H^{0}(\mathcal{A})\longrightarrow\operatorname{D}^{\mathrm{c}}(\mathcal{A})$

identifies $H^{0}(\mathcal{A})$ with a $d\mathbb{Z}$-cluster tilting
subcategory of $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$.

###### Proof.

Notice that, since $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ is generated
as a triangulated category with split idempotents by the free DG
$\mathcal{A}$-modules, the finiteness assumption on $H^{0}(\mathcal{A})$
implies that $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ has finite-
dimensional morphism spaces. The equivalence between the two conditions then
follows from Theorem 2.2.4. ∎

######  Definition 3.2.5.

Let $\mathcal{F}$ be a $(d+2)$-angulated category.

  1. (1)

A _(DG) enhancement of $\mathcal{F}$_ is a pre-$(d+2)$-angulated DG category
$\mathcal{A}$ such that there exists an equivalence of $(d+2)$-angulated
categories

$\mathcal{F}\simeq H^{0}(\mathcal{A}),$

where the $(d+2)$-angulation on the right-hand side is induced by the
triangulation on $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ as in Theorem
2.2.1. If $\mathcal{F}$ has an enhancement we say that it is _algebraic._

  2. (2)

We say that _$\mathcal{F}$ admits a unique (DG) enhancement_ if it admits an
enhancement and any two enhancements of $\mathcal{F}$ are quasi-equivalent
(that is isomorphic in $\operatorname{Ho}\left(\operatorname{dgcat}\right)$).

The following two questions are natural.

######  Question 3.2.6.

Let $\mathcal{F}$ be a $(d+2)$-angulated category. Does $\mathcal{F}$ admit an
enhancement? If it does, under which conditions is the enhancement unique?

######  Definition 3.2.7.

Let $(\mathcal{C},\Sigma)$ be a pair consisting of an additive category
$\mathcal{C}$ and an autoequivalence
$\Sigma\colon\mathcal{C}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{C}$. We
say that a pre-$(d+2)$-angulated DG category $\mathcal{A}$ _induces a (DG)
enhanced $(d+2)$-angulated structure on $(\mathcal{C},\Sigma)$_ if there
exists an equivalence
$F\colon\mathcal{C}\stackrel{{\scriptstyle\sim}}{{\to}}H^{0}(\mathcal{A})$
such that the diagram

${\mathcal{C}}$${H^{0}(\mathcal{A})}$${\mathcal{C}}$${H^{0}(\mathcal{A})}$$\scriptstyle{F}$$\scriptstyle{\Sigma}$$\scriptstyle{[d]}$$\scriptstyle{F}$

commutes up to natural isomorphism. In this case, we say that the pair
$(\mathcal{C},\Sigma)$ _admits a (DG) enhanced $(d+2)$-angulated structure_.

######  Question 3.2.8.

Let $(\mathcal{C},\Sigma)$ be a pair consisting of an additive category
$\mathcal{C}$ and an automorphism
$\Sigma\colon\mathcal{C}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{C}$. Does
the pair admit an enhanced $(d+2)$-angulated structure? If it does, is the
enhancement of the underlying $(d+2)$-angulated category unique?

######  Remark 3.2.9.

Theorems 4.5.8 and 5.1.3 provide a complete answer to Questions 3.2.6 and
3.2.8 in the setting of Theorems 2.2.15 and 2.2.20.

######  Remark 3.2.10.

Let $(\mathcal{F},\Sigma,{\pentagon})$ be a $(d+2)$-angulated category and
$n\geq 1$ an integer. Recall from Example 2.1.9 that the pair
$(\mathcal{F},\Sigma^{n})$ is endowed with an $n(d+2)$-angulation
${\pentagon}^{n}$ obtained by extending the sequences in ${\pentagon}$
periodically up to iterated applications of $\Sigma$. In general, we do not
know if the existence of an enhancement of $(\mathcal{F},\Sigma,{\pentagon})$
implies the existence of an enhancement of
$(\mathcal{F},\Sigma^{n},{\pentagon}^{n})$. Notwithstanding, as explained in
the introduction, Theorem 5.1.3 provides a positive answer when
$\mathcal{F}=\operatorname{proj}(\Lambda)$ for some twisted $(d+2)$-periodic
algebra with respect to an algebra automorphism
$\sigma\colon\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\Lambda$ and
${\pentagon}={\pentagon}_{\eta}$ is the AL $(d+2)$-angulation associated to an
exact sequence of $\Lambda$-bimodules

$\eta\colon\qquad 0\to{}_{1}{\Lambda}_{\sigma}\to P_{d+1}\to\cdots\to P_{1}\to
P_{0}\to\Lambda\to 0$

with projective-injective middle terms. Indeed, the algebra $\Lambda$ is
twisted $n(d+2)$-periodic with respect to $\sigma^{n}$ and the class
${\pentagon}^{n}$ is precisely the AL $n(d+2)$-angulation associated to the
exact sequence obtained by splicing together the exact sequences

$\eta,\ \Sigma(\eta),\dots,\Sigma^{n-1}(\eta).$

Therefore, in this case, the $n(d+2)$-angulated category
$(\mathcal{F},\Sigma^{n},{\pentagon}^{n})$ admits an enhancement by Theorem
5.1.3. However, the precise relationship between enhancements of
$(\mathcal{F},\Sigma,{\pentagon})$ and
$(\mathcal{F},\Sigma^{n},{\pentagon}^{n})$ is not clear (even if the
semisimple algebra $\Lambda/J_{\Lambda}$ is separable, so that we can
guarantee that enhancements are unique). For example, can one
construct—explicitly—an enhancement of
$(\mathcal{F},\Sigma^{n},{\pentagon}^{n})$ from a given enhancement of
$(\mathcal{F},\Sigma,{\pentagon})$?

##  4\. Higher structures and $(d+2)$-angulations

###  4.1. Toda brackets and standard $(d+2)$-angulations

######  Setting 4.1.1.

We fix a triangulated category $\mathcal{T}$ with shift functor $[1]$ in this
subsection.

We aim to characterise standard $(d+2)$-angles (Definition 2.2.2) in terms of
higher Toda brackets. We begin by recalling the necessary definitions.

######  Definition 4.1.2.

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{3}\xrightarrow{f_{3}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

The _Toda family $\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$_ is the
set of pairs $(\beta,\alpha)$ such that there exists a commutative diagram of
the form

(4.1.3)
${x_{3}}$${x_{2}}$${x_{1}}$${x_{0}}$${c}$${x_{2}}$${x_{1}}$${c[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{u}$$\scriptstyle{f_{2}}$$\scriptstyle{v}$$\scriptstyle{\beta[1]}$

in which the horizontal row is an exact triangle in $\mathcal{T}$.

######  Remark 4.1.4.

In the setting of Definition 4.1.2, notice that the Toda family
$\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$ is non-empty if and only
if $f_{1}f_{2}=0$ and $f_{2}f_{3}=0$.

######  Remark 4.1.5.

An alternative (perhaps more suggestive) way to depict the condition for the
pair $(\beta,\alpha)$ to lie in
$\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$ is as a commutative
diagram

${x_{2}}$${x_{1}}$${x_{3}}$${c}$${x_{0}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{+1}$$\scriptstyle{f_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{-1}$

in which $\beta$ is a morphism of degree $-1$. Here, the oriented triangle is
an exact triangle in $\mathcal{T}$ and the un-oriented triangles commute. In
particular, the pair $(\beta,\alpha)$ yields the well-defined composite

${x_{3}[1]}$${c[1]}$${x_{0}.}$$\scriptstyle{\alpha[1]}$$\scriptstyle{\beta[1]}$

######  Definition 4.1.6.

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

The _Toda bracket $\langle{f_{1},\dots,f_{d+2}}\rangle$_ is the subset of
$\mathcal{T}(x_{d+2}[d],x_{0})$ defined inductively as follows:

  1. (1)

If $d=0$, then $\langle{f_{1},f_{2}}\rangle\coloneqq\left\\{f_{1}\circ
f_{2}\right\\}$.

  2. (2)

If $d\geq 1$, then

$\langle{f_{1},\dots,f_{d+2}}\rangle\coloneqq\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}\langle{\beta[1],\alpha[1],f_{4}[1],\cdots,f_{d+2}[1]}\rangle$

is the union of Toda brackets of length $d+1$.

######  Remark 4.1.7.

Definition 4.1.6 is equivalent to that in [Shi02], which is itself based on
that in [Coh68], see [CF17, Prop. 5.8]. We also note that Definition 4.1.2 is
equivalent to the desuspension of that in [CF17, Sec. 5].

######  Remark 4.1.8.

The Toda bracket is not quite a $(d+2)$-ary operation as its output is a _set_
of morphisms rather than a single map (compare with Proposition 4.1.31).

######  Example 4.1.9.

The Toda bracket of a triple of composable morphisms

$x_{3}\xrightarrow{f_{3}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

is the subset of $\mathcal{T}(x_{3}[1],x_{0})$ given by

$\langle{f_{1},f_{2},f_{3}}\rangle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}\langle{\beta[1],\alpha[1]}\rangle=\left\\{(\beta\circ\alpha)[1]\right|(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)\\}.$

######  Example 4.1.10.

The Toda bracket of the morphisms in an exact triangle

$x_{3}\xrightarrow{f_{3}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{3}[1]$

satisfies $\mathbf{1}_{x_{3}[1]}\in\langle{f_{1},f_{2},f_{3}}\rangle$. Indeed,
in view of Example 4.1.9, it is enough to observe that the tautological
diagram

${x_{3}}$${x_{2}}$${x_{1}}$${x_{3}[1]}$${x_{3}}$${x_{2}}$${x_{1}}$${x_{3}[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\mathbf{1}_{x_{3}}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{\mathbf{1}_{x_{3}}[1]}$

exhibits the membership
$(\mathbf{1}_{x_{3}},\mathbf{1}_{x_{3}})\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$.

The following classical result, which is a refinement of Example 4.1.10,
illustrates the usefulness of Toda brackets.

######  Theorem 4.1.11 ([Hel68, Thm. 13.2], see also [CF17, Thm. B.1]).

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{3}\xrightarrow{f_{3}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{3}[1].$

The following statements are equivalent:

  1. (1)

The above sequence is an exact triangle in $\mathcal{T}$.

  2. (2)

The following two conditions are satisfied:

    1. (a)

For each $y\in\mathcal{T}$, The induced sequence of $\mathbf{k}$-modules

$\mathcal{T}(y,x_{3})\to\mathcal{T}(y,x_{2})\to\mathcal{T}(y,x_{1})\to\mathcal{T}(y,x_{3}[1])\to\mathcal{T}(y,x_{2}[1])$

is exact.

    2. (b)

We have $\mathbf{1}_{x_{3}[1]}\in\langle{f_{1},f_{2},f_{3}}\rangle$.

The objective of this section is to prove the following $(d+2)$-angulated
analogue of Theorem 4.1.11 (compare also with Theorem 2.2.1). As is customary,
given a subset $X$ of an abelian group, we let

$-X\coloneqq\left\\{-x\right|x\in X\\}.$

######  Theorem 4.1.12.

Let $\mathcal{C}\subseteq\mathcal{T}$ be a $d$-rigid subcategory such that
${\mathcal{C}=\operatorname{add}(c)}$ for some basic object $c$,
${\mathcal{C}[d]=\mathcal{C}}$ and
$\operatorname{thick}(\mathcal{C})=\mathcal{T}$. Suppose given a sequence of
morphisms in $\mathcal{C}$ of the form

${c_{d+2}}$${c_{d+1}}$${\cdots}$${c_{2}}$${c_{1}}$${c_{d+2}[d],}$$\scriptstyle{f_{d+2}}$$\scriptstyle{f_{d+1}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$

with $d\geq 1$. The following statements are equivalent:

  1. (1)

The above sequence fits as the spine of a commutative diagram of the form

${c_{d+1}}$${\cdots}$${c_{3}}$${c_{2}}$${c_{d+2}}$${x_{d.5}}$${\cdots}$${x_{3.5}}$${x_{2.5}}$${c_{1}}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$

in which the oriented triangles are exact triangles in $\mathcal{T}$ and such
that the connecting morphism $c_{0}\to c_{d+2}[d]$ is given by the obvious
(shifted) composite along the bottom row of the diagram.

  2. (2)

The following two conditions are satisfied:

    1. (a)

For each $y\in\mathcal{C}$, the induced sequence of vector spaces

${\mathcal{T}(y,c_{d+2})}$${\cdots}$${\mathcal{T}(y,c_{2})}$${\mathcal{T}(y,c_{1})}$${\mathcal{T}(y,c_{d+2}[d])}$${\mathcal{T}(y,c_{d+1}[d])}$

is exact.

    2. (b)

We have

$\mathbf{1}_{c_{d+2}[d]}\in(-1)^{1+\sum_{i=1}^{d+1}i}\langle{f_{1},f_{2},\dots,f_{d+2}}\rangle.$

Computing higher Toda brackets can be rather intricate. We establish a few
computation rules that are needed for the proof of Theorem 4.1.12 as well as
elsewhere in the article.

######  Lemma 4.1.13.

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}$

and let $\varphi\colon x_{1}\to x_{1}^{\prime}$ be an isomorphism. Then, there
is an equality of Toda brackets

$\langle{f_{1},f_{2},\dots,f_{d+2}}\rangle=\langle{f_{1}\varphi^{-1},\varphi
f_{2},\dots,f_{d+2}}\rangle.$

###### Proof.

The claim is obvious for $d=0$ for we have

$\langle{f_{1},f_{2}}\rangle=\left\\{f_{1}\circ
f_{2}\right\\}=\left\\{(f_{1}\varphi^{-1})\circ(\varphi\circ
f_{2})\right\\}=\langle{f_{1}\varphi^{-1},\varphi f_{2}}\rangle.$

To prove the claim for $d\geq 1$ it is enough to show that

$\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)=\operatorname{T}\\!\left(f_{1}\varphi^{-1},\varphi
f_{2},f_{3}\right).$

Moreover, it is enough to prove that

$\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)\subseteq\operatorname{T}\\!\left(f_{1}\varphi^{-1},\varphi
f_{2},f_{3}\right)$

for the reverse induction is obtained by replacing $f_{1}$ and $f_{2}$ by
$f_{1}\varphi^{-1}$ and $\varphi f_{2}$, respectively, and $\varphi$ by its
inverse. Indeed, given a pair
$(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$ there is
a commutative diagram in $\mathcal{T}$ of the form

${x_{3}}$${x_{2}}$${x_{1}}$${x_{0}}$${c}$${x_{2}}$${x_{1}}$${c[1]}$${x_{3}}$${x_{2}}$${x_{1}^{\prime}}$${x_{0}}$${c}$${x_{2}}$${x_{1}^{\prime}}$${c[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{\varphi}$$\scriptstyle{u}$$\scriptstyle{f_{2}}$$\scriptstyle{v}$$\scriptstyle{\beta[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{\varphi
f_{2}}$$\scriptstyle{f_{1}\varphi^{-1}}$$\scriptstyle{u}$$\scriptstyle{\varphi
f_{2}}$$\scriptstyle{v\varphi^{-1}}$$\scriptstyle{\varphi}$$\scriptstyle{\beta[1]}$

in which the rows of the front face are exact triangles in $\mathcal{T}$ (the
top layer exists by assumption). The bottom face of the diagram exhibits the
membership
$(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1}\varphi^{-1},\varphi
f_{2},f_{3}\right)$, which is what we needed to prove. ∎

######  Notation 4.1.14.

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{3}\xrightarrow{f_{3}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}$

and let $S$ be a set of exact triangles of the form

$c\to x_{2}\xrightarrow{f_{2}}x_{1}\to c[1].$

We let
$\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}\subseteq\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$
be the subset of all pairs $(\beta,\alpha)$ that are part of a commutative
diagram of the form (4.1.3) in which the bottom horizontal row belongs to the
set $S$.

######  Lemma 4.1.15.

Let $d\geq 1$ and suppose given a sequence of morphisms in $\mathcal{T}$ of
the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

Let $S$ be a non-empty class of triangles with middle morphism $f_{2}$ as in
Notation 4.1.14. Then, in the definition of the Toda bracket
$\langle{f_{1},\dots,f_{d+2}}\rangle$, we can replace
$\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$ by its subset
$\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}$.

###### Proof.

Given an arbitrary exact triangle in $\mathcal{T}$ of the form

$c^{\prime}\xrightarrow{u^{\prime}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{v^{\prime}}c^{\prime}[1],$

there exists an isomorphism of exact triangles

${c^{\prime}}$${x_{2}}$${x_{1}}$${c^{\prime}[1]}$${c}$${x_{2}}$${x_{1}}$${c[1]}$$\scriptstyle{u^{\prime}}$$\scriptstyle{\varphi}$$\scriptstyle{f_{2}}$$\scriptstyle{v^{\prime}}$$\scriptstyle{\varphi[1]}$$\scriptstyle{u}$$\scriptstyle{f_{2}}$$\scriptstyle{v}$

where the bottom row is an exact triangle in $S$ (chosen arbitrarily). Thus,
given a commutative diagram in $\mathcal{T}$ of the form

${x_{3}}$${x_{2}}$${x_{1}}$${x_{0}}$${c^{\prime}}$${x_{2}}$${x_{1}}$${c^{\prime}[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{u^{\prime}}$$\scriptstyle{f_{2}}$$\scriptstyle{v^{\prime}}$$\scriptstyle{\beta[1]}$

there exists an extended commutative diagram

${x_{3}}$${x_{2}}$${x_{1}}$${x_{0}}$${c^{\prime}}$${x_{2}}$${x_{1}}$${c^{\prime}[1]}$${x_{3}}$${x_{2}}$${x_{1}}$${x_{0}}$${c}$${x_{2}}$${x_{1}}$${c[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{u^{\prime}}$$\scriptstyle{\varphi}$$\scriptstyle{f_{2}}$$\scriptstyle{v^{\prime}}$$\scriptstyle{\beta[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\varphi\alpha}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{u}$$\scriptstyle{f_{2}}$$\scriptstyle{v}$$\scriptstyle{(\beta\varphi^{-1})[1]}$$\scriptstyle{\varphi[1]}$

Finally, since by Lemma 4.1.13 there is an equality of Toda brackets

$\langle{\beta[1],\alpha[1],f_{4}[1],\dots,f_{d+2}[1]}\rangle=\langle{(\beta\varphi^{-1})[1],(\varphi\alpha)[1],f_{4}[1],\dots,f_{d+2}[1]}\rangle,$

we see that we may go without of the pair
$(\beta,\alpha)\in\mathcal{T}(f_{1},f_{2},f_{3})$ by considering the pair
$((\beta\varphi^{-1})[1],(\varphi\alpha)[1])\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}$
instead. This finishes the proof. ∎

We now describe the action of the shift functor on the Toda brackets.

######  Definition 4.1.16 ([CF17, p. 2709]).

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{3}\xrightarrow{f_{3}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

The _negative Toda family
$-\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$_ is the set

$-\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)\coloneqq\left\\{(\beta,-\alpha)\right|(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)\\}$

(notice the unusual convention for the negative of a set).

######  Remark 4.1.17.

In the setting of Definition 4.1.16, notice that

$-\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)=\operatorname{T}\\!\left(f_{1},-f_{2},f_{3}\right).$

Indeed, given a diagram of the form (4.1.3) exhibiting the membership relation
$(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$, there is
a commutative diagram in $\mathcal{T}$ of the form

${x_{3}}$${x_{2}}$${x_{1}}$${x_{0}}$${c}$${x_{2}}$${x_{1}}$${c[1]}$${x_{3}}$${x_{2}}$${x_{1}}$${x_{0}}$${c}$${x_{2}}$${x_{1}}$${c[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{-1}$$\scriptstyle{f_{2}}$$\scriptstyle{-1}$$\scriptstyle{f_{1}}$$\scriptstyle{u}$$\scriptstyle{f_{2}}$$\scriptstyle{v}$$\scriptstyle{\beta[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{-\alpha}$$\scriptstyle{-f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{-u}$$\scriptstyle{-f_{2}}$$\scriptstyle{-1}$$\scriptstyle{v}$$\scriptstyle{\beta[1]}$

that exhibits the pair
$(\beta,-\alpha)\in-\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$ as a
member of $\operatorname{T}\\!\left(f_{1},-f_{2},f_{3}\right)$. The converse
is then also clear.

######  Lemma 4.1.18.

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

Then, there is an equality of Toda brackets

$-\langle{f_{1},f_{2},f_{3},\dots,f_{d+2}}\rangle=\langle{f_{1},-f_{2},f_{3},\dots,f_{d+2}}\rangle.$

###### Proof.

We proceed by induction on the number $n=2$ of morphisms in the sequence. If
$n=2$, then the claim is obvious. Suppose then that the claim holds for all
sequences of $n$ composable morphisms in $\mathcal{T}$ for some $n\geq 2$. Let
$f_{1},\dots,f_{n},f_{n+1}$ be a sequence of composable morphisms in
$\mathcal{T}$. Then, in view of Remark 4.1.17 and the inductive hypothesis,

$\displaystyle\langle{f_{1},-f_{2},f_{3},\dots,f_{n},f_{n+1}}\rangle$
$\displaystyle=\bigcup_{(\beta,\gamma)\in\operatorname{T}\\!\left(f_{1},-f_{2},f_{3}\right)}\langle{\beta[1],\gamma[1],f_{4}[1],\dots,f_{n}[1],f_{n+1}[1]}\rangle$
$\displaystyle=\bigcup_{(\beta,\gamma)\in-\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}\langle{\beta[1],\gamma[1],f_{4}[1],\dots,f_{n}[1],f_{n+1}[1]}\rangle$
$\displaystyle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}\langle{\beta[1],-\alpha[1],f_{4}[1],\dots,f_{n}[1],f_{n+1}[1]}\rangle$
$\displaystyle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}-\langle{\beta[1],\alpha[1],f_{4}[1],\dots,f_{n}[1],f_{n+1}[1]}\rangle$
$\displaystyle=-\langle{f_{1},f_{2},f_{3},\dots,f_{n},f_{n+1}}\rangle.$

The claim follows. ∎

######  Notation 4.1.19.

Given a set $X$ of morphisms in $\mathcal{T}$, we let

$X[1]\coloneqq\left\\{f[1]\right|f\in X\\},$

and similarly for sets of pairs of morphisms in $\mathcal{T}$ (we are thinking
of Toda families as in Definition 4.1.2).

######  Lemma 4.1.20 ([CF17, Lemma 5.12]).

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{3}\xrightarrow{f_{3}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

Then, there is an equality

$\operatorname{T}\\!\left(f_{1}[1],f_{2}[1],f_{3}[1]\right)=-\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)[1].$

######  Lemma 4.1.21.

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

Then, the following relation between Toda brackets holds:

$\langle{f_{1},f_{2},\dots,f_{d+2}}\rangle[1]=(-1)^{d}\langle{f_{1}[1],f_{2}[1],f_{3}[1],\dots,f_{d+2}[1]}\rangle.$

###### Proof.

We proceed by induction on the number $n$ of morphisms in the sequence. If
$n=2$, then the claim follows from the functoriality of the shift:

$\langle{f_{1},f_{2}}\rangle[1]=\left\\{f_{1}\circ
f_{2}\right\\}{[1]}=\left\\{(f_{1}\circ
f_{2})[1]\right\\}=\left\\{f_{1}[1]\circ
f_{2}[1]\right\\}=\langle{f_{1}[1],f_{2}[1]}\rangle.$

Suppose that the claim holds for all sequences of $n$ composable morphisms in
$\mathcal{T}$ for some $n\geq 2$. Let $f_{1},\dots,f_{n},f_{n+1}$ be a
sequence of composable morphisms in $\mathcal{T}$. Then, in view of Lemmas
4.1.18 and 4.1.20 and the inductive hypothesis, the Toda bracket

$\langle{f_{1},f_{2},\dots,f_{n},f_{n+1}}\rangle[1]$

equals

$\displaystyle\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}\langle{\beta[1],\alpha[1],f_{4}[1],\dots,f_{n}[1],f_{n+1}[1]}\rangle[1]$
$\displaystyle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}(-1)^{n-2}\langle{\beta[2],\alpha[2],f_{4}[2],\dots,f_{n}[2],f_{n+1}[2]}\rangle$
$\displaystyle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}(-1)^{n-1}\langle{\beta[2],-\alpha[2],f_{4}[2],\dots,f_{n}[2],f_{n+1}[2]}\rangle$
$\displaystyle=\bigcup_{(\beta,-\alpha)\in-\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}(-1)^{n-1}\langle{\beta[2],-\alpha[2],f_{4}[2],\dots,f_{n}[2],f_{n+1}[2]}\rangle$
$\displaystyle=\bigcup_{(\beta[1],-\alpha[1])\in-\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)[1]}(-1)^{n-1}\langle{\beta[2],-\alpha[2],f_{4}[2],\dots,f_{n}[2],f_{n+1}[2]}\rangle$
$\displaystyle=\bigcup_{(\beta[1],-\alpha[1])\in\operatorname{T}\\!\left(f_{1}[1],f_{2}[1],f_{3}[1]\right)}(-1)^{n-1}\langle{\beta[2],-\alpha[2],f_{4}[2],\dots,f_{n}[2],f_{n+1}[2]}\rangle$
$\displaystyle=(-1)^{n-1}\langle{f_{1}[1],\dots,f_{n}[1],f_{n+1}[1]}\rangle.$

The claim follows. ∎

We combine the above results into the following equivalent way to compute the
Toda bracket of a sequence of morphisms.

######  Proposition 4.1.22.

Let $d\geq 1$ and suppose given a sequence of morphisms in $\mathcal{T}$ of
the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}.$

Then, there is an equality

$\langle{f_{1},\dots,f_{d+2}}\rangle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}(-1)^{d-1}\langle{\beta,\alpha,f_{4},\dots,f_{d+2}}\rangle[1].$

###### Proof.

According to Definition 4.1.6 and Lemma 4.1.21, there are equalities

$\displaystyle\langle{f_{1},\dots,f_{d+2}}\rangle$
$\displaystyle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}\langle{\beta[1],\alpha[1],f_{4}[1],\cdots,f_{d+2}[1]}\rangle$
$\displaystyle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}(-1)^{d-1}\langle{\beta,\alpha,f_{4},\dots,f_{d+2}}\rangle[1].\qed$

######  Proposition 4.1.23.

Let $d\geq 1$ and suppose given a sequence of morphisms in $\mathcal{T}$ of
the form

${x_{d+2}}$${x_{d+1}}$${\cdots}$${x_{2}}$${x_{1}}$${x_{d+2}[d].}$$\scriptstyle{f_{d+2}}$$\scriptstyle{f_{d+1}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$

Suppose that the above sequence fits as the spine of a commutative diagram of
the form

${x_{d+1}}$${\cdots}$${x_{3}}$${x_{2}}$${x_{d+2}}$${x_{d.5}}$${\cdots}$${x_{3.5}}$${x_{2.5}}$${x_{1}}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$

in which the oriented triangles are exact triangles in $\mathcal{T}$ and the
morphism $f_{1}$ is the apparent (shifted) composite along the bottom row of
the diagram. Then,

$\mathbf{1}_{x_{d+2}[d]}\in(-1)^{1+\sum_{i=1}^{d+1}i}\langle{f_{1},f_{2},\dots,f_{d+2}}\rangle.$

###### Proof.

We proceed by induction on $d$. The case $d=1$ follows from Theorem 4.1.11.
Suppose then that the claim holds for $d\geq 1$ and consider a sequence of
morphisms $f_{1},f_{2},\dots,f_{d+2},f_{d+3}$ that satisfies the conditions in
the statement of the proposition. Thus, there are exact triangles

${x_{i+1.5}}$${x_{i+1}}$${x_{i.5}}$${x_{i+1.5}[1],\qquad{i=1,\dots,d+1},}$$\scriptstyle{h_{i+1}}$$\scriptstyle{g_{i+1}}$$\scriptstyle{u_{i}}$

such that the triangles

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commute, where we set $x_{1.5}\coloneqq x_{1}$ and $x_{d+2.5}\coloneqq
x_{d+3}$, so that also $g_{2}=f_{2}$ and $h_{d+2}=f_{d+3}$. We also set

$f_{1}^{(d)}=(u_{d+1}[d-1])\circ\cdots\circ(u_{3}[1])\circ u_{2}$

and notice that $f_{1}^{(d)}\colon x_{2.5}\to x_{d+3}[d]$ and
$f_{1}=(f_{1}^{(d)}[1])\circ u_{1}$. Having fixed the necessary notation, the
induction hypothesis implies that

$\mathbf{1}_{x_{d+3}[d]}\in(-1)^{1+\sum_{i=1}^{d+1}i}\langle{f_{1}^{(d)},g_{3},f_{4},\dots,f_{d+2},f_{d+3}}\rangle.$

We claim that
$(f_{1}^{(d)},g_{3})\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$ and
therefore, according to Proposition 4.1.22,

$\displaystyle\langle{f_{1},f_{2},\dots,f_{d+2},f_{d+3}}\rangle$
$\displaystyle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}(-1)^{d}\langle{\beta,\alpha,f_{3},\dots,f_{d+2},f_{d+3}}\rangle[1]$
$\displaystyle\supseteq(-1)^{d}\langle{f_{1}^{(d)},g_{3},f_{4},\dots,f_{d+2},f_{d+3}}\rangle[1]$
$\displaystyle\ni(-1)^{d+2}((-1)^{1+\sum_{i=1}^{d+1}i}\mathbf{1}_{x_{d+3}[d]})[1]$
$\displaystyle=(-1)^{1+\sum_{i=1}^{d+2}i}\mathbf{1}_{x_{d+3}[d+1]}.$

To prove the claim, it suffices observe that the following diagram commutes:

${x_{3}}$${x_{2}}$${x_{1}}$${x_{d+3}[d+1]}$${x_{2.5}}$${x_{2}}$${x_{1}}$${x_{2.5}[1].}$$\scriptstyle{f_{3}}$$\scriptstyle{g_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{h_{2}}$$\scriptstyle{f_{2}}$$\scriptstyle{u_{1}}$$\scriptstyle{f_{1}^{(d)}[1]}$

This finishes the proof. ∎

We also need the following lemma.

######  Lemma 4.1.24.

Suppose given a sequence of morphisms in $\mathcal{T}$ of the form

$x_{3}\xrightarrow{f_{3}}x_{2}\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{3}[1].$

The following statements are equivalent:

  1. (1)

We have $\mathbf{1}_{x_{3}[1]}\in\langle{f_{1},f_{2},f_{3}}\rangle$.

  2. (2)

There exist an object $y\in\mathcal{T}$ and morphisms $g\colon y\to x_{2}$ and
$h\colon x_{1}\to y[1]$ such that the sequence

${x_{3}\oplus{y}}$${x_{2}}$${x_{1}}$${(x_{3}\oplus{y})[1]}$$\scriptstyle{\left(\begin{smallmatrix}f_{3}&g\end{smallmatrix}\right)}$$\scriptstyle{f_{2}}$$\scriptstyle{\left(\begin{smallmatrix}f_{1}\\\
h\end{smallmatrix}\right)}$

is an exact triangle in $\mathcal{T}$.

###### Proof.

By definition, $\mathbf{1}_{x_{2}[1]}\in\langle{f_{0},f_{1},f_{2}}\rangle$ if
and only if there exists a commutative diagram in $\mathcal{T}$ of the form

${x_{3}}$${x_{2}}$${x_{1}}$${x_{3}[1]}$${c}$${x_{2}}$${x_{1}}$${c[1]}$$\scriptstyle{f_{3}}$$\scriptstyle{\alpha}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{u}$$\scriptstyle{f_{2}}$$\scriptstyle{v}$$\scriptstyle{\beta[1]}$

in which the bottom row is an exact triangle in $\mathcal{T}$ and
$(\beta\alpha)[1]=\mathbf{1}_{x_{3}[1]}$ or, equivalently
$\beta\alpha=\mathbf{1}_{x_{3}}$. In particular, $c\cong x_{3}\oplus y$ for
some object $y\in\mathcal{T}$. If we identify $c$ with $x_{3}\oplus y$ we see
that the morphism $u$ has components $f_{3}\colon x_{3}\to x_{2}$ and $g\colon
y\to x_{2}$ for some morphism $g$. Similarly, under the identification
$c[1]=(x_{3}\oplus y)[1]=x_{3}[1]\oplus y[1]$ the morphism $v$ has components
$f_{1}\colon x_{1}\to x_{3}[1]$ and $h\colon x_{1}\to y[1]$ for some morphism
$h$. The claim follows. ∎

In the proof of Theorem 4.1.12 we will use elementary properties of Poincaré
polynomials of periodic graded modules.

######  Definition 4.1.25.

Let $A$ be a graded algebra with a degree $d$ unit. The _Poincaré polynomial_
of a degree-wise finite-dimensional $A$-module $M$ is

$p(M)=\sum_{i=0}^{d-1}\dim_{\mathbf{k}}M^{i}\cdot
t^{i}\in\mathbb{Z}[t]/(t^{d}-1).$

We now show some elementary properties of these Poincaré polynomials. All
proofs are simple, albeit somewhat tedious, exercises, so we only sketch them.

######  Proposition 4.1.26.

Given a graded algebra $A$ with a degree $d$ unit $u\in A^{d}$, the Poincaré
polynomials of degree-wise finite-dimensional $A$-modules satisfy the
following properties:

  1. (1)

$p(M\oplus N)=p(M)+p(N)$.

  2. (2)

Given a short exact sequence $0\to M\to N\to P\to 0$, $p(N)=p(M)+p(P)$.

  3. (3)

$p(M)=0\Leftrightarrow M=0$.

  4. (4)

$p(M)\in\mathbb{Z}$ if and only if $M^{i}=0$ for $i\notin d\mathbb{Z}$.

  5. (5)

If $p(M)\in\mathbb{Z}$ and $N\subset M$ then $p(N)\in\mathbb{Z}$.

  6. (6)

$p(M(-1))=t\cdot p(M)$.

  7. (7)

$p(M(d))=p(M)$.

  8. (8)

Given an exact sequence

${N}$${M}$${P}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{+1}$$\scriptstyle{h}$

if $K=\ker g=\operatorname{im}f$ then

$p(M)+tp(N)-tp(P)=(t+1)p(K).$

###### Proof.

The first five properties are obvious. For (6), we use the following three
facts:

  * •

$M(-1)^{i}=M^{i-1}$,

  * •

multiplication by $u$ induces an isomorphism $M(-1)^{0}=M^{-1}\cong M^{d-1}$,

  * •

$t^{d}=1$ in $\mathbb{Z}[t]/(t^{d}-1)$.

Property (7) is a direct consequence of (6). For (8), we break the exact
sequence

$0\to K(-1)\to N(-1)\stackrel{{\scriptstyle
g}}{{\to}}P(-1)\stackrel{{\scriptstyle h}}{{\to}}M\to K\to 0$

into three short exact sequences and apply (2) and (6). ∎

######  Proposition 4.1.27.

Let $d\geq 1$. Given a graded algebra $A$ with a degree $d$ unit and an exact
sequence of degree-wise finite-dimensional $A$-modules,

${C_{d+1}}$${\cdots}$${C_{2}}$${C_{d+2}}$${C_{1}}$$\scriptstyle{f_{d+1}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{d+2}}$$\scriptstyle{+d}$$\scriptstyle{f_{d+2}}$

if $K=\ker f_{2}$ then the following relation between Poincaré polynomials
holds,

$(-1)^{d}\left(p(C_{2})-p(C_{1})\right)+\sum_{i=3}^{d+2}(-1)^{i}p(C_{i})=\left((-1)^{d}-1\right)p(K).$

###### Proof.

Consider $K_{i}=\ker f_{i}$ and the short exact sequences

$\begin{array}[]{rr}0\to K_{i}\to C_{i}\to K_{i-1}\to 0,&2\leq i\leq d+2,\\\
0\to K_{1}\to C_{1}\to K_{d+2}(d)\to 0.\end{array}$

Now it is just a matter of applying (2) and (7) from Proposition 4.1.26. ∎

######  Proposition 4.1.28.

Let $d\geq 1$. Given a graded algebra $A$ with a degree $d$ unit and exact
sequences of degree-wise finite-dimensional $A$-modules,

${M_{d+1}}$${M_{d}}$${\cdots}$${M_{2}}$${M_{d+2}}$${X_{d.5}}$${X_{d-1.5}}$${\cdots}$${X_{2.5}}$${M_{1}}$$\scriptstyle{g_{d+1}}$$\scriptstyle{g_{d}}$$\scriptstyle{g_{2}}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$

if $K_{i}=\ker g_{i}$ then

$p(M_{2})+p(M_{d+2})-p(M_{1})+\sum_{i=3}^{d+1}p(M_{i})t^{d+2-i}=\\\
p(K_{2})+p(K_{d+1})+\sum_{i=3}^{d+1}\left(p(K_{i})+p(K_{i-1})\right)t^{d+2-i}$

###### Proof.

This follows from Proposition 4.1.26 (8). ∎

######  Corollary 4.1.29.

In the setting of Proposition 4.1.28, if the $M_{i}$ are concentrated in
degrees $d\mathbb{Z}$ then

$(-1)^{d}\left(p(M_{2})-p(M_{1})\right)+\sum_{i=3}^{d+2}(-1)^{i}p(M_{i})=\left((-1)^{d}-1\right)p(K_{2}).$

###### Proof.

By Proposition 4.1.26 (4) and (5), $p(M_{i}),p(K_{i})\in\mathbb{Z}$. In the
polynomial equation of Proposition 4.1.28, both polynomials must have the same
coefficients in the same degrees. This translates into $d$ equations involving
$p(M_{i}),p(K_{i})\in\mathbb{Z}$. From these equations, it is easy to deduce
the equation in the statement. ∎

We are ready to prove Theorem 4.1.12.

###### Proof of Theorem 4.1.12.

Let $\mathcal{C}\subseteq\mathcal{T}$ be a $d$-rigid subcategory such that
$\operatorname{add}(\mathcal{C})=\mathcal{C}$ and
$\mathcal{C}[d]=\mathcal{C}$, and

(4.1.30)
${c_{d+2}}$${c_{d+1}}$${\cdots}$${c_{2}}$${c_{1}}$${c_{d+2}[d]}$$\scriptstyle{f_{d+2}}$$\scriptstyle{f_{d+1}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$

a sequence of morphisms in $\mathcal{C}$ with $d\geq 1$.

(1)$\Rightarrow$(2) Suppose that the sequence (4.1.30) satisfies the
conditions in Theorem 4.1.12(1). The fact, for each $y\in\mathcal{C}$, the
induced sequence of vector spaces

$\mathcal{T}(y,c_{d+2})\to\cdots\to\mathcal{T}(y,c_{2})\to\mathcal{T}(y,c_{1})\to\mathcal{T}(y,c_{d+2}[d])\to\mathcal{T}(y,c_{d+1}[d])$

is exact follows from standard arguments using that
$\mathcal{C}\subseteq\mathcal{T}$ is a $d$-rigid subcategory, see for example
the first paragraph in the proof of [Lin19, Lemma 4.6]. That

$\mathbf{1}_{c_{d+2}[d]}\in(-1)^{1+\sum_{i=1}^{d+1}}\langle{f_{1},f_{2},\dots,f_{d+2}}\rangle$

follows from Proposition 4.1.23.

(2)$\Rightarrow$(1) If $d=1$, then we are precisely in the setting of Theorem
4.1.11; hence, we may assume that $d>1$. Inductively, we shall construct a
commutative diagram

${c_{d+1}}$${\cdots}$${c_{3}}$${c_{2}}$${c_{d+2}}$${x_{d.5}}$${\cdots}$${x_{3.5}}$${x_{2.5}}$${c_{1}}$$\scriptstyle{g_{d+1}}$$\scriptstyle{g_{3}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{d+2}}$$\scriptstyle{u_{d}}$$\scriptstyle{+1}$$\scriptstyle{h_{3}}$$\scriptstyle{h_{2}}$$\scriptstyle{u_{2}}$$\scriptstyle{+1}$$\scriptstyle{u_{1}}$$\scriptstyle{+1}$

with the following properties:

  * •

The triangles

$x_{k.5}\xrightarrow{h_{k}}c_{k}\xrightarrow{g_{k}}x_{k-1.5}\xrightarrow{u_{k-1}}x_{k.5}[1],\qquad
2\leq k\leq d,$

are exact, where $x_{1.5}=c_{1}$ and $p_{2}=f_{2}$;

  * •

We have $\mathbf{1}_{c_{d+2}[1]}\in\langle{u_{d},g_{d+1},f_{d+2}}\rangle$,
although the triangle

$c_{d+2}\xrightarrow{f_{d+2}}c_{d+1}\xrightarrow{g_{d+1}}x_{d.5}\xrightarrow{u_{d}}c_{d+2}[1]$

is not (yet) known to be exact.

  * •

There are equalities $f_{k}=h_{k-1}\circ g_{k}$, for all $3\leq k\leq d+1$.

  * •

There is an equality $f_{1}=u_{d}[d-1]\circ\cdots\circ u_{2}[1]\circ u_{1}$.

In the first step we notice that, since the morphism
$(-1)^{1+\sum_{i=1}^{d+1}i}\mathbf{1}_{c_{d+2}[d]}$ lies in

$\langle{f_{1},f_{2},\dots,f_{d+2}}\rangle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)}(-1)^{d-1}\langle{\beta,\alpha,f_{4},\dots,f_{d+2}}\rangle[1],$

there exist a pair of morphisms
$(f_{1}^{(d)},g_{3})\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$ such
that

$(-1)^{1+\sum_{i=1}^{d+1}i}\mathbf{1}_{c_{d+2}[d]}\in(-1)^{d-1}\langle{f_{1}^{(d)},g_{3},f_{4},\dots,f_{d+2}}\rangle[1]$

or, equivalently,

$(-1)^{1+\sum_{i=1}^{d}i}\mathbf{1}_{c_{d+2}[d-1]}\in\langle{f_{1}^{(d)},g_{3},f_{4},\dots,f_{d+2}}\rangle.$

In particular, there exists a diagram of the form

${c_{2}}$${c_{1}}$${c_{3}}$${x_{2.5}}$${c_{d+2}[d]}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{u_{1}}$$\scriptstyle{+1}$$\scriptstyle{f_{3}}$$\scriptstyle{g_{3}}$$\scriptstyle{f_{1}^{(d)}}$$\scriptstyle{h_{2}}$$\scriptstyle{-1}$

in which the oriented triangle is exact. Thus, we obtain a diagram of the form

${c_{d+1}}$${\cdots}$${c_{3}}$${c_{2}}$${c_{d+2}}$${x_{2.5}}$${c_{1}}$$\scriptstyle{f_{d+1}}$$\scriptstyle{f_{4}}$$\scriptstyle{g_{3}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{d+2}}$$\scriptstyle{h_{2}}$$\scriptstyle{f_{1}^{(d)}}$$\scriptstyle{+(d-1)}$$\scriptstyle{u_{1}}$$\scriptstyle{+1}$

We may then continue the inductive procedure with the sequence
$f_{1}^{(d)},g_{3},f_{4},\dots,f_{d+2}$

$(-1)^{1+(1+2)}\mathbf{1}_{c_{d+2}[1]}=\mathbf{1}_{c_{d+2}[1]}\in\langle{u_{d},g_{d+1},f_{d+2}}\rangle,$

as required). It remains to prove that the sequence

$c_{d+2}\xrightarrow{f_{d+2}}c_{d+1}\xrightarrow{g_{d+1}}x_{d.5}\xrightarrow{u_{d}}c_{d+2}[1]$

is an exact triangle. Since
$\mathbf{1}_{c_{d+2}[1]}\in\langle{u_{d},g_{d+1},f_{d+2}}\rangle$, Lemma
4.1.24 shows that there exists an object $y\in\mathcal{T}$ and an exact
triangle of the form

$y\oplus
c_{d+2}\xrightarrow{\left(\begin{smallmatrix}f_{d+2}&g\end{smallmatrix}\right)}c_{d+1}\xrightarrow{g_{d+1}}x_{d.5}\xrightarrow{\left(\begin{smallmatrix}u_{d}\\\
h\end{smallmatrix}\right)}(y\oplus c_{d+2})[1].$

We need to prove that $y=0$; since
$\operatorname{thick}(c)=\operatorname{thick}(\mathcal{C})=\mathcal{T}$ it
suffices to prove that $\mathcal{T}(c[i],y)=0$ for all $i\in\mathbb{Z}$.
Moreover, since $\mathcal{C}[d]=\mathcal{C}=\operatorname{add}(c)$ and $c$ is
basic, $c[d]\cong c$ and the graded endomorphism algebra $A$ of $c$ in
$\mathcal{T}$ is of the form $\Lambda(\sigma,d)$ in Section 4.5.1, so it has a
degree $d$ unit. Moreover,
$M_{x}=\bigoplus_{i\in\mathbb{Z}}\mathcal{T}(c[i],x)$ is a degree-wise finite-
dimensional $A$-module for any $x\in\mathcal{T}$. We have to show that
$M_{y}=0$. We will denote $p(x)=p(M_{x})$, and we will show that $p(y)=0$,
which suffices by Proposition 4.1.26 (3).

Applying $\bigoplus_{i\in\mathbb{Z}}\mathcal{T}(c[i],-)$ to

${c_{d+1}}$${\cdots}$${c_{2}}$${c_{d+2}}$${c_{1}}$$\scriptstyle{f_{d+1}}$$\scriptstyle{f_{3}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{d+2}}$$\scriptstyle{+d}$$\scriptstyle{f_{d+2}}$

and

${c_{d+1}}$${c_{d}}$${\cdots}$${c_{2}}$${y\oplus
c_{d+2}}$${x_{d.5}}$${x_{d-1.5}}$${\cdots}$${x_{2.5}}$${c_{1}}$$\scriptstyle{g_{d+1}}$$\scriptstyle{g_{d}}$$\scriptstyle{f_{2}}$$\scriptstyle{\left(\begin{smallmatrix}f_{d+2}&g\end{smallmatrix}\right)}$$\scriptstyle{h_{d}}$$\scriptstyle{+1}$$\scriptstyle{\left(\begin{smallmatrix}u_{d+2}\\\
h\end{smallmatrix}\right)}$$\scriptstyle{+1}$$\scriptstyle{u_{d-1}}$$\scriptstyle{h_{2}}$$\scriptstyle{+1}$$\scriptstyle{u_{1}}$

we obtain diagrams of $A$-modules as in Proposition 4.1.27 and Proposition
4.1.28. By Proposition 4.1.27, if
$K=\ker\bigoplus_{i\in\mathbb{Z}}\mathcal{T}(c[i],f_{2})$,

$(-1)^{d}\left(p(c_{2})-p(c_{1})\right)+\sum_{i=3}^{d+2}(-1)^{i}p(c_{i})=\left((-1)^{d}-1\right)p(K).$

Moreover, since $c_{i}\in\mathcal{C}$ and $\mathcal{C}$ is $d$-rigid,
$M_{c_{i}}$ is concentrated in $d\mathbb{Z}$. Hence, by Corollary 4.1.29,

$(-1)^{d}\left(p(c_{2})-p(c_{1})\right)+\sum_{i=3}^{d+1}(-1)^{i}p(c_{i})+p(y)+p(c_{d+2})=\left((-1)^{d}-1\right)p(K).$

Both equations are identical, except for the fact that the second one has an
extra term, $p(y)$. Therefore, $p(y)=0$, as we wanted to prove. ∎

We conclude this section with the following result.

######  Proposition 4.1.31 ([Sag08, Prop. 4.10]).

Let $\mathcal{C}\subseteq\mathcal{T}$ a $d$-rigid subcategory such that
$\operatorname{add}(\mathcal{C})=\mathcal{C}$ and
$\mathcal{C}[d]=\mathcal{C}$. Suppose given a sequence of morphisms in
$\mathcal{C}$ of the form

$c_{d+2}\xrightarrow{f_{d+2}}c_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}c_{1}\xrightarrow{f_{1}}c_{0}$

Then, the Toda bracket $\langle{f_{1},\dots,f_{d+2}}\rangle$ is an element of
the quotient

$\operatorname{Toda}\left(f_{d+2},f_{1}\right)\coloneqq\frac{\mathcal{T}(c_{d+2}[d],c_{0})}{f_{1}\cdot\mathcal{T}(c_{d+2}[d],c_{1})+\mathcal{T}(c_{d+1}[d],c_{0})\cdot
f_{d+2}[d]}.$

###  4.2. Massey products versus Toda brackets

Let $\mathcal{A}$ be a DG category. We recall that, similarly to the
cohomology of a space, the graded category $H^{\bullet}(\mathcal{A})$ is
endowed with so-called Massey products [BK90, Sec. 5.A]. Massey products are
closely related to Toda brackets. In fact, it is stated in _loc. cit._ that

> “It can be verified that for an enhanced triangulated category the Massey
> products coincide with the Toda brackets.”

As it turns out, this statement indeed holds, but only up to a suitable sign.
Since this agreement is crucial to our main results, we include a proof for
the convenience of the reader (see also [Bod14] for an alternative proof that
involves the language of twisted complexes).

We begin by recalling the definition of the Massey products, see for example
[May69].

######  Definition 4.2.1.

Let $\mathcal{A}$ be a DG category and suppose given a sequence in
$H^{\bullet}(\mathcal{A})$ of the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}$

such that $f_{i}$ is homogeneous of degree $|f_{i}|$, where $d\geq 0$.

  1. (1)

A _defining system (for the above sequence)_ is a set

$\left\\{g_{ij}\colon x_{j}\to x_{i}\right|0\leq i<j\leq d+2,\ j-i<d+2\\}$

of morphisms in $\mathcal{A}$ that are homogeneous of degree

$|g_{ij}|=i-j+1+\sum_{i<k\leq j}|f_{k}|$

with the following two properties:

    1. (a)

The equality

$\partial(g_{ij})=\sum_{i<k<j}(-1)^{|g_{ik}|-1}g_{ik}g_{kj}$

holds (notice that $|g_{ik}|-1=i-k+\sum_{i<\ell\leq k}|f_{\ell}|$). In
particular,

$\partial(g_{i-1,i})=0$

so that $g_{i-1,i}$ is a cocycle of degree $|f_{i}|$.

    2. (b)

The morphism $g_{i-1,i}$ represents $f_{i}$ in $H^{\bullet}(\mathcal{A})$.

  2. (2)

The _Massey product_ is the subset of

$\llangle{f_{1},\dots,f_{d+2}}\rrangle\subseteq
H^{-d+\sum_{i=1}^{d+2}|f_{i}|}(\mathcal{A}(x_{d+2},x_{0}))$

consisting of the classes of the cocycles

$\sum_{0<k<d+2}(-1)^{|g_{0k}|-1}g_{0k}g_{k,d+2}=\sum_{0<k<d+2}(-1)^{-k+\sum_{0<\ell\leq
k}|f_{\ell}|}g_{0k}g_{k,d+2}$

for all possible defining systems.

######  Remark 4.2.2.

In the setting of Definition 4.2.1, the degree of the Massey product
$\llangle{f_{1},\dots,f_{d+2}}\rrangle$ is more recognisable by letting
$n=d+2$ so that $\llangle{f_{1},\dots,f_{n}}\rrangle$ is a subset of the
cohomology of $\mathcal{A}(x_{n},x_{0})$ in degree $2-n+\sum_{i}|f_{i}|$.

######  Remark 4.2.3.

The Massey product $\llangle{f_{1},f_{2}}\rrangle$ of a pair of (homogeneous)
composable morphisms in $H^{\bullet}(\mathcal{A})$ is always defined and is
given by the singleton

$\llangle{f_{1},f_{2}}\rrangle=\left\\{(-1)^{|f_{1}|-1}f_{1}\circ
f_{2}\right\\}.$

We see that, if $|f_{1}|=0$, the Massey product of $f_{1}$ and $f_{2}$ is the
sign-twisted composite $-f_{1}\circ f_{2}$ (compare with Definition 4.1.6 and
Theorem 4.2.6).

######  Remark 4.2.4.

In the setting of Definition 4.2.1, if $d\geq 1$, then the Massey product is
defined (i.e. non-empty) if and only if

$\forall 0\leq i<j\leq d+2,\;j-i<d+2,\qquad
0\in\llangle{f_{i},\dots,f_{j}}\rrangle.$

######  Remark 4.2.5.

Let $\mathcal{A}$ be a small DG category. Suppose given a sequence of degree
$0$ morphisms

$M_{d+2}\xrightarrow{f_{d+2}}M_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}M_{1}\xrightarrow{f_{1}}M_{0}$

in the triangulated category
$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$. By definition, their
Massey product satisfies

$\llangle{f_{1},\dots,f_{d+2}}\rrangle\subseteq
H^{-d}(\operatorname{hom}_{\mathcal{A}}\\!\left(M_{d+2},M_{0}\right))\cong
H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))(M_{d+2},M_{0}[-d]),$

while their Toda bracket satisfies

$\langle{f_{1},\dots,f_{d+2}}\rangle\subseteq
H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))(M_{d+2}[d],M_{0}).$

Of course, we may compare both brackets by means of the isomorphism

$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))(M_{d+2},M_{0}[-d])\cong
H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))(M_{d+2}[d],M_{0})$

induced by the $d$-fold shift functor.

We shall prove the following agreement result for Massey products and Toda
brackets.

######  Theorem 4.2.6.

Let $\mathcal{A}$ be a small DG category. Suppose given a sequence of degree
$0$ morphisms

$M_{d+2}\xrightarrow{f_{d+2}}M_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}M_{1}\xrightarrow{f_{1}}M_{0}$

in the triangulated category
$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$. Then, their Massey
product and their Toda bracket are related by the formula

$\llangle{f_{1},\dots,f_{d+2}}\rrangle[d]=(-1)^{\sum_{i=1}^{d+1}i}\langle{f_{1},\dots,f_{d+2}}\rangle.$

######  Remark 4.2.7.

Let $\mathcal{A}$ be a small DG category. Since
$\operatorname{D}(\mathcal{A})\subseteq
H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$ is a full triangulated
subcategory, Theorem 4.2.6 also establishes a relationship between Massey
products and Toda brackets in $\operatorname{D}(\mathcal{A})$.

For the proof of Theorem 4.2.6, we need the following observation. Given a
small DG category $\mathcal{A}$ and a morphism $f\colon M\to N$ in
$\operatorname{dgMod}(\mathcal{A})$, recall that the _cone of $f$_, denoted by
$\operatorname{cone}\left(f\right)$, is the DG $\mathcal{A}$-module whose
underlying graded $\mathcal{A}$-module is $M(1)\oplus N$ equipped with the
differential

$d_{\operatorname{cone}\left(f\right)}\coloneqq d_{M[1]}\oplus
d_{N}+\left(\begin{smallmatrix}0&0\\\
f&0\end{smallmatrix}\right)=\left(\begin{smallmatrix}-d_{M}&0\\\
f&d_{N}\end{smallmatrix}\right);$

above, $M\mapsto M(1)$ denotes the shift of the underlying graded
$\mathcal{A}$-module of $M$. The definition above ensures that the canonical
split short exact sequence of graded $\mathcal{A}$-modules

${0}$${N}$${\operatorname{cone}\left(f\right)}$${M[1]}$${0}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$

is in fact a (not necessarily split) short exact sequence of DG
$\mathcal{A}$-modules. The above sequence yields an exact triangle

${M}$${N}$${\operatorname{cone}\left(f\right)}$${M[1]}$$\scriptstyle{f}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$

in the triangulated category
$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$.

######  Proposition 4.2.8.

Let $\mathcal{A}$ be a small DG category. Suppose given a sequence of degree
$0$ morphisms

$M_{d+2}\xrightarrow{f_{d+2}}M_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}M_{1}\xrightarrow{f_{1}}M_{0}$

in the triangulated category
$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$, with $d\geq 1$. Let
$S$ be the set consisting of the exact triangles of the precise form

${\operatorname{cone}\left(g_{12}\right)[-1]}$${M_{2}}$${M_{1}}$${\operatorname{cone}\left(g_{12}\right),}$$\scriptstyle{\pi[-1]}$$\scriptstyle{f_{2}}$$\scriptstyle{-\iota}$

where $g_{12}$ ranges over all degree $0$ cocycles $M_{2}\to M_{1}$ in
$\operatorname{dgMod}(\mathcal{A})$ that represent the morphism $f_{2}$. Then,
the Massey product $\llangle{f_{1},\dots,f_{d+2}}\rrangle$ satisfies the
formula

$\llangle{f_{1},\dots,f_{d+2}}\rrangle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}}\llangle{\beta,\alpha,f_{4},\dots,f_{d+2}}\rrangle.$

###### Proof.

Let

$\left\\{g_{ij}\colon M_{j}\to M_{i}\right|0\leq i<j\leq d+2,\ j-i<d+2\\}$

be a defining system for the Massey product
$\llangle{f_{1},\dots,f_{d+2}}\rrangle$. Define the morphisms in
$\operatorname{dgMod}(\mathcal{A})$

$\displaystyle\alpha$ $\displaystyle\coloneqq\begin{pmatrix}g_{23}\\\
g_{13}\end{pmatrix}\colon
M_{3}\longrightarrow\operatorname{cone}\left(g_{12}\right)[-1],$
$\displaystyle\beta$
$\displaystyle\coloneqq\begin{pmatrix}g_{02}&-g_{01}\end{pmatrix}\colon\operatorname{cone}\left(g_{12}\right)[-1]\longrightarrow
M_{0}[-1].$

Notice that $|\alpha|=0=|\beta|$. We claim that the pair formed by the
cohomology classes of these morphisms
$(\left\\{\beta\right\\},\left\\{\alpha\right\\})\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}$.
Notice first that $\alpha$ and $\beta$ are indeed cocycles since, in view of
the equations satisfied by a defining system, we have

$\displaystyle\partial(\alpha)$ $\displaystyle=\begin{pmatrix}d_{M_{2}}&0\\\
-g_{12}&-d_{M_{1}}\end{pmatrix}\circ\begin{pmatrix}g_{23}\\\
g_{13}\end{pmatrix}-\begin{pmatrix}g_{23}\\\ g_{13}\end{pmatrix}\circ
d_{M_{3}}$ $\displaystyle=\begin{pmatrix}d_{M_{2}}\circ g_{23}-g_{23}\circ
d_{M_{3}}\\\ -g_{12}g_{23}-(d_{M_{1}}\circ g_{13}+g_{13}\circ
d_{M_{3}})\end{pmatrix}$ $\displaystyle=\begin{pmatrix}\partial(g_{23})\\\
\partial(g_{13})-\partial(g_{13})\end{pmatrix}=0$

and

$\displaystyle\partial(\beta)$
$\displaystyle=-d_{M_{0}}\circ\begin{pmatrix}g_{02}&-g_{01}\end{pmatrix}-\begin{pmatrix}g_{02}&-g_{01}\end{pmatrix}\circ\begin{pmatrix}d_{M_{2}}&0\\\
-g_{12}&-d_{M_{1}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-(d_{M_{0}}\circ g_{02}+g_{02}\circ
d_{M_{2}})-g_{01}g_{12}&d_{M_{0}}\circ g_{01}-g_{01}\circ
d_{M_{1}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-\partial(g_{02})+\partial(g_{02})&\partial(g_{01})\end{pmatrix}=0.$

Here we use that $d_{M_{0}[-1]}=-d_{M_{0}}$. Thus, to exhibit the membership
relation
$(\left\\{\beta\right\\},\left\\{\alpha\right\\})\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)$,
it suffices to observe that the diagram (see Remark 4.1.5)

${M_{2}}$${M_{1}}$${M_{3}}$${\operatorname{cone}\left(g_{12}\right)[-1]}$${M_{0}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{-\iota}$$\scriptstyle{+1}$$\scriptstyle{f_{3}}$$\scriptstyle{\left\\{\alpha\right\\}}$$\scriptstyle{\left\\{\beta\right\\}}$$\scriptstyle{-1}$$\scriptstyle{\pi[-1]}$

commutes in $H^{\bullet}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$,
for $g_{01}$ and $g_{23}$ represent the morphisms $f_{1}$ and $f_{3}$ and
$\pi[-1]$ and $\iota$ are the apparent projection and inclusion into the
corresponding direct summands.

We wish to use the given defining system for the Massey product
$\llangle{f_{1},\dots,f_{d+2}}\rrangle$ to construct a defining system for the
Massey product
$\llangle{\left\\{\beta\right\\},\left\\{\alpha\right\\},f_{4},\dots,f_{d+2}}\rrangle$.
We claim that the collection of morphisms

$\displaystyle\overline{g}_{01}$
$\displaystyle\coloneqq\beta=\begin{pmatrix}g_{02}&-g_{01}\end{pmatrix}\colon\operatorname{cone}\left(g_{12}\right)[-1]\longrightarrow
M_{0};$ $\displaystyle\overline{g}_{0j}$ $\displaystyle\coloneqq
g_{0,j+1}\colon M_{j+1}\longrightarrow M_{0}[-1],\qquad d+2>j\geq 2;$
$\displaystyle\overline{g}_{1j}$
$\displaystyle\coloneqq\begin{pmatrix}g_{2,j+1}\\\
g_{1,j+1}\end{pmatrix}\colon
M_{j+1}\longrightarrow\operatorname{cone}\left(g_{12}\right)[-1],\qquad
d+2>j\geq 2;$ $\displaystyle\overline{g}_{ij}$ $\displaystyle\coloneqq
g_{i+1,j+1}\colon M_{j+1}\longrightarrow M_{i+1},\qquad d+2>j>i\geq 2;$

is such a defining system. Notice that

  * •

$\overline{g}_{01}=\beta$ represents $\left\\{\beta\right\\}$;

  * •

$\overline{g}_{12}=\alpha$ represents $\left\\{\alpha\right\\}$;

  * •

$\overline{g}_{i-1,i}=g_{i,i+1}\colon M_{i+1}\to M_{i}$ represents $f_{i+1}$.

We now analyse the differentials $\partial(\overline{g}_{i,j})$ in the four
cases:

  * •

We have already shown that $\partial(\overline{g}_{01})=\partial(\beta)=0$.

  * •

If $j\geq 2$, then

$\displaystyle\partial(\overline{g}_{0j})$
$\displaystyle=-\partial(g_{0,j+1})$
$\displaystyle=-\sum_{0<\ell<j+1}(-1)^{-\ell}g_{0\ell}g_{\ell,j+1}$
$\displaystyle=-\begin{pmatrix}g_{02}&-g_{01}\end{pmatrix}\begin{pmatrix}g_{2,j+1}\\\
g_{1,j+1}\end{pmatrix}+\sum_{2<\ell<j+1}(-1)^{1-\ell}g_{0\ell}g_{\ell,j+1}$
$\displaystyle=-\overline{g}_{01}\overline{g}_{1j}+\sum_{2<\ell<j+1}(-1)^{1-\ell}\overline{g}_{0,\ell-1}\overline{g}_{\ell-1,j}$
$\displaystyle=\sum_{0<k<j}(-1)^{-k}\overline{g}_{0k}\overline{g}_{kj},$

where the equality $\partial(\overline{g}_{0j})=-\partial(g_{0,j+1})$ stems
from the fact that the target of $\overline{g}_{0j}$ is the DG
$\mathcal{A}$-module $M_{0}[-1]$, which has the differential $-d_{M_{0}}$.

  * •

If $i=1$ and $j\geq 2$, then

$\displaystyle\partial(\overline{g}_{1j})$
$\displaystyle=\begin{pmatrix}d_{M_{2}}&0\\\
-g_{12}&-d_{M_{1}}\end{pmatrix}\circ\begin{pmatrix}g_{2,j+1}\\\
g_{1,j+1}\end{pmatrix}-(-1)^{|\overline{g}_{1j}|}\begin{pmatrix}g_{2,j+1}\\\
g_{1,j+1}\end{pmatrix}\circ d_{M_{j+1}}$
$\displaystyle=\begin{pmatrix}d_{M_{2}}\circ
g_{2,j+1}-(-1)^{|g_{2,j+1}|}g_{2,j+1}\circ d_{M_{j+1}}\\\
-g_{12}g_{2,j+1}-(d_{M_{1}}\circ g_{1,j+1}-(-1)^{|g_{1,j+1}|}g_{1,j+1}\circ
d_{M_{j+1}})\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\partial(g_{2,j+1})\\\
-g_{12}g_{2,j+1}-\partial(g_{1,j+1})\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\sum_{2<\ell<j+1}(-1)^{2-\ell}g_{2\ell}g_{\ell,j+1}\\\
-\sum_{2<\ell<j+1}(-1)^{1-\ell}g_{1\ell}g_{\ell,j+1}\end{pmatrix}$
$\displaystyle=\sum_{2<\ell<j+1}(-1)^{2-\ell}\begin{pmatrix}g_{2\ell}\\\
g_{1\ell}\end{pmatrix}g_{\ell,j+1}$
$\displaystyle=\sum_{2<\ell<j+1}(-1)^{2-\ell}\overline{g}_{1,\ell-1}\overline{g}_{\ell-1,j}$
$\displaystyle=\sum_{1<k<j}(-1)^{1-k}\overline{g}_{1k}\overline{g}_{kj}.$

  * •

If $i\geq 2$, then

$\displaystyle\partial(\overline{g}_{ij})=\partial(g_{i+1,j+1})$
$\displaystyle=\sum_{i+1<k+1<j+1}(-1)^{(i+1)-(k+1)}g_{i+1,k+1}g_{k+1,j+1}$
$\displaystyle=\sum_{i<k<j}(-1)^{i-k}\overline{g}_{ik}\overline{g}_{kj}.$

This shows that the collection of morphisms
$\left\\{\overline{g}_{ij}\right\\}$ is indeed a defining system for the
Massey product
$\llangle{\left\\{\beta\right\\},\left\\{\alpha\right\\},f_{4},\dots,f_{d+2}}\rrangle$.

Finally, the above process is clearly reversible: Given
$(\left\\{\beta\right\\},\left\\{\alpha\right\\})\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}$
and a defining system for the Massey product
$\llangle{\left\\{\beta\right\\},\left\\{\alpha\right\\},f_{4},\dots,f_{d+2}}\rrangle$,
one obtains a defining system for the Massey product
$\llangle{f_{1},\dots,f_{d+2}}\rrangle$ by means of the above formulas. This
finishes the proof. ∎

We are ready to prove Theorem 4.2.6.

###### Proof of Theorem 4.2.6.

Given a sequence of degree $0$ morphisms

$M_{d+2}\xrightarrow{f_{d+2}}M_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}M_{1}\xrightarrow{f_{1}}M_{0}$

in the triangulated category
$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$, for $\mathcal{A}$ a
small DG category, we need to prove that

$\llangle{f_{1},\dots,f_{d+2}}\rrangle[d]=(-1)^{\sum_{i=1}^{d+1}i}\langle{f_{1},\dots,f_{d+2}}\rangle.$

If $d=0$, then the claim follows from the discussion in Remark 4.2.3. For
$d\geq 1$, the claim follows from a straightforward induction using
Propositions 4.1.22 and 4.2.8 and Lemma 4.1.15. Indeed, suppose that the claim
holds for all sequences of $n$ composable morphisms for some $n\geq 2$. Let
$f_{1},\dots,f_{n},f_{n+1}$ be a sequence of composable homogeneous morphisms
in $H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$. Then, the
$(n-1)$-fold shifted Massey product

$\llangle{f_{1},\dots,f_{n},f_{n+1}}\rrangle[n-1]$

equals

$\displaystyle\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}}\llangle{\beta,\alpha,f_{4},\dots,f_{n},f_{n+1}}\rrangle[n-1]$
$\displaystyle=\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}}(-1)^{\sum_{i=1}^{n-1}i}\langle{\beta,\alpha,f_{4},\dots,f_{n},f_{n+1}}\rangle[1]$
$\displaystyle=(-1)^{\sum_{i=1}^{n-1}i}(-1)^{n}\bigcup_{(\beta,\alpha)\in\operatorname{T}\\!\left(f_{1},f_{2},f_{3}\right)_{S}}(-1)^{n-2}\langle{\beta,\alpha,f_{4},\dots,f_{n},f_{n+1}}\rangle[1]$
$\displaystyle=(-1)^{\sum_{i=1}^{n}i}\langle{f_{1},\dots,f_{n},f_{n+1}}\rangle,$

where $S$ is as in Proposition 4.2.8. The claim follows. ∎

###  4.3. Massey products and minimal $A_{\infty}$-structures

In this subsection we relate (under some conditions) higher operations in a
minimal $A_{\infty}$-category and Massey products. We begin by recalling the
necessary definitions.

####  4.3.1. Reminder on $A_{\infty}$-categories

We recall the most basic aspects of the theory of $A_{\infty}$-categories and
its relationship to that of DG categories. We refer the reader to [Kel01, LH]
for details.

An _$A_{\infty}$ -category_ is a categorical structure $\mathcal{A}$
consisting of a class of objects and graded vector spaces of morphisms

$\mathcal{A}(x,y),\qquad x,y\in\mathcal{A}.$

Moreover, $\mathcal{A}$ is equipped with higher composition operations

$m_{n}\colon\mathcal{A}(x_{1},x_{0})\otimes\mathcal{A}(x_{2},x_{1})\otimes\cdots\otimes\mathcal{A}(x_{n},x_{n-1})\longrightarrow\mathcal{A}(x_{n},x_{0}),\qquad
n\geq 1$

given by graded morphisms of degree $2-n$ that must satisfy a certain infinite
system of equations that we do not recall here. We only mention that, in
particular, these equations imply the following:

  * •

For each $x,y\in\mathcal{A}$, the pair $(\mathcal{A}(x,y),m_{1})$ is a cochain
complex, that is $m_{1}\circ m_{1}=0$.

  * •

The differential $m_{1}$ is a graded derivation with respect to the
composition operation

$m_{2}\colon\mathcal{A}(y,z)\otimes\mathcal{A}(x,y)\longrightarrow\mathcal{A}(x,z),$

(that is, it satisfies the graded Leibniz rule).

  * •

Although the composition operation $m_{2}$ is not necessarily associative, the
failure of the associativity is controlled by the ternary operation $m_{3}$ in
a precise sense. In particular, as in the case of DG algebras, $\mathcal{A}$
has an associated graded category $H^{\bullet}(\mathcal{A})$ as well as an
ordinary category $H^{0}(\mathcal{A})$, whose (associative) composition laws
are induced by $m_{2}$.

We also note that DG categories can be identified with the
$A_{\infty}$-categories with $m_{n}=0$ for all $n\geq 3$; in particular, the
composition operation $m_{2}$ is associative in this case. An
$A_{\infty}$-category $\mathcal{A}$ is _minimal_ if $m_{1}=0$; in this case
the composition operation is also associative.

Let $\mathcal{A}$ and $\mathcal{B}$ be $A_{\infty}$-categories. An
_$A_{\infty}$ -functor_ $F\colon\mathcal{A}\to\mathcal{B}$ consists of the
following data:

  * •

A map
$F_{0}\colon\operatorname{ob}\left(\mathcal{A}\right)\to\operatorname{ob}\left(\mathcal{B}\right)$
between the classes of objects of $\mathcal{A}$ and $\mathcal{B}$.

  * •

Graded morphisms

$F_{n}\colon\mathcal{A}(x_{1},x_{0})\otimes\mathcal{A}(x_{2},x_{1})\otimes\cdots\otimes\mathcal{A}(x_{n},x_{n-1})\longrightarrow\mathcal{B}(F_{0}(x_{n}),F_{0}(x_{0})),$

for $n\geq 1$, of degree $1-n$.

Moreover, the above morphisms are required to satisfy a further infinite
system of equations that imply that the morphisms

$F_{1}\colon\mathcal{A}(x,y)\longrightarrow\mathcal{B}(F_{0}(x),F_{0}(y)),\qquad
x,y\in\mathcal{A},$

are cochain maps and, moreover, $F$ induces a graded functor

$H^{\bullet}(F)\colon H^{\bullet}(\mathcal{A})\longrightarrow
H^{\bullet}(\mathcal{B})$

via the apparent formulas

$x\longmapsto F_{0}(x),\quad
x\in\operatorname{ob}\left(\mathcal{A}\right),\qquad\text{and}\qquad\left\\{f\right\\}\longmapsto\left\\{F_{1}(f)\right\\},\quad
f\in\operatorname{Z}(\mathcal{A}(x,y)),$

and in particular an ordinary functor

$H^{0}(F)\colon H^{0}(\mathcal{A})\longrightarrow H^{0}(\mathcal{B}).$

An $A_{\infty}$-functor $F\colon\mathcal{A}\to\mathcal{B}$ is a _quasi-
equivalence_ if the induced graded functor

$H^{\bullet}(F)\colon H^{\bullet}(\mathcal{A})\longrightarrow
H^{\bullet}(\mathcal{B})$

is an equivalence of graded categories.

######  Remark 4.3.1.

There are different notions of ‘unitality’ for the composition operation in an
$A_{\infty}$-category. First of all, the existence of units (identities) may
not be required, in which case the corresponding $A_{\infty}$-categories are
termed _non-unital_. Second, one may require that the graded category
$H^{\bullet}(\mathcal{A})$ is unital, in which case one speaks of
_cohomologically unital_ $A_{\infty}$-categories. Finally, one may require
that the (higher) composition operations $m_{n}$ are _strictly unital_ in a
precise sense. All these variants of unitality extend to
$A_{\infty}$-functors. We do not wish to elaborate on how the various notions
of unitality relate to each other, but only remark that strict unitality
implies cohomological unitality, and that every cohomologically unital
$A_{\infty}$-category is quasi-equivalent to a strictly unital
$A_{\infty}$-category, see for example [Sei08, Sec. I.2a]. The
$A_{\infty}$-categories in this article are cohomologically unital.

Let $\mathcal{A}$ be a DG category. By the Homotopy Transfer Theorem [Kad80],
the graded category $H^{\bullet}(\mathcal{A})$ admits a (non-unique) minimal
$A_{\infty}$-structure

$(H^{\bullet}(\mathcal{A}),m_{3},m_{4},\dots)$

whose composition operation $m_{2}$ is the composition law in the graded
category $H^{\bullet}(\mathcal{A})$ (and is implicit in the notation), and
such that there is a quasi-equivalence of $A_{\infty}$-categories

$F\colon(H^{\bullet}(\mathcal{A}),m_{3},m_{4},\dots)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{A}$

that is the identity on the objects: $F_{0}=\mathbf{1}$. In this case,
$(H^{\bullet}(\mathcal{A}),m_{3},m_{4},\dots)$ is called a _minimal
$A_{\infty}$-model of $\mathcal{A}$_. Up to quasi-equivalence of
$A_{\infty}$-categories, the DG category structure on $\mathcal{A}$ is
determined by the additional structure on its cohomology
$H^{\bullet}(\mathcal{A})$. This is the main motivation for the use of
$A_{\infty}$-categories in this article (see the proof of Theorem 5.1.3 to see
why this is important for us).

####  4.3.2. Massey products in DG categories with sparse cohomology

Let $\mathcal{A}$ be a DG category. It was a longstanding belief that the
higher Massey products in the graded category $H^{\bullet}(\mathcal{A})$ were
related to the higher operations in a minimal model for $\mathcal{A}$ via the
formula

$\pm m_{n}(f_{1},\dots,f_{n})\in\llangle{f_{1},\dots,f_{n}}\rrangle.$

However, counter-examples were found recently by Buijs, Moreno-Fernández and
Murillo in [BMFM20]. There, the authors give sufficient conditions for the
above formula to hold; however, these conditions are not suitable for our
purposes. Below we show that the above relationship between Massey products
and higher operations holds under a different sufficient condition that is
better adapted to our framework.

The following definition is motivated by Remark 3.2.3.

######  Definition 4.3.2.

A graded category $\mathcal{C}$ is _$d$ -sparse_ if

$\forall x,y\in\mathcal{C},\ \forall i\not\in
d\mathbb{Z},\qquad\mathcal{C}(x,y)^{i}=0.$

Similarly, a DG category $\mathcal{A}$ has _$d$ -sparse cohomology_ or is
_cohomologically $d$-sparse_ if the graded category $H^{\bullet}(\mathcal{A})$
is $d$-sparse. Explicitly, $\mathcal{A}$ is cohomologically $d$-sparse if

$\forall x,y\in\mathcal{A},\qquad\forall i\not\in d\mathbb{Z},\qquad
H^{i}(\mathcal{A}(x,y))=0.$

######  Remark 4.3.3.

For $d=1$, the sparseness condition on a (DG) category is, of course, vacuous.

######  Proposition 4.3.4.

Let $\mathcal{A}$ be DG category with $d$-sparse cohomology, for some $d\geq
1$. Suppose given a sequence of composable morphisms in
$H^{\bullet}(\mathcal{A})$ of the form

$x_{k+2}\xrightarrow{f_{k+2}}x_{k+1}\xrightarrow{f_{k+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}$

with $k\not\in d\mathbb{Z}$. Let
$(H^{\bullet}(\mathcal{A}),m_{3},m_{4},\dots)$ be a minimal $A_{\infty}$-model
of $\mathcal{A}$. Then,

$m_{k+2}(f_{1},f_{2},\dots,f_{k+2})=0.$

In other words, the higher operation $m_{k+2}$ is identically $0$.

###### Proof.

By definition, $m_{k+2}$ is a homogeneous morphism of degree $2-(k+2)=-k$.
Since $\mathcal{A}$ is cohomologically $d$-sparse by assumption and

$|m_{k+2}(f_{1},f_{2},\dots,f_{k+2})|=-k+\sum_{i=1}^{k+2}|f_{i}|,$

we must have $m_{k+2}(f_{1},f_{2},\dots,f_{k+2})=0$, because either some
$|f_{i}|\notin d\mathbb{Z}$ and hence $f_{i}=0$ or all $|f_{i}|\in
d\mathbb{Z}$ and then $-k+\sum_{i=1}^{k+2}|f_{i}|\not\in d\mathbb{Z}$. ∎

######  Notation 4.3.5.

Let $(\mathcal{A},m_{3},m_{4},\dots)$ be a minimal $A_{\infty}$-category whose
underlying graded category $\mathcal{A}=H^{\bullet}(\mathcal{A})$ is
$d$-sparse. In view of Proposition 4.3.4, we write

$(\mathcal{A},m_{d+2},m_{2d+2},\dots)=(\mathcal{A},m_{3},m_{4},\dots)$

since the higher operations $m_{i+2}$ with $i\not\in d\mathbb{Z}$ must vanish.

The following proposition shows that, in a DG category with $d$-sparse
cohomology, the lowest non-trivial higher operation of any minimal
$A_{\infty}$-model, that is $m_{d+2}$, determines the Massey products of the
corresponding length.

######  Proposition 4.3.6.

Let $\mathcal{A}$ be DG category with $d$-sparse cohomology, for some $d\geq
1$, and suppose given a sequence of composable morphisms in
$H^{\bullet}(\mathcal{A})$ of the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}$

and such that $f_{i}f_{i+1}=0$ for each $0<i<d+2$. Then, for every minimal
$A_{\infty}$-model $(H^{\bullet}(\mathcal{A}),m_{d+2},m_{2d+2},\dots)$ of
$\mathcal{A}$, we have

$-(-1)^{\sum_{0<k\leq
d+2}(d+2-k)|f_{k}|}m_{d+2}(f_{1},\dots,f_{d+2})\in\llangle{f_{1},\dots,f_{d+2}}\rrangle.$

###### Proof.

Let $(H^{\bullet}(\mathcal{A}),m_{d+2},m_{2d+2},\dots)$ be a minimal
$A_{\infty}$-model of $\mathcal{A}$ and choose a quasi-equivalence

$F\colon(H^{\bullet}(\mathcal{A}),m_{d+2},m_{2d+2},\dots)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{A}$

with $F_{0}=\mathbf{1}$. For $0\leq i<j\leq d+2$ and $j-i<d+2$, we define

$g_{ij}\coloneqq(-1)^{\sum_{i<k\leq
j}(j-k)|f_{k}|}F_{j-i}(f_{i+1},\dots,f_{j}).$

We claim that the $g_{ij}$’s form a defining system for computing the Massey
product $\llangle{f_{1},\dots,f_{d+2}}\rrangle$. Indeed, in view of the
equations satisfied by the $A_{\infty}$-functor $F$ [Kel01, Sec. 3.4], the
morphism $d(F_{j-i}(f_{i+1},\dots,f_{j}))$ equals

$-\sum_{i<k<j}(-1)^{(k-i-1)+(j-k-1)\sum_{i<\ell\leq
k}|f_{\ell}|}F_{k-i}(f_{i+1},\dots,f_{k})F_{j-k}(f_{k+1},\dots,f_{j})$

and the claim follows by multiplying by the sign $(-1)^{\sum_{i<k\leq
j}(j-k)|f_{k}|}$ in the definition of $g_{ij}$.

Finally, the morphism $d(F_{d+2}(f_{1},\dots,f_{d+2}))$ equals

$\displaystyle F_{1}(m_{d+2}(f_{1},\dots,f_{d+2}))$
$\displaystyle-\sum_{0<k<d+2}(-1)^{(k-1)+(d+1-k)\sum_{0<\ell\leq
k}|f_{\ell}|}F_{k}(f_{1},\dots,f_{k})F_{d+2-k}(f_{k+1},\dots,f_{d+2}).$

From this, it follows that the morphism

$-(-1)^{\sum_{0<k\leq d+2}(d+2-k)|f_{k}|}m_{d+2}(f_{1},\dots,f_{d+2})$

represents the same cohomology class in
$\llangle{f_{1},\dots,f_{d+2}}\rrangle$ as the one induced by the above
defining system. ∎

###  4.4. The universal Massey product

In this subsection we introduce the universal Massey product of length $d+2$
associated to a cohomologically $d$-sparse DG category (Definition 4.4.5).

####  4.4.1. Reminder on Hochschild cohomology

We refer the reader to [Mur22, Sec. 1] for more details on the material in
this section. Let $\mathcal{C}$ be a category and
$\mathcal{C}^{e}\coloneqq\mathcal{C}\otimes\mathcal{C}^{\mathrm{op}}$ its
enveloping category. As usual, we identify the abelian category of
$\mathcal{C}$-bimodules with that of (right) $\mathcal{C}^{e}$-modules; in
particular, the category $\mathcal{C}$ yields the _diagonal
$\mathcal{C}$-bimodule_

$(x,y)\longmapsto\mathcal{C}(x,y),\qquad x,y\in\mathcal{C},$

that, with some abuse of notation, we also denote by $\mathcal{C}$. Recall
that the _Hochschild cohomology of $\mathcal{C}$ with coefficients in a
$\mathcal{C}$-bimodule $M$_, also known as the _Hochschild–Mitchell
cohomology_ [Mit72], is the graded vector space

$\operatorname{HH}^{\bullet}\\!\left(\mathcal{C},M\right)\coloneqq\operatorname{Ext}_{\mathcal{C}^{e}}^{\bullet}\\!\left(\mathcal{C},M\right).$

In particular, the Hochschild cohomology
$\operatorname{HH}^{n}\\!\left(\mathcal{C},M\right)$ vanishes if $n$ is
negative. If $M=\mathcal{C}$ the diagonal $\mathcal{C}$-bimodule, we write

$\operatorname{HH}^{\bullet}\\!\left(\mathcal{C}\right)\coloneqq\operatorname{HH}^{\bullet}\\!\left(\mathcal{C},\mathcal{C}\right).$

If $\mathcal{C}$ is a graded category, then the category of
$\mathcal{C}$-bimodules is also graded and the Hochschild cohomology is then a
bi-graded vector space
$\operatorname{HH}^{\bullet,*}\\!\left(\mathcal{C},M\right)$ and

$\operatorname{HH}^{p}\\!\left(\mathcal{C},M\right)=\prod_{q\in\mathbb{Z}}\operatorname{HH}^{p,q}\\!\left(\mathcal{C},M\right),\qquad
p\geq 0.$

Here, $p$ is the _Hochschild (=horizontal) degree_ and $q$ is the _internal
(=vertical) degree_. Hochschild cohomology is Morita invariant and suitably
functorial with respect to graded functors in the first variable and bimodule
morphisms in the second variable, see for example [Mur06].

The Hochschild cohomology of a graded category can be computed by means of the
_bar complex_ , which is the projective resolution of the diagonal bimodule
whose component in degree $n$ is

$B_{n}(\mathcal{C})\coloneqq\bigoplus_{x_{0},x_{1},\dots,x_{n}}\mathcal{C}(x_{0},-)\otimes\mathcal{C}(x_{1},x_{0})\otimes\mathcal{C}(x_{2},x_{1})\otimes\cdots\otimes\mathcal{C}(x_{n},x_{n-1})\otimes\mathcal{C}(-,x_{n})$

and whose differential $d\colon B_{n}(\mathcal{C})\to B_{n-1}(\mathcal{C})$
and augmentation $d\colon B_{0}(\mathcal{C})\to\mathcal{C}$ act on an
elementary tensor by the formula

$d(f_{0}\otimes f_{1}\otimes\cdots\otimes f_{n}\otimes
f_{n+1})=\sum_{i=0}^{n}(-1)^{i}f_{0}\otimes f_{1}\otimes\cdots\otimes
f_{i}f_{i+1}\otimes\dots\otimes f_{n}\otimes f_{n+1}$

that uses the composition operation in $\mathcal{C}$. The _normalised bar
complex_ $\bar{B}_{\bullet}(\mathcal{C})$, consisting of replacing
$\mathcal{C}(x,x)$ with $\mathcal{C}(x,x)/\mathbf{k}\cdot\mathbf{1}_{x}$ above
whenenever $x_{i+1}=x_{i}=x\in\mathcal{C}$, is also a projective resolution of
the diagonal bimodule. The _Hochschild cochain complex of $\mathcal{C}$_ is
the the cochain complex

$\operatorname{C}^{\bullet,*}\\!\left(\mathcal{C},M\right)\coloneqq\operatorname{Hom}_{\mathcal{C}^{e}}^{*}\\!\left(B_{\bullet}(\mathcal{C}),M\right).$

Explicitly, in Hochschild degree $n$,

$\operatorname{C}^{n,*}\\!\left(\mathcal{C},M\right)=\prod_{x_{0},x_{1},\dots,x_{n}}\operatorname{Hom}_{\mathbf{k}}^{*}\\!\left(\mathcal{C}(x_{1},x_{0})\otimes\mathcal{C}(x_{2},x_{1})\otimes\cdots\otimes\mathcal{C}(x_{n},x_{n-1}),M(x_{n},x_{0})\right)$

and the differential $\varphi\mapsto d(\varphi)$, of bidegree $(1,0)$, is
given by the formula

$\displaystyle d(\varphi)(f_{1}\otimes\cdots\otimes f_{n+1})$
$\displaystyle=(-1)^{|\varphi|_{v}|f_{1}|}f_{1}\cdot\varphi(f_{2}\otimes\cdots\otimes
f_{n+1})$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i}\varphi(f_{1}\otimes\cdots\otimes
f_{i}f_{i+1}\otimes\dots\otimes f_{n}\otimes f_{n+1})$
$\displaystyle+(-1)^{n}\varphi(f_{1}\otimes\cdots\otimes f_{n})\cdot f_{n+1},$

where $|\varphi|_{v}$ denotes the vertical degree. We can alternatively use
the _normalised Hochschild cochain complex,_

$\bar{\operatorname{C}}^{\bullet,*}\\!\left(\mathcal{C},M\right)\coloneqq\operatorname{Hom}_{\mathcal{C}^{e}}^{*}\\!\left(\bar{B}_{\bullet}(\mathcal{C}),M\right),$

which is the subcomplex formed by the Hochschild cochains which vanish when at
least one entry is an identity morphism in $\mathcal{C}$. If $M=\mathcal{C}$
the diagonal $\mathcal{C}$-bimodule, we also write

$\operatorname{C}^{\bullet}\\!\left(\mathcal{C}\right)\coloneqq\operatorname{C}^{\bullet}\\!\left(\mathcal{C},\mathcal{C}\right).$

Essentially by definition, the Hochschild cohomology

$\operatorname{HH}^{0,*}\\!\left(\mathcal{C},M\right)=\operatorname{Hom}_{\mathcal{C}^{e}}^{*}\\!\left(\mathcal{C},M\right)$

is isomorphic to the so-called _end_ of the bimodule $M$, see [Kel05b, Sec.
2.1] for the definition. In particular,

$\operatorname{HH}^{0,*}\\!\left(\mathcal{C}\right)=\operatorname{Hom}_{\mathcal{C}^{e}}^{*}\\!\left(\mathcal{C},\mathcal{C}\right)\cong
Z(\mathcal{C})$

is isomorphic to the end of the diagonal bimodule, that is the _graded centre_
of $\mathcal{C}$.

If $\mathcal{C}$ is an ungraded category (=graded category concentrated in
degree $0$), then the Hochschild cohomology of $\mathcal{C}$ with coefficients
in a graded (!) $\mathcal{C}$-bimodule can be computed in terms of ungraded
Hochschild cohomology by means of the formula

$\operatorname{HH}^{p,q}\\!\left(\mathcal{C},M\right)\cong\operatorname{HH}^{p}\\!\left(\mathcal{C},M^{q}\right),\qquad
p\geq 0,\quad q\in\mathbb{Z}.$

The Hochschild cohomology

$\operatorname{HH}^{\bullet,*}\\!\left(\mathcal{C}\right)=\operatorname{Ext}_{\mathcal{C}^{e}}^{\bullet,*}\\!\left(\mathcal{C},\mathcal{C}\right)$

of a graded category has the structure of a _Gerstenhaber algebra_ [Ger63] and
[Mur20a, Sec. 1]. Thus,
$\operatorname{HH}^{\bullet,*}\\!\left(\mathcal{C}\right)$ is equipped with a
graded commutative product

$\operatorname{HH}^{p,q}\\!\left(\mathcal{C}\right)\otimes\operatorname{HH}^{r,s}\\!\left(\mathcal{C}\right)\longrightarrow\operatorname{HH}^{p+r,q+s}\\!\left(\mathcal{C}\right),\qquad
a\otimes b\longmapsto a\cdot b,$

called the _cup product_ ; up to sign, it agrees with the Yoneda product on
the corresponding extension spaces. Thus, the cup product satisfies the
identities

$\displaystyle(a\cdot b)\cdot c$ $\displaystyle=a\cdot(b\cdot c),$
$\displaystyle a\cdot b$ $\displaystyle=(-1)^{|a||b|}b\cdot a.$

Here $|x|=p+q$ is the _total degree_ of an element
$x\in\operatorname{HH}^{p,q}\\!\left(\mathcal{C}\right)$. Moreover, there is a
degree $-1$ Lie bracket

$\operatorname{HH}^{p,q}\\!\left(\mathcal{C}\right)\otimes\operatorname{HH}^{r,s}\\!\left(\mathcal{C}\right)\longrightarrow\operatorname{HH}^{p+r-1,q+s}\\!\left(\mathcal{C}\right),\qquad
a\otimes b\longmapsto[a,b],$

that satisfies the following graded variants of the usual Lie algebra
identities:

$\displaystyle[a,a]$ $\displaystyle=0,\qquad|a|\text{ is odd},$
$\displaystyle[a,b]$ $\displaystyle=-(-1)^{(|a|-1)(|b|-1)}[b,a],$
$\displaystyle[a,[b,c]]$
$\displaystyle=[[a,b],c]+(-1)^{(|a|-1)(|b|-1)}[b,[a,c]],$
$\displaystyle[a,[a,a]]$ $\displaystyle=0,\qquad|a|\text{ is even}.$

The cup product and the Lie bracket are compatible in the sense that

$[a,b\cdot c]=[a,b]\cdot c+(-1)^{(|a|-1)|b|}b\cdot[a,c].$

Finally, there is a _Gerstenhaber squaring operation_ or _Gerstenhaber square_

$\operatorname{HH}^{p,q}\\!\left(\mathcal{C}\right)\longrightarrow\operatorname{HH}^{2p-1,2q}\\!\left(\mathcal{C}\right),\qquad
a\longmapsto\operatorname{Sq}\\!\left(a\right),$

defined whenever $p+q$ is even or $\operatorname{char}(\mathbf{k})=2$. The
Gerstenhaber square satisfies the following compatibility relations with
respect to the cup product and the Lie bracket:

$\displaystyle\operatorname{Sq}\\!\left(a+b\right)$
$\displaystyle=\operatorname{Sq}\\!\left(a\right)+\operatorname{Sq}\\!\left(b\right)+[a,b],$
$\displaystyle\operatorname{Sq}\\!\left(a\cdot b\right)$
$\displaystyle=\operatorname{Sq}\\!\left(a\right)\cdot b^{2}+a\cdot[a,b]\cdot
b+a^{2}\cdot\operatorname{Sq}\\!\left(b\right),$
$\displaystyle[\operatorname{Sq}\\!\left(a\right),b]$
$\displaystyle=[a,[a,b]].$

If $\operatorname{char}(\mathbf{k})\neq 2$, then the Gerstenhaber square is in
fact determined by the Lie bracket according to the formula

$\operatorname{Sq}(a)=\frac{1}{2}[a,a],$

which follows from the previous relations. In particular, if
$\operatorname{char}(\mathbf{k})\neq 2$, then the relations that involve the
Gerstenhaber square are redundant.

######  Remark 4.4.1.

For a graded algebra $A$ (that we can regard as a graded category with only
one object), the automorphism group $\operatorname{Aut}(A)$ acts on the right
of the cochain complex $\operatorname{C}^{\bullet,*}\\!\left(A\right)$ by
conjugation,

$\displaystyle\varphi^{f}(x_{1},\dots,x_{n})$
$\displaystyle=f^{-1}\varphi(f(x_{1}),\dots,f(x_{n})),\qquad\varphi\in\operatorname{C}^{n,*}\\!\left(A\right),\quad
f\in\operatorname{Aut}(A).$

This induces a right action of $\operatorname{Aut}(A)$ on
$\operatorname{HH}^{\bullet,*}\\!\left(A\right)$ which is determined by

$x^{f}=(f^{-1})_{*}f^{*}(x).$

Here
$f^{*}\colon\operatorname{HH}^{\bullet,*}\\!\left(A,A\right)\to\operatorname{HH}^{\bullet,*}\\!\left(A,{}_{f}A_{f}\right)$
is induced by the algebra automorphism $f$ in the first variable and
$f^{-1}_{*}\colon\operatorname{HH}^{\bullet,*}\\!\left(A,{}_{f}A_{f}\right)\to\operatorname{HH}^{\bullet,*}\\!\left(A,A\right)$
is induced by the $A$-bimodule morphism $f^{-1}\colon{}_{f}A_{f}\to A$ in the
second variable.

The following result is probably well known but we have not found any
reference.

######  Proposition 4.4.2.

The right action of $\operatorname{Aut}(A)$ on
$\operatorname{HH}^{\bullet,*}\\!\left(A\right)$ (Remark 4.4.1) factors
through the outer automorphism group $\operatorname{Out}(A)$.

###### Proof.

We have to show that the inner automorphism $f_{u}\colon
A\stackrel{{\scriptstyle\sim}}{{\to}}A$,
$f_{u}(x)\coloneqq(-1)^{|u||x|}uxu^{-1}$, induced by a unit $u\in A$ acts
trivially on $\operatorname{HH}^{\bullet,*}\\!\left(A\right)$. The sign in the
definition of the outer automorphism comes from the fact that $\mathbf{1}_{A}$
and $f_{u}$ can be regarded as endofunctors of the graded category with one
object $A$ and multiplication by $u$ can then be regarded as a degree $|u|$
natural isomorphism $u\colon\mathbf{1}_{A}\Rightarrow f_{u}$. We will produce
a null-homotopy for the map

$\displaystyle\operatorname{C}^{\bullet,*}\\!\left(A\right)$
$\displaystyle\longrightarrow\operatorname{C}^{\bullet,*}\\!\left(A\right),$
$\displaystyle\varphi$ $\displaystyle\longmapsto\varphi^{f_{u}}.$

The null-homotopy is the map

$\displaystyle h\colon\operatorname{C}^{\bullet,*}\\!\left(A\right)$
$\displaystyle\longrightarrow\operatorname{C}^{\bullet-1,*}\\!\left(A\right)$

defined by

$\displaystyle h(\varphi)(x_{1},\dots,x_{n-1})$
$\displaystyle=\sum_{i=0}^{n-1}(-1)^{i+|u||\varphi|_{v}}u^{-1}\varphi(ux_{1}u^{-1},\dots,ux_{i}u^{-1},u,x_{i+1},\dots,x_{n-1}).$

A tedious but straightforward computation shows that

$dh+hd=\mathbf{1}_{\operatorname{C}^{\bullet,*}\\!\left(A\right)}-(-)^{f_{u}}.\qed$

######  Remark 4.4.3.

Proposition 4.4.2 can be generalized from algebras $A$ to categories
$\mathcal{C}$. In theory, the role of $\operatorname{Aut}(A)$ should be
replaced by the self-equivalences of $\mathcal{C}$, but these do not form a
group (taking pseudo-inverses is not well defined). Nevertheless,
$\operatorname{Out}(A)$ can be replaced by the group of natural isomorphism
classes of self-equivalences of $\mathcal{C}$, which happens to act on
$\operatorname{HH}^{\bullet,*}\\!\left(\mathcal{C}\right)$ by the same
formulas as above. This can be checked by using the functoriality properties
of Hochschild cohomology of categories, compare [Mur06].

####  4.4.2. The universal Massey product of lenght $d+2$

Let $(\mathcal{A},m_{3},m_{4},\dots)$ be a minimal $A_{\infty}$-category. We
are interested in the Hochschild cohomology of the graded category
$\mathcal{A}=H^{\bullet}(\mathcal{A})$ (recall that $m_{1}=0$ by definition
and $m_{2}$ is the composition law in $\mathcal{A}$). Firstly, notice that by
definition the higher operations are Hochschild cochains:

$m_{n}\in\operatorname{C}^{n,2-n}\\!\left(\mathcal{A}\right),\qquad n\geq 3.$

Similarly, if

$F\colon(\mathcal{A},m_{3},m_{4},\dots)\longrightarrow(\mathcal{B},m_{3},m_{4},\dots)$

is an $A_{\infty}$-functor between minimal $A_{\infty}$-categories, then
$F_{0}$ and $F_{1}$ define a graded functor

$F_{0,1}\colon\mathcal{A}\longrightarrow\mathcal{B}.$

between the underlying graded categories, and the rest of components are
Hochschild cochains:

$F_{n}\in\operatorname{C}^{n,1-n}\\!\left(\mathcal{A},\mathcal{B}(F_{0,1},F_{0,1})\right),\qquad
n\geq 2.$

The following result is a straightforward consequence of results of Lefèvre-
Hasegawa [LH].

######  Proposition 4.4.4.

Let $(\mathcal{A},m_{d+2},m_{2d+2},\dots)$ be a minimal $A_{\infty}$-category
whose underlying graded category $\mathcal{A}$ is $d$-sparse. The following
statements hold:

  1. (1)

The first non-trivial higher operation (see Proposition 4.3.4)

$m_{d+2}\in\operatorname{C}^{d+2,-d}\\!\left(\mathcal{A}\right)$

is a Hochschild cocycle.

  2. (2)

Let $(\mathcal{A},m_{d+2}^{\prime},m_{2d+2}^{\prime},\dots)$ be another
minimal $A_{\infty}$-category with the same underlying graded category
$\mathcal{A}$ and

$F\colon(\mathcal{A},m_{d+2},m_{2d+2},\dots)\longrightarrow(\mathcal{A},m_{d+2}^{\prime},m_{2d+2}^{\prime},\dots)$

an $A_{\infty}$-functor with $F_{0}=\mathbf{1}$ and $F_{1}=\mathbf{1}$. Then,

$\left\\{m_{d+2}\right\\}=\left\\{m_{d+2}^{\prime}\right\\}\quad\text{in}\quad\operatorname{HH}^{d+2,-d}\\!\left(\mathcal{A}\right).$

###### Proof.

(1) Since $\mathcal{A}$ is $d$-sparse, the higher operations $m_{i+2}$ and
$m_{i+2}^{\prime}$ vanish whenever $i\not\in d\mathbb{Z}$ (Proposition 4.3.4).
The claim follows from [LH, Lemma B.4.1], which shows that the differential
$d(m_{d+2})$ of $m_{d+2}$ in the Hochschild complex
$\operatorname{C}^{\bullet,*}\\!\left(\mathcal{A}\right)$ equals an expression
of the form

$\sum\pm m_{i+2}(\cdots),$

where in particular $0<i<d$, and hence the above expression must vanish.

(2) Similarly as in the previous statement, since $\mathcal{A}$ is $d$-sparse,
the higher components $F_{i+1}$ of the $A_{\infty}$-functor $F$ must vanish
whenever $i\not\in d\mathbb{Z}$ as, by definition, $|F_{i+1}|=-i$. The claim
follows from [LH, Lemma B.4.2], which shows that the differential $d(F_{d+1})$
of $F_{d+1}$ in the Hochschild complex
$\operatorname{C}^{\bullet,*}\\!\left(\mathcal{A}\right)$ equals an expression
of the form

$m_{d+2}-m_{d+2}^{\prime}+\sum\pm F_{i+1}(\cdots)-\sum\pm
m_{i+2}^{\prime}(\cdots),$

where in particular $0<i<d$, and hence the two right-most terms in the
expression must vanish. This finishes the proof. ∎

######  Definition 4.4.5.

Let $\mathcal{A}$ be a cohomologically $d$-sparse DG category and
$H^{\bullet}(\mathcal{A})$ is graded cohomology category. The _universal
Massey product of length $d+2$_ is the Hochschild cohomology class

$\left\\{m_{d+2}\right\\}\in\operatorname{HH}^{d+2,-d}\\!\left(H^{\bullet}(\mathcal{A})\right)$

represented by the operation $m_{d+2}$ in any minimal $A_{\infty}$-model of
$\mathcal{A}$.

######  Remark 4.4.6.

Given a cohomologically $d$-sparse DG category, Proposition 4.4.4 shows that
the universal Massey product of length $d+2$ is well defined and, moreover, is
natural with respect to quasi-equivalences: if
$F\colon\mathcal{A}\to\mathcal{B}$ is a quasi-equivalence between
cohomologically $d$-sparse DG categories, then the induced isomorphism

$\operatorname{HH}^{d+2,-d}\\!\left(H^{\bullet}(\mathcal{A})\right)\cong\operatorname{HH}^{d+2,-d}\\!\left(H^{\bullet}(\mathcal{B})\right)$

takes the universal Massey product of length $d+2$ of $\mathcal{A}$ to that of
$\mathcal{B}$. The universal Massey product of length $d+2$ is also considered
in [Sei15, Ch. 3]. For $d=1$, it has been investigated for example in [BKS04,
Kad82] and plays a crucial role in [Mur22] as well.

######  Remark 4.4.7.

The formula in Proposition 4.3.6 still holds if we replace $m_{d+2}$ by any
other Hochschild cochain $m$ of bidegree $(d+2,-d)$ representing the universal
Massey product of length $d+2$ of $\mathcal{A}$, i.e. given a sequence of
composable morphisms in $H^{\bullet}(\mathcal{A})$ of the form

$x_{d+2}\xrightarrow{f_{d+2}}x_{d+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}x_{1}\xrightarrow{f_{1}}x_{0}$

and such that $f_{i}f_{i+1}=0$ for each $0<i<d+2$, then

$-(-1)^{\sum_{0<k\leq
d+2}(d+2-k)|f_{k}|}m(f_{1},\dots,f_{d+2})\in\llangle{f_{1},\dots,f_{d+2}}\rrangle.$

This is straightforward to check, see [BKS04, Lem. 5.14] for the case $d=1$.
Compare also [BD89, Thm. 3.3] in topology.

######  Remark 4.4.8.

Occasionally, we consider, more generally, the universal Massey product

$\left\\{m_{d+2}\right\\}\in\operatorname{HH}^{d+2,-d}\\!\left(\mathcal{A}\right)$

associated to a minimal $A_{\infty}$-category
$\mathcal{A}=H^{\bullet}(\mathcal{A})$ whose underlying graded category is
$d$-sparse (here, $\operatorname{HH}^{d+2,-d}\\!\left(\mathcal{A}\right)$
denotes the Hochschild cohomology of the underlying graded category of
$\mathcal{A}$).

###  4.5. The restricted universal Massey product

In this subsection we relate universal Massey products to standard and
Amiot–Lin $(d+2)$-angulations (Theorem 4.5.8). The key ingredient for
establishing the desired relationship is the restricted universal Massey
product (Definition 4.5.4).

######  Setting 4.5.1.

We fix a small DG category $\mathcal{A}$ with the following properties:

  1. (1)

The DG category $\mathcal{A}$ is cohomologically $d$-sparse. We fix a minimal
$A_{\infty}$-model

$(H^{\bullet}(\mathcal{A}),m_{d+2},m_{2d+2},\dots)$

of $\mathcal{A}$ (see Notation 4.3.5).

  2. (2)

The essential image of $H^{0}(\mathcal{A})$ under the canonical fully faithful
functor

$H^{0}(\mathbf{h})\colon
H^{0}(\mathcal{A})\longrightarrow\operatorname{D}^{\mathrm{c}}(\mathcal{A})$

is closed under the action of the $d$-fold shift and its inverse. In
particular, there is a commutative diagram

${H^{0}(\mathcal{A})}$${\operatorname{D}^{\mathrm{c}}(\mathcal{A})}$${H^{0}(\mathcal{A})}$${\operatorname{D}^{\mathrm{c}}(\mathcal{A})}$$\scriptstyle{[d]}$$\scriptstyle{[d]}$

where $[d]\colon
H^{0}(\mathcal{A})\stackrel{{\scriptstyle\sim}}{{\to}}H^{0}(\mathcal{A})$ is
an autoequivalence such that

$\forall x,y\in\mathcal{A},\qquad H^{0}(\mathcal{A}(x,y[d]))\cong
H^{0}(\mathcal{A}(x,y)[d])\cong H^{d}(\mathcal{A}(x,y))$

as vector spaces (functorially on $x$ and $y$).

  3. (3)

The category $H^{0}(\mathcal{A})$ is additive and has split idempotents and
finite-dimensional morphism spaces.

  4. (4)

There exists a basic object $c\in H^{0}(\mathcal{A})$ such that
$\operatorname{add}(c)=H^{0}(\mathcal{A})$. In particular, the Yoneda functor

$H^{0}(\mathcal{A})\longrightarrow\operatorname{mod}(\Lambda),\qquad
x\longmapsto H^{0}(\mathcal{A})(c,x)$

restricts to an equivalence of categories

$H^{0}(\mathcal{A})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{proj}(\Lambda),$

where $\Lambda\coloneqq H^{0}(\mathcal{A}(c,c))$.

######  Notation 4.5.2.

Since the finite-dimensional algebra $\Lambda$ is basic, there exists an
algebra automorphism
$\sigma\colon\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\Lambda$ (unique up to
inner automorphisms) such that the diagram

${H^{0}(\mathcal{A})}$${\operatorname{proj}(\Lambda)}$${H^{0}(\mathcal{A})}$${\operatorname{proj}(\Lambda)}$$\scriptstyle{\sim}$$\scriptstyle{[d]}$$\scriptstyle{-\otimes_{\Lambda}{}_{\sigma}{\Lambda}_{1}}$$\scriptstyle{\sim}$

commutes up to natural isomorphism. Explicitly, we fix an isomorphism
$\varphi\colon c\stackrel{{\scriptstyle\sim}}{{\to}}c[d]$ in
$H^{0}(\mathcal{A})$ (which exists since $c$ is a basic additive generator of
$H^{0}(\mathcal{A})$, which is closed under $[d]$). The algebra automorphism

$\sigma=\sigma_{\varphi}\colon
H^{0}(\mathcal{A})(c,c)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(\mathcal{A})(c,c),\qquad
a\longmapsto\varphi^{-1}(a[d])\varphi,$

has the desired property. In particular, the autoequivalence

$\Sigma\coloneqq-\otimes_{\Lambda}{}_{\sigma}{\Lambda}_{1}\colon\operatorname{proj}(\Lambda)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{proj}(\Lambda)$

can be identified with the restriction of scalars $P\longmapsto
P_{\sigma^{-1}}$ along $\sigma^{-1}$.

####  4.5.1. The $d$-sparse graded algebra $\Lambda(\sigma,d)$

The isomorphism $\varphi\colon c\stackrel{{\scriptstyle\sim}}{{\to}}c[d]$
(Notation 4.5.2) can be interpreted as an invertible cohomology class

$\varphi\in H^{0}(\mathcal{A})(c,c[d])\cong H^{d}(\mathcal{A}(c,c));$

under this identification, the algebra automorphism $\sigma=\sigma_{\varphi}$
of $\Lambda$ acts by conjugation $a\mapsto\varphi^{-1}a\varphi$. Notice also
that $\varphi$ induces isomorphisms of vector spaces

$H^{0}(\mathcal{A}(c,c))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{di}(\mathcal{A}(c,c)),\qquad
b\longmapsto\varphi^{i}b,$

for each $i\in\mathbb{Z}$. Moreover, as a bimodule over
$\Lambda=H^{0}(\mathcal{A}(c,c))$, the cohomology space
$H^{di}(\mathcal{A}(c,c))$ identifies with the twisted $\Lambda$-bimodule
${}_{\sigma^{i}}{\Lambda}_{1}$. Indeed, for $a\in H^{0}(\mathcal{A}(c,c))$ and

$b\in
H^{0}(\mathcal{A}(c,c))\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{di}(\mathcal{A}(c,c))$

we have

$\displaystyle b\cdot a=\varphi^{-i}((\varphi^{i}b)a)=ba\qquad\text{and}\qquad
a\cdot b=\varphi^{-i}(a(\varphi^{i}b))=\sigma^{i}(a)b.$

More generally, we introduce the $d$-sparse graded algebra with underlying
graded vector space

$\Lambda(\sigma,d)\coloneqq\bigoplus_{di\in
d\mathbb{Z}}{}_{\sigma^{i}}{\Lambda}_{1}$

and multiplication law

$a\cdot b\coloneqq\sigma^{j}(a)b\in\Lambda(\sigma,d)^{d(i+j)},\qquad
a\in{}_{\sigma^{i}}{\Lambda}_{1},\ b\in{}_{\sigma^{j}}{\Lambda}_{1}.$

It is straightforward to verify that the linear map

$H^{\bullet}(\mathcal{A})(c,c)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\Lambda(\sigma,d),\qquad
a\longmapsto\varphi^{-i}a,\qquad a\in H^{di}(\mathcal{A})(c,c)$

is an isomorphism of graded algebras and, therefore, we obtain an induced
isomorphism of Gerstenhaber algebras

$\operatorname{HH}^{\bullet,*}\\!\left(H^{\bullet}(\mathcal{A})(c,c)\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{HH}^{\bullet,*}\\!\left(\Lambda(\sigma,d)\right),$

which we treat as an identification in what follows. Finally, since the object
$c\in H^{0}(\mathcal{A})$ is an additive generator there is also a canonical
isomorphism of Gerstenhaber algebras

$\operatorname{HH}^{\bullet,*}\\!\left(H^{\bullet}(\mathcal{A})(c,c)\right)\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{HH}^{\bullet,*}\\!\left(H^{\bullet}(\mathcal{A})\right)$

and, consequently, a further isomorphism of Gerstenhaber algebras

$\operatorname{HH}^{\bullet,*}\\!\left(H^{\bullet}(\mathcal{A})\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{HH}^{\bullet,*}\\!\left(\Lambda(\sigma,d)\right).$

######  Remark 4.5.3.

There is a further isomorphism of ($d$-sparse) graded algebras

$\Lambda(\sigma,d)\cong\frac{\Lambda\langle\imath^{\pm 1}\rangle}{(\imath
a-\sigma(a)\imath)_{a\in\Lambda}},\qquad|\imath|=-d,$

induced by the $\Lambda$-linear map

$\Lambda\langle\imath^{\pm
1}\rangle\longrightarrow\Lambda(\sigma,d),\qquad\imath\longmapsto
1\in\Lambda(\sigma,d)^{-d}={}_{\sigma^{-1}}{\Lambda}_{1}.$

Indeed, for $a\in\Lambda=\Lambda(\sigma,d)^{0}$ and
$1\in\Lambda(\sigma,d)^{-d}$ we have

$\sigma(a)\cdot 1=\sigma^{-1}(\sigma(a))1=a=1a=\sigma^{0}(1)a=1\cdot a.$

We treat this isomorphism as an identification simply for the purpose of
removing the ambiguities that arise when considering homogeneous elements of
the graded algebra $\Lambda(\sigma,d)$ (whose non-zero components are all
equal to $\Lambda$ as vector spaces).

####  4.5.2. The restricted universal Massey product

The inclusion $j\colon\Lambda\hookrightarrow\Lambda(\sigma,d)$ of the degree
$0$ part induces a restriction morphism

$j^{*}\colon\operatorname{HH}^{\bullet,*}\\!\left(\Lambda(\sigma,d),\Lambda(\sigma,d)\right)\longrightarrow\operatorname{HH}^{\bullet,*}\\!\left(\Lambda,\Lambda(\sigma,d)\right)$

that, in bidegree $(d+2,-d)$, yields a morphism

$j^{*}\colon\operatorname{HH}^{d+2,-d}\\!\left(\Lambda(\sigma,d),\Lambda(\sigma,d)\right)\longrightarrow\operatorname{HH}^{d+2,-d}\\!\left(\Lambda,\Lambda(\sigma,d)\right)\cong\operatorname{HH}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right),$

since

$\Lambda(\sigma,d)^{-d}={}_{\sigma^{-1}}{\Lambda}_{1}\cong{}_{1}{\Lambda}_{\sigma}$

is the degree $-d$ part of $\Lambda(\sigma,d)$ and

$\operatorname{HH}^{d+2,-d}\\!\left(\Lambda,\Lambda(\sigma,d)\right)=\operatorname{HH}^{d+2}\\!\left(\Lambda,\Lambda(\sigma,d)^{-d}\right)=\operatorname{HH}^{d+2}\\!\left(\Lambda,{}_{\sigma^{-1}}{\Lambda}_{1}\right);$

notice also that the target of $j^{*}$ is (isomorphic to) the extension space

$\operatorname{HH}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)=\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right).$

Using the graded Morita invariance of Hochschild cohomology, we may identify
$j^{*}$ with the restriction morphism

$j^{*}\colon\operatorname{HH}^{d+2,-d}\\!\left(H^{\bullet}(\mathcal{A}),H^{\bullet}(\mathcal{A})\right)\longrightarrow\operatorname{HH}^{d+2}\\!\left(H^{0}(\mathcal{A}),H^{-d}(\mathcal{A})\right),$

where $H^{-d}(\mathcal{A})$ is the ungraded $H^{0}(\mathcal{A})$-bimodule

$H^{-d}(\mathcal{A})\colon(x,y)\longmapsto H^{-d}(\mathcal{A}(x,y)).$

######  Definition 4.5.4.

The _restricted universal Massey product of length $d+2$_ is the class

$j^{*}\left\\{m_{d+2}\right\\}\in\operatorname{HH}^{d+2}\\!\left(H^{0}(\mathcal{A}),H^{-d}(\mathcal{A})\right),$

where

$\left\\{m_{d+2}\right\\}\in\operatorname{HH}^{\bullet,*}\\!\left(H^{\bullet}(\mathcal{A}),H^{\bullet}(\mathcal{A})\right)$

is the universal Massey product of length $d+2$ associated to the minimal
$A_{\infty}$-model

$(H^{\bullet}(\mathcal{A}),m_{d+2},m_{2d+2},\dots)$

of the $d$-sparse DG category $\mathcal{A}$.

######  Remark 4.5.5.

In view of the isomorphism

$\operatorname{HH}^{\bullet,*}\\!\left(H^{\bullet}(\mathcal{A})\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{HH}^{\bullet,*}\\!\left(\Lambda(\sigma,d)\right),$

the universal Massey product $\left\\{m_{d+2}\right\\}$ can be identified with
a class

$\left\\{m_{d+2}\right\\}\in\operatorname{HH}^{\bullet,*}\\!\left(\Lambda(\sigma,d),\Lambda(\sigma,d)\right)$

and therefore

$j^{*}\left\\{m_{d+2}\right\\}\in\operatorname{HH}^{d+2,-d}\\!\left(\Lambda,\Lambda(\sigma,d)\right)\cong\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$

can be represented by an exact sequence of $\Lambda$-bimodules.

######  Remark 4.5.6.

Occasionally, we consider, more generally, the restricted universal Massey
product

$j^{*}\left\\{m_{d+2}\right\\}\in\operatorname{HH}^{d+2,-d}\\!\left(\mathcal{A}^{0},\mathcal{A}\right)=\operatorname{HH}^{d+2}\\!\left(\mathcal{A}^{0},\mathcal{A}^{-d}\right)$

associated to a minimal $A_{\infty}$-category
$\mathcal{A}=H^{\bullet}(\mathcal{A})$ whose underlying graded category is
$d$-sparse, where $\mathcal{A}^{0}$ is the degree $0$ part of $\mathcal{A}$
(in particular,
$\operatorname{HH}^{d+2,-d}\\!\left(\mathcal{A}^{0},\mathcal{A}\right)$
denotes the Hochschild cohomology of the ungraded category $\mathcal{A}^{0}$
with coefficients in the graded $\mathcal{A}^{0}$-bimodule $\mathcal{A}$).

The cohomology class investigated in [BD89] is a topological version of the
restricted universal Massey products considered here in the case $d=1$. It
computes Toda brackets, which are the topological counterpart of Massey
products. The connection is explained in [Mur20b].

####  4.5.3. Standard $(d+2)$-angles versus Amiot–Lin $(d+2)$-angulations

######  Notation 4.5.7.

Choose an exact sequence

$\eta\colon\qquad
0\to{}_{1}{\Lambda}_{\sigma}\stackrel{{\scriptstyle\iota}}{{\to}}P_{d+1}\to\cdots\to
P_{1}\to P_{0}\to\Lambda\to 0$

that represents $j^{*}\left\\{m_{d+2}\right\\}$ in
$\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$;
we may and we will assume that $P_{0},P_{1},\dots,P_{d}$ are projective
$\Lambda$-bimodules. We let

${\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}\coloneqq{\pentagon}_{\eta}$

be the class of $(d+2)$-angles in $\operatorname{proj}(\Lambda)$ obtained as
in Construction 2.2.13. Note, however, that we do not claim that
$j^{*}\left\\{m_{d+2}\right\\}$ can be represented by a extension in which
_all_ middle terms are projective $\Lambda$-bimodules, and therefore _a
priori_ the class ${\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$ is not known
to be a $(d+2)$-angulation of $(\operatorname{proj}(\Lambda),\Sigma)$.

We aim to prove the following theorem. We remind the reader of our standing
assumptions (Setting 4.5.1).

######  Theorem 4.5.8.

Suppose that the DG category $\mathcal{A}$ is pre-$(d+2)$-angulated. Then, the
following statements hold:

  1. (1)

The class of standard $(d+2)$-angles on

$(\operatorname{proj}(\Lambda),\Sigma)\simeq(H^{0}(\mathcal{A}),[d])$

coincides with the class
$(-1)^{\Sigma_{i=1}^{d+2}i}{\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$ from
Notation 4.5.7 (see also Remark 2.1.6). In particular, the class
$(-1)^{\Sigma_{i=1}^{d+2}i}{\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$ does
not depend on the choice of an exact sequence representing
$j^{*}\left\\{m_{d+2}\right\\}$ in
$\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$.

  2. (2)

Suppose that $\Lambda/J_{\Lambda}$ is separable. Let

$0\to{}_{1}{\Lambda}_{\sigma}\to P_{d+1}\to\cdots\to P_{1}\to
P_{0}\to\Lambda\to 0$

be an exact sequence that represents $j^{*}\left\\{m_{d+2}\right\\}$ in
$\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$
such that the $\Lambda$-bimodules $P_{i}$, $0\leq i<d+1$, are projective-
injective $\Lambda$-bimodules. Then, $P_{d+1}$ is also a projective-injective
$\Lambda$-bimodule.

In particular, if $\Lambda/J_{\Lambda}$ is separable, the standard
$(d+2)$-angulation and the AL $(d+2)$-angulation of
$(\operatorname{proj}(\Lambda),\Sigma)$ coincide.

######  Notation 4.5.9.

To alleviate the notation, we denote by
$\mathcal{C}\subseteq\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ be the
essential image of $H^{0}(\mathcal{A})$ under the canonical fully faithful
functor

$H^{0}(\mathbf{h})\colon
H^{0}(\mathcal{A})\longrightarrow\operatorname{D}^{\mathrm{c}}(\mathcal{A}).$

Under the induced equivalence of categories
$H^{0}(\mathcal{A})\simeq\mathcal{C}$, the $H^{0}(\mathcal{A})$-bimodule

$H^{di}(\mathcal{A})\colon(x,y)\longmapsto H^{di}(\mathcal{A}(x,y))\cong
H^{0}(\mathcal{A})(x,y[di]),\qquad i\in\mathbb{Z},$

identifies with the twisted $\mathcal{C}$-bimodule

${}_{[di]}{\mathcal{C}}_{1}\colon(M,N)\longmapsto\mathcal{C}(M,N[di]),\qquad
i\in\mathbb{Z}.$

Similarly, the autoequivalence $[d]\colon
H^{0}(\mathcal{A})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H^{0}(\mathcal{A})$
induces the autoequivalence

$\Sigma\coloneqq-\otimes_{\mathcal{C}}{}_{[d]}{\mathcal{C}}_{1}\colon\operatorname{Mod}(\mathcal{C})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Mod}(\mathcal{C})$

with quasi-inverse

$\Sigma^{-1}\coloneqq-\otimes_{\mathcal{C}}{}_{[-d]}{\mathcal{C}}_{1}\colon\operatorname{Mod}(\mathcal{C})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Mod}(\mathcal{C}).$

Working towards the proof of Theorem 4.5.8, we first investigate the
relationship between the restricted universal Massey product and extensions.
The reader might find it useful to compare the discussion below with
Construction 2.2.13. We begin with a general observation. Let $\mathcal{X}$ be
a category and consider an exact sequence of $\mathcal{X}$-bimodules of the
form

$0\to M\to E_{d+1}\to\cdots\to E_{1}\to E_{0}\to\mathcal{X}\to 0,$

where $E_{0},E_{1},\dots,E_{d}$ are projective $\mathcal{X}$-bimodules. Notice
that for each $x\in\mathcal{X}$ the sequence of _left_ $\mathcal{X}$-modules

$0\to M(x,-)\to E_{d+1}(x,-)\to\cdots\to E_{1}(x,-)\to
E_{0}(x,-)\to\mathcal{X}(x,-)\to 0$

is split-exact (=contractible as a complex), for the representable left
$\mathcal{X}$-module $\mathcal{X}(x,-)$ as well as $E_{i}(x,-)$,
$i=0,1,\dots,d$, are projective as left $\mathcal{X}$-modules. From this
follows that, for every (right) $\mathcal{X}$-module $N$, the sequence of
(right) $\mathcal{X}$-modules

$N\otimes_{\mathcal{X}}(0\to M\to E_{d+1}\to\cdots\to E_{1}\to
E_{0}\to\mathcal{X}\to 0)$

is exact. Thus, in view of the canonical isomorphism
$N\otimes_{\mathcal{X}}\mathcal{X}\cong N$, there is a well-defined morphism

$\Phi_{N}\colon\operatorname{Ext}_{\mathcal{X}^{e}}^{\bullet}\\!\left(\mathcal{X},M\right)\longrightarrow\operatorname{Ext}_{\mathcal{X}}^{\bullet}\\!\left(N,N\otimes_{\mathcal{X}}M\right),$

which is an algebra morphism with respect to the Yoneda product if
$M=\mathcal{X}$ the diagonal $\mathcal{X}$-bimodule.

Returning to the context of Setting 4.5.1, if we let $\mathcal{X}=\mathcal{C}$
and $M={}_{[q]}{\mathcal{C}}_{1}$ in the previous discussion, for each
$\mathcal{C}$-module $N$ we obtain a morphism

$\Phi_{N}\colon\operatorname{Ext}_{\mathcal{C}^{e}}^{d+2}\\!\left(\mathcal{C},{}_{[q]}{\mathcal{C}}_{1}\right)\longrightarrow\operatorname{Ext}_{\mathcal{C}}^{d+2}\\!\left(N,N\otimes_{\mathcal{C}}{}_{[q]}{\mathcal{C}}_{1}\right).$

In particular, if $q=-d$, we may consider the class

$\Phi_{N}(j^{*}\left\\{m_{d+2}\right\\})\in\operatorname{Ext}_{\mathcal{C}}^{d+2}\\!\left(N,N\otimes_{\mathcal{C}}{}_{[-d]}{\mathcal{C}}_{1})\right)\cong\operatorname{Ext}_{\mathcal{C}}^{d+2}\\!\left(N,\Sigma^{-1}N\right)$

induced by the restricted universal Massey product of length $d+2$. Thus,
given an exact sequence of $\mathcal{C}$-bimodules

$\eta\colon\qquad 0\to{}_{[-d]}{\mathcal{C}}_{1}\to P_{d+1}\to\cdots\to
P_{1}\to P_{0}\to\mathcal{C}\to 0$

that represents the restricted universal Massey product
$j^{*}\left\\{m_{d+2}\right\\}$, we see that the class
$\Phi_{N}(j^{*}\left\\{m_{d+2}\right\\})$ is represented by the exact sequence

$N\otimes_{\mathcal{C}}(0\to{}_{[-d]}{\mathcal{C}}_{1}\to P_{d+1}\to\cdots\to
P_{1}\to P_{0}\to\mathcal{C}\to 0).$

Suppose given a sequence of composable morphisms in $\mathcal{C}$ of the form

$M_{d+2}\xrightarrow{f_{d+2}}M_{f+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}M_{1}\xrightarrow{f_{1}}M_{0}$

and such that $f_{i}f_{i+1}=0$ for all $0\leq i<d+2$, so that the Massey
product $\llangle{f_{1},\dots,f_{d+2}}\rrangle$ in $H^{\bullet}(A)$ is non-
empty (see Remark 4.2.4). Let

$N\coloneqq\operatorname{\operatorname{coker}}\mathcal{C}(-,f_{1})\colon\mathcal{C}^{\mathrm{op}}\longrightarrow\operatorname{mod}\mathbf{k}.$

In view of Proposition 4.1.31 and Theorem 4.2.6, the Massey product is an
element of the vector space $\operatorname{Massey}\left(f_{d+2},f_{1}\right)$
defined as the quotient

$\frac{\operatorname{Hom}_{\operatorname{D}^{\mathrm{c}}(\mathcal{A})}\\!\left(M_{d+2},M_{0}[-d]\right)missing}{f_{1}[-d]\cdot\operatorname{Hom}_{\operatorname{D}^{\mathrm{c}}(\mathcal{A})}\\!\left(M_{d+2},M_{1}[-d]\right)+\operatorname{Hom}_{\operatorname{D}^{\mathrm{c}}(\mathcal{A})}\\!\left(M_{d+1},M_{0}\right)\cdot
f_{d+2}}.$

We wish to construct a further morphism

$\Psi\colon\operatorname{Ext}_{\mathcal{C}}^{d+2}\\!\left(N,\Sigma^{-1}N\right)\longrightarrow\operatorname{Massey}\left(f_{d+2},f_{1}\right).$

For this, let $B_{\bullet}(\mathcal{C})$ be the bar complex of $\mathcal{C}$,
so that $N\otimes_{\mathcal{C}}B_{\bullet}(\mathcal{C})$ is a projective
resolution of $N$ that we use to compute the extension space
$\operatorname{Ext}_{\mathcal{C}}^{d+2}\\!\left(N,\Sigma^{-1}N\right)$ as

$\operatorname{Ext}_{\mathcal{C}}^{d+2}\\!\left(N,\Sigma^{-1}N\right)=H^{d+2}(\operatorname{Hom}_{\mathcal{C}}\\!\left(N\otimes_{\mathcal{C}}B_{\bullet}(\mathcal{C}),\Sigma^{-1}N\right)).$

Following essentially the same arguments as in [Mur20b, Sec. 5], one can check
that under the above isomorphism the class
$\Phi_{N}(j^{*}\left\\{m_{d+2}\right\\})$ is represented by the class of the
morphism

$N\otimes_{\mathcal{C}}B_{d+2}(\mathcal{C})\longrightarrow\Sigma^{-1}N$

that corresponds, under the Yoneda embedding, to the morphism

$\displaystyle\bigoplus_{L_{0},L_{1},\dots,L_{d+2}}N(L_{0})\otimes\mathcal{C}(L_{1},L_{0})\otimes\cdots\otimes\mathcal{C}(L_{d+2},L_{d+1})$
$\displaystyle\longrightarrow\Sigma^{-1}N(L_{d+2})$

given by

(4.5.10) $[h]\otimes g_{1}\otimes\cdots\otimes g_{d+2}\longmapsto[(h[-d])\circ
m_{d+2}(g_{1},\dots,g_{d+2})],$

where we use that

$m_{d+2}(g_{1},\dots,g_{d+2})\in\operatorname{Hom}_{\operatorname{D}^{\mathrm{c}}(\mathcal{A})}\\!\left(L_{d+2},L_{0}[-d]\right)$

and therefore the composite

${L_{d+2}}$${L_{0}[-d]}$${N[-d]=\Sigma^{-1}N}$$\scriptstyle{m_{d+2}(g_{1},\dots,g_{d+2})}$$\scriptstyle{h[-d]}$

indeed represents an element in $\Sigma^{-1}N(L_{d+2})$. We refer the reader
to the last paragraph of the proof of [Mur20b, Prop. 5.2] as well as [Mur20b,
Prop. 5.6 and Rmks. 5.4 and 5.7] for details.

Continuing with the construction of the morphism $\Psi$, consider the morphism
of augmented chain complexes

${\cdots}$${0}$${\mathcal{C}(-,M_{d+2})}$${\cdots}$${\mathcal{C}(-,M_{1})}$${\mathcal{C}(-,M_{0})}$${N}$${\cdots}$${N\otimes_{\mathcal{C}}B_{d+3}(\mathcal{C})}$${N\otimes_{\mathcal{C}}B_{d+2}(\mathcal{C})}$${\cdots}$${N\otimes_{\mathcal{C}}B_{1}(\mathcal{C})}$${N\otimes_{\mathcal{C}}B_{0}(\mathcal{C})}$${N.}$$\scriptstyle{p}$

that corresponds, under the Yoneda embedding, to the elements

(4.5.11) $(-1)^{\sum_{j=1}^{i}j}p\otimes f_{1}\otimes\cdots\otimes
f_{i}\otimes\mathbf{1}_{M_{i}}\in(N\otimes_{\mathcal{C}}B_{i}(\mathcal{C}))(M_{i}),$

compare with the proof of [Mur20b, Prop. 5.9]. Here $p$ is the natural
projection onto the cokernel. The morphism $\Psi$ is obtained by applying the
functor $\operatorname{Hom}_{\mathcal{C}}\\!\left(-,\Sigma^{-1}N\right)$ to
the above morphism of chain complexes and passing to cohomology in degree
$d+2$, while keeping in mind the isomorphisms

$\operatorname{Hom}_{\mathcal{C}}\\!\left(\mathcal{C}(-,M_{d+2}),\Sigma^{-1}N\right)\cong\Sigma^{-1}N(M_{d+2})\cong\frac{\mathcal{C}(M_{d+2},M_{0}[-d])}{(f_{1}[-d])\cdot\mathcal{C}(M_{d+2},M_{1}[-d])}.$

As a consequence of the above discussion, we obtain the following relationship
between the restricted universal Massey product
$j^{*}\left\\{m_{d+2}\right\\}$ and the Massey product
$\llangle{f_{1},\dots,f_{d+2}}\rrangle$.

######  Proposition 4.5.12.

Suppose given a sequence of composable morphisms in $\mathcal{C}$ of the form

$M_{d+2}\xrightarrow{f_{d+2}}M_{f+1}\xrightarrow{f_{d+1}}\cdots\xrightarrow{f_{2}}M_{1}\xrightarrow{f_{1}}M_{0}$

and such that $f_{i}f_{i+1}=0$ for all $0\leq i<d+2$. Set
$N=\operatorname{\operatorname{coker}}\mathcal{C}(-,f_{1})$. Then,

$(\Psi\circ\Phi_{N})(j^{*}\left\\{m_{d+2}\right\\})=-(-1)^{\sum_{i=1}^{d+2}i}\llangle{f_{1},\dots,f_{d+2}}\rrangle$

in $\operatorname{Massey}\left(f_{d+2},f_{1}\right)$.

###### Proof.

In view of (4.5.10) and (4.5.11),

$\displaystyle(\Psi\circ\Phi_{N})(j^{*}\left\\{m_{d+2}\right\\})$
$\displaystyle=(-1)^{\sum_{i=1}^{d+2}i}\left\\{m_{d+2}(f_{1},\dots,f_{d+2})\right\\}\in\operatorname{Massey}\left(f_{d+2},f_{1}\right).$

The claim then follows from Proposition 4.3.6, which in this case says that

$-m_{d+2}(f_{1},\dots,f_{d+2})\in\llangle{f_{1},\dots,f_{d+2}}\rrangle.\qed$

######  Remark 4.5.13.

In view of Theorem 4.2.6, under the assumptions of Proposition 4.5.12,

$(\Psi\circ\Phi_{N})(j^{*}\left\\{m_{d+2}\right\\})[d]=(-1)^{d+1}\langle{f_{1},\dots,f_{d+2}}\rangle\in\operatorname{Toda}\left(f_{d+2},f_{1}\right).$

The following proposition shows that the standard $(d+2)$-angles in
$\mathcal{C}$ are detected by the restricted universal Massey product of
length $d+2$. In the statement we do not assume $\mathcal{A}$ to be
pre-$(d+2)$-angulated.

######  Proposition 4.5.14.

Let $(H^{\bullet}(\mathcal{A}),m_{d+2},m_{2d+2},\dots)$ be a minimal
$A_{\infty}$-model of $\mathcal{A}$. Suppose given a sequence of composable
morphisms

(4.5.15) ${M_{d+2}}$${M_{d+1}}$${\cdots}$${M_{1}}$${\Sigma
M_{d+2}.}$$\scriptstyle{f_{d+2}}$$\scriptstyle{f_{d+1}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$

in $(\mathcal{C},\Sigma)\simeq(H^{0}(\mathcal{A}),[d])$. Then, the following
statements are equivalent:

  1. (1)

The above sequence satisfies the conditions in Theorem 4.1.12.

  2. (2)

The above sequence lies in
$(-1)^{\Sigma_{i=1}^{d+2}i}{\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$ (see
Notation 4.5.7 and Remark 2.2.14).

###### Proof.

For simplicity, we identify (4.5.15) with a sequence of projective
$\Lambda$-modules. In view of Theorem 4.2.6, statement (1) is equivalent to
the following two conditions:

  * •

The sequence

${M_{d+2}}$${M_{d+1}}$${\cdots}$${M_{1}}$${\Sigma M_{d+2}}$${\Sigma
M_{d+1}}$$\scriptstyle{f_{d+2}}$$\scriptstyle{f_{d+1}}$$\scriptstyle{f_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{\Sigma
f_{d+2}}$

is exact.

  * •

We have $-\mathbf{1}_{M_{d+2}}\in\llangle{f_{1},\dots,f_{d+2}}\rrangle$.

The first condition is precisely the first condition that a sequence must
satisfy to be a member of the class
$(-1)^{\Sigma_{i=1}^{d+2}i}{\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$.

On the other hand, the second (and last) condition that a sequence must
satisfy to be a member of the class
$(-1)^{\Sigma_{i=1}^{d+2}i}{\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$ says
the following: Let $N=\operatorname{\operatorname{coker}}{f_{1}}$, so that
there is an exact sequence

$0\to\Sigma^{-1}N\xrightarrow{i}M_{d+1}\to\cdots\to
M_{1}\xrightarrow{f_{1}}\Sigma M_{d+2}\xrightarrow{p}N\to 0$

with $f_{d+2}=i\circ\Sigma^{-1}p$. Then, the above sequence must represent the
class

$\Phi_{N}((-1)^{\Sigma_{i=1}^{d+2}i}j^{*}\left\\{m_{d+2}\right\\})\in\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(N,\Sigma^{-1}N\right)$

that, by definition, is represented by the exact sequence

$N\otimes_{\Lambda}(0\to{}_{1}{\Lambda}_{\sigma}\stackrel{{\scriptstyle(-1)^{\Sigma_{i=1}^{d+2}i}\iota}}{{\longrightarrow}}P_{d+1}\to\cdots\to
P_{1}\to P_{0}\to\Lambda\to 0)$

(the $\Lambda$-bimodule extension on the right of the tensor product
represents the class
$(-1)^{\Sigma_{i=1}^{d+2}i}j^{*}\left\\{m_{d+2}\right\\}$, see Notation 4.5.7
and Remark 2.2.14). The extension space can be computed by means of the
projective resolution

$\cdots\to\Sigma^{-1}M_{1}\xrightarrow{\Sigma^{-1}f_{1}}M_{d+2}\xrightarrow{}M_{d+1}\to\cdots\to
M_{1}\xrightarrow{f_{1}}\Sigma M_{d+2}\xrightarrow{p}N\to 0$

and, in these terms, we see that the former sequence is represented by the
class

$[\Sigma^{-1}p]\in\frac{\ker(?\circ\Sigma^{-1}f_{1})}{\operatorname{Hom}_{\Lambda}\\!\left(M_{d+1},\Sigma^{-1}N\right)\cdot
f_{d+2}}\cong\operatorname{Ext}_{\Lambda}^{d+2}\\!\left(N,\Sigma^{-1}N\right).$

Here

$?\circ\Sigma^{-1}f_{1}\colon\operatorname{Hom}_{\Lambda}\\!\left(M_{d+2},\Sigma^{-1}N\right)\longrightarrow\operatorname{Hom}_{\Lambda}\\!\left(\Sigma^{-1}M_{1},\Sigma^{-1}N\right)$

is the morphism given by pre-composition with $\Sigma^{-1}f_{1}$. Thus, to
summarise, the second condition for membership in the class
$(-1)^{\Sigma_{i=1}^{d+2}i}{\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$ says
that

$\left[\Sigma^{-1}p\right]=\Phi_{N}((-1)^{\Sigma_{i=1}^{d+2}i}j^{*}\left\\{m_{d+2}\right\\})$

in $\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(N,\Sigma^{-1}N\right)$.

Now, the morphism $\Psi$ is injective by construction and identifies the
extension space
$\operatorname{Ext}_{\Lambda}^{d+2}\\!\left(N,\Sigma^{-1}N\right)$ with the
subspace

$\frac{\ker(?\circ\Sigma^{-1}f_{1})}{\operatorname{Hom}_{\Lambda}\\!\left(M_{d+1},\Sigma^{-1}N\right)\cdot
f_{d+2}}\subseteq\frac{\operatorname{Hom}_{\Lambda}\\!\left(M_{d+2},\Sigma^{-1}N\right)missing}{\operatorname{Hom}_{\Lambda}\\!\left(M_{d+1},\Sigma^{-1}N\right)\cdot
f_{d+2}}\cong\operatorname{Massey}\left(f_{d+2},f_{1}\right).$

Thus, to conclude the proof it is enough to prove that

$\Psi(\Phi_{N}((-1)^{\Sigma_{i=1}^{d+2}i}j^{*}\left\\{m_{d+2}\right\\}))=\Psi(\left[\Sigma^{-1}p\right])$

if and only if
$-\mathbf{1}_{M_{d+2}}\in\llangle{f_{1},\dots,f_{d+2}}\rrangle$. Indeed,

$\Psi\colon\left[\Sigma^{-1}p\right]\longmapsto\left[\mathbf{1}_{M_{d+2}}\right]\in\operatorname{Massey}\left(f_{d+2},f_{1}\right).$

and

$\Psi\colon\Phi_{N}((-1)^{\Sigma_{i=1}^{d+2}i}j^{*}\left\\{m_{d+2}\right\\})\longmapsto-\llangle{f_{1},\dots,f_{d+2}}\rrangle$

by Proposition 4.5.12. The claim follows. ∎

The graded algebra structure on $\Lambda(\sigma,d)$ endows the Hochschild
cohomology

$\operatorname{HH}^{\bullet,*}\\!\left(\Lambda,\Lambda(\sigma,d)\right)=\operatorname{Ext}_{\Lambda^{e}}^{\bullet}\\!\left(\Lambda,\Lambda(\sigma,d)\right)$

with the structure of a bigraded algebra, where the internal grading is
induced by the grading of $\Lambda(\sigma,d)$. In particular,

$\operatorname{HH}^{\bullet,0}\\!\left(\Lambda,\Lambda(\sigma,d)\right)=\operatorname{Ext}_{\Lambda^{e}}^{\bullet}\\!\left(\Lambda,\Lambda(\sigma,d)^{0}\right)=\operatorname{Ext}_{\Lambda^{e}}^{\bullet}\\!\left(\Lambda,\Lambda\right)$

is the Yoneda algebra of the diagonal $\Lambda$-bimodule. Similarly, since
$\Lambda^{e}$ is also self-injective, we may consider the _Hochschild–Tate
cohomology_

$\underline{\operatorname{HH}}^{\bullet,*}\\!\left(\Lambda,\Lambda(\sigma,d)\right)\coloneqq\underline{\operatorname{Ext}}_{\Lambda^{e}}^{\bullet,*}\left(\Lambda,\Lambda(\sigma,d)\right),$

which is again a bigraded algebra, and note that there are isomorphisms

$\underline{\operatorname{HH}}^{p,q}\\!\left(\Lambda,\Lambda(\sigma,d)\right)\cong\operatorname{HH}^{p,q}\\!\left(\Lambda,\Lambda(\sigma,d)\right)=\operatorname{Ext}_{\Lambda^{e}}^{p}\\!\left(\Lambda,{}_{\sigma^{q}}{\Lambda}_{1}\right),\qquad
p>0,\quad q\in\mathbb{Z}.$

We refer the reader to [Mur22, Sec. 5] for details.

######  Remark 4.5.16.

An extension

$0\to{}_{\sigma}{\Lambda}_{1}\to P_{d+1}\to\cdots\to P_{1}\to
P_{0}\to\Lambda\to 0$

of $\Lambda$-bimodules such that $P_{i}$, $0\leq i<d+1$, is a projective-
injective $\Lambda$-bimodule represents a unit in the Hochschild–Tate
cohomology
$\underline{\operatorname{HH}}^{\bullet,*}\\!\left(\Lambda,\Lambda(\sigma,d)\right)$
if and only if $P_{d+1}$ is also a projective-injective $\Lambda$-bimodule,
see [Mur22, Rmk. 5.9].

We are finally ready to prove Theorem 4.5.8.

###### Proof of Theorem 4.5.8.

Statement (1) follows from Proposition 4.5.14. We prove statement (2). Choose
an exact sequence

$\eta\colon\qquad
0\to{}_{1}{\Lambda}_{\sigma}\stackrel{{\scriptstyle\iota}}{{\to}}P_{d+1}\to\cdots\to
P_{1}\to P_{0}\to\Lambda\to 0$

that represents the restricted universal Massey product

$j^{*}\left\\{m_{d+2}\right\\}\in\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$

and such that $P_{i}$, $0\leq i<d+1$, is a projective-injective
$\Lambda$-bimodule (such a representative always exists). We need to prove
that the $\Lambda$-bimodule $P_{d+1}$ is also projective-injective. It is
enough to prove that $N\otimes P_{d+1}$ is projective for every finite-
dimensional $\Lambda$-module $N$, see [AR91, Thm. 3.1] and notice that the
proof remains valid under the assumption of $\Lambda/J_{\Lambda}$ being
separable, since in this case the finite-dimensional algebra
$(\Lambda/J_{\Lambda})\otimes(\Lambda/J_{\Lambda})^{\mathrm{op}}$ is
semisimple [ERZ57, Cor. 18].

Thus, let $N$ be a finite-dimensional $\Lambda$-module and choose a projective
presentation

$Q_{1}\stackrel{{\scriptstyle f_{1}}}{{\longrightarrow}}\Sigma Q_{d+2}\to N\to
0.$

Using axioms ($d$-TR1) and ($d$-TR2), complete the morphism $f_{1}$ to a
standard $(d+2)$-angle

$Q_{d+2}\to Q_{d+1}\to\cdots\to Q_{1}\xrightarrow{f_{1}}\Sigma Q_{d+2}.$

In view of the validity of statement (1), the above $(d+2)$-angle lies in the
class $\pm{\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$. In particular, the
exact sequence

$N\otimes_{\Lambda}(0\to{}_{1}{\Lambda}_{\sigma}\stackrel{{\scriptstyle\pm\iota}}{{\to}}P_{d+1}\to\cdots\to
P_{1}\to P_{0}\to\Lambda\to 0)$

is equivalent to an exact sequence of the form

$0\to\Sigma^{-1}N\xrightarrow{}Q_{d+1}\to\cdots\to Q_{1}\xrightarrow{}\Sigma
Q_{d+2}\xrightarrow{}N\to 0.$

Notice that all of the middle terms in the latter exact sequence are
projective-injective $\Lambda$-bimodules and, therefore, the class of this
sequence in the Hochschild–Tate cohomology
$\underline{\operatorname{HH}}^{\bullet,*}\\!\left(\Lambda,\Lambda(\sigma,d)\right)$
is a unit. Consequently, the former (equivalent) sequence also represents a
unit in
$\underline{\operatorname{HH}}^{\bullet,*}\\!\left(\Lambda,\Lambda(\sigma,d)\right)$,
and hence the $\Lambda$-module ${N\otimes_{\Lambda}P_{d+1}}$ must be
projective-injective, which is what we needed to prove. This finishes the
proof of the theorem. ∎

Theorem 4.5.8 and Proposition 4.5.14 yield the following important
corollaries.

######  Corollary 4.5.17.

Let $\mathcal{A}$ be a small DG category satisfying the assumptions in Setting
4.5.1. Consider the following statements:

  1. (1)

The DG category $\mathcal{A}$ is pre-$(d+2)$-angulated.

  2. (2)

The canonical fully faithful functor

$H^{0}(\mathbf{h})\colon
H^{0}(\mathcal{A})\longrightarrow\operatorname{D}^{\mathrm{c}}(\mathcal{A})$

identifies $H^{0}(\mathcal{A})$ with a $d\mathbb{Z}$-cluster tilting
subcategory of $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$.

  3. (3)

The restricted universal Massey product

$j^{*}\left\\{m_{d+2}\right\\}\in\operatorname{Ext}_{\Lambda}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$

is represented by an extension all of whose middle terms are projective-
injective $\Lambda$-bimodules.

In general (1)$\Leftrightarrow$(2) and (1)$\Leftarrow$(3). If
$\Lambda/J_{\Lambda}$ is separable, then (1)$\Rightarrow$(3).

###### Proof.

The equivalence (1)$\Leftrightarrow$(2) is Proposition 3.2.4.

(1)$\Rightarrow$(3) This is precisely Theorem 4.5.8(2), which requires the
algebra $\Lambda/J_{\Lambda}$ to be separable.

(3)$\Rightarrow$(1) By Proposition 4.5.14 and Remark 2.2.14, the Amiot–Lin
$(d+2)$-angulation associated to a representative of
$(-1)^{\Sigma_{i=1}^{d+2}i}j^{*}\left\\{m_{d+2}\right\\}$ coincides with the
class of sequences that satisfy the conditions in Theorem 4.1.12, which
therefore forms a $(d+2)$-angulation. Consequently, the canonical fully
faithful functor

$H^{0}(\mathbf{h})\colon
H^{0}(\mathcal{A})\longrightarrow\operatorname{D}^{\mathrm{c}}(\mathcal{A})$

identifies the left-hand side with a standard $(d+2)$-angulated subcategory of
the right-hand side, which is precisely the definition of $\mathcal{A}$ being
pre-$(d+2)$-angulated. ∎

######  Remark 4.5.18.

Suppose for simplicity that the field $k$ is perfect. In the context of
Corollary 4.5.17, it follows from [Dug12, Thm. 3.2] that there is a stable
bimodule isomorphism
$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)\cong{}_{1}\Lambda_{\sigma}$. An important
aspect of Corollary 4.5.17 is that this stable bimodule isomorphism is
witnessed by the restricted universal Massey product of $\mathcal{A}$.

Recall that the Nakayama automorphism $\eta$ of a Frobenius algebra $\Lambda$
is characterised (up to inner automorphisms) by the existence of an
isomorphism of $\Lambda$-bimodules $D\Lambda\cong{}_{1}\Lambda_{\eta}$.

######  Corollary 4.5.19 ([Dug12, Prop. 3.3], [GKO13, Sec. 5.4], [IO13, Cor.
4.6]).

Let $\mathcal{A}$ be a small DG category satisfying the assumptions in Setting
4.5.1 and such that the canonical fully faithful functor

$H^{0}(\mathbf{h})\colon
H^{0}(\mathcal{A})\longrightarrow\operatorname{D}^{\mathrm{c}}(\mathcal{A})$

identifies $H^{0}(\mathcal{A})$ with a $d\mathbb{Z}$-cluster tilting
subcategory of $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$. If
$H^{0}(\mathcal{A})$ is $ds$-Calabi–Yau, $\ell\in\mathbb{Z}$, in the sense
that there are functorial isomorphisms

$H^{0}(A)(y,x[d\ell])\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}DH^{0}(A)(x,y),\qquad
x,y\in H^{0}(A),$

then $\sigma^{-\ell}$ is a Nakayama automorphism for $\Lambda$. In particular,
the stable module category $\operatorname{\underline{mod}}\Lambda$ is
$((d+2)\ell-1)$-Calabi–Yau.

###### Proof.

The argument is standard; we reproduce it for the convenience of the reader.
The functorial isomorphism

$H^{0}(A)(c,c[d\ell])\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}DH^{0}(A)(c,c)$

can be interpreted as an isomorphism of $\Lambda$-bimodules
${}_{1}\Lambda_{\sigma^{-\ell}}\cong D\Lambda$ that exhibits $\sigma^{-\ell}$
as a Nakayama automorphism for $\Lambda$. This proves the first claim. To
prove the second claim, observe that Corollary 4.5.17 yields the existence of
an isomorphism

$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)\cong{}_{1}\Lambda_{\sigma},$

in $\operatorname{\underline{mod}}\Lambda^{e}$ that in turn induces an
isomorphism of functors

$\Omega_{\Lambda}^{(d+2)\ell}\cong\nu^{-1}$

on $\operatorname{\underline{mod}}\Lambda$, where
$\nu=-\otimes_{\Lambda}D\Lambda$ is the Nakayama functor. The claim follows
from the fact that $\nu\Omega_{\Lambda}$ is a Serre functor on
$\operatorname{\underline{mod}}\Lambda$, see [ES06, Prop. 1.2 and Cor. 1.3]
and compare also with [IV14, Thm. 1.8]. This finishes the proof. ∎

######  Corollary 4.5.20.

Let $\Lambda$ be a finite-dimensional self-injective algebra that is twisted
$(d+2)$-periodic with respect to an algebra automorphism $\sigma$ and let

$\Sigma\coloneqq-\otimes_{\Lambda}{}_{\sigma}{\Lambda}_{1}\colon\operatorname{proj}(\Lambda)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{proj}(\Lambda).$

If $\Lambda/J_{\Lambda}$ is separable, then, up to equivalence, the pair
$(\operatorname{proj}(\Lambda),\Sigma)$ admits at most one algebraic
$(d+2)$-angulation.

###### Proof.

Indeed, Theorem 4.5.8 shows that any two algebraic $(d+2)$-angulations of
$(\operatorname{proj}(\Lambda),\Sigma)$ coincide with the AL
$(d+2)$-angulations given by any choice of exact sequences with projective-
middle terms in
$\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$
that represent the corresponding restricted universal Massey products of the
enhancements, up to a sign depending on $d$, see also Remark 2.2.14. But such
AL $(d+2)$-angulations are unique up to equivalence by Proposition 2.2.18. ∎

######  Remark 4.5.21.

Notice that Corollary 4.5.20 _does not_ prove that any two enhancements are