Marion Boucrot

Morphisms of pre-Calabi-Yau categories and morphisms of cyclic A ∞ -categories

Keywords: Mathematics subject classification 2020: 16E45, 18G70, 14A22 A ∞ -categories, pre-Calabi-Yau categories

In this article we prove that there exists a relation between d-pre-Calabi-Yau morphisms introduced by M. Kontsevich, A. Takeda and Y. Vlassopoulos and cyclic A∞-morphisms, extending a result proved by D. Fernández and E. Herscovich. This leads to a functor between the category of d-pre-Calabi-Yau structures and the partial category of A∞-categories of the form A⊕A * [d-1] with A a graded quiver and whose morphisms are the data of an A∞-structure on

Introduction

Pre-Calabi-Yau algebras were introduced by M. Kontsevich and Y. Vlassopoulos in the last decade. These structures have also appeared under different names, such as V ∞ -algebras in [START_REF] Tradler | Algebraic string operations[END_REF], A ∞algebras with boundary in [START_REF] Seidel | Fukaya A∞-structures associated to Lefschetz fibrations. II, Algebra, geometry[END_REF], or weak Calabi-Yau structures in [START_REF] Kontsevich | Conference on Homological Mirror Symmetry[END_REF] for example. These references show that pre-Calabi-Yau structures play an important role in homological algebra, symplectic geometry, string topology, noncommutative geometry and even in Topological Quantum Field Theory.

In the finite dimensional case, pre-Calabi-Yau algebras are strongly related to A ∞ -algebras. Actually, for d ∈ Z, a d-pre-Calabi-Yau structure on a finite dimensional vector space A is equivalent to a cyclic A ∞ -structure on A ⊕ A * [d -1] that restricts to A. The definition of pre-Calabi-Yau morphisms first appeared in [START_REF] Kontsevich | Pre-Calabi-Yau algebras and topological quantum field theories[END_REF] and then in [START_REF] Leray | Pre-Calabi-Yau algebras and homotopy double poisson gebras[END_REF], in the properadic setting. A natural question is then about the link between pre-Calabi-Yau morphisms and A ∞ -morphisms of the corresponding boundary construction. D. Fernández and E. Herscovich studied this link in [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] at the level of double Poisson dg algebras and a restricted class of pre-Calabi-Yau algebras, when the multiplications m n vanish for n ≥ 4. In this paper, we study the relation between A ∞ -morphisms and pre-Calabi-Yau morphisms in a larger generality. The main result of this paper is the existence of a functor from the category of d-pre-Calabi-Yau structures pCY d to the partial category A ∞d whose objects are A ∞categories of the form A ⊕ A * [d -1] and whose morphisms are the data of an A ∞ -structure on A ⊕ B * [d -1] together with a diagram of the form

A[1] ⊕ B * [d] A[1] ⊕ A * [d] B[1] ⊕ B * [d] (1.1)
where each of the arrows are A ∞ -morphisms.

We also show that this functor restricts to a functor between a subcategory of pCY d and the partial subcategory of A ∞d whose objects are those of A ∞d and whose morphisms are the data of an almost cyclic A ∞ -structure on A ⊕ B * [d -1] together with a diagram of the form (1.1) where the arrows are A ∞ -morphisms.

Let us briefly present the contents of the article. In Section 2, we fix the notations and conventions we use in this paper and in Section 3, we recall the notions related to A ∞ -categories. Section 4 is devoted to present the notion of discs and diagrams as well as the notion of pre-Calabi-Yau structures based on the necklace bracket introduced in [START_REF] Kontsevich | Pre-Calabi-Yau algebras and topological quantum field theories[END_REF], which is given as the commutator of a necklace product, and their link with A ∞ -structures in the case of a Hom-finite graded quiver. We incidentally show that the necklace product for a graded quiver A is in fact equivalent to the usual Gerstenhaber circle product on A ⊕ A * [d -1] (see Proposition 4.23), which does not seem to have been observed in the literature so far. In Section 5, we recall the definitions of pre-Calabi-Yau morphisms and of the category pCY d given in [START_REF] Kontsevich | Pre-Calabi-Yau algebras and topological quantum field theories[END_REF].

Section 6 is the core of the article. In Subsection 6.1, we prove that given d-pre-Calabi-Yau categories A and B and a strict d-pre-Calabi-Yau morphism A → B, we can produce a cyclic A ∞ -structure on A ⊕ B * [d -1] and a diagram of the form (1.1) whose arrows are cyclic strict A ∞morphisms. We summarize these results in Corollary 6.14. In Subsection 6.2, we prove that given d-pre-Calabi-Yau categories A and B and any d-pre-Calabi-Yau morphism A → B, we can produce an A ∞ -structure on A ⊕ B * [d -1] and a diagram of the form (1.1) where the arrows are A ∞morphisms. Moreover, with an additional assumption on the pre-Calabi-Yau morphism A → B, the A ∞ -structure on A ⊕ B * [d -1] is almost cyclic. We summarize this in Corollary 6.20 in terms of functors.

Acknowledgements. This paper is part of a PhD thesis directed by Estanislao Herscovich and Hossein Abbaspour. The author thanks them for the useful discussions and advices. She also thanks David Fernández for comments on the paper and Johan Leray for useful discussions about pre-Calabi-Yau morphisms. This work is supported by the French National Research Agency in the framework of the "France 2030" program (ANR-15-IDEX-0002) and by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).

Notations and conventions

In what follows k will be a field of characteristic 0 and to simplify we will denote ⊗ for ⊗ k . We will denote by N = {0, 1, 2, . . . } the set of natural numbers and we define N * = N \ {0}. For i, j ∈ N, we define the interval of integers i, j = {n ∈ N|i ≤ n ≤ j}.

Recall that if we have a (cohomologically) graded vector space V = ⊕ i∈Z V i , we define for n ∈ Z the graded vector space V [n] given by V [n] i = V n+i for i ∈ Z and the map s V,n : V → V [n] whose underlying set theoretic map is the identity. Moreover, if f : V → W is a morphism of graded vector spaces, we define the map f [n] : V [n] → W [n] sending an element s V,n (v) to s W,n (f (v)) for all v ∈ V . We will denote s V,n simply by s n when there is no possible confusion, and s 1 just by s.

We now recall the Koszul sign rules, that are the ones we use to determine the signs appearing in this paper. If V, W are graded vector spaces, we have a map τ V,W : V ⊗ W → W ⊗ V defined as τ V,W (v ⊗ w) = (-1) |w||v| w ⊗ v where v ∈ V is a homogeneous element of degree |v| and w ∈ W is a homogeneous element of degree |w|. More generally, given graded vector spaces V 1 , . . . , V n and σ ∈ S n , we have a map

τ σ V1,...,Vn : V 1 ⊗ • • • ⊗ V n → V σ -1 (1) ⊗ • • • ⊗ V σ -1 (n) defined as τ σ V1,...,Vn (v 1 ⊗ • • • ⊗ v n ) = (-1) ϵ (v σ -1 (1) ⊗ • • • ⊗ v σ -1 (n) ) with ϵ = i>j σ -1 (i)<σ -1 (j) |v σ -1 (i) ||v σ -1 (j) | where v i ∈ V i is a homogeneous element of degree |v i | for i ∈ 1, n .
Throughout this paper, when we consider an element v of degree |v| in a graded vector space V , we mean a homogeneous element v of V . Also, we will denote by id the identity map of every space of morphisms, without specifying it. All the products in this paper will be products in the category of graded vector spaces. Given graded vector spaces (V i ) i∈I , we thus have

i∈I V i = n∈Z i∈I V n i
where the second product is the usual product of vector spaces.

Given graded vector spaces V, W we will denote by Hom k (V, W ) the vector space of k-linear maps f : V → W and by hom d k (V, W ) the vector space of homogeneous k-linear maps f : V → W of degree d, i.e. f (v) ∈ W n+d for all v ∈ V n . We assemble them in the graded vector space Hom k (V, W ) = d∈Z hom d k (V, W ) ⊆ Hom k (V, W ). We define the graded dual of a graded vector space V = n∈Z V n as the graded vector space V * = Hom k (V, k). Moreover, given graded vector spaces V , V ′ , W , W ′ and homogeneous elements f ∈ Hom k (V, V ′ ) and g ∈ Hom k (W, W ′ ), we have that

(f ⊗ g)(v ⊗ w) = (-1) |g||v| f (v) ⊗ g(w)
for homogeneous elements v ∈ V and w ∈ W . Recall that given graded vector spaces V 1 , . . . , V n and d ∈ Z we have a homogeneous linear isomorphism of degree 0

H j : ( n i=1 V i )[d] → V 1 ⊗ • • • ⊗ V j-1 ⊗ V j [d] ⊗ V j+1 ⊗ • • • ⊗ V n (2.1) sending an element s d (v 1 ⊗ • • • ⊗ v n ) to (-1) d(|v1|+•••+|vj-1|) v 1 ⊗ • • • ⊗ v j-1 ⊗ s d v j ⊗ v j+1 ⊗ • • • ⊗ v n .
Moreover, given graded vector spaces V and W and an integer d ∈ Z, we have homogeneous linear isomorphisms of degree 0

Hom k (V, W )[d] → Hom k (V, W [d]) (2.2) sending s d f ∈ Hom k (V, W )[d] to the map sending v ∈ V to s d (f (v))
and

Hom k (V, W )[d] → Hom k (V [-d], W ) (2.3) sending s d f ∈ Hom k (V, W )[d] to the map sending s -d v ∈ V [-d] to (-1) d|f | f (v).
Recall that a graded quiver A consists of a set of objects O together with graded vector spaces y A x for every x, y ∈ O. A dg quiver A is a graded quiver such that y A x is a dg vector space for every x, y ∈ O. Given a quiver A, its enveloping graded quiver is the graded quiver A e = A op ⊗A whose set of objects is O × O and whose space of morphisms from an object (x, y) to an object (x ′ , y ′ ) is defined as the graded vector space

(x ′ ,y ′ ) (A op ⊗ A) (x,y) = x A x ′ ⊗ y ′ A y . A
O p1 × • • • × O pn
where T n = N n for n > 1 and T 1 = N * . Given x = (x 1 , . . . , x n ) ∈ Ō we define its length as lg(x) = n, its left term as lt(x) = x 1 and right term as rt(x) = x n . For i ∈ 1, n , we define x≤i = (x 1 , . . . , x i ), x≥i = (x i , . . . , x n ) and for j > i, x i,j = (x i , x i+1 , . . . , x j ). One can similarly define x<i and x>i . Moreover, given x = (x 1 , . . . , xn ) ∈ Ō we define its length as lg( x) = n, its left term as lt(x) = x1 and its right term as rt(x) = xn . For x = (x 1 , . . . , x n ) ∈ Ō, we will denote

A ⊗x = x1 A x2 ⊗ x2 A x3 ⊗ • • • ⊗ x lg(x)-1 A x lg(x)
and we will often denote an element of A ⊗x as a 1 , a 2 , . . . , a lg(x)-1 instead of

a 1 ⊗ a 2 ⊗ • • • ⊗ a lg(x)-1 for a i ∈ xi A xi+1 , i ∈ 1, lg(x) -1 .
Moreover, given a tuple x = (x 1 , . . . , xn ) ∈ Ō we will denote

A ⊗ x = A ⊗x 1 ⊗ A ⊗x 2 ⊗ • • • ⊗ A ⊗x n .
Given tuples x = (x 1 , . . . , x n ), ȳ = (y 1 , . . . , y m ) ∈ Ō, we define their concatenation as x ⊔ ȳ = (x 1 , . . . , x n , y 1 , . . . , y m ). We also define the inverse of a tuple x = (x 1 , . . . , x n ) ∈ Ō as x-1 = (x n , x n-1 , . . . , x 1 ). If σ ∈ S n and x = (x 1 , . . . , x n ) ∈ O n , we define x • σ = (x σ(1) , x σ(2) , . . . , x σ(n) ). Moreover, given σ ∈ S n and x = (x 1 , . . . , xn ) ∈ Ōn , we define x • σ = (x σ(1) , xσ(2) , . . . , xσ(n) ). We denote by C n the subgroup of S n generated by the cycle σ = (12 . . . n) which sends i ∈ 1, n -1 to i + 1 and n to 1.

A ∞ -categories

In this section, we recall the notion of (cyclic) A ∞ -categories and (cyclic) A ∞ -morphisms as well as the definition of the natural bilinear form associated to a graded quiver. We also introduce a bilinear form on categories of the form A ⊕ B * where A and B are graded quivers related by a morphism A → B. We refer the reader to [START_REF] Seidel | Fukaya A∞-structures associated to Lefschetz fibrations. II, Algebra, geometry[END_REF] for the definitions of modules and bimodules over a category.

Definition 3.1. Given a graded quiver A with set of objects O, we define the graded vector space

C(A) = p≥1 x∈O p Hom k (A[1] ⊗x , lt(x) A rt(x) ) Given x = (x 1 , . . . , x n ) ∈ O n and a map F x : A[1] ⊗x → lt(x) A rt(x)
, we associate to F x a disc with several incoming arrows and one outgoing arrow (see Figure 3.1).

x 1 x 2 x 3 x 4 x n-1 x n F Figure 3.1: A disc representing a map F x : A[1] ⊗x → lt(x) A rt(x) , where x = (x 1 , . . . , x n )
To simplify, we will often omit the objects and assemble the incoming arrows in a big arrow (see Figure 3.2).

F

F x : A[1] ⊗x → lt(x) A rt(x) is the tuple x ∈ Ō.
Definition 3.3. Let A be a graded quiver. By the isomorphism (2.2), an element sF ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] sending an element (sa 1 , . . . , sa n-1 ) to s(F x(sa 1 , . . . , sa n-1 ))

induces maps A[1] ⊗x → lt(x) A rt(x)
for x = (x 1 , . . . , x n ) ∈ Ō, a i ∈ xi A xi+1 , i ∈ 1, n -1 .
To a homogeneous element sF ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF], we thus associate disc with a bold outgoing arrow (see Figure 3.3) to indicate that the output of sF is an element in A [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF]. [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] with a disc associated to a map

F

G (xi,y1,...,ym,xi+1) : A[1] ⊗(xi,y1,...,ym,xi+1) → xi A xi+1
F x : A[1] ⊗x → x1 A xn [1] with x = (x 1 , . . . , x n ). The type of this diagram is x≤i ⊔ ȳ ⊔ x>i . This diagram is associated to the map F x • (id ⊗i-1 ⊗G (xi,y1,...,ym,xi+1) ⊗ id ⊗n-i ).
Definition 3.5. Given a dg bimodule M over a dg category A with set of objects O, its naturalization is the chain complex M nat = (

x∈O x M x )/(f g -(-1) |g||f | gf ) where f ∈ y A x and g ∈ x M y .
Definition 3.6. Given dg category A with differential d A and product µ, we define the dg bimodule Bar(A) as Bar(A) = ( x ′ Bar(A) x ) x,x ′ ∈O , where

x ′ Bar(A) x = p≥0 (x0,...,xp)∈O p x ′ A x0 ⊗ x0 A x1 [1] ⊗ ... ⊗ xp-1 A xp [1] ⊗ xp A x whose differential restricted to x ′ Bar(A) x is given by x∈ Ō d x 0 + d x 1
where

d x 0 (f 0 ⊗ sf 1 ⊗ ... ⊗ sf n ⊗ f n+1 ) = n+1 i=0 (-1) i-1 j=0 (|fj |+1) f 0 ⊗ sf 1 ⊗ ... ⊗ s(d A (f i )) ⊗ ... ⊗ sf n ⊗ f n+1 and d x 1 (f 0 ⊗ sf 1 ⊗ ... ⊗ sf n ⊗ f n+1 ) = (-1) ϵ1 f 0 f 1 ⊗ sf 2 ⊗ ... ⊗ sf n ⊗ f n+1 + n-1 i=1 (-1) ϵi f 0 ⊗ sf 1 ⊗ ... ⊗ s(f i f i+1 ) ⊗ ... ⊗ sf n ⊗ f n+1 + (-1) ϵn f 0 ⊗ sf 1 ⊗ ... ⊗ sf n-1 ⊗ f n f n+1 for all x = (x 0 , x 1 , . . . , x n ), f 0 ∈ x ′ A x0 , f n+1 ∈ xn A x and f i ∈ xi-1 A xi , i ∈ 1, n , with ϵ i = |f 0 | + i-1 j=1 (|f j | + 1)
where we have written f i f i+1 instead of µ(f i , f i+1 ) for i ∈ 0, n to denote the composition of A.

The bimodule structure of Bar(A) is given by ρ = ( (x ′ ,y ′ ) ρ (x,y) ) x,y,x ′ ,y ′ ∈O where

(x ′ ,y ′ ) ρ (x,y) : (x ′ ,y ′ ) A e (x,y) ⊗ y Bar(A) x → y ′ Bar(A) x ′ sends (f ⊗g)⊗(a⊗sa 0 ⊗sa 1 ⊗• • •⊗sa n ⊗a ′ ) to (-1) |f |(|a|+|b|+ n i=0 |sai|) (ga⊗sa 0 ⊗sa 1 ⊗• • •⊗sa n ⊗a ′ f ) for all (f ⊗ g) ∈ (x ′ ,y ′ ) A e (x,y) , (x 0 , . . . , x n+1 ) ∈ Ōn+1 , a ∈ y A x0 , b ∈ xn+1 A
x and a i ∈ xi A xi+1 for i ∈ 0, n . Moreover, in that case, we have a quasi-isomorphism of dg bimodules Bar(A) → A whose restriction to Bar(A) p vanishes for p ≥ 1 and whose restriction to Bar(A) 0 is µ. Definition 3.7. Given a dg category A, the dual bimodule of the dg bimodule Bar(A) is given by Bar(A) ∨ =

x,y∈O y Bar(A) ∨

x , with

y Bar(A) ∨ x = Hom -A e ( y ′ Bar(A) x ′ , y ′ A x ⊗ y A x ′ )
where the subscript -A e of the Hom indicates that we consider the space of morphisms of right A e -modules.

In particular, y Bar ∨ x is a right graded A e -module for each x, y ∈ O for the left A e -structure of A e which is the inner one.

More precisely, Bar(A) is a A-bimodule with the action given by ρ = ( (x",y") ρ (x,y) ) x,y,x",y"∈O where (x",y") ρ (x,y) : (x",y") A e (x,y) ⊗ y Bar(A) ∨ x → y" Bar(A) ∨

x"

sends (f ⊗ g) ⊗ y ′ y Φ x ′ x to the map y ′ y" Ψ x ′ x" : y" Bar(A) x" → y ′ A x" ⊗ y" A x ′ defined by y ′ y" Ψ x ′ x" (a ⊗ sa 0 ⊗ sa 1 ⊗ • • • ⊗ sa n ⊗ a ′ ) = (-1) (|f |+|g|)|Φ (1) | Φ (1) f ⊗ gΦ (2)
where we have written

y ′ y Φ x ′ x (a ⊗ sa 0 ⊗ sa 1 ⊗ • • • ⊗ sa n ⊗ a ′
) as a tensor product Φ (1) ⊗ Φ (2) , for all (f ⊗ g) ∈ (x ′ ,y ′ ) A e (x,y) , (x 0 , . . . , x n+1 ) ∈ Ōn+1 , a ∈ y A x0 , a ′ ∈ xn+1 A x and a i ∈ xi A xi+1 for i ∈ 0, n .

Remark 3.8. If A is a dg category, we have a map

Bar(A) ∨ nat → p≥1 x∈O p Hom k (A[1] ⊗x , lt(x) A rt(x) ) sending z y Φ z x ∈ Hom -A e ( z Bar(A) z , z A x ⊗ y A z ) to the collection of k-linear maps Ψ : p≥0 (x0,...,xp)∈O p x0 A x1 [1] ⊗ ... ⊗ xp-1 A xp [1] → x0 A xp
given by Ψ(sa 0 , . . . , sa p-1 ) = µ(τ

• x0 x0 Φ x0 xp (1 x0 , sa 0 , . . . , sa p-1 , 1 x0 )) for a i ∈ xi A xi+1
and where 1 x0 is the identity of x0 A x0 . Moreover, this map is an isomorphism of graded vector spaces. Definition 3.9. Given a graded quiver A with set of objects O, the Gerstenhaber product of elements sF, sG ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] is defined as the element sF • G sG ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] given by

(sF • G sG) x = 1≤i<j≤lg(x) sF x≤i ⊔x ≥j • G,i,j sG x i,j
for x ∈ Ō where sF x≤i ⊔x ≥j • G,i,j sG x i,j (sa 1 , . . . , sa n-1 ) = (-1) ϵi sF x≤i ⊔x ≥j (sa 1 , . . . , sa i-1 , sG x i,j (sa i , . . . , sa j-1 ), sa j , . . . , sa n-1 )

for a i ∈ xi A xi+1 , with ϵ = (|G| + 1) i-1 r=1 |sa r |.
The map (sF • G sG) x is by definition the sum of the maps associated to diagrams of type x and of the form F G Definition 3.10. Given a graded quiver A with set of objects O, the Gerstenhaber bracket is the graded Lie bracket [-, -] G on C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] defined for elements sF, sG ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] as the element [sF, sG] G ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] given by

[sF, sG] x G = (sF • G sG) x -(-1) (|F|+1)(|G|+1) (sG • G sF) x for x ∈ Ō.
Definition 3.11. An A ∞ -structure on a graded quiver A is a homogeneous element sm A ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] of degree 1 satisfying the Maurer-Cartan equation [sm A , sm A ] G = 0.

Remark 3.12. An A ∞ -structure on a graded quiver A is tantamount to the data of a homogeneous element sm A ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] satisfying the following identities

A x [1] ⊗ x A y [1] → k of degree d + 2.
Definition 3.16. A bilinear form Γ on a graded quiver A is nondegenerate if the induced map

y A x [1] → ( y A x [1]) *
sending an element sa ∈ y A x [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] to the map sending sb ∈ x A y [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] to y Γ x (sa, sb) is an isomorphism.

Example 3.17. Consider two graded quivers A and B and a morphism (Φ 0 , Φ) : A → B of graded quivers. Define the bilinear form

Γ Φ : (A[1] ⊕ B * [d]) ⊗2 → k of degree d + 2 by y Γ Φ x (tf, sa) = -(-1) |sa||tf | x Γ Φ y (sa, tf ) = (-1) |tf |+1 (f • x Φ y )(a) for f ∈ Φ0(y) B * Φ0(x)
, a ∈ x A y , where t stands for the shift morphism B * → B * [d] and

y Γ Φ x (tf, tg) = y Γ Φ x (sa, sb) = 0 for f ∈ Φ0(y) B * Φ0(x) , g ∈ Φ0(x) B * Φ0(y)
, a ∈ y A x and b ∈ x A y . This bilinear form Γ Φ will be called the Φ-mixed bilinear form.

Example 3.18. If B = A, the bilinear form Γ id of the previous example is called the natural bilinear form on A and will be denoted Γ A .

Remark 3.19. The natural bilinear form on a Hom-finite graded quiver A is nondegenerate, whereas the Φ-mixed bilinear form is not in general. Definition 3.20. An A ∞ -structure sm A ∈ C(A) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] on a graded quiver A is almost cyclic with respect to a homogeneous bilinear form Γ : A[1] ⊗2 → k if the following holds:

xn Γ x1 (sm x A (sa 1 , . . . , sa n-1 ), sa n ) = (-1) |san|( n-1 i=1 |sai|) xn-1 Γ xn (sm x•σ -1 A (sa n , sa 1 , . . . , sa n-2 ), sa n-1 ) (3.1) for each n ∈ N * , x = (x 1 , . . . , x n ) ∈ Ō, σ = (12 . . . n) with a i ∈ xi A xi+1 for i ∈ 1, n -1 and a n ∈ xn A x1 . An almost cyclic A ∞ -category is an A ∞ -category whose A ∞ -
F x : A[1] ⊗x → F0(lt(x)) B F0(rt(x)
) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] is a map of degree 0, that satisfies 1≤i<j≤lg(x)

F x≤i ⊔x ≥j • (id ⊗x ≤i ⊗sm x i,j A ⊗ id ⊗x ≥j ) = 1≤i1<•••<in≤lg(x) sm ȳ B (F x≤i 1 ⊗ F x i 1 ,i 2 ⊗ • • • ⊗ F x in ,lg(x) ) (MI)
for every x ∈ ŌA and with ȳ = (F 0 (x 1 ), F 0 (x i1 ), . . . , F 0 (x lg(x) )). Note that given x ∈ O A the terms in both sums are sums of maps associated with diagrams of type x that are respectively of the form

F and m B m A F F F
The following definition was introduced in [START_REF] Kajiura | Noncommutative homotopy algebras associated with open strings[END_REF] by H. Kajiura in the cyclic case. Definition 3.22. An A ∞ -morphism (F 0 , F) between almost cyclic A ∞ -categories (A, sm A ) and (B, sm B ) with respect to bilinear forms γ and Γ is cyclic if

F0(y) Γ F0(x) (F 1 (sa), F 1 (sb)) = y γ x (sa, sb) for x, y ∈ O, a ∈ y A x and b ∈ x A y and for n ≥ 3 x∈Z ȳ∈Z ′ lt(z) Γ rt(z) (F x(sa 1 , . . . , sa i ), F ȳ (sa i+1 , . . . , sa n )) = 0
for z ∈ Ō, (sa 1 , . . . , sa i ) ∈ A [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] ⊗x and (sa i+1 , . . . , sa n ) ∈ A [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] ⊗ȳ where

Z = {x ∈ Ō | lt(x) = lt(z), rt(x) = rt(z)} and Z ′ = {ȳ ∈ Ō | lt(ȳ) = rt(z), rt(ȳ) = lt(z)}

Pre-Calabi-Yau categories

In this section, we present the diagrammatic calculus and recall the definition of d-pre-Calabi-Yau structures, d ∈ Z, appearing in [START_REF] Kontsevich | Pre-Calabi-Yau algebras and topological quantum field theories[END_REF] and [START_REF] Yeung | Pre-Calabi-Yau structures and moduli of representations[END_REF] as well as their relation with A ∞ -structures when the graded quiver considered is Hom-finite.

Diagrammatic calculus

In this subsection, we define discs and diagrams and we explain how to evaluate and compose them.

Following [START_REF] Yeung | Pre-Calabi-Yau structures and moduli of representations[END_REF] we define the following graded vector space.

Definition 4.1. Given a graded quiver A with set of objects O, we define the graded vector space

Multi • (Bar(A)[d]) = n∈N * Multi n (Bar(A)[d]) = n∈N * x∈ Ōn Multi x(Bar(A)[d])
where Multi x(Bar(A)[d]) is the graded vector space consisting of sums of homogeneous k-linear maps of the form

F x1 ,...,x n : A[1] ⊗x 1 ⊗A[1] ⊗x 2 ⊗• • •⊗A[1] ⊗x n → lt(x 1 ) A rt(x 2 ) [-d]⊗ lt(x 2 ) A rt(x 3 ) [-d]⊗• • •⊗ lt(x n ) A rt(x 1 ) [-d] for x = (x 1 , . . . , xn ). The action of σ = (σ n ) n∈N * ∈ n∈N * C n on an element F = (F x) x∈ Ō ∈ Multi • (Bar(A)[d]) is the element σ • F ∈ Multi • (Bar(A)[d]) given by (σ • F) x = τ σ -1 lt(x 1 ) A rt(x 2 ) [-d], lt(x 2 ) A rt(x 3 ) [-d],..., lt(x n ) A rt(x 1 ) [-d] • F x•σ • τ σ A[1] ⊗x 1 ,A[1] ⊗x 2 ,...,A[1] ⊗x n for x ∈ Ō. We will denote by Multi • (Bar(A)[d]) C lg(•) the space of elements of Multi • (Bar(A)[d]) that are invariant under the action of n∈N * C n . Remark 4.2. If A is a dg category, Multi n (Bar(A)[d]
) is the dg vector space

Hom (A e ) ⊗n (Bar(A)[d], id (A ⊗n ) σ )
where id (A ⊗n ) σ denotes the A ⊗n -bimodule A ⊗n whose structure is given by

(y1,...,yn) (A ⊗n ) (x1,...,xn) = n i=1 yi A xi+1
with the convention that

x n+1 = x 1 .
The action on morphisms is given by

(g 1 ⊗ • • • ⊗ g n ) • ω • (f 1 ⊗ • • • ⊗ f n ) = (-1) |f1|(|f2|+•••+|fn|) (g 1 ⊗ • • • ⊗ g n ) • ω • (f n ⊗ f 2 ⊗ • • • ⊗ f n-1 ⊗ f 1 )
for objects

x i , x ′ i , y i , y ′ i ∈ O, i ∈ 1, n , ω ∈ n i=1 yi A xi+1 , f i ∈ xi A x ′ i and g i ∈ yi A x ′ i where • denotes the usual bimodule structure of Hom (A e ) ⊗n (Bar(A)[d], A ⊗n ).

Definition 4.3.

A disc D is a circle with distinguished set of points which are either incoming or outgoing points. An incoming (resp. outgoing) point will be pictured as an incoming (resp. outgoing) arrow (see Figure 4.1). The size of the disc D is the number of outgoing arrows, and it will be denoted by |D|. The type of the decorated disc is the tuple of the form (x 1 , . . . , xn ) where xi is the tuple formed by objects of O, read in counterclockwise order, between the outgoing arrows i -1 and i, with the convention that the arrow 0 is the arrow n (see Figure 4.2). Definition 4.6. A marked disc is a decorated disc with a bold arrow (see Figure 4.3).

x 1 3 3 x 1 2 x 1 1 1 x 2 4 x 2 3 x 2 2 x 2 1 x 3 3 2 x 3 2 x 3 1 Figure 4.2: A decorated disc of type x = (x 1 , x2 , x3 ) where x1 = (x 1 1 , x 1 2 , x 1 3 ), x2 = (x 2 1 , x 2 2 , x 2 3 , x 2 4 ) and x3 = (x 3 1 , x 3 2 , x 3
x 1 3 x 1 2 x 1 1 x 2 4 x 2 3 x 2 2 x 2 1 x 3 3 x 3 2 x 3 1 3 1 2 Figure 4.3: A marked disc Definition 4.7. Consider s d+1 F x ∈ Multi x(Bar(A)[d])[d + 1] with x = (x 1 , . . . , xn ) ∈ Ōn . Given (a, b) ∈ ({i} × 1, lg(x 1 ) + • • • + lg(x n ) ) ⊔ ({o} × 1, n )
, by the isomorphisms (2.1), (2.2) and (2.3), s d+1 F x induces a morphism of the form

A[1] ⊗x 1 ⊗ • • • ⊗ A[1] ⊗x j-1 ⊗ A[1] ⊗x j ≤b ′ ⊗ x j b ′ A[-d] x j b ′ +1 ⊗ A[1] ⊗x j >b ′ ⊗ A[1] ⊗x n → lt(x 1 ) A rt(x 2 ) [-d] ⊗ • • • ⊗ lt(x n ) A rt(x 1 ) [-d] (4.1)
given by (-1)

(d+1)|F x| F x•(id ⊗(lg(x 1 )+•••+lg(x j-1 )+b ′ -j) ⊗s -d-1 ⊗id ⊗(lg(x j )-b ′ +lg(x j+1 )+•••+lg(x n )-n+j) ) if a = i and b = lg(x 1 ) + • • • + lg(x j ) + b ′ with j ∈ 1, n , b ′ ∈ 1, lg(x j ) -1 ,

and a morphism of the form

A[1] ⊗x 1 ⊗ A[1] ⊗x 2 ⊗ • • • ⊗ A[1] ⊗x n → lt(x 1 ) A rt(x 2 ) [-d] ⊗ • • • ⊗ lt(x b-1 ) A rt(x b ) [-d] ⊗ lt(x b ) A rt(x b+1 ) [1] ⊗ • • • ⊗ lt(x n ) A rt(x 1 ) [-d] (4.2)
given by (id

⊗(b-1) ⊗s d+1 ⊗ id ⊗(n-b) ) • F x if a = o. To s d+1 F x
s d+1 F ∈ Multi • (Bar(A)[d])[d + 1], the evaluation of E(D, s d+1 F x) at elements ( sa 1 , sa 2 , sa 3 ) is obtained by first compute F x( sa 1 , sa 2 , sa 3 ) = lt(x 1 ) F -d rt(x 2 ) ⊗ lt(x 2 ) F -d rt(x 3 ) ⊗ lt(x 3 ) F -d rt(x 3 )
, and then apply the shift s d+1 to the third tensor factor of the result since the bold arrow is the third outgoing arrow of the disc. The result is then

(-1) ϵ lt(x 1 ) F -d rt(x 2 ) ⊗ lt(x 2 ) F -d rt(x 3 ) ⊗ lt(x 3 ) F 1 rt(x 3 ) with ϵ = (d + 1)(| lt(x 1 ) F -d rt(x 2 ) | + | lt(x 2 ) F -d rt(x 3 ) |).
To simplify, from now on we will omit the objects when drawing a marked disc as well as the label of the outgoing arrows. By convention, if the bold arrow of the marked disc (of size n) is outgoing, it denotes the n-th outgoing arrow, and if the bold arrow is incoming, then the clockwise preceding outgoing arrow of the marked disc is the one labeled by n. Moreover, we will draw a big incoming arrow instead of several consecutive incoming arrows (see

A i for i ∈ 1, n , together with a subset R ⊆ (⊔ n i=1 A i ) 2 satisfying that (D.1) if (α, β) ∈ R ∩ (A i × A j ), then i ̸ = j
and α is an incoming arrow and β is an outgoing arrow;

(D.2) if ( y ′ α x ′ , y β x ) ∈ R, then x ′ = y and x = y ′ ;
where we use the notation introduced in Definition 4.4. We will represent a pair (α, β) ∈ R ∩ (A i × A j ) by connecting the outgoing arrow β of D j with the incoming arrow α of D i (see Figure 4.6), and we will say that the disc D i shares an arrow with D j or that α and β are connected.

An incoming (resp. outgoing) arrow of the diagram (D, R) is an arrow α of one of the discs D i such that there is no arrow β satisfying that (α, β) ∈ R (resp. (β, α) ∈ R). A diagram (D, R) has a distinguished object of O between any couple of consecutive arrows of (D, R) given by the decoration of the discs D 1 , . . . , D n .

(A.1) ∀i ∈ 1, n , ∃ α i ∈ D i such that (α i , β j ) ∈ R for some j ̸ = i, β j ∈ D j ;
(A.2) for (x, y) ∈ R, either x or y is a bold arrow, but not both;

(A.3) there is no family of arrows {x 1 , . . . , x k } such that (x i , x i+1 ) ∈ R for all i ∈ 1, k and

x k = x 1 .
Note that there is precisely one disc in D = {D 1 , . . . , D n } whose bold arrow is also an incoming or outgoing arrow of the diagram (D, R), i.e. an admissible diagram (D, R) always has either an incoming or outgoing bold arrow.

The size of an admissible diagram

(D = {D 1 , . . . , D n }, R) is n i=1 |D i | -n + 1
, and it will be denoted by |D|. We will relabel the outgoing arrows of (D, R) in clockwise direction from 1 to |D|, such that the outgoing arrow labeled by |D| is precisely the bold arrow of (D, R) if the latter arrow is outgoing, and it is the outgoing arrow preceding the bold arrow of (D, R) in clockwise sense if the bold arrow of (D, R) is incoming. Given an admissible diagram (D = {D 1 , . . . , D n }, R) and a tuple (s d+1 F 1 , . . . , s d+1 F n ) with

F i ∈ Multi xi (Bar(A)[d]) such that xi is the type of D i for all i ∈ 1, n , we will define a map E((D, R), s d+1 F 1 , . . . , s d+1 F n ) ∈ Multi x(Bar(A)[d])[d + 1]
where x is the type of (D, R), as follows. First, we suppose that the bold arrow is on a sink. Definition 4.14. Let (D = {D 1 , . . . , D n }, R) be a diagram and let (s d+1 F 1 , . . . , s d+1 F n ) be a tuple of homogeneous elements with F i ∈ Multi xi (Bar(A) [d]) such that xi is the type of D i for all i ∈ 1, n . We will define the evaluation of E((D, R), s d+1 F 1 , . . . , s d+1 F n ) at an element sa x of A[1] ⊗ x, where x is the type of (D, R) by induction on n as follows. We fix a source D s of (D, R).

(E.1) We place each element sa i j ∈ x i j A x i j+1 [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] in the incoming arrow x i j α x i j+1 of (D, R) in counterclockwise order beginning at the bold arrow. This will create a sign, as follows : if an element of degree m turns around the source representing a map of degree m ′ , we add a sign (-1) mm ′ and if an element of degree ℓ passes through an element of degree ℓ ′ to go to its place, we add a sign (-1) ℓℓ ′ . Here, turning around the source means passing through all of its inputs.

(E.2) We evaluate E(D s , s d+1 F s ) at the elements sa i j ∈ x i j A x i j+1 [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] corresponding to the incoming arrows of D s , to obtain an element of the form [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] corresponding to the incoming arrows of (D, R) between the last outgoing arrow of (D, R) and the bold arrow of D s connected with an arrow of the rest of the diagram in the clockwise order. Recall that the elements b i for i ∈ 1, |D s | -1 are labeled with an index from 1, |D| coming from the corresponding outgoing arrow of (D, R).

(-1) (d+1)(|b 1 |+•••+|b |Ds |-1 |) b 1 ⊗ • • • ⊗ b |Ds|-1 ⊗ s d+1 b |Ds| ∈ y ′ 1 A y1 [-d] ⊗ • • • ⊗ y ′ |Ds| A y |Ds| [1] (E.3) We add a Koszul sign coming from transposing b 1 ⊗ • • • ⊗ b |Ds|-1 with all of the elements sa i j ∈ x i j A x i j+1
(E.4) We consider the diagram (D ′ , R ′ ) given by removing the disc D s from (D, R) and place s d+1 b |Ds| at the incoming arrow of (D ′ , R ′ ) that was previously connected to the bold outgoing arrow of D s . By induction we evaluate E((D ′ , R ′ ), s d+1 F 1 , . . . , s d+1 Fs , . . . , s d+1 F n ) at the elements sa i j ∈ x i j A x i j+1 [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] corresponding to the incoming arrows of D ′ . This evaluation carries a sign

(-1) b |Ds|(| sa u |+| sa v |)+| sa u || sa v |
where | sa u | and | sa v | are the tuples of objects corresponding to the incoming arrows of D ′ that precede and follow the element b |Ds| without any outgoing arrow between them. Recall that the tensor factors in the evaluation of E((D ′ , R ′ ), s d+1 F 1 , . . . , s d+1 Fs , . . . , s d+1 F n ) at the elements sa i j ∈ x i j A x i j+1 [1] stated before are labeled with an index from 1, |D| coming from the corresponding outgoing arrow of (D, R).

(E.5) We reorder the tensor factors obtained in steps (E.4) and (E.5) according to the labeling of the outgoing arrows of (D, R), and add the respective Koszul sign.

We illustrate the previous procedure with an admissible diagram D consisting of two discs, given as follows

D 1 D 2
In this case, the evaluation of E(D, s d+1 G ȳ , s d+1 F x) at ( sa 1 , . . . , sa p-1 , sb First, we place each tensor factor sa i j and sb i j of sa i and sb i in the corresponding incoming arrow, as explained in (E.1), adding a sign (-1)

(|G ȳ |+d+1)(| sa 1 |+•••+| sa p-1 |+| sa p >q |)
. Also, we add a sign

(-1) | sb 1 || sa p >q |+| sb m || sa p ≤q |
for the permutation of the corresponding elements. We picture this as follows

D 1 D 2 sa 1 sa n sa p >q sb 1 sa p ≤q sb m In step (E.2), we compute E(D 1 , s d+1 G ȳ )( sb 1 , . . . , sb m ) = (-1) (d+1)(|ϵ1|+•••+|ϵm-1|) ϵ 1 ⊗• • •⊗s d+1 ϵ m .
After step (E.3) we have gained a total sign (-1) ∆ with

∆ = (|G ȳ | + d + 1)(| sa p ≤q | + | sa p+1 | + • • • + | sa n |) + (|G ȳ | + | sa 1 | + • • • + | sa p >q |)(|ϵ 1 | + • • • + |ϵ j-i-1 |), multiplying the element ϵ 1 ⊗• • •⊗ϵ j-i-1 ⊗E(D 2 , s d+1 F x)( sa 1 ⊗• • •⊗ sa p >q ⊗s d+1 ϵ j-i ⊗ sa p ≤q ⊗• • •⊗ sa n ). In (E.4), we compute E(D 2 , s d+1 F x)( sa 1 ⊗ • • • ⊗ sa p ≤q ⊗ s d+1 ϵ j-i ⊗ sa p >q ⊗ • • • ⊗ sa n ) = δ 1 ⊗ • • • ⊗ δ i-1 ⊗ δ j ⊗ • • • ⊗ s d+1 δ n
and add a sign (-1) ∆ ′ to the final result, with

∆ ′ = | sa p >q |(d+1+|ϵ j-i |+| sa p ≤q |)+| sa p ≤q |(d+1+|ϵ j-i |)+(d+1)(|δ 1 |+• • •+|δ i-1 |+|δ j |+• • •+|δ n-1 |)
In (E.5), we reorder the outputs with a sign, giving finally

E(D, s d+1 G ȳ , s d+1 F x)( sa 1 , . . . , sa n ) = (-1) ∆+∆ ′ +∆ ′′ δ 1 ⊗• • •⊗δ i-1 ⊗ϵ 1 ⊗• • •⊗ϵ j-i-1 ⊗δ j ⊗• • •⊗s d+1 δ n where ∆ ′′ = (|δ 1 | + . . . |δ i-1 |)(|ϵ 1 | + • • • + |ϵ j-i-1 |).
Now, supose that the bold arrow is on a source. Definition 4.15. Let (D = {D 1 , . . . , D n }, R) be a diagram and let (s d+1 F 1 , . . . , s d+1 F n ) be a tuple of homogeneous elements with F i ∈ Multi xi (Bar(A) [d]) such that xi is the type of D i for all i ∈ 1, n . We will define the evaluation E((D, R), s d+1 F 1 , . . . , s d+1 F n ) at an element sa x of A[1] ⊗ x, where x is the type of (D, R) by induction on n as follows. Suppose that the bold arrow is an arrow of the source D s of (D, R) and that the p-th outgoing arrow α of D s is connected with another disc.

(F.1) We place each element sa i j ∈ x i j A x i j+1 [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] in the incoming arrow x i j α x i j+1 of (D, R) in counterclockwise order beginning at the bold arrow. This will create a sign, as follows : if an element of degree m turns around a source representing a map of degree m ′ , we add a sign (-1) mm ′ and if an element of degree ℓ passes through an element of degree ℓ ′ to go to his place, we add a sign (-1) ℓℓ ′ . Again, turning around a source means passing through all of its inputs.

(F.2) We transpose all the elements that do not correspond to incoming arrows of D s with the elements corresponding to incoming arrows of D s between α and the bold arrow. We add the sign coming from this transposition. We evaluate E(D s , s d+1 F s ) at the elements sa i j ∈ x i j A x i j+1 [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] corresponding to the incoming arrows of D s , to obtain an element of the form (F.4) We then consider the diagram (D ′ , R ′ ) given by removing the disc D s from (D, R). Its bold arrow is the one previously connected to α. Note that the element b p is associated with this incoming arrow of (D ′ , R ′ ).

s d+1 (b 1 ⊗ • • • ⊗ b |Ds| ) ∈ ( y ′ 1 A y1 [-d] ⊗ • • • ⊗ y ′ |Ds | A y |Ds| [-d])[d + 1] (F.
(F.5) We evaluate E((D ′ , R ′ ), s d+1 F 1 , . . . , s d+1 Fs , . . . , s d+1 F n ) at the elements sa i j ∈ x i j A x i j+1 [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] corresponding to the incoming arrows of D ′ . Recall that the tensor factors in this evaluation are labeled with an index from 1, |D| coming from the corresponding outgoing arrow of (D, R).

We illustrate the previous procedure with an admissible diagram D consisting of two discs, given as follows

D 1 D 2
In this case, the evaluation of E(D, s d+1 F x, s d+1 G ȳ ) at ( sa 1 , . . . , sa p-1 , sb sa n

In step (F.2), we place all the elements sb i after the others and multiply by a sign

(-1) (| sb 1 |+•••+| sb m |)(| sa p+1 |+•••+| sa n |)
We then compute

E(D 1 , s d+1 F x)( sa 1 , . . . , sa n ) = (-1) (d+1)(|ϵ1|+•••+|ϵn-1|) ϵ 1 ⊗ • • • ⊗ s d+1 ϵ n . After (F.
3), we end with

ϵ 1 ⊗ • • • ⊗ ϵ p-1 ⊗ (E, s d+1 G ȳ )(ϵ p ⊗ sb 1 ≤q ⊗ • • • ⊗ sb m ⊗ sb 1 >q ) ⊗ ϵ p+1 ⊗ • • • ⊗ s d+1 ϵ n
preceded by a sign (-1) ∆ ′ with

∆ ′ = | sb 1 ≤q || sa p | + | sb 1 >q || sa p+1 | + ∆ + (| sb 1 ≤q | + • • • + | sb m | + | sb 1 >q |)(|ϵ p+1 | + • • • + |ϵ n | + d + 1) + (|G ȳ | + d + 1)(ϵ 1 + • • • + ϵ p-1 )
In (F.5), we order the elements and compute E(D 2 , s d+1 G ȳ )( sb Finally, we get

1 ≤q ⊗ ϵ p ⊗ sb
E(D, s d+1 F x, s d+1 G ȳ )( sa 1 , . . . , sa n ) = (-1) ∆ ′′ ϵ 1 ⊗ • • • ⊗ ϵ p-1 ⊗ δ 1 ⊗ • • • ⊗ δ m ⊗ ϵ p+1 ⊗ • • • ⊗ s d+1 ϵ n
where

∆ ′′ = ∆ ′ + | sb 1 >q |(|ϵ i | + | sb 2 | + • • • + | sb m |) + (d + 1)|G ȳ |. Definition 4.16. A filled diagram is an admissible diagram (D = {D 1 , . . . , D n }, R) together with ele- ments s d+1 F i ∈ Multi • (Bar(A)[d])[d + 1] for i ∈ 1, n . To each filled diagram D = {(D = {D 1 , . . . , D n }, R), (s d+1 F i ) i∈ 1,n } we associate the element E( D) = E((D, R), s d+1 F x1 , . . . , s d+1 F xn ) ∈ Multi x(Bar(A)[d])[d + 1]
where x is the type of (D, R) and xi is the type of D i for i ∈ 1, n . We depict a filled diagram as a diagram replacing the names of the discs by the corresponding maps (see Example 4.17. Consider the following filled diagram.

x 1 1 x 1 2 x 1 p 1 -2 x 1 p 1 -1 x 1 p 1 x 3 <i x 3 ≥i x 3 i x 3 i-1 y 2 1 y 2 q 2 y 2 q 2 -1 y 1 1 y 1 2 y 1 q 1 G F
It represents a map

s d+1 F x : A[1] ⊗x 1 ⊗ A[1] ⊗x 2 ⊗ A[1] ⊗x 3 → x 1 1 A x 2 p 2 [-d] ⊗ x 2 1 A x 3 p 3 [-d] ⊗ x 3 1 A x 1 p 1 [1]
where x = (x 1 , x2 , x3 ) and xj = (x j 1 , . . . , x j lg(x i ) ) for j ∈ 1, 3 that takes as an input the last output of a map

s d+1 G (ȳ 1 ,ȳ 2 ) : A[1] ⊗ȳ 1 ⊗ A[1] ⊗ȳ 2 → y 1 1 A y 2 m [-d] ⊗ x 3 i-1 A x 3 i [1] with ȳ1 = (y 1 1 , y 1 2 , . . . , y 1 n , x 3 i ), ȳ2 = (x 3 i-1 , y 2 1 , . . . , y 2 m ). The type of the diagram is (x 1 , x2 , ȳ1 ⊔ x3 ≥i , x3 <i ⊔ ȳ2
), meaning that it represents a map that can be evaluated at ( sa 1 , sa 2 , sb 1 ⊗ sa 3 ≥i , sa 3 <i ⊗ sb 2 ) where sa i ∈ A[1] ⊗x i and sb i ∈ A ⊗ȳ i . This evaluation goes in 5 steps.

1. We place the elements around the diagram, which creates a sign (-1) ϵ with

ϵ = |s d+1 G (ȳ 1 ,ȳ 2 ) |(| sa 1 | + | sa 2 | + | sa 3 ≥i |) + | sa ≥i || sb 1 | + | sb 2 || sa >i | 2. We evaluate E(D 1 , s d+1 G (ȳ 1 ,ȳ 2 ) ) at ( sb 1 , s b 2 )
. The result of this evaluation is an element of the shifted tensor product

( y 1 1 A y 2 m [-d]⊗ x 3 i-1 A x 3 i [-d])[d+1] that we will write s d+1 ( y 1 1 G -d y 2 m ⊗ x 3 i-1 G -d x 3 i
). We then use the isomorphism (2.1) to obtain the tensor product

(-1) ϵ+(d+1)| y 1 1 G -d y 2 m | y 1 1 G -d y 2 m ⊗ x 3 i-1 G 1 x 3 i ∈ y 1 1 A y 2 m [-d] ⊗ x 3 i-1 A x 3 i [1]
3. We put the first output before and get

(-1) ϵ+(d+1)| y 1 1 G -d y 2 m |+ϵ ′ y 1 1 G -d y 2 m ⊗ E(D 2 , s d+1 F x( sa 1 , sa 2 , sa ≥i , sa <i ⊗ x 3 i-1 G 1 x 3 i )) with ϵ ′ = y 1 1 G -d y 2 m |(|F | + d + 1 + | sa 1 | + | sa 2 | + | sa ≥i |) 4. We evaluate F x at ( sa 1 , sa 2 , sa <i ⊗ x 3 i-1 G 1 x 3 i ⊗ sa ≥i ) which add a sign (-1) | sa ≥i |(| sa<i|+| x 3 i-1 G 1 x 3 i |)
to the result for the transposition of the corresponding elements.

5. We order the outputs according to the labeling of their corresponding outgoing arrow and finally get (-1)

ϵ+(d+1)| y 1 1 G -d y 2 m |+ϵ ′ +ϵ ′′ x 1 1 F -d x 2 p 2 ⊗ x 1 2 F -d x 3 p 3 ⊗ y 1 1 G -d y 2 m ⊗ x 3 1 F 1 x 1 p 1
with ϵ ′′ = (-1)

(| y 1 1 G -d y 2 m |+1)(| x 1 1 F -d x 2 p 2 |+| x 2 1 F -d x 3 p 3 |)+(d+1)(| x 1 1 F -d x 2 p 2 |+•••+| x 1 2 F -d x 3 p 3 |+•••+| y 1 1 G -d y 2 m |)
.

The necklace graded Lie algebra

In order to recall what a pre-Calabi-Yau structure on A is, one first defines a graded Lie algebra, called the necklace graded Lie algebra and appearing in [START_REF] Kontsevich | Pre-Calabi-Yau algebras and topological quantum field theories[END_REF]. As a graded vector space, this graded Lie algebra is Multi

• (Bar(A)[d]) C lg(•) [d + 1].
In order to define a graded Lie bracket on this space, we first define a new operation as follows.

Definition 4.18. Consider a graded quiver A with set of objects O as well as tuples of elements of Ō given by x = (x 1 , ..., xn ), ȳ = (ȳ 1 , ..., ȳm ) ∈ Ō such that rt(ȳ 1 ) = x v j and lt(ȳ m ) = x v j-1 for some v ∈ 1, n and j ∈ 1, lg(x v ) . The inner necklace composition at v,j of elements given by x = (x 1 , ..., xn ), ȳ = (ȳ 1 , ..., ȳm ) ∈ Ō such that lt(ȳ v ) = x 1 j-1 and rt(ȳ v+1 ) = x 1 j for some v ∈ 1, m and j ∈ 1, lg(x 1 ) . The outer necklace composition at v,j of elements

s d+1 F x ∈ Multi x(Bar(A)[d])[d + 1] and s d+1 G ȳ ∈ Multi ȳ (Bar(A)[d])[d + 1] is given by s d+1 F x • nec,v,j inn s d+1 G ȳ = E( D) ∈ Multi x ⊔ v,j,inn ȳ (Bar(A)[d])[d + 1] with x ⊔ v,
s d+1 F x ∈ Multi x(Bar(A)[d])[d + 1] and s d+1 G ȳ ∈ Multi ȳ (Bar(A)[d])[d + 1] is given by s d+1 F x • nec,v,j out s d+1 G ȳ = E( D) ∈ Multi x ⊔ v,j,out ȳ (Bar(A)[d])[d + 1] with x ⊔ v,
s d+1 F, s d+1 G ∈ Multi • (Bar(A)[d]) C lg(•) [d + 1] is the element s d+1 F • nec s d+1 G ∈ Multi • (Bar(A)[d])[d + 1]
given by

(s d+1 F • nec s d+1 G) z = ( x, ȳ,v,j)∈Iinn s d+1 F x • nec,v,j inn s d+1 G ȳ + ( x, ȳ,v,j)∈Iout s d+1 F x • nec,v,j out s d+1 G ȳ
for all z ∈ Ō, where

I inn = {( x, ȳ, v, j) ∈ Ō × Ō × 1, lg( x) × 1, lg(x v ) | x ⊔ v,j,inn ȳ = z} I out = {( x, ȳ, v, j) ∈ Ō × Ō × 1, lg( ȳ) × 1, lg(x 1 ) | x ⊔ v,j,out ȳ = z}
Definition 4.21. Given a graded quiver A with set of objects O, the necklace bracket of two elements

s d+1 F, s d+1 G ∈ Multi • (Bar(A)[d]) C lg(•) [d + 1] is defined as the element [s d+1 F, s d+1 G] nec ∈ Multi • (Bar(A)[d])[d + 1]
where

[s d+1 F, s d+1 G] z nec = (s d+1 F • nec s d+1 G) z -(-1) (|F|+d+1)(|G|+d+1) (s d+1 G • nec s d+1 F) z
for every z ∈ Ō.

Lemma 4.22.

Let A be a graded quiver with set of objects O. Then, we have an injective map

j : Multi • (Bar(A)[d]) C lg(•) [d + 1] → C(A ⊕ A * [d -1]
) [1] sending

s d+1 ϕ ∈ Multi • (Bar(A)[d])[d + 1] to sψ x given by (π A •ψ x)( sa n , tf n-1 , sa n-2 , tf n-2 , ..., sa 2 , tf 2 , sa 1 ) = (-1) ϵ n-1 i=1 (f i •s d )⊗s d ϕ x( sa 1 , sa 2 , ..., sa n ) with ϵ = n-1 i=1 |tf i | n j=i+1 | sa j | + (|ϕ| + 1) n-1 i=1 |tf i | + d(n -1) + 1≤i<j≤n | sa i || sa j | + 1≤i<j≤n-1 |tf i ||tf j |
and by

(π A * • ψ x)( sa n , tf n-1 , sa n-2 , tf n-2 , ..., sa 2 , tf 2 , sa 1 )(s 1-d b) = (-1) δ n-1 i=1 (f i • s d ) ϕ x′ ( sa 1 ⊗ sb ⊗ sa n , sa 2 , ..., sa n-1 ) for x = (x 1 , . . . , xn ), x′ = (x 1 ⊔ xn , x2 , . . . , xn-1 ) sa i ∈ A[1] ⊗x i for i ∈ 1, n , sb ∈ rt(x 1 ) A[1] lt(x n ) and tf i ∈ rt(x i+1 ) A * lt(x i ) [d] for i ∈ 1, n -1 with δ = n-1 i=1 |tf i | n j=i+1 | sa j | + (|ϕ| + 1) n-1 i=1 |tf i | + d(n -1) + 1≤i<j≤n | sa i || sa j | + 1≤i<j≤n-1 |tf i ||tf j | + (d + 1) n i=1 | sa i | + (| sa n | + |sb|) n-1 i=2 | sa i | + | sa n ||sb| where π A (resp. π A * ) is the canonical projection A ⊕ A * [d -1] → A (resp. A ⊕ A * [d -1] → A * [d -1]).
We have the following relation between the necklace product and the usual Gerstenhaber circle product, which does not seem to have been observed in the literature so far.

Proposition 4.23. Let F, G be elements in Multi

• (Bar(A)[d]) C lg(•) . Then, we have jx ⊔ v,j,inn ȳ (s d+1 F x • nec,v,j inn s d+1 G ȳ ) = jx(s d+1 F x) • G,p,q π A (jȳ(s d+1 G ȳ )) (4.3) for x, ȳ ∈ Ō, v ∈ 1, lg( x) and j ∈ 1, lg(x v ) and jx ⊔ v,j,out ȳ (s d+1 F x • nec,v,j out s d+1 G ȳ ) = -(-1) (|F|+d+1)(|G|+d+1) jȳ(s d+1 G ȳ ) • G,p,q π A * (jx(s d+1 F x)) (4.4)
for v ∈ 1, lg( ȳ) and j ∈ 1, lg(x 1 ) , where p = lg(x 1 ) + ... + lg(x v ) + j + 2 and q = p + lg( ȳ) i=1 lg(ȳ i ).

Proof. We will first show the identity (4.3). Given x = (x 1 , . . . , xn ), ȳ = (ȳ 1 , . . . , ȳm ) ∈ Ō, both the compositions

s d+1 F x • nec,v,j inn s d+1 G ȳ and jx(s d+1 F x) • G,p,q π A (jȳ(s d+1 G ȳ ))
are zero if there are no u, v ∈ 1, n and j ∈ 1, lg(x v ) such that lt(ȳ u ) = x v j and rt(ȳ u+1 ) = x v j+1 with the convention that xn+1 = x1 . We will thus assume that there exist such u, v ∈ 1, n and j ∈ 1, lg(x v ) . We can further suppose that u = n because of the invariance under the action of C n . To simplify the expressions, we will denote the result of applying F x on any argument as a tensor product lt(x

1 ) F -d rt(x 2 ) ⊗ • • • ⊗ lt(x n-1 ) F -d rt(x n ) ⊗ lt(x n ) F -d rt(x 1 )
, where we omit those arguments. We have

jx ⊔ v,j,inn ȳ (s d+1 F x • nec,v,j inn s d+1 G ȳ ) ( sa n , tf n-1 , . . . , sa v+1 , tf v , sa v >j , sb m , tg m-1 , . . . , sb 2 , tg 1 , sb 1 , sa v ≤j , tf v-1 , . . . , sa 2 , tf 1 , sa 1 ) = (-1) ϵ ((f 1 • s d ) ⊗ • • • ⊗ (f v • s d ) ⊗ (g 1 • s d ) ⊗ • • • ⊗ (g m-1 • s d ) ⊗ (f v+1 • s d ) ⊗ • • • ⊗ (f n-1 • s d ) ⊗ id) (s d+1 F x • nec,v,j inn s d+1 G ȳ )( sa 1 , sa 2 , . . . , sa v-1 , sa v ≤j , sb 1 , . . . , sb m-1 , sb m , sa v >j , . . . , sa n-1 , sa n ) = (-1) ϵ+ϵ ′ ((f 1 • s d ) ⊗ • • • ⊗ (f v • s d ) ⊗ (g 1 • s d ) ⊗ • • • ⊗ (g m-1 • s d ) ⊗ (f v+1 • s d ) ⊗ • • • ⊗ (f n-1 • s d ) ⊗ id) s d+1 ( lt(x 1 ) F -d rt(x 2 ) , . . . lt(x v-1 ) F -d rt(x v ) , lt(ȳ 1 ) G -d rt(ȳ 2 ) , ..., lt(ȳ m-1 ) G -d rt(ȳ m ) , lt(x v ) F -d rt(x v+1 ) . . . lt(x n ) F -d rt(x 1 ) )
for elements

sa i ∈ A[1] ⊗x i , sb i ∈ A[1] ⊗ȳ i , tf i ∈ rt(x i+1 ) A * lt(x i ) [d] and tg i ∈ rt(ȳ i+1 ) A * lt(ȳ i ) [d], with ϵ = n-1 i=v |tf i | n k=i+1 | sa k | + m-1 i=1 |tg i |( n k=v+1 | sa k | + | sa v >j | + m k=i+1 | sb k |) + v-1 i=1 |tf i |( n k=v+1 | sa k | + | sa v >j | + m k=1 | sb k | + | sa v ≤j | + v-1 k=i+1 | sa k |) + d(n + m) + (|F| + |G| + d)( n-1 i=1 |tf i | + m-1 i=1 |tg i |) + n-1 i=1 |tf i | m-1 i=1 |tg i | + 1≤i<k≤n-1 |tf i ||tf k | + 1≤i<k≤m-1 |tg i ||tg k | + 1≤i<k≤n | sa i || sa k | + | sa v >j || sa v ≤j | + 1≤i<k≤m | sb i || sb k | + n i=1 | sa i | m i=1 | sb i | ϵ ′ = (|G| + d + 1)( v-1 i=1 | sa i | + | sa v ≤j |) + m-1 i=1 | lt(ȳ i ) G -d rt(ȳ i+1 ) |(| sa v >j | + n i=v+1 | sa i | + n i=v | lt(x i ) F -d rt(x i+1 ) | + | lt(ȳ m ) G 1 rt(ȳ 1 ) | + d + 1)
On the other hand, we have that

jx(s d+1 F x) • G,p,q π A (jȳ(s d+1 G ȳ )) ( sa n , tf n-1 , . . . , sa v+1 , tf v , sa v >j , sb m , tg m-1 , . . . , sb 2 , tg 1 , sb 1 , sa v ≤j , tf v-1 , . . . , sa 2 , tf 1 , sa 1 ) = (-1) δ jx(s d+1 F x)( sa n , tf n-1 , . . . , sa v+1 , tf v , sa v >j , π A ( m-1 i=1 (g i • s d ) ⊗ id)s d+1 G ȳ ( sb 1 , . . . , sb m ) , sa v ≤j , tf v-1 , . . . , sa 2 , tf 1 , sa 1 ) = (-1) δ+δ ′ ((f 1 • s d ) ⊗ • • • ⊗ (f v-1 • s d ) ⊗ (f v • s d ) ⊗ • • • ⊗ (f n-1 • s d ) ⊗ id) s d+1 ( lt(x 1 ) F -d rt(x 2 ) , . . . lt(x v-1 ) F -d rt(x v ) , λ G , lt(x v ) F -d rt(x v+1 ) . . . lt(x n ) F -d rt(x 1 ) )
where

λ G = m-1 i=1 (g i • s d ))( lt(ȳ 1 ) G -d rt(ȳ 2 ) , ..., lt(ȳ m-1 ) G -d rt(ȳ m ) ) ∈ k and δ = (|G| + d + 1)( n i=v+1 | sa i | + | sa v >j | + n-1 i=v |tf i |) + m-1 i=1 |tg i | m k=i+1 | sb k | + d(m -1) + (|G| + d) m-1 i=1 |tg i | + 1≤i<k≤m-1 |tg i ||tg k | + 1≤i<k≤m | sb i || sb k | δ ′ = 1≤i<k≤n-1 |tf i ||tf k | + 1≤i<k≤n | sa i || sa k | + | sa v ≤j || sa v >j | + n i=1 | sa i || lt(ȳ m ) G 1 rt(ȳ 1 ) | + v-1 i=1 |tf i |( n k=i+1 | sa k | + | lt(ȳ m ) G 1 rt(ȳ 1 ) |) + n-1 i=v |tf i | n k=i+1 | sa k | + |λG|( v-1 i=1 | sa i | + | sa v ≤j | + | lt(ȳ m ) G 1 rt(ȳ 1 ) | + v-1 i=1 |tf i | + n i=v lt(x i ) F -d rt(x i+1 ) |) + d(n -1) + (|F| + 1) n-1 i=1 |tf i | + (d + 1) m-1 i=1 | lt(ȳ i ) G -d rt(ȳ i+1 ) |
Therefore, we have

jx(s d+1 F x) • G,p,q πA jȳ(s d+1 G ȳ ) ( sa n , tf n-1 , . . . , sa v+1 , tf v , sa v >j , sb m , tg m-1 , . . . , sb 2 , tg 1 , sb 1 , sa v ≤j , tf v-1 , . . . , sa 2 , tf 1 , sa 1 ) = (-1) δ+δ ′ +δ ′′ ((f 1 • s d ) ⊗ • • • ⊗ (f v • s d ) ⊗ (g 1 • s d ) ⊗ • • • ⊗ (g m-1 • s d ) ⊗ (f v+1 • s d ) ⊗ • • • ⊗ (f n-1 • s d ) ⊗ id) s d+1 ( lt(x 1 ) F -d rt(x 2 ) , . . . lt(x v-1 ) F -d rt(x v ) , lt(ȳ 1 ) G -d rt(ȳ 2 ) , ..., lt(ȳ m-1 ) G -d rt(ȳ m ) , lt(x v ) F -d rt(x v+1 ) . . . lt(x n ) F -d rt(x 1 ) ) with δ ′′ = m-1 i=1 |tg i |( n-1 i=v |tf i | + v-1 i=1 lt(x i ) F -d rt(x i+1 )
). One can easily check that ϵ + ϵ ′ = δ + δ ′ + δ ′′ mod 2. Then, the first identity is proved.

We now prove the identity (4.4). Given x = (x 1 , . . . , xn ), ȳ = (ȳ 1 , . . . , ȳm ) ∈ Ō, both the compositions

s d+1 F x • nec,v,j out s d+1 G ȳ and jȳ(s d+1 G ȳ ) • G,p,q π A * (jx(s d+1 F x))
are zero if there are no v ∈ 1, m and j ∈ 1, lg(x 1 ) such that x1 j-1 = lt(ȳ v ) and x1 j = rt(ȳ v+1 ). We will thus assume that such v ∈ 1, m and j ∈ 1, lg(x 1 ) exist. Then, we have

jx ⊔ v,j,out ȳ (s d+1 F x • nec,v,j out s d+1 G ȳ ) ( sb m , tg m-1 , . . . , sb v+1 , sa 1 ≥j , tf n , sa n , tf n-1 , sa n-1 , . . . , tf 1 , sa 1 <j , sb v , tg v-1 , . . . , sb 1 ) = (-1) ϵ ((g 1 • s d ) ⊗ • • • ⊗ (g v-1 • s d ) ⊗ (f 1 • s d ) ⊗ • • • ⊗ (f n • s d ) ⊗ (g v+1 • s d ) ⊗ • • • ⊗ (g m-1 • s d ) ⊗ id) (s d+1 F x • nec,v,j out s d+1 G ȳ )( sb 1 , . . . , sb v , sa 1 <j , . . . , sa n , sa 1 ≥j , sb v+1 , . . . , sb m ) = (-1) ϵ+ϵ ′ ((g 1 • s d ) ⊗ • • • ⊗ (g v-1 • s d ) ⊗ (f 1 • s d ) ⊗ • • • ⊗ (f n • s d ) ⊗ (g v+1 • s d ) ⊗ • • • ⊗ (g m-1 • s d ) ⊗ id) s d+1 ( lt(ȳ 1 ) G -d rt(ȳ 2 ) , . . . , lt(ȳ v-1 ) G -d rt(ȳ v ) , lt(x 1 ) F -d rt(x 2 ) , . . . , lt(x n ) F -d rt(x 1 ) , lt(ȳ v+1 ) G -d rt(ȳ v+2 ) , . . . , lt(ȳ m ) G -d rt(ȳ 1 )
)

with ϵ = (|F| + |G| + d)( m-1 i=1 |tg i | + n i=1 |tf i |) + d(n + m) + m-1 i=v+1 |tg i | m k=i+1 | sb k | + | sa 1 ≥j || sa 1 <j | + 1≤i<k≤n | sa i || sa k | + n i=1 |tf i |( n k=i+1 | sa k | + | sa 1 ≥j | + m i=v+1 | sb i |) + v-1 i=1 |tg i |( n i=1 | sa i | + m k=i+1 | sb k |) + 1≤i<k≤m | sb i || sb k | + 1≤i<k≤m-1 |tg i ||tg k | + 1≤i<k≤n |tf i ||tf k | + m-1 i=1 |tg i | n i=1 |tf i | + m i=1 | sb i | n i=1 | sa i | ϵ ′ = n i=1 | sa i |( m i=v+1 | sb i | + m i=v+1 | lt(ȳ i ) G -d rt(ȳ i+1 ) |) + (|F| + d + 1) v-1 i=1 | lt(ȳ i ) G -d rt(ȳ i+1 ) | + | sa 1 ≥j | n i=2 | sa i | + | sa 1 <j || lt(ȳ v ) G 1 rt(ȳ v+1 ) |
On the other hand, we have that

jȳ(s d+1 G ȳ ) • G,p,q π A * (jx(s d+1 F x)) ( sb m , tg m-1 , . . . , sb v+1 , sa 1 ≥j , tf n , sa n , tf n-1 , sa n-1 , . . . , sa 1 <j , sb v , tg v-1 , . . . , sb 1 ) = (-1) δ jȳ(s d+1 G ȳ )( sb m , tg m-1 , . . . , sb v+1 , tg, sb v , tg v-1 , . . . , sb 1 )
where tg = π A * (jx(s d+1 F x))( sa 1 ≥j , tf n , sa n , tf n-1 , sa n-1 , . . . , sa 1 <j ) ∈ A * [d] and

δ = (|F| + d + 1)( m i=v+1 | sb i | + m-1 i=v+1 |tg i |)
Therefore, we have that

jȳ(s d+1 G ȳ ) • G,p,q π A * (jx(s d+1 F x)) ( sb m , tg m-1 , . . . , sb v+1 , tg v+1 , sa 1 ≥j , sa n , tf n-1 , sa n-1 , . . . , sa 1 <j , sb v , tg v-1 , . . . , sb 1 ) = (-1) δ+δ ′ ((g 1 • s d ) ⊗ • • • ⊗ (g v-1 • s d ) ⊗ (g • s d ) ⊗ (g v+1 • s d ) ⊗ • • • ⊗ (g m • s d ) ⊗ id) s d+1 ( lt(ȳ 1 ) G -d rt(ȳ 2 ) , . . . , lt(ȳ v-1 ) G -d rt(ȳ v ) , lt(ȳ v ) G -d rt(ȳ v+1 ) , . . . , lt(ȳ m ) G -d rt(ȳ 1 )
) where

δ ′ = |tg| m k=v+1 | sb k | + m-1 i=1 |tg i | m k=i+1 | sb k | + 1≤i<k≤m | sb i || sb k | + 1≤i<k≤m-1 |tg i ||tg k | + m-1 i=1 |tg i ||tg| + (|G| + 1)( m-1 i=1 |tg i | + |tg|) + d(m -1)
Furthermore, by definition, we have that

(g • s d )( lt(ȳ v ) G -d rt(ȳ v+1 ) ) = (-1) ∆ ( n i=1 (f i • s d ))( lt(x 1 ) F -d rt(x 2 ) ⊗ • • • ⊗ lt(x n ) F -d rt(x 1 ) )
where

∆ =|tg| + 1 + 1≤i<k≤n | sa i || sa k | + | lt(ȳ v ) G 1 rt(ȳ v+1 ) | n i=2 | sa i | + 1≤i<k≤n |tf i ||tf k | + d(n -1) + (|F| + 1) n i=1 |tf i | + n-1 i=1 |tf i | n k=i+1 | sa k | + | sa 1 ≥j |( n i=2 | sa i | + | sa 1 <j | + | lt(ȳ v ) G 1 rt(ȳ v+1 ) |)
Finally, we have that

jȳ(s d+1 G ȳ ) • G,p,q π A * (jx(s d+1 F x)) ( sb m , tg m-1 , . . . , sb v+1 , tg v+1 , sa 1 ≥j , tf n , sa n , tf n-1 , sa n-1 , . . . , sa 1 <j , sb v , tg v-1 , . . . , sb 1 ) = (-1) γ ((g 1 • s d ) ⊗ • • • ⊗ (g v-1 • s d ) ⊗ (f 1 • s d ) ⊗ • • • ⊗ (f n • s d ) ⊗ (g v+1 • s d ) ⊗ • • • ⊗ (g m-1 • s d ) ⊗ id) s d+1 ( lt(ȳ 1 ) G -d rt(ȳ 2 ) , . . . , lt(ȳ v-1 ) G -d rt(ȳ v ) , lt(x 1 ) F -d rt(x 2 ) , . . . , lt(x n ) F -d rt(x 1 ) , lt(ȳ v+1 ) G -d rt(ȳ v+2 ) , . . . , lt(ȳ m ) G -d rt(ȳ 1 ) )
where

γ = δ + δ ′ + |tg|( m-1 i=v+1 |tg i | + d + 1 + v-1 i=1 | lt(ȳ i ) G -d rt(ȳ i+1 ) |) + ∆ + n i=1 |tf i |( v-1 i=1 | lt(ȳ i ) G -d rt(ȳ i+1 ) | + m-1 i=v+1 |tg i |) It is straightforward to check that ϵ + ϵ ′ + γ = 1 + (|F| + d + 1)(|G| + d + 1) mod 2.
Corollary 4.24. The necklace bracket [-, -] nec introduced in Definition 4.21 gives a graded Lie algebra structure on Multi

• (Bar(A)[d]) C lg(•) [d + 1]. Proof. For F, G ∈ Multi • (Bar(A)[d]) C lg(•) , we have that j([s d+1 F, s d+1 G] nec ) = [j(s d+1 F), j(s d+1 G)] G
Moreover, using that j is injective, we have that [-, -] nec is a graded Lie bracket.

Pre-Calabi-Yau structures

Definition 4.25. A d-pre-Calabi-Yau structure on a graded quiver A is an element

s d+1 M A ∈ Multi • (Bar(A)[d]) C lg(•) [d + 1]
of degree 1, solving the Maurer-Cartan equation

[s d+1 M A , s d+1 M A ] nec = 0
Note that, since s d+1 M A has degree 1, this is tantamount to requiring that

s d+1 M A • nec s d+1 M A = 0.
We now recall the following result of [START_REF] Kontsevich | Pre-Calabi-Yau algebras and topological quantum field theories[END_REF] which states the link between a d-pre-Calabi-Yau structure on a Hom-finite graded quiver A and a cyclic A ∞ -structure on A ⊕ A * [d -1]. We denote by Γ the natural bilinear form on the boundary quiver

∂ d-1 A = A ⊕ A * [d -1], defined in Example 3.18. Proposition 4.26. A d-pre-Calabi-Yau structure on a graded quiver A induces an A ∞ -structure on A ⊕ A * [d -1]
that restricts to A and that is almost cyclic with respect to Γ. Moreover, if the graded quiver A is Hom-finite, then the data of a d-pre-Calabi-Yau structure on A is equivalent to the data of a cyclic

A ∞ -structure on A ⊕ A * [d -1] that restricts to A. Proof. Consider an element s d+1 M A ∈ Multi • (Bar(A)[d]) C lg(•) [d + 1] of degree 1.
We then define sm A⊕A * = j(s d+1 M A ) By Proposition 4.23, s d+1 M A defines a d-pre-Calabi-Yau structure if and

only if sm A⊕A * defines an A ∞ -structure on A ⊕ A * [d -1]
. Moreover, it is straightforward to show that this A ∞ -structure is almost cyclic with respect to Γ.

If A is Hom-finite, the bijectivity of j tells us that the collection of maps m A⊕B * are in correspondence with maps of the form M A , which shows the equivalence.

5 Pre-Calabi-Yau morphisms

The mixed necklace graded Lie algebra

One can also define a "mixed" necklace bracket, which will be useful in the next section. As we did for the necklace bracket, we first define the following graded vector space.

B•(A[1], B[-d]) = n∈N x∈ Ōn A Hom k n i=1 A[1] ⊗x i , n-1 i=1 Φ(lt(x i )) B Φ(rt(x i+1 )) [-d] ⊗ ( lt(x n ) A[-d] rt(x 1 ) ⊕ Φ(lt(x n )) B * [-1] Φ(rt(x 1 )) ) Definition 5.2. Consider s d+1 F x ∈ Bx(A[1], B[-d])[d + 1] with x = (x 1 , . . . , xn ) ∈ Ōn . Then s d+1 F
x induces a morphism of the form

A[1] ⊗x 1 ⊗ Φ(rt(x 1 )) B Φ(lt(x n )) [-d] ⊗ A[1] ⊗x n ⊗ • • • ⊗ A[1] ⊗x n-1 → n-1 i=1 Φ(lt(x i )) B Φ(rt(x i+1 )) [-d] (5.1)
sending ( sa 1 , s -d b, sa n , . . . , sa n-1 ) to

(-1) ϵ Φ(lt(x 1 )) F -d Φ(rt(x 2 )) ⊗ • • • ⊗ Φ(lt(x n-1 )) F -d Φ(rt(x n )) ⊗ Φ(lt(x n )) F 1 Φ(rt(x 1
)) (sb) where we have written

(id ⊗(n-1) ⊗π B ) F x1 ,...,x n ( sa 1 , . . . , sa n ) = Φ(rt(x 1 )) F -d Φ(lt(x 2 )) ⊗ • • • ⊗ Φ(rt(x n-1 )) F -d Φ(lt(x n )) ⊗ Φ(rt(x n )) F 1 Φ(lt(x 1 ))
where π B denotes the canonical projection

lt(x n ) A[-d] rt(x 1 ) ⊕ Φ(lt(x n )) B * [-1] Φ(rt(x 1 )) → Φ(lt(x n )) B * [-1] Φ(rt(x 1 )) for sa i ∈ A[1] ⊗x i with ϵ = (d + 1)(| sa 1 | + |F x|) + (|sb| + | sa n |) n-1 i=2 | sa i | + |sb|| sa n | + (d + 1) n-1 i=1 | Φ(lt(x i )) F -d Φ(rt(x i+1 )) |
and a morphism of the form

A[1] ⊗x 1 ⊗ A[1] ⊗x 2 ⊗ • • • ⊗ A[1] ⊗x n → n-1 i=2 Φ(lt(x i )) B Φ(rt(x i+1 )) [-d] ⊗ lt(x n ) A[1] rt(x 1 ) (5.2)
given by id ⊗(n-1) ⊗(s d+1 • π A ) • F x1 ,...,x n where π A denotes the canonical projection

lt(x n ) A[-d] rt(x 1 ) ⊕ Φ(lt(x n )) B * [-1] Φ(rt(x 1 )) → lt(x n ) A[-d] rt(x
B : B • (A[1], B[-d])[d + 1] ≃ -→ B A • (A[1], B[-d]) ⊕ B B • (A[1], B[-d]) where B A • (A[1], B[-d]) = n∈N x∈ Ōn A Hom k n i=1 A[1] ⊗x i , n-1 i=1 Φ(lt(x i )) B Φ(rt(x i+1 )) [-d] ⊗ lt(x n ) A[1] rt(x 1 )
and

B B • (A[1], B[-d]) = n∈N x∈ Ōn A Hom k A[1] ⊗x 1 ⊗ Φ(rt(x 1 )) B[-d] Φ(lt(x n )) ⊗ A[1] ⊗x n ⊗ • • • ⊗ A[1] ⊗x n-1 , n-1 i=1 Φ(lt(x i )) B Φ(rt(x i+1 )) [-d] sending an element s d+1 F ∈ B • (A[1], B[-d])[d + 1]
O A → O B . Given s d+1 F, s d+1 G ∈ B • (A[1], B[-d])[d + 1], we define their Φ-mixed necklace product as the element s d+1 F • Φ nec s d+1 G ∈ B • (A[1], B[-d])[d + 1] given by (s d+1 F • Φ nec s d+1 G) x = E( D) + E( D ′ ) ∈ B • (A[1], B[-d])[d + 1]
where the sums are over all the filled diagrams D and D ′ of type x and of the form

F G D = D ′ = F G
More precisely, each of D and D ′ can be pictured as two different filled diagrams by put in bold the arrow corresponding to the last output of F which is either an element in A [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF]

or in B * [d]. Thus, D is one of the following F G D 1 = D 2 = F G
and D ′ is one of the following

F G D ′ 1 = D ′ 2 = F G
Remark 5.6. The diagrams of Definition 5.5 are filled with F and G seen as elements of Multi

• (Bar(Q ′ Φ )[d]) where Q ′
Φ is the graded quiver whose set of objects is O A and whose spaces of morphisms are y (Q ′ Φ ) x = y A x ⊕ Φ(y) B Φ(x) . Definition 5.7. Given graded quivers A and B with respective sets of objects O A and O B and a map Φ : O A → O B , the Φ-mixed necklace bracket is the graded Lie bracket which is defined for elements

F, G ∈ B • (A[1], B[-d]) by [s d+1 F, s d+1 G] Φ nec = s d+1 F • Φ nec s d+1 G -(-1) (|F|+d+1)(|G|+d+1) s d+1 G • Φ nec
d+1 F ∈ Multi • (A[1], B[-d])[d + 1]
of degree 0, the multinecklace composition of s d+1 M A and s d+1 F is the element

s d+1 F • multinec s d+1 M A ∈ Multi • (A[1], B[-d])[d + 1] given by (s d+1 F • multinec s d+1 M A ) x = E( D) for x ∈ O A ,
s d+1 M B • pre s d+1 F ∈ Multi • (A[1], B[-d])[d + 1]
given by

(s d+1 M B • pre s d+1 F) x = E(D ′ ) for x ∈ O A ,
s d+1 M A ) → (B, s d+1 M B ) is a map F 0 : O A → O B together with an element s d+1 F ∈ Multi • (A[1], B[-d]) C lg(•) [d + 1] of degree 0 satisfying the following equation (s d+1 F • multinec s d+1 M A ) x = (s d+1 M B • pre s d+1 F) x (5.3)
for all x ∈ ŌA . Note that the left member and right member of the previous identity belong to

Hom k n i=1 A[1] ⊗x i , n-1 i=1 F0(lt(x i )) B F0(rt(x i+1 )) [-d] ⊗ F0(lt(x n )) B F0(rt(x 1
)) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] We now recall how to compose d-pre-Calabi-Yau morphisms.

(G 0 • F 0 , s d+1 F • pCY s d+1 G)
where

s d+1 F • pCY s d+1 G ∈ Multi • (A[1], B[-d]) C lg(•) [d + 1]
is of degree 0 and is given by Proof. We only have to check that the composition is associative and that the composition of any two d-pre-Calabi-Yau morphisms F and G is a d-pre-Calabi-Yau morphism. The associativity of the composition is clear. Now, consider two d-pre-Calabi-Yau morphisms F and G. Their composition is the sum of diagrams of the form (5.4). Therefore, the multinecklace composition of this composition and the pre-Calabi-Yau structure s d+1 M A is a sum of diagrams of the form

(s d+1 G • pCY s d+1 F ) x = E( D)
M A F F G F G F F G G F F (5.5)
Given x ∈ ŌA , we have to sum over all diagrams of such type and we have several possibilities for the type of the inner diagram, defined as the subdiagram consisting of the disc filled with M A together with all those discs directly connected to the one filled with M A . Note that if we fix the type of the outer diagram given as the complement of the inner diagram, the type of the inner one is fixed. Moreover, changing the inner diagram for one of same type does not change the type of the whole diagram. Therefore, taking the sum over all diagrams of type x ∈ ŌA is the same as taking the sum over all the possible types for the outer diagram and for each of those, taking the sum over all the suitable types for the inner one. This second sum allows us to use that F is a pre-Calabi-Yau morphism to replace the inner diagram by one consisting of a discs filled with M B whose incoming arrows are connected with outgoing arrows of discs filled with F. Then, the sum of all the diagrams of type x of the form (5.5) is equal to the sum of all the diagrams of type x of the form

G F G F G G F F M B F F (5.6)
and we now define the inner diagram as filled diagram consisting of the disc filled with M B and of all the discs connected to it. The previous remarks on the types of the inner and outer diagrams still hold. Thus, the sum over all possible types for the whole diagram is again the sum over all the possible types for the outer diagram and for each of those, taking the sum over all the suitable types for the inner one. G being a pre-Calabi-Yau morphism, one can again use (5.3) and say that the sum of all the diagrams of type x of the form (5.6) is now equal to the sum of all the diagrams of type x of the form

G F F F G M C G Therefore, s d+1 G • pCY s d+1 F is a pre-Calabi-Yau morphism.
The following class of morphisms will be useful in the next subsection. Note that this condition is not closed under the pre-Calabi-Yau composition. We thus restrict this notion of good morphisms and give the following definition.

The relation between pre-Calabi-Yau morphisms and A ∞ -morphisms

Recall that given a Hom-finite graded quiver A, we have an equivalence between the data of a d-pre-Calabi-Yau structure on A and a cyclic A ∞ -structure on A ⊕ A * [d -1] that restricts to A. In this section, we study the relation between d-pre-Calabi-Yau morphisms and A ∞ -morphisms.

The case of strict morphisms

In this subsection, we study the relation between strict d-pre-Calabi-Yau morphisms and A ∞morphisms. We first recall the notion of strict d-pre-Calabi-Yau morphism.

(Φ lt(x 1 ),rt(x 2 ) ⊗• • •⊗Φ lt(x n ),rt(x 1 ) )•s d+1 M x1 ,...,x n A = s d+1 M x1 ,...,x n B •(Φ ⊗ lg(x 1 )-1 ⊗• • •⊗Φ ⊗ lg(x n )-1 ) for each n ∈ N * , (x 1 , . . . , xn ) ∈ Ōn A .
For simplicity, we will omit the elements when writing the map Φ. We will denote

Φ 0 (x) = (Φ 0 (x 1 ), . . . , Φ 0 (x n )) for x = (x 1 , . . . , x n ) ∈ O n A and Φ 0 (x) = (Φ 0 (x 1 ), . . . , Φ 0 (x n )) for x = (x 1 , . . . , xn ) ∈ Ōn A . Definition
lt(x 1 ) A rt(x n ) ⊕ Φ0(lt(x 1 )) B * Φ0(rt(x n )) [d -1] → lt(x 1 ) A rt(x n )
is given by the map

m x1 ,...,x n A⊕B * →A : n-1 i=1 (A[1] ⊗x i ⊗ Φ0(rt(x i+1 )) B * [d] Φ0(lt(x i )) ) ⊗ A[1] ⊗x n → lt(x 1 ) A rt(x n ) defined by m x1 ,...,x n A⊕B * →A = m x1 ,...,x n A⊕A * →A • (id ⊗ lg(x 1 )-1 ⊗Φ * ⊗ id ⊗ lg(x 2 )-1 ⊗ • • • ⊗ id ⊗ lg(x n-1 )-1 ⊗Φ * ⊗ id ⊗ lg(x n )-1 )
and the composition of m x1 ,...,x n A⊕B * with the canonical projection

lt(x 1 ) A rt(x n ) ⊕ Φ0(lt(x 1 )) B * Φ0(rt(x n )) [d -1] → Φ0(lt(x 1 )) B * Φ0(rt(x n )) [d -1]
is given by the map m x1 ,...,x n A⊕B * →B * :

n-1 i=1 (A[1] ⊗x i ⊗ Φ0(rt(x i+1 )) B * [d] Φ0(lt(x i )) ) ⊗ A[1] ⊗x n → Φ0(lt(x 1 )) B * Φ0(rt(x n )) [d -1] defined by m x1 ,...,x n A⊕B * →B * = m Φ0(x 1 ),...,Φ0(x n ) B⊕B * →B * • (Φ ⊗ lg(x 1 )-1 ⊗ id ⊗Φ ⊗ lg(x 2 )-1 ⊗ • • • ⊗ id ⊗Φ ⊗ lg(x n )-1 ) Proposition 6.4. The element sm A⊕B * ∈ C(A ⊕ B * [d -1])[1] defines an A ∞ -structure on A ⊕ B * [d -1]
that is almost cyclic with respect to the Φ-mixed bilinear form Γ Φ , defined in Example 3.17.

Proof. Using the Proposition 5.9, we have that the equality sm

A⊕B * • G sm A⊕B * = 0 is tantamount to s d+1 M A⊕B * • Φ nec s d+1 M A⊕B * = 0 where s d+1 M A⊕B * ∈ B • (A[1], B[-d])[d + 1] is uniquely deter- mined by M x A⊕B * = (Φ ⊗(n-1) ⊗ id) • s d+1 M x A ∈ B A x (A[1], B[-d]) and M Φ0(x) A⊕B * = s d+1 M Φ0(y) B •(Φ ⊗ lg(x 1 )-1 ⊗id ⊗Φ ⊗ lg(x n )-1 ⊗Φ ⊗ lg(x 2 )-1 ⊗• • •⊗Φ ⊗ lg(x n-1 )-1 ) ∈ B B x (A[1], B[-d])
for x = (x 1 , . . . , xn ) ∈ Ōn and ȳ = (x 1 ⊔ xn , x2 , . . . , xn-1 ). Moreover,

π A (s d+1 M A⊕B * • Φ nec s d+1 M A⊕B * ) x = E( D) + E( D ′ ) where π A is the canonical projection B • (A[1], B[-d])[d + 1] → B A • (A[1], B[-d])
and where the sums are over all the filled diagrams D and D ′ of type x that are of the form

M A M A Φ Φ Φ Φ Φ and M A Φ Φ Φ Φ M B
respectively. Now, note that the second diagram can be cut into two as follows.

M A Φ Φ Φ Φ M B
Using the morphism identity satisfied by Φ, the diagram on the right can be replaced by one with a disc filled with M A whose outgoing arrows are connected with the unique incoming arrow of discs of size 1 filled with Φ. We thus obtain that

π A (s d+1 M A⊕B * • G s d+1 M A⊕B * ) x = E( D) + E( D ′ )
where the sums are over all the filled diagrams D and D ′ of type x that are of the form

M A M A Φ Φ Φ Φ Φ and M A M A Φ Φ Φ Φ Φ respectively. Moreover, E( D) + E( D ′ ) = 0 since s d+1 M A is a d-pre-Calabi-Yau structure.
Thus, if we show that this structure satisfies the cyclicity condition (3.1), sm A⊕B * satisfies the Stasheff identities (SI).

Using the definition of Γ Φ and sm B⊕B * and since sm B⊕B * is cyclic with respect to Γ B , we have that

Γ Φ (sm x1 ,...,x n A⊕B * →B * ( sa 1 , tf 1 , . . . , sa n-1 , tf n-1 , sa n ), sb) = Γ B (sm Φ0(x 1 ),...,Φ0(x n ) B⊕B * →B * (Φ ⊗ lg(x 1 )-1 ( sa 1 ), tf 1 , . . . , tf n-1 , Φ ⊗ lg(x n )-1 ( sa n )), Φ(sb)) = (-1) ϵ Γ B (sm Φ0(x n )⊔Φ0(x 1 ),Φ0(x 2 ),...,Φ0(x n-1 ) B⊕B * →B Φ ⊗ lg(x n )-1 ( sa n ) ⊗ Φ(sb) ⊗ Φ ⊗ lg(x 1 )-1 ( sa 1 ), tf 1 , . . . , tf n-2 , Φ ⊗ lg(x n-1 )-1 ( sa n-1 ) , tf n-1 ) = (-1) ϵ+δ Γ B ((f n-2 • s d ) ⊗ • • • ⊗ (f 1 • s d ) ⊗ id) (M Φ0(x n-1 ),...Φ0(x n )⊔Φ0(x 1 ) B (Φ ⊗ lg(x n-1 ) ( sa n-1 ), . . . , Φ ⊗ lg(x n ) ( sa n ) ⊗ Φ(sb) ⊗ Φ ⊗ lg(x 1 ) ( sa 1 )), tf n-1 (6.1) for s a i ∈ A[1] ⊗x i , tf i ∈ Φ0(rt(x i+1 )) B * [d] Φ0(lt(x i )) , with ϵ = (| sa n | + |sb|)( n-1 i=1 (| sa i | + |tf i |) and δ = (| sa 1 | + |sb| + | sa n |)( n-1 i=1 | sa i | + n-2 i=1 |tf i |) + 2≤i≤j≤n-2 | sa i ||tf j | + dn + 2≤i<j≤n-1 | sa i || sa j | + 1≤i<j≤n-2 |tf i ||tf j |
Moreover, using that Φ is a d-pre-Calabi-Yau morphism, we have that the last member of (6.1) is

(-1) ϵ+δ Γ A ( n-1 i=2 (f n-i • Φ) ⊗ id) • s d+1 M xn-1 ,...,x n ⊔x 1 A ( sa n-1 , . . . , sa 2 , sa n ⊗ sb ⊗ sa 1 ), tf n-1 • Φ = (-1) ϵ Γ A sm xn ⊔x 1 ,x 2 ,...,x n-1 A⊕A * →A ( sa n ⊗ sb ⊗ sa 1 , tf 1 • Φ, sa 2 , . . . , tf n-2 • Φ, sa n-1 ), tf n-1 • Φ (6.2)
where Φ denotes the morphism Φ[d -1]. Thus, comparing (6.1) and (6.2) we get that Γ B (sm

Φ0(x n )⊔Φ0(x 1 ),Φ0(x 2 ),...,Φ0(x n-1 ) B⊕B * →B Φ ⊗ lg(x n )-1 ( sa n ) ⊗ Φ(sb) ⊗ Φ ⊗ lg(x 1 )-1 ( sa 1 ), tf 1 , . . . , tf n-2 , Φ ⊗ lg(x n-1 )-1 ( sa n-1 ) , tf n-1 ) = Γ A (sm xn ⊔x 1 ,x 2 ,...,x n-1 A⊕A * →A ( sa n ⊗ sb ⊗ sa 1 , tf 1 • Φ, sa 2 , . . . , tf n-2 • Φ, sa n-1 ), tf n-1 • Φ) (6.3)
Finally, we have that

(-1) ϵ Γ A (sm xn ⊔x 1 ,x 2 ,...,x n-1 A⊕A * →A ( sa n ⊗ sb ⊗ sa 1 , tf 1 • Φ, sa 2 , . . . , tf n-2 • Φ, sa n-1 ), tf n-1 • Φ) = (-1) ϵ Γ Φ (sm xn ⊔x 1 ,x 2 ,...,x n-1 A⊕B * →A ( sa n ⊗ sb ⊗ sa 1 , tf 1 , sa 2 , . . . , tf n-2 , sa n-1 ), tf n-1 ) Therefore, A ⊕ B * [d -1] together with sm A⊕B * is an A ∞ -
M A ) → (B, s d+1 M B ).
We define maps of graded vector spaces Proof. We only check that φ A is a morphism since the case of φ B is similar. We only have to verify that A⊕B * →A ( sa 1 , tf 1 , sa 2 , . . . , sa n-1 , tf n-1 , sa n ) = sm x1 ,...,x n A⊕A * →A ( sa 1 , φ A (tf 1 ), sa 2 , . . . , sa n-1 , φ A (tf n-1 ), sa n ) for sa i ∈ A[1] ⊗x i and tf i ∈ Φ0(rt(x i+1 )) B * [d] Φ0(lt(x i )) , so that (6.4) holds.

φ x,y A : x A[1] y ⊕ Φ0(x) B * [d] Φ0(y) → x (A[1] ⊕ A * [d]) y and φ x,y B : x A[1] y ⊕ Φ0(x) B * [d] Φ0(y) → Φ0(x) (B[1] ⊕ B * [d]) Φ0(y) given by φ x,y A (sa) = sa, φ x,y B (sa) = Φ x,y (sa), φ x,y A (tf ) = tf • Φ y,x [d] and φ x,y B (tf ) = tf for x, y ∈ O A , sa ∈ x A[1] y and tf ∈ Φ0(x) B * [d] Φ0(y) .
Moreover, we also have φ x,y A • sm x1 ,...,x n A⊕B * →B * ( sa 1 , tf 1 , sa 2 , . . . , sa n-1 , tf n-1 , sa n ) = sm x1 ,...,x n A⊕B * →B * ( sa 1 , tf 1 , sa 2 , . . . , sa n-1 , tf n-1 , sa n ) • Φ x,y and given sb ∈ x A y , we have that sm x1 ,...,x n A⊕B * →B * ( sa 1 , tf 1 , sa 2 , . . . , sa n-1 , tf n-1 , sa n )(Φ x,y (sb)) = Γ Φ (sm x1 ,...,x n A⊕B * →B * ( sa 1 , tf 1 , sa 2 , . . . , sa n-1 , tf n-1 , sa n ), sb) On the other hand, we have ( sa 1 ) ⊗ φ A (tf 1 ) ⊗ φ

⊗ lg(x 2 )-1 A ( sa 2 ) • • • ⊗ φ ⊗ lg(x n )-1 A ( sa n )), sb)
Using the identity (6.3), we thus get (6.5).

It remains to show that the morphisms are cyclic. To prove it, we note that y Γ A x (φ y,x A (tf ), φ x,y A (sa)) = y Γ A x (t(f • Φ x,y ), sa) = f (Φ x,y (sa)) = y Γ Φ x (tf, sa)

as well as Definition 6.9. A partial category is an A ∞ -pre-category as defined in [START_REF] Kontsevich | Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry[END_REF] where the multiplications m n vanish for n > 2. We now summarize the results of Propositions 6.4 and 6.6. The minus in the identity (6.7) comes from the fact that the discs filled with M A change their place, in the sense that the order of the labeling of their first outgoing arrow changes. Since s d+1 M A is of degree 1, this create a minus sign. Moreover, E( D ′ ) = E( D 2 ) so that it remains to show that E( D) + E( D 1 ) = 0. This is the case since s d+1 M A is a pre-Calabi-Yau structure. Indeed, the sum of these evaluations of diagrams is the composition of

s d+1 M A • nec s d+1 M A

Figure 3 . 2 :Definition 3 . 2 .

 3232 Figure 3.2: The representation of a homogeneous element in C(A) Definition 3.2. The type of a disc representing a map F x : A[1] ⊗x → lt(x) A rt(x) is the tuple x ∈ Ō.

Figure 3 . 3 :G

 33 Figure 3.3: The representation of a homogeneous element in C(A)[START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF]

Figure 4 . 1 : A disc of size 3 Definition 4 . 4 .

 41344 Figure 4.1: A disc of size 3

3 ) 4 . 5 .

 345 Figure 4.2: A decorated disc of type x = (x 1 , x2 , x3 ) where x1= (x 1 1 , x 1 2 , x 1 3 ), x2 = (x 2 1 , x 2 2 , x 2 3 , x24 ) and x3 = (x 3 1 , x 3 2 , x 3 3 ) Definition 4.5. Given x ∈ Ō we associate to a map F x ∈ Multi x(Bar(A[d])) the unique decorated diagram of type x.

Example 4 . 8 .

 48 and (a, b) as before, we associate the marked diagram which is decorated as the one of F x, where the bold arrow is the b-th incoming arrow if a = i and the b-th outgoing arrow if a = o where the incoming arrows are numbered in clockwise order, the first being the incoming arrow following the last outgoing one. Given a disc D of type x, we will denote by E(D, s d+1 F x) the map (4.1) if a = i and (4.2) if a = o. Given the marked disc in Figure 4.3 and an element

FFigure 4 . 4 :FFigure 4 . 5 :

 4445 Figure 4.4: A disc representing a map of the form (4.2)

Figure 4 .

 4 Figure 4.6: A diagram

Definition 4 . 11 .

 411 The type of an admissible diagram (D = {D 1 , . . . , D n }, R) is the tuple x = (x 1 , . . . , xm ) ∈ Ō where m = |D| and xi ∈ Ō is the tuple composed of the objects of (D, R), placed in counterclockwise order, that we can read between the outgoing arrows i -1 and i of (D, R). Definition 4.12. A source (resp. sink) of an admissible diagram is a disc which shares none of its incoming (resp. outgoing) arrows with another one. Remark 4.13. An admissible diagram has at least one source and one sink.

3 )

 3 We add a Koszul sign coming from transposing b p+1 ⊗ • • • ⊗ b |Ds| with all of the elements that do not correspond to an incoming arrow of D s . Recall that the elements b i for i ∈ 1, |D s | \ {p} are labeled with an index from 1, |D| coming from the corresponding outgoing arrow of (D, R).

1 >q , sb 2 ,

 2 . . . , sb m ) and multiply the result by (-1) | sb 1 >q |(|ϵi|+| sb 2 |+•••+| sb m |)+(d+1)|G ȳ |

Figure 4 Figure 4 . 7 :

 447 Figure 4.7: A filled diagram

Definition 4 . 19 .

 419 j,inn ȳ = (x 1 , . . . , xv-1 , ȳ1 ⊔ xv <j , ȳ2 , . . . , ȳm-1 , xv >j-1 ⊔ ȳm , xv+1 , . . . , xn ) and where D is the filled diagram of type x ⊔ Consider a graded quiver A with set of objects O as well as tuples of elements of Ō

Definition 4 . 20 .

 420 j,out ȳ = (ȳ 1 , . . . , ȳv-1 , x1 <j-1 ⊔ ȳv , x2 , . . . , xn , ȳv+1 ⊔ x1 >j , ȳv+2 , . . . , ȳm )and where D is the filled diagram of type x ⊔ Given a graded quiver A with set of objects O, the necklace product of elements

Definition 5 . 1 .

 51 Given graded quivers A and B with respective sets of objects O A and O B and a map Φ : O A → O B , consider the graded vector space

1 ) 5 . 3 .

 153 Lemma Given two graded quivers A and B with respective sets of objects O A and O B and a map Φ : O A → O B , we have an isomorphism

Definition 5 . 4 .

 54 to the elements defined in (5.2) and (5.1). Let A and B be graded quivers with respective sets of objects O A and O B and consider a map Φ : O A → O B . We define the graded quiver Q Φ whose set of objects is O A and whose spaces of morphisms are y (Q Φ ) x = y A x ⊕ Φ(y) B * Φ(x) [d -1] for x, y ∈ O A . Definition 5.5. Let A and B be graded quivers with respective sets of objects O A and O B and consider a map Φ :

s d+1 F Definition 5 . 11 .

 511 Given d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) with respective sets of objects O A and O B and an element s

  where the sum is over all the filled diagrams D of type x of the form F F F M A and where we have omitted the bold arrow, meaning that it is any of the outgoing arrows. Definition 5.12. Given d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) with respective sets of objects O A and O B and an element s d+1 F ∈ Multi • (A[1], B[-d])[d + 1] of degree 0, the pre composition of s d+1 F and s d+1 M B is the element

  where the sum is over all the filled diagrams D ′ of type x of the form M B F F F and where we have omitted the bold arrow, meaning that it is any of the outgoing arrows. Definition 5.13. Given d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) with respective sets of objects O A and O B a d-pre-Calabi-Yau morphism (F 0 , F) : (A,

Definition 5 .

 5 14. Let (A, s d+1 M A ), (B, s d+1 M B ) and (C, s d+1 M C ) be d-pre-Calabi-Yau categories with respective sets of objects O A , O B and O C and let (F 0 , F) : (A, s d+1 M A ) → (B, s d+1 M B ) and (G 0 , G) : (B, s d+1 M B ) → (C, s d+1 M C ) be d-pre-Calabi-Yau morphisms. The composition of s d+1 F and s d+1 G is the pair

  where the sum is over all filled diagrams D of type x ∈ ŌA of the form G and where we have omitted the bold arrow, meaning that it any of the outgoing arrows. Proposition 5.15. For d ∈ Z, d-pre-Calabi-Yau categories and d-pre-Calabi-Yau morphisms together with the composition given in Definition 5.14 define a category, denoted as pCY d . Given a graded quiver A with set of objects O, the identity morphism Id : (A, s d+1 M A ) → (A, s d+1 M A ) is given by Id x = id x Ax for x ∈ O and Id x1 ,...,x n = 0 for (x 1 , . . . , xn ) ∈ Ōn such that n ̸ = 1 or n = 1 and lg(x 1 ) ̸ = 1.

Definition 5 .

 5 16. Given d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) with respective sets of objects O A and O A and a d-pre-Calabi-Yau morphism F = (F 0 , F) : (A, s d+1 M A ) → (B, s d+1 M B ), we say that F is good if E( D) = E( D ′ ) where the sums are over all the filled diagrams D and D ′ of type x of the form

Definition 5 .

 5 17. Given d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) with respective sets of objects O A and O A and a d-pre-Calabi-Yau morphism F = (F 0 , F) : (A, s d+1 M A ) → (B, s d+1 M B ), we say that F is nice if E( D) = E( D ′ ) where the sums are over all the filled diagrams D and D ′ of type x of the form

Definition 6 . 1 .

 61 Let (A, s d+1 M A ), (B, s d+1 M B ) be d-pre-Calabi-Yau categories with respective sets of objects O A and O B . A d-pre-Calabi-Yau morphism (Φ 0 , Φ) : (A, s d+1 M A ) → (B, s d+1 M B ) is strict if Φ x vanishes for each x ∈ Ōn A with n > 1 and lg(x 1 ) ̸ = 2 if n = 1. Equivalently, a strict d-pre-Calabi-Yau morphism between d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) is the data of a map between their sets of objects Φ 0 : O A → O B together with a collection Φ = (Φ x,y : x A y [1] → x B y [1]) x,y∈O A of maps of degree 0 that satisfies

6 . 2 . 1 AΦ0Definition 6 . 3 .

 62163 We denote by SpCY d the subcategory of pCY d whose objects are d-pre-Calabi-Yau categories and whose morphisms are strict d-pre-Calabi-Yau morphisms. Given d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) with respective sets of objects O A and O B and a strict d-pre-Calabi-Yau morphism (Φ 0 , Φ) : (A, s d+1 M A ) → (B, s d+1 M B ), we now construct an A ∞ -structure on A ⊕ B * [d -1]. For x ∈ ŌA , we denote by sm x A⊕A * →A the composition of jx-1(s d+1 Mx-) and the canonical projection on A[START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF]. Similarly, we will denote by sm Φ0(x) B⊕B * →B * the composition of j canonical projection on B * [d]. We define sm A⊕B * ∈ C(A⊕B * [d-1])[1] as the unique element such that the composition of m x1 ,...,x n A⊕B * with the canonical projection

Proposition 6 . 6 .

 66 Let (A, s d+1 M A ), (B, s d+1 M B ) be d-pre-Calabi-Yau categories with respective sets of objects O A and O B . Consider a strict d-pre-Calabi-Yau morphism (Φ 0 , Φ) : (A, s d+1 M A ) → (B, s d+1 M B ) and the A ∞ -category (A⊕B * [d-1], sm A⊕B * ) where sm A⊕B * ∈ C(A⊕B * [d-1])[1] is given in Definition 6.3. The maps φ A and φ B defined in Definition 6.5 are cyclic A ∞ -morphisms, in the sense of Definition 3.22.

  φ x,y A • sm x1 ,...,x n A⊕B * →A = sm x1 ,...,x n A⊕A * →A • (φ ⊗ lg(x 1 )-1 A ⊗ φ A ⊗ φ ⊗ lg(x 2 )-1 A • • • ⊗ φ ⊗ lg(x n )-1 A ) (6.4) and φ x,y A • sm x1 ,...,x n A⊕B * →B * = sm x1 ,...,x n A⊕A * →A * • (φ ⊗ lg(x 1 )-1 A ⊗ φ A ⊗ φ ⊗ lg(x 2 )-1 A • • • ⊗ φ ⊗ lg(x n )-1 A ) (6.5) for x, y ∈ O A , (x 1 , . . . , xn ) ∈ Ōn A such that lt(x n ) = x, rt(x 1 ) = y. First,note that φ x,y A (sm x1 ,...,x n A⊕B * →A ( sa 1 , tf 1 , sa 2 , . . . , sa n-1 , tf n-1 , sa n )) = sm x1 ,...,x n

  sm x1 ,...,x n A⊕A * →A * (φ ⊗ lg(x 1 )-1 A ( sa 1 ) ⊗ φ A (tf 1 ) ⊗ φ ⊗ lg(x 2 )-1 A ( sa 2 ) • • • ⊗ φ ⊗ lg(x n )-1 A ( sa n ))(sb) = Γ A (sm x1 ,...,x n A⊕A * →A * (φ ⊗ lg(x 1 )-1 A

2 . 6 . 7 .

 267 y Γ Bx (φ y,x B (tf ), φ x,y B (sa)) = y Γ B x (tf, Φ x,y (sa)) = f (Φ x,y (sa)) = y Γ Φ x (tf, sa)for sa ∈ x A[1] y , tf ∈ Φ0(y) B * [d] Φ0(x). The second condition to be a cyclic morphism is obviously satisfied since φ x A and φ x B vanish forx ∈ O n with n > Definition Let (A ⊕ A * [d -1], sm A⊕A * ), (B ⊕ B * [d -1], sm B⊕B * ) be A ∞ -categories. A hat morphism from A ⊕ A * [d -1] to B ⊕ B * [d -1] is a triple (sm A⊕B * , φ A , φ B ) where sm A⊕B * is an A ∞ -structure on A ⊕ B * [d -1] and φ A , φ B are A ∞ -morphisms A[1] ⊕ B * [d] A[1] ⊕ A * [d] B[1] ⊕ B * [d]

Definition 6 . 8 .

 68 Let (A⊕A * [d-1], sm A⊕A * ), (B⊕B * [d-1], sm B⊕B * ) and (C⊕C * [d-1], sm C⊕C * ) be A ∞categories. Two hat morphisms (sm A⊕B * , φ A , φ B ) : A ⊕ A * [d -1] → B ⊕ B * [d -1], (sm B⊕C * , ψ B , ψ C ) : B ⊕ B * [d -1] → C ⊕ C * [d -1] are composable if there exist a triple (sm A⊕C * , χ A , χ C ) where χ A : A ⊕ C * [d -1] → A ⊕ B * [d -1] and χ C : A ⊕ C * [d -1] → B ⊕ C * [d -1] are A ∞ -morphisms and such that φ B • χ A = ψ B • χ C . The composition of (sm A⊕B * , φ A , φ B ) and (sm B⊕C * , ψ B , ψ C ) is then given by (sm A⊕C * , φ A • χ A , ψ C • χ C ).

Definition 6 . 10 .

 610 The A ∞ -hat category is the partial category A ∞d whose objects are A ∞ -categories of the form A ⊕ A * [d -1] and whose morphisms are hat morphisms. Definition 6.11. A functor between partial categories A and B with respective sets of objects O A and O B is the data of a map F 0 : O A → O B together with a family F = ( y F x ) x,y∈O A sending a morphism f : x → y to a morphism y F x (f ) : F 0 (x) → F 0 (y) such that if two morphisms f : x → y and g : y → z are composable, then y F x (f ) and z F y (g) are composable and their composition is given by the morphismz F y (g) • y F x (f ) = z F x (g • f ).Definition 6.12. We define the partial subcategory cyc A ∞d of A ∞d whose objects are almost cyclic A ∞categories of the formA ⊕ A * [d -1] and whose morphisms A ⊕ A * [d -1] → B ⊕ B * [d -1] are the data of an almost cyclic A ∞ -structure on A ⊕ B * [d -1]together with a diagram of the form (6.6) where φ A and φ B are A ∞ -morphisms. Definition 6.13. We define the partial subcategory Scyc A ∞d of cyc A ∞d whose objects are the ones of cyc A ∞d and whose morphisms are strict cyclic morphisms of cyc A ∞d .

Proposition 6 . 16 .

 616 The element sm A⊕B * ∈ C(A⊕B * [d-1])[START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF] defines an A ∞ -structure on A⊕B * [d-1]. Moreover, if the morphism Φ is good, sm A⊕B * satisfies the cyclicity condition (3.1).Proof. Using Proposition 5.9, it suffices to show thats d+1 M A⊕B * • Φ nec s d+1 M A⊕B * = 0. We have that π A (s d+1 M A⊕B * • Φ nec s d+1 M A⊕B * ) = 0 is tantamount to E( D) + E( D ′ ) + E( D ′′ ) = 0where the sums are over all the filled diagrams D, D ′ and D ′′ of type x of the form third diagram can be cut into two as follows Using that Φ is a pre-Calabi-Yau morphism, the left side can be changed into a diagram consisting of a disc filled with M A whose outgoing arrows are shared with discs filled with Φ. We thus get that E( D ′′ ) = E( D 1 ) -E( D 2 ) (6.7)where D 1 and D 2 are filled diagrams of the form

FFF

  with a tensor product composed of maps of the collection Φ and of the identity map id in the last tensor factor. Therefore, the element sm A⊕B * satisfies the Stasheff identities (SI).It is clear that if the morphism Φ is good, then theA ∞ -structure on A ⊕ B * [d -1] is almost cyclic with respect to Γ 1 . Indeed, Γ Φ 1 • (sm A⊕B * →A ⊗ id A * ) = E( D)where the sum is over all the filled diagrams E( D) of the formM A On the other hand, Γ Φ 1 • (sm A⊕B * →B * ⊗ id B ) = E( D ′ )where the sum is over all the filled diagrams E( D ′ ) of the form

  graded quiver A with set of objects O is said to be Hom-finite if y A x is finite dimensional for every x, y ∈ O. Given graded quivers A and B with respective sets of objects O A and O B , a morphism of graded quivers (Φ 0 , Φ) : A → B is the data of a map Φ 0 : O A → O B between the sets of objects together with a collection Φ = ( y Φ x ) x,y∈O A of morphisms of graded vector spaces y Φ x : y A x → Φ0(y) B Φ0(x) for every x, y ∈ O A . In this paper we will only consider small graded quivers and small categories.

	We will denote	Ō =
	n∈N Ō =
	n∈N

*

O n and more generally, we will denote by Ō the set formed by all finite tuples of elements of Ō, i.e. * Ōn = n>0 (p1,...,pn)∈Tn

n . These are called the Stasheff identities and were first introduced in [8] by J. Stasheff. Example 3.13. If A is a dg category with differential d A and product

	sm x≤i ⊔x ≥j A	• (id ⊗x ≤i ⊗sm A x i,j	⊗ id ⊗x ≥j ) = 0	(SI)
	1≤i<j≤n
	for every n ∈ N µ, it carries a natural A ∞ -structure
	sm A ∈ C(A)[1] with m 1 A = d A , m 2 A = µ and m n A = 0 for n ≥ 3.

* and x ∈ O

Definition 3.14. Given

  a graded quiver A with set of objects O its graded dual quiver is the quiver A * whose set of objects is O and for x, y ∈ O, the space of morphisms from x to y is defined as y A * x = ( x A y ) * .

Definition 3.15. A bilinear form of degree d on

  a graded quiver A is a collection Γ = ( y Γ x ) x,y∈O of homogeneous k-linear maps y Γ x : y

Definition 3.21. An A ∞ -morphism between

  structure is almost cyclic with respect to a fixed homogeneous bilinear form. An almost cyclic A ∞ -category with respect to a nondegenerate bilinear form is called a cyclic A ∞ -category. A ∞ -categories (A, sm A ) and (B, sm B ) with respective sets of objects O A and O B is a map F 0 : O A → O B together with a collection F = (F x) x∈ ŌA , where

	The following definition was first introduced by M. Sugawara in [9].

  1 , sa p >q , sb 2 . . . , sb m-1 , sa p ≤q , sb m , sa p+1 . . . , sa n ) with sa i ∈ A[1] ⊗x i and sb i ∈ A[1] ⊗ȳ i , where x = (x 1 , . . . , xn ) and ȳ = (ȳ 1 , . . . , ȳm ) are the respective types of D 1 and D 2 , is detailed as follows.

1

  ≤q , sa p , sb 2 . . . , sb m , sa p+1 , sb 1 >q , sa p+2 . . . , sa n ) with sa i ∈ A[1] ⊗x i and sb i ∈ A[1] ⊗ȳ i , where x = (x 1 , . . . , xn ) and ȳ = (ȳ 1 , . . . , ȳm ) are the respective types of D 1 and D 2 , is detailed as follows. First, we place each tensor factor sa i j and sb i j of sa i and sb i in the corresponding incoming arrow, as explained in (F.1), adding a sign (-1) | sb 1 ≤q || sa p |+| sb 1 >q || sa p+1 | for the permutation of the corresponding elements.

		sa p
	sa 1	sb 1 ≤q
	D 1	D 2
		sb 1 >q sa p+1

  category that is almost cyclic with respect to Γ Φ . Let (A, s d+1 M A ), (B, s d+1 M B ) be d-pre-Calabi-Yau categories with respective sets of objects O A and O B . Consider a strict d-pre-Calabi-Yau morphism (Φ 0 , Φ) : (A, s d+1

	Definition 6.5.

 [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF]sending s d+1 ϕ ∈ Bx(A [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF], B[-d])[d + 1] to sψx where ψx :

is given by ψx( sa n , tf n-1 , sa n-2 , tf n-2 , ..., sa 2 , tf 2 , sa 1 ) = (-1) ϵ n i=1

Proposition 5.9. Let A and B be graded quivers with respective sets of objects O A and O B and consider a map Φ :

Pre-Calabi-Yau morphisms (after M. Kontsevich, A. Takeda and Y. Vlassopoulos)

Following the article [START_REF] Kontsevich | Pre-Calabi-Yau algebras and topological quantum field theories[END_REF], we recall the definition of the category of d-pre-Calabi-Yau categories.

The action of σ

given by

for x ∈ Ō. We will denote by Multi

) that are invariant under the action of C lg(•) .

Corollary 6.14. There exists a functor SpCY

given in Definition 6.5.

General case

We now present the relation between not necessarily strict d-pre-Calabi-Yau morphisms and A ∞morphisms. Consider d-pre-Calabi-Yau categories (A, s d+1 M A ), (B, s d+1 M B ) as well as a (Φ 0 , Φ) :

Definition 6.15. We define

where the sum is over all the filled diagrams D of type x and of the form

where the sum is over all the filled diagrams D ′ of type x and of the form

) [START_REF] Fernández | Cyclic A∞-algebras and double Poisson algebras[END_REF]. We will denote by m A⊕B * →A (resp. m A⊕B * →B * ) the composition of m A⊕B * with the canonical projection on

Lemma 6.17. Consider d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) as well as a d-pre-Calabi-Yau morphism (Φ 0 , Φ) : (A, s d+1 M A ) → (B, s d+1 M B ). Then, Φ induces morphisms

and a morphism

defined by

( sa n , . . . , sa 1 ))

and for each n ∈ N * , x = (x 1 , . . . , xn ) ∈ Ōn A .

Definition 6.18. Consider d-pre-Calabi-Yau categories (A, s d+1 M A ) and (B, s d+1 M B ) as well as a dpre-Calabi-Yau morphism (Φ 0 , Φ) : (A, s d+1 M A ) → (B, s d+1 M B ). We define maps of graded vector spaces φ

Proposition 6.19. The maps φ A and φ B are morphisms of A ∞ -categories.

Proof. The part of the identity (MI) for φ A that takes place in

is clearly satisfied. Moreover, by definition of the A ∞ -structure sm A⊕B * , the part of the identity (MI) that takes place in

where the sums are over all the filled diagrams D 1 , D 2 and D 3 of type x of the form

and respectively. Using that Φ is a pre-Calabi-Yau morphism, we thus have that φ A is an A ∞ -morphism. The case of φ B is similar.

We now summarize the results of Propositions 6.16 and 6.19.