$\displaystyle\varphi_{2}:\operatorname{Co-}t_{(\mathcal{A},\mathcal{B})}(\mathscr{D})$
$\displaystyle\leftrightarrow$
$\displaystyle\operatorname{cCP}(\mathscr{C}):\psi_{2},$

where

$\varphi_{2}(\mathcal{A}^{\prime},\mathcal{B}^{\prime})=(\mathcal{B}\cap\Sigma\mathcal{A}^{\prime},\mathcal{B}^{\prime}\cap\Sigma^{2}\mathcal{A})\text{
and}$
$\psi_{2}(\mathcal{X},\mathcal{Y})=\left(\operatorname{add}(\Sigma^{-1}\mathcal{A}\ast\Sigma^{-1}\mathcal{X}),\operatorname{add}(\mathcal{Y}\ast\Sigma^{2}\mathcal{B})\right).$

######  Remark 5.6.

The bijective correspondence $\varphi_{2}$ moreover sends cohearts to cores:
with the notations of the theorem, let
$(\mathcal{A}^{\prime},\mathcal{B}^{\prime})\in\operatorname{Co-}t_{(\mathcal{A},\mathcal{B})}(\mathscr{D})$
and let
$(\mathcal{X},\mathcal{Y})=\varphi_{2}(\mathcal{A}^{\prime},\mathcal{B}^{\prime})$.
Then
$\mathcal{X}\cap\mathcal{Y}=\mathcal{S}^{\prime}=\Sigma\mathcal{A}^{\prime}\cap\mathcal{B}^{\prime}$.

Note that this result has the following nice corollary. It might be known to
experts, but we were not able to find this statement in the literature.

######  Corollary 5.7.

Let $\mathscr{D}$ be a triangulated category and let
$(\mathcal{A},\mathcal{B})$ be a co-$t$-structure on $\mathcal{D}$ with
coheart $\mathcal{S}$. Then for each co-$t$-structure
$(\mathcal{A}^{\prime\prime},\mathcal{B}^{\prime\prime})$ on
$\mathscr{T}:=\mathrm{thick}(\mathcal{S})$ intermediate with respect to the
restriction $(\mathcal{A}\cap\mathscr{T},\mathcal{B}\cap\mathscr{T})$ of
$(\mathcal{A},\mathcal{B})$, there exists a unique co-$t$-structure
$(\mathcal{A}^{\prime},\mathcal{B}^{\prime})\in\operatorname{Co-}t_{(\mathcal{A},\mathcal{B})}(\mathscr{D})$
extending $(\mathcal{A}^{\prime\prime},\mathcal{B}^{\prime\prime})$.

###### Proof.

Note that $\mathscr{C}\subset\mathscr{T}$. Therefore, both collections
$\operatorname{Co-}t_{(\mathcal{A},\mathcal{B})}(\mathscr{D})$ and
$\operatorname{Co-}t_{(\mathcal{A}\cap\mathscr{T},\mathcal{B}\cap\mathscr{T})}(\mathscr{T})$
are in bijective correspondences with $\operatorname{cCP}(\mathscr{C})$, hence
they are themselves in a bijective correspondence. It remains to show that the
bijection
$\operatorname{Co-}t_{(\mathcal{A},\mathcal{B})}(\mathscr{D})\to\operatorname{Co-}t_{(\mathcal{A}\cap\mathscr{T},\mathcal{B}\cap\mathscr{T})}(\mathscr{T})$
is given by the restriction. Let $(\mathcal{X},\mathcal{Y})$ be the cotorsion
pair corresponding to
$(\mathcal{A}^{\prime\prime},\mathcal{B}^{\prime\prime})$. Then

$(\mathcal{A}^{\prime},\mathcal{B}^{\prime}):=\left(\operatorname{add}(\Sigma^{-1}\mathcal{A}\ast\Sigma^{-1}\mathcal{X}),\operatorname{add}(\mathcal{Y}\ast\Sigma^{2}\mathcal{B})\right)$

is the corresponding co-$t$-structure on $\mathcal{D}$, while we have

$(\mathcal{A}^{\prime\prime},\mathcal{B}^{\prime\prime})=\left(\operatorname{add}(\Sigma^{-1}(\mathcal{A}\cap\mathscr{T})\ast\Sigma^{-1}\mathcal{X}),\operatorname{add}(\mathcal{Y}\ast\Sigma^{2}(\mathcal{B}\cap\mathscr{T}))\right).$

Observe that we have
$\mathcal{A}^{\prime\prime}\subset\mathcal{A}^{\prime}\cap\mathscr{T}$ and
$\mathcal{B}^{\prime\prime}\subset\mathcal{B}^{\prime}\cap\mathscr{T}$. Thus,
$(\mathcal{A}^{\prime\prime},\mathcal{B}^{\prime\prime})=(\mathcal{A}^{\prime}\cap\mathscr{T},\mathcal{B}^{\prime}\cap\mathscr{T})$.
∎

######  Remark 5.8.

Note that Corollary 5.7 ensures the existence and uniqueness of extensions of
co-t-structures intermediate with respect to the restriction
$(\mathcal{A}\cap\mathscr{T},\mathcal{B}\cap\mathscr{T})$, to co-t-structures
intermediate with respect to $(\mathcal{A},\mathcal{B})$. A similar result is
true for co-t-structures “intermediate” in a broader sense, i.e. such that
$\Sigma^{a}\mathcal{A}\subseteq\mathcal{A}^{\prime}\subseteq\Sigma^{b}\mathcal{A}$
for finite $a<b$. This follows from [AT22, Corollary 3.9] applied twice as in
the proof of Corollary 5.7. [AT22, Corollary 3.9] generalizes Theorem 5.5 to
suitable $(b-a)$-Auslander extriangulated categories.

The “intermediate” assumptions, however, cannot be dropped completely. Here is
an elementary explanation: each non-zero triangulated category $\mathscr{D}$
admits at least two distinct co-t-structures $(0,\mathscr{D})$ and
$(\mathscr{D},0)$. For either of them, the coheart is the zero subcategory.
They are both unbounded and the largest thick subcategory which admits a
bounded co-t-structure restricted from either of them is again the zero
subcategory $0=\mathrm{thick}(0)$. The restricted co-t-structures, of course,
both equal $(0,0)$. In other words, the co-t-structure $(0,0)$ on the zero
subcategory admits two distinct extensions to $\mathscr{D}$. We do not get a
contradiction with Corollary 5.7 precisely because $(0,\mathscr{D})$ is not
intermediate with respect to $(\mathscr{D},0)$ and vice versa.

Note also that there exists a similar result for t-structures with finite
distance from a given one, see [CLZ22] and references to Keller [Kel05] and
Marks-Zvonareva [MZ22] therein.

######  Lemma 5.9.

Let $\mathscr{D}$ be an essentially small $R$-linear triangulated category and
let $(\mathcal{A},\mathcal{B})$ be a co-$t$-structure with associated coheart
$\mathcal{S}$ and extended coheart $\mathscr{C}$. Then the extriangulated
category $\mathscr{C}$ is reduced 0-Auslander, with projective objects
$\mathcal{S}$ and injective objects $\Sigma\mathcal{S}$.

###### Proof.

Since $\mathscr{C}=\mathcal{S}\ast\Sigma\mathcal{S}$, it is enough to show
that $\mathcal{S}$ is projective in $\mathscr{C}$. Let $S\in\mathcal{S}$ and
$C\in\mathscr{C}$. Then $S$ belongs to $\Sigma\mathcal{A}\cap\mathcal{B}$
while $\Sigma C$ lies in $\Sigma^{3}\mathcal{A}\cap\Sigma\mathcal{B}$. Because
$\Sigma\mathcal{A}$ is left Hom-orthogonal to $\Sigma\mathcal{B}$, we have
$\mathscr{D}(S,\Sigma C)=0$. ∎

######  Corollary 5.10.

Let $\mathscr{D}$ be an essentially small $R$-linear triangulated category and
let $(\mathcal{A},\mathcal{B})$ be a co-$t$-structure with associated extended
coheart $\mathscr{C}$. There are mutually inverse bijective correspondences

$\displaystyle\varphi_{3}:\operatorname{Co-}t_{(\mathcal{A},\mathcal{B})}(\mathscr{D})$
$\displaystyle\leftrightarrow$
$\displaystyle\operatorname{Tilt}(\mathscr{C}):\psi_{3},$

given by

$\varphi_{3}(\mathcal{A}^{\prime},\mathcal{B}^{\prime})=\Sigma\mathcal{A}^{\prime}\cap\mathcal{B}^{\prime}\text{
and}$
$\psi_{3}(\mathcal{R})=(\left(\operatorname{add}(\Sigma^{-1}\mathcal{A}\ast\Sigma^{-1}(\operatorname{copr}\mathcal{R}),\operatorname{add}((\operatorname{pr}\mathcal{R})\ast\Sigma^{2}\mathcal{B})\right),$

where $\operatorname{pr}\mathcal{R},\operatorname{copr}\mathcal{R}$ are
considered in $\mathscr{C}$.

###### Proof.

Combine Lemma 5.9 with Theorems 5.2 and 5.5. ∎

######  Corollary 5.11.

_[Mutation of intermediate co- $t$-structures]_ Let $\mathscr{D}$ be an
essentially small $R$-linear triangulated category and let
$(\mathcal{A},\mathcal{B})$ be a co-$t$-structure with extended coheart
$\mathscr{C}$. Let
$(\mathcal{A}^{\prime},\mathcal{B}^{\prime})\in\operatorname{Co-}t_{(\mathcal{A},\mathcal{B})}(\mathscr{D})$
with coheart $\mathcal{S}^{\prime}$. Assume that the category $\mathscr{C}$
satisfies the assumptions of Theorem 4.26 and let $X\in\mathcal{S}^{\prime}$
be indecomposable. Then there is a unique intermediate co-$t$-structure
$(\mathcal{C},\mathcal{D})\in\operatorname{Co-}t_{(\mathcal{A},\mathcal{B})}(\mathscr{D})$
whose coheart $\mathcal{T}$ satisfies
$\mathcal{T}\cap\mathcal{S}^{\prime}=\operatorname{add}\\{X^{\prime}\in\mathcal{S}^{\prime},X^{\prime}\text{
\emph{indecomposable and} }X^{\prime}\ncong X\\}$. Moreover, there is an
indecomposable object $Y\in\mathcal{T}$ such that

$\operatorname{add}\\{Y^{\prime}\in\mathcal{T},Y^{\prime}\text{
\emph{indecomposable and} }Y^{\prime}\ncong
Y\\}=\operatorname{add}\\{X^{\prime}\in\mathcal{S}^{\prime},X^{\prime}\text{
\emph{indecomposable and} }X^{\prime}\ncong X\\}.$

###### Proof.

Combine Theorem 4.26, Lemma 5.9 and Corollary 5.10. ∎

######  Remark 5.12.

We note that the assumptions of Theorem 4.26 are satisfied for example if
$\mathscr{C}$ is Krull–Schmidt and any almost complete rigid subcategory is
functorially finite in $\mathscr{C}$, as in Condition (F) in [AI12, Secion
2.4]. Note though that our assumptions concern the subcategory $\mathscr{C}$
and not the entire category $\mathscr{D}$ which might have no silting
subcategories.

The theory of mutations of bounded co-$t$-structures is implicit in [AI12] and
stated explicitly in certain generality in [KY14], see also [Jør18, Section
6]. It goes through the bijection of bounded co-$t$-structures on
$\mathscr{D}$ and silting subcategories in $\mathscr{D}$ mentioned in Remark
5.4 and silting mutations from [AI12]. Mutations of co-$t$-structures,
intermediate with respect to a certain bounded co-$t$-structure (and so also
bounded), are considered in [BY13, Corollary 4.3]. It follows from the results
in this section that mutations of intermediate co-$t$-structures can be
canonically defined via mutations of silting subcategories in the extended
coheart of the original co-$t$-structure, which do not have to be silting in
the entire $\mathscr{D}$.

######  Proposition 5.13.

  * (i)

Mutations in Corollary 5.11 recover known mutations of co-$t$-structures,
intermediate with respect to a bounded one.

  * (ii)

Mutations in Corollary 5.11 commute with the restriction to and extension from
$\mathrm{thick}(\mathcal{S}).$

###### Proof.

Part (i) follows from part (ii), which in turn is direct consequence of
Corollary 5.7 and its proof. ∎

######  Remark 5.14.

The upshot of Corollary 5.11 and Proposition 5.13 is that the theory of
mutation for bounded co-t-structures implies that for non-bounded ones by
restricting to and extending from thick subcategories. The fact that the
extension of a mutated restriction exists and is unique follows from Corollary
5.7 and seems to be novel. See also the discussion in Remark 5.8.

######  Remark 5.15.

Another result of Pauksztello–Zvonareva [PZ20, Theorem 3.6] provides a
bijection between complete cotorsion pairs in extended cohearts
$\mathcal{S}\ast\Sigma\mathcal{S}$ of co-$t$-structures and functorially
finite torsion classes in $\operatorname{mod}\mathcal{S}$, under certain
natural conditions. By adapting the proofs in [PZ20, Section 3] to the setting
of $\mathrm{Hom}$-finite Krull-Schmidt 0-Auslander categories with noetherian
$\operatorname{mod}\overline{\mathcal{P}}$ and using our Proposition 3.11, we
can obtain a bijection between complete cotorsion pairs in such 0-Auslander
categories and functorially finite torsion classes in
$\operatorname{mod}\overline{\mathcal{P}}$. A similar bijection was given in
[LZ20] for a certain class of 0-Auslander subcategories (defined in terms of
2-rigid subcategories $\mathcal{W}$ with $\mathcal{I}\subsetneq\mathcal{W}$)
in extriangulated categories; there, support $\tau$-tilting subcategories were
considered instead of functorially finite torsion classes. Via these
bijections, Theorem 4.26 recovers mutations of functorially finite torsion
classes and support $\tau$-tilting subcategories in $\tau$-tilting theory
[AIR14, IR14, Tre21, IJY14].

Generalizing [PZ20, Theorem 3.6], the functor inducing the equivalence in
Proposition 3.11 sends cotorsionfree classes to functorially finite torsion
classes. A maximal green sequence for a finite-dimensional algebra $\Lambda$
is a maximal chain of functorially finite torsion classes in the lattice of
torsion classes in $\operatorname{mod}\Lambda$. In our setup, this motivates a
(dual, in some sense) definition of maximal green sequences for 0-Auslander
categories as a maximal chain in the poset of cotorsionfree classes in
complete cotorsion pairs. We do not investigate such sequences further in this
paper.

##  6\. 0-Auslander categories from gentle algebras

Inspired by [PPP21, Figure 30], it was suggested in [INP18] that the
$\tau$-tilting theory for gentle algebras might be viewed as a mutation theory
for maximal rigid objects in a certain exact category constructed from the
module category over the blossoming gentle algebra. In [loc. cit.], an
explicit small example was considered. In this section, we carry out this
strategy for all gentle algebras. We define two exact subcategories of the
module category over the blossoming algebra and show that they are equivalent.
The first subcategory, studied in Section 6.2, is algebraically more
convenient, while the second (see Section 6.3) is more obviously related to
the combinatorics of non-kissing walks [McC17, BDM+20, PPP21]. We prove that
those categories are 0-Auslander and their maximal rigid objects are in
bijection with support $\tau$-tilting modules [AIR14] over the gentle algebra,
respectively with non-kissing facets. Under this identification, straight
walks bijectively correspond to indecomposable projective-injective objects.
The ideal quotients of these categories by the projective-injective objects
are no longer exact, but are reduced 0-Auslander extriangulated categories.
The mutation theory developed in Section 4 thus recovers $\tau$-tilting
mutation for gentle algebras, and flips of non-kissing facets.

###  6.1. Reminder on gentle algebras and non-kissing

Fix an algebraically closed field $K$. If $\alpha$ and $\beta$ are two arrows
in a quiver with $t(\alpha)=s(\beta)$, we write $\alpha\beta$ for their
composition.

####  6.1.1. Gentle bound quivers

######  Definition 6.1.

[BR87] A gentle algebra is a basic, finite-dimensional algebra $A=KQ/I$ where:

  * •

Each vertex in $Q$ is the source of at most two arrows and the target of at
most two arrows.

  * •

The ideal $I$ is generated by paths of length two.

  * •

For each arrow $\alpha$ in $Q$ with target vertex $i$, there is at most one
arrow $\beta$ and at most one arrow $\gamma$ with source $i$, such that
$\alpha\beta\notin I$ and $\alpha\gamma\in I$.

  * •

For each arrow $\alpha$ in $Q$ with source vertex $j$, there is at most one
arrow $\beta$ and at most one arrow $\gamma$ with target $j$, such that
$\beta\alpha\notin I$ and $\gamma\alpha\in I$.

####  6.1.2. String and band modules

Indecomposable representations of a gentle bound quiver $(Q,I)$ are
parameterized by strings and bands. A string for $(Q,I)$ is a non-oriented
path in $Q$, written as a word $\alpha_{1}^{\pm 1}\cdots\alpha_{r}^{\pm 1},$
where:

  * •

for every subpath $\alpha_{i}\alpha_{i+1},$ we have
$t(\alpha_{i})=s(\alpha_{i+1})$ and $\alpha_{i}\alpha_{i+1}\notin I$;

  * •

for every subpath $\alpha_{i}^{-1}\alpha_{i+1}^{-1},$ we have
$s(\alpha_{i})=t(\alpha_{i+1})$ and $\alpha_{i+1}\alpha_{i}\notin I$;

  * •

for every subpath $\alpha_{i}\alpha_{i+1}^{-1},$ or
$\alpha_{i}^{-1}\alpha_{i+1}$ we have $\alpha_{i}\neq\alpha_{i+1}$.

Note that strings of length 0 are allowed. For each vertex $p\in Q_{0}$, we
write $\varepsilon_{p}$ for the string of length 0 at $p$, and also for the
associated idempotent in $KQ/I$. Strings are usually considered unoriented
(e.g. the strings $\alpha\beta^{-1}$ and $\beta\alpha^{-1}$ are implicitly
identified).

With each string $\sigma$, one can associate an indecomposable representation
$M_{\sigma}$ of $(Q,I)$: The vertices appearing in the string encode basis
vectors, while the arrows show which basis vector is sent to which other basis
vector under the corresponding linear map. The representation associated with
the string $\varepsilon_{p}$ is the simple object concentrated at vertex $p$.

######  Example 6.2.

The quiver below, with the ideal of relations generated by $\beta\epsilon$ and
$\delta\gamma$, is gentle. On the right, the string
$\epsilon^{-1}\delta^{-1}\alpha\beta\gamma$ is drawn.:

${1}$${2}$${3}$${2}$${1}$${3}$${5}$${{\epsilon^{-1}\delta^{-1}\alpha\beta\gamma:}}$${5}$${3}$${4}$${4}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\scriptstyle{\delta}$$\scriptstyle{\epsilon}$$\scriptstyle{\delta}$$\scriptstyle{\epsilon}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$

The associated indecomposable representation is:

${K}$${K}$${{K^{2}}}$${K}$${K.}$$\scriptstyle{1}$$\scriptstyle{\left[{}^{1}_{0}\right]}$[1
0]$\scriptstyle{\left[{}^{0}_{1}\right]}$[0 1]

This is the indecomposable projective cover of the simple module concentrated
at vertex 1.

A band is a string of length at least one, whose source and target coincide,
and such that

  * •

its square is a string;

  * •

it is not the power of a smaller string.

Given a band, an element $\lambda\in K^{\times}$ and a positive integer $d$,
there is an associated indecomposable representation. The vector space at each
vertex appearing in the band is $K^{d}$ (note that the source and target
vertices of the band are identified), all linear maps are identities except
for one which is given by a Jordan block of size $d$ with eigenvalue
$\lambda$.

All indecomposable representations of a gentle bound quiver are given by
string and band modules. We refer to [BR87, p.161] for a more detailed
statement.

####  6.1.3. Blossoming

Let $(Q,I)$ be a gentle bound quiver. In order to simplify computations (of
the Avella-Alaminos–Geiß invariant [AAG08] in [Asa12] or of the
Auslander–Reiten translation in [BDM+20, PPP21]), it is convenient to expand
$(Q,I)$ by making all vertices in $Q_{0}$ 4-valent by adding as few arrows as
possible. The resulting gentle bound quiver is called the blossoming bound
quiver of $(Q,I)$ and is denoted $(Q^{\text{\char 96}},I^{\text{\char 96}})$.
It is characterised (up to isomorphism) by the fact that:

  * •

it is gentle;

  * •

its vertices are either leaves (i.e. sinks or sources) or 4-valent;

  * •

by deleting its leaves, one recovers $(Q,I)$.

The leaves of $Q^{\text{\char 96}}$ are called its blossoming vertices, or
simply its blossoms.

######  Example 6.3.

The blossoming bound quiver of the gentle bound quiver appearing in Example
6.2 is:

${2}$${1}$${3}$${5}$${4}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\scriptstyle{\delta}$$\scriptstyle{\epsilon}$

We denote by $A^{\text{\char 96}}$ the algebra $KQ^{\text{\char
96}}/I^{\text{\char 96}}.$ A walk on $(Q,I)$ is defined as a maximal string
for $(Q^{\text{\char 96}},I^{\text{\char 96}})$. A cohook is a string of the
form $\alpha_{1}\cdots\alpha_{r}\beta^{-1}$. By adding left and a right
cohooks to a string for $(Q,I)$, one obtains a walk on $(Q,I)$: If $\sigma$ is
a string for $(Q,I)$ and if $\omega$ is the associated walk, then we have
$M_{\omega}=\tau_{A^{\text{\char 96}}}M_{\sigma}$. This yields a bijection
between strings for $(Q,I)$ and walks on $(Q,I)$ that do not correspond to
injective string modules over $(Q^{\text{\char 96}},I^{\text{\char 96}})$. It
is shown in [BDM+20, PPP21] that the $\tau$-rigidity of representations of
$(Q,I)$ can be described purely combinatorially by making use of this
bijection. The compatibility relation between walks corresponding to the
$\tau$-rigidity is called non-kissing and was introduced (for grids) in
[McC17]. Maximal sets of pairwise non-kissing walks on $(Q,I)$, called non-
kissing facets, are thus in bijection with isomorphism classes of basic
$\tau$-tilting pairs (in the sense of [AIR14]) over $A$. By [McC17, BDM+20,
PPP21], there is a notion of flip for non-kissing facets, similar to the
notion of flip for triangulations. Under the bijection explained above, flips
correspond to $\tau$-tilting mutations.

###  6.2. Exact blossoming of the module category of a gentle algebra

######  Definition 6.4.

Let $\mathscr{E}\subseteq\operatorname{mod}A^{\text{\char 96}}$ be the full
subcategory consisting of those $E\in\operatorname{mod}A^{\text{\char 96}}$
that satisfy the following conditions.

  * (e1)

$\mathrm{Hom}_{A^{\text{\char 96}}}(\operatorname{soc}P,E)=0$ for any
$P\in\operatorname{proj}A^{\text{\char 96}}$.

  * (e2)

$\mathrm{pd}_{A^{\text{\char 96}}}E\leq 1$.

It is straightforward to check that
$\mathscr{E}\subseteq\operatorname{mod}A^{\text{\char 96}}$ is closed under
isomorphisms, extensions, and direct summands.

######  Remark 6.5.

For any $E\in\operatorname{mod}A^{\text{\char 96}}$, the following holds.

  1. (1)

$E$ satisfies (e1) if and only if it satisfies $\mathrm{Hom}_{A^{\text{\char
96}}}(S,E)=0$ for any simple projective $S\in\operatorname{mod}A^{\text{\char
96}}$, if and only if it does not contain any non-zero simple projective in
its composition series.

  2. (2)

If $E$ satisfies (e2), then $\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(E,\operatorname{Fac}J)=0$ holds for any
$J\in\operatorname{inj}A^{\text{\char 96}}$. This is true in any abelian
category.

  3. (3)

$E$ belongs to $\mathscr{E}$ if and only if there exists a short exact
sequence

(6.1) $0\to P_{1}\overset{\iota}{\longrightarrow}P_{0}\to E\to 0$

in $\operatorname{mod}A^{\text{\char 96}}$ satisfying
$P_{0},P_{1}\in\operatorname{proj}A^{\text{\char 96}}$ and
$\iota(\operatorname{soc}P_{1})=\operatorname{soc}P_{0}$.

######  Proposition 6.6.

The following holds.

  1. (1)

For any $P\in\operatorname{proj}A^{\text{\char 96}}$, we have
$P/\operatorname{soc}P\in\operatorname{proj}\mathscr{E}$.

  2. (2)

$\mathscr{E}$ has enough projectives. Moreover, any $E\in\mathscr{E}$ has a
projective resolution of the form

$0\to P_{1}/\operatorname{soc}P_{1}\to P_{0}/\operatorname{soc}P_{0}\to E\to
0.$

In particular, we have $\mathrm{pd}_{\mathscr{E}}E\leq 1$.

  3. (3)

Any object in $\operatorname{proj}\mathscr{E}$ is a direct summand of an
object of the form $P/\operatorname{soc}P$ for some
$P\in\operatorname{proj}A^{\text{\char 96}}$.

###### Proof.

(1) Remark that we have
$\operatorname{soc}P\in\operatorname{proj}A^{\text{\char 96}}$. Thus,
$P/\operatorname{soc}P\in\mathscr{E}$ is a direct consequence of Remark 6.5
(3). From the short exact sequence $0\to\operatorname{soc}P\to P\to
P/\operatorname{soc}P\to 0$ and the assumption of
$\mathrm{Hom}_{A^{\text{\char 96}}}(\operatorname{soc}P,E)=0$, we also have
$\mathrm{Ext}^{1}_{A^{\text{\char 96}}}(P/\operatorname{soc}P,E)=0$ for any
$E\in\mathscr{E}$. This shows
$P/\operatorname{soc}P\in\operatorname{proj}\mathscr{E}$.

(2) This follows from (1) and Remark 6.5 (3).

(3) This follows immediately from (1) and (2). ∎

######  Lemma 6.7.

The following holds.

  1. (1)

If $E\in\mathscr{E}$ belongs to $\operatorname{Fac}J$ for some
$J\in\operatorname{inj}A^{\text{\char 96}}$, then
$E\in\operatorname{inj}\mathscr{E}$.

  2. (2)

If a short exact sequence in $\operatorname{mod}A^{\text{\char 96}}$

$0\to P\overset{\iota}{\longrightarrow}H\to I\to 0$

satisfies $P\in\operatorname{proj}A^{\text{\char 96}}$,
$H\in(\operatorname{proj}A^{\text{\char
96}})\cap(\operatorname{inj}A^{\text{\char 96}})$ and
$\iota(\operatorname{soc}P)=\operatorname{soc}H$, then we have
$I\in\operatorname{inj}\mathscr{E}$.

  3. (3)

For any $P\in\operatorname{proj}A^{\text{\char 96}}$, there exists a short
exact sequence $0\to P\overset{\iota}{\longrightarrow}H\to I\to 0$ in
$\operatorname{mod}A^{\text{\char 96}}$ satisfying the conditions in (2).
Remark that this also induces the following short exact sequence in
$\mathscr{E}$,

$0\to
P/\operatorname{soc}P\overset{\iota}{\longrightarrow}H/\operatorname{soc}H\to
I\to 0$

in which we have $P/\operatorname{soc}P\in\operatorname{proj}\mathscr{E}$,
$H/\operatorname{soc}H\in(\operatorname{proj}\mathscr{E})\cap(\operatorname{inj}\mathscr{E})$,
and $I\in\operatorname{inj}\mathscr{E}$.

###### Proof.

(1) This follows from Remark 6.5 (2).

(2) This follows from (1) and Remark 6.5 (3).

(3) Since $\operatorname{soc}P\in\operatorname{proj}A^{\text{\char 96}}$, we
just have to take an injective hull $P\overset{\iota}{\longrightarrow}H$ and
put $I=\operatorname{Cok}\iota$. ∎

######  Corollary 6.8.

The category $\mathscr{E}$ is $0-$Auslander.

###### Proof.

This is a combination of Proposition 6.6.(2) and Lemma 6.7.(3): these state
that $\mathrm{pd}(\mathscr{E})\leq 1$ and that
$\mathrm{dom.dim}(\mathscr{E})\geq 1,$ respectively. ∎

By Proposition 3.6, we find that $\mathscr{E}$ also has enough injectives and
that $\mathrm{id}(\mathscr{E},\mathbb{E})\leq
1\leq\mathrm{codom.dim}(\mathscr{E})$. For the sake of completeness, we
describe the injectives and write this more explicitly.

######  Proposition 6.9.

The following holds.

  1. (1)

For any $E\in\mathscr{E}$, there exists
$H\in(\operatorname{proj}A^{\text{\char
96}})\cap(\operatorname{inj}A^{\text{\char 96}})$ and a short exact sequence

(6.2) $0\to E\to I^{0}\to I^{1}\to 0$

satisfying $I^{0},I^{1}\in\mathscr{E}\cap\operatorname{Fac}H$.

  2. (2)

$\mathscr{E}$ has enough injectives. More precisely, for any
$E\in\mathscr{E}$, the sequence $(\ref{InjResolE})$ gives an injective
resolution of $E$ in $\mathscr{E}$. In particular, we have
$\mathrm{id}_{\mathscr{E}}E\leq 1$.

  3. (3)

Any object in $\operatorname{inj}\mathscr{E}$ belongs to
$\mathscr{E}\cap\operatorname{Fac}H$ for some
$H\in(\operatorname{proj}A^{\text{\char
96}})\cap(\operatorname{inj}A^{\text{\char 96}})$. Moreover for any
$I\in\operatorname{inj}\mathscr{E}$, there is a short exact sequence in
$\operatorname{mod}A^{\text{\char 96}}$

$0\to P\overset{\iota}{\longrightarrow}H\to I\to 0$

satisfying conditions in Lemma 6.7 (2). Remark that this also induces the
following short exact sequence in $\mathscr{E}$,

$0\to
P/\operatorname{soc}P\overset{\iota}{\longrightarrow}H/\operatorname{soc}H\to
I\to 0$

in which we have $P/\operatorname{soc}P\in\operatorname{proj}\mathscr{E}$,
$H/\operatorname{soc}H\in(\operatorname{proj}\mathscr{E})\cap(\operatorname{inj}\mathscr{E})$,
and $I\in\operatorname{inj}\mathscr{E}$.

###### Proof.

(1) By Remark 6.5 (3) and Lemma 6.7 (3), we have short exact sequences in
$\operatorname{mod}A^{\text{\char 96}}$

$0\to P_{1}\overset{\iota_{1}}{\longrightarrow}P_{0}\to E\to 0,\quad 0\to
P_{0}\overset{\iota_{2}}{\longrightarrow}H\to I^{1}\to 0$

satisfying $P_{0},P_{1}\in\operatorname{proj}A^{\text{\char 96}}$,
$H\in(\operatorname{proj}A^{\text{\char
96}})\cap(\operatorname{inj}A^{\text{\char 96}})$, and

$\iota_{1}(\operatorname{soc}P_{1})=\operatorname{soc}P_{0},\quad\iota_{2}(\operatorname{soc}P_{0})=\operatorname{soc}H.$

Thus we obtain the following commutative diagram in
$\operatorname{mod}A^{\text{\char 96}}$ for some
$I^{0}\in\operatorname{mod}A^{\text{\char 96}}$,

$\textstyle{0}$$\textstyle{0}$$\textstyle{P_{1}}$$\textstyle{P_{1}}$$\textstyle{0}$$\textstyle{P_{0}}$$\textstyle{H}$$\textstyle{I^{1}}$$\textstyle{0}$$\textstyle{0}$$\textstyle{E}$$\textstyle{I^{0}}$$\textstyle{I^{1}}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in which rows and columns are exact. Remark 6.5 (3) shows
$I^{0},I^{1}\in\mathscr{E}$.

(2) This follows from (1), since Lemma 6.7 (2) shows
$I^{0},I^{1}\in\operatorname{inj}\mathscr{E}$.

(3) By (2), any $I\in\operatorname{inj}\mathscr{E}$ is a direct summand of an
object in $\mathscr{E}\cap\operatorname{Fac}H$ for some
$H\in(\operatorname{proj}A^{\text{\char
96}})\cap(\operatorname{inj}A^{\text{\char 96}})$. Since
$\mathscr{E}\cap\operatorname{Fac}H\subseteq\operatorname{mod}A^{\text{\char
96}}$ is closed under isomorphisms and direct summands, it follows
$I\in\mathscr{E}\cap\operatorname{Fac}H$.

Take an epimorphism $H\to I$ in $\operatorname{mod}A^{\text{\char 96}}$. Since
$\mathrm{pd}_{A^{\text{\char 96}}}I\leq 1$, it gives a short exact sequence
$0\to P\overset{\iota}{\longrightarrow}H\to I\to 0$ with
$P\in\operatorname{proj}A^{\text{\char 96}}$, which necessarily satisfies
$\iota(\operatorname{soc}P)=\operatorname{soc}H$ since
$\mathrm{Hom}_{A^{\text{\char 96}}}(\operatorname{soc}H,I)=0$. ∎

Let $\varepsilon_{p}\in A^{\text{\char 96}}$ denote the idempotent element
corresponding to each $p\in Q_{0}^{\text{\char 96}}$. Define $\varepsilon\in
A^{\text{\char 96}}$ to be

$\varepsilon=\sum_{p\in Q_{0}^{\text{\char 96}}\setminus
Q_{0}}\varepsilon_{p},$

namely, the sum of idempotents corresponding to blossom vertices. Denote the
two-sided ideal $A^{\text{\char 96}}\varepsilon A^{\text{\char 96}}\subseteq
A^{\text{\char 96}}$ by $\mathfrak{a}$.

######  Definition 6.10.

Denote the functor tensoring with $A=A^{\text{\char 96}}/\mathfrak{a}$ by

$G\colon\operatorname{mod}A^{\text{\char 96}}\to\operatorname{mod}A\ ;\
M\mapsto M/\mathfrak{a}M.$

The following proposition can be seen as a consequence of Proposition 3.11 and
Corollary 6.8; we nonetheless give a representation-theoretic proof.

######  Proposition 6.11.

The functor $G$ induces an equivalence of categories
$\overline{G}\colon\mathscr{E}/\operatorname{inj}\mathscr{E}\overset{\simeq}{\longrightarrow}\operatorname{mod}A$.

###### Proof.

Since $G$ obviously satisfies $G(H)=0$ for any
$H\in(\operatorname{proj}A^{\text{\char
96}})\cap(\operatorname{inj}A^{\text{\char 96}})$, Proposition 6.9 (3) and the
right-exactness of $G$ shows $G(\operatorname{inj}\mathscr{E})=0$. Thus $G$
naturally induces an additive functor
$\overline{G}\colon\mathscr{E}/\operatorname{inj}\mathscr{E}\to\operatorname{mod}A$.
Let us show $\overline{G}$ is an equivalence of categories.

Let $E,F\in\mathscr{E}$ be any pair of objects. By Proposition 6.9, we have a
short exact sequence

$0\to E\to I^{0}\to I^{1}\to 0$

in $\operatorname{mod}A^{\text{\char 96}}$, satisfying
$I^{0},I^{1}\in\operatorname{inj}\mathscr{E}$. Since $F$ does not contain any
non-zero simple projective in its composition series, so does $\mathfrak{a}F$.
This implies that
$\mathrm{top}(\mathfrak{a}F)\in\operatorname{mod}A^{\text{\char 96}}$ is a
direct sum of simples corresponding to source blossom vertices, and thus
$\mathfrak{a}F$ has a projective cover $H\to\mathfrak{a}F$ for some
$H\in(\operatorname{proj}A^{\text{\char
96}})\cap(\operatorname{inj}A^{\text{\char 96}})$. Thus by Remark 6.5 (2), we
have $\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(I^{1},\mathfrak{a}F)=\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(E,\mathfrak{a}F)=0$, and obtain the following exact sequences.

$0\to\mathrm{Hom}_{A^{\text{\char
96}}}(E,\mathfrak{a}F)\to\mathrm{Hom}_{A^{\text{\char
96}}}(E,F)\to\mathrm{Hom}_{A^{\text{\char 96}}}(E,F/\mathfrak{a}F)\to 0,$
$\mathrm{Hom}_{A^{\text{\char
96}}}(I^{0},\mathfrak{a}F)\to\mathrm{Hom}_{A^{\text{\char
96}}}(E,\mathfrak{a}F)\to 0.$

They naturally induce the isomorphism

$(\mathscr{E}/\operatorname{inj}\mathscr{E})(E,F)\cong\mathrm{Hom}_{A^{\text{\char
96}}}(E,F/\mathfrak{a}F)\cong\mathrm{Hom}_{A}(E/\mathfrak{a}E,F/\mathfrak{a}F),$

which shows that $\overline{G}$ is fully faithful.

It remains to show that $\overline{G}$ is essentially surjective. Remark that
for any $v\in Q_{0}$, we have $G(\varepsilon_{v}A^{\text{\char
96}})=\varepsilon_{v}A^{\text{\char
96}}/\operatorname{soc}(\varepsilon_{v}A^{\text{\char
96}})\cong\varepsilon_{v}A$. Let $M\in\operatorname{mod}A$ be any object, and
take its projective presentation

$Q_{1}\overset{q}{\longrightarrow}Q_{0}\to M\to
0\quad(Q_{0},Q_{1}\in\operatorname{proj}A)$

in $\operatorname{mod}A$. By the remark above, we may find
$P_{0},P_{1}\in\operatorname{proj}A^{\text{\char 96}}$ and epimorphisms in
$\operatorname{mod}A^{\text{\char 96}}$

$\pi_{i}\colon P_{i}\to Q_{i}\quad(i=1,2)$

which give $G(P_{i})\cong Q_{i}$. Here, $Q_{0},Q_{1}\in\operatorname{mod}A$
are regarded as objects in $\operatorname{mod}A^{\text{\char 96}}$ through the
fully faithful embedding
$\operatorname{mod}A\hookrightarrow\operatorname{mod}A^{\text{\char 96}}$
induced by the restriction of the action along the surjective homomorphism of
$K$-algebras $A^{\text{\char 96}}\to A^{\text{\char 96}}/\mathfrak{a}\cong A$.
Since $P_{1}\in\operatorname{proj}A^{\text{\char 96}}$, we may find
$p\in\mathrm{Hom}_{A^{\text{\char 96}}}(P_{1},P_{0})$ which makes

$\textstyle{P_{1}}$$\textstyle{P_{0}}$$\textstyle{Q_{1}}$$\textstyle{Q_{0}}$$\scriptstyle{p}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{0}}$$\scriptstyle{q}$$\scriptstyle{\circlearrowright}$

commutative in $\operatorname{mod}A^{\text{\char 96}}$. Since $G$ is right
exact, this gives $G(\operatorname{Cok}p)\cong M$ in $\operatorname{mod}A$.

We will modify $p\colon P_{1}\to P_{0}$ to make the cokernel belong to
$\mathscr{E}$. Take an injective hull $P_{1}\overset{i}{\longrightarrow}H$,
with $H\in(\operatorname{proj}A^{\text{\char
96}})\cap(\operatorname{inj}A^{\text{\char 96}})$. Then

$p^{\prime}=\begin{bmatrix}p\\\ i\end{bmatrix}\colon P_{1}\to P_{0}\oplus H$

is a monomorphism in $\operatorname{mod}A^{\text{\char 96}}$. By adding the
semisimple projective $S=\operatorname{soc}(P_{0}\oplus
H)/p^{\prime}(\operatorname{soc}P_{1})$ to $P_{1}$, we obtain a monomorphism

$\iota=\left[\begin{array}[]{cc}p&\ast\\\ i&\ast\end{array}\right]\colon
P_{1}\oplus S\to P_{0}\oplus H$

which satisfies $\iota(\operatorname{soc}(P_{1}\oplus
S))=\operatorname{soc}(P_{0}\oplus H)$. Since we have $G(S)=G(H)=0$, we have
$G(\operatorname{Cok}\iota)\cong G(\operatorname{Cok}p)\cong M$. By Remark 6.5
(3), we have $\operatorname{Cok}\iota\in\mathscr{E}$. ∎

###  6.3. The category of walks

The exact category $\mathscr{E}$ of Section 6.2 is convenient to define and
study from a purely algebraic point of view, but is not well-adapted to the
combinatorics of walks (i.e. maximal strings in the blossoming bound quiver).
In this section, we exchange this subcategory for an equivalent one, which
will be more adapted to walks.

######  Definition 6.12.

The exact category of walks is defined as the full subcategory $\mathscr{F}$
of $\operatorname{mod}A^{\text{\char 96}}$ whose objects $M$ satisfy
$\mathrm{Hom}_{A^{\text{\char 96}}}(\operatorname{soc}P,\tau M)=0$
(equivalently $\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(M,\operatorname{soc}P)=0$) for any
$P\in\operatorname{proj}A^{\text{\char 96}}$, $M$ has projective dimension at
most one, and has no simple projective summands.

######  Remark 6.13.

The subcategory $\mathscr{F}$ is extension-closed in
$\operatorname{mod}A^{\text{\char 96}}$, and is thus an exact category.

######  Remark 6.14.

The subcategory $\mathscr{F}$ contains all band modules, since they are all of
projective dimension one, and not supported at sinks or sources.

######  Example 6.15.

In Figure 1, taken from [INP18], we highlight the indecomposable
$A^{\text{\char 96}}$-modules that belong to the subcategories $\mathscr{F}$
and $\mathscr{E}$. Those in $\mathscr{F}$ precisely coincide with the modules
whose support contains two different blossoming vertices. Moreover, we see
that $\mathscr{F}$ is stable under the action of $\tau_{A^{\text{\char
96}}}^{2}$. Those two facts are instances of Proposition 6.21 and Corollary
6.26 respectively. One can also notice that the full subquivers of this
Aulsander–Reiten quiver given by $\mathscr{E}$ and $\mathscr{F}$ are
isomorphic. This is explained by the equivalence of exact categories proven in
Proposition 6.16.

$2$$h$$b$$f$$a$$\begin{smallmatrix}3\\\
f\;h\end{smallmatrix}$$\begin{smallmatrix}a\;b\\\
1\end{smallmatrix}$$g$$e$$\begin{smallmatrix}3\\\
f\end{smallmatrix}$$\begin{smallmatrix}a\\\
1\end{smallmatrix}$$\begin{smallmatrix}3\\\
h\end{smallmatrix}$$\begin{smallmatrix}b\\\
1\end{smallmatrix}$$\begin{smallmatrix}2\\\ 3\;g\\\
f\;\phantom{\;g}\end{smallmatrix}$$\begin{smallmatrix}a\;\phantom{\;e}\\\
1\;e\\\ 2\end{smallmatrix}$$3$$1$$\begin{smallmatrix}d\\\ 3\\\
h\end{smallmatrix}$$\begin{smallmatrix}b\\\ 1\\\
c\end{smallmatrix}$$\begin{smallmatrix}2\\\ 3\\\
f\end{smallmatrix}$$\begin{smallmatrix}a\\\ 1\\\
2\end{smallmatrix}$$\begin{smallmatrix}2\\\
3\;g\end{smallmatrix}$$\begin{smallmatrix}1\;e\\\
2\end{smallmatrix}$$\begin{smallmatrix}d\\\
3\end{smallmatrix}$$\begin{smallmatrix}1\\\
c\end{smallmatrix}$$\begin{smallmatrix}e\\\ 2\\\ 3\\\
f\end{smallmatrix}$$\begin{smallmatrix}a\\\ 1\\\ 2\\\
g\end{smallmatrix}$$\begin{smallmatrix}2\\\
3\end{smallmatrix}$$\begin{smallmatrix}1\\\
2\end{smallmatrix}$$\begin{smallmatrix}d\phantom{3}2\\\
\phantom{d\;}\,3\phantom{2}g\end{smallmatrix}$$\begin{smallmatrix}\phantom{c\;}1\phantom{2}e\\\
c\phantom{1}2\end{smallmatrix}$$\begin{smallmatrix}e\\\ 2\\\
3\end{smallmatrix}$$\begin{smallmatrix}1\\\ 2\\\
g\end{smallmatrix}$$\begin{smallmatrix}d\phantom{3}2\\\
3\end{smallmatrix}$$\begin{smallmatrix}1\\\
c\;\,2\end{smallmatrix}$$\begin{smallmatrix}2\\\
g\end{smallmatrix}$$\begin{smallmatrix}e\\\
2\end{smallmatrix}$$c$$d$$\begin{smallmatrix}\phantom{d3}e\\\ d\phantom{3}2\\\
3\end{smallmatrix}$$\begin{smallmatrix}1\\\ c\phantom{3}2\\\
\phantom{c3}g\end{smallmatrix}$ Figure 1. The Auslander–Reiten quiver of
$\operatorname{mod}A^{\text{\char 96}}$: The indecomposable objects in
$\mathscr{E}$ are circled in blue; the projective (resp. injective) modules
are highlighted by a red vertical line to their left (resp. to their right)
and similarly for the relative projectives (resp. injectives) in $\mathscr{E}$
by using dashed blue vertical lines. The indecomposable objects in
$\mathscr{F}$ are squared in green. Projectives (resp. injectives) in
$\mathscr{F}$ are the projective (resp. injective) $A^{\text{\char
96}}$-modules that belong to $\mathscr{F}$.

######  Proposition 6.16.

There is an equivalence of exact categories
$F:\mathscr{F}\xrightarrow{\simeq}\mathscr{E}$, sending an $A^{\text{\char
96}}$-module $M\in\mathscr{F}$ to the cokernel $FM$ of
$\varepsilon.\operatorname{soc}M\to M$, where $\varepsilon$ is the sum of the
idempotents $e_{t}$, over all sinks $t\in Q_{0}^{\text{\char 96}}$. A quasi-
inverse is constructed as follows: For an $A^{\text{\char 96}}$-module
$N\in\mathscr{E}$, consider its minimal projective resolution $0\to
S_{1}\oplus P_{1}\to P_{0}\to N$, where $S_{1}$ is semisimple and
$P_{0},P_{1}$ have no simple summands. Then the cokernel $G^{\prime}N$ of
$P_{1}\to P_{0}$ belongs to $\mathscr{F}$.

###### Proof.

The assignment $M\mapsto FM$ is functorial by construction.

$\bullet$ For any $M\in\mathscr{F}$, $FM$ belongs to $\mathscr{E}$: The short
exact sequence $\varepsilon.\operatorname{soc}M\rightarrowtail
M\twoheadrightarrow FM$ induces, for any simple projective $S$, an exact
sequence

$0\to(S,\varepsilon.\operatorname{soc}M)\xrightarrow{f}(S,M)\xrightarrow{g}(S,FM)\to\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(S,\varepsilon.\operatorname{soc}M)$

where we have written $(L,N)$ for $\mathrm{Hom}_{A^{\text{\char 96}}}(L,N)$.
By definition of $\varepsilon$, the map $f$ is bijective, hence $g$ is zero.
Moreover, $S$ being projective, we have $\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(S\varepsilon.\operatorname{soc}M)=0$. Thus $\mathrm{Hom}_{A^{\text{\char
96}}}(S,M)=0$.

Because $M$ has projective dimension at most one, there is a short exact
sequence $P_{1}\rightarrowtail P_{0}\twoheadrightarrow M$, with $P_{0},P_{1}$
projective $A^{\text{\char 96}}$-modules. There is a diagram

${{P_{1}}}$${{P_{1}}}$${{P_{1}\oplus K}}$${{P_{0}}}$${FM}$${K}$${M}$${FM}$

(where $K=\varepsilon.\operatorname{soc}M$) showing that $FM$ also has
projective dimension at most one.

$\bullet$ For any $N\in\mathscr{E}$, $G^{\prime}N$ belongs to $\mathscr{F}$:
By definition, $G^{\prime}N$ has projective dimension at most one and there is
a diagram

${{P_{1}}}$${{P_{1}}}$${{S_{1}\oplus
P_{1}}}$${{P_{0}}}$${N}$${{S_{1}}}$${G^{\prime}N}$${N}$

Because $P_{0}$ has no simple summands, $G^{\prime}N$ has no simple projective
summands. Moreover, for any simple projective $A^{\text{\char 96}}$-module
$S$, there is an induced exact sequence

$(S_{1},S)\xrightarrow{h}\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(N,S)\to\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(G^{\prime}N,S)\to\mathrm{Ext}^{1}_{A^{\text{\char 96}}}(S_{1},S)$

where the last term vanishes since $S_{1}$ is projective, and where the map
$h$ is an isomorphism (because $\mathrm{Hom}_{A^{\text{\char
96}}}(P_{1},S)=0=\mathrm{Hom}_{A^{\text{\char 96}}}(P_{0},S)$). This shows
that $\mathrm{Ext}^{1}_{A^{\text{\char 96}}}(G^{\prime}N,S)=0$

$\bullet$ The funtor $F$ is dense: Let $N\in\mathscr{E}$. The short exact
sequence $S_{1}\rightarrowtail G^{\prime}N\twoheadrightarrow N$ induces an
isomorphism $\operatorname{soc}G^{\prime}N\cong\operatorname{soc}N\oplus
S_{1}$, where $\operatorname{soc}N$ has no projective summands. This implies
$FG^{\prime}N\cong N$.

$\bullet$ The functor $F$ is full: Let $M,N\in\mathscr{F}$ and let
$FM\xrightarrow{f}FN$ be any morphism. The composition $M\twoheadrightarrow
FM\to FN$ factors through $N\twoheadrightarrow FN$, as in the diagram below,
because $\mathrm{Ext}^{1}_{A^{\text{\char
96}}}(M,\varepsilon.\operatorname{soc}N)=0$:

${M}$${FM}$${{\varepsilon.\operatorname{soc}N}}$${N}$${FN}$$\scriptstyle{f}$$\scriptstyle{g}$

We thus have $Fg=f$.

$\bullet$ The functor $F$ is faithful: Given a commutative diagram

${M}$${FM}$${{\varepsilon.\operatorname{soc}N}}$${N}$${FN}$$\scriptstyle{0}$$\scriptstyle{i_{N}}$$\scriptstyle{g}$$\scriptstyle{h}$

there is a morphism $h$ such that $i_{N}h=g$. However, $M$ has no simple
projective summands. This implies that $h$, hence $g$, vanishes.

$\bullet$ The functor $F$ is exact: Let $L\rightarrowtail M\twoheadrightarrow
N$ be a short exact sequence with all three terms in $\mathscr{F}$. There is
an induced split exact sequence
$\varepsilon.\operatorname{soc}L\to\varepsilon.\operatorname{soc}M\to\varepsilon.\operatorname{soc}N$.
By the snake lemma, the induced sequence between the cokernels $FL\to FM\to
FN$ is short exact. ∎

######  Corollary 6.17.

An $A^{\text{\char 96}}$-module belongs to $\mathscr{F}$ if and only if it is
the cokernel of an injection between projective $A^{\text{\char 96}}$-modules
having no simple summands.

###### Proof.

By Proposition 6.16, any module in $\mathscr{F}$ has projective dimension one
and has a projective presentation by projective modules having no simple
summands. Conversely, let $P_{1}\rightarrowtail P_{0}\twoheadrightarrow M$ be
short exact with $P_{0},P_{1}$ projective without any simple summand. Then $M$
is not simple projective and has projective dimension at most one. Moreover,
for any simple projective $A^{\text{\char 96}}$-module $S$,
$\mathrm{Ext}^{1}_{A^{\text{\char 96}}}(M,S)$ is given by the cokernel of the
induced map $\mathrm{Hom}_{A^{\text{\char
96}}}(P_{1},S)\to\mathrm{Hom}{A^{\text{\char 96}}}(P_{0},S)$ whose domain and
codomain both vanish. Hence $\mathrm{Ext}^{1}_{A^{\text{\char 96}}}(M,S)=0$
and $M$ belongs to $\mathscr{F}$. ∎

######  Lemma 6.18.

For any $v\in Q_{0}$, there is a short exact sequence in
$\operatorname{mod}A^{\text{\char 96}}$

$0\to P_{v}\to I_{a}\oplus I_{b}\to I_{v}\to 0,$

where $\operatorname{soc}P_{v}=S_{a}\oplus S_{b}$.

###### Proof.

This is immediate from string combinatorics:

${v}$${0}$${{\hskip 30.0pt\oplus}}$${0}$${a}$${a}$${{b\hskip
25.0pt}}$${v}$${b}$$\scriptstyle{v}$$\scriptstyle{v}$

∎

######  Corollary 6.19.

The projective objects of $\mathscr{F}$ are precisely the projective
$A^{\text{\char 96}}$-modules having no simple summands, and the injective
objects of $\mathscr{F}$ are precisely the injective $A^{\text{\char
96}}$-modules having no simple summands.

###### Proof.

Any non-simple projective $A^{\text{\char 96}}$-module belongs to
$\mathscr{F}$ and is projective in $\mathscr{F}$. Conversely, let
$M\in\mathscr{F}$ and let $P_{1}\rightarrowtail P_{0}\twoheadrightarrow M$ be
short exact with $P_{0},P_{1}$ projective $A^{\text{\char 96}}$-modules having
no simple summands. If $M$ is projective in $\mathscr{F}$, then this sequence
splits and $M$ is a summand of $P_{0}$, hence a projective in
$\operatorname{mod}A^{\text{\char 96}}$ having no simple summands.

Let $I$ be indecomposable injective in $\operatorname{mod}A^{\text{\char
96}}$. If $I$ is also projective, then it belongs to $\mathscr{F}$ and is thus
injective in $\mathscr{F}$. Otherwise, $I$ has simple socle not supported at a
sink, hence $I$ belongs to $\mathscr{E}$ and
$\varepsilon.\operatorname{soc}I=0$, showing that $I=GI$ belongs to
$\mathscr{F}$.

Conversely, let $M$ be injective in $\mathscr{F}$. Then, by Proposition 6.16,
$FM$ is injective in $\mathscr{E}$, which implies that $FM$ belongs to
$\operatorname{Fac}H$ by Lemmas 6.7 and 6.9. From the short exact sequence
$\varepsilon.\operatorname{soc}M\rightarrowtail M\twoheadrightarrow FM$, one
easily deduce that $FM$ belongs to $\operatorname{Fac}H$ if and only if $M$
belongs to $\operatorname{Fac}H$ (using that $M$ has no simple projective
summands). It follows that $M$ admits a projective resolution of the form
$P_{1}\rightarrowtail H_{0}\twoheadrightarrow M$, where $P_{1}$ is projective
having no simple summands and $H_{0}$ is projective-injective. By Lemma 6.18,
there is a short exact sequence $P_{1}\rightarrowtail H_{1}\twoheadrightarrow
I_{1}$ with $H_{1}$ projective-injective and $I_{1}$ injective having no
simple summands. Therefore, $M$ is a summand of $H_{0}\oplus I_{1}$, hence an
injective with no simple summands. ∎

######  Lemma 6.20.

Let $\sigma$ be a string for $(Q^{\text{\char 96}},I^{\text{\char 96}})$. Then
$M_{\sigma}$ is of projective dimension at most one if and only if one of the
following three conditions is satisfied:

  1. (1)

$M_{\sigma}$ is simple projective over $A^{\text{\char 96}}$.

  2. (2)

$M_{\sigma}$ is simple, supported at a non-blossoming vertex, and is
projective as an $A$-module.

  3. (3)

$\sigma$ contains at least one arrow or inverse arrow and

    1. (a)

starts at a blossom or is such that the target of the unique arrow $\alpha$,
such that $\alpha^{-1}\sigma$ is a string, is a sink and

    2. (b)

ends in a blossom or is such that the target of the unique arrow $\beta$, such
that $\sigma\beta$ is a string, is a sink.

###### Proof.

Let $\sigma$ be a string for $(Q^{\text{\char 96}},I^{\text{\char 96}})$.

$\bullet$ Assume first that $M_{\sigma}=S_{i}$ is simple. If $i$ is a sink,
then $S_{i}$ is projective, hence of projective dimension at most one.
Otherwise, its projective presentation is of the form $P_{j}\oplus P_{k}\to
P_{i}\twoheadrightarrow S_{i}$, where the quiver $Q^{\text{\char 96}}$
contains the subquiver $j\leftarrow i\to k$. Because each non-blossoming
vertex has two successors, the morphism $P_{j}\oplus P_{k}\to P_{i}$ is
injective if and only if both $j$ and $k$ are sinks, if and only if $i$ is a
sink in $Q$.

$\bullet$ Assume that $M_{\sigma}$ is not simple, and let $P_{1}\to P_{0}$ be
a projective presentation of $M_{\sigma}$.

If the starting vertex of $\sigma$ is not a blossom, then $P_{1}$ contains
$P_{t(\alpha)}$ as a summand, where $\alpha^{-1}\sigma$ is a string. The
restriction $P_{t(\alpha)}\to P_{0}$ of $P_{1}\to P_{0}$ is injective if and
only if $t(\alpha)$ is not the source of two distinct arrows, if and only if
$t(\alpha)$ is a blossoming vertex. A similar argument applies at the endpoint
of $\sigma$. By string combinatorics (see for example [PPP21, Proposition
1.43]), we have that $P_{1}\to P_{0}$ is injective if and only if
$P_{t(\alpha)}\oplus P_{t(\beta)}\to P_{0}$ is injective (with the convention
that $P_{t(\alpha)}$ is zero if $\sigma$ starts at a blossoming vertex, and
similarly for $P_{t(\beta)}$). ∎

The following proposition justifies the name of the exact category
$\mathscr{F}$.

######  Proposition 6.21.

Let $\sigma$ be a string for the gentle bound quiver $(Q^{\text{\char
96}},I^{\text{\char 96}})$. Then the string module $M_{\sigma}$ belongs to
$\mathscr{F}$ if and only if $\sigma$ is a maximal string for $(Q^{\text{\char
96}},I^{\text{\char 96}})$, hence a walk for $(Q,I)$.

###### Proof.

Let $\sigma$ be a string for $(Q^{\text{\char 96}},I^{\text{\char 96}})$.

$\bullet$ Assume that $M_{\sigma}$ belongs to $\mathscr{F}$, i.e. that
$M_{\sigma}$ is not simple projective over $A^{\text{\char 96}}$, has
projective dimension at most one, and $\mathrm{Ext}_{A^{\text{\char
96}}}^{1}(M_{\sigma},S)=0$ (equivalently $\mathrm{Hom}_{A^{\text{\char
96}}}(S,\tau M_{\sigma})=0$) for each simple projective $A^{\text{\char
96}}$-module $S$. Notice first that Lemma 6.20 implies that $M_{\sigma}$
cannot be simple. Indeed, case (1) of Lemma 6.20 is excluded by definition of
$\mathscr{F}$ and case (2) is excluded because every vertex that is a sink in
$Q$ is the source, in $Q^{\text{\char 96}}$, of an arrow with target a
blossoming vertex $p$, thus has a non-split extension with the simple
projective $S_{p}$. Hence, $\sigma$ is of the form given in condition (3) of
Lemma 6.20. The requirement on vanishing of extensions says that there cannot
exist an arrow $\alpha$ as in (3)(a) nor an arrow $\beta$ as in (3)(b). Thus
$\sigma$, which has length at least one, starts and ends at a blossom, meaning
that it is a walk on $(Q,I)$.

$\bullet$ Assume that $\sigma$ is a walk on $(Q,I)$. Then $\sigma$ starts at
and ends in a blossoming vertex, and $M_{\sigma}$ is not a simple module.
Hence, by Lemma 6.20, $M_{\sigma}$ has projective dimension at most one.
Moreover, because $\sigma$ is maximal, $\tau M_{\sigma}$ is the indecomposable
module associated with the string $\sigma^{\prime}$ obtained from $\sigma$ by
removing a left and a right cohook. In particular, $\sigma^{\prime}$ is not
supported on any blossom, showing that $\mathrm{Hom}_{A^{\text{\char
96}}}(S,\tau M_{\sigma})=0$ for each simple projective $A^{\text{\char
96}}$-module $S$. ∎

Recall from [PPP21] that:

  * •

A projective walk is a walk of the form
$\alpha_{1}^{-1}\cdots\alpha_{r}^{-1}\beta_{1}\cdots\beta_{s}$, hence the
string of an indecomposable projective $A^{\text{\char 96}}$-module at a non-
blossoming vertex.

  * •

A shifted projective walk is a walk of the form
$\alpha_{1}\cdots\alpha_{r}\beta_{1}^{-1}\cdots\beta_{s}^{-1}$, hence the
string of an indecomposable injective $A^{\text{\char 96}}$-module at a non-
blossoming vertex.

  * •

A straight walk is a walk that is a path in $Q^{\text{\char 96}}$, hence the
string of a projective-injective $A^{\text{\char 96}}$-module.

######  Proposition 6.22.

Let $\omega$ be a maximal string for $(Q^{\text{\char 96}},I^{\text{\char
96}})$. The following properties holds:

  1. (1)

The string module $M_{\omega}$ is projective and non-injective in
$\mathscr{F}$ if and only if $\omega$ is a projective walk.

  2. (2)

The string module $M_{\omega}$ is injective and non-projective in
$\mathscr{F}$ if and only if $\omega$ is a shifted projective walk.

  3. (3)

The string module $M_{\omega}$ is projective and injective in $\mathscr{F}$ if
and only if $\omega$ is a straight walk.

###### Proof.

This is a reformulation of Corollary 6.19. ∎

Although the following corollary is also immediate from Propositions 6.16 and
6.8, we make it explicit for future references.

######  Corollary 6.23.

The exact category $\mathscr{F}$ is 0-Auslander.

######  Lemma 6.24.

Let $\sigma,\sigma^{\prime}$ be two strings for $(Q,I)$ and let
$\omega,\omega^{\prime}$ be the associated walks. Then, we have

$\mathrm{Hom}_{A}(M_{\sigma},M_{\sigma^{\prime}})\cong\overline{\mathscr{F}}(M_{\omega},M_{\omega^{\prime}}),$

where $\overline{\mathscr{F}}$ denotes the quotient by the ideal of morphisms
factoring through an injective object.

###### Proof.

Noting that $M_{\sigma}=GFM_{\omega}$, the statement follows from combining
Propositions 6.16 and 6.11 ∎

######  Lemma 6.25.

Let $\sigma,\sigma^{\prime}$ be two strings for $(Q,I)$, and let
$\omega,\omega^{\prime}$ be the associated walks. Assume that
$M_{\sigma^{\prime}}=\tau_{A}M_{\sigma}$. Then
$M_{\omega^{\prime}}=\tau_{A^{\text{\char 96}}}^{2}M_{\omega}$.

###### Proof.

This is [PPP21, Proposition 2.49]. ∎

######  Corollary 6.26.

The subcategory $\mathscr{F}$ of $\operatorname{mod}A^{\text{\char 96}}$ is
$\tau^{2}$-stable.

###### Proof.

This follows from Propositions 6.21 and 6.25. ∎

######  Proposition 6.27.

The subcategory $\mathscr{F}$ of $\operatorname{mod}A^{\text{\char 96}}$ is
functorially finite.

###### Proof.

Recall that middle terms of almost-split sequences in
$\operatorname{mod}A^{\text{\char 96}}$ have precisely one or two
indecomposable summands. Meshes in the Auslander–Reiten quiver of
$A^{\text{\char 96}}$ are thus either squares or triangles. Moreover, arrows
in this Auslander–Reiten quiver are given by adding or deleting one hook or
cohook. All this will be used implicitly in the proof.

Let $M\in\operatorname{mod}A^{\text{\char 96}}$ be indecomposable not in
$\mathscr{F}$. We construct a right $\mathscr{F}$-approximation of $M$, the
construction of a left approximation being similar. Note that $M$ cannot be a
band module. Thus it is a string module, associated with some string $\sigma$,
at most one of whose endpoints is a blossom. Let us consider three cases.

$\bullet$ If $\sigma$ is reduced to a sink. Then $M$ is simple projective and
its right approximation is 0.

$\bullet$ If $\sigma$ has length at most one, and has one endpoint a blossom.
Then at the other endpoint of $\sigma$ (which we may assume is its starting
point, and which might be the blossom itself if $\sigma$ is reduced to a
source) it is possible to add a cohook $c$. Since every cohook in
$(Q^{\text{\char 96}},I^{\text{\char 96}})$ ends in a blossom, $c\sigma$ is a
walk, hence $M_{c\sigma}\in\mathscr{F}$ by Proposition 6.21. Moreover, in the
ray of the Auslander–Reiten quiver of $A^{\text{\char 96}}$ ending in
$M_{\sigma}$ and not containing $M_{c\sigma}$, all indecomposables are
associated with strings starting at the same endpoint as $\sigma$, hence
strings that are not walks. The string module $M_{c\sigma}$ is thus the right
approximation of $M_{\sigma}$.

$\bullet$ If $\sigma$ has length at most one, and does not contain any
blossom. Then if $N$ is an indecomposable $A^{\text{\char 96}}$-module sitting
in one of the two rays of the Auslander–Reiten quiver of $A^{\text{\char 96}}$
ending in $M_{\sigma}$, then its string is obtained by a succession of addings
of cohooks or removings of hooks, always on the same side of the string
$\sigma$. Hence one of its endpoints is an endpoint of $\sigma$, thus in
$Q_{0}$. Using Proposition 6.21, this implies that $N$ does not belong to
$\mathscr{F}$. Therefore, any morphism from a module in $\mathscr{F}$ to
$M_{\sigma}$ factors through $\tau_{A^{\text{\char 96}}}M_{\sigma}$. The
string $\rho$ associated with $\tau_{A^{\text{\char 96}}}M_{\sigma}$ is
obtained from $\sigma$ by adding a cohook on each side of $\sigma$. It follows
that $\tau_{A^{\text{\char 96}}}M_{\sigma}$ belongs to $\mathscr{F}$ (using
Proposition 6.21 again) and that it is the right approximation of
$M_{\sigma}$. ∎

######  Corollary 6.28.

The category $\mathscr{F}$ has almost-split extensions and, for any
$M\in\mathscr{F}$, we have $\tau_{\mathscr{F}}M=\tau_{A^{\text{\char
96}}}^{2}M$.

###### Proof.

First note that $\mathscr{F}$ is $\tau_{A^{\text{\char 96}}}^{2}$-stable by
Corollary 6.26, so that the statement makes sense. Since $\mathscr{F}$ is
functorially finite (Proposition 6.27), it has almost-split extensions.
Moreover, for any indecomposable module $M\in\mathscr{F}$, its
Auslander–Reiten translation $\tau_{\mathscr{F}}M$ is a summand of the right
$\mathscr{F}$-approximation of $\tau_{A^{\text{\char 96}}}M$. If $M$ is a band
module, then $\tau_{A^{\text{\char 96}}}M=M$ and we have
$\tau_{\mathscr{F}}M=\tau_{A^{\text{\char 96}}}M=\tau_{A^{\text{\char
96}}}^{2}M$. If $M$ is projective in $\operatorname{mod}A^{\text{\char 96}}$,
then it is projective in $\mathscr{F}$ by Corollary 6.19 and we have
$\tau_{\mathscr{F}}M=0=\tau_{A^{\text{\char 96}}}^{2}M$. Otherwise, $M$ is a
string module associated with a non-projective walk $\omega$, and the string
associated with $\tau_{A^{\text{\char 96}}}M$ is obtained from that walk by
removing a hook at each side of $\omega$. We are thus in the third case of the
proof of Proposition 6.27, and the right $\mathscr{F}$-approximation of
$\tau_{A^{\text{\char 96}}}M$ is $\tau_{A^{\text{\char 96}}}^{2}M$, which is
indecomposable. Therefore $\tau_{\mathscr{F}}M=\tau_{A^{\text{\char
96}}}^{2}M$. ∎

######  Notation 6.29.

If $M$ belongs to $\operatorname{mod}A$, there is a unique (up to isomorphism)
object $X_{M}$ in $\mathscr{F}$ such that

  * •

$GFX_{M}=M$, where $F$ and $G$ are the functors from Propositions 6.11 and
6.16, and

  * •

$X_{M}$ has no non-zero injective summands.

In other words we have
$X_{M_{\sigma}}=M_{\omega}\in\operatorname{mod}A^{\text{\char 96}}$, where
$\omega$ is the walk associated with $\sigma$ if $\sigma$ is a string and
where $\omega=\sigma$ if $\sigma$ is a band. If $v\in Q_{0}$ and $P=P_{v}$ is
the associated indecomposable projective $A$-module, let $X_{(0,P)}$ be the
indecomposable injective $A^{\text{\char 96}}$-module $I_{v}^{\text{\char
96}}$ associated with vertex $v$. Let $X_{(M,P_{v_{1}}\oplus\cdots\oplus
P_{v_{r}})}=X_{M}\oplus X_{(0,P_{v_{1}})}\oplus\cdots\oplus
X_{(0,P_{v_{r}})}$.

Following [AIR14], for two pairs
$(M,P),(N,Q)\in\operatorname{mod}A\times\operatorname{proj}A$ we write
$(M,P)\perp_{\tau}(N,Q)$ if and only if $\mathrm{Hom}_{A}(M,\tau N)=0$ and
$\mathrm{Hom}_{A}(Q,M)=0$.

######  Theorem 6.30.

Assume that $(Q,I)$ is gentle, and let $A=KQ/I$.

  1. (1)

Let $M,N\in\operatorname{mod}A\times\operatorname{proj}A$. Then
$M\perp_{\tau}N$ if and only if $\mathbb{E}(X_{N},X_{M})=0$ in $\mathscr{F}$.

  2. (2)

Let $\omega,\omega^{\prime}$ be two walks for $(Q,I)$. Then
$\mathbb{E}(M_{\omega^{\prime}},M_{\omega})\neq 0$ in $\mathscr{F}$ if and
only if the walk $\omega$ kisses the walk $\omega^{\prime}$.

###### Proof.

The second part of the statement follows from (1) and [PPP21, Theorem 2.46].
Let us prove (1): Recall from Proposition 6.27 that the category $\mathscr{F}$
is functorially finite in $\operatorname{mod}A^{\text{\char 96}}$ and hence
has Auslander–Reiten–Serre duality. Moreover, for any walk or band $\omega$,
we have $\tau_{\mathscr{F}}M_{\omega}=\tau_{A^{\text{\char
96}}}^{2}M_{\omega}$ by Corollary 6.28.

$\bullet$ Assume first that $\omega$ and $\omega^{\prime}$ are the walks
associated with some strings $\sigma,\sigma^{\prime}$. Let
$\sigma^{\prime\prime}$ be such that
$M_{\sigma^{\prime\prime}}=\tau_{A}M_{\sigma^{\prime}}$, and let
$\omega^{\prime\prime}$ be the walk associated with $\sigma^{\prime\prime}$.
By Lemmas 6.24, 6.25 and 6.28 and Auslander–Reiten–Serre duality, we have

$\displaystyle\mathrm{Hom}_{A}(M_{\sigma},\tau_{A}M_{\sigma^{\prime}})$
$\displaystyle=$
$\displaystyle\mathrm{Hom}_{A}(M_{\sigma},M_{\sigma^{\prime\prime}})$
$\displaystyle\cong$
$\displaystyle\overline{\mathscr{F}}(M_{\omega},\tau_{A^{\text{\char
96}}}^{2}M_{\omega^{\prime}})$ $\displaystyle\cong$
$\displaystyle\overline{\mathscr{F}}(M_{\omega},\tau_{\mathscr{F}}M_{\omega^{\prime}})$
$\displaystyle\cong$ $\displaystyle
D\mathbb{E}(M_{\omega^{\prime}},M_{\omega}).$

$\bullet$ We note that the sequence of isomorphisms in the first point above
is still valid if $\sigma$ or $\sigma^{\prime}$ are bands, with the following
convention: the walk associated with a band for $(A,I)$ is the same band,
viewed as a band for $(Q^{\text{\char 96}},I^{\text{\char 96}})$. We also note
that $\sigma^{\prime\prime}=\sigma^{\prime}$ when $\sigma^{\prime}$ is a band.

$\bullet$ Let $v\in Q_{0}$, let $\sigma$ be a string or a band for $(Q,I)$ and
let $\omega$ be the associated walk (with the convention that $\omega=\sigma$
is $\sigma$ is a band). Let $c,d$ be the blossoming vertices appearing in the
top of the injective $A^{\text{\char 96}}$-module $I_{v}^{\text{\char 96}}$
(with $d=\emptyset$ if the top of $I_{v}^{\text{\char 96}}$ is simple), and
consider the short exact sequence

$0\to P_{v}^{\text{\char 96}}\to P_{c}^{\text{\char 96}}\oplus
P_{d}^{\text{\char 96}}\to I_{v}^{\text{\char 96}}\to 0,$

of Lemma 6.18, with the convention that $P_{d}=0$ if $d=\emptyset$. There is
an induced exact sequence

$\mathrm{Hom}_{A^{\text{\char 96}}}(P_{c}^{\text{\char 96}}\oplus
P_{d}^{\text{\char
96}},M_{\omega})\xrightarrow{\varphi}\mathrm{Hom}_{A^{\text{\char
96}}}(P_{v}^{\text{\char 96}},M_{\omega})\to\mathbb{E}(I_{v}^{\text{\char
96}},M_{\omega})\to 0.$

If $\sigma$ is a band, then $\sigma=\omega$ does not contain any blossoming
vertex and the first term of the exact sequence vanishes, showing that
$\mathrm{Hom}_{A}(P_{v},M\sigma)\cong\mathrm{Hom}_{A^{\text{\char
96}}}(P_{v}^{\text{\char 96}},M_{\omega})\cong\mathbb{E}(I_{v}^{\text{\char
96}},M_{\omega})$. If $\sigma$ is a string, then a basis for
$\mathrm{Hom}_{A^{\text{\char 96}}}(P_{v}^{\text{\char 96}},M_{\omega})$ is
parametrized by the occurrences of $v$ in $\omega$ and a basis for the image
of $\varphi$ is given by those morphisms that are parametrized by the
occurrences of $v$ in $\omega$ that do not belong to $\sigma$ (indeed, only
the endpoints of $\omega$ are blossoms). Hence $\mathbb{E}(I_{v}^{\text{\char
96}},M_{\omega})\cong\mathrm{Hom}_{A^{\text{\char 96}}}(P_{v}^{\text{\char
96}},M_{\omega})/\operatorname{Im}\varphi$ is non-zero if and only if $v$
occurs in $\sigma$, if and only if $\mathrm{Hom}_{A}(P_{v},M_{\sigma})\neq 0$.

The remaining cases being trivially satisfied, this finishes the proof. ∎

By combining Theorems 4.26, 6.23 and 6.30, we obtain the following:

######  Corollary 6.31.

For any finite-dimensional gentle algebra, the flip of non-kissing facets is
an instance of mutations of maximal rigid objects in 0-Auslander
extriangulated categories.

##  7\. Negative extensions in hereditary extriangulated categories

In [GNP21], we defined covariant and contravariant left derived functors of
the functor $\mathrm{Hom}$ in essentially small extriangulated categories
having enough projective, respectively enough injective morphsims. We proved
that they agree under certain conditions; in this case, they form a bivariant
$\delta-$functor satisfying natural universal properties, whose components we
called _balanced negative extensions_. We conjectured that one can define
balanced negative extensions in each extriangulated category. In this section,
we give a partial confirmation in the hereditary case. Namely, we define
balanced negative extensions $\mathbb{E}_{\mathrm{PB}}^{\bullet}$ in
hereditary categories satisfying certain restrictive conditions $(\rm{N+})$.
Also, we construct another bivariant $\delta-$functor
$\mathbb{E}_{\mathrm{emb}}^{\bullet}$ for each $0-$Auslander category embedded
into a triangulated category in a special way. These two functors do not agree
in general, but we have nontrivial examples when they do agree. The latter
$\delta-$functor a priori depends on the embedding, but for categories linear
over fields and having finite-dimensional $\mathrm{Hom}-$ and
$\mathbb{E}-$spaces, dimensions of its components are independent of this
choice. Moreover, we expect that whenever a $0-$Auslander category admits
universal balanced negative extensions, these are given by
$\mathbb{E}_{\mathrm{emb}}^{\bullet}$, and so
$\mathbb{E}_{\mathrm{emb}}^{\bullet}$ is the correct replacement of
$\mathbb{E}_{\mathrm{PB}}$ when the condition $(\rm{N+})$ is not satisfied. We
will work in the following generality, which, in particular, allows for a
relatively easy presentation of relevant results from [GNP21].

######  Assumption 7.1.

In this section, we always assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$
is an essentially small hereditary $R$-linear extriangulated category which
has enough projective objects and enough injective objects.

By 7.1, to each object $X\in\mathscr{C}$ we can assign a pair of
$\mathfrak{s}$-triangles

(7.1)
$Q_{X}\overset{q_{X}}{\longrightarrow}P_{X}\overset{p_{X}}{\longrightarrow}X\overset{\omega_{X}}{\dashrightarrow},$
(7.2)
$X\overset{i_{X}}{\longrightarrow}I_{X}\overset{j_{X}}{\longrightarrow}J_{X}\overset{\iota_{X}}{\dashrightarrow}$

in which $P_{X}\in\mathscr{C}$ is projective and $I_{X}\in\mathscr{C}$ is
injective. These $\mathfrak{s}$-triangles are automatically dominant (resp.
codominant) in the sense of [GNP21], meaning that the morhpisms

$\omega_{X}^{\sharp}:\mathscr{C}(Q_{X},-)\to\mathbb{E}(X,-)\quad\mbox{and}\quad{\iota_{X}}_{\sharp}:\mathscr{C}(-,J_{X})\to\mathbb{E}(-,X)$

are epimorphic.

Remark that since $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is hereditary,
$Q_{X}$ is projective and $J_{X}$ is injective, so these
$\mathfrak{s}$-triangles can be thought of as a projective resolution (resp.
an injective coresolution) of $X$. For each $X$, we fix a pair of
$\mathfrak{s}$-triangles (7.1) and (7.2) and use these in our constructions.
Note though that the connected sequences of functors we define will not depend
on such a choice.

###  7.1. Reminder on negative extensions

Recall that we defined in [GNP21] a  covariant connected sequence of functors
as a pair
$(F^{\bullet}=\\{F^{n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathrm{Mod}R\\}_{n\in[a,b]},\delta_{\sharp})$,
where

  * •

$a\in\left\\{-\infty\right\\}\cup\mathbb{Z}$ and
$b\in\mathbb{Z}\cup\left\\{+\infty\right\\}$ such that $a\leq b$;

  * •

$F^{\bullet}=\\{F^{n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathrm{Mod}R\\}_{n\in[a,b]}$
a sequence of $R$-bilinear functors;

  * •

$\delta_{\sharp}=\delta_{F,\sharp}^{n}\in\mathscr{C}(F^{n}(-,C),F^{n+1}(-,A))$
is a collection of morphisms in $\mathrm{Mod}\mathscr{C}$, for any
$\mathbb{E}$-extension $\delta\in\mathbb{E}(C,A)$ and $n\in[a,b],$ natural
with respect to morphisms of $\mathbb{E}$-extensions.

Contravariant connected sequences were defined dually. A bivariant connected
sequence of functors is a triplet
$(F^{\bullet},\delta_{\sharp},\delta^{\sharp})$ where
$(F^{\bullet},\delta_{\sharp})$ and $(F^{\bullet},\delta^{\sharp})$ form a
covariant, respectively a contravariant connected sequence. As in [GNP21], we
use the shorthand notation $f_{\ast}=F^{n}(-,f)$ and $f^{\ast}=F^{n}(f,-)$ for
any morphism $f$ in $\mathscr{C}$ and any $F^{n}$ in any connected sequence.

A covariant partial $\delta$-functor is a connected sequence
$(F^{\bullet}=\\{F^{n}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathrm{Mod}R\\}_{n\in[a,b]},\delta_{\sharp})$
such that for any $\mathfrak{s}$-triangle
$A\overset{x}{\longrightarrow}B\overset{y}{\longrightarrow}C\overset{\delta}{\dashrightarrow}$,
the complex

(7.3)
$F^{a}(-,A)\overset{F^{a}(-,x)}{\longrightarrow}F^{a}(-,B)\overset{F^{a}(-,y)}{\longrightarrow}F^{a}(-,C)\overset{\delta^{a}_{F,\sharp}}{\longrightarrow}F^{a+1}(-,A)\overset{F^{a+1}(-,x)}{\longrightarrow}\cdots\overset{F^{b}(-,y)}{\longrightarrow}F^{b}(-,C)$

is exact except at the end-terms. It is a $\delta$-functor if
$a=-\infty,b=+\infty$. For a connected sequence
$(F^{\bullet},\delta_{\sharp})$, we say that an $\mathfrak{s}$-triangle is
$(F^{\bullet},\delta_{\sharp})$-acyclic if the associated unbounded complex
(7.3) is exact. In other words, a sequence $(F^{\bullet},\delta_{\sharp})$ is
a $\delta$-functor if and only if each $\mathfrak{s}$-triangle in
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ is
$(F^{\bullet},\delta_{\sharp})$-acyclic.

Contravariant and bivariant (also called balanced) (partial) $\delta$-functors
are defined similarly.

The bifunctors $\mathrm{Hom}$ and $\mathbb{E}$ are half-exact in either
argument and, therefore, are bivariant partial $\delta$-functors. Moreover,
the sequence $(\mathbb{E}^{\bullet}=\\{\mathbb{E}^{n}\\}_{n\geq
0},\delta_{\sharp},\delta^{\sharp})$ is also a bivariant partial
$\delta$-functor, where we denote $\mathbb{E}^{0}=\mathrm{Hom}_{\mathscr{C}}$,
see [GNP21, Example 5.8] (under 7.1, $\mathbb{E}^{n}=0$ for $n\geq 2$).

We are interested in connected sequences $(F^{\bullet},\delta_{\sharp})$ such
that $F^{n}=\mathbb{E}^{n}$, 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
$n\geq 0,\,A,C\in\mathscr{C}$. We then say that
$(F^{\bullet},\delta_{\sharp})$ has $F^{n}=\mathbb{E}^{n}$ with the canonical
connecting morphisms for $n\geq 0$. Similarly for contravariant and bivariant
sequences. To prove that the connected sequences we construct later are
$\delta$-functors, we will apply the following useful result.

######  Lemma 7.2.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies 7.1. Then a
covariant $(F^{\bullet},\delta_{\sharp})$ having $F^{n}=\mathbb{E}^{n}$ with
the canonical connecting morphisms for $n\geq 0$ is a $\delta$-functor if and
only if for each $X$, some (equivalently, each) dominant
$\mathfrak{s}$-triangle (7.1) is $(F^{\bullet},\delta_{\sharp})$-acyclic.
Dually, a contravariant $(F^{\bullet},\delta^{\sharp})$ having
$F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for $n\geq 0$
is a $\delta$-functor if and only if for each $X$, some (equivalently, each)
dominant $\mathfrak{s}$-triangle (7.2) is
$(F^{\bullet},\delta^{\sharp})$-acyclic.

###### Proof.

This is a direct application of [GNP21, Proposition 4.22] and its dual version
for contravariant functors to our setting and generality, since the dominance
of $\mathfrak{s}$-triangle (7.1) implies that it can be taken as $\theta$ in
the condition [GNP21, Proposition 4.22.(2)] for any
$\delta\in\mathbb{E}(X,Y)$, for any $Y\in\mathscr{C}$. ∎

######  Remark 7.3.

In fact, the same result holds (with the same proof) for any category
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ of finite (positive) global dimension
and having enough projective morphisms for $(F^{\bullet},\delta_{\sharp})$
(resp. enough injective morphisms for $(F^{\bullet},\delta^{\sharp})$), in the
sense of [GNP21].

A balanced partial $\delta$-functor
$(F^{n})_{n\geq-1},\delta_{\sharp},\delta^{\sharp})$ having
$F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for $n\geq 0$
(more precisely, its part with $n\in[-1,0]$) is the same as a _negative first
extension structure_ on $(\mathscr{C},\mathbb{E},\mathfrak{s})$ in the sense
of [AET21]. We are interested in _universal_ negative extensions, where the
natural universal property was introduced in [GNP21]:

A contravariant $\delta$-functor $(F^{n})_{n\in\mathbb{Z}},\delta_{\sharp})$
is universal among $\delta$-functors having $F^{n}=\mathbb{E}^{n}$ with the
canonical connecting morphisms for $n\geq 0$ if for each
$((G^{n}),n\in\mathbb{Z};\delta_{\sharp})$ having $G^{n}=\mathbb{E}^{n}$ with
the canonical connecting morphisms for $n\geq 0$, there exists a sequence of
natural transformations $\\{\varphi^{n}\colon G^{n}\to
F^{n}\\}_{n\in\mathbb{Z}}$ 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

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

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

If such universal $\delta$-functor $(F^{n})_{n\in\mathbb{Z}},\delta_{\sharp})$
exists, $F^{-n}$ is called the $n$-th left contravariant derived functor of
the bifunctor $\mathrm{Hom}$, or the universal contravariant negative $n$-th
extension.

Similar definitions apply for covariant and for bivariant (= balanced)
$\delta$-functors. Note that balanced negative extensions are considered
universal if they are universal among the balanced $\delta$-functors (having
$F^{n}~{}=~{}\mathbb{E}^{n}$ with the canonical connecting morphisms for
$n\geq 0$), which is a weaker property than being a bivariant $\delta$-functor
being universal both among covariant and contravariant $\delta$-functors.

In [GNP21, Section 5], for categories having enough projective, respectively
enough injective morphisms, we defined universal covariant
$(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$ and contravariant
$(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$ $\delta$-functors
having
$\mathbb{E}_{\mathrm{I}}^{n}~{}=~{}\mathbb{E}_{\mathrm{I\\!I}}^{n}~{}=~{}\mathbb{E}^{n}$
with the canonical connecting morphisms for $n\geq 0$. Their definition
involves the dualities

$(-)^{\vee}\colon\mathscr{C}\mathrm{Mod}\overset{\leftarrow}{\rightarrow}\mathrm{Mod}\mathscr{C}\colon(-)^{\vee},$

see [GNP21, Section 5.1]. In particular, we have

$\mathbb{E}_{\mathrm{I}}^{-n}(-,X)=(\mathbb{E}^{n}(X,-))^{\vee},\quad\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,-)=(\mathbb{E}^{n}(-,X))^{\vee}.$

In our special case - that is, under 7.1, - their components
$\mathbb{E}_{\mathrm{I}}^{-n},\mathbb{E}_{\mathrm{I\\!I}}^{-n}$ admit
alternative easier recursive descriptions given in [GNP21, Remark 5.23].
Namely, we have

$\mathbb{E}_{\mathrm{I}}^{-n}(-,X)\cong\mathrm{Ker}\big{(}\mathbb{E}_{\mathrm{I}}^{-(n-1)}(-,Q_{X})\overset{(q_{C})_{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-(n-1)}(-,P_{X})\big{)},$

for any $\mathfrak{s}$-triangle of the form (7.1), for each $X\in\mathscr{C}$.
This is well-defined up to unique natural isomorphisms, independently of the
choice of $\mathfrak{s}$-triangles. Dually for
$\mathbb{E}_{\mathrm{I\\!I}}^{\bullet}$, where one can use any
$\mathfrak{s}$-triangle (7.2) for the first argument. In what follows we will
not use the precise definition of connecting morphisms, only their existence;
the reader is referred to [GNP21, Definition 5.3].

######  Remark 7.4.

  * •

If $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is exact, then
$\mathbb{E}_{\mathrm{I}}^{-n}(X,Y)=\mathbb{E}_{\mathrm{I\\!I}}^{-n}(X,Y)=0$.

  * •

It is clear from the definition via dualities that for a category of
(positive) global dimension $n$, we have
$\mathbb{E}_{\mathrm{I}}^{-m}=\mathbb{E}_{\mathrm{I\\!I}}^{-m}=0,$ for each
$m>n$. In particular, under 7.1, we have
$\mathbb{E}_{\mathrm{I}}^{-m}=\mathbb{E}_{\mathrm{I\\!I}}^{-m}=0,$ for each
$m\geq 2$ (this can also be seen directly from the alternatve description).

The universality of $(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$ and
$(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$ is proved in [GNP21,
Proposition 5.19, Corollary 5.21, Corollary 5.22]. It means that
$(\mathbb{E}_{\mathrm{I}}^{-n}),n>0$ 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
$n>0$.

A special case [GNP21, Corollary 5.39] says that if
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies 7.1 and conditions (NI$+$)
and (NII+) defined below, 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 $n\geq 0$.

######  Condition 7.5.

  * (NI$+$)

$((\omega_{Y})_{\sharp})^{-1}((\mathcal{I})(X,Q_{Y}))=0$ holds in
$\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)$ for any $X,Y\in\mathscr{C}$ and
$\omega_{Y}$ in any $\mathfrak{s}$-triangle of the form (7.1) for $Y$.

  * (NII$+$)

$((\iota_{X})^{\sharp})^{-1}((\mathcal{P})(J_{X},Y))=0$ holds in
$\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)$ for any $X,Y\in\mathscr{C}$ and
$\iota_{X}$ in any $\mathfrak{s}$-triangle of the form (7.2) for $X$.

######  Example 7.6.

Conditions (NI$+$), (NII$+$) are satisfied if
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ is an exact category or the category
$K^{[-1,0]}(\operatorname{proj}\Lambda)$, where $\Lambda$ is a finite-
dimensional algebra, see [GNP21, Remark 5.32, Example 5.34].

######  Remark 7.7.

Certain weaker conditions (NI) and (NII) for categories which are
$\mathrm{Hom}$-finite over a field $k$ were shown in [GNP21, Proposition 5.30]
to be sufficient to ensure that

$\mbox{dim}_{k}\mathbb{E}_{\mathrm{I}}^{-n}=\mbox{dim}_{k}\mathbb{E}_{\mathrm{I\\!I}}^{-n},n>0.$

Under 7.1, we can reformulate these conditions as follows by using [GNP21,
Corollary 5.9, Corollary 5.11]:

  * (NI)

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

  * (NII)

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

Consider the following properties a bivariant $\delta$-functor
$(E^{\bullet},\delta_{\sharp},\delta^{\sharp})$ might have:

  * (U)

(Universality) $(E^{\bullet},\delta_{\sharp},\delta^{\sharp})$ is universal
among $\delta$-functors having $F^{n}=\mathbb{E}^{n}$ with the canonical
connecting morphisms for $n\geq 0$;

  * (C)

(Compatibility) $(E^{\bullet},\delta_{\sharp},\delta^{\sharp})$ has
$F^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for $n\geq 0$
and

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

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

Note that under conditions (NI+) and (NII+) on
$(\mathscr{C},\mathbb{E},\mathfrak{s})$, the $\delta$-functor
$(\mathbb{E}_{\mathrm{I}}^{n}\cong\mathbb{E}_{\mathrm{I\\!I}}^{n})_{n\in\mathbb{Z}}$
satisfies both properties (U) and (C). This motivates the following questions
(cf. [GNP21, Section 5.5]).

######  Problem 7.8.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies 7.1.

  * (a)

Does $(\mathscr{C},\mathbb{E},\mathfrak{s})$ admits a bivariant
$\delta$-functor $(\mathbb{E}^{\bullet},\delta_{\sharp},\delta^{\sharp})$
having property (C)?

  * (b)

Does $(\mathscr{C},\mathbb{E},\mathfrak{s})$ admit
$(\mathbb{E}^{\bullet},\delta_{\sharp},\delta^{\sharp})$ satisfying property
(U)?

In fact, we expect that if a bivariant $\delta$-functor has property (U), it
has property (C), so a positive answer 7.8.(b) would also give the positive
answer to 7.8.(a). Moreover, we have the following easy observation.

######  Lemma 7.9.

Under 7.1, if $(\mathscr{C},\mathbb{E},\mathfrak{s})$ admits a pair of
bivariant $\delta$-functors
$(\mathbb{E}_{un}^{\bullet},\delta_{\sharp},\delta^{\sharp})$ having both
properties $\rm{(U)}$ and $\rm{(C)}$ and
$(E^{\bullet},\delta_{\sharp},\delta^{\sharp})$ having property $\rm{(C)}$.
Then we have
$(\mathbb{E}_{un}^{\bullet},\delta_{\sharp},\delta^{\sharp})\overset{\sim}{\to}(\mathbb{E}^{\bullet},\delta_{\sharp},\delta^{\sharp})$.

###### Proof.

The universality of
$(\mathbb{E}_{un}^{\bullet},\delta_{\sharp},\delta^{\sharp})$ implies the
existence of a morphism of $\delta$-functors
$(\mathbb{E}_{un}^{\bullet},\delta_{\sharp},\delta^{\sharp})\to(\mathbb{E}^{\bullet},\delta_{\sharp},\delta^{\sharp})$.
Since both $\delta$-functors have property (C), they agree on pairs $(I,-)$
and $(-,P)$. That the morphisms between components are in fact isomorphisms
for each pair of objects then follows by the five lemma from the existence of
(co)resolutions (7.1) and (7.2) for each object in $\mathscr{C}$. ∎

  * •

As explained above, for $(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfying
conditions (NI+) and (NII+), it follows from [GNP21] that the answer to both
7.8.(a) and (b) is positive and the $\delta$-functor in question is given by
$(\mathbb{E}_{\mathrm{I}}^{n}\cong\mathbb{E}_{\mathrm{I\\!I}}^{n})_{n\in\mathbb{Z}}$
with suitable connecting morphisms.

  * •

In Section 7.3, for $(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfying a
certain condition (N+), we construct a bivariant $\delta$-functor
$\mathbb{E}_{\mathrm{PB}}^{\bullet}$ giving the positive answer both to
7.8.(a) and (b);

  * •

In Section 7.4, for a class of $0$-Auslander categories embedded in
triangulated categories, we construct bivariant $\delta$-functors
$\mathbb{E}_{\mathrm{emb}}^{\bullet}$ satisfying property (C).

In fact, under 7.1, property (C) defines the bifunctor $\mathbb{E}^{-2}$
uniquely up to natural isomorphism and implies that

$E^{-m}=0,\quad\forall m\geq 3,$

as will be explained in the next subsection. The nontrivial problems are then
to find $\mathbb{E}^{-1}$, $\delta_{\sharp}$, and $\delta^{\sharp}$ completing
it to a $\delta$-functor, and to check if the result is universal.

###  7.2. Definition of $\mathbb{E}^{-2}$

In [GNP21], for each $X,Y\in\mathscr{C}$ and each choice of
$\mathfrak{s}$-triangles (7.1) and (7.2), we studied the following two
complexes:

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

and

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

We proved the following.

######  Fact 7.10.

  * (i)

[GNP21, Lemma 5.26] The rightmost morphisms in $C_{\mathrm{I}}^{\bullet}$ and
$C_{\mathrm{I}\\!\mathrm{I}}^{\bullet}$, i.e.
$\iota_{X}^{\sharp}\circ\omega_{Y\sharp}$ and
$\omega_{Y\sharp}\circ\iota_{X}^{\sharp}$, have the same image.

  * (ii)

[GNP21, Proposition 5.28] For any $X,Y\in\mathscr{C}$, the cohomologies of the
complexes $C_{\mathrm{I}}^{\bullet}$ and
$C_{\mathrm{I}\\!\mathrm{I}}^{\bullet}$ are isomorphic.

Note that 7.10.(ii) implies, in particular, that the morphisms
$\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)\overset{j_{X}^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I_{X},Y)$
and
$\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y})\overset{q_{Y\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P_{Y})$
have isomorphic kernels. These kernels define the bifunctor $\mathbb{E}^{-2}$.
More precisely, by taking kernels in the commutative square

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
64.54933pt\hbox{{\hbox{\kern-64.54933pt\raise 17.07156pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathscr{C}(J_{X},Q_{Y})}$}}}}}{\hbox{\kern
15.16621pt\raise 17.07156pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathscr{C}(J_{X},P_{Y})}$}}}}}{\hbox{\kern-63.88614pt\raise-17.07156pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathscr{C}(I_{X},Q_{Y})}$}}}}}{\hbox{\kern
15.82942pt\raise-17.07156pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathscr{C}(I_{X},P_{Y})}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>22.26184pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{q_{Y\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 15.16621pt\raise
17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.09029pt\hbox{$\scriptstyle{j_{X}^{\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern-39.83365pt\raise-9.07156pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.83365pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.09029pt\hbox{$\scriptstyle{j_{X}^{\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
39.83365pt\raise-9.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{q_{Y\ast}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
15.82942pt\raise-17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\circlearrowright}$}}}}}\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces}}}},$

we can define $\mathbb{E}^{-2}(X,Y)\in\mathrm{Mod}R$,
$\eta^{\mathrm{I}}_{X,Y}\in(\mathrm{Mod}R)(\mathbb{E}^{-2}(X,Y),\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y))$
and
$\eta^{\mathrm{I}\\!\mathrm{I}}_{X,Y}\in(\mathrm{Mod}R)(\mathbb{E}^{-2}(X,Y),\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y}))$
so that they fit in the following commutative diagram

(7.4)
$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\mathbb{E}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y})}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P_{Y})}$$\textstyle{0}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)}$$\textstyle{\mathscr{C}(J_{X},Q_{Y})}$$\textstyle{\mathscr{C}(J_{X},P_{Y})}$$\textstyle{0}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(I_{X},Y)}$$\textstyle{\mathscr{C}(I_{X},Q_{Y})}$$\textstyle{\mathscr{C}(I_{X},P_{Y})}$$\scriptstyle{\eta^{\mathrm{I}\\!\mathrm{I}}_{X,Y}}$$\scriptstyle{q_{Y\ast}}$$\scriptstyle{\eta^{\mathrm{I}}_{X,Y}}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{q_{Y\ast}}$$\scriptstyle{j_{X}^{\ast}}$$\scriptstyle{j_{X}^{\ast}}$$\scriptstyle{j_{X}^{\ast}}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{q_{Y\ast}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

whose rows and columns are exact.

######  Lemma 7.11.

Take any $s\in\mathscr{C}(J_{X^{\prime}},J_{X})$ and
$t\in\mathscr{C}(Q_{Y},Q_{Y^{\prime}})$ satisfying
$x_{\ast}\iota_{X^{\prime}}=s^{\ast}\iota_{X}$ and
$y^{\ast}\omega_{Y^{\prime}}=t_{\ast}\omega_{Y}$, which exist since
$I_{X}\in\mathscr{C}$ is injective and $P_{Y}\in\mathscr{C}$ is projective.
Then there exists a unique morphism
$\mathbb{E}^{-2}(x,y)\colon\mathbb{E}^{-2}(X,Y)\to\mathbb{E}^{-2}(X^{\prime},Y^{\prime})$
which makes

(7.5)
$\textstyle{\mathbb{E}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)}$$\textstyle{\mathbb{E}^{-2}(X^{\prime},Y^{\prime})}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X^{\prime}},Y^{\prime})}$$\scriptstyle{\eta^{\mathrm{I}}_{X,Y}}$$\scriptstyle{\mathbb{E}^{-2}(x,y)}$$\scriptstyle{s^{\ast}y_{\ast}}$$\scriptstyle{\eta^{\mathrm{I}}_{X^{\prime},Y^{\prime}}}$$\scriptstyle{\circlearrowright}$
and
$\textstyle{\mathbb{E}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y})}$$\textstyle{\mathbb{E}^{-2}(X^{\prime},Y^{\prime})}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X^{\prime},Q_{Y^{\prime}})}$$\scriptstyle{\eta^{\mathrm{I}\\!\mathrm{I}}_{X,Y}}$$\scriptstyle{\mathbb{E}^{-2}(x,y)}$$\scriptstyle{x^{\ast}t_{\ast}}$$\scriptstyle{\eta^{\mathrm{I}\\!\mathrm{I}}_{X^{\prime},Y^{\prime}}}$$\scriptstyle{\circlearrowright}$

commutative simultaneously. Remark that either of these commutativity
conditions determines $\mathbb{E}^{-2}(x,y)$ uniquely. Since the left square
does not involve $t$ while the right one does not involve $s$, this in
particular implies that $\mathbb{E}^{-2}(x,y)$ does not depend on the choices
of $s$ and $t$.

###### Proof.

By the universality of kernel, there exists a unique morphism
$\mathbb{E}^{-2}(x,y)$ which makes the left square of (7.5) commutative. It is
not difficult to show that this morphism makes the right diagram also
commutative, by using the commutativity of (7.4) for $X^{\prime},Y^{\prime}$
and the monomorphicity of morphisms appearing in the left-upper square in it.
∎

######  Proposition 7.12.

$\mathbb{E}^{-2}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathrm{Mod}R$
is an $R$-bilinear functor. Moreover, $\mathbb{E}^{-2}$ is uniquely determined
up to natural isomorphism, independently of the choices
$(\ref{sTri_Chosen_proj})$ and $(\ref{sTri_Chosen_inj})$.

###### Proof.

This follows from Lemma 7.11 by the usual, straightforward argument. ∎

######  Remark 7.13.

Note that the top left commutative square in the diagram (7.4) means that we
have

(7.6)
$\mathbb{E}^{-2}(X,Y)=\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)\times_{\mathcal{C}(J_{X},Q_{Y})}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y}).$

######  Proposition 7.14.

Assume that $(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies 7.1. Let
$(E^{\bullet},\delta_{\sharp},\delta^{\sharp})$ be a $\delta$-functor having
$E^{n}=\mathbb{E}^{n}$ with the canonical connecting morphisms for $n\geq 0$.
If it satisfies property (C), then $E^{-2}$ is naturally isomorphic to
$\mathbb{E}^{-2}$ and

$E^{-m}=0,\quad\forall m\geq 3.$

###### Proof.

This follows immediately from the long exact sequences for $E^{n}$ applied to
$\mathfrak{s}$-triangles of the form (7.1) for the first argument or,
equivalently, to $\mathfrak{s}$-triangles of the form (7.2) for the second
argument. Indeed, they show that $E^{-2}$ agrees with the kernel of
$\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)\overset{j_{X}^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I_{X},Y)$
and with the kernel
$\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y})\overset{q_{Y\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P_{Y})$,
and that $E^{-n}=0$ for $n\geq 3$. ∎

###  7.3. Negative extensions via pullbacks

In this subsection, we complete $\mathbb{E}^{-2}$ to a bivariant connected
sequence of functors in an extriangulated category
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfying 7.1. Under certain
condition, this sequence is a $\delta$-functor having property (U), so its
components give the (universal) balanced negative extensions in
$(\mathscr{C},\mathbb{E},\mathfrak{s})$ (see Theorem 7.24 below).

######  Definition 7.15.

Let $X,Y\in\mathscr{C}$ be any pair of objects. Using the complexes
$C_{\mathrm{I}}^{\bullet},C_{\mathrm{I\\!I}}^{\bullet}$ in 7.10, we define
$\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)\in\mathrm{Mod}R$ and
$\wp^{\mathbf{i}}_{X,Y}\in(\mathrm{Mod}R)(\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y),\mathbb{E}^{-1}_{\mathbf{i}}(X,Y))$
$(\mathbf{i}=\mathrm{I},\mathrm{I}\\!\mathrm{I})$ by a pullback diagram taken
as below.

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}(J_{X},Q_{Y})}$$\scriptstyle{\wp^{\mathrm{I}}_{X,Y}}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{X,Y}}$$\scriptstyle{\iota_{X}^{\sharp}\circ\omega_{Y\sharp}}$$\scriptstyle{\omega_{Y\sharp}\circ\iota_{X}^{\sharp}}$$\scriptstyle{\circlearrowright}$

Remark that by 7.10.(i), $\iota_{X}^{\sharp}\circ\omega_{Y\sharp}$ and
$\omega_{Y\sharp}\circ\iota_{X}^{\sharp}$ appearing in the diagram have the
same image, hence $\wp^{\mathbf{i}}_{X,Y}$ is surjective for
$\mathbf{i}=\mathrm{I},\mathrm{I}\\!\mathrm{I}$.

######  Lemma 7.16.

Let $x\in\mathscr{C}(X^{\prime},X)$ and $y\in\mathscr{C}(Y,Y^{\prime})$ be any
pair of morphisms. Then there exists a unique morphism
$\mathbb{E}_{\mathrm{PB}}^{-1}(x,y)\colon\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)\to\mathbb{E}_{\mathrm{PB}}^{-1}(X^{\prime},Y^{\prime})$
which makes

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathbf{i}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X^{\prime},Y^{\prime})}$$\textstyle{\mathbb{E}_{\mathbf{i}}^{-1}(X^{\prime},Y^{\prime})}$$\scriptstyle{\wp^{\mathbf{i}}_{X,Y}}$$\scriptstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(x,y)}$$\scriptstyle{x^{\ast}y_{\ast}}$$\scriptstyle{\wp^{\mathbf{i}}_{X^{\prime},Y^{\prime}}}$$\scriptstyle{\circlearrowright}$

commutative for $\mathbf{i}=\mathrm{I},\mathrm{I}\\!\mathrm{I}$.

###### Proof.

This follows from the commutativity of

$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}(J_{X},Q_{Y})}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X^{\prime},Y^{\prime})}$$\textstyle{\mathbb{E}(J_{X^{\prime}},Q_{Y^{\prime}})}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X^{\prime},Y^{\prime})}$$\scriptstyle{\iota_{X}^{\sharp}\circ\omega_{Y\sharp}}$$\scriptstyle{\omega_{Y\sharp}\circ\iota_{X}^{\sharp}}$$\scriptstyle{x^{\ast}y_{\ast}}$$\scriptstyle{s^{\ast}t_{\ast}}$$\scriptstyle{x^{\ast}y_{\ast}}$$\scriptstyle{\iota_{X^{\prime}}^{\sharp}\circ\omega_{Y^{\prime}\sharp}}$$\scriptstyle{\omega_{Y^{\prime}\sharp}\circ\iota_{X^{\prime}}^{\sharp}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

and the universality of the pullback.

∎

We set $\mathbb{E}_{\mathrm{PB}}^{-2}:=\mathbb{E}^{-2}$. In what follows, we
abbreviate $\mathbb{E}_{\mathrm{PB}}^{-1}(x,y)$ and
$\mathbb{E}_{\mathrm{PB}}^{-2}(x,y)$ simply to $x^{\ast}y_{\ast}$, as before.

######  Proposition 7.17.

With the above definition,
$\mathbb{E}_{\mathrm{PB}}^{-1}\colon\mathscr{C}^{\mathrm{op}}\times\mathscr{C}\to\mathrm{Mod}R$
is an $R$-bilinear functor, and
$\wp^{\mathbf{i}}\colon\mathbb{E}_{\mathrm{PB}}^{-1}\to\mathbb{E}_{\mathbf{i}}^{-1}$
is a natural transformation for
$\mathbf{i}=\mathrm{I},\mathrm{I}\\!\mathrm{I}$. Moreover,
$\mathbb{E}_{\mathrm{PB}}^{-1}$ is uniquely determined up to natural
isomorphism compatible with $\wp^{\mathbf{i}}$ for
$\mathbf{i}=\mathrm{I},\mathrm{I}\\!\mathrm{I}$, independently of the choices
$(\ref{sTri_Chosen_proj})$ and $(\ref{sTri_Chosen_inj})$.

###### Proof.

This follows from Lemma 7.16 by the usual, straightforward argument. ∎

######  Corollary 7.18.

The following holds.

  1. (1)

If $I$ is injective, then
$\wp^{\mathrm{I}}_{I,-}\colon\mathbb{E}_{\mathrm{PB}}^{-1}(I,-)\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I,-)$
is an isomorphism, $\mathbb{E}_{\mathrm{PB}}^{-2}(I,-)=0$.

  2. (2)

If $P$ is projective, then
$\wp^{\mathrm{I}\\!\mathrm{I}}_{-,P}\colon\mathbb{E}_{\mathrm{PB}}^{-1}(-,P)\overset{\cong}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(-,P)$
is an isomorphism, $\mathbb{E}_{\mathrm{PB}}^{-2}(-,P)=0$.

###### Proof.

(1) follows from the independency shown in Proposition 7.17, since we may
choose an $\mathfrak{s}$-triangle satisfying $J_{X}=0$ if $X$ is injective.
Similarly for (2). ∎

######  Lemma 7.19.

Let $\delta\in\mathbb{E}(W,X)$ be any element. Remark that there exists
$w\in\mathscr{C}(W,J_{X})$ such that $w^{\ast}\iota_{X}=\delta$ since
$I_{X}\in\mathscr{C}$ is injective. For the composition $\xi_{\delta}$ of

$\mathbb{E}_{\mathrm{PB}}^{-2}(X,-)\overset{\eta^{\mathrm{I}}_{X,-}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},-)\overset{w^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(W,-),$

the following holds.

  1. (1)

$\xi_{\delta}$ is independent of the choice of $w$.

  2. (2)

There exists a unique
$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-2}(X,-)\to\mathbb{E}_{\mathrm{PB}}^{-1}(W,-)$
which makes

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,-)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(W,-)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(W,-)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(W,-)}$$\scriptstyle{0}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\xi_{\delta}}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{W,-}}$$\scriptstyle{\wp^{\mathrm{I}}_{W,-}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

commutative.

###### Proof.

(1) If $w^{\prime}\in\mathscr{C}(W,J_{X})$ also satisfies
$w^{\prime\ast}\iota_{X}=\delta$, then there exists $l\in\mathscr{C}(W,I_{X})$
such that $w-w^{\prime}=j_{X}\circ l$ by the exactness of
$\mathscr{C}(W,I_{X})\overset{j_{X}\circ-}{\longrightarrow}\mathscr{C}(W,J_{X})\overset{(\iota_{X})_{\sharp}}{\longrightarrow}\mathbb{E}(W,X)$.
Then it follows

$w^{\ast}\circ\eta^{\mathrm{I}}_{X,Y}=w^{\prime\ast}\circ\eta^{\mathrm{I}}_{X,Y}+l^{\ast}\circ
j_{X}^{\ast}\circ\eta^{\mathrm{I}}_{X,Y}=w^{\prime\ast}\circ\eta^{\mathrm{I}}_{X,Y}$

since the composition of
$\mathbb{E}_{\mathrm{PB}}^{-2}(X,-)\overset{\eta^{\mathrm{I}}_{X,-}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},-)\overset{j_{X}^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I_{X},-)$
is zero by the definition of $\eta^{\mathrm{I}}_{X,-}$.

(2) Let $Y\in\mathscr{C}$ be any object. It is enough to show that

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(W,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(W,Y)}$$\textstyle{\mathbb{E}(J_{W},Q_{Y})}$$\scriptstyle{(\xi_{\delta})_{Y}}$$\scriptstyle{0}$$\scriptstyle{\iota_{W}^{\sharp}\circ\omega_{Y\sharp}}$$\scriptstyle{\omega_{Y\sharp}\circ\iota_{W}^{\sharp}}$$\scriptstyle{\circlearrowright}$

is commutative, namely that the equation

(7.7) $\iota_{W}^{\sharp}\circ\omega_{Y\sharp}\circ(\xi_{\delta})_{Y}=0$

holds. Indeed if this is shown, we obtain a unique morphism
$(\delta^{\sharp})_{Y}$ which makes

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(W,Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(W,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(W,Y)}$$\scriptstyle{0}$$\scriptstyle{(\delta^{\sharp})_{Y}}$$\scriptstyle{(\xi_{\delta})_{Y}}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{W,Y}}$$\scriptstyle{\wp^{\mathrm{I}}_{W,Y}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

commutative, which is easily shown to be natural in $Y$, by using the
universality of pullbacks.

It remains to show (7.7). Since $(\mathscr{C},\mathbb{E},\mathfrak{s})$ is
hereditary, $J_{X}$ is injective and we have $\mathbb{E}(J_{W},J_{X})=0$. In
particular $w_{\ast}\iota_{W}=0$ holds for $w\in\mathscr{C}(W,J_{X})$. Thus
(7.7) follows from the commutativity of the following diagram.

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)}$$\textstyle{\mathscr{C}(J_{X},Q_{Y})}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(W,Y)}$$\textstyle{\mathscr{C}(W,Q_{Y})}$$\textstyle{\mathbb{E}(J_{W},Q_{Y})}$$\scriptstyle{\eta^{\mathrm{I}}_{X,Y}}$$\scriptstyle{(\xi_{\delta})_{Y}}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{w^{\ast}}$$\scriptstyle{w^{\ast}}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{(w_{\ast}\iota_{W})^{\sharp}=0}$$\scriptstyle{\iota_{W}^{\sharp}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

∎

######  Remark 7.20.

In Lemma 7.19, in particular if $\delta=\iota_{X}$, then we have
$\xi_{\delta}=\eta^{\mathrm{I}}_{X,-}$ since we can choose $w=\mathrm{id}$,
and

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,-)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(J_{X},-)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},-)}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\eta^{\mathrm{I}}_{X,-}}$$\scriptstyle{\wp^{\mathrm{I}}_{J_{X},-}}$$\scriptstyle{\circlearrowright}$

becomes commutative.

######  Definition 7.21.

For any extension $\delta\in\mathbb{E}(C,A)$, we define as follows.

  1. (1)

Define
$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-1}(A,-)\to\mathscr{C}(C,-)$
to be the composition of
$\mathbb{E}_{\mathrm{PB}}^{-1}(A,-)\overset{\wp^{\mathrm{I}\\!\mathrm{I}}_{A,-}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(A,-)\overset{\delta^{\sharp}}{\longrightarrow}\mathscr{C}(C,-)$.

  2. (2)

Define
$\delta_{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-1}(-,C)\to\mathscr{C}(-,A)$
to be the composition of
$\mathbb{E}_{\mathrm{PB}}^{-1}(-,C)\overset{\wp^{\mathrm{I}}_{-,C}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(-,C)\overset{\delta_{\sharp}}{\longrightarrow}\mathscr{C}(-,A)$.

  3. (3)

Define
$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-2}(A,-)\to\mathbb{E}_{\mathrm{PB}}^{-1}(C,-)$
to be the unique morphism which makes

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(A,-)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(C,-)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(C,-)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{A},-)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(C,-)}$$\scriptstyle{0}$$\scriptstyle{\eta^{\mathrm{I}}_{A,-}}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{w^{\ast}}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{C,-}}$$\scriptstyle{\wp^{\mathrm{I}}_{C,-}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

commutative for any $w\in\mathscr{C}(C,J_{A})$ such that
$w^{\ast}\iota_{A}=\delta$, by using Lemma 7.19.

  4. (4)

Define
$\delta_{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-2}(-,C)\to\mathbb{E}_{\mathrm{PB}}^{-1}(-,A)$
to be the unique morphism which makes

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(-,C)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(-,A)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(-,A)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(-,Q_{C})}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(-,A)}$$\scriptstyle{0}$$\scriptstyle{\eta^{\mathrm{I}\\!\mathrm{I}}_{-,C}}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{t_{\ast}}$$\scriptstyle{\wp^{\mathrm{I}}_{-,A}}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{-,A}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

commutative for any $t\in\mathscr{C}(Q_{C},A)$ such that
$t_{\ast}\omega_{C}=\delta$, by using the dual of Lemma 7.19.

######  Proposition 7.22.

The following forms a bivariant connected sequence of functors
$(\mathbb{E}_{\mathrm{PB}}^{\bullet},\delta_{\sharp},\delta^{\sharp})$.

  * •

$\mathbb{E}_{\mathrm{PB}}^{n}~{}=~{}\mathbb{E}^{n}$ with the canonical
connecting morphisms for $n\geq 0$.

  * •

$\mathbb{E}_{\mathrm{PB}}^{-1}$ is the one obtained in Proposition 7.17.
$\mathbb{E}_{\mathrm{PB}}^{-2}=\mathbb{E}^{-2}$. Also, we put
$\mathbb{E}_{\mathrm{PB}}^{-n}=0$ for $n>2$.

  * •

For any $\delta\in\mathbb{E}(C,A)$, those $\delta^{\sharp},\delta_{\sharp}$ on
$\mathbb{E}_{\mathrm{PB}}^{-1}$ and $\mathbb{E}_{\mathrm{PB}}^{-2}$ are those
in Definition 7.21.

###### Proof.

It remains to show the naturality of each of

  * (i)

$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-1}(A,-)\to\mathscr{C}(C,-)$,
$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-2}(A,-)\to\mathbb{E}_{\mathrm{PB}}^{-1}(C,-)$,
and

  * (ii)

$\delta_{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-1}(-,C)\to\mathscr{C}(-,A)$,
$\delta_{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-2}(-,C)\to\mathbb{E}_{\mathrm{PB}}^{-1}(-,A)$

with respect to morphisms of $\mathbb{E}$-extensions. By duality, it is enough
to show (i). Naturality of
$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-1}(A,-)\to\mathscr{C}(C,-)$
follows from the fact that
$(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$ is a contravariant
$\delta$-functor and the naturality of
$\wp^{\mathrm{I}\\!\mathrm{I}}\colon\mathbb{E}_{\mathrm{PB}}^{-1}\to\mathbb{E}_{\mathrm{I\\!I}}^{-1}$.

Let us show the naturality of
$\delta^{\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-2}(A,-)\to\mathbb{E}_{\mathrm{PB}}^{-1}(C,-)$.
Let
$\delta\in\mathbb{E}(C,A),\delta^{\prime}\in\mathbb{E}(C^{\prime},A^{\prime})$
be any pair of extensions, and let $(a,c)\colon\delta\to\delta^{\prime}$ be
any morphism. Take
$s\in\mathscr{C}(C,J_{A}),s^{\prime}\in\mathscr{C}(C^{\prime},J_{A^{\prime}}),u\in\mathscr{C}(J_{A},J_{A^{\prime}})$
such that

$s^{\ast}\iota_{A}=\delta,\ \
s^{\prime\ast}\iota_{A^{\prime}}=\delta^{\prime},\ \
a_{\ast}\iota_{A}=u^{\ast}\iota_{A^{\prime}}$

arbitrarily. Then for
$a_{\ast}\delta=c^{\ast}\delta^{\prime}\in\mathbb{E}(C,A^{\prime})$, we have
$(s^{\prime}\circ c)^{\ast}\iota_{A^{\prime}}=(u\circ
s)^{\ast}\iota_{A^{\prime}}=a_{\ast}\delta$. Thus by Lemma 7.19 (1),
compositions of

$\displaystyle\mathbb{E}_{\mathrm{PB}}^{-2}(A^{\prime},-)\overset{\eta^{\mathrm{I}}_{A^{\prime},-}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(J_{A^{\prime}},-)\overset{(s^{\prime}\circ
c)^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(C,-),$
$\displaystyle\mathbb{E}_{\mathrm{PB}}^{-2}(A^{\prime},-)\overset{\eta^{\mathrm{I}}_{A^{\prime},-}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(J_{A^{\prime}},-)\overset{(u\circ
s)^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(C,-)$

are equal, which we denoted by $\xi_{a_{\ast}\delta}$. Then it follows that
either of $\delta^{\sharp}\circ
a^{\ast},c^{\ast}\circ\delta^{\prime\sharp}\colon\mathbb{E}_{\mathrm{PB}}^{-2}(A^{\prime},-)\to\mathbb{E}_{\mathrm{PB}}^{-1}(C,-)$
makes the following diagram commutative,

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(A^{\prime},-)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(C,-)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(C,-)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(C,-)}$$\scriptstyle{0}$$\scriptstyle{\xi_{a_{\ast}\delta}}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{C,-}}$$\scriptstyle{\wp^{\mathrm{I}}_{C,-}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

hence $\delta^{\sharp}\circ
a^{\ast}=(a_{\ast}\delta)^{\sharp}=c^{\ast}\circ\delta^{\prime\sharp}$ follows
from (the uniqueness in) Lemma 7.19 (2). In particular the diagram

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(A^{\prime},-)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(C^{\prime},-)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(A,-)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(C,-)}$$\scriptstyle{\delta^{\prime\sharp}}$$\scriptstyle{a^{\ast}}$$\scriptstyle{c^{\ast}}$$\scriptstyle{\delta^{\sharp}}$$\scriptstyle{\circlearrowright}$

is commutative, which shows the naturality as desired. ∎

Consider the following condition:

  * (N$+$)

For any $X,Y\in\mathscr{C}$, complex $C_{\mathrm{I}}^{\bullet}$ in 7.10
associated with (7.1) and (7.2)

$\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)\overset{j_{X}^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I_{X},Y)\overset{i_{X}^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)\overset{\iota_{X}^{\sharp}\circ\omega_{Y\sharp}}{\longrightarrow}\mathbb{E}(J_{X},Q_{Y})$

is exact at the both middle terms.

######  Remark 7.23.

By 7.10.(ii), condition (N$+$) is equivalent to asking the complex
$C_{\mathrm{I}\\!\mathrm{I}}^{\bullet}$

$\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y})\overset{q_{Y\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,P_{Y})\overset{p_{Y\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\overset{\omega_{Y\sharp}\circ\iota_{X}^{\sharp}}{\longrightarrow}\mathbb{E}(J_{X},Q_{Y})$

to be exact at the both middle terms for any $X,Y\in\mathscr{C}$. In this
sense, condition (N$+$) is self-dual.

######  Theorem 7.24.

The following are equivalent.

  1. (1)

$(\mathscr{C},\mathbb{E},\mathfrak{s})$ satisfies (N$+$).

  2. (2)

$\omega_{X}$ is $(\mathbb{E}_{\mathrm{PB}}^{\bullet},\delta_{\sharp})$-acyclic
for any $X\in\mathscr{C}$, where $\omega_{X}$ is the extension chosen in
(7.1).

  3. (3)

$\iota_{X}$ is $(\mathbb{E}_{\mathrm{PB}}^{\bullet},\delta^{\sharp})$-acyclic
for any $X\in\mathscr{C}$, where $\iota_{X}$ is the extension chosen in (7.2).

  4. (4)

$(\mathbb{E}_{\mathrm{PB}}^{\bullet},\delta_{\sharp})$ is a covariant
$\delta$-functor.

  5. (5)

$(\mathbb{E}_{\mathrm{PB}}^{\bullet},\delta^{\sharp})$ is a contravariant
$\delta$-functors.

  6. (6)

$(\mathbb{E}_{\mathrm{PB}}^{\bullet},\delta_{\sharp},\delta^{\sharp})$ is a
bivariant $\delta$-functor.

Moreover, if one of these equivalent conditions is satisfied, then
$(\mathbb{E}_{\mathrm{PB}}^{\bullet},\delta_{\sharp},\delta^{\sharp})$
satisfies properties (U) and (C).

###### Proof.

By definition, (6) means that both of (4) and (5) hold. Equivalence (2)
$\Leftrightarrow$ (4) follows from [GNP21, Proposition 4.22]. Dually for (3)
$\Leftrightarrow$ (5).

Since (1) $\Leftrightarrow$ (3) can be shown dually, it suffices to show (1)
$\Leftrightarrow$ (2). Let $X,Y\in\mathscr{C}$ be any pair of objects. By the
definitions so far,

(7.8)
$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(J_{X},Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(I_{X},Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)}$$\textstyle{\mathscr{C}(J_{X},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(I_{X},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}(J_{X},Q_{Y})}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{j_{X}^{\ast}}$$\scriptstyle{i_{X}^{\ast}}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{X,Y}}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\wp^{\mathrm{I}}_{J_{X},Y}}$$\scriptstyle{\cong}$$\scriptstyle{\wp^{\mathrm{I}}_{I_{X},Y}}$$\scriptstyle{\cong}$$\scriptstyle{\wp^{\mathrm{I}}_{X,Y}}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{\eta^{\mathrm{I}}_{X,Y}}$$\scriptstyle{j_{X}^{\ast}}$$\scriptstyle{i_{X}^{\ast}}$$\scriptstyle{\iota_{X}^{\sharp}\circ\omega_{Y\sharp}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

is commutative. Remark that the sequence

$0\to\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)\overset{\eta^{\mathrm{I}}_{X,Y}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)\overset{j_{X}^{\ast}}{\longrightarrow}\mathbb{E}_{\mathrm{I}}^{-1}(I_{X},Y)$

is always exact by the definition of $\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)$.

In the rightmost square, since
$\iota^{\sharp}_{X}\colon\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)\to\mathscr{C}(J_{X},Y)$
is monomorphic and since

$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)}$$\textstyle{\mathscr{C}(J_{X},Q_{Y})}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{X,Y}}$$\scriptstyle{\wp^{\mathrm{I}}_{X,Y}}$$\scriptstyle{\omega_{Y\sharp}\circ\iota_{X}^{\sharp}}$$\scriptstyle{\iota_{X}^{\sharp}\circ\omega_{Y\sharp}}$$\scriptstyle{\circlearrowright}$

is a pullback by the definition of $\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)$, we
obtain an isomorphism $\kappa$ for kernels which makes

(7.9)
$\textstyle{0}$$\textstyle{\mathrm{Ker}(\iota_{X}^{\sharp})}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)}$$\textstyle{\mathscr{C}(J_{X},Y)}$$\textstyle{0}$$\textstyle{\mathrm{Ker}(\iota_{X}^{\sharp}\circ\omega_{Y\sharp})}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}(J_{X},Q_{Y})}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\cong}$$\scriptstyle{\kappa}$$\scriptstyle{\wp^{\mathrm{I}}_{X,Y}}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{\iota_{X}^{\sharp}\circ\omega_{Y\sharp}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

commutative. Thus $(\ref{DiagUnivBivProl1})$ and $(\ref{DiagUnivBivProl2})$
give an isomorphism of sequences

$\textstyle{0}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(J_{X},Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(I_{X},Y)}$$\textstyle{\mathrm{Ker}(\iota_{X}^{\sharp})}$$\textstyle{0}$$\textstyle{0}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(I_{X},Y)}$$\textstyle{\mathrm{Ker}(\iota_{X}^{\sharp}\circ\omega_{Y\sharp})}$$\textstyle{0}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{j_{X}^{\ast}}$$\scriptstyle{i_{X}^{\ast}}$$\scriptstyle{\wp^{\mathrm{I}}_{J_{X},Y}}$$\scriptstyle{\cong}$$\scriptstyle{\wp^{\mathrm{I}}_{I_{X},Y}}$$\scriptstyle{\cong}$$\scriptstyle{\kappa}$$\scriptstyle{\cong}$$\scriptstyle{\eta^{\mathrm{I}}_{X,Y}}$$\scriptstyle{j_{X}^{\ast}}$$\scriptstyle{i_{X}^{\ast}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in which we denoted the morphisms induced by $i_{X}^{\ast}$ to kernels by the
same symbols. Consequently, its lower row is exact if and only if the upper
row is exact. This is nothing but (1) $\Leftrightarrow$ (2).

Property (C) holds by Corollary 7.18.

It remains to show the universality. Let
$(F^{\bullet},\delta_{\sharp},\delta^{\sharp})$ be any bivariant
$\delta$-functor having $F^{n}~{}=~{}\mathbb{E}^{n}$ with the canonical
connecting morphisms for $n\geq 0$. By [GNP21, Corollary 5.21, Corollary
5.22], there exists a unique morphism
$\varphi^{\bullet}\colon(F^{\bullet},\delta_{\sharp})\to(\mathbb{E}_{\mathrm{I}}^{\bullet},\delta_{\sharp})$
of covariant $\delta$-functors and a unique morphism
$\psi^{\bullet}\colon(F^{\bullet},\delta^{\sharp})\to(\mathbb{E}_{\mathrm{I\\!I}}^{\bullet},\delta^{\sharp})$
of contravariant $\delta$-functors.

Construction of $\lambda^{(-1)}\colon F^{-1}\to\mathbb{E}_{\mathrm{PB}}^{-1}$.

For any $X,Y\in\mathscr{C}$, the commutativity of

$\textstyle{F^{-1}(X,Y)}$$\textstyle{\mathscr{C}(X,Q_{Y})}$$\textstyle{\mathscr{C}(J_{X},Y)}$$\textstyle{\mathbb{E}(J_{X},Q_{Y})}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{\circlearrowright}$

implies that of

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
63.45863pt\hbox{{\hbox{\kern-63.45863pt\raise 17.07156pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{F^{-1}(X,Y)}$}}}}}{\hbox{\kern
18.03783pt\raise 17.07156pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)}$}}}}}{\hbox{\kern-61.97394pt\raise-17.07156pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)}$}}}}}{\hbox{\kern
15.3263pt\raise-17.07156pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathbb{E}(J_{X},Q_{Y})}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>23.40073pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.18472pt\hbox{$\scriptstyle{\varphi_{X,Y}^{(-1)}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 18.03783pt\raise
17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.10834pt\hbox{$\scriptstyle{\psi_{X,Y}^{(-1)}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern-39.83365pt\raise-8.50269pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.83365pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.11806pt\hbox{$\scriptstyle{\iota_{X}^{\sharp}\circ\omega_{Y\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
39.83365pt\raise-9.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.11806pt\hbox{$\scriptstyle{\omega_{Y\sharp}\circ\iota_{X}^{\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
15.3263pt\raise-17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\circlearrowright}$}}}}}\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces}}}}.$

Thus we obtain a unique homomorphism $\lambda^{(-1)}_{X,Y}\colon
F^{-1}(X,Y)\to\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)$ which makes

$\textstyle{F^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Y)}$$\scriptstyle{\varphi^{(-1)}_{X,Y}}$$\scriptstyle{\lambda^{(-1)}_{X,Y}}$$\scriptstyle{\psi^{(-1)}_{X,Y}}$$\scriptstyle{\wp^{\mathrm{I}}_{X,Y}}$$\scriptstyle{\wp^{\mathrm{I}\\!\mathrm{I}}_{X,Y}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

commutative, by the universality of pullback. It is straightforward to check
that $\lambda^{(-1)}=\\{\lambda^{(-1)}_{X,Y}\\}_{X,Y\in\mathscr{C}}$ forms a
natural transformation $\lambda^{(-1)}\colon
F^{-1}\to\mathbb{E}_{\mathrm{PB}}^{-1}$, again by using the universality of
pullbacks.

Construction of $\lambda^{(-2)}\colon F^{-2}\to\mathbb{E}_{\mathrm{PB}}^{-2}$.

Remark that the left-upper square in $(\ref{Comm_Added})$ becomes a pullback
square. Let us see that $\lambda^{(-2)}$ can be obtained in a similar way as
for $\lambda^{(-1)}$ using this square. Indeed, for any $X,Y\in\mathscr{C}$,
the commutativity of

$\textstyle{F^{-2}(X,Y)}$$\textstyle{F^{-1}(X,Q_{Y})}$$\textstyle{F^{-1}(J_{X},Y)}$$\textstyle{\mathscr{C}(J_{X},Q_{Y})}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\iota_{X}^{\sharp}}$$\scriptstyle{\omega_{Y\sharp}}$$\scriptstyle{\circlearrowright}$

implies that of

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
75.53587pt\hbox{{\hbox{\kern-69.14915pt\raise 17.07156pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{F^{-2}(X,Y)}$}}}}}{\hbox{\kern
21.19722pt\raise 17.07156pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y})}$}}}}}{\hbox{\kern-68.85597pt\raise-17.07156pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)}$}}}}}{\hbox{\kern
20.80849pt\raise-17.07156pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathscr{C}(J_{X},Q_{Y})}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>24.48212pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.95056pt\hbox{$\scriptstyle{\psi_{X,Q_{Y}}^{(-1)}\circ\omega_{Y\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 21.19722pt\raise
17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.02695pt\hbox{$\scriptstyle{\varphi_{J_{X},Y}^{(-1)}\circ\iota_{X}^{\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern-45.52417pt\raise-8.50269pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
45.52417pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.51805pt\hbox{$\scriptstyle{\iota_{X}^{\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
45.52417pt\raise-9.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.42361pt\hbox{$\scriptstyle{\omega_{Y\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
20.80849pt\raise-17.07156pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\circlearrowright}$}}}}}\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces}}}},$

hence we obtain a unique homomorphism $\lambda^{(-2)}_{X,Y}\colon
F^{-2}(X,Y)\to\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)$ which makes

$\textstyle{F^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(J_{X},Y)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,Y)}$$\textstyle{\mathbb{E}_{\mathrm{I\\!I}}^{-1}(X,Q_{Y})}$$\scriptstyle{\varphi^{(-1)}_{J_{X},Y}\circ\iota_{X}^{\sharp}}$$\scriptstyle{\lambda^{(-2)}_{X,Y}}$$\scriptstyle{\psi^{(-1)}_{X,Q_{Y}}\circ\omega_{Y\sharp}}$$\scriptstyle{\eta^{\mathrm{I}}_{X,Y}}$$\scriptstyle{\eta^{\mathrm{I}\\!\mathrm{I}}_{X,Y}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

commutative, by the universality of pullback. We can check that
$\lambda^{(-2)}=\\{\lambda^{(-2)}_{X,Y}\\}_{X,Y\in\mathscr{C}}$ forms a
natural transformation $\lambda^{(-2)}\colon
F^{-2}\to\mathbb{E}_{\mathrm{PB}}^{-2}$, in a straightforward way.

So far we have obtained $\lambda^{\bullet}\colon
F^{\bullet}\to\mathbb{E}_{\mathrm{PB}}^{\bullet}$. It remains to show its
compatibility with $\delta_{\sharp}$ and $\delta^{\sharp}$. By duality, it is
enough to show the compatibility with $\delta_{\sharp}$.

Let $\delta\in\mathbb{E}(C,A)$ be any element. It is enough to show the
commutativity of

(7.10)
$\textstyle{F^{-2}(X,C)}$$\textstyle{F^{-1}(X,A)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,C)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,A)}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\lambda^{(-2)}_{X,C}}$$\scriptstyle{\lambda^{(-1)}_{X,A}}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\circlearrowright}$
and
$\textstyle{F^{-1}(X,C)}$$\textstyle{\mathscr{C}(X,A)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,C)}$$\textstyle{\mathscr{C}(X,A)}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\lambda^{(-1)}_{X,C}}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\circlearrowright}$

for any $X\in\mathscr{C}$. By definition, commutativity of the right square of
$(\ref{CommD})$ follows immediately from that of

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
125.19147pt\hbox{{\hbox{\kern-108.90016pt\raise 22.76208pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{F^{-1}(X,C)}$}}}}}{\hbox{\kern
64.62866pt\raise 22.76208pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathscr{C}(X,A)}$}}}}}{\hbox{\kern-109.94879pt\raise-22.76208pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,C)}$}}}}}{\hbox{\kern-10.33214pt\raise-22.76208pt\hbox{\hbox{\kern
3.0pt\raise-3.06888pt\hbox{$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,C)}$}}}}}{\hbox{\kern
64.62866pt\raise-22.76208pt\hbox{\hbox{\kern
3.0pt\raise-2.5pt\hbox{$\textstyle{\mathscr{C}(X,A)}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>29.27597pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.34723pt\hbox{$\scriptstyle{\delta_{\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 64.62866pt\raise
22.76208pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.10834pt\hbox{$\scriptstyle{\lambda^{(-1)}_{X,C}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern-85.35782pt\raise-14.1932pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}\ignorespaces{\hbox{\hbox{\kern
1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>6.32916pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.18472pt\hbox{$\scriptstyle{\varphi^{(-1)}_{X,C}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern-6.81944pt\raise-14.1932pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
43.0361pt\raise-29.27596pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.34723pt\hbox{$\scriptstyle{\delta_{\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
64.62866pt\raise-22.76208pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.11806pt\hbox{$\scriptstyle{\wp^{\mathrm{I}}_{X,C}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern-10.33214pt\raise-22.76208pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-5.33333pt\raise-48.47603pt\hbox{{}\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.34723pt\hbox{$\scriptstyle{\delta_{\sharp}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{}{}{{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}}{\hbox{\kern
65.1854pt\raise-30.7648pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\kern-59.62744pt\raise-8.53578pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\circlearrowright}$}}}}}\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\kern
22.88512pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\circlearrowright}$}}}}}\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\kern-2.72212pt\raise-34.14302pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\circlearrowright}$}}}}}\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces}}}}.$

Commutativity of the left square of $(\ref{CommD})$ follows from the
commutativity of

$\textstyle{F^{-2}(X,C)}$$\textstyle{F^{-1}(X,A)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-2}(X,C)}$$\textstyle{\mathbb{E}_{\mathrm{PB}}^{-1}(X,A)}$$\textstyle{\mathbb{E}_{\mathrm{I}}^{-1}(X,A)}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{\lambda^{(-2)}_{X,C}}$$\scriptstyle{\lambda^{(-1)}_{X,A}}$$\scriptstyle{\delta_{\sharp}}$$\scriptstyle{0}$$\scriptstyle{\varphi^{(-1)}_{X,A}}$$\scriptstyle{\wp^{\mathrm{I}}_{X,A}}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$$\scriptstyle{\circlearrowright}$

in which $\varphi^{(-1)}_{X,A}\circ\delta_{\sharp}=0$ holds since
$\mathbb{E}_{\mathrm{I}}^{-2}(X,C)=0$, and of