Serre Functors and Graded Categories

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras.


Introduction
This paper relates two algebraic structures: Frobenius algebras and categories with Serre duality. It's well-known that they are related, but we set out the relationship quite carefully in order to prove some statements in the representation theory of quivers.
We want to show that the Nakayama automorphism of a preprojective algebra has finite order if and only if the derived category of the corresponding quiver is fractional Calabi-Yau. The Nakayama automorphism and the Serre functor agree in certain situations so, up to some technical considerations about gradings, this is about them both having finite order.
To give a precise statement we use graded categories, which live in a 2-category, and each one is enriched in a fixed monoidal category. We give equivalences between such structures, so we are really studying higher category theory, but we only use 2-categories in this paper.

Frobenius algebras and Calabi-Yau categories
Given a tensor category, such as vector spaces, one can look for algebra objects and coalgebra objects. A Frobenius algebra is an object with both algebra and coalgebra structures simultaneously, satisfying certain axioms of a topological nature [Abr96]. This reflects the fact that commutative Frobenius algebras classify 2-dimensional topological quantum field theories: see [Koc04] for a nice explanation. If our tensor category has duals then we can express the Frobenius algebra axioms in a form that would look more recognisable to a classically trained algebraist, using nondegenerate forms or module isomorphisms. Reformulating the definitions in this way has the advantage of exhibiting an automorphism of our object known classically as the Nakayama automorphism. If this automorphism is the identity, one calls the algebra a symmetric algebra.
We might want a "many-object" version of a Frobenius algebra. This is a category C with Serre duality. It comes with a Serre functor Ë : C ∼ → C, an autoequivalence which replaces the Nakayama automorphism. In situations of interest our categories often have extra structure (e.g., a triangulated structure) and come with a suspension functor Σ : C ∼ → C. Then automorphisms of C should really be equivariant, i.e., they should come equipped with natural isomorphisms witnessing a weak commutation with Σ.
If the Serre functor is naturally isomorphic to an nth power of Σ, C is said to be a Calabi-Yau category of dimension n [Kon98]. Calabi-Yau categories, and more generally Calabi-Yau A ∞ -categories, are also connected to TQFTs [Cos07] as well as to homological mirror symmetry. There is a subtlety in their definition: we should really have an equivariant isomorphism of equivariant functors between Ë and Σ n . Calabi-Yau categories are abundant: as well as the examples coming from geometry which motivate the terminology, there is a formal construction which, from a dg-algebra, produces a new dg-algebra whose derived category is Calabi-Yau [KVdB11].
One could weaken the definition of a Calabi-Yau category and look for categories with Serre duality where we have a natural isomorphism between some power Ë m of the Serre functor and Σ N : these are known as fractional Calabi-Yau categories of dimension N/m [Kon98]. At first the definition may seem surprising, but there are interesting examples of such categories in many parts of mathematics. Perhaps the simplest is quiver representations [MY01], but they also appear in algebraic geometry, matrix factorisations, theoretical physics, and other parts of representation theory [Ta,AA13,KLM13,HIMO,Kuz19].
The aim of this paper is to establish methods to prove that certain categories are fractional Calabi-Yau. We do this in two steps: first we change the equivariant structure to one inspired by classical Auslander-Reiten theory, thereby moving to a situation which is often better understood in the representation theory of algebras. Then we move back to a one-object setting, so we can test the fractional Calabi-Yau property via the Nakayama automorphism of a well-known algebra called the preprojective algebra.
When investigating Calabi-Yau properties, we are asking about isomorphims of functors. This is a 2-categorical notion, and we believe it is easier if we embrace this from the start. Although this seems reasonable to us, it is less common within the representation theory of finite-dimensional algebras, so we take the opportunity to formulate 2-categorical versions of some well-known results.

Quivers and preprojective algebras
Let Q be a quiver and let Q be its path algebra. Assume Q has no oriented cycles, then Q is finite-dimensional. According to Gabriel's theorem, the representation theory of the algebra Q behaves very differently depending on the underlying graph of Q: if it is an ADE Dynkin graph then Q has finitely many indecomposable modules; otherwise it has infinitely many.
There is another algebra we can associate to Q: its preprojective algebra Π(Q). An explicit presentation of this is given by doubling the arrows of Q and quotienting out by certain relations; a basis-free description of Π(Q) was given by Baer, Geigle, and Lenzing [BGL87]. Whether or not the underlying graph is Dynkin is reflected in the algebra Π(Q) itself: in the Dynkin case it is finite-dimensional; otherwise it is infinite-dimensional.
The preprojective algebra arose out of an attempt to better understand Gabriel's theorem. The proof of Gabriel's theorem by Bernstein, Gelfand, and Ponomarev introduced and made use of Coxeter functors on the category of Q-modules, then Gelfand and Ponomarev introduced model algebras to understand the image of the projective Q-modules under these functors. In the modern approach, Coxeter functors are replaced by Auslander-Reiten functors τ − and model algebras by preprojective algebras.
In the Dynkin case, Π(Q) has another nice property: it is self-injective. This was a folklore result for a long time. A careful proof, depending on complicated case-by-case checks, was written down by Brenner, Butler, and King [BBK02]. In fact, they showed more than just self-inectivity: they proved that Π(Q) is a Frobenius algebra and gave an explicit formula for its Nakayama automorphism. In particular, it squares to the identity. Using the derived category interpretation of the preprojective algebra, the Nakayama automorphism should correspond to the Serre functor Ë: this has previously been explained on the level of the module category [Gra20]. So we expect Ë 2 to act trivially on the orbit category; on D b ( Q), it should be a power of τ − . This can be deduced on the Grothendieck group K 0 ( Q) from work of Gabriel. The derived functor of τ − is realised as the shifted inverse Serre functor Ë − Σ. So a natural isomorphism between Ë 2 and τ −N should correspond to a natural isomorphism between Ë N+2 and Σ N . This is the fractional Calabi-Yau property of D b ( Q), which was introduced by Kontsevich and proved in general by Miyachi and Yekutieli [Kon98,MY01].
Cluster algebras, introduced by Fomin and Zelevinsky, have been hugely influential in modern representation theory. The construction of the cluster category by Buan, Marsh, Reineke, Reiten, and Todorov [BMRRT06] involves starting with the derived category of Q-modules and constructing a new category whose objects are the orbits of an endofunctor. Iyama and Oppermann realised that the abstract construction of the preprojective algebra given by Baer, Geigle, and Lenzing could be interpreted using orbit categories: Π(Q) is an endomophism algebra of a generator of D b ( Q)/τ − . With this perspective, they were able to give a very general explanation for the fact that Π(Q) is self-injective: it follows from the existence of a Serre functor on the derived category of Q [IO13].
The work of Iyama and Oppermann is more general than described above: they work with drepresentation finite algebras Λ, for which there exists a good higher dimensional analogue of the Auslander-Reiten functor [Iya07]. Around the same time, Herschend and Iyama noticed that examples of d-representation finite algebras had the fractional Calabi-Yau property, and were able to show that this property always holds up to some twists [HI11a].
At this point, the picture seems clear: d-representation finite algebra ←→ (d + 1)-preprojective algebra Serre duality on the derived category ←→ self-injectivity of preprojective algebra fractional Calabi-Yau property ←→ finite order Nakayama automorphism However, there are two problems: • Even in the classical d = 1 case, for Dynkin quivers, things don't seem to match up exactly. For a quiver of type D 4 , the derived category is Calabi-Yau of dimension 2/3 [MY01], which suggests the Nakayama automorphism of the corresponding preprojective algebra should have order 3 − 2, i.e., it should be the identity. But we know that this automorphism is non-trivial, even in the group of outer automorphisms [BBK02].
• The results of Herschend and Iyama aren't as strong as one might hope. They only obtain twisted fractional Calabi-Yau properties, and are only able to make a precise connection to the higher preprojective algebra under a "homogeneous" assumption which is quite restrictive.
In this article we address these problems as follows: • We carefully analyse the relationship between Serre functors and Nakayama automorphisms in a graded setting and expose a mismatch of signs in the naive correspondence between them. When taking account of these signs, the results of Miyachi-Yekutieli and Brenner-Butler-King do match. Based on this, we propose new graded Nakayama automorphism of the preprojective algebra, which is not related to the classical Nakayama automorphism by an inner automorphism. From the perspective of derived categories, we see this graded automorphism as more fundamental than the classical Nakayama automorphism.
• We prove a general result which states that the derived category of a d-representation finite algebra is Calabi-Yau if and only if the appropriate graded Nakayama automorphism of the preprojective algebra is of finite order.
This also allows us to connect two other results in the literature: the higher preprojective algebras of type A have finite order Nakayama automorphism [HI11a] and the iterated higher Auslander algebras of type A are fractional Calabi-Yau [DJW19,DJL21].

Description of main results
Given suitable finiteness and linearity assumptions, there is a correspondence between algebras and categories. We discuss this in Section 2.

algebras (A) ←→ categories (C)
Given a group G , there are two notions of G -structure on a category C: either the hom-spaces can be G -graded, or we can have an action of G by functors C → C and work equivariantly. We discuss this in Section 4. graded (G) ←→ equivariant (E) In all these situations, there is a notion of Serre duality where we have a Serre functor Ë : C → C with quasi-inverse Ë − (see Section 3.) If our category is triangulated, we also have a suspension Σ : C → C . We could use the Serre functor to replace the suspension with a translation Ì = Ë − Σ.
We discuss this in Section 5. Partly due to technical convenience and partly due to historical accident, some of these combinations are more well-studied than others. Two corners are particularly well studied. To the top left corner of our diagram, translated graded algebras (TGA), belongs the preprojective algebra of a quiver. To the bottom right corner, suspended equivariant categories (SEC), belongs the derived category of a quiver. We study how to move between these corners, keeping track of Serre duality.
We would like to swap the fractional Calabi-Yau relation Ë m ∼ = Σ N in a triangulated category for the relation Ë m−N ∼ = Ì N , but this is too naive: we need to keep track of the suspended structure and record this in a new translated structure. To give a correct technical formulation, we note that Serre functors come with canonical commutation transformations. In particular, if (D, F ) is aequivariant category, such as a triangulated category with F = Σ, we have maps ζ F : ËF ∼ → F Ë.
The study of commutation transformations for triangulated functors goes back to Verdier [Ve67], and their importance in studying Calabi-Yau structures was emphasised in work of Keller [Kel08,Kel09].
With this insight, we can state our main result (see Theorems 5.13 and 5.14) as follows: Theorem A. Let D be a category with automorphism F : D → D and let χ : → × be a character.

Then we have an F -equivariant isomorphism of functors
if and only if we have a Ì-equivariant isomorphism of functors Moreover, if the orbit category C = D/Ì has finitely many isomorphism classes of objects, we construct a graded Frobenius algebra A. Then the natural isomorphisms above exist if and only if the χ-Nakayama automorphism of A is the identity map up to a shift by N.
We apply this in the case of d-representation algebras and higher preprojective algebras. Suppose Λ is a basic d-representation finite algebra, so it has a d-cluster tilting module (see [IO13]). Let Π denote its higher preprojective algebra, which has a natural grading coming from its definition as a tensor algebra. Then Π is Frobenius, so it has a (classical) Nakayama automorphism α. Define β(a) = (−1) dp for a ∈ Π a degree p homogeneous element of Π. Then, by Theorem 6.14 and Corollary 6.16, we have:

Summary of contents
We now outline the contents of this article.
Section 2 starts with the basic 2-dimensional category theory we will use, including adjunctions, equivalences, and 2-groupoids. We describe idempotent completion as a 2-functor. We then discuss monoidal categories, including duals, pivotal structures, and the Drinfeld centre. Next we explain the Picard group of a monoidal category. We then give basic facts about -linear categories and a 2-categorical treatment of indecomposable objects. Finally we discuss the connection betweencategories and -algebras, and give the definition of a base algebra (Definition 2.47) which will play an important role later.
In Section 3 we meet the definition of a Serre structure on a category enriched in a monoidal category with pivotal structure. We discuss uniqueness and transfer of structure across equivalences of categories. We see the compatibility lemma: roughly, this says that if we know how a Serre functor acts on objects, and we have the structural isomorphisms in the definition of a Serre structure, then we can recover the action of the Serre functor on morphisms. We revise the well-known result that Serre functors Ë commute with autoequivalences, note that this in fact gives us an element of the Drinfeld centre of the monoidal category of automorphisms, and prove a technical result on the commutation map ËË − → Ë − Ë which will be useful later. Finally, we discuss how Serre structures on -categories are related to Frobenius structures on -algebras.
In Section 4 we see the two notions of grading on a category. The first we call an equivariant structure: this is where a group acts on our category. We discuss strict and weak actions, and the equivariant centre. The second we call a hom-graded structure: this is where the hom spaces form a "graded algebra with several objects". Then we introduce graded Serre structures (in both settings) which depend on a character of our group. In the hom-graded setting, they have appeared in the physics literature [Laz07] but don't seem to have been used much. This author finds working with an arbitrary character conceptually easier than keeping track of minus signs. We carefully prove uniqueness results and make a connection to graded Frobenius structures. Then we consider moving between the two notions of grading using the well-known operations of orbit categories and smash products. We use Asashiba's theorem that these are 2-functors and they give an equivalence of 2-categories [Asa17]. We show that both operations preserve Serre structures in the appropriate graded setting. In the case of orbit categories, this is related to a result of Dugas [Dug12].
We restrict to the -graded setting in Section 5. We start by discussing triangulated categories and triangulated functors, and recall results of Bondal-Kapranov and Van den Bergh on triangulated Serre functors. Then we define an abstract (inverse) Auslander-Reiten functor Ì on a -equivariant category. We study "change of action", where the -equivariant structure on a category is modified by an element of the Drinfeld centre, such a Serre functor. Keeping the triangulated situation in mind, this allows us to prove a nice compatibility result for the Auslander-Reiten functor: the map ÌÌ → ÌÌ we obtain by change of action is just the identity map. Next we study a formal, or "synthetic", version of the fractional Calabi-Yau property. Given three functors F , Ë, and Ì = Ë − F , a relation can be expressed in three equivalent ways: We provide an equivariant version of this basic idea, and observe that Ë k = Ì N can be checked on the orbit category or, given a finiteness condition, on the graded base algebra.
Finally, we apply our theory in Section 6. First we outline how tensor products of chain complexes give graded functors between derived categories and use this to connect the strong Calabi-Yau property and the bimodule Calabi-Yau property. Then we discuss Dynkin quivers. After recalling the background information we quote the fractional Calabi-Yau result of Miyachi and Yekutieli, and the Nakayama automorphism result of Brenner, Butler, and King. We show how these results are related and, largely, determine each other. Then we explain how the theory is applied to the d-representation finite algebras which appear in Iyama's higher homological algebra: these are algebras of higher global dimension 0 ≤ d < ∞ to which many parts of the representation theory of Dynkin quivers can be generalised. We explain how to use Herschend and Iyama's calculation of the Nakayama automorphism for the preprojective algebras of "higher type A algebras" to recover Dyckerhoff, Jasso, and Walde's result that these algebras are fractional Calabi-Yau. We finish by considering algebras constructed from Postnikov diagrams by Pasquali [Pas19]: these 2-representation finite algebras are always fractional Calabi-Yau and we show in an explicit example how to calculate their Calabi-Yau dimension.

2-dimensional category theory
We recall some standard facts from 2-dimensional category theory, partly to fix notations and terminology. Precise definitions can be found in [Lei] or [JY21], and our discussion is also motivated by [nLab].

Bicategories and 2-categories
A bicategory is a weak 2-dimensional category. Bicategories B have 0-cells x, y , ... (also called objects), 1-cells f , g : x → y between 0-cells, and 2-cells α : f → g between 1-cells. We write ob B for the objects of B. Between any two objects x and y there is a hom category B(x, y ) whose objects are the 1-cells and whose morphisms are the 2-cells. B comes with (horizontal) composition functors which are associative up to specified natural isomorphism and with unit functors which are unital up to specified natural isomorphism. In particular, each object x has an identity 1-cell 1 x , and each 1-cell f : x → y has an identity 2-cell 1 f .
Example 2.1. There is a bicategory of bimodules. Its 0-cells are rings, its 1-cells are R-S-bimodules, and its 2-cells are bimodule maps. Horizontal composition is given by tensor product. Note that the weak structure in the definition of a bicategory is necessary here: for an R-S-bimodule M, we have A (strict, locally small) 2-category C is a category enriched in categories. This means that between each ordered pair of objects we have a morphism category C(x, y ), and the composition is functorial. 2categories are bicategories where the natural isomorphisms for associativity and unitality are identities. Note that the 0-and 1-cells of C form a (classical) 1-category, called the underlying category of C.
Example 2.2. A category C is called small if ob C is a set. There is a 2-category Cat whose 0-cells are small categories, 1-cells are functors, and 2-cells are natural transformations.

2-functors
Let B and C be bicategories. A weak 2-functor F : B → C is a function ob B → ob C, which we also denote F, together with a collection of functors which commute with the composition and unit functors up to specified natural isomorphisms.
Let B and C be 2-categories. A 2-functor F : B → C is a weak 2-functor which strictly preserves units and horizontal composition of 2-cells (see [JY21, Proposition 4.1.8]).
Given two 2-functors F, G : B → C, a 2-transformation α : F → G is a function sending each x ∈ ob B to a 1-cell α Example 2.3. There is a 2-category 2Cat whose 0-cells are small 2-categories, 1-cells are 2-functors, and 2-cells are 2-transformations. (As for categories, a 2-category C is small if ob C is a set.) Given any property P of functors, we say a 2-functor F : B → C is locally P if for all x, y ∈ ob B, the component functor F x,y has property P. For example, F is locally an equivalence if every functor F x,y is an equivalence of categories.
Definition 2.4. Given a 2-category B and a subset S ⊂ ob B, the full sub-2-category of B on S is the 2-category B| S with 0-cells S and morphism categories B| S (x, y ) = B(x, y ).
Note that every full sub-2-category comes with an inclusion 2-functor B| S → B which is locally an equivalence.
Lemma 2.5. Let F : B → C be a 2-functor and let S ⊂ ob B and T ⊂ ob C. If F(S) ⊆ T then F restricts to a 2-functor F | S : B| S → C| T .

Equivalences and adjunctions
A 2-cell is called iso (or an isomorphism) if it has a two-sided inverse. If there exists an iso 2-cell α : f → g we write f ≃ g . A 1-cell f : x → y is an equivalence if there exists g : y → x such that gf ≃ 1 x and fg ≃ 1 y , and we say g is a quasi-inverse of f . If such an equivalence exists, we say x and y are equivalent. A weak 2-functor F : B → C is a biequivalence if every functor F x,y is an equivalence of categories and every object of B is equivalent to F x for some x ∈ ob B.
The following simple fact is quite useful.
Lemma 2.6. Every equivalence f : x → y in a bicategory B induces a biequivalence F : B → B such that Fx = y .
Proof. This is a standard construction. Define a weak 2-functor F which acts on 0-cells by sending x to y and fixing other objects. For w , z = x let F w ,z be the identity functor. If one or both of w , z are x, define F w ,z by pre-or post-composing with f or its quasi-inverse.
An adjunction in a 2-category is the data (f , g , η, ε) where f : x → y and g : y → x are 1-cells and η : 1 x → gf and ε : fg → 1 y are 2-cells satisfying the triangle identities: f is called the left adjoint and g the right adjoint. We write f ⊣ g .
An adjoint equivalence is an adjunction such that η and ε are both isomorphisms. Any equivalence f : x ∼ → y can be upgraded to an adjoint equivalence (f , g , η, ε).

2-groupoids
Definition 2.7. A 2-category is called a 2-groupoid if all its 1-cells are equivalences and all its 2-cells are isomorphisms.
The full sub-2-category of 2Cat on the 2-groupoids is called the 2-category of 2-groupoids, and is denoted 2Gpd.
Every 2-category C has a maximal sub-2-groupoid, which we denote core C. This construction is functorial: if F : B → C is a 2-functor we get a new 2-functor core F : core B → core C. It is not 2-functorial on 2Cat, because an arbitrary 2-transformation involves component 1-cells which may not be isomorphisms, but it is 2-functorial on the sub-2-category Cat 2,1 of 2Cat which contains only iso 2-transformations.
The 2-category core(Cat), which we denote Cat 2,0 , will be particularly important for us. Its 0-cells are categories, its 1-cells are equivalences of categories, and its 2-cells are natural isomorphisms.

Idempotent completion
Let C be a category and let e ∈ C(x, x) be an idempotent, i.e., e 2 = e. We say that e splits if C has an object y and and morphisms f : x → y and g : y → x such that e = gf and 1 y = fg . If all idempotents in C split we say that C is idempotent complete. We have a full sub-2-category ic Cat of Cat on the indempotent complete categories.
Any category C has an idempotent completion ic C. Its objects are pairs (x, e) where x ∈ ob C and e 2 = e ∈ C(x, x). The morphism set from (x, e) to (y , d) is dC(x, y )e = {dfe | f ∈ C(x, y )}. Note ic C is idempotent complete, and we have a fully faithful inclusion functor C ֒→ ic C.
Lemma 2.8. If C is idempotent complete then C ֒→ ic C is dense, so C and ic C are equivalent.
This construction extends to a 2-functor ic : Cat → Cat as follows. Given a functor F : C → D we define ic F on objects by (x, e) → (Fx, Fe) and on maps it is just F . For a natural transformation α : F → G , define components by The full subcategory of ic C on objects (x, 1 x ) is equivalent to C. By considering the components of natural transformations on these objects, we get the following: Lemma 2.9. The 2-functor ic : Cat → Cat is locally fully faithful.

Monoidal categories
Monoidal categories will appear in at least two places: as enriching categories (e.g., a monoidal category with vector spaces as objects) and as endomorphism categories (e.g., a monoidal category with endofunctors as objects).
We outline just what we need; precise definitions can be found in [EGNO15].

Monoidal categories and monoidal transformations
A monoidal category M is a category with a tensor product bifunctor − ⊗ − : M × M → M and a unit object ½. We often omit associator isomorphisms, pretending M is strictly associative; by Mac Lane's coherence theorem we can do this without loss of generality. We do not assume M is symmetric or even braided.
One-object bicategories correspond to monoidal categories by a process known as "looping".
Example 2.10. Any object x in a bicategory B has an endomorphism category B(x, x) which has monoidal structure given by composition.
Let M, N be two monoidal categories. A monoidal functor from M to N is a pair (F , J) consisting of a functor F : M → N and a binatural isomorphism J : ) satisfying unit and associativity conditions. If J is the identity, we say F is a strict monoidal functor. We get a category Mon of monoidal categories and monoidal functors.
A monoidal transformation α : (F , J) → (G , K ) between monoidal functors is a natural transformation such that, for all X , Y ∈ M, the following diagram commutes: We get a 2-category of monoidal categories, monoidal functors, and monoidal transformations. If F is an equivalence of categories then we say (F , J) is a monoidal equivalence. These are precisely the 1-cells which are equivalences.

Duals
A dual pair in M is a 4-tuple (L, R, c, e) where L and R are objects in M and c : ½ → R ⊗ L ("coevaluation") and e : L ⊗ R → ½ ("evaluation") are maps in M which satisy the usual triangle identities: We say L is left dual to R, written L = R ∨ , and R is right dual to L, written R = ∨ L. So the left dual is written on the left in an evaluation map. Note that the left dual is written with a superscript on the right, and vice versa. However, left matches left in the following: Example 2.11. If f : x → x is left adjoint to g : x → x in a bicategory B, then f is left dual to g in the monoidal category B(x, x).
Left and right duals are defined up to canonical isomorphism: if L 1 and L 2 are both left duals of R, so we have two dual pairs (L 1 , R, c 1 , e 1 ) and (L 2 , R, c 2 , e 2 ), then one can check that the map is iso. Duals are preserved by monoidal functors. The unit object ½ is self-dual.
We will mainly work with left duals, and will sometimes add a subscript to the maps c and e as follows: From now on we assume M is left rigid, i.e., every object has a left dual. A category which is both left and right rigid is simply called rigid (or, in some other parts of the literature, autonomous).
Given a map f : The following lemma is stated in [Se11, Lemma 4.9] and credited to Saavedra Rivano [Sa72, Prop. 5.2.3]. The reference isn't easy to access, so we give a proof.
Lemma 2.14. If α : (F , J) → (G , K ) is a monoidal transformation then the following diagram commutes: In particular, every monoidal transformation between functors on rigid categories is an isomorphism.
Proof. We want to show that α ∨ X α X ∨ = 1 is the evaluation for FX . This follows from monoidality, naturality, and unitality, as in the following commutative diagram:

A technical lemma
Given a map f : X ⊗ Y → Z , we can dualize to get f ∨ : Z ∨ → Y ∨ ⊗ X ∨ . Then we define two maps f ℓ and f r as follows: The following lemma is immediate from the triangle identities and the definition of f ∨ : Lemma 2.15. The following diagrams commute (we omit the ⊗ sign): As a corollary, we get a useful technical lemma: Lemma 2.16. The following diagram commutes: Proof. Apply Lemmas 2.12 and 2.15 and use the definitions of f ℓ and f r .

Pivotal categories
A pivotal structure on a rigid category M is a natural isomorphism of monoidal functors from the identity functor on M to the double dual. Therefore, for any object X , we have ι ∨ X = ι −1 X ∨ by Lemma 2.14.
Unwinding the definition of a pivotal structure, for each object X ∈ M we have an isomorphism ι X : X ∼ → X ∨∨ satisfying ι X ⊗Y = ι X ⊗ ι Y and, for any map f : X → Y , the following diagram commutes: A pivotal category (sometimes called a sovereign category) is a rigid category with a specified pivotal structure. Pivotal categories have a useful graphical calculus: see [Se11] or [TV17] for surveys.
Example 2.17. The category fVec of finite dimensional vector spaces over a field is left rigid, with Example 2.18. Given a bicategory B and an object x ∈ B, the full subcategory Autoeq B (x) ⊆ B(x, x) of equivalences f : x → x is left rigid: left duals f ∨ are precisely left adjoints of f . As f is an equivalence, its left and right adjoints are canonically isomorphic. This gives a pivotal structure on Autoeq B (x).
Note that a pivotal structure identifies left and right duals. In fact, such an identification is equivalent to the existence of a pivotal structure, and in some sources it is taken as the definition.

The centre
Let M be a monoidal category.
Definition 2.19. The centre (or Drinfeld centre) of M is the following category Z(M). Its objects are pairs (Z , z) where Z ∈ ob M and z is a natural isomorphism Z ⊗ − Z(M) is a monoidal category, with tensor product (Z , z) ⊗ (W , w ) = (Z ⊗ W , (z ⊗ 1) • (1 ⊗ w )).
Suppose (Z , z) ∈ Z(M) and Z has left dual Z ∨ , with evaluation and coevaluation maps e : Z ∨ ⊗Z → ½ and c : ½ → Z ⊗ Z ∨ . Then we define a natural transformation z † : Z ∨ ⊗ − ∼ → − ⊗ Z ∨ with components constructed as follows: The following is easy to check: see [TV17, Section 5.2].
Lemma 2.20. (Z ∨ , z † ) is left dual to (Z , z), with evaluation and coevaluation maps e and c.
Similarly, a pivotal structure on M induces a pivotal structure on Z(M).
The following result is also useful. Its proof follows from the naturality of z: see [TV17, Exercise 5.1.5]. Let (Z , z) ∈ Z(M).
Lemma 2.21. If z ½ = 1 Z and Z has a right dual ∨ Z then z X has two-sided inverse z † ( ∨ Z ) .
Therefore, if (i) z ½ = 1 Z , (ii) M is rigid, and (iii) we have a pivotal structure ι X : X → X ∨∨ , then the inverse of z X is the following map:

Picard groups
Let M be a monoidal category. We say that an object X ∈ M is invertible if there exists another object Y ∈ M such that X ⊗ Y and Y ⊗ X are both isomorphic to ½. Note that invertible objects are dualizable, and their evaluation and coevaluation maps are isomorphisms.
The following definition was given in [HPS97, Definition A.2.7] in the case of a closed symmetric monoidal category, based on earlier work in algebraic topology (e.g., [HMS94]). As suggested at the start of [May01, Section 2], it also works in the non-symmetric case, and this is useful for algebra.
Equivalently, one could take the maximal subgroup of the monoid given by isomorphism classes of all objects.
Definition 2.22 generalizes the following classical situation.
Example 2.23. Let (X , O X ) be a ringed space and let Sh X be the monoidal category of sheaves on X , with monoidal structure given by the usual tensor product of sheaves. Then Pic(Sh X ) is the usual Picard group Pic(X ). (See Prop II.6.12 of [Har77].) The following construction gives many examples of Picard groups.
Example 2.24. If x is an object in a bicategory B, define PicEnd B (x) to be the Picard group of the monoidal category B(x, x).
One could define the Picard groupoid of a bicategory, but we will not need this.
There are other examples coming from algebra.
Example 2.25. Let R be a ring, which we do not assume is commutative. Let R -mod-R be the monoidal category of finitely generated R-R-bimodules, with monoidal structure given by tensor product over R. Then Pic(R -mod-R) is the group of invertible finitely generated R-R-bimodules, sometimes called the Picard group of R. Note that, by Morita theory, we can recover this group from functors between categories of left modules: Example 2.26. Let R be a ring and let D b (R-R) be the derived category of R-R-bimodules, with monoidal structure given by derived tensor product over R. Then Pic(D b (R-R)) is the derived Picard group of R (see [Yek99] and [RZ03]). Let Tri denote the 2-category of triangulated categories and let D b (R) be the derived category of R-modules. Then, by Rickard's theory [Ric91], we have an embedding of groups Pic Functoriality immediately implies that isomorphic monoidal categories have isomorphic Picard groups.
The following result is easy. Note that injectivity of Pic(F , J) does not imply that F is full or that F is faithful.
Proof. We set F = Φ C,C and J is given by the natural isomorphism which compares composition in C(C, C) and B(ΦC, ΦC).
Together, Proposition 2.30 and Corollary 2.29 imply: So, from Proposition 2.30 and Lemma 2.6, we get: Corollary 2.32. Equivalent objects in a bicategory have isomorphic PicEnd groups.

-linear and additive categories
Much of the material here can be found in Chapters 2 and 3 of Gabriel and Roiter's book [GR92].
Fix a field . Throughout, we work with -linear categories, also called -categories, in which all hom spaces are -modules. Note that products and coproducts in such categories coincide: they are biproducts, given by the direct sum. In general we do not assume our categories are additive, so biproducts may not exist. We always work with -linear functors, which automatically preserve biproducts: see [MacL71, Section VIII.2]. These are the 0-and 1-cells of a 2-category Cat with arbitrary natural transformations as 2-cells. As in Section 2.1.4, we have variants Cat 2,1 and Cat 2,0 where cells of dimension ≥ 2 and ≥ 1, respectively, are equivalences.
Note that idempotent completion restricts to a locally fully faithful 2-functor on -linear categories.
A -category is called additive if it has all finite biproducts. Given any -category C, we can form a new -category Mat C whose objects are formal finite direct sums of objects in C. Maps are given by matrices with entries in C. For example, if C is the one-object category with endomorphism ring then Mat C is equivalent to the category of -vector spaces.
Lemma 2.33. Mat C is additive. If C is additive then C and Mat C are equivalent.
The construction of Mat C extends to a 2-functor Mat : Cat → Cat as follows. Given two -categories C and D and a functor F : C → D between them, define Mat F : Mat C → Mat D on objects by (Mat F )(x ⊕y ) = Fx ⊕Fy . On maps, just apply F termwise to the matrix. Given a natural transformation η : F → G with components η x : Fx → Gx, define (Mat η) x⊕y : Fx ⊕ Fy → Gx ⊕ Gy as the diagonal matrix with entries η x and η y .
The full subcategory of Mat C on the objects x ∈ C is isomorphic to C. So, similarly to Lemma 2.9, we get the following: Lemma 2.34. The 2-functor Mat : Cat → Cat is locally fully faithful.
Remark 2.35. One might wonder if Lemmas 2.9 and 2.34 are both consequences of some formal result in category theory about free 2-functors or Cauchy completion (see [Law74]).
Note that it is possible for a morphism in Mat C to be an isomorphism even if none of its matrix components is an isomorphism in C.
Example 2.36. Let fVec be the category of finite dimensional vector spaces and let V i be a vector space of dimension i. Let C be the full subcategory of fVec on the objects V 2 and V 3 . Then, in Mat C, there is an isomorphism between V 2 ⊕ V 2 ⊕ V 2 and V 3 ⊕ V 3 .
In the previous example, things go wrong because C, and hence also Mat C, is not idempotent complete.
Definition 2.37. Let C ∈ ob Cat. We say that an object x ∈ ob C is indecomposable if it is nonzero and there do not exist nonzero objects y , z ∈ ob C such that x ∼ = y ⊕ z.

Proposition 2.38. Taking the full subcategory of indecomposable objects defines a 2-functor
Ind : Cat 2,0 → Cat 2,0 and this 2-functor is locally fully faithful on the full sub-2-category of Cat 2,0 on categories where every object is a finite direct sum of indecomposable objects.
Proof. Equivalences of categories preserve limits, such as direct sums, and therefore preserve indecomposable objects. So equivalences F : C → D do restrict to equivalences Ind F : Ind C → Ind D. As isomorphisms preserve indecomposable objects, components of natural isomorphisms also restrict to indecomposable objects. Now we restrict to the full sub-2-category where every object is a finite direct sum of indecomposable objects. By naturality, natural transformations between functors F , G : C → D are determined by the components of indecomposable objects. So Ind C,D is faithful. And given a natural transformation γ : ic F → ic G , we defineγ : F → G by extending to any direct sums in the obvious way. Then Indγ = γ, so Ind C,D is full.
If F : C → D does not preserve indecomposables then Ind F is not defined, so we cannot extend Ind to a 2-functor on Cat 2,1 . Similar to Lemma 2.33, we have: Lemma 2.39. If every object of C is indecomposable then C and Ind C are equivalent.
We say an idempotent e = e 2 is primitive when it cannot be written as the sum of two other nonzero idempotents. The following result is straightforward.
Lemma 2.40. An object (x, e) ∈ ic C is indecomposable precisely when e is a primitive idempotent.
Also, directly from the definitions, we have the following.
The following terminology is used in [GR92, Section 3.5].
Definition 2.42. We say a -category C is finite if: • C has finitely many isomorphism classes of indecomposable objects, and • C is hom-finite, i.e., all hom spaces C(x, y ) are finite-dimensional.
Lemma 2.43. If C is finite then every object of C is isomorphic to a direct sum of finitely many indecomposable objects.
Proof. Suppose not, then we have an object Each object x i has an identity morphism 1 i so, as hom functors preserve limits, the infinite dimensional vector space with basis . So C is not hom-finite. Definition 2.44. We define the following full sub-2-categories of Cat 2,0 : • BaseCat is the full sub-2-category on the idempotent complete -categories where every object is indecomposable; • AddCat is the full sub-2-category on the additive idempotent complete -categories.
We define fBaseCat and fAddCat as the full sub-2-categories of BaseCat and AddCat, respectively, on the finite -categories.
By Lemma 2.41, we have: Using Lemma 2.5 again, the 2-functors restrict to the finite sub-2-categories. If C is an additive category where every obejct is a direct sum of finitely many indecomposables then ε C : Mat Ind C → C is iso, so we do get an adjoint equivalence in 2Gpd.

Algebras
There is a category of (associative, unital) -algebras whose morphisms are unital -linear functions which preserve the multiplication. In fact, this is a truncation of a 2-category Alg. Given 1-cells f , g : A → B, the 2-cells from f to g are by definition The composition is as in B. Note that f and g are isomorphic 1-cells if they are related by an inner automorphism of B.
Given a -algebra A, we can form the one-object -linear category C A whose hom-space is just A. This defines a 2-functor B : Alg → Cat (which we may think of as delooping).
Lemma 2.46. B is locally an equivalence.
Proof. The functors B A,B : Alg(A, B) → Cat(C A , C B ) are in fact isomorphisms of categories, by construction.
An algebra A is called basic if the summands of the regular left A-module are all non-isomorphic. We have a full sub-2-category fBasicAlg of Alg on the finite-dimensional basic algebras.
If C is a -category with finitely many objects then we can form the following -algebra: with multiplication of f : w → x and h : y → z given by Let skel D denote a skeletal subcategory of D.
Definition 2.47. If C is a finite -category, the base category of C is D = Ind ic C and the base algebra of C is B skel D .
By construction, B D is a basic algebra, and C determines B D up to isomorphism. If A is an algebra of finite representation type, i.e., A -mod is finite, then the base algebra of A -mod is just the Auslander algebra of A.
Note that, once we move away from single object categories, the construction of B C is not functorial on Cat: a morphism C → D of categories (i.e., functor) will not in general induce a unital map B C → B D of algebras.
Now we take the core of B, which we denote B 2,0 .
Proposition 2.48. There exists a 2-functorB : fBasicAlg 2,0 ∼ −→ fBaseCat which is a biequivalence and which makes the following diagram commute: Proof. Let F = Ind ic B 2,0 denote the composite functor Alg 2,0 → BaseCat. We want to restrict F to full sub-2-categories, so we use Lemma 2.5. Let A ∈ ob fBasicAlg 2,0 . As A is finite-dimensional, B 2,0 A is hom-finite, and so F A is also hom-finite. Also, as A is finite-dimensional, it decomposes as a finite direct sum of left ideals, and thus it has finitely many primitive idempotents. So F A has finitely many indecomposable objects. Therefore we get the 2-functorB.
We know that B 2,0 and ic are locally fully faithful by Lemmas 2.9 and 2.46. As A has finitely many idempotents, we know by Lemma 2.40 that every object of ic B 2,0 A is a direct sum of finitely many indecomposables. So F is locally fully faithful by Proposition 2.38. Now given C ∈ ob fBaseCat, let A = B C be its base algebra. Then F A is isomorphic to skel C by construction. So F is 2-dense and therefore a biequivalence. Now, by Propositions 2.45 and 2.48, we have two biequivalences:

Serre structures
Our main interest is in -linear categories, but we work with categories enriched in a pivotal monoidal category V. We define and study Serre functors in this setting. This will be useful later when we consider nontrivial pivotal structures on the monoidal category of graded finite dimensional vector spaces.
Enriched Serre functors have been considered previously by Lyubashenko and Manzyuk in the case where V is symmetric [LM08]. The setup we develop here, where V is pivotal, applies even when V does not admit a braiding. Our approach is similar to work of Fuchs and Stigner [FuSt08] for Frobenius algebras.

Definitions and duals
We recall parts of V-enriched category theory, following [JY21, Section 1.3]. Throughout, we assume that V is a pivotal monoidal category (see Section 2.2.4) but do not assume that V is braided.
Let C be a V-enriched category: for all x, y ∈ C we have an object C(x, y ) ∈ V, and for all x, y , z ∈ C we have unit maps and composition maps m : If C and D are two V-enriched categories, a V-functor F : C → D is a function F : ob C → ob D and, for all y , z ∈ C, a map F yz : C(y , z) → D(Fy , Fz) in V such that, for all x ∈ C, the following diagram commutes: Recall that the pivotal structure is denoted ι. By naturality of ι, the following diagram commutes for any V-functor F : C → D: Gy ) such that the following diagram commutes for all y , z ∈ C: We will use V-modules for monoidal categories V. Our reference is [GaSh16], which is written in the generality of monoidal bicategories, but the results apply immediately to our setting by considering V to be a monoidal bicategory with only identity 2-cells.
Let C be a V-enriched category. A right C-module M is a function M : ob C → ob V and, for all x, y ∈ C, a map M xy : My ⊗ C(x, y ) → Mx in V satisfying unitality and associativity axioms. The standard example is M = C(−, z) for any z ∈ ob C, with M xy given by composition. This defines a fully faithful Yoneda embedding, which acts on objects by sending z ∈ ob C to the right C-module C(−, z) [GaSh16, Proposition 9.6].
Similarly, a left C-module N is a function N : ob C → ob V and, for all x, y ∈ C, a map N xy : There is also a notion of a C-D-bimodule, which comes with a map ob D × ob C → ob V. The standard example is the hom bimodule C(−, −).
If M is a right D-module and F : C → D is a V-functor, then we obtain a right C-module MF with object map MF : ob C → ob V given by composition and, for x, y ∈ C, action map given by: . Explicitly, we have: However, if M is a right C-module, we need to use the pivotal structure to put a left C-module structure on the (left) dual M ∨ . The action map is constructed using ι and (M yx ) ℓ as follows: Now, following the classic definition of Bondal and Kapranov [BK89], we are able to give our main definition.
Serre structure for C, we say that C has Serre duality and that Ë is a Serre functor.
The above definition says that, for all x, y ∈ ob C we have an invertible map κ x,y which is both a left Unravelling the definition, the bimodule morphism condition says that two diagrams commute. Let w , x, y , z ∈ ob C. The left module morphism condition gives "left naturality" of κ: and the right module condition gives "right naturality" of κ: In formulas, these say Remark 3.2. One can also specify Serre structures using trace maps or pairings: see, for example, [RVdB02, Proposition I.1.4]. We show how this matches our conventions, but we won't pursue it further. Given a Serre structure as in Definition 3.1, we obtain a pairing by: and given a trace map C(x, Ëx) we construct a pairing p x,y : C(y , Ëx)⊗C(x, y ) → ½ by precomposing with the composition in C, then construct the following map: To obtain κ, we dualize and precompose with the pivotal structure map: Remark 3.3. Traditionally, the definition of a Serre functor involves a binatural transformation of functors, not a morphism of bimodules. It may comfort the reader(/author) to note that modules do induce V-functors, by the following construction. As V is rigid, it is left and right closed, i.e., it has left and right internal homs where ∨ Y denotes the right dual of Y . These satisfy the adjunction formulas Using these, we define two V-enriched categories.
The first, called V r , has hom object V r (X , So, given a left C-module N, we have a map N xy : C(x, y )⊗Nx → Ny which, by the second adjunction formula, determines a map C(x, y ) → [Nx, Ny ] r in V and thus determines a V-functor N : C → V r .
The second V-enriched category is suggestively called (V ℓ ) rev , with hom object V ℓ (X , Y ) = [Y , X ] ℓ (note the reversed order) and multiplication map Then, given a right C-module M, we have a map M xy : My ⊗ C(x, y ) → Mx which, by the first adjunction formula, determines a map C(x, y ) → [My , Mx] r in V and thus determines a V-functor Remark 3.4. Our definition of a Serre structure uses enriched categories with dualizable hom-spaces.
In the case of vector spaces, this means hom-finite categories. This may seem restrictive, as the classical definition of a Serre functor uses arbitrary -linear categories, but in fact the finite dimensional condition is forced: κ and (κ * ) −1 give an isomorphism between C(x, y ) and C(Ëx, Ëy) * * , and Ë is an autoequivalence, so we have a -linear isomorphism between C(x, y ) and its double dual. Thus any -linear category which has Serre duality must be hom-finite.

Uniqueness
It is well-known that Serre functors are unique up to isomorphism [BK89, Proposition 3.4(b)]: Proposition 3.5. If (Ë, κ) and (Ë ′ , κ ′ ) are two Serre structures on C then there exists a natural isomorphism α : Ë ∼ → Ë ′ such that the following diagram commutes: We can extend this idea. Let C i be a V-category with Serre structure (Ë i , κ i ), for i = 1, 2.
is a natural isomorphism such that the following diagram commutes: / / F y y r r r r r r r r r r If F is an equivalence then we say (F , α) is an equivalence of Serre structures.
Proposition 3.7. Let C 1 and C 2 be V-categories.
(b) Given Serre structures (Ë i , κ i ) on C i and an equivalence of V-categories F : C 1 ∼ → C 2 , F can be upgraded to an equivalence of Serre structures (F , α).
(b) We have four invertible morphisms in the diagram of Definition 3.6, so they define a map By the Yoneda lemma, this determines a natural transformation α : We record two useful results. The first is a direct consequence of Proposition 3.7.
Corollary 3.8. Let B be a skeletal subcategory of C. Then B has a Serre structure if and only if C does.
Lemma 3.9. If C has a Serre structure (Ë, κ) and B is a full subcategory of C which is closed under Proof. This is clear: (Ë, κ) restricts to B.
The assumption ËB ⊆ B in Lemma 3.9 is necessary: consider a -category C with object set /3 and C(n, n + 1) = but C(n, n − 1) = 0. Then C has a Serre structure but the full subcategory on the objects 0 and 1 does not.
We now put categories with Serre structures into a 2-categorical setting.
Consider the following 2-category of V-categories with Serre structures, denoted ËV Cat: • The objects are triples (C, Ë, κ), where C is a V-category and (Ë, κ) is a Serre structure on C.
Therefore we have the following result, which loosely says that we can regard a Serre structure as a property rather than a structure: Corollary 3.13. The 2-groupoid ËV Cat 2,0 of V-categories with Serre structures is biequivalent to the full sub-2-groupoid of V Cat 2,0 on the V-categories which admit a Serre structure.

The compatibility lemma
In their original definition, Bondal and Kapranov included an extra compatibility condition which was later shown to follow from the naturality of κ: see [RVdB02, Lemma I.1.1]. We state and prove this in the pivotal enriched setting.
Lemma 3.14. The following diagram commutes: Proof. From the right naturality of κ we get: Then by the naturality and monoidality of ι we get: By the dual of Lemma 2.16, the following diagram commutes: Now from left naturality of κ we get: Tensoring with (κ ∨∨ ) −1 , we get: By definition of m ℓ (see Section 2.2.3) we have: Lemma 2.13 says that the following diagram commutes: Therefore, by tensoring with ι, dualising, and using Lemma 2.12, the following diagram commutes: Combining diagrams (3), (4), (5), and (6) gives: Putting (1), (2), and (7) together gives the following commutative diagram: The partial diagram on the left involving the curved arrow commutes by the naturality of ι. The one on the right commutes by Lemma 2.14.
Finally, taking 1 y ∈ C(y , y ) and using unitality properties gives

Commutation
Bondal and Orlov showed that Serre functors commute with autoequivalences [BO01, Proposition 1.3]. We give a proof of this fact, as it is both short and instructive.
Proposition 3.15. If F : C ∼ → C is an autoequivalence and C has Serre structure (Ë, κ), then there is a canonical natural isomorphism ËF ∼ → F Ë.
Proof. Assume F has quasi-inverse G . Then we have isomorphisms: for any x, y . Dualizing and using the pivotal structure gives an isomorphism C(x, ËFy) ∼ → C(x, F Ëy). As the Yoneda embedding is fully faithful this determines the natural isomorphism ËF ∼ → F Ë.
Given F : C ∼ → C, we write ζ F : ËF ∼ → F Ë for the above natural transformation.
The following lemma is immediate from the proof of Proposition 3.15.
Now we give a stronger version of Proposition 3.15.
Proposition 3.17. If C has a Serre structure (Ë, κ) then (Ë, ζ) belongs to the Drinfeld centre of the monoidal category of autoequivalences of C.
Proof. From Definition 2.19 we must check two properties. First we check naturality of ζ. Given two autoequivalences F , G : C ∼ → C and a natural transformation α : F → G , we must show that the following diagram commutes: Let F ! and G ! denote the left adjoints of F and G , respectively. Then α induces a natural transformation We break our diagram up into smaller squares: The first and third commute by naturality of κ (from the Serre structure), and the second and fourth commute by an exercise in adjunctions and the fact that F !! ∼ = F naturally (as F is an equivalence).
Next, we must check that the following diagram commutes: By the Yoneda lemma, it is enough to check that the following diagram commutes: First we note that the following diagram commutes by naturality of adjunctions: Therefore we just need the following diagram to commute: This is easy to see by considering the equalities indicated by dashed lines.
Lemma 3.18. The natural transformation ζ Ë − is just the composition Proof. In fact we'll prove that ζ −1 Ë − = η 2 ε 1 , which is equivalent. The natural transformation is represented by By Lemma 3.14 and by taking duals, the following squares commute: So the following diagram is commutative: Then by naturality of ι we have Now use naturality of κ, and Lemma 3.14 again, to get: Tidy this up to get: As (η 2 , ε 1 ) is a unit-counit adjunction pair, the following square commutes: ? ?
as required. Definition 3.19. A Frobenius structure on A is a pair (ϕ, α) where α ∈ Aut(A) and ϕ : A ∼ → (A * ) α is an isomorphism of A-A-bimodules. If α belongs to a Frobenius structure for C then we say it is a Nakayama automorphism for C.

Frobenius algebras
We say that A is a Frobenius algebra if there exists a Frobenius structure on A.
Note that a given algebra A can have different Frobenius structures, with different Nakayama automorphisms, but the Nakayama automorphism of a Frobenius algebra is well-defined up to inner automorphisms (i.e., the 1-cell in Alg is defined up to isomorphism).
Remark 3.20. A Frobenius algebra is just a looping of a one-object fVec-enriched category with Serre duality: see Example 2.17 and note that the left and right bimodule actions on A * match the maps m ℓ and m r described in Section 3.1. In fact we could consider Frobenius algebra objects in a pivotal monoidal category V.
We want to relate Frobenius algebras to categories with Serre duality. To start, we will show that Serre structures pass through many of the constructions introduced in Section 2.
Let C be a -linear category. The first part of the following Lemma, on ic C, is [Che17, Lemma 3.4].
Lemma 3.21. If C has a Serre structure then so do the categories ic C, Mat C, and Ind C.
Next we claim Mat Ë is a Serre functor for Mat C. Given X , Y ∈ Mat C we can write X = X 1 ⊕· · ·⊕X n and Y = Y 1 ⊕ · · · ⊕ Y m with X i , Y i ∈ C. As vector spaces, To construct Mat κ X ,Y we apply a map κ Xi ,Yj to each summand and use the isomorphism Finally, as Serre functors respect biproducts, we obtain a Serre structure on Ind C by Lemma 3.9.
Example 3.22. The converse statements of Lemma 3.21 are all false. Let D be a -linear category with precisely two objects x and y and a Serre functor which interchanges them. Consider the full subcategory C 1 of Mat D on the objects x and x ⊕ y . Then ic C 1 has a Serre structure but C 1 does not. Similarly, consider the full subcategory C 2 of Mat D on the objects x, y , and x ⊕ x. Then Mat C 2 and Ind C 2 have Serre structures but C 2 does not.
Let C be a category with finitely many objects. We say that a Serre structure (Ë, κ) is strict if Ë is an automorphism (not just an autoequivalence). Recall the algebra A C constructed in Section 2.5.

Proposition 3.23. There is a bijection between Frobenius structures on A C and strict Serre structures on C.
Proof. Let A = A C . If (Ë, κ) is a strict Serre structure on C then Ë induces an automorphism α on A and κ induces an isomorphism ϕ : A ∼ → (A * ) α of A-A-bimodules. Now suppose (ϕ, α) is a Frobenius structure on A. The algebra A has a primitive idempotent e x for each object x ∈ ob C. As α is an algebra automorphism, it preserves idempotents and the identity 1 A = x∈ob C e x and therefore permutes the idempotents. This defines an action of Ë on ob C. Then α extends this object action to a genuine automorphism of C, and ϕ provides the form κ.
Recall the 2-functor B from Section 2.5. Considering one-object categories gives the following.

Corollary 3.24. There is a bijection between Frobenius structures on A and Serre structures on B A.
Similarly to Section 3.2, there is a 2-category FAlg whose objects are Frobenius algebras. We don't give the definition explicitly because, as with Proposition 3.12 and Corollary 3.13, we have 2-functor U : FAlg → Alg whose core is locally an equivalence. Therefore we can just work with the full sub-2-groupoid of Alg 2,0 given by the algebras which admit Frobenius structures.
Theorem 3.25. A Frobenius structure on an algebra A induces a Serre structure on the -category P A. Therefore, the biequivalence P : fBasicAlg 2,0 ∼ → fAddCat restricts to a biequivalence Moreover, a Serre structure on a finite -category C induces a Frobenius structure on its base algebra B C .
Proof. We get the first statement by combining Lemma 3.21 and Corollary 3.24. So by Lemma 2.5 we get the 2-functor fBasicFAlg 2,0 ∼ → fAdd Ë Cat, and it is a biequivalence by Corollary 3.13. To get the final statement, use Corollary 3.8 and Lemma 3.21. Then, noting that every Serre structure on a skeletal category is strict, the result follows by Proposition 3.23.

Graded categories
We largely follow Asashiba [Asa17], though our conventions are slightly different: see Remark 4.30 below. Orbit categories rose in prominence after their use in categorifying cluster algebras by Buan, Marsh, Reineke, Reiten, and Todorov [BMRRT06] and related work of Keller [Kel05]. They were first systematically studied by Cibils and Marcos [CM06].
From now on, we specialise from the pivotal enhanced setting and work in -categories, so we work with traditional linear algebra duals (−) * = Hom (−, ) instead of abstract duals (−) ∨ . But we will use the pivotal enhanced theory from Section 4.2.2, where our pivotal structure will depend on a fixed character.

Equivariant and hom-graded categories
Fix a group G . In applications, G will be abelian, so we write the composition of p, q ∈ G additively, as p + q, but everything would work for an arbitrary group.

Equivariant categories
A G -equivariant category is a -category D together with a G -action: a group homomorphism from G to the group Autom(D) of automorphisms of D.
G -equivariant categories are the objects of the following 2-category, which we denote G -Cat: • The 0-cells are G -equivariant categories: pairs (D, F ) where D is a -category and F : G → Autom(D) is a group homomorphism. We write F p = F (p).

. Composition of 2-cells, both vertical and horizontal, is as usual for natural transformations
Note that any -linear category D can be given a trivial G -category structure: we write ∆D = (D, F ) for D equipped with the group homomorphism F sending every p ∈ G to the identity functor on D.
Remark 4.1. Our main application involves G = . In this case, we can specify less data to determine a G -equivariant category. Given an automorphism F 1 : D ∼ → D, we define F p = (F 1 ) p for p ∈ : this explains our use of superscripts for G -indexing. We denote this -equivariant category (D, F 1 ). Given a functor Φ : D → D ′ and a natural isomorphism φ : ΦF 1 ∼ → F ′1 Φ, we define φ p as the composition for n > 0, and use a similar definition for n < 0. This respects the group composition by construction. Similarly, for 2-cells, we only need to check commutativity of the p = 1 square.

Strictification
According to our definition, G -equivariant categories are equipped with strict G -actions: G should act by automorphisms. However, in practice one often meets weak G -actions, where the elements of G act by autoequivalences. We would like to replace these by strict G -actions. This isn't essential, but reduces the technical complexity of the paper.
A weak G -equivariant category is a -category D together with a weak G -action: a group homomorphism from G to the group Autoeq(D) of autoequivalences of D. This just means that there exist natural isomorphisms F p+q ∼ → F p F q . A coherent G -equivariant category (D, F , f ) is a weak G -equivariant category (D, F ), F : G → Autoeq(D), equipped with a specified natural isomorphisms f p,q : F p+q ∼ → F p F q satisfying the obvious associativity axiom f pq,r • f p,q F r = f p,qr • F p f p,r .
Definition 4.3. A strictification of (D ′ , F ′ , f ) is a G -equivariant category (D, F ) together with an equivalence ε : D ′ ∼ → D and a collection α = (α p : εF ′p ∼ → F p ε) p∈G of natural isomorphisms such that There exist at least two ways to strictify, which involve making or category smaller or bigger. One is to take a skeleton, so all autoequivalences must be automorphisms. The other is to consider the discrete monoidal -linear category ΩG with objects given by elements of G , then to replace D ′ with a category of weakly G -equivariant functors from ΩG to D ′ . In the case G = , D ′ is given explicitly by sequences ( We will use this implicitly from now on. In particular, we use it whenever we have a coherent G -equivariant category but we want to apply Theorem 4.29 (below).

The equivariant centre
The following result is useful. Later we will combine it with Proposition 3.15.
We break it up as follows: Then the triangles and the square commute by the assumption that (Z , z) ∈ ZEnd(D).

Hom-graded categories
A G -hom-graded category is a -linear category whose hom spaces are G -graded, with composition respecting this grading.
G -hom-graded categories are the objects of the following 2-category, which we denote Cat G : • The 0-cells are G -hom-graded categories C, so each hom space has a direct sum decomposition C(x, y ) = p∈G C p (x, y ). If we have a homogeneous map f ∈ C p (x, y ) then we say f has degree p and write deg f = p. The composition should respect the grading: if g ∈ C q (y , z) then deg(gf ) = p + q.
Given p ∈ G , let p : ob C 1 → G be the constant function with image p. In particular, 0 sends every object to the identity of G . We say H is a strict degree-preserving functor if (H,0) is degree-preserving.
As in the ungraded case, we have 2-functors We set BaseCat G = Ind ic Cat G which we could use to upgrade Ë to a G -equivariant functor. But for applications, especially with triangulated categories, it will be useful to consider more general commutation maps.
Let χ : G → × be a character of G . It is immediate from the definition that: if a Serre structure exists for D then, for every character χ, a χ-equivariant Serre structure exists on D. Therefore there is some redundancy in the previous definition, but we still find it useful to record the natural isomorphism s explicitly. Note also that: if (Ë, s, κ) is a χ-equivariant Serre structure for (D, F ) then the pair (Ë, s) is automatically an equivariant autoequivalence of (D, F ).
Example 4.9. We continue Example 4.2. Let χ = sgn be the character with sgn(1) = −1. Define ) is a Serre structure on D. Our pivotal structure identifies double duals with the identity functor so, following Proposition 3.15, we get maps: We will upgrade the existence and uniqueness results of Section 3.1 to the χ-equivariant setting. Let F − be a quasi-inverse of F . Then, by functoriality, F α : F Ë → F Ë ′ is represented by: Recall that s = χ(g )ζ F g , where ζ F g is defined by Proposition 3.15, so is represented by the following composition: ËFy) and ζ ′ F g is defined similarly.
We need to check that α is a morphism of equivariant functors. So, by factoring out χ(g ), we just need to check that the following diagram commutes: We can check this on the representing morphisms, giving the outside rectangle of the following diagram: We have indicated how to break this into smaller diagrams, each of which clearly commutes.
Proof. By Proposition 3.7 we know that (Ë 2 , κ 2 ) exists, and s 2 exists automatically, so we have (Ë 2 , s 2 , κ 2 ). By Proposition 3.7 again, we know that α exists and that (Φ, α) is an equivalence of Serre structures. So we just need to check that α is a morphism of G -equivariant functors, i.e., that the following diagram commutes: defined by the adjunction.
The diagram we need to draw is similar to the proof of Proposition 4.10, but much bigger. We suggest the interested reader draws this on a large piece of paper; here we only sketch the details.
After some simplification, it reduces to a diagram between hom-spaces for D 2 of the following form: : : After removing the top-right and bottom-left corners, and contracting the equalities, we are left with four squares which need to commute. Going from top-left to bottom-right, they commute because of: the definition of φ − , the naturality of κ 1 , the dual of the definition of φ − , and the naturality of κ 2 .
As with Corollary 3.13, we can define a 2-category G -Ë χ Cat whose objects are G -equivariant categories equipped with χ-equivariant Serre structures and the above uniqueness results show that it is biequivalent to the full sub-2-category of G -Cat on the G -equivariant categories admitting such structures.

Hom-graded Serre structures
Let fVec G denote the monoidal category of G -graded -vector spaces V = g ∈G V g , where V g ∈ fVec. This category is left rigid, with Let χ : G → × be a character of G . Recall the notion of a pivotal structure from Section 2.2. We define a pivotal structure ι χ V : V ∼ → V * * on fVec G by sending the homogeneous vector v ∈ V g to χ(g ) ev v . With this pivotal structure, we denote our monoidal category fVec χ .
Again, we can define a 2-category Ë χ Cat G whose objects are G -hom-graded categories equipped with χ-hom-graded Serre structures and the above uniqueness results show that it is biequivalent to the full sub-2-category of Cat G on the hom-graded categories admitting such structures.

Graded Frobenius algebras 4.3.1 Graded algebras
Graded algebras are much more well-studied than graded categories. A graded algebra is a unital algebra A = p∈G A g with homogeneous composition. A map of graded algebras is an algebra map which preserves degree. These form the objects and 1-cells of a 2-category Alg G , with 2-cells given by conjugation of degree 0 elements, but the resulting 2-functor B G : Alg G → Cat G is not locally an equivalence: it is only locally fully faithful. Cat G has more 1-cells than Alg G : the 1-cells in Alg G correspond to the strict 1-cells in Cat G . Given an algebra A, let prim(A) denote its set of primitive idempotents. By Lemma 2.40, the indecomposable objects of ic B A are indexed by prim(A). If A is basic then these are pairwise nonisomorphic. Notice that if A is graded then prim(A) ⊂ A 0 .
Given a graded algebra A, let uA denote its underlying (ungraded) algebra.
Definition 4.19. The 2-category fBasicAlg G is as follows: • The 0-cells are basic finite-dimensional G -graded -algebras.
• The 1-cells A → B are degree-adjusted morphisms: pairs (f , γ) where f : uA → uB is a map of -algebras and γ : prim(A) → G is a function such that a ∈ A p implies f (dae) ∈ A p+γ(e)−γ(d) .
By construction of fBasicAlg G we have an analogue of Proposition 2.48:

Graded Nakayama automorphisms
Let A, B be G -graded algebras and let M = g ∈G M g be a graded A-B-bimodule. Let χ : G → be a character on G . We define the χ-dual χ M * to be M * as a graded vector space, with B-A-bimodule structure given by the formulas (ξa)(m) = ξ(am) and (bξ)(m) = χ(g )ξ(mb), for ξ ∈ χ M * , a ∈ A, b ∈ B g , and m ∈ M.
We say (σ, ℓ) is a χ-graded Nakayama automorphism for C. Let C be a hom-graded category with finitely many objects. Then, just as for ordinary categories in Section 2.5, we construct a graded algebra A C . We repeat Proposition 3.23 in the graded setting: Proposition 4.23. There is a bijection between χ-graded Frobenius structures on A C and strict χ-hom-graded Serre structures on C.
There is a 2-category whose objects are G -graded algebras equipped with χ-graded Frobenius structures. On restricting to basic algebras and taking the core, we get a 2-groupoid which is biequivalent to the full sub-2-groupoid of the bicategory of fBasicAlg G 2,0 on algebras admitting graded Frobenius structures.
We have a G -graded biequivalence The following is a graded generalization of Theorem 3.25, and is proved in the same way.
Theorem 4.25. A χ-graded Frobenius structure on a G -graded algebra A induces a χ-hom-graded Serre structure on the G -graded -category P G A. Therefore, the biequivalence P G : fBasicAlg G 2,0 ∼ → fAddCat G restricts to a biequivalence Moreover, a χ-hom-graded Serre structure on a finite G -graded -category C induces a χ-graded Frobenius structure on its G -graded base algebra B C .
Let tr denote the trivial character G → × which sends every g ∈ G to 1 ∈ . Given a linear map f : A → A, let f χ : A → A be the map defined on homogeneous elements a ∈ A p by f χ (a) = χ(p)f (a). As there is lots of existing knowledge of tr-graded Frobenius structures on graded algebras, the following straightforward result is useful.
• The G -category C#G has objects ob C × G , which we write as either (x, p) or x p depending on context. Homs are given by C#G (x p , y q ) = C q−p (x, y ). The G -action F : G → Autom(C#G ) on objects is the obvious one: for r ∈ G we have F r (x p ) = x p+r . On morphisms it is trivial: C#G (x g , y h ) and C#G (x p+r , y q+r ) are both copies of the same set C q−p (x, y ), so F r takes f : x p → y q to f : x p+r → y q+r using the identity map.
Given a morphism f ∈ C#G (x p , y q ) = C q−p (x, y ) we send it to Hf : Hx → Hy , which has degree γ(y ) + q − p − γ(x) and is thus a morphism in C#G (x p+γ(x) , y q+γ(y) ). Note that this action is strict: the G -action commutes with the functor, so the natural isomorphism of our equivariant functor is just the identity.
Example 4.27. Let C be the -hom-graded category from Example 4.6. Then C# has objects x p , with x ∈ {1, 2} and p ∈ . Its nonzero morphism spaces are 1-dimensional, with bases α p : 1 p → 2 p and β p : 2 p → 1 p+1 , for as well as the identity morphisms. The -action is given by Let D denote the -equivariant category from Example 4.2. We have an equivariant equivalence (Φ, φ) : C# → D where Φ(x p ) = 2p + x on objects, and Φ(α p ) = f 2p+1 and F (β p ) = f 2p+2 on maps. The two compositions ΦF 1 and F 1 Φ are equal: both send x p to 2p + 2 + x and act in the obvious way on maps. The natural transformation φ : ΦF 1 → F 1 Φ is the identity, and it clearly satisfies the necessary commutative diagram. • The G -graded category D/G is the orbit category : it has the same objects as D and its homogeneous morphism spaces are (D/G ) p (x, y ) = D(x, F p y ). The composition of f ∈ (D/G ) p (x, y ) and g ∈ (D/G ) q (y , z) is F p (g ) • f .
• The degree-preserving functor (Φ, φ)/G is strict (its degree adjuster is zero). It is just Φ on objects, and it sends a degree p morphism f : x → F p y to the composite φ y • Φf : ΦF p y φy 9 9 s s s s • The morphism α/G of degree-preserving functors is just α.
If (D, F ) is a G -equivariant category, note that G acts on (D, F ) (not just D) in the following way: The grading of D/G encodes the group action of D in the following way: Lemma 4.28. Let C = D/G . There is an isomorphism of degree-preserving endofunctors of C: Proof. For x ∈ C, define the degree q map θ x : F q x → x by θ x = 1 F q x ∈ D(F q x, F q x) = C q (F q x, x). As it's the identity, it's clearly natural. Its inverse is the degree −q map 1 x ∈ D(x,

Asashiba's biequivalence
The main result is the following [Asa17, Theorem 7.5]: Theorem 4.29 (Asashiba). Taking orbit categories is a biequivalence with quasi-inverse given by taking smash products: Remark 4.30. Our categories are defined oppositely to [Asa17], where the natural transformations given as part of an equivariant structure are in the opposite direction. The orbit category and smash product category are defined with opposite signs, which correspond to our opposite conventions: see [Asa11, Proposition 2.11]. In [Asa17, Section 7.1], the orbit 2-functor is defined by the existence of 1-and 2-cells which fit in commutative diagrams. It is straightforward to check that, under the above identifications, the constructions of 1-and 2-cells given above do make these diagrams commutative.
So, combining Proposition 2.30 and Theorem 4.29, we get: Corollary 4.31. If D is a G -category then there is a group isomorphism
Let (Ë, s, κ) be a χ-equivariant Serre structure on (D, F ). We want to show that taking the orbit category sends (Ë, s, κ) to a χ-graded Serre structure (Ë C , α, κ C ) on C = D/G . The orbit 2-functor gives us a degree-preserving functor (Ë C , ℓ) = (Ë, s)/G which is strict, so Ë C is Ë on objects, ℓ = 0 sends every object to 0 ∈ G , and a map f : So we need to define a map which preserves the degree of morphisms. Our map has components: Note the scaling by χ(g ).
Proof. We just need to check that (Ë C , 0, κ C ) is a (fVec G , ι χ )-enriched Serre structure. Fix maps f : w → x, g : x → y , and h : y → z of degrees p, q, and r , respectively. Then we want the following diagram to commute: κw,z C r (y , z) * * ⊗ C q (y , Ëx) * ⊗ C p (Ëw , Ëx) / / C p+q+r (z, Ëw) * This says that κ(hgf ) = ι(h)κ(g )Ë(f ). We split this check into two halves, multiplying on the left or right of g . The "left" diagram check is: The "right" diagram check is: Let's do the "left" check first. Writing out the diagram carefully, using C q (x, y ) = D(x, F q y ), we get: The top left square commutes by naturality of the pivotal structure ι χ . The top right square commutes by ι χ = χ(r )ι and naturality of κ. The commutativity of the bottom hexagon is an exercise in taking duals of maps in a rigid tensor category.
Proof. First we check naturality of κ D . This follows from ι χ -naturality for κ and the definition, as the following diagrams show: It remains to check s p = χ(p)ζ F g . Recall that ζ F g is defined by the following diagram: , so ζ F g is multiplication by χ(−p), so s p = 1 = χ(p)χ(−p) = χ(p)ζ F g , as required.
Fix G and χ : G → × . Immediately from the uniqueness results (Propositions 4.10 and 4.15) we get:

Calabi-Yau categories
From now on we set G = and work with -equivariant categories.

Triangulated Calabi-Yau categories
A triangulated category is a triple (D, Σ, ∆) where D is a -linear category, Σ : D ∼ → D is an autoequivalence, and ∆ is a distinguished subset of the set of "triangles" Triangulated categories should satisfy some well-known axioms.
If Σ is in fact an automorphism of categories (as is sometimes specified in the definition) then (D, Σ) is a -equivariant category. If not, we can strictify as in [KV87, Section 2] to get an equivalent triangulated category (D ′ , Σ ′ ) which is -equivariant. Hereafter, we will assume that our triangulated categories come equipped with automorphisms.
There is a 2-category Tri whose 0-cells are triangulated categories. Φf / / Φy If a triangulated category has Serre duality, it comes with two canonical triangulated autoequivalences. It is natural to ask whether there is any relation between them. Kontsevich noted that relations do exist in at least two cases: for derived categories of Calabi-Yau varieties and for derived categories of some quivers. He called these categories Calabi-Yau and fractional Calabi-Yau, respectively [Kon98].
The existence of a natural isomorphism between powers of Ë and Σ might be called a weak Calabi-Yau condition. We ask for a strong Calabi-Yau condition, as in [Kel08], using the whole graded structure.
Definition 5.3. Suppose (D, Σ, ∆) has a sgn-equivariant Serre structure (Ë, s, κ). D is called: • Calabi-Yau of dimension n (or n-CY ) if (Ë, s) ∼ = (Σ, −1 Σ 2 ) n in -Cat; We repeat the standard warning with this definition: N/m should be treated as a pair of integers and not as a rational number.
Example 5.4. The derived category of the path algebra of a quiver of type A 3 is fractional Calabi-Yau of dimension 2/4, but is not fractional Calabi-Yau of dimension 1/2.

Auslander-Reiten functors
We now restrict to G = . Hereafter, F will denote an element of Autom(D) instead of a map G → Autom(D): see Remark 4.1 for more details.
Now we show that (Z , z) ⋆ is an isomorphism of categories. Let Z − be a right adjoint quasi-inverse of Z with counit natural isomorphism η : 1 D ∼ → Z − Z . We define an inverse functor which sends the This map on objects naturally extends to a functor M D,ZF → M D,F , and one checks that it is a two-sided inverse to (Z , z) ⋆ by using standard properties of adjunctions.

The Auslander-Reiten functor
Fix χ : → × and let (Ë, s, κ) be a χ-equivariant Serre structure on (D, F ). We will concentrate on s 1 but from now on we will drop the superscript 1, just writing s : ËF → F Ë.
We might call (Ë − , s − ) a χ-equivariant inverse Serre functor for (D, F ). We use it to define another important equivariant functor. Remark 5.8. When D = D b ( Q), Ì is the derived functor of the inverse Auslander-Reiten translate τ − . For us, Ì is more fundamental than its quasi-inverse. Now we apply the change of action isomorphism: we use Proposition 5.5 with (Z , z) = (Ë − , ζ † ).
The image of (Ì, t) under this isomorphism could be called the χ-equivariant AR functor for (D, Ì). Proof. First, by definition, t : ÌF ∼ → F Ì is defined as We want to understand the maps in this composition.
From the definition of ζ † we have: By Proposition 3.17 we know and by Lemma 3.18 we know that ζ Ë − is the following composition: Putting the last four diagrams together gives: Therefore t ′ : ÌÌ → ÌÌ is the identity natural transformation.

Transfer of equivariant Serre structures
We now want to compare Serre structures on the equivariant categories (D, F ) and (D, Ì). Recall that we have natural transformations Proof. Our quasi-inverse Ë − is both left and right adjoint to the Serre functor Ë, so this follows from Proof. The assumptions ensure that (Ë, κ) is a Serre structure on D, so this reduces to proving the "two implies three" property for the following list of statements: By Proposition 3.17 we know that ζ Ì = (Ë − ζ F ) • (ζ Ë − F ), so by Lemma 5.10 we have So as all maps are isomorphisms this is clear.

Synthetic Calabi-Yau categories
Now we prove a result which will give a characterisation of fractional Calabi-Yau triangulated categories. However, we consider a general equivariant category (D, F ) with Serre duality, as even when D is triangulated it can be useful to choose an automorphism F other than the automorphism Σ coming from the triangulated structure: for example, in Section 6.3 below, F will be a power of Σ. So, in general, we think of these relations as "synthetic" (not "genuine") fractional Calabi-Yau relations. (c) the -equivariant category (D, F ) has a χ-equivariant Serre functor (Ë, s) and Proof. "(a) ⇒ (b)": Suppose (D, F ) has a χ-equivariant Serre functor (Ë, s) and we have an isomorphism of equivariant functors As Ë is an equivalence we have an isomorphism By Lemma 5.6 we have an isomorphism which we use repeatedly. Composing these gives an isomorphism of equivariant endofunctors of (D, F ). Applying the change of structure isomorphism (Ë − , ζ † ) ⋆ and using Lemma 5.9 gives an isomorphism and we know (Ë, s ′ ) is χ-equivariant by Proposition 5.11.
where (Ë, s ′ ) is χ-equivariant for (D, Ì). Applying (Ì, 1 Ì 2 ) k to the right and using the methods above gives an isomorphism Now apply (Ë, ζ) ⋆ . The map F Ì → ÌF above is sent to: To get an endomorphism in (D, F ) we use η and η −1 . Using Proposition 5.11 and the zigzag equations, this leaves us with χ(1)1 F 2 : FF → FF . A similar calculation shows that and so using η and η −1 leaves us with ζ † We compose with (F , χ(1)1 F 2 ) m on the right to get Now we move between equivariant and hom-graded categories.
Moreover, if C has finitely many isoclasses of indecomposable objects and A denotes the -graded base algebra of C, then (b) is equivalent to: (c) A is a χ-graded Frobenius algebra with Nakayama automorphism (α, ℓ) satisfying Proof. "(a) ⇐⇒ (b)": Given an isomorphism (Ë, s ′ ) k ∼ → (Ì, 1 Ì 2 ) N , apply the orbit 2-functor −/G . By Proposition 4.32, (Ë, s ′ )/G is a χ-hom-graded Serre functor and, by Lemma 4.28, ((Ì, 1 Ì 2 ) N )/G is isomorphic to (1 C , N). By Theorems 4.29 and 4.35 these steps are reversible. Now suppose C has finitely many isoclasses of indecomposable objects. "(b) ⇐⇒ (c)": Use Theorem 4.25. 6 Applications 6.1 Derived categories Let A, B, C be -algebras. Given chain complexes X of C -B-bimodules and Y of B-A-bimodules, we get a chain complex X ⊗ B Y of C -A-bimodules defined as follows. Its degree n term is Given an algebra Λ, let D b (Λ) denote the bounded derived category of left Λ-modules. This is a triangulated category with automorphism Σ which shifts cochain complexes one place to the left and changes the sign of all differentials. Similarly, let D b (Γ-Λ) denote the bounded derived category of Γ-Λ-bimodules.
Let ffBim denote the following bicategory: • The objects are finite-dimensional algebras Λ, Γ of finite global dimension.
Composition of 1-cells is by derived tensor product.
Recall the 2-category Tri of triangulated categories from Section 5.1. We have a weak 2-functor D : ffBim → Tri which sends the object Λ ∈ Tri to D b (Λ). On 1-cells, it sends X ∈ D b (Γ-Λ) to the triangulated is a bounded complex of projective left Λ-modules then the commutation isomorphism is given by summing the following maps: Lemma 6.1. The weak 2-functor D : ffBim → Tri is locally fully faithful.
Proof. The action of D on 1-cells has an inverse which sends the natural transformation α : To see this is a bijection one uses naturality of α for maps Λ → M i .
So by Proposition 2.28 we get: The following result says that our "strong" fractional Calabi-Yau condition is equivalent to the "bimodule" Calabi-Yau condition.
Proof. From Lemma 6.1 and Definition 5.3, all that we need to check is that D(Λ[1]) ∼ = (Σ, −1 Σ 2 ). We know that D(Λ[1]) ∼ = (Λ[1]⊗ Λ , φ) and so we get a diagram and we need to show that the bottom arrow is multiplication by −1. But from the formula for φ M above, applied when X = Λ[1] which is concentrated in degree −1, we get that φ M is −1. So, as both maps α are 1, we are done.
6.2 Dynkin quivers 6.2.1 Dynkin diagrams We revise some standard Lie theory, referring to [Hum90] for details.
The simply laced Dynkin diagrams are those of "ADE type": they are simple (unoriented) graphs belonging to the following list: A n for n ≥ 1, D n for n ≥ 4, E 6 , E 7 , and E 8 . The subscript denotes the number of vertices in the graph and is called the rank. For each Dynkin diagram Γ we have a graph automorphism ρ : Γ → Γ defined on the vertices of Γ as follows: • Type A n : ρ swaps vertices i ↔ n + 1 − i.
• Type D n , n is even: ρ is the identity.
• Type D n , n is odd: ρ swaps vertices n − 1 ↔ n and fixes the others.
• Types E 7 and E 8 : ρ is the identity.
Notice that ρ has order 1 or 2, depending on the type.
Each Dynkin diagram has an associated finite root system Φ which is the disjoint union Φ = Φ + ∪ Φ − of positive and negative roots. Let R = |Φ + | = |Φ − | denote the number of positive roots. (This is traditionally denoted N, but we use this symbol elsewhere.) Each type has a Coxeter number, denoted h, defined as the order of the Coxeter element of the associated reflection group. This is related to the number R of positive roots by the following formula: Let be an algebraically closed field. We refer to [Gra20] for more details and for references.
Let Q be an acyclic quiver and let Λ = Q denote its path algebra. Let Q 0 denote the vertices of Q, then we have a canonical bijection Q 0 ∼ → prim(Λ) sending the vertex i to the length zero path e i . Write P i = Λe i . The left Λ-modules P i , i ∈ Q 0 , form a complete list of indecomposable projective Λ-modules up to isomorphism.
We say that Q is a Dynkin quiver if the underlying graph of Q is of ADE type. Let Λ -mod denote the category of finite-dimensional left Λ-modules.
Theorem 6.5 (Gabriel). The category Q -mod has finitely many isomorphism classes of indecomposable objects if and only if Q is a Dynkin quiver. In this case, the set of indecomposable objects up to isomorphism is in canonical bijection with the positive roots Φ + of the Dynkin diagram.
Let τ and τ − denote the classical Auslander-Reiten translate and its inverse.
Theorem 6.6 (Platzeck-Auslander, Gabriel). If Q is Dynkin then every indecomposable Λ-module is isomorphic to τ −p P i for some i ∈ Q 0 and p ≥ 0.
The algebra Λ = Q is hereditary, so its global dimension is ≤ 1. It is exactly 1 if Q has at least one arrow.
Let D = D b (Λ). As Λ is hereditary, every indecomposable X ∈ D is of the form Σ n M where n ∈ and M ∈ Λ -mod ֒→ D under the embedding taking a module to a stalk complex in degree 0. The derived functor of τ − is Ë − Σ, so using the same notation as with the module category, we have τ − (e i Λ) * ∼ = ΣP i . Thus, by Theorem 6.6,

Preprojective algebras
The preprojective algebra Π(Q) of the quiver Q was defined classically using generators and relations.
We claim that γ is a constant function. Write m i = γ(i) + 2. By Theorem 5.13 we have that, for each P i , τ −mi P i ∼ = Σ 2 P i . Now, following an argument from [HI11a, Section 4.1], suppose Hom Λ (P i , P j ) = 0. Then apply Ë mi mj = Ë mj mi : we see Ë mi mj P i = Σ 2mj P i and Σ 2mi P j . As there is a nonzero map between them, they must be concentrated in the same degree. Hence m i = m j so, as Q is connected, γ is constant. By Theorem 6.5 Λ has R indecomposable modules, so as τ −m ∼ = Σ 2 we must have nm = 2R. So, by Proposition 6.4, we have m = h. So by Theorem 5.13, D is fractional Calabi-Yau of dimension (h − 2)/h.
For the cases with ρ = 1 we have that β acts as −1 on the elements of Π of (tensor) degree 1, so α acts as 1 and so, arguing as above, in these cases D is fractional Calabi-Yau of dimension ((h/2) − 1)/(h/2).
6.3 Higher representation finite algebras 6.3.1 Higher preprojective algebras Following [IO13], we consider the following full subcategory of D b (Λ): Then U is a d-cluster tilting subcategory of D b (Λ) in the sense of Iyama [IY08, Section 3]. Note that Ì = Ë − d .
Set C = U/Ë − d . Then the object Λ ∈ C generates C (by which we mean the functor Mat ic Λ ֒→ C is an equivalence of graded categories, where Λ denotes the full subcategory of C on the object Λ).
We may assume without loss of generality that Λ is basic. Note that, by definition, C has as many objects as there are summands of Λ, which is finite because Λ is finite-dimensional.
Definition 6.10. The preprojective algebra of Λ, denoted Π, is the -graded base algebra of C.
Lemma 6.15 can be useful in practice to calculate explicit Calabi-Yau dimensions. Combined with Theorem 6.14, it also gives a theoretical result. We say an algebra is fractional Calabi-Yau if it is p/q-fCY for some p, q ∈ .
Corollary 6.16. Let Λ be a d-representation finite algebra and let (α, ℓ) denote the sgn-graded Nakayama automorphism of its higher preprojective algebra. Then α has finite order if and only if Λ is fractional Calabi-Yau.
In [HI11a, Remark 1.6], Herschend and Iyama ask: is every d-representation-finite algebra fractionally Calabi-Yau? Keeping the notation above, we translate this into the following: Question 6.17. Does α have finite order?

Higher type A algebras
Recall the higher type A preprojective algebras as studied in [Iya11,IO11]. In type A d s , they are given as kQ/I where Q has vertex set: The arrows are of the form α i ,x : x → x + f i , x, x + f i ∈ Q 0 for 1 ≤ i ≤ d + 1 where Define a permutation ω 0 of Q 0 by: ω 0 : (x 1 , x 2 , ... , x d+1 ) → (x d+1 , x 1 , ... , x d ).
Let G = d+1 with generating set e 1 , ... , e d+1 . The algebra Π is graded with arrows α x,i in degree e i . If p : x → y is a path from x to y of degree δ ∈ G , we have the relation y − x = i δ i f i : see the proof of [HI11a, Theorem 3.5]. In components, this says: Let P G denote the free hom-graded -linear category on Q modulo the relations from I , so A P G ∼ = Π. Note that the vertices are already elements of d+1 ≥0 ⊂ G , so we have a natural degree adjuster n : ob P G → G sending x ∈ Q 0 to x ∈ G . Lemma 6.19. (σ, n) is a 1-cell P G → P G in Cat G .
Proof. We want the following equation to hold: If deg f = δ ∈ G then deg σ(f ) = ω(δ). So, in degree i, this says which is true by the formula given above.
We have a group homomorphism ϕ : G → which projects onto the last component, so λ i e i is sent to λ. Therefore we get a -graded category P = ϕ * (P G ), and (σ, ℓ) is a 1-cell P → P, where ℓ(x) = x d+1 is the last term of x.
If we cut Π at the arrows f d+1 (or, equivalently, take the degree 0 subalgebra with respect to the -grading) we get the d-representation finite algebra of type A d s , denoted Λ d s .
The following result was first proved by Dyckerhoff, Jasso, and Walde [DJW19, Remark 2.29]. Their proof uses the language of ∞-categories and a description of the algebras Λ d s from [JK19]. It was also proved by Dyckerhoff, Jasso, and Lekili using symplectic geometry [DJL21, Remark 2.5.2].
Proof of Theorem 6.21. We use Proposition 6.13. If d is even then (sgn) d = tr. If d is odd then we use the fact that we take an even power σ d+1 together with Lemma 4.26.

Example: an algebra coming from a Postnikov diagram
Given a quiver with potential, i.e., a formal sum of cycles up to cyclic permutation, one obtains an algebra by formally differentiating the potential with respect to the arrows: this is known as the Jacobi algebra. Herschend and Iyama studied self-injective Jacobi algebras and showed that they always arise as preprojective algebras of 2-representation finite algebras; moreover, one can recover the underlying 2-representation finite algebra by taking a cut of the quiver [HI11b].
Pasquali studied Jacobi algebras associated to Postnikov diagrams [Pas19]. These algebras are obtained by taking stable endomorphism algebras of tilting modules for some other algebras, which were constructed by Jensen, King, and Su to categorify the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-dimensional space [JKS16]. Pasquali showed that the Jacobi algebra is self-injective precisely when the Postnikov diagram is symmetric, i.e., invariant under a rotation by 2kπ/n. In this case, the rotation of the Postnikov diagram induces the Nakayama automorphism of the corresponding Jacobi algebra [Pas19, Theorem 8.2 and Corollary 8.3].
In particular, it has finite order. Pasquali notes the connection to fractionally Calabi-Yau algebras, but the theory of [HI11a] requires the quiver to have a cut which is fixed by the Nakayama automorphism in order for the 2-representation finite algebra to satisfy be homogeneous. This rarely happens in examples.
Using Pasquali's description of the Nakayama automorphism together with Corollary 6.16, we obtain: Theorem 6.22. Every 2-representation finite algebra obtained as a cut of a self-injective quiver with potential coming from a Postnikov diagram is fractional Calabi-Yau.
Computing the Calabi-Yau dimension explicitly takes more work. We show how to do this in an explicit example. Consider the following quiver Q: It is a cobweb quiver and comes from a (4, 10)-Postnikov diagram: see [Pas19, Proposition 10.2]. It has an obvious potential W given by the sum of the clockwise cycles minus the sum of the anticlockwise cycles. The Jacobi algebra A = J(Q, W ) has Nakayama automorphism induced by the rotation of 2 5 2π, i.e., by the unique graph automorphism which acts on vertices by Note that σ 5 = 1 Π .
Any cut must contain exactly one arrow from each cycle, so by considering the central pentagon we see this quiver has no cut which is invariant under the Nakayama permutation. Consider the following cut, consisting of all but one outer arrow and one arrow from the central pentagon: This defines a 2-representation finite algebra Λ, with quiver given by the arrows not in the cut. It also induces a grading on the preprojective algebra Π = A of Λ, with arrows in the cut having degree 1.

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