Protoperads II: Koszul duality

In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of bialgebras with diagonal symmetries, like double Lie algebras (DLie). We give a criterion to show that a binary quadratic protoperad is Koszul and we apply it successfully to the protoperad DLie. As a corollary, we deduce that the properad DPois which encodes double Poisson algebras is Koszul. This allows us to describe the homotopy properties of double Poisson algebras which play a key role in non commutative geometry.


Introduction
This paper develops the Koszul duality theory for protoperads, defined in [Ler19], which are an analog of properads (see [Val03,Val07]) with more symmetries. The main application of this theory is the proof of the Koszulness of the properad which encodes double Lie algebras, from which it follows that the properad encoding double Poisson algebras is Koszul.
The motivation for this work is to determine what is a double Poisson bracket up to homotopy. A double Poisson structure, as defined by Van den Bergh in [Van08a], gives a Poisson structure in noncommutative algebraic geometry (see [Gin05,Van08b]) under the Kontsevich-Rosenberg principle, i.e., if A is a double Poisson algebra, then the associated affine representation schemes Rep n (A) have (classical) Poisson structures.
In order to determine the homotopical properties of a family of algebras, we use the classical strategy, which was already used to understand, for example, the homotopical properties of Gerstenhaber algebras (and also the homotopic properties of associative, commutative, Lie, Poisson, etc, algebras). The idea is to go to the upper level and understand the homological properties of the algebraic object that encodes the structure, such as the operad Gerst for Gerstenhaber algebras. In the good case where the operad (or the properad) satisfies good properties, we can use Koszul duality in order to have a minimal cofibrant replacement of our operad. We can then go down to the level of algebras. Thanks to this cofibrant replacement (in the case of Gerstenhaber algebras, the operad G ∞ ), we obtain the associated notion of algebra up to homotopy: for example, Gerstenhaber algebras up to homotopy are encoded by G ∞ (see [Gin04] or [GCTV12, §2.1]). This structure has a good homotopical behaviour at the algebras' level, the homotopy transfer theorem (see [LV12,§10.3] for algebras over an operad), etc.
Double Poisson structures are properadic in nature as they are made up of operations with multiple inputs and multiples outputs. They are encoded by the properad DPois, which is constructed with the properads As and DLie (see Lemma 5.9), where the properad DLie encodes double Lie structure and the properad As encodes associative algebra structure. The properad DLie is a quadratic properad defined by generators and relations, with the generator V DLie concentrated in arity (2, 2): Thus double Lie bracket on a chain complex A is given by a morphism of properads DLie → End A where End A is the properad of endomorphisms of A (see [Val07] for the definition).
The theory of properads is the good general algebraic framework to encode operations with several inputs and outputs. In certain cases, this framework can be simplified. For example, algebraic structures with several inputs and one output, like associative, commutative or Lie algebras, are encoded by operads (see [LV12]). In a certain sense, the operadic framework is the minimal one to study such structures. In this smaller framework, homotopical properties are much easier to study.
Similarly, protoperads form a special class of properads, which provide the appropriate framework for studying the double Lie properads. In the first article [Ler19], we have developed this minimal framework, such that there exists a protoperad DLie which encodes the double Lie structure. In [Ler19], we proved the existence of the free protoperad functor and gave an explicit combinatorial description of this, in terms of bricks and walls. An important property of protoperads is their compatibility with properads via the induction functor (see Definition/Proposition 1.16).
In this paper, we develop the homological algebra for protoperads. With the monoidal exact functor of induction, we prove the existence of a bar-cobar adjunction in the case of protoperads: Ω : coprotoperads coaug k protoperads aug k : B. We obtain also the following theorem, the protoperadic analogue of the criterion of Koszul of the properads [Val03,Th. 149], [Val07].
Theorem (see Theorem 2.25). -Let P be a connected weight-graded protoperad. The following are equivalent: (1) the inclusion P ¡ → BP is a quasi-isomorphism, i.e., the protoperad P is Koszul; (2) the morphism of protoperads ΩP ¡ → P is a quasi-isomorphism, where P ¡ is the Koszul dual of P (see Theorem 2.29) We give a useful criterion to show that a binary quadratic protoperad (i.e., a quadratic protoperad generated by a S-module concentrated in arity 2) is Koszul. Take a binary quadratic protoperad P given by generators and relations. We associate to P a family of associative algebras A (P, n), for n 2. The algebra A (P, n) is constructed so that its bar construction splits and such that one of these factors is the n-th arity of the normalized simplicial bar construction of the protoperad P.
Theorem (see Theorem 4.3). -Let P be a binary quadratic protoperad. If, for all integers n 2, the quadratic algebra A (P, n) is Koszul, then the protoperad P is Koszul.
This is a useful criterion because the study of the Koszulness of algebras is easier than for pro(to)perads. Many tools are available, such as PBW or Gröbner bases, or rewriting methods (see [LV12,Chap. 4

]).
We use this criterion to show that the protoperad DLie is Koszul. As the functor of induction is exact and preserves the weight, so preserves the Koszulness, the properad DLie, which encodes double Lie algebras, is also Koszul. This theorem is very important: it is the first example of a Koszul properad with a generator not in arity (1, 2) or (2, 1). And so, with an argument of distributive law, we deduce the main theorem of this paper.
In an future article, we will explain the homotopy transfer theorem for properadic algebras and we will use this in an other future work, where we will study the implications of Theorem 5.11 in derived noncommutative algebraic geometry à la Berest et al. (see [BCER12,BFP + 17, BFR14,CEEY17]). In particular, we will link it to pre-Calabi Yau structures as in [Yeu18,IK18]. We will also look at the cohomological theory of double Poison algebras. Indeed, the work of Merkulov and Vallette gives the notion of deformation theory of P-algebras, for P a properad. We want to link the deformation complex defined in [MV09b] with the work of Pichereau et al. who defined the cohomology of differential double Poisson algebra (see [PVdW08]).
Organization of the paper. -After a review of definitions and some properties of protoperads (see [Ler19]) in Section 1, following the results on properads (see [Val03,Val07,MV09a]), we introduce the notion of shuffle protoperads in Section 1.3. In Section 2, we define the Koszul duality of protoperads. We transpose a part of the results on properads obtained by Vallette in [Val03,Val07] to the protoperatic framework thanks to the exactness of the induction functor Ind (see [Ler19,Prop. 4.4.]). In Section 3, we define the simplicial bar construction and the normalized one for protoperads and we described the levelisation morphism (see Definition/Proposition 3.6). In Section 4, we give a criterion to prove that a binary quadratic protoperad is Koszul and we use it to prove that the protoperad DLie is Koszul. Finally, in Section 5, we use results of Vallette on distributive laws to prove that the properad DPois is Koszul.
Notations. -We write N * for the set N {0}. In all this paper, k is a field with characteristic 0. We denote by Fin, the category with finite sets as objects and bijections as morphisms and Set, the category of all sets and all maps. For two integers a and b, we note by [[a, b]] the set [a, b] ∩ Z, and, for n ∈ N * , S n is the automorphism group of [[1, n]], i.e., S n = Aut Fin ([[1, n]]). We denote by Ch k the category of Z-graded chain complexes over the field k.
The goal of this paper is to study the Koszulness of the properad DPois, using the protoperad DLie. These two objects are not to be confused with D(Pois) and D(Lie) where D is the polydifferential functor on the category of props introduce by Merkulov and Willwacher.

Recollections on pro(to)perads
We briefly recall the definition of protoperads and some results of [Ler19]. We denote by S-mod red k , the category of contravariant functors from Fin to the category of chain complexes Ch k such that P (∅) = 0.
1.1. Combinatorial functors. -We recall two important functorial combinatorial constructions which are described in [Ler19,§1]: the functors W conn and X conn .
Notation. -For a poset (P, P ), we denote by Succ(P ), the set of pairs (r, s) ∈ P ×P such that r < P s and there does not exist t ∈ P such that r < P t < P s.
Definition 1.1 (The functor of walls). -The functor W n : Fin op → Fin op is defined, for every finite sets S, as follows. An element W of W n (S) is a collection {W α } α∈A of non-empty subsets of S, indexed by some finite set A of cardinality n, such that (1) the union of these subsets is S, i.e., α∈A W α = S; (2) for every s ∈ S, the set Γ(W, s) := {W α | s ∈ W α } is totally ordered by s ; (3) these orders are compatible on the intersection of the sets Γ: for every s and t in S, and W a , in W n is induced by the right action on S, i.e., where W α · σ := σ −1 (W α ) and the order σ is induced by the total orders of Γ W ·σ s·σ := {W α · σ|s · σ ∈ W α · σ}. Using the collection of functors W n , one define the functor of walls by with the partial order W a < W c . We represent this wall by ∼ on W as follows. For two elements a and b of A, we say W a conn.
∼ W b if there exist an integer n 2 and a sequence W 0 , W 1 , . . . , W n−1 , W n of elements of W with W 0 = W a and W n = W b such that, for all i in [[0, n − 1]], J.É.P. -M., 2020, tome 7 Definition 1.3 (Projection K). -We define the natural projection K as follows: for a finite set S, we have where π is the projection of W to its quotient by conn.
∼ . Definition We also define the functor An element W of W conn (S) is called a connected wall over S, and an element of a wall W is called a brick of W .
Non-example 1.6. -The wall of Example 1.2 is not connected.
Hence, we have other important subfunctors of W conn .
-The functor Y : Fin op → Set op is defined, for every finite set S, by An element of Y(S) is a non-ordered partition of S, i.e., a wall composed with one row of bricks over S.
-The functor X = Y × Y is defined, for every finite set S, by An element K of X(S) is an ordered pair of unordered partitions of the finite set S, so we also denote by (I, J) such a K. Graphically, an element of X has the form -The functor X conn is the subfunctor of X of connected walls, defined by The functor X conn encodes a new monoidal structure on the category of reduced S-modules, the connected composition product, as we will see in Definition/Proposition 1.12. This bi-additive bifunctor gives S-mod red a symmetric monoidal structure, with identity I , defined, for all non empty sets S, by I (S) := k concentrated in degree 0.
Definition/Proposition 1.8 (Concatenation product). -The concatenation product is the bifunctor defined, for all finite sets S and all reduced S-modules P and Q, by: This product is symmetric monoidal without unit (since we are working with reduced S-modules).
Definition/Proposition 1.9 (see [Ler19, §2.2.1]). -We denote by S, the functor which sends a reduced S-module V to the free symmetric monoid without unit S(V ) for the concatenation product. Moreover, it is isomorphic to Remark 1.10. -We can extend the concatenation product: (1) − ⊗ conc − : S-mod k × S-mod red k −→ S-mod red k . This extension is induced by the equivalence of categories S-mod k ∼ = Ch k × S-mod red k , by the injection (−) S : Ch k → S-mod k defined, for all chain complexes C and all finite sets S, by (C) S (∅) = C and (C) S (S) = 0 when |S| > 0, and by the action of the category Ch k on S-mod red k defined, for all chain complexes C and all finite sets S, by This extension allows us to define the suspension of a S-module. -Let Σ (respectively Σ −1 ) be the chain complex k concentrated in degree 1 (resp. in degree −1). For V a reduced S-module, the suspension of V (resp. desuspension of V ) is the reduced S-module ΣV not.
Definition/Proposition 1.12 (Connected composition product of S-modules (see The connected composition product of reduced S-modules is the bifunctor defined, for all reduced S-modules P , Q and for all non empty finite sets S, by: for all σ in S r , τ in S s with (−1) |σ(p)| , (−1) |τ (q)| , the Koszul signs induced by permutations. We also denote by I , the S-module given by which is the unit of the product c . The category (S-mod red k , c , I ) is a (symmetric) monoidal category. The monoids for this product are called protoperads.
We have a compatibility between these monoidal structures.

.19]))
Let P and Q be two reduced S-modules. There is a natural isomorphism of S-modules: In particular, for a protoperad P, the S-module SP is a monoid for the product .
We have a notion of free protoperad. The combinatorics of the free protoperad is described by the functor of connected walls W conn .  The notion of protoperad is compatible with the notion of properad, defined by Vallette in [Val03,Val07] and [MV09a], via the induction functor.
Definition/Proposition 1.15 (Properad -Free properad (see [Val03,Val07])) The category of reduced S-bimodules, i.e., the category of functors P : Fin × Fin op −→ Ch k such that, for all finite set S, P (S, ∅) = 0 = P (∅, S), is monoidal for the connected composition product denoted by Val c . The monoids for this product are called properads. We have the free properad functor, which is denoted by F Val which is the left adjoint to the forgetful functor: We define a monoidal adjunction between these categories.

.1]))
We define the induction functor Ind : S-mod red k → S-bimod red k which is given, for all reduced S-modules V and, for all finite sets S and E, by: This functor is exact, has a right adjoint which is the functor of restriction Res, and is monoidal. Hence, that induces the functor Ind : protoperads −→ properads.
Moreover, the induction functor commutes with the free monoid constructions, formally by adjunction, i.e., we have the natural isomorphism of reduced S-bimodules: Then, for a protoperad P defined by generators and relations, i.e., P = F (V )/ R , the properad Ind(P) is given by The functor of induction also commutes with the cofree conilpotent comonoids constructions: because the underlying functor of F c (−) (resp. F c,Val ) is the same of the one of F (−) (resp. F Val ), and Ind respects the coproduct, because it is monoidal.
Remark 1.17. -In the rest of this paper, we use the same name for the induction functor from S-mod red k to S-bimod red k , and the one from protoperads to properads.
1.3. Shuffle protoperad. -Here, we introduce shuffle protoperads. This notion is very similar to the notion of protoperad, without the actions of the symmetric groups; these notions are related by the functor defines in Proposition 1.24. In the operadic framework, shuffle operads are very useful to study Kozsulness. That permits to define the notion of PBW-basis or Gröbner basis (see [Hof10,DK10] [Ler17], the proof of a similar PBW theorem does not hold directly in the protoperadic case. In fact, the analogous of the spectral sequence appearing in [Hof10,§4] does not collapse in page 1, because the underlying combinatorial of protoperads is more complicated. Nevertheless, the notion of shuffle protoperad still holds some appeal, as we will see (see Remark 3.8).
We denote Ord, the category of totally ordered finite sets, with bijections. Analogously to [Ler19, §1.2], we define the combinatorial functors, Y sh : Ord op → Set op and X sh : Set op → Fin op which encode shuffle protoperads. The shuffle framework corresponds to choosing a representative for each wall. We define the functor Y sh as follows: for all finite, totally ordered sets S, we set We have the natural isomorphism of functors X sh ∼ = Y sh × Y sh . As in the unshuffle case (see Definition/Proposition 1.9), using the functor Y sh , one can define the following functor.
Definition 1.18 (The functor S sh ). -Let V : Ord op → Ch k be a functor. We denote by S sh (V ), the functor S sh (V ) : Ord op → Ch k , given, for a totally ordered set S, by The functors Y sh and X sh are compatible with their unshuffled analoguous through the forgetful functor (−) : Ord → Fin. We will define the shuffle-analoguous of the functor X conn .
Definition 1.20 (Projection K sh ). -We define the projection K sh as follows: for a totally ordered finite set (S, < S ), we have where π is the projection of W to its quotient by conn.
∼ (cf. [Ler19, §1.4]), and the set K sh S (W ) is totally ordered by the order < defined as follows: for B α and B β in As for K (cf. [Ler19, Lem. 1.10]), the product K sh on Y sh is associative. Let M and S be two totally ordered finite sets. Every monotone injection j : Let M, N and S be three totally ordered finite sets and ϕ be the diagram of monotone injections ϕ : given by the union of the images by i and j of the partitions of M and N , extended by singletons, i.e., defined by the following composition We have the following commutative diagram: Finally, we define the functor X conn,sh : Ord op → Fin op , for all totally ordered finite sets S, by X conn,sh (S) : for two objects P and Q of Func(Ord op , Ch k ) and a finite totally ordered S, by Proposition 1.22. -The product ¡ c is associative. Also, for all objects A and B in the category Func(Ord op , Ch k ), the endofuncteur is an abelian (symmetric) monoidal category and the monoidal product preserves reflexive coequalizors and sequential colimits.
Proof. -Let S be a totally ordered finite set and P and Q be two reduced S-modules.
We have the following isomorphisms: As for the case of protoperads, we have a combinatorial description of the free shuffle protoperad.
The functor F ¡ is the left adjoint to the forgetful functor protoperads sh k : For.
By Proposition 1.24, we have the following.
Moreover, if P is a weight graded protoperad, then so is the shuffle protoperad P sh .

Koszul duality of protoperads
In this section, we adapt the constructions of [MV09a,§3] and [Val03,Val07] for properads to the protoperadic framework. -An augmentation of a protoperad P is a morphism of protoperads ε : P → I , where I is the unit of the product c . A protoperad with an augmentation is called augmented. We denote by protoperads aug k , the category of augmented protoperads. To an augmented protoperad (P, ε), we associate its augmentation ideal P, defined as the kernel of the augmentation ε, i.e., P := Ker(ε).
For two reduces S-modules M and P , the S-module P c (P ⊕ M ) has a weightgrading, which we denote Let (P, ε) be an augmented protoperad. Then, we have the isomorphism of reduced S-modules P ∼ = I ⊕ P. Moreover, by the bigrading given by [Ler19, Lem. 5.16], we can decompose the connected composition product Definition 2.2 (Partial composition product). -Let (P, ε) be an augmented protoperad. The partial composition product is the restriction of the product µ : P c P → P to Using the partial composition, we introduce the notion of an infinitesimal bimodule over a protoperad.
has two morphisms of S-modules, respectively called the left and right actions: such that the following compatibility diagrams commute: (1) associativity of the left action λ: (2) associativity of the right action ρ: (3) the left and right actions commute: Remark 2.4. -We also have the dual definitions of co-augmented coprotoperad, partial coproduct and infinitesimal cobimodule.(see [MV09a] for properadic definition).
compatible with the product µ of P, i.e., the following diagram commutes: This compatibility and associativity of the product µ allows the extension of the morphism λ to a morphism λ : We have a similar equivalence for ρ and ρ .
For the definition of infinitesimal bimodule in the properadic case, which is similar to the protoperadic case, the reader can refer to [MV09a].
Lemma 2.6. -Let P be a protoperad and M be an infinitesimal P-bimodule. The S-bimodule Ind(M ) is an infinitesimal Ind(P)-bimodule.
Proof. -The functor Ind is monoidal for the products c and ⊗ conc (see [Ler19,Prop. 4.7, Th. 4.16]) and is additive, i.e., Ind(V ⊕W ) ∼ = Ind(V )⊕Ind(W ), so preserves the weight grading: Definition 2.7 (Derivation, coderivation). -Let (P, ε) be an augmented protoperad and (M, λ, ρ) be an infinitesimal P-bimodule. A morphism of S-modules d : P → M of homological degree n is called a homogeneous derivation if the following diagram commutes: , for all p and q in P: We denote Der n (P, M ), the k-module of derivations from P to M of homological degree n and the derivation complex is denoted by Der • (P, M ), with the differential Let (C, ν) be a coaugmented coprotoperad and (N, λ, ρ), an infinitesimal C-cobimodule. A morphism of S-modules d : N → C of homological degree n is a homogeneous coderivation if the following diagram commutes: We denote Coder n (C, N ), the k-module of homogeneous coderivations from C to N of degree n and Coder • (C, N ), the coderivation complex.
By Lemma 2.10, we can associate to sµ 2 , a homogeneous coderivation of homological degree −1. We consider the coderivation with ∂ P the coderivation induced by the internal differential of P. We show that ∂ 2 = 0, which is equivalent to showing that ∂ P d sµ2 + d sµ2 ∂ P + d 2 sµ2 = 0, because ∂ P is a differential. By Proposition 2.8, Ind(d sµ2 ) is a coderivation of homological degree −1 (in the properadic sense). As the functor Ind commutes with the cofree conilpotent comonoid functor F c (−), with the suspension and Ind is exact (so commutes with the functor (−)), we have the following isomorphism Ind F c (ΣP), ∂ P + d sµ2 ∼ = F c (ΣInd(P)), ∂ Ind(P) + Ind(d sµ2 ) .
As the coderivation d sµ2 is the suspension of the partial product µ (1,1) , and the functor Ind is compatible with the weight-bigrading in P of P c P and commutes with the suspension, then Ind(d sµ P 2 ) is equal to d sµ Ind(P) 2 , the coderivation induced by the partial product of the properad Ind(P).
This lends to the definition of the bar construction of a protoperad.
Definition/Proposition 2.13 (Bar construction). -Let (P, µ, ∂ P , ε) be an augmented protoperad. The bar construction of P is the following quasi-cofree coaugmented coprotoperad: which gives the functor B : protoperads aug k → coprotoperads coaug k . Moreover, the respective bar constructions commute with the induction functor: where the functor B Val is the bar construction for properads defined in [Val03,Val07].
Proposition 2.14. -Let V be a reduced S-module concentrated in homological degree 0. Then the homology of the chain complex given by the bar construction of the free protoperad over V is acyclic, i.e., where Σ is the shifting of homological degree one.
Proof. -For this proof, we use the notion of coloring of a wall W and the coloring complex associated to W , defined in [Ler19,§6]. Let S be a totally ordered finite set. We have the following isomorphisms of chain complexes: where the differential on the right side acts on the coloring as in the coloring complex. So we have: then, by [Ler19, Th. 6.15], BF (V ) ΣV .
We also have the cobar construction.
-Let (C, ∆, ∂ C , ν) be a coaugmented coprotoperad. The cobar construction of C is the following quasi-free augmented protoperad: which gives the functor Ω : coprotoperads coaug k → protoperads aug k . Moreover, the respective cobar constructions commute with the induction functor: where the functor Ω Val is the cobar construction for properads defined in [Val03,Val07].
By the exactness of the functor Ind, we directly have the adjunction between bar and cobar construction. as the results are very similar, we try to use the same notation as in [Val03].
2.3.1. Definition of the Koszul dual. -Let (P, µ, ε) be an augmented protoperad, with a weight grading, P = n∈N P [n] . This grading induced a new one on the bar construction of P: where B (r) P = F (r) (ΣP) is the grading described in [Ler19, Th. 5.21]. We interpret r as the number of elements of P and ρ as the total weight induced by the weight of each element of P. As the product µ of P respects the weight grading, d sµ2 respects the induced grading on BP; so we have Thus we have the following lemma.
Definition 2.18 (Koszul dual). -Let P (respectively C) be a weight-graded, connected protoperad (resp. coprotoperad). We define the Koszul dual of P (resp. of C), denoted by P ¡ (resp. C ¡ ) by the weight-graded S-module: Remark 2.19. -One can remark that a weight-graded connected pro(to)perad P is augmented, the augmentation given by the projection on the weight 0 which is isomorphic to I . Similarly, a weight-graded connected copro(to)perad C is coaugmented.
By Lemma 2.17, we have the equalities: Moreover, if the protoperad P is concentrated in homological degree 0, then we have The dual coprotoperad P ¡ is not concentrated in degree 0, but satisfies: Proposition 2.21. -Let P = n P [n] (resp. C = n C [n] ) be a weight-graded, connected protoperad (resp. coprotoperad). Then the Koszul dual of P is a sub weight-graded, connected, coaugmented coprotoperad of F c (ΣP [1] ) (respectively, the Koszul dual of C is a connected, weight-graded, augmented protoperad quotient of F (Σ −1 C [1] )).

Koszul resolution
Definition 2.22 (Koszul protoperad, coprotoperad). -Let P and C be respectively a protoperad and a coprotoperad, each weight-graded and connected. The protoperad P is Koszul if the inclusion P ¡ → BP is a quasi-isomorphism. Dually, the coprotoperad C is Koszul if the projection ΩC C ¡ is a quasi-isomorphism.
Proposition 2.23. -If P is a weight-graded, connected protoperad which is Koszul, then its dual P ¡ is a Koszul coprotoperad, and P ¡¡ = P. (1) the complex P ¡ c P, ∂ = ∂ P + d r ∆ , where the differential d r ∆ is induced by the homogeneous morphism of homological degree −1: where the right morphism is induced by the isomorphism (P ¡ ) [1] ∼ = P [1] ; (2) the complex P c P ¡ , ∂ = ∂ P + d l ∆ , where the differential d l ∆ is induced by the homogeneous morphism of degree −1: As in the properadic case, we have the following Koszul criterion: Theorem 2.25 (Koszul criterion). -Let P be a connected weight-graded protoperad.
The following are equivalent: (1) the inclusion P ¡ → BP is a quasi-isomorphism, i.e., the protoperad P is Koszul; (2) the Koszul complex P ¡ c P, ∂ = ∂ P + d r ∆ is acyclic; (3) the Koszul complex P c P ¡ , ∂ = ∂ P + d l ∆ is acyclic; (4) the morphism of protoperads ΩP ¡ → P is a quasi-isomorphism. 2.3.3. The case of quadratic protoperads. -This subsection is strongly inspired by [Val08, §2] which described the notion of a quadratic properad. We adapt the notion to the protoperadic framework. Let V be a S-module and R ⊂ F (2) (V ): such a pair (V, R) is called a quadratic datum.
As the underlying S-modules of the free protoperad F (V ) and the cofree coprotoperad F c (V ) are isomorphic, we consider the following morphisms of S-modules: where the isomorphism ( * ) is an isomorphism of S-modules. Using this, we naturally define a quotient protoperad of F (V ) or a sub-coprotoperad of F c (V ).
Definition 2.27 (Quadratic (co)protoperad). -The (homogeneous) quadratic protoperad generated by V and R is the quotient protoperad of F (V ) by the ideal generated by R ⊂ F (2) (V ). We denote this protoperad by P(V, R) := F (V )/ R . Dually, the (homogeneous) quadratic coprotoperad cogenerated by V and R is the subcoprotoperad of F c (V ) cogenerated by F c,(2) (V ) R. We denote this coprotoperad by C (V, R).

Simplicial bar construction for protoperads
We construct the simplicial bar complex for protoperads, as in the properadic case (see [Val07,§6]). Recall that Σ n denotes the homological suspension of degree n (see Definition 1.11). One can check that ∂ 2 C(P) = 0. This chain complex is called the (reduced) simplicial bar construction of P.
Definition 3.2 (Normalized bar construction). -The normalized bar construction is given by the quotient of the simplicial bar construction by the image of the degeneracy maps. We denote by N(P) the following graded S-module, given in grading n, by: where K S is the natural projection defined in Definition 1.3 (see also [Ler19, Def. 1.9]). We denote the label of the number of levels by n ↑, because W conn,lev n↑ (S) is also weightgraded by the number of bricks: an element (W 1 , . . . , W n ) lives in W conn,lev n↑,b (S) with b = |W 1 | + · · · + |W n |. The graded functor of level connected wall is denoted by Remark that the unlevelisation morphism projects the functor of n-leveled connected wall with n bricks to W conn n . We denote by π n↑ the restriction of the unlevelisation morphism to W conn,lev n↑,n π n↑ : W conn,lev n↑,n −→ −→ W conn n Proposition 3.3. -Let P be an augmented protoperad, and its augmentation ideal P, and S be a finite set. We have the following isomorphism  where the functors C Val and N Val are respectively, the reduced simplicial bar construction and the normalized simplicial bar construction for properads (see [Val07]).
Proof. -The functor Ind is monoidal and exact (see Definition/Proposition 1.16).
Proposition 3.5 (1) The simplicial bar construction and the normalized bar construction preserve quasi-isomorphisms.
(2) Let P be a quasi-free protoperad on a weight-graded S-module V , i.e., P has underlying S-module F (V ), such that V (0) = 0 and concentrated in homological degree 0. The natural projection N(P) → ΣV is a quasi-isomorphism.
the map e sends each element of α∈A ΣP(W α ) to the sum of representatives (with signs induced by the Koszul sign of the symmetry). We apply the functor Res to this map, which is an exact functor, and which satisfies Res•Ind = id, then the map e is a quasi-isomorphism. We just use the same arguments that for the properadic case (see [Val07,§6]).
Remark 3.8 (About shuffle protoperads). -All these constructions, the bar construction, the simplicial bar construction and the normalized simplicial bar construction, have their shuffle analoguous.
-As we have a free shuffle protoperad F ¡ (−) and an associated cofree conilpotent shuffle coprotoperad F c ¡ (−), one has the shuffle bar construction B ¡ . Moreover, let P be a quadratic protoperad. For every ordered finite set S, we have the isomorphism of chain complexes These isomorphisms justify the construction of shuffle protoperads (see the proof of Theorem 4.3).
-As we have a shuffle product ¡ c , one can construct the shuffle simplicial bar construction C ¡ and the shuffle normalized simplicial bar construction N ¡ .
Moreover, these constructions commute with the functor (−) sh , by Corollary 1.26 and Proposition 1.24.

Studying Koszulness of binary quadratic protoperad
In this section, we describe a criterion to study the Koszulness of binary quadratic protoperad, which are protoperads given by a quadratic datum (V, R) such that V is concentrated in arity 2, V (S) = 0 for all finite sets S with |S| = 2.

4.1.
A useful criterion. -We give an algebraic criterion for a binary quadratic protoperads concentrated in homological degree 0 to be Koszul. Let P be a binary quadratic protoperad concentrated in homological degree 0, given by the quadratic datum (V, R), then V = V 0 (2). We associate to P, a family of quadratic algebras {A (P, n)} n 2 , defined by where S sh is defined in Definition 1.18. We will see that the algebras A (P, n) are quadratic. Fix n 2, we consider the decomposition of V in irreducible representations: where V ν = k · v ν is the trivial representation or the signature representation of S 2 (recall that the characteristic of k is different to 2). To V , we associate the set V (P, n) of generators of {A (P, n)} n 2 : Thus V (P, n) corresponds to the generators of S(P)([[1, n]]) as algebra for the product µ (see Proposition 1.13), i.e., As P is binary and quadratic, the set of relations R is concentrated in arity 2 and 3. Each relation in R(2) is given by a linear combination of terms as 1 2 where each brick is labeled by a generator v ν . To a such relation r in R(2), we associate a family of quadratic relations {r ij } 1 i<j n in terms of V (P, n), where r ij is given by replacing a monomial indexed by v α for the bottom brick and v β for the upper brick, with v α and v β two generators, by the monomial (v α ) ij (v β ) ij in V (P, n) ⊗2 , as in Figure 1. We denote by R(2) ij , the set of relations in V (P, n) ⊗2 which are obtained , where each brick is labeled by a generator v ν . If n 3, for all relation r in R(3), we associate a family of quadratic relations {r ijk } 1 i<j<k n with r ijk ∈ V (P, n) ⊗2 , where r ijk is given by replacing all monomial indexed by v α for the bottom brick and v β for the upper brick, with v α and v β two generators, as in Figure 2. We denote by R(3) ijk , the set of relations in V (P, n) ⊗2 which are obtained by the labeled procedure ijk . We consider the quadratic algebra The new relations given by the commutator [(v α ) ij , (v β ) ab ] correspond to the "parallelism commutativity" which is present in the protoperadic structure:  As V (P, n) is, by construction, a set of generators of the algebra S(F sh (V ))([[1, n]]), we have the following morphism of algebras T (V (P, n)) → S(F sh (V ))([[1, n]]), which factorizes as follows: the isomorphism ϕ induces the isomorphism (3). All relations in A (P, n) are given by r ab and r ijk for 1 i < j < k n, 1 a < b n, r in R(2) and r in R(3). So, as we see in Figure 2, any choice of representative m for m gives us the same partition, then the partition p does not depend of the choice of the representative m. By the same argument, as the differential of B Alg (A (P, n)) is induced by the product of A (P, n), the bar complex splits: (A (P, n)) where each level w i of w is sent to a monomial m i in A (P, n), as in Figure 3. It is clear that this application is an isomorphism (A(P, 5)) (A (P, n)).
As the algebras A (P, n) are Koszul by hypothesis, for all n 2, then the homology of B Alg,(p) (A (P, n)) is concentrated in degree p. As this complex splits (see Equation (5)), then the homology of B Alg,(p) p0 (A (P, n)) is also concentrated in degree p. Then, by the isomorphism in Equation (6), the homology of N  and, for n 4,  This corollary is very important: it is the first example of a Koszul properad with a generator not in arity (1, 2) or (2, 1).

DPois is Koszul
In this section, we study the Koszul dual of the protoperad DLie, which is called DCom, by analogy of the case of operads Lie and Com.
5.1. The Koszul dual of DLie. -To the protoperad DLie, we associate its Koszul dual, which we will called DCom: Then, as in the case of the protoperad DLie, we can diagrammatically interpret We also have the following relations: Suppose that, for a fixed integer n, we have the following equality: Proof.
-We prove this result by induction on the weight of monomials, i.e., the number of vertices of the underlying graph. By Lemma 5.2, we have that this lemma holds for a monomial of weight 2. Let n be an integer strictly greater than 2. Suppose the lemma holds for every monomial of weight w < n. We consider Φ, a monomial of weight n and we denote by Φ, its underlying non-oriented graph. As DCom is a properad, the graph Φ is connected: we label its n vertices by v 1 , . . . , v n . There exists α in [[1, n]] such that the subgraph Φ * := Φ v α is connected. By the induction hypothesis, we can rewrite Φ * as a stairway, then we can rewrite Φ as a one of these two following monomials: or and, by invariance of stairways under the cyclic group action, we have our result.
Lemma 5.4. -Every monomial of DCom such that the underlying non-oriented graph has a cycle is null. Proof.
-We prove the result by induction on the weight of monomials. We limit ourselves to considering only monomials whose underlying non-oriented graph is a cycle, i.e., monomials whose each elementary block is linked by two edges to another block. We have the relation = 0 which initialize our induction. Suppose that every cycle of weight n − 1 is null. We consider a cycle Φ of weight n and, we isolate one of the blocs v α in the cycle (i.e., one of the vertex of the underlying graph) such that its two outputs are linked with an other bloc. In a cycle, such a bloc already exists. We denote by Φ * , the monomial obtained by the forgetfulness of the bloc v α in the initial cycle. The monomial Φ * does not contain a cycle, then, by Lemma 5.3, Φ * can be rewriting in a stairway. Finally, the monomial Φ can be rewrite as one of the two following monomials in Figure 4 and Figure 5. By the invariance of stairways under the diagonal action of the cyclic i j Figure 4. Monomial of form 1 Figure 5. Monomial of form 2 group (cf. Lemma 5.1), a monomial with the form 2 (see Figure 5) can be rewrite as a monomial with the form 1 (see Figure 4). Then, Φ can be rewrite as a monomial which contains a smaller cycle, then Φ is null. := sgn(Z/nZ) ⊗n = sgn(Z/nZ) if n even, triv(Z/nZ) if n odd.
We exhibit the action of the differential on generators of degree 2, 3 and 4.
As the properads As and Ind(DLie) are Koszul (see [LV12,Chap. 9] for the case of As), we obtain the main theorem of this paper.

Appendix. The algebras A (DLie, n) are Koszul
In this section, DLie is the protoperad of double Lie algebras. We consider the family of quadratic algebras A (DLie, n), for n 2, given by the quadratic datum (V (DLie, n), R(DLie, n)), with V (DLie, n) = {x ij | 1 i < j n} and, for n in N, x ab x uv − x uv x ab 1 i < j < k n 1 u < v n Proof. -The algebra A (DLie, 2) is isomorphic to k[x], which is Koszul. We denote by W n , the Koszul dual of A (DLie, n); this quadratic algebra is given by the quadratic datum (V (DLie, n) ∨ , R(DLie, n) ⊥ ) : We prove that the algebra W n is Koszul by the rewriting method; we will follow [LV12, Chap. 4, Sect 4.1].
Step 1. -We totally order the set of generators of W n by the right lexicographic order on indices: x ij < x k if j < or j = and i < k.
Step 2. -We extend this order to the set of monomials by the left lexicographic order.
-We obtain the following rewriting rules: Step 4. -We test the confluence of rewriting rules for all critical monomials. Recall that a critical monomial is a monomial x ij x k x uv such that monomials x ij x k and x k x uv can be rewrite by rewriting rules. Any critical monomial gives an oriented graph under the rewriting rules which is confluent if it has only one terminal vertex. We denote by α-β the confluence diagram associated to the monomial x ij x k x uv where x ij x k is the leading term (the term of the left side) of the rewriting rule α and x k x uv , the leading term of the rewriting rule β. We adopt the following notation: for a monomial x ij x k x uv , when we use the rewriting rule α on x ij x k , we denote that by x ab x cd x uv and when we use the rewriting rule α on x k x uv , we denote that by We start with the case of 1 i < j < k n and x uv < x ij to study diagrams of the form 1-β: Similarly, all diagrams α-1 are confluent. Now, we study the diagrams for a critical monomial with the leading term of 2 on the left. We start with 2-2: let u < i < j < k: For 2-3, there are three cases: we begin with i < j < u < k: −x jk x iu x uk 6 / / x iu x jk x uk 5 , , x jk x ik x iu for the case i < j = u < k, we have: