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On the functors associated with beaded open Jacobi diagrams

Morphisms in the linear category A of Jacobi diagrams in handlebodies give rise to interesting contravariant functors on the category gr of finitely-generated free groups, encoding part of the composition structure of the category A. These functors correspond, via an equivalence of categories given by Powell, to functors given by beaded open Jacobi diagrams. We study the polynomiality of these functors and whether they are outer functors. These results are inspired by and generalize previous results obtained by Katada.


Introduction
In [HM21], Habiro and Massuyeau extend the Kontsevich integral to construct a functor from the category of bottom tangles in handlebodies to the linear category A of Jacobi diagrams in handlebodies. This category has N as objects and for n, m ∈ N, a generator of the vector space A(n, m) can be represented by a Jacobi diagram whose edges are oriented, such that each univalent vertex is embedded into the interior of the 1-manifold X m , consisting of m arcs and where we have beads coloured with elements of the free group of rank n on edges. For example, for F 3 = x 1 , x 2 , x 3 , the following is a non-zero element of A(3, 2): The definition of the composition in the linear category A is natural from the geometric point of view. However, it is quite complicated to understand it algebraically and this paper sheds some light on this.
To study the composition in a category, we can fix an object n in the category and look at For n = 0, these functors have been studied by Katada in [Kata, Katb]. Katada shows that A d (0, −) is a polynomial functor of degree 2d which is an outer functor: i.e. for all m ∈ N, the inner automorphisms act trivially on A d (0, m). She also gives the complete structure of the functors A d (0, −) for d ∈ {1, 2, 3} and, for general d, a decomposition of the functor A d (0, −) into indecomposables.
The aim of this paper is to study, more generally, the functors A d (n, −) : gr op → K-Mod for d, n ∈ N.
Our first result shows that the polynomiality of the functors A d (0, −) is exceptional.
Using the gr op -graduation of A(n, m) introduced by Habiro and Massuyeau in [HM21], we obtain, in Proposition 6.1, a subfunctor A(n, −) 0 of A(n, −), satisfying A(0, −) 0 = A(0, −). The generators of A(n, m) 0 are those of A(n, m) which can be represented by a Jacobi diagram on X m without beads on X m (but there may be beads on the Jacobi diagram). For example, for F 3 = x 1 , x 2 , x 3 , the following represents a non-zero element of A(3, 2) 0 : The graduation by the degree d of the Jacobi diagrams defines a subfunctor A d (n, −) 0 of A(n, −) 0 . Considering the subspace of A d (n, m) 0 generated by the Jacobi diagrams having at least t trivalent vertices, we obtain subfunctors A t d (n, −) 0 of A d (n, −) 0 defining a filtration: ) 0 corresponding, for n = 0, to the filtration considered by Katada in [Kata].
These functors satisfy the following: Theorem 2 (Theorem 6.9). For n ∈ N and d ≥ 1, the functor A d (n, −) 0 : gr op → K-Mod is polynomial of degree 2d and the filtration (1.1) corresponds to the polynomial filtration.
However, contrary to the result of Katada, the functors A d (n, −) 0 are rarely outer functors: Theorem 3 (Theorem 6.13). For d, n ∈ N, the functor A d (n, −) 0 is an outer functor iff n = 0 or d = 0.
We obtain a description of the functor A 1 (n, −) 0 generalizing that of the functor A 1 (0, −) given in [Kata,Section 4]. Let a : gr → K-Mod be the abelianization functor, P 2 : gr → K-Mod the second Passi functor (see Section 3) and (−) # : F (gr; K) op → F (gr op ; K) the duality functor, we have: Proposition 4. (Proposition 6.12) For n ∈ N, we have a natural equivalence where the action of S 2 on K[F n ] is given by taking the inverse in F n : v → v −1 and the action of S 2 on P # 2 is given in Section 3. In particular, we have A 1 (0, −) ≃ S 2 • a # . In Section 6.7 we give another proof, based on [PV18], of [Katb,Theorem 10.1], giving a direct sum decomposition of the functor A d (0, −) in the category of functors on gr op .
One of the main ingredients of this paper is the use of the equivalence of categories given by Powell in [Powa]: where F ω (gr op ; K) is the category of analytic functors on gr op and F Lie is the category of Klinear functors from the linear PROP associated with the operad Lie to K-Mod (see Section 2.4 for further details). It turns out that the polynomial filtration of a functor is easier to understand in the category F Lie than in the category F ω (gr op ; K) (see [Powa] and Section 2.4). It is also easier to show that the action of inner automorphisms is trivial in the category F Lie (see [Powb] and Section 2.5). The proofs of the previous results are based on the computation of the functor α −1 (A d (n, −) 0 ). In Section 6.2, we introduce the K-vector space J Fn d (m) which is the quotient by the AS and the IHX relations, of the K-vector space generated by equivalence classes of open Jacobi diagrams D whose edges are oriented and labelled by The generators of J Fn d (m) are called F n -beaded open Jacobi diagrams. The correspondance between the AS relation and the antisymmetry relation for Lie algebras and the IHX relation and the Jacobi relation for Lie algebras implies that this defines a functor J Fn d in F Lie (see Proposition 6.6). We have the following: Theorem 5 (Theorem 6.7). For n, d ∈ N, we have an equivalence of functors in F Lie : The title of this paper reflects the fact that the functors J Fn d are much easier to study than the functors A d (n, −) 0 . A more in-depth study of the functors J Fn d will be given in another paper.
Acknowledgement: This work was inspired by Mai Katada's papers [Kata, Katb], where the natural appearance of outer polynomial functors intrigued the author of this paper and motivated her to take an interest in the category of Jacobi diagrams in handlebodies. The author is grateful to Mai Katada for communicating her papers. The author is also grateful to Geoffrey Powell for discussions on his papers [Powb, Powa] and helpful comments on the previous versions of the manuscript. Notation. Denote by: K a field of characteristic 0. K-Mod the category of K-vector spaces. For n ≥ 0:

Contents
• n = {1, . . . , n}; • S n is the symmetric group on n letters; • X n is the oriented 1-manifold consisting of n arc components; • F n = x 1 , . . . , x n is the free group of rank n. The trivial group is denoted by {1}.
For generalities on Jacobi diagrams we refer the reader to [CDM12, Chapter 5].

Functors on gr op
2.1. Generalities on gr. Let gr be the category of finitely-generated free groups. This category is essentially small, with skeleton given by N, where n ∈ N corresponds to the free group F n of rank n. For clarity, we will sometimes denote the object n by F n . The object 0 = F 0 = {1} is a null-object in gr. For n, m objects of gr, we denote by 0 : n → m the composition n → 0 → m. Explicitly, 0 is the homomorphism F m → F n sending each generator to 1. The category gr is a PROP for the symmetric strict monoidal structure given by the free product.
By Pirashvili's result [Pir02], the PROP gr is isomorphic to the free symmetric monoidal category generated by a commutative Hopf monoid. In other words, the morphisms of gr are generated by the permutations groups in gr(n, n) for n ∈ N and the following homomorphisms: (1) m 1 : 1 → 0 corresponding to F 1 → {1}; (2) m 2 : 1 → 2 corresponding to F 1 → F 2 sending the generator x of F 1 to x 1 x 2 ; (3) m 3 : 0 → 1 corresponding to {1} → F 1 ; (4) m 4 : 1 → 1 corresponding to F 1 → F 1 sending the generator x of F 1 to x −1 ; (5) m 5 : 2 → 1 corresponding to F 2 → F 1 sending x 1 to x and x 2 to x (this is the folding map).
2.2. Generalities on functors on gr and gr op . We denote by F (gr; K) (resp. F (gr op ; K)) the category of functors from gr (resp. gr op ) to K-Mod. These categories are abelian.
Composition with vector space duality functor (−) # : K-Mod → K-Mod op gives rise to a pair of adjoint functors, named the duality functors For k ∈ {1, . . . , n}, let i n k : F n−1 → F n be the homomorphism given by The n-th cross-effect of a functor N : gr op → K-Mod is a functor cr n (N ) : (gr op ) ×n → K-Mod. Its evaluation on F 1 in each variable cr n (N )(1, . . . , 1) is equal to the kernel of the natural homomorphism In the examples, it is easier to compute cross-effects using the following equivalent description using a cokernel instead of a kernel. For k ∈ {1, . . . , n}, let r n k : F n → F n−1 be the homomorphism given by For d ∈ N, a functor N : gr op → K-Mod is polynomial of degree at most d if cr d+1 (N )(1, . . . , 1) = 0. Let F d (gr op ; K) be the full subcategory of polynomial functors of degree at most d. The forgetful functor F d (gr op ; K) → F (gr op ; K) has a right adjoint denoted by p d . For N : gr op → K-Mod, the functor p d (N ) is the largest subfunctor of N polynomial of degree d. Hence, a functor N : gr op → K-Mod admits a natural filtration, called the polynomial filtration of N : A functor N : gr op → K-Mod is analytic if it is the colimit of its subfunctors p d (N ). Let For N : gr op → K-Mod there is a natural short exact sequence Fix a generator µ ∈ Lie(2). For n ∈ N and i ∈ {1, . . . , n}, let µ n+1 i ∈ CatLie(n + 1, n) be the morphism given by the set map s n+1 i : n + 1 → n defined by s n+1 i (j) = j for j < n + 1 and s n+1 i (n+ 1) = i and taking 1 ∈ Lie(1) for the fibres of cardinal one and µ ∈ Lie(2) for the fiber of cardinal 2. The K-linear category CatLie is generated (via linear combination and composition) by the morphisms µ n+1 i ∈ CatLie(n + 1, n) and CatLie(n, n) ≃ K(S n ) for n ∈ N. Note that a pointed version of CatLie with a shuffle condition on fibers intervenes in [HV15]. Let F Lie be the category of K-linear functors from CatLie to K-Mod. For n ∈ N, CatLie(n, −) : CatLie → K-Mod is a linear functor. By the enriched Yoneda lemma, for F : CatLie → K-Mod a K-linear functor, we have an isomorphism: We deduce that the functors CatLie(n, −), for n ≥ 0, are projective generators of F Lie .
For F ∈ F Lie and i ∈ N, since CatLie(m, n) = 0 for m < n, F admits a subfunctor F ≤i given by truncation, i.e.
if n > i It follows that a functor F ∈ F Lie admits a natural filtration: In [Powa], Powell gives an equivalence of categories between F Lie and F ω (gr op ; K). In particular, Powell constructs explicit exact functors: giving this equivalence. By [Powa,Corollary 8.9], for F ∈ F ω (gr op ; K) we have an isomorphism: The category CatLie is easier to understand than the category gr op . For example, we have CatLie(i, j) = 0 for i < j. It follows that it is easier to work with CatLie-modules than with functors on gr op . In particular, the polynomiality of functors in F ω (gr op ; K) has an easy interpretation in CatLie-modules: a functor F ∈ F ω (gr op ; K) is polynomial of degree equal to d iff α(F )(d) = 0 and α(F )(k) = 0 for k > d. Via the equivalence of categories, the polynomial filtration of a functor in F ω (gr op ; K) corresponds to the filtration given by the truncations of CatLie-modules. More precisely, for N : gr op → K-Mod and i ∈ N we have: To prove Theorem 6.7, we will need the following explicit description of the functor α given in [Powa,Theorem 9.17]. Let CatAss u be the linear PROP associated with unital associative algebras; CatAss u is the K-linear category such that Ob(CatAss u ) = N and where Fin is the category of finite sets. More explicitly, a generator of CatAss u (m, n) is represented by a set map f ∈ Fin(m, n) and an order of the elements of each fiber of f . We denote by (s n+1 i , i < n + 1) (resp. (s n+1 i , n + 1 < i)) the morphism in CatAss(n + 1, n) given by the set map s n+1 i : n + 1 → n and the order i < n + 1 (resp. n + 1 < i) on the fiber of cardinal 2. The morphism of operads Lie → Ass u induces a functor CatLie → CatAss u sending the morphism µ n+1 i ∈ CatLie(n + 1, n) to (s n+1 i , i < n + 1) − (s n+1 i , n + 1 < i) ∈ CatAss(n + 1, n) By [Powa, Proposition 9.13] the sets of morphisms in CatAss u define a functor: In [Powa, Lemma A.2], the functor CatAss u (i, −) : gr op → K-Mod, for i an object of (CatLie) op , is described explicitly on the generators of gr recalled in Section 2.1. By [Powa,Theorem 9.17], α = CatAss u ⊗ CatLie −. Let Σ be the category of finite sets and bijections and F (Σ; K) the category of functors from Σ to K-Mod. The obvious functor Σ → gr op induces a functor F (gr op ; K) → F (Σ; K). By [Powa,Remark 9.18], for an object F of F Lie and d an object of gr op , the functor in F (Σ; K) associated with CatAss u ⊗ CatLie F is given explicitly by: A K[S d ]-module M defines an object of F Lie which is 0 for n = d and M on d. Such CatLie-module will be said atomic and will be denoted by M [d].
For a K[S d ]-module M , by [Powa,Example 9.20], we have: This is in the image of the faithful embedding F ω (ab op ; K) ֒→ F ω (gr op ; K), where ab is the category of finitely-generated free abelian groups. The category of analytic functors F ω (ab op ; K) is semi-simple. More precisely we have an equivalence of categories: The functor α −1 extends this equivalence of categories in the sense that we have a commutative diagram: Remark 2.2. For N : gr op → K-Mod, by Proposition 2.1 we can consider the graded functor gr(N ) associated with the filtered functor N , obtained by considering the polynomial filtration.
is not, in general, the direct sum of atomic functors. This illustrates the fact that, considering the graded associated with a functor, we lose much of the structure.
2.5. Outer CatLie-modules. In [Powb], Powell gives a characterization of CatLie-modules corresponding to outer functors via the equivalence of categories given in the previous section. These CatLie-modules will be called outer CatLie-modules. We briefly recall Powell's result. Let τ : F Lie → F Lie be the shifting functor given by precomposition with − + 1 : CatLie → CatLie. Let µ : τ → Id be the natural transformation defined as follows: Lie be the full subcategory of F Lie of functors such that µ F = 0. By [Powb,Theorem 4.16], under the equivalence of categories F ω (gr op ; K) ≃ F Lie , the full subcategory F Out Lie be the functor given by F µ := coker(µ F ). By [Powb, Proposition 2.17], (−) µ is the left adjoint to the inclusion F µ Lie ֒→ F Lie and so corresponds to the functor Ω : F (gr op ; K) → F Out (gr op ; K) via the equivalence of categories F ω (gr op ; K) ≃ F Lie .

On the second Passi functor P 2
The contents of this Section will be used in the proof of Proposition 6.12. As the results of this Section are of independent interest, we chose to dedicate a separate Section to them. The reader can skip this section on first reading.
Let P 2 : gr → K-Mod be the functor defined by: P 2 (F n ) = IF n /(IF n ) 3 . The functor P 2 is called the second Passi functor (see [HPV15,Ves18,PV18]). It is the largest quotient of P 1 that is polynomial of degree 2.
The group S 2 acts on P 2 by the following way: by (2.1), the element [ . Since σ 2 = 1, σ defines an action of S 2 on P 1 . By composition with P 1 ։ P 2 we obtain where the last isomorphism is given by adjunction, so that S 2 acts on P 2 .
In the following lemma we give an explicit description of this action of S 2 on P 2 . Recall that a(G) ≃ IG/(IG) 2 , for G an object of gr.
Lemma 3.1. The natural transformation σ : P 2 → P 2 restricts to a natural transformation σ |a ⊗2 : a ⊗2 → a ⊗2 given by the place permutation. The induced natural transformation σ : a → a is given by σ(x) = −x for G an object of gr and x an element in a(G).
For that of σ, we have: The action of S 2 on P 2 corresponds to an action of S 2 on P # 2 . In order to describe the functor A 1 (n, −) 0 in Proposition 6.12, we need the following results on the second Passi functor.
Corollary 3.4. For n ∈ N, we have a natural equivalence: where the action of S 2 on K[F n ] is given by taking the inverse in F n : v → v −1 and the action of S 2 on P # 2 is described in Lemma 3.1.

4.
Habiro-Massuyeau's category 4.1. Definition. In [HM21, Section 4.1], Habiro and Massuyeau consider Jacobi diagrams on a 1-manifold coloured by elements of a group (see also [GL01,ST04]). In order to avoid the confusion with the fact that we will also consider Jacobi diagrams where the univalent vertices are "coloured" by a set, we replace the terminology used by Habiro and Massuyeau by beaded Jacobi diagrams (following, for example, [GR04]). For d ≥ 0, let X d be the oriented 1-manifold consisting of d arc components. Recall that a Jacobi diagram D on X d is a uni-trivalent graph such that each trivalent vertex is oriented, the set of univalent vertices is embedded into the interior of X d and each connected component of D contains at least one univalent vertex. When a Jacobi diagram D on X d is drawn in the plane, we draw the 1-manifold X d with solid lines, the Jacobi diagram part D with dashed lines and we assume counterclockwise orientation for the trivalent vertices of D.
For G a group, a G-beaded Jacobi diagram on X d is a Jacobi diagram D on X d whose graph edges are oriented and a G-valued function on a finite subset of (Int(X d ) ∪ D) \ Vert(D). This function labels the oriented edges of D and the arcs of X d , by elements in G. In figures, the labels are encoded by "beads" coloured with elements of G.
Two G-beaded Jacobi diagrams on X d are said to be equivalent if they are related by the following sequence of local moves (see [HM21, (4.1) and p.618]), where w, x ∈ G: For example, these two G-beaded Jacobi diagrams on X 2 , where w 1 , w 2 , w 3 ∈ G, are equivalent: In particular, each G-beaded Jacobi diagram on X d is equivalent to a G-beaded Jacobi diagram of the form: where w 1 , . . . , w d ∈ G and where we can have beads on the Jacobi diagram represented by the dashed part in the figure.
In [HM21, Section 4.2] Habiro and Massuyeau define the linear category A of Jacobi diagrams in handlebodies. This category has N as objects and for n, m ∈ N, A(n, m) is the vector space generated by the equivalence classes of F n -beaded Jacobi diagrams on X m modulo the STU relation. The composition in the category A is quite complicated and we refer the reader to [HM21, Section 4.2] for its definition.
By [HM21,Section 4.3], the linear category A admits a symmetric monoidal structure given on objects by the addition of integers. We denote this monoidal structure by ⊙.

4.2.
Two gradings and sub-(semi)-categories. In [HM21, Section 4.4] the authors define two gradings on the morphisms of A. The first one is a N-grading given by the degree of the Jacobi diagram: for m, n ∈ N, A(n, m) can be decomposed as a direct sum with respect to the degree d of the Jacobi diagrams This grading is compatible with the composition in the category A giving maps The second grading is a gr op -grading: the homotopy class of an (m, n)-Jacobi diagram D on X n is the homomorphism h(D) : F n → F m that maps each generator x j to the product of the beads along the j th -oriented component of X n . We have Note that the identity morphism in A(n, n) is in the homotopy class of the identity homomorphism F n → F n . This grading is compatible with the composition in the category A. (4.4) • : A(m, n ′ ) g × A(n, m) f → A(n, n ′ ) g•f for f ∈ gr op (n, m) = gr(m, n) and g ∈ gr op (m, n ′ ) = gr(n ′ , m).
Using these gradings we can consider the following subcategory and sub-semicategory of A. Recall that a semicategory is defined as a category without the condition on the existence of identity morphisms (see [Mit72,Section 4]).
Taking degree d = 0, by (4.2) we have maps Hence A has a subcategory, denoted by A 0 , such that Obj(A 0 ) = Obj(A) and the morphisms in This isomorphism comes from the fact that A 0 (n, m) is generated by F n -beaded empty Jacobi diagrams on X m . So we have only beads on the arcs of X m . Such a choice of beads corresponds to a homomorphism: F n → F m sending x i to the bead on the i-th arc of X m . Via the isomorphism h given in (4.5), the generators (m 1 , m 2 , m 3 , m 4 , m 5 ) of gr recalled in Section 2.1 correspond to the morphisms (η, µ, ε, S, ∆) given in [HM21, (5.28)].
Recall that 0 ∈ gr op (n, m) = gr(m, n) is the composition m → 0 → n in gr. By Section 4.1, a F n -beaded Jacobi diagram D on X m in the homotopy class of 0, is represented by a F n -beaded Jacobi diagram without beads on X m (but there may be beads on D).
By (4.4) we have maps We deduce that A has a sub-semicategory, denoted by A(−, −) 0 , such that Obj(A(−, −) 0 ) = Obj(A) and the morphisms in A(−, −) 0 are given by beaded Jacobi diagrams in the homotopy class of 0.

Projective generators on Habiro-Massuyeau's category
Let A-Mod be the category of K-linear functors from A to the category K-Mod. For n ∈ N, A(n, −) : A → K-Mod is a linear functor. By the enriched Yoneda lemma, for F : A → K-Mod a K-linear functor, we have an isomorphism: We deduce that the functors A(n, −), for n ≥ 0, are projective generators of A-Mod.
Note that, for d ∈ N, A d (n, −) do not define subfunctors of A(n, −) since the degree of Jacobi diagrams is not preserved by composition. However, for n ∈ N, by restriction to A 0 , we have linear functors A(n, −) : A 0 ≃Kgr op → K-Mod and so functors Then, by (4.2), the grading by the degree of the Jacobi diagrams defines subfunctors Remark 5.1. In [Kata, Katb], Katada studies the functor A(0, −) : gr op → K-Mod and, for d ∈ N, its subfunctors A d (0, −) denoted by A d in [Kata, Katb]. In [Kata], she proves that A d (0, −) is a polynomial functor of degree 2d which is an outer functor. She also gives a complete description of the functor A 1 (0, −) and the more complicated case of the functor A 2 (0, −). In [Katb, Theorem 10.1], she gives a direct decomposition of the functor A d (0, −) for d ≥ 1 (see also Proposition 6.16 for another proof) and obtains in [Katb,Proposition 10.2] that this is an indecomposable decomposition.
Remark 5.2. More generally, A(−, −) : A op × A → K-Mod is a linear functor and, by restriction, we have a functor A(−, −) : gr × gr op → K-Mod. The study of these bifunctors will be done elsewhere.
5.1. Generalities on the functors A(n, −) and A d (n, −). The first result of this section shows that the functors A(n, −) are connected to each other by injective natural transformations. Let ε ∈ A 0 (1, 0) = A(1, 0) be the morphism corresponding, via the isomorphism h of (4.5) to the morphism m 3 given in Section 2.1. For n ≥ 1 we have A(n, 0) ≃ K[ε ⊙n ] ≃ K and A(0, 0) = A 0 (0, 0) = K. So 0 is a terminal object in the K-linear category A. We deduce that, for n ≥ 1, the functors A(n, −) : gr op → K-Mod are not reduced. We denote by A(n, −) the reduced subfunctor of A(n, −). In particular we have: A(n, −) ≃ K ⊕ A(n, −). Note that 0 is far from being an initial object in A and that A(0, −) is reduced. The proof of this proposition is based on the following lemmas.
Lemma 5.6. For d ≥ 1 and n ≥ 1 the functor A d (n, −) : gr op → K-Mod is not polynomial.
By Section 2.3, cr k (A d (n, −))(1, . . . , 1) is the cokernel of the following homomorphism: is the F n -beaded Jacobi diagram D on X k obtained from D by inserting the F n -beaded arc: 1 between the (i − 1)-th and the i-th arc of D. So the following F n -beaded Jacobi diagram on X k is a non-zero element in cr k (A d (n, −))(1, . . . , 1): In [Kata], Katada considers the filtration

The functors A(n, −) 0 and beaded open Jacobi diagrams
For d ∈ N and n ≥ 1, by Proposition 5.4, A d (n, −) is not polynomial. In this section we introduce a subfunctor of A d (n, −), which is polynomial and which coincides, for n = 0, with A d (0, −). Proof. For g ∈ gr op (m, n ′ ) and h ∈ gr op (m, n ′ ) we have where 0 ∈ gr op (n, m) = gr(m, n) is the homomorphism F m → F n sending each generator to 1.
Remark 6.2. For n, m ∈ N, the generators of A(n, m) 0 are those of A(n, m) which can be represented by an F n -beaded Jacobi diagrams on X m , without beads on X m . Corollary 6.3. For d, n ∈ N, the precomposition with Id n ⊙ ε ∈ A 0 (n + 1, n) gives injective natural transformations Note that, for n = 0, we have Similarly to Proposition 5.7 we have the following result: Proposition 6.4. For d, n ∈ N, the functors A(n, −) 0 and A d (n, −) 0 have a filtration given by the subfunctors: Remark 6.5. We have A 0 (n, −) 0 = K and A 0 (n, −) ≃ P n ≃ A 0 (n, −) 0 ⊕ P n . For G a group, a G-beaded open Jacobi diagram is an open Jacobi diagram whose graph edges are oriented and a map from a finite subset of D \ V ert(D) to G which labels oriented edges of D by elements in G. In figures, the labels are encoded by "beads" coloured with elements of G.
Two G-beaded open Jacobi diagrams are said to be equivalent if they are related by the following local moves where w, x ∈ G: For G a group and Z a set, J G (Z) is the quotient by the AS and the IHX relations, of the K-vector space generated by equivalence classes of Z-labelled, G-beaded, open Jacobi diagrams. Let J G d (Z) be the subspace of J G (Z) generated by the Jacobi diagram having 2d vertices. A generator in CatLie(n, m) can be viewed as a m + n-labelled, F 0 = {1}-beaded open Jacobi diagram. In this case the orientation of the edges can be taken arbitrarily (by the last relation given in (6.1)).
Proposition 6.6. For d ∈ N, n → J Fm d (n) has the structure of a K-linear functor on CatLie Proof. By the description of the category CatLie given in Section 2.4, it is sufficient to define J Fm d on the generators σ ∈ CatLie(n, n) ≃ K[S n ] and µ n i ∈ CatLie(n, n − 1). Let D be a generator in J Fm d (n), that is D is represented by a n-labelled, F m -beaded, open Jacobi diagram.
The action of CatLie(n, n) ≃ K[S n ] on D is given by the permutation of the labels of univalent vertices.
To define J which is compatible with the action of the symmetric group S l . By Section 2.1, the PROP gr is generated by the permutations and the homomorphisms m i for i ∈ {1, 2, 3, 4, 5}. To prove that the linear maps f l define a natural transformation of functors on gr op , it is sufficient to prove the naturality for these five homomorphisms. Using the explicit description of the functor CatAss u (c, −) : gr op → K-Mod, for c ∈ N, given in [Powa,Lemma A.2] and the definition of the composition in the category A given in [HM21], we obtain that the maps f l define a natural transformation: For example, for m 5 : F 2 → F 1 , the induced map CatAss u (c, 1) → CatAss u (c, 2) sends a set map f : c → 1 with an order of c to the sum of all the maps c → 2 obtained by shuffles and the map A d (n, −) 0 (1) → A d (n, −) 0 (2) sends a Jacobi diagram on X 1 to the sum of the Jacobi diagrams on X 1 and X 2 obtained by a shuffle of the univalent vertices. This corresponds in A to the box notation used to define the composition.
Since CatLie is generated by the morphisms µ c+1 i ∈ CatLie(c+1, c), to prove that f •L = f •R it is sufficient to prove this relation on these generators. Let [ε] be a generator in CatAss u (c, l) represented by a set map ε : c → l and a given order on each of its fiber. We denote by E i the ordered fiber of ε(i) by ε: explicitly we have: We consider the following ordered sets We have: where [(ε • s c+1 i , E i<m+1 )] is the generator in CatAss u (c, l) represented by the set map ε • s c+1 i : c + 1 → l and the order on the fibers over j, is the same that for ε for j = i and is E i<m+1 for j = i. The generator [(ε • s c+1 i , E m+1<i )] is defined similarly. We have: By the STU relation we obtain that: and we deduce that f l define a natural transformation By the isomorphism given in (2.4), we obtain that this natural transformation is a natural equivalence. By (2.3), Theorem 6.7 and Corollary 6.8 we have: Remark 6.10. The polynomiality of the functors A d (n, −) 0 can be proved without using the equivalence of categories F ω (gr op ; K) ≃ F Lie , using similar arguments as in the proof of Lemma 5.6. As this seems instructive to us, we give this alternative proof below. We want to prove that cr 2d+1 (A d (n, −) 0 ) = 0. By Section 2.3, cr 2d+1 (A d (n, −) 0 )(1, . . . , 1) is the cokernel of the following homomorphism: A generator of A d (n, −) 0 (F 2d+1 ) is represented by a F n -beaded Jacobi diagram D on X 2d+1 having 2d vertices and without bead on X 2d+1 . Since the Jacobi diagram has 2d univalent vertices, at least one of the (2d + 1)-arc components of X 2d+1 has no univalent vertex. Assume that this is the case for the k-th arc of X 2d+1 . Denote by Dk the generator of A d (n, −) 0 (F 2d ) obtained from D by forgetting the k-th arc of X 2d+1 , then: We deduce that the cokernel of the previous map is zero.
, as a special case we obtain the following result, originally due to Katada. 6.5. On the functors A 1 (n, −) 0 . Recall that, in Section 3, we define the functor P 2 and study the action of S 2 on it.
Proposition 6.12. For n ∈ N, we have a natural equivalence where the action of S 2 on K[F n ] is given by taking the inverse in F n : v → v −1 and the action of S 2 on P # 2 is given in Section 3. In particular, we have A 1 (0, −) ≃ S 2 • a # . The second part of the statement corresponds to a result of Katada given in [Kata,Section 4].
Since ρ 1 and ρ 2 are isomorphisms, ρ is a natural equivalence.  Theorem 6.13. For d, n ∈ N, the functor A d (n, −) 0 is an outer functor iff n = 0 or d = 0.
By Theorem 6.7 and Section 2.5, A d (n, −) 0 is an outer functor iff J Fn d belongs to F µ Lie . The proof of this theorem relies on the following result.
• For k < 2d − 1. By functoriality of J d on CatLie, we have morphisms We will prove that these maps are surjective. Let D be a generator of J d (k + 1); since k + 1 < 2d, D has at least one connected component which is not of the form . We can chose one of these connected components having the form (which could be non-connected). By iteration we obtain that the morphism CatLie(2d, k + 1) ⊗ J d (2d) → J d (k + 1) is surjective. By the naturality of µ J d and the fact that (µ J d ) 2d−1 = 0, we deduce that (µ J d ) k = 0.
Proof of Theorem 6.13. For d = 0, by (6.5) A 0 (n, −) 0 ≃ K which is obviously an outer functor. For n = 0, A d (0, −) 0 is an outer functor by Proposition 6.14. If n = 0 and d ≥ 1, we prove that J Fn Since n ≥ 1, for w 1 = 1 ∈ F n , (µ J     where for µ 2d 1 (D) we use the AS relation and for µ 2d d+1 (D) we use the relation in F n -beaded Jacobi diagrams.
By Corollary 6.8, α −1 (A i d (0, −)) ≃ (J F0 d ) ≤2d−i , so α −1 A i d (0, −)/A i+1 d (0, −) is the atomic functor concentrated in 2d−i, where it is equal to the vector space D 2d−i which is the quotient by AS and IHX relations of the K-vector space generated by the (2d − i)-labelled Jacobi diagrams of degree d. The symmetric group S 2d−i acts on D 2d−i by the permutation of the labels of univalent vertices.
By (2.5) we obtain: corresponding to the description of the functor B d,i given by Katada in [Kata,(3)].
The decomposition of B d,0 given by Katada in [Katb,Proposition 7.7] is functorial. In other words, denoting by S λ the Schur functor associated with the partition λ ⊢ d, we have, for d ≥ 0: where, for λ = (λ 1 , . . . , λ l ) ⊢ d, 2λ is the partition (2λ 1 , . . . , 2λ l ) ⊢ 2d. By [Ves18, Theorem 4.2], Ext 1 F (gr;K) (F, S 2d • a) = 0, for F a polynomial functor so S 2d • a is an injective object in the category of polynomial functors on gr, so S 2d • a # is a projective object in the category of polynomial functors on gr op . This allows us to give another proof of [Katb,Theorem 10.1].