# Positive and negative extensions in extriangulated categories

Mikhail Gorsky  Institut für Algebra und Zahlentheorie, Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany<EMAIL_ADDRESS>stuttgart.de ,  Hiroyuki Nakaoka  Graduate School of Mathematics, Nagoya
University, Furocho, Chikusaku, Nagoya 464-8602, Japan
<EMAIL_ADDRESS>and  Yann Palu  LAMFA, Université
Picardie Jules Verne, 80039 Amiens, France<EMAIL_ADDRESS>
###### Abstract.

We initiate the study of derived functors in the setting of extriangulated
categories. By using coends, we adapt Yoneda’s theory of higher extensions to
this framework. We show that, when there are enough projectives or enough
injectives, thus defined extensions agree with the ones defined earlier via
projective or injective resolutions. For categories with enough projective or
enough injective morphisms, we prove that these are right derived functors of
the Hom-bifunctor in either argument.

Since Hom is only half-exact in each argument, it is natural to expect
“negative extensions”, i.e. its left derived functors, to exist and not
necessarily vanish. We define negative extensions with respect to the first
and to the second argument and show that they give rise to universal
$\delta$-functors, when there are enough projective or injective morphisms,
respectively. In general, they are not balanced. However, for topological
extriangulated categories, the existence of a balanced version of negative
extensions follows from combining the work of Klemenc on exact
$\infty$-categories with results of the second and third authors. We discuss
various criteria under which one has the balance of the above bifunctors on
the functorial or on the numerical levels. This happens, in particular, in the
cases of exact or triangulated categories, and also in the case of the
category of $n$-term complexes with projective components over a finite-
dimensional algebra.

Given a connected sequence of functors on an extriangulated category
$(\mathscr{C},\mathbb{E},\mathfrak{s})$, we determine the maximal relative
extriangulated structure, with respect to which the sequence is a
$\delta$-functor. We also find several equivalent criteria for the existence
of enough projective or injective morphisms in a given extriangulated
category.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgments
    2. 1.2 Notation
  2. 2 Preliminaries
    1. 2.1 Extriangulated categories
    2. 2.2 Projective deflations
    3. 2.3 Recollection of results on defects
  3. 3 Positive extensions
    1. 3.1 Definition of positive extensions
    2. 3.2 Description of elements in the coends
    3. 3.3 The notion of trivialization
    4. 3.4 Exactness for positive extensions
    5. 3.5 Positive global dimension
  4. 4 Connected sequences and $\delta$-functors
    1. 4.1 Connected sequences of functors
    2. 4.2 $\delta$-functors
    3. 4.3 Cohomology and relation to relative theories
  5. 5 Negative extensions
    1. 5.1 Negative extensions given by $\mathscr{C}$-duals
    2. 5.2 Projective deflations and acyclicity
    3. 5.3 Universality of $\mathbb{E}_{\mathrm{I}}^{\bullet}$ and $\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$
    4. 5.4 Comparison of $\mathbb{E}_{\mathrm{I}}^{\bullet}$ and $\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$
    5. 5.5 Conjectures
  6. 6 Projective resolutions for defects

##  1\. Introduction

The notion of _extriangulated categories_ was recently introduced by the
second and the third authors [NP1] as a unification of exact categories (in
the sense of Quillen [Buh, Kel, Q]) and of triangulated categories [V]. One
can also consider it as an axiomatization of full extension-closed
subcategories of triangulated categories, although it is not known whether
every extriangulated category admits an embedding (as a full extension-closed
subcategory) into a triangulated category. The datum defining an
extriangulated category consists of an additive category ${\mathcal{A}}$, a
bifunctor
$\mathbb{E}\colon{\mathcal{A}}^{\mathrm{op}}\times{\mathcal{A}}\to\mbox{Ab}$
and a so-called _realization_ correspondence $\mathfrak{s}$ that essentially
defines the class of _conflations_. This datum must then satisfy certain
axioms that simultaneously generalize axioms of exact and of triangulated
categories.

The class of extriangulated categories is stable under several natural
operations. An extension-closed subcategory of an extriangulated category
becomes extriangulated with a naturally induced structure [NP1]. Any closed
sub-bifunctor of $\mathbb{E}$ induces a _relative_ extriangulated structure,
or extriangulated _substructure_ [HLN, ZH]. An ideal quotient of an
extriangulated category by an ideal generated by morphisms that factor through
projective-injective objects [NP1] or, more generally, by an ideal generated
by morphisms that factor through morphisms with injective domain and
projective codomain [FGPPP], has an induced extriangulated structure.

Barwick [Bar] defined (additive) _exact $\infty$-categories_ as a higher
categorical framework where higher algebraic $K$-theory is well-defined and
well-behaved. Recently, the second and the third authors [NP2] proved that the
homotopy category of an exact $\infty$-category admits a natural
extriangulated structure. They called extriangulated categories admitting such
$\infty$-categorical enhancements _topological_. Klemenc [Kl] proved that each
exact $\infty$-category has a _stable hull_ , that is, it embeds into a stable
$\infty$-category, and this embedding is universal. Since the homotopy
category of a stable $\infty$-category is triangulated, this indicates that
each topological extriangulated category admits an embedding as an extension
closed subcategory into a triangulated category. Therefore, topological
extriangulated categories inherit a notion of higher positive and negative
extensions from this triangulated embedding. We study existence of such
extensions in extriangulated categories, without reference to any specific
embedding.

In Subsection 2.1 we recall the definitions and basic properties of
extriangulated categories. In particular in Subsection 2.3, we summarize known
results concerning the category of defects.

By using coends, in Section 3 we define a version of Yoneda’s theory of higher
(positive) extensions ([Buch, Buh, BH, M, Y]) for extriangulated categories.
We prove the existence of long exact sequences in both arguments and prove
that this definition agrees with the notion of higher extensions defined by
Herschend, Liu, and the second author [HLN, Section 5.1] via projective or
injective resolutions. As the main theorem of Section 3, we show the existence
of long exact sequences in the positive direction.

We use this to introduce the notion of _positive global dimension of
extriangulated categories_. For exact categories, this recovers the usual
definition. Each non-zero triangulated category has infinite global dimension.
A non-trivial example of an extriangulated category of global dimension $n$ is
given by the category of $n$-term complexes in the bounded homotopy category
$K^{b}(\operatorname{proj}\Lambda)$ of finitely generated projective modules
over a finite-dimensional algebra $\Lambda$. More general extriangulated
categories of global dimension $n$ appeared in the recent work by Liu and Zhou
[LZ].

The case of positive global dimension $1$ is quite special. This condition is
equivalent to asking the bifunctor $\mathbb{E}$ to be right exact in each
argument. We will initiate the detailed study of such categories, which we
call _hereditary_ , in the upcoming sequel [GNP]. We expect this notion to be
of great importance in representation theory of finite-dimensional algebras
and beyond, in particular, in $\tau$-tilting theory. The reason is that
$K^{[-1,0]}(\operatorname{proj}\Lambda)$ (where $\Lambda$ is a finite-
dimensional algebra) carries a natural hereditary extriangulated structure,
such that $\operatorname{\mathrm{mod}}\nolimits\Lambda$ is its costable
category. It is important to note that the (abelian) extriangulated structure
on $\operatorname{\mathrm{mod}}\nolimits\Lambda$ is not induced by this
hereditary extriangulated structure on
$K^{[-1,0]}(\operatorname{proj}\Lambda)$ whenever $\Lambda$ is not hereditary.
More generally, _extended cohearts of co-t-structures_ introduced by
Pauksztello and Zvonareva [PZ] are hereditary. Further examples of hereditary
extriangulated categories appeared in [PPPP]: these are given by
$2$-Calabi–Yau triangulated categories with a cluster tilting object $T$,
endowed with the relative extriangulated structure making $T$ projective. More
examples will be discussed in [GNP].

Generalizing standard notions from the setting of exact categories to that of
extriangulated ones, in Section 4 we define _connected sequences of functors_
and _$\delta$ -functors_ in this framework. We also consider _partial
$\delta$-functors_, for which the corresponding long sequences are bounded at
least on one side and exact everywhere except at the end terms. For example,
each half-exact functor is a partial $\delta$-functor. More precisely, we
define all these notions for sequences of bifunctors which are contravariant
in one argument and covariant in the second and considered as functors in one
of the arguments. This will be sufficient for our studies of positive and
negative extensions.

These notions clearly depend on the choice of an extriangulated structure on
an additive category. Given a connected sequence of functors for an
extriangulated category $(\mathscr{C},\mathbb{E},\mathfrak{s})$, we find in
Section 4.3 the maximal relative structure with respect to which the sequence
becomes a $\delta$-functor. Each partial $\delta$-functor can be extended in
one or two directions. For each new term, one can pose a natural universality
condition. As usual, the $n$-th term of the universal $\delta$-functor having
a half-exact functor $F$ as the degree zero term is called its $(-n)$-th left,
resp. $n$-th right derived functor, depending of the sign of $n$. If $F$ is
right or left exact, its derived functors on one side vanish.

In the abelian or even the exact setting, there are several ways to find
derived functors of a given functor $F$. One of the construction was first
considered by Cartan and Eilenberg [CE] under the name of _satellite_ functors
of $F$. This notion actually predates the notion of derived functors, but
satisfies the same universality properties. Satellite functors are defined by
an iterative procedure. The classical definition uses projective covers or
injective envelopes of a functor [BH, CE, M], see also [MR] and references
therein. They can be also defined in more general situations [J]. A slightly
more conceptual way to understand derived functors is to define them via
projective resolutions of one argument or injective resolutions of the other.
An even more conceptual approach to derived functors is to consider them as
cohomologies of objects in derived categories with respect to natural
$t$-structures.

In each extriangulated category, the Hom\- and the $\mathbb{E}$-functors are
at least half-exact in each argument (with respect to the extriangulated
structure). In this paper, we look for derived functors of the bifunctor Hom.
For categories having enough projective, resp. enough injective morphisms, we
prove that $\mathbb{E}^{n}$ is the $n$-th right derived functor of Hom
considered as a functor in the first or in the second variable, for $n>0$. The
definition of $\mathbb{E}^{n}$ in [HLN] follows the second strategy above and
uses projective or injective resolutions. Note that $\mathbb{E}$ is the first
derived functor of Hom precisely with respect to the extriangulated structure
defined by $\mathbb{E}$, this bifunctor is not the first derived functor of
Hom with respect to an arbitrary extriangulated structure on the underlying
additive category $\mathscr{C}$.

Remark that if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ does not come from an
exact category, the Hom-functor is only half-exact in each argument. That
means that the inflations $A\overset{f}{\longrightarrow}B$ are not necessarily
monomorphic, so $\mathscr{C}(-,f)\colon\mathscr{C}(-,A)\to\mathscr{C}(-,B)$
may have a non-zero kernel, and dually for deflations. On the other hand, if
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ corresponds to a triangulated
category, there naturally exist exact sequences in negative direction given by
$\mathscr{C}(-,-[n])$ $(n<0)$. In Section 5, we investigate the question of
the existence of ‘natural’ negative extensions in general extriangulated
categories.

There might be many possibilities to complete the partial covariant and
contravariant $\delta$-functors $(\mathbb{E}^{n},n>0;\delta_{\sharp})$ and
$(\mathbb{E}^{n},n>0;\delta^{\sharp})$ to actual $\delta$-functors. We discuss
some natural options in Section 4. One of them, first considered in [INP] has
projectively, resp. injectively stable morphisms as the degree zero terms. As
explained in [MR], in the case of module categories, these can be seen as the
$0$-th derived functors of the functor Hom; more precisely, as the $1$-st left
derived functors of the functor $\mathbb{E}$. We expect that the same happens
in the general extriangulated case and these functors satisfy the relevant
universal property. Let us emphasize that while $\mathbb{E}$ is the $1$-st
right derived functor of Hom (with respect to the structure
$(\mathscr{C},\mathbb{E},\mathfrak{s})$), the latter is not the $1$-st left
derived functor of $\mathbb{E}$.

In the present paper, we are more interested in the problem of defining left
derived functors of the bifunctor Hom, which we call the _negative
extensions_. In Section 5, we define a covariant and a contravariant connected
sequences of functors in negative degrees by taking the duals of the functors
$\mathbb{E}$ with respect to the first, resp. to the second argument. For
categories having enough projective, resp. injective morphisms, we prove that
they are actually $\delta$-functors and prove their universality. Thus, in a
certain reasonable generality, these give the covariant and the contravariant
version of negative extensions.

The nontrivial problem in our setting is the question of _balance_. The
functors Hom\- and $\mathbb{E}$-are in general only half-exact. There is
therefore no reason to expect their derived functors in the first argument to
agree with their derived functors in the second argument. Also, derived
categories of extriangulated categories have not yet been defined (and the
naïve definition via categories of complexes fails).

Our definition of positive extensions ensures that they give derived functors
in either variable, so right derived functors of Hom and of $\mathbb{E}$ are
_balanced_ , i.e. one has a bivariant partial $\delta$-functor in non-negative
degrees, which is universal both as a covariant and as a contravariant
$\delta$-functor among those with Hom as the degree zero term.

For the negative extensions defined above, however, we do not have such a
balance in general. We study in detail the question of the existence of the
balance. We formulate and prove precise conditions under which our
construction gives balanced derived functors. This happens, in particular, in
all exact and all triangulated categories, but also in the case of $n$-term
complexes of finitely generated projectives over a finite-dimensional algebra
(Example 3.33). One can find some natural non-examples via non-positively
graded differential graded algebras.

The work of Klemenc suggests that at least for topological extriangulated
categories $(\mathscr{C},\mathbb{E},\mathfrak{s})$, there must exist an
embedding as an extension-closed subcategory into a certain triangulated
category $\mathcal{T}$, namely into the homotopy category of the stable hull
of the exact $\infty-$categorical enhancement of
$(\mathscr{C},\mathbb{E},\mathfrak{s})$. While it is not a priori unique or
universal on the level of additive categories, it still makes sense to expect
such an embedding to behave particularly nicely, compared to arbitrary
embeddings of $(\mathscr{C},\mathbb{E},\mathfrak{s})$ as full extension-closed
subcategories into triangulated categories. The restriction of shifted Hom-
functors
$(\mathcal{T}(-,-[n])|_{\mathscr{C}^{\mathrm{op}}\times\mathscr{C}},n\in\mathbb{Z})$,
together with natural connecting morphisms, defines a bivariant
$\delta$-functor on $(\mathscr{C},\mathbb{E},\mathfrak{s})$, which has
$\mathscr{C}(-,-)$ as the degree zero term. We expect that this
$\delta$-functor is universal _among the bivariant $\delta$-functors having
$\mathscr{C}(-,-)$ as the degree zero term_. It is thus natural to expect that
its terms in negative degrees give _balanced negative extensions_ in the
category $(\mathscr{C},\mathbb{E},\mathfrak{s})$. Moreover, in Section 5.5, we
conjecture that balanced negative extensions exist even beyond the topological
case and give conjectural bounds on the degrees of the non-vanishing negative
extensions in categories of finite positive global dimension. In general, we
cannot expect the balanced negative extensions to agree with their non-
balanced covariant and contravariant counterparts. We describe a conjectural
relation between them. Explicit calculations of balanced negative extensions
in special cases will appear in the upcoming sequel [GNP]. The partial
$\delta$-functors $(\mathbb{E}^{n},n>0;\delta_{\sharp})$ and
$(\mathbb{E}^{n},n>0;\delta^{\sharp})$ and their completions to bivariant
$\delta$-functors will play an important role in the definition of Hall
algebras of extriangulated categories in the forthcoming work of the first
author [Gor]. Namely, they will be used to define so-called _defect-counting_
bilinear forms on the split Grothendieck group of the category $\mathscr{C}$
used as a part of the definition of these algebras, and also to define various
bilinear forms on the Grothendieck group of the category
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ yielding twists of the multiplication.

Balanced partial $\delta-$functors play a crucial role in the recent work by
Adachi, Enomoto, and Tsukamoto [AET]. For an extriangulated category with a
certain extra datum provided by a bifunctor and a pair of connecting
morphisms, they define _$s-$ torsion pairs_, which form a poset in a natural
way, and study intervals in this poset. They construct certain bijections
generalizing those induced by HRS-tilts of $t-$structures and certain
bijections related to $\tau-$tilting reduction. They call this extra datum of
a bifunctor and a pair of morphisms _a negative first extension_. Let us
comment on a minor difference in the notation.
$(F^{-1},\delta_{\sharp},\delta^{\sharp})$ is a ‘negative first extension’ in
their terminology if and only if $(F^{-1},\mbox{Hom})$ (or, equivalently,
$(F^{-1},\mbox{Hom},\mathbb{E})$) together with connecting morphisms form a
‘bivariant partial $\delta-$functor’ in our terminology. In other words, they
require the partial $\delta-$functor
$(F^{-1},\mbox{Hom},\delta_{\sharp},\delta^{\sharp})$ to be balanced, but do
not require it to be universal. In particular, an extriangulated category may
admit many different ‘negative first extensions’, in their terminology. This
might explain why their datum is called _a_ negative first extension, while
our derived functors are called _the_ negative extensions. Their work can be
seen as a non-trivial indication of the importance of balanced, but not
necessarily universal, ‘negative first extensions’, or, in general, of not
necessarily universal bivariant (partial) $\delta-$functors.

As mentioned above, we prove the universality for positive extensions and both
the acyclicity and the universality for negative extensions for categories
_having enough projective (resp. injective) morphisms_. These notions appeared
earlier in ideal approximation theory ([BM, FGHT, ZH]). They are slightly
weaker requirements than the existence of enough projective (resp. injective)
objects. We further explore these conditions and prove that, quite
surprisingly, they are equivalent to many interesting properties of our
category. Indeed, by Corollaries 3.26 and 6.5, the following are equivalent:

  * •

$(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective morphisms.

  * •

$\mathbb{E}(C,-)$ is finitely generated for any $C\in\mathscr{C}$.

  * •

$\mathbb{E}^{n}(C,-)$ are finitely presented for any $C\in\mathscr{C}$ and any
$n\in\mathbb{N}_{\geq 0}$.

  * •

The inclusion
$\iota\colon\operatorname{\mathrm{def}}\nolimits\mathbb{E}\hookrightarrow\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$,
of the category of defects into the category of finitely presented functors,
has a left adjoint.

In fact, the authors first noticed the importance of some of these conditions
from the perspective of the present work, and on the very late stages of work
on the paper realized that they are equivalent to a previously known notion.

###  1.1. Acknowledgments

Parts of this work were done during the first and the third authors’
participation at the Junior Trimester Program ”New Trends in Representation
Theory” at the Hausdorff Institute for Mathematics in Bonn. They are very
grateful to the Institute for the perfect working conditions. We are grateful
to Thomas Brüstle, Bernhard Keller, and Gustavo Jasso for useful discussions
and providing important references. The first author would like to thank
Steffen Koenig for his support. The second author is supported by JSPS KAKENHI
Grant Number JP20K03532. He is grateful to Norihiro Hanihara for his comments
on the $2$nd extension groups. The third author is supported by the French ANR
grant CHARMS (ANR-19-CE40-0017).

###  1.2. Notation

Throughout this article, we fix a commutative ring $R$ with $1$, and
$\mathscr{C}$ denotes an essentially small additive $R$-linear category.

##  2\. Preliminaries

###  2.1. Extriangulated categories

Let us briefly recall the definition of extriangulated categories from [NP1].

######  Definition 2.1.

Suppose $\mathscr{C}$ is equipped with an $R$-bilinear functor
$\mathbb{E}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mbox{Mod\,}R$.
For any pair of objects $A,C\in\mathscr{C}$, an element
$\delta\in\mathbb{E}(C,A)$ is called an $\mathbb{E}$-extension. In particular,
for any $A,C\in\mathscr{C}$, the zero element $0\in\mathbb{E}(C,A)$ is called
the split $\mathbb{E}$-extension.

For any $\mathbb{E}$-extension $\delta\in\mathbb{E}(C,A)$, any
$a\in\mathscr{C}(A,A^{\prime})$ and any $c\in\mathscr{C}(C^{\prime},C)$, we
abbreviate

$\mathbb{E}(C,a)(\delta)\in\mathbb{E}(C,A^{\prime})\ \ \text{and}\ \
\mathbb{E}(c,A)(\delta)\in\mathbb{E}(C^{\prime},A)$

simply to $a_{\ast}\delta$ and $c^{\ast}\delta$. Remark that we have
$\mathbb{E}(c,a)(\delta)=c^{\ast}a_{\ast}\delta=a_{\ast}c^{\ast}\delta$ in
$\mathbb{E}(C^{\prime},A^{\prime})$.

######  Definition 2.2.

Let
$\delta\in\mathbb{E}(C,A),\delta^{\prime}\in\mathbb{E}(C^{\prime},A^{\prime})$
be any pair of $\mathbb{E}$-extensions. A morphism
$(a,c)\colon\delta\to\delta^{\prime}$ of $\mathbb{E}$-extensions is a pair of
morphisms $a\in\mathscr{C}(A,A^{\prime})$ and $c\in\mathscr{C}(C,C^{\prime})$
in $\mathscr{C}$, satisfying $a_{\ast}\delta=c^{\ast}\delta^{\prime}$.

######  Definition 2.3.

Let $\delta_{1}=(A_{1},\delta_{1},C_{1}),\delta_{2}=(A_{2},\delta_{2},C_{2})$
be any pair of $\mathbb{E}$-extensions. Let

$C_{1}\overset{\iota_{1}}{\longrightarrow}C_{1}\oplus
C_{2}\overset{\iota_{2}}{\longleftarrow}C_{2}\ \text{and}\
A_{1}\overset{p_{1}}{\longleftarrow}A_{1}\oplus
A_{2}\overset{p_{2}}{\longrightarrow}A_{2}$

be coproduct and product in $\mathscr{C}$, respectively. By the additivity of
$\mathbb{E}$, we have a natural isomorphism $\mathbb{E}(C_{1}\oplus
C_{2},A_{1}\oplus
A_{2})\cong\mathbb{E}(C_{1},A_{1})\oplus\mathbb{E}(C_{1},A_{2})\oplus\mathbb{E}(C_{2},A_{1})\oplus\mathbb{E}(C_{2},A_{2})$
of $R$-modules.

Let $\delta_{1}\oplus\delta_{2}\in\mathbb{E}(C_{1}\oplus C_{2},A_{1}\oplus
A_{2})$ be the element corresponding to $(\delta_{1},0,0,\delta_{2})$ through
this isomorphism. Namely, $\delta_{1}\oplus\delta_{2}$ is the unique element
which satisfies

$\mathbb{E}(\iota_{i},p_{j})(\delta_{1}\oplus\delta_{2})=\begin{cases}\delta_{i}&\text{if}\
i=j,\\\ 0&\text{if}\ i\neq j.\end{cases}$

######  Definition 2.4.

Let $A,C\in\mathscr{C}$ be any pair of objects. Two sequences of morphisms
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C$ and
$A\overset{x^{\prime}}{\longrightarrow}B^{\prime}\overset{y^{\prime}}{\longrightarrow}C$
in $\mathscr{C}$ are said to be equivalent if there exists an isomorphism
$b\in\mathscr{C}(B,B^{\prime})$ such that $b\circ x=x^{\prime}$,
$y^{\prime}\circ b=y$. The equivalence class of
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C$ is denoted by
$[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C]$.

We also use the following notations.

  1. (1)

For any $A,C\in\mathscr{C}$, we denote as

$0=[A\overset{\Big{[}\raise
4.30554pt\hbox{\leavevmode\vtop{\halign{#\cr{$\scriptstyle{1}$}\crcr{$\scriptstyle{0}$}\crcr}}}\Big{]}}{\longrightarrow}A\oplus
C\overset{[0\ 1]}{\longrightarrow}C].$

  2. (2)

For any $[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C]$ and
$[A^{\prime}\overset{x^{\prime}}{\longrightarrow}B^{\prime}\overset{y^{\prime}}{\longrightarrow}C^{\prime}]$,
we denote as

$[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C]\oplus[A^{\prime}\overset{x^{\prime}}{\longrightarrow}B^{\prime}\overset{y^{\prime}}{\longrightarrow}C^{\prime}]=[A\oplus
A^{\prime}\overset{x\oplus x^{\prime}}{\longrightarrow}B\oplus
B^{\prime}\overset{y\oplus y^{\prime}}{\longrightarrow}C\oplus C^{\prime}].$

######  Definition 2.5.

Let $\mathscr{C}$ and $\mathbb{E}$ be as before.

  1. (1)

A realization of $\mathbb{E}$ is an association which associates an
equivalence class
$\mathfrak{s}(\delta)=[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C]$
to each $\mathbb{E}$-extension $\delta\in\mathbb{E}(C,A)$, which satisfies the
following condition $(\ast)$.

     * $(\ast)$

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

$\mathfrak{s}(\delta)=[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C]\text{
and
}\mathfrak{s}(\delta^{\prime})=[A^{\prime}\overset{x^{\prime}}{\longrightarrow}B^{\prime}\overset{y^{\prime}}{\longrightarrow}C^{\prime}].$

Then, for any morphism $(a,c)\colon\delta\to\delta^{\prime}$, there exists
some $b\in\mathscr{C}(B,B^{\prime})$ such that $b\circ x=x^{\prime}\circ a$,
$y^{\prime}\circ b=c\circ y$.

  2. (2)

A realization $\mathfrak{s}$ of $\mathbb{E}$ is additive if it satisfies the
following conditions.

     * •

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

     * •

For any pair of $\mathbb{E}$-extensions $\delta\in\mathbb{E}(C,A)$ and
$\delta^{\prime}\in\mathbb{E}(C^{\prime},A^{\prime})$, we have
$\mathfrak{s}(\delta\oplus\delta^{\prime})=\mathfrak{s}(\delta)\oplus\mathfrak{s}(\delta^{\prime})$.

######  Definition 2.6.

Let $\mathfrak{s}$ be an additive realization of $\mathbb{E}$. A sequence
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C$ is said to
realize $\delta$, whenever it satisfies
$\mathfrak{s}(\delta)=[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C]$.
We also use the following terms.

  * •

An $\mathfrak{s}$-triangle is a pair
$(A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C,\delta)$
consisting of $\delta\in\mathbb{E}(C,A)$ and a sequence
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C$ which realizes
$\delta$. We denote such a pair abbreviately by

(2.1)
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}.$

  * •

A morphism $x\in\mathscr{C}(A,B)$ is called an $\mathfrak{s}$-inflation if it
admits some $\mathfrak{s}$-triangle $(\ref{sTriabb})$.

  * •

A morphism $y\in\mathscr{C}(B,C)$ is called an $\mathfrak{s}$-deflation if it
admits some $\mathfrak{s}$-triangle $(\ref{sTriabb})$.

  * •

A sequence $A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C$ is
called an $\mathfrak{s}$-deflation if it admits some $\mathfrak{s}$-triangle
$(\ref{sTriabb})$.

When $\mathfrak{s}$ is clear from the context, we often simply say inflation
instead of $\mathfrak{s}$-inflation. Similarly for deflation and conflation.
We also abbreviately say extriangle, instead of $\mathfrak{s}$-triangle.

######  Definition 2.7.

A triplet $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is called an $R$-linear
extriangulated category if it satisfies the following conditions.

  * (ET1)

$\mathbb{E}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mbox{Mod\,}R$
is an $R$-bilinear functor.

  * (ET2)

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

  * (ET3)

For any pair of $\mathbb{E}$-extensions $\delta\in\mathbb{E}(C,A)$,
$\delta^{\prime}\in\mathbb{E}(C^{\prime},A^{\prime})$ with

$\mathfrak{s}(\delta)=[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C],\
\
\mathfrak{s}(\delta^{\prime})=[A^{\prime}\overset{x^{\prime}}{\longrightarrow}B^{\prime}\overset{y^{\prime}}{\longrightarrow}C^{\prime}]$

and any commutative square

(2.2)
$\textstyle{A}$$\textstyle{B}$$\textstyle{C}$$\textstyle{A^{\prime}}$$\textstyle{B^{\prime}}$$\textstyle{C^{\prime}}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{x^{\prime}}$$\scriptstyle{y^{\prime}}$$\scriptstyle{\circlearrowright}$

in $\mathscr{C}$, there exists a morphism
$(a,c)\colon\delta\to\delta^{\prime}$ satisfying $c\circ y=y^{\prime}\circ b$.

  * (ET3)op

Dual of (ET3).

  * (ET4)

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

$A\overset{f}{\longrightarrow}B\overset{f^{\prime}}{\longrightarrow}D\ \
\text{and}\ \
B\overset{g}{\longrightarrow}C\overset{g^{\prime}}{\longrightarrow}F$

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

(2.3)
$\textstyle{A}$$\textstyle{B}$$\textstyle{D}$$\textstyle{A}$$\textstyle{C}$$\textstyle{E}$$\textstyle{F}$$\textstyle{F}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g}$$\scriptstyle{d}$$\scriptstyle{h}$$\scriptstyle{h^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{e}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

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

    * (i)

$D\overset{d}{\longrightarrow}E\overset{e}{\longrightarrow}F$ realizes
$f^{\prime}_{\ast}\delta^{\prime}$,

    * (ii)

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

    * (iii)

$f_{\ast}\delta^{\prime\prime}=e^{\ast}\delta^{\prime}$.

  * (ET4)op

Dual of (ET4).

###  2.2. Projective deflations

######  Definition 2.8.

A morphism $f\in\mathscr{C}(X,Y)$ is said to be projective if
$\mathbb{E}(f,-)=f^{\ast}$ is zero. Projective morphisms form an ideal of
$\mathscr{C}$ ([INP, Definition 1.22]). We will denote this ideal by
$\mathcal{P}$, and the quotient $\mathscr{C}/\mathcal{P}$ will be denoted by
$\underline{\mathscr{C}}$ in the rest.

Dually, a morphism $f\in\mathscr{C}(X,Y)$ is injective if
$\mathbb{E}(-,f)=f_{\ast}$ is zero. We denote the ideal of injective morphisms
by $\mathcal{I}$, and the quotient $\mathscr{C}/\mathcal{I}$ by
$\overline{\mathscr{C}}$.

######  Remark 2.9.

The following holds. Dually for injectives.

  1. (1)

If either of $X,Y\in\mathscr{C}$ is a projective object, then any
$f\in\mathscr{C}(X,Y)$ is a projective morphism.

  2. (2)

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective objects, then
a morphism is projective if and only if it factors through a projective
object. In particular $\underline{\mathscr{C}}$ agrees with the usual stable
category of $\mathscr{C}$ in this case.

######  Definition 2.10.

A projective morphism in $\mathscr{C}$ which is also a deflation is called a
projective deflation. We say that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has
enough projective morphisms if any $C\in\mathscr{C}$ admits a projective
deflation $G\overset{g}{\longrightarrow}C$.

######  Remark 2.11.

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective objects, then
it has enough projective morphisms.

Conversely, the following holds.

######  Proposition 2.12.

If a projective deflation $G\overset{g}{\longrightarrow}C$ is right minimal,
then $G$ is a projective object.

###### Proof.

Suppose that $G\overset{g}{\longrightarrow}C$ is a right minimal projective
deflation. If $Y\overset{y}{\longrightarrow}G$ is any deflation, then $g\circ
y$ is also a deflation. By projectivity of $g$, there exists
$s\in\mathscr{C}(G,Y)$ such that $g\circ y\circ s=g$. Since $g$ is right
minimal, it follows that $y\circ s$ is an isomorphism, and that $y$ is a
retraction. Hence $G$ is projective. ∎

######  Corollary 2.13.

If every (indecomposable) $C\in\mathscr{C}$ admits a right minimal projective
deflation $G\to C$, then $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough
projective objects.

###### Proof.

This is immediate from Proposition 2.12. ∎

On the other hand, remark that if $p\colon P\to C$ is a deflation in
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ from a projective object $P$, then
$[p\ 0]\colon P\oplus X\to C$ is a projective deflation for any
$X\in\mathscr{C}$. This observation leads us to the following example in non-
Krull–Schmidt case.

######  Example 2.14.

Let $K$ be an algebraically closed field, and let $Q$ be the quiver
$1\leftarrow 2$ of type $A_{2}$. Let
$\mathcal{A}=\operatorname{\mathrm{mod}}\nolimits KQ$ be the abelian category
of finite dimensional modules over the path algebra $KQ$. Put
$P_{1}=1,P_{2}=\begin{smallmatrix}2\\\ 1\end{smallmatrix},S_{2}=2$ and
$M(l,m,n)=P_{1}^{\oplus l}\oplus P_{2}^{\oplus m}\oplus S_{2}^{\oplus n}$ for
$l,m,n\in\mathbb{N}_{\geq 0}$ for simplicity, and let $p\colon P_{2}\to S_{2}$
denote the canonical epimorphism taking the quotient by
$\operatorname{rad}P_{2}$.

Define $\mathscr{C}\subseteq\mathcal{A}$ to be the extension-closed (hence
exact) subcategory consisting of those $X\in\mathcal{A}$ whose dimension
vector $\dim X=[c_{1}\ c_{2}]$ satisfy $c_{1}\leq c_{2}$. Equivalently, if we
decompose $X$ into $X\cong M(l,m,n)$ in $\mathcal{A}$, then $X$ belongs to
$\mathscr{C}$ if and only if $n\geq l$. Especially we have
$P_{2},S_{2},P_{1}\oplus S_{2}\in\mathscr{C}$, but $P_{1}\notin\mathscr{C}$.
We see the following for $M(l,m,n)\in\mathscr{C}$.

  1. (1)

$M(0,m,0)\in\mathscr{C}$ is projective in $\mathcal{A}$, hence so is in
$\mathscr{C}$ for any $m\geq 0$.

  2. (2)

If $n\geq l>0$, then $M(l,m,n)\in\mathscr{C}$ is not projective in
$\mathscr{C}$, since
$\mbox{Ext}^{1}_{\mathcal{A}}(M(l,m,n),M(1,0,2))\cong\mbox{Ext}^{1}_{\mathcal{A}}(S_{2},P_{1})^{\oplus
n}\neq 0$.

  3. (3)

If $n\geq l>0$, then $M(l,m,n)\in\mathscr{C}$ does not admit any deflation
from a projective object in $\mathscr{C}$. Indeed, there is no epimorphism
from any $M(0,m^{\prime},n^{\prime})$ already in $\mathcal{A}$, since
$\operatorname{top}M(0,m^{\prime},n^{\prime})=S_{2}^{m^{\prime}+n^{\prime}}$.

  4. (4)

For any $n\geq l\geq 0$, there is a conflation

$M(n,0,u)\to M(l,m+n,u)\overset{q}{\longrightarrow}M(l,m,n)$

in $\mathscr{C}$ where we put $u=\max(l,n)$, isomorphic to the direct sum of
the short exact sequences

$\displaystyle 0\to P_{1}^{\oplus n}\to P_{1}^{\oplus l}\oplus P_{2}^{\oplus
m}\oplus P_{2}^{\oplus n}\overset{\mathrm{id}\oplus\mathrm{id}\oplus p^{\oplus
n}}{\longrightarrow}P_{1}^{\oplus l}\oplus P_{2}^{\oplus m}\oplus
S_{2}^{\oplus n}\to 0,$ $\displaystyle 0\to S_{2}^{\oplus
u}\overset{\mathrm{id}}{\longrightarrow}S_{2}^{\oplus u}\to 0\to 0$

in $\mathcal{A}$. In particular, the deflation $q$ is of the form

$[p^{\prime}\ 0]\colon P\oplus S\to M(l,m,n)$

for a projective module $P=P_{1}^{\oplus l}\oplus P_{2}^{\oplus m}\oplus
P_{2}^{\oplus n}$ in $\mathcal{A}$, and $q$ can be shown to be a projective
morphism in $\mathscr{C}$.

The above (1)$,\ldots$,(4) shows that $\mathscr{C}$ has enough projective
morphisms, while it does not have enough projective objects.

######  Remark 2.15.

Let $\mathcal{U}\subseteq\mathscr{C}$ be a full additive subcategory
consisting of projective-injective objects, and let $\mathscr{C}/\mathcal{U}$
denote the quotient of $\mathscr{C}$ by $\mathcal{U}$. Let
$(\mathscr{C}/\mathcal{U},\mathbb{E}/\mathcal{U},\mathfrak{s}/\mathcal{U})$ be
the induced extriangulated category ([NP1, Proposition 3.30]). Then any
projective deflation $f$ in $(\mathscr{C},\mathbb{E},\mathfrak{s})$ induces a
projective deflation in
$(\mathscr{C}/\mathcal{U},\mathbb{E}/\mathcal{U},\mathfrak{s}/\mathcal{U})$.
In particular if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms, then so does
$(\mathscr{C}/\mathcal{U},\mathbb{E}/\mathcal{U},\mathfrak{s}/\mathcal{U})$.

######  Definition 2.16.

An $\mathfrak{s}$-triangle
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$
is called dominant if $g$ is a projective deflation. Also we call an
$\mathbb{E}$-extension $\vartheta$ dominant if it is realized by a dominant
$\mathfrak{s}$-triangle. Remark that $\vartheta$ is dominant if and only if
$\vartheta^{\sharp}\colon\mathscr{C}(F,-)\to\mathbb{E}(C,-)$ is epimorphic.

Dually, an extension $\iota\in\mathbb{E}(C,A)$ is called codominant if
$\iota_{\sharp}\colon\mathscr{C}(-,C)\to\mathbb{E}(-,A)$ is epimorphic, and an
$\mathfrak{s}$-triangle is codominant if it realizes a codominant extension.

###  2.3. Recollection of results on defects

We recall the definitions of _Auslander’s defects_ [Aus], closely related to
_effaceable functors_ [Gr]. In the setting of extriangulated categories, they
were introduced by Ogawa [Og].

######  Definition 2.17.

A _right $\mathscr{C}$-module_ is a contravariant additive functor
$\mathscr{C}^{\mathrm{op}}\to\mbox{Mod\,}R$ to the category of $R$-modules.
Dually, a _left $\mathscr{C}$-module_ is a covariant additive functor to
$R$-modules. Right and left $\mathscr{C}$-modules form abelian categories
$\mbox{Mod\,}\mathscr{C}$ and $\mbox{Mod\,}\mathscr{C}^{\mathrm{op}}$,
respectively.

These categories have enough projectives, which are given by direct summands
of direct sums of representable functors
$\mathscr{C}(-,A)=\mbox{Hom}_{\mathscr{C}}(-,A)\in\mbox{Mod\,}\mathscr{C}$,
resp.
$\mathscr{C}(A,-)=\mbox{Hom}_{\mathscr{C}}(A,-)\in\mathscr{C}\mbox{Mod\,}$.

######  Definition 2.18.

A $\mathscr{C}$-module $M$ is _finitely generated_ if admits an epimorphism
$\mbox{Hom}_{\mathscr{C}}(-,X)\twoheadrightarrow M$ from a representable
functor. It is _finitely presented_ if it is a cokernel of a morphism of
representable functors. A finitely presented module is called _coherent_ if
each of its finitely generated submodule is also finitely presented.

We have the following embeddings of full additive subcategories:

$\operatorname{\mathrm{coh}}(\mathscr{C})\hookrightarrow\operatorname{\mathrm{}{fp}}\nolimits(\mathscr{C})\hookrightarrow\operatorname{\mathrm{fg}}\nolimits(\mathscr{C})\hookrightarrow\mbox{Mod\,}\mathscr{C}.$

Here the first three categories are the categories of coherent, finitely
presented and finitely generated right $\mathscr{C}$-modules, respectively.

The category of coherent modules is always abelian and closed under kernels,
cokernels and extensions in $\mbox{Mod\,}\mathscr{C}$ ([Her, Proposition 1.5],
see also [F, Appendix B]). For
$\operatorname{\mathrm{}{fp}}\nolimits(\mathscr{C})$ this holds if and only if
$\mathscr{C}$ has weak kernels.

######  Definition 2.19.

([Og, Definition 2.4]) The _contravariant defect_ of a conflation
$X\overset{f}{\rightarrowtail}Y\overset{g}{\twoheadrightarrow}Z$ is the
cokernel of the morphism
$\mbox{Hom}_{\mathscr{C}}(-,g)\colon\mbox{Hom}_{\mathscr{C}}(-,Y)\to\mbox{Hom}_{\mathscr{C}}(-,Z)$
in $\mbox{Mod\,}\mathscr{C}$. Note that it is also the kernel of
$\mathbb{E}(-,f)\colon\mathbb{E}(-,X)\to\mathbb{E}(-,Y)$.

For an extriangulated category $(\mathscr{C},\mathbb{E},\mathfrak{s})$, let
$\operatorname{\mathrm{def}}\nolimits\mathbb{E}$ be the full subcategory of
right $\mathscr{C}$-modules isomorphic to defects of conflations in
$(\mathscr{C},\mathbb{E},\mathfrak{s})$. We will also use the following
notations in this article, especially in Section 6.

######  Definition 2.20.

For any extension $\delta\in\mathbb{E}(C,A)$, put
$\Theta_{\delta}=\mbox{Im}\big{(}\delta_{\sharp}\colon\mathscr{C}(-,C)\to\mathbb{E}(-,A)\big{)}$.
This is nothing but the contravariant defect of the sequence realizing
$\delta$. We have a commutative diagram

$\textstyle{\mathscr{C}(-,C)}$$\textstyle{\underline{\mathscr{C}}(-,C)}$$\textstyle{\Theta_{\delta}}$$\textstyle{\mathbb{E}(-,A)}$$\scriptstyle{\underline{(-)}}$$\scriptstyle{\underline{\wp}_{\delta}}$$\scriptstyle{\underline{\delta}_{\sharp}}$$\scriptstyle{\wp_{\delta}}$$\scriptstyle{i_{\delta}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

such that $i_{\delta}\circ\wp_{\delta}=\delta_{\sharp}$ holds.

For any morphism $(a,c)\colon{}_{A}\delta_{C}\to{}_{B}\rho_{D}$ of
$\mathbb{E}$-extensions, there exists a unique morphism
$\eta=\eta_{(a,c)}\colon\Theta_{\delta}\to\Theta_{\rho}$ which makes

$\textstyle{\mathscr{C}(-,C)}$$\textstyle{\Theta_{\delta}}$$\textstyle{\mathbb{E}(-,A)}$$\textstyle{\mathscr{C}(-,D)}$$\textstyle{\Theta_{\rho}}$$\textstyle{\mathbb{E}(-,B)}$$\scriptstyle{\wp_{\delta}}$$\scriptstyle{i_{\delta}}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\wp_{\rho}}$$\scriptstyle{i_{\rho}}$$\scriptstyle{\rho_{\sharp}}$$\scriptstyle{c\circ-}$$\scriptstyle{\eta_{(a,c)}}$$\scriptstyle{a_{\ast}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

commutative in $\mbox{Mod\,}\mathscr{C}$. Especially, $\eta$ is an epimorphism
if $c$ is a split epimorphism, while $\eta$ is a monomorphism if $a$ is a
split monomorphism.

######  Remark 2.21.

We can also regard as
$\operatorname{\mathrm{def}}\nolimits(\mathbb{E})\subseteq\operatorname{\mathrm{}{fp}}\nolimits(\underline{\mathscr{C}})$.
Indeed, exact sequence

$\mathscr{C}(-,B)\overset{y\circ-}{\longrightarrow}\mathscr{C}(-,C)\overset{\wp_{\delta}}{\longrightarrow}\Theta_{\delta}\to
0$

in $\mbox{Mod\,}\mathscr{C}$ induces an exact sequence

$\underline{\mathscr{C}}(-,B)\overset{\underline{y}\circ-}{\longrightarrow}\underline{\mathscr{C}}(-,C)\overset{\wp_{\delta}}{\longrightarrow}\Theta_{\delta}\to
0$

in $\mbox{Mod\,}\underline{\mathscr{C}}$ for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$,
since we have $\Theta_{\delta}|_{\mathcal{P}}=0$.

The following results are known.

######  Theorem 2.22.

([En, Proposition 2.9], [Og, Proposition 2.5]) For any extriangulated category
$(\mathscr{C},\mathbb{E},\mathfrak{s})$, the category
$\operatorname{\mathrm{def}}\nolimits(\mathbb{E})$ is a Serre subcategory of
$\operatorname{\mathrm{coh}}(\mathscr{C})$.

######  Lemma 2.23.

([Og, Proposition 2.16], see also [INP, Section 4]) If
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective objects, then
the stable category $\underline{\mathscr{C}}$ has weak kernels and there exist
isomorphisms between

$\operatorname{\mathrm{def}}\nolimits\mathbb{E}\overset{\sim}{\to}\operatorname{\mathrm{}{fp}}\nolimits(\underline{\mathscr{C}})\overset{\sim}{\to}\operatorname{\mathrm{coh}}(\underline{\mathscr{C}}).$

##  3\. Positive extensions

###  3.1. Definition of positive extensions

Since $(\mbox{Mod\,}R,\otimes_{R},R)$ is a complete and cocomplete symmetric
monoidal closed category, we have the following bicategory of bimodules.

######  Definition 3.1.

Bicategory $\mathbb{B}$ is given by the following.

  * •

$0$-cells are small $R$-linear categories.

  * •

For a pair $\mathscr{A},\mathscr{B}$ of $0$-cells, the morphism category from
$\mathscr{A}$ to $\mathscr{B}$ is the category of
$\mathscr{B}$-$\mathscr{A}$-bimodules

$\mathbb{B}(\mathscr{A},\mathscr{B})=\mathscr{B}\mbox{Mod\,}\mathscr{A},$

namely, the category of $R$-linear functors
$\mathscr{A}^{\mathrm{op}}\otimes_{R}\mathscr{B}\to\mbox{Mod\,}R$. Remark that
$R$-linear functors
$\mathscr{A}^{\mathrm{op}}\otimes_{R}\mathscr{B}\to\mbox{Mod\,}R$ naturally
correspond to $R$-bilinear functors
$\mathscr{A}^{\mathrm{op}}\times\mathscr{B}\to\mbox{Mod\,}R$, bijectively.

  * •

For any $G\in\mathbb{B}(\mathscr{B},\mathscr{C})$ and
$F\in\mathbb{B}(\mathscr{A},\mathscr{B})$, their composition $G\diamond
F\in\mathbb{B}(\mathscr{A},\mathscr{C})$ is defined by using coends as

$(G\diamond
F)(X,Y)=G(-,Y)\otimes_{\mathscr{B}}F(X,-)=\int^{B\in\mathscr{B}}G(B,Y)\otimes_{R}F(X,B)$

for each $X\in\mathscr{A},Y\in\mathscr{C}$. By the universality of coends,
this gives the composition functor

$\diamond\colon\mathbb{B}(\mathscr{B},\mathscr{C})\times\mathbb{B}(\mathscr{A},\mathscr{B})\to\mathbb{B}(\mathscr{A},\mathscr{C})$

for any triplet of $0$-cells $\mathscr{A},\mathscr{B},\mathscr{C}$ in
$\mathbb{B}$.

In particular for each small $R$-linear category $\mathscr{C}$, the category
$\mathbb{B}(\mathscr{C},\mathscr{C})=\mathscr{C}\mbox{Mod\,}\mathscr{C}$ is
equipped with the structure of a monoidal category
$(\mathscr{C}\mbox{Mod\,}\mathscr{C},\diamond,\mathscr{C}(-,-))$. For any
$F\in\mathscr{C}\mbox{Mod\,}\mathscr{C}$, we define $F^{\diamond
n}\in\mathscr{C}\mbox{Mod\,}\mathscr{C}$ inductively by

$F^{\diamond n}=F^{\diamond(n-1)}\diamond F$

for any $n\in\mathbb{N}_{\geq 0}$.

In particular if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is extriangulated,
then we obtain an $R$-bilinear functor $\mathbb{E}^{n}:=\mathbb{E}^{\diamond
n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mbox{Mod\,}R$ for any
$n>0$. This is an extriangulated version of the construction given in [Y].
Also, we put $\mathbb{E}^{0}=\mbox{Hom}_{\mathscr{C}}$.

######  Remark 3.2.

By definition, if $\mathbb{E}^{n}=0$, we also have $\mathbb{E}^{k}=0$ for any
$k\geq n$.

######  Remark 3.3.

Let $\mathscr{C}$ and $F$ be as above. For any $s,t\in\mathbb{N}_{\geq 0}$, we
have a natural isomorphism $F^{\diamond(s+t)}\cong F^{\diamond s}\diamond
F^{\diamond t}$.

######  Remark 3.4.

Later in Corollary 3.21, we will see that if
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective objects or
enough injective objects, then $\mathbb{E}^{n}$ defined above becomes
isomorphic to those defined in [HLN] and [LN].

The aim of this section is to show the following.

######  Theorem 3.5.

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be a small $R$-linear
extriangulated category. Let
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$
be any $\mathfrak{s}$-triangle.

  1. (1)

We have a long exact sequence

$\displaystyle\mathscr{C}(-,A)\to\mathscr{C}(-,B)\to\mathscr{C}(-,C)\to\mathbb{E}(-,A)\to\cdots$
$\displaystyle\cdots\to\mathbb{E}^{n-1}(-,C)\to\mathbb{E}^{n}(-,A)\to\mathbb{E}^{n}(-,B)\to\mathbb{E}^{n}(-,C)\to\cdots$

in $\mbox{Mod\,}\mathscr{C}$.

  2. (2)

Dually, we have a long exact sequence

$\displaystyle\mathscr{C}(C,-)\to\mathscr{C}(B,-)\to\mathscr{C}(A,-)\to\mathbb{E}(C,-)\to\cdots$
$\displaystyle\cdots\to\mathbb{E}^{n-1}(A,-)\to\mathbb{E}^{n}(C,-)\to\mathbb{E}^{n}(B,-)\to\mathbb{E}^{n}(A,-)\to\cdots$

in $\mathscr{C}\mbox{Mod\,}$.

Since (2) can be shown dually, we will mainly deal with (1) in the rest.

###  3.2. Description of elements in the coends

Let $\mathscr{C}$ be a small $R$-linear category.

######  Definition 3.6.

Let $E\in\mbox{Mod\,}\mathscr{C}$ and $F\in\mathscr{C}\mbox{Mod\,}$ be any
pair of objects.

  1. (1)

Define a set $S=S(E,F)$ by

$S(E,F)=\coprod_{M\in\mathscr{C}}\Big{(}E(M)\times F(M)\Big{)}.$

  2. (2)

Define $\sim$ to be the equivalence relation on $S$, generated by

$(\kappa,F(f)(\lambda))\sim(E(f)(\kappa),\lambda)$

where $f\in\mathscr{C}(M,N),\kappa\in E(N),\lambda\in F(M)$ are arbitrary.

We denote by $\overline{(\rho,\lambda)}$ the equivalence class to which
$(\rho,\lambda)\in E(M)\times F(M)$ belongs.

######  Remark 3.7.

The quotient set $S/\\!\sim$ is nothing but the coend

$\int^{M\in\mathscr{C}}(U\circ E)(M)\times(U\circ F)(M)$

taken over $\mathrm{Set}$, where $U\colon\mbox{Mod\,}R\to\mathrm{Set}$ denotes
the forgetful functor.

######  Lemma 3.8.

Let $E\in\mbox{Mod\,}\mathscr{C}$ and $F\in\mathscr{C}\mbox{Mod\,}$ be any
pair of objects. Assume $\mathscr{C}$ has finite direct sums. Then the
following holds.

  1. (1)

$S/\\!\sim$ has the following structure of an $R$-module.

     * •

For $(\rho_{i},\lambda_{i})\in E(M_{i})\times F(M_{i})$ for $i=1,2$, the sum
of their equivalence classes is defined by

$\overline{(\rho_{1},\lambda_{1})}+\overline{(\rho_{2},\lambda_{2})}=\overline{([\rho_{1}\
\rho_{2}],\left[\begin{smallmatrix}\lambda_{1}\\\
\lambda_{2}\end{smallmatrix}\right])}$

where $[\rho_{1}\ \rho_{2}]=E(p_{1})(\rho_{1})+E(p_{2})(\rho_{2})\in
E(M_{1}\oplus M_{2})$ and $\left[\begin{smallmatrix}\lambda_{1}\\\
\lambda_{2}\end{smallmatrix}\right]=F(\iota_{1})(\lambda_{1})+F(\iota_{2})(\lambda_{2})\in
F(M_{1}\oplus M_{2})$ are given by using the direct sum

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     * •

For $(\rho,\lambda)\in E(M)\times F(M)$ and $r\in R$, define the
multiplication by $r$ by

$r\overline{(\rho,\lambda)}=\overline{(r\rho,\lambda)}.$

Remark that we also have

$\overline{(r\rho,\lambda)}=\overline{(E(r\cdot\mathrm{id}_{M})(\rho),\lambda)}=\overline{(\rho,F(r\cdot\mathrm{id}_{M})(\lambda))}=\overline{(\rho,r\lambda)}$

by the $R$-linearity of $E$ and $F$. Consequently, any $\rho\in E(M)$ or
$\lambda\in F(M)$ give the zero element
$0=\overline{(\rho,0)}=\overline{(0,\lambda)}$ in $S/\\!\sim$.

  2. (2)

For any $M\in\mathscr{C}$, the canonical map

$\gamma_{M}\colon E(M)\times F(M)\to S/\\!\sim\ ;\
(\rho,\lambda)\mapsto\overline{(\rho,\lambda)}$

is $R$-bilinear.

###### Proof.

(1) For $(\rho_{i},\lambda_{i})\in E(M_{i})\times F(M_{i})$ for $i=1,2$, the
twisting morphism $t=\left[\begin{array}[]{cc}0&1\\\
1&0\end{array}\right]\in\mathscr{C}(M_{1}\oplus M_{2},M_{2}\oplus M_{1})$
gives

$\displaystyle\overline{([\rho_{1}\
\rho_{2}],\left[\begin{smallmatrix}\lambda_{1}\\\
\lambda_{2}\end{smallmatrix}\right])}$ $\displaystyle=$
$\displaystyle\overline{(E(t)([\rho_{2}\
\rho_{1}]),\left[\begin{smallmatrix}\lambda_{1}\\\
\lambda_{2}\end{smallmatrix}\right])}$ $\displaystyle=$
$\displaystyle\overline{([\rho_{2}\
\rho_{1}],F(t)(\left[\begin{smallmatrix}\lambda_{1}\\\
\lambda_{2}\end{smallmatrix}\right]))}=\overline{([\rho_{2}\
\rho_{1}],\left[\begin{smallmatrix}\lambda_{2}\\\
\lambda_{1}\end{smallmatrix}\right])}.$

For any $f\in\mathscr{C}(M,N),\kappa\in E(N),\lambda\in F(M)$ and
$(\rho,\nu)\in E(L)\times F(L)$, the morphism
$f\oplus\mathrm{id}_{L}\in\mathscr{C}(M\oplus L,N\oplus L)$ gives

$([E(f)(\kappa)\ \rho],\left[\begin{smallmatrix}\lambda\\\
\nu\end{smallmatrix}\right])\sim([\kappa\
\rho],\left[\begin{smallmatrix}F(f)(\lambda)\\\ \nu\end{smallmatrix}\right]).$

One can easily check the well-definedness of the addition using these
properties. The other conditions can be also confirmed easily.

(2) For any $\rho,\rho^{\prime}\in E(M)$ and $\lambda\in F(M)$, we have

$(\rho+\rho^{\prime},\lambda)=(E(\Delta)([\rho\
\rho^{\prime}]),\lambda)\sim([\rho\ \rho^{\prime}],F(\Delta)(\lambda))=([\rho\
\rho^{\prime}],\left[\begin{smallmatrix}\lambda\\\
\lambda\end{smallmatrix}\right])$

for the diagonal morphism $\Delta\in\mathscr{C}(M,M\oplus M)$, which shows
$\overline{(\rho+\rho^{\prime},\lambda)}=\overline{(\rho,\lambda)}+\overline{(\rho^{\prime},\lambda)}$.
The other conditions are obviously satisfied. ∎

######  Proposition 3.9.

Let $E\in\mbox{Mod\,}\mathscr{C}$ and $F\in\mathscr{C}\mbox{Mod\,}$ be any
pair of objects, and assume that $\mathscr{C}$ has finite direct sums. Equip
$S/\\!\sim$ with the structure of an $R$-module given in Lemma 3.8. Then the
map

$\varphi\colon S/\\!\sim\ \to\int^{M\in\mathscr{C}}E(M)\otimes_{R}F(M)$

which sends $\overline{(\rho,\lambda)}$ to the element represented by
$\rho\otimes\lambda$, gives an isomorphism of $R$-modules.

###### Proof.

$\varphi$ is obtained as a map of sets given by the universality of the coend
$S/\\!\sim$. By Lemma 3.8 (2), we also have an $R$-linear map
$\int^{M\in\mathscr{C}}E(M)\otimes_{R}F(M)\to S/\\!\sim$, which gives the
inverse of $\varphi$. In particular $\varphi$ becomes $R$-linear. ∎

###  3.3. The notion of trivialization

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be a small $R$-linear
extriangulated category, and $F\in\mathscr{C}\mbox{Mod\,}$ be any object,
throughout this subsection. We may identify
$\int^{M\in\mathscr{C}}E(M)\otimes_{R}F(M)$ with $S/\\!\sim$ through the
isomorphism given in Proposition 3.9 in the rest.

Remark that the composition in $\mathbb{B}$ induces functors

$-\diamond
F\colon\mathbb{B}(\mathscr{C},\mathscr{C})\to\mathbb{B}(\mbox{Mod\,}R,\mathscr{C}),\quad-\diamond
F\colon\mathbb{B}(\mathscr{C},\mbox{Mod\,}R)\to\mathbb{B}(\mbox{Mod\,}R,\mbox{Mod\,}R).$

Through the equivalences

$\mathscr{C}\mbox{Mod\,}(\mbox{Mod\,}R)\simeq\mathscr{C}\mbox{Mod\,},\quad(\mbox{Mod\,}R)\mbox{Mod\,}(\mbox{Mod\,}R)\simeq\mbox{Mod\,}R$

respectively, they correspond to $R$-linear functors

$-\diamond
F\colon\mathscr{C}\mbox{Mod\,}\mathscr{C}\to\mathscr{C}\mbox{Mod\,},\quad-\diamond
F\colon\mbox{Mod\,}\mathscr{C}\to\mbox{Mod\,}R,$

which we denote by the same symbol. We also use the following notation.

######  Definition 3.10.

Put $\mathbb{E}F=\mathbb{E}\diamond F\in\mathscr{C}\mbox{Mod\,}$ for
simplicity. By definition, $\mathbb{E}F\colon\mathscr{C}\to\mbox{Mod\,}R$ is
an $R$-linear functor defined by coends, satisfying

$\mathbb{E}F(A)=\int^{M\in\mathscr{C}}\mathbb{E}(M,A)\otimes_{R}F(M)$

for any object $A\in\mathscr{C}$.

For any pair $(\rho,\lambda)\in\mathbb{E}(M,A)\times F(M)$, we denote its
equivalence class in $\mathbb{E}F(A)$ by
$\rho\cup\lambda=\overline{(\rho,\lambda)}$ in the following argument.

######  Claim 3.11.

For any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$,
the sequence

(3.1)
$F(A)\overset{F(x)}{\longrightarrow}F(B)\overset{F(y)}{\longrightarrow}F(C)\overset{\delta\cup-}{\longrightarrow}\mathbb{E}F(A)\overset{\mathbb{E}F(x)}{\longrightarrow}\mathbb{E}F(B)\overset{\mathbb{E}F(y)}{\longrightarrow}\mathbb{E}F(C)$

is a complex in $\mbox{Mod\,}R$.

###### Proof.

Remark that we have an exact sequence

$\mathscr{C}(-,A)\overset{x\circ-}{\longrightarrow}\mathscr{C}(-,B)\overset{y\circ-}{\longrightarrow}\mathscr{C}(-,C)\overset{\delta_{\sharp}}{\longrightarrow}\mathbb{E}(-,A)\overset{x_{\ast}}{\longrightarrow}\mathbb{E}(-,B)\overset{y_{\ast}}{\longrightarrow}\mathbb{E}(-,C)$

in $\mbox{Mod\,}\mathscr{C}$. Applying the additive functor $-\diamond
F\colon\mbox{Mod\,}\mathscr{C}\to\mbox{Mod\,}R$, we obtain a complex in
$\mbox{Mod\,}R$ which is naturally isomorphic to $(\ref{Complex_sTriangle})$.
∎

######  Remark 3.12.

Later in Lemma 3.18, we will show that $(\ref{Complex_sTriangle})$ becomes
exact whenever $F$ is half-exact (Definition 3.14) on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$.

The following definition generalizes the notion of _a trivialization of a
2-extension in an abelian category_ considered by Dyckerhoff in [D, Section
2.4.2].

######  Definition 3.13.

Let $F\in\mathscr{C}\mbox{Mod\,}$ be any object. For any pair of objects
$A,M\in\mathscr{C}$, an element $(\rho,\lambda)\in\mathbb{E}(M,A)\times F(M)$
is said to have a trivialization if it satisfies the following condition.

  * •

For an $\mathfrak{s}$-triangle which realizes $\rho$

(3.2)
$A\overset{p}{\longrightarrow}X\overset{q}{\longrightarrow}M\overset{\rho}{\dashrightarrow},$

there exists $\mu\in F(X)$ such that $\lambda=F(q)(\mu)$.

Obviously, this condition does not depend on the choice of an
$\mathfrak{s}$-triangle $(\ref{sTri_rho})$ realizing $\rho$. Abbreviately, we
call $\mu$ a trivialization of the pair $(\rho,\lambda)$ with respect to
$(\ref{sTri_rho})$.

######  Definition 3.14.

([Og, Definition 2.7]) An object $F\in\mathscr{C}\mbox{Mod\,}$ is said to be
half-exact on $(\mathscr{C},\mathbb{E},\mathfrak{s})$ if

$F(A)\overset{F(x)}{\longrightarrow}F(B)\overset{F(y)}{\longrightarrow}F(C)$

is exact for any $\mathfrak{s}$-conflation
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C$.

######  Lemma 3.15.

Assume that $F\in\mathscr{C}\mbox{Mod\,}$ is half-exact on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$. Let $f\in\mathscr{C}(M,N)$ be any
morphism, and let $\rho\in\mathbb{E}(N,A)$ and $\lambda\in F(M)$ be any pair
of elements. Then $(f^{\ast}\rho,\lambda)\in\mathbb{E}(M,A)\times F(M)$ has a
trivialization if and only if $(\rho,F(f)(\lambda))\in\mathbb{E}(N,A)\times
F(N)$ has a trivialization.

###### Proof.

Realize $\rho$ and $f^{\ast}\rho$ to obtain a morphism of
$\mathfrak{s}$-triangles

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3.0pt}}}}}}\ignorespaces{\hbox{\kern-7.53471pt\raise
17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}\ignorespaces{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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14.34918pt\raise 0.00015pt\hbox{\hbox{\kern
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so that

$X\overset{\left[\begin{smallmatrix}-q\\\
g\end{smallmatrix}\right]}{\longrightarrow}M\oplus Y\overset{[f\
t]}{\longrightarrow}N\overset{p_{\ast}\rho}{\dashrightarrow}$

becomes an $\mathfrak{s}$-triangle (see [LN, Proposition 1.20] or [NP1,
Corollary 3.16]). Assumption on $F$ implies that

$\textstyle{F(X)}$$\textstyle{F(M)}$$\textstyle{F(Y)}$$\textstyle{F(N)}$$\scriptstyle{F(q)}$$\scriptstyle{F(g)}$$\scriptstyle{F(f)}$$\scriptstyle{F(t)}$$\scriptstyle{\circlearrowright}$

is a weak pullback. Thus the existence of $\mu\in F(X)$ such that
$F(q)(\mu)=\lambda$ is equivalent to the existence of $\nu\in F(Y)$ such that
$F(t)(\nu)=F(f)(\lambda)$. ∎

######  Definition 3.16.

Assume that $F\in\mathscr{C}\mbox{Mod\,}$ is half-exact on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$. We say that
$\alpha=\rho\cup\lambda\in\mathbb{E}F(A)$ has a trivialization if its
representative $(\rho,\lambda)\in\mathbb{E}(M,A)\times F(M)$ has a
trivialization. By Lemma 3.15, this does not depend on the choice of a
representative.

######  Proposition 3.17.

Assume that $F\in\mathscr{C}\mbox{Mod\,}$ is half-exact on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$. For any $A,M\in\mathscr{C}$ and any
$\alpha\in\mathbb{E}F(A)$, the following are equivalent.

  1. (1)

$\alpha=0$ in $\mathbb{E}F(A)$.

  2. (2)

$\alpha$ has a trivialization.

###### Proof.

$(1)\Rightarrow(2)$ If $\alpha=0$, then we may choose
$(0,0)\in\mathbb{E}(0,A)\times F(0)$ as a representative of $\alpha$, which
has a trivialization in the obvious way.

$(2)\Rightarrow(1)$ Take a representative
$(\rho,\lambda)\in\mathbb{E}(M,A)\times F(M)$ of $\alpha$. Realize $\rho$ to
obtain an $\mathfrak{s}$-triangle
$A\overset{p}{\longrightarrow}X\overset{q}{\longrightarrow}M\overset{\rho}{\dashrightarrow}$.
If $\alpha$ has a trivialization, there exists $\mu\in F(X)$ such that
$F(q)(\mu)=\lambda$. This gives

$(\rho,\lambda)=(\rho,F(q)(\mu))\sim(q^{\ast}\rho,\mu)=(0,\mu),$

which shows $\alpha=0\cup\mu=0$. ∎

###  3.4. Exactness for positive extensions

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ and $F$ be as in Subsection 3.3.

######  Lemma 3.18.

Assume that $F\in\mathscr{C}\mbox{Mod\,}$ is half-exact on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$. Then for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$,
the sequence

$F(A)\overset{F(x)}{\longrightarrow}F(B)\overset{F(y)}{\longrightarrow}F(C)\overset{\delta\cup-}{\longrightarrow}\mathbb{E}F(A)\overset{\mathbb{E}F(x)}{\longrightarrow}\mathbb{E}F(B)\overset{\mathbb{E}F(y)}{\longrightarrow}\mathbb{E}F(C)$

is exact in $\mbox{Mod\,}R$.

###### Proof.

This is a complex by Claim 3.11, which is exact at $F(B)$ by assumption.

Exactness at $F(C)$ Let $\lambda\in F(C)$ be any element. Suppose
$\delta\cup\lambda=0$. By Proposition 3.17, this means that
$(\delta,\lambda)\in\mathbb{E}(C,A)\times F(A)$ has a trivialization, hence
there exists $\mu\in F(B)$ such that $F(y)(\mu)=\lambda$.

Exactness at $\mathbb{E}F(A)$ Let $M\in\mathscr{C}$ be any object, and let
$(\rho,\lambda)\in\mathbb{E}(M,A)\times F(M)$ be any element. Suppose that
$\mathbb{E}F(x)(\rho\cup\lambda)=(x_{\ast}\rho)\cup\lambda$ is zero in
$\mathbb{E}F(B)$. Realize $\rho$ to obtain an $\mathfrak{s}$-triangle
$A\overset{p}{\longrightarrow}X\overset{q}{\longrightarrow}M\overset{\rho}{\dashrightarrow}$.

By [NP1, Proposition 3.15], we obtain a diagram made of morphisms of
$\mathfrak{s}$-triangles

$\textstyle{A}$$\textstyle{B}$$\textstyle{C}$$\textstyle{X}$$\textstyle{Y}$$\textstyle{C}$$\textstyle{M}$$\textstyle{M}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{\delta}$$\scriptstyle{p}$$\scriptstyle{y^{\prime}}$$\scriptstyle{p_{\ast}\delta}$$\scriptstyle{q}$$\scriptstyle{q^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{x_{\ast}\rho}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

satisfying $y^{\prime\ast}\delta+q^{\prime\ast}\rho=0$. Since
$(x_{\ast}\rho,\lambda)\in\mathbb{E}(M,B)\times F(M)$ has a trivialization by
Proposition 3.17, there exists $\mu\in F(Y)$ such that
$F(q^{\prime})(\mu)=\lambda$. Then $-F(y^{\prime})(\mu)=F(-y^{\prime})(\mu)\in
F(C)$ gives

$\delta\cup(F(-y^{\prime})(\mu))=(-y^{\prime\ast}\delta)\cup\mu=(q^{\prime\ast}\rho)\cup\mu=\rho\cup(F(q^{\prime})(\mu))=\rho\cup\lambda$

in $\mathbb{E}F(A)$.

Exactness at $\mathbb{E}F(B)$ Let $M\in\mathscr{C}$ be any object, and let
$(\rho,\lambda)\in\mathbb{E}(M,B)\times F(M)$ be any element. Suppose that
$\mathbb{E}F(y)(\rho\cup\lambda)=(y_{\ast}\rho)\cup\lambda$ is zero in
$\mathbb{E}F(C)$. Realize $\rho$ to obtain an $\mathfrak{s}$-triangle
$B\overset{p}{\longrightarrow}X\overset{q}{\longrightarrow}M\overset{\rho}{\dashrightarrow}$.
By (ET4), we obtain a diagram made of morphisms of $\mathfrak{s}$-triangles

$\textstyle{A}$$\textstyle{B}$$\textstyle{C}$$\textstyle{A}$$\textstyle{X}$$\textstyle{Y}$$\textstyle{M}$$\textstyle{M}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{\delta}$$\scriptstyle{p}$$\scriptstyle{p^{\prime}}$$\scriptstyle{p\circ
x}$$\scriptstyle{y^{\prime}}$$\scriptstyle{\delta^{\prime}}$$\scriptstyle{q}$$\scriptstyle{q^{\prime}}$$\scriptstyle{\rho}$$\scriptstyle{y_{\ast}\rho}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

satisfying $x_{\ast}\delta^{\prime}=q^{\prime\ast}\rho$. Since
$(y_{\ast}\rho,\lambda)\in\mathbb{E}(M,C)\times F(M)$ has a trivialization,
there is $\mu\in F(Y)$ such that $F(q^{\prime})(\mu)=\lambda$. Then
$\delta^{\prime}\cup\mu\in\mathbb{E}F(A)$ gives

$\mathbb{E}F(x)(\delta^{\prime}\cup\mu)=(x_{\ast}\delta^{\prime})\cup\mu=(q^{\prime\ast}\rho)\cup\mu=\rho\cup(F(q^{\prime})(\mu))=\rho\cup\lambda$

in $\mathbb{E}F(B)$. ∎

Now we prove (1) in Theorem 3.5.

###### Proof of Theorem 3.5 (1).

It suffices to show that

(3.3)
$\mathbb{E}^{n-1}(-,C)\overset{\delta\cup-}{\longrightarrow}\mathbb{E}^{n}(-,A)\overset{x_{\ast}}{\longrightarrow}\mathbb{E}^{n}(-,B)\overset{y_{\ast}}{\longrightarrow}\mathbb{E}^{n}(-,C)$

is exact for any $n\in\mathbb{N}_{>0}$ by induction. Remark that the case
$n=1$ follows from [NP1, Corollary 3.12].

Suppose this was shown for $n-1$. Then in particular
$\mathbb{E}^{n-1}(X,-)\in\mathscr{C}\mbox{Mod\,}$ is half-exact on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$, for any $X\in\mathscr{C}$. Lemma 3.18
applied to $F=\mathbb{E}^{n-1}(X,-)$ shows that

(3.4)
$F(C)\overset{\delta\cup-}{\longrightarrow}\mathbb{E}F(A)\overset{\mathbb{E}F(x)}{\longrightarrow}\mathbb{E}F(B)\overset{\mathbb{E}F(y)}{\longrightarrow}\mathbb{E}F(C)$

is exact in $\mbox{Mod\,}R$. Since
$\mathbb{E}^{n}\cong\mathbb{E}\diamond\mathbb{E}^{n-1}$ in
$\mathscr{C}\mbox{Mod\,}\mathscr{C}$, we have a natural isomorphism
$\mathbb{E}F\cong\mathbb{E}^{n}(X,-)$ in $\mathscr{C}\mbox{Mod\,}$. Through
this isomorphism, the sequence $(\ref{ExSeqFor_nX})$ becomes isomorphic to

$\mathbb{E}^{n-1}(X,C)\overset{\delta\cup-}{\longrightarrow}\mathbb{E}^{n}(X,A)\overset{x_{\ast}}{\longrightarrow}\mathbb{E}^{n}(X,B)\overset{y_{\ast}}{\longrightarrow}\mathbb{E}^{n}(X,C).$

As $X\in\mathscr{C}$ was arbitrary, this shows the exactness of
$(\ref{ExSeqFor_n})$. ∎

Let us see relations with projective deflations. In the rest, to make the
notations concise, we will often write

$\delta_{\sharp}=\delta\cup-\colon\mathbb{E}^{n}(-,C)\to\mathbb{E}^{n+1}(-,A)\
\ \text{and}\ \
\delta^{\sharp}=-\cup\delta\colon\mathbb{E}^{n}(A,-)\to\mathbb{E}^{n+1}(C,-)$

for any $\delta\in\mathbb{E}(C,A)$ and any $n\in\mathbb{N}_{>0}$.

######  Corollary 3.19.

Let
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$
be a dominant $\mathfrak{s}$-triangle. Then for any $n\in\mathbb{N}_{>0}$,

(3.5)
$\mathbb{E}^{n-1}(G,-)\overset{f^{\ast}}{\longrightarrow}\mathbb{E}^{n-1}(F,-)\overset{\vartheta^{\sharp}}{\longrightarrow}\mathbb{E}^{n}(C,-)\to
0$

is exact in $\mathscr{C}\mbox{Mod\,}$. Namely $\mathbb{E}^{n}(C,-)$ together
with $\vartheta^{\sharp}$ gives a cokernel of
$\mathbb{E}^{n-1}(G,-)\overset{f^{\ast}}{\longrightarrow}\mathbb{E}^{n-1}(F,-)$.

###### Proof.

This is immediate from Theorem 3.5, since $g^{\ast}=\mathbb{E}^{n}(g,-)=0$
holds for any $n\in\mathbb{N}_{>0}$ by definition. ∎

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective morphisms,
then $\\{\mathbb{E}^{n},\delta_{\sharp}\\}_{n\geq 0}$ is universal among
sequences of functors $\\{E^{n},\varepsilon^{n}\\}_{n\geq 0}$ of similar kind.
More precisely, the following holds (Below, $\\{E^{n}\\}$ is an arbitrary
sequence, for which we do not require $E^{n}=E^{\diamond n}$. For a more
general treatment of such sequences, see Subsection 4).

######  Proposition 3.20.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms. Let $E^{\bullet}=\\{E^{n}\\}_{n\geq 0}$ be a sequence of objects
$E^{n}\in\mathscr{C}\mbox{Mod\,}\mathscr{C}$ satisfying
$E^{0}=\mathscr{C}(-,-)$, equipped with a collection of morphisms
$\varepsilon^{n}_{\delta}\in\mathscr{C}(E^{n}(A,-),E^{n+1}(C,-))$ in
$\mathscr{C}\mbox{Mod\,}$ for any $\mathbb{E}$-extension
$\delta\in\mathbb{E}(C,A)$ and $n\geq 0$, which is natural with respect to
morphisms of $\mathbb{E}$-extensions. The following holds.

  1. (1)

For any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$,
there is an associated complex

(3.6)
$\cdots\overset{\varepsilon^{n-1}_{\delta}}{\longrightarrow}E^{n}(C,-)\overset{E^{n}(y,-)}{\longrightarrow}E^{n}(B,-)\overset{E^{n}(x,-)}{\longrightarrow}E^{n}(A,-)\overset{\varepsilon^{n}_{\delta}}{\longrightarrow}\cdots$

in $\mathscr{C}\mbox{Mod\,}$ starting from $\mathscr{C}(C,-)$ and continuing
to the right.

  2. (2)

The pair $\mathbb{E}^{\bullet}=\\{\mathbb{E}^{n}\\}_{n\geq 0}$ equipped with
$\delta^{\sharp}$ is universal among such pairs, in the sense that there
exists a sequence of natural transformations
$\\{\varphi^{n}\colon\mathbb{E}^{n}\to E^{n}\\}_{n\geq 0}$ which satisfies the
following conditions.

     * (i)

$\varphi^{0}=\mathrm{id}$.

     * (ii)

For any $n\geq 0$, the following diagram

$\textstyle{\mathbb{E}^{n}(A,-)}$$\textstyle{\mathbb{E}^{n+1}(C,-)}$$\textstyle{E^{n}(A,-)}$$\textstyle{E^{n+1}(C,-)}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\varphi^{n}_{A,-}}$$\scriptstyle{\varphi^{n+1}_{C,-}}$$\scriptstyle{\varepsilon_{\delta}^{n}}$$\scriptstyle{\circlearrowright}$

is commutative in $\mathscr{C}\mbox{Mod\,}$ for any
$\delta\in\mathbb{E}(C,A)$.

  3. (3)

Suppose that $E^{\bullet}$ also satisfies $E^{n}(p,-)=0$ for any $n>0$ and any
projective deflation $p$, and that it makes $(\ref{CPX0})$ exact for any
$\mathfrak{s}$-triangle. Then $\varphi^{n}\colon\mathbb{E}^{n}\to E^{n}$ is an
isomorphism for any $n\geq 0$.

Dually for the case with enough injective morphisms.

###### Proof.

(1) The naturality with respect to morphisms of extensions show that
$\varepsilon_{(a_{\ast}c^{\ast}\delta)}^{n}\colon E^{n}(A^{\prime},-)\to
E^{n+1}(C^{\prime},-)$ is equal to the composition of

$E^{n}(A^{\prime},-)\overset{E^{n}(a,-)}{\longrightarrow}E^{n}(A,-)\overset{\varepsilon_{\delta}^{n}}{\longrightarrow}E^{n+1}(C,-)\overset{E^{n+1}(c,-)}{\longrightarrow}E^{n+1}(C^{\prime},-)$

for any
$\delta\in\mathbb{E}(C,A),a\in\mathscr{C}(A,A^{\prime}),c\in\mathscr{C}(C^{\prime},C)$
and any $n\geq 0$. This shows that $\varepsilon_{0}^{n}=0$ holds for
$0\in\mathbb{E}(C,A)$, and that $(\ref{CPX0})$ indeed becomes a complex since
we have
$E^{n+1}(y,-)\circ\varepsilon_{\delta}^{n}=\varepsilon_{(y^{\ast}\delta)}^{n}=0$
and $\varepsilon_{\delta}^{n}\circ
E^{n}(x,-)=\varepsilon_{(x_{\ast}\delta)}^{n}=0$.

(2) This can be shown in the same way as for the universality of usual derived
functors on abelian or exact categories. Since we will give a similar proof in
detail for negative extensions later in Lemma 5.16 in Subsection 5.3, here we
only give a brief explanation of how to construct $\varphi^{n}$ inductively.

For $n=0$, we just put $\varphi^{0}=\mathrm{id}$. Let $n>0$, and suppose that
we have constructed $\varphi^{k}$ for $k<n$ satisfying (i),(ii). For any
$C\in\mathscr{C}$, there exists a dominant $\mathfrak{s}$-triangle
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$
by assumption. Then by the hypothesis of induction and by the universality of
cokernel, there exists $\varphi^{n}_{C,-}\colon\mathbb{E}^{n}(C,-)\to
E^{n}(C,-)$ unique so that

$\textstyle{\mathbb{E}^{n-1}(G,-)}$$\textstyle{\mathbb{E}^{n-1}(F,-)}$$\textstyle{\mathbb{E}^{n}(C,-)}$$\textstyle{0}$$\textstyle{E^{n-1}(G,-)}$$\textstyle{E^{n-1}(F,-)}$$\textstyle{E^{n}(C,-)}$$\scriptstyle{f^{\ast}}$$\scriptstyle{\vartheta^{\sharp}}$$\scriptstyle{\varphi^{n-1}_{G,-}}$$\scriptstyle{\varphi^{n-1}_{F,-}}$$\scriptstyle{\varphi^{n}_{C,-}}$$\scriptstyle{E^{n-1}(f,-)}$$\scriptstyle{\varepsilon_{\vartheta}^{n-1}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

becomes commutative. It is straightforward to check that
$\varphi^{n}=\\{\varphi_{C,-}^{n}\\}_{C\in\mathscr{C}}$ gives a natural
transformation $\varphi^{n}\colon\mathbb{E}^{n}\to E^{n}$ and that
$\varphi=\\{\varphi^{n}\\}_{n\geq 0}$ satisfies the required conditions
(i),(ii).

(3) This is obvious, since the above proof suggests that $E^{\bullet}$
possesses the same universality as $\mathbb{E}^{\bullet}$. ∎

######  Corollary 3.21.

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective objects and
enough injective objects, then $\mathbb{E}^{n}$ for $n>0$ are isomorphic to
those in [HLN, Section 5.1] or in [LN, Section 5.1].

###### Proof.

This is a particular case of Proposition 3.20 (3) and its dual. ∎

######  Remark 3.22.

As the above argument suggests, we may also harmlessly define functors
$\mathbb{E}^{n}$ for $n>0$ equipped with $\delta^{\sharp}$ for any essentially
small extriangulated category $(\mathscr{C},\mathbb{E},\mathfrak{s})$ with
enough projective morphisms, inductively by

$\mathbb{E}^{n}(C,-):=\operatorname{Cok}\big{(}\mathbb{E}^{n-1}(G,-)\overset{f^{\ast}}{\longrightarrow}\mathbb{E}^{n-1}(F,-)\big{)}$

by choosing dominant $\mathfrak{s}$-triangle
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$
for each $C\in\mathscr{C}$. The usual argument (similar to the above
proposition) shows that this is well-defined up to unique natural
isomorphisms, independently of the choices of dominant
$\mathfrak{s}$-triangles. Dually for the case with enough injective morphisms.
This alternative definition is essentially the iterative construction of
_(co)satellites_ of Hom (considered as a functor in the first, resp. in the
second argument), see [CE, M, MR] and references therein.

Proposition 3.20 also implies the following (the reader might find it
interesting to treat the case $n=2$ by direct computation):

######  Corollary 3.23.

Let $\mathscr{C}$ be a triangulated category with shift functor $\Sigma$. View
$\mathscr{C}$ as an extriangulated category with
$\mathbb{E}=\mathscr{C}(-,\Sigma-)$. Then, for any $n\geq 0$, we have
$\mathbb{E}^{n}\cong\mathscr{C}(-,\Sigma^{n}-)$.

######  Lemma 3.24.

For any $C\in\mathscr{C}$, the following are equivalent.

  1. (1)

There is a projective deflation $G\overset{g}{\longrightarrow}C$.

  2. (2)

There is a dominant $\mathbb{E}$-extension $\vartheta\in\mathbb{E}(C,F)$.

  3. (3)

There is a dominant $\mathfrak{s}$-triangle
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$.

  4. (4)

$\mathbb{E}(C,-)$ is finitely presented.

  5. (5)

$\mathbb{E}(C,-)$ is finitely generated.

###### Proof.

Equivalences (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3) are obvious. (3)
$\Rightarrow$ (4) follows from Corollary 3.19. (4) $\Rightarrow$ (5) is
trivial. It remains to show (5) $\Rightarrow$ (2). Suppose that there is an
epimorphism $\mathscr{C}(F,-)\overset{\wp}{\longrightarrow}\mathbb{E}(C,-)$.
By Yoneda lemma, $\vartheta=\wp_{F}(\mathrm{id}_{F})\in\mathbb{E}(C,F)$
satisfies $\wp=\vartheta_{\sharp}$, hence is dominant. ∎

######  Proposition 3.25.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms. Then $\mathbb{E}^{n}(C,-)$ is finitely presented for any
$C\in\mathscr{C}$ and any $n\in\mathbb{N}_{\geq 0}$.

###### Proof.

We proceed by an induction on $n\in\mathbb{N}_{\geq 0}$. This is obvious for
$n=0$ (and also shown for $n=1$ in Lemma 3.24). Assume $n>0$, and suppose that
we have shown the statement for $n-1$. Take a dominant $\mathfrak{s}$-triangle
$F\to G\to C\overset{\vartheta}{\dashrightarrow}$. By Corollary 3.19, this
induces a right exact sequence

$\mathbb{E}^{n-1}(G,-)\to\mathbb{E}^{n-1}(F,-)\overset{\vartheta^{\sharp}}{\longrightarrow}\mathbb{E}^{n}(C,-)\to
0$

in $\mathscr{C}\mbox{Mod\,}$. Since $\mathbb{E}^{n-1}(F,-)$ and
$\mathbb{E}^{n-1}(G,-)$ are finitely presented by the hypothesis, so is
$\mathbb{E}^{n}(C,-)$. ∎

######  Corollary 3.26.

For any $(\mathscr{C},\mathbb{E},\mathfrak{s})$, the following are equivalent.

  1. (1)

$(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective morphisms.

  2. (2)

$\mathbb{E}(C,-)$ is finitely generated for any $C\in\mathscr{C}$.

  3. (3)

$\mathbb{E}^{n}(C,-)$ is finitely presented for any $C\in\mathscr{C}$ and any
$n\in\mathbb{N}_{\geq 0}$.

###### Proof.

This is immediate from Lemma 3.24 and Proposition 3.25. ∎

As a corollary, existence of projective deflations is ensured for example in
the following situation.

######  Corollary 3.27.

Assume that $\mathscr{C}$ is Krull–Schmidt with a finite number of isoclasses
of indecomposables. Let $C\in\mathscr{C}$ be any object. If
$\underset{X\in\operatorname{ind}(\mathscr{C})}{\oplus}\mathbb{E}(C,X)$ is
finitely generated over $R$, then $\mathbb{E}(C,-)$ is finitely presented.
Here, $\operatorname{ind}(\mathscr{C})$ denotes a set of representatives of
the isoclasses of indecomposable objects in $\mathscr{C}$.

In particular if
$\underset{X\in\operatorname{ind}(\mathscr{C})}{\oplus}\mathbb{E}(C,X)$ is
finitely generated over $R$ for any $C\in\operatorname{ind}(\mathscr{C})$,
then $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective morphisms.

###### Proof.

Let $\\{\theta_{i}\in\mathbb{E}(C,X_{i})\mid 1\leq i\leq s\\}$ be a set of
generators of
$\underset{X\in\operatorname{ind}(\mathscr{C})}{\oplus}\mathbb{E}(C,X)$. Put
$F=\underset{1\leq i\leq s}{\oplus}X_{i}$. Realize
$\vartheta=\left[\begin{smallmatrix}\theta_{1}\\\ \vdots\\\
\theta_{s}\end{smallmatrix}\right]$ to obtain an $\mathfrak{s}$-triangle
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$.
By Lemma 3.24, it suffices to show $\mathbb{E}(g,X)=0$ for any
$X\in\mathscr{C}$. Since $\mathscr{C}$ is Krull–Schmidt, we may assume that
$X$ is indecomposable. Let $\sigma\in\mathbb{E}(C,X)$ be any element. By the
definition of $\vartheta$, there exists $x\in\mathscr{C}(F,X)$ such that
$x_{\ast}\vartheta=\sigma$. Thus we obtain
$g^{\ast}\sigma=x_{\ast}g^{\ast}\vartheta=0$, as desired. ∎

###  3.5. Positive global dimension

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category
which has $\mathbb{E}^{n}$ for $n\geq 0$. More precisely, assume that
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ is small, or essentially small with
enough projective morphisms or enough injective morphisms.

######  Definition 3.28.

We say that an extriangulated category $(\mathscr{C},\mathbb{E},\mathfrak{s})$
has _positive global dimension_ $m$, for some $m\in\mathbb{Z}_{\geq 0}$, if
$\mathbb{E}^{m}\neq 0$ and $\mathbb{E}^{m+1}=0$. Equivalently, if
$\mathbb{E}^{m}\neq 0$ and $\mathbb{E}^{n}=0$ for any $n>m$. If such $m$ does
not exist, we say that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has _infinite
positive global dimension_.

######  Remark 3.29.

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is extriangulated and if
$\mathcal{D}\subseteq\mathscr{C}$ is an extension-closed category, then
$\mathcal{D}$ can be equipped with an induced structure of an extriangulated
category $(\mathcal{D},\mathbb{E}_{\mathcal{D}},\mathfrak{s}_{\mathcal{D}})$.
Though it satisfies $\mathbb{E}_{\mathcal{D}}(X,Y)=\mathbb{E}(X,Y)$ for any
$X,Y\in\mathcal{D}$ by definition, it does not necessarily satisfy
$(\mathbb{E}_{\mathcal{D}})^{n}(X,Y)\cong\mathbb{E}^{n}(X,Y)$ for $n\geq 2$ .

Remark that while $(\mathbb{E}_{\mathcal{D}})^{m}=0$ for some
$m\in\mathbb{N}_{>0}$ should imply $(\mathbb{E}_{\mathcal{D}})^{n}=0$ for all
$n\geq m$ by definition, restrictions of
$\\{\mathbb{E}^{n}\\}_{n\in\mathbb{N}_{>0}}$ to $\mathcal{D}$ do not have this
property in general. In fact, there are plenty of examples of extension-closed
subcategories $\mathcal{D}$ of a triangulated category $\mathcal{T}$ which
satisfies $\mathcal{T}(\mathcal{D},\mathcal{D}[m])=0$ and
$\mathcal{T}(\mathcal{D},\mathcal{D}[n])\neq 0$ for $n>m>0$, as the following
example shows.

######  Example 3.30.

Let ${\mathcal{A}}$ be an abelian category. It is well known that
${\mathcal{A}}$ is the heart of the canonical $t$-structure on its bounded
derived category $\mathcal{D}^{b}({\mathcal{A}})$, and
$\mbox{Ext}_{\mathcal{A}}^{n}(A,B)={\mathcal{D}}^{b}({\mathcal{A}})(A,B[n])$
holds for all $n\in\mathbb{N},A,B\in{\mathcal{A}}$. If we now consider an
$m$-periodic derived category ${\mathcal{D}}_{m}({\mathcal{A}})$ of
${\mathcal{A}}$ (defined as the Verdier quotient of the category of
$m$-periodic complexes up to homotopy by the category of $m$-periodic acyclic
complexes), it does not have any $t$-structure. Since this category is
periodic, we have
${\mathcal{D}}_{m}({\mathcal{A}})(A,B[km])={\mathcal{D}}_{m}({\mathcal{A}})(A,B)={\mathcal{A}}(A,B)$
for any $k\in\mathbb{Z}$. In general, this does not coincide with
$\mbox{Ext}^{km}(A,B)$.

######  Example 3.31.

  * (i)

For an exact category, Definition 3.28 agrees with the usual definition of the
global dimension.

  * (ii)

Let $\mathcal{T}$ be a triangulated category and $A,B$ be a pair of objects
with $\mathcal{T}(A,B)\neq 0$. We have

$\mathbb{E}_{\mathcal{T}}^{n}(A[n],B)=\mathcal{T}(A[n],B[n])\cong\mathcal{T}(A,B)\neq
0$

for any $n>0$. Thus, any non-zero triangulated category has infinite global
dimension.

######  Example 3.32.

[LZ] Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is an extriangulated
category with enough projectives and enough injectives. Let
$(\mathcal{U},\mathcal{V})$ be a hereditary cotorsion pair on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ and
$\mathcal{W}=\mathcal{U}\cap\mathcal{V}$. Denote $\mathcal{W}=\mathcal{W}_{0}$
and define by induction
$\mathcal{W}_{i+1}:=\mbox{Cone}(\mathcal{W}_{i},\mathcal{W})$, for $i\geq 0$.
By [LZ, Lemma 5.4], the category $\mathcal{W}_{i}$ has positive global
dimension at most $i$. For $i=1,$ this notion generalizes _extended cohearts
of co- $t-$structures on triangulated categories_, considered by Pauksztello
and Zvonareva in [PZ].

######  Example 3.33.

Let $\Lambda$ be a finite-dimensional algebra and let
$\operatorname{proj}\Lambda$ be the category of finitely generated projectives
over $\Lambda$. The category $K^{[-n,0]}(\operatorname{proj}\Lambda)$ of
$n$-term complexes with projective components in
$K^{b}(\operatorname{\mathrm{mod}}\nolimits\Lambda)$ is an extriangulated
category of positive global dimension $n$. It has enough projectives given by
$\mbox{add}\,\Lambda$ and enough injectives given by $\mbox{add}\,\Lambda[n]$.

##  4\. Connected sequences and $\delta$-functors

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $R$-linear extriangulated
category, as before.

###  4.1. Connected sequences of functors

######  Definition 4.1.

Let
$F^{\bullet}=\\{F^{n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mbox{Mod\,}R\\}_{n\in\mathbb{Z}}$
be a sequence of $R$-bilinear functors. We define as follows.

  1. (1)

A covariant connected sequence of functors $(F^{\bullet},\delta_{\sharp})$ is
a pair of $F^{\bullet}$ and a collection of morphisms
$\delta_{\sharp}=\delta_{F,\sharp}^{n}\in\mathscr{C}(F^{n}(-,C),F^{n+1}(-,A))$
in $\mbox{Mod\,}\mathscr{C}$ for any $\mathbb{E}$-extension
$\delta\in\mathbb{E}(C,A)$ and $n\in\mathbb{Z}$, which is natural with respect
to morphisms of $\mathbb{E}$-extensions.

  2. (2)

A contravariant connected sequence of functors $(F^{\bullet},\delta^{\sharp})$
is a pair of $F^{\bullet}$ and a collection of morphisms
$\delta^{\sharp}=\delta_{F}^{\sharp,n}\in\mathscr{C}(F^{n}(A,-),F^{n+1}(C,-))$
in $\mbox{Mod\,}\mathscr{C}$ for any $\mathbb{E}$-extension
$\delta\in\mathbb{E}(C,A)$ and $n\in\mathbb{Z}$, which is natural with respect
to morphisms of $\mathbb{E}$-extensions.

  3. (3)

A bivariant connected sequence of functors
$(F^{\bullet},\delta_{\sharp},\delta^{\sharp})$ is a triplet in which
$(F^{\bullet},\delta_{\sharp})$ is a covariant, and
$(F^{\bullet},\delta^{\sharp})$ is a contravariant connected sequence,
respectively.

We can also consider bounded (at least on one side) sequences of functors. For
$a\in\left\\{-\infty\right\\}\cup\mathbb{Z}$ and
$b\in\mathbb{Z}\cup\left\\{+\infty\right\\}$ such that $a<b$, we define
covariant connected sequences of functors
$(F^{\bullet}=\\{F^{n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mbox{Mod\,}R\\}_{n\in[a,b]},\delta_{\sharp})$
in the same way as above. We will also denote it by
$((F^{n}),n\in[a,b],\delta_{\sharp})$ or
$((F^{n})_{n\in[a,b]},\delta_{\sharp})$ in the sequel. Similarly, we define
bounded (above, below, or on both sides) contravariant and bivariant connected
sequences. In the sequel, we abbreviately write $f_{\ast}=F^{n}(-,f)$ and
$f^{\ast}=F^{n}(f,-)$ for any morphism $f$ in $\mathscr{C}$, similarly as for
$\mathbb{E}^{\bullet}$.

######  Proposition 4.2.

The following holds.

  1. (1)

If $(F^{\bullet},\delta_{\sharp})$ is a covariant connected sequence of
functors, then

(4.1)
$\cdots\overset{\delta_{\sharp}}{\longrightarrow}F^{n}(-,A)\overset{x_{\ast}}{\longrightarrow}F^{n}(-,B)\overset{y_{\ast}}{\longrightarrow}F^{n}(-,C)\overset{\delta_{\sharp}}{\longrightarrow}\cdots$

becomes a complex in $\mbox{Mod\,}\mathscr{C}$ for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$.

  2. (2)

Similarly, if $(F^{\bullet},\delta^{\sharp})$ is a contravariant connected
sequence of functors, then

(4.2)
$\cdots\overset{\delta^{\sharp}}{\longrightarrow}F^{n}(C,-)\overset{y^{\ast}}{\longrightarrow}F^{n}(B,-)\overset{x^{\ast}}{\longrightarrow}F^{n}(A,-)\overset{\delta^{\sharp}}{\longrightarrow}\cdots$

becomes a complex in $\mathscr{C}\mbox{Mod\,}$ for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$.

###### Proof.

This can be shown in a similar way as in Proposition 3.20 (1). ∎

######  Remark 4.3.

In this article, we will mainly consider connected sequences
$(F^{\bullet},\delta_{\sharp})$ that also satisfy

  * •

$F^{n}=\mathbb{E}^{n}$, and $\delta_{\sharp}\colon F^{n}(-,C)\to F^{n+1}(-,A)$
agrees with $\delta_{\sharp}\colon\mathbb{E}^{n}(-,C)\to\mathbb{E}^{n+1}(-,A)$
for each $A,C\in\mathscr{C}$

for any $n>0$ (or for any $n\geq 0$). In such a case, we simply say that
$(F^{\bullet},\delta_{\sharp})$ _has_ $F^{n}=\mathbb{E}^{n}$ _with the
canonical connecting morphisms_ for $n>0$ (resp. for $n\geq 0$). Similarly for
contravariant/bivariant ones.

In the rest, what we state about covariant connected sequences has its
contravariant counterpart, and vice versa.

######  Notation 4.4.

Let $(F^{\bullet},\delta_{\sharp})$ be a covariant connected sequence of
functors.

Since the complex $(\ref{CPX1})$ is determined uniquely up to isomorphisms by
the extension $\delta$ and does not depend on the choice of representative of
$\mathfrak{s}(\delta)=[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C]$,
we will often abbreviate it to $\Gamma(\delta)$. Especially, its acyclicity
does not depend on the choice of representatives.

###  4.2. $\delta$-functors

######  Definition 4.5.

Let $(F^{\bullet},\delta_{\sharp})$ be an unbounded covariant connected
sequence. We say that an $\mathbb{E}$-extension $\delta$ is
$(F^{\bullet},\delta_{\sharp})$-acyclic or simply $F^{\bullet}$-acyclic if
$\Gamma(\delta)$ is acyclic (see Notation 4.4). We say that
$(F^{\bullet},\delta_{\sharp})$ is a $\delta$-functor on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ (or $\mathfrak{s}$-acyclic, or just
acyclic) if any $\mathbb{E}$-extension $\delta$ is
$(F^{\bullet},\delta_{\sharp})$-acyclic.

Similarly, an unbounded contravariant connected sequence of functors
$(F^{\bullet},\delta^{\sharp})$ is a $\delta$-functor if $(\ref{CPX2})$ is
acyclic for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$.

A bivariant connected sequence $(F^{\bullet},\delta_{\sharp},\delta^{\sharp})$
is called a bivariant $\delta$-functor if both $(F^{\bullet},\delta_{\sharp})$
and $(F^{\bullet},\delta^{\sharp})$ are $\delta$-functors.

The following is obvious from the definition.

######  Remark 4.6.

Let $(F^{\bullet},\delta_{\sharp})$ be a covariant sequence having
$F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for $n\geq 0$
which is acyclic on $(\mathscr{C},\mathbb{E},\mathfrak{s})$. Acyclicity
inherits to relative theories and extension-closed subcategories in the
following way.

  1. (1)

For any closed subfunctor $\mathbb{F}\subseteq\mathbb{E}$, if we define
$(F^{\prime\bullet},\delta^{\prime}_{\sharp})$ by replacing positive parts by
$\mathbb{F}^{n}$, then $(F^{\prime\bullet},\delta^{\prime}_{\sharp})$ is also
acyclic on $(\mathscr{C},\mathbb{F},\mathfrak{s}_{\mathbb{F}})$.

  2. (2)

For any extension-closed subcategory $\mathcal{D}\subseteq\mathscr{C}$, the
restriction of $(F^{\bullet},\delta_{\sharp})$ naturally gives an acyclic
covariant connected sequence of functors
$(F^{\bullet},\delta_{\sharp})|_{\mathcal{D}^{\mathrm{op}}\times\mathcal{D}}$
on $(\mathcal{D},\mathbb{E}_{\mathcal{D}},\mathfrak{s}_{\mathcal{D}})$.

Similarly for contravariant/bivariant ones.

It is natural to define versions of $\delta$-functors for bounded (below,
above, or on both sides) connected sequences (that have a chance to be
completed to actual $\delta$-functors).

######  Definition 4.7.

For $a\in\left\\{-\infty\right\\}\cup\mathbb{Z}$ and
$b\in\mathbb{Z}\cup\left\\{+\infty\right\\}$ such that $a<b$, we say that a
covariant connected sequences of functors
$(F^{\bullet}=\\{F^{n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mbox{Mod\,}R\\}_{n\in[a,b]},\delta_{\sharp})$
is a _partial $\delta$-functor_ if for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$
the complex

$F^{a}(-,A)\overset{F^{a}(-,x)}{\longrightarrow}F^{a}(-,B)\overset{F^{a}(-,y)}{\longrightarrow}F^{a}(-,C)\overset{\delta^{a}_{F,\sharp}}{\longrightarrow}F^{a+1}(-,A)\overset{F^{a+1}(-,x)}{\longrightarrow}\cdots\overset{F^{b}(-,y)}{\longrightarrow}F^{b}(-,C)$

is exact except at the end-terms. Similarly, we define contravariant and
bivariant partial $\delta$-functors.

######  Example 4.8.

  * •

Each bifunctor which is half-exact in one of the arguments is a partial
$\delta$-functor. In particular, Hom and $\mathbb{E}$ are half-exact in either
argument, and so they are both bivariant partial $\delta$-functors.

  * •

More generally, by the results in the previous section, each interval in the
connected sequence $(\mathbb{E}^{\bullet}=\\{\mathbb{E}^{n}\\}_{n\geq
0},\delta_{\sharp},\delta^{\sharp})$ is a bivariant partial $\delta$-functor.

######  Example 4.9.

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be as before. The following are
examples of acyclic bivariant connected sequence of functors
$(F^{\bullet},\delta_{\sharp},\delta^{\sharp})$ of $\mathbb{E}$ satisfying
$F^{0}=\mbox{Hom}_{\mathscr{C}}$.

  1. (1)

Suppose that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is exact, in the sense
that any inflation is monomorphic and any deflation is epimorphic. If we
define $F^{\bullet}$ by

$F^{n}=\begin{cases}\mathbb{E}^{n}&n>0\\\ \mbox{Hom}_{\mathscr{C}}&n=0\\\
0&n<0\end{cases},$

then it has a natural structure of acyclic bivariant connected sequence of
functors of $\mathbb{E}$, i.e. of a bivariant $\delta$-functor.

  2. (2)

Suppose that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is an extension-closed
subcategory of a triangulated category $(\mathcal{T},[1],\triangle)$. If we
define $F^{\bullet}$ by

$F^{n}=\mathcal{T}(-,-[n])|_{\mathscr{C}^{\mathrm{op}}\times\mathscr{C}}\quad(\forall
n\in\mathbb{Z}),$

then as in Remark 4.6 (2), it has a natural structure of a bivariant
$\delta$-functor on $\mathbb{E}$.

######  Example 4.10.

The following is a consequence of [INP, Theorem 1.26].

  1. (1)

Suppose that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
objects. If we define $F^{\bullet}$ by

$F^{n}=\begin{cases}\mathbb{E}^{n}&n>0\\\
\underline{\mathscr{C}}(-,\Omega^{n}(-))&n\leq 0\end{cases},$

then it has a natural structure of acyclic covariant connected sequence of
functors of $\mathbb{E}$. Here
$\Omega\colon\underline{\mathscr{C}}\to\underline{\mathscr{C}}$ denotes the
syzygy functor ([INP, Definition-Proposition 1.25]).

  2. (2)

Suppose that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough injective
objects. If we define $F^{\bullet}$ by

$F^{n}=\begin{cases}\mathbb{E}^{n}&n>0\\\
\overline{\mathscr{C}}(\Sigma^{n}(-),-)&n\leq 0\end{cases},$

then it has a natural structure of acyclic contravariant connected sequence of
functors on $\mathbb{E}$. Here
$\Sigma\colon\overline{\mathscr{C}}\to\overline{\mathscr{C}}$ denotes the
cosyzygy functor.

Given a partial $\delta$-functor, we have natural universality conditions on
its continuations. Note that the directions of natural transformations are
different for continuations to the left and to the right.

######  Definition 4.11.

Let $((F^{n}),n\in[a,b];\delta^{\sharp})$ be a partial $\delta$-functor. Let
$((E^{n}),n\in[a,c];\delta^{\sharp})$, with $c>b$, be a partial
$\delta$-functor such that $E^{i}=F^{i}$ for $i\in[a,b]$. We say that
$((E^{n}),n\in[a,c];\delta^{\sharp})$ is universal among such partial
$\delta$-functors if for each $((G^{n}),n\in[a,c];\delta^{\sharp})$ such that
$E^{i}=F^{i}$ for $i\in[a,b]$, there exists a unique sequence of natural
transformations $\\{\varphi^{n}\colon E^{n}\to G^{n}\\}_{n\in[a,c]}$ which
satisfies the following conditions.

  * (i)

$\varphi^{n}=\mathrm{id}$ for any $n\in[a,b]$.

  * (ii)

For any $n\in[a,c]$, the following diagram

$\textstyle{E^{n}(A,-)}$$\textstyle{E^{n+1}(C,-)}$$\textstyle{G^{n}(A,-)}$$\textstyle{G^{n+1}(C,-)}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\varphi^{n}_{A,-}}$$\scriptstyle{\varphi^{n+1}_{C,-}}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\circlearrowright}$

is commutative in $\mathscr{C}\mbox{Mod\,}$ for any
$\delta\in\mathbb{E}(C,A)$.

Similar definitions apply for covariant and for bivariant partial
$\delta$-functors.

######  Definition 4.12.

Let $(F^{n}),n\in[a,b];\delta^{\sharp})$ be a partial $\delta$-functor. Let
$((E^{n}),n\in[d,b];\delta^{\sharp})$, with $d<a$, be a partial
$\delta$-functor such that $E^{i}=F^{i}$ for $i\in[a,b]$. We say that
$((E^{n}),n\in[d,b];\delta^{\sharp})$ is universal among such partial
$\delta$-functors if for each $((G^{n}),n\in[d,b];\delta^{\sharp})$ such that
$E^{i}=F^{i}$ for $i\in[a,b]$, there exists a sequence of natural
transformations $\\{\varphi^{n}\colon G^{n}\to E^{n}\\}_{n\in[d,b]}$ which
satisfies the following conditions.

  * (i)

$\varphi^{n}=\mathrm{id}$ for any $n\in[a,b]$.

  * (ii)

For any $n\in[d,b]$, the following diagram

$\textstyle{G^{n}(A,-)}$$\textstyle{G^{n+1}(C,-)}$$\textstyle{E^{n}(A,-)}$$\textstyle{E^{n+1}(C,-)}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\varphi^{n}_{A,-}}$$\scriptstyle{\varphi^{n+1}_{C,-}}$$\scriptstyle{\varepsilon_{\delta}^{n}}$$\scriptstyle{\circlearrowright}$

is commutative in $\mathscr{C}\mbox{Mod\,}$ for any
$\delta\in\mathbb{E}(C,A)$.

Similar definitions apply for covariant and for bivariant partial
$\delta$-functors.

######  Remark 4.13.

Note that a bivariant partial $\delta$-functor extending a given bivariant
partial $\delta$-functor
$((F^{n})_{n\in[a,b]},\delta_{\sharp},\delta^{\sharp})$ is said to be
universal if it is universal among such partial bivariant $\delta$-functors.
It is not required to be universal among covariant partial $\delta$-functors
extending $(F^{n},\delta_{\sharp})$ or universal among contravariant partial
$\delta$-functors extending $(F^{n},\delta^{\sharp})$.

######  Definition 4.14.

Let $F$ be a bifunctor on $(\mathscr{C},\mathbb{E},\mathfrak{s})$ which is
half-exact in one of the arguments. We can consider it as a partial
$\delta$-functor with $a=b=0$.

Let $((E^{n})_{n\geq 0},\delta_{\sharp})$ (resp. $((E^{n})_{n\geq
0},\delta^{\sharp})$) be universal among partial $\delta$-functors in non-
negative degrees having $F$ in degree $0$. We say that $E^{n}$ is the _$n$ -th
right derived functor_ of $F$.

Similarly, let $((E^{n})_{n\leq 0},\delta_{\sharp})$ (resp. $((E^{n})_{n\leq
0},\delta^{\sharp})$) be universal among partial $\delta$-functors in non-
positive degrees having $F$ in degree $0$. We say that $E^{-n}$ is the _$n$
-th left derived functor_ of $F$.

######  Definition 4.15.

Let $F$ be a bifunctor which is half-exact in each argument. Let
$((E^{n})_{n\geq 0},\delta_{\sharp},\delta^{\sharp})$ be universal among
partial bivariant $\delta$-functors in non-negative degrees having $F$ in
degree $0$. We say that $E^{n}$ is the _balanced $n$-th right derived functor_
of $F$, or _bivariant $n$-th right derived functor_ of $F$.

Similarly, let $((E^{n})_{n\leq 0},\delta_{\sharp},\delta^{\sharp})$ be
universal among partial bivariant $\delta$-functors in non-positive degrees
having $F$ in degree $0$. We say that $E^{-n}$ is the _balanced $n$-th left
derived functor_ of $F$, or _bivariant $n$-th left derived functor_ of $F$.

######  Example 4.16.

By Proposition 3.20, $\mathbb{E}^{n},n>0$, is the $n$-th right derived functor
of Hom. More precisely, it is both the covariant and the contravariant derived
functor, which is then the balanced derived functor.

Later on, we will discuss left derived functors of the bifunctor Hom. We will
define its covariant and contravariant left derived functors. In general, they
do not coincide, and so do not provide the balanced derived functors. We do
expect, however, that for the bifunctor Hom, there exist balanced versions of
left derived functors. We cannot expect this for an arbitrary bifunctor which
is half-exact in each argument, but for Hom and $\mathbb{E}$, the existence of
such a balance seems probable.

###  4.3. Cohomology and relation to relative theories

Throughout this subsection, let $(F^{\bullet},\delta_{\sharp})$ be a covariant
connected sequence of functors.

######  Definition 4.17.

As in Definition 4.5, to any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$,
we may associate a complex
$\Gamma_{F}(A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow})$
in $\mbox{Mod\,}\mathscr{C}$

$\cdots\overset{\delta_{\sharp}}{\longrightarrow}F^{n}(-,A)\overset{x_{\ast}}{\longrightarrow}F^{n}(-,B)\overset{y_{\ast}}{\longrightarrow}F^{n}(-,C)\overset{\delta_{\sharp}}{\longrightarrow}F^{n+1}(-,A)\overset{x_{\ast}}{\longrightarrow}\cdots$

with $F^{0}(-,A),F^{0}(-,B),F^{0}(-,C)$ sitting in degrees $-2,-1,0$,
respectively. As before, we will often denote it abbreviately by
$\Gamma_{F}(\delta)$ in what follows.

Denote the $n$-th cohomology of $\Gamma_{F}(\delta)$ by $H_{F}^{n}(\delta)$.
By definition it is a functor
$H_{F}^{n}(\delta)\colon\mathscr{C}^{\mathrm{op}}\to\mbox{Mod\,}R$ which sends
each $X\in\mathscr{C}$ to $(H_{F}^{n}(\delta))(X)\in\mbox{Mod\,}R$, the $n$-th
cohomology of the complex of $R$-modules $(\Gamma_{F}(\delta))(X)$.

######  Remark 4.18.

If
$\mathfrak{s}(\delta)=[A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C]$,
we have

$\displaystyle H_{F}^{3n}(\delta)$ $\displaystyle\cong$
$\displaystyle\mbox{Ker}(\delta_{F,\sharp}^{n})\,/\,\mbox{Im}(F^{n-1}(-,y)),$
$\displaystyle H_{F}^{3n-1}(\delta)$ $\displaystyle\cong$
$\displaystyle\mbox{Ker}(F^{n}(-,y))\,/\,\mbox{Im}(F^{n}(-,x)),$
$\displaystyle H_{F}^{3n-2}(\delta)$ $\displaystyle\cong$
$\displaystyle\mbox{Ker}(F^{n}(-,x))\,/\,\mbox{Im}(\delta_{F,\sharp}^{n-1})$

for any $n\in\mathbb{Z}$ by definition.

######  Proposition 4.19.

For any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$,
the following holds.

  1. (1)

Let

(4.3)
$\textstyle{A}$$\textstyle{B}$$\textstyle{C}$$\textstyle{A^{\prime}}$$\textstyle{B^{\prime}}$$\textstyle{C}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{\delta}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{x^{\prime}}$$\scriptstyle{y^{\prime}}$$\scriptstyle{a_{\ast}\delta}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

be a morphism of $\mathfrak{s}$-triangles such that

(4.4) $A\overset{\left[\begin{smallmatrix}x\\\
a\end{smallmatrix}\right]}{\longrightarrow}B\oplus A^{\prime}\overset{[b\
-x^{\prime}]}{\longrightarrow}B^{\prime}\overset{y^{\prime\ast}\delta}{\dashrightarrow}$

is an $\mathfrak{s}$-triangle. Then we have a long exact sequence

$\cdots\to H_{F}^{n-1}(a_{\ast}\delta)\to H_{F}^{n}(y^{\prime\ast}\delta)\to
H_{F}^{n}(\delta)\to H_{F}^{n}(a_{\ast}\delta)\to\cdots\quad(n\in\mathbb{Z})$

of cohomologies.

  2. (2)

Let

(4.5)
$\textstyle{A}$$\textstyle{B^{\prime\prime}}$$\textstyle{C^{\prime\prime}}$$\textstyle{A}$$\textstyle{B}$$\textstyle{C}$$\scriptstyle{x^{\prime\prime}}$$\scriptstyle{y^{\prime\prime}}$$\scriptstyle{c^{\ast}\delta}$$\scriptstyle{b^{\prime}}$$\scriptstyle{c}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{\delta}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

be a morphism of $\mathfrak{s}$-triangles such that

(4.6) $B^{\prime\prime}\overset{\left[\begin{smallmatrix}-y^{\prime\prime}\\\
b^{\prime}\end{smallmatrix}\right]}{\longrightarrow}C^{\prime\prime}\oplus
B^{\prime}\overset{[c\
y]}{\longrightarrow}C^{\prime}\overset{x^{\prime\prime}_{\ast}\delta}{\dashrightarrow}$

is an $\mathfrak{s}$-triangle. Then we have a long exact sequence

$\cdots\to H_{F}^{n-1}(x^{\prime\prime}_{\ast}\delta)\to
H_{F}^{n}(c^{\ast}\delta)\to H_{F}^{n}(\delta)\to
H_{F}^{n}(x^{\prime\prime}_{\ast}\delta)\to\cdots\quad(n\in\mathbb{Z})$

of cohomologies.

###### Proof.

Since (1) can be shown in a similar way as (2), it suffices to show (2). If we
are given $(\ref{t2})$, it yields a morphism of complexes
$\varphi\colon\Gamma_{F}(c^{\ast}\delta)\to\Gamma(\delta)$. Its mapping cone
$C(\varphi)$ is naturally quasi-isomorphic to
$\Gamma_{F}(x^{\prime\prime}_{\ast}\delta)$. ∎

######  Corollary 4.20.

For any $\mathbb{E}$-extension $\delta\in\mathbb{E}(C,A)$, the following
holds.

  1. (1)

For any morphism $a\in\mathscr{C}(A,A^{\prime})$, there exists some deflation
$y^{\prime}\in\mathscr{C}(B^{\prime},C)$ which induces an exact sequence
$(\ref{e1})$.

  2. (2)

For any morphism $c\in\mathscr{C}(C^{\prime\prime},C)$, there exists some
inflation $x^{\prime\prime}\in\mathscr{C}(A,B^{\prime\prime})$ which induces
an exact sequence $(\ref{e2})$.

###### Proof.

(1) The given data gives rise to a morphism of $\mathfrak{s}$-triangle
$(\ref{t1})$ such that $(\ref{e1})$ becomes an $\mathfrak{s}$-triangle. Thus
Proposition 4.19 (1) can be applied. Similarly for (2). ∎

######  Proposition 4.21.

Let $\delta\in\mathbb{E}(C,A)$ be any $\mathbb{E}$-extension. The following
are equivalent.

  1. (1)

$x_{\ast}\delta$ is $F^{\bullet}$-acyclic for any inflation $x$ from $A$.

  2. (2)

$y^{\ast}\delta$ is $F^{\bullet}$-acyclic for any deflation $y$ to $C$.

###### Proof.

Since either of (1),(2) in particular implies that $\delta$ is
$F^{\bullet}$-acyclic, this is immediate from Corollary 4.20. ∎

######  Proposition 4.22.

Suppose that $(F^{\bullet},\delta_{\sharp})$ is given, as before. Assume that
there exists $m\in\mathbb{Z}$ such that $H^{n}(\delta)=0$, for all $n>m$ and
for all $\delta$. The following are equivalent.

  1. (1)

$F^{\bullet}$ is a $\delta$-functor on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$.

  2. (2)

For any $\mathbb{E}$-extension $\delta$, there exists an $F^{\bullet}$-acyclic
$\mathbb{E}$-extension $\theta$ and a morphism $a$ in $\mathscr{C}$ such that
$\delta=a_{\ast}\theta$.

###### Proof.

$(1)\Rightarrow(2)$ is trivial. Conversely, suppose that $(2)$ is satisfied.
Then by using Corollary 4.20, we can show that

  * •

$H^{n}(\delta)=0$ for any $\mathbb{E}$-extension $\delta$

holds for any $n\in\mathbb{Z}$ by a descending induction on $n$, since we have
$H^{n}(\delta)=0$ for $n>m$. ∎

The dual statement to Proposition 4.22 holds for
$(F^{\bullet},\delta_{\sharp})$ and $m$ such that $H^{n}(\delta)=0$, for all
$n<m$ and for all $\delta$.

The following is immediate from the above discussion.

######  Corollary 4.23.

Suppose that $(F^{\bullet},\delta_{\sharp})$ is a connected sequence of
functors. Assume that there exists $m\in\mathbb{Z}$ such that either
$((F^{n}),n>m;\delta_{\sharp})$, or $((F^{n}),n<m;\delta_{\sharp})$ is a
partial $\delta$-functor. Define $\mathbb{F}\subseteq\mathbb{E}$ by

$\displaystyle\mathbb{F}(C,A)$ $\displaystyle=\\{\delta\in\mathbb{E}(C,A)\mid
x_{\ast}\delta\ \text{is}\ F^{\bullet}\text{-acyclic for any inflation $x$
from $A$}\\}$ $\displaystyle=\\{\delta\in\mathbb{E}(C,A)\mid y^{\ast}\delta\
\text{is}\ F^{\bullet}\text{-acyclic for any deflation $y$ to $C$}\\}.$

Then $\mathbb{F}\subseteq\mathbb{E}$ is a closed sub-bifunctor. The resulting
$(\mathscr{C},\mathbb{F},\mathfrak{s}|_{\mathbb{F}})$ is the largest relative
theory on which $(F^{\bullet},\delta_{\sharp})$ becomes a $\delta$-functor.

######  Remark 4.24.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies condition (WIC)
of [NP1], meaning that if $g\circ f$ is an inflation, then $f$ is an inflation
and dually for deflations. Then
$(\mathscr{C},\mathbb{F},\mathfrak{s}|_{\mathbb{F}})$ also satisfies condition
(WIC).

##  5\. Negative extensions

In this section, we introduce covariant and contravariant versions of negative
extensions, obtained from $\mathbb{E}^{n}$ by taking $\mathscr{C}$-duals. We
discuss their balance under certain conditions and their universality. We
conclude the section by some conjectures on the existence of balanced negative
extensions in the general situation when these conditions are not necessarily
satisfied.

For simplicity, we use the following notations.

  * •

$\mathbb{E}^{0}=\mbox{Hom}_{\mathscr{C}}$ as before, hence
$\mathbb{E}^{0}(X,Y)=\mathscr{C}(X,Y)$ for any $X,Y\in\mathscr{C}$.

  * •

$f_{\ast}=f\circ-\colon\mathbb{E}^{0}(-,A)\to\mathbb{E}^{0}(-,B)$ and
$f^{\ast}=-\circ f\colon\mathbb{E}^{0}(B,-)\to\mathbb{E}^{0}(A,-)$ for any
$f\in\mathscr{C}(A,B)$.

###  5.1. Negative extensions given by $\mathscr{C}$-duals

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

######  Definition 5.1.

For each $n\in\mathbb{N}_{\geq 0}$, we define $R$-bilinear functors
$\mathbb{E}_{\mathrm{I}}^{-n},\mathbb{E}_{\mathrm{I\\!I}}^{-n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mbox{Mod\,}R$
in the following way.

  1. (1)

Using the $\mathscr{C}$-dual
$(-)^{\vee}\colon\mathscr{C}\mbox{Mod\,}\to\mbox{Mod\,}\mathscr{C}$, define
$\mathbb{E}_{\mathrm{I}}^{-n}$ by the following.

     * •

For any $A\in\mathscr{C}$, define
$\mathbb{E}_{\mathrm{I}}^{-n}(-,A)\in\mbox{Mod\,}\mathscr{C}$ by
$\mathbb{E}_{\mathrm{I}}^{-n}(-,A)=\big{(}\mathbb{E}^{n}(A,-)\big{)}^{\vee}$.

     * •

For any morphism $a\in\mathscr{C}(A,A^{\prime})$, define
$a_{\ast}=\mathbb{E}_{\mathrm{I}}^{-n}(-,a)\colon\mathbb{E}_{\mathrm{I}}^{-n}(-,A)\to\mathbb{E}_{\mathrm{I}}^{-n}(-,A^{\prime})$
by
$\mathbb{E}_{\mathrm{I}}^{-n}(-,a)=\big{(}\mathbb{E}^{n}(a,-)\big{)}^{\vee}$.

  2. (2)

Using the $\mathscr{C}$-dual
$(-)^{\vee}\colon\mbox{Mod\,}\mathscr{C}\to\mathscr{C}\mbox{Mod\,}$, define
$\mathbb{E}_{\mathrm{I\\!I}}^{-n}$ by the following.

     * •

For any $C\in\mathscr{C}$, define
$\mathbb{E}_{\mathrm{I\\!I}}^{-n}(C,-)\in\mathscr{C}\mbox{Mod\,}$ by
$\mathbb{E}_{\mathrm{I\\!I}}^{-n}(C,-)=\big{(}\mathbb{E}^{n}(-,C)\big{)}^{\vee}$.

     * •

For any morphism $c\in\mathscr{C}(C^{\prime},C)$, define
$c^{\ast}=\mathbb{E}_{\mathrm{I\\!I}}^{-n}(c,-)\colon\mathbb{E}_{\mathrm{I\\!I}}^{-n}(C,-)\to\mathbb{E}_{\mathrm{I\\!I}}^{-n}(C^{\prime},-)$
by
$\mathbb{E}_{\mathrm{I\\!I}}^{-n}(c,-)=\big{(}\mathbb{E}^{n}(-,c)\big{)}^{\vee}$.

We also put
$\mathbb{E}_{\mathrm{I}}^{n}=\mathbb{E}_{\mathrm{I\\!I}}^{n}=\mathbb{E}^{n}$
for positive integer $n$.

######  Remark 5.2.

Let $n\in\mathbb{N}_{\geq 0}$ be any non-negative integer. By definition,
$\mathbb{E}_{\mathrm{I}}^{-n}$ satisfies

$\mathbb{E}_{\mathrm{I}}^{-n}(C,A)=(\mathscr{C}\mbox{Mod\,})\big{(}\mathbb{E}^{n}(A,-),\mathscr{C}(C,-)\big{)}$

for any $A,C\in\mathscr{C}$. For any
$a\in\mathscr{C}(A,A^{\prime}),c\in\mathscr{C}(C^{\prime},C)$ and any
$\varphi\in\mathbb{E}_{\mathrm{I}}^{-n}(C,A)$, the element
$a_{\ast}c^{\ast}\varphi=c^{\ast}a_{\ast}\varphi\in\mathbb{E}_{\mathrm{I}}^{-n}(C^{\prime},A^{\prime})$
is given the composition of

$\mathbb{E}^{n}(A^{\prime},-)\overset{a^{\ast}}{\longrightarrow}\mathbb{E}^{n}(A,-)\overset{\varphi}{\longrightarrow}\mathscr{C}(C,-)\overset{-\circ
c}{\longrightarrow}\mathscr{C}(C^{\prime},-).$

It can be easily checked that
$\mathbb{E}_{\mathrm{I}}^{-n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mbox{Mod\,}R$
indeed forms an $R$-bilinear functor. Also remark that we naturally have
$\mathbb{E}_{\mathrm{I}}^{0}=\mathbb{E}^{0}$. Similarly for
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$.

The following completes
$\mathbb{E}_{\mathrm{I}}^{\bullet},\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$ to
connected sequences of functors. In non-negative degrees, these sequences
coincide with the covariant, resp. contravariant partial $\delta$-functors
given by $(\mathbb{E}^{n}),n\geq 0$ together with the canonical connecting
morphisms.

######  Definition 5.3.

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be as above.

  1. (1)

For any $\delta\in\mathbb{E}(C,A)$ and any $n\in\mathbb{N}_{\geq 0}$, define
$\delta_{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,C)\to\mathbb{E}_{\mathrm{I}}^{-n}(-,A)$
by

$\delta_{\sharp}=\big{(}\delta^{\sharp}\colon\mathbb{E}^{n}(A,-)\to\mathbb{E}^{n+1}(C,-)\big{)}^{\vee}$

using the $\mathscr{C}$-dual
$(-)^{\vee}\colon\mathscr{C}\mbox{Mod\,}\to\mbox{Mod\,}\mathscr{C}$. Together
with those $\delta_{\sharp}$ which we already have for positive parts, it
gives a covariant connected sequence
$(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$.

  2. (2)

For any $\delta\in\mathbb{E}(C,A)$ and any $n\in\mathbb{N}_{\geq 0}$, define
$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{I\\!I}}^{-(n+1)}(A,-)\to\mathbb{E}_{\mathrm{I\\!I}}^{-n}(C,-)$
by

$\delta^{\sharp}=\big{(}\delta_{\sharp}\colon\mathbb{E}^{n}(-,C)\to\mathbb{E}^{n+1}(-,A)\big{)}^{\vee}$

using the $\mathscr{C}$-dual
$(-)^{\vee}\colon\mbox{Mod\,}\mathscr{C}\to\mathscr{C}\mbox{Mod\,}$. Similarly
as in (1), it gives a contravariant connected sequence
$(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$.

In the rest we mainly deal with $\mathbb{E}_{\mathrm{I}}$, since
$\mathbb{E}_{\mathrm{I\\!I}}$ can be dealt in a similar way.

The above definitions match well with the naive ones in the following specific
cases.

######  Proposition 5.4.

For any $n\in\mathbb{N}_{>0}$ and any $X,Y\in\mathscr{C}$, the following
holds.

  1. (1)

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ corresponds to a triangulated
category $(\mathscr{C},[1],\triangle)$, then
$\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)\cong\mathscr{C}(X,Y[-n])$.

  2. (2)

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is exact, or more generally if any
inflation is monomorphic, then $\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)=0$.

###### Proof.

(1) Indeed we have

$\displaystyle\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)$ $\displaystyle=$
$\displaystyle(\mathscr{C}\mbox{Mod\,})(\mathbb{E}^{n}(Y,-),\mathscr{C}(X,-))$
$\displaystyle\cong$
$\displaystyle(\mathscr{C}\mbox{Mod\,})(\mathscr{C}(Y[-n],-),\mathscr{C}(X,-))\cong\mathscr{C}(X,Y[-n])$

by Yoneda lemma.

(2) Let
$\varphi\in\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)=(\mathscr{C}\mbox{Mod\,})(\mathbb{E}^{n}(Y,-),\mathscr{C}(X,-))$
be any element. It suffices to show that $\varphi_{Z}(\alpha)=0$ holds for any
$Z\in\mathscr{C}$ and any $\alpha\in\mathbb{E}^{n}(Y,Z)$. As before, by
Proposition 3.9, there exists $M\in\mathscr{C}$, $\rho\in\mathbb{E}(M,Z)$ and
$\theta\in\mathbb{E}^{n-1}(Y,M)$ such that $\alpha=\rho\cup\theta$. Realize
$\rho$ to obtain an $\mathfrak{s}$-triangle $Z\overset{z}{\longrightarrow}Q\to
M\overset{\rho}{\dashrightarrow}$, in which $z$ is a monomorphism. By the
naturality of $\varphi$,

$\textstyle{\mathbb{E}^{n}(Y,Z)}$$\textstyle{\mathscr{C}(X,Z)}$$\textstyle{\mathbb{E}^{n}(Y,Q)}$$\textstyle{\mathscr{C}(X,Q)}$$\scriptstyle{\varphi_{Z}}$$\scriptstyle{z_{\ast}}$$\scriptstyle{z\circ-}$$\scriptstyle{\varphi_{Q}}$$\scriptstyle{\circlearrowright}$

is commutative. In particular
$z\circ\varphi_{Z}(\alpha)=\varphi_{Q}(z_{\ast}\alpha)=0$ holds (because
$z_{\ast}\alpha=(z_{\ast}\rho)\cup\theta=0\cup\theta=0$), hence
$\varphi_{Z}(\alpha)=0$ follows from the monomorphicity of $z$. ∎

###  5.2. Projective deflations and acyclicity

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be as in the previous subsection.
The aim of this subsection is to show that the existence of enough projective
morphisms implies that $(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$
is a $\delta$-functor (Theorem 5.8).

######  Lemma 5.5.

Let
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$
be a dominant $\mathfrak{s}$-triangle. Then for any $n\in\mathbb{N}_{>0}$,

(5.1)
$0\to\mathbb{E}_{\mathrm{I}}^{-n}(-,C)\overset{\vartheta_{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-(n-1)}(-,F)\overset{f_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-(n-1)}(-,G)$

is exact in $\mbox{Mod\,}\mathscr{C}$.

###### Proof.

Sequence $(\ref{SeqNega})$ is the $\mathscr{C}$-dual of $(\ref{SeqPosi})$ in
Corollary 3.19, hence exact by the left exactness of
$(-)^{\vee}\colon\mathscr{C}\mbox{Mod\,}\to\mbox{Mod\,}\mathscr{C}$. ∎

######  Proposition 5.6.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms. Then for any morphism $x\in\mathscr{C}(X,X^{\prime})$, the
following are equivalent.

  1. (1)

$\mathbb{E}_{\mathrm{I}}^{-1}(x,-)=0$.

  2. (2)

$\mathbb{E}_{\mathrm{I}}^{-n}(x,-)=0$ for any $n\in\mathbb{N}_{>0}$.

In particular if we take $x=\mathrm{id}_{X}$ for an object $X\in\mathscr{C}$,
we see that $\mathbb{E}_{\mathrm{I}}^{-1}(X,-)=0$ holds if and only if
$\mathbb{E}_{\mathrm{I}}^{-n}(X,-)=0$ holds for any $n>0$.

###### Proof.

(2) $\Rightarrow$ (1) is trivial. Let us show the converse. Let
$C\in\mathscr{C}$ be any object. By assumption, there exists a dominant
extension $\vartheta\in\mathbb{E}(C,F)$. Then

$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(X^{\prime},C)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-(n-1)}(X^{\prime},F)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(X,C)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-(n-1)}(X,F)}$$\scriptstyle{\vartheta_{\sharp}}$$\scriptstyle{x^{\ast}}$$\scriptstyle{x^{\ast}}$$\scriptstyle{\vartheta_{\sharp}}$$\scriptstyle{\circlearrowright}$

is commutative, in which the horizontal arrows are monomorphic for any $n\geq
1$ by Lemma 5.5. Thus (2) follows from (1) by an induction on $n$. ∎

######  Corollary 5.7.

Assume that $R=K$ is a field, and that $\mathscr{C}$ is Hom-finite over $K$.
If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective morphisms,
then $\dim_{K}\mathbb{E}_{\mathrm{I}}^{n}(X,Y)<\infty$ holds for any
$X,Y\in\mathscr{C}$ and any $n\in\mathbb{Z}$.

###### Proof.

For non-negative $n$, this follows from Proposition 3.25, since
$\mathbb{E}_{\mathrm{I}}^{n}=\mathbb{E}^{n}$ by definition. For negative $n$,
this can be shown using the exactness of $(\ref{SeqNega})$ in Lemma 5.5, by an
induction on $n$. ∎

######  Theorem 5.8.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms. Then $(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$ is a
$\delta$-functor on $(\mathscr{C},\mathbb{E},\mathfrak{s})$.

###### Proof.

Consider the following statements $\mathrm{(P_{\mathit{n}})}$ and
$\mathrm{(Q_{\mathit{n}})}$ for each $n\in\mathbb{N}_{\geq 0}$.

  * $\mathrm{(P_{\mathit{n}})}$

Sequence

$\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,C)\overset{\delta_{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n}(-,A)\overset{x_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n}(-,B)$

is exact for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$.

  * $\mathrm{(Q_{\mathit{n}})}$

Sequence

$\mathbb{E}_{\mathrm{I}}^{-n}(-,A)\overset{x_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n}(-,B)\overset{y_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n}(-,C)\overset{\delta_{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-(n-1)}(-,A)$

is exact for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$.

Remark that $\mathrm{(Q_{\mathit{0}})}$ holds. By an induction on $n$, it
suffices to show the following (1) and (2) for any $n\in\mathbb{N}_{\geq 0}$.

  1. (1)

$\mathrm{(Q_{\mathit{n}})}$ implies $\mathrm{(P_{\mathit{n}})}$.

  2. (2)

$\mathrm{(P_{\mathit{n}})}$ and $\mathrm{(Q_{\mathit{n}})}$ imply
$\mathrm{(Q}_{n+1}\mathrm{)}$.

First we show (1). Suppose that $\mathrm{(Q_{\mathit{n}})}$ holds. Let
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$
be any $\mathfrak{s}$-triangle. By assumption, there exists a dominant
$\mathfrak{s}$-triangle $F\to G\to C\overset{\vartheta}{\dashrightarrow}$.
Since $g^{\ast}\delta=0$ by projectivity, we obtain a morphism

$\textstyle{F}$$\textstyle{G}$$\textstyle{C}$$\textstyle{A}$$\textstyle{B}$$\textstyle{C}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\vartheta}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{\delta}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

of $\mathfrak{s}$-triangles. This induces a commutative diagram

$\textstyle{0}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,C)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,F)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,G)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,C)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,C)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,A)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,B)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,C)}$$\scriptstyle{\vartheta_{\sharp}}$$\scriptstyle{f_{\ast}}$$\scriptstyle{g_{\ast}}$$\scriptstyle{a_{\ast}}$$\scriptstyle{b_{\ast}}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{x_{\ast}}$$\scriptstyle{y_{\ast}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in $\mbox{Mod\,}\mathscr{C}$, whose top row is exact by
$\mathrm{(Q_{\mathit{n}})}$ and Lemma 5.5. Also
$\mathbb{E}_{\mathrm{I}}^{-n}(-,A)\overset{x_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n}(-,B)\overset{y_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n}(-,C)$
is exact by $\mathrm{(Q_{\mathit{n}})}$ in the bottom row. Then we can show
easily that the middle square becomes a weak pullback, and that
$\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,C)\overset{\delta_{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n}(-,A)\overset{x_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n}(-,B)$
becomes exact as desired.

It remains to show (2). Suppose that $\mathrm{(P_{\mathit{n}})}$ and
$\mathrm{(Q_{\mathit{n}})}$ holds. Let
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$
be any $\mathfrak{s}$-triangle. Take a dominant $\mathfrak{s}$-triangle
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}B\overset{\vartheta}{\dashrightarrow}$.
By $\mathrm{(ET4)^{\mathrm{op}}}$, we obtain a commutative diagram

$\textstyle{F}$$\textstyle{F}$$\textstyle{E}$$\textstyle{G}$$\textstyle{C}$$\textstyle{A}$$\textstyle{B}$$\textstyle{C}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{x^{\prime}}$$\scriptstyle{y^{\prime}}$$\scriptstyle{\delta^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{g}$$\scriptstyle{x}$$\scriptstyle{y}$$\scriptstyle{\delta}$$\scriptstyle{x^{\ast}\vartheta}$$\scriptstyle{\vartheta}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

made of $\mathfrak{s}$-triangles. This induces a commutative diagram

$\textstyle{0}$$\textstyle{0}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,A)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,B)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,C)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,A)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-(n+1)}(-,A)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,F)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,E)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,A)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,G)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(-,G)}$$\scriptstyle{x_{\ast}}$$\scriptstyle{y_{\ast}}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\upsilon_{\sharp}}$$\scriptstyle{\delta^{\prime}_{\sharp}}$$\scriptstyle{(x^{\ast}\vartheta)_{\sharp}}$$\scriptstyle{f^{\prime}_{\ast}}$$\scriptstyle{g^{\prime}_{\ast}}$$\scriptstyle{f_{\ast}}$$\scriptstyle{x^{\prime}_{\ast}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in $\mbox{Mod\,}\mathscr{C}$. The lower row containing
$(x^{\ast}\vartheta)_{\sharp}$ is exact by $\mathrm{(P_{\mathit{n}})}$ and
$\mathrm{(Q_{\mathit{n}})}$. Two columns in the middle are exact by Lemma 5.5,
since $g$ and hence $y^{\prime}$ are projective. Thus the upper row also
becomes exact, which shows $\mathrm{(Q}_{n+1}\mathrm{)}$. ∎

######  Corollary 5.9.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms, and enough injective objects. Then the following are equivalent.

  1. (1)

$\mathbb{E}_{\mathrm{I}}^{-1}(I,-)=0$ holds for any injective object
$I\in\mathscr{C}$.

  2. (2)

$\mathbb{E}_{\mathrm{I}}^{-n}(I,-)=0$ holds for any injective object
$I\in\mathscr{C}$ and any $n>0$.

  3. (3)

$\mathbb{E}_{\mathrm{I}}^{-n}(i,-)=0$ holds for any injective inflation $i$
and any $n>0$.

  4. (4)