$\mathscr{C}(I,x)\colon\mathscr{C}(I,A)\to\mathscr{C}(I,B)$ is monomorphic for
any injective object $I\in\mathscr{C}$ and any inflation
$A\overset{x}{\longrightarrow}B$.

###### Proof.

(1) $\Leftrightarrow$ (2) follows from Proposition 5.6. (2) $\Rightarrow$ (3)
is obvious, since any injective inflation factors through an injective object
by assumption, as in Remark 2.9. Also (3) $\Rightarrow$ (2) is immediate, if
we take $i=\mathrm{id}_{I}$ in (3).

(1) $\Rightarrow$ (4) follows from the exactness of
$\mathbb{E}_{\mathrm{I}}^{-1}(I,C)\to\mathscr{C}(I,A)\overset{x\circ-}{\longrightarrow}\mathscr{C}(I,B)$
for an $\mathfrak{s}$-triangle $A\overset{x}{\longrightarrow}B\to
C\dashrightarrow$ associated to $x$, shown in Theorem 5.8.

It remains to show (4) $\Rightarrow$ (1). For any $C\in\mathscr{C}$, take a
dominant $\mathfrak{s}$-triangle $F\overset{f}{\longrightarrow}G\to
C\dashrightarrow$. Then
$0\to\mathbb{E}_{\mathrm{I}}^{-1}(I,C)\to\mathscr{C}(I,F)\overset{f\circ-}{\longrightarrow}\mathscr{C}(I,G)$
is exact by the same theorem or Lemma 5.5. Thus the monomorphicity of
$\mathscr{C}(I,F)\overset{f\circ-}{\longrightarrow}\mathscr{C}(I,G)$ implies
$\mathbb{E}_{\mathrm{I}}^{-1}(I,C)=0$. ∎

Counterparts of Theorem 5.8 and Corollary 5.9 for
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$ become as follows.

######  Theorem 5.10.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough injective
morphisms, defined dually as in Definition 2.10. Then
$(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$ is a
$\delta$-functor on $(\mathscr{C},\mathbb{E},\mathfrak{s})$.

######  Corollary 5.11.

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

  1. (1)

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

  2. (2)

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

  3. (3)

$\mathbb{E}_{\mathrm{I\\!I}}^{-n}(-,p)=0$ holds for any projective deflation
$p$ and any $n>0$.

  4. (4)

$\mathscr{C}(y,P)\colon\mathscr{C}(C,P)\to\mathscr{C}(B,P)$ is monomorphic for
any projective object $P\in\mathscr{C}$ and any deflation
$B\overset{y}{\longrightarrow}C$.

###### Proof.

These can be shown in a similar way as Theorem 5.8 and Corollary 5.9. ∎

Regarding Corollaries 5.9 and 5.11, we define conditions (NI) and (NII) as
follows.

######  Condition 5.12.

For $(\mathscr{C},\mathbb{E},\mathfrak{s})$, consider the following
conditions.

  * (NI)

$\mathscr{C}(I,x)\colon\mathscr{C}(I,A)\to\mathscr{C}(I,B)$ is monomorphic for
any injective object $I\in\mathscr{C}$ and any inflation
$A\overset{x}{\longrightarrow}B$.

  * (NII)

$\mathscr{C}(y,P)\colon\mathscr{C}(C,P)\to\mathscr{C}(B,P)$ is monomorphic for
any projective object $P\in\mathscr{C}$ and any deflation
$B\overset{y}{\longrightarrow}C$.

######  Remark 5.13.

Remark that conditions (NI), (NII) are trivially satisfied if
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ corresponds to an exact category or a
triangulated category.

Let us conclude this subsection with the following immediate corollary of
Theorem 5.8, which relates the (alternating sums of) dimensions of
$\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)$ and
$\underline{\mathscr{C}}(X,\Omega^{n}Y)$.

######  Corollary 5.14.

Assume that $R=K$ is a field and $\mathscr{C}$ is Hom-finite over $K$, and
that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective morphisms.
Suppose that $Y\in\mathscr{C}$ admits a complex in $\mathscr{C}$

(5.2)
$P_{n}\overset{d_{n}}{\longrightarrow}P_{n-1}\overset{d_{n-1}}{\longrightarrow}\cdots\overset{d_{1}}{\longrightarrow}P_{0}\overset{d_{0}}{\longrightarrow}Y$

which satisfies the following conditions.

  * •

$P_{k}$ is a projective object for any $0\leq k\leq n$.

  * •

There exist conflations

$\Omega^{k}Y\overset{m_{k}}{\longrightarrow}P_{k-1}\overset{p_{k-1}}{\longrightarrow}\Omega^{k-1}Y$

for $1\leq k\leq n$, such that $\Omega^{0}Y=Y$, $\Omega^{n}Y=P_{n}$, and
$d_{k}=m_{k}\circ p_{k}$ $(1\leq k\leq n-1)$, $d_{0}=p_{0}$, $d_{n}=m_{n}$.

Also, we put $\Omega^{k}Y=0$ for $k>n$. Then the equation

$\sum_{k\geq 0}(-1)^{k}\dim_{K}\mathbb{E}_{\mathrm{I}}^{-k}(X,Y)=\sum_{k\geq
0}(-1)^{k}\dim_{K}\underline{\mathscr{C}}(X,\Omega^{k}Y)+\sum_{0\leq k\leq
n}(-1)^{k}\dim_{K}\mathscr{C}(X,P_{k})$

holds for any $X\in\mathscr{C}$.

###### Proof.

This follows from the exactness of

$0\to\mathbb{E}_{\mathrm{I}}^{-1}(X,\Omega^{k}Y)\to\mathscr{C}(X,\Omega^{k+1}Y)\to\mathscr{C}(X,P_{k})\to\mathscr{C}(X,\Omega^{k}Y)\to\underline{\mathscr{C}}(X,\Omega^{k}Y)\to
0$

and the existence of isomorphisms

$\mathbb{E}_{\mathrm{I}}^{-1}(X,\Omega^{k}Y)\cong\mathbb{E}_{\mathrm{I}}^{-2}(X,\Omega^{k-1}Y)\cong\cdots\cong\mathbb{E}_{\mathrm{I}}^{-(k+1)}(X,Y)$

for any $0\leq k\leq n$, both of which are the consequences of Theorem 5.8.
Remark also that we have $\mathbb{E}_{\mathrm{I}}^{-k}(X,Y)=0$ and
$\underline{\mathscr{C}}(X,\Omega^{k}Y)=0$ for any $k\geq n+1$. ∎

######  Remark 5.15.

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective objects and
if $\mathbb{E}^{n+1}=0$, then any $Y\in\mathscr{C}$ admits a sequence
$(\ref{ProjResolY})$ with the required conditions.

###  5.3. Universality of $\mathbb{E}_{\mathrm{I}}^{\bullet}$ and
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be as in the previous subsection.
A question one might have will be that whether the definition of
$\mathbb{E}_{\mathrm{I}}^{\bullet}$ (respectively
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$) is natural enough. In this
subsection, we answer it with Proposition 5.19 by showing that it is universal
among covariant (resp. contravariant) connected sequence of functors
$F^{\bullet}$ having $F^{n}=\mathbb{E}^{n}$ with the canonical connecting
morphisms for $n\geq 0$, i.e. these are covariant (resp. contravariant)
derived functors of the bifunctor Hom, under the assumption of the existence
of enough projective morphisms (resp. enough injective morphisms).

######  Lemma 5.16.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms. Let $(F^{\bullet},\delta_{\sharp})$ and
$(G^{\bullet},\delta_{\sharp})$ be two covariant connected sequences of
functors. Suppose that they satisfy $F^{n}=G^{n}$ and
$\delta^{n}_{F,\sharp}=\delta^{n}_{G,\sharp}$ for any $n\in\mathbb{N}_{\geq
0}$ and any $\delta\in\mathbb{E}(C,A)$. If $(G^{\bullet},\delta_{\sharp})$
satisfies the condition

  * $(\star)$

$(G^{\bullet},\delta_{\sharp})$ is a $\delta$-functor, and $G^{-n}(-,p)=0$
holds for any projective deflation $p$ and any $n>0$,

then there exists a unique _morphism_
$\varphi^{\bullet}\colon(F^{\bullet},\delta_{\sharp})\to(G^{\bullet},\delta_{\sharp})$
_of covariant connected sequence of functors_ , given by identities in non-
negative degrees. That is, a sequence of natural transformations
$\\{\varphi^{n}\colon F^{n}\to G^{n}\\}_{n\in\mathbb{Z}}$ which satisfies the
following conditions.

  * (i)

$\varphi^{n}=\mathrm{id}$ for any $n\geq 0$.

  * (ii)

For any $n\in\mathbb{Z}$, the following diagram $\mathbf{D}(n;\delta)$

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

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

###### Proof.

By (i), it suffices to construct $\varphi^{n}\colon F^{n}\to G^{n}$ for any
$n\leq-1$ by a descending induction on $n$, in order that they satisfy (ii).
Assume $n\leq-1$, and suppose that we have obtained $\varphi^{k}$ for all
$k\geq n+1$. It suffices to show the following.

######  Claim 5.17.

Suppose that we have $\\{\varphi^{k}\\}_{k\geq n+1}$ satisfying (ii) for any
$k\geq n+1$. Then the following holds.

  1. (1)

For any $C\in\mathscr{C}$, choose any dominant $\mathfrak{s}$-triangle

(5.3)
$Q\overset{q}{\longrightarrow}P\overset{p}{\longrightarrow}C\overset{\omega}{\dashrightarrow}.$

Then there exists a unique morphism $\varphi^{n}_{-,C}\colon F^{n}(-,C)\to
G^{n}(-,C)$ in $\mbox{Mod\,}\mathscr{C}$ such that $\mathbf{D}(n;\omega)$
becomes commutative with respect to the pre-existing $\varphi^{n+1}$.

  2. (2)

The morphisms $\varphi^{n}_{-,C}\colon F^{n}(-,C)\to G^{n}(-,C)$ obtained in
(1) form a natural transformation $\varphi^{n}\colon F^{n}\to G^{n}$, and is
independent of the choices of $(\ref{sDeflChosen})$.

  3. (3)

The natural transformation $\varphi^{n}$ makes $\mathbf{D}(n;\delta)$
commutative for any $\delta\in\mathbb{E}(C,A)$.

Indeed it is obvious that the claim makes the induction work. It remains to
show Claim 5.17.

###### Proof of Claim 5.17.

(1) This is obvious from the existence of the following commutative diagram

$\textstyle{F^{n}(-,C)}$$\textstyle{F^{n+1}(-,Q)}$$\textstyle{F^{n+1}(-,P)}$$\textstyle{0}$$\textstyle{G^{n}(-,C)}$$\textstyle{G^{n+1}(-,Q)}$$\textstyle{G^{n+1}(-,P)}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{F^{n+1}(-,p)}$$\scriptstyle{\varphi^{n+1}_{-,Q}}$$\scriptstyle{\varphi^{n+1}_{-,P}}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{G^{n+1}(-,p)}$$\scriptstyle{\circlearrowright}$

in which the top row is a complex, and the bottom row is an exact sequence by
assumption.

(2) Let $c\in\mathscr{C}(C,C^{\prime})$ be any morphism, and let
$Q\overset{q}{\longrightarrow}P\overset{p}{\longrightarrow}C\overset{\omega}{\dashrightarrow}$
and
$Q^{\prime}\overset{q^{\prime}}{\longrightarrow}P^{\prime}\overset{p^{\prime}}{\longrightarrow}C^{\prime}\overset{\omega^{\prime}}{\dashrightarrow}$
be any pair of dominant $\mathfrak{s}$-triangles. The morphisms
$\varphi^{n}_{-,C}$ and $\varphi^{n}_{-,C^{\prime}}$ obtained in (1) makes
$\mathbf{D}(n;\omega)$ and $\mathbf{D}(n;\omega^{\prime})$ commutative
respectively, by construction. Since $\omega$ is dominant, there exists
$a\in\mathscr{C}(Q,Q^{\prime})$ such that
$a_{\ast}\omega=c^{\ast}\omega^{\prime}$. Since $\varphi^{n+1}$ is natural by
the hypothesis, we have a commutative diagram

$\textstyle{F^{n+1}(-,Q)}$$\textstyle{F^{n+1}(-,Q^{\prime})}$$\textstyle{G^{n+1}(-,Q)}$$\textstyle{G^{n+1}(-,Q^{\prime})}$$\scriptstyle{F^{n+1}(-,a)}$$\scriptstyle{\varphi^{n+1}_{-,Q}}$$\scriptstyle{\varphi^{n+1}_{-,Q^{\prime}}}$$\scriptstyle{G^{n+1}(-,a)}$$\scriptstyle{\circlearrowright}$

in $\mbox{Mod\,}\mathscr{C}$. Commutativity of these diagrams and the
monomorphicity of $\omega^{\prime}_{\sharp}\colon G^{n}(-,C^{\prime})\to
G^{n+1}(-,Q^{\prime})$ shows that

$\textstyle{F^{n}(-,C)}$$\textstyle{F^{n}(-,C^{\prime})}$$\textstyle{G^{n}(-,C)}$$\textstyle{G^{n}(-,C^{\prime})}$$\scriptstyle{F^{n}(-,c)}$$\scriptstyle{\varphi^{n}_{-,C}}$$\scriptstyle{\varphi^{n}_{-,C^{\prime}}}$$\scriptstyle{G^{n}(-,c)}$$\scriptstyle{\circlearrowright}$

is commutative. This means that $\varphi^{n}_{-,C}$ is natural in
$C\in\mathscr{C}$, hence we obtain a natural transformation $\varphi^{n}$.
Moreover, if we apply the above argument to $c=\mathrm{id}_{C}$, it shows the
uniqueness of $\varphi^{n}$.

(3) Let $\delta\in\mathbb{E}(C,A)$ be any element. Take a dominant extension
$\omega\in\mathbb{E}(C,Q)$. Then there is $a\in\mathscr{C}(Q,A)$ such that
$a_{\ast}\omega=\delta$. then we have the following commutative diagram

$\textstyle{F^{n}(-,C)}$$\textstyle{F^{n+1}(-,Q)}$$\textstyle{F^{n+1}(-,A)}$$\textstyle{G^{n}(-,C)}$$\textstyle{G^{n+1}(-,Q)}$$\textstyle{G^{n+1}(-,A)}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{F^{n+1}(-,a)}$$\scriptstyle{\varphi^{n}_{-,C}}$$\scriptstyle{\varphi^{n+1}_{-,Q}}$$\scriptstyle{\varphi^{n+1}_{-,A}}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{G^{n+1}(-,a)}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in which the left square is $\mathbf{D}(n;\omega)$, that is commutative by the
construction. Thus the outer square $\mathbf{D}(n;\delta)$ also becomes
commutative. ∎

∎

######  Remark 5.18.

The above condition $(\star)$ is satisfied by the following examples we have
seen so far.

  * •

The covariant part of Example 4.9 (1) (exact categories).

  * •

The covariant part of Example 4.9 (2) when $\mathscr{C}=\mathcal{T}$
(triangulated categories).

  * •

$(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$ in Definition 5.3.

  * •

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective objects, the
one in Example 4.10 (1) (stable Hom).

######  Proposition 5.19.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms. Then $(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$ is
universal among covariant connected sequence of functors having
$F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for $n\geq 0$.

###### Proof.

This is immediate from Theorem 5.8 and Lemma 5.16. ∎

######  Corollary 5.20.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms. Then $\mathbb{E}_{\mathrm{I}}^{-n}$ is the $n$-th left covariant
derived functor of the bifunctor Hom.

######  Corollary 5.21.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms. For any covariant $\delta$-functor $(F^{\bullet},\delta_{\sharp})$
having $F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for
$n\geq 0$, the following are equivalent.

  1. (1)

$(F^{\bullet},\delta_{\sharp})$ satisfies $(\star)$.

  2. (2)

$(F^{\bullet},\delta_{\sharp})$ is universal among covariant connected
sequence of functors having $F^{n}=\mathbb{E}^{n}$ with the canonical
connecting morphisms for $n\geq 0$.

  3. (3)

$(F^{\bullet},\delta_{\sharp})$ is isomorphic to
$(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$ as a covariant
$\delta$-functor.

###### Proof.

This is immediate from Lemma 5.16 and Proposition 5.19. ∎

Its counterpart for contravariant ones becomes as follows.

######  Corollary 5.22.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough injective
morphisms. Then $\mathbb{E}_{\mathrm{I\\!I}}^{-n}$ is the $n$-th left
contravariant derived functor of the bifunctor Hom. For any contravariant
$\delta$-functor $(F^{\bullet},\delta^{\sharp})$ having $F^{n}=\mathbb{E}^{n}$
with the canonical connecting morphisms for $n\geq 0$, the following are
equivalent.

  1. (1)

$(F^{\bullet},\delta^{\sharp})$ satisfies $F^{n}(i,-)=0$ for any injective
inflation $i$ and any $n<0$ .

  2. (2)

$(F^{\bullet},\delta^{\sharp})$ is universal among contravariant connected
sequence of functors having $F^{n}=\mathbb{E}^{n}$ with the canonical
connecting morphisms for $n\geq 0$.

  3. (3)

$(F^{\bullet},\delta^{\sharp})$ is isomorphic to
$(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$ as a contravariant
$\delta$-functor.

The above results show that we should think of the partial $\delta$-functors
$(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$ and
$(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$ as of the
(collections of) derived functors of the covariant, resp. contravariant
functor Hom with one argument active and the other inert. In other words,
$(\mathbb{E}_{\mathrm{I}}^{-n}),n>0$ and
$(\mathbb{E}_{\mathrm{I\\!I}}^{-n}),n>0$ should be seen as negative extensions
in the extriangulated category $(\mathscr{C},\mathbb{E},\mathfrak{s})$.

In general, thus defined negative extensions are not balanced: given an
arbitrary extriangulated category, we often do not have
$\mathbb{E}_{\mathrm{I}}^{-n}\cong\mathbb{E}_{\mathrm{I\\!I}}^{-n}$ for all
$n>0$.

As Remark 4.6 suggests, on $(\mathscr{C},\mathbb{E},\mathfrak{s})$ there can
exist various (non-canonical or non-intrinsic) balanced $\delta$-functors
having $F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for
$n\geq 0$. Indeed, there are plenty of such examples, e.g., any extension-
closed subcategory of a triangulated category $\mathcal{T}$ can be equipped
with such a $\delta$-functor by restrictions of that for $\mathcal{T}$. It may
depend on the embedding (see Example 5.24) and in general is not universal on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$.

In the case where $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough
projective morphisms and enough injective morphisms, since we do have
universal covariant $(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$ and
contravariant $(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$, it is
natural to look for precise conditions on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ forcing them to agree and constitute a
balanced universal $\delta$-functor
$(\mathbb{E}_{\mathrm{I}}^{\bullet}\cong\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta_{\sharp},\delta^{\sharp})$.
We will investigate such conditions in the next subsection.

######  Remark 5.23.

As the above argument suggests, we may also harmlessly define functors
$\mathbb{E}_{\mathrm{I}}^{-n}$ 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

$\mathbb{E}_{\mathrm{I}}^{-n}(-,C):=\mbox{Ker}\big{(}\mathbb{E}^{-(n-1)}(-,F)\overset{f_{\ast}}{\longrightarrow}\mathbb{E}^{-(n-1)}(-,G)\big{)}$

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 Lemma
5.16) shows that this is well-defined up to unique natural isomorphisms,
independently of the choices of dominant $\mathfrak{s}$-triangles. Dually for
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$, in the case with enough injective
morphisms. This again can be seen as a definition of (left) satellites of the
functor Hom, see [CE, M, MR] and references therein.

We end this subsection with the following example.

######  Example 5.24.

Let $K$ be an algebraically closed field, and let $Q$ be the quiver
$1\leftarrow 2\leftarrow 3$ of type $A_{3}$. For each $m>0$, put

$\mathscr{C}_{(m)}=\mbox{add}\,(1[1]\oplus\begin{smallmatrix}3\\\
2\end{smallmatrix}[-m])\subseteq D^{b}(\operatorname{\mathrm{mod}}\nolimits
KQ)=\mathcal{T}.$

Then $\mathscr{C}_{(m)}\subseteq\mathcal{T}$ is extension-closed, hence has an
induced extriangulated structure
$(\mathscr{C}_{(m)},\mathbb{E}_{(m)},\mathfrak{s}_{(m)})$. Indeed, we see
easily that it splits, i.e. $\mathbb{E}_{(m)}=0$ holds. In particular, those
$\mathscr{C}_{(m)}$ for $m>0$ are all equivalent as extriangulated categories.

For each $m>0$, there are two natural covariant connected sequence of functors
of $\mathbb{E}_{(m)}$ given by $(\mathbb{E}_{(m)})_{\mathrm{I}}^{-n}$ and
$\mathcal{T}(-[n],-)|_{\mathscr{C}_{(m)}}$. Obviously we have
$(\mathbb{E}_{(m)})_{\mathrm{I}}^{-n}=0$ for any $n>0$ since
$\mathbb{E}_{(m)}=0$. On the other hand, we have

$\mathcal{T}((\begin{smallmatrix}3\\\
2\end{smallmatrix}[-m])[n],1[1])\begin{cases}\neq 0&n=m\\\ =0&n\neq
m\end{cases}.$

This example shows that

  * (i)

For an extriangulated category $(\mathscr{C},\mathbb{E},\mathfrak{s})$
equipped with an embedding $\mathscr{C}\hookrightarrow\mathcal{T}$ as an
extension-closed subcategory into a triangulated category, we do not
necessarily have
$\mathbb{E}_{\mathrm{I}}^{-n}\cong\mathcal{T}(-[n],-)|_{\mathscr{C}}$.

  * (ii)

Besides $\mathcal{T}(-[n],-)|_{\mathscr{C}}$ can differ, depending on the
embedding $\mathscr{C}\hookrightarrow\mathcal{T}$.

A similar construction as above can be also performed to obtain non-splitting
examples which suggest (i),(ii), for example by taking

$\mathscr{C}_{(m)}=\mbox{add}\,\big{(}\begin{smallmatrix}3\\\
2\end{smallmatrix}\oplus\\{M[k]\mid
M\in\operatorname{ind}KQ,k>0\\}\big{)}\subseteq
D^{b}(\operatorname{\mathrm{mod}}\nolimits KQ)$

for $m>0$ for the same quiver $Q$.

###  5.4. Comparison of $\mathbb{E}_{\mathrm{I}}^{\bullet}$ and
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$

Let us compare $\mathbb{E}_{\mathrm{I}}^{\bullet}$ and
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$ under the existence of enough
projective morphisms and enough injective morphisms. By definition, we have
$\mathbb{E}_{\mathrm{I}}^{-n}(-,f)=0$ for any $f\in\mathcal{P}$ and any $n>0$,
while we have $\mathbb{E}_{\mathrm{I\\!I}}^{-n}(f,-)=0$ for any
$f\in\mathcal{I}$ and any $n>0$. In this subsection, we will see that
conversely these properties determine $\mathbb{E}_{\mathrm{I}}^{\bullet}$ and
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$ to some extent.

This property suggests us the following example. In fact, it shows that there
are indeed some cases where $\mathbb{E}_{\mathrm{I}}^{\bullet}$ does not
satisfy $\mathbb{E}_{\mathrm{I}}^{-n}(I,-)=0$ for injective object $I$, while
$\mathbb{E}_{\mathrm{I\\!I}}^{-n}(I,-)=0$ always holds by definition.

######  Example 5.25.

Let $K$ be an algebraically closed field, and let $Q$ be the quiver
$1\leftarrow 2\leftarrow 3\leftarrow 4$ of type $A_{4}$. Put

$\mathscr{C}=\mbox{add}\,(3[-1]\oplus 2\oplus\begin{smallmatrix}4\\\ 3\\\
2\end{smallmatrix}\oplus\begin{smallmatrix}4\\\ 3\end{smallmatrix})\subseteq
D^{b}(\operatorname{\mathrm{mod}}\nolimits KQ)=\mathcal{T}.$

It is extension-closed, hence has an induced extriangulated structure
$(\mathscr{C},\mathbb{E},\mathfrak{s})$. We see that
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ is hereditary, with enough projective
objects and enough injective objects. Since $C=3[-1]$ is injective, we have
$\mathbb{E}_{\mathrm{I\\!I}}^{-1}(C,-)=0$. On the other hand, if we put
$A=\begin{smallmatrix}4\\\ 3\end{smallmatrix}$, then it has an
$\mathfrak{s}$-conflation $2\to\begin{smallmatrix}4\\\ 3\\\
2\end{smallmatrix}\to A$ in which $\begin{smallmatrix}4\\\ 3\\\
2\end{smallmatrix}$ is projective in $\mathscr{C}$, hence

$0\to\mathbb{E}_{\mathrm{I}}^{-1}(C,A)\to\mathscr{C}(C,2)\to\mathscr{C}(C,\begin{smallmatrix}4\\\
3\\\ 2\end{smallmatrix})$

is exact by Lemma 5.5. Since
$\mathscr{C}(C,2)\cong\mbox{Ext}_{KQ}^{1}(3,2)\neq 0$ and
$\mathscr{C}(C,\begin{smallmatrix}4\\\ 3\\\
2\end{smallmatrix})\cong\mbox{Ext}_{KQ}^{1}(3,\begin{smallmatrix}4\\\ 3\\\
2\end{smallmatrix})=0$, this shows $\mathbb{E}_{\mathrm{I}}^{-1}(C,A)\neq 0$.

In view of Corollaries 5.21 and 5.22, if one expects for
$\mathbb{E}_{\mathrm{I}}^{\bullet}$ and
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$ to agree, an obviously necessary
condition should be that $\mathbb{E}_{\mathrm{I}}^{-n}(i,-)=0$ for any
injective inflation and $\mathbb{E}_{\mathrm{I\\!I}}^{-n}(-,p)=0$ for any
projective inflation, for any $n>0$. In the sequel, we will see that this
condition is in fact essentially sufficient to give a numerical equation for
their dimensions, while a slightly stronger condition ensures that they agree
as functors. See Proposition 5.30 and Theorem 5.37 for detail.

######  Lemma 5.26.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms and enough injective morphisms. Let $X,Y\in\mathscr{C}$ be any pair
of objects, and take arbitrary $\mathfrak{s}$-triangles

(5.4)
$X\overset{i}{\longrightarrow}I\overset{j}{\longrightarrow}J\overset{\iota}{\dashrightarrow}\
\ \text{and}\ \
Q\overset{q}{\longrightarrow}P\overset{p}{\longrightarrow}Y\overset{\omega}{\dashrightarrow}$

in which $i$ is an injective inflation and $p$ is a projective deflation.

Then the morphisms

$\iota^{\sharp}\circ\omega_{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{\omega_{\sharp}}{\longrightarrow}\mathscr{C}(X,Q)\overset{\iota^{\sharp}}{\longrightarrow}\mathbb{E}(J,Q)$

and

$\omega_{\sharp}\circ\iota^{\sharp}\colon\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\overset{\iota^{\sharp}}{\longrightarrow}\mathscr{C}(J,Y)\overset{\omega_{\sharp}}{\longrightarrow}\mathbb{E}(J,Q),$

have the same image in $\mathbb{E}(J,Q)$. Equivalently, there is an
isomorphism $\mu$ which makes the following diagram commutative.

(5.5)
$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)/(\omega_{\sharp})^{-1}(\mathcal{I}(X,Q))}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)/(\iota^{\sharp})^{-1}(\mathcal{P}(J,Y))}$$\textstyle{\mathbb{E}(J,Q)}$$\scriptstyle{\cong}$$\scriptstyle{\mu}$$\scriptstyle{\iota^{\sharp}\circ\omega_{\sharp}}$$\scriptstyle{\omega_{\sharp}\circ\iota^{\sharp}}$$\scriptstyle{\circlearrowright}$

Here, we denoted the morphisms induced by $\iota^{\sharp}\circ\omega_{\sharp}$
and $\omega_{\sharp}\circ\iota^{\sharp}$ by the same symbols, and
$(\omega_{\sharp})^{-1}(\mathcal{I}(X,Q))$ denotes the preimage of
$\mathcal{I}(X,Q)$ by the monomorphism
$\omega_{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{\omega_{\sharp}}{\longrightarrow}\mathscr{C}(X,Q)$.
Similarly for $(\iota^{\sharp})^{-1}(\mathcal{P}(J,Y))$.

###### Proof.

Since
$\mathscr{C}(I,Q)\overset{i^{\ast}}{\longrightarrow}\mathscr{C}(X,Q)\overset{\iota^{\sharp}}{\longrightarrow}\mathbb{E}(J,Q)$
is exact, we have
$\mbox{Ker}\big{(}\iota^{\sharp}\colon\mathscr{C}(X,Q)\to\mathbb{E}(J,Q)\big{)}=\mathcal{I}(X,Q)$,
and thus
$\mbox{Ker}\big{(}\iota^{\sharp}\circ\omega_{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\to\mathbb{E}(J,Q)\big{)}=(\omega_{\sharp})^{-1}(\mathcal{I}(X,Q))$.
Similarly we have
$\mbox{Ker}\big{(}\omega_{\sharp}\circ\iota^{\sharp}\colon\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\to\mathbb{E}(J,Q)\big{)}=(\iota^{\sharp})^{-1}(\mathcal{P}(J,Y))$.
Thus indeed it suffices to show the existence of isomorphism $\mu$ which makes
$(\ref{mucomm})$ commutative.

By Theorems 5.8 and 5.10, sequences
$0\to\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{\omega_{\sharp}}{\longrightarrow}\mathscr{C}(X,Q)\overset{\mathscr{C}(X,q)}{\longrightarrow}\mathscr{C}(X,P)$
and
$0\to\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\overset{\iota^{\sharp}}{\longrightarrow}\mathscr{C}(J,Y)\overset{\mathscr{C}(j,Y)}{\longrightarrow}\mathscr{C}(I,Y)$
are exact. Put
$L_{X,Y}=\mbox{Ker}\big{(}\mathscr{C}(X,q)\big{)}=\omega_{\sharp}\big{(}\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\big{)}$
and
$M_{X,Y}=\mbox{Ker}\big{(}\mathscr{C}(j,Y)\big{)}=\iota^{\sharp}\big{(}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\big{)}$
for simplicity. Since
$\omega_{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\to\mathscr{C}(X,Q)$
and $\iota^{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\to\mathscr{C}(J,Y)$
are monomorphic, it suffices to give an isomorphism

(5.6)
$L_{X,Y}/\big{(}L_{X,Y}\cap\mathcal{I}(X,Q)\big{)}\overset{\cong}{\longrightarrow}M_{X,Y}/\big{(}M_{X,Y}\cap\mathcal{P}(J,Y)\big{)}$

compatibly with $\iota^{\sharp}$ and $\omega_{\sharp}$.

For any $s\in L_{X,Y}$, we obtain a morphism of $\mathfrak{s}$-triangles

(5.7)
$\textstyle{X}$$\textstyle{I}$$\textstyle{J}$$\textstyle{Q}$$\textstyle{P}$$\textstyle{Y}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{\iota}$$\scriptstyle{s}$$\scriptstyle{0}$$\scriptstyle{t}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{\omega}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

for some $t\in M_{X,Y}$ by (ET3). Conversely, any $t\in M_{X,Y}$ gives such a
morphism of $\mathfrak{s}$-triangles $(\ref{Morph_s_t})$ for some $s\in
L_{X,Y}$ by (ET3)op. Moreover, in $(\ref{Morph_s_t})$ we have

$s\in\mathcal{I}(X,Q)\ \Leftrightarrow\ s\ \text{factors through}\ i\
\Leftrightarrow\ t\ \text{factors through}\ p\ \Leftrightarrow\
t\in\mathcal{P}(J,Y),$

hence obtain an isomorphism
$L_{X,Y}/\big{(}L_{X,Y}\cap\mathcal{I}(X,Q)\big{)}\overset{\cong}{\longrightarrow}M_{X,Y}/\big{(}M_{X,Y}\cap\mathcal{P}(J,Y)\big{)}$
as desired. By definition of a morphism of $\mathfrak{s}$-triangles, any $s\in
L_{X,Y}$ and $t\in M_{X,Y}$ corresponding through this isomorphism satisfies
$\iota^{\sharp}(s)=\omega_{\sharp}(t)$. ∎

######  Remark 5.27.

Let $\mathcal{T}$ be a triangulated category and let
$\mathcal{D}\subseteq\mathscr{C}\subseteq\mathcal{T}$ be additive full
subcategories. If $(\mathscr{C},\mathscr{C})$ is an $\mathcal{D}$-mutation
pair in the sense of [IY, Definition 2.5], then by definition
$\mathscr{C}\subseteq\mathcal{T}$ is an extension-closed subcategory, hence
extriangulated. In this case $\mathscr{C}$ is Frobenius whose subcategory of
projective-injectives agrees with $\mathcal{D}$, as seen in [NP1, Example
7.2], and the ideal quotient $\mathscr{C}/\mathcal{D}$ is known to be
triangulated by [IY, Theorem 4.2]. In particular there is an isomorphism by
shifting $(\mathscr{C}/\mathcal{D})(X,\Omega
Y)\overset{\cong}{\longrightarrow}(\mathscr{C}/\mathcal{D})(\Sigma X,Y)$ for
any $X,Y\in\mathscr{C}$, where $\Sigma$ (respectively $\Omega$) denotes the
positive (resp. negative) shift by $1$ in $\mathscr{C}/\mathcal{D}$. The above
isomorphism $(\ref{IsomLM})$ can be viewed as a restriction of this
isomorphism.

######  Proposition 5.28.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms and enough injective morphisms. Let $X,Y\in\mathscr{C}$ be any pair
of objects, and take arbitrary $\mathfrak{s}$-triangles
$(\ref{sTri_Comparison})$ in which $i$ is an injective inflation and $p$ is a
projective deflation. Then the cohomologies of the complexes

$C_{\mathrm{I}}^{\bullet}\colon\quad\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)\overset{j^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I,Y)\overset{i^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{\iota^{\sharp}\circ\omega_{\sharp}}{\longrightarrow}\mathbb{E}(J,Q)$

and

$C_{\mathrm{I}\\!\mathrm{I}}^{\bullet}\colon\quad\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)\overset{q_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P)\overset{p_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\overset{\omega_{\sharp}\circ\iota^{\sharp}}{\longrightarrow}\mathbb{E}(J,Q)$

are isomorphic.

###### Proof.

Put
$\mathbf{V}_{(\iota,\omega)}=\Theta_{\omega}(J)\cap\Theta^{\iota}(P)=\mbox{Ker}\big{(}\mathbb{E}(J,q)\big{)}\cap\mbox{Ker}\big{(}\mathbb{E}(j,Q)\big{)}$
in $\mathbb{E}(J,Q)$, for simplicity. Consider the following double complex
$C^{\bullet,\bullet}$ induced from $\mathbb{E}^{n}$ $(n\geq 0)$, concentrated
in the lower-right quadrant.

$\textstyle{\mathscr{C}(J,Q)}$$\textstyle{\mathscr{C}(J,P)}$$\textstyle{\mathscr{C}(J,Y)}$$\textstyle{\mathbb{E}(J,Q)}$$\textstyle{\mathbb{E}(J,P)}$$\textstyle{\cdots}$$\textstyle{\mathscr{C}(I,Q)}$$\textstyle{\mathscr{C}(I,P)}$$\textstyle{\mathscr{C}(I,Y)}$$\textstyle{\mathbb{E}(I,Q)}$$\textstyle{\mathbb{E}(I,P)}$$\textstyle{\cdots}$$\textstyle{\mathscr{C}(X,Q)}$$\textstyle{\mathscr{C}(X,P)}$$\textstyle{\mathscr{C}(X,Y)}$$\textstyle{\mathbb{E}(X,Q)}$$\textstyle{\mathbb{E}(X,P)}$$\textstyle{\cdots}$$\textstyle{\mathbb{E}(J,Q)}$$\textstyle{\mathbb{E}(J,P)}$$\textstyle{\mathbb{E}(J,Y)}$$\textstyle{\mathbb{E}^{2}(J,Q)}$$\textstyle{\vdots}$$\textstyle{\mathbb{E}(I,Q)}$$\textstyle{\mathbb{E}(I,P)}$$\textstyle{\mathbb{E}(I,Y)}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\textstyle{\vdots}$$\scriptstyle{q_{\ast}}$$\scriptstyle{p_{\ast}}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{q_{\ast}}$$\scriptstyle{j^{\ast}}$$\scriptstyle{i^{\ast}}$$\scriptstyle{\iota^{\sharp}}$$\scriptstyle{j^{\ast}}$

If we take the kernels of the leftmost horizontal arrows, we obtain a complex

$C_{\mathrm{I}}^{\prime\bullet}\colon\quad\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)\overset{j^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I,Y)\overset{i^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\to\mbox{Ker}\big{(}\mathbb{E}(J,q)\big{)}\overset{a}{\longrightarrow}\mbox{Ker}\big{(}\mathbb{E}(I,q)\big{)}\to\cdots$

by Lemma 5.5 for $n=1$. Similarly, if we take the kernels vertically, we
obtain a complex

$C_{\mathrm{I\\!I}}^{\prime\bullet}\colon\quad\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)\overset{q_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P)\overset{p_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\to\mbox{Ker}\big{(}\mathbb{E}(j,Q)\big{)}\overset{b}{\longrightarrow}\mbox{Ker}\big{(}\mathbb{E}(j,P)\big{)}\to\cdots.$

Here, $a$ and $b$ are the unique morphisms given by the restriction of
$\mathbb{E}(j,Q)$ and $\mathbb{E}(J,q)$, respectively. In particular, we have
$\mbox{Ker}a=\mbox{Ker}b=\mathbf{V}_{(\iota,\omega)}$. By the acyclic assembly
lemma [W, 2.7.3], we have quasi-isomorphisms

(5.8)
$C_{\mathrm{I}}^{\prime\bullet}\overset{\text{qis.}}{\longrightarrow}\operatorname{Tot}(C^{\bullet,\bullet})\overset{\text{qis.}}{\longleftarrow}C_{\mathrm{I\\!I}}^{\prime\bullet}$

between $C_{\mathrm{I}}^{\prime\bullet},C_{\mathrm{II}}^{\prime\bullet}$ and
the total complex $\operatorname{Tot}(C^{\bullet,\bullet})$ of
$C^{\bullet,\bullet}$. By truncating $(\ref{Qis_Seq})$ we obtain complexes

$C_{\mathrm{I}}^{\prime\prime\bullet}\colon\quad\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)\overset{j^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I,Y)\overset{i^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{u}{\longrightarrow}\mathbf{V}_{(\iota,\omega)}$

and

$C_{\mathrm{I\\!I}}^{\prime\prime\bullet}\colon\quad\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)\overset{q_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P)\overset{p_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\overset{v}{\longrightarrow}\mathbf{V}_{(\iota,\omega)},$

which are connected by a pair of quasi-isomorphisms
$C_{\mathrm{I}}^{\prime\prime\bullet}\overset{\text{qis.}}{\longrightarrow}D^{\bullet}\overset{\text{qis.}}{\longleftarrow}C_{\mathrm{I\\!I}}^{\prime\prime\bullet}$,
with $D^{\bullet}$ given by a truncation of
$\operatorname{Tot}(C^{\bullet,\bullet})$. Thus, the assertion follows from
Lemma 5.26. ∎

######  Corollary 5.29.

Assume that $R=K$ is a field and $\mathscr{C}$ is Hom-finite over $K$, and
that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective morphisms
and enough injective morphisms. Suppose that the following conditions are
satisfied.

  * (i)

$\mathbb{E}_{\mathrm{I}}^{-1}(i,-)=0$ for any injective inflation.

  * (ii)

$\mathbb{E}_{\mathrm{I\\!I}}^{-1}(-,p)=0$ for any projective deflation.

Then
$\dim_{K}\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)$
and
$\dim_{K}\mathbb{E}_{\mathrm{I}}^{-2}(X,Y)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-2}(X,Y)$
hold for any $X,Y\in\mathscr{C}$.

###### Proof.

Take any pair of $\mathfrak{s}$-triangles $(\ref{sTri_Comparison})$ as in
Lemma 5.26. By assumption, the resulting complexes
$C_{\mathrm{I}}^{\bullet},C_{\mathrm{I\\!I}}^{\bullet}$ split in the middle.
Hence the following $2$-term complexes

(5.9)
$\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{\iota^{\sharp}\circ\omega_{\sharp}}{\longrightarrow}\mathbb{E}(J,Q)\quad\text{and}\quad\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\overset{\omega_{\sharp}\circ\iota^{\sharp}}{\longrightarrow}\mathbb{E}(J,Q)$

have isomorphic cohomologies. Similarly, so do

(5.10)
$\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)\overset{j^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I,Y)\quad\text{and}\quad\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)\overset{q_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P).$

From $(\ref{2termIsoCoh1})$, we obtain
$\dim_{K}\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)$.

Since $X,Y\in\mathscr{C}$ were arbitrary, we may also apply this equality to
$I,J$ and $P,Q$, to obtain

$\displaystyle\dim_{K}\mathbb{E}_{\mathrm{I}}^{-1}(X,Q)-\dim_{K}\mathbb{E}_{\mathrm{I}}^{-1}(X,P)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)-\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P)$
$\displaystyle\underset{(\ref{2termIsoCoh2})}{=}\dim_{K}\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)-\dim_{K}\mathbb{E}_{\mathrm{I}}^{-1}(I,Y)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(J,Y)-\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(I,Y)$

from $(\ref{2termIsoCoh2})$. Since
$0\to\mathbb{E}_{\mathrm{I}}^{-2}(X,Y)\to\mathbb{E}_{\mathrm{I}}^{-1}(X,Q)\to\mathbb{E}_{\mathrm{I}}^{-1}(X,P)\to
0$ and
$0\to\mathbb{E}_{\mathrm{I\\!I}}^{-2}(X,Y)\to\mathbb{E}_{\mathrm{I\\!I}}^{-1}(J,Y)\to\mathbb{E}_{\mathrm{I\\!I}}^{-1}(I,Y)\to
0$ are exact by Theorems 5.8 and 5.10, this means
$\dim_{K}\mathbb{E}_{\mathrm{I}}^{-2}(X,Y)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-2}(X,Y)$.
∎

If we assume the existence of enough projective objects and injective objects,
then we obtain equalities of dimensions in any degree. Moreover, such
equalities are shown to be equivalent to (NI),(NII) in Condition 5.12 as
follows.

######  Proposition 5.30.

Assume that $R=K$ is a field and $\mathscr{C}$ is Hom-finite over $K$, and
that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective objects and
enough injective objects. Then the following are equivalent.

  1. (1)

$(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies conditions (NI) and (NII).

  2. (2)

$\dim_{K}\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)$
holds for any $X,Y\in\mathscr{C}$.

  3. (3)

$\dim_{K}\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)$
holds for any $X,Y\in\mathscr{C}$ and any $n>0$.

###### Proof.

Remark that Corollaries 5.9 and 5.11 can be applied, by assumption. (1)
$\Rightarrow$ (2) is shown in Corollary 5.29. (2) $\Rightarrow$ (1) is
obvious, since we always have $\mathbb{E}_{\mathrm{I}}^{-1}(-,P)=0$ for any
projective object $P\in\mathscr{C}$ and
$\mathbb{E}_{\mathrm{I\\!I}}^{-1}(I,-)=0$ for any injective object
$I\in\mathscr{C}$ by definition. (3) $\Rightarrow$ (2) is trivial.

It remains to show that (2) ($\Leftrightarrow$ (1)) implies (3). Assume that
(2) holds. We show

$\dim_{K}\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)=\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)\quad(\forall
X,Y\in\mathscr{C})$

by an induction on $n\in\mathbb{N}_{>0}$. By Corollary 5.29, this holds for
$n=1,2$. Assume $n\geq 3$, and suppose that we have shown for all $1\leq k\leq
n-1$. Let $X,Y\in\mathscr{C}$ be arbitrary, and take $\mathfrak{s}$-triangles

$X\to I\to X^{\prime}\overset{\iota}{\dashrightarrow}\quad\text{and}\quad
Y^{\prime}\to P\to Y\overset{\omega}{\dashrightarrow}$

in which $P\in\mathscr{C}$ is projective and $I\in\mathscr{C}$ is injective.
Then we have isomorphisms

$\omega_{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-k}(-,Y)\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-(k-1)}(-,Y^{\prime})\quad\text{and}\quad\iota^{\sharp}\colon\mathbb{E}_{\mathrm{I\\!I}}^{-k}(X,-)\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-(k-1)}(X^{\prime},-)$

for any $k\geq 2$ by Theorems 5.8 and 5.10. Thus we obtain

$\displaystyle\dim_{K}\mathbb{E}_{I}^{-n}(X,Y)$ $\displaystyle=$
$\displaystyle\dim_{K}\mathbb{E}_{\mathrm{I}}^{-(n-1)}(X,Y^{\prime})\ =\
\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-(n-1)}(X,Y^{\prime})$ $\displaystyle=$
$\displaystyle\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-(n-2)}(X^{\prime},Y^{\prime})\
=\ \dim_{K}\mathbb{E}_{\mathrm{I}}^{-(n-2)}(X^{\prime},Y^{\prime})$
$\displaystyle=$
$\displaystyle\dim_{K}\mathbb{E}_{\mathrm{I}}^{-(n-1)}(X^{\prime},Y)\ =\
\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-(n-1)}(X^{\prime},Y)\ =\
\dim_{K}\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)$

by the hypothesis of the induction. ∎

Remark that the above (2),(3) in Proposition 5.30 are only numerical equations
between $\mathbb{E}_{\mathrm{I}}$ and $\mathbb{E}_{\mathrm{I\\!I}}$. However,
if we assume a slightly stronger condition than (NI),(NII) on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$, we obtain isomorphisms
$\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)\cong\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)$
as in Theorem 5.37 below.

Regarding Lemma 5.26, we define Conditions (NI$+$) and (NII$+$) as follows.

######  Condition 5.31.

For $(\mathscr{C},\mathbb{E},\mathfrak{s})$, consider the following
conditions.

  * (NI$+$)

$(\omega_{\sharp})^{-1}(\mathcal{I}(X,Q))=0$ holds in
$\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)$ for any $X,Y\in\mathscr{C}$ and any
dominant extension $\omega\in\mathbb{E}(Y,Q)$.

  * (NII$+$)

$(\iota^{\sharp})^{-1}(\mathcal{P}(J,Y))=0$ holds in
$\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)$ for any $X,Y\in\mathscr{C}$ and any
codominant extension $\iota\in\mathbb{E}(J,X)$.

######  Remark 5.32.

Similarly to (NI),(NII), Conditions (NI$+$),(NII$+$) are also trivially
satisfied if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ corresponds to an exact
category or a triangulated category.

######  Lemma 5.33.

  * (i)

Assume that the category $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough
injective objects. If we have $\mathscr{C}(I,Q)=0$ for all $I$ injective and
all $Q$ appearing as the first terms in dominant $\mathfrak{s}$-triangles,
then the condition (NI$+$) holds in $(\mathscr{C},\mathbb{E},\mathfrak{s})$.

  * (ii)

Assume that the category $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough
projective objects. If we have $\mathscr{C}(J,P)=0$ for all $P$ projective and
all $J$ appearing as the third terms in codominant $\mathfrak{s}$-triangles,
then the condition (NII$+$) holds in $(\mathscr{C},\mathbb{E},\mathfrak{s})$.

###### Proof.

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough injectives, the ideal
$\mathcal{I}(X,Q)$ is formed precisely by the maps that factor through an
injective object. The assumption in part (i) then implies that
$\mathcal{I}(X,Q)=0$ for any $X,Y\in\mathscr{C}$ and any dominant extension
$\omega\in\mathbb{E}(Y,Q)$. This implies the condition (NI$+$). Part (ii) is
proved in a dual way. ∎

######  Example 5.34.

Conditions (NI$+$),(NII$+$) are satisfied for the category
$K^{[-n,0]}(\operatorname{proj}\Lambda)$ of complexes of length $(n+1)$ of
projectives $\Lambda$-modules up to homotopy, where $n\geq 0$ and $\Lambda$ is
a finite-dimensional algebra. Indeed, the case $n=0$ is covered by Remark
5.32, since the category is simply $\operatorname{proj}\Lambda$. If $n>0$,
since $0$ is a projective-injective object, it follows that the projective
objects of $K^{[-n,0]}(\operatorname{proj}\Lambda)$ are given precisely by
complexes concentrated in degree $0$, and the injectives are given by
complexes concentrated in degree $-n$.

An object $Q$ appears as the first term of a dominant extension if and only if
it is isomorphic to a complex in $K^{b}(\operatorname{proj}\Lambda)$ whose
$-n$-th term is zero (and thus it has an inflation to $0$). But in
$K^{[-n,0]}(\operatorname{proj}\Lambda)$, there are no non-zero maps from
complexes concentrated in degree $-n$ to complexes whose $-n$-th terms are
zero. By 5.33(i), the condition (NI+) is satisfied. By dual arguments, the
condition (NII+) is also satisfied.

######  Lemma 5.35.

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

  1. (1)

(NI$+$) holds if and only if
$\iota^{\sharp}\circ\omega_{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\to\mathbb{E}(J,Q)$
is monomorphic for any $X,Y\in\mathscr{C}$, any dominant
$\omega\in\mathbb{E}(Y,Q)$ and any codominant $\iota\in\mathbb{E}(J,X)$.

  2. (2)

(NII$+$) holds if and only if
$\omega_{\sharp}\circ\iota^{\sharp}\colon\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\to\mathbb{E}(J,Q)$
is monomorphic for any $X,Y\in\mathscr{C}$, any dominant
$\omega\in\mathbb{E}(Y,Q)$ and any codominant $\iota\in\mathbb{E}(J,X)$.

  3. (3)

(NI$+$) implies (NI).

  4. (4)

(NII$+$) implies (NII).

  5. (5)

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies (NI) and (NII), then
(NI$+$) holds if and only if (NII$+$) holds.

###### Proof.

(1) is obvious from
$\mbox{Ker}\big{(}\iota^{\sharp}\circ\omega_{\sharp}\big{)}=(\omega_{\sharp})^{-1}(\mathcal{I}(X,Q))$.
Similarly for (2).

(3) In (NI$+$), if $X=I$ is an injective object, then we have

$\mathbb{E}_{\mathrm{I}}^{-1}(I,Y)=(\omega_{\sharp})^{-1}(\mathcal{I}(I,\Omega
Y))=0$

since $\mathcal{I}(I,\Omega Y)=\mathscr{C}(I,\Omega Y)$. Thus (NI$+$) implies
(NI).

(4) Dually to (3), condition (NII) is nothing but condition (NII$+$)
restricted to the case where $Y$ is projective.

(5) Let $X,Y\in\mathscr{C}$ be any pair of objects, and let

(5.11)
$X\overset{i}{\longrightarrow}I\overset{j}{\longrightarrow}J\overset{\iota}{\dashrightarrow}\
\ \text{and}\ \
Q\overset{q}{\longrightarrow}P\overset{p}{\longrightarrow}Y\overset{\omega}{\dashrightarrow}$

be any pair of $\mathfrak{s}$-triangles in which $i$ is an injective inflation
and $p$ is a projective deflation. Under the assumption of (NI) and (NII), the
morphisms

$\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{\iota^{\sharp}\circ\omega_{\sharp}}{\longrightarrow}\mathbb{E}(J,Q)\quad\text{and}\quad\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\overset{\omega_{\sharp}\circ\iota^{\sharp}}{\longrightarrow}\mathbb{E}(J,Q)$

have isomorphic kernels by Proposition 5.28, as seen in the proof of Corollary
5.29. Thus (5) follows from (1) and (2). ∎

######  Lemma 5.36.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
objects and enough injective objects. Let
$X\overset{i}{\longrightarrow}I\overset{j}{\longrightarrow}J\overset{\iota}{\dashrightarrow}$
and
$Q\overset{q}{\longrightarrow}P\overset{p}{\longrightarrow}Y\overset{\omega}{\dashrightarrow}$
be any pair of $\mathfrak{s}$-triangles in which $I\in\mathscr{C}$ is an
injective object and $P\in\mathscr{C}$ is a projective object. The following
holds.

  1. (1)

Assume (NI$+$) and (NII$+$). Then there is a unique isomorphism
$\psi^{(-1)}_{X,Y}\colon\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)$
which makes

$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}(J,Q)}$$\scriptstyle{\psi^{(-1)}_{X,Y}}$$\scriptstyle{\cong}$$\scriptstyle{\iota^{\sharp}\circ\omega_{\sharp}}$$\scriptstyle{\omega_{\sharp}\circ\iota^{\sharp}}$$\scriptstyle{\circlearrowright}$

commutative.

  2. (2)

Assume (NI) and (NII). Then there is a unique isomorphism
$\alpha\colon\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)$
which makes

$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)}$$\textstyle{\mathscr{C}(J,Q)}$$\scriptstyle{\alpha}$$\scriptstyle{\cong}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{\iota^{\sharp}}$$\scriptstyle{\circlearrowright}$

commutative.

###### Proof.

(1) This is immediate from Lemma 5.26.

(2) Though this can be deduced from Proposition 5.28, let us give a direct
construction, for the sake of clarity. In the commutative square

$\textstyle{\mathscr{C}(J,Q)}$$\textstyle{\mathscr{C}(I,Q)}$$\textstyle{\mathscr{C}(J,P)}$$\textstyle{\mathscr{C}(I,P)}$$\scriptstyle{\mathscr{C}(j,Q)}$$\scriptstyle{\mathscr{C}(J,q)}$$\scriptstyle{\mathscr{C}(I,q)}$$\scriptstyle{\mathscr{C}(j,P)}$$\scriptstyle{\circlearrowright}$

morphisms $\mathscr{C}(I,q)$ and $\mathscr{C}(j,P)$ are monomorphic by
Corollaries 5.9 and 5.11. Since

$0\to\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)\overset{\omega_{\sharp}}{\longrightarrow}\mathscr{C}(J,Q)\overset{\mathscr{C}(J,q)}{\longrightarrow}\mathscr{C}(J,P)$

and

$0\to\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)\overset{\iota^{\sharp}}{\longrightarrow}\mathscr{C}(J,Q)\overset{\mathscr{C}(j,Q)}{\longrightarrow}\mathscr{C}(I,Q)$

are exact by Theorems 5.8 and 5.10, we obtain the desired isomorphism for
kernels. ∎

######  Theorem 5.37.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
objects and enough injective objects. If
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies (NI$+$) and (NII+), then
there exists a unique sequence of natural isomorphisms
$\\{\psi^{(n)}\colon\mathbb{E}_{\mathrm{I}}^{n}\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{n}\\}_{n\in\mathbb{Z}}$
which satisfies the following (i) and (ii).

  * (i)

$\psi^{(n)}=\mathrm{id}$ for any $n\geq 0$.

  * (ii)

For any $n\in\mathbb{Z}$, the following diagram $\mathbf{D}(n;\delta,\rho)$

$\textstyle{\mathbb{E}_{\mathrm{I}}^{n}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{n+1}(X,Z)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{n+1}(X,Z)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{n+2}(W,Z)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{n}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{n+1}(W,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{n+1}(W,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{n+2}(W,Z)}$$\scriptstyle{\rho_{\sharp}}$$\scriptstyle{\psi_{X,Z}^{(n+1)}}$$\scriptstyle{\cong}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\psi^{(n)}_{X,Y}}$$\scriptstyle{\psi^{(n+2)}_{W,Z}}$$\scriptstyle{\cong}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\psi_{W,Y}^{(n+1)}}$$\scriptstyle{\cong}$$\scriptstyle{\rho_{\sharp}}$$\scriptstyle{\circlearrowright}$

is commutative for any $\delta\in\mathbb{E}(W,X)$ and any
$\rho\in\mathbb{E}(Y,Z)$.

###### Proof.

This can be shown in a similar way as Lemma 5.16. As in (i), we already have
$\psi^{(n)}$ for $n\geq 0$, which obviously satisfies (ii) for any $n\geq 0$.
Let us construct $\psi^{(-n)}$ for $n\in\mathbb{N}_{\geq 1}$ inductively. It
suffices to show the following lemma.

######  Lemma 5.38.

Suppose that we have obtained natural isomorphisms
$\psi^{(k)}\colon\mathbb{E}_{\mathrm{I}}^{k}\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{k}$
for all $k\geq-n+1$, such that (ii) holds for any $k\geq-n+1$. Then the
following holds.

  1. (1)

For any $X,Y\in\mathscr{C}$, choose $\mathfrak{s}$-triangles

(5.12)
$X\overset{i}{\longrightarrow}I\overset{j}{\longrightarrow}J\overset{\iota}{\dashrightarrow}\
\ \text{and}\ \
Q\overset{q}{\longrightarrow}P\overset{p}{\longrightarrow}Y\overset{\omega}{\dashrightarrow}$

arbitrarily, so that $I\in\mathscr{C}$ is an injective object and
$P\in\mathscr{C}$ is a projective object. Then there is a unique isomorphism
$\psi^{(-n)}_{X,Y}\colon\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)$
which makes $\mathbf{D}(-n;\iota,\omega)$ commutative.

  2. (2)

The isomorphisms $\psi^{(-n)}_{X,Y}$ obtained in (1) form a natural
isomorphism
$\psi^{(-n)}\colon\mathbb{E}_{\mathrm{I}}^{-n}\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-n}$,
which is independent of the choices of $(\ref{sTri_IP})$.

  3. (3)

The natural isomorphism $\psi^{(-n)}$ obtained in (2) makes
$\mathbf{D}(-n;\delta,\rho)$ commutative for any $\delta\in\mathbb{E}(W,X)$
and any $\rho\in\mathbb{E}(Y,Z)$, with respect to the pre-existing
$\psi^{(-n+1)}$ and $\psi^{(-n+2)}$.

Indeed, it is obvious that this lemma makes the induction work. It remains to
show Lemma 5.38.

###### Proof of Lemma 5.38.

(1) Remark that $\mathfrak{s}$-triangles of the form $(\ref{sTri_IP})$ exist
by assumption. The top and the bottom rows of $\mathbf{D}(-n;\iota,\omega)$
are

$\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)\overset{\omega_{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n+1}(X,Q)\underset{\cong}{\overset{\psi^{(-n+1)}_{X,Q}}{\longrightarrow}}\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(X,Q)\overset{\iota^{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-n+2}(J,Q)$

and

(5.13)
$\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)\overset{\iota^{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(J,Y)\underset{\cong}{\overset{\psi^{(-n+1)}_{J,Y}}{\longleftarrow}}\mathbb{E}_{\mathrm{I}}^{-n+1}(J,Y)\overset{\omega_{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n+2}(J,Q),$

respectively. Each of their compositions is

  * •

an isomorphism if $n\geq 3$ by Theorems 5.8 and 5.10,

  * •

a monomorphism if $n=2$ by the same theorems,

  * •

a monomorphism if $n=1$ by Lemma 5.35.

Therefore the isomorphism $\psi^{(-n)}_{X,Y}$ is unique in either case, and
exists

  * •

obviously as a composition of relevant isomorphisms if $n\geq 3$,

  * •

by Lemma 5.36 (2) if $n=2$, indeed we may take $\psi^{(-2)}_{X,Y}$ to be the
composition of the isomorphisms
$\mathbb{E}_{\mathrm{I}}^{-2}(X,Y)\underset{\cong}{\overset{\omega_{\sharp}}{\longrightarrow}}\mathbb{E}_{\mathrm{I}}^{-1}(X,Q)\underset{\cong}{\overset{\psi^{(-1)}_{X,Q}}{\longrightarrow}}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q)\underset{\cong}{\overset{\alpha}{\longleftarrow}}\mathbb{E}_{\mathrm{I}}^{-1}(J,Y)\underset{\cong}{\overset{\psi^{(-1)}_{J,Y}}{\longrightarrow}}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(J,Y)\underset{\cong}{\overset{\iota^{\sharp}}{\longleftarrow}}\mathbb{E}_{\mathrm{I\\!I}}^{-2}(X,Y),$

  * •

by Lemma 5.36 (1) if $n=1$.

Thus (1) is shown.

(2) For $X,Y\in\mathscr{C}$, let $(\ref{sTri_IP})$ and $\psi^{(-n)}_{X,Y}$ be
as in (1). Let $x\in\mathscr{C}(X,X^{\prime})$ be any morphism and let
$X^{\prime}\overset{i^{\prime}}{\longrightarrow}I^{\prime}\overset{j^{\prime}}{\longrightarrow}J^{\prime}\overset{\iota^{\prime}}{\dashrightarrow}$
be any $\mathfrak{s}$-triangle in which $I^{\prime}\in\mathscr{C}$ is an
injective object. By (1), there also exists a unique
$\psi^{(-n)}_{X^{\prime},Y}$ which makes
$\mathbf{D}(-n;\iota^{\prime},\omega)$ commutative. We may obtain a morphism
of $\mathfrak{s}$-triangles as follows.

$\textstyle{X}$$\textstyle{I}$$\textstyle{J}$$\textstyle{X^{\prime}}$$\textstyle{I^{\prime}}$$\textstyle{J^{\prime}}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{\iota}$$\scriptstyle{x}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{i^{\prime}}$$\scriptstyle{j^{\prime}}$$\scriptstyle{\iota^{\prime}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

This induces commutative diagrams

$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(X^{\prime},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+1}(X^{\prime},Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(X^{\prime},Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+2}(J^{\prime},Q)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+1}(X,Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(X,Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+2}(J,Q)}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{\psi^{(-n+1)}_{X^{\prime},Q}}$$\scriptstyle{\cong}$$\scriptstyle{(\iota^{\prime})^{\sharp}}$$\scriptstyle{x^{\ast}}$$\scriptstyle{x^{\ast}}$$\scriptstyle{x^{\ast}}$$\scriptstyle{g^{\ast}}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{\psi^{(-n+1)}_{X,Q}}$$\scriptstyle{\cong}$$\scriptstyle{\iota^{\sharp}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

and

$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X^{\prime},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(J^{\prime},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+1}(J^{\prime},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+2}(J^{\prime},Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(J,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+1}(J,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+2}(J,Q)}$$\scriptstyle{(\iota^{\prime})^{\sharp}}$$\scriptstyle{\psi^{(-n+1)}_{J^{\prime},Y}}$$\scriptstyle{\cong}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{x^{\ast}}$$\scriptstyle{g^{\ast}}$$\scriptstyle{g^{\ast}}$$\scriptstyle{g^{\ast}}$$\scriptstyle{\iota^{\sharp}}$$\scriptstyle{\psi^{(-n+1)}_{J,Y}}$$\scriptstyle{\cong}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

and also

$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+2}(J^{\prime},Q)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+2}(J^{\prime},Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+2}(J,Q)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+2}(J,Q)}$$\scriptstyle{\psi^{(-n+2)}_{J^{\prime},Q}}$$\scriptstyle{\cong}$$\scriptstyle{g^{\ast}}$$\scriptstyle{g^{\ast}}$$\scriptstyle{\psi^{(-n+2)}_{J,Q}}$$\scriptstyle{\cong}$$\scriptstyle{\circlearrowright}$

by the naturality of $\psi^{(k)}$ for $k\geq-n+1$. Pasting them with
$\mathbf{D}(-n;\iota,\omega)$ and $\mathbf{D}(-n;\iota^{\prime},\omega)$, we
obtain the commutativity of

$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(X^{\prime},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X^{\prime},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)}$$\scriptstyle{\psi^{(-n)}_{X^{\prime},Y}}$$\scriptstyle{\cong}$$\scriptstyle{x^{\ast}}$$\scriptstyle{x^{\ast}}$$\scriptstyle{\psi^{(-n)}_{X,Y}}$$\scriptstyle{\cong}$$\scriptstyle{\circlearrowright}$

since the composition of $(\ref{ComposMonom})$ is monomorphic. This shows the
naturality of $\psi^{(-n)}_{X,Y}$ in $X\in\mathscr{C}$. Moreover if we apply
the above argument to $x=\mathrm{id}_{X}$, this shows that $\psi^{(-n)}_{X,Y}$
is independent of the choices of $\mathfrak{s}$-triangles
$X\overset{i}{\longrightarrow}I\overset{i}{\longrightarrow}J\overset{\iota}{\dashrightarrow}$.
In a similar way we can also show the naturality in $Y\in\mathscr{C}$ and that
$\psi_{X,Y}$ is independent of the choices of
$Q\overset{q}{\longrightarrow}P\overset{p}{\longrightarrow}Y\overset{\omega}{\dashrightarrow}$.

(3) Let $\delta\in\mathbb{E}(W,X)$ and $\rho\in\mathbb{E}(Y,Z)$ be any pair of
$\mathbb{E}$-extensions, and let $(\ref{sTri_IP})$ be arbitrary as in (1).
Then there exist $w\in\mathscr{C}(W,J)$ and $z\in\mathscr{C}(Q,Z)$ such that
$w^{\ast}\iota=\delta$ and $z_{\ast}\omega=\rho$. They give a commutative
diagram

$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+1}(X,Z)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(X,Z)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+2}(J,Z)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+2}(W,Z)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+1}(X,Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(X,Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+2}(J,Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(J,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+1}(J,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+2}(J,Q)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(W,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+1}(W,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+2}(W,Q)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-n+2}(W,Z)}$↻$\scriptstyle{\rho_{\sharp}}$$\scriptstyle{\psi^{(-n+1)}_{X,Z}}$$\scriptstyle{\cong}$$\scriptstyle{\iota^{\sharp}}$$\scriptstyle{w^{\ast}}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{\cong}$$\scriptstyle{\psi^{(-n+1)}_{X,Q}}$$\scriptstyle{\iota^{\sharp}}$$\scriptstyle{w^{\ast}z_{\ast}}$$\scriptstyle{\psi^{(-n)}_{X,Y}}$$\scriptstyle{\cong}$$\scriptstyle{\iota^{\sharp}}$$\scriptstyle{\psi^{(-n+1)}_{J,Y}}$$\scriptstyle{\cong}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{w^{\ast}z_{\ast}}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\psi^{(-n+1)}_{W,Y}}$$\scriptstyle{\cong}$$\scriptstyle{\omega_{\sharp}}$$\scriptstyle{z_{\ast}}$$\scriptstyle{\cong}$$\scriptstyle{\psi^{(-n+2)}_{W,Z}}$$\scriptstyle{\cong}$$\scriptstyle{\psi^{(-n+2)}_{J,Q}}$$\scriptstyle{z_{\ast}}$$\scriptstyle{z_{\ast}}$$\scriptstyle{z_{\ast}}$$\scriptstyle{w^{\ast}}$$\scriptstyle{w^{\ast}}$$\scriptstyle{w^{\ast}}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\rho_{\sharp}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in which, the middle rectangle is nothing but $\mathbf{D}(-n;\iota,\omega)$
that is commutative by the construction. Thus we obtain the commutativity of
$\mathbf{D}(-n;\delta,\rho)$, which appears as the outer rectangle. ∎

∎

######  Corollary 5.39.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
objects and enough injective objects. If
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies (NI$+$) and (NII+), then
$\\{\mathbb{E}_{\mathrm{I}}^{n}\cong\mathbb{E}_{\mathrm{I\\!I}}^{n}\\}_{n\in\mathbb{Z}}$
gives a universal bivariant $\delta$-functor having $\mathbb{E}^{n}$ with the
canonical connecting morphisms for $n\geq 0$. It is unique by Corollary 5.21.
More precisely, if we define

  * •

$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{I}}^{-n}(A,-)\to\mathbb{E}_{\mathrm{I}}^{-n+1}(C,-)$
to be the composition of

$\mathbb{E}_{\mathrm{I}}^{-n}(A,-)\underset{\cong}{\overset{\psi^{(-n)}_{A,-}}{\longrightarrow}}\mathbb{E}_{\mathrm{I\\!I}}^{-n}(A,-)\overset{\delta^{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(C,-)\underset{\cong}{\overset{\psi^{(-n+1)}_{C,-}}{\longleftarrow}}\mathbb{E}_{\mathrm{I}}^{-n+1}(C,-),$

  * •

$\delta_{\sharp}\colon\mathbb{E}_{\mathrm{I\\!I}}^{-n}(-,C)\to\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(-,A)$
to be the composition of

$\mathbb{E}_{\mathrm{I\\!I}}^{-n}(-,C)\underset{\cong}{\overset{\psi^{(-n)}_{-,C}}{\longleftarrow}}\mathbb{E}_{\mathrm{I}}^{-n}(-,C)\overset{\delta_{\sharp}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-n+1}(-,A)\underset{\cong}{\overset{\psi^{(-n+1)}_{-,A}}{\longrightarrow}}\mathbb{E}_{\mathrm{I\\!I}}^{-n+1}(-,A)$

for any $\delta\in\mathbb{E}(C,A)$ and any $n$, then both
$(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp},\delta^{\sharp})$ and
$(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta_{\sharp},\delta^{\sharp})$ are
bivariant $\delta$-functors for which
$\psi^{\bullet}=\\{\psi^{(n)}\colon\mathbb{E}_{\mathrm{I}}^{n}\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{n}\\}_{n\in\mathbb{Z}}$
gives an isomorphism.

###### Proof.

This is immediate from Theorem 5.37 and the construction. ∎

###  5.5. Conjectures

An extriangulated category is called _topological_ if it is equivalent to the
homotopy category of an exact $\infty$-category, see [NP2]. Klemenc [Kl]
constructed a _stable hull_ for each exact $\infty$-category. That is, he
proved that each exact $\infty$-category embeds into a stable
$\infty$-category, and there is an embedding satisfying a reasonable universal
property. Since the homotopy category of a stable $\infty$-category is
triangulated, this shows (using [NP2, Prop 4.28]) that for each topological
extriangulated category $(\mathscr{C},\mathbb{E},\mathfrak{s})$, there is a
triangulated category $\mathcal{T}$ and an embedding
$f\colon(\mathscr{C},\mathbb{E},\mathfrak{s})\hookrightarrow\mathcal{T}$ as an
extension-closed subcategory. This in turn indicates that
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ has a balanced negative extensions
given by $\mathbb{E}^{-n}(A,B):=\mathcal{T}(f(A)[n],f(B))$ which may have some
kind of universality. The construction of Klemenc uses quasi-categorical
enhancements, and thus this morphism spaces are not easy to calculate. We make
the following conjectures on the existence of balanced $\mathbb{E}^{-n}$ and
their relation to $\mathbb{E}_{\mathrm{I}}^{\bullet}$ and
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$.

######  Conjecture 5.40.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is a topological
extriangulated category. Then it admits a bivariant $\delta$-functor
$(\mathbb{E}^{\bullet},\delta_{\sharp},\delta^{\sharp})$ having
$F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for $n\geq 0$
and universal among such bivariant $\delta$-functors. Moreover, we have

$\mathbb{E}^{-n}(I,-)\cong\mathbb{E}_{\mathrm{I}}^{-n}(I,-);\quad\mathbb{E}^{-n}(-,P)\cong\mathbb{E}_{\mathrm{I\\!I}}^{-n}(-,P),$

for all $I$ injective, $P$ projective, $n>0$.

######  Conjecture 5.41.

Assume that Conjecture 5.40 holds for $(\mathscr{C},\mathbb{E},\mathfrak{s})$
and $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has positive global dimension $n$.
Then $\mathbb{E}^{-m}=0$ for all $m>2n$.

For extriangulated categories for which the statement of Conjecture 5.41
holds, one might be able to define the _negative global dimension_.

######  Example 5.42.

  1. (i)

The negative global dimension of an exact category is $0$.

  2. (ii)

Each non-zero triangulated category has infinite positive and negative global
dimensions.

  3. (iii)

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has positive global dimension $n$
and is such that
$\mathbb{E}_{\mathrm{I}}^{m}\cong\mathbb{E}_{\mathrm{I\\!I}}^{m}$, for all
$m$, then by definition, $\mathbb{E}^{-m}=0$ for all $m>n$. Thus, the negative
global dimension of $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is $-n$.

  4. (iv)

As a special case of the previous point, in the category
$K^{[-n,0]}(\operatorname{proj}\Lambda)$ of $n$-term complexes with finitely
generated projective components over a finite-dimensional algebra, we have
$\mathbb{E}_{\mathrm{I}}^{m}\cong\mathbb{E}_{\mathrm{I\\!I}}^{m}$, for all
$m$. Thus, the negative global dimension is precisely $-n$.

The reader may be surprised by the bound $2n$ instead of $n$. The next example
explains its appearance in the conjecture.

######  Example 5.43.

Let $A$ be a non-positively graded dg-algebra. Consider the full extension-
closed subcategory $\mathscr{C}:=A\ast A[1]\ast A[2]\ast\cdots\ast A[n]$ of
$\operatorname{\mathrm{per}}\nolimits A$ with the induced extriangulated
structure. This is an example of an extriangulated category of positive global
dimension $n$. Its projective objects are given by $\mbox{add}A$ and its
injective objects are given by $\mbox{add}A[n]$. Straightforward calculations
show that any bivariant $\delta$-functor $F^{\bullet}$ of having
$F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for $n\geq 0$
has to satisfy

$F^{-2n}(A,A[n])=\mbox{Hom}(A[n],A).$

For an arbitrary $A$, this might be non-zero. This means the that the negative
global dimension of $\mathscr{C}$ can be $-2n$.

Conjectures 5.40 and 5.41 do not seem to follow easily from Klemenc’s work
[Kl]. Note though that Conjecture 5.40 clearly holds for all triangulated
categories, including those that do not admit enhancements, i.e. non-
topological. This suggests that there is a chance that one would be able to
define universal bivariant (=balanced) negative extensions for all
extriangulated categories.

##  6\. Projective resolutions for defects

The $\delta$-functor in Example 4.10 (2) gives projective resolutions of
defects _in the category $\operatorname{\mathrm{def}}\nolimits\mathbb{E}$_.
Note that it is not a projective resolution in
$\operatorname{\mathrm{coh}}(\mathbb{E})$.

In this section, we will show that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has
enough projective morphisms if and only if the inclusion
$\operatorname{\mathrm{def}}\nolimits\mathbb{E}\hookrightarrow\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$
has a left adjoint (Corollary 6.5). As a result, we will see that any object
in $\operatorname{\mathrm{def}}\nolimits\mathbb{E}$ admits a natural
projective resolution in $\operatorname{\mathrm{def}}\nolimits\mathbb{E}$
whenever $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective
morphisms (Corollary 6.6). We use the notations in Subsection 2.2.

Let us recall the definition of _reflection_ from [Bor, Definition 3.1.1]. For
the inclusion functor
$\iota\colon\operatorname{\mathrm{def}}\nolimits\mathbb{E}\hookrightarrow\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$,
it becomes as follows.

######  Definition 6.1.

Let
$\iota\colon\operatorname{\mathrm{def}}\nolimits\mathbb{E}\hookrightarrow\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$
denote the inclusion. For an object
$\mathbf{F}\in\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$, a reflection
of $\mathbf{F}$ along $\iota$ is a pair
$(\Gamma_{\mathbf{F}},\gamma_{\mathbf{F}})$ of
$\Gamma_{\mathbf{F}}\in\operatorname{\mathrm{def}}\nolimits\mathbb{E}$ and
$\gamma_{\mathbf{F}}\in(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathbf{F},\Gamma_{\mathbf{F}})$
such that

$-\circ\gamma_{\mathbf{F}}\colon(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Gamma_{\mathbf{F}},\Theta)\to(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathbf{F},\Theta)$

is an isomorphism for any
$\Theta\in\operatorname{\mathrm{def}}\nolimits\mathbb{E}$.

######  Proposition 6.2.

Let $C\in\mathscr{C}$ be any object, and let
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$
be any dominant $\mathfrak{s}$-triangle. Then the following holds.

  1. (1)

$\underline{\wp}_{\vartheta}\colon\underline{\mathscr{C}}(-,C)\to\Theta_{\vartheta}$
is an isomorphism in
$\operatorname{\mathrm{}{fp}}\nolimits\underline{\mathscr{C}}$. In particular
we have
$\underline{\mathscr{C}}(-,C)\in\operatorname{\mathrm{def}}\nolimits\mathbb{E}$.

  2. (2)

$(\Theta_{\vartheta},\wp_{\vartheta})$ is a reflection of $\mathscr{C}(-,C)$
along $\iota$.

  3. (3)

$\Theta_{\vartheta}$ is projective in
$\operatorname{\mathrm{def}}\nolimits\mathbb{E}$.

###### Proof.

(1) This follows from
$\mbox{Im}\big{(}\mathscr{C}(-,G)\overset{g\circ-}{\longrightarrow}\mathscr{C}(-,C)\big{)}=\mathcal{P}(-,C)$.

(2) Let $\Theta\in\operatorname{\mathrm{def}}\nolimits\mathbb{E}$ be any
object. In the commutative diagram

$\textstyle{(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathscr{C}(-,C),\Theta)}$$\textstyle{(\operatorname{\mathrm{}{fp}}\nolimits\underline{\mathscr{C}})(\underline{\mathscr{C}}(-,C),\Theta)}$$\textstyle{(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta_{\vartheta},\Theta)}$$\scriptstyle{-\circ\underline{(-)}}$$\scriptstyle{\cong}$$\scriptstyle{-\circ\underline{\wp}_{\vartheta}}$$\scriptstyle{-\circ\wp_{\vartheta}}$$\scriptstyle{\circlearrowright}$

the bottom row is an isomorphism by (1), while

(6.1)
$-\circ\underline{(-)}\colon(\operatorname{\mathrm{}{fp}}\nolimits\underline{\mathscr{C}})(\underline{\mathscr{C}}(-,C),\Theta)\overset{\cong}{\longrightarrow}(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathscr{C}(-,C),\Theta)$

is an isomorphism since $\Theta|_{\mathcal{P}}=0$. Thus it follows that

$-\circ\wp_{\vartheta}\colon(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta_{\vartheta},\Theta)\to(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathscr{C}(-,C),\Theta)$

is also an isomorphism.

(3) This is immediate from the isomorphism of functors

$(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathscr{C}(-,C),-)|_{\operatorname{\mathrm{def}}\nolimits\mathbb{E}}\cong(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta_{\vartheta},-)\colon\operatorname{\mathrm{def}}\nolimits\mathbb{E}\to\mbox{Mod\,}R$

obtained in (2), and the projectivity of $\mathscr{C}(-,C)$ in
$\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$.

∎

######  Corollary 6.3.

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

  1. (1)

$\delta$ is dominant.

  2. (2)

$(\Theta_{\delta},\wp_{\delta})$ is a reflection of $\mathscr{C}(-,C)$ along
$\iota$.

###### Proof.

(1) $\Rightarrow$ (2) is shown in Proposition 6.2 (2).

(2) $\Rightarrow$ (1) Realize $\delta$ to obtain an $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$.
Let $\rho\in\mathbb{E}(C,X)$ be any extension. If (2) holds, then there exists
$\eta\in(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta_{\delta},\Theta_{\rho})$
such that $\eta\circ\wp_{\delta}=\wp_{\rho}$. Then $y^{\ast}\delta=0$ implies
$y^{\ast}\rho=0$, hence $y$ is a projective deflation. ∎

######  Proposition 6.4.

Let $C\in\mathscr{C}$ and
$\Theta\in\operatorname{\mathrm{def}}\nolimits\mathbb{E}$ be any pair of
objects. For any morphism
$\varphi\in(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathscr{C}(-,C),\Theta)$,
the following holds.

  1. (1)

There exist $\delta\in\mathbb{E}(C,A)$ and a monomorphism
$\eta\in(\mbox{Mod\,}\mathscr{C})(\Theta_{\delta},\Theta)$ such that
$\eta\circ\wp_{\delta}=\varphi$.

  2. (2)

Let $\delta$ and $\eta$ be arbitrary as in (1). If $(\Theta,\varphi)$ is a
reflection of $\mathscr{C}(-,C)$ along $\iota$, then $\eta$ is an isomorphism.
In particular, $\varphi$ becomes an epimorphism in $\mbox{Mod\,}\mathscr{C}$
in this case.

  3. (3)

Let $\delta$ and $\eta$ be arbitrary as in (1). Then $\delta$ is dominant
whenever $(\Theta,\varphi)$ is a reflection of $\mathscr{C}(-,C)$ along
$\iota$.

###### Proof.

(1) Replacing through an isomorphism, we may assume that there exists
$\rho\in\mathbb{E}(D,A)$ such that $\Theta=\Theta_{\rho}$. By the projectivity
of $\mathscr{C}(-,C)$, there exists $c\in\mathscr{C}(-,C)$ which makes

$\textstyle{\mathscr{C}(-,C)}$$\textstyle{\mathscr{C}(-,D)}$$\textstyle{\Theta_{\rho}}$$\scriptstyle{c\circ-}$$\scriptstyle{\wp_{\rho}}$$\scriptstyle{\varphi}$$\scriptstyle{\circlearrowright}$

commutative. If we put $\delta=c^{\ast}\rho\in\mathbb{E}(C,A)$, then
$\eta=\eta_{(\mathrm{id}_{A},c)}\in(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta_{\delta},\Theta_{\rho})$
satisfies the required properties.

(2) If $(\Theta,\varphi)$ is a reflection of $\mathscr{C}(-,C)$ along $\iota$,
then there should exist
$\xi\in(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta,\Theta_{\delta})$
such that $\xi\circ\varphi=\wp_{\delta}$. Then, since both
$\mathrm{id}_{\Theta},\eta\circ\xi\in(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta,\Theta)$
satisfy $\mathrm{id}_{\Theta}\circ\varphi=\varphi$ and
$(\eta\circ\xi)\circ\varphi=\varphi$, it follows
$\eta\circ\xi=\mathrm{id}_{\Theta}$. In particular $\eta$ is a split
epimorphism, hence an isomorphism since it is monomorphic by (1).

(3) Suppose that $(\Theta,\varphi)$ is a reflection of $\mathscr{C}(-,C)$
along $\iota$. Then $\eta$ becomes an isomorphism by (2), hence
$(\Theta_{\delta},\wp_{\delta})$ also becomes a reflection of
$\mathscr{C}(-,C)$ along $\iota$. Thus $\delta$ becomes dominant by Corollary
6.3. ∎

######  Proposition 6.5.

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

  1. (1)

The inclusion
$\iota\colon\operatorname{\mathrm{def}}\nolimits\mathbb{E}\hookrightarrow\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$
has a left adjoint.

  2. (2)

Any object $\mathbf{F}\in\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$ has
a reflection $(\Gamma_{\mathbf{F}},\gamma_{\mathbf{F}})$ along $\iota$.

  3. (3)

For any $C\in\mathscr{C}$, the object
$\mathscr{C}(-,C)\in\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$ has a
reflection along $\iota$.

  4. (4)

$\underline{\mathscr{C}}(-,C)\in\operatorname{\mathrm{def}}\nolimits\mathbb{E}$
holds for any $C\in\mathscr{C}$.

  5. (5)

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

###### Proof.

The equivalence (1) $\Leftrightarrow$ (2) is well-known. Indeed if (2) holds,
then a formal argument shows that $\mathbf{F}\mapsto\Gamma_{\mathbf{F}}$
constitutes a functor
$\Gamma\colon\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}\to\operatorname{\mathrm{def}}\nolimits\mathbb{E}$
left adjoint to $\iota$, whose unit
$\gamma\colon\mathrm{Id}_{\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}}\to\iota\circ\Gamma$
is given by
$\gamma=\\{\gamma_{\mathbf{F}}\\}_{\mathbf{F}\in\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}}$.

(2) $\Rightarrow$ (3) is trivial. The equivalence (3) $\Leftrightarrow$ (5)
follows from Corollary 6.3 and Proposition 6.4. (3) $\Rightarrow$ (4) follows
from Proposition 6.2 (1). Conversely if (4) holds, then for any
$C\in\mathscr{C}$, the morphism
$\underline{(-)}\colon\mathscr{C}(-,C)\to\underline{\mathscr{C}}(-,C)$ gives a
reflection of $\mathscr{C}(-,C)$ along $\iota$ since $(\ref{IsoQuot})$ is an
isomorphism.

It remains to show (3) $\Rightarrow$(2). Suppose that (3) holds, and let
$\mathbf{F}\in\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$ be any object.
By definition, there exists a right exact sequence

$\mathscr{C}(-,C^{\prime})\overset{c\circ-}{\longrightarrow}\mathscr{C}(-,C)\overset{\pi}{\longrightarrow}\mathbf{F}\to
0$

in $\mbox{Mod\,}\mathscr{C}$. Take dominant extensions
$\vartheta\in\mathbb{E}(C,F)$ and
$\vartheta^{\prime}\in\mathbb{E}(C^{\prime},F^{\prime})$. Realize them to
obtain $\mathfrak{s}$-triangles
$F\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}C\overset{\vartheta}{\dashrightarrow}$
and
$F^{\prime}\overset{f^{\prime}}{\longrightarrow}G^{\prime}\overset{g^{\prime}}{\longrightarrow}C^{\prime}\overset{\vartheta^{\prime}}{\dashrightarrow}$.
Since $\vartheta^{\prime}$ is dominant, we obtain a morphism of
$\mathfrak{s}$-triangles

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which induces a commutative square

$\textstyle{\mathscr{C}(-,C^{\prime})}$$\textstyle{\Theta_{\vartheta^{\prime}}}$$\textstyle{\mathscr{C}(-,C)}$$\textstyle{\Theta_{\vartheta}}$$\scriptstyle{\wp_{\vartheta^{\prime}}}$$\scriptstyle{c\circ-}$$\scriptstyle{\eta=\eta_{(a,c)}}$$\scriptstyle{\wp_{\vartheta}}$$\scriptstyle{\circlearrowright}$

in $\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$. If we put
$\Gamma_{\mathbf{F}}=\operatorname{Cok}\eta$, we obtain a commutative diagram

(6.2)
$\textstyle{\mathscr{C}(-,C^{\prime})}$$\textstyle{\mathscr{C}(-,C)}$$\textstyle{\mathbf{F}}$$\textstyle{0}$$\textstyle{\Theta_{\vartheta^{\prime}}}$$\textstyle{\Theta_{\vartheta}}$$\textstyle{\Gamma_{\mathbf{F}}}$$\textstyle{0}$$\scriptstyle{c\circ-}$$\scriptstyle{\pi}$$\scriptstyle{\wp_{\vartheta^{\prime}}}$$\scriptstyle{\wp_{\vartheta}}$$\scriptstyle{\gamma_{\mathbf{F}}}$$\scriptstyle{\eta}$$\scriptstyle{\operatorname{cok}\eta}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in $\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C}$. Let us to show that
$(\Gamma_{\mathbf{F}},\gamma_{\mathbf{F}})$ gives a reflection of $\mathbf{F}$
along $\iota$.

Take any $\Theta\in\operatorname{\mathrm{def}}\nolimits\mathbb{E}$. If we
apply $(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(-,\Theta)$ to
$(\ref{MorSeq1})$, then we obtain

$\textstyle{0}$$\textstyle{(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathbf{F},\Theta)}$$\textstyle{(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathscr{C}(-,C),\Theta)}$$\textstyle{(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathscr{C}(-,C^{\prime}),\Theta)}$$\textstyle{0}$$\textstyle{(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Gamma_{\mathbf{F}},\Theta)}$$\textstyle{(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta_{\vartheta},\Theta)}$$\textstyle{(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta_{\vartheta^{\prime}},\Theta)}$$\scriptstyle{-\circ\pi}$$\scriptstyle{-\circ\gamma_{\mathbf{F}}}$$\scriptstyle{-\circ\wp_{\vartheta}}$$\scriptstyle{\cong}$$\scriptstyle{-\circ\wp_{\vartheta^{\prime}}}$$\scriptstyle{\cong}$$\scriptstyle{-\circ\operatorname{cok}\eta}$$\scriptstyle{-\circ\eta}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in which the rows are exact, the vertical morphisms in the middle and on the
right are isomorphisms by Proposition 6.2 (2). Hence

$-\circ\gamma_{\mathbf{F}}\colon(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Gamma_{\mathbf{F}},\Theta)\to(\operatorname{\mathrm{}{fp}}\nolimits\mathscr{C})(\mathbf{F},\Theta)$

becomes an isomorphism as desired. ∎

######  Corollary 6.6.

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ has enough projective morphisms,
then $\operatorname{\mathrm{def}}\nolimits\mathbb{E}$ has enough projectives.

###### Proof.

Let $\Theta\in\operatorname{\mathrm{def}}\nolimits\mathbb{E}$ be any object.
We may assume that there exists $\delta\in\mathbb{E}(C,A)$ such that
$\Theta=\Theta_{\delta}$. If we take a dominant extension
$\vartheta\in\mathbb{E}(C,F)$, then there exists
$\eta\in(\operatorname{\mathrm{def}}\nolimits\mathbb{E})(\Theta_{\vartheta},\Theta_{\delta})$
such that $\eta\circ\wp_{\vartheta}=\wp_{\delta}$. Then $\eta$ is epimorphic,
and $\Theta_{\vartheta}$ is projective by Proposition 6.2 (3). ∎

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