# The Derived Auslander–Iyama Correspondence

Gustavo Jasso  Lund University, Centre for Mathematical Sciences, Slvegatan
18A, 221 00 Lund, Sweden<EMAIL_ADDRESS>https://gustavo.jasso.info
Université Paris Cité, UFR de Mathématiques, Case 7012, Bâtiment Sophie
Germain, 8 place Aurélie Nemours,75013 Paris Cedex 13, France
<EMAIL_ADDRESS>https://webusers.imj-prg.fr/ bernhard.keller/ and
Fernando Muro  Universidad de Sevilla, Facultad de Matemáticas, Departamento
de Álgebra, Calle Tarfia s/n, 41012 Sevilla, Spain<EMAIL_ADDRESS>https://personal.us.es/fmuro/ With an appendix by Bernhard Keller

###### Abstract.

We work over a perfect field. Recent work of the second-named author
established a Derived Auslander Correspondence that relates finite-dimensional
self-injective algebras that are twisted 3-periodic to algebraic triangulated
categories of finite type. Moreover, the aforementioned work also shows that
the latter triangulated categories admit a unique differential graded
enhancement. In this article we prove a higher-dimensional version of this
result that, given an integer $d\geq 1$, relates twisted $(d+2)$-periodic
algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster
tilting object. We also show that the latter triangulated categories admit a
unique differential graded enhancement. Our result yields recognition theorems
for interesting algebraic triangulated categories, such as the Amiot cluster
category of a self-injective quiver with potential in the sense of Herschend
and Iyama and, more generally, the Amiot–Guo–Keller cluster category
associated with a $d$-representation finite algebra in the sense of Iyama and
Oppermann. As an application of our result, we obtain infinitely many
triangulated categories with a unique differential graded enhancement that is
not strongly unique. In the appendix, B. Keller explains how—combined with
crucial results of August and Hua–Keller—our main result yields the last key
ingredient to prove the Donovan–Wemyss Conjecture in the context of the
Homological Minimal Model Program for threefolds.

###### Key words and phrases:

Triangulated categories; differential graded categories;
$A_{\infty}$-categories; cluster tilting; twisted periodicity; Massey
products; Hochschild cohomology; Toda brackets.

###### 2020 Mathematics Subject Classification:

Primary: 18G80; secondary: 18N40.

###### Contents

  1. 1 Cluster tilting subcategories
  2. 2 $(d+2)$-angulated categories
  3. 3 Enhanced $(d+2)$-angulated categories
  4. 4 Higher structures and $(d+2)$-angulations
  5. 5 Existence and uniqueness of enhancements
  6. 6 Recognition theorems
  7. A The Donovan–Wemyss conjecture, by Bernhard Keller

## Introduction

We work over a field $\mathbf{k}$. Triangulated categories, introduced by
Verdier in the late 1960s [Ver96] (see also Puppe’s [Pup62]) are central
objects of study in homological algebra [HJR10]. Typical examples of
triangulated categories include derived categories of rings and schemes,
stable categories of representations of finite groups, and the stable homotopy
category (of spectra). Despite their ubiquity and usefulness, it has long been
recognised that triangulated categories lack a number of desirable _formal
properties_. For example, limits of triangulated categories in general do not
exist, nor is there a reasonable tensor product operation on triangulated
categories; moreover, most triangulated categories do not admit interesting
(co)limits, such as pushouts and pullbacks, see [Toë11, Sec. 2.2] for a more
detailed discussion of these shortcomings.

In the seminal article [BK90], it is proposed to address the above issues by
endowing the triangulated categories that appear in practice (at least in
algebra and geometry) with what nowadays is called a ‘differential graded
enhancement.’ Recall that a differential graded (=DG) category [Kel65] is a
category-like structure in which the morphisms between two given objects form
a cochain complex, and whose composition law is compatible with the
differentials in a suitable sense. A DG category $\mathcal{A}$ has an
associated homotopy category $H^{0}(\mathcal{A})$, obtained by passing to the
$0$-th cohomology of its morphism complexes; when the DG category
$\mathcal{A}$ has the property of being _pre-triangulated_ (Definition 3.1.1),
the category $H^{0}(\mathcal{A})$ is canonically a triangulated category.
Given a triangulated category $\mathcal{T}$, we say that a pre-triangulated DG
category $\mathcal{A}$ is a _DG enhancement of $\mathcal{T}$_ if there is an
equivalence of triangulated categories $\mathcal{T}\simeq H^{0}(\mathcal{A})$.
Interpreting these in the appropriate manner, pre-triangulated DG categories
have all the expected formal properties that triangulated categories lack and,
moreover, all triangulated categories that arise naturally in algebra and
geometry (but, in general, not in topology) admit a DG enhancement, see
[Kel06] for details. For completeness, we mention that there are alternative
frameworks for enhancing triangulated categories, such as (stable) Quillen
model categories [HPS97, Hov99], spectral categories [SS03], triangulated
$A_{\infty}$-categories [LH, BLM08], stable $\infty$-categories [Lur17], etc.

Given a triangulated category $\mathcal{T}$, two natural questions arise: Does
$\mathcal{T}$ admit a DG enhancement? and if it does, is the enhancement
unique up to the appropriate notion of equivalence? In general, the first
question has a negative answer, for the stable homotopy category does not
admit a DG enhancement [Sch10], and there are even triangulated categories
that are linear over a field and do not admit a DG enhancement [RVdB20] (there
also exist triangulated categories that do not even admit ‘topological’
enhancements [MSS07]). The second question is also rather delicate. Indeed,
there exist triangulated categories that admit inequivalent DG enhancements
[DS07, Sch02], and there are even such triangulated categories that are linear
over a field [Kaj13, RVdB19]. Notwithstanding, there are several results that
show that important classes of triangulated categories, such as derived
categories of modules or of quasi-coherent sheaves, admit a unique DG
enhancement [LO10, CS18, CNS22]. We note that there are also several important
results on the uniqueness of ‘topological’ enhancements [Ant, Sch01, Sch07,
SS02].

On the other hand, recent work of the second-named author established an
existence and uniqueness result for a quite different class of triangulated
categories. We say that a triangulated category $\mathcal{T}$ that is linear
over a field is of _finite type_ if it has finite-dimensional morphism spaces
and split idempotents and, moreover, there exists an object $c\in\mathcal{T}$
such that every other object of $\mathcal{T}$ is a direct summand of a finite
direct sum of copies of $c$. This means, in particular, that there are only
finitely many indecomposable objects in $\mathcal{T}$ up to isomorphism (in
fact, the underlying additive category of $\mathcal{T}$ is equivalent to the
category of finitely-generated projective $\mathcal{T}(c,c)$-modules). We also
need the following terminology: Given an algebra $\Lambda$, recall that a
$\Lambda$-bimodule $I$ is _invertible_ if there exists a $\Lambda$-bimodule
$I^{-1}$ such that there are isomorphisms

$I\otimes_{\Lambda}I^{-1}\cong\Lambda\cong I^{-1}\otimes_{\Lambda}I$

of $\Lambda$-bimodules. A finite-dimensional basic self-injective algebra
$\Lambda$ is _twisted $n$-periodic_ if the $n$-th syzygy
$\Omega_{\Lambda^{e}}^{n}(\Lambda)$ of the diagonal $\Lambda$-bimodule is
stably isomorphic to an invertible $\Lambda$-bimodule, where $\Lambda^{e}$
denotes the enveloping algebra of $\Lambda$. Finally, we denote the category
of finitely-generated projective (right) $\Lambda$-modules by
$\operatorname{proj}(\Lambda)$. The following theorem summarises the main
results in [Mur22]. Recall that a field is perfect if it has characteristic
$0$ or if it has characteristic $p>0$ and every element has a $p$-th root.

######  Theorem (Derived Auslander Correspondence).

Suppose that $\mathbf{k}$ is a perfect field. There is a bijective
correspondence between the following:

  1. (1)

Quasi-isomorphism classes of DG algebras $A$ such that the perfect derived
category $\operatorname{D}^{\mathrm{c}}(A)$ is of finite type.

  2. (2)

Equivalence classes of algebraic triangulated categories $\mathcal{T}$ of
finite type.

  3. (3)

Equivalence classes of pairs $(\Lambda,I)$ consisting of

     * •

a finite-dimensional basic self-injective algebra $\Lambda$ that is twisted
$3$-periodic and

     * •

an invertible $\Lambda$-bimodule $I$ such that
$\Omega_{\Lambda^{e}}^{3}(\Lambda)\cong I$ in the stable category of
$\Lambda$-bimodules.

The correspondences are given by

$A\longmapsto(\operatorname{D}^{\mathrm{c}}(A),A)\qquad\text{and}\qquad(\mathcal{T},c)\longmapsto(\mathcal{T}(c,c),\mathcal{T}(c,c[-1])).$

Moreover, the algebraic triangulated categories in (2) admit a unique DG
enhancement.

In the context of the above theorem, two pairs $(\Lambda,I)$ and
$(\Lambda^{\prime},I^{\prime})$ are _equivalent_ if the pairs
$(\operatorname{proj}(\Lambda),-\otimes_{\Lambda}I)$ and
$(\operatorname{proj}(\Lambda),-\otimes_{\Lambda^{\prime}}I^{\prime})$ are
equivalent as pairs in the natural sense (Definition 2.1.10). In particular,
if the pairs $(\Lambda,I)$ and $(\Lambda^{\prime},I^{\prime})$ are equivalent,
then the algebras $\Lambda$ and $\Lambda^{\prime}$ are Morita equivalent (even
isomorphic, since they are basic). The restriction to basic algebras is
technically convenient since, under these assumptions, the Picard group of
invertible $\Lambda$-bimodules and the group of outer automorphisms of
$\Lambda$ are isomorphic. In particular, up to isomorphism, every invertible
$\Lambda$-bimodule is of the form ${}_{\sigma}{\Lambda}_{1}$ for some algebra
automorphism $\sigma\colon\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\Lambda$,
where ${}_{\sigma}{\Lambda}_{1}$ is the $\Lambda$-bimodule with underlying
right $\Lambda$-module $\Lambda_{\Lambda}$ but with left action twisted by
$\sigma$. We also observe that if the algebra $\Lambda$ is connected and non-
separable, then the invertible $\Lambda$-bimodule $I$ is uniquely determined
up to bimodule isomorphism by the algebra $\Lambda$, see [Mur22, Prop. 9.8]
(this observation generalises to arbitrary periods, see Proposition 2.2.6).

We have stated the main results in [Mur22] in the above form in order to
emphasise the analogy with a classical result in representation theory of
algebras: the famous Auslander correspondence [Aus71]. Recall that a finite-
dimensional algebra $A$ is said to have finite representation type if it
admits an additive generator, that is a finite-dimensional $A$-module $M$ such
that every other finite-dimensional $A$-module is a direct summand of a finite
direct sum of copies of $M$.

######  Theorem (Auslander Correspondence).

Let $\mathbf{k}$ be a field. There is a bijective correspondence between the
following:

  1. (1)

Morita equivalence classes of finite-dimensional algebras $A$ of finite
representation type and

  2. (2)

Morita equivalence classes of Auslander algebras, that is finite-dimensional
algebras $\Gamma$ with global dimension at most $2$ and dominant dimension
[Tac64] at least $2$.

The correspondence is given by the formula
$A\mapsto\operatorname{Hom}_{A}\\!\left(M,M\right)$, where $M$ is an additive
generator of the category of finite-dimensional $A$-modules.

Thus, the Derived Auslander Correspondence can be interpreted both as a result
concerning the general theory of triangulated categories and their DG
enhancements, and as a result in representation theory of finite-dimensional
algebras. Notice, moreover, the following parallel between the triangulated
and the classical Auslander Correspondence: in both cases, the second class of
algebras that is considered (twisted $3$-periodic algebras in the triangulated
case and Auslander algebras in the classical case) is part of an infinite
hierarchy of finite-dimensional algebras. Indeed, as part of his foundational
work on higher-dimensional Auslander–Reiten theory, Iyama introduced in
[Iya07a] the class of _$d$ -Auslander algebras_, obtained by replacing the
number $2$ by the number $d+1$ in the definition of Auslander algebra.
Remarkably, Iyama was also able to extend the classical Auslander
Correspondence to $d$-Auslander algebras by introducing the important concept
of a $d$-cluster tilting module [Iya07b] (see Definition 6.2.2). We state
Iyama’s higher-dimensional Auslander Correspondence below as it helps us
position the main result in this article in an appropriate context.

######  Theorem (Auslander–Iyama correspondence).

Let $\mathbf{k}$ be a field and $d\geq 1$ an integer. There is a bijective
correspondence between the following:

  1. (1)

Equivalence classes of pairs $(A,M)$ consisting of

     * •

a finite-dimensional algebra $A$ and

     * •

a $d$-cluster tilting $A$-module $M$.

  2. (2)

Morita equivalence classes of $d$-Auslander algebras.

The correspondence is given by the formula
$(A,M)\mapsto\operatorname{Hom}_{A}\\!\left(M,M\right)$.

In the context of the previous theorem, two pairs $(A,M)$ and
$(A^{\prime},M^{\prime})$ are _equivalent_ if there exists an equivalence

$\operatorname{mod}(A)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{mod}(A^{\prime})$

between the corresponding categories of finite-dimensional modules that
restricts to an equivalence between the closures of $M$ and $M^{\prime}$ under
finite direct sums and direct summands. In particular, if the pairs $(A,M)$
and $(A^{\prime},M^{\prime})$ are equivalent, then the algebras $A$ and
$A^{\prime}$ are Morita equivalent.

At this stage, it is worth pointing out that a $1$-cluster tilting module is
simply an additive generator of the ambient module category. In particular,
for $d=1$, the Auslander–Iyama Correspondence specialises to the classical
Auslander Correspondence. We also mention that there are other results in
representation theory of algebras that follow a similar pattern, such as the
the Auslander–Iyama–Solberg Correspondence [IS18] that extends the
correspondence in [AS93]. If $d=2$, other than in higher Auslander–Reiten
theory, $2$-cluster tilting modules (and their triangulated counter parts
[IY08]) play a crucial role in the categorification of Fomin–Zelevinsky
cluster algebras [FZ02] by means of $2$-Calabi–Yau triangulated categories
[Kel08], see for example [Ami09, BMR+06, BIRS09, DK08, GLS06, Kel13, Pal08]
among many others. Cluster tilting modules also appear in commutative algebra,
(possibly non-commutative) algebraic geometry and symplectic geometry, see for
example [AIR15, DJL21, HIMO, IT13, IW14].

### Our main result

Inspired by the above pattern, the aim of this article is to prove a suitable
higher-dimensional extension of the Derived Auslander Correspondence
established in [Mur22]. For the definition of $d\mathbb{Z}$-cluster tilting
object, see Definition 1.3.1 (and compare with Remark 1.3.5); here we only
mention that such objects are in particular $d$-step generators of the ambient
category in the sense of Rouquier [Rou08], but they generate in a stronger
sense (see Theorem 1.3.7).

######  Theorem A (Derived Auslander–Iyama Correspondence).

Suppose that $\mathbf{k}$ is a perfect field and $d\geq 1$ an integer. There
are bijective correspondences between the following:

  1. (1)

Quasi-isomorphism classes of DG algebras $A$ that satisfy the following:

    1. (a)

The ordinary algebra $H^{0}(A)$ is a basic finite-dimensional algebra.

    2. (b)

The free DG $A$-module $A\in\operatorname{D}^{\mathrm{c}}(A)$ is a
$d\mathbb{Z}$-cluster tilting object, that is $A$ is a $d$-cluster tilting
object that satisfies $A\cong A[d]$.

  2. (2)

Equivalence classes of pairs $(\mathcal{T},c)$ consisting of

    1. (a)

an algebraic triangulated category $\mathcal{T}$ with finite-dimensional
morphism spaces and split idempotents and

    2. (b)

a basic $d\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$, that is a
$d$-cluster tilting object such that $\mathcal{T}(c,c[i])=0$ for all $i\not\in
d\mathbb{Z}$.

  3. (3)

Equivalence classes of pairs $(\Lambda,I)$ consisting of

    1. (a)

a basic finite-dimensional self-injective algebra $\Lambda$ that is twisted
$(d+2)$-periodic and

    2. (b)

an invertible $\Lambda$-bimodule $I$ such that
$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)\cong I$ in the stable category of
$\Lambda$-bimodules.

The correspondences are given by

$A\longmapsto(\operatorname{D}^{\mathrm{c}}(A),A)\qquad\text{and}\qquad(\mathcal{T},c)\longmapsto(\mathcal{T}(c,c),\mathcal{T}(c,c[-d])).$

Moreover, the algebraic triangulated categories in (2) admit a unique DG
enhancement.

In the context of A, two pairs $(\mathcal{T},c)$ and
$(\mathcal{T}^{\prime},c^{\prime})$ as in (2) are _equivalent_ if there exists
an equivalence of triangulated categories

$F\colon\mathcal{T}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{T}^{\prime}$

such that $F(c)\cong c^{\prime}$. Similarly, two pairs $(\Lambda,I)$ and
$(\Lambda^{\prime},I^{\prime})$ as in (3) are equivalent if there exists an
algebra isomorphism
$\varphi\colon\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\Lambda^{\prime}$
such that $\varphi^{*}(I^{\prime})\cong I$. Notice also that a devisage
argument shows that $\operatorname{D}^{\mathrm{c}}(A)$ has finite-dimensional
morphism spaces if and only if for all $i\in\mathbb{Z}$ the vector space
$H^{i}(A)\cong\operatorname{Hom}_{\operatorname{D}(A)}\\!\left(A,A[i]\right)$
is finite dimensional, which is the case for the DG algebras considered in the
theorem. We also mention that the compact derived category of a non-
contractible DG algebra satisfying the conditions in A cannot be equivalent to
the derived category of an abelian or exact category: Indeed, the finite-
dimensionality of the morphism spaces implies that we would be dealing with a
bounded derived category, and such categories do not admit non-zero objects
that are isomorphic to any of its non-zero shifts. In particular, the results
in [CNS22] cannot be used to establish uniqueness of DG enhancements for the
triangulated categories in A(2).

As before, we note that a $1\mathbb{Z}$-cluster tilting object is simply an
additive generator of the ambient triangulated category (in particular, the
notions of $1$\- and $1\mathbb{Z}$-cluster tilting object coincide). Hence,
for $d=1$, A specialises to the Derived Auslander Correspondence from [Mur22].
In fact, our proof of A shares some methods with the second-named author’s
proof of the Derived Auslander Correspondence; in particular, we rely
crucially on techniques of homotopy theory as well as on the obstruction
theory developed by the second-named author in [Mur20b]. As we explain in more
detail later in this introduction, the central problem that arises is to
interpret the property of an object being $d\mathbb{Z}$-cluster tilting in
terms of the Hochschild(–Tate) cohomology of a certain graded algebra.

A shows a deep connection between $d\mathbb{Z}$-cluster tilting objects and
twisted $(d+2)$-periodic algebras. Indeed, the fact that the endomorphism
algebra of a $d\mathbb{Z}$-cluster tilting object in an triangulated category
is twisted $(d+2)$-periodic was observed first by Dugas [Dug12] in the
algebraic case and in general by Chan, Darpö, Iyama and Marczinzik [CDIM]
where they also investigate the connection between twisted periodicity and
higher Auslander–Reiten theory focusing on the fundamental class of trivial
extensions of finite-dimensional algebras. From the viewpoint promoted in this
article, this is a consequence of the relationship between
$d\mathbb{Z}$-cluster tilting objects and $(d+2)$-angulated categories and a
general characterisation of twisted periodicity [GSS03, Han20], see
Proposition 2.2.9. For more information on (twisted) periodic algebras and
their importance in representation theory and elsewhere in mathematics, we
refer the reader to [ES08, CDIM] and the references therein.

We wish to highlight that A is closely related to the theory of
$(d+2)$-angulated categories in the sense of Geiss, Keller and Oppermann
[GKO13] and their DG enhancements (which we introduce in this article).
Indeed, by the main result in [GKO13], the additive closure of a
$d\mathbb{Z}$-cluster tilting object in a triangulated category inherits the
structure of a $(d+2)$-angulated category (we recall this result as Theorem
2.2.1). One of the main results in this article (Theorem 5.1.3) establishes
the existence and uniqueness of enhancements for a class of $(d+2)$-angulated
categories introduced by Amiot [Ami07] in the case $d=1$ and by Lin [Lin19] in
the general case $d\geq 1$. From this perspective, A can be regarded as a
contribution to the emerging subject of higher-dimensional homological
algebra, where $(d+2)$-angulated categories and their exact/abelian variants
[Jas16] play a central role.

Notice that A cannot be generalised to triangulated categories with
$d$-cluster tilting objects that are not $d\mathbb{Z}$-cluster tilting, at
least not without further assumptions. On the one hand, for $d>2$, Ladkani
[Lad] has shown that _every_ basic finite-dimensional algebra $A$ over an
algebraically closed field arises as the endomorphism algebra of a
distinguished $d$-cluster tilting object in an algebraic triangulated category
constructed explicitly from a presentation of $A$ in terms of a quiver with
relations. This means that the endomorphism algebras of $d$-cluster tilting
objects do not admit any useful characterisation (homological or otherwise) in
this case. Moreover, still for $d>2$, Ladkani also has shown that there are
_non-equivalent_ algebraic triangulated categories with $d$-cluster tilting
objects whose endomorphism algebras are isomorphic. In both instances,
Ladkani’s constructions involve $d$-cluster tilting objects that are not
$d\mathbb{Z}$-cluster tilting, see Remark 6.1.7 for details and precise
references. On the other hand, for $d=2$, Keller and Lie have announced [KL23]
a general (and stronger) result for algebraic $2$-Calabi–Yau triangulated
categories that requires additional assumptions on the enhancements which are
necessary to apply a result of Van den Bergh [VdB15] in the course of the
proof, see Remark 6.3.4 for details. We also mention recent work of Hanihara
[Han22] that deals with ‘hereditary’ $d$-Calabi-Yau triangulated categories
with a $d$-cluster tilting object.

### Existence and uniqueness of $A_{\infty}$-structures

Our approach to the proof of A suggests a further interpretation in terms of
the existence and uniqueness of $A_{\infty}$-structures on certain graded
algebras that are $d$-sparse in the sense that their non-zero components are
concentrated in degrees that are multiples of $d$ (see Theorem 5.1.2 for a
precise statement). Indeed, to prove that the correspondence in A is
surjective we construct, given a basic twisted $(d+2)$-periodic algebra
$\Lambda$ with respect to an algebra automorphism
$\sigma\colon\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\Lambda$, a minimal
$A_{\infty}$-structure on the $d$-sparse graded algebra $\Lambda(\sigma,d)$
with non-zero components

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

with the property that its associated perfect derived category admits a
$d\mathbb{Z}$-cluster tilting object with graded endomorphism algebra
isomorphic to $\Lambda(\sigma,d)$. As explained previously in this
introduction, we establish the existence of such an $A_{\infty}$-algebra
structure by leveraging the obstruction theory developed in [Mur20a].

On the other hand, the uniqueness of enhancements for the triangulated
categories in A(2) (which is essentially equivalent to the injectivity of the
correspondences) can be explained in terms of Hochschild cohomology of
$d$-sparse graded algebras. For the sake of comparison, recall that a graded
algebra $A$ is intrinsically formal if every DG algebra (or, more generally,
$A_{\infty}$-algebra) $B$ such that $H^{\bullet}(B)\cong A$ as graded algebras
is already quasi-isomorphic to $A$ (equipped with the trivial differential and
vanishing higher operations). The following criterion for intrinsic formality,
due to Kadeishvili, is well known.

######  Theorem ([Kad88], see also [ST01, Thm. 4.7]).

Let $A$ be a graded algebra such that

$\operatorname{HH}^{p+2,-p}\\!\left(A,A\right)=0,\qquad p\geq 1.$

Then, the following statements hold:

  1. (1)

The graded algebra $A$ is intrinsically formal.

  2. (2)

If $B$ is a DG algebra with $H^{\bullet}(B)=A$, then $B$ is quasi-isomorphic
to $A$ (viewed as a DG algebra with vanishing differential) via a quasi-
isomorphism that is the identity in cohomology.

Kadeishvili’s Theorem can be applied, for example, to show that the algebra of
Laurent polynomials $\mathbf{k}[\imath^{\pm 1}]$ on a variable of
cohomological degree $-d$ is intrinsically formal (more generally,
Kadeishvili’s Theorem can be used to prove that the Laurent polynomial algebra
$K[\imath^{\pm}]=K\otimes_{\mathbf{k}}\mathbf{k}[\imath^{\pm}]$ is
intrinsically formal if $K$ is a finite-dimensional algebra of projective
dimension at most $d$ as a $K$-bimodule [Sai20, Cor. 4.2]). At this juncture
it is interesting to observe that the $d$-sparse graded algebra
$\Lambda(\sigma,d)$ introduced above can be identified with the twisted
Laurent polynomial algebra

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

However, in general $\Lambda(\sigma,d)$ is _far from being intrinsically
formal_ and, indeed, has a very rich Hochschild cohomology.

Nonetheless, we leverage a key insight (already present in [Mur20a, Mur22] in
the case $d=1$): Given a _$d$ -sparse Massey algebra_, which is a pair $(A,m)$
consisting $d$-sparse graded algebra $A$ and a class
$m\in\operatorname{HH}^{d+2,-d}\\!\left(A\right)$ whose Gerstenhaber square
$\operatorname{Sq}(m)$ vanishes (when the characteristic of $\mathbf{k}$ is
different from $2$, this is equivalent to the familiar Maurer–Cartan equation

$\frac{1}{2}[m,m]=0$

with respect to the usual Lie bracket on
$\operatorname{HH}^{\bullet,*}\\!\left(A\right)$), there is a bigraded cochain
complex $\operatorname{C}^{\bullet,*}\\!\left(A,m\right)$ whose vector spaces
of cochains are given by the Hochschild cohomology
$\operatorname{HH}^{\bullet,*}\\!\left(A\right)$ of the graded algebra $A$ and
whose differential of bidegree $(d+1,-d)$ is given by

$\partial\colon x\longmapsto\pm[m,x]$

almost everywhere (Definition 5.2.5). The Hochschild cohomology of the
$d$-sparse Massey algebra $(A,m)$ is the cohomology of this complex, denoted
by $\operatorname{HH}^{\bullet,*}\\!\left(A,m\right)$. The uniqueness of
enhancements for the triangulated categories in A(2) is closely related to the
following generalisation of Kadeishvili’s intrinsic formality theorem, which
is of independent interest (see the proofs of Theorem 5.3.3(1) and Theorem
5.1.2(2)).

######  Theorem B.

Let $(A,m)$ be a $d$-sparse Massey algebra $A$. Suppose that

$\operatorname{HH}^{p+2,-p}\\!\left(A,m\right)=0,\qquad p>d.$

If $B$ and $C$ are minimal $A_{\infty}$-algebras with
$H^{\bullet}(B)=H^{\bullet}(C)=A$ as graded algebras such that

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

then $B$ and $C$ are $A_{\infty}$-isomorphic through an
$A_{\infty}$-isomorphism with identity linear part.

Kadeishvili’s theorem is an immediate consequence of the case $d=1$ of B, see
Corollary 5.2.8. Although B likely has a direct proof, we deduce it from the
vanishing of certain terms in an extended spectral sequence of Bousfield–Kan
type that is the main technical tool in the obstruction theory developed in
[Mur20a]. Results similar to B have appeared in other contexts; for example,
in symplectic geometry, Lekili and Perutz have shown that the
$A_{\infty}$-structures on certain graded algebras arising from the Fukaya
category of the punctured torus are controlled by a _pair_ of Hochschild
cohomology classes, see [LP11] for details.

### Applications

Although A has a theoretical character, it has interesting consequences and
useful applications. First of all, notice that a twisted $(d+2)$-periodic
algebra $\Lambda$ is also twisted $(d+2)k$-periodic for every $k\geq 1$.
Choose an invertible $\Lambda$-bimodule $I$ such that
$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)\cong I$ in the stable category of
$\Lambda$-bimodules and, consequently,
$\Omega_{\Lambda^{e}}^{(d+2)k}(\Lambda)\cong I^{k}$ in the stable category of
$\Lambda$-bimodules for every $k\geq 1$. Thus, we may apply A to the infinite
sequence of pairs

$(\Lambda,I),\ (\Lambda,I^{\otimes 2}),\ (\Lambda,I^{\otimes 3}),\ \dots$

to obtain an infinite sequence of algebraic triangulated categories

$\mathcal{T}_{1},\ \mathcal{T}_{2},\ \mathcal{T}_{3},\ \cdots$

equipped with an $n_{k}\mathbb{Z}$-cluster tilting object
$c_{k}\in\mathcal{T}_{k}$ such that
$\mathcal{T}_{k}(c_{k},c_{k})\cong\Lambda$, where $n_{k}=dk+2k-2$. It would be
interesting to investigate whether there is a relationship between these
triangulated categories (see also Remark 3.2.10 and Theorem 5.1.3).

Our approach to the proof of A yields a more precise statement concerning the
equivalence classes of strong enhancements of the (algebraic)
$(d+2)$-angulated categories we consider (Theorem 5.4.10). As a consequence we
obtain, to our knowledge, the first examples of triangulated categories that
admit a unique enhancement but not a unique strong enhancement (that is, an
enhancement that is not ‘strongly unique’ in the sense of [LO10]), see
Corollary 5.4.12.

#### Recognition theorems

As for further applications, the bijectivity of the correspondence in A yields
recognition theorems for interesting classes of algebraic triangulated
categories that admit $d\mathbb{Z}$-cluster tilting objects, from which we
single out the following (we refer the reader to Section 6 for details and
definitions). Below, we restrict to connected non-separable algebras for
simplicity.

Following Iyama and Oppermann [IO11], we say that a finite-dimensional algebra
$A$ is _( $d$-hereditary) $d$-representation finite_ if $A$ has global
dimension at most $d$ and there exists a $d$-cluster tilting $A$-module (which
turns out to be unique up to multiplicity of its indecomposable direct
summands [Iya11]). For example, if $d=1$, then $A$ is a hereditary algebra of
finite representation type and, if the ground field is moreover algebraically
closed, then $A$ is Morita equivalent to the path algebra of a Dynkin quiver.
Recall that the the category of finitely-generated projective modules over the
Gelfand–Ponomarev preprojective algebra [GP79] of a Dynkin quiver admits an
algebraic (1-Calabi–Yau) triangulated structure [AR96] that is essentially
unique by the main result in [Mur22]. A yields the following analogous
recognition theorem.

######  Theorem (Theorem 6.2.7).

Suppose that $\mathbf{k}$ is a perfect field. The $d$-Calabi–Yau
Amiot–Guo–Keller cluster category of a connected, non-separable
$d$-representation finite algebra has a unique enhancement and is uniquely
characterised among algebraic triangulated categories by the existence of a
$d\mathbb{Z}$-cluster tilting object whose endomorphism algebra is isomorphic
to the $(d+1)$-preprojective algebra $\Pi_{d+1}(A)$ of $A$ in the sense of
[IO13].

In the case $d=1$ of the above theorem, the algebra $\Pi_{d+1}(A)=\Pi_{2}(A)$
is the preprojective algebra of $A$ as defined in [BGL87].

As a further application of A, we obtain the following recognition theorem for
the Amiot cluster category of a self-injective quiver with potential. This
result complements the Recognition Theorem for acyclic cluster categories of
Keller and Reiten [KR08], which deals with $2$-Calabi–Yau categories with a
cluster tilting object whose endomorphism algebra has acyclic Gabriel quiver.

######  Theorem (Theorem 6.3.1).

Let $\mathbf{k}$ be an arbitrary field. The $2$-Calabi–Yau Amiot cluster
category of a connected, non-separable self-injective quiver with potential
$(Q,W)$ has a unique enhancement and is uniquely characterised among algebraic
triangulated categories by the existence of a $2\mathbb{Z}$-cluster tilting
object whose endomorphism algebra is isomorphic to the (completed) Jacobian
algebra of $(Q,W)$.

Both of the above results are special cases of a more general recognition
theorem for the Amiot–Guo–Keller cluster category of a (pseudo-compact)
homologically smooth DG algebra that satisfies a number of technical
conditions that guarantee, in particular, the existence of a
$d\mathbb{Z}$-cluster tilting object, see Theorem 6.1.5.

#### Proof of the Donovan–Wemyss Conjecture

As a final application, in the appendix, B. Keller explains how—combined with
crucial results of August [Aug20] and Hua–Keller [HK]—our main result yields
the last key ingredient to prove the Donovan–Wemyss Conjecture ([DW16, Conj.
1.4] and [Aug20, Conj. 1.3]) in the context of the Homological Minimal Model
Program for threefolds [Wem18].

Let $X=\operatorname{Spec}R$ be the formal neighbourhood of an isolated
compound Du Val (=cDV) singularity over the field of complex numbers, that is
$R$ is a complete local three-dimensional hypersurface with an isolated
singularity whose generic hyperplane section is a Du Val/Kleinian singularity.
The class of cDV singularities was introduced by Reid [Rei83] who was
motivated by their natural occurrence in the minimal model program for three-
folds [KM98]. Suppose now that $X$ admits a crepant resolution (=smooth
minimal model) $p\colon\tilde{X}\to X$. To this geometric setup, Donovan and
Wemyss [DW16] associate a basic finite-dimensional algebra
$\Lambda=\Lambda(p)$ – the contraction algebra – that controls the non-
commutative deformations of the exceptional fibre of $p$, see [Wem23] for a
survey of the state of the art. Donovan and Wemyss made the following
remarkable conjecture [DW16, Aug20]:

> Let $R_{1}$ and $R_{2}$ be isolated cDV singularities that admit crepant
> resolutions $p_{1}\colon\tilde{X}_{1}\to X_{1}$ and
> $p_{2}\colon\tilde{X}_{2}\to X_{2}$ with corresponding contraction algebras
> $\Lambda(p_{1})$ and $\Lambda(p_{2})$. Then, the $\mathbb{C}$-algebras
> $R_{1}$ and $R_{2}$ are isomorphic if and only if the contraction algebras
> $\Lambda(p_{1})$ and $\Lambda(p_{2})$ are derived equivalent.

The Donovan–Wemyss Conjecture is considered the most important open problem in
the context of the Homological Minimal Model Program [Wem18]. It is known that
derived contraction algebras of a given isolated compound Du Val singularity
are derived equivalent [Wem18, Dug10]. In the appendix, Keller explains how to
reduce the remaining implication of the Donovan–Wemyss Conjecture to the
validity of the following statement:

> Let $R$ be an isolated cDV singularity that admits a crepant resolution
> $p\colon\tilde{X}\to X$ and
> $\mathrm{sg}(R)=\operatorname{D}^{\mathrm{b}}(\operatorname{mod}R)/\operatorname{K}^{\mathrm{b}}(\operatorname{proj}R)$
> its singularity category with its canonical DG enhancement
> $\mathrm{sg}_{dg}(R)$. Then, up to quasi-equivalence of DG categories,
> $\mathrm{sg}_{dg}(R)$ is uniquely determined by the following properties:
>
>   * •
>
> The algebraic triangulated category
> $H^{0}(\mathrm{sg}_{dg}{R})=\mathrm{sg}(R)$ has finite-dimensional morphism
> spaces and split idempotents.
>
>   * •
>
> There exists a $2\mathbb{Z}$-cluster tilting object in $\mathrm{sg}(R)$
> whose graded endomorphism algebra is isomorphic to the algebra of Laurent
> polynomials $\Lambda(p)[\imath^{\pm 1}]$, $|\imath|=-2$.
>
>

The above statement is a special case of A, and hence the remaining
implication of the Dononvan–Wemyss Conjecture follows, see the appendix for
details.

### Outline of the proof of A and structure of the article

The proof of A is involved and occupies the vast majority of the article. For
this reason, we provide a rough outline of the argument that we hope will help
the reader have a clear overview of the proof and of the article as a whole.

In Section 1 we recall the definition and basic aspects related to $d$-cluster
tilting subcategories. Most of the material in this section is well known to
specialists, but Theorem 1.3.9 (a characterisation of $d\mathbb{Z}$-cluster
tilting subcategories) is new.

In Section 2 we sum up the essentials of the theory of $(d+2)$-angulated
categories. Again, most of the material in this section is known to experts,
but Theorem 2.2.4 (essentially a reformulation of Theorem 1.3.9 in terms of
$(d+2)$-angulated structures) is new. In particular, Proposition 2.2.10 shows
that the correspondence in A is well defined. Moreover, we recall the
construction of $(d+2)$-angulated structures on the category of finitely-
generated projective modules over a twisted $(d+2)$-periodic algebra $\Lambda$
given by Amiot [Ami07] in the case $d=1$ and Lin [Lin19] in the general case
$d\geq 1$. In a nutshell, a choice of an exact sequence of $\Lambda$-bimodules

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

with projective-injective middle terms exhibiting the twisted periodicity of
$\Lambda$ defines a class of $(d+2)$-angles ${\pentagon}_{\eta}$ on
$\operatorname{proj}(\Lambda)$ that, up to equivalence, depends only on the
class of $\sigma$ in the outer automorphism group of $\Lambda$ and not on the
choice of $\eta$ (see Proposition 2.2.18, which is also new). We call these
structures Amiot–Lin (AL) $(d+2)$-angulations. In Section 3 we review the
theory of DG categories and their derived categories, and introduce DG
enhancements of $(d+2)$-angulated categories (Definition 3.2.5) as
straightforward generalisations of DG enhancements of triangulated categories.

We prove A by providing an inverse to the map
$(\mathcal{T},c)\mapsto(\Lambda,\sigma)$ that associates to an algebraic
triangulated category (with the relevant finiteness assumptions) with a
$d\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$ the corresponding
twisted $(d+2)$-periodic algebra $\Lambda=\mathcal{T}(c,c)$ together with the
algebra automorphism $\sigma$ induced by the action of the $d$-fold shift of
$\mathcal{T}$. Thus, given a pair $(\Lambda,\sigma)$ consisting of a twisted
$(d+2)$-periodic algebra with respect to the algebra automorphism $\sigma$, we
need to construct an _algebraic_ triangulated category $\mathcal{T}$ with a
$d\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$ together with a fully
faithful functor

$\displaystyle\operatorname{proj}(\Lambda)$
$\displaystyle\longrightarrow\mathcal{T}$ $\displaystyle\Lambda$
$\displaystyle\longmapsto c$

that is compatible with the actions of the functors
$-\otimes_{\Lambda}{}_{\sigma}{\Lambda}_{1}$ and $[d]$ on the source and the
target category, respectively. For this, we observe that in fact $\mathcal{T}$
must be equivalent to the perfect derived category
$\operatorname{D}^{\mathrm{c}}(A)$ of a DG algebra $A$ whose cohomology is
(isomorphic to) the $d$-sparse graded algebra

$\Lambda(\sigma,d)=\bigoplus_{di\in\mathbb{Z}}{}_{\sigma^{i}}{\Lambda}_{1},\qquad
a\cdot b=\sigma^{j}(a)b,\quad|b|=dj.$

Thus, we need to construct a DG algebra $A$ such that
$H^{\bullet}(A)\cong\Lambda(\sigma,d)$ and
${A\in\operatorname{D}^{\mathrm{c}}(A)}$ is a $d\mathbb{Z}$-cluster tilting
object. Equivalently, by Kadeishvili’s Homotopy Transfer Theorem, we need to
endow $\Lambda(\sigma,d)$ with a minimal $A_{\infty}$-algebra structure

$A=(\Lambda(\sigma,d),m_{d+2},m_{2d+2},m_{3d+2},\cdots),$

where $m_{i+2}=0$ if $i\not\in d\mathbb{Z}$ due to the fact that
$\Lambda(\sigma,d)$ is $d$-sparse (Proposition 4.3.4), in such a way that
$A\in\operatorname{D}^{\mathrm{c}}(A)$ is a $d\mathbb{Z}$-cluster tilting
object. Moreover, in order to obtain an inverse to the correspondence in A,
such a minimal $A_{\infty}$-algebra $A$ should be unique up to
$A_{\infty}$-quasi-isomorphism. In particular, we need to translate the
property ‘$A\in\operatorname{D}^{\mathrm{c}}(A)$ is a $d\mathbb{Z}$-cluster
tilting object’ into a property of the minimal $A_{\infty}$-algebra that can
be leveraged using techniques of homological and homotopical algebra.

Suppose first that $A$ is a minimal $A_{\infty}$-algebra with
$H^{\bullet}(A)=\Lambda(\sigma,d)$ and observe that, since the latter graded
algebra is $d$-sparse, the first non-trivial higher operation

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

is in fact a Hochschild _cocycle_ (Proposition 4.4.4). Thus, associated to $A$
there is a well-defined class in Hochschild cohomology

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

that we call its _universal Massey product of length $d+2$_. As it turns out,
most relevant to our purpose is the so-called _restricted universal Massey
product_

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

obtained by restricting $\left\\{m_{d+2}\right\\}$ along the morphism

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

induced by the inclusion $j\colon\Lambda\to\Lambda(\sigma,d)$ of the degree
$0$ part. Corollary 4.5.17 then establishes the following remarkable fact:
$A\in\operatorname{D}^{\mathrm{c}}(A)$ is a $d\mathbb{Z}$-cluster tilting
object if and only if the restricted universal Massey product
$j^{*}\left\\{m_{d+2}\right\\}$ is represented by an extension all of whose
middle terms are projective-injective $\Lambda$-bimodules. Furthermore, the
latter property is equivalent to the class $j^{*}\left\\{m_{d+2}\right\\}$
being a _unit_ in the Hochschild–Tate cohomology
$\underline{\operatorname{HH}}^{\bullet,*}\\!\left(\Lambda,\Lambda(\sigma,d)\right)$,
see Remark 4.5.16.

The proof of Corollary 4.5.17, which is at the core of the argument, relies on
a careful analysis of the relationship between Toda brackets, Massey products,
minimal $A_{\infty}$-structures, Hochschild cohomology and
$(d+2)$-angulations; we study these relationships in Section 4, the results of
which can be regarded as $(d+2)$-angulated analogues of results in [Mur20b,
Mur22] and of classical results for triangulated categories. To understand why
this technical excursion is essential in our approach, notice that if the
restricted universal Massey product
$j^{*}\left\\{m_{d+2}\right\\}\in\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$
is represented by an extension with projective-injective middle terms, then
there is a well-defined AL $(d+2)$-angulation
${\pentagon}_{j^{*}\left\\{m_{d+2}\right\\}}$ on
$\operatorname{proj}(\Lambda)$. On the other hand, a general theorem of Geiss,
Keller and Oppermann (Theorem 2.2.1) shows that

$\operatorname{proj}(\Lambda)\simeq\operatorname{add}(A)\subseteq\operatorname{D}^{\mathrm{c}}(A)$

is endowed with the structure of $(d+2)$-angulated category—which we call
standard—that is induced by the canonical triangulation of
$\operatorname{D}^{\mathrm{c}}(A)$, provided that
$A\in\operatorname{D}^{\mathrm{c}}(A)$ is a $d\mathbb{Z}$-cluster tilting
object. The proof of Corollary 4.5.17 relies on the fact that the AL
$(d+2)$-angulation and the standard $(d+2)$-angulation of
$\operatorname{proj}(\Lambda)$ coincide up to sign; we demonstrate this in
Theorem 4.5.8. Toda brackets enter the proof of Theorem 4.5.8 as these can be
used to detect standard $(d+2)$-angles (Theorem 4.1.12), and a certain
agreement between Toda brackets and Massey products on
$H^{\bullet}(A)=\Lambda(\sigma,d)$ (Theorem 4.2.6) turns out to be essential
for computations. Due to its technical nature, the reader may want to skip
Section 4 on a first reading and refer to it as necessary throughout their
reading of Section 5 (familiarity with Definitions 4.4.5 and 4.5.4 will
largely suffice).

Having established all the necessary technical preliminaries, in Section 5 we
prove A and B. We deduce A from a more precise statement, Theorem 5.1.2, which
allows us to construct the inverse of the correspondence: There exists an
essentially unique minimal $A_{\infty}$-algebra structure $A$ on
$\Lambda(\sigma,d)$ whose restricted universal Massey product is a unit in the
corresponding Hochschild–Tate cohomology, that is such that
$A\in\operatorname{D}^{\mathrm{c}}(A)$ is a $d\mathbb{Z}$-cluster tilting
object. The proofs of B and Theorem 5.1.2 utilise the same homotopical
techniques (which we discuss in the relevant subsections) leveraged in [Mur22]
to establish A in the case $d=1$ of triangulated categories of finite type; in
particular the obstruction theory developed in [Mur20a] by the second-named
author plays a crucial role in this part of the article. In Section 5.5 we
illustrate the necessity for the hypotheses in A and Theorem 5.1.3 by means of
examples and, in particular, show that there exist triangulated categories
with a $d\mathbb{Z}$-cluster tilting object that admit inequivalent
topological enhancements although they admit a unique ($\mathbf{k}$-linear) DG
enhancement (see Section 5.5.4). Finally, Section 6 includes several
recognition theorems that we deduce from A.

### Conventions

We fix a positive integer $d\geq 1$ once and for all (we occasionally let
$d=-1,0$, but this is always clear from the context). All algebras and
categories in this article are assumed to be linear over a ground field
$\mathbf{k}$, except for a few obvious examples; unless explicitly noted
otherwise, all modules are right modules. The Jacobson radical of an algebra
$\Lambda$ is denoted by $J_{\Lambda}$. Our main results require $\mathbf{k}$
to be a perfect field, and this assumption is indicated whenever relevant.
Unless noted otherwise, all (ordinary) categories considered in this article
are additive; accordingly, all subcategories are assumed to be (strictly) full
and closed under the formation of finite direct sums in the ambient category.
Given a collection $X$ of objects in a category $\mathcal{C}$ with split
idempotents (=Karoubian), we denote by $\operatorname{add}(X)$ the smallest
(additive) subcategory of $\mathcal{C}$ that contains $X$ and is closed under
direct summands; if $X=\left\\{x\right\\}$ consists of a single object, we
write $\operatorname{add}(x)=\operatorname{add}(X)$. We say that an object $c$
in a category in which the Krull–Remak–Schmidt Theorem holds (for example, in
an additive category with finite-dimensional morphism spaces and split
idempotents [Kra15, Cor. 4.4]) is _basic_ if in any decomposition
$c=c_{1}\oplus c_{2}\oplus\cdots\oplus c_{n}$ into indecomposable objects,
there is an isomorphism $c_{i}\cong c_{j}$ if and only if $i=j$. A finite-
dimensional algebra $A$ is _basic_ if the regular representation of $A$ is
basic as an object of its category of finite-dimensional (projective) modules.
We usually denote the shift/suspension functor in a triangulated category by
$[1]$, although the notation $\Sigma$ is used when dealing with general
$(d+2)$-angulated categories. If $\mathcal{T}$ is a triangulated category with
split idempotents and $X$ is a class of objects in $\mathcal{T}$, we denote by
$\operatorname{thick}(X)$ the smallest triangulated subcategory of
$\mathcal{T}$ that contains $X$ and is closed under direct summands; if
$X=\left\\{x\right\\}$ consists of a single object, we write
$\operatorname{thick}(x)=\operatorname{thick}(X)$. Finally, we compose
morphisms in a category as functions: the composite of morphisms $f\colon x\to
y$ and $g\colon y\to z$ is the morphism $gf=g\circ f\colon x\to z$.

### Acknowledgements

The authors thank Peter Jørgensen, Zhengfang Wang and the Uppsala
Representation Theory Group for comments on a previous versions of this
article, as well as Yankı Lekili for bringing references [Sei15, LP11] to our
attention. They are particularly grateful to Julian Külshammer for his
numerous comments and suggestions. The authors also wish to thank A. Lorenzin
for a question that lead to the inclusion of Section 5.4. Finally, the authors
thank Martin Herschend for discussions in relation to extending the scope of
our main results, see Remark 2.2.19.

### Financial support

G. J. was partially supported by the Deutsche Forschungsgemeinschaft (German
Research Foundation) under Germany’s Excellence Strategy – GZ 2047/1, Projekt-
ID 390685813 as well by the Swedish Research Council (Vetenskapsrådet)
Research Project Grant ‘Higher structures in higher-dimensional homological
algebra.’ F. M. was partially supported by grants PID2020-117971GB-C21 funded
by MCIN/AEI/10.13039/501100011033, US-1263032 (US/JUNTA/FEDER, UE), and
P20_01109 (JUNTA/FEDER, UE).

##  1\. Cluster tilting subcategories

######  Setting 1.0.1.

We fix a triangulated category $\mathcal{T}$ with finite-dimensional morphism
spaces and split idempotents as well as a subcategory
$\mathcal{C}\subseteq\mathcal{T}$ that satisfies
${\mathcal{C}=\operatorname{add}(\mathcal{C})}$.

###  1.1. Functorially finite subcategories

We begin by recalling a classical definition due to Auslander and Smalø, and
independently to Enochs.

######  Definition 1.1.1 ([AS80, AS81] and [Eno81]).

The subcategory $\mathcal{C}\subseteq\mathcal{T}$ is _contravariantly finite
in $\mathcal{T}$_ if the following condition is satisfied:

  1. (1)

For each object $x\in\mathcal{T}$ there exist an object $c\in\mathcal{C}$ and
a morphism $f\colon c\to x$ with the following property: For each morphism
$g\colon c^{\prime}\to x$ with $c^{\prime}\in\mathcal{C}$, there exists a (not
necessarily unique) morphism $h\colon c^{\prime}\to c$ such that $g=f\circ h$.

${c^{\prime}}$${c}$${x}$$\scriptstyle{\forall g}$$\scriptstyle{\exists
h}$$\scriptstyle{f}$

Equivalently, the induced linear map

$\mathcal{T}(c^{\prime},c)\longrightarrow\mathcal{T}(c^{\prime},x),\qquad
h\longmapsto f\circ h,$

is surjective (but not necessarily injective). Such a morphism $f$ is called a
_right $\mathcal{C}$-approximation of $x$_.

Dually, the subcategory $\mathcal{C}\subseteq\mathcal{T}$ is _covariantly
finite in $\mathcal{T}$_ if the following condition is satisfied:

  1. (1)

For each object $x\in\mathcal{T}$ there exist an object $c\in\mathcal{C}$ and
a morphism $f\colon x\to c$ with the following property: For each morphism
$g\colon x\to c^{\prime}$ with $c^{\prime}\in\mathcal{C}$, there exists a (not
necessarily unique) morphism $h\colon c\to c^{\prime}$ such that $g=h\circ f$.

${x}$${c}$${c^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\forall
g}$$\scriptstyle{\exists h}$

Equivalently, the induced linear map

$\mathcal{T}(c,c^{\prime})\longrightarrow\mathcal{T}(x,c^{\prime}),\qquad
h\longmapsto h\circ f,$

is surjective (but not necessarily injective). Such a morphism $f$ is called a
_left $\mathcal{C}$-approximation of $x$_.

Finally, the subcategory $\mathcal{C}\subseteq\mathcal{T}$ is _functorially
finite in $\mathcal{T}$_ if it is both contravariantly finite and covariantly
finite in $\mathcal{T}$.

######  Remark 1.1.2.

Contravariantly finite subcategories are also called ‘pre-covering’ and right
approximations are also called ‘pre-covers’. Similarly, covariantly finite
subcategories are also called ‘pre-enveloping’ and left approximations are
also called ‘pre-envelopes,’ see [Eno81].

######  Example 1.1.3.

Suppose that $\mathcal{C}$ is a right admissible subcategory of $\mathcal{T}$,
that is the inclusion $\mathcal{C}\subseteq\mathcal{T}$ admits a right adjoint
$R\colon\mathcal{T}\to\mathcal{C}$. Let $x\in\mathcal{T}$ and $c=R(x)$. The
co-unit $\varepsilon\colon c\to x$ is a right $\mathcal{C}$-approximation of
$x$, for it induces natural isomorphisms

$\mathcal{T}(c^{\prime},c)=\mathcal{T}(c^{\prime},R(x))\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathcal{T}(c^{\prime},x),\qquad
c^{\prime}\in\mathcal{C}.$

In particular, one may think of contravariantly finite subcategories as mild
generalisations of right admissible subcategories. Similarly, covariantly
finite subcategories are mild generalisations of left admissible
subcategories, that is subcategories such that the inclusion admits a left
adjoint.

The following basic example of a functorially finite subcategory is the most
relevant for the purposes of this article.

######  Example 1.1.4.

Let $c\in\mathcal{T}$ and $\mathcal{C}=\operatorname{add}(c)$. Then, the
subcategory $\mathcal{C}$ is functorially finite in $\mathcal{T}$. Indeed,
$\mathcal{C}$-approximations of an object $x\in\mathcal{T}$ are easily
constructed from a decomposition $c=c_{1}\oplus\cdots\oplus c_{k}$ into
indecomposable objects and bases of the (finite-dimensional) vector spaces
$\mathcal{T}(c_{i},x)$ and $\mathcal{T}(x,c_{i})$, $1\leq i\leq k$.

######  Remark 1.1.5.

Suppose that $\mathcal{C}$ is contravariantly finite in $\mathcal{T}$. For
each object $x\in\mathcal{T}$ there exists a commutative diagram, constructed
inductively, of the form

${\cdots}$${c_{2}}$${c_{1}}$${c_{0}}$${\cdots}$${x_{2}}$${x_{1}}$${x_{0}=x}$$\scriptstyle{+1}$$\scriptstyle{+1}$

in which each of the morphisms $c_{i}\to x_{i}$ is a right
$\mathcal{C}$-approximation and, as is customary, the triangles

${c_{i}}$${x_{i+1}}$${x_{i}}$$\scriptstyle{+1}$

represent exact triangles

$x_{i+1}\longrightarrow c_{i}\longrightarrow x_{i}\longrightarrow x_{i+1}[1].$

Dually, if $\mathcal{C}$ is covariantly finite in $\mathcal{T}$, then for each
object $x$ of $\mathcal{T}$ there exists a commutative diagram of the form

${c^{0}}$${c^{1}}$${c^{2}}$${\cdots}$${x=x^{0}}$${x^{1}}$${x^{2}}$${\cdots}$$\scriptstyle{+1}$$\scriptstyle{+1}$

in which each of the morphisms $x^{i}\to c^{i}$ is a left
$\mathcal{C}$-approximation and with corresponding exact triangles

$x^{i}\longrightarrow c^{i}\longrightarrow x^{i+1}\longrightarrow x^{i}[1].$

From the point of view of this article, the existence of such
‘$\mathcal{C}$-(co)resolutions’ is one of the main motivations for considering
functorially finite subcategories.

###  1.2. The Verdier product

The following elementary operation on subcategories of a triangulated category
is needed in the proof of Theorem 1.3.9 below. All subcategories
$\mathcal{X}\subseteq\mathcal{T}$ considered in this section are assumed to
satisfy $\operatorname{add}(\mathcal{X})=\mathcal{X}$.

######  Definition 1.2.1.

Let $\mathcal{X},\mathcal{Y}\subseteq\mathcal{T}$ be subcategories. The
_Verdier product of $\mathcal{X}$ with $\mathcal{Y}$_ is the full subcategory
$\mathcal{X}*\mathcal{Y}$ of $\mathcal{T}$ spanned by the objects
$t\in\mathcal{T}$ such that there exists an exact triangle

$x\longrightarrow t\longrightarrow y\longrightarrow x[1]$

with $x\in\mathcal{X}$ and $y\in\mathcal{Y}$.

######  Notation 1.2.2.

Given subcategories $\mathcal{X},\mathcal{Y}\subseteq\mathcal{T}$, we let
$\mathcal{X}\vee\mathcal{Y}$ be the smallest additive subcategory of
$\mathcal{T}$ containing $\mathcal{X}$ and $\mathcal{Y}$. Obviously,
$\mathcal{X}\vee\mathcal{Y}\subseteq\mathcal{X}*\mathcal{Y}$ and
$\mathcal{X}\vee\mathcal{Y}=\mathcal{Y}\vee\mathcal{X}$.

######  Lemma 1.2.3.

Let $\mathcal{X},\mathcal{Y},\mathcal{Z}\subseteq\mathcal{T}$ be
subcategories. The following statements hold:

  1. (1)

The Verdier product is associative:
$(\mathcal{X}*\mathcal{Y})*\mathcal{Z}=\mathcal{X}*(\mathcal{Y}*\mathcal{Z})$.

  2. (2)

Suppose that

$\forall x\in\mathcal{X},\ \forall
y\in\mathcal{Y},\qquad\mathcal{T}(y,x[1])=0.$

Then,
$\mathcal{X}*\mathcal{Y}=\mathcal{X}\vee\mathcal{Y}\subseteq\mathcal{Y}*\mathcal{X}$.

  3. (3)

If

$\forall x\in\mathcal{X},\ \forall y\in\mathcal{Y},\qquad\mathcal{T}(x,y)=0,$

then $\mathcal{X}*\mathcal{Y}$ is closed under direct summands.

###### Proof.

(1) This is an immediate consequence of the octahedral axiom.

(2) The equality $\mathcal{X}*\mathcal{Y}=\mathcal{X}\vee\mathcal{Y}$ follows
from the fact that every triangle

$x\to z\to y\to x[1]$

with $x\in\mathcal{X}$ and $y\in\mathcal{Y}$ splits, for the morphism $y\to
x[1]$ vanishes by assumption. The inclusion
$\mathcal{X}\vee\mathcal{Y}=\mathcal{Y}\vee\mathcal{X}\subseteq\mathcal{Y}*\mathcal{X}$
is obvious.

(3) See [IY08, Prop. 2.1(1)]. ∎

######  Example 1.2.4.

Let $\mathcal{C}\subseteq\mathcal{T}$ be a subcategory. The reader can easily
verify, using the rotation axiom, that the Verdier product
$\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]$ coincides with the full
subcategory of $\mathcal{T}$ spanned by those objects $x\in\mathcal{T}$ for
which there exists a commutative diagram of the form

${c_{d-2}}$${\cdots}$${c_{1}}$${c_{0}}$${c_{d-1}}$${x_{d-2}}$${\cdots}$${x_{2}}$${x_{1}}$${x_{0}=x}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$

in which $c_{i}\in\mathcal{C}$, $0\leq i<d$. In other words,
$\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]$ is the full subcategory
of $\mathcal{T}$ spanned by those objects that admit a
$\mathcal{C}$-resolution of length (at most) $d$.

###  1.3. Cluster tilting subcategories

The following remarkable class of subcategories of a triangulated category,
introduced by Iyama and Yoshino (see also [Iya07a, Iya07b, IJ17] for the case
of abelian categories), is the main focus of this article.

######  Definition 1.3.1 ([IY08, Sec. 3]).

The subcategory $\mathcal{C}$ is _$d$ -cluster tilting_ in $\mathcal{T}$ if
the following three conditions are satisfied:

  1. (1)

The subcategory $\mathcal{C}$ is functorially finite in $\mathcal{T}$.

  2. (2)

An object $x\in\mathcal{T}$ lies in $\mathcal{C}$ if and only if

$\forall c\in\mathcal{C},\ \forall 0<i<d,\qquad\mathcal{T}(x,c[i])=0.$

  3. (3)

An object $x\in\mathcal{T}$ lies in $\mathcal{C}$ if and only if

$\forall c\in\mathcal{C},\ \forall 0<i<d,\qquad\mathcal{T}(c,x[i])=0.$

The subcategory $\mathcal{C}$ is _$d\mathbb{Z}$ -cluster tilting in
$\mathcal{T}$_ if, in addition to the three conditions above, $\mathcal{C}$
satisfies the following condition:

  1. (4)

The equality $\mathcal{C}=\mathcal{C}[d]$ holds.

We say that an object $c\in\mathcal{T}$ is _$d$ \- or $d\mathbb{Z}$-cluster-
tilting in $\mathcal{T}$_, if the subcategory
$\mathcal{C}=\operatorname{add}(c)$ is $d$\- or $d\mathbb{Z}$-cluster tilting
in $\mathcal{T}$, respectively.

######  Remark 1.3.2.

Let $d=1$. By vacuity, a subcategory $\mathcal{C}\subseteq\mathcal{T}$ is
$1$-cluster tilting if and only if $\mathcal{C}=\mathcal{T}$. Consequently,
the notions of $1$-cluster tilting and $1\mathbb{Z}$-cluster tilting coincide.
In particular, $c\in\mathcal{T}$ is a $1$-cluster tilting object
(=$1\mathbb{Z}$-cluster tilting object) if and only if
$\operatorname{add}(c)=\mathcal{T}$, that is $c$ generates $\mathcal{T}$ under
finite direct sums and direct summands.

######  Definition 1.3.3.

A subcategory $\mathcal{C}\subseteq\mathcal{T}$ is _$d$ -rigid_ if

$\forall c,c^{\prime}\in\mathcal{C},\ \forall
0<i<d,\qquad\mathcal{T}(c,c^{\prime}[i])=0.$

We say that $\mathcal{C}$ is _$d\mathbb{Z}$ -rigid_ if, moreover, for each
$c,c^{\prime}\in\mathcal{C}$ and each $i\in\mathbb{Z}$,

$i\not\in
d\mathbb{Z}\qquad\Longrightarrow\qquad\mathcal{T}(c,c^{\prime}[i])=0.$

######  Remark 1.3.4.

By definition, $d$-cluster tilting subcategories are $d$-rigid.

######  Remark 1.3.5.

Let $\mathcal{C}\subseteq\mathcal{T}$ be a $d$-cluster tilting subcategory. It
follows immediately from the definition of $d$-cluster tilting subcategory
that the following conditions are equivalent:

  1. (1)

The equality $\mathcal{C}=\mathcal{C}[d]$ holds.

  2. (2)

The subcategory $\mathcal{C}$ is $d\mathbb{Z}$-rigid.

In other words, the $d\mathbb{Z}$-cluster tilting subcategories are precisely
the $d$-cluster tilting subcategories that are $d\mathbb{Z}$-rigid.

######  Lemma 1.3.6.

Let $\mathcal{C}\subseteq\mathcal{T}$ be a $d$-rigid subcategory. For all
$0\leq j<i<d$ such that $(j,i)\neq(0,d-1)$, the inclusion

$\mathcal{C}[i]*\mathcal{C}[j]\subseteq\mathcal{C}[j]*\mathcal{C}[i]$

holds.

###### Proof.

Given that $\mathcal{C}\subseteq\mathcal{T}$ is a $d$-rigid subcategory, the
assumption on $i$ and $j$ implies that

$\forall c,c^{\prime}\in\mathcal{C},\qquad\mathcal{T}(c,c^{\prime}[i-j+1])=0.$

Thus, by Lemma 1.2.3(2) the inclusion
$\mathcal{C}[i-j]*\mathcal{C}\subseteq\mathcal{C}*\mathcal{C}[i-j]$ holds.
Since

$\mathcal{C}[i]*\mathcal{C}[j]\subseteq\mathcal{C}[j]*\mathcal{C}[i]\qquad\Longleftrightarrow\qquad\mathcal{C}[i-j]*\mathcal{C}\subseteq\mathcal{C}*\mathcal{C}[i-j],$

the claim follows. ∎

The class of $d$-cluster tilting subcategories admits the following useful
characterisation, due to Iyama and Yoshino ((1)$\Rightarrow$(2)) and to
Beligiannis ((2)$\Rightarrow$(1)).

######  Theorem 1.3.7 ([IY08, Thm. 3.1], [Bel15, Thm. 5.3]).

Let $\mathcal{C}\subseteq\mathcal{T}$ be a subcategory that satisfies
$\mathcal{C}=\operatorname{add}(\mathcal{C})$. The following statements are
equivalent:

  1. (1)

The subcategory $\mathcal{C}$ is $d$-cluster tilting in $\mathcal{T}$.

  2. (2)

The following conditions are satisfied:

    1. (a)

The subcategory $\mathcal{C}$ is $d$-rigid.

    2. (b)

The equality $\mathcal{T}=\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]$
holds.

###### Proof.

The implication (1)$\Rightarrow$(2) is proven in [IY08, Thm. 3.1] while the
converse implication (2)$\Rightarrow$(1) is part of [Bel15, Thm. 5.3], which
contains many more equivalent conditions and requires the subcategory
$\mathcal{C}\subseteq\mathcal{T}$ to be contravariantly finite. We prove that
the latter property in fact follows from the assumptions in part (2) of the
statement theorem. The argument is well known to experts (compare with [GKO13,
Lemma 4.3]); we include it for the sake of completeness.

Suppose that the conditions in part (2) of the statement of the theorem are
satisfied. If $d=1$, then $\mathcal{C}=\mathcal{T}$ and there is nothing to
prove; thus, we may assume that $d\geq 2$. By Example 1.2.4, for every object
$x\in\mathcal{T}$ there exists a commutative diagram of the form

${c_{d-2}}$${\cdots}$${c_{1}}$${c_{0}}$${c_{d-1}}$${x_{d-2}}$${\cdots}$${x_{2}}$${x_{1}}$${x=x_{0}}$$\scriptstyle{+1}$$\scriptstyle{+1}$$\scriptstyle{+1}$

in which $c_{i}\in\mathcal{C}$, $0\leq i<d$. We claim that

$\forall c\in\mathcal{C},\ \forall 0<i\leq j\leq
d-2,\qquad\mathcal{T}(c,x_{j}[i])=0.$

We prove the claim by reverse induction on $j$. If $j=d-2$, then for each
$0<i\leq d-2$ and each $c\in\mathcal{C}$ there is an induced exact sequence

$0=\mathcal{T}(c,c_{d-2}[i])\longrightarrow\mathcal{T}(c,x_{d-2}[i])\longrightarrow\mathcal{T}(c,c_{d-1}[i+1])=0$

in which the outer terms vanish since the subcategory $\mathcal{C}$ is
$d$-rigid; this proves the claim in this case. Suppose now that the claim
holds for some $1<j\leq d-2$. Then, for each $0<i\leq j-1$ and each
$c\in\mathcal{C}$ there is an induced exact sequence

$0=\mathcal{T}(c,c_{j-1}[i])\longrightarrow\mathcal{T}(c,x_{j-1}[i])\longrightarrow\mathcal{T}(c,x_{j}[i+1])=0$

in which the right-most term vanishes by the inductive hypothesis and the
left-most terms since the subcategory $\mathcal{C}$ is $d$-rigid; this
finishes the induction step. We claim now that the morphism $c_{0}\to x_{0}$
is in fact a right $\mathcal{C}$-approximation of $x=x_{0}$. Indeed, there is
an induced exact sequence

$\mathcal{T}(c,c_{0})\longrightarrow\mathcal{T}(c,x_{0})\longrightarrow\mathcal{T}(c,x_{1}[1])=0$

in which the term on the right vanishes, as shown above. This proves the
claim. Since the object $x\in\mathcal{T}$ was chosen arbitrarily, this shows
that $\mathcal{C}$ is contravariantly finite in $\mathcal{T}$, which is what
we needed to prove. ∎

######  Remark 1.3.8.

Let $\mathcal{C}\subseteq\mathcal{T}$ be a $d$-cluster tilting subcategory.
Then, $\mathcal{C}$ generates $\mathcal{T}$ as a triangulated category:
indeed, $\mathcal{T}=\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]$ and
the right-hand side is of course contained in the triangulated closure of
$\mathcal{C}$. Thus, in particular, $\mathcal{C}$ is a $d$-step generator in
the sense of [Rou08]. However, it is remarkable that every object in
$\mathcal{T}$ admits a $\mathcal{C}$-resolution of length at most $d$ (not
involving any shifts).

The following characterisation of $d\mathbb{Z}$-cluster tilting subcategories
is quite natural from the point of view of $(d+2)$-angulated categories, whose
definition we recall in Definition 2.1.1.

######  Theorem 1.3.9.

Let $\mathcal{C}\subseteq\mathcal{T}$ be a subcategory of $\mathcal{T}$ that
satisfies $\mathcal{C}=\operatorname{add}(\mathcal{C})$. The following
statements are equivalent:

  1. (1)

The subcategory $\mathcal{C}$ is $d\mathbb{Z}$-cluster tilting in
$\mathcal{T}$.

  2. (2)

The following conditions are satisfied:

    1. (a)

The subcategory $\mathcal{C}$ is $d$-rigid.

    2. (b)

The equality $\mathcal{C}=\mathcal{C}[d]$ holds.

    3. (c)

The inclusion
$(\mathcal{C}[-1]*\mathcal{C})\subseteq(\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1])$
holds.

    4. (d)

The subcategory $\mathcal{C}$ satisfies
$\operatorname{thick}(\mathcal{C})=\mathcal{T}$.

###### Proof.

The implication (1)$\Rightarrow$(2) follows immediately from the definition of
$d\mathbb{Z}$-cluster tilting subcategory and Theorem 1.3.7.

(2)$\Rightarrow$(1) Notice that the case $d=1$ is trivial, for assumptions
(2b) and (2c) imply that $\mathcal{C}$ is closed under (co)cones and, in view
of assumption (2d), we must have
$\mathcal{C}=\operatorname{add}(\mathcal{C})=\mathcal{T}$ (assumption (2a) is
vacuous in this case).

Suppose now that $d\geq 2$. Since $\mathcal{C}=\mathcal{C}[d]$ by assumption,
we only need to prove that $\mathcal{C}$ is a $d$-cluster tilting subcategory
of $\mathcal{T}$; for this, we shall verify the remaining condition in Theorem
1.3.7(2), namely that

$\mathcal{T}=\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1].$

Let $\operatorname{thick}(\mathcal{C})$ be the smallest thick subcategory of
$\mathcal{T}$ containing $\mathcal{C}$. Clearly [AI12, Lemma 2.15(a)]

$\mathcal{T}=\operatorname{thick}(\mathcal{C})=\bigcup_{\ell>0,\
n_{1},\dots,n_{\ell}\in\mathbb{Z}}\operatorname{smd}(\mathcal{C}[n_{1}]*\cdots*\mathcal{C}[n_{\ell}]),$

where $\mathcal{X}\mapsto\operatorname{smd}\mathcal{X}$ denotes the passage to
the closure under direct summands (recall that
$\operatorname{thick}(\mathcal{C})=\mathcal{T}$ by assumption (2d)). We make
the following observations:

  * •

Since $\mathcal{C}=\mathcal{C}[d]$, for each $n\in\mathbb{Z}$ there exists an
integer $0\leq i<d$ such that $\mathcal{C}[n]=\mathcal{C}[i]$. Consequently,

$\mathcal{T}=\operatorname{thick}(\mathcal{C})=\bigcup_{\ell>0,\ 0\leq
n_{1},\dots,n_{\ell}<d}\operatorname{smd}(\mathcal{C}[n_{1}]*\cdots*\mathcal{C}[n_{\ell}]).$

  * •

Since $\mathcal{C}$ is a $d$-rigid subcategory of $\mathcal{T}$, Lemma
1.2.3(3) readily implies that

$\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]=\operatorname{smd}(\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]).$

Thus, it is enough to prove that, given $\ell>0$ and $0\leq
n_{1},\dots,n_{\ell}<d$, the inclusion

$\mathcal{C}[n_{1}]*\cdots*\mathcal{C}[n_{\ell}]\subseteq\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]$

holds. We prove this claim by induction on $\ell$. For $\ell=1$ the claim is
obvious, so suppose that the claim holds for some $\ell>0$. Let $0\leq
n_{1},\dots,n_{\ell},n<d$ and notice that, by the inductive hypothesis, the
inclusion

$(\mathcal{C}[n_{1}]*\cdots*\mathcal{C}[n_{\ell}])*\mathcal{C}[n]\subseteq(\mathcal{C}*\mathcal{C}[1]*\cdots\mathcal{C}[d-1])*\mathcal{C}[n].$

holds. We distinguish two cases:

$n=0$:

Since $\mathcal{C}[d-1]=(\mathcal{C}[d])[-1]=\mathcal{C}[-1]$, the equality

$\mathcal{C}[d-1]*\mathcal{C}[n]=\mathcal{C}[-1]*\mathcal{C}$

holds. Moreover, by assumption,

$(\mathcal{C}[-1]*\mathcal{C})\subseteq(\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]).$

Consequently,

$\displaystyle(\mathcal{C}[n_{1}]*\cdots*\mathcal{C}[n_{\ell}])*\mathcal{C}[n]$
$\displaystyle\subseteq(\mathcal{C}*\mathcal{C}[1]*\cdots\mathcal{C}[d-1])*\mathcal{C}[n]$
$\displaystyle=(\mathcal{C}*\cdots*\mathcal{C}[d-2])*(\mathcal{C}[d-1]*\mathcal{C}[n])$
$\displaystyle=(\mathcal{C}*\cdots*\mathcal{C}[d-2])*(\mathcal{C}[-1]*\mathcal{C})$
$\displaystyle\subseteq(\mathcal{C}*\cdots\mathcal{C}[d-2])*(\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1])$
$\displaystyle\subseteq(\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-2])*\mathcal{C}[d-1],$

where the last inclusion is obtained from the inclusions

$\mathcal{C}[i]*\mathcal{C}[j]\subseteq\mathcal{C}[j]*\mathcal{C}[i]$

for $0\leq j<i<d$ with $(j,i)\neq(0,d-1)$ (see Lemma 1.3.6) and the equality

$\mathcal{C}[i]*\mathcal{C}[i]=\mathcal{C}[i]\vee\mathcal{C}[i]=\mathcal{C}[i],$

(see Lemma 1.2.3(2)) which stems from the fact that there are no non-trivial
self-extensions between the objects of $\mathcal{C}[i]$ (recall that $d\geq 2$
by assumption). This proves the claim in this case.

$n\neq 0$:

A simpler version of the argument used in the case $n=0$ shows that there are
inclusions

$\displaystyle\mathcal{C}[n_{1}]*\cdots*\mathcal{C}[n_{\ell}]*\mathcal{C}[n]$
$\displaystyle\subseteq(\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1])*\mathcal{C}[n]$
$\displaystyle\subseteq\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1],$

where in the last inclusion we use Lemma 1.3.6 and Lemma 1.2.3(2) and the
assumption that $n\neq 0$.

In view of Theorem 1.3.7, this finishes the proof of the theorem. ∎

######  Remark 1.3.10.

In the context of Theorem 1.3.9, condition (2d) is not essential: We may
replace $\mathcal{T}$ by its full subcategory
$\operatorname{thick}(\mathcal{C})$, noticing that conditions (2a)–(2c) only
depend on the latter subcategory.

##  2\. $(d+2)$-angulated categories

###  2.1. Definition and basic properties

Although we do not use most axioms in the definition (at least not
explicitly), we include the complete definition of a $(d+2)$-angulated
category since it plays an important conceptual role in our main results.
Recall that $d\geq 1$ is a fixed integer.

######  Definition 2.1.1 ([GKO13, Def. 2.1] and [BT13, Thm. 4.2]).

Let $(\mathcal{F},\Sigma)$ be a pair consisting of an additive category
$\mathcal{F}$ and an automorphism111As in the case of triangulated categories,
one may assume instead that $\Sigma$ is merely an auto-_equivalence_ , but
every autoequivalence can be strictified by replacing $\mathcal{F}$ by an
equivalent category, see for example [KV87, Sec. 2]. We mostly ignore this
distinction as it is not crucial to our results.
$\Sigma\colon\mathcal{F}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{F}$. A
_$(d+2)$ -angulation_ of $(\mathcal{F},\Sigma)$ is a class ${\pentagon}$ of
sequences

$x_{1}\to x_{2}\to\cdots\to x_{d+2}\to\Sigma(x_{1})$

that satisfy the following axioms:

  1. ($d$-TR1)

    1. (a)

The class ${\pentagon}$ is closed under direct sums and direct summands.

    2. (b)

For each object $x\in\mathcal{F}$, the sequence

$x\xrightarrow{1}x\to 0\to\cdots\to 0\to\Sigma(x)$

($d$ consecutive zeroes) lies in ${\pentagon}$.

    3. (c)

Every morphism $x\to y$ in $\mathcal{F}$ is the _leftmost_ morphism in a
sequence that lies in ${\pentagon}$.

  2. ($d$-TR2)

A sequence

$x_{1}\xrightarrow{f}x_{2}\to\cdots x_{d+1}\to\Sigma(x_{1})$

lies in ${\pentagon}$ if and only if its _left rotation_

$x_{2}\to x_{3}\to\cdots\to
x_{d+2}\to\Sigma(x_{1})\xrightarrow{(-1)^{d}\Sigma(f)}\Sigma(x_{2})$

lies in ${\pentagon}$ (notice that the sign in the rotation depends on $d$).

  3. ($d$-TR3)

Given a solid diagram in $\mathcal{F}$ of the form

${x_{1}}$${x_{2}}$${x_{3}}$${\cdots}$${x_{d+1}}$${x_{d+2}}$${\Sigma(x_{1})}$${y_{1}}$${y_{2}}$${y_{3}}$${\cdots}$${y_{d+1}}$${y_{d+2}}$${\Sigma(y_{1}),}$$\scriptstyle{u}$$\scriptstyle{\Sigma(u)}$

in which the leftmost square commutes and both rows lie in ${\pentagon}$,
there exist morphisms $x_{i}\to y_{i}$, $3\leq i\leq d+2$, rendering the
diagram commutative.

  4. ($d$-TR4)

Given the solid part of the diagram

${x_{1}}$${x_{2}}$${x_{3}}$${x_{4}}$${\cdots}$${x_{d+1}}$${x_{d+2}}$${\Sigma(x_{1})}$${x_{1}}$${y_{2}}$${y_{3}}$${y_{4}}$${\cdots}$${y_{d+1}}$${y_{d+2}}$${\Sigma(x_{1})}$${x_{1}}$${y_{2}}$${z_{3}}$${z_{4}}$${\cdots}$${z_{d+1}}$${z_{d+2}}$${\Sigma(x_{1})}$$\scriptstyle{u}$$\scriptstyle{f}$$\scriptstyle{\Sigma(f)}$$\scriptstyle{v}$

whose rows lie in ${\pentagon}$, a collection of dotted morphisms rendering
the diagram commutative exists, and with the further property that the natural
complex (see [BT13, Sec. 4] for the precise signs)

$\displaystyle x_{3}$ $\displaystyle\to{x_{4}\oplus y_{3}}\to{x_{5}\oplus
y_{4}\oplus z_{3}}\to\cdots$ $\displaystyle\cdots$
$\displaystyle\to{x_{d+2}\oplus y_{d+1}\oplus z_{d}}\to y_{d+2}\oplus
z_{d+1}\to z_{d+2}\xrightarrow{\Sigma(u)\circ v}\Sigma(x_{3})$

lies in ${\pentagon}$.

In this case, the triple $(\mathcal{F},\Sigma,{\pentagon})$ is called a
_$(d+2)$ -angulated category_. Similarly, we say that ${\pentagon}$ is a _pre-
$(d+2)$-angulation_ if it satisfies axioms ($d$-TR1)–($d$-TR3), in which case
the triple $(\mathcal{F},\Sigma,{\pentagon})$ is a _pre- $(d+2)$-angulated
category_.

######  Notation 2.1.2.

Let $(\mathcal{F},\Sigma,{\pentagon})$ be a (pre-)$(d+2)$-angulated category.
If there is no risk of confusion, we often abuse the notation and identify the
triple $(\mathcal{F},\Sigma,{\pentagon})$ with its underlying additive
category $\mathcal{F}$.

######  Remark 2.1.3.

In Definition 2.1.1 we use the equivalent axioms given by Bergh and Thaule
[BT13], that is, we use a suitable analogue of octahedral axiom instead of a
higher analogue of Neeman’s mapping cone axiom for triangulated categories
[Nee01, Def. 1.3.13] used in the original definition of Geiss, Keller and
Oppermann. The higher analogue of Neeman’s mapping cone axiom is used in
Example 2.1.9 below, so we include it here:

  * ($d$-TR$4^{\prime}$)

In the situation of ($d$-TR3), the morphisms $x_{i}\to y_{i}$, $3\leq i\leq
d+2$, can be chosen such that the usual mapping cone (see [GKO13, 2.1] for the
precise form of the morphisms)

$x_{2}\oplus y_{1}\to{x_{3}\oplus y_{2}}\to\cdots\to{\Sigma x_{1}\oplus
y_{d+2}}\to\Sigma(x_{2}\oplus y_{1})$

lies in ${\pentagon}$.

######  Remark 2.1.4.

A $3$-angulated category is nothing else but a triangulated category. Indeed,
for $d=1$, axioms ($d$-TR1)–($d$-TR4) are identical to axioms (TR1)–(TR4) in
the usual definition of a triangulated category.

######  Remark 2.1.5.

We do not know examples of pre-$(d+2)$-angulated categories that do not
satisfy axiom ($d$-TR4) for any $d\geq 1$. In fact, no such examples are known
even in the classical case of triangulated categories.

######  Remark 2.1.6.

Given a $(d+2)$-angulated category $(\mathcal{F},\Sigma,{\pentagon})$ and a
unit $u\in\mathbf{k}^{\times}$ in the ground field, we can form a new
$(d+2)$-angulation $u\cdot{\pentagon}$ of $(\mathcal{F},\Sigma)$ as follows:

$x_{1}\xrightarrow{f}x_{2}\to\cdots x_{d+1}\to\Sigma(x_{1})$

belongs to ${\pentagon}$ if and only if

$x_{1}\xrightarrow{u^{-1}\cdot f}x_{2}\to\cdots x_{d+1}\to\Sigma(x_{1})$

belongs to $u\cdot{\pentagon}$ (see [Bal02] for the case $d=1$ of ordinary
triangulations).

######  Remark 2.1.7.

For the purposes of this article, it is enough to consider only
$(d+2)$-angulated categories whose underlying additive category has split
idempotents. This is not a serious restriction, since $(d+2)$-angulations can
be extended to idempotent completions in an essentially unique way, see [BS01]
for the case $d=1$ of triangulated categories and [Lin21] for the general
case.

The following result is a $(d+2)$-angulated analogue of a classical result of
Freyd [Fre66] and of Heller’s parametrisation of the pre-triangulations on a
pre-triangulated category [Hel68].

######  Proposition 2.1.8 ([GKO13, Prop. 2.5 and Prop. 3.4]).

Let $(\mathcal{F},\Sigma)$ be a pair consisting of an additive category and an
automorphism
$\Sigma\colon\mathcal{F}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{F}$.
Suppose that $(\mathcal{F},\Sigma)$ admits a pre-$(d+2)$-angulation. Then, the
following statements hold:

  1. (1)

The category $\operatorname{mod}(\mathcal{F})$ of finitely-presented functors
$\mathcal{F}^{\mathrm{op}}\to\operatorname{Mod}\mathbf{k}$ to vector spaces is
a Frobenius abelian category. Moreover, if idempotents split in $\mathcal{F}$,
then the Yoneda embedding $x\mapsto\mathcal{F}(-,x)$ induces an equivalence
between $\mathcal{F}$ and the full subcategory of
$\operatorname{mod}(\mathcal{F})$ spanned by the projective-injective objects.

  2. (2)

The automorphism of
$\Sigma\colon\operatorname{mod}(\mathcal{F})\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{mod}(\mathcal{F})$
induced by
$\Sigma\colon\mathcal{F}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{F}$ is an
exact functor. In particular, there is an induced natural isomorphism

$\sigma\colon\Sigma\Omega_{\mathcal{F}}^{-1}\stackrel{{\scriptstyle\sim}}{{\Longrightarrow}}\Omega_{\mathcal{F}}^{-1}\Sigma,$

where
$\Omega_{\mathcal{F}}^{-1}\colon\operatorname{\underline{mod}}(\mathcal{F})\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{\underline{mod}}(\mathcal{F})$
is a choice of cosyzygy functor on the stable category, that makes the pair
$(\Sigma,\sigma)$ into an exact functor on
$\operatorname{\underline{mod}}(\mathcal{F})$.

  3. (3)

There is a (canonical) bijective map from the class of pre-$(d+2)$-angulations
on $(\mathcal{F},\Sigma)$ to the class of isomorphisms of exact functors

$(\Sigma,\sigma)\stackrel{{\scriptstyle\sim}}{{\Longrightarrow}}(\Omega_{\mathcal{F}}^{-(d+2)},(-1)^{d}\mathbf{1}_{\Omega_{\mathcal{F}}^{-(d+2)-1}})$

on $\operatorname{\underline{mod}}(\mathcal{F})$. In particular, if the pair
$(\mathcal{F},\Sigma)$ admits a pre-$(d+2)$-angulation, then there are natural
isomorphisms

$\Sigma\cong\Omega_{\mathcal{F}}^{-(d+2)}\qquad\text{and}\qquad\Sigma^{-1}\cong\Omega_{\mathcal{F}}^{d+2}$

of exact functors on $\operatorname{\underline{mod}}(\mathcal{F})$.

The following example, that to our knowledge has not yet appeared in the
literature, provides a simple method to constructing ‘longer’ angulations from
‘shorter’ ones.

######  Example 2.1.9.

Let $(\mathcal{F},\Sigma,{\pentagon})$ be a $(d+2)$-angulated category and
$n\geq 1$ an integer. The pair $(\mathcal{F},\Sigma^{n})$ is endowed with an
$n(d+2)$-angulation ${\pentagon}^{n}$ consisting of the sequences

${x_{1}}$${x_{2}}$${\cdots}$${x_{d+2}}$${\Sigma(x_{1})}$${\Sigma(x_{2})}$${\cdots}$${\Sigma(x_{d+2})}$${\cdots}$${\cdots}$${\Sigma^{n-1}(x_{1})}$${\Sigma^{n-1}(x_{2})}$${\cdots}$${\Sigma^{n-1}(x_{d+2})}$${\Sigma^{n}(x_{1})}$

in which the first sequence

$x_{1}\to x_{2}\to\cdots\to x_{d+2}\to\Sigma(x_{1})$

lies in ${\pentagon}$ and the subsequent rows of the diagram are obtained from
the first one by iterated application of the suspension functor (without any
additional signs). Utilising the fact that ${\pentagon}$ is a
$(d+2)$-angulation of $(\mathcal{F},\Sigma)$, it is straightforward to verify
that ${\pentagon}^{n}$ is an $n(d+2)$-angulation of
$(\mathcal{F},\Sigma^{n})$. Indeed, in terms of Proposition 2.1.8(3), if the
class ${\pentagon}$ is classified by an isomorphism
$\Sigma\cong\Omega_{\mathcal{F}}^{d+2}$ of exact functors on
$\operatorname{\underline{mod}}(\mathcal{F})$, then the class
${\pentagon}^{n}$ is the pre-$n(d+2)$-angulation of $(\mathcal{F},\Sigma^{n})$
classified by the induced isomorphism
$\Sigma^{n}\cong\Omega_{\mathcal{F}}^{n(d+2)}$. Thus, to prove that
${\pentagon}^{n}$ is a $(d+2)$-angulation, it suffices to verify that the
higher analogue of Neeman’s mapping cone axiom is satisfied (Remark 2.1.3). We
leave the details of this verification to the reader.

We record the apparent notion of equivalence between (pre-)$(d+2)$-angulated
categories for later use. For this, we also introduce the following auxiliary
definition.

######  Definition 2.1.10.

Let $(\mathcal{C},\Phi_{\mathcal{C}})$ and $(\mathcal{D},\Phi_{\mathcal{D}})$
be pairs consisting of categories $\mathcal{C}$ and $\mathcal{D}$ and
automorphisms
$\Phi_{\mathcal{C}}\colon\mathcal{C}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{C}$
and
$\Phi_{\mathcal{D}}\colon\mathcal{D}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}$.
A _functor of pairs
$(\mathcal{C},\Phi_{\mathcal{C}})\to(\mathcal{D},\Phi_{\mathcal{D}})$_ is a
pair $(F,F^{\sharp})$ consisting of a functor
$F\colon\mathcal{C}\to\mathcal{D}$ and a natural isomorphism $F^{\sharp}\colon
F\Phi_{\mathcal{C}}\Rightarrow\Phi_{\mathcal{D}}F$. A functor of pairs is an
_equivalence_ if its underlying functor is an equivalence of categories.

######  Remark 2.1.11.

Observe that the collection of pairs (as in Definition 2.1.10) form a
$2$-category in a natural way. In particular, the notion of equivalence of
pairs is simply the corresponding notion of equivalence in this $2$-category.

######  Notation 2.1.12.

In the context of Definition 2.1.10, we always identify a functor of pairs
with the functor between the corresponding underlying categories.

######  Definition 2.1.13.

Let $(\mathcal{F},\Sigma,{\pentagon})$ and
$(\mathcal{F}^{\prime},\Sigma^{\prime},{\pentagon}^{\prime})$ be two
pre-$(d+2)$-angulated categories. A _pre- $(d+2)$-angulated equivalence_
between $(\mathcal{F},\Sigma,{\pentagon})$ and
$(\mathcal{F}^{\prime},\Sigma^{\prime},{\pentagon}^{\prime})$ is an
equivalence of pairs

$F\colon(\mathcal{F},\Sigma)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(\mathcal{F}^{\prime},\Sigma^{\prime})$

such that, for each $(d+2)$-angle

$x_{1}\to x_{2}\to\cdots\to x_{d+2}\xrightarrow{\eta}\Sigma(x_{0})$

in ${\pentagon}$, the sequence

$F(x_{1})\to F(x_{2})\to\cdots\to
F(x_{d+2})\xrightarrow{\eta^{\prime}}\Sigma(F(x_{0}))$

lies in ${\pentagon}^{\prime}$, where
$\eta^{\prime}=F^{\sharp}_{x_{0}}F(\eta)$. If ${\pentagon}$ and
${\pentagon}^{\prime}$ are in fact $(d+2)$-angulations, we say that $F$ is a
_$(d+2)$ -angulated equivalence_ or an _equivalence of $(d+2)$-angulated
categories_.

###  2.2. Constructions of $(d+2)$-angulated categories

We recall the two constructions of $(d+2)$-angulated categories that are most
relevant for the purposes of this article.

####  2.2.1. Standard $(d+2)$-angulated categories

The following theorem of Geiss, Keller and Oppermann is one of the main
sources of $(d+2)$-angulated categories; in fact, this theorem was the main
motivation for introducing the notion of $(d+2)$-angulated category, and it
also motivated the definition of a $d\mathbb{Z}$-cluster tilting subcategory
(Definition 1.3.1).

######  Theorem 2.2.1 ([GKO13, Thm. 1]).

Let $\mathcal{C}\subseteq\mathcal{T}$ be a $d\mathbb{Z}$-cluster tilting
subcategory of $\mathcal{T}$. Then, the pair $(\mathcal{C},[d])$ is endowed
with a $(d+2)$-angulation whose $(d+2)$-angles are those sequences

$c_{d+2}\to c_{d+1}\to\cdots\to c_{2}\to c_{1}\to c_{d+2}[d]$

in $\mathcal{C}$ which fit 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.

######  Definition 2.2.2.

We call a $(d+2)$-angulated category
$(\mathcal{F},\Sigma,{\pentagon}_{\mathcal{F}})$ _standard_ if there exists a
fully faithful functor of pairs
$\iota\colon(\mathcal{F},\Sigma)\hookrightarrow(\mathcal{T},[d])$ with the
following properties:

  1. (1)

The category $\mathcal{T}$ is triangulated with suspension functor $[1]$.

  2. (2)

The essential image $\iota(\mathcal{F})$ of $\iota$ is a $d$-rigid subcategory
of $\mathcal{T}$.

  3. (3)

The image of the $(d+2)$-angulation of $\mathcal{F}$ under $\iota$ coincides
with the class of sequences considered in Theorem 2.2.1. More precisely, the
closure under isomorphisms of the class of sequences

$F(x_{1})\to F(x_{2})\to\cdots\to
F(x_{d+2})\xrightarrow{F_{x_{1}}^{\sharp}F(\eta)}\Sigma(F(x_{1}))$

in $\iota(\mathcal{F})$ that are the image of a $(d+2)$-angle

$x_{1}\to x_{2}\to\cdots\to x_{d+2}\xrightarrow{\eta}\Sigma(x_{1})$

that lies in the class ${\pentagon}_{\mathcal{F}}$ coincides with the class of
sequences considered in Theorem 2.2.1.

In this case, we say that $\iota(\mathcal{F})\subseteq\mathcal{T}$ is a
_standard $(d+2)$-angulated subcategory_ and call the class
${\pentagon}_{\mathcal{F}}$ a _standard $(d+2)$-angulation_.

######  Remark 2.2.3.

Let $(\mathcal{F},\Sigma,{\pentagon})$ be a standard $(d+2)$-angulated
category with respect to a fully faithful functor
$\iota\colon\mathcal{F}\hookrightarrow\mathcal{T}$ into a triangulated
category $\mathcal{T}$. Since $\iota$ is a functor of pairs
$\iota(\mathcal{F})[d]=\iota(\mathcal{F})$ and since moreover
$\iota(\mathcal{F})$ is $d$-rigid then $\iota(\mathcal{F})$ is actually a
$d\mathbb{Z}$-rigid subcategory of $\mathcal{T}$.

Theorems 1.3.9 and 2.2.1 combine into the following result.

######  Theorem 2.2.4.

Suppose that $\mathcal{T}$ has finite-dimensional morphism spaces and split
idempotents. Let $\mathcal{C}\subseteq\mathcal{T}$ be a subcategory that
satisfies $\operatorname{add}(\mathcal{C})=\mathcal{C}$ and
$\iota\colon\mathcal{C}\hookrightarrow\mathcal{T}$ the inclusion functor.
Then, the following statements are equivalent:

  1. (1)

The subcategory $\mathcal{C}$ is a $d\mathbb{Z}$-cluster tilting subcategory
of $\mathcal{T}$.

  2. (2)

$\operatorname{thick}(\mathcal{C})=\mathcal{T}$ and the functor $\iota$
exhibits $\mathcal{C}$ as a standard $(d+2)$-angulated category; in other
words, the pair $(\mathcal{C},[d])$ admits a $(d+2)$-angulation whose
$(d+2)$-angles are given as in Theorem 2.2.1.

###### Proof.

The implication (1)$\Rightarrow$(2) is precisely Theorem 2.2.1. We prove the
converse implication (2)$\Rightarrow$(1). By assumption, $\mathcal{C}$ is a
$d\mathbb{Z}$-rigid subcategory such that
$\operatorname{thick}(\mathcal{C})=\mathcal{T}$ and
$\mathcal{C}=\mathcal{C}[d]$. In view of Theorem 1.3.9, it is enough to prove
that the inclusion

$(\mathcal{C}[-1]*\mathcal{C})\subseteq(\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1])$

holds. Indeed, let $x\in\mathcal{C}[-1]*\mathcal{C}$ and choose an exact
triangle in $\mathcal{T}$ of the form

$x\to c_{2}\xrightarrow{f}c_{1}\to x[1]$

with $c_{1},c_{2}\in\mathcal{C}$. By axioms ($d$-TR1) and ($d$-TR2), there
exists a $(d+2)$-angle in $\mathcal{C}$ of the form

$c_{d+2}\to c_{d+1}\to\cdots\to c_{2}\xrightarrow{f}c_{1}\to c_{d+2}[d].$

Thus, since $\iota$ exhibits $\mathcal{C}$ as a standard $(d+2)$-angulated
category, there must exist 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}$

where $x=x_{2.5}$. This shows that
$x\in\mathcal{C}*\mathcal{C}[1]*\cdots*\mathcal{C}[d-1]$, as required, see
Example 1.2.4. This finishes the proof of the theorem. ∎

######  Remark 2.2.5.

In the context of Theorem 2.2.4, the condition

$\operatorname{thick}(\mathcal{C})=\mathcal{T}$

is not essential: We may replace $\mathcal{T}$ by its full subcategory
$\operatorname{thick}(\mathcal{C})$, noticing that statement (2) only depends
on the latter subcategory (compare with Theorem 1.3.9(2a)).

####  2.2.2. Amiot–Lin $(d+2)$-angulations

In addition to Theorem 2.2.1, an important source of $(d+2)$-angulated
categories is the following ‘higher-dimensional’ version of a theorem of
Amiot, who proved the result in the case $d=1$. We need some preliminaries to
state the result.

Given an algebra $\Lambda$, we let
$\Lambda^{e}=\Lambda\otimes\Lambda^{\mathrm{op}}$ be its enveloping algebra
and identify the category of $\Lambda$-bimodules with that of (right)
$\Lambda^{e}$-modules in the canonical way. Recall that a finite-dimensional
self-injective algebra $\Lambda$ is _twisted $n$-periodic_ if there exist an
algebra automorphism
$\sigma\colon\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\Lambda$ and an
isomorphism

$\Omega_{\Lambda^{e}}^{n}(\Lambda)\cong{}_{1}{\Lambda}_{\sigma}$

in the stable category of $\Lambda$-bimodules; here,
${}_{1}{\Lambda}_{\sigma}$ denotes the diagonal $\Lambda$-bimodule with right
action twisted by $\sigma$. Equivalently, $\Lambda$ is twisted $n$-periodic if
there exists an exact sequence of $\Lambda$-bimodules

$0\to{}_{1}{\Lambda}_{\sigma}\to P_{n}\to\cdots\to P_{2}\to P_{1}\to\Lambda\to
0$

whose middle terms are projective $\Lambda$-bimodules. Similarly, $\Lambda$ is
_$n$ -periodic_ if it is twisted $n$-periodic with
$\sigma=\mathbf{1}_{\Lambda}$.

On the other hand, let $\Lambda$ be a basic self-injective algebra. By [Bol84,
Prop. 3.8], the map

$\operatorname{Out}(\Lambda)\longrightarrow\operatorname{Pic}(\Lambda),\qquad[\sigma]\longmapsto[{}_{1}{\Lambda}_{\sigma}],$

yields an isomorphism between the outer automorphism group
$\operatorname{Out}(\Lambda)$ of $\Lambda$ and the Picard group
$\operatorname{Pic}(\Lambda)$ of invertible $\Lambda$-bimodules. In
particular, every autoequivalence of the additive category
$\operatorname{proj}(\Lambda)$ is of the form

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

for some automorphism $\sigma$ of $\Lambda$; notice also that the above
autoequivalence is isomorphic to the restriction of scalars

$(-)_{\sigma}\colon P\longmapsto P_{\sigma}$

along $\sigma$. In particular, $\Lambda$ is twisted $n$-periodic if and only
if the $\Lambda$-bimodule $\Omega_{\Lambda^{e}}^{n}(\Lambda)$ is stably
isomorphic to an invertible $\Lambda$-bimodule.

We record the following observation for later use. Recall that the syzygy
$\Omega(M)$ of a module $M$ is the kernel of its projective cover.

######  Proposition 2.2.6.

Let $\Lambda$ be a finite-dimensional self-injective algebra that is twisted
$(d+2)$-periodic. Suppose that $\Lambda$ is connected and non-separable. Then,
the following statements hold:

  1. (1)

Every invertible $\Lambda$-bimodule $I$ that is isomorphic to
$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)$ in
$\operatorname{\underline{mod}}(\Lambda^{e})$ is already isomorphic to
$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)$ as a $\Lambda$-bimodule.

  2. (2)

The $(d+2)$-syzygy $\Omega_{\Lambda^{e}}^{d+2}(\Lambda)$ is an invertible
$\Lambda$-bimodule.

###### Proof.

The proof of [Mur22, Prop. 9.8] applies verbatim. ∎

Recall that a finite-dimensional algebra $\Lambda$ is _separable_ if it is
isomorphic to a finite product of matrix algebras over skew fields, whose
centres are separable field extensions of the ground field $\mathbf{k}$; in
particular, if $\mathbf{k}$ is perfect, then $\Lambda$ is separable if and
only if it is semisimple (see for example [Wei94, Sec. 9.2.1]). The following
theorem of Green, Snashall and Solberg (with some simplifications due to
Hanihara) gives a convenient characterisation of the class of twisted periodic
algebras, see also the comments in the proof of [CDIM, Prop. 3.5].

######  Theorem 2.2.7 ([GSS03, Thm. 1.4] and [Han20, Cor. 2.2]).

Let $\Lambda$ be a finite-dimensional self-injective algebra and $n\geq 1$ an
integer; suppose, moreover, that $\Lambda/J_{\Lambda}$ is separable. The
following statements are equivalent:

  1. (1)

The algebra $\Lambda$ is twisted $n$-periodic.

  2. (2)

There exist an algebra automorphism
$\sigma\colon\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\Lambda$ and a natural
isomorphism

$\Omega_{\Lambda}^{n}\cong(-)_{\sigma}$

of exact functors
$\operatorname{\underline{mod}}(\Lambda)\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{\underline{mod}}(\Lambda)$.

######  Remark 2.2.8.

Suppose that the equivalent conditions in Theorem 2.2.7 are satisfied and that
$\Lambda$ is connected and non-separable. Proposition 2.2.6 shows that
$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)$ is an invertible $\Lambda$-bimodule but
it need not be isomorphic to ${}_{1}\Lambda_{\sigma}$. Although these
bimodules induce isomorphic functors on
$\operatorname{\underline{mod}}\Lambda$, this only permits us to conclude that
$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)\cong{}_{1}\Lambda_{\sigma\gamma}$ where
$\gamma$ is a _stable_ inner automorphism, that is an algebra automorphism
such that the restriction of scalars along $\gamma$ is isomorphic to the
identity functor of the stable category
$\operatorname{\underline{mod}}\Lambda$ (compare with [IV14, Thm. 1.8] for
example).

Our interest in twisted periodic algebras in the context of $(d+2)$-angulated
categories stems from the following observation, which is essentially a
reformulation of Proposition 2.1.8.

######  Proposition 2.2.9.

Let $(\mathcal{F},\Sigma,{\pentagon})$ be a $(d+2)$-angulated category with
finite-dimensional morphism spaces and split idempotents. Suppose that there
exists a basic object $x\in\mathcal{F}$ such that
$\operatorname{add}(x)=\mathcal{F}$ and set
$\Lambda\coloneqq\mathcal{F}(x,x)$. If $\Lambda/J_{\Lambda}$ is separable,
then $\Lambda$ is self-injective and twisted $(d+2)$-periodic.

###### Proof.

The proof is an adaptation of the proof of [Han20, Prop. 3.1], which treats
the case $d=1$ of triangulated categories. By assumption, the functor

$\mathcal{F}(x,-)\colon\mathcal{F}\longrightarrow\operatorname{proj}(\Lambda),\qquad
y\longmapsto\mathcal{F}(x,y)$

is an equivalence of categories; hence the finite-dimensional algebra
$\Lambda$ is self-injective by Proposition 2.1.8(1). In particular, there are
commutative diagrams of equivalences of categories

${\mathcal{F}}$${\operatorname{proj}(\Lambda)}$${\mathcal{F}}$${\operatorname{proj}(\Lambda)}$$\scriptstyle{\sim}$$\scriptstyle{\Sigma^{-1}}$$\scriptstyle{(-)_{\sigma}}$$\scriptstyle{\sim}$

and
${\operatorname{\underline{mod}}(\mathcal{F})}$${\operatorname{\underline{mod}}(\Lambda)}$${\operatorname{\underline{mod}}(\mathcal{F})}$${\operatorname{\underline{mod}}(\Lambda),}$$\scriptstyle{\sim}$$\scriptstyle{\Sigma^{-1}}$$\scriptstyle{(-)_{\sigma}}$$\scriptstyle{\sim}$

where $\sigma\colon\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\Lambda$ is an
algebra automorphism (compare [Bol84, Prop. 3.8]) and the functors in the
diagram on the right are exact with respect to the induced triangulated
structures. Proposition 2.1.8(3) yields the existence of a natural isomorphism
$\Sigma^{-1}\cong\Omega_{\mathcal{F}}^{d+2}$ of exact functors on
$\operatorname{\underline{mod}}(\mathcal{F})$ and, consequently, there is also
a natural isomorphism $(-)_{\sigma}\cong\Omega_{\Lambda}^{d+2}$ of exact
functors on $\operatorname{\underline{mod}}(\Lambda)$. Theorem 2.2.7 then
implies that $\Lambda$ is twisted $(d+2)$-periodic, which is what we needed to
prove. ∎

Combining Theorem 2.2.1 and Proposition 2.2.9, we obtain an alternative proof
of the following result that emphasises the role of $(d+2)$-angulations in
this context.

######  Proposition 2.2.10 ([Dug12], [CDIM, Prop. 8.5(a)]).

Let $\mathcal{T}$ be a triangulated category with finite-dimensional morphism
spaces and split idempotents and such that there exists a
$d\mathbb{Z}$-cluster tilting object $c\in\mathcal{T}$. Set
$\Lambda\coloneqq\mathcal{T}(c,c)$. If $\Lambda/J_{\Lambda}$ is separable,
then $\Lambda$ is self-injective and twisted $(d+2)$-periodic.

###### Proof.

Indeed, Theorem 2.2.1 shows that the additive category
$\mathcal{C}\coloneqq\operatorname{add}(c)\subseteq\mathcal{T}$ is endowed
with the structure of a $(d+2)$-angulated category with suspension functor
$[d]\colon\mathcal{C}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{C}$. Both
claims then follows from Proposition 2.2.9. ∎

######  Remark 2.2.11.

In [CDIM, Prop. 8.5(a)], the authors assume that the ambient triangulated
category admits a Serre functor in the sense of [BK89], but this fact is not
used in their proof.

######  Remark 2.2.12.

Proposition 2.2.10 shows that the correspondence in A is well defined.

######  Construction 2.2.13.

Let $\Lambda$ be a basic finite-dimensional self-injective algebra and
$\sigma$ an automorphism. Notice that $\Lambda$ and $\Lambda^{\mathrm{op}}$
are Frobenius algebras [Zim14, Prop. 4.5.7] and therefore so is their tensor
product $\Lambda^{e}$; in particular the projective $\Lambda$-bimodules are
also injective $\Lambda$-bimodules. Consider an exact sequence of
$\Lambda$-bimodules

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

with $P_{i}$ projective-injective as $\Lambda$-bimodules for $0\leq i<d+1$. In
particular $\eta$, regarded as a complex of left $\Lambda$-modules, is
contractible222It suffices to assume that $P_{i}$ is left projective for
$0\leq i<d+1$ (as a consequence $P_{d+1}$ too), so that the tensored sequence
in (2) remains exact. This hypothesis is missing in [Mur20b, Prop. 5.6 and
Rmk. 5.7]. Nevertheless, any extension has representatives satisfying this
assumption, such as the representative considered in the proof of [Mur20b,
Prop. 5.6].. We will be mainly interested in the case that $\Lambda$ is
twisted $(d+2)$-periodic with respect to $\sigma$ and $P_{d+1}$ is also a
projective-injective $\Lambda$-bimodule, so that $\eta$ exhibits the existence
of an isomorphism
$\Omega_{\Lambda^{e}}^{d+2}(\Lambda)\cong{}_{1}{\Lambda}_{\sigma}$ in
$\operatorname{\underline{mod}}(\Lambda^{e})$. Let

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

and

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

its quasi-inverse. We define a class ${\pentagon}_{\eta}$ of _exact
$(d+2)$-angles_ in $\operatorname{proj}(\Lambda)$ as follows: A complex of
finitely generated projective-injective $\Lambda$-modules

$Q_{d+2}\xrightarrow{f}Q_{d+1}\to\cdots\to Q_{1}\xrightarrow{g}\Sigma Q_{d+2}$

lies in ${\pentagon}_{\eta}$ if it has the following two properties:

  1. (1)

The extended complex

$Q_{d+2}\xrightarrow{f}Q_{d+1}\to\cdots\to Q_{1}\xrightarrow{g}\Sigma
Q_{d+2}\xrightarrow{\Sigma f}\Sigma Q_{d+1}$

is exact (notice that this condition only depends on the pair
$(\operatorname{proj}(\Lambda),\Sigma)$).

  2. (2)

Denote the $\Lambda$-module $\operatorname{\operatorname{coker}}{g}$ by $N$.
By construction, there is an exact sequence

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

such that $f=i\circ\Sigma^{-1}p$. We require the above exact sequence to be
equivalent to the exact sequence

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

in the extension space
$\operatorname{Ext}_{\Lambda}^{d+2}\\!\left(N,\Sigma^{-1}N\right)$. The latter
sequence is exact because, as we have pointed out above, $\eta$ is
contractible as a complex of left $\Lambda$-modules.

######  Remark 2.2.14.

In connection with Remark 2.1.6, given an exact sequence of
$\Lambda$-bimodules

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

and a unit $u\in\mathbf{k}^{\times}$ in the ground field the class

$u\cdot[\eta]\in\operatorname{Ext}_{\Lambda^{e}}^{d+2}\\!\left(\Lambda,{}_{1}{\Lambda}_{\sigma}\right)$

is represented by the exact sequence of $\Lambda$-bimodules

$u\cdot\eta\colon\quad
0\to{}_{1}{\Lambda}_{\sigma}\xrightarrow{u^{-1}\cdot\iota}P_{d+1}\to\cdots\to
P_{1}\to P_{0}\to\Lambda\to 0.$

Therefore, the class ${\pentagon}_{u\cdot\eta}$ given by Construction 2.2.13
coincides with $u\cdot{\pentagon}_{\eta}$.

We are ready to state the theorem(s) of Amiot and Lin.

######  Theorem 2.2.15 ([Ami07, Thm. 8.1] and [Lin19, Thm. 1.3]).

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

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

Then, the class ${\pentagon}_{\eta}$ of exact $(d+2)$-angles from Construction
2.2.13 associated to an extension $\eta$ with projective-injective middle
$\Lambda$-bimodules endows the pair $(\operatorname{proj}(\Lambda),\Sigma)$
with the structure of a $(d+2)$-angulated category.

######  Remark 2.2.16.

Notice that Theorem 2.2.15 is _a priori_ quite different from Theorem 2.2.1.
Indeed, the former result does not require the existence of an ambient
triangulated category in which to embed $\operatorname{proj}(\Lambda)$.
However, A and Theorem 4.5.8 show that, if $\mathbf{k}$ is perfect,
$\operatorname{proj}(\Lambda)$ embeds as a $d\mathbb{Z}$-cluster tilting
subcategory in an essentially unique algebraic triangulated category in such a
way that the $(d+2)$-angulated structure from Theorem 2.2.15 coincides with
that from Theorem 2.2.1.

######  Definition 2.2.17.

In the context of Theorem 2.2.15, we call the class ${\pentagon}_{\eta}$ an
_Amiot–Lin (AL) $(d+2)$-angulation_ of the pair
$(\operatorname{proj}(\Lambda),\Sigma)$.

The following result shows that the $(d+2)$-angulated structure from Theorem
2.2.15 is independent of the choice of (truncated) projective resolution of
the diagonal bimodule.

######  Proposition 2.2.18.

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

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

Choose exact sequences

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

and

$\eta^{\prime}\colon\quad 0\to{}_{1}{\Lambda}_{\sigma}\to Q_{d+1}\to\cdots\to
Q_{1}\to Q_{0}\to\Lambda\to 0$

of $\Lambda$-bimodules with projective-injective middle terms. Then, there
exists a morphism of exact sequences

${\eta\colon}$${0}$${{}_{1}{\Lambda}_{\sigma}}$${P_{d+1}}$${\cdots}$${P_{1}}$${P_{0}}$${\Lambda}$${0}$${\eta^{\prime}\colon}$${0}$${{}_{1}{\Lambda}_{\sigma}}$${Q_{d+1}}$${\cdots}$${Q_{1}}$${Q_{0}}$${\Lambda}$${0}$$\scriptstyle{u}$

where $u\in Z(\Lambda)^{\times}$ is a unit in the centre of $\Lambda$. In
particular, the $(d+2)$-angulations ${\pentagon}_{\eta}$ and
${\pentagon}_{\eta^{\prime}}$ of $(\operatorname{proj}(\Lambda),\Sigma)$ are
equivalent in the sense of Definition 2.1.13.

###### Proof.

The Comparison Lemma for projective resolutions yields a morphism of complexes

${\eta\colon}$${0}$${{}_{1}{\Lambda}_{\sigma}}$${P_{d+1}}$${\cdots}$${P_{1}}$${P_{0}}$${\Lambda}$${0}$${\eta^{\prime}\colon}$${0}$${{}_{1}{\Lambda}_{\sigma}}$${Q_{d+1}}$${\cdots}$${Q_{1}}$${Q_{0}}$${\Lambda}$${0}$$\scriptstyle{\varphi}$

in which the morphism $\varphi$ is an isomorphism in the stable category of
$\Lambda$-bimodules. Since the algebra $\Lambda$ is finite-dimensional, the
morphism $\varphi$ can be replaced by an isomorphism in the category of
$\Lambda$-bimodules without modifying the rows and the other vertical
morphisms in the diagram (except for possibly $P_{d+1}\to Q_{d+1}$), see
[Che21, Cor. 2.3]. Also, it is easily verified that there are isomorphisms of
algebras

$\operatorname{End}_{\Lambda^{e}}\\!\left({}_{1}{\Lambda}_{\sigma}\right)=\operatorname{End}_{\Lambda^{e}}\\!\left(\Lambda\right)\cong
Z(\Lambda),$

where the rightmost isomorphism is classical. In particular, the isomorphism
$\varphi$ is given by right multiplication by a unit $u\in
Z(\Lambda)^{\times}$, which we interpret as a natural isomorphism
$u\colon\Sigma\Rightarrow\Sigma$ in the usual way. It readily follows that the
pair $(\mathbf{1}_{\operatorname{proj}(\Lambda)},u)$ yields an equivalence
between the $(d+2)$-angulations ${\pentagon}_{\eta^{\prime}}$ and
${\pentagon}_{\eta}$ on the pair $(\operatorname{proj}(\Lambda),\Sigma)$. ∎

######  Remark 2.2.19.

We mention that Proposition 2.2.18 holds, more generally, for locally bounded
categories (or even more general objects) for which one can lift stable
isomorphisms between twists of the diagonal bimodule to actual bimodule
isomorphisms. We thank Martin Herschend for informing us that the latter
property holds for Krull–Schmidt categories whose quiver of irreducible maps
is connected, for such categories have local centre.

The following theorem is a $(d+2)$-angulated analogue of [Han20, Thm. 1.2],
which treats the case $d=1$.

######  Theorem 2.2.20.

Let $\Lambda$ be a basic finite-dimensional algebra such that
$\Lambda/J_{\Lambda}$ is separable. The following statements are equivalent:

  1. (1)

The algebra $\Lambda$ is twisted $(d+2)$-periodic.

  2. (2)

The additive category $\operatorname{proj}(\Lambda)$ admits the structure of a
$(d+2)$-angulated category.

###### Proof.

The proof is a straightforward adaptation of the proof of [Han20, Thm. 1.2].
If $\Lambda$ is twisted $(d+2)$-periodic, then Theorem 2.2.15 shows that the
additive category $\operatorname{proj}(\Lambda)$ admits the structure of a
$(d+2)$-angulated category. The converse is precisely Proposition 2.2.9. ∎

####  2.2.3. Twisted periodic algebras and higher Auslander–Reiten Theory

Although not needed in the sequel, the following discussion provides further
motivation for the study of twisted periodic algebras from the perspective of
Iyama’s higher Auslander–Reiten theory, see the introduction to [Iya07b] for
the analogous discussion in the ($d$-)abelian case.

Let $\Lambda$ be a basic Frobenius algebra; for simplicity, assume that
$\Lambda$ is connected and not semisimple. The _Nakayama functor_

$\nu\coloneqq-\otimes_{\Lambda}D\Lambda\colon\operatorname{proj}\Lambda\stackrel{{\scriptstyle\sim}}{{\to}}\operatorname{proj}\Lambda$

is a Serre functor in the sense that there are natural isomorphisms

$\operatorname{Hom}_{\Lambda}\\!\left(Y,\nu
X\right)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}D\\!\operatorname{Hom}_{\Lambda}\\!\left(X,Y\right),\qquad
X,Y\in\operatorname{proj}\Lambda,$

where $V\mapsto DV$ denotes the passage to the $\mathbf{k}$-linear dual, see
[BO01, Def. 1.2].

Suppose now that $\Lambda$ is twisted $(d+2)$-periodic with respect to an
algebra automorphism $\sigma$ and let

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

Fix an arbitrary Amiot–Lin $(d+2)$-angulation on the pair
$(\operatorname{proj}\Lambda,\Sigma)$. Following [IY08], we set

$\nu_{d}\coloneqq\Sigma^{-d}\nu\colon\operatorname{proj}\Lambda\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{proj}\Lambda.$

Let $P^{0}$ be an indecomposable projective $\Lambda$-module. Extend the map
${f\colon P^{0}\to\nu P^{0}}$ that maps the simple top of $P^{0}$ to the
simple socle of $\nu P^{0}$ to a $(d+2)$-angle

$\nu_{d}P^{0}\to P^{-d}\to\cdots\to P^{-1}\to P^{0}\stackrel{{\scriptstyle
f}}{{\to}}\nu P^{0},$

which we may assume is minimal in the sense that all the morphisms lie in the
radical of $\operatorname{proj}\Lambda$. Since $(d+2)$-angles are exact
[GKO13, Prop. 2.5(a)], the above $(d+2)$-angle is the first part of a minimal
projective resolution of the simple top $S$ of $P^{0}$ and, simultaneously, of
a minimal injective co-resolution of the simple socle
$\Omega_{\Lambda}^{d+2}(S)\cong\sigma^{*}(S)$ of $\nu_{d}P^{0}$. Following the
paradigm of classical Auslander–Reiten theory [AR75, AR78], the above
$(d+2)$-angle can be regarded as the prototypical example of an _almost-split
$(d+2)$-angle_ in the sense of [IY08, 3.8], see also [Fed19, Def. 5.1]. For
the equivalence between the existence of a Serre functor and the existence of
almost-split $(d+2)$-angles see [RVdB02, Thm. I.2.4] for the case $d=1$ and
[Zho22, Thm. 1.5] for the general case (see also [Ami07, Prop. 1.2.] for a
related statement).

##  3\. Enhanced $(d+2)$-angulated categories

###  3.1. Pre-triangulated differential graded categories

We begin by recalling the notion of a pre-triangulated differential graded
category in the sense of [BK90]. We need a few preliminaries.

####  3.1.1. Reminder on differential graded categories

We recall basic aspects of the theory of differential graded categories that
are needed in the sequel; we refer the reader to [Kel06] and the references
therein for details.

A _differential graded (=DG) category_ is a category enriched in the symmetric
monoidal category
$\operatorname{C}(\mathbf{k})=\operatorname{C}(\operatorname{Mod}(\mathbf{k}))$
of cochain complexes of $\mathbf{k}$-modules (endowed with the usual tensor
product of cochain complexes). Thus, the morphisms between two objects in a DG
category form a cochain complex and the composition law satisfies the graded
Leibniz rule. A _DG functor_ is simply an enriched functor between DG
categories. A _graded category_ is a DG category whose cochain complexes of
morphisms have vanishing differential, and a _graded functor_ is a DG functor
between graded categories. For example, given a ($\mathbf{k}$-linear) additive
category $\mathcal{C}$, there is a DG category
$\operatorname{C}_{\mathrm{dg}}(\mathcal{C})$ whose objects are the cochain
complexes in $\mathcal{C}$ and for
$X,Y\in\operatorname{C}_{\mathrm{dg}}(\mathcal{C})$ we let
$\operatorname{hom}_{\mathcal{C}}^{\bullet}\\!\left(X,Y\right)$ be the complex
whose component of degree $i$ is the $\mathbf{k}$-module

$\operatorname{hom}_{\mathcal{C}}^{i}\\!\left(X,Y\right)\coloneqq\prod_{k\in\mathbb{Z}}\mathcal{C}(X^{k},Y^{k+i})$

of degree $i$ morphisms of graded objects $X\to Y$ and differential

$\partial\colon f\longmapsto d_{Y}f-(-1)^{i}fd_{X},\qquad
f\in\operatorname{hom}_{\mathcal{C}}^{i}\\!\left(X,Y\right).$

Note also that the usual shift of cochain complexes

$(X,d_{X})\longmapsto X[1]\coloneqq(X(1),-d_{X}),$

where $X\mapsto X(1)$ denotes the shift of graded objects, can be promoted to
a DG functor

$[1]\colon\operatorname{C}_{\mathrm{dg}}(\mathcal{C})\longrightarrow\operatorname{C}_{\mathrm{dg}}(\mathcal{C}),\qquad
X\longmapsto X[1]$

whose action on morphisms is given by

$[1]\colon f\longmapsto(-1)^{i}f,\qquad
f\in\operatorname{hom}_{\mathcal{C}}^{i}\\!\left(X,Y\right).$

A DG category $\mathcal{A}$ has two associated categories whose objects are
the same as those of $\mathcal{A}$: the _underlying category
$\operatorname{Z}^{0}(\mathcal{A})$_ and the _$0$ -th cohomology category
$H^{0}(\mathcal{A})$_. As the notation suggests, the morphisms in
$\operatorname{Z}^{0}(\mathcal{A})$ are the $\mathbf{k}$-modules of cocycles

$\operatorname{Z}^{0}(\mathcal{A})(x,y)\coloneqq\operatorname{Z}^{0}(\mathcal{A}(x,y)),\qquad
x,y\in\operatorname{Z}^{0}(\mathcal{A}),$

while the morphisms in $H^{0}(\mathcal{A})$ are the cohomology
$\mathbf{k}$-modules

$H^{0}(\mathcal{A})(x,y)\coloneqq H^{0}(\mathcal{A}(x,y)),\qquad x,y\in
H^{0}(\mathcal{A}).$

Similarly, $\mathcal{A}$ has an associated graded category
$H^{\bullet}(\mathcal{A})$ whose objects are the same as those of
$\mathcal{A}$ and with graded $\mathbf{k}$-modules of morphisms

$H^{\bullet}(\mathcal{A})(x,y)\coloneqq H^{\bullet}(\mathcal{A}(x,y)),\qquad
x,y\in H^{\bullet}(\mathcal{A}).$

A DG functor $F\colon\mathcal{A}\to\mathcal{B}$ induces apparent functors

$\operatorname{Z}^{0}(F)\colon\operatorname{Z}^{0}(\mathcal{A})\longrightarrow\operatorname{Z}^{0}(\mathcal{B})\qquad\text{and}\qquad
H^{0}(F)\colon H^{0}(\mathcal{A})\longrightarrow H^{0}(\mathcal{B})$

as well as a graded functor

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

A DG functor $F$ is a _quasi-equivalence_ if the induced functor
$H^{\bullet}(F)$ is an equivalence of graded categories.

For example, if $\mathcal{C}$ is an additive category then

$\operatorname{Z}^{0}(\operatorname{C}_{\mathrm{dg}}(\mathcal{C}))=\operatorname{C}(\mathcal{C})$

while

$H^{0}(\operatorname{C}_{\mathrm{dg}}(\mathcal{C}))=\operatorname{K}(\mathcal{C})$

is the homotopy category of cochain complexes in $\mathcal{C}$. Similarly, the
graded category $H^{\bullet}(\operatorname{C}_{\mathrm{dg}}(\mathcal{C}))$ has
graded $\mathbf{k}$-modules of morphisms

$H^{i}(\operatorname{hom}_{\mathcal{C}}^{\bullet}\\!\left(X,Y\right))=\operatorname{K}(\mathcal{C})(X,Y[i]),\qquad
X,Y\in\operatorname{C}(\mathcal{C}),\ i\in\mathbb{Z}.$

Under these identifications, the shift (DG) functor on
$\operatorname{C}_{\mathrm{dg}}(\mathcal{C})$ induces the usual shift functors
on $\operatorname{C}(\mathcal{C})$ and $\operatorname{K}(\mathcal{C})$.

####  3.1.2. The homotopy category of small DG categories

As with quasi-isomorphisms between cochain complexes, quasi-equivalences
between DG categories need not admit a quasi-inverse given by a DG functor.
However, the category $\operatorname{dgcat}$ of small DG categories admits a
cofibrantly generated Quillen model category structure, called the _Tabuada
model structure_ , whose weak equivalences are the quasi-equivalences (we do
not need to recall here what are the classes of (co)fibrations) [Tab05]. We
denote the homotopy category of the Tabuada model structure by
$\operatorname{Ho}\left(\operatorname{dgcat}\right)$, which is, by the general
theory of Quillen model categories, equivalent to the localisation of
$\operatorname{dgcat}$ at the class of quasi-equivalences. In particular, two
small DG categories $\mathcal{A}$ and $\mathcal{B}$ are isomorphic in
$\operatorname{Ho}\left(\operatorname{dgcat}\right)$ if and only if they are
connected by a finite zig-zag of quasi-equivalences. Furthermore, the set of
morphisms $\mathcal{A}\to\mathcal{B}$ in
$\operatorname{Ho}\left(\operatorname{dgcat}\right)$ is in bijection with the
set of quasi-isomorphism classes of DG $\mathcal{A}$-$\mathcal{B}$-bimodules
$X$ such that, for each $a\in\mathcal{A}$, the DG $\mathcal{B}$-module
$X(-,a)$ is quasi-isomorphic to a representable DG $\mathcal{B}$-module, see
[Toë11, Cor. 4.8].

####  3.1.3. The derived category of a DG category

We recall the construction of the derived category of a (small) DG category.
We refer the reader [Kel94] for details. Let $\mathcal{A}$ be a small DG
category, that is the objects in $\mathcal{A}$ form a set. The _opposite DG
category $A^{\mathrm{op}}$ of $\mathcal{A}$_ is the DG category with the same
objects as $\mathcal{A}$ and graded morphism spaces

$\mathcal{A}^{\mathrm{op}}(x,y)\coloneqq\mathcal{A}(y,x),\qquad
x,y\in\mathcal{A}^{\mathrm{op}};$

the composition law in $\mathcal{A}^{\mathrm{op}}$ is given in terms of the
composition law in $\mathcal{A}$ by the formula

$g\circ^{\mathrm{op}}f\coloneqq(-1)^{ij}f\circ g$

whenever $f$ and $g$ are homogeneous of degree $i$ and $j$, respectively. The
_DG category of (right) DG $\mathcal{A}$-modules_ is the DG category

$\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A})\coloneqq\operatorname{Fun}_{\mathrm{dg}}\\!\left(\mathcal{A}^{\mathrm{op}},\operatorname{C}_{\mathrm{dg}}(\mathbf{k})\right)$

of DG functors
$\mathcal{A}^{\mathrm{op}}\to\operatorname{C}_{\mathrm{dg}}(\mathbf{k})=\operatorname{C}_{\mathrm{dg}}(\operatorname{Mod}\mathbf{k})$.
We denote the morphism complexes in
$\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A})$ by

$\operatorname{hom}_{\mathcal{A}}\\!\left(M,N\right),\qquad
M,N\in\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}).$

The DG Yoneda embedding

$\mathbf{h}\colon\mathcal{A}\longrightarrow\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}),\qquad
x\longmapsto\mathcal{A}(-,x).$

identifies $\mathcal{A}$ with full DG subcategory of
$\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A})$.

The underlying category

$\operatorname{dgMod}(\mathcal{A})\coloneqq\operatorname{Z}^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$

is a Grothendieck abelian category. Moreover,
$\operatorname{dgMod}(\mathcal{A})$ is a Frobenius exact category in which a
short exact sequence

$0\to L\stackrel{{\scriptstyle i}}{{\to}}M\to N\to 0$

of DG $\mathcal{A}$-modules is admissible (=a conflation) if there exists a
morphism $p\colon M\to L$ of graded (!) $\mathcal{A}$-modules such that
$pi=\mathbf{1}_{L}$; a DG $\mathcal{A}$-module is projective-injective in this
exact structure if and only if it is _contractible_ , that is if it is a zero
object in the homotopy category
$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$. In particular, the
stable category of the Frobenius exact category
$\operatorname{dgMod}(\mathcal{A})$ equals
$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$ and is a triangulated
category with suspension functor induced by the shift functor on
$\operatorname{C}_{\mathrm{dg}}(\mathbf{k})$ by means of the formula

$[1]\colon M\longmapsto M[1]\coloneqq[1]\circ M,\qquad
M\in\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}),$

keeping in mind that a DG $\mathcal{A}$-module is a DG functor
$\mathcal{A}^{\mathrm{op}}\to\operatorname{C}_{\mathrm{dg}}(\mathbf{k})$. By
construction, there are canonical isomorphisms

$H^{i}(\operatorname{hom}_{\mathcal{A}}\\!\left(M,N\right))\cong
H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))(M,N[i]),\qquad
M,N\in\operatorname{dgMod}(\mathcal{A})$

and, as a consequence of the DG Yoneda Lemma,

$H^{i}(\operatorname{hom}_{\mathcal{A}}\\!\left(\mathbf{h}_{x},M\right))\cong
H^{i}(M_{x}),\qquad x\in\mathcal{A},\ M\in\operatorname{dgMod}(\mathcal{A}).$

A DG $\mathcal{A}$-module $N$ is _acyclic_ if for each $x\in\mathcal{A}$ the
cochain complex of $\mathbf{k}$-modules $N_{x}$ is acyclic. The _derived
category $\operatorname{D}(\mathcal{A})$ of $\mathcal{A}$_ is the full
subcategory of the triangulated category
$H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$ spanned by the DG
$\mathcal{A}$-modules that are _DG projective_ , that is the DG
$\mathcal{A}$-modules $P$ such that every epimorphism $M\to P$ with acyclic
kernel is split. The derived category of $\mathcal{A}$ is a triangulated
subcategory of $H^{0}(\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A}))$ that
is closed under small coproducts and retracts and is compactly generated by
the _free DG $\mathcal{A}$-modules_

$\mathbf{h}_{x}=\mathcal{A}(-,x)\colon y\longmapsto\mathcal{A}(y,x),\qquad
x\in\mathcal{A}.$

The _perfect derived category $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ of
$\mathcal{A}$_ is the full subcategory of $\operatorname{D}(\mathcal{A})$
spanned by the compact objects; equivalently,
$\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ is the thick subcategory of
$\operatorname{D}(\mathcal{A})$ generated by the free DG $\mathcal{A}$-modules
[Kel94, Thm. 5.3]. In particular, $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$
is a triangulated category in which idempotents split.

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a DG functor. The restriction of
scalars

$F^{*}\colon\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{B})\longrightarrow\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A})$

induces an exact functor

$F^{*}\colon\operatorname{D}(\mathcal{B})\longrightarrow\operatorname{D}(\mathcal{A})$

that admits an exact left adjoint

$\mathbb{L}F_{!}\colon\operatorname{D}(\mathcal{A})\longrightarrow\operatorname{D}(\mathcal{B})$

that preserves small co-products (constructed explicitly as a left derived
functor). Moreover, the functor $\mathbb{L}F_{!}$ preserves compact objects
(since it preserves the free DG modules) and hence restricts to an exact
functor

$\mathbb{L}F_{!}\colon\operatorname{D}^{\mathrm{c}}(\mathcal{A})\longrightarrow\operatorname{D}^{\mathrm{c}}(\mathcal{B})$

between the corresponding perfect derived categories.

####  3.1.4. Pre-triangulated DG categories

The following definitions motivate our definition of DG enhancement of a
$(d+2)$-angulated category (see Definitions 3.2.1 and 3.2.5).

######  Definition 3.1.1 ([BK90]).

A small DG category $\mathcal{A}$ is _pre-triangulated_ if the 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 triangulated subcategory of
$\operatorname{D}^{\mathrm{c}}(\mathcal{A})$. Similarly, $\mathcal{A}$ is
_Karoubian pre-triangulated_ if the above functor is an equivalence of
categories.

######  Remark 3.1.2.

The term _pre-triangulated_ is also used for triangulated categories which
need not satisfy the octahedral axiom [Nee01, Def. 1.1.2], see also Definition
2.1.1 and Remark 2.1.5. Despite the clash of terminology, it is always clear
from the context which notion is in use. In particular, notice that the $0$-th
cohomology category of a pre-triangulated DG category is a triangulated
category (and therefore satisfies the octahedral axiom).

######  Definition 3.1.3 ([BK90]).

Let $\mathcal{T}$ be a triangulated category.

  1. (1)

A _(DG) enhancement of $\mathcal{T}$_ is a pre-triangulated DG category
$\mathcal{A}$ such that there exists an equivalence of triangulated categories

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

(notice that we only require the existence of such an equivalence, and not the
datum of a preferred such).

  2. (2)

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

####  3.1.5. Morita equivalences

The category of small DG categories and DG functors between them admits a
cofibrantly generated Quillen model category structure, called the Morita
model structure [Tab05, Thm. 5.3], see also [Tab06, Tab07a], whose weak
equivalences are the _Morita equivalences_ , that is the DG functors
$F\colon\mathcal{A}\to\mathcal{B}$ such that the induced exact functor

$\mathbb{L}F_{!}\colon\operatorname{D}^{\mathrm{c}}(\mathcal{A})\longrightarrow\operatorname{D}^{\mathrm{c}}(\mathcal{B})$

is an equivalence of triangulated categories. A DG category is fibrant in the
Morita model structure if and only if it is Karoubian pre-triangulated; more
generally, the DG Yoneda embedding

$\mathcal{A}\longrightarrow\operatorname{D}^{\mathrm{c}}_{\mathrm{dg}}(\mathcal{A})$

is a fibrant replacement (that is a Morita equivalence with fibrant target)
for every small DG category $\mathcal{A}$, where
$\operatorname{D}^{\mathrm{c}}_{\mathrm{dg}}(\mathcal{A})$ is the _perfect
derived DG category of $\mathcal{A}$_, that is the full DG subcategory of
$\operatorname{dgMod}_{\mathrm{dg}}(\mathcal{A})$ spanned by the DG
$\mathcal{A}$-modules that are DG projective and compact in
$\operatorname{D}(\mathcal{A})$. The _Morita category of small DG categories_
is the homotopy category $\operatorname{Hmo}$ associated to the Morita model
structure. Two DG categories are _Morita equivalent_ if they are isomorphic in
$\operatorname{Hmo}$. In particular, every small DG category is Morita
equivalent to its perfect derived DG category.

######  Remark 3.1.4.

If $\mathcal{A}$ and $\mathcal{B}$ are small DG categories that are Morita
equivalent, then the perfect derived categories
$\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ and
$\operatorname{D}^{\mathrm{c}}(\mathcal{B})$ are equivalent as triangulated
categories. The converse, however, is false [DS07, Sch02, Kaj13, RVdB19].

######  Remark 3.1.5.

Every quasi-equivalence between small DG categories is also a Morita
equivalence. Closely related to this is the fact that $\operatorname{Hmo}$ is
equivalent to the full subcategory of
$\operatorname{Ho}\left(\operatorname{dgcat}\right)$ spanned by the Karoubian
pre-triangulated DG categories. In particular, two Karoubian pre-triangulated
DG categories are quasi-equivalent if and only if they are Morita equivalent.

######  Theorem 3.1.6 ([Kel06, Thm. 3.8]).

Let $\mathcal{X}$ be a Karoubian pre-triangulated DG category and
$\mathcal{A}\subseteq\mathcal{X}$ a full DG subcategory such that
$\operatorname{thick}(H^{0}(\mathcal{A}))=H^{0}(\mathcal{X})$. Then, the
inclusion functor $\mathcal{A}\hookrightarrow\mathcal{X}$ is a Morita
equivalence and, in particular, there are canonical equivalences of
triangulated categories

$\operatorname{D}^{\mathrm{c}}(\mathcal{A})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{D}^{\mathrm{c}}(\mathcal{X})\stackrel{{\scriptstyle\sim}}{{\longleftarrow}}H^{0}(\mathcal{X}).$

######  Example 3.1.7.

Given a DG algebra $A$, there are canonical morphisms of DG algebras (=DG
functors between DG categories with a single object)

$H^{0}(A)\longleftarrow\tau^{\leq 0}A\longrightarrow A,$

where

$(\tau^{\leq 0}A)^{i}\coloneqq\begin{cases}A^{i}&i<0,\\\
\operatorname{Z}^{0}(A)&i=0,\\\ 0&i>0.\end{cases}$

By construction, the map $\tau^{\leq 0}A\to A$ is a quasi-isomorphism if and
only if the cohomology of $A$ is concentrated in non-positive degrees. In
particular, if $H^{\bullet}(A)=H^{0}(A)$ is concentrated in degree $0$, then
both of the above maps are quasi-isomorphisms of DG algebras and thus also
Morita equivalences. Combining this observation with Theorem 3.1.6, we
conclude that the (compact) derived category of an ordinary algebra admits a
unique enhancement in the sense of Definition 3.1.3.

######  Remark 3.1.8.

In contrast to the situation described in Example 3.1.7 for the derived
category of an ordinary algebra, establishing the uniqueness of enhancements
for more general triangulated categories is a subtle endeavour. Moreover, the
majority of results of this kind that are available rely on the existence of
$t$-structures in a crucial way, see for example [LO10, CS18, CNS22]. Similar
to the situation in [Mur22], in this article we consider triangulated
categories that need not admit any non-trivial $t$-structures (such as
$2$-Calabi–Yau triangulated categories with a $2$-cluster tilting object, see
Theorem 6.1.5 and compare with [ZZ, Thm. 4.1]).

###  3.2. Pre-$(d+2)$-angulated differential graded categories

The following are straightforward extensions of Definitions 3.1.1 and 3.1.3 to
the context of $(d+2)$-angulated categories.

######  Definition 3.2.1.

A small DG category $\mathcal{A}$ is _pre- $(d+2)$-angulated_ if the fully
faithful functor

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

identifies identifies $H^{0}(\mathcal{A})$ with a standard $(d+2)$-angulated
subcategory of $\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ (see Definition
2.2.2). Similarly, $\mathcal{A}$ is _Karoubian pre- $(d+2)$-angulated_ if it
is pre-$(d+2)$-angulated and idempotents split in $H^{0}(\mathcal{A})$.

######  Remark 3.2.2.

A (Karoubian) pre-3-angulated DG category is precisely a (Karoubian) pre-
triangulated DG category.

######  Remark 3.2.3.

Let $\mathcal{A}$ be a pre-$(d+2)$-angulated DG category. Since, by
definition, $H^{0}(\mathcal{A})$ is equivalent to a $d$-rigid subcategory of
$\operatorname{D}^{\mathrm{c}}(\mathcal{A})$ closed under the action of the
$d$-fold suspension and its inverse, the standard isomorphisms

$\operatorname{Hom}_{\operatorname{D}(\mathcal{A})}\\!\left(\mathbf{h}_{x},\mathbf{h}_{y}{[i]}\right)\cong
H^{i}(\mathcal{A}(x,y)),\qquad x,y\in\mathcal{A},$

imply that

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

This is a strong restriction on the cohomology of $\mathcal{A}$ (see also
Definition 4.3.2).

######  Proposition 3.2.4.

Let $\mathcal{A}$ be a small DG category such that $H^{0}(\mathcal{A})$ is an
additive category with finite-dimensional morphism spaces and split
idempotents. The following statements are equivalent:

  1. (1)

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

  2. (2)

The fully faithful functor

$H^{0}(\mathbf{h})\colon