}}}$${{X^{\prime}}}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g}$$\scriptstyle{\definecolor{.}{rgb}{0.2,0.23921568627451,1}\color[rgb]{0.2,0.23921568627451,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.23921568627451,1}d}$$\scriptstyle{\delta}$$\scriptstyle{h}$$\scriptstyle{\definecolor{.}{rgb}{0.2,0.23921568627451,1}\color[rgb]{0.2,0.23921568627451,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.23921568627451,1}h^{\prime}}$$\scriptstyle{\definecolor{.}{rgb}{0.2,0.23921568627451,1}\color[rgb]{0.2,0.23921568627451,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.23921568627451,1}\delta^{\prime\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\delta^{\prime}}$$\scriptstyle{\definecolor{.}{rgb}{0.2,0.23921568627451,1}\color[rgb]{0.2,0.23921568627451,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.2,0.23921568627451,1}e}$$\scriptstyle{f^{\prime}_{\ast}\delta^{\prime}}$

and an $\mathbb{E}$-extension
$\delta^{\prime\prime}\in\mathbb{E}(Y^{\prime},X)$ realized by
$X\overset{h}{\longrightarrow}Z\overset{h^{\prime}}{\longrightarrow}Y^{\prime}$,
which satisfy the following compatibilities.

    1. (i)

$Z^{\prime}\overset{d}{\longrightarrow}Y^{\prime}\overset{e}{\longrightarrow}X^{\prime}$
realizes $f^{\prime}_{\ast}\delta^{\prime}$,

    2. (ii)

$d^{\ast}\delta^{\prime\prime}=\delta$,

    3. (iii)

$f_{\ast}\delta^{\prime\prime}=e^{\ast}\delta^{\prime}$ (equivalently, $(f,e)$
is a morphism from $\delta^{\prime\prime}$ to $\delta^{\prime}$ realised by
$(f,1_{B},e)$).

  * (ET4)op

Dual of (ET4).

######  Remark 1.12.

Note that the dual of an extriangulated category is again extriangulated. Thus
the proof of a statement also implies the dual statement.

######  Terminology 1.13.

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category.

  1. (1)

A sequence $X\xrightarrow{i}Y\xrightarrow{p}Z$ is called a conflation if it
realizes some $\mathbb{E}$-extension in $\mathbb{E}(Z,X)$; in which case the
morphism $X\xrightarrow{i}Y$ is called an inflation, written $X\rightarrowtail
Y$, and the morphism $Y\xrightarrow{p}Z$ is called a deflation, witten
$Y\twoheadrightarrow Z$.

  2. (2)

An extriangle is a diagram
$X\overset{i}{\rightarrowtail}Y\overset{p}{\twoheadrightarrow}Z\overset{\delta}{\dashrightarrow}$
where $X\overset{i}{\rightarrowtail}Y\overset{p}{\twoheadrightarrow}Z$ is a
conflation realizing the $\mathbb{E}$-extension $\delta\in\mathbb{E}(Z,X)$.

  3. (3)

Similarly, we call morphism of extriangles (or morphism of conflations) any
diagram

${X}$${Y}$${Z}$${A}$${B}$${C}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{\delta}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{\zeta}$

where $(f,h):\delta\to\delta^{\prime}$ is a morphism of
$\mathbb{E}$-extensions realized by $(f,g,h)$.

######  Definition 1.14 ([BTS21, Definition 2.32]).

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ and
$(\mathscr{D},\mathbb{F},\mathfrak{t})$ be two extriangulated categories. An
exact functor, or extriangulated functor, from $\mathscr{C}$ to $\mathscr{D}$
is the data of an additive functor $F:\mathscr{C}\to\mathscr{D}$ and of a
natural transformation $\Gamma:\mathbb{E}\to\mathbb{F}(F-,F-)$ such that, for
any extriangle
$X\overset{i}{\rightarrowtail}Y\overset{p}{\twoheadrightarrow}Z\overset{\delta}{\dashrightarrow}$
in $\mathscr{C}$, the diagram
$FX\overset{Fi}{\rightarrowtail}FY\overset{Fp}{\twoheadrightarrow}FZ\overset{\Gamma_{Z,X}(\delta)}{\dashrightarrow}$
is an extriangle in $\mathscr{D}$.

There is also a notion of extriangulated natural transformations.
Extriangulated categories thus form a 2-category (see [BTHSS22, Definition 4.1
and Corollary 4.15] for $n$-exangulated categories), which gives a notion of
extriangulated equivalence.

######  Proposition 1.15 ([NOS22, Proposition 2.13] (see also [BTHSS22,
Proposition 4.11] and [HHZ22, Proposition 2.14])).

An extriangulated funtor $(F,\Gamma):\mathscr{C}\to\mathscr{D}$ is an
extriangulated equivalence if and only if $F$ is an equivalence of categories
and $\Gamma$ is a natural isomorphism.

######  Definition 1.16 ([Hau21, Definition 3.7]).

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category. An
extriangulated subcategory of $\mathscr{C}$ is the data of an extriangulated
category $(\mathscr{C}^{\prime},\mathbb{E}^{\prime},\mathfrak{s}^{\prime})$
and an exact functor $(F,\Gamma):\mathscr{C}^{\prime}\to\mathscr{C}$ such that
$\mathscr{C}^{\prime}$ is a strictly full subcategory of $\mathscr{C}$, $F$ is
the inclusion functor and for all $X,Z\in\mathscr{C}^{\prime}$, $\Gamma_{Z,X}$
is an inclusion.

######  Example 1.17.

Any strictly full, extension-closed subcategory of an extriangulated category
is canonically an extriangulated subcategory (see Section 2.1.3 below). In
particular, given a $t$-structure on a triangulated category $\mathscr{T}$,
its heart is an extriangulated subcategory of $\mathscr{T}$. And given a
co-$t$-structure on $\mathscr{T}$, its coheart and its extended coheart are
extriangulated subcategories. By [Hau21, Proposition 3.9 and Theorem 5.1],
there is a bijection between the subgroups of the Grothendieck group of an
extriangulated category and certain of its extriangulated subcategories (that
are called complete and dense). Many more examples of extriangulated
subcategories and of extriangulated functors are given in [BTHSS22, Section
5].

###  1.3. First properties

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category.

The axioms above ensure that any extriangle induce six-term exact sequences
after application of some covariant or contravariant Hom-functor. In
particular in any conflation, the inflation is a weak kernel of the deflation,
and the deflation is a weak cokernel of the inflation.

######  Proposition 1.18 ([NP19, Propositions 3.3 & 3.11]).

Assume that
$X\xrightarrow{x}Y\xrightarrow{y}Z\overset{\delta}{\dashrightarrow}$ is an
extriangle. Then the following sequences of natural transformations are exact:

$\mathscr{C}(Z,-)\overset{-\circ
y}{\longrightarrow}\mathscr{C}(Y,-)\overset{-\circ
x}{\longrightarrow}\mathscr{C}(X,-)\overset{\delta^{\sharp}}{\longrightarrow}\mathbb{E}(Z,-)\overset{y^{\ast}}{\longrightarrow}\mathbb{E}(Y,-)\overset{x^{\ast}}{\longrightarrow}\mathbb{E}(X,-),$
$\mathscr{C}(-,X)\overset{x\circ-}{\longrightarrow}\mathscr{C}(-,Y)\overset{y\circ-}{\longrightarrow}\mathscr{C}(-,Z)\overset{\delta_{\sharp}}{\longrightarrow}\mathbb{E}(-,X)\overset{x_{\ast}}{\longrightarrow}\mathbb{E}(-,Y)\overset{y_{\ast}}{\longrightarrow}\mathbb{E}(-,Z),$

where $\delta^{\sharp}(f)=f_{\ast}\delta$ and
$\delta_{\sharp}(g)=g^{\ast}\delta$.

The following, and its dual, are often used as replacements for the non-
existent push-outs or pull-backs in extriangulated categories:

######  Proposition 1.19 ([Che23, Lemma A.11]).

Any two extriangles
$A\overset{i}{\rightarrowtail}B\overset{p}{\twoheadrightarrow}C\overset{\delta}{\dashrightarrow}$
and
$A\overset{i^{\prime}}{\rightarrowtail}B^{\prime}\overset{p^{\prime}}{\twoheadrightarrow}C^{\prime}\overset{\delta^{\prime}}{\dashrightarrow}$
with same source can be completed to a commutative diagram of extriangles

${A}$${B}$${C}$${{B^{\prime}}}$${{\color[rgb]{0.36078431372549,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}E}}$${C}$${{C^{\prime}}}$${{C^{\prime}}}$$\scriptstyle{i^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{\delta^{\prime}}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{\delta}$$\scriptstyle{\definecolor{.}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}\color[rgb]{0.36078431372549,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}j^{\prime}}$$\scriptstyle{\definecolor{.}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}\color[rgb]{0.36078431372549,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}q^{\prime}}$$\scriptstyle{i^{\prime}_{\ast}\delta}$$\scriptstyle{\definecolor{.}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}\color[rgb]{0.36078431372549,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}j}$$\scriptstyle{\definecolor{.}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}\color[rgb]{0.36078431372549,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}q}$$\scriptstyle{i_{\ast}\delta^{\prime}}$

where $q^{\ast}\delta^{\prime}=-(q^{\prime})^{\ast}\delta$ and
$A\overset{\left[{}^{\;\;i}_{-i^{\prime}}\right]}{\rightarrowtail}B\oplus
B^{\prime}\overset{[j\;j^{\prime}]}{\twoheadrightarrow}E\overset{q^{\ast}\delta^{\prime}}{\dashrightarrow}$
is an extriangle.

######  Definition 1.20.

An object $P\in\mathscr{C}$ is called projective if, for any
$X\in\mathscr{C}$, we have $\mathbb{E}(P,X)=0$. The extriangulated category
$\mathscr{C}$ has enough projectives if, for any $X\in\mathscr{C}$, there is a
deflation $P\twoheadrightarrow X$ with $P$ projective. There are obvious dual
notions for injectives.

######  Definition 1.21.

An extriangulated category $\mathscr{C}$ is Frobenius if it has enough
projectives, enough injectives and its injective and projective objects
coincide. The stable category $\underline{\mathscr{C}}$ of a Frobenius
extriangulated category is the ideal quotient by those morphisms that factor
through some projective object.

Any Frobenius exact category is Frobenius as an extriangulated category. If
$\mathcal{R}$ is a functorially finite rigid subcategory of a 2-Calabi–Yau
triangulated category $\mathscr{T}$, then
$\mathscr{C}=\mathcal{R}^{\perp_{1}}$ is a Frobenius extriangulated category.
Indeed, $\mathscr{C}$ is the largest extension-closed full subcategory of
$\mathscr{T}$ containing $\mathcal{R}$ and for which each object in
$\mathcal{R}$ is both projective and injective. Because $\mathcal{R}$ is
functorially finite, $\mathscr{C}$ has enough projectives and enough
injectives. Therefore, the following proposition unifies Theorems 1.1 and 1.2
mentionned in Section 1.1.

######  Proposition 1.22 ([NP19, Remark 7.5]).

If $\mathscr{C}$ is a Frobenius extriangulated category, then its stable
category is triangulated and the quotient functor is exact.

###### Proof.

The triangulated structure is constructed as follows: For each
$X\in\mathscr{C}$, fix an extriangle

$X\overset{\iota_{X}}{\rightarrowtail}I_{X}\overset{\pi_{X}}{\twoheadrightarrow}\Sigma
X\overset{\delta_{X}}{\dashrightarrow}$

with $I_{X}$ injective. For any morphism $X\xrightarrow{f}Y$, there is a
morphism of extriangles

${X}$${{I_{X}}}$${{\Sigma X}}$${Y}$${{I_{Y}}}$${{\Sigma
Y}}$$\scriptstyle{\delta_{X}}$$\scriptstyle{\delta_{Y}}$$\scriptstyle{f}$$\scriptstyle{\sigma_{f}}$

and letting $\Sigma\underline{f}$ be $\underline{\sigma_{f}}$ defines an
autoequivalence $\Sigma:\underline{\mathscr{C}}\to\underline{\mathscr{C}}$
which, up to natural isomorphism, does not depend on the choices made above.
For each extension $\sigma\in\mathbb{E}(Z,X)$, there is a morphism of
extriangles

${{(\ast\ast)}}$${X}$${Y}$${Z}$${X}$${{I_{X}}}$${{\Sigma
X}}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{\delta_{X}}$$\scriptstyle{d}$$\scriptstyle{\delta}$

and the triangles in $\underline{\mathscr{C}}$ are all those diagrams that are
isomorphic to some diagram of the form
$X\xrightarrow{\underline{i}}Y\xrightarrow{\underline{p}}Z\xrightarrow{\underline{d}}\Sigma
X$, with $i,p,d$ as in $(\ast\ast)$ above.

That $\underline{\mathscr{C}}$ is triangulated is an easy adaptation of
Happel’s proof for exact categories. An alternative proof using homotopical
algebra, which thus only applies to weakly idempotent complete Frobenius
extriangulated categories, can be found in [NP19, Corollary 7.4]. We thus only
prove that the quotient functor is exact. Define
$\Gamma_{Z,X}:\mathbb{E}(Z,X)\to\underline{\mathscr{C}}(Z,\Sigma X)$, by
$\Gamma_{Z,X}(\delta)=\underline{d}$ with the notation of $(\ast\ast)$. The
diagram

${X}$${{Y^{\prime}}}$${Z}$${X}$${Y}$${Z}$${X}$${{I_{X}}}$${{\Sigma
X}}$$\scriptstyle{i}$$\scriptstyle{p}$$\scriptstyle{\delta_{X}}$$\scriptstyle{d}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{i^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{\cong}$

shows that $\underline{d}$ does not depend on the choice of a realisation of
$\delta$. It only remains to prove that $\Gamma$ is a natural transformation.
Let $X\xrightarrow{x}X^{\prime}$ and $Z^{\prime}\xrightarrow{z}Z$ be morphisms
in $\mathscr{C}$. We have to show that the two diagrams

${{(1)}}$${{\mathbb{E}(Z,X)}}$${{\underline{\mathscr{C}}(Z,\Sigma
X)}}$${{(2)}}$${{\mathbb{E}(Z,X)}}$${{\underline{\mathscr{C}}(Z,\Sigma
X)}}$${{\mathbb{E}(Z,X^{\prime})}}$${{\underline{\mathscr{C}}(Z,\Sigma
X^{\prime})}}$${{\mathbb{E}(Z^{\prime},X)}}$${{\underline{\mathscr{C}}(Z^{\prime},\Sigma
X)}}$$\scriptstyle{\Gamma_{Z,X}}$$\scriptstyle{x_{\ast}}$$\scriptstyle{\Sigma\underline{x}\circ-}$$\scriptstyle{\Gamma_{Z,X^{\prime}}}$$\scriptstyle{\Gamma_{Z,X}}$$\scriptstyle{z^{\ast}}$$\scriptstyle{-\circ\underline{z}}$$\scriptstyle{\Gamma_{Z^{\prime},X}}$

commute. Choose morphisms of extriangles

${X}$${Y}$${Z}$${X}$${Y}$${Z}$${{(a)}}$${X}$${{I_{X}}}$${{\Sigma
X}}$${{(b)}}$${{X^{\prime}}}$${{Y^{\prime}}}$${Z}$${{X^{\prime}}}$${{I_{X^{\prime}}}}$${{\Sigma
X^{\prime}}}$${{X^{\prime}}}$${{I_{X^{\prime}}}}$${{\Sigma
X^{\prime}}}$${X}$${{I_{X}}}$${{\Sigma
X}}$${{(c)}}$${X}$${E}$${{Z^{\prime}}}$${X}$${Y}$${Z}$$\scriptstyle{x}$$\scriptstyle{d_{1}}$$\scriptstyle{\sigma_{x}}$$\scriptstyle{\delta}$$\scriptstyle{\delta_{X}}$$\scriptstyle{\delta_{X^{\prime}}}$$\scriptstyle{x}$$\scriptstyle{\delta}$$\scriptstyle{d_{2}}$$\scriptstyle{x_{\ast}\delta}$$\scriptstyle{\delta_{X^{\prime}}}$$\scriptstyle{z}$$\scriptstyle{d_{3}}$$\scriptstyle{\delta_{X}}$$\scriptstyle{z^{\ast}\delta}$$\scriptstyle{\delta}$

By definition, we have
$\Gamma_{Z,X^{\prime}}(x_{\ast}\delta)=\underline{d_{2}}$,
$\Sigma\underline{x}\circ\Gamma_{Z,X}(\delta)=\underline{\sigma_{x}\circ
d_{1}}$, $\Gamma_{Z,X}(\delta)\circ\underline{z}=\underline{d_{1}\circ z}$ and
$\Gamma_{Z^{\prime},X}(z^{\ast}\delta)=\underline{d_{3}}$. Because the
diagrams (a) and (b) are morphisms of extriangles, we have:

$d_{2}^{\ast}\delta_{X^{\prime}}=x_{\ast}\delta=x_{\ast}d_{1}^{\ast}\delta_{X}=d_{1}^{\ast}x_{\ast}\delta_{X}=d_{1}^{\ast}\sigma_{x}^{\ast}\delta_{X^{\prime}}=(\sigma_{x}d_{1})^{\ast}\delta_{X^{\prime}}.$

Hence, $(d_{2}-\sigma_{x}d_{1})^{\ast}\delta_{X^{\prime}}=0$. By the long
exact sequence of Proposition 1.18, this implies that $d_{2}-\sigma_{x}d_{1}$
factors through $I_{X^{\prime}}$. We thus have
$\Gamma_{Z,X^{\prime}}(x_{\ast}\delta)=\underline{d_{2}}=\underline{\sigma_{x}\circ
d_{1}}=\Sigma\underline{x}\circ\Gamma_{Z,X}(\delta)$ and (1) commutes.
Similarly, it follows from the equalities

$d_{3}^{\ast}\delta_{X}=z^{\ast}\delta=z^{\ast}d_{1}^{\ast}\delta_{X}=(d_{1}z)^{\ast}\delta_{X}$

that (2) also commutes. ∎

We conclude this section by recalling an extriangulated version of the nine
lemma.

######  Proposition 1.23 ([NP19, Lemma 5.9]).

Assume that $\mathscr{C}$ is weakly idempotent complete, i.e. that all
sections have a kernel. Then any commutative square for which two opposite
edges are inflations and the other two are deflations can be completed to a
diagram made of morphisms of conflations of the form:

${K}$${L}$${{\color[rgb]{0.36078431372549,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}X}}$${A}$${B}$${C}$${{A^{\prime}}}$${{B^{\prime}}}$${{C^{\prime}}}$

######  Remark 1.24.

By Remark 1.12, the duals of Propositions 1.19 and 1.23 also hold.

##  2\. Examples of extriangulated categories

###  2.1. Some examples arising in representation theory

####  2.1.1. Exact categories

Let $\mathscr{E}$ be an exact category such that, for any $X,Z\in\mathscr{E}$,
$\operatorname{Ext}^{1}(Z,X)$ is a set. This is true for example if
$\mathscr{E}$ is small, or if it has enough injectives or enough projectives.
We let $\mathbb{E}$ be the bifunctor $\operatorname{Ext}^{1}$, and define
$X\overset{i}{\rightarrowtail}Y\overset{p}{\twoheadrightarrow}Z\overset{\delta}{\dashrightarrow}$
to be an extriangle if and only if $(i,p)$ is a conflation in $\mathscr{E}$
whose equivalence class in $\operatorname{Ext}^{1}(Z,X)$ is $\delta$. This
gives $\mathscr{E}$ the structure of an extriangulated category whose
conflations are precisely those of the exact structure on $\mathscr{E}$.

Conversely, it is shown in [NP19, Corollary 3.18] that if
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ is an extriangulated category all of
whose inflations are monomorphisms and all of whose deflations are
epimorphisms, then its conflations form an exact structure on $\mathscr{C}$.

####  2.1.2. Triangulated categories

Let $(\mathscr{T},\Sigma,\Delta)$ be a triangulated category. For all
$X,Z\in\mathscr{T}$, let $\mathbb{E}(Z,X)=\mathscr{T}(Z,\Sigma X)$ and define
$X\overset{i}{\rightarrowtail}Y\overset{p}{\twoheadrightarrow}Z\overset{\delta}{\dashrightarrow}$
to be an extriangle if and only if
$X\xrightarrow{i}Y\xrightarrow{p}Z\xrightarrow{\delta}\Sigma X$ is a triangle
in $\Delta$. This makes $\mathscr{T}$ into an extriangulated category whose
extriangles canonically identify with the triangles in $\Delta$.

Conversely, any extriangulated category $\mathscr{C}$ for which $\mathbb{E}$
is given by $\mathscr{C}(-,\Sigma-)$ for some autoequivalence
$\Sigma:\mathscr{C}\to\mathscr{C}$ is triangulated, with triangles given by
the extriangles as above [NP19, Proposition 3.22].

####  2.1.3. Extension-closed subcategories

If $\mathscr{T}$ is a triangulated category and if $\mathscr{C}$ is a full,
extension-closed subcategory, then one can endow $\mathscr{C}$ with the
bifunctor
$\mathbb{E}=\mathscr{T}(-,\Sigma-)|_{\mathscr{C}^{\text{op}}\times\mathscr{C}}$
and define extriangles as in Section 2.1.2. Then $\mathscr{C}$ becomes
extriangulated. Some form of converse holds for topological extriangulated
categories ([Kle22], see also [Che23] for algebraic extriangulated categories,
or Section 2.2).

An important example is the homotopy category of two-term complexes: If
$\Lambda$ is an Artin algebra, the full subcategory
$K^{[-1,0]}(\operatorname{proj}\Lambda)$ of the perfect derived category
$K^{b}(\operatorname{proj}\Lambda)$ consisting of complexes concentrated in
degree -1 and 0 (with cohomological conventions) is extension-closed, hence
extriangulated.

More generally, any extriangulated structure restricts to extension-closed,
full subcategories by [NP19, Remark 2.18].

####  2.1.4. Relative structures

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category.

######  Definition 2.1 ([HLN21, Proposition 3.16]).

A biadditive subfunctor $\mathbb{F}\subseteq\mathbb{E}$ is closed if
$(\mathscr{C},\mathbb{F},\mathfrak{s}|_{\mathbb{F}})$ is extriangulated.

If $\mathbb{F}$ is a closed subfunctor of $\mathbb{E}$, then
$(\mathscr{C},\mathbb{F},\mathfrak{s}|_{\mathbb{F}})$ is an extriangulated
subcategory of $(\mathscr{C},\mathbb{E},\mathfrak{s})$. If $\mathbb{F}$ is any
biadditive subfunctor of $\mathbb{E}$, then a conflation is called an
$\mathfrak{s}|_{\mathbb{F}}$-conflation if it realises an element
$\delta\in\mathbb{F}(Z,X)$, for some $X,Z\in\mathscr{C}$.

######  Proposition 2.2 ([HLN21, Proposition 3.11 & Lemma 3.15]).

Let $\mathbb{F}\subseteq\mathbb{E}$ be a biadditive subfunctor. Then
$\mathbb{F}$ is closed if and only if $\mathfrak{s}|_{\mathbb{F}}$-inflations
are stable under composition if and only if
$\mathfrak{s}|_{\mathbb{F}}$-deflations are stable under composition.

######  Example 2.3.

Let $\mathscr{T}$ be a $\mathbb{K}$-linear, Hom-finite, 2-Calabi–Yau
triangulated category with a cluster tilting object $T$, and let
$\mathbb{E}=\mathscr{T}(-,\Sigma-)$. Define $\mathbb{F}=[\Sigma
T](-,\Sigma-)$, where $[\Sigma T]$ denotes the ideal of morphisms factoring
through $\operatorname{add}\Sigma T$. Then
$\mathfrak{s}|_{\mathbb{F}}$-deflations are stable under composition because
$\mathscr{T}(T,f)=0$ if and only if $f\in[\Sigma T]$. Thus $\mathbb{F}$ is a
closed subfunctor and $(\mathscr{T},\mathbb{F},\mathfrak{s}|_{\mathbb{F}})$ is
extriangulated. This extriangulated structure will appear in Section 4 where
it is used to categorify $g$-vectors of cluster algebras.

######  Proposition 2.4 ([HLN21, Proposition 3.19]).

Let $\mathcal{R}$ be a full additive subcategory of $\mathscr{C}$, and define
$\mathbb{E}_{\mathcal{R}}(Z,X)=\\{\delta\in\mathbb{E}(Z,X)\;|\;\forall\,R\in\mathcal{R},(\delta_{\sharp})_{R}=0\\}$.
Then $\mathbb{E}_{\mathcal{R}}$ is a closed subfunctor of $\mathbb{E}$ and
gives the biggest extriangulated substructure for which all objects in
$\mathcal{R}$ are projective. There is a dual version, denoted by
$\mathbb{E}^{\mathcal{R}}$.

The closed subfunctor $\mathbb{F}$ of Example 2.3 coincides with
$\mathbb{E}_{\mathcal{R}}$, where $\mathcal{R}=\operatorname{add}T$.

######  Example 2.5.

Let $\mathcal{R}$ be a rigid, full, additive subcategory of some triangulated
category $\mathscr{T}$. Then, $\mathcal{R}\ast\Sigma\mathcal{R}$ might not be
extension-closed in $\mathscr{T}$. However, it is extension-closed in
$(\mathscr{T},\mathbb{E}_{\mathcal{R}})$, whence also extriangulated.

####  2.1.5. Quotients by projectives-injectives

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category, and
let $\mathcal{Q}$ be a full subcategory each of whose object is projective-
injective. Because $\mathbb{E}$ vanishes on
$\mathcal{Q}^{\text{op}}\times\mathscr{C}$ and on
$\mathscr{C}^{\text{op}}\times\mathcal{Q}$, it induces a bifunctor
$\mathbb{E}:(\mathscr{C}/[\mathcal{Q}])^{\text{op}}\times(\mathscr{C}/[\mathcal{Q}])\to
Ab$. For $X,Z\in\mathscr{C}/[\mathcal{Q}]$ and $\delta\in\mathbb{E}(Z,X)$ let
$\mathfrak{t}(\delta)=[X\xrightarrow{\underline{i}}Y\xrightarrow{\underline{p}}Z]$,
where $\mathfrak{s}(\delta)=[X\xrightarrow{i}Y\xrightarrow{p}Z]$ and
$\underline{f}$ denotes the class of a morphism $f\in\mathscr{C}$ in
$\mathscr{C}/[\mathcal{Q}]$. Then
$(\mathscr{C}/[\mathcal{Q}],\mathbb{E},\mathfrak{t})$ is extriangulated [NP19,
Proposition 3.30].

This construction is useful for additive categorification of cluster algebras
with coefficients. Indeed, acyclic cluster algebras with principal
coefficients cannot be categorified by Frobenius exact categories (see [FK10,
Theorem 5.6] and [Pre17, Proposition 2.3] for precise statements). However,
Matthew Pressland constructs a Frobenius extriangulated category, obtained by
killing half of the projective-injectives of a Frobenius exact category and
proves that it categorifies acyclic cluster algebras with principal
coefficients [Pre17, Corollary 1].

####  2.1.6. Categories of Cohen–Macaulay dg modules

Let $A$ be dg algebra over a field, and assume that $A$ is connective
($\operatorname{H}^{>0}(A)=0$), proper (its cohomology is of finite total
dimension) and Gorenstein (the thick subcategory of the derived category
$\mathcal{D}A$ generated by the dual of $A$ coincide with the perfect derived
category $\operatorname{per}A$ of $A$).

######  Definition 2.6 ([Jin20, Definition 2.1]).

A dg $A$-module $M$ is Cohen-Macaulay if $M\in\mathcal{D}^{\text{b}}A$,
$\operatorname{H}^{>0}(M)=0$ and $\operatorname{Hom}_{\mathcal{D}A}(M,A[i])=0$
for all $i>0$.

######  Theorem 2.7 ([Jin20, Theorem 2.4]).

The category $\operatorname{CM}A$ of Cohen-Macaulay dg-modules is a Frobenius
extriangulated category with subcategory of projective-injectives
$\operatorname{add}A$. Moreover, there is a triangle equivalence
$\underline{\operatorname{CM}}A\simeq\mathcal{D}^{\text{b}}A/\operatorname{per}A.$

####  2.1.7. Higgs categories

In [Wu21], Yilin Wu introduces the relative cluster category
$\mathcal{C}_{n}(A,B)$ associated with a morphism $f:B\to A$ of dg algebras
satisfying some technical conditions (involving the positive integer $n$). It
is related to Amiot–Guo’s cluster categories via the exact sequence of
triangulated categories

$0\to\operatorname{per}(eAe)\to\mathcal{C}_{n}(A,B)\to\mathcal{C}_{n}(\overline{A})\to
0$

where $e=f(1_{B})$, $\overline{A}$ is the homotopy cofiber of $f$ and
$\mathcal{C}_{n}(\overline{A})$ is Amiot–Guo’s generalised $n$-Calabi–Yau
cluster category.

######  Definition 2.8 ([Wu21, Theorem 5.42]).

The Higgs category $\mathscr{H}$ is the full subcategory
${}^{\perp_{>0}}(eA)\cap(eA)^{\perp_{>0}}$ of $\mathcal{C}_{n}(A,B)$.

######  Theorem 2.9 ([Wu21, Theorem 5.46]).

The Higgs category $\mathscr{H}$ is a Frobenius extriangulated category with
subcategory of projective-injectives $\operatorname{add}(eA)$. Moreover, its
stable category is triangle equivalent to $\mathcal{C}_{n}(\overline{A})$.

###  2.2. Enhancements

We briefly mention two enhancements for extriangulated categories that are
known so far. They would deserve more details and we refer the interested
reader to [Bar15, Kle22, BT21] for exact $\infty$-categories and [Che23] for
exact dg categories.

####  2.2.1. Topological extriangulated categories

Clark Barwick introduced an analogue of exact categories for
$\infty$-categories in [Bar15] (see also [DK19] for the non-additive case) by
interpreting the axioms in an infinity-categorical meaning. For example,
kernel-cokernel pairs should be interpreted as fiber-cofiber sequences.

Exact $\infty$-categories can be seen as a form of enhancement for
extriangulated categories:

######  Theorem 2.10 ([NP20, Kle22]).

Let $\mathscr{E}$ be a small exact $\infty$-category. Then its homotopy
category is naturally extriangulated.

An extriangulated category which is equivalent to the homotopy category of an
exact $\infty$-category is called a topological extriangulated category. The
following result is an $\infty$-categorical version of the Gabriel–Quillen
embedding theorem.

######  Theorem 2.11 ([Kle22]).

Let $\mathscr{E}$ be a small exact $\infty$-category. Then there is a stable
$\infty$-category $\mathcal{H}^{\text{st}}(\mathscr{E})$ and a fully faithful
functor $\mathscr{E}\to\mathcal{H}^{\text{st}}(\mathscr{E})$ exhibiting (the
essential image of) $\mathscr{E}$ as a full, extension-closed subcategory.
Moreover, a diagram in $\mathscr{E}$ is a conflation if and only if its image
in $\mathcal{H}^{\text{st}}(\mathscr{E})$ is a fiber-cofiber sequence.

In fact, $\mathcal{H}^{\text{st}}$ defines a functor that is left adjoint to
the inclusion functor of stable $\infty$-categories into exact
$\infty$-categories, and the functor
$\mathscr{E}\to\mathcal{H}^{\text{st}}(\mathscr{E})$ is the unit of the
adjunction. The stable hull enjoys some universal property: For any stable
$\infty$-category $\mathscr{C}$, the unit induces an equivalence between the
$\infty$-categories of exact functors
$\operatorname{Fun}^{\text{ex}}(\mathcal{H}^{\text{st}}(\mathscr{E}),\mathscr{C})$
and $\operatorname{Fun}^{\text{ex}}(\mathscr{E},\mathscr{C})$.

######  Corollary 2.12.

Any topological extriangulated category is extension-closed in some
triangulated category.

######  Remark 2.13.

In [BT21], Erlend Due Børve and Paul Trygsland define an $\Omega$-spectrum of
higher extensions for exact $\infty$-categories and show that it enhances
higher extensions of extriangulated categories.

####  2.2.2. Algebraic extriangulated categories

An algebraic version of Theorem 2.11 was proven in Xiaofa Chen’s PhD thesis
[Che23]. He defines exact dg categories by interpreting the axioms of an exact
category in the dg setting. In particular, conflations do not compose to zero,
but come equipped with a specified null-homotopy (see [Che23, Definition 3.49
and Definition 5.1] for more details). Exact dg categories are algebraic
analogues of exact $\infty$-categories and enhance extriangulated categories.

######  Theorem 2.14 ([Che23, Theorem 6.19]).

Let $\mathscr{A}$ be an exact dg category. Then its homotopy category
$\operatorname{H}^{0}\\!\mathscr{A}$ is canonically extriangulated.

An extriangulated category which is equivalent to the homotopy category of an
exact dg category is called an algebraic extriangulated category.

######  Proposition 2.15 ([Che23, Definition-Proposition 6.20]).

For an extriangulated category $\mathscr{C}$, the following are equivalent:

  1. (i)

It is an algebraic extriangulated category.

  2. (ii)

It is equivalent to an extension-closed, full subcategory of an algebraic
triangulated category.

  3. (iii)

It is the quotient of an exact category by some class of projective-injective
objects.

The proof of the first implication is based on:

######  Theorem 2.16 ([Che23, Theorem 6.1]).

Let $\mathscr{A}$ be a connective exact dg category. Then there is a
pretriangulated dg category $\mathcal{D}^{b}_{dg}(\mathscr{A})$ and an exact
morphism $\mathscr{A}\to\mathcal{D}^{b}_{dg}(\mathscr{A})$ (in the homotopy
category of the category of small dg categories for the Dwyer–Kan model
structure) inducing a quasi-equivalence between $\tau_{\leq 0}\mathscr{A}$ and
$\tau_{\leq 0}\mathscr{D}$, for some extension-closed dg subcategory
$\mathscr{D}$ of $\mathcal{D}^{b}_{dg}(\mathscr{A})$.

A consequence of the following result is that any algebraic extriangulated
category is topological extriangulated:

######  Theorem 2.17 ([Che23, Theorem 5.26]).

Let $\mathscr{A}$ be an exact dg category. Then its dg nerve
$N_{dg}(\mathscr{A})$ inherits the structure of an exact $\infty$-category.

###  2.3. Localisation of extriangulated categories

A theory of localisation for extriangulated categories is developed in
[NOS22]. Given an extriangulated category
$(\mathscr{C},\mathbb{E},\mathfrak{s})$, and a class $\mathcal{W}$ of
morphisms in $\mathscr{C}$ to be inverted, Theorem 3.5 in [NOS22] gives
explicit conditions so that the localisation $\mathscr{C}[\mathcal{W}^{-1}]$
inherits an extriangulated structure from $\mathscr{C}$. Its extriangles are
certain equivalence classes of diagrams of the form

$A\xrightarrow{w}X\overset{i}{\rightarrowtail}Y\overset{p}{\twoheadrightarrow}Z\xleftarrow{w^{\prime}}C\overset{\delta}{\dashrightarrow},$

where $w,w^{\prime}\in\mathcal{W}$, and
$X\overset{i}{\rightarrowtail}Y\overset{p}{\twoheadrightarrow}Z\overset{\delta}{\dashrightarrow}$
is an extriangle in $\mathscr{C}$.

The following specific case applies to many localisations arising in practice.

######  Definition 2.18.

A full subcategory $\mathcal{N}$ of $\mathscr{C}$ is thick if it is closed
under summands and isomorphisms and if, for any conflation $X\rightarrowtail
Y\twoheadrightarrow Z$ with any two of $X,Y,Z$ in $\mathcal{N}$, the third
also belongs to $\mathcal{N}$. A thick subcategory $\mathcal{N}$ is
biresolving if every object $X\in\mathscr{C}$ admits a deflation
$N\twoheadrightarrow X$ and an inflation $X\rightarrowtail N^{\prime}$ with
$N,N^{\prime}\in\mathcal{N}$. A thick subcategory is percolating if any
morphism that factors through $\mathcal{N}$ can be factored as a deflation
with codomain in $\mathcal{N}$ followed by an inflation.

Fix a thick subcategory $\mathcal{N}$ of $\mathscr{C}$ and define
$\mathcal{W}_{\mathcal{N}}$ to be the class of morphisms in $\mathscr{C}$ that
are finite compositions of inflations with cones in $\mathcal{N}$ and
deflations with cocones in $\mathcal{N}$.

######  Theorem 2.19 ([NOS22, Corollary 4.27]).

If the thick subcategory is biresolving, then the localisation
$\mathscr{C}[\mathcal{W_{\mathcal{N}}}^{-1}]$ is triangulated.

This result generalises Verdier quotients for triangulated categories and
[Rum21, Theorem 5] for exact categories. A version of this result at the level
of exact $\infty$-categories follows from the work in progress [JKPW23],
mentionned in Remark 3.15.

######  Theorem 2.20 ([NOS22, Corollary 4.27]).

If the thick subcategory is percolating (and under some mild condition which
is always satisfied if $\mathscr{C}$ is weakly idempotent complete, or if it
is exact or triangulated), then the localisation
$\mathscr{C}[\mathcal{W_{\mathcal{N}}}^{-1}]$ is extriangulated.

Moreover, when $\mathcal{N}$ is percolating, [NOS22, Corollary 4.42] gives
sufficient condition for the localisation to be exact or even abelian. Theorem
2.20 thus generalises Serre quotients for abelian categories and [HvR19,
Theorem 8.1], attributed to Manuel Cardenas.

##  3\. Hovey’s correspondence

Hovey’s correspondence [Hov02, Hov07] is a device for constructing model
category structures on abelian categories. It is inspired by the somewhat
canonical model category structure on a Frobenius category, but with two
cotorsion pairs mimicking the role played by the projectives and the
injectives. A similar, but weaker correspondence, due to Apostolos Beligiannis
and Idun Reiten and involving only one cotorsion pair appeared independently
in [BR07, Theorem 4.2] (we note however, that their Theorem 3.5 suggests that
some stronger statement should hold). Hovey’s correspondence was generalised
to exact categories by James Gillespie [Gil11, Gil15, Gil16]. He also
discovered that this correspondence was a powerful tool for constructing model
structures on categories of complexes, with the derived category as its
homotopy category. The strategy of passing from a cotorsion pair on an exact
category to a model structure on the category of complexes has been
considered, and beautifully generalised, by Henrik Holm and Peter Jørgensen
[HJr19]. In that article, the category of complexes is thought of as a
category of representations of the $A^{\infty}_{\infty}$ quiver with radical
square zero relations. This point of view allow them to generalise Gillespie’s
result to categories of representations of self-injective bound quivers, thus
including categories of $n$-term complexes, of periodic complexes and many
more. We also note the use of Gillespie’s result in order to define the stable
category of an arbitrary ring in [BGH14]

Hovey’s correpondence has also been generalised in a different direction: the
proof generalises to triangulated categories, with short exact sequences being
replaced by triangles [Yan15]. Not so surprisingly, it also generalises to
extriangulated categories.

###  3.1. Exact model structures

A model structure is equivalently described as two “intertwined” weak
factorisation systems.

######  Definition 3.1.

A weak factorisation system on a category $\mathscr{C}$ is a pair
$(\mathcal{L},\mathcal{R})$ of classes of morphisms in $\mathscr{C}$
satisfying:

  1. (1)

The classes $\mathcal{L}$ and $\mathcal{R}$ are stable under retracts;

  2. (2)

Any morphism $f\in\mathscr{C}$ factorises as $f=r\circ l$, for some
$r\in\mathcal{R}$ and $l\in\mathcal{L}$;

  3. (3)

The classes $\mathcal{L}$ and $\mathcal{R}$ are weakly orthogonal in the sense
that any commutative square

${A}$${X}$${B}$${Y}$$\scriptstyle{l}$$\scriptstyle{\alpha}$$\scriptstyle{r}$$\scriptstyle{\beta}$$\scriptstyle{h}$

with $l\in\mathcal{L}$ and $r\in\mathcal{R}$, admits a lifting $h$ making both
triangles commute.

######  Notation 3.2.

We write $l\,\square\,r$ if, for any morphisms $\alpha,\beta$ such that the
square above commutes, there is a lifting $h$. We also write
$\mathcal{L}\,\square\,\mathcal{R}$ when $l\,\square\,r$ for any
$(l,r)\in\mathcal{L}\times\mathcal{R}$.

The following lemma shows a strong relation between lifting properties and
$\operatorname{Ext}^{1}$-vanishing conditions (see [Šťo13, Lemma 5.6] for a
converse, and more precise, statement).

######  Lemma 3.3.

Let $X\stackrel{{\scriptstyle i}}{{\rightarrowtail}}Y\twoheadrightarrow Z$,
$A\rightarrowtail B\stackrel{{\scriptstyle p}}{{\twoheadrightarrow}}C$ be two
conflations in an extriangulated category. If $\mathbb{E}(Z,A)=0$, then $i$
has the left lifting property with respect to $p$.

###### Proof.

It follows from a straightforward diagram chase in the diagram:

${{\mathbb{E}(Z,A)}}$${{(Y,A)}}$${{(Y,B)}}$${{(Y,C)}}$${{\mathbb{E}(Y,A)}}$${{(X,A)}}$${{(X,B)}}$${{(X,C)}}$${{\mathbb{E}(X,A)}}$${{\mathbb{E}(Z,A)}}$

∎

######  Remark 3.4.

In a work in progress with Peter Jørgensen, we proved a version adapted to
higher homological algebra: Let $\mathscr{C}$ be an $n$-angulated category
with suspension functor $\Sigma$, and let $A\overset{f}{\to}B\to
C^{\bullet}\to\Sigma A$ and $X\overset{g}{\to}Y\to Z^{\bullet}\to\Sigma X$ be
two $n$-angles. Then $f\,\square\,g$ if and only if any morphism of complexes
$C^{\bullet}\to Z^{\bullet}$ is null-homotopic.

######  Definition 3.5.

A complete cotorsion pair in an extriangulated category $\mathscr{C}$ is a
pair $(\mathcal{U},\mathcal{V})$ of full subcategories such that, for any
$X\in\mathscr{C}$, the following holds:

  1. (1)

We have $X\in\mathcal{V}$ if and only if for any $U\in\mathcal{U}$,
$\mathbb{E}(U,X)=0$.

  2. (2)

We have $X\in\mathcal{U}$ if and only if for any $V\in\mathcal{V}$,
$\mathbb{E}(X,V)=0$.

  3. (3)

There are conflations $V_{X}\rightarrowtail U_{X}\twoheadrightarrow X$ and
$X\rightarrowtail V^{X}\twoheadrightarrow U^{X}$ with
$U_{X},U^{X}\in\mathcal{U}$ and $V_{X},V^{X}\in\mathcal{V}$.

The following is mostly a consequence of the previous lemma.

######  Proposition 3.6.

Let $(\mathcal{U},\mathcal{V})$ be a cotorsion pair on an extriangulated
category $\mathscr{C}$. Define $\mathcal{L}$ to be the class of all those
inflations whose cokernel belongs to $\mathcal{U}$. Dually, define
$\mathcal{R}$ to be the class of all those deflations whose kernel belongs to
$\mathcal{V}$. Then $(\mathcal{L},\mathcal{R})$ is a weak factorisation
system.

######  Definition 3.7.

Let $\mathscr{C}$ be a category with finite products and coproducts. Then a
model structure on $\mathscr{C}$ can be defined as the data of three classes
of morphisms
$(\mathcal{F}\\!\mathit{ib},\mathcal{C}\\!\mathit{of},\mathbb{W})$ such that
$(\mathcal{C}\\!\mathit{of}\cap\mathbb{W},\mathcal{F}\\!\mathit{ib})$ and
$(\mathcal{C}\\!\mathit{of},\mathcal{F}\\!\mathit{ib}\cap\mathbb{W})$ are weak
factorisation systems and the class $\mathbb{W}$ satisfies the two-out-of-
three condition.

The fact that $\mathbb{W}$ is automatically closed under retracts was first
noticed by André Joyal and Myles Tierney when $\mathscr{C}$ either has push-
outs or has pull-backs (see also [MP12, Lemma 14.2.5]). The general case
follows from [Egg06] and the fact that weak equivalences are precisely those
morphisms that become isomorphisms in the localisation
$\mathscr{C}[\mathbb{W}^{-1}]$. This is explained in detail in Pierre Cagne’s
PhD Thesis [Cag18, Section 2.2].

######  Question 3.8.

Let $(\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V})$ be two cotorsion
pairs on an extriangulated category $\mathscr{C}$. We obtain, by Proposition
3.6, two weak factorisation systems
$(w\mathcal{C}\\!\mathit{of},\mathcal{F}\\!\mathit{ib})$,
$(\mathcal{C}\\!\mathit{of},w\mathcal{F}\\!\mathit{ib})$. Define
$\mathbb{W}=w\mathcal{F}\\!\mathit{ib}\circ w\mathcal{C}\\!\mathit{of}$. Under
which conditions on the two cotorsion pairs is
$(\mathcal{F}\\!\mathit{ib},\mathcal{C}\\!\mathit{of},\mathbb{W})$ a model
structure on $\mathscr{C}$?

We note that if
$(\mathcal{F}\\!\mathit{ib},\mathcal{C}\\!\mathit{of},\mathbb{W})$ is an exact
(as in Definition 3.10 below) model structure on $\mathscr{C}$, then an object
is cofibrant (resp. fibrant, trivially cofibrant, trivially fibrant) if and
only if it belongs to $\mathcal{U}$ (resp. to $\mathcal{T}$, $\mathcal{S}$,
$\mathcal{V}$). Part of the answer to 3.8 is then rather immediate: First, it
is necessary that $\mathcal{S}\subseteq\mathcal{U}$ and
$\mathcal{V}\subseteq\mathcal{T}$ (those two conditions are equivalent).
Second, both $\mathcal{U}\cap\mathcal{V}$ and $\mathcal{S}\cap\mathcal{T}$
should be the full subcategory of trivially cofibrant and trivially fibrant
objects, hence it is necessary that
$\mathcal{U}\cap\mathcal{V}=\mathcal{S}\cap\mathcal{T}$. The last condition is
less obvious, and arises when considering the following question: Is it
possible to read the two-out-of-three property for $\mathbb{W}$ directly on
the two cotorsion pairs? Some very specific instance of the two-out-of-three
property implies that, for any object $X$ in $\mathscr{C}$, the morphism $0\to
X$ is a weak equivalence if and only if so is $X\to 0$. This easily translates
to a condition on $(\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V})$. It
turns out that this condition (condition (3) below) implies the two-out-of-
three property for $\mathbb{W}$ in full generality.

######  Definition 3.9.

Two cotorsion pairs $(\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V})$ on
an extriangulated category $\mathscr{C}$ are called:

  1. (1)

Twin cotorsion pairs if $\mathcal{S}\subseteq\mathcal{U}$ (or equivalently
$\mathcal{V}\subseteq\mathcal{T}$);

  2. (2)

Concentric twin cotorsion pairs if moreover
$\mathcal{U}\cap\mathcal{V}=\mathcal{S}\cap\mathcal{T}$;

  3. (3)

Hovey twin cotorsion pairs if moreover, for any object $X$, the existence of a
conflation $V\rightarrowtail S\twoheadrightarrow X$ with $V\in\mathcal{V}$,
$S\in\mathcal{S}$ is equivalent to the existence of a conflation
$X\rightarrowtail V^{\prime}\twoheadrightarrow S^{\prime}$ with
$V^{\prime}\in\mathcal{V}$, $S^{\prime}\in\mathcal{S}$.

Not all model structures on $\mathscr{C}$ can be expected to come from twin
cotorsion pairs (for counter-examples, see the model structures arising from
rigid objects in [JM17]). However, all those that interact nicely with the
extriangulated structure do.

######  Definition 3.10.

A model structure on $\mathscr{C}$ is called an exact model structure if its
cofibrations are precisely those inflations whose cone is cofibrant and its
fibrations are precisely those deflations whose cocone is fibrant.

In an exact model structure, the acyclic cofibrations coincide with the
inflations having a trivially cofibrant cone, and the acyclic fibrations
coincide with the deflations having a trivially fibrant cocone.

###  3.2. Hovey’s correspondence for extriangulated categories

In order to prove that the weak equivalences defined by some choice of Hovey
twin cotorsion pairs satisfy the two-out-of-three property, some technical
assumption is needed:

######  Definition 3.11.

An additive category is weakly idempotent complete if every section has a
cokernel or, equivalently, if every retraction has a kernel.

We note that abelian categories and triangulated categories are weakly
idempotent complete as additive categories.

######  Proposition 3.12 (see [Büh10, Section 7] for exact categories, [Kla22,
Proposition 2.7], [Tat22, Section II.1.3] and [Msa22, Proposition 3.33] for
extriangulated categories).

Let $\mathscr{C}$ be an extriangulated category. The following are equivalent:

  1. (1)

The additive category $\mathscr{C}$ is weakly idempotent complete.

  2. (2)

Every section is an inflation.

  3. (3)

Every retraction is a deflation.

  4. (4)

If a composition $g\circ f$ is an inflation, then $f$ is an inflation.

  5. (5)

If a composition $g\circ f$ is a deflation, then $g$ is a deflation.

The following result is originaly due to Mark Hovey [Hov02, Hov07] for abelian
categories. It was proven in different generalities in [BR07] (specific case
using one cotorsion pair only), in [Gil11] for exact categories, in [Yan15]
for triangulated categories and in [NP19] for extriangulated categories.

######  Theorem 3.13 (Hovey’s correspondence).

Let $\mathscr{C}$ be a weakly idempotent complete extriangulated category.
Then there is a bijective correspondence between Hovey twin cotorsion pairs in
$\mathscr{C}$ and exact model structures on $\mathscr{C}$.

The correspondence sends a model structure to the cotorsion pairs
$(\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V})$, where the four
subcategories are given respectively by the trivially cofibrant, the fibrant,
the cofibrant and the trivially fibrant objects.

We note the immediate corollary:

######  Corollary 3.14.

Let $\mathscr{C}$ be a weakly idempotent complete extriangulated category, and
let $(\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V})$ be Hovey twin
cotorsion pairs on $\mathscr{C}$. Then the inclusion
$\mathcal{T}\cap\,\mathcal{U}\to\mathscr{C}$ induces an equivalence of
categories

$(\mathcal{T}\cap\,\mathcal{U})\,/\,[\mathcal{S}\cap\mathcal{V}]\overset{\simeq}{\longrightarrow}\mathscr{C}[\mathbb{W}^{-1}]$

expressing the localisation of $\mathscr{C}$ at the class $\mathbb{W}$ as an
ideal subquotient of $\mathscr{C}$.

######  Remark 3.15.

The work in progress [JKPW23] with Gustavo Jasso, Sondre Kvamme and Tashi
Walde, considers a version of Hovey’s correspondence for fibration categories,
in the setting of exact $\infty$-categories.

###  3.3. Triangulated homotopy categories

Corollary 3.14 gives a strategy for studying the structure of the homotopy
category $\mathscr{C}[\mathbb{W}^{-1}]$.

######  Definition 3.16.

A cotorsion pair $(\mathcal{X},\mathcal{Y})$ of an extriangulated category
$\mathscr{C}$ is called hereditary if $\mathcal{X}$ is stable under taking
kernels of deflations (between objects in $\mathcal{X}$) and $\mathcal{Y}$ is
stable under taking cokernels of inflations (between objects in
$\mathcal{Y}$).

######  Proposition 3.17 (James Gillespie [Gil16]).

Let $\mathscr{E}$ be an exact category with some Hovey twin cotorsion pairs
$(\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V})$. Assume that
$(\mathcal{S},\mathcal{T})$ and $(\mathcal{U},\mathcal{V})$ are hereditary.
Then the homotopy category $\mathscr{E}[\mathbb{W}^{-1}]$, for the exact model
structure obtained by Hovey’s correspondence, is triangulated.

###### Proof.

When $(\mathcal{S},\mathcal{T})$ and $(\mathcal{U},\mathcal{V})$ are
hereditary, the extension-closed subcategory $\mathcal{T}\cap\,\mathcal{U}$ of
$\mathscr{E}$ is Frobenius exact, with projective-injective objects
$\mathcal{S}\cap\mathcal{V}$. Hence
$(\mathcal{T}\cap\,\mathcal{U})\,/\,[\mathcal{S}\cap\mathcal{V}]$ is
triangulated, and so is $\mathscr{C}[\mathbb{W}^{-1}]$ by Corollary 3.14. ∎

We give a generalisation of this result, which seems to be new already in the
case of exact categories.

######  Theorem 3.18.

_[NP19, Theorem 6.20]_ For any weakly idempotent complete extriangulated
category $\mathscr{C}$ and any Hovey twin cotorsion pairs
$(\mathcal{S},\mathcal{T}),(\mathcal{U},\mathcal{V})$ on $\mathscr{C}$, the
associated homotopy category $\mathscr{C}[\mathbb{W}^{-1}]$ is triangulated.

######  Remark 3.19.

Under the assumptions of Theorem 3.18, the model structure on $\mathscr{C}$ is
not stable _stricto sensu_ because $\mathscr{C}$ has too few limits and
colimits. However, the equivalent of the octahedral axiom gives specific
choices of weak bicartesian squares that compensate for the lack of
(co)limits.

A specific case of Theorem 3.18 is Proposition 1.22 that generalises Theorems
1.1 and 1.2 giving homotopical content to these Theorems.

We conclude this section by remarking that Corollary 3.14 also generalises the
following result, which is nicely motivated in the introduction of [IY20].

######  Theorem 3.20 ([Nak18, Corollary 6.13] and [IY20, Theorem 1.1]).

Let $\mathscr{T}$ be a triangulated category, $\mathscr{S}$ be a thick
subcategory and $(\mathcal{A},\mathcal{A}^{\perp_{1}})$ and
$(^{\perp_{1}}\mathcal{B},\mathcal{B})$ be complete cotorsion pairs in
$\mathscr{T}$ with $\mathcal{A},\mathcal{B}\subseteq\mathscr{S}$. Assume that
$(\mathcal{A},\mathcal{B})$ is a complete cotorsion pair in $\mathscr{S}$.
Then there is an equivalence of categories
$\left(\mathcal{A}^{\perp_{1}}\cap\,^{\perp_{1}}\mathcal{B}\right)/[\mathcal{A}\cap\mathcal{B}]\xrightarrow{\simeq}\mathscr{T}/\mathscr{S}$.

This result (stated here with cotorsion pairs rather than with torsion pairs
as in [IY20]) is equivalent to [Nak18, Corollary 6.13] and also follows from
Corollary 3.14. Indeed, the assumptions of Theorem 3.20 can be equivalently
expressed by asking $(\mathcal{A},\mathcal{A}^{\perp_{1}})$ and
$(^{\perp_{1}}\mathcal{B},\mathcal{B})$ to be Hovey twin cotorsion pairs and
letting $\mathscr{S}$ be
Cone$(\mathcal{A},\mathcal{B})=\mathcal{B}\ast\Sigma\mathcal{A}$, which is a
thick subcategory of $\mathscr{T}$ by [Nak18, Proposition 6.8] (this also
holds if $\mathscr{T}$ is extriangulated by [NP19, Proposition 5.3]).

##  4\. Polytopal realisations of g-vector fans

###  4.1. ABHY’s construction

Motivated by the theory of scattering amplitudes in theoretical physics,
Arkani-Hamed, Bai, He and Yan [AHBHY18] introduced a new realization of the
classical associahedron. Fix a family of positive real numbers
$\underline{c}=(c_{i\,j})$, for $2\leq i\leq j-2\leq n+1$, and fix variables
(coordinates) $q_{i\,j}$ for $1\leq i\leq j-2\leq n+1$, but $(1,n+3)$. Write
$N=\frac{n(n+3)}{2}$, and consider the affine subspace $E_{\underline{c}}$ of
$\mathbb{R}^{N}$ given by the following equations:

$q_{i\,j}+q_{i+1\,j+1}=q_{i\,j+1}+q_{i+1\,j}+c_{i+1\,j+1}$

for each $1\leq i\leq j-2\leq n$. Let

$\displaystyle\pi:\mathbb{R}^{N}$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{R}^{n}$ $\displaystyle(q_{i\,j})$
$\displaystyle\longmapsto$ $\displaystyle(q_{2,n+3},\ldots,q_{n+1\,n+3})$

and define
$\mathbb{A}_{\underline{c}}=\pi\left(E_{\underline{c}}\cap\mathbb{R}_{>0}^{N}\right)$.

For $n=2$, $E_{\underline{c}}$ is thus given by the system of equations:

$E_{\underline{c}}:\left\\{\begin{array}[]{ccl}q_{1\,3}+q_{2\,4}&=&q_{1\,4}+c_{2\,4}\hskip
28.45274pt(1)\\\ q_{1\,4}+q_{2\,5}&=&q_{2\,4}+c_{2\,5}\hskip 28.45274pt(2)\\\
q_{2\,4}+q_{3\,5}&=&q_{2\,5}+c_{3\,5}\hskip 28.45274pt(3)\end{array}\right.$

which is equivalent to

$E_{\underline{c}}:\left\\{\begin{array}[]{ccl}q_{1\,3}&=&c_{2\,4}+c_{2\,5}-q_{2\,5}\hskip
28.45274pt(1)+(2)\\\ q_{1\,4}&=&c_{2\,5}+c_{3\,5}-q_{3\,5}\hskip
28.45274pt(2)+(3)\\\ q_{2\,4}&=&c_{3\,5}+q_{2\,5}-q_{3\,5}\hskip
28.45274pt(3)\end{array}\right.$

Intersecting it with the positive orthant and only remembering the last two
coordinates, we obtain

$\displaystyle\mathbb{A}_{\underline{c}}$ $\displaystyle=$
$\displaystyle\\{(q_{2\,5},q_{3\,5})\in\mathbb{R}_{>0}^{2}\;|\;q_{1\,3}>0,q_{1\,4}>0,q_{2\,4}>0\\}$
$\displaystyle=$
$\displaystyle\\{(q_{2\,5},q_{3\,5})\in\mathbb{R}_{>0}^{2}\;|\;q_{2\,5}<c_{2\,4}+c_{2\,5},\;q_{3\,5}<c_{2\,5}+c_{3\,5},\;q_{3\,5}-q_{2\,5}<c_{3\,5}\\}.$

Figure 1. ABHY’s construction for $n=2$ gives the associahedron of type
$A_{2}$.

######  Theorem 4.1 ([AHBHY18]).

For all positive $\underline{c}$, $\mathbb{A}_{\underline{c}}$ is an
associahedron of type $A_{n}$.

######  Question 4.2.

Is it possible to understand this result by using representation theory?

That is precisely what a team from the LaCIM did, allowing them to generalise
the result to other Dykin types.

###  4.2. Representation theoretic interpretation

The team from the LaCIM alluded to above was formed by: Véronique Bazier-
Matte, Nathan Chapelier-Laget, Guillaume Douville, Kaveh Mousavand, Hugh
Thomas and Emine Yıldırım [BMCLD+18]. This subsection is inspired from Hugh
Thomas’ talk at the RIMS in 2019.

Let $Q$ be a Dynkin quiver of type $A,D$ or $E$. Denote by $\mathcal{D}_{Q}$
the full subcategory
$\operatorname{add}(\\{M\in\operatorname{mod}\mathbb{K}Q,M\text{indecomposable}\\}\cup\\{I_{j}[-1],j\in
Q_{0}\\})$ of $\mathcal{D}^{\text{b}}(\mathbb{K}Q)$. For each
$M\in\operatorname{ind}\mathcal{D}_{Q}$, fix a coordinate $q_{M}$. For each
$M\in\operatorname{ind}\mathbb{C}Q$ fix some positive real number $c_{M}>0$.
Let $n=|Q_{0}|$ and $N=|\operatorname{ind}\mathcal{D}_{Q}|$.

######  Example 4.3.

If $Q=1\leftarrow 2$, then $\operatorname{ind}\mathbb{C}Q$ contains the
following three isomorphism classes.

$P_{1}=S_{1}=\mathbb{C}\leftarrow 0$,
$P_{2}=I_{1}=\mathbb{C}\stackrel{{\scriptstyle 1}}{{\leftarrow}}\mathbb{C}$,
and $S_{2}=I_{2}=0\leftarrow\mathbb{C}$. The Auslander–Reiten quiver of the
subcategory $\mathcal{D}_{Q}$ is given by:

${{I_{2}[-1]}}$${{I_{1}}}$${{q_{1\,4}}}$${{c_{2\,5}}}$${{q_{2\,5}}}$${{I_{1}[-1]}}$${{P_{1}}}$${{I_{2}}}$${{q_{1\,3}}}$${{c_{2\,4}}}$${{q_{2\,4}}}$${{c_{3\,5}}}$${{q_{3\,5}}}$

For $c=(c_{M})_{M\in\operatorname{ind}\mathcal{D}_{Q}}$, define
$\mathbb{E}_{c}$ to be the affine subspace of $\mathbb{R}^{N}$ given by the
$N-n=|\operatorname{ind}\mathbb{C}Q|$ equations $q+t=r_{1}+r_{2}+r_{3}+c$ for
each mesh

${{r_{1}}}$${q}$${c}$${t}$${{r_{2}}}$${{r_{3}}}$

in the Auslander–Reiten quiver of $\mathcal{D}_{Q}$, with the convention that
$r_{3}$ and $r_{2}$ are 0 if the mesh has fewer middle terms. Consider the
projection

$\displaystyle\pi:\mathbb{R}^{N}$ $\displaystyle\longrightarrow$
$\displaystyle\mathbb{R}^{n}$
$\displaystyle(q_{M})_{M\in\operatorname{ind}\mathcal{D}_{Q}}$
$\displaystyle\longmapsto$ $\displaystyle(q_{I_{j}})_{j\in Q_{0}}$

and define $\mathbb{A}_{c}$ by
$\mathbb{A}_{c}=\pi\left(\mathbb{E}_{c}\cap\mathbb{R}_{>0}^{N}\right)$.

######  Theorem 4.4 (Bazier-Matte–Chapelier-
Laget–Douville–Mousavand–Thomas–Yıldırım).

For any positive $c$, $\mathbb{A}_{c}$ is a generalised associahedron of the
same Dynkin type as $Q$.

This beautiful approach using representations of quivers has some drawbacks.
First, the authors have to work with parameters $c_{M}$ in $\mathbb{Q}$, and
deduce the result over $\mathbb{R}$ by density. Second, they have to work over
an acyclic intial seed (this was later lifted in [BM21]). Third, non-simply
laced Dynkin types are deduced from simply-laced types by a non-trivial
folding technique [AHHL21]. More arguably, one might be tempted to say that
their methods gives an interpretation of ABHY’s construction allowing for more
general versions but does not explain why ABHY’s construction works.

Figure 2. The cases treated by the two teams, in Dynkin type $A$: ABHY
considers linear orientation (i.e. fan triangulations), LaCIM considers
acyclic initial seeds (i.e. triangulations with no internal triangles). We aim
at treating the general case (and at extending to dissections)

###  4.3. Reformulation using McMullen’s type cone

Our aim is to give another interpretation of ABHY’s construction that avoids
the drawbacks mentionned above, while explaining why this construction is
natural. The tools we will make use of are McMullen’s type cone and cluster
categories.

####  4.3.1. Polyhedral cones

A subset of $\mathbb{R}^{n}$ is a polyhedral cone if it is given as the non-
negative span of finitely many vectors. A cone is simplicial if it is
generated by a set of linearly independent vectors.

Let $X\subset\mathbb{R}^{n}$. A supporting hyperplane of $X$ is a hyperplane
$H\subset\mathbb{R}^{n}$ that intersects $X$ and such that $X$ is entirely
contained in one of the half-spaces defined by $H$. The faces of a cone $C$
are the intersections of $C$ with its supporting hyperplanes. Faces of
dimension one are called rays while faces of codimension one are called
facets.

####  4.3.2. Polyhedral fans

A polyhedral fan is a collection $\mathcal{F}$ of polyhedral cones satisfying
the following two conditions:

  * •

For any cone $C$ in $\mathcal{F}$, each face of $C$ is in $\mathcal{F}$.

  * •

For any two cones $C,C^{\prime}\in\mathcal{F}$, $C\cap C^{\prime}$ is both a
face of $C$ and of $C^{\prime}$.

Given a fan $\mathcal{F}$, we will always fix a generator for each ray, and we
will denote cones by $\mathbb{R}_{\geq 0}\mathcal{R}$, where $\mathcal{R}$ is
the set of the chosen generators of the rays of the cone.

A fan $\mathcal{F}$ is

  * •

simplicial if all its cones are simplicial;

  * •

complete if $\mathbb{R}^{n}=\cup\mathcal{F}$;

  * •

essential if $\\{0\\}$ is one of its cones.

####  4.3.3. Polytopes and their normal fans

A polytope is the convex hull of finitely many points in $\mathbb{R}^{n}$. It
can be equivalently defined as the bounded intersection of finitely many
closed affine halfspaces. Similarly as for cones, we call faces of a polytope
its intersections with some supporting hyperplane and its facets are its
codimension one faces. Its 0-dimensional and 1-dimensional faces are
respectively called vertices and edges.

The (outer) normal cone of a face $F$ of a polytope $P$ is the cone generated
by the outer normal vectors of the facets of $P$ containing $F$. The normal
fan of $P$ is the collection of the normal cones of all its faces. If a
complete polyhedral fan $\mathcal{F}$ in $\mathbb{R}^{n}$ is the normal fan of
a polytope $P$ of $\mathbb{R}^{n}$, we say that it is polytopal, and $P$ is
then called a polytopal realization of $\mathcal{F}$.

Figure 3. On the left hand side, we picked a good choice of normal hyperplanes
and we obtain a realisation of the classical associahedron of type $A_{2}$. On
the right hand side, a bad choice of normal hyperplanes.

####  4.3.4. McMullen’s type cone

We fix an essential, complete, simplicial fan $\mathcal{F}$ in
$\mathbb{R}^{n}$ with $N$ rays arbitrarily ordered. The type cone of
$\mathcal{F}$, introduced in [McM73], morally parametrizes all polytopal
realisations of $\mathcal{F}$.

For any two adjacent maximal cones $\mathbb{R}_{\geq 0}\mathcal{R}$ and
$\mathbb{R}_{\geq 0}\mathcal{R}^{\prime}$ of $\mathcal{F}$ with
${\mathcal{R}\smallsetminus\\{r\\}=\mathcal{R}^{\prime}\smallsetminus\\{r^{\prime}\\}}$,
there is, up to scalar multiplication, a unique linear dependency of the form

$\alpha
r+\alpha^{\prime}r^{\prime}=\sum_{s_{i}\in\mathcal{R}\cap\mathcal{R}^{\prime}}\alpha_{i}s_{i}$

where we may and will assume $\alpha,\alpha^{\prime}>0$. We will also make the
following choice: $\alpha+\alpha^{\prime}=2$ (this convention is arbitrary,
but naturally arises for $g$-vector fans from categorification of cluster
algebras).

Figure 4. Two adjacent cones in $\mathbb{R}^{3}$ giving rise to a relation
$\alpha r+\alpha^{\prime}r^{\prime}=\alpha_{1}s_{1}+\alpha_{2}s_{2}$.

######  Definition 4.5.

The type cone of $\mathcal{F}$ is the cone

$\mathbb{TC}(\mathcal{F})=\left\\{h\in\mathbb{R}^{N}\;|\;\alpha
h_{r}+\alpha^{\prime}h_{r^{\prime}}>\sum_{s_{i}\in\mathcal{R}\cap\mathcal{R}^{\prime}}\alpha_{i}h_{s_{i}}\;\begin{array}[]{l}\text{for
any adjacent maximal}\\\ \text{cones }\mathbb{R}_{\geq 0}\mathcal{R}\text{ and
}\mathbb{R}_{\geq 0}\mathcal{R}^{\prime}\text{ of
}\mathcal{F}\end{array}\right\\}.$

We note that this version of the definition comes from [CFZ02, Lemma 2.1].

Let $G$ be the $N\times n$-matrix whose rows are the coordinates of the
vectors defining the rays of $\mathcal{F}$. For any height vector
$h\in\mathbb{R}^{N}$, we define the polytope

$P_{h}=\\{x\in\mathbb{R}^{n}\;|\;Gx\leq h\\},$

where $x\leq y$ if the inequality holds coordinate-wise.

######  Lemma 4.6.

[CFZ02] Let $h\in\mathbb{R}^{N}$. Then, $P_{h}$ is a polytopal realization of
$\mathcal{F}$ if and only if $h\in\mathbb{TC}(\mathcal{F})$.

####  4.3.5. From the type cone to ABHY’s construction

The key input from part I of [PPPP19], is to realise that ABHY’s construction
is, in good cases, a convenient reformulation of the type cone.

######  Theorem 4.7.

[PPPP19] Assume that the type cone $\mathbb{TC}(\mathcal{F})$ of the fan
$\mathcal{F}$ is simplicial (equivalently: has $N-n$ facets). Let $K$ be the
$(N-n)\times N$ matrix whose rows are the coordinates of the inner normal
vectors of the facets of $\mathbb{TC}(\mathcal{F})$. Then, for any positive
$c\in\mathcal{R}_{>0}^{N-n}$, the polytope

$Q_{c}=\left\\{q\in\mathbb{R}^{N}\;|\;Kq=c\text{ and }q\geq 0\right\\}$

is a polytopal realisation of $\mathcal{F}$. Moreover, the polytopes $Q_{c}$
describe all polytopal realisations of $\mathcal{F}$.

######  Remark 4.8.

Let $h\in\mathbb{R}^{N}$ be such that $Kh=c$. Then the polytope $Q_{c}$ is
obtained from $P_{h}$, via a standard method, by some well-chosen affine
transformation, and the condition $h\in\mathbb{TC}(\mathcal{F})$ is equivalent
to $c>0$. Going from $P_{h}$ to $Q_{c}$ can be thought of as going from the
type cone to ABHY’s construction. We thus have an explanation of why ABHY’s
construction works.

Our aim is now to apply Theorem 4.7 to the $g$-vector fan
$\mathcal{F}_{\Sigma}$ of a cluster algebra of finite type with initial seed
$\Sigma$ in order to recover ABHY’s construction and to generalise it to any
initial seed. In order to do so, we have to show that $\mathcal{F}_{\Sigma}$
satisfies the unique exchange relation property, that its type cone is
simplicial and that the equations defining the polytope $Q_{c}$ coincide with
those appearing in the LaCIM’s interpretation of ABHY’s construction. That is
where we start using representation theory and where:

###  4.4. Cluster categories come into play

Let $\Delta$ be a Dynkin diagram and let $\mathcal{A}_{\Delta}$ be the
associated cluster algebra of finite type. Fix any initial seed $\Sigma_{0}$,
and let $\mathcal{F}_{\Sigma_{0}}$ be the fan given by the $g$-vectors
relative to $\Sigma_{0}$.

######  Definition 4.9.

The simplicial fan $\mathcal{F}$ has the unique exchange relation property if
the linear dependencies $\alpha
r+\alpha^{\prime}r^{\prime}=\sum_{s_{i}\in\mathcal{R}\cap\mathcal{R}^{\prime}}\alpha_{i}s_{i}$
only depend on $r$ and $r^{\prime}$ and not on the choice of the cones
$\mathbb{R}_{\geq 0}\mathcal{R}$ and $\mathbb{R}_{\geq
0}\mathcal{R}^{\prime}$.

######  Theorem 4.10.

[PPPP19] The $g$-vector fan $\mathcal{F}_{\Sigma_{0}}$ has the unique exchange
relation property and its type cone is simplicial. Moreover, the associated
polytopes $Q_{c}$ in Theorem 4.7 are precisely given by ABHY’s construction in
a re-interpretation similar to that of Bazier-Matte–Chapelier-
Laget–Douville–Mousavand–Thomas–Yıldırım.

Sketch of proof

_First step_ : Replace the full subcategory $\mathcal{D}_{Q}$ by the cluster
category $\mathscr{C}_{\Delta}$ of [BMR+06].

It is the orbit category of the bounded derived category of $\Delta$ with
respect to the action of the autoequivalence $\tau^{-1}[1]$ (see Figure 5 for
an example). By [Kel05], the cluster category $\mathscr{C}_{\Delta}$ inherits
a triangulated structure from the derived category. Its shift functor we
denote by $\Sigma$. The functor $\tau^{-1}[1]$ is chosen so that, in the
cluster category, $\Sigma$ is naturally isomorphic to $\tau$. In other words,
$\mathscr{C}_{\Delta}$ is 2-Calabi–Yau.

Figure 5. The Auslander–Reiten quiver of the cluster category of type $A_{3}$.

_Second step_ : Replace the triangulated structure of $\mathscr{C}_{\Delta}$
by some relative extriangulated structure.

More precisely, there is a cluster tilting object $T\in\mathscr{C}_{\Delta}$
associated with the seed $\Sigma_{0}$. Let $\Delta_{T}$ be the collection of
all those triangles $X\to Y\to Z\xrightarrow{\delta}\Sigma X$ fo which the
morphism $\delta$ factors through an object isomorphic to $\Sigma T^{\oplus
r}$ for some $r$. Then by [HLN21, Proposition 3.19] (see Section 2.1.4)
$\mathscr{C}_{\Delta}$ endowed with $\Delta_{T}$ is an extriangulated
category. Even though this extriangulated structure has the drawback of not
being 2-Calabi–Yau anymore, it has some interest in categorifying $g$-vectors:
The Grothendieck group of the triangulated $\mathscr{C}_{\Delta}$ has torsion
(we even have K${}_{0}(A_{2})=0$) whereas $g$-vectors naturally live in the
Grothendieck group of the extriangulated $\mathscr{C}_{\Delta}$. Let

$\operatorname{K}(\mathscr{C}_{\Delta};T)=\operatorname{K}^{\text{split}}_{0}(\mathscr{C}_{\Delta})/\langle[L]+[N]-[M]\;|\;L\to
M\to N\to\Sigma L\text{ in }\Delta_{T}\rangle$

be the Grothendieck group of $\mathscr{C}_{\Delta}$ endowed with the
extriangulated structure given by $\Delta_{T}$, and let

$\mathfrak{g}:\operatorname{K}^{\text{split}}_{0}(\mathscr{C}_{\Delta})\to\operatorname{K}_{0}(\mathscr{C}_{\Delta};T)$

be the quotient map.

_Third step_ : Compute the Grothendieck group
K${}_{0}(\mathscr{C}_{\Delta};T)$.

For any object $X\in\mathscr{C}_{\Delta}$, there is a triangle $T_{1}\to
T_{0}\to X\to\Sigma T_{1}$. The assignement
$\operatorname{ind}_{T}:X\mapsto[T_{0}]-[T_{1}]$ induces a well-defined group
homomorphism
$\operatorname{K}^{\text{split}}_{0}(\mathscr{C}_{\Delta})\to\operatorname{K}^{\text{split}}_{0}(\operatorname{add}T)$.
One can deduce from [Pal08, Proposition 2.2] that there is a commutative
diagram

${{\operatorname{K}^{\text{split}}_{0}(\mathscr{C}_{\Delta})}}$${{\operatorname{K}_{0}(\mathscr{C}_{\Delta};T)}}$${{\operatorname{K}^{\text{split}}_{0}(\operatorname{add}T)}}$$\scriptstyle{\mathfrak{g}}$$\scriptstyle{\operatorname{ind}_{T}}$$\scriptstyle{\varphi}$

where $\varphi$ is a group isomorphism.

_Fourth step_ : Study the kernel of the quotient map $\mathfrak{g}$.

Write $T=T_{1}\oplus\cdots\oplus T_{n}$ where the $T_{i}$’s are indecomposable
and $T$ is assumed basic. We prove in [PPPP19, Corollary 3.9] that
$\operatorname{Ker}\mathfrak{g}$ is generated, as a monoid, by the relations
$[L]+[N]-[M]$, for each Auslander–Reiten triangle $L\to M\to N\to\Sigma L$ in
$\mathscr{C}_{\Delta}$ with $L$ not isomorphic to any $\Sigma T_{i}$. Those
are precisely the Auslander–Reiten sequences (see [INP18, Proposition 5.10])
of the extriangulated category $(\mathscr{C}_{\Delta},\Delta_{T})$.

_Last step_ : The type cone is simplicial

There are $N$ Auslander–Reiten triangles in $\mathscr{C}_{\Delta}$, hence
$N-n$ Auslander–Reiten sequences in $(\mathscr{C}_{\Delta},\Delta_{T})$. One
thus deduces from the above that $\mathbb{TC}(\mathscr{F}_{\Delta})$ is
simplicial. Moreover, from the form of the Auslander–Reiten sequences, one
also deduces that using the type cone construction recovers the ABHY
construction in the LaCIM’s version for simply laced types.

∎

######  Remark 4.11.

The results in [PPPP19] on $(\mathscr{C}_{\Delta},\Delta_{T})$ are proven in
greater generality, by considering some hereditary extriangulated categories.
They can thus be applied to other categories such as the categories
$K^{[-1,0]}(\operatorname{proj}\Lambda)$ of 2-term complexes of projectives
over a finite-dimensional algebra. It follows that ABHY’s construction also
works for realising $g$-vector fans of (brick, 2-acyclic) gentle algebras of
finite representation type. Due to their relation with dissections on oriented
surfaces, those polytopal realisations turned out to be useful in theoretical
physics (see [JL22, Chh19, ABJ+19] for instance).

##  5\. Mutation

This section, based on [GNP23], aims at explaining how extriangulated
categories form a convenient setting for studying mutations in representation
theory.

###  5.1. 0-Auslander extriangulated categories

As illustrated by the previous section, the extriangulated categories
$K^{[0,1]}(\operatorname{proj}\Lambda)$ and
$(\mathscr{C}_{\Delta},\Delta_{T})$ may be used to categorify $g$-vector fans.
Both those categories enjoy some nice properties which motivates the
following:

######  Definition 5.1.

An extriangulated category $\mathscr{C}$ is 0-Auslander if

  1. (a)

for any $X\in\mathscr{C}$, there is a conflation $P_{1}\rightarrowtail
P_{0}\twoheadrightarrow X$ with $P_{0}$ and $P_{1}$ projective and

  2. (b)

for any projective $P\in\mathscr{C}$, there is a conflation $P\rightarrowtail
Q\twoheadrightarrow I$ where $Q$ is projective-injective and $I$ is injective.

######  Remark 5.2.

Conditions (a) and (b) can be interpreted as
$\operatorname{gl.dim}\mathscr{C}\leq 1\leq\operatorname{dom.dim}\mathscr{C}$,
whence the name 0-Auslander.

######  Remark 5.3.

One easily shows that Definition 5.1 is self-dual. Indeed, it is equivalent to

  1. (a’)

for any $X\in\mathscr{C}$, there is a conflation $X\rightarrowtail
I_{0}\twoheadrightarrow I_{1}$ with $I_{0}$ and $I_{1}$ injective and

  2. (b’)

for any injective $I\in\mathscr{C}$, there is a conflation $P\rightarrowtail
Q\twoheadrightarrow I$ where $Q$ is projective-injective and $P$ is
projective.

######  Example 5.4.

The following are examples of 0-Auslander extriangulated categories.

  1. ($\alpha$)

The category of finite-dimensional representations
$\operatorname{rep}_{\mathbb{K}}\overrightarrow{A_{n}}$ of a quiver of type
$A_{n}$ with linear orientation, over a field $\mathbb{K}$. If $\mathbb{K}$ is
algebraically closed, this is the only non-semisimple example of a 0-Auslander
module category [Tac64].

  2. ($\beta$)

Let $Q$ be a quiver of type $A_{n}$, with any orientation, and let
$\mathbb{K}$ be a field. There is an exact structure $\mathscr{E}$ on
$\operatorname{rep}_{\mathbb{K}}Q$ whose Auslander–Reiten sequences are
precisely those of $\operatorname{rep}_{\mathbb{K}}Q$ that have non-
indecomposable middle term. Then $\mathscr{E}$ is a 0-Auslander exact category
[BHRS23].

  3. ($\gamma$)

Let $\mathscr{C}$ be a cluster category [BMR+06], a generalised cluster
category [Ami09] or, more generally, any Hom-finite 2-Calabi–Yau triangulated
category with some cluster tilting object $T$. There is a maximal
extriangulated structure $\mathbb{E}^{T}$ on $\mathscr{C}$ making $T$
projective [HLN21, Proposition 3.19]. Then $(\mathscr{C},\mathbb{E}^{T})$ is
0-Auslander.

  4. ($\delta$)

Let $\mathcal{A}$ be an additive category (our motivating example being
$\mathcal{A}=\operatorname{proj}\Lambda$, for an Artin algebra $\Lambda$).
Then the category of two-term complexes $C^{[-1,0]}(\mathcal{A})$ and the
homotopy category two-term complexes $K^{[-1,0]}(\mathcal{A})$ are 0-Auslander
[BSZ05].

  5. ($\varepsilon$)

If $\Lambda$ is a finite-dimensional gentle algebra over an algebraically
closed field, there is an associated oriented surface with dissection [BCS21,
OPS18, PPP19]. Some specific arcs on the surface, called accordions, play a
key role in describing combinatorially the $\tau$-tilting theory of the gentle
algebra $\Lambda$. There is an exact category $\mathscr{W}$ whose
indecomposable objects are the accordions and whose non-split extensions
categorify crossings of accordions [GNP23, Section 6]. The category
$\mathscr{W}$ is 0-Auslander. See Figure 6 for an example.

  6. ($\zeta$)

If $(\mathcal{A},\mathcal{B})$ is a co-$t$-structure [Pau08, Bon10] in a
triangulated category $\mathscr{T}$, then its extended coheart
$\mathscr{C}=(\Sigma^{2}\mathcal{A})\cap\mathcal{B}$ is a 0-Auslander
extriangulated category.

######  Remark 5.5.

In the examples $(\mathscr{C},\mathbb{E}^{T})$ and $K^{[-1,0]}(\mathcal{A})$
above, the only projective-injectives are the zero objects. Such 0-Auslander
extriangulated categories are called reduced 0-Auslander. If $\mathscr{C}$ is
any 0-Auslander extriangulated category then the ideal quotient of
$\mathscr{C}$ by the morphisms factoring through a projective-injective object
is a reduced 0-Auslander extriangulated category.

Figure 6. The Auslander–Reiten quiver of the category $\mathscr{W}$ associated
with the gentle bound quiver
$(Q,I)=(1\xrightarrow{\alpha}2\xrightarrow{\beta}3,\alpha\beta)$. Its vertices
are in bijection with the accordions (curvy and purple) with respect to the
base dissection (straight and green) corresponding to $(Q,I)$.

###  5.2. Mutation of maximal rigid objects

Fix a field $\mathbb{K}$ and a $\mathbb{K}$-linear, Krull–Schmidt, Hom-finite,
0-Auslander extriangulated category $\mathscr{C}$.

Recall that an object $R\in\mathscr{C}$ is rigid if $\mathbb{E}(R,R)=0$.

######  Definition 5.6.

Let $R\in\mathscr{C}$ be rigid. We say that $R$ is

  1. (i)

maximal rigid if $R\oplus X$ is rigid only if $X\in\operatorname{add}R$;

  2. (ii)

$\mathbb{E}$-tilting if the conditions $\mathbb{E}(R,X)=0=\mathbb{E}(X,R)$
imply $X\in\operatorname{add}R$;

  3. (iii)

tilting if for any projective $P\in\mathscr{C}$, there is a conflation
$P\rightarrowtail R_{0}\twoheadrightarrow R_{1}$ with
$R_{0},R_{1}\in\operatorname{add}R$;

  4. (iv)

silting if $\operatorname{thick}R=\mathscr{C}$, where $\operatorname{thick}R$
is the smallest thick subcategory of $\mathscr{C}$ containing $R$;

  5. (v)

complete rigid if $\mathscr{C}$ has a projective generator $P$ and $|R|=|P|$,
where $|X|$ denote the number of isomorphism classes of indecomposable
summands of $X$.

The following results are proven in [GNP23, Section 4].

######  Proposition 5.7.

Let $R\in\mathscr{C}$ be rigid. We have:

  1. (1)

Conditions _(i)_ to _(iv)_ in Definition 5.6 are equivalent.

  2. (2)

If $\mathscr{C}$ has a projective generator, they are also equivalent to
condition _(v)_.

  3. (3)

$R$ can be completed to a silting object via some version of Bongartz
completion.

######  Theorem 5.8.

Let $R$ be a basic silting object in $\mathscr{C}$ and let $X$ be an
indecomposable summand of $R$ which is not projective-injective. Write
$R=\overline{R}\oplus X$. Then there is a unique up to isomorphism
indecomposable $Y\in\mathscr{C}$ such that $\overline{R}\oplus Y$ is silting.
Moreover there is an exchange conflation $X\rightarrowtail E\twoheadrightarrow
Y$ or $Y\rightarrowtail E^{\prime}\twoheadrightarrow X$ with
$E,E^{\prime}\in\operatorname{add}\overline{R}$, but not both.

######  Remark 5.9.

This theorem is proven with less assumptions on $\mathscr{C}$ and allows $R$
to be a full additive subcategory. In the independant article [AT23], Takahide
Adachi and Mayu Tsukamoto study mutations of silting objects in more general
extriangulated categories. Theorem 5.8 can be though of as a two-term version
of their result. The next subsection shows that our setting is presumably the
right level of generality for studying mutations in representation theory.

######  Example 5.10.

Theorem 5.8 applies to the examples listed in Example 5.4

  1. ($\alpha$)

For $\mathscr{C}=\operatorname{rep}_{\mathbb{K}}\overrightarrow{A_{n}}$, the
silting objects are the tilting representations. An almost complete tilting
representation has one or two complements to a tilting representation. It has
two complements precisely when it is sincere, which holds if and only if it
contains the projective-injective. Those classical results are consequences of
Theorem 5.8.

  2. ($\beta$)

For $\mathscr{C}=\mathscr{E}$ as in (ii) of Example 5.4, the silting objects
are the maximal almost-rigid modules of [BGMS23]. This suggests that using
0-Auslander exact categories might help proving mutation of maximal almost-
rigid modules over other classes of algebras.

  3. ($\gamma$)

For $\mathscr{C}$ a cluster category (or a 2-Calabi–Yau triangulated category)
with the extriangulated structure $\mathbb{E}^{T}$ given by some cluster
tilting object $T$, the silting objects are the cluster tilting objects and
Theorem 5.8 recovers classical results by [BMR+06, IY08]. For non 2-Calabi–Yau
triangulated (or extriangulated) categories, a similar approach recovers
mutation of relative tilting objects.

  4. ($\delta$)

For $\mathscr{C}=K^{[-1,0]}(\operatorname{proj}\Lambda)$ the silting objects
are the two-term silting objects of $K^{b}(\operatorname{proj}\Lambda)$ whose
mutations were studied by [AI12, Kim20]. The importance of two-term silting
complexes stems from the bijection, induced by $H^{0}$, with support
$\tau$-tilting modules [AIR14].

  5. ($\varepsilon$)

For $\mathscr{C}=\mathscr{W}$, the silting objects are the maximal collection
of pairwise non-crossing accordions or, equivalently, the non-kissing facets.
Their mutation thus categorifies the flip of non-kissing facets [McC17, PPP21]
and the quotient of $\mathscr{W}$ by its projective-injectives categorifies
the reduced non-kissing complex.

  6. ($\zeta$)

Let $\mathscr{C}$ be the extended coheart of a co-$t$-structure
$(\mathcal{A},\mathcal{B})$. Combining [AT22, Theorem 5.5] and [PZ20, Theorem
2.1] we obtain a bijection between the silting objects of $\mathscr{C}$ and
those co-$t$-structures $(\mathcal{A}^{\prime},\mathcal{B}^{\prime})$ which
are intermediate with respect to $(\mathcal{A},\mathcal{B})$, i.e. such that
$\mathcal{A}\subseteq\mathcal{A}^{\prime}\subseteq\Sigma\mathcal{A}$. Theorem
5.8 thus extends from [BY13] the applicability of mutation for intermediate
co-$t$-structures.

## Conclusion

Since most, if not all, extriangulated categories arising in representation
theory are topological (or even algebraic), they are extension-closed, full
subcategories of triangulated categories. One might thus argue that the
axiomatics of extriangulated categories is dispensable. However extriangulated
structures are very versatile as illustrated by the examples encountered in
representation theory. The following are all examples of extriangulated
categories naturally arising:

  1. (a)

exact categories;

  2. (b)

triangulated categories;

  3. (c)

extension-closed subcategories of triangulated categories;

  4. (d)

quotients of exact categories by some projective-injective objects;

  5. (e)

triangulated categories endowed with some relative extriangulated structure;

  6. (f)

quotients of exact categories by morphisms with injective domains and
projective codomains.

Note that, for points (d), (e), (f), no ambiant triangulated category in which
the extriangulated category is ext-closed is given a priori. Even when it is
possible to construct one, this ambiant triangulated category might not enjoy
some of the desirable properties (being Hom-finite, Krull–Schmidt, weakly
idempotent complete, etc…) satisfied by the categories we started with.

As a final comment, Figure 7 illustrates the place of extriangulated
categories among other relevant categorical structures appearing in
representation theory and beyond.

Figure 7.

## References

  * [ABJ+19] P. B. Aneesh, Pinaki Benerjee, Mrunmay Jagadale, Renjan Rajan John, Alok Laddha, and Sujoy Mahato.  On positive geometries of quartic interactions II : Stokes polytopes, lower forms on associahedra and worldsheet forms.  Preprint, arXiv:1911.06008, 2019.
  * [AHBHY18] Nima Arkani-Hamed, Yuntao Bai, Song He, and Gongwang Yan.  Scattering forms and the positive geometry of kinematics, color and the worldsheet.  J. High Energy Phys., (5):096, front matter+75, 2018.
  * [AHHL21] Nima Arkani-Hamed, Song He, and Thomas Lam.  Cluster configuration spaces of finite type.  SIGMA Symmetry Integrability Geom. Methods Appl., 17:Paper No. 092, 41, 2021.
  * [AI12] Takuma Aihara and Osamu Iyama.  Silting mutation in triangulated categories.  Journal of the London Mathematical Society, 85(3):633–668, 2012\.
  * [AIR14] Takahide Adachi, Osamu Iyama, and Idun Reiten.  $\tau$-tilting theory.  Compos. Math., 150(3):415–452, 2014.
  * [Ami09] Claire Amiot.  Cluster categories for algebras of global dimension 2 and quivers with potential.  Ann. Inst. Fourier (Grenoble), 59(6):2525–2590, 2009.
  * [AT22] Takahide Adachi and Mayu Tsukamoto.  Hereditary cotorsion pairs and silting subcategories in extriangulated categories.  J. Algebra, 594:109–137, 2022.
  * [AT23] Takahide Adachi and Mayu Tsukamoto.  An assortment of properties of silting subcategories of extriangulated categories.  Preprint, arXiv:2303.08125, 2023.
  * [Bar15] Clark Barwick.  On exact $\infty$-categories and the theorem of the heart.  Compositio Mathematica, 151(11):2160–2186, 2015.
  * [BCS21] Karin Baur and Raquel Coelho Simões.  A geometric model for the module category of a gentle algebra.  Int. Math. Res. Not. IMRN, (15):11357–11392, 2021.
  * [BGH14] Daniel Bravo, James Gillespie, and Mark Hovey.  The stable module category of a general ring.  Preprint, arXiv:1405.5768, 2014.
  * [BGMS23] Emily Barnard, Emily Gunawan, Emily Meehan, and Ralf Schiffler.  Cambrian combinatorics on quiver representations (type $\mathbb{A}$).  Adv. in Appl. Math., 143:Paper No. 102428, 37, 2023.
  * [BHRS23] Thomas Brüstle, Eric Hanson, Sunny Roy, and Ralf Schiffler.  Private communication in Oberwolfach.  2023\.
  * [BM21] Véronique Bazier-Matte.  Combinatoire des algèbres amassées.  PhD thesis, Université du Québec À Montréal, 2021.
  * [BMCLD+18] Véronique Bazier-Matte, Nathan Chapelier-Laget, Guillaume Douville, Kaveh Mousavand, Hugh Thomas, and Emine Yıldırım.  ABHY Associahedra and Newton polytopes of ${F}$-polynomials for finite type cluster algebras.  Preprint, arXiv:1808.09986, 2018.
  * [BMR+06] Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov.  Tilting theory and cluster combinatorics.  Adv. Math., 204(2):572–618, 2006.
  * [Bon10] M. V. Bondarko.  Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general).  J. K-Theory, 6(3):387–504, 2010.
  * [BR07] Apostolos Beligiannis and Idun Reiten.  Homological and homotopical aspects of torsion theories.  Mem. Amer. Math. Soc., 188(883):viii+207, 2007.
  * [BSZ05] Raymundo Bautista, Maria Jose Souto Salorio, and Rita Zuazua.  Almost split sequences of complexes of fixed size.  J. Algebra, 287(1):140–168, 2005.
  * [BT21] Erlend D. Børve and Paul Trygsland.  A theorem of Retakh for exact $\infty$-categories and higher extension functors.  Preprint, arXiv:2110.05138, 2021.
  * [BTHSS22] Raphael Bennett-Tennenhaus, Johanne Haugland, Mads Hustad Sandøy, and Amit Shah.  The category of extensions and a characterisation of $n$-exangulated functors.  Preprint, arXiv:2205.03097, 2022.
  * [BTS21] Raphael Bennett-Tennenhaus and Amit Shah.  Transport of structure in higher homological algebra.  J. Algebra, 574:514–549, 2021.
  * [Büh10] Theo Bühler.  Exact categories.  Expo. Math., 28(1):1–69, 2010.
  * [BY13] Thomas Brüstle and Dong Yang.  Ordered exchange graphs.  In Advances in representation theory of algebras, EMS Ser. Congr. Rep., pages 135–193. Eur. Math. Soc., Zürich, 2013.
  * [Cag18] Pierre Cagne.  Towards a homotopical algebra of dependent types.  PhD thesis, Université Paris Diderot, 2018.
  * [CFZ02] Frédéric Chapoton, Sergey Fomin, and Andrei Zelevinsky.  Polytopal realizations of generalized associahedra.  Canad. Math. Bull., 45(4):537–566, 2002.
  * [Che23] Xiaofa Chen.  On exact dg categories.  PhD thesis, arXiv:2306.08231, 2023.
  * [Chh19] Saroj Chhatoi.  A note on convex realization of halohedron.  Preprint, arXiv:1910.13786, 2019.
  * [DK19] Tobias Dyckerhoff and Mikhail Kapranov.  Higher Segal spaces, volume 2244 of Lecture Notes in Mathematics.  Springer, Cham, 2019.
  * [Egg06] J.M. Egger.  Quillen model categories without equalisers or coequalisers.  Preprint, arXiv:0609808, 2006.
  * [FK10] Changjian Fu and Bernhard Keller.  On cluster algebras with coefficients and 2-Calabi-Yau categories.  Transactions of the American Mathematical Society, 362(2):859–895, 2010.
  * [Gil11] James Gillespie.  Model structures on exact categories.  J. Pure Appl. Algebra, 215(12):2892–2902, 2011.
  * [Gil15] James Gillespie.  How to construct a Hovey triple from two cotorsion pairs.  Fund. Math., 230(3):281–289, 2015.
  * [Gil16] James Gillespie.  Hereditary abelian model categories.  Bull. Lond. Math. Soc., 48(6):895–922, 2016.
  * [GNP23] Mikhail Gorsky, Hiroyuki Nakaoka, and Yann Palu.  Hereditary extriangulated categories: silting objects, mutation, negative extensions.  Preprint, arXiv:2303.07134, 2023.
  * [Hap88] Dieter Happel.  Triangulated categories in the representation theory of finite-dimensional algebras, volume 119 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 1988.
  * [Hau21] Johanne Haugland.  The Grothendieck group of an $n$-exangulated category.  Appl. Categ. Structures, 29(3):431–446, 2021.
  * [HHZ22] Jian He, Jing He, and Panyue Zhou.  Localization of $n$-exangulated categories.  Preprint, arXiv:2205.07644, 2022.
  * [HJr19] Henrik Holm and Peter Jø rgensen.  Model categories of quiver representations.  Adv. Math., 357:106826, 46, 2019.
  * [HLN21] Martin Herschend, Yu Liu, and Hiroyuki Nakaoka.  $n$-exangulated categories (I): Definitions and fundamental properties.  J. Algebra, 570:531–586, 2021.
  * [Hov02] Mark Hovey.  Cotorsion pairs, model category structures, and representation theory.  Math. Z., 241(3):553–592, 2002.
  * [Hov07] Mark Hovey.  Cotorsion pairs and model categories.  In Interactions between homotopy theory and algebra, volume 436 of Contemp. Math., pages 277–296. Amer. Math. Soc., Providence, RI, 2007\.
  * [HvR19] Ruben Henrard and Adam-Christian van Roosmalen.  Localizations of (one-sided) exact categories.  Preprint, arXiv:1903.10861, 2019.
  * [INP18] Osamu Iyama, Hiroyuki Nakaoka, and Yann Palu.  Auslander–Reiten theory in extriangulated categories.  Preprint, arXiv:1805.03776, to appear in _Trans. Amer. Math. Soc._ , 2018.
  * [IY08] Osamu Iyama and Yuji Yoshino.  Mutation in triangulated categories and rigid Cohen–Macaulay modules.  Inventiones mathematicae, 172(1):117–168, 2008.
  * [IY20] Osamu Iyama and Dong Yang.  Quotients of triangulated categories and equivalences of Buchweitz, Orlov, and Amiot-Guo-Keller.  Amer. J. Math., 142(5):1641–1659, 2020.
  * [Jin20] Haibo Jin.  Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations.  Adv. Math., 374:107338, 59, 2020.
  * [JKPW23] Gustavo Jasso, Sondre Kvamme, Yann Palu, and Tashi Walde.  Homotopical algebra in exact $\infty$-categories.  Work in progress, 2023.
  * [JL22] Mrunmay Jagadale and Alok Laddha.  Towards positive geometry of multi scalar field amplitudes. Accordiohedron and effective field theory.  J. High Energy Phys., (4):Paper No. 100, 37, 2022.
  * [JM17] Lucie Jacquet-Malo.  Model categories structures from rigid objects in exact categories.  Preprint, arXiv:1706.06530, 2017.
  * [Kel05] Bernhard Keller.  On triangulated orbit categories.  Documenta Mathematica, 10:551–581, 2005.
  * [Kim20] Yuta Kimura.  Tilting theory of noetherian algebras.  Preprint, arXiv:2006.01677, 2020.
  * [Kla22] Carlo Klapproth.  $n$-extension closed subcategories of $n$-exangulated categories.  Preprint, arXiv:2209.01128, 2022.
  * [Kle22] Jona Klemenc.  The stable hull of an exact $\infty$-category.  Homology, Homotopy and Applications, 24(2):195–220, 2022.
  * [McC17] Thomas McConville.  Lattice structure of Grid-Tamari orders.  J. Combin. Theory Ser. A, 148:27–56, 2017.
  * [McM73] Peter McMullen.  Representations of polytopes and polyhedral sets.  Geometriae Dedicata, 2:83–99, 1973.
  * [MP12] J. Peter May and Kathleen Ponto.  More concise algebraic topology.  Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2012.  Localization, completion, and model categories.
  * [Msa22] Dixy Msapato.  The Karoubi envelope and weak idempotent completion of an extriangulated category.  Appl. Categ. Structures, 30(3):499–535, 2022.
  * [Nak18] Hiroyuki Nakaoka.  A simultaneous generalization of mutation and recollement of cotorsion pairs on a triangulated category.  Appl. Categ. Structures, 26(3):491–544, 2018.
  * [NOS22] Hiroyuki Nakaoka, Yasuaki Ogawa, and Arashi Sakai.  Localization of extriangulated categories.  Journal of Algebra, 611:341–398, 2022.
  * [NP19] Hiroyuki Nakaoka and Yann Palu.  Extriangulated categories, Hovey twin cotorsion pairs and model structures.  Cah. Topol. Géom. Différ. Catég., LX, 2019.
  * [NP20] Hiroyuki Nakaoka and Yann Palu.  External triangulation of the homotopy category of an exact quasi-category.  Preprint, arXiv:2004.02479, 2020.
  * [OPS18] Sebastian Opper, Pierre-Guy Plamondon, and Sibylle Schroll.  A geometric model for the derived category of gentle algebras.  Preprint, arXiv:1801.09659, 2018.
  * [Pal08] Yann Palu.  Cluster characters for 2-Calabi-Yau triangulated categories.  Ann. Inst. Fourier (Grenoble), 58(6):2221–2248, 2008.
  * [Pau08] David Pauksztello.  Compact corigid objects in triangulated categories and co-$t$-structures.  Cent. Eur. J. Math., 6(1):25–42, 2008.
  * [PPP19] Yann Palu, Vincent Pilaud, and Pierre-Guy Plamondon.  Non-kissing and non-crossing complexes for locally gentle algebras.  Journal of Combinatorial Algebra, 3(4):401–438, 2019.
  * [PPP21] Yann Palu, Vincent Pilaud, and Pierre-Guy Plamondon.  Non-kissing complexes and tau-tilting for gentle algebras.  Mem. Amer. Math. Soc., 274(1343):vii+110, 2021.
  * [PPPP19] Arnau Padrol, Yann Palu, Vincent Pilaud, and Pierre-Guy Plamondon.  Associahedra for finite type cluster algebras and minimal relations between g-vectors.  Preprint, arXiv:1906.06861, to appear in _Proc. London Maht. Soc._ , 2019.
  * [Pre17] Matthew Pressland.  A categorification of acyclic principal coefficient cluster algebras.  Preprint, arXiv:1702.05352, 2017.
  * [PZ20] David Pauksztello and Alexandra Zvonareva.  Co-$t$-structures, cotilting and cotorsion pairs.  Preprint, arXiv:2007.06536, 2020.
  * [Rum21] Wolfgang Rump.  The acyclic closure of an exact category and its triangulation.  J. Algebra, 565:402–440, 2021.
  * [Šťo13] Jan Šťovíček.  Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves.  In Advances in representation theory of algebras, EMS Ser. Congr. Rep., pages 297–367. Eur. Math. Soc., Zürich, 2013.
  * [Tac64] Hiroyuki Tachikawa.  On dominant dimensions of QF-3 algebras.  Transactions of the American Mathematical Society, 112(2):249–266, 1964.
  * [Tat22] Aran Tattar.  Torsion structures, subobjects and unique filtrations in non-abelian categories.  PhD thesis, Universität zu Köln, 2022.
  * [Wu21] Yilin Wu.  Relative cluster categories and higgs categories.  Preprint, arXiv:2109.03707, 2021.
  * [Yan15] Xiaoyan Yang.  Model structures on triangulated categories.  Glasg. Math. J., 57(2):263–284, 2015.